CHAPTER
1
History of Shock Waves PETER KREHL Ernst-Mach-Institut, Fraunhofer-Institutfur Kurzzeitdynamik, Eckerstr. 4, D-79104 Freiburg, Germany
A shock wave is a surface of discontinuity propagating in a gas at which density and velocity experience abrupt changes. One can imagine two types of shock waves: (positive) compression shocks which propagate into the direction where the density of the gas is a minimum, and (negative) rarefaction waves which propagate into the direction of maximum density. 1 Gy6zy Zempl~n University of Budapest 1905
1.1 1.2 1.3 1.4
1.5 1.6 1.7 1.8 1.9
Introduction Shock Waves: Definition and Scope Early Percussion Research Evolution of Shock Waves 1.4.1 Natural Supersonic Phenomena and Early Speculations 1.4.2 Shock Waves in Gases 1.4.3 Shock Waves in Liquids 1.4.4 Shock Waves in Solids Evolution of Detonation Physics Milestones in Early High-Speed Diagnostics Further Reading Chronology of Milestones Notes
1This modern and concise definition of a shock wave was first given by the young Hungarian physicist Dr. G Zempl~n [C. R. Acad. Sci. Paris 141:710 (1905)]. Visiting on a fellowship G6tingen and France (1904-1906), his interest in shock waves was obviously stimulated by Felix Klein, and Pierre Duhhem and Jacques Hadamard, respectively. Handbook of Shock Waves, Volume 1 Copyright ~ 2001 by P. Krehl. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
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1.1 I N T R O D U C T I O N This chapter illuminates the history of shock wave physics in terms of a Chronology of Milestones. To get a realistic picture of this complex evolution process--i.e., to reflect on previous states of knowledge, motivations, speculations, and achievementsmmajor results in the progress of not only shock wave research but also closely related fields such as percussion, explosion, and detonation are covered. In addition, some of the most important milestones in the advancement of high-speed diagnosticsma key technology that has heavily determined the progress of shock wave physics in the past and the p r e s e n t ~ have been included. Some general remarks on the historical background and interrelations between the different disciplines that are not obvious from the Chronology are given in this chapter.
1.2 SHOCK WAVES: DEFINITION AND SCOPE Shock waves 2 are mechanical waves of finite amplitudes and arise when matter is subjected to a rapid compression. Compared to acoustic waves, which are 2 In the 19th century the shock wave phenomenon, a puzzle for early researchers, had a different meaning than today and designated a tidal wave resulting from an earth- or seaquake. Euler (1759), without yet coining a term, addressed the "size of disturbance" of a sound wave, meaning its amplitude. Poisson (1808) described intense sound as the case "where the molecule velocities can no longer be regarded as very small." Stokes (1848) used the term surface of discontinuity, and Airy (1848-1849) described the wave as an "interruption of continuity of particles of air." Riemann (1859) already used the modern terms shock compression [Verdichtungssprung] and compression wave [Verdichtungswelle] to illustrate the jumplike steepening of the wave front. Earnshaw (1860) used the terms positive wave, to illustrate that the motion of particles are in the direction of wave transmission, and wave of condensation, to characterize the increase in density. Toepler (1864) was the first to use the term shock wave [Stoj~welle] in the present sense; he originated a shock wave from a spark discharge and first visualized it subjectively using a stroboscopic method. He also used the terms spark wave [Funkenwelle] and air percussion wave [Lufterschfitterungswelle] interchangeably, but incorrectly used the term sound wave [Schallwelle]. Rankine (1870) used the terms abrupt disturbance and wave offinite longitudinal disturbance, and Hugoniot (1885) the term discontinuity [discontinuit~ de la vitesse du gaz et de sa pression]. Mach and coworkers (1875-1885) used the terms shock wave, Riemann wave [Riemann'sche Welle], bang wave [KnaUwelle], and explosion wave [Explosionswelle]. In the specific case of a supersonic projectile, Mach and Salcher (1887) used the terms head wave or bow wave [Kopfwelle] and tail wave [Achterwelle]. Von Oettingen and yon Gernet (1888), studying oxyhydrogen explosions, called the detonation front Sto~welle. In France the term shock wave [onde de choc] was first used by Vieille and Hadamard (1898), and later by Duhem (1901) and Jouguet (1904). Duhem also used the terms partition wave [onde-cloison], true Hugoniot wave, surface slope [surface de glissement], and quasi shock wave to characterize special types. The term shock wave was not immediately taken up by encyclopedias. For example, in the German encyclopedia Meyers Konversationslexikon (1929), a shock wave was still defined as a "tidal wave originated by an earthquake", a wave type that we designate today as a tsunami. The 1962 edition of the Encyclopaedia Britannica does not even list the term shock wave.
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waves of very small, almost infinitesimal amplitudes, shock waves can be characterized by four unusual properties: (i) a pressure-dependent, supersonic velocity of propagation; (ii) the formation of a steep wave front with abrupt change of all thermodynamic quantities; (iii) for nonplanar shock waves, a strong decrease of the propagation velocity with increasing distance from the center of origin; and (iv) nonlinear superposition (reflection and interaction) properties. Shock wave effects have been observed in all four states of matter and also in media composed of multiple phases. It is now generally recognized that shock waves play a dominant role in most mechanical high-rate phenomena. Shock waves can assume manifold geometry and exist in all proportions, ranging from the microscopic regime to cosmic dimensions. This has led to an avalanche of new shock-wave-related fields in physics, chemistry, materials science, engineering, military technology, medicine, etc. Even before World War I some new disciplines were in the process of being established, such as supersonics, cavitation, detonation, blasting technique, and underwater explosions. In the period between the two world wars, these disciplines were further extended to gas dynamics, seismology, high-speed combustion, plasma p!,ysics, chemical kinetics, thermochemistry, aeroballistics, nonlinear acoustics, transonic flows, etc. The largest expansion of shock wave physics certainly occurred during and after World War II, which created such new disciplines as hypersonic aerodynamics, nuclear explosions, detonics, exploding wires, rarefied gas dynamics, superaerodynamics, aerothermodynamics, magnetofluid dynamics, cosmic gas dynamics, reentry, laser-supported detonation, implosions, impact physics, fracture mechanics, high-rate materials dynamics, shock synthesis, laser fusion, shock lithotripsy, and explosive working. Because the literature is scattered throughout many disciplines, it has become quite difficult even for the specialist to get a survey of the present state of the art. In addition, many investigations on shock waves and detonation are classified or published as company or institute reports and not listed in public library catalogues. This enormous breadth of shock-wave-related disciplines has led also to a wealth of new technical terms that make communications among shock scientists more difficult than during the pioneering "good old days" of legendary all-round knowledge. Modern aerodynamicists, for example, accustomed to working with gases and thinking in terms of mean-free path lengths, viscosity effects, boundary and shock layers, vorticity, slipstreams, Mach and Reynolds numbers, etc. can nowadays barely communicate with solid-state shock physicists who treat shock waves in terms of Huginiot elastic limit, elastic precursor, plastic wave, spallation, lattice compression, shock polymorphism, etc. However, it should be remembered that shock waves, independent of the state of matter of the applied medium, have a common root and are based on the mighty mechanical principle of collision (percussion, impact),
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which has also become the fundament of such eminent fields of science as plasma physics and particle physics. The Chronology in Section 1.8, illuminating the historical evolution of shock wave physics in terms of milestones, emphasizes the phenomenological aspects. In a tabular form it specifies the contributor's affiliation and motivation of his research, discloses preceding work and cross connections with similar studies elsewhere, and comments on the achievements under the present point of view. This rather encyclopedic approach is certainly arbitrary and was influenced by the author's years of diagnosing a diversity of shock wave phenomena in all states of matter. It is hoped that this form of presenting historic milestones may render a better survey than a lengthy narrative description to the historically interested reader. Because of space limitations, the Chronology omits the beginning of percussion research and does not start until 1759. This was apparently the year of the earliest published reference on the reflection of the possible properties of shock waves, then considered by Euler as waves with "disturbances of large size." The Chronology ends in 1945 due to the magnitude of shock-wave-related research that has taken place since then. In the following chapters of this Handbook reference is made mostly to works published after 1945, and this complements--although presented in a different style--the Chronology. Those who are interested in a more extended chronology will find it in Krehl's monograph. 3
1.3 EARLY P E R C U S S I O N
RESEARCH
Widely used by primitive man to produce tools and weapons, and practiced in an almost unchanged manner throughout a period of several 100,000 years, percussion was a fundament of civilization. However, the basic laws of percussion were not discovered until the 17th century, only recently compared to the long history of its application. Many prominent naturalists of that century contributed to the understanding of percussion, such as Galilei (1638), Marci (1639), Descartes (1644), Wallis (1668), Wren (1668), Huygens (1669), Mariotte (1671), and Newton (1687). Percussion studies started with the use of tangible bodies like billiard balls or cannonballs and were mainly based on the observation of their velocities and directions before and after collision (central and eccentric collision). Early ballistic impact studies had revealed that the observed effects strongly depend on the hardness of the collision partners (elastic and inelastic collision) and that in the case of inelastic 3p. Krehl. A historical perspective on percussion, explosion and shock wave research. (SpringerVerlag, Heidelberg, in progress).
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collision the kinetic energy is partly transformed into heat. Since the very short moment of contact and deformation during collision were not yet accessible, neither experimentally nor theoretically, Newton's and Huygens' percussion theories relinquished from the beginning the difficult task of evaluating the enormous instantaneous force. In his Principia (1687), Newton suggested the first corpuscular model of percussion on an atomic level to illustrate that the propagation of sound occurs via percussion from one particle to another. His model, in a way representing the archetype of early shock wave models, stimulated other naturalists to explain the propagation of other mechanical waves, such as seismic shocks (Desmarest 1756), in the same manner. Newton's model was also used by Bernoulli in his Hydrodynamica (1738), in which he first expressed the phenomenon of heat by the average mean square velocity of the colliding atoms, thus initiating the first thermodynamic theory of heat (KrOnig 1856; Maxwell 1860-1866). Navier (1822) used the corpuscular model to derive the laws of motion of continuous media. The multiple-percussion pendulum, today also known as Newton's cradle, soon became a spectacular apparatus for demonstrating chain percussion. The ballistic pendulum was invented by Cassini, Jr. (1707) and is based on the law of the conservation of impulse, one of the basic findings of 17th-century percussion research. Introduced into ballistics by Robins (1746), the ballistic pendulum allowed the first quantitative determination of the velocity of a projectile. Furthermore, he used this simple but most efficient apparatus to study projectile drag as a function of its velocity, thus creating aeroballistics. Robins' remarkable supersonic experiments up to a velocity of 1700ft/sec (M ~ 1.5) revealed a considerable increase of air drag when approaching the sound velocity. Those experiments were repeated and analyzed more recently with modern means by Hoerner (1958) and proved that Robins indeed must have reached supersonic velocities in his gun shots. Percussion research reached its next climax in the second half of the 19th century. Neumann (1856-1857), De Saint-Venant (1866-1867), and Hertz (1882) developed (partly contradicting) percussion theories in which they included Hooke's law of deformation. This also allowed the determination of the instantaneous stress distribution or percussion force. Hertz theoretically demonstrated that the stress distribution in a plate, impacted by a hard sphere, has a conical geometry (the Hertzian cone) that extends from the surface into the impacted plate which can result in conical cracks. This important result explained not only previous observations but also confirmed various hypotheses of prehistorians about how hand axes, arrowheads, knives, and other objects from flint stone or other very hard minerals were produced by primitive man (Kerkhof and Muller-Beck4). Contact times during percussion were first 4E Kerkhof and H. M{iller-Beck: Zur bruchmechanischen Deutung der Schlagmarken an Steingeraten. Glastech.Bet. 42:439--448 (1969).
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measured electrically by Sabine (1876). Those measurements revealed that the contact times are indeed of very short duration--somewhere in the microsecond regime, depending on the mass of the percussion partners and their initial velocities. Tait's percussion machine (1890-1895) allowed for the first time the continuous recording of the contact time of percussion. Using a graphical method he also evaluated the duration of percussion for various examples, such as contact times between a golf ball and a club, one billiard ball with another, and a hammer and a nail.
1.4 E V O L U T I O N
OF SHOCK
WAVES
The large number of disciplines that now fall in the category of shock waves did not evolve along a straight path into the present state. Rather, they emerged from complex interactions among shock-wave-related disciplines or independently from other branches of science. One practical means of getting a useful survey on the development of shock wave physics is to classify the large number of milestones in terms of states of mattermi.e., shock waves in gases, liquids, solids, and plasmas. The following paragraphs will refer to the first three states of matter only.
1.4.1 NATURAL S U P E R S O N I C P H E N O M E N A AND EARLY SPECULATIONS Shock waves are a common phenomenon on Earth and under certain conditions are produced during volcanic eruptions and earthquakes. The most striking natural shock wave phenomena are certainly thunder and the fall of meteors. Earnshaw (1851) was probably the first who reflected on the possibility that thunder would propagate with supersonic velocity as he noticed that the time delay between lightning and thunder was less than one would expect when assuming that thunder propagates with the velocity of sound. Nine years later, Montigny, Hirn, and Raillard, independently of each other, resumed the problem, thereby partly assuming unrealistically high propagation velocities of thunder. Accounts of the famous KAigle Fall (Blot 1803), a meteorite shower that spawned a barrage of reports, and the Washington Meteor (1873) provoked disputes among contemporary scientists on the possible cause of observed shock phenomena (Abbe 1877). Ernst Mach, renowned together with Peter Salcher as the discoverer of the head wave phenomenon (1887), correctly explained this phenomenon likewise by the supersonic motion of the meteor (E. Mach and Doss 1893). Cosmic shock
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phenomena compared to terrestrial ones reach enormous dimensions, such as the solar wind (Parker 1958), which is a stream of ionized gas particles emitted from the sun's corona that is accelerated in Earth's magnetic field and produces a bow wave similar to that ahead of a supersonically moving blunt object (Axford and Kellog 1962). Much larger shock dimensions are generated during stellar explosions (supernovae). Earliest accounts of these date back to Chinese and Swiss annals (A.D. 1006). The shock wave of largest imaginable dimensions would be the "Big Bang," the "shock of all shocks," which, according to the big bang theory, resulted about 10 to 20 billion years ago from a gigantic explosion of a highly concentrated mass of gaseous matter at a single point in space. The relic radiation field resulting from the fireball of the Big Bang eventmpredicted by Alpher, Herman, and Gamow (1948-1949) to be around 5 K--was recorded by Penzias and Wilson (1965) as a residual blackbody radiation of 3 K. Until the advent of gunpowder, the only means available to man for producing shock waves was whip cracking, probably used since antiquity. However, it was scarcely used by early scientists as a subject of investigation because the mechanism of shock generation and its analysis are rather complicated. Lummer (1905) first speculated that the shock might be caused by supersonic motion of the whip tip. The solution of this puzzle required ambitious diagnostics and was not uncovered until the advent of sophisticated high-speed photographic recording techniques. 5 Black powder (gunpowder) was invented in China and first described in Europe by Roger Bacon (1267) for incendiary and explosive applications. Since it can only burn rapidly and cannot detonate, it cannot be used to generate shock waves. However, applied in fire arms, which were in use in Europe since the early 14th century, the hot gases of the reacting gunpowder are initially confined in the barrel but are suddenly released at the moment when the projectile leaves the muzzle, which generates the impressive muzzle blast, a shock wave. After the inventions of the electrostatic generator (von Guericke 1663) and the Leiden jar (von Kleist and Cuneus 1745), it became possible for the first time to store considerable electric charges and to discharge them in a very short time. The discharge is accompanied by a spectacular flash and a sharp report, an impressive demonstration that was often shown in university lectures and private circles and that stimulated discussions on the nature of lightning and thunder. The electric spark proved to be not only useful to generate shock waves at any time, in any space, and of any desired geometry, but was also precisely triggerable in time with an electric light source (Knochenhauer 1858), in most case a second spark discharge confined to a pointlike geometry to meet the requirement of illumination for the shadow or 5cf. ref. 357 in Chronology.
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schlieren method. Furthermore, the alternative method of generating shock waves by chemical explosives allowed the differentiation between electrical and chemical secondary effects of observed shock phenomena, an important advantage that facilitated the interpretation of Mach reflection (E. Mach and Wosyka 18 7 5). It is quite possible that early acousticians also reflected on unusual phenomena associated with intense sound. Prof. Sir Richard Southwell, 6 commemorating at the University of Glasgow the centennial of Rankine's appointment to the Queen Victoria Chair of Civil Engineering and Mechanics, made the interesting annotation that the voice, kept down to a mannerly noise volume, will get through a speaking-tube unaltered but becomes increasingly distorted when the volume is raised. Early experiments on the velocity of sound at very low temperatures--i.e., in air of perfect dryness--were performed in the North Pole regions by the famous Arctic explorers Parry and Ross (1821--1825). The fact that the report of a gun was heard at their further station before the command to fire was heard suggested also the idea that intense air waves travel more quickly than weaker waves. The puzzling shock wave, characterized by a stepped wave front, was difficult to accept by early naturalists because it involved the abandonment of the principle Natura non facit saltus, i.e., the denial of the continuity of dynamic effects. Surprisingly, however, the problem was successfully tackled neither by experimentalists nor philosophers, but rather by mathematical physicists. Jouguet 7 wrote: "The shock wave represents a phenomenon of rare peculiarity such that it has been uncovered by the pen of mathematicians, first by Riemann, then by Hugoniot. The experiments followed not until afterwards." Riemann and Hugoniot, however, were not the only pioneers. As shown in the Chronology, they had a surprisingly large number of predecessors who substantially contributed to this new field, thus paving the way for understanding discontinuous wave propagation.
1.4.2
S H O C K WAVES IN GASES
The impetuous development of experimental shock wave physics started with studies in gases primarily for the following reasons: (i) In the 17th century the elastic nature of air was studied experimentally and already used in practice-such as in the wind-gun and pneumatic lighter, which revealed the adiabatic properties of quickly compressed air. The first scientific milestone was the 6R. Southwell.W.J. M. Rankine: A commemorativelecture delivered on 12 December, 1955, in Glasgow. Proc. Inst. Civ. Eng. (London) 5:177 (1956). 7E. Jouguet: R~sum~des theories sur la propagation des explosions. La science agrienne 3(No. 2): 138-155 (1934).
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determination of the isothermal equation of state (Boyle and Townley 1660). (ii) The relatively low sound velocity of air in comparison to a liquid or solid-for example, smaller by a factor of about 5 and 20 in the case of water and iron, respectivelymwas advantageous for early experimentalists, when high-speed diagnostics were still in their infancy. (iii) All three optical methods (schlieren, shadowgraph, and interferometry) are light transmission techniques, i.e., require a translucent medium, and therefore are ideally suited for studies in gases. (iv) In practice, the majority of shock wave applications, then and now, take place in air. Early ballisticians already noticed the importance of air resistance and its dependency on projectile geometry and velocity. Up to the 18th century the resistance of bodies was measured by the timing of free fall, the mounting of the body on a pendulum, and suspension of the body in the flow. Systematic aeroballistic studies at substantial velocities were performed by Robins (1746) in his sensational ballistic experiments. He also devised a rotating-arm machine that allowed rotation of the test object in a reproducible manner by means of a falling weight. Von Karm~in8 (1932), who coined the term wave drag for a new type of drag at supersonic velocity, appropriately called these pioneering studies of early ballisticians "the theoretical-empirical preschool of supersonic aerodynamics." Early attempts at measuring the sound velocity, both in air (Mersenne 1636, Cassini, Jr. et al. 1738) and water (Colladon 1826), used a long baseline to compensate for the limited accuracy of available clocks. This method, however, was not directly transferable to the crucial test of whether waves of intense sound would propagate faster than sound velocity, because the pressure rapidly decreases with distance from the source; i.e., the region of supersonic velocity would be limited only to the near field of the explosion source. Regnault (1863), widely known for his careful measurements and sophisticated methods, originally had in mind to measure sound velocities in various gases and liquids. To secure a long baseline, he performed his experiments in the public sewage channels and gas pipe lines of Paris, which advantageously confined the sound within two dimensions. To secure sufficient sound intensity at the receiver station, he generated the sound at the tube entrance with small amounts of explosives, at first not being aware that he applied shock (blast) waves rather than sound waves. His remarkable results, published in various international journals but today almost forgotten, obviously proved quantitatively the existence of supersonic velocities for the first time and certainly must have encouraged contemporaries from other countries to tackle this subject further.
8 T. von KCtrman: H6her, schneller und heisser. Interavia 11:407 (1956).
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At low speeds the air behaves like an incompressible fluid. The classical theory of hydrodynamics, which involves no viscosity and is concerned with irrotational motion, predicts that a body moving steadily will experience no resistance or lift. At higher speeds, however, energy is increasingly dissipated so that bodies moving at speeds faster than that of sound have a considerable resistance. Ernst Mach, who held the chair of experimental physics at the German Karl-Ferdinand Universitat of Prague (1867-1895), was interested in physical and physiological acoustics. He was supported by a team of coworkers, later including also his son Ludwig Mach, and had the opportunity to systematically continue his research in this particular field throughout a period of almost 28 years, certainly a peculiarity in the research scenery of the 19th century. E. Mach even began his gas dynamics studies with one of the most difficult subjects of shock wave physics, the oblique interaction of shock waves (E. Mach and Wosyka 1875)--a curiosity in the evolution of gas dynamics. Later called the Mach effect by yon Neumann (1943), this interaction is a complex nonlinear superposition phenomenon and has remained even today a challenging subject of continuous research. Subsequently, E. Mach and Sommer (1877) proved experimentally that indeed a shock wave propagates with supersonic velocity but rapidly approaches sound velocity at increasing distance from the source, thus confirming on a laboratory scale Regnault's previous result. E. Mach and Salcher (1887) first showed that a projectile flying supersonically produces a hyperbolic shock wave, the so-called head wave, which moves stationary with the projectile. These pioneering experimental investigations of Ernst Mach and his team, together with theoretical studies in England, France and Germany established the basic knowledge of supersonic flows in the late 1880s. Practical aerodynamics, however, was still in its infancy, and the first flight of man (von Lilienthal 1891) had not yet been achieved. It appears that studies on the exhaust of compressed gas from an orifice originated from malfunctions of Papin's safety valve (1679). These valves were quite frequently applied in steam engines but often had too narrow an outflow diameter and could not quickly reduce dangerous overpressures, thus causing disastrous steam boiler explosions with many casualties and great damage to neighboring equipment (Arago 1830). This problem prompted not only engineers but also scientists to studies that became substantial roots of early supersonic research (de Saint-Venant and Wantzel 1839; Napier 1866; Reynolds 1885; Emden Bros. 1899; Stodola 1903; Prandtl 1904-1907). The invention of the Laval nozzle (de Laval 1888), a nozzle of convergent-divergent geometry, first allowed supersonic exit velocities. Soon an important device in engineering, such as for increasing the efficiency of steam turbines, this nozzle had also an enormous impact on supersonic flows and the development of aerodynamics. Progress in this field was immediately fructified by progress of high-speed photography. After successful visualization and interpretation of
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the flow phenomena in front of a Laval nozzle (Salcher and Whitehead 1889; L. Mach 1897) and later also of those in its interior (Prandtl 1904; Meyer and Prandtl 1908), the nozzle was adapted in England in the world's first supersonic wind tunnel at the National Physics Laboratory, Teddington, which reached supersonic flow velocities up to M = 2 (Stanton 1920-1926). The first hypersonic velocities (Erdmann 1944) were reached at the large supersonic wind tunnel facility at Heeresversuchsstelle Peenemf~nde, the main center of German rocketry during World War II. The shock tube, invented in France by Vieille (1899) as a by-product of his detonation studies, became the most important measuring and testing device of gas dynamics. He applied the shock tube to demonstrate that shock waves generated by the detonation of explosives propagate essentially in the same manner as shock waves generated by the bursting diaphragm of the highpressure section that formed one end of his tube. The basic theory of the shock tube was laid down by Kobes (1910), Hildebrandt (1927), and Schardin (1932). Kobes and Hildebrandt had a rather curious approach to gas dynamics: they investigated whether it would be possible to improve the performance of air suction brakes on long railway trains by using shock waves. The shock tube, rediscovered during World War II by Bleakney (1949) and associates at Princeton University, soon proved its excellent applicability for quantitatively investigating propagation and interaction phenomena of shock waves within a large range of gas dynamics parameters. Furthermore, it was introduced worldwide in other laboratories for the study of shock wave interactions with scaled architectural structures such as model houses, plants, shelters, and vehicles. Then in the long period of the Cold War such interactions were of great practical concern because of the constant threat of nuclear blast to civil and military installations. That the shock tube was also useful for generating high temperatures in gases was first recognized and exploited in high-speed spectroscopic studies by Laporte (1953). The theoretical approach of treating shock waves can be traced back as far as Newton's Principia (1687). Assuming incorrectly that sound is an isothermal process, he made a crude calculation of sound velocity in air. Laplace (1816), noticing a discrepancy of almost 20% between Newton's theoretical result and already-existing measured data, improved the theory by assuming that sound is an adiabatic process. Prior to this, Poisson (1808), stimulated by Laplace in this subject, had mathematically tackled the sound velocity problem in a paper published in the Journal de l'Ecole Polytechnique. Under the heading "Onedimensional movement of air in the case that the velocities of the molecules are no longer very small" [Mouvement d'une ligne d'air dans le cas of~ les vitesses des molecules ne sont pas supposees tr~s-petites], he also touched the basic question of how to solve the wave equation in the case of noninfinitesimal amplitudes, thus laying the foundation for the first shock wave theory. Most noteworthy,
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this happened at a time when an experimental verification of such discontinuities, propagating as a wave throughout the medium, was still pending. Poisson's early approach, first resumed in England by Challis (1848), was quickly extended by Airy (1848), Stokes (1848-1849), Rankine (1858, 1870), Earnshaw (1858-1860), Riemann (1859), Christoffel (1877), Hugoniot (18851887), Tumlirz (1887), Burton (1893), Hadamard (1898-1905), H. Weber (1901), Duhem (1901-1909), Jouguet (1901-1910), Zemplen (1905), Lummer (1905), Lord Rayleigh (1910), G. I. Taylor (1910), etc. However, the transition to present-day shock wave theory, largely a result of many international contributions, was not straightforward, and their disputes and cumbersome struggles for understanding the shock wave puzzle may be dimmed in light of things we now take for granted. Details of this gradual process of understanding may be found in the Chronology. In this context, some remarks concerning their motivations seem worthwhile. Airy, Challis, and Jouguet first studied tidal waves, which, steepened in shallow water into hydraulic jumps, propagate rather slowly and are clearly observable with the naked eye. The analogy between the reflection properties of a hydraulic jump and a shock wave in a gas is indeed striking (Jouguet 1920) and was later applied in water table experiments (Preiswerk 1938; Einstein 1948; Crossley, Jr. 1949). Earnshaw treated the great solitary wave (1845) before he had his key experience with thunder (1851). Riemann's interest in "air waves of finite amplitude" [Luftwellen von endlicher Schwingungsweite] did not arise from a purely mathematical curiosity. Stimulated by von Helmholtz, he treated the problem of combination tones, a phenomenon of the nonlinearity of the ear and observable only at sufficiently high sound levels. Rankine and Hugoniot approached the shock wave phenomenon from the thermodynamic point of view, Lord Rayleigh from the acoustic. Heinrich Weber, treating shock waves also as a mathematical problem to find solutions for various types of partial differential equations, edited Riemann's lectures on mathematical physics and extended his theoretical studies on shock waves by numerous examples and comments.
1.4.3
S H O C K WAVES IN LIQUIDS
Liquids were regarded for a long time as incompressible matter, until Canton (1762) first demonstrated its very low compressibility. In shock wave physics, liquids and gases are both treated as compressible fluids. Liquids, however, are much more difficult to compress than gases and, as a consequence, typical shock wave properties such as wave-steepening effects and supersonic propagation are clearly observable only at significantly higher shock pressures. Furthermore, shock-compressed liquids may show unusual properties (high
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viscosity, phase transformations) and generate complicated side effects (cavitation). Shock waves in liquids, particularly in water, were hardly treated until the beginning of World War I. However, a few remarkable contributions, described in more detail in the Chronology, should be emphasized here. Water hammer, a steep-fronted pressure wave that is felt as a sharp hammerlike blow, is caused by the sudden retardation or acceleration of flow in a long pipe, for example when a valve is closed sufficiently rapidly. Montgolfier and Argand (1796) applied this phenomenon successfully in constructing a hydraulic pump they called a "hydraulic ram" [belier hydraulique]. Generally, however, this effect is detrimental in pipe systems because the pressure pulse can propagate to remote areas and destroy tubes, valves, and other installations. Kareljkich and Zhukovsky (1898-1900) in Moscow first scientifically treated the problem of water hammer or hydraulic shocks in water supply lines. At the turn to the 20th century, this problem also became important in other countries when large water pipe systems had to be built to satisfy the increasing water requirements of fast-growing urban communities. The water hammer can also be generated by an object impacting and penetrating a liquid and in this modification was probably the earliest observed shock wave effect in a liquid. Carr~ (1705) observed the curious phenomenon that a bullet shot into a wooden box filled with water blew up the box. The impacting bullet, transferring a large amount of momentum to the water, generates a shock wave that ruptures the walls. Since the first air battles of World War I this effect has been a constant menace to military aircraft, whose fuel tanks cannot fully be armored against gun shots. 9 Other shock wave effects in liquids were also observed in military applications. For example Abbot in the United States (1881) and Blochmann (1898) in Germany studied underwater explosion phenomena of submarine mines, a subject of increasing interest to the navy since the invention of the torpedo in the 1860s. During World War II, research on underwater explosions was pushed forward by the United States and England on a large scale. Their UNDEX Reports, published shortly after the end of war, include a wealth of data on underwater explosion phenomena and their analytical treatment, and even today are a rich source of information. I~ Water ricochets, a now well-known percussion phenomenon, was studied by Marci (1639), who threw a stone on a pond's surface at a low angle and explained the effect with the law of reflection. This phenomenon gained new
9 R. Yurkovich. "Hydraulic ram: a fuel tank vulnerability study." Rept. No. G964, McDonnell Douglas Corporation, St. Louis, MO (Sept. 1969). 10Underwater Explosion Research (UNDEX). A Compendium of British and American Reports. 3 vols., ed. by G. K. Hartmann, U.S. Naval Ordnance Laboratory, and E. G. Hill, British Admiralty. The Library of Congress, Photoduplication Service, Washington, DC (1950).
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interest with the advent of seaplanes and the need for them to land at high speed or on rough sea. Investigations performed in various countries, such as the United States (Von K~irman and Wattendorf 1929), Germany (Wagner 1932), and the former Soviet Union. (Sedov and Wladimirow 1942), revealed that this skipping effect is a complicated combination of gliding and periodic bouncing that also generates finite-amplitude waves in the water. Cavitation damage was first observed shortly after the first use of steam turbines. The central implosion of cavitation bubbles, accompanied by the emission of shock waves, results in material destruction. At the beginning of the age of steam turbines in the 1880s, erosion effects caused by cavitation were observed not only on the blade tips of turbine wheels but also on marine propellers that were initially driven at very high revolutions to avoid loss involving high gear reduction between turbine and propeller. Studies on cavitation phenomena were initiated both from the engineering (Thornycroft and Barnaby 1895; Cook 1928) and scientific point of view (Lord Rayleigh 1917; Prandtl 1925; Jouguet 1927; Ackeret 1938). Cavitation and associated shock pressure effects can now be generated in a very wide spatial/temporal range, covering meters/milliseconds down to nanometers/femtoseconds. An example for the upper limit is the gas sphere of an underwater explosion, which can be regarded as a single, huge bubble. An example for the lower limit, or micro-cavitation, is the irradiating of biological tissue with femtosecond laser pulses, which results in ultrashort shock pulses (the photodisruption effect). This procedure has been applied in femtosecond laser nanosurgery as a "nanoscalpel" to cut nanometer-sized particles, such as chromosomes in a living cell. 11 The electrohydraulic effect, first observed in England by Singer and Crosse (1815) and later rediscovered in the former Soviet Union, 12 uses a powerful electric discharge fed into a thin wire or spark gap submersed in water to generate shock waves. This effect was made famous by the Latvian urologist Goldberg, 13 who first successfully applied it to the disintegration of bladder stones in man (shock lithotripsy). Later the electrohydraulic effect was also used in production technology for forming metal sheets. 1.4.4
S H O C K WAVES IN SOLIDS
The pioneers of classical shock wave theory did not limit their analyses to fluids only, but had also reflected on the peculiarities of shock waves in solids. 11K. K6nig, I. Riemann, P. Fischer, and K. J. Halbhuber. Intracellular nanosurgerywith near infrared femtosecond laser pulses. Cell. & Mol. Biol. 45:195-201 (1999). 12L. A. Yutkin. Elektrogidravliceskij effekt. Masgiz, Moskva (1955). 13V. Goldberg. Zur Geschichte der Urologie: Eine neue Methode der Harnsteinzertn3mmerung--elektrohydraulische Lithotripsie. Urologe [B] 19:23-27 (1979).
History of Shock Waves
15
In his treatise On the thermodynamic theory of waves of finite longitudinal disturbance, Rankine (1869) clearly states that his derived relations are valid "for any substance, gaseous, liquid or solid." Christoffel (1877), Hugoniot (1889), Duhem (1903), Hadamard (1903), and Jouguet (1920) addressed the solid state in more detail. Other contributions that could not immediately be verified by contemporary experimentalists who lacked the diagnostic means, later stimulated the evolution of shock wave physics in solids. Prominent examples include: (i) various theories of percussion derived by early naturalists; (ii) Maxwell's theory of elasticity (1850); (iii) a theory on the equation of state for solid matter based on the lattice vibration theory derived by Mie (1903) and Gruneisen (1912); and (iv) theories on the dynamic plasticity of metals such as proposed by G. I. Taylor (1942), yon K~irman (1942), and Rakhmatulin (1945). Contrary to the rapid and steady progress of shock wave physics in gaseous matter since the 1870s, research in solids has evolved slowly. The main reason was certainly the very challenging high-speed diagnostics, which require submicrosecond resolution and thus were not available until after World War II. However, using simple experiments early researchers did study the dynamic properties of solids, particularly their rate-dependent strength. J. Hopkinson (1872) measured the strength of a steel wire when the wire was suddenly stretched by a falling weight, and he made the important observation that the strength is much greater under rapid loading than in the static case--a phenomenon that was later studied in more detail by his son (B. Hopkinson 1905). The latter also discovered the fracture phenomenon of back spalling from an explosive-loaded metal plate (B. Hopkinson 1912). In the 19th century, Parsons and Moisson (1892) attempted to use shock waves to induce polymorphic phase transformations in solids, particularly in carbon to produce artificial diamonds. However, their efforts did not give clear evidence and were just too ambitious for their time. An important step toward this goal were the later results of static high-pressure investigations on a large number of liquid and solid substances carried out by Bridgman (1903-1961) in a long-lasting campaign that formed the foundation for understanding matter under high pressures. Those results gave modern shock physicists their first clues to the static compressibility of solids at high pressures and to the stressdependent plasticity of metals, thus arousing their curiosity about how substances would behave under dynamic pressures. This promoted also various other spectacular investigations, for example on shock-induced polymorphic transitions in iron (Bancroft e t al.14), on possible ice modifications of
14D. Bancroft,E. L. Peterson,and S. Minshall. Polymorphismof iron at high pressures.J. Appl.
Phys. 27:291-298 (1956).
16
P. Krehl
shock-compressed water (Rice and Walsh; 15 Al'tshuler et a1.16), and on shockinduced transformation of graphite into diamond (DeCarli and Jamieson17). Early suppositions that craters might have been generated by meteorite impact in the geologic past had to battle against established cryptovolcanic hypotheses, but first systematic studies of meteorite material collected from various craters around the world (Spencer 1933) and the discovery of curious striated conical geologic structures (shatter cones, Butcher 1933, Boon and Albritton 1938) supported the impact theory. The famous Meteor Crater, Arizona was first brought to notice in 1891 by the discovery od many masses of meteoric iron scattered around the crater and the finding of diamond in this iron. Eventually, the sensational discovery of quartz high-pressure, shock-induced polymorphs in meteorite craters--beginning in Meteor Crater by Chao, Shoemaker and MadsenlS--constituted evidence for meteorite impact scars (socalled astroblemes) and significantly promoted knowledge on the geological history of Earth, Moon, and other planets. 19 Solid-state shock wave physics, partly an outgrowth of nuclear weapons research imposed by the Manhattan Project, did not start until 1945 and therefore is beyond the scope of this survey. Obviously, however, modern testing methods for studying materials response under shock loading have close roots to percussion. To a large extent they are based on the planar impact of two rodlike bodies, a basic arrangement treated previously (Euler 1745; Neumann 1857-1858; de Saint-Venant 1867; Ramsauer 1909; Donnell 1930) and used today in high-rate materials testing such as the Hopkinson pressure bar (B. Hopkinson 1914), which was further developed into the split Hopkinson pressure bar or the Kolsky bar (Kolsky 1949); the Taylor test (Taylor and Whiffin 1948); the flyer plate method (McQueen and Marsh 1960); and the planar impact by a high-velocity projectile (Hughes and Gourley 1961). Modern investigations of shock waves in solids revealed rather complex behavior in comparison to gases and liquids, and theories describing the solid state under shock loading, taking structural properties into account also, are still in development. The impressive advancement of solid-state shock wave physics
15 M. H. Rice and J. M. Walsh. Dynamic compression of liquids from measurements on strong shock waves. J. Chem. Phys. 26:815-823 (1957). 16 L. V. Al'tshuler, A. A. Bakanova and R. E Trunin. Phase transition of water compressed by strong shock waves. Sov. Phys. Dokl. 3:761-763 (1958). 17p. S. DeCarli and J. C. Jamieson. Formation of diamond by explosive shock. Science 133:1821-1822 (1961). 18 E. C. T. Chao, E. H. Shoemaker and B. M. Madsen: First natural occurence of coesite. Science 132:220-222 (1960) 19 B. M. French and N. M. Short (eds.). Proc. 1st Conference on Shock Metamorphism of Natural Materials. NASA Goddard Space Flight Center, Greenbelt, MD (1966). Mono Book Corporation, Baltimore (1968).
17
History of Shock Waves
was mainly based on (i) the generation of well-defined shock wave profiles, (ii) the advance of submicrosecond measurements and visualization techniques, and (iii) computational analysis employing refined thermodynamic equations of state and using more accurate dynamic materials parameters. Today this specific branch of high-pressure physics provides not only a rich source of equation-ofstate data for all kinds of solids but also information on most shock compression and diagnostic techniques used in such disciplines as impact physics, geology, seismology, fracture mechanics, laser fusion, and materials science.
1.5 E V O L U T I O N
OF DETONATION
PHYSICS
Historically, investigation on the nature of shock waves was closely related to the puzzle of detonation, a high-transient thermochemical wave phenomenon. The terms explosion and detonation were not always used as they are today. 2~ Today the more general term explosion can be defined as a process causing a rapid increase of pressure, which can steepen into a shock wave. An explosion does not necessarily have to be connected with the exothermic reaction of a chemical explosive; for example, in the case of a steam boiler the explosion is the sudden rupture of the boiler walls, or in the case of a shock tube it is the bursting membrane. Conversely, a detonation is a violent explosion related to high explosives in which the rate of heat release is great enough for the explosion to be propagated through the explosive as a steep shock front, the so-called detonation wave. The discovery and correct interpretation of this term was not achieved until the period from 1880 to 1905, which was almost 300 years after the invention of gold fulminate, the first high explosive, by the alchemist Croll (1608). The following short survey focuses on some of the circumstances in the step-by-step process of investigating the nature of detonation. Since Croll's discovery an impressive number of high explosives were invented, the most important being silver fulminate (Bertollet 1708), picric acid (Hausmann 1788), mercury fulminate (Howard 1799), guncotton (Sch0nbein 1846), nitroglycerine (Sobrero 1847), and dynamite (A. Nobel 1867) tetryl (Michler and Meyer 1879), TNT (Haeussermann 1891) and lead azide (Hyronimus 1907). Although by 1880 some of these substances were already being used for military and civil purposesmfor example, in 1835 France alone 2o The terms detonation and explosion are of Latin origin and had been used interchangeably since the late 17th century. However, the term detonation was apparently first used in the presentday meaning in England by Abel (1869) who studied detonation effects in guncotton. In France Berthelot (1870) first called the detonation front "shock" [choc] and in 1881 "explosive wave" [l'onde explosive], which apparently was taken up in England by Dixon (1893) and Chapman (1899). The term detonation was later adopted by Vieille (1900).
18
P. Krehl
produced 800 million percussion caps employing mercury fulminatemthe physicochemical processes of explosion/detonation were not yet uncovered. The first measurements of the detonation velocity in long, confined charges of various high explosives, carried out by Abel (1869-1874) in England, revealed unusual high velocities in the range of some 1000m/s. Abel stated that "the detonation of gun-cotton travels more rapidly than any other known medium with the exception of light and electricity." Studying the behavior of unconfined and confined charges, he speculated that detonation in a high explosive might be transmitted by means of some "synchronous vibrations" (1869). Shortly thereafter Berthelot (1870) in France was the first to correctly assume that detonation might be caused by a traveling mechanical shock, but an experimental proof had yet to come. At this point it is useful to look back on previous attempts at understanding explosion in gases. The discovery of oxyhydrogen and its violent explosive properties (Turquet de Mayerne 1620; Cavendish 1760) stimulated not only a crude theory on the origin of earthquakes (L~mery 1700; Kant 1756) but also turned the interest of naturalists to other explosive gaseous mixtures, particularly to firedamp, which had been a hazard to miners since the beginning of hard coal mining in the 12th century. Davy (1816) analyzed the explosivity of firedamp and discovered that the critical mixture for explosion is 9% methane and 91% air. The "Davy lamp," his famous invention to avoid firedamp explosions in mines, could only partly mitigate the risk of explosion accidents; there were other sources of open fire, such as explosives used for blasting purposes, one of the oldest and most important civil applications of explosives since the 16th century. When Bunsen (1867) measured the strength and rate of combustion of various explosive gaseous mixtures such as oxygen, he used an experimental setup that could not provoke explosion. Dust explosions, certainly the oldest kind of man-made explosion, frequently occurred in flour mills and bakeries and later in metal powder works. Such explosions stimulated hypotheses (Faraday and Lyell 1845; Rankine and Macadam 1872) that in an analogous manner coal-dust-laden air might cause explosions in coal mines. Eventually, a series of tragic firedamp explosions in the French coal mining industry (1876) led to the foundation of the French Fire-Damp Commission (1878) to investigate possible causes of these explosions from a scientific viewpoint and to reflect on possible countermeasures. Additional mining accidents in England, France, and the United States soon afterward---some of them probably produced by the presence of coal dust--initiated the foundations of similar national institutions. In France, Mallard and Le Ch~telier, from the Ecole des Mines, were asked to examine the best means of guarding against explosions of firedamp in mines. This led to a series of investigations on the specific heat of gases at high temperatures, the temperatures of ignition, and the velocities of propagation of flame in gaseous
History of Shock Waves
19
mixtures (1880-1882). In addition, similar studies were carried out in Paris at the College de France by Berthelot, who worked together with Vieille at the Laboratoire Central, Service des Pouclres et 5alpetres. These investigations revealed (i) that an explosive wave, later generally termed a detonation wave, exists in explosive gaseous mixtures and propagates at a tremendous speed of up to 2500m/s, and (ii) that the propagation velocity only depends on the mixture composition not on the tube diameter as long as that diameter is not too small (1878-1883). Mallard and Le Chatelier (1883), who first recorded the propagation of the explosive wave in long tubes with a drum camera, observed that the transition from combustion into detonation occurs suddenly, and that the detonation velocity is comparable to the sound velocity of the burnt detonation products. These results promoted the Chapman-Jouguet theory, which evolved in England from independent contributions by Schuster (1893), Dixon (1893), and Chapman (1899); in Russia by Mikhel'son (1890); and in France by Berthelot (1891) and Jouguet (1904). The theory assumes that the hot products of the combustion wave act as an expanding hot-gas piston that accelerates the unburnt mixture ahead, thereby forming the explosive wave, which is a shock wave. In comparison to a normal shock wave with its discontinuous transition of uncompressed to compressed gas across the shock front, however, the detonation front also separates two chemically different states of unburnt and burnt gases. However, various experimental studies later revealed that the detonation front is not necessarily a homogeneous zone of reaction but can exhibit a periodic cell structure (Bone et al. 1936). In addition, the assumption of a sharp detonation front is an idealization, and the Chapman-Jouguet theory was later refined by introducing a three-layer model of the detonation front. This model was independently advanced by Zeldovich (1940) in the Soviet Union, von Neumann (1942) in the United States, and Doring (1943) in Germany, today known as the ZND theory. The study of the classic chlorine-hydrogen explosion--a puzzling photochemical-induced reaction discovered by Gay-Lussac and Thenard (1809) and investigated in more detail by Chapman (1909-1933), Bodenstein (1913), and Nernst (1918)mrevealed that detonation is not an instantaneous, single-stage chemical reaction but rather occurs in a chain reaction [Kettenreaktion], thereby passing through various short-lived intermediate states. 21 Their findings stimulated the evolution of chemical kinetics (Semenov and.Hinshelwood 1928; Nobel Prize of Chemistry 1956), which quickly became a new exciting branch of physical chemistry.
2~M. Bodenstein: 100Jahre Photochemie des Chlorknallgases. Bet. Dtsch. Chem. Gesellsch. 75A: 119-125 (1942).
20
v. Krehl
1.6 MILESTONES IN EARLY HIGH-SPEED DIAGNOSTICS The advancement of high-speed diagnostics--encompassing appropriate fast methods of measurement, visualization, and recordingmhas always been essential for a detailed analysis of shock and detonation effects and their proper applications. In this regard Ernst Mach's scientific way of experimenting was very successful and directive to his contemporaries. He was an eminent philosopher who also can be regarded as the first gas dynamicist and highspeed photographer of his time. He carried out his shock and explosion research according to the motto "Seeing is understanding." Within the short period of time from 1864 to 1891, the three principal optical techniques of flow visualization (i.e., the schlieren method, shadowgraphy, and Mach-Zehnder interferometry) had been invented. From the historical point of view it is remarkable that one of the first applications of the schlieren method, which was invented by A. Toepler (1864) at BonnPoppelsdorf, was the visualization of a propagating spark wave, a weak shock wave. The famous ballistic experiments by E. Mach and Salcher (1887) also used the schlieren method to visualize the head wave generated by a supersonic projectile. The shadowgraph technique was invented at the University of Agram [now Zagreb] by Dvof~ik (1880), who was one of E. Mach's assistants (1871-1875). Applied by Boys (1890) in England, this technique considerably simplified the visualization of supersonic flows in ballistic testing ranges. E. Mach and L. Mach proved the great potential of interferometry for flow visualization, for example by quantitatively analyzing the density jump at the shock front (E. Mach and Weltrubsky 1878) and the flow around a supersonic bullet (E. Mach and L. Mach 1889). The Mach-Zehnder interferometer is particularly appropriate for flow visualization studies in ballistic tunnels, shock tubes, and wind tunnels because it allows a large distance between object and reference beam. These three optical methods gave early shock researchers their first insights into an abundance of completely new supersonic flow phenomena. The challenge of recording gas dynamic events largely inspired the development of new high-speed photographic equipment, which in turn enabled the discovery of new shock phenomena. In the pioneering period, gas dynamicists were often also high-speed photographers who invented, developed, and built their own equipment. Snapshot photography of a dynamic event was first demonstrated by FoxTalbot (1851). With the advent of gelantin dry plates (Maddox 1871) which later could be improved significantly in sensitivity (1878-1880) and of electric spark light sources of high intensity and short duration, it became possible for the first time to both stop the motion of propagating shock waves with practically no motion blur and obtain a sufficient exposure density on photographic film. The first photographed shock wave was generated by the
History of Shock Waves
21
discharge of a Leiden jar, visualized with the schlieren method, and photographed on a high-sensitive gelantin dry plate (E. Mach and Wentzel 1884). The evolution from single-shot photography to high-speed cinematography is a story of its own, 22 but a few milestones can be illuminated here. The rotating mirror, a mechanical device for resolving the motion of an object in one dimension, was apparently first used by Wheatstone (1834). However, it took almost a hundred years before the mirror was modified into a practical streak camera for resolving the propagation and reflection of detonation waves (Payman 1931). High-speed cinematography reached a first climax with the invention of the Multiple Spark Camera (Cranz and Schardin 1929). Based on a principle of recording that allows one to realize any desired frame rate and to use any type of "light source" even beyond the visible spectrum, it was also modified later for flash X-ray and neutron cinematography. The ambitious U.S. program of atomic weapons development and testing during and after World War II resulted in further developments of mechanical framing cameras with ultrahigh frame rates. Simultaneously, the various requirements of dynamic plasma diagnosing in numerous fusion research programs stimulated new developments of ultrafast image tube cameras, particularly in the United States, England, France, and the Soviet Union. With the advent of the microchannel plate (MCP) in the 1980s~an American invention based on the electron multiplier (Farnsworth 1930)~a new optoelectronic device with excellent gating capability and of high light intensification became available that could be combined very successfully with the already existing charge-coupled device (CCD). These so-called ICCDs (intensified CCDs), applied in a multiple arrangement with optical image splitting, created a new generation of ultrafast multiple digital framing cameras that are well suited for recording all kinds of shock wave phenomena. Flash radiography, invented simultaneously in Germany (Steenbeck 1938) and the United States (Kingdon and Tanis 1938), immediately became an important diagnostic tool that particularly stimulated detonics. Contrary to optical methods, flash radiography is insensitive to self-luminous events that accompany all detonation processes, and smoke resulting from detonation products cannot obscure the test object. In addition, X rays promote insight into the interior of shock-loaded solids and the measurement of temporal shock front positions. These particular properties of flash X rays allowed, for the first time, the visualization of shock waves emerging from exploding wires, shaped charges, and detonation fronts in liquid and solid explosives. Furthermore, it also became possible to visualize shock wave propagation and interaction phenomena in optically opaque media, which represent the majority of solids, to quantitatively determine the density jump behind the shock front using photo densitometry, and to measure the lattice compression of shock-compressed crystals using flash X-ray diffraction. 22 S. F. Ray: High-Speed photography and photonics. Focal Press, Oxford (1997).
22
P. Krehl
FIGURE 1 Life spans of renowned percussion, explosion and shock wave researchers. The begin of the shock wave era, marked above by the broken line, can be attributed to Simeon Denis Poisson who first mathematically treated "waves of which the velocities of the molecules are not supposed to be very small" (1808).
1.7 FURTHER READING 1. L. V. Al'tshuler, and V. A. Simonenko, History and prospects of shock wave physics. High Pressure Research 5:813-815 (1990). 2. J. D. Anderson, Jr. Modern compressible flow, with historical perspective. McGraw-Hill, New York (1990).
History of Shock Waves
FIGURE 1
23
(Continued)
3. R. Assehton: History of explosives. Institute of Makers of Explosives, New York (1940). 4. P. A. Bauer, E. K. Dabora, and N. Manson, Chronology of early research on detonation wave. In Dynamics of detonations and explosions: Detonations, A. L. Kuhl, J. C. Leyer, A. A. Borisov, and W. A. Sirignano, eds. Progr. Astro- & Aeronautics (AIAA, Wash. D.C.) 133:3-18 (1991). 5. A. Busemann, Compressible flow in the thirties. Ann. Rev. Fluid Mech. 3:1-12 (1971). 6. R. Ch~ret, The life and work of Pierre-Henri Hugoniot. Shock Waves 2:1-4 (1992). 7. D. H. Clark, and E R. Stephenson, The historical supernovae. Pergamon Press, Oxford (1977). 8. H. Dryden, Supersonic travel within the last two hundred years. Scient. Monthly 78:289-295 (May 1954).
24
P. Krehl
9. G.E. Duvall, Shock wave research: yesterday, today and tomorrow. Proc. 4th Conf. on Shock Waves in Condensed Matter, Y. M. Gupta, ed. Spokane, WA (July 1985). Plenum Press, New York (1986), pp. 1-12. 10. R.J. Emrich, Early development of the shock tube and its role in current research. Proc. 5th Int. Shock Tube Symp., Z. I. Slawsky, J. F. Moulton, Jr., and W. S. Filler, eds. White Oak, Silver Spring, MA (April 1965), pp. 1-10. 11. E Fischer, Zur Geschichte der Dampfkesselexplosionen. Dingler's Polytechn. J. 213:296-308 (1874). 12. K. L. Goin, The history, evolution, and use of wind tunnels. AIAA StudentJ. (Febr. 1971): 3-13. 13. A. Hertzberg, Shock tube research, past, present and future. Proc. 7th Int. Shock Tube Symp., I. I. Glass, ed. Toronto (June 1969), Univ. of Toronto Press (1970), pp. 3-5. 14. J. N. Johnson, and R. Ch6ret, Shock waves in solids: an evolutionary perspective. Shock Waves 9:193-200 (1999). 15. T. von K~irm~in, Aerodynamics. Selected topics in the light of their historical development. Cornell University Press, Ithaca, NJ (1954). 16. T. yon K~irman and L. Edson: The wind and beyond. Theodore von Karm~in pioneer and pathfinder in space. Little, Brown & Co., Boston & Toronto (1967). 17. P. Krehl, and M. van der Geest, The discovery of the Mach reflection effect and its demonstration in an auditorium. Shock Waves 1:3-15 (1991). 18. P. Krehl, and S. Engemann, August Toepler, the first who visualized shock waves. Shock Waves 5:1-18 (1995). 19. N. Manson, Historique de la d6couverte de l'onde de d~tonation. [Colloque C4.] J. de Physique 48:7-37 (1987). 20. N. Manson, and E. K. Dabora, Chronology of research on detonation waves: 1920-1950. In Dynamic aspects of detonations, A. L. Kuhl, J.-C. Leyer, A. A. Borisov, and W. A. Sirignano, eds. Progr. Astro- & Aeronautics (AIAA, Wash. D.C.) 153:3-39 (1993). 21. L. M~dard, Histoire de la thermochimie. Publ. de l'Univ, de Provence, Aix-en-Provence (1994). 22. L. M~dard, Eoeuvre scientifique de Paul Vieille (1854-1934). Rev. Hist. Sci. 47:381--404 (1994). 23. W. E Merzkirch, Mach's contribution to the development of gas dynamics; Seeger, R. J., On Mach's curiosity about shock waves. In Ernst Mach, physicist and philosopher, R. S. Cohen and R. J. Seeger, eds. Boston Studies in the Philosophy of Science 6:42-67 (1970). 24. E. Oeser, Historical earthquake theories from Aristotle to Kant. Abhandl. Geolog. Bundesanstalt Wien 48:11-31 (1992). 25. H. Reichenbach, Contributions of Ernst Mach to fluid mechanics. Ann. Rev. Fluid Mech. 15:128 (1983). 26. E. W. E. Rogers, Aerodynamicsmretrospect and prospect. Aeronaut. J. 86:43-67 (1982). 27. I. Szab6, Geschichte der Theorie des Stot~es. Humanismus und Technik 17:14-44, 128-144 (1973). 28. I. Szab6, Geschichte der mechanischen Principien und ihre wichtigsten Anwendungen. Birkhauser, Basel etc. (1977). See also: Geschichte der Stoj~wellen, pp. 281-314; Geschichte der Stoj~theorie, pp. 425-479. 29. S. P. Timoshenko, History of strength of materials. McGraw-Hill, New York etc. (1953). 30. C. A. Truesdell, The mechanical foundations of elasticity and fluid dynamics. J. Ration. Mech. Annual 1, 125-171, 173-300 (1952). 31. C. A. Truesdell, Rational fluid mechanics. In Leonardi Euleri Opera Omnia XII [II]. Teubner, Leipzig etc. (1954). See also editor's introduction, pp. I-CXXV.
History of Shock Waves
25
1.8 CHRONOLOGY OF MILESTONES* 1759
Royal Academy of Sciences, Berlin I
1760s Private
1762
1770
laboratory, Great Marlborough Street, London Royal Society, London
University of Basel
Euler I addresses in a letter to Lagrange the possibility that the propagation of sound might depend on the "size of the disturbances," which expressed in modern terms would mean size of the displacement or intensity of sound. He writes to Lagrange, "It is very remarkable that the propagation of sound actually takes place more rapidly than the theory indicates, and at present I renounce the opinion I had formerly that the following disturbances could accelerate the propagation of the preceding ones, in such a way that the higher is the sound the greater is its speed, as possibly you have seen in our latest memoirs. It has also come into my mind that the size of the disturbances might cause some acceleration, since in the calculation they have been supposed infinitely small, and it is plain that [finite] size would change the calculation and render it intractable. But, in so far as I can discern, it seems to me that this circumstance would rather diminish the speed." m Euler's hypothesis was indeed correct insofar as the amplitude of sound (the "size of disturbance") might influence the speed of sound. However, he incorrectly assumed that the velocity would diminish with increasing amplitude. Cavendish 2 experiments with a mixture of "fixed air" and "inflammable air" and "inflammable air," i.e., oxyhydrogen, and its ignition by an electric spark. He constructs a "measurer of explosions of inflammable air" to quantify the released mechanical energy of a confined volume of that gas. Canton 3 demonstrates the small compressibility of liquids, which have hitherto been regarded as being incompressible. He places the test liquid in a thermometer-like arrangement and, compressing the bulb, obtains an observable magnification of the change of volume in the capillary. Daniel Bernoulli 4 treats the problem of collision by applying the theory of elasticity and develops the first wave theory of percussion. Assuming a freely suspended rod hit perpendicular to its axis, he calculates the loss of kinetic energy as a result of harmonic elastic vibrations.
* Covering not only shock waves but also related high-rate phenomena, such as percussion, blast, explosion, and implosion, as well as milestones in the development of basic shock diagnostic techniques. Dates years provided in the text by an arrow (for example --~ 1803) refer the reader to other milestones listed under these years and notes following the symbol 9 comment the milestone from the today's point of view.
26
V. Krehl
1770s Ecole Royale du Gdnie, Mdzidres, France
Monge 5 enters a field of study that will hold his interest for many years: Lagrange's theory of general partial differential equations and his own method of geometric construction of particular solutions, which he calls "method of characteristics." Starting from a first-order partial differential equation, he gives a geometric interpretation of the method of the variation of parameters and writes, "In the following I will use, as I have always done, different characteristics for the different ways of differentiating; this method is more practical, as it is not necessary to find a fractional form in order to present a partial differential." He concludes: "This memoir contains the constructions of integrals of partial differential equations, more generally than the ones I had constructed until now, and there I demonstrate that the geometrical places of these integrals generally satisfy their partial differential equations, which is what I had thought to myself". His geometric construction of a particular solution of the equations under consideration allows him to determine the general nature of the arbitrary function involved in the solution of a partial differential equation ~ Monge's graphical method was widely applied and further developed by subsequent mathematicians and physicists who introduced such basic notations as characteristic curve (or Monge's curve), characteristic cone (or Monge's cone), trajectory of characteristics, and characteristic variable etc. A detailed discussion of his work was given by Taton. 6 Earnshaw (1858) and Riemann (1859), independently of one another, first applied Monge's classic work of characteristics on gas dynamics, particularly on the propagation of the shock wave. Their studies were continued and greatly extended by various other prominent shock wave pioneers, such as Hugoniot (--~ 1887), Hadamard (--. 1903), and Prandtl and Busemann ( - , 1929). Reviews on the progress of further developments of this important mathematical tool and its application in gas dynamics and hydrodynamics were given, for example, by Oswatitsch, y Courant and Friedrichs, s and Abbott. 9
1774
London
Nairne, 1~ a British mechanic and experimentalist, studies the electrical explosion of thin wires. Discharging a battery of Leiden jars, initially charged up to a high voltage, through a 1-m-long iron wire 0.15 mm in diameter, he observes that "it flew about the room in innumerable red hot balls, on examining these balls, they were in general hollow, and seemed to be nothing but scoria."
1780
Diocese Chiemsee, Bavaria
Wisshofer, a German priest and naturalist, publishes in
Salzburg a pamphlet on the design of an electric gun that uses the discharge of a Leiden jar (1745) to ignite an inflammable gas for propelling the projectile. 11
History of Shock Waves
27
Royal Academy of Sciences, Berlin
Lagrange 12 publishes a treatise on the motion of fluids. In the last chapter entitled Du mouvement d'un fluide contenue dans
1781
un canal peu profond et presque horizontal, et en particulier du mouvement des ondes he considers a surface wave of infinitesimal height in shallow water in a canal of finite length and dervies the famous formula for the propagation velocity as v = (gh) ~/2 where g is the gravitational acceleration and h the water depth of the liquid at rest. Referring to the known formula describing the velocity v of a free-falling body, v=(2gh) 1/2 and drawing a comparison between sound waves in air and gravity waves, he states, "Thus, as the velocity of propagation of sound is found equal to that which a weight would require in falling from the height of the atmosphere (assumed homogeneous), the velocity of propagation of waves will be the same as what which a weight would acquire in falling from a height equal to half the depth of water in the canal." He refers to measurements previously performed by de la Hire in France who observed a velocity of 0.46 m/s [1.412 pied par seconde] in a water depth of about 2.2cm [ 8/,0 pouce], thus essentially confirming his theoretical result. 9 In the case of hydraulic jumps ("shooting" flow) the velocit~ v is calculated by the formula v [g(h 1 + h2)h2/2hl]/2, where h I is the water depth ahead of the jump and h2 is the water depth behind the jump. 13 For increasingly weaker jumps ("streaming" flow) h 2 approaches h 1, eventually converging with Lagrange's solution.
1783
Chair of Mathematics, Royal Military Academy, Woolwich
Hutton, 14 performing ballistic experiments in the years 1775 and 1783-1786 using the ballistic pendulum (Cassini, Jr. 1707, Robins 1740), measures supersonic muzzle velocities of cannon shots (charge: up to 16oz of gunpowder; projectile: iron cannon ball, 1.96 inch in diameter, weight up to 16oz, 13 dr,) up to 2030ft/s. His studies first confirm that supersonic velocities are not only obtainable for small caliber guns (Robins 1746) but also for larger ones. In another treatise HUTTON15 investigates the drag of projectiles within a wide range of velocities up to supersonic speeds (20--2000ft/sec) "to show according to what power of the velocity, at every point, the resistance increases." He observes that "commencing with the 2nd power or square of the velocity, at the very beginning or slow motion [5ft/sec], the exponent of the power gradually increases, till at the velocity of 1500 or 1600 ft/sec, it arrives at the 2.153 power of the same . . . . After the 1600 feet velocity, where the exponent (2.153) is greatest, it gradually decreases again to the end [towards 2000ft/sec]." He explains the velocity-dependent drag of a projectile by a vacuum generated at its rear, "The
28
P. Krehl circumstance of the variable and increasing exponent in the ratio of the resistance is owing chiefly to the increasing degree of vacuity left behind the ball, in its flight through the air, and to the condensation of the air before it. It is well known, that air can only rush into a vacuum with a certain degree of velocity, viz., about 1200 or 1400 feet in a second of time; therefore, as the ball moves through the air, there is always left behind a kind of vacuum, either partial or complete; that as the velocity is greater, the degree of vacuity behind goes on increasing, till at length, when the ball moves as rapidly as the air can rush in and follow it, the vacuum behind the ball is complete, and to complete ever after, as the ball continues to move with all greater degrees of velocity. Now the resistance, which the ball continues to move with all greater degrees of velocity. Now the resistance, which the ball suffers in its flight, is of a triple nature; one part of it being in consequence of the vis inertia of the particles of air, which the ball strikes in its course; another part from the accumulation of the elastic air before the ball; and the third part arises from the continued pressure of the air on the forepart of the ball, when the velocity of this is such as to leave a vacuum behind it in its flight, either wholly or in p a r t . . . A s soon as the motion of the ball becomes equal to that of the air, and always when greater [i.e., at supersonic speeds], then the ball has to sustain the whole pressure of the atmosphere on its forepart, without having any aid from a counter-pressure behind . . . . " Hutton's explanation well illustrates the attempts of early supersonic pioneers to find a plausible reason for the puzzling phenomenon of the strong increase of drag in the transonic regime.
1784
Private laboratory, Great Marlborough Street, London
Cavendish, 16 hearing about recent experiments on oxyhydrogen explosions performed by Warhire (1776) and Priestley (1781), resumes his own investigations on this subject (Cavendish -+1760s), resulting in his famous paper on the synthesis of water. He observes that mixtures of "common air" and "inflammable air," enclosed in a vessel and electrically fired, are converted into a deposit inside the vessel of dew that is pure water, whereby all of the inflammable air but only about four-fifths of the common air is converted. - Lavoisier who repeated the experiment, later termed "inflammable air" hydrogen [from the Greek hydrogenium, meaning water-former].
1785
Turin, Italy
Count Morozzo 17 reports on a flour-dust explosion in a Turin flour warehouse, probably the earliest account of such a phenomenon. He describes the circumstances as follows: "On the 14th of December, 1785, about six o'clock in the evening, there took place in the house of Mr. Giacomelli,
History of Shock Waves
29 baker in this city, an explosion which threw down the windows and window-frames of his shop, which looked into the street; the noise was as loud as that of a large cracker, and was heard at a considerable distance. At the moment of the explosion, a very bright flame, which lasted only a few seconds, was seen in the shop; and it was immediately observed, that the inflammation proceeded from the flourwarehouse, which was situated over the back shop, and where a boy was employed in stirring some flour by the light of a lamp. The boy had his face and arms scorched by the explosion; his hair was burnt, and it was more than a fortnight before his burns were healed . . . . " Count Morozzo speculated that the flour might have produced "inflammable air" by fermentation (such as observed on dampened hay) which, mixed with air and dispersed flour dust, was ignited by the light of the lamp, thus leading to this violent inflammation. However, he critically remarked that model experiments did not prove this hypothesis and that the flour, upon examination, was extraordinarily dry and originated from the Piedmont area, which had had no rain for five or six months.
Academy of Sciences, Turin
Lagrange 18 studies the percussion force of a water jet impinging perpendicularly or obliquely on a plane. This basic problem of hydrodynamics--a term coined by D. Bernoulli ~9 to analytically cover hydrostatic as well as hydraulic (i.e., dynamic) phenomena by a single method---anticipates the difficult task of evaluating the flow of water impacting the blade of a water turbine. Measurements of the percussion force performed by Krafft (1973) gave a much smaller value than predicted theoretically by D. Bernoulli (1736), D'Alembert (1769), and Bossut (1772). Assuming a simple model of fluid flow with a central core of stagnated liquid surrounded by a shell of streaming and laterally deviated liquid, Lagrange derived a simple formula for the percussion force that better matched experimental results.
1786
York, England
Goodricke 2~ publishes a report on observations in which he discloses that a star, positioned near the head of Cepheus, changes its brightness periodically. A few years before, he had observed similar phenomena in the head of Medusa (so-called Algol) and of/~ Lyrae. 9 Variable stars, also called cepheids, are found in various regions of our galaxy and in other galaxies. The change in the brightness is caused by the change in the temperature and radius of the photosphere, which also creates shock waves. 21
1792
Munich
Von Baader, 22 a German mining engineer [Bergrat] interested in the application of explosives in the mining industry, observes that the energy of a blast can be focused on a
30
P. Krehl small area by forming a hollow in the charge that increases the explosive effect and saves powder (the "cavity effect"). 9 Seven years later he makes the observation that the surface relief of an explosive is reproduced on a closely facing steel plate by the focusing of explosion products ("explosive engraving"). Von Baader's publication was aFparently read and put into practice in Norway and for a short time also in the Harz Mines. 23 However, since he used black powder, which is not capable of detonation, his arrangement was not a shaped charge device in the modem sense.
1794
Ecole Polytechnique, Paris
Originally established as Ecole Centrale des Travaux Publics by order of Napoleon under the direction of Gaspard Monge (1746-1818), the school is renamed Ecole Polytechnique the following year. 24 9 Many prominent French pioneers of fluid mechanics, percussion, explosion, and shock physics studied and/or taught here, e.g., Jean B. C. Bdanger (1790-1874), Dominique E Arago (1776--1853), Jean B. Blot (1774-1862), Jacques S. Hadamard (1865-1963), Henri-Pierre Hugoniot (1851-1887),Joseph L. Lagrange (1736-1813), Pierre-Simon Laplace (1749-1827), Henry Le Chatelier (1850-1936), Louis Navier (1785-1836), Simeon D. Poisson (1781-1840), Adhemar de Saint-Venant (1797-1886), Victor Regnault (18101878), Emile Sarrau (1837-1904), Paul Vieille (1854-1934), and Pierre L. Wantzel (1814-1848).
Wittenberg, Germany
Chladni, 25 more known to later generations of physicists by his contributions to acoustics, starts his 30-year campaign on researching meteorites ("fire balls") and first proposes their extraterrestrial origin. 9 His result was not widely accepted until the EAigle Fall of stony meteorites (--+ 1803).
Brussels, Belgium
Mons 26 reports in a letter to Prof. Gren, editor of the Journal der Physik, on experiments by Parcieux, who observed in a dark room in the moment of explosion or implosion of thinwalled glass spheres "a vivid flame similar to an electric spark"
["eine lebhafte Flamme gleich einem electrischen Funken"]. Parcieux produced (i) an explosion by using a sealed glass sphere filled with air of atmospheric pressures that, positioned in a recipient, exploded after evacuation, and (ii) an implosion by evacuating a glass recipient that, not capable of withstanding the atmospheric pressure, imploded during evacuation. 9 Similar experiments on imploding and exploding glass spheres were performed more recently by Glass 27 and associates. Using high-speed schlieren visualization, the latter researchers also noticed that in the case of implosion the glass fragments form a jet, like in a shaped charge.
History of Shock Waves
31
1796
Paris
Montgolfier, 28 13 years after his sensational hot-air balloon ascents, invents with the assistance of Argand the hydraulic ram [bdier hydraulique], a water pump that uses the kinetic energy from a copious flow of running water under a small head to force a small portion of that water to a higher level.. Hydraulic shocks are detrimental in common water pipelines (Karelijkich and Zhukovsky--~ 1898), however, when used in this type of pump they should be as strong as possible to provoke efficient pumping. Since the hydraulic ram does not require any additional source of energy and is very simple in construction, it is still in current use in the mountains. For example, modern ram pumps 29 can deliver 700 L / m i n up to a height of 300 m.
1802
Coll~gede France, Paris
Biot, 3~ mathematician and physicist, publishes the first "theory of sound" and acknowledges the assistance of Laplace. He advances physical arguments in favor of p - Kp:', which results in a sound velocity a - - ( ~ p o / P o ) 1/2, where Po and Po are the pressure and density at rest, respectively, and 7 = %/6, is the ratio of specific heats at constant pressure and constant volume.
1803
KAigle, Normandy, France
Biot 31 gives his famous account on the "EAigle fall of stony meteorites": "On Tuesday, April 26, 1802, about one in the afternoon, the weather being serene, there was observed from Caen, Pont-Audenen and the environs of Alenqon, Falaise and Verneuil, a fiery globe of a very brilliant splendor, which moved in the atmosphere with great rapidity. Some moments after there was heard at EAigle, and in the environs of that city to the extent of more than thirty leagues in every direction, a violent explosion, which lasted five or six minutes. At first there were three or four reports like those of a cannon, followed by a kind of discharge which resembled a firing of musketry; after which there was heard a dreadful rumbling like the beating of a drum, a multitude of mineral masses were seen to fall .... " 9 From the number of shock waves or explosions felt by the observer, it was concluded that the meteorite must have broken into a large number of pieces on striking the dense atmosphere at low levels. Later about 3000 meteor fragments were collected, ranging in mass from 9 g to 8 kg. 32
1807
Paris
The Niepce Brothers obtain from Napoleon a patent for their piston engine that, supposed to be driven by spark-ignited lycopodium dust explosions, was to be used as a drive motor for ships. 33 9 W h e n experiments with lycopodium dust failed, they
Pyr~olophore, an explosion-driven
32
R Krehl successfully used petroleum. With their engine, a forerunner of the Diesel engine, it became possible in 1816 to power barges on the Seine. Turning in 1813 to the problem of photography, they succeeded in 1826 to make the first photograph [h(.liographie].
1808
Ecole Polytechnique, Paris
Poisson 34 presents his "theory of sound," and at the end of his treatise he extends his theory to the special case that "the velocities of the molecules in an air column are not supposed to be very small." He assumes that disturbances with finite amplitudes propagate in an ideal gas in one (positive) direction x and applies the law a 2 = dpo/dpo, with a being the sound velocity, i.e., the limit to which the velocity of propagation of the wave approximates when the particle velocity becomes indefinitely small. He arrives at the following general equation: dtp/dt 4- a dq~/dx 4- ~1. dq~2/dx 2 = 0. Here q~ is the velocity function and dq~/dx the velocity of disturbance (or particle velocity) at time t of a particle whose distance from the origin is x. His exact solution for a wave traveling in one (positive) direction reveals that the particle velocity dq~/dx behind a pressure disturbance can be expressed as dq~/dx = F { x - a - d q ~ / d x t}, where F denotes an arbitrary function. His solution differs from the equation given previously by D'Alembert (1747) only in that dq~/dx also appears in the argument of the function E Poisson notes that Lagrange 35 had already obtained a very similar result given by dq~/dx = f { x - a t - t . f ( x - at)}. ,, Poisson's mathematical result obviously indicates the quicker propagation of the parts of the wave where the disturbance is forward (that is, the compressed parts) and the slower propagation of the parts where the disturbance is backward (that is, the dilated parts). This leads to a change of the wave profile during wave propagation, an important feature that will be first recognized by Challis (--+ 1848), and qualitatively worked out and illustrated by Stokes (-+ 1848). Retrospectively, Poisson's treatise can be regarded as the root of shock wave theory and theoretical nonlinear acoustics as well. He also coined the symbol 7 for the ratio of the specific heats, which, later resumed by Rankine (--~ 1869), is used even to this day.
1809
Ecole Polytechnique, Paris
Gay-Lussac and ThCnard 36 expose a 1:1 mixture of hydrogen and chlorine to diffuse daylight and gradually obtain hydrogen chloride [acide muriatique]. On the other hand, when being exposed to direct sunlight this gaseous mixture ("chlorine detonating gas") explodes violently and destroys the glass balloon. 9 Compared to an oxyhydrogen explosion (Turquet de Mayerne (1620; Cavendish--, 1760s-+ 1784), the
History of Shock Waves
33 released heat of a chlorine-hydrogen explosion amounts to only about 2/3. The chlorine-hydrogen reaction, today a classic example of a photochemical reaction, long remained a puzzle to chemists and physicists. Work on his puzzle eventually led to the discovery of chain reaction (Bodenstein--+ 1913).
1815
Meeting of the Royal Society, London
Singer and Crosse 3r report on the effects of wire explosions carried out by De Nelis, who used an exploding lead wire placed in axial direction of a thin-walled metallic cylinder filled with water and pulsed from a large battery of Leiden jars. The cylinder itself is expanded more or less in proportion to its power of resistance, usually becoming undulated on the surface or burst open. Generating more violent wire explosions by using a larger battery, they observe that even iron cylinders with a thickness greater than that of the strongest muskets are heavily damaged by cracks. They state that "the expansive power of electricity acting in this way is therefore vastly superior to the most potent g u n p o w d e r . " . Their remarkable results anticipated the electrohydraulic effect that Yutkin (1950) rediscovered in the Soviet Union.
1816
EAcademie des Sciences, Paris
Laplace 38 publishes his previous hypothesis (Biot-+1802) that a sound wave is an adiabatic process and states, without demonstration, a correction of Newton's formula that was published in his Principia (1687).
Grand Duke's Laboratory, Florence, Italy
Davy 39 investigates the nature of firedamp. Studying test samples of firedamp collected together with Faraday on a journey in the Apennines, he observes that "1 part of gas inflamed with 6 parts of air in a similar bottle, produced a slight whistling sound: 1 part of gas with 8 parts of air, rather a louder sound; 1 part with 10, 11, 12, 13 and 14 parts, still inflamed, but the violence of combustion diminished. In 1 part of gas and 15 parts of air, the candle burnt without explosion with a greatly enlarged flame .... " He correctly states that firedamp (methane) consists of "4 proportions of hydrogen in weight 4, and 1 proportion of charcoal in weight 11.5." Using gas from the distillation of coal mixed with eight times its volume of air, he also determines the rate at which an explosion of gases propagates in a tube and makes the first rough experiment on the temperature reached in an explosion. The flame, fired in a 1-foot-long tube 1 inch in diameter, takes more than a second to traverse the tube. He also observes that the same mixture that burns in a wide tube may not support flame propagation in a narrow tube with a diameter less than a certain "critical diameter." This phenomenon will eventually lead to his construction of a safe mining
34
P. Krehl lamp (the Davy lamp) in which a copper mesh with small openings prevents flame propagation from the inside of the lamp to the atmosphere of the mine. 9 Archduke John of Austria, who visited Davy in 1815, was one of the first promoters of the Davy lamp, which shortly after was introduced in the Styrian mining industry. 4~
1822
Ecole des Ponts et Chauss4es, Paris
Navier 41 presents a paper on the law of motion of continuous media. He considers fluids and solids to be made up of particles that are close to each other and act upon each other by attraction or repulsion, resulting from the caloric heat. He derives partial differential equations to which he applies Fourier's method to find particular solutions. 9 His result, later refined by Poisson 42 (1829), de Saint-Venant 43 (1845), and Stokes 44 (1845) was called the "Navier-Stokes equations." Maxwell, 45 supplementing his dynamical theory of gases, derived the Navier-Stokes equations by assuming his distribution function of gas molecules. A first attempt to solve the Navier-Stokes equations for shock waves was first successfully achieved by Becker (-~ 1921).
1823
Ecole Polytechnique, Paris
Poisson 46 reviews the present state of treating the problem of calculating the velocity of sound. He writes, "The sound velocity in air, derived from a formula given by Newton [Principia, Book II, Scholium (after Proposition)I, differs from the observed velocity and surpasses the calculated velocity by a fifth [20%]. When Lagrange [Misc. Taur. 1, I-X, 1-112 (1759)] in his first studies on the theory of sound arrived at the same formula; he tried of course to explain this discrepancy between calculation and observation. His analysis was based on two suppositions: the minuteness of the air vibrations, and the proportionality of the elastic force with density. He proved first, against Euler's opinion [M~'m. Acad. Sci. Berlin 15 (1759)], that the amplitude of vibration does not influence the magnitude of the sound velocity; in addition he mentioned that one could coincide this velocity with the result of observation by supposing that the elastic force increases to a larger extent than the density; but he could not give any particular reason for this increase of elasticity which cannot be described by the general law of compression for air. It is not less true that this increase is really due to the air motion: It is Mr. Laplace [Ann. Chim. & Phys. 3,238-241 (1816)] who pointed out the true reason who fully explained and eliminated the difference between Newton's formula and the measurement. This cause is the release of heat which always occurs during the compression of air or the production of coldness which goes along with dilatation, likewise . . . . " Poisson, 47 stimulated by Laplace's approach, derives
History of Shock Waves
35 in the same year the two gas laws for adiabatic compression. A gas with a ratio of specific heats k, initially at pressure p, density p, and temperature O and then adiabatically compressed, reaches a new state (p', p', O') which is given by p' - p(p,/p)K and O' = (266.67 + O)(p'/p) k-1. He writes, "These equations contain the elasticity and temperature laws of gases, which are compressed or expanded without a variation in their heat quantity; which will happen when the gases are in the heat-proof glass container, or when the compression, as with the sound phenomenon, will be so fast that one can assume that the heat loss is negligible . . . . " . Poisson assumed an absolute zero at -266.67~ which we have to replace today by-273.15~ His equation of state for adiabatic compression, applicable only for sound waves of infinitesimal amplitudes and later be called the Poisson isentrope or static adiabate, was a major achievement toward the dynamic adiabate of a shockcompressed gas by Hugoniot ( - , 1887).
1824-1825
Port Bowen, Prince Regent's Inlet, Canada
Capt. Parry 48 of the Royal Navy, famous polar explorer who searches for a Northwest Passage, spends the winter in Port Bowen studying the Eskimos and, while waiting for the ice to break through, gathers scientific data. Together with Lieut. Foster, participating in the expedition as an assistant surveyor, he performs experiments on the velocity of sound "to determine the rate at which sound travels at various temperatures and pressures of the atmosphere." The measurements at very low temperatures, i.e., in air of perfect dryness, are considered to be of particular interest because they avoid any corrections of sound velocity data caused by the humidity of the atmosphere. Using a six-pounder brass gun placed on the beach at the head of Port Bowen and fired on signal from the "Hecla", Parry and Foster, carefully noting the interval elapsed between the flash and report at a distance of about 3.9 km by the beats of a pocket-chronometer held at the ear of each observer, notice an anomalous high velocity of sound. 9 Parry 49 had already made similar observations during his second polar voyage. In the appendix of his report, obviously written by Foster, it reads: "The experiments on the 9th February 1822, were attended with a singular circumstance, which was--the officers' word of command "Fire", was several times distinctly heard both by Captain Parry and myself [Foster], about one beat of the chronometer after the report of the gun; from which it would appear, that the velocity of sound depended in some measure upon its intensity . . . . " Contemporary naturalists attributed their unusual findings to possible influences of humidity and wind that they had not
36
P. Krehl precisely recorded, but Parry and Foster 5~ replied, "it was certainly far from our intention to oppose our opinions on these points to those of Newton and Laplace. We considered our remark at the time, as a fair deduction from our own experiments, without at all considering with what theory it might be at variance: our only wish being, to furnish data for philosophers to arrive at such laws as will make the computed and observed velocities of sound agree more exactly with each other, than appears to be the case, in the present state of our information of all the modifying circumstances to which the motion of sound is subjected." Parry's unusual observations, supported also by those of Ross who participated in the expeditions, were later cited by Earnshaw (--+1858) as an experimental proof of his mathematical theory of sound that intense air waves travel more quickly than weaker waves.
1826
Lake of Geneva, Switzerland
Colladon, a Swiss apothecary, measures quite accurately the sound velocity in water. As a strong source of sound he uses a bell placed under water and triggered simultaneously with a cannon, and for sound detection a long ear-trumpet submerged 5 m under the surface. To get a high accuracy he chooses the broadest part of the Lake of Geneva (about 8 km), and to better visualize the cannon fire he performs the measurements at night.
1828
Geneva
Colladon and Sturm, 51 the latter being a Swiss private tutor, publish data on the compressibility of various substances and on the measurements of the sound velocity in the Lake of Geneva. They show that putting the measured data of compressibility of water into Poisson's formula for the speed of sound yields a value of 1437.8m/s, which is in close agreement with the measured value of 1435 m / s in the Lake of Geneva. They also report on the measurement of heat emitted by liquids following the application of strong and sudden pressures. Their results earn them the prize set by the Paris Academy.
Compagnie du canal des Ardennes, France
Bdanger 52 investigates in a pioneering study the behavior of water flow in an open channel and high-speed shooting with sudden changes in depth, known as a hydraulic jump, the oldest known type of discontinuous wave motion and well resolvable with the naked eye. He derives a formula for the height Ah = h 2 - h 1 of a hydraulic jump in terms of the initial water depth h 1 and the velocity v of the jump, given by Ah = 0.Se - hi 4- (0.25g 2 q- hi) 1/2, with g = v2/2 g. 9 His remarkable study is an early attempt to characterize the propagation speed of a discontinuous wave front by its
History of Shock Waves
37 strength,--in his case of a hydraulic jump propagating in (incompressible) shallow water--by its step height. Analogically, in the case of a shock wave advancing in a (compressible) fluid in a layer of invariant thickness this would correspond to a step increase in density at the shock front. Jouguet 53 showed that, using his classic theory, the loss of internal energy [perte de charge] of a hydraulic jump can be described in terms of the difference in water heights, which is only a particular case of Hugoniot's law of the dynamic adiabate (---~1887) when the water is considered as an adiabatically moving "hydraulic gas" with 7 = 2. The analogy between a hydraulic jump and a shock wave has fascinated shock wave researchers from the early times to now (Preiswerk --~ 1938).
1830
Ecole Polytechnique, Paris
Arago 54 discusses in detail the possible causes of frequent boiler explosions of steam engines, which typically result in many casualties and heavy damage to adjacent facilities. Particularly addressing the dangers emanating from the use of Papin's safety valve (1679), he points out that many valve constructions are too narrow to allow a quick release when the internal boiler pressure suddenly increases (Airy--~1863)--a dangerous phenomenon for which he mentions various causes. 9 The limitation of the outflow of fluid through small openings became a much-discussed subject among engineers as well as scientists (Bernoulli 1738; De Saint-Venant and Wantzel--~ 1839; Napier--~ 1866; Reynolds--~1885; De Laval--~1888; Salcher and Whitehead--~ 1889; L. Mach-~ 1897; the Emden Brothers---~ 1899; Stodola--~ 1903; Prandtl---~ 1904; etc.). The topic stimulated the evolution of supersonic flows and promoted the effective operation of steam turbines.
1834
Chair of Physics, Kings College, London
Wheatstone 55 first uses a rotating mirror as a diagnostic tool to resolve high-speed phenomena. 9 34 years later Sabine, 56 then president of the Royal Society, states at his presentation of the Copley Medal to Wheatstone, "But no series of his researches have shown more originality and ingenuity than those by which he succeeded in measuring the velocity of the electric current and the duration of the spark. The principle of the rotating mirror employed in these experiments, and by which he was enabled to measure time to the millionth part of a second, admits of application in ways so varied and important that it may be regarded as having placed a new instrument of research in the hands of those employed in delicate physical inquiries of this order." The rotating mirror, subsequently used by Foucauh 57 (1850) and Feddersen 58 (1858) in sensational experiments, became an important
38
P. Krehl element in the later technique of high-speed rotating-mirror cameras. Stimulated after 1945 by the need to study shock wave effects involved in the development of nuclear weapons, work with these cameras resulted in very sophisticated ultrahigh-speed cameras incorporating helium-driven turbines. 59
1838
ILAcadCmie des Sciences, Paris
Arago 6~ publishes an essay on thunder for the Annuaire du Bureau des Longitudes at Paris, in which he gives a masterly historical sketch of the real facts that have hitherto been accumulated. From these he deduces the inferences, scientific and practical, that may legitimately be drawn. He discusses also ball lightning and analyzes a number of evidently reliable observations, pointing out that an observer, viewing the descent of the ball at an angle from the side, is not subject to the optical illusion described. Shortly after, Faraday 61 will give essentially the same explanation, stating that the optical illusion is caused by an afterimage perceived by eyes that just have seen the dazzling flash of an ordinary bolt. 9 Ball lightning has a diameter somewhere between a golf ball and a large beach ball, moves horizontally at low speed and can decay silently or explode violently. Ball lightning has been well documented since the Middle Ages as a natural but rare phenomenon associated with thunder, but still is an enigma to modem science. 62
1839
Ecole des Ponts et Chauss~es, Paris
De Saint-Venant and Wantze163 study compressible flow in a duct of changing area and the exhaust of compressed air from a small opening. Using Poisson's adiabatic law ( - , 1823) and Bernoulli's energy equation, they assume compressible flowm i.e., p = p(p)--and express the difference of enthalpy by the pressure integral. This leads to their famous fundamental formula relating outflow velocity V at given pressure p in the pressure reservoir by V2 = [1 - (p/po)"]2po/mpo with m - - ( • - 1)/~'. For an outflow into a vacuum (p = 0), the maximum outflow velocity is given by Vma• -(2po/mpo) 1/2, which, in the case of air (to = 1.405), amounts to 757 m/s.
1840
Chair of Physics, Kings College, London
Wheatstone 64 invents the first electric chronoscope to measure projectile velocities by employing an electromagnetically controlled mechanical stopwatch.
1842
Meeting of the Association for the Advancement of Science, Manchester
Russell 65 coins the expressions great solitary wave (or wave of the first order or wave of translation), a single hump of constant shape and constant speed which, moving on the surface of an inviscid incompressible fluid, is capable of traveling in a uniform channel a considerable distance with almost no change. Referring to his former studies (18331840), he reports on the reflection of such a wave type at a
History of Shock Waves
39 solid boundary, "when the angle of the ridge with the surface is small, not greater than 30 ~ the reflexion is complete in angle and in quantity. When the ridge of the wave makes an angle greater than 30 ~ the angle of reflexion is still equal to the angle of incidence, but the refected wave is less in quantity than the incident wave.., when the angle of the ridge of the wave is within 15 ~ or 20 ~ of being perpendicular to the plane [i.e., at an angle of incidence within 75 ~ or 70 ~, annotation by the author], reflexion ceases, the size of the wave near the point of incidence and its velocity rapidly increases, and it moves forward rapidly with a high crest at right angles to the resisting surface. Thus at different angles we have the phenomenon of total reflexion, partial reflexion, and, non-reflexion and lateral accumulation; phenomena analogous in name, but dissimilar in condition from the reflexion of heights, &c." 9 His "lateral accumulation" of the reflected wave front, merging with the front of the incident wave, creates a new wave front that extends at a right angle to the boundary. This phenomenon of irregular wave reflection, found for the interaction of hydraulic jumps, was rediscovered by Mach and Wosyka (-+ 1875) for the case of interacting aerial shock waves. It was brought again to light by von Neumann (---~1943), who called it "Mach reflection."
1844
Conservatoire des Arts-etMetiers, Paris
Pouillet 6~ describes an electric circuit to measure the duration of short current pulses by studying its action on the magnetic needle of a galvanometer, a method based on the principle of the ballistic pendulum and later renowned as the ballistic galvanometer. He considers a precise time measurement as essential for the better understanding of high speed events, such as the ignition process of gunpowder and the contact duration of impacting bodies. His concept, later refined by Ramsauer (-+1909), will allow even the measurement of times in microseconds.
1845
Haswell Collieries, Durham District, England
Faraday and Lyell,67 investigating possible causes of a serious explosion at Haswell Collieries, observe many signs of the coal dust being partly burned and partly subjected to a charring or shocking action. Their conclusions that coal dust adds considerably to the disastrous effects of firedamp explosions and that proper ventilation is an effective means of preventing similar accidents will be confirmed later by leading French mining engineers (--+ 1890).
Royal Observatory, Greenwich
Airy, 68 Astronomer Royal, analytically treats the motion of "waves of finite amplitudes" in a uniform water canal of rectangular section and finds, by the method of successive
40
P. Krehl approximation, that in a progressive wave different parts travel with different velocities. In particular, he makes important statement that the crests tend to gain upon hollows so that the anterior slopes become steeper steeper.
1847
1848
will the the and
Chair of Chemistry, University of Basel
Sch6nbein experiments with the nitration of cellulose and invents nitrocellulose. 69 He communicates his process to John Taylor, 7~ who in the following year will be granted an English patent. The detonation of guncotton, however, is difficult to control, and it will take more than 40 years to convert nitrocellulose into a reliable gun propellant. 9 In 1862 von Lenk in Austria will try to use guncotton as an explosive. However, acid residues in guncotton, originating from the production process, provoked dangerous self-ignitions. This problem was not solved until 1866 by Abel 71 in England.
Technical State Academy, Prague
Doppler 72 speculates that the propagation velocity of sound should increase with increasing intensity and reflects on acoustical consequences of his discovered "Doppler effect." He discusses what might happen to a disturbance that, propagating with a velocity u, moves faster than the sound velocity a of the surrounding medium. Assuming a sequence of explosionlike emissions of disturbances, he graphically constructs the cone geometry for the three cases of disturbances propagating with a constant, an increasing, and a decreasing supersonic velocity. Doppler shows that, for a constant velocity u, the half-cone angle ~ of this cone allows one to determine the velocity u by the simple relation sin ~ = a/u. 9 His purely theoretical results were confirmed 40 years later by Mach and Salcher's ballistic experiments (--~ 1887). Mach designated the envelope of this cone as the "head wave" and Prandtl (---~1913) the cone angle ~ as the "Mach angle."
Chair of Applied Chemistry, University of Turin
Sobrero, 73 a former student of von Liebig and Pelouze, reports in a letter to Prof. Pelouze on his discovery of nitroglycerine. The increasing number of new explosive substances discovered in the following years will stimulate physicists and chemists to uncover the puzzle of detonation, which is felt to be somehow closely related to rapidly propagating mechanical waves.
Royal Observatory, Greenwich
Airy 74 shows that "the velocity does not depend on the absolute pressure of the air in its normal state of density, but upon the proportion of the change of pressure to the change of density. This is increased by the suddenness of
History of Shock Waves
41 condensation in one part, which, when the elastic force is great, makes it still greater--and by the suddenness of rarefaction in another part, which, when the elastic force is small, makes it still smaller,--thus in both ways increasing the change of pressure."
Cambridge Observatory
Challis 75 resumes the classical analytical problem of the velocity of disturbances. Obviously not knowing Poisson's previous work (--+ 1808) but commenting on Airy's remarks (---, 1848) on his theory of sound, he finds that for waves of finite amplitudes propagating in a perfect gas the velocity of propagation alters as it advances and tends ultimately to become a series of sudden compressions followed by gradual dilatations. The velocity of propagation is greater than the sound velocity and certain faster parts in the wave profile will take over the slower ones, thus leading to ambiguous mathematical solutions (the "Challis paradox").
Pembroke College, Cambridge
Stokes 76 replies to Challis' claim of a contradiction in the commonly accepted theory of sound. Stokes, assuming an isothermal gas, introduces surfaces of discontinuity in the velocity and density of the medium, thereby eluding the Challis paradox. He indicates that small pressure disturbances might create compression waves with discontinuous fronts, because each subsequent sound wave will propagate in a medium with a slightly higher sound velocity. He writes, "Of course, after the instant at which the expression (A) becomes infinite, some motion or other will go on, and we might wish to know what the nature of the motion was. Perhaps the most natural supposition to make for trial is, that a surface of discontinuity is formed, in passing across which there is an abrupt change of density and velocity. The existence of such a surface will presently be shown to be possible . . . . The strange results at which I have arrived appear to be fairly deducible from the two hypotheses already mentioned. It does not follow that the discontinuous motion considered can ever take place in nature, for we have all along been reasoning on an ideal elastic fluid which does not exist in nature. In the first place, it is not true that the pressure varies as the density, in consequence of the heat and cold produced by condensation and rarefaction respectively. But it will be easily seen that the discontinuous motion remains possible when we take account of the variation of temperature due to condensation and rarefaction, neglecting, however, the communication of heat from one part of the fluid to another. Indeed, so far as the possibility of discontinuity is concerned, it is immaterial according to what law the pressure may increase with the
42
P. Krehl density . . . . " He also first derives the conservation relations of mass and momentum that are now usually attributed to Rankine and Hugoniot.
1849
Royal Observatory, Greenwich
Airy 77 first points out the analogy between the velocity change in waves of sound of finite amplitudes and that which takes place in sea waves when they roll into shallow water.
1850
University of Edinburgh
Maxwell 7s publishes a paper on the theory of elasticity. He shows that two elastic constants are necessary to describe the elastic behavior of an isotropic solid, and completely develops the technique of photoelastic stress analysis. 9 Because it permits the determination of the entire stress field, his method proved very useful in the study of impact-induced shock wave propagation in 2-D birefringent solid specimens. 79
1851
Private laboratory at Lacock Abbey, U.K.
Fox-Talbot s~ performs the first microsecond snapshot photo from a page of the London Times rotating at high speed on a revolving disk by using an electric spark from a Leiden jar as a flash light source. He states, "it is in our power to obtain the pictures of all moving objects, no matter in how rapid motion they may be, provided we have the means of sufficiently illuminating them with a sudden electric flash." To obtain the necessary high sensitivity, he uses albumine plates, which he exposes immediately after sensitization. 9 This experiment was an important step toward single-shot photography because it first proved the excellent property of film to freeze high-speed events for a later detailed analysis.
Consultant Engineering, Edinburgh
Rankine, sl civil engineer and independant scholar, addresses the sound problem and previous arguments given by Laplace (-+ 1816) and Airy (-+ 1848), and states: "Now the velocity with which a disturbance of density is propagated is proportional to the square root, not of the total pressure divided by the total density, but of the variation of pressure divided by the variation of density . . . . It is therefore greater than the result of Newton's calculation, and this, whether the disturbance is a condensation or a dilatation, or compounded of both."
Parish of Sheffield
Earnshaw, mathematician and chaplain, observes unusual sound phenomena that he later s2 will describe as follows: "a thunder-storm which lasted about half an hour was terminated by a flash of lightning of great vividness, which was instantly (i.e., without any appreciable interval between) followed by an awful crash, that seemed as if by atmospheric concussion alone it would crush the cottages to ruins. Every one in the village had felt at the moment of the crash that the electric fluid had certainly fallen somewhere in the village . . . .
History of Shock Waves
43 But, to the surprise of everybody, it turned out that no damage had been done in the village, but that that flash of lightning had killed three sheep, knocked down a cow, and injured the milkmaid at a distance of more than a mile from the village... " . Since sound needs about 41 seconds to cover an English mile ( = 1523m) and Earnshaw noticed that lightning flash and thunder was felt almost simultaneously even though the strike happened more than a mile away, he correctly stated that intense sound, such as originating from a thunderclap, must propagate with supersonic velocity. Seven years later he will present his "theory of sound of finite amplitudes" (Earnshaw---~ 1858).
Pembroke College, Cambridge
Stokes s3 submits the view that during the propagation of pulses in an elastic fluid compressions and expansions of the particles take place so rapidly that there is no time for any appreciable transmission of heat between different particles, thus showing that Challis' supposition 84 that the developed heat is lost by radiation is untenable, and that Laplace's view (--, 1816) has a real physical foundation.
1854
Bethelehem Zinc Works, PA
First unusual explosion accident of finely powdered zinc ("metal-dust explosion"). 85
1856
Ecole Polytechnique, Paris
Jamin 86 invents the first optical interferometer, the archetype of many subsequent interferometer constructions, and applies it to measure the refractive index of gases. 9 Already 14 years later A. Toepler and Boltzmann 87 introduced optical interferometry in fluid dynamics (acoustics) to determine the amplitude at the threshold of heating, a masterpiece of experimental physics.
KOnigliche Realschule, Berlin
Kr6nig 88 publishes the first theory of gases. Following D. Bernoulli's model (1738), he assumed that a gas consists of discrete particles (molecules), each of which behaves according to universal mechanical laws. 9 Subsequently, Clausius 89 (1857), Maxwell 9~ (1859-1879), and Bohzmann 91 (18681904) made important improvements and today are regarded as the main founders of the kinetic theory of gases. 92
Private laboratory in Joule's brewery at Salford, Manchester & Chair of Physics, University of Glasgow
Joule and Thomson 93 treat thermal effects of bodies moving through air (aerodynamic heating) and conclude "that a body round which air is flowing rapidly acquires a higher temperature than the average temperature of the air close to it all round." In addition, they note that "the same phenomenon must take place universally whenever air flows against a solid or a solid is carried through air. If the velocity of 1780 feet per second in the foregoing experiment gave 137~ difference of temperature between the air and
44
e. Krehl the solid, how probable is it that meteors moving at from six to thirty miles per second, even through a rarefied atmosphere, really acquire, in accordance with the same law, all the heat which they manifest! On the other hand, it seemed worth while to look for the same kind of effect on a much smaller scale in bodies moving at moderate velocities through the ordinary atmosphere.., we have tried and found, with thermometers of different sizes and variously shaped bulbs, whiled through the air at the end of a string, with velocities of from 80 to 120 feet per second, temperatures always higher than when the same thermometers are whirled in exactly the same circumstances at smaller velocities." In the case where the velocity of translation of the body, v, is a small fraction of the velocity of sound, a (-- 1115 ft/sec at 17~ they estimate for the "hot spots" at the body's surface--i.e., at those points where the flow velocity is slowed down to zeroma temperature increase AO [~C] -- 58.8(v/a) 2. For a bulb thermometer moving at v - 183ft/sec, they measure a temperature rise of AO = I~ Their theoretical value, according to the formula given above, would yield a temperature increase of 1.5~ 9 Aerothermodynamics (G. A. Crocco--,1931), at Joule's time rather a subject of academic curiosity, became immediately important after World War II when supersonic flight could be realized not only in the military realm but later also in civil aviation (Tupolev Tu-144 and Concorde).
1857
Allegheny Arsenal, PA
Rodman 94 invents his "indentation gauge" to measure the maximum internal pressure in a gun. It consists of a piston working in a hole bored into the wall of a gun and acting on an indenting tool, for the purpose of determining the pressure in the bore at different points. With the help of this gauge, he discovers that the maximum pressure in a gun decreases with increasing grain size of the gunpowder (termed "mammoth powder" or "Rodman powder"). This finding becomes important for large-caliber guns to reduce the danger of damaging the barrel.. 15 years later, Noble 9~ who first critically studied the pressure data obtained by the Rodman gauge, stated; "It is curious that so distinguished an artillerist as Major Rodman should never have taken the trouble to calculate what energies the pressure which his instrument gave would have generated in a projectile; had he done so he would have found that many of the results indicated by his instrument were not only improbable but were absolutely impossible." Contrary to Rodman, Noble first correlated measured pressure data in the bore with theory using measured kinematic data of the projectile (Noble -+ 1872).
History of Shock Waves 1858
45
Herzogliche Realschule, Meiningen, Saxony
Knochenhauer 96 studies an electric discharge circuit that consists of two Leiden jars coupled to each other. It will be modified later by A. Toepler (---~1864) and further improved by E. Mach (---~1878) to control a delay pulse in the microsecond regime, an important requirement to stop motion of shock waves within a given field of view for recording purposes.
British Association for the Advancement of Science, Leeds
Earnshaw 97 presents on November 20 his famous theory of sound of finite amplitudes, which, published two years later, is the most complete. About the objective of his work he writes, "I consider this article as tending to account for the discrepancy between the calculated and observed velocities (which most experimentalists have remarked and wondered at), when allowance is made (as will be done in a future part of this paper) for change of temperature .... " He improves Poisson's one-dimensional theory of finite amplitude disturbances (---~1808), putting the equations into a form, in which the motions of particular particles are followed (Lagrangian coordinates). Using the adiabatic law, he obtains a complete solution for a wave progressing in one direction in a medium in which the pressure is any function of waves of the density, and observes that the differential equations of motion need not necessarily possess a unique solution for the velocity. He assumes that in a real fluid heat conduction and viscosity might prevent the true formation of a discontinuity, and speculates: "I have defined a bore to be a tendency to discontinuity of pressure; and it has been shown that as a wave progresses such a tendency necessarily arises. As, however, discontinuity of pressure is a physical impossibility, it is certain Nature has a way of avoiding its actual occurrence. To examine in what way she does this, let us suppose a discontinuity to have actually occurred at the point A, in a wave which is moving forwards. Imagine a film of fluid at A forming a section at right angles to the tube. Then on the back of this film there is a certain pressure which is discontinuous with respect to the pressure on its front. To restore continuity of pressure, the film at A will rush forward with a sudden increase of velocity, the pressure in the front of the film not being sufficient to preserve continuity of velocity. In so doing the film will play the part of a piston generating a bit of wave in front, and a small regressive wave behind. The result will be a prolongation of the wave's front, thereby increasing the original length of the wave, and producing simultaneously a feeble regressive wave of a negative character . . . . " He draws the important conclusion that "the velocity with which a
46
p. Krehl sound is transmitted through the atmosphere depends on the degree of violence with which it was produced . . . . The report
of fire-arms will travel sensibly faster than a gentle sound, such as the human voice. . . . " . The transactions of this meeting 98 later read: "Fortunately, it transpired at the Meeting, that in Captain Parry's Expedition to the North, whilst making experiments on sound, during which it was necessary to fire a cannon at the word of command given by an officer, it was found that the persons stationed at the distance of three miles to mark the arrival of the report of the gun, always heard the report of the gun before they heard the command to fire; thus proving that the sound of the gun's report had outstripped the sound of the officer's voice; and confirming in a remarkable manner the result of the author's mathematical investigation, that the velocity of sound depends in some degree on its intensity." It was James C. Ross, later becoming a famous South Pole explorer and carrying out important arctic and antarctic magnetic surveys, who was in command of the cannon during Parry's expedition (--+1824-
]825). 1859
Royal Scientific Society, GOttingen
Riemann 99 presents on November 22 his "theory of waves of finite amplitudes," which, not limited to a single progressive wave as was Earnshaw's solution (--+ 1858), is put on a more general basis and suited to calculate the propagation of plane waves of finite amplitude proceeding in both directions. Limiting his study to a steady two-dimensional flow and considering motions occurring at a fixed point in the gas (Eulerian coordinates), he assumes a pressure-density relation p = p(p) that depends only on density and holds for all particles and all time, even across shocks, i.e., limits to adiabatic motion in the case of weak shocks. To find the essential propagation properties of waves of finite amplitudes, he integrates the partial differential equations using Monge's "method of characteristics" (--+ 1770s) which simplifies under the assumption that the sound speed is a function of density alone ("Riemann invariants"). He shows that an original disturbance splits into two opposite waves: the rarefaction wave grows thicker, and the condensation wave (a shock wave) thinner which he calls a "compression shock" [Verdichtungsstofg]., His results formed an important step toward a mathematical treatment of shock wave steepening and formation. However, using the "static adiabate" he incorrectly assumed that the entropy remains unchanged through the shock wave (isentropic process). The total energy content
History of Shock Waves
47 (enthalpy) remains unchanged, whereas the entropy always increases through a shock wave; this was first recognized by Rankine (-+1869) and later, independently, by Hugoniot (-+ 1887).
Private observatory, Redhill, Surrey, England
Carrington, 1~176 using a telescope, observes a violent and rapid eruption near a large sunspot. At that very moment modest, but very marked disturbances of three magnetic elements are observed at Kew Observatory, affecting all the elements simultaneously and commencing quite abruptly. He reports, "While engaged in the forenoon of Thursday, September 1, in taking my customary observation of the forms and positions of the solar spots, an appearance was witnessed which I believe to be exceedingly rare. The image of the sun's disk was, as usual with me, projected on to a plate of glass coated with distemper of a pale straw color, and at a distance and under a power which presented a picture of about 11 inches diameter. I had secured diagrams of all the groups and detached spots, and was engaged at the time in counting from the chronometer and recording the contacts of the spots with the crosswires used in the observation, when within the area of the great north group (the size of which had previously excited great remark), two patches of intensely bright and white light broke out, in the positions indicated in Fig. 1 . . . . My first impression was that by some chance a ray of light had penetrated a hole in the screen attached to the object glass, for the brilliancy was fully equal to that of direct sun-light; but by at once interrupting the current observation, and causing the image to move... I saw I was an unprepared witness of a very different affair. I therefore noted down the time by the chronometer, and seeing the outburst to be very rapidly on the increase, and being somewhat flurried by the surprise, I hastily ran to call some one to witness the exhibition with me, and on returning within 60 seconds, was mortified to find that it was already much changed and enfeebled. Very shortly afterwards the last trace was gone. In this lapse of 5 minutes, the two patches of light traversed a space of about 35,000 miles." m About 17 hours later other researchers observed considerable magnetic disturbances, a so-called "magnetic storm" [magnetisches Ungewitter], a term which was introduced by A. yon Humbold 1~ the year before. Carrington's unique observation made evident for the first time the enormous dimensions and dynamics of solar flare explosions (Chapman and Ferraro--+1931; Gold 1949; Parker 1961) and started much discussion on the coincidence of solar eruptions and magnetic disturbances.
48 1860
E Krehl
Parish of Sheffield, England
Earnshaw 1~ discusses the problem of whether violent sounds would propagate more rapidly than gentle sounds. He distinguishes three kinds of waves, all propagating with different velocities v in regard to the sound velocity a: minute waves (v = 0 to 0.Sa), ordinary waves, (v---0.Sa to a), and violent waves (v -- a to oo). Although this arbitrary classification is rather hypothetical, he draws a very important conclusion: "If the theory here advanced be true, the report of fire-arms
should travel faster than the human voice, and the crash of thunder faster than the report of a cannon." 9 Earnshaw obtained a memoir from Montigny, 1~ professor at Antwerp, who observed that in the case of a thunderclap sound is sometimes propagated with a velocity far greater than the ordinary sound velocity, a phenomenon that Earnshaw (---~1851) had already noticed and that was much discussed among scientists. For example, in the same year Hirn, 1~ French autodidact and independent scholar at Colmar, speculated on possible reasons why the velocity of sound depends on intensity and assumed a pressure-dependent ratio of the specific heats at constant pressure and volume. Raillard, 1~ French abbot and amateur naturalist, referred to Biot, who had previously had a discussion with Poisson on irregular propagation phenomena of thunder. The latter, however, although essentially supporting this idea, did not resume it in his M~noires sur la theorie du son (1808). Raillard speculated also on the propagation velocity of thunder but estimated abnormally high velocities (5000-6600m/s). He wrote, "I heard the first outbursts of thunder three or four seconds after the lightning had appeared; however, according to the delay of the reinforcement of the noise originating from the stem of the lightning, and to its orientation, I estimated that the fire was lit in the vicinity of Gray, about 20 km from Courchamp where I was . . . . " 1862
Nitroglycerin Ltd., Heleneborg, Sweden
Nobel 1~ applies for a patent on the improvements in the process of manufacturing Nitroglycerin which is called Pyroglycerin, then Glonoine Oil, and later Nobel's Blasting Oil. He erects works at Heleneborg, an isolated area outside Stockholm, where nitroglycerine is manufactured for the first time on a commercial scale. 9 Two years later they were entirely wrecked by an explosion which cost the lives of Nobel's youngest brother and his chemist Hertzmann.
1863
Ecole Pyrotechnique, Brttxelles
Le Boulenge 1~ invents an electrically triggerable dropping weight timing system (the Le-Bouleng~-chronograph) that after some improvements will become a robust and accurate chronograph with a temporal resolution of less than i ms. It
History of Shock Waves
49 will prove its applicability even for ballistic "open range" measurements.
University of Cambridge
Airy 1~ reviews previous theoretical and experimental attempts to calculate the destructive energy of steam boiler explosions. He concludes that one cubic foot of water at 60psi is equal to the destructive energy of one pound of gunpowder. 9 Various hypotheses for possible causes of steam boiler explosions 1~176 were (i) generation and ignition of oxyhydrogen when, at low water level in the boiler, the water chemically reacts with the overheated iron walls; (ii) sudden destruction of the initial isolation of water from the boiler walls, nullifying the protecting "Leidenfrost layer"; (iii) reduction of mechanical strength of the boiler material at high temperature; (iv) sudden generation of large quantities of steam by the phenomenon of "delay of boiling"; and (v) increasing unyieldingness of the boiler walls when firing sulfurous coal. The accidents prompted engineers and metallurgists to study dynamic material behavior under thermal and mechanical stress and to improve production technology. They also provoked the foundations of the first official safety inspection authorities.
Colli~ge de France, Paris
Regnault 111 begins a five-year campaign of measuring the sound velocities in air and other gases. To exclude negative side effects such as wind he performs his experiments in long pipes with lengths up to 20 km and diameters ranging from 0.1 to 1.1 m, using the gas pipeline and sewage channel system of Paris. This allows long base lines to compensate for the low accuracy of available chronoscopes. Discharging a small quantity of gunpowder (about i g) at the pipe entrance, he determines the average blast velocity by mechanically recording the arriving pressure signal at the pipe end using a membrane microphone, combined with a rotating drum chronograph. He is the first to confirm experimentally that the sound velocity also depends on the sound intensity, thereby touching an essential feature of a shock wave. Since his remarkable achievements have barely been acknowledged by the modem shock physics community, he is cited here in more detail: "the theoretical calculation assumes that the excess of compression which exists in the wave is infinitely small compared with the barometric pressure supported by the gas. But the experiments made to determine the rate of sound in free air have been hitherto made by means of a cannon, and the wave has been reckoned from its source, namely the cannon's mouth. Now this wave as it leaves the cannon is under enormous compressionma compression, it is
50
P. Krehl true, which diminishes very rapidly as the wave spreads spherically through space; but during the first part of its course it cannot be supposed that its compression is infinitely small. When the excess of compression in the wave is a sensible fraction of the compression of the gaseous medium at rest, we can no longer employ Laplace's formula, but must have recourse to a more complex formula embracing the true elements of the problem. Even the formula which I have given in my Memoir [MCm. Acad. Roy. Paris 37 (1868)] is only an approximation; for it implicitly admits Mariotte's law and all its consequences. In short, the mathematical theory has as yet only touched upon the propagation of waves in a perfect gasuthat is to say, in an ideal fluid possessing all the properties which had been introduced hypothetically into the calculation. It is therefore not surprising that the results of my experiments often disagree from theory . . . . " 9 His remarkable result, however, that intense sound propagates faster than with sound velocity, was not immediately accepted. 112 E. Mach and Sommer (---~1877) first confirmed Regnauh's observations.
1864
Royal Agricultural Academy, Poppelsdorf, Germany
Toepler 113 publishes his "schlieren method." Although its principle was previously discovered by Hooke 114 (1665) and Foucauh 115 (1859), Toepler uses an arrangement that will prove extremely useful in the study of compressible flow. He directly visualizes the propagation and reflection of shock waves in air and first notices the sharp wave front, but is at first confused by the appearance of several shock fronts: "Apart from the envelope and little clouds, the spark seems to be surrounded by concentric spheroids a b c with rather sharp boundaries. They are never disrupted or bulged; with increasing size they approach a spherical geometry. Closely to the spark they resemble a cylinder which is bounded by two hemispheres. Operating the induction coil at high repetition rate they give the impression of soap bubbles which, formed around the spark, immediately disappear again. It makes one believe that always several, usually three or four, are visible simultaneously in the field of view. However, in the case that the coil is working at the lowest possible rate so that the ear is capable of clearly differentiating between each stroke, it is obvious that each discharge corresponds to only a single one of the above described spheroids, but that from spark to spark, the phenomenon strongly varies in size and formation." To illustrate this discontinuous wave phenomenon, he first uses the correct terms shock wave [Stoj~welle] and air percussion wave [Lufterschfitterungswelle], but likewise also the incorrect term sound wave [Schallwelle]. Since high-
History of Shock Waves
51 sensitive films are not yet available to him, he studies the shocks subjectively by using a sophisticated stroboscopic arrangement and a modification of Knochenhauer's circuit (---~ 1858) to delay the illumination spark relative to the spark generating the shock wave. He also inspects the spark channel and notices that it is not a homogeneous cylindrical plasma column but rather is pinched and shows constrictions in the axial direction.
Heleneborg, Sweden
Nobel 116 finds that nitroglycerine (Sobrero--~1847) can be fired by an initial explosion such as can be produced by a small charge of gunpowder, and soon experiments with small metal receptacles loaded with fulminate of mercury mixed with gunpowder or nitrate of potash. His invention of the blasting cap ("detonator") initiates the explosive reaction in a column of explosive by percussion, or the local heat of an electric spark or an electrically heated w i r e . , The introduction of the initial ignition principle, using a strong blast wave rather than heating, was a significant achievement in the technique of blasting. Ten years after having perfected his famous invention, Nobel 117 stated with plain words: "but the real era of nitroglycerine opened with the year 1864, when a charge of pure nitroglycerine was first set off by means of a minute charge of gunpowder."
Chair of Natural Philosophy, South Carolina College, Columbia
Le Conte 118 reviews the large body of international literature relating to the obvious discrepancy between the velocity of sound as given by the physical theory and by direct experiment. He addresses also the theories of violent sound given by Airy (--+ 1849), Earnshaw (--+ 1858), Challis (--+ 1848,1851), Stokes (--+1848,1851), and Parry's experiments (-->18241825). Previous observations on thunder by Earnshaw (---->1851,1860) and Montigny (--+1860) he considers as a psychological illusion. Rejecting all hypotheses of wave propagation attributed to the peculiarities of large amplitudes, he writes, "It is true there may be nothing a priori improbable in the assumption that the velocity of sound might be related to the violence of the disturbance; but the fact that the analytical investigations conduct to such extreme results as to set at nought all our physical conceptions, originate a strong presumption that they belong to that class of mathematical fictions which have frequently sharpened the ingenuity and brightened the imagination of some of the most eminent geometers." He supports Laplace's view "...that the accuracy of the physical reasoning upon which Laplace's formula is based has not been invalidated by the recent discussions on the mathematical theory of sound." 9 The paper is very interesting from the historical point of view,
52
P. Krehl because the large number of reasons discussed illustrates not only the keen interest of contemporary naturalists in this subject, but also reveals the difficulties to accept hitherto unknown mechanisms of generating supersonic velocities of aerial waves.
1865
1866
Athenaeum, Deventer, The Netherlands
SchrOder van der Kolk, 119 correctly assuming that intense sound propagates faster than weak sound, tries to derive a formula for the sound velocity s in terms of the ratio of the specific heats, 7; the sound velocity at infinitely small amplitude, So; and the specific volume reduction, AV = V0 - Va, caused by the intense sound. - Since he assumed compression along the static adiabate (Poisson's law) and not along the dynamic adiabate (Hugoniot curve)--which is steeper in the p,V-diagrammhis equation gives too small a velocity increase. The problem was first solved in a general manner by Hugoniot (--+ 1887) and later put in a practicable equation known today as the "Hugoniot relation" by Vieille (--~ 1900).
Stockholm
Nobel 12~ addressing the advantage and multi-purpose applications of nitroglycerine in the mining industry, writes: "The greatest advantage of nitroglycerine consists in the fact that when it is used a force can be introduced into the blast-hole of a mine ten times as great as when powder is used. Hence arises a great economy in manual labor, the importance of which is understood when it is remembered that the labor of the miner represents, according to the hardness of the rock, from five to twenty times the value of the powder required, a saving therefore which will often amount to 50 per cent. The use of this substance is very simple. If the blast-hole of the mine is fissured, it must be lined with clay in order to render it tight. Nitroglycerine is then poured in, and the upper part of the hole is filled with water; in the nitroglycerine is then introduced a safety-match of suitable length, at the end of which is pressed a strong percussion-cap. The operation is finished, and it is only necessary to put fire to the match."
The Napier Brothers, Glasgow
Napier, 121 a Scottish mechanical engineer, studies the flow characteristics of a gas, from a vessel in which it is compressed, through an orifice into the atmosphere. He observes that the rate of discharge increases as the ratio of the receiver pressure to the initial pressure diminishes from unity to about 0.5, but that when the latter stage is reached, a further reduction in the receiver pressure has no effect on the rate of discharge, which remains constant (the "choking effect").. An important step in the theory of orifice discharge was made not until 1885 by Reynolds, a22 who assumed a continuous fall of pressure along the axis of the jet.
History of Shock Waves 1867
53
Humboldt UniversitFzt, Berlin
Magnus receives Toepler's paper on schlieren observations of spark waves (--~ 1864). He criticizes Toepler's use of the term sound wave [Schallwelle] for the visualized spark (--shock) wave. In a letter 123 to Toepler he states, "I was never in doubt about the correctness of your observations...however, I have declared myself against the expression 'sound wave' as, I suppose, I already did previously. Now you state in your kind letter that the air is expanded by the spark which causes a compression propagating with the speed of sound: this is clear and nobody will contest it, just as little as this compression was reflected. However, visible is not the sound, but rather the air which, heated and perhaps colored by the spark, expands from the position of the spark and is reflected, because the compressed air itself is visible with your apparatus . . . . A designation can easily give reason for a misinterpretation. Who will not imagine waves emitted by a sounding body when hearing about 'sound w a v e s ' ? . . . " . This stimulated ToepleP 24 to give a more detailed definition in his next paper: "the electric spark is a very favorable source of sound; it can be used to provide single shocks which [at increasing repetition rate] can be driven up to the generation of a tone. The expression 'sound' has been used for any perceptible impression to the sense of hearing, likewise the word 'sound wave,' also in case that the air particles do not experience a full oscillation .... "
Nitroglycerin Ltd., Heleneborg, Stockholm
Nobel 125 invents Dynamite [after the Greek dynamis, meaning power], which he also calls Guhr Dynamite. It is a mixture of nitroglycerine and a suitable nonexplosive porous absorbent [Kieselguhr], which fully establishes nitroglycerine as the leading blasting agent. To explode it safely and under all conditions, it is ignited by a detonator cap (A. Nobel---~ 1864). 9 The invention of Dynamite was a large commercial success: From 1867 to 1874 Nobel founded 15 factories worldwide, which increased dynamite production from 11 tons in 1867 to 3120 tons in 1874.126
Chair of Chemistry [Allgemeine Experimentalchemie], University of Heidelberg
Bunsen 127 determines the explosion pressure of oxyhydrogen in a closed vessel to be around 9.5 atm. He speculates that in a gaseous explosion the total gaseous mass does not explode at once, but rather successively in discontinuous partial explosions that propagate stepwise through the gas ["e/ne discontinuierliche, gleichsam stufenweise erfolgende Verbrennung"]. To measure the rate at which an explosion is propagated in a gas he first releases highly pressurized gas from a narrow opening only 1.2 mm in diameter and ignites it in free air. Then he slowly reduces the pressure until the flame backfires, which he regards as a criterion that the explosion velocity has just
54
e Krehl surpassed the outflow velocity. He determines an explosion velocity for oxyhydrogen of only 34 m/s and for a mixture of CO-O 2 of only i m/s. 9 His puzzling results stimulated Berthelot and Vieille (--+1881), Mallard and Le Ch~telier (-+ 1881), and von Oettingen and von Gernet (--+ 1888). Their results revealed that Bunsen did not provoke an explosion but rather a deflagration, which explained his obtained low velocities.
1868
1869
4th International World Fair, Paris
Toepler 128 displays his improved schlieren apparatus and demonstrates the propagation of spark (=shock) waves to the public.
Private study at VilleporcherSaintOuen/VendOme, France
De Saint-Venant ~29 treats the longitudinal impact of elongated bars, which, for simplicity, are assumed to be of the same material and thickness but of different length. He shows that, except when the lengths are equal, a considerable fraction of the original energy takes the form of vibrations in the longer bar so that the translational velocities after impact are less than those calculated by Newton for bodies that he calls "perfect elastic." He observes that after impact the short bar will take the initial velocity of the longer bar and becomes free of tension.
St. Petersburg, Russia
The International Treatise of Petersburg is proposed on December 11, with the goal to ban the use of explosive bullets for small arms. It is signed by all European and North American countries.
University of Heidelberg
Kirchhoff, 13~ after having read Toepler's paper, on the visualization of shock waves, 131 writes to him: "without doubt the expression 'sound wave' as you use it is justified, and an air quake [Lufierschfaterung] makes an impression on the ear even if it is of only very short duration but of sufficiently high intensity."
Royal Military Academy, Woolwich, Arsenal, London
AbeP 32 shows that unconfined charges of guncotton, nitroglycerine, dynamite, and mercury fulminate only burn if ignited by a flame or a hot-wire, but detonate if subjected to an impulsive force such as applied by a hammer blow, a detonator cap, or the impact of a projectile. 9 It appears that Abel was the first who used the term detonation in the modem sense. Hitherto the terms explosion and detonation, in use since the late 17th century, 133 were applied interchangeably.
Chair of Civil Engineering & Mechanics, University of Glasgow
Rankine TM submits a paper to the London Royal Society on "adiabatic waves, that is waves of longitudinal disturbance in which there is no transfer of heat..." and on the problem of how "to determine the relations which must exist between the laws of the elasticity and heat of any substance, gaseous,
History of Shock Waves
55 liquid or solid, and those of the wave-like propagation of a finite longitudinal disturbance in that substance." His significant achievements can be summarized as follows: (i) Treating the shock wave as a two-dimensional discontinuity, he assumes a dissipative fluidmi.e., conductive but nonviscous--and applies the conservation laws of mass, momentum and energy to both states far up- and downstream from the shock front, thus obtaining three equations which, however, are only equivalent to those of Hugoniot (--~ 1887) in the case of a perfect gas, later to be referred to as the Rankine-Hugoniot equations (or conditions). (ii) He coins the expression adiabatic [derived from the Greek ~ ~ i v ~ z v - to pass through] to characterize a change in the volume and pressure of the contents of an enclosure without exchange between the enclosure and its surroundings. He also uses the term adiabatic curve for a p(v)-diagram, obtained by plotting the pressure p against the specific volume v in the adiabatic equation. In contrast, GIBBS135 will shortly after propose the expression isotropic curve, since in an adiabatic process the entropy remains constant. (iii) RAN~INE addresses also the rarefaction wave phenomenon which he calls sudden rarefaction. Referring to a discussion with Sir Thomson, he annotates: "Sir William Thomson has pointed out to the author, that a wave of sudden rarefaction, though mathematically possible, is an unstable condition of motion; any deviation from absolute suddenness tending to make the disturbance become more and more gradual. Hence the only wave of sudden disturbance whose permanency of type is physically possible, is one of sudden compression; and this is to be taken into account in connexion with all that is stated in the paper respecting such waves." (iv) Rankine also measures the ratio of specific heats, ~/ and finds that "7 is nearly 1.41 for air, oxygen, nitrogen and hydrogen, and for steam-gas nearly 1.3." Asked by the editor of the Proceedings of the Royal Society to give credit also to previous investigators on waves of finite disturbances and to point out to what extent the results arrived at his paper are identical with their researches, Rankine 136 cites the works of Poisson, Stokes, Airy, and Earnshaw, and claims: "The new results, then, obtained in the present paper may be considered to be the following: the conditions as to transformation and transfer of heat which must be fulfilled, in order that permanence of type may be realized, exactly or approximately; the types of wave which enable such conditions to be fulfilled, with a given law of the conduction of heat; and the velocity of advance of such waves. The method of investigation in the present paper, by
56
P. Krehl the aid of mass-velocity to express the speed of advance of a wave, is new, so far as I know; and it seems to me to have great advantages in point of simplicity." War
Department, U.K.
Abel 137 reports on his observation that one detonating dynamite cartridge can trigger another that is positioned in the vicinity and speculates that detonation is transmitted by means of some "synchronous vibrations." He states, "The vibrations produced by a particular explosion, if synchronous with those which would result from the explosion of a neighboring substance which is in a state of high chemical tension, will, by their tendency to develop those vibrations, either determine the explosion of that substance, or at any rate greatly aid the disturbing effect of mechanical force suddenly applied, while, in the case of another explosion which produces vibrations of different character, the mechanical force applied by its agency has to operate with little or no aid . . . . " 9 E. Mach and Wentzel (--+1885) refused Abel's "queer" hypothesis and correctly attributed this phenomenon to the mechanical effect of the shock wave.
Chair of Organic Chemistry, College de France, Paris
Berthelot 138 defines the "strength" of condensed and gaseous explosives and emphasizes the role of a mechanical shock, which during detonation propagates "from layer to layer," thus anticipating an important assumption of the ChapmanJouguet theory (Jouguet-+1905). He writes, "In order to transmit the transformation of a detonating bulk which is not subjected in all parts to the same action, it is necessary that the same conditions of temperature, pressure etc. which have provoked the phenomenon in one point propagate successively, layer to layer [couche par couche], through all parts of the bulk . . . . "
1871
England
Maddox 139 prepares emulsions of silver bromide in essentially the same manner as that used for making colloidal emulsions but replaces collodion by gelatin. Further improvements of this new photographic process were made by Bennet (1878), van Monkhoven (1879), and Eder (1880) which, leading to the so-called "high sensitive photo-gelantine dry plate" in the 1880s, were a basic requirement to make the first photograph of a shock wave (E. Mach and Wentzel--> 1884).
1872
Royal Academy of Sciences, Brussels
Melsens 14~ reports on the severe injuries observed in the recent Prussian-French War (1870-1871). He comes to the conclusion that they were not caused by explosive projectiles banned by the St. Petersburg Treatise (--+1868), but rather by air compression phenomena in front of the projectile. 9 Nine years later, he addressed the same subject in a lecture
History of Shock Waves
57 presented at Paris. His hypothesis inspired E. Mach to investigate these possible ballistic phenomena in more detail. TM However, he had to wait almost six years before his photographic technique was matured enough to catch the motion of high-speed bullets in flight (E. Mach and Salcher--~ 1887).
Expedition of H.M.S. "Challenger," England
A prolonged oceanographic exploration 142 is carried out by the British Admiralty and the Royal Society from 1872-1876. The expedition performs scientific research such as depth sounding, dredging, and measuring currents, depths, and contours of the ocean basin. Tait (-~ 1878) participates and measures deep-sea temperatures.
Trinity College, Cambridge
John Hopkinson 143 measures the strength of steel wires when they are suddenly stretched by a falling weight. He observes that the minimum height from which a weight has to be dropped to break the wire is independent of the size of the weight. He explains this surprising result in terms of the propagation of elastic waves up and down the wire.
Elswick Ordnance Company, U.K.
Noble 144 first reports on his "crusher gauge," a by-product of his investigation into the behavior of explosives and artillery that will render his name famous. Introductorily he compares previous estimations on the elastic force of fired gunpowder in cannons, which not only widely range in historic studies between 100 and 100,000 atmospheresmfor example, 100 by John Bernoulli; 145 1000 by Robins (1743); 2000 by Hutton; 10,000 by Daniel Bernoulli; 12,400 by Rodman (1857-1859); and 100,000 by Rumford (1797)rebut also in modern "reliable" handbooks between 2200 by Bloxam (1867) and Owen (1871), and 29,000 atmospheres by Plobert (1859). 146 With the help of his crusher gauge, a modification and improvement of the indentation gauge (Rodman-~ 1857), Noble determines the maximum pressure produced when a charge of gunpowder is exploded in a confined space (such as in a cannon) and finds a value of about 5500 atmospheres which well correlates with theoretical estimations based on measurements of in-barrel projectile velocities. 9 In the early period of shock and explosion research the crusher gauge was the instrument most used to evaluate the maximum pressure of high-rate thermodynamic phenomena such as explosion and detonation. Since the gauge is simple in construction, inexpensive, and insensitive to electromagnetic radiation, it saw a renaissance in World War II: Penney 147 used his "five-gallon-can blast pressure gauges" to map the overpressure in the "Mach stem" region of an atomic explosion. Blast pressures were computed from the degrees of
58
P. Krehl crushing in the cans. To measure maximum pressures generated in underwater explosions, Abboth 148 used crusher gauges in which a steel piston acted on a small lead cylinder fixed on a massive support.
Woolwich, Arsenal, London
Abel and Brown, ]49 using the Noble-chronograph, measure the detonation velocity of guncotton to be around 20,000ft/sec and state: "Recent experiment has shown that the rapidity with which gun-cotton detonates is altogether unprecedented, the swiftness of the action being truly marvelous. Indeed, with the exception of light and electricity, the detonation of gun-cotton travels faster than anything else we are cognizant of . . . . "
Laboratoire Central des Poudres, Paris
Roux and Sarrau 15~ confirm Abel's observation (---~1869) and differentiate between an "explosion of the first kind" (detonation) and an "explosion of the second kind" (deflagration). 9 In a deflagrationma rapid combustion process that gives off heat and light--the flame speed is below the velocity of sound in the burnt gases; in a detonation, burning takes place at or above the velocity of sound in the burnt gases. A detonation is always associated with a high-pressure and high-temperature shock wave that is sustained by the liberated energy via shock compression rather than via heat transfer as in the case of combustion. The energy of this reaction maintains constant conditions at the front of the detonation wave, thus leading to a constant detonation velocity. Detonation and deflagration are words derived from the Latin verbs detonare (to thunder out) and deflagrare (to burn down), respectively.
1874
Realschule Kaschau, AustroHungarian Empire
Antolik TM publishes his "soot method" and records with this method strange interference patterns in the vicinity of gliding spark discharges. He observes that conically shaped branches [kegelartige "AuslFzufer"] originate from the concave parts of a spark path, which disappear when the discharge occurs in a vacuum. Antolik explains this phenomenon by the behavior of the gliding spark, which in a vacuum prefers to follow a straight rather than a given crooked path. In reality, however, the soot figures are the very first records of irregular interactions of shock waves.
1875
Chair of Experimental Physics, KarlFerdinandUniversitFzt, Prague
E. Mach, 152 together with his student Wosyka, immediately repeats Antolik's experiments (---~1874). They verify that his soot pictures are indeed of acoustic and not of electric origin. Mach and Wosyka are the first to study Mach reflection. They record the trajectories of the triple point (Mach funnel) and stop the Mach disk by using two oppositely facing V-shaped gliding sparks. 153 They also arrive at the following important
History of Shock Waves
59 conclusion: "It should be pointed out that Antolik's simple and ingenious method of preliminary tracing of the spark enables various applications in the field of acoustics, because it can be used to create intense sound waves with an arbitrary initial shape.". The soot method, although in principle very simple, is somewhat tricky to handle, particularly to provide a homogeneous and well-adhering soot layer reliable enough to obtain a high spatial resolution and wide dynamic range of pressure recording. By increasing the adhesion of the soot layer on the glass plate, it is also possible to record even double Mach reflection, which results in two concentric Mach funnels. 154 The soot method was also used to record periodic cell structures in gaseous detonations (Shchelkin and Troshin 1965; Schultz-Grunow 1969).
Royal Society for the Encouragement of Arts, Manufactures & Commerce, London
Nobel 155 reads before the Society a paper entitled "Modem blasting agents." Giving information regarding his invention of Dynamite and the difficulties of its introduction into practical use, he states, "the concentration of power, velocity of explosion, and immunity from danger, are the three points on which mainly depend the success or non-success of a new explosive substrate." Speaking of gunpowder he says, "That old mixture possesses a truly admirable elasticity which permits its adaptation to purposes of the most varied nature. Thus, in a mine it is wanted to blast without propelling; in a gun to propel without blasting; in a shell it serves both purposes combined; in a fuse, as in fireworks, it burns quite slowly without exploding. Its pressure exercised in those numerous operations, varies between 1 oz. (more or less) to the square inch in a fuse, and 85,000 lb. to the square inch in a shell. But like a servant for all work, it lacks perfection in each department, and modem science armed with better tools, is gradually encroaching on its old domain."
Ordnance Company, Elswick; Woolwich Arsenal, London
Noble and Abel 156 start an ambitious program on researches on gunpowder and its explosive effects in guns with the following goals: (i) To ascertain the products of combustion of gunpowder fired under circumstances similar to those which exist when it is exploded in guns or mines; (ii) to ascertain the "tension" of the products of combustion at the moment of explosion, and to determine the law according to which the tension varies with the gravimetric density of the powder; (iii) to ascertain whether any, and if so what, well defined variation in the nature or proportions of the products accompanies a change in the density or size of grains of the powder; (iv) to determine whether any, and if so what, influence is exerted on the nature of the metamorphosis by
60
P. Krehl the pressure under which the gunpowder is fired; (v) to determine the volume of permanent gases liberated by the explosion; (vi) to compare the explosion of gunpowder fired in a close vessel with that of similar gunpowder when fired in the bore of a gun; (vii) to determine the heat generated by the combustion of gunpowder, and thence to deduce the temperature at the instant of explosion; and (viii) to determine the work which the gunpowder is capable of performing on a shot in the bore of a gun, and thence to ascertain the total theoretical work if the bore be supposed of indefinite length. 9 Their results became the basis of modern internal ballistics.
1876
1877
Jabin de SaintEtienne, Graissessac
Two serious firedamp explosions in the French hard coal mining industry (231 miners killed) will initiate the foundation of a governmental research commission (---~1878).
British Telegraph Manufactory, London
Sabine, 157 chief engineer, measures the shock contact time of elastic bodies using an ingenious electric method that allows the measurement of the time between two successive mechanical movements with a considerable degree of accuracy. The method is based on the fact that a charged capacitor can only be discharged at a certain definite rate through a given circuit. For the duration of a blow of a light hammer (weighing about 1 oz) against a steel anvil, he finds contact times around 50~ts. Further experiments reveal that the contact time decreases with increasing impact velocity. 9 His important results reliably proved for the first time that the contact time of impacting elastic bodies is indeed extremely short, which was later found theoretically by Hertz (-~ 1882) and reconfirmed experimentally by Tait (-~ 1892).
KarlFerdinandUniversiti~t, Prague
Rosicky,15s a coworker of E. Mach, visualizes shock-focusing phenomena in an elliptic reflector.. Today ellipsoidal reflectors are also used for focusing spark-generated shock waves in extracorporeal shock wave lithotripsy. 159"16~
Cambridge University
Lord Rayleigh, being in the final phase of his book The Theory of 5ound, has a controversy 161 with Stokes who previously published a paper on sounds of finite amplitudes. Rayleigh writes to him on June 2, 1877: "In consequence of our conversation the other evening I have been looking at your paper 'On a difficulty in the theory of sound,' Phil. Mag. Nov. 1848. The latter half of the paper appears to me to be liable to an objection, as to which (if you have time to look at the matter) I should be glad to hear your opinion . . . . It would appear therefore that on the hypotheses made, no discontinuous change is possible . . . . " Stokes admits that Thomson
History of Shock Waves
61 (later Lord Kelvin) had already made similar objections that the proposed motion would violate the conservation of energy. Avoiding a confrontation with his former student, he kindly replies to Lord Rayleigh on June 5, 1877: "It seemed, however, hardly worth while to write a criticism on a passage in a paper which was buried among other scientific antiquities. PS: You will observe I wrote somewhat doubtfully about the possibility of the queer motion . . . . "
Karl-Ferdinand Universitat, Prague
E. Mach and Sommer 162 measure the propagation of shock waves on a laboratory scale and confirm that indeed a shock wave travels faster than a sound wave and that the shock velocity increases with shock strength. Their results confirm previous large-scale measurements by Regnauh (-~1863), thus reaching therewith an important milestone in shock wave physics. Using a linear percussion model--a row of gas molecules arranged two by two along a straight line--they illustrate that the velocity of percussion must also increase when the velocity of sound increases, such as in the case of violent sound, and state, "It does not contradict the theory to assume that the velocity of sound increases with the intensity of the impulse. Only for very small vibrations does the velocity of sound not depend on the amplitude. But this is not valid for vibrations of finite amplitude as has been proved by Riemann in his paper Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, 1860. 'Velocity of sound' receives hereby a quite different meaning; it is different at every point of the wave and alters during the wave motion. It appears that we deal in our experiments with such waves as described by Riemann."
Chair of Mathematics, University of Straj~burg
Christoffe1163 extends Riemann's theory on shock waves (--~1859) to the case of a three-dimensional propagation. He also treats the propagation of percussion through an elastic solid medium.
Philosophical Society of Washington, DC
A committee formed by the society publishes 164 the observations of 48 people who witnessed the fall of the Washington Meteor on Dec. 24, 1873. Descriptions of observed acoustic phenomena range between "no sound" and "very violent sound" (for example, "short, hard reports like heavy cannon and continued resounding"). The committee concludes that the sound could have been generated by sound focusing and the Doppler effect. 9 Later E. Mach and Doss (-,1893), referring to this "queer theory," explained the loud report not as the result of any explosion or focusing effects, but due to the head wave phenomenon alone.
62 1878
e. Krehl
Karl-Ferdinand-Universitat, Prague
E. Mach and Weltrubsky 165 use the "Jamin interferometer" (Jamin--~ 1856) and first record the density jump at the shock front. E. Mach, Tumlirz, and K6gler 166 measure thoroughly the velocity-distance profile of a blast wave, which they generate by an electric spark discharge. They confirm previous observations by E. Mach and Sommer (--+ 1877) that the blast wave velocity approaches the sound velocity at greater distance from its center of origin. E. Mach and Gruss 167 study Mach reflection in a V-shaped gliding-spark arrangement E. Mach and Wosyka (--+ 1875) in more detail. They give an ingenious interpretation of the propagating and temporarily increasing width of the Mach disk. Their publication contains correct drawings of the interaction phenomenon, but not photographs.
Chair of Natural Philosophy, University of Edinburgh
TaR begins his research on the corrections of deep-sea temperatures obtained during the "Challenger" expedition (--+ 1872). His research will lead to important experimental studies of the compressibility of liquids (TAR-+ 1888) and the behavior of solid materials under impact (Tait-+ 1892).
Paris
Foundation of the French Fire-Damp Commission [Commis-
sion d'~tude des moyens propres a prevenir les explosions du grisou] to uncover the so-far mysterious phenomenon of detonation and to investigate how firedamp explosions can be prevented. Prominent members are Berthelot (president), Le Chatelier, Mallard, and Vieille (secretary). Shortly after, similar commissions will be formed in Prussia, Belgium, and England (president Abel). 9 The results of the investigations initiated by the French commission became an important milestone toward a theory of detonation (Berthelot-+ 1881).
Cambridge University
Lord Rayleigh publishes his monograph The Theory of Sound. In vol. 2 he returns to the point of his previous controversy with Stokes (-+ 1877) and writes, "but it would be improper to pass over in silence an error on the subject of discontinuous motion into which Riemann and other writers have fallen. It has been held that a state of motion is possible in which the fluid is divided into two parts by a surface of discontinuity . . . . " There follows a proof of the impossibility of such motions based on a "violation" of the conservation of energy. 168 9 Rayleigh held to this statement in the 2nd edition (1896) of his book, although at that time the existence of a sharply pronounced discontinuity at the shock front had been proven by using both the schlieren (Toepler--+ 1865; E. Mach and Wentzel-+ 1884) and the interferometer techniques (E. Mach and Wehrubsky-+ 1878).
History of Shock Waves
63
1880
University of Agram, Kingdom of CroatiaSlovenia [now Zagreb, Croatia]
Dvof~ik 169 describes "a simple kind of schlieren observation," which, later called shadowgraphy, does not require any lenses or concave mirrors and allows a large field of view.- The change of illumination is roughly proportional to the rate of change of density gradient. For this reason the method is sometimes superior to the schlieren method (TOEPLER-+1864)--in which the change of illumination is roughly proportional to the density gradient--for observing certain flow phenomena. 17~
1881
Ecole des Mines, Paris
Mallard and Le Chfitelier 171 measure the propagation of the flame front in gaseous mixtures of H2-O2, CO-O2, and CH 402 using the Deprez-chronograph.172 In short tubes with a length of 1.35 m, they observe a speed of 5 7 0 m / s in oxyhydrogen, which reduces to only 7 0 m / s when the length is 1 However, similar to Davy (-+ 1816) they reduced to about 5" notice that the flame is not propagated in narrow tubes.
Russian Artillery Academy, St. Petersburg
Maiyevskii, 173 studying the ballistics of cannon shells, publishes a work on the resistance of projectiles at high speeds, w, that exceed the sound velocity, a. For velocities up to w/a < 1.1 he finds the formula for the ballistic resistance Cx: Cx = 0.19211 + 3.34(w/a)2].. Although subsequent measurements have shown that the air resistance does not steadily increase from transonics to supersonics, it is interesting to note that Maiyevskii, very similar to his forerunner Euler (1745), considered the ratio w/a--later termed the Mach number (Ackeret --+ 1929)--an important quantity governing air resistance at high speed. 174
U.S. Army Corps of Engineers, Washington, DC
Abbot 175 reports on strange surface phenomena of an underwater explosion, which he observed 0.1 s after the explosion. He writes, "The surface of the water around the torpedo over a distance of 200 feet is covered by a misty spray resembling rain, which has been thrown upward from the surface by the shock. Over the torpedo appears a dome of water of which the diameter is about 100 feet and the extreme height about 20 feet. The surface of this dome is a fleecy texture, and through the top are bursting upward many spearlike jets, which cover a space about 50 feet in diameter and attain in the middle an extreme height of 105 feet.". A correct explanation of this phenomenon was first given by Blochmann (--+ 1898).
Coll~ge de France, Paris
Berthelot 176 studies the propagation of the flame speed in essentially the same gaseous mixtures as Mallard and Le Chfitelier (-+ 1881), however, contrary to them he announces the discovery of enormous velocities of explosion. Berthelot, using long tubes (up to 5 m in length, 8 mm in diameter),
64
P. Krehl measures flame velocities of up to 2500m/s. He makes the important observation that the initially slow flame speed approaches a characteristic high limiting value after propagating a sufficiently long distance in the tube. That value is independent of the pressure of the gases, the material of the tube, and the tube's diameter above a small limit, but is a constant of each gaseous mixture. Calling this new thermochemical phenomenon an "explosion wave" [l'onde explosive], he does not explain this supersonic velocity of combustion by any thermal conductivity and diffusion process that governs the propagation of a slow flame, but rather by a transfer of gas compression from layer to layer (Berthelot---~1870), herewith reaching an important milestone toward a theory of detonation. He also states that its velocity could be predicted if the heat of combustion and the density and specific heat of the products are known. Berthelot 177 first demonstrates that a shock wave, generated by an explosive (mercury fulminate), can chemically decompose a gas into its elements. Using the example of acetylene (C2H2), he shows that the violent explosion that is accompanied by a flash emission transforms the initial gas into fine carbon particles dispersed in a hydrogen atmosphere. He draws the important conclusion that "these phenomena give evidence that direct thermodynamic relations exist between chemical and mechanical actions.", More recent studies by Aten and Greene 178 using infrared analysis of quenched products have shown that diacetylene (C4H 2) is the most important product of acetylene pyrolysis by shock heating: it may form over 5% of the total and decomposes into hydrogen and carbon (C4H2--~ C4 + H2).
Royal Commission on Accidents in Mines, U.K.
Abel and associates, stimulated by the calamitous accident in the Seaham Colliery at Durham in the autumn of 1880, show that the finely divided particles suspended in air are a source of danger similar to that occasionally experienced in flour mills. They begin to carry out explosion experiments with coal dust in large mine galleries and demonstrate that, with a very highly inflammable dust suspended in the air in which no trace of hydrocarbon gas (firedamp) is present, a blownout shot can produce ignitions that extend as far as the mixture of air with sufficient dust to maintain flame extends. 9 Some years later their results were thoroughly confirmed and also considerably extended by large-scale experiments carried out by the Prussian Fire-Damp Commission at Neunkirchen in the Saarbrucken District.
History of Shock Waves
65
Service des Poudres et Salp(.tres, Paris
Foundation of the journal Memorial des Poudres et Salp~tres ( 1 . 1 8 8 2 / 8 3 - 17.1913/14; continued as Memorial des Poudres, 18.1921 + 29.1939, 30.1948 + 46/47.1964/65) by the order of a ministerial decree. The editorial board consists of E. Sarrau, Ch. Arnould and E. Desortaux. 9 It was the first professional journal exclusively dedicating to the communication of results on the research of explosives, and its applications for civil and military purposes. Initially created with the intension to improve communications between French researchers on explosion technology, it soon advanced to an international scientific forum.
Physikalisches Institut, Humboldt UniversitFlt, Berlin
Hertz 179 treats the collision of bodies analytically and applies the potential theory to calculate stress and strain as a result of the acting force in the contact area. He first makes the important assumption that the duration of collision is much longer than the time that the elastic wave needs to travel the colliding body. Later this will be confirmed experimentally by Berger (--+ 1924). For straight, elastic collision of two spheres colliding with the relative velocity v, he derives simple formula for the maximum pressure P and duration of collision T, given by P - k iv 6/5 and T - k2 v-1/5. Here the coefficients k 1 and k 2 depend on mass, radius, and E-module of the spheres. For steel spheres 3 cm in diameter colliding at 100 m/s, the contact time T is about 36 ~ts, which is a multiple of the transit time of the excited longitudinal waves in the spheres. Hertz measures the outline of the surface of contact by covering one of the colliding bodies with soot, thus giving an experimental proof of his theory. He also calculates the elastic stress distribution for contact of a hard sphere on a plate, which, for low impact velocities, results in conical cracks (Hertzian cone fracture).. The Hertzian cone is best observed in glass and can be generated either statically by pressing a hard sphere on the surface or dynamically by impact (Kerkhof and Mailer-Beck 1969). In glass it extends from the point of contact under a cone angle of about 130 ~. However, it exists also in flint stone, obsidian, and other hard (and rather isotropic) materials. In prehistoric times, stone fragments [Absplif~] split off by Hertzian cone fracture were used for further processing into handaxes, arrowheads, knives, etc.
Laboratoire Central, Service des Poudres et Salp~tres, Paris
Vieille 18~ first applies a dynamic method to determine the accelerating force of an explosion. Arranging a small piston in the wall of the test vessel, he records its displacement-time profile under the action of the expanding gases on a sootcovered rotating drum and obtains the acceleration by double differentiation.
1882
66
1883
p Krehl
Coll~ge de France & Laboratoire Central, Service des Poudres et Salpttres, Paris
Berthelot and Vieille, 181 using the Desprez-chronograph (Maillard and Le Chatelier-~ 1881) and later the Le-Bouleng4 chronograph (Le Boulenge--~ 1863), measure the detonation velocity in about 50 mixtures of fuels and oxidizers, diluted by different amounts of nitrogen. They observe a uniform detonation velocity that only depends on the mixture composition, not on the tube material and diameter (as long as the latter is not too small). 9 Both electromechanical chronographs require electrical pulses for start/stop activation. The start pulse was provided by simultaneously igniting the gaseous mixture via a spark, and the stop pulse was generated by suspending perpendicularly to the tube axis a thin foil strip that, covered with a small amount of fulminate, exploded at the moment of arrival of the detonation front, thus breaking the holding current in the chronograph.
Nitrocellulose Fabrik Wolff & Co., Walsrode, Germany
Von Foerster, 182 chief engineer of the company, rediscovers the hollow cavity effect for high explosives without inlet (yon Baader-~ 1 7 9 2 ) . . The important discovery of the hollow charge effect, in Europe also called the von Foerster effect, can be ascribed to various inventors of different nationalities. A retrospect on this phenomenon was published in Germany by Freiwald 183 during World War II and more recently in the U.S.A. by Kennedy 184 to commemorate the 100th anniversary of von Foerster's discovery of the shaped charge effect. The shaped charge lined cavity effect, for both military and civil applications being of greater importance than the unlined cavity, was discovered much later (Thomanek--~ 1938).
France
Moisson, 185 naval captain and ballistician, investigates theoretically the air resistance of projectiles of cylindrical, spherical, and ogival geometry at speed u as a function of the ratio u/a = 0.2 to 2, where a is the sound velocity. This ratio will later be called the Mach number (Ackeret--~ 1929).
Ecole des Mines, Paris
Mallard and Le Chatelier 186 study detonation in gases and make streak records of flames from explosions in glass tubes 10-20ram in diameter and 1-3m in length. They use a rotating drum covered with film upon which the flame image is projected. Depending on the tube length and boundary conditions at the tube exit, they observe (i) a constant slow propagation velocity (deflagration), (ii) a phase of vibrations (intermediate state), and (iii) a rapid wave propagation (explosive wave).
Krakatao Island, Sunda Strait, Java
Explosion of volcano Perbuatan on August 27. The enormous mass of spilled lava into the ocean produces the greatest steam explosion in history. 187 9 The mighty blast came in as a
History of Shock Waves
67 "roar" at Batavia, 160 km away, and was said to be still audible at a distance of 3500km in Australia. Tidal waves were observed on four continents and residues of the blast wave recorded as barometric fluctuations around the globe.
1884
Stockholm
De Lava118s invents the first steam turbine. He uses steam jets, which, issuing from nozzles, give up their energy by impulsive action on a moving vane (reaction turbine). This principle, based on the idea of Heron's steam ball [aeolipile, A.D. 100], requires steam jets of very high velocity and will eventually lead to the invention of the "Laval nozzle" (---~1888), which will considerably stimulate supersonic aerodynamics.
Clark, Chapman, Parsons & Co., Gateshead, U.K.
Parsons invents the first multistage steam turbine, which is based on the principle of the reaction that a steam jet exerts on the orifice from which it issues. His machine utilizes several stages in series. In each stage, the expansion of the steam is restricted to allow the greatest extraction of kinetic energy without causing the turbine blades to overspeed. Immediate applications for marine propulsion purposes, however, will uncover serious limitations set by cavitation problems at the propeller, which will stimulate systematic studies of cavitation phenomena. 189
Ecole d'Artillerie de la Marine, Lorient, France
Hugoniot and Sebert 19~ examine a one-dimensional discontinuous gas flow and assume that the flow parameters before and after the discontinuity behave adiabatically (Poisson's law). 9 These studies--later significantly improved by Hugoniot, who assumed a steeper equation of state (his "dynamic adiabate," later called by others the Hugoniot curve)--led to the first general shock theory (Hugoniot---~ 1887).
Karl-Ferdinand Universitat, Prague
E. Mach and Wentze1191 succeed in making the first photograph of a shock wave. They select the most sensitive silver bromide gelantin dry plates then commercially available (Maddox--+1871). As a first test, they generate a shock wave by discharging a Leiden jar and use the spark of a second one, which is fired with a delay of about 20 ~s, as a light source. 9A large number of Mach's original photo plates have survived. They were donated to the Ernst-Mach-Institut at Freiburg/Breisgau by Karma Mach, Ernst Mach's daughter inlaw. Together with his notebooks and correspondence they are now kept at the archives of the Deutsches Museum, Mfmchen.
Brighton, U.K.
Phillips 192 constructs the first wind tunnel. Instead of being operated by a ventilator, it is operated by a steady stream of vapor emerging from a system of fine nozzles. The wind tunnel has a cross section of 0.43 x 0.43 m 2 and consists of a 1.83-m-long test section, which, most remarkably, is followed
68
e. Krehl by a 1.83-m-long diffuser. Using a mechanical balance, he measures for the first time the resistance of curved air foils up to a velocity of 18m/s.
1885
College de France, Paris
Berthelot and Vieille 193 invent the "bomb calorimeter" and measure with this new thermochemical method the specific heat of various gases up to above 2000~ with an accuracy hitherto unattainable.
Laboratoire Central, Service des Poudres et Salp~.tres, Paris
u
Karl-
E. Mach and Wentze1195 publish a study on blast waves originated from chemical explosions. Setting up a pair of parallel line charges of silver fulminate, they record, with the help of the soot method (Antolik --+ 1875), the interference of the two head waves drawn by the detonation fronts and find a detonation velocity ranging from 1700 to 2000m/s. It is interesting to note that they use here the correct term shock wave [Stofiwellel to appropriately describe the observed abrupt pressure increase and write, "The propagation of the shock wave can be felt by the hand, and optically (using the schlieren method) it can be proved that this wave consists of a single shock (without periodicity).". Before that time, Mach and coworkers had experimented with spark discharges and chemical explosives, thus using the terms spark wave [Funkenwelle] and explosion wave [Explosionswelle], as well as the terms percussion wave [Knallwelle] and compression shock [Verdichtungsstofg], the latter having adopting from Riemann (--+ 1859).
Ferdinand, Universitat, Prague
1887
invents "smokeless powder" [poudre B = poudre blanche, meaning white powder whereas poudre N = poudre noir, black powder]. In December of the same year test shots
with a 65 mm cannon are performed to show that the new powder permits the ballistic effect of black powder to be secured with the same pressure and with the charge reduced to only about a third, thus allowing a significant increase of the power of fire arms. Details of his invention were not published in the open literature until six years later. 194
University of KOnigsberg, Germany
Von Neumann 196 publishes a treatise on the longitudinal impact of two thin cylindrical rods, a subject on which he lectured previously (1857-1858). His approach, based on D'Alembert's solution of the wave equation, allows one to evaluate the normal velocity and axial stress in the rods as a function of time.
Private laboratory at Terling Place, Essex, U.K.
Lord Rayleigh 197 shows theoretically that waves passing along the surface of an elastic body probably play an important part in earthquakes, inasmuch as, spreading only in two dimensions, their intensity will gain the upper hand at great
History of Shock Waves
69 distances compared with waves spreading through the interior of the Earth. 9 His surmise was fully confirmed (and called Rayleigh waves). For example, in the great Messina earthquake (1908), Prince Galitzin 198 at Petrograd traced the seismic surface shocks that had traveled around in the Earth in opposite directions. His measured data of the surface wave velocity showed good agreement with Rayleigh's theory. Generally, seismographic records show three separate groups of waves: (i) longitudinal waves with the highest velocity of propagation velocity; (ii) distortion waves with mainly transverse motion; and (iii) surface or Rayleigh waves with the smallest propagation velocity but largest amplitudes.
Karl-
Ferdinand, Universitat, Prague, & Imperial AustroHungarian Marine Academy, Fiume [now Rijeka], Croatia
E. Mach and Salcher 199 start a long series of very successful and unique ballistic experiments. Salcher, who performs the experiments at the Adriatic Naval Test Station in Fiume together with his colleague Riegler, discusses with Mach via correspondence the progress of his work. Using two supersonic infantry riflesmthe Austrian Werndl (438 m/s) and later the Portuguese Guedes (530m/s)mSalcher photographs for the first time supersonic projectiles in flight. The photos reveal that a supersonic projectile, similar to a "bow wave" [Bugwelle] of a ship, produces a hyperbolic-like "head wave" [Kopfwelle] (a shock wave), which is followed by a "tail wave" [Achterwelle] (an expansion wave) and, depending on the projectile geometry, a series of intermediate waves. Addressing also the analogy to the motion of a body in water, they write, "It is possible to reproduce this phenomenon if we take a rod of cross section AB in a large water tank and move it at a velocity which exceeds the velocity of wave propagation." To find the density distribution around the supersonic projectile, they also propose Nobili-Guebhard's electrolytic method by using "a silver-coated copper sheet on the bottom of a container filled with an electrolyte, placing a non-conducting model projectile on the sheet and dipping metal probes connected to a battery to find the equipotentials." ,, This method was indeed used successfully by Taylor and Sharman 2~176 to investigate the field of flow of a compressible fluid past a cylinder. Unfortunately, Mach and Salcher did not consider themselves obliged to cite Doppler (---~1847) as the spiritual originator of the head wave phenomenon. In the 20th century the cone geometry was termed Mach cone and the head wave Mach wave. In the case of the sonic boom, resulting for example from an aircraft flying with supersonic speed, the region outside of the Mach cone was be called the
zone of silence.
70
P. Krehl Tumlirz, 2~ a coworker of E. Mach, presents his shock wave theory, which is based on Riemann's mathematical model (--~ 1859) and assumes an adiabatic law. To avoid Riemann's error, he explicitly uses the principle of energy applicable to continuous motion, in place of the principle of momentum. He concludes that as soon as a discontinuity is formed, it immediately disappears again, this effect being accompanied by a lengthening of the wave and a more rapid advance of the disturbance. He takes this process to be an explanation for the increased velocity of the wave.
Ecole Polytechnique, Paris
Hugoniot 2~ obviously not knowing Rankine's previous work (---~1869), formulates a general theory of discontinuous flow and, following the behavior of a point in the fluid (Lagrangian coordinates), being initially at rest, he uses the laws of conservation of mass, momentum and energy. His most remarkable results an be summarized as follows: (i) He shows that Riemann's assumption that a shock wave is an isentropic process is not correct and that the p,v data are not positioned along the static adiabate [Poisson's law, p - po(Vo/V);], but rather along a dynamic adiabate, later called the Hugoniot curve. Under the consideration that, at the discontinuity, kinetic energy is transformed into internal energy, he derives the famous relation el - e0 1 -~(Po + Pl)(Vo- vl). Later to be called the Rankine-Hugoniot relation (Rankine--~ 1869), it contains no velocity terms, only thermodynamic quantities. Here e0, P0, v0 and e l, Pl, Vl denote the thermodynamic states before and after such a discontinuity, respectively. (ii) Hugoniot shows that for a perfect gas of constant ratio of specific heats, m, the maximum shock compression is given by (m + 1 ) / ( m - 1), which for air with m = 1.4 is equal to 6. He uses the letter m instead of ~, which, coined by Poisson (--~ 1808) and also used by Rankine (---~1869), will later generally be adopted throughout. (iii) Hugoniot also addresses the propagation of shock waves in solids. Considering the conditions at the contact area of two colliding bodies, he states, "It is doubtful whether the discontinuities which are described by the theory of wave propagation are only a simplified analytical fiction or whether they correspond to the physical reality. This is an open question which is difficult to answer at the present state of science.". For an ideal gas the dynamic adiabate, not explicitly given by him in his memoirs, was later derived by Hadamard (--+1903). If the initial state is the standard laboratory state (25~ and 1 bar), the Hugoniot curve is called the principal Hugoniot. Sometimes the term Hugoniot
History of Shock Waves
71
relation is also used in the literature, meaning the dependency of the shock front velocity in terms of the overpressure at the shock front. Apparently, it was first derived by Vieille (--->1900). In the modern literature the Rankine-Hugoniot equations (conservation of mass, momentum and energy) are often given in the Eulerian representation which is preferable both from a mathematical and from a physical point of view: (i) (U - up)v o = Uv; (ii) p - P0 = Uup/vo; and (iii) e - e0 = 1/2(p + po)(Vo - v). Here e, e0; p, P0; and v, v0 are the specific internal energy, pressure and specific volume at the disturbed and undisturbed state, respectively; U is the shock front velocity and u r the particle velocity to which the shock-compressed material is accelerated. From (i) and (ii) the following expressions for U and up can be derived: u = v 0 [ ( p - po)/(Vo - v)] ~/'-" up = [ ( p - po)/(Vo - v)] ~/~. Depending on the particular application, the Lagrangian representation might be more convenient. 2~
1888
University of Edinburgh
Tait 2~ suggests his empirical isothermal equation of state (the "Tait equation") to fit data for the compressibility of sea water up to 500 bar. 9 Kirkwood and Richardson 2~ modified the Tait equation of sea-water for use from initial conditions (r0, P0) up to 25 kbar. Their form (p + B)/( po + B) = (p/po) A, resembles an isentropic in a perfect gas. Here A and B are two empirical functions of temperature, and p and p are pressure and density, respectively. The Tait equation has also been used to describe the p(p)-relation of organic liquids. 2~ Tait and Lord Kelvin publish their book Treatise on Natural Philosophy, in which they also treat the collision of spherical bodies. They introduce a "restitution coefficient," which they define as the quotient of velocities after and before impact.
Ecole Normale, Tir du Champ de Chalon; Artillerie de la Marine
Journee, 2~ De Labouret, 2~ and Sebert 2~ perform supersonic ballistic experiments that essentially confirm Mach and Salcher's observed head wave phenomenon (--+ 1887). They do not use high-speed photography, but rather measure the front velocity of the head (shock) wave along a line perpendicular to the Mach cone periphery and compare the data with Doppler's cone model of wave propagation (---~1847).
U.S. Naval Torpedo Station, Newport, RI
Munroe 21~ discovers by accident how to shape explosives to concentrate energy. He observes that, by increasing the depth of the cavity in the explosive, greater and greater effects on a metal plate facing the explosive can be generated. The phenomenon, later to be called the Munroe effect and caused by an oblique collision of explosive waves, will be used by him to imprint designs on iron plates by interposing a
72
P. Krehl stencil between the explosive and plates of iron (explosive engraving). 9 His discovery was partly a rediscovery of the hollow cavity effect of an unlined shaped charge (von Baader---~ 1792; von Foerster--+ 1883).
University of Dorpat, Russia [now Tarpu, Estonia]
Von Oettingen and yon Gernet 211 resume Bunsen's hypothesis (---~1867) on the discontinuous nature of oxyhydrogen explosions. Using a high-speed rotating mirror and a still camera, they produce time-resolved records from the propagation of the flame front originating from an oxyhydrogen explosion in a straight, 40-cm-long tube. To visualize the otherwise dark oxyhydrogen explosion, they add a small quantity of salt. With this unique streak diagnostics they determine an initial explosion velocity of 2560m/s, thus confirming Berthelot's previous result (--* 1881). After several reflections the shock wave diminished to a velocity of 600m/s. They call the explosion wave the "main wave" [Hauptwelle] or "Berthelot wave" and its reflection at the tube end the "shock wave" [Stoj~welle]. They also observe various secondary waves, which they call "Bunsen waves".
AB Separator Company, Stockholm
De Lava1212 receives a Swedish patent for his "Laval nozzle". Apparently knowing that previous studies on straight nozzles had shown that the gas can be expanded best with sound velocity (critical speed, de Saint Venant and Wantzel---~ 1839) and following his intuition, he uses a convergent-divergent nozzle geometry. This expands the gas isentropically from subsonic to supersonic speeds, thus increasing the efficiency of steam turbines. In his patent he claims: "At rotating steam engines, a steam inlet channel, having cross sections in the vicinity of the rotating part of the steam engine that are increasing in the direction of the said rotating part with the objective to expand the steam in that way that the steam will achieve its highest possible speed before its contact with the rotating, working part of the steam engine.", De Laval had experimented with nozzles 12 years before. As recorded in his personal notes of 1876, he observed that the severe shock behind a nozzle consumes a lot of energy. 213 It appears that he resumed this subject when he experimented with the Sshaped turbine (Heron's steam ball). In his notebook of 1886, there are entries on a 10-inch turbine "with conically widened nozzles at the orifice." However, the idea did not appear to be new: Traupe1214 annotates in his book on thermal turbines that K6rting, owner of a factory for steam apparatus in Hannover, already used this principle in 1878 for steam ejectors.
History of Shock Waves
73
Imperial AustroHungarian Navy Academy, Fiume, Croatia
Salcher and Whitehead 215 study the discharge parameters of a "free-air jet" exhausting from a pressure reservoir through a small opening and compare their experimental data with various existing theories. Salcher performs the experiments at Whitehead's torpedo factory at Flume. Whitehead gained much experience in the generation and storage of highpressure gases, because the torpedoes are propelled by pressurized air up to 100 bar. Illuminating the free air jet with a flash light source of short duration (such as an electric spark) or of long duration (such as by the Geisler discharge tube or using even sun light), which allows the visualization of nonstationary or stationary flow characteristics, respectively, they first make the startling observation that a jet emerging from a pressurized nozzle contains a crossed wave pattern. Since this interference pattern reminds Salcher of an ancient Greek harp, he calls it a lyre [Lyra] in a letter to E. Mach. 216 They correctly interpret this as a superposition of reflected shock waves.. Later this structure--a sequence of pairs of oblique shock fronts, each irregularly interacting and creating a sequence of Mach disks--was coined shock diamonds. Today this is a frequently observable phenomenon in the exhaust of jet engines. The jet experiments described above were later resumed by L. Mach (-~1897), who also first obtained interferograms of excellent quality.
Navy Academy, Fiume & KarlFerdinandUniversitat, Prague
The study of free-air jets inspires Salcher to suggest a supersonic blow-down wind tunnel with the air flowing and the test body being at rest. In a paper written with E. Mach, 217 he says, "On the occasion of the experiments on projectiles Salcher hit upon the idea of likewise investigating the inverse case of the flow of air against a body at rest in order to confirm the results already obtained." They confirm that the inverse case is indeed possible, but with the existing equipment head wave studies of model projectiles were not practicable because of the available small jet diameter. Obviously, Huguenard and Sainte kague in France were the first who realized Salcher's idea for drag measurements of projectiles at supersonic speeds (kangevin and Chilowsky --~ 1918).
Karl-Ferdinand Universitdt, Prague
E. Mach, together with his son L. Mach, first applies interferometry to visualize the flow field around a supersonic bullet. 218 They also apply schlieren photography to visualize the interaction phenomena of two shock waves emerging from two closely spaced point sparks and get the first schlieren photos of the Mach disk. Their experiments fully confirm the triple-point model that they had only assumed hitherto on the basis of soot records (E. Mach and Wosyka
1889
74
P. Krehl --~1875, E. Mach and Gruss - ~ 1 8 7 8 ) . . 54 years later, Campbell, Spitzer, and Price (-~ 1943), using two detonator caps in a very similar geometry, first proved that Mach reflection also exists in water.
1890
French FireDamp Commission, Paris
Charpy 219 and Le Ch~telier 22~ review the first results of experiments the commission had performed to study possible causes of firedamp explosions [le grisou] in coal mines. To avoid such explosions the commission recommends: (i) provision of an effective ventilation system to prevent sudden outbursts of firedamp, to reduce the concentration of methane in the air below 5%, particularly in all higher gallery sections; (ii) use of safe explosives; (iii) avoidance of open fire, sparks, etc.; and (iv) exclusive use of such miner's lamps as remain safe even at higher air speeds. 9 Pure coaldust explosions (without any presence of firedamp), hitherto frequently observed in England but only rarely by French mining engineers, were not yet considered a real hazard.
1891
BerlinLichterfelde
Man's longest flight (300m) to date is performed by von Lilienthal. At this date the basics of supersonics are understood, but practical aviation is still in its infancy.
Chemische Fabrih Griesheim, Germany
Haeussermann 221 discovers the explosive properties of trinitrotoluene (TNT), a substance that had already been synthesized by Wilbrand (1865) by the nitration of toluene with mixed acid. 222 9 Haeussermann first suggested the military use of TNT in shells and undertook its manufacture on an industrial scale. TNT, indeed, gained great military importance in both world wars and remains important today.
Newport Torpedo Station, RI
Munroe 223 invents a smokeless powder. He calls it Indurite because the final powder when dried is exceedingly hard.
Universiti~t Wf~rzburg & Karl-Ferdinand Universitiit
Zehnder 224 and L. Mach 225 independently invent a special type of interferometer that will later be called the MachZehnder interferometer. Consisting of two beam splitters and two mirrors, it divides the source beam into two different parallel light paths (object beam and reference beam) of arbitrary distance. 9 This optical setup proved to be most worthwhile to measure variations of refractive index in compressible gas flow. Zehnder 226 invented and applied this new type of interferometer prior to L. Mach in his Ph.D. thesis at the University of Worzburg under the guidance of R6ntgen in order to investigate the pressure dependency of the refractive index of water. L. Mach, first using it in nonstationary gas dynamics, commercialized his invention and became a wealthy man.
University of Edinburgh
To measure the shock duration between an impinging block and the material to be studied, T a i t 227 builds a simple but very
1892
History of Shock Waves
75 effective percussion apparatus that he humorously calls the "guillotine." The impactor, a block sliding freely between vertical guide rails (precisely like the axe of a guillotine), is attached with a pointer to continuously record the block movement on a revolving plate-glass wheel that is coated with soot. For time measurement he uses a tuning fork that simultaneously produces a second trace on the revolving plate. He also estimates the duration of impact between hammer and nail (200 Its) and the associated time-average force (300 lb-wt). 9 One year later Tait 228 wrote to Hertz: "Some months ago, I was told by Lord Kelvin that you had brilliantly attacked the problem of the impact of elastic spheres. Being very busy at the time, I glanced over your paper in Crellos 92 [J. Reine & Angew. Math. 92 (1881)], but did not attempt to read it. I had been working for some years at direct experiments on impact, but I used a mass of 2; 4; and 8 kg falling through i m or so, and the elastic body on which it fell was a cylinder whose upper surface was very slightly convex. The amount of longitudinal distortion was, in some cases, as much as 30mm. I found, by a graphical method, that the force called into play was at the power 3 of the distortion thus measured. On lately reading your paper with some care, I found to my great surprise that this is the same law which you have theoretically deduced for spherical bodies . . . . " Unfortunately, we do not know today whether Hertz answered this letter. Tait's records, now kept at the Archives of the University of Edinburgh, do not contain any such letters.
Ecole
Sup~rieure de Pharmacie, Paris & C. A. Parsons Co., Newcastleupon-Tyne, U.K.
Royal College of Science, South Kensington, U.K.
The first attempts are made to produce artificial diamonds. However, there is no clear evidence of any incipient transformation of carbon into diamond. Moissan 229 at Paris uses a solution of carbon in a suitable molten metal at high temperature, which he quenches rapidly in water. 9 Later Parsons, 23~ used a 0.303-inch caliber rifle to fire steel bullets at 1500m/sec into an armored press steel house filled with graphite powder. All his attempts, however, gave negative results. Moissan and his contemporaries believed that diamonds could be synthesized successfully by this method, but later investigations rejected this conclusion. TM The spectacular shock synthesis of diamonds did not succeed until about 70 years later by DeCarli and Jamieson at Stamford Research Institute, Menlo Park, CA and the Dept. of Geology of the University of Chicago, IL, respectively. 232 Boys 233 studies flow about bullets and interaction processes of multiple shock waves by using the shadow method (Dvorak--+ 1880). He succeeds also in measuring the spin rate of a shot by using photography. Since a professional ballistic range is not available to him, he performs his shot
76
P. Krehl experiments in a long public hallway in his institute. Boys, who repeated E. Mach and Salcher's ballistic experiments (--> 1887) and promoted the spreading of their method in England, writes 234 to Mach, "I am much obliged to you for your kindly sending me copies of your papers and the two photographs. I have when speaking on bullet photography thoroughly recognized that the whole credit of bullet photography is yours, as you were the first, to carry it out successfully and that your apparatus answers perfectly i.e. so far as I can judge from your account of them. In the English papers were inaccurate reports and in one case of a scientific paper ! corrected it, as stated, what you had done. The daily papers are always so untrustworthy that it is absurd to credit them. I do not think I have failed to appreciate or to recognize what you have done . . . . If you should think I have not properly recognized your work, I am exceedingly sorry, that it should be so, but I am sure if you had heard what I have said at the Royal Society and elsewhere, that you would not think so... " . In the same year E. Mach 23~ wrote, "Boys' method is certainly a simplification when using it merely for demonstration purposes in a lecture. However, I suppose that everybody who wants to study this matter in more detail, will prefer an optical image which allows to estimate the condensation by its shading, rather than a mere silhouette which only reveals the contours of the air waves . . . . Nevertheless, I am grateful to Mr. Boys that he has taken over this assignment hitherto not touched by others, and I hope that he intends to continue it in future." Today, however, in most outdoor ballistic facilities shadowgraphy is used more frequently than the schlieren method because of its simplicity and minor sensitivity toward temperature fluctuations.
Washington, DC
1893
University of Manchester
U.S. President Harrison states 236 in his farewell message to Congress, "I consider one of the great achievements of my administration the invention of smokeless powder by Charles E. Munroe." 9 However, the chief obstacle that eventually prevented the general employment of Indurite (---~ 1891) by the U.S. military was its inconsistency of composition due to the use of improperly nitrated guncotton and to difficulties in removing the residual solvent. Schuster 237 derives a simple formula to calculate the velocity V of the detonation front, which, based on Riemann's theory (-+ 1859), is given by V - [(P/Po)(P - Po)/(P - P0)]1/2He observes a good agreement with experimentally determined rates of explosion in various explosive gaseous mixtures and writes, "Lord Rayleigh criticizing his
History of Shock Waves
77 [Riemann's] investigation, draws attention to the fact that a steady wave is only possible for a particular relation between the pressure and density of the gas, which is different from the one actually holding. In the case of the explosion-waves it seems possible, however, that the temperature, pressure, and density of the gas should so adjust themselves as to make Riemann's equations applicable. In fact, they must do so if the front of the wave keeps its type, which it probably does when the velocity has become constant . . . . In the strict sense of the word I do not think the explosion-wave can be steady, because if the motion is, as assumed, linear, compression must precede the explosion, and Lord Rayleigh's objection would hold for the front part of the wave in which no combination takes place. But it seems possible to me that the motion may not strictly be a linear one, and that yet taking the average velocities over a cross-section of the tube the ordinary equations would apply. It seems probable that jets of hot gases are projected bodily forward from that part of the wave in which the combination takes place, and that these jets, which would correspond to the spray of a breaking wave really fire the mixture.". His correct supposition of a steadily moving detonation wave, made previously in a similar manner by Berthelot 238 and in the same year worked out in more detail by Dixon (-+1893), led to the first theory of detonation (Chapman--+ 1899).
England
Burton 239 resumes Lord Rayleigh's critiques on Riemann's theory (see also his Theory of Sound, vol. II, p. 41, -+ 1878). He also tackles the difficult problem of whether in the absence of viscosity the motion of spherical waves of finite amplitude can become discontinuous, as in the case of plane waves.
Karl-Ferdinand Universitat, Prague, & Polytechnikum Riga
E. Mach and Doss 240 assume that the sharp bang of a meteorite approaching the Earth is a supersonic phenomenon, thus creating a head wave. Mach's motivation to treat the phenomenon of meteoric showers was a letter by C. Abbe, an employee at the Washington Weather Bureau who belonged to a committee of the Philosophical Society of Washington, which had analyzed the fall of the Washington Meteor (-+ 1877). Abbe claims to have already given in 1877 a "true theory of thunder and meteorite explosions" and states, "We are disposed to consider the so-called 'explosion', and subsequent 'rumbling' not as due to a definite explosion of the meteor, but as a result of the concentration at the observer's ear of the vast volume of sound emanating, almost simultaneously, from a large part of the meteor's path, being, in that respect, not dissimilar to
78
p. Krehl ordinary thunder." Abbe then tries to explain the violent sound by the Doppler effect and concludes: "we may remark that it requires only comparatively feeble noises distributed along the entire path of the meteor to produce, by their concentration at the observer's station, a sound equal to that of loud thunder." Mach, rejecting Abbe's theory and his prior claim, replies that only the head wave phenomenon is the true cause of the explosionlike sound effects. 9 Mach's interpretation was indeed correct; however, although pioneering supersonics and being far-sighted, he could not yet realize that the head wave of meteoroite which enter the Earth's atmosphere at speeds up to several 10 km/s, is closely wrapped around the meteorite and forms the so-called "hypersonic boundary layer," thus creating hitherto unknown surface heating and erosion effects.
World Colombian Exposition, Chicago
De Laval displays his reversible single-stage steam turbine. The engine (15 hp at 16,000rpm) is designed for marine use and has been tested on Lake MOlaren in the vicinity of Stockholm to drive a launch. Its novelty is that the turbine blades are driven by a stream of hot, high-pressure steam emerging from a series of unique convergent-divergent nozzles (De Laval-+ 1888). 9 Today his turbine is part of the collection of the Smithsonian Institution and on display in the History of Technology Building at Washington, DC.
University of Moscow
Mikhel'son 241 first proposes a linear law in his theory of detonation and assumes steady propagation of the reaction products--i.e., equal velocities at which any of the intermediate states propagate. Starting from the equations of mass and momentum, he derives the elementary relation P--Po 4-(U/vo)2(Vo- v), which, also derived in the same year in England (Schuster-+ 1893), represents a straight line in the p, v-plane. Here U denotes the shock front velocity, and v and vo are the specific volume at pressure p and P0, respectively. 9 In the western world this line is called Rayleigh line, referring to the work of Lord Rayleigh (--~ 1910) on aerial shock waves. Zeldovich 242 coined this line the Mikhel'son-line in honor of Mikhel'son's early contribution to the theory of detonation, which was unknown among contemporary scientists outside Russia. The theory of detonation was established independently six years later by Chapman (--+ 1899).
Chair of Chemistry, Owens College, Manchester
reports on his observations of the high velocity of explosions in gases. He put forth the view that the detonation wave travels with the velocity of sound in the burning gases, essentially supporting Schuster's view of an unsteady motion of the detonation front (Schuster-~ 1893). Using a coiled-up D i x o n 243
History of Shock Waves
79 lead pipe (length, 55 m; inner diameter, 8 mm), he measures in oxyhydrogen a velocity of 2821 m/s, thus essentially confirming Berthelot's previous measurements (Berthelot-+ 1881).
1895
1896
Institution of Naval Architects, London
Thornycroft 244 and Barnaby 245 investigate reasons for the failure of a British destroyer to meet its design speed. They observe that a marine screw propeller, if turned too fast, might waste its effort by creating vacuous spaces in the water, which afterward suddenly collapse. They also coin this phenomenon cavitation. 9 The systematic search for the origin of erosion by cavitation bubbles was initiated by the finding of severe destructive effects on the propellers of the British ocean liners "Lusitania" and "Mauretania". 246 A committee was appointed in 1915 by the British Admiralty to determine the cause of erosion of propeller blades which resulted in pioneering results (Lord Rayleigh-+1917, Cook--+ 1928).
Komaishi,
Tsunamis originating from a seaquake in the Pacific destroy the coastal town of Komaishimabout 27,000 people die and 5000 are w o u n d e d . . Then this puzzling wave phenomenon was a subject of much discussion some years before this disaster, Rudolph 247 had reviewed previous hypotheses on the origin of seaquakes and associated tidal waves, and had speculated that they are caused by submarine gaseous detonations at the sea bottom. Rottok, 248 another German scientist, had assumed that they might be caused by submarine volcanic eruptions. In the past, tsunamis were often referred to as tidal waves in the English literature. However, they are not created by gravitation as are tidal waves, but rather by tectonic displacements associated with earthquakes. Occasionally, tsunamis can also be generated when a huge body of water is displaced impulsively, e.g., by exploding islands (Krakatao--+ 1883), landslides, and underwater explosions of nuclear devices. Tsunamis cannot be felt aboard ships on the open sea. When they approach the coastline and enter shallow water, their velocity diminishes and their wave amplitudes can increase to heights of up to 30 m. Tsunamis then become very similar to hydraulic jumps of large amplitudes. The term tsunami is a Japanese word composed of two characters meaning "harbor" [tsul and "long wave" [nami], or "long-wave-in-harbor." The term was adapted in the 1960s for general use, in preference to either of the terms tidal wave or seismic sea wave. An equivalent phenomenon, encountered in rivers and confined waters and known as bores, attracted many early naturalists such as Airy, Challis, Earnshaw, Jouguet, Lord Rayleigh, Russell, and de Saint-Venant.
Japan
80
1897
1898
P. Krehl
Pressburg, Hungary
Siersch, 249 director of the Dynamite AG, Wien and concerned about the safe use of explosive in coal mines, applies photography to classify the nature and intensity of the flash emitted by an exploding charge. [He concludes that the shape and dimensions of the flash afford a clue to the eventual security of the explosive, since the smaller the flash the greater the relative security of an explosive for use in the mining industry. Using a still camera and photographing the flash with an open shutter during night, he observes that the flash intensity from an explosive depends on the geometry, the mode of stemming, and the density and admixtures]. 9 On the whole, this straight-forward method proved to be useful, however, he was not yet aware that shock wave reflection and interaction phenomena can also contribute considerably to the geometry and intensity of the flash (Michel-L~vy and Muraour--~ 1934).
C.A. Parsons Co., Newcastleupon-Tyne, U.K.
Parsons 25~ begins a three-decade study of marine propulsion. High propeller speeds are generally advantageous for the steam turbine, but if too high, they lead to much cavitation. With the help of flow visualization he minimizes cavitation effects, thus also improving the propulsive efficiency. He ascribes cavitation to the "water-hammer of collapsing vortices" and compares this phenomenon to whip cracking "whereby nearly all the energy of the arm that swings the whip is finally concentrated in the tag.". The first mathematical treatment of cavitation was performed by Lord Rayleigh (---~1917).
KarlFerdinandUniversitF~t, Prague
L. Mach, 25~ resuming previous experiments by his father and Salcher (--~1889), visualizes free air jets emerging from nozzles of various exit geometry. He applies not only the schlieren but also the interferometer technique, and makes the important observation that with increasing driving pressure (i) the jet diameter surmounts the nozzle diameter, and (ii) the reflected wave fronts no longer intersect in a point (regular reflection) but rather form a new wave, which later will be called Mach reflection (von Neumann---~ 1943).
Laboratoire Central, Service des Poudres et Salp(.tres, Paris
Vieille 252 ignites small amounts of explosives at one end of an air-filled tube with a length of 4 m and a diameter of 22 mm. He measures the shock propagation velocity using chronography and obtains supersonic velocities for both gunpowder (337-1268 m/s) and mercuric fulminate (359-1138m/s). His measurements of a fast-propagating discontinuity confirm theoretical models provided by Riemann (---~1859) and Hugoniot (---~1885, --~ 1887) as well as observations by E.
History of Shock Waves
81 Mach, who generated supersonic waves from both explosives (E. Mach--+1877) and electric sparks (E. Mach--,1875, --,1878).
Aleksejew Water Line Station, Moscow
Kareljskich and associates study the propagation of hydraulic shocks in water pipes. They use water pipes with diameters of up to 6in. and lengths of up to 2494ft. The pipes are connected with Moscow's main water line (24-in. in diameter) via a fast-closing valve. For pipes ranging between 2-in. and 6-in. in diameter, they record pressure jumps between 3 and 4 bar. With the aid of an electrical chronograph they measure speeds ranging between 4200 and 3290ft/s, regardless of whether the shock is generated by a sudden opening or closing of the valve. Zhukovsky, 253 supervising and analyzing the experiments, notices that water hammer waves in plumbing systems are related to shock discontinuities that propagate with constant speed, being dependent only on the wall material and thickness of the pipe and independent of the shock intensity. He draws the following conclusions: (i) The pressure jump p can be estimated by the simple relation p = p0c0V, where P0 and co represent the ambient density and sound velocity of the liquid, respectively, and V is the velocity of the discontinuity carried by the wave, which he assumes is moving with the acoustic speed. (ii) Reflected shocks can generate detrimental periodic oscillations in the pipe system. (iii) At transitions from large to small pipe diameters, the shock intensity can double and, under unfavorable reflection conditions, even further increase up to a fatal level of shock loading. (iv) Hydraulic shocks can be prevented by using slowly closing valves, with a closing time proportional to the length of the water pipe, and by installing wind tanks in the vicinity of the valves. 9 The problem of water hammer, a steady companion in the extension of urban infrastructures, was also tackled in the 1890s in England by Church 254 and in the United States by Carpenter. 25~ Zhukovsky and associates, however, were the first to thoroughly treat this subject both experimentally and theoretically.
German Imperial Navy, Torpedo Inspection Organization, Kiel
Blochmann 256 first correlates the numerous underwater explosion phenomena under local, temporal, and causal aspects. Based on pressure-time profiles recorded with a mechanical apparatus [Dynamometer], he develops a theory of underwater explosion that, along with gas bubble oscillation, allows the prediction of the shock pressure in the water at any distance from the explosive. His analytical results explain hitherto strange surface phenomena (Abbot-+ 1881), such as the dome of spray thrown up from the surface
82
P. Krehl (which he explains by reflection of the incident shock wave at the surface) and, shortly after, the formation of spearlike plumes of spray (which occur during breakthrough of the gaseous explosion products).
1899
Physical Laboratory, University of Wisconsin, Madison
Wood 257 repeats Toepler's shock propagation and reflection experiments (---~1864), but instead of observing the phenomena through a schlieren telescope he uses film for recording. In the introduction he states, "I have always felt that the very beautiful method derived in 1867 by Toepler for the study of 'schlieren' or stri~e, is not as well known outside of Germany as it deserves to be, and trust that the photographs illustrating this paper are sufficient excuse for bringing it before the readers of the Philosophical Magazine. Sound waves in air were observed by Toepler, but they have never to my knowledge been photographed. When seen subjectively, the wavefronts, if at all complicated, cannot be very carefully studied, as they are only illuminated for an instant, and appear in rapid succession in different parts of the fields of the viewingtelescope.". Wood failed to notice that 15 years before E. Mach and Wentzel (---~1884) had successfully photographed the shock wave emerging from a spark discharge. Nevertheless, Wood, who extended his experiments in the following years, obtained interesting results of the reflection, refraction, and diffraction of spark (weak shock) waves, which made widely known the great potentials of shock wave photography to the Anglo-American public.
Universities of Leipzig and Munich
The Emden Brothers, 258 studying in detail gas jets emerging from orifices, make the important observation that stationary waves are generated in the jet as soon as the driving pressure exceeds a critical value for provoking a gas flow propagating with sound velocity.
Laboratoire Central, Service des Poudres et Salpdtres, Paris
Vieille 259 constructs the first bursting diaphragm shock tube to demonstrate that a shock wave propagates with a speed greater than the speed of sound. His device consists of steel tubes with a constant cross section 22 mm in diameter. The driver section has a length of 2 m, followed by a 4-m-long expansion tube. As diaphragms he uses collodion, paper, glass, and steel. The diaphragm ruptures automatically at reaching a certain overpressure, for example, at 35 bar for a 1.5-mm-thick glass plate. In air he achieves shocks with Mach numbers up to M -- 2 and concludes with plain words that "explosives do not play any essential role in phenomena of propagation at great speeds," meaning that the phenomenon of supersonics is not limited to the use of explosives, but for
History of Shock Waves
83 example can be generated also by a bursting membrane. ,, The "shock tube", a term coined much later (Bleakney, Wimar and Fletcher 1949) became an important diagnostic tool for a variety of scientific disciplines, such as for aerodynamic purposes, to study the kinetics of chemical reactions and vibrational and rotational energy transfer, for plasma spectroscopy, to investigate vapor bubble dynamics in two-phase flow, and even for fertilizer production. 26~
University of Manchester
Chapman, 261 assistant to Prof. Dixon, treats an unsupported detonation by assuming that, once the maximum velocity is reached, the detonation frontmi.e., the front of the explosive wave--is of such a character that (i) it moves steadily; (ii) the flow is planar; (iii) the chemical reaction occurs instantaneously; and (iv) the flow, following immediately behind the discontinuous shock, is exactly sonic. He analyzes the solutions to the shock jump conditions for explosive gases and observes that the minimum-wave-speed solution agrees with experimental measurements previously made by Dixon (41893). He concludes, "When an explosion starts, its character and velocity are continually changing until it becomes a wave permanent in type and of uniform velocity. I think it is reasonable to assume that this wave--i.e., the wave of which the velocity has been measured by Prof. Dixon--is that steady wave which possesses minimum velocity; for, once it has become a permanent wave with uniform velocity, no reason can be discovered for its changing to another permanent wave having a greater uniform velocity and a greater maximum pressure . . . . ",, His lasting contributions, which were later independently made by Jouguet (--~1905), have been commemorated by the state of the exploded gas immediately behind the explosion wave being called the Chapman-Jouguet state.
Ballistic Test Range, BerlinCummersdorf
Wolf 262 first investigates large-scale explosions by order of the Prussian Ministry of War. He (i) measures the velocity of the spherical blast wave emerging from the explosion of large quantities of trinitrophenol and uses an electrical contactmicrophone, which triggers a "Le-Bouleng~-chronograph"; (ii) studies the blast response on structures; and (iii) records the pressure-time profile of blast waves using a thin rubber membrane directly coupled to a drum chronograph. His observations fully confirm Mach's previous observations that the blast wave is supersonic close to the charge but rapidly decreases with increasing distance. For a charge weight of 1500kg, he measures a velocity of 858m/s at a distance of 10m. His mechanically recorded pressure-time
84
v. Krehl profiles of blast waves show all the typical characteristics, such as the steep rise, the rapid decay, and the phase of negative pressure.
1900
Laboratoire Central, Service des Poudres et Salp~.tres, Paris
Vieille, 263 starting from Hugoniot's theory (---~1887), first derives a relation between the shock front velocity, V, as a function of the overpressure, (p - P0), at the shock front. This relation, V -- a [1 4- (m 4- 1)/2m x (p - po)/Po] 1/2, will later be known as the Hugoniot relation. Here P0 and a denote the pressure and sound velocity at rest, respectively, and m is the constant ratio of specific heats. He also confirms this relationship experimentally. In another study, Vieille 264 speculates on hypersonic flight, predicting stagnation pressure and temperature for flight in ideal air at speeds up to M ~ 30, and associated surface phenomena such as incandescence and erosion, leading for example in meteorite falls to thermal ruptures. He concludes, "Without admitting these numbers an absolute value, one can imagine that the incandescence of meteorites, the erosion of the surface and the rupture which accompanies their passage through our atmosphere are explicable by pressures and temperatures predictable by the law of the propagation of discontinuities, even when taking into account of the rarefaction of the medium passing through."
University of Wisconsin, Madison
Wood 26~ photographs focused spark (weak shock) waves by using spherical, parabolic, and elliptical mirrors.
Royal Institution, London
In an evening lecture entitled Some Modern Explosives, Noble 266 reports on the physical and chemical effects of detonating explosives in the bore of a gun and states, "I am not without hope that the experiments I have been describing may, in some small degree, add to our knowledge of the kinetic theory of gas . . . . The kinetic theory of gases has, however, for us artillerists a special charm, because it indicates that the velocity communicated to a projectile in the bore of a gun is due to the bombardment of that projectile by myriads of small projectiles moving at enormous speeds, and parting with the energy they possess by impact to the projectile . . . . But in the particular gun under discussion, when the charge was exploded there were no less than 20,500 cubic centimetres of gas, and each centimetre at the density of explosion contained 580 times the quantity of gas, that is, 580 times the number in the exploded charge is 8 31 quadrillions, or let us say approximately for the total number eight-followed by twenty-four cyphers . . . . "
History of Shock Waves
85
1901
Chair of Mathematics, University of Stra~burg
Weber 267 presents his revised edition of Riemann's lectures on mathematics, which he had delivered in the period 18551866 at the University of G6ttingen. He extends Riemann's theory (--->1859) and treats shock waves in two chapters entitled "Propagation of Shocks in a Gas" and "Aerial Vibrations of Finite Amplitude." Returning to Lord Rayleigh's previous objection (---~1878) on Riemann's theory (see also his Theory of Sound, vol. II, p. 41), he demonstrates that Riemann's theory is indeed correct and compatible with the law of energy. 268 Lord Rayleigh (-->1910) will resume this problem in his classic review paper on the evolution of shock wave theories.
1903
Coll~.ge de France, Paris
Hadamard 269 treats discontinuities mathematically and in a general form. For an ideal gas he derives the "Hugoniot curve" [loi adiabatique dynamique] as 1/2(p I + po)(Vo - vl) = (ply1-poVo)/(7 - 1) which, plotted in the p,v-plane, is steeper than Poisson's adiabatic law [loi adiabatique statique]. While studying the works of Riemann (-~ 1859) and Hugoniot (--~ 1887), he noticed that the shock front problem can be considered separately and can be mathematically transformed by a particular simple procedure not connected with any specific problems and that can be fully described by the so-called "identity and kinematic conditions" and their derivations. He postulates, "If a function of the coordinates and of time, together with all its derivatives, is defined both outside of and at the surface of discontinuity, then the rule for compound differentiation can be applied to it at the surface of discontinuity . . . . " . Referring to Hadamand's theorem, v o n Karman 27~ later annotated, "According to his theorem, a vortex-free flow ahead of a shock wave can remain vortexfree after passing through the shock only when the wave is straight. If the shock wave is curved, it produces vorticity. This is a fact which makes the analysis of motion behind a shock wave rather complicated." Hadamard used also the terms shock wave [onde de choc] to illustrate the wave-type character of this phenomenon and acceleration wave [onde d'accglgration] to elucidate the steepening process behind the shock. He distinguished the characteristics as propagation paths of vanishingly small shock waves, as the energy defect across them becomes zero.
Institut fftr Thermische Maschinen, ETH Zftrich
Stodola 271 publishes his famous book on steam turbines, which contains the first studies of flow characteristics through a supersonic (Laval) nozzle. He measures the pressure distribution along the nozzle axis at different back pressures and, noticing a sequence of steep pressure increases,
86
P. Krehl states, "I see in these extraordinary heavy increases of pressure a realization of the 'compression shock' theoretically derived by Riemann, because steam particles of great velocity strike against a slower moving steam mass and are therefore compressed to a higher degree . . . . " 9 Each zone of maximum pressure is visible in a photograph as a vertical line in respect to the nozzle axis, which was called by Cranz 272 the "barrier line" [Staulinie]. A historical review on the outflow of gases and steam from orifices was given by Prandtl. 273
1904
Kitty Hawk, North Carolina
The Wright Brothers start their wind tunnel experiments to optimize the design of wings and propeller blades. On December 17, Wilbur Wright performs the first controlled motor flight over a distance of 260m; total duration is 59 s (i.e., average velocity 4.4 m/s). 9 Later v o n K a r m a n 274 stated in his memoirs, "The peak event of this part of my visit to the U.S.A. was my meeting in Dayton, Ohio, with Orville Wright . . . . To my surprise and enormous interest, I found that Orville Wright was familiar with the fundamentals of aerodynamic theory. He told me that before the historic flight at Kitty Hawk, he and his brother spen~almost two thousand hours with their small wind tunnel, studying the relative merits of various wing shapes."
Technische Hochschule Dresden
M. Toepler 275 visualizes and photographs spark (weak shock) waves by using the schlieren method of his father (A. Toepler --+ 1864).
Owens College, Manchester
Lamb 276 solves the theoretical problem of surface waves excited by impulsive line or point loads. He finds that the surface disturbance may be divided roughly into two parts: (i) a minor tremor, composed of both longitudinal and transverse waves, which starts with some abruptness and may be described as a long undulation leading up to the main shock and decaying gradually after this has passed; and (ii) the main shock propagating as a solitary wave with the velocity found by Lord Rayleigh (--+1887). Lamb's contribution is of fundamental significance to theoretical seismology.
Institut far angewandte Mechanik, Universitiit G6ttingen
Prandt1277 begins a study on wave propagation phenomena inside and ouside of nozzles of various geometry when stored high pressure air is exhausted through them. Starting from Riemann's theory (---~1859), he gives a quantitative explanation on the periodic formation of stationary waves in free jets (Salcher---~ 1889): Expansion waves originating at the edge of the outlet are reflected at the boundary of the free jet as compression waves, which in turn are reflected as expansion waves. This process repeats periodically, thus resulting in
History of Shock Waves
87 crossed lines (later to be called "shock diamonds"). He also deduces the "wavelength" of the crossed wave pattern in the photograph from the ratio c/w (w - supersonic flow velocity along the axis, c = sound velocity at that state) which can be estimated with sufficient accuracy from the inclination of the characteristic lines with respect to the axis of the jet using Mach's law sin a = c/w. Later he will coin this angle 0~ the "Mach angle" ( P r a n d t l ~ 1913).
1905
University of Bordeaux, France
Jouguet 27s derives an expression for the entropy change in a small-amplitude shock wave in terms of the second derivative (~2v/~p2)s. Since the adiabatic curve p(v) in the pressurevolume diagram is concave down for practically all substances--i.e., this expression is always positivemJouguet concludes that a rarefaction shock is impossible. Zeldovich 279, however, theoretically showed that rarefaction shocks are indeed possible, which later was also proven experimentally by Kutateladze 2s~ at the Institute of Heat Physics, Novosibirsk.
III. Internationaler MathematikerKongref~, Heidelberg
Prandtl 2sl proposes his concept of a "boundary layer" [Grenzschicht] near the surface of a body moving through a fluid. This concept will prove extraordinarily fruitful in the development of fluid m e c h a n i c s . . During World War II, some aerodynamicists considered the removal of a part of the boundary layer air by suction through a porous surface or a number of slots to increase the laminar stability, to delay transition, and to reduce drag. 2s2 The flow in a boundary layer may likewise be laminar or turbulent, and the flow pattern and location of shock waves are dependent on the type of flow in the boundary layer. 283
Breslau University, Germany
Lummer TM publishes his shock theory [Theorie des Knalls], which, outlined only briefly by him, approaches the shock problem by using the Huygens principle of wave front propagation and referring to the Doppler principle. Lummer also first speculates on whip cracking as being a supersonic phenomenon. First successful attempts to tackle this puzzle experimentally were undertaken in France by Carri/~re (--~ 1927). 9 It seems that von Neumann (-->1942) had a similar approach of modeling shock wave propagation in mind, but obviously did not follow it up.
Humboldt Universitat, Berlin
Nernst, 2s5 an avid automobile fan and in the late 1890s the owner of one of the first automobiles in G6ttingen, indicates that the shock phenomenon of knocking [Klopfen, Schlagzandung] in reciprocating internal combustion engines might be due to the buildup of a detonation wave. 9 His correct
88
P. Krehl hypothesis later initiated a long period of international research on this important practical problem.
University of Bordeaux, France
Jouguet 286 after having studied detonation in more detail, concludes, independelty of Chapman (-+ 1899), that (i) the chemical reaction at the detonation front occurs instantaneously from unburnt into burnt gas, (ii) the detonation products propagate at constant velocity; and (iii) behind the detonation front the velocity of detonation products with respect to this front is equal to the local velocity of sound. JOUGUET correctly postulates that a detonation wave comprises a shock wave followed by a combustion wave and arrives at very similar conclusions as Chapman (thus the Chapman-Jouguet hypothesis 1899/1905). Assuming that the detonation products are at thermodynamic equilibrium and using previously measured data of heat capacities at high temperatures, Jouguet calculates the velocity of the detonation wave for various gaseous mixtures and obtains good agreement with previously measured values (Berthelot and Vielle---~ 1883; Dixon--~ 1893). 9 The Chapman-Jouguet model assumes a homogeneous layer of reaction; however, most surprisingly, later studies of detonation waves in gases rather showed complicated patterns, such as a "spin" structure (Bone, Fraser and Wheeler-+1936) or a "periodic cell" strucutre (Shchelkin and Troshin 1965).
New York City
Percy Maxim, 287 an American gunsmith, invents the first silencer for small fire arms. He founds the Maxim Silent Firearms Company and will obtain a German patent in 1910. His design is based on the modern concept of a multiple baffle arrangement, which is screwed onto the barrel. In his legendary indoor demonstrations, he proves the efficient reduction of muzzle blast. 9 However, the commercial success failed, because the interest of military circles to introduce "silent firearms" into the army was small--perhaps since in the use of common firearms the sound of explosion is also an effective physiological factor.
Cambridge Engineering School, University of Cambridge, U.K.
Hopkinson 288 repeats previous experiments of his father John Hopkinson (-~ 1872) and reconfirms that the tensile strength of metal wires is indeed much greater under rapid conditions than when measured statically. 9 This important result stimulated research on the dynamic elastic behavior of solids as well as that of shock-loaded materials, and initiated the "onedimensional finite-amplitude theory" on dynamic plasticity of metals derived by Taylor 289 (1942), von Karman zg~ (1942), and Rakhmatulin 291 (1945).
History of Shock Waves
1906
89
University of Gi~ttingen
Zemplen 292 considers an ideal gas with constant specific heat and shows that entropy changes in a shock wave: It rises with increasing pressure and falls with decreasing pressure. From this he concludes that a rarefaction shock is impossible (the Zempl~n theorem). In his paper he gives the first concise and modern definition of a shock wave: "A shock wave is a surface of discontinuity propagating in a gas at which density and velocity experience abrupt changes. One can imagine two types of shock waves: (positive) compression shocks which propagate into the direction where the density of the gas is a minimum, and (negative) rarefaction waves which propagate into the direction of maximum density."
Institut fur angewandte Mechanik, Universiti~t GOttingen
Prandt1293 obtains a first estimate of the shock front thickness for an ideal gas of constant viscosity and heat conductivity. Starting from heat conduction processes in the transition layer, he calculates for an aerial shock wave with a pressure jump of 0.2 atm a shock front thickness of 0.5 ~tm and states that "the thickness of shock layers range within the wavelengths of visible light.".
University of Bordeaux, France
Duhem 294 demonstrates that true shock waves--i.e., waves having a discontinuous front according to Riemann's and Hugoniot's theory--are only stable in perfect fluids. In real fluids, however, only "quasi shock waves" are possible.
Lehmanns Verlag, M~nchen
Foundation of the German journal Zeitschrift far das gesamte SchieJ~- und Slprengstoffwesen ( 1 . 1 9 0 6 - 3 9 . 1 9 4 4 ) with the goal to "improve and promote the communication between science and industry, and to advance the development and application of explosives and propellants." Editor-in-chief is R. Escales. 9 It was the second international journal that exclusively dedicated to the quickly growing field of explosives, ballistics and shock waves (cf. Memorial des Poudres & Salp~tres-~ 1882).
Laboratoire de la Commission des Substances Explosives, Paris
Dautriche 295 describes a simple method of measuring the detonation velocity of a test explosive. His "difference method" uses a match [cordeau] of known detonation velocity (6500m/s) placed on a lead plate, their two ends being inserted into the test cartridge at a known distance are ignited subsequently by the passage of the detonation wave in the cartridge. When the two waves in the cordeau meet, they make a sharp furrow in the lead plate that, shifted from the midpoint of cordeau, is a measure of the detonation velocity in the cartridge. For confined dynamite he measures detonation velocities ranging from 1991 to 6794 m/s, depending on the initial density of the cartridges. 9 His method much
90
P. Krehl resembles E. Mach and Sommer's interference method (---~1877), which they used to determine the propagation of velocity of explosion waves.
1907
1908
EcoleNationale des Mines, Saint-Etienne, France
Crussard 296 first applies the Rankine-Hugoniot equations (--~1887) on a reactive fluid, thereby using a graphical representation. He shows that the explosion wave is composed of a shock and a combustion wave that propagates with a velocity equal to the speed of sound in the medium that follows them, thus anticipating the supplementary Chapman-Jouguet relation (Jouguet---~ 1917).
Institut ffir angewandte Mechanik, Universitdt GOttingen
Prandt1297 resumes his previous studies (---~1904) on supersonic wave propagation of gases and steam exhausting from nozzles. The use of plane nozzles confined between two glass plates allows the visualization of wave phenomena of the free jet as well as of the nozzle interior. Schlieren photography clearly shows the formation of a shock wave inside the nozzle, indicating the need for considering area ratio distribution to obtain uniform supersonic flow.
Institutfflr angewandte Mechanik, University of GOttingen
Meyer, 298 one of Prandtl's Ph.D. students, visualizes the propagation of Mach waves inside the divergent section of a supersonic nozzle and theoretically treats the oblique interaction of shock waves. He presents shock wave tables of pressure ratios at various angles of incidence and reflection and derives the "Prandtl-Meyer function."
Podkamennaja Tunguska, Siberia, Russia
On June 30, at about 7 a.m. local time, an asteroid, impacts the Earth's surface in the Stony Tunguska, about 3400 km east of Moscow. It generates a huge blast wave, equivalent to the energy liberated by the explosion of about 107 tons of TNT and devastates an unpopulated, 720-square-mile area of forests, but does not form a crater. 299 Accounts from the town of Kansk (about 600 km south from the impact site) and from Kuriski-Popovich Village, District of Kansk state that "a first shock caused the doors, windows and votive lamp to shake, a minute later a second shock followed, accompanied by subterranean rumbling,", and "a severe earthquake and two loud bursts, like the firing of a large caliber gun, were observed in the vicinity," respectively. 9 Seismic shocks and air pressure waves were recorded as far away as Central Europe, but the fall of the meteor was not brought immediately to the notice of the scientific world, although strange barometric phenomena were recorded in England.3~176 Napier, discussing wave motion at the meeting of the British Association at Dublin in the same year, showed microbarograms of a series of waves which were taken on the day of the
History of Shock Waves
91 meteorite fall and remarked, "the succession of four undulations, commencing with a range of about five thousandths of an inch, lasting about a quarter of an hour and then violently interrupted by a sudden, though slight explosive disturbance, which set up different, and much faster oscillations for a similar interval . . . . It would seem that the disturbance, if not simultaneous at the different places, traveled faster than 100 miles per hour."
1909
University of Jena, Germany
Auerbach, 3~ reviewing the present state of the art of physical acoustics, addresses also the enormous progress achieved in supersonics since Antolik's soot experiments. Under the headings "Aeromechanics" [Aeromechanik] and "Anomalies of the Propagation Velocity" [Anomalien der Fortpflanzungsgeschwindigkeit], he discussed many most notable early contributions to gas dynamics., Subsequent handbook articles covering this rapidly growing field of compressible flows were given by Prandtl 3~ (1905), Prandtl 3~ (1913), Ackeret TM (1927), and Busemann 3~ (1931). It appears that the term gas dynamics [Gasdynamik], to some extent forming a link between thermodynamics and hydrodynamics, 3~ was first used in Ackeret's handbook article.
Pittsburgh, PA
Several severe accidents in the American coal mining industry in the previous year, partly attributed to dust explosions, result in the establishment of the U.S. Bureau of Mines. Focusing also on the nature of dust explosions, it will study over a period of 60 years the properties of hundreds of different powders.
Compagnie des Omnibus, Paris
Lorin, 3~ a French engineer, obtains a patent on a "propulsive duct" [propulseur a reaction], a compressor-less jet engine that, shortly after, he proposes for use in aeronautics. 3~ Based on the ram effect and later to be called ramjet, it derives its thrust by the addition and combustion of fuel with air compressed solely as a result of forward speed. 9 At that time, only five years after the first motor flight (Wright Bros. -~ 1903), the application of this engine type was far out of sight. The first successful application of ramjet to flight was not made until 1945, when supersonic flight was maintained by a ramjet developed by the Applied Physics Laboratory at Johns Hopkins University and associated contractors under the sponsorship of the U.S. Navy Bureau of Ordnance. A review of early ramjet developments was given by Avery. 3~ The principle of ramjet was later extended to scramjet (Billig 1959).
University of Heidelberg
Ramsauer 31~ investigates the phenomenon of percussion and shows that discrepancies with de Saint-Venant's theory
92
e Krehl (--+ 1867) are due to the nonperfect elastic behavior. He also shows that the resulting complex interaction process can be divided into a shock due to the actual collision and another (impairing) shock due to oscillation.
1910
Technische Hochschule Wien
Kobes 311 investigates the question of whether the application of shock waves could improve the performance of railway airsuction brakes, then an important practicle problem particularly on long trains to avoid overrunning by the last cars. Kobes publishes the first shock tube theory. 9 His "shock tube," [a term not yet coined until Bleakney et al. 312 (1949)] was not a laboratory-type, smooth and straight pipe, but rather consisted of a test train with 71 cars (total length 746m) with the common arrangement of brake hoses, elbows, joints, and valves. He determined an average shock wave velocity of 370m/s from the measured shock arrival time at the last car.
Militartechnische Akademie, Berlin
Bensberg and Cranz 313 provide a series of quantified drag measurements on projectiles that reveal that the drag coefficient gradually decreases after passing the sound barrier.
Royal Colonial Institute, London
Cornish, 314 publishes photographs of all kinds of wave phenomena in nature--such as the oblique interaction of hydraulic jumps in very shallow water--thereby also observing by chance Mach reflection. 9 In addition, he published propagation phenomena of snow and sand waves and waves in rivers, which apparently are barely known among modern fluid dynamicists.
Private laboratory at Terling Place, Essex, U.K.
Lord Rayleigh 315 thoroughly reviews and comments on previous theories of "sound of finite amplitudes." Starting from the Navier-Stokes equations, he investigates possible influences of heat conduction on the shape of the discontinuity. He resumes his earlier critiques (-~ 1877, --~ 1878) and states, "The problem now under discussion is closely related to one which has given rise to a serious difference of opinion. In his paper of 1848 Stokes considered the sudden transition from one constant velocity to another, and concluded that the necessary conditions for a permanent regime could be satisfied . . . . Similar conclusions were put forward by Riemann in 1860. Commenting on these results in the Theory of Sound (1878), I pointed out that, although the conditions of mass and momentum were satisfied, the condition of energy was violated, and that therefore the motion was not possible; and in republishing this paper Stokes admitted the criticism, which had indeed already been made privately
History of Shock Waves
93 by Kelvin. On the other hand, Burton and H. Weber maintain, at least to some extent, the original view . . . . Inasmuch as they ignored the question of energy, it was natural that Stokes and Riemann made no distinction between the cases where energy is gained or lost. As I understand, Weber abandons Riemann's solution for the discontinuous wave (or bore, as it is sometimes called for brevity) of rarefaction, but still maintains it for the case of the bore of condensation. No doubt there is an important distinction between the two cases; nevertheless, I fail to understand how a loss of energy can be admitted in a motion which is supposed to be the subject to the isothermal or adiabatic laws, in which no dissipative action is contemplated. In the present paper the discussion proceeds upon the supposition of a gradual transition between the two velocities or densities. It does not appear how a solution which violates mechanical principles, however rapid the transition, can become valid when the transition is supposed to become absolutely abrupt. All that I am able to admit is that under these circumstances dissipative forces (such as viscosity) that are infinitely small may be competent to produce a finite effect . . . . " He derives a simple formula to estimate the shock front thickness x, which is on the order of p/pu, where u is the velocity of the wave and/~/p is the specific gas viscosity. He writes, "For the present purpose we may take u as equal to the usual velocity of sound, i.e., 3 x 104 cm per second. For air under ordinary conditions the value of pip in C.G.S. measure is 0.13; so that x is of the order ~1 x 10 -5 cm. That the transitional layer is in fact extremely thin is proved by such photographs as those of Boys, of the aerial wave of approximate discontinuity which advances in front of a m o d e m rifle bullet; but that according to calculation this thickness should be well below the microscopic limit may well occasion surprise."
Cavendish Laboratory, Cambridge
G. I. Taylor 316 investigates the thermodynamic conditions at the shock front. He extends Lord Rayleigh's approach (---> 1910) by including not only heat conduction, but viscosity as well, and establishes theoretically that a propagating sharp transition layer of permanent type is possible only when the pressure increases across the layer and when diffusion processes operate in its interior. To obtain an estimate of the thickness of a shock wave, he set up the continuum equations for the perfect gas with constant viscosity p and heat conductivity ~ and shows that they can be solved exactly if either p - 0 or ~c = 0 and approximately if the velocity jump across the layer is relatively small.
94
P. Krehl
1911
Harvard University, Cambridge, MA
Bridgman 317 describes a gauge for the measurement of static pressures based on the electrical properties of manganin, an alloy consisting of Cu (84%), Mn (12%), and Ni (4%). The resistance of manganin is shown to be a linear function of pressure up to 12 kbar and, by extrapolation, enables pressure measurements to be made with some certitude to 20 kbar. Later he will extend 31s the linear gauge response up to 30kbar. 318 9 After World War II manganin gauges became an important diagnostic tool to measure the pressure (or stress) in shock-loaded samples (Hauver 1960).
1912
Royal Institution, London
In an evening discussion with Lord Rayleigh in the chair, B. Hopkinson 31~ reports on new fracture phenomena that occur in metal specimens when small quantities of explosive are detonated in contact with them. Using charges of guncotton placed upon steel plates, he observes that for plates with a thickness greater than 1 inch a circular disk of metal from the opposite side of the plate is broken away and thrown off; he calls this "scabbing.". This phenomenon of separation, today also called "back spalling" or the "Hopkinson effect", occurs when a strong compressive shock of short duration is reflected from the back surface of a body, thus producing a tensile wave.
1913
Point Hawkins, MD
The British cargo ship "Alum Chine", destined to transport explosives to the Panama Canal for use in blasting operations and having 285 tons of dynamite on board, explodes during loading of freight at Point Hawkins (about 6.5 km southeast of Baltimore). The disaster leaves 62 dead and 60 wounded. The explosion is felt as a blast and/or seismic shock, depending on the distance and the direction from the origin of the explosion. 9 Analyzing a large number of observations at distances ranging from 6.4 to 160km, Munroe 32~ later concluded that these differences are attributed to the acoustic phenomenon of zones of silence and the orientation of the ship at the moment of the explosion. 321 This study is important as it was the first detailed documentation on the destructive effects of a large-yield explosion.
Elektrochemisches Institut, TH, Hannover
Bodenstein, 322 studying the photochemical-induced chlorinehydrogen explosion (Gay-Lussac and Thenard--~ 1807), finds that the reaction velocity is proportional to the square of the chlorine concentration and inversely proportional to the oxygen concentration. Through the concept of a chain reaction he correctly explains this law and, simultaneously, the fact that the photochemical yield exceeds the Einstein law of equivalents by a factor of 104. The concept of atomic "chain
History of Shock Waves
95 reaction" [Kettenreaktion] will gain great importance in detonation and combustion physics. 9 In the example of a detonating chlorine-hydrogen mixture, a long-time puzzle to physicochemists [Nernst (1918): "die boshafte Chlorknallgasexplosion"], the chain reaction occurs in three steps: initiation propagation and termination. 323
Institut ff~r angewandte Mechanik, Universitat G6ttingen
PRANDTL,324 reviewing the progress of gas dynamics and supersonics, coins the term Mach angle" [Machscher Winkel].
1914
Cambridge Engineering School, University of Cambridge
B. Hopkinson 325 describes a novel and ingenious variant of the ballistic pendulum to analyze the force and time of a blow. Instead of a pendulum, he uses a long, thin steel rod of high strength (the "Hopkinson pressure bar"), divided by a transverse joint into a long and a short portion. The rod takes the blow longitudinally and transmits it as a wave of elastic compression, which proceeds from the long piece to the short one. At the extreme end of the short piece, the wave of compression is reflected back along the rod as a wave of tension. When the reflected wave reaches the joint, the short piece flies off and carries with it a fraction of the whole momentum, which depends on the length of the short piece. This enables the length of the pressure wave to be determined, and from that the duration of the blow is readily inferred. Moreover, by using a very short length for the detachable piece, the maximum pressure is also measured. He examines the blows given by a bullet striking the end of the rod normally and by the detonation of guncotton positioned at or close to one end of the rod.
1915
University of Cambridge
B. Hopkinson 326 proposes his cube-root law (the "Hopkinson law") for scaling the blast field about conventional explosive charges under sea-level conditions. He states that self-similar blast (shock) waves are produced at identical scaled distances when two explosive charges of similar geometry and the same explosive but different sizes are detonated in the same atmosphere.
1917
Private laboratory at Terling Place, Essex, U.K.
Lord Rayleigh 327 solves the problem of the collapse of a spherical empty cavity in a large mass of liquid and calculates the velocity of contraction. Introductorily he writes, "I learned from Sir C. Parsons that he also was interested in the same question in connection with cavitation behind screw-propellers, and that at his instigation Mr. S. Cook, on the basis of an investigation by Besant, had calculated the pressure devel-
96
P. Krehl oped when the collapse is suddenly arrested by impact against a rigid concentric obstacle . . . . It appears that before the cavity is closed these pressures may rise very high in the fluid near the inner boundary." To find the pressure in the interior of the fluid during the collapse, he extends Besant's calculation and shows that the final volume is extremely small when the initial pressure of the gas is only a small fraction of that of the surrounding fluid. In reality, however, the bubble undergoes isentropic compression, and a high temeprature as well as a high dynamic pressure should be reached. In the same paper he also considers the problem that the cavity contains a small amount of gas, which is isothermally compressed and converts the energy of collapse into the pressure of this imprisoned gas. (cf. also Parsons---~1884, 1897; and Cook--+ 1928). 9 Rayleigh's famous paper stimulated many subsequent researches on caivtation. The first experimental evidence of the high-pressure pulse originating from a collapsing bubble was given by Harrison (1952) using acoustic diagnostics and by Giith (1954) using optical schlieren technique. Bubble jet formation, a result of unstable, asymmetric bubble wall collapsing, was suggested by Kornfeld and Suvorov (1944) as a possible damaging mechanism in cavitation erosion. It was first experimentally confirmed by Naud~ and Ellis (1961). A review of cavitation-generated erosion phenomena was given recently by Philipp and Lauterborn. 328 Ecole
Polytechnique, Paris
1918
Mac Cook Field, Dayton, OH
Jouguet, 329 resuming previous studies on detonation (Chapman--1899; Jouguet-~1905) and referring to Crussard's graphical method (--~ 1907), assumes that the RankineHugoniot relation is not only valid to describe discontinuities (shock waves) propagating in the same fluid but also to describe two separate, chemically distinct environments (reactive waves). Jouguet explains the mechanism of detonation at constant speed by considering the detonation front as a shock wave of a special kind to which the RankineHugoniot relation can be applied by inducing in the energy balance the part due to chemical reaction (the "supplementary Chapman-Jouguet condition"). Caldwell and Fales 33~ design and build the first American high-speed wind tunnel (14-inch diameter, 200m/s) and study compressibility effects on airfoils. They note that at a "critical speed" there is a large decrease in lift coefficient accompanied by a large increase in drag coefficient.. Their wind tunnel can be seen in the USAF Museum at Dayton, OH.
History of Shock Waves
1919
97
Moscow
Tupolev organizes, together with Zhukovsky (the "Father of Russian Aviation"), the Centralized Aerohydrodynamic Institute [LIAFId]. In 1922 he will become head of the institute's design bureau.
Coll~ge de France Governmental Laboratory, France
Langevin and Chilowsky suggest the first supersonic wind tunnel using a high-speed current of air at supersonic velocity emerging from a Laval nozzle to test a new type of projectile. Hugenard and Sainte Lagu/~ carry out such experiments on a stationary high-speed current of air at velocities greater than the velocity of sound TM. Initially using a Laval working section 8cm in diameter obtain a Mach number barely above 1 (M = 1.07). Later, however, after changing the diffuser angle, they will reach Mach numbers up to 1.4. The drag is measured by a torsion balance. 332 9A British ballistic commission, paying a visit to them, initiated similar studies at the National Physical Laboratory (NPL) in Teddington. 333
U.K.
British researchers begin to study propeller tip phenomena and recognize that a propeller is a wing whose flow characteristics and, therefore, propulsion efficiency vary along the span. They observe a loss in thrust and a large increase in blade drag when the rotational speed of the blade tips approaches or even exceeds the sound velocity. Contemporary theories of the airscrew, however, are still limited to subsonic tip speeds, and even seven years later Glauert TM will write, "Little is known on the effect of the compressibility of the air on the characteristics of an aerofoil moving with high velocity and further progress, both in theory and in experiment, is necessary before the theory of the airscrew can be modified to take account of this effect." This important practical problem is rather complex and can be divided in three cases: (i) subsonic tip velocity; (ii) subsonic flight and supersonic tip velocity; and (iii) supersonic flight velocity. 9 Doppler (--, 1847) had already shown that a body moving in a circle at supersonic speed produces a rotating Mach wave. This was later investigated theoretically by Prandt1335 and experimentally by Hihon. 336 A theory of high-speed propellers, referring to all three cases mentioned above, was first worked out by Frankl. 337
1920s National
Physics Laboratory, Teddington, U.K.
Stanton 338 and coworkers set up a supersonic wind channel, probably the first in the world with a Mach number significantly above 1. The mini blow-down facility, having only a diameter of 0.8 in. but approaching M = 2, is used initially to investigate the drag, lift, and upsetting moment of projectile models of diameters not exceeding 0.09 in. Later Stanton 339
98
P. Krehl will study airfoils of various geometric configurations at supersonic speed in a more advanced, continuously driven wind channel with a 3.07-in. diameter and speeds up to M - 3.25. 9 This wind tunnel has survived and is now kept in the NPL Museum at Teddington.
1920
1921
Ecole Polytechnique, Paris
Jouguet 34~ discusses the similarity between shooting channel flow and supersonic compressible flow and suggests this analogy to study two-dimensional gas flows by means of experiments with a rectangular water channel. An extension for three-dimensional motion will be given later by Riabouchinsky 341 (1932).
France
Fauchon-Villepl~e 342 obtains a German patent on his "electromagnetic rail gun," which he invented during World War I by order of the Minist~re de l'armement et des fabrications de guerre (1916-1918). His idea was resumed after World War II by various research institutions to possibly generate ultrahigh velocities.
Mount Wilson Observatory, CA
Anderson, 343 father of scientific exploding wire research, uses exploding wires to produce temperatures in excess of the 3000~ available at that time for high temeprature spectroscopic studies. He investigates the pressure shift of spectral lines in exploding wires and concludes that the brilliant flash has an intrinsic intensity that corresponds to a temperature of about one hundred times the intrinsic brilliancy of the sun. Using a rotating mirror, he visualizes the dynamics of the flash size, apparently the emitted shock wave, and measures a speed of propagation in open air of about 3300m/s. 9 The visualization of the shock and flow field around exploding wires, a small-size phenomenon that requires optical magnification and therefore virtually increases the velocity on the film plane, requires ambitious ultrahigh-speed diagnostics 344 and was not realized until the late 1950s.
Humboldt Universit~t, Berlin
Becker 345 presents his thesis entitled Stoj~welle und Detonation for the certificate of habilitation. Published one year later, this thesis will become renowned for its clarity. His achievements can be summarized as follows: (i) To illustrate on a qualitative basis how shock waves in gases are formed, he proposes his simple "Becker piston model" which, assuming a stepwise motion of a piston in a tube and the coalescence of pressure pulses, explains on a qualitative basis how shock waves in fluids are formed; (ii) Treating the Navier-Stokes equations for non-weak shocks, he obtains the first solution involving both viscosity and heat conduction. (iii) He calculates the thickness of a shock front in air
History of Shock Waves
99 by assuming constant values for the transport coefficient and the specific heats. For air (1 bar, 0~ he finds for a shock pressure of 8 bar that the front thickness becomes already smaller than the mean free path length (about 90nm) and for strong shocks at 2000 bar even remains under the mean distance of two molecules (about 3.3 nm). He concludes that classical kinetic theory is inapplicable to very intense shock waves in gases, because in the shock front the temperature increase is the result of only a few collision processes. (iv) Starting from the Tamman equation of state, he also estimates shock wave data in liquids. For example, for a strong shock wave propagating in ethyl ether at 10 kbar the calculated shock front thickness amounts to 0.65nm which is comparable to 0.55 nm, the mean distance of two molecules. (v) He calculates the detonation velocity in gases and essentially confirms Jouguet's theory (---~1905). For liquid and solid explosives, however, similar calculations are not yet possible, because the equation of state of the hot burnt gases are still unknown.
1922
Lynn Works, General Electric Company, United States
Briggs, Hall, and Dryden 346 begin with measurements of the characteristics of wing sections at sonic and supersonic speeds, with the wing sections corresponding to tip sections of propeller blades. They do not use a wind tunnel but rather open-air jets from 2 to 12 in. in diameter, thus following Salcher's suggested principle of supersonic testing (Mach and Salcher--. 1889).
1923
University of Danzig, Germany
Ramsauer 347 studies systematically full-scale underwater explosions with charges of gun-cotton up to 2 kg fired at depths up to 30 ft in 40 ft of water and first determines the position of the gas bubble boundary. He uses an ingenious "electrolytic probe method" which consists of an arrangement of electrodes supported at suitable distances from the charge by a rigid frame and together with a common electrode forms conducting circuits with the seawater acting as an electrolyte. The bubble radius expansion, interrupting the electrode circuits successively, is recorded with a mechanical chronograph. He finds that the maximum radius rmax (M/P)1/3, with M being the mass of the explosive and P the total static pressure at the depth of the explosion. He also makes the important observation that the bubble migrates upwards. 9 His method was limited to the recording of the bubble expansion, but could not detect its oscillation which during World War II was recognized as a further source of underwater shock waves endangering submarines (--.1941). "~
100
P. Krehl
1924
SiemensSchuckert Werke, Wien
Berger 348 investigates collision phenomena in solid bodies and proves experimentally that the shocked contact surface of collision moves impulsively. This phenomenon is attributed to the rarefaction wave, which is created by reflection of the compression wave at the free surface (surface shock unloading).
1925
Harvard University, Cambridge, MA
Bridgman 349 makes first systematic measurements of the piezoresistivity of metals and recognizes the tensor nature of this effect in crystals. In the 1960s this "piezoresistive effect" will become important in solid-state shock physics as a diagnostic tool to measure the response of shock-loaded materials.
Aerodynamische Versuchsanstalt, GOttingen
Ackeret 35~ publishes his famous "two-dimensional linearized wing theory," where he considers a thin wing exposed to a uniform and parallel supersonic flow. According to his theory, the deflection of the stream causes a pressure increase at a concave corner and a pressure drop at a convex corner. Consequently, in the case of supersonic flow, a shock wave emanates from the concave corner and an expansion (rarefaction) wave from the convex corner. Both wave types had already been observed by Mach and Salcher (--+ 1887) in their pioneering supersonic ballistic experiments.
1926
PhysikalischTechnische Reichsanstalt, Berlin
Grfineisen 351 proposes an equation of state for solid matter based on his "lattice vibration theory," which he derived from previous investigations by Mie 352 and himself. 353 The so-called "Mie-Gruneisen equation" can be written as pv + G ( v ) - F(v)e, where e is the specific internal energy, v is the specific volume, F(v) is the Gruneisen coefficient for the material, and G(v) is related to the lattice potential. First applied to shock-compressed solids by Walsh, Rice, McQueen, and Yarger, 354 it proved most worthwhile to describe the thermodynamic state of shock-compressed matter.
1927
Institut Catholique de Toulouse, France
Carri/~re 355 studies the phenomenon of whip cracking, the oldest means of man to generate shock waves. He uses a machine-driven whip and high-speed schlieren photography 9 The first pictures showing the shock wave emerging from a real whip were not obtained until 1958 by Bernstein et al. at the U.S. Naval Research Laboratory. 356 More recent investigations using high-speed videography and laser stroboscopy, however, revealed that for generating strong shocks the supersonic motion of the tuft is only a conditio sine qua non, but that the essential mechanism of shock generation occurs in the final stage of acceleration and is due to the abrupt flapping of the tuft at the turning point. 3~7
History of Shock Waves
101
TH BerlinCharlottenburg
Hildebrandt, 358 investigating the nonstationary flow in long lines of railway airbrake systems, extends Kobes' shock tube theory (--~ 1910).
General Electric Company, Schenectady, NY
Langmuir, who had studied gas discharges in electron tubes with his colleagues since the early 1920s, coins the term plasma for a quasi-neutral system of ionized gas that he considers to be the fourth state of matter, consisting of neutrals, ions, and electrons. Mott-Smith, 359 a coworker of Langmuir for many years, will later remember, "We noticed the similarity of the discharge structures [between mercury vapor discharges, Geissler tubes and gas-filled thermionic tubes] they revealed. Langmuir pointed out the importance and probable wide bearing of this fact. We struggled to find a name for it. For all members of the team realized that the credit for a discovery goes not to the man who makes it, but to the man who names it . . . . We tossed around names.., but one day Langmuir came in triumphantly and said he had it. He pointed out that the 'equilibrium' part of the discharge acted as a sort of sub-stratum carrying particles of special kinds, like high-velocity electrons from thermionic filaments, molecules and ions of gas impurities. This reminds him of the way blood plasma carries around red and white corpuscles and germs . . . . So he proposed to call our 'uniform discharge' a "plasma".... Langmuir 36~ will not use the word plasma in a scientific paper until 1929. 9 Shock waves propagating in a plasma without an applied external magnetic field behave much like shock waves propagating in a neutral gas, because submicroscopic forces among the electrically charged particles can be neglected in the first order. In the case of strong shock waves, collision processes at the shock front provoke an ionization of the neutral gas particles leading to luminous phenomena (Michel-L~vy, Muraour, and Vassy-~ 1941).
Swampscott, MA
The first Symposium on Combustion is held at the 76th Meeting of the American Chemical Society with the goal "to emphasize the practical significance of combustion research, particularly in the area of high-output combustion in aviation power plants." General chairman is
1928
Brown.
Royal Aircraft Establishment, Farnborough, U.K.
Glauert 361 gives the first interpretation of Prandtrs aerodynamic theory of airfoils in England and thereupon derives a rule, the so-called "Prandtl-Glauert rule", that enables designers to calculate the amount of lift needed at high speed---up to the speed of sound but not beyond.
102
1929
P. Krehl
Parsons Marine Steam Turbine Co., Wallsendon-Tyne, U.K.
C o o k 362 reports on his investigations of the hydrodynamic properties of collapsing cavities in an incompressible fluid and his calculations of the pressures that might arise from the collapsing vortices of cavitating propellers. His studies, verified by experimental methods, convince the Committee of Erosion Research that the deterioration of propeller blades of cruisers and destroyers by erosion is indeed caused by the water-hammer effect, i.e., is resulting from cavitation, thus essentially confirming Parsons' previous hypothesis (-->1897).
Budapest
Fono, 363 a Hungarian engineer, obtains a German Patent for a propulsive device. Furnishing a design for a convergentdivergent inlet, he describes its use specifically for supersonic flight. ~ His engine is clearly recognizable as a prototype of today's ramjet.
KaiserWilhelmInstitut fur StrOmungsforschung, GOttingen
Prandtl and Busemann 364 develop a graphical method based on the method of characteristics to approximately determine smooth supersonic flows at arbitrary initial and boundary conditions. Their method replaces the stationary two-dimensional supersonic potential flow by a crossing system of stationary sound waves. 9 Busemann later mentioned that Prandtl had already used a primitive form of the characteristics method as early as 1906 to shape the exit of his Laval nozzles for parallel supersonic jets. 365
ETH Zurich & Escher Wyss AG, Zf~rich
Ackeret 366 delivers his inaugural lecture [Privatdozent], thereby coining the term Mach number [Machsche Zahl] with the argument "since the well-known physicist E. Mach clearly recognized the fundamental significance of this ratio in our field and confirmed it by clever experimental methods." 9 The term Mach number denotes the ratio of the velocity of a body or disturbance to the velocity of sound and was introduced into the English literature in the late 1930s. However, it was not immediately accepted by the Russians, who at one time preferred Bairstow 367 number, or the French, who proposed Moisson 368 number 369 (-~Moisson 1883). Nowadays Mach's name is used by almost anyone describing something that is very fast. In fact, Mach is more known for this than for his numerous contributions to the philosophy of science.
University of Cambridge
G. I. Taylor 37~ investigates long gravity waves in the atmosphere. These waves can result from extraordinary strong blast waves, such as those observed during the huge volcanic eruption of Krakatao (--~ 1883) that traveled at great height with a velocity of about 320m/s around the Earth. Taylor's
History of Shock Waves
103 results, presented at the 4th Pacific Science Congress in Java, show good agreement with recorded data of 1883.
Institut fiir Technische Physik, Berlin
Cranz and Schardin 371 invent the "Cranz-Schardin multiple spark camera," the prototype of which provides 8 frames of excellent quality at a maximum frame rate of 3 x 105 frames per second. They immediately apply the camera to record head waves and the oblique interaction of shock waves (Mach reflection). They also investigate the well-known phenomenon of why an implosion is always accompanied by a sharp report and show that it is not caused by the rarefaction wave itself but rather by the blast wave created shortly after the implosion. They demonstrate this phenomenon by using a 34-m-long evacuated tube that they suddenly open at one end. The air masses, then rushing in violently and being reflected at the other end of the tube, return to the open end of the tube after about 0.2 s, thereby perceptible as a report. Zornig from the United States, later becoming Colonel of the U.S. Army and involved in the Manhattan Project (---~1943), apparently discusses with Schardin the historic soot recording experiments of irregular shock reflection (Mach and Wosyka---~1875), their possible interpretation, and related problems of nonacoustic reflection of air shocks. 372 9 This subject became important for selecting the optimum height of burst (von Neumann---~1943, 1945) of the two nuclear explosions in Japan (---~1945).
1930
Mount Wilson Observatory, Pasadena, CA
Hubble 373 publishes a plot of the "Doppler shift" of light versus distance for 22 galaxies and reports that all distant galaxies recede from us and more distant galaxies recede faster (Hubble's l a w ) . . Two years later, after having compiled more data on velocities of nebulae, he stated 374 (together with Humason), "The relation [between radial velocity and distance] is a linear increase in the velocity amounting to about +500km/sec per million parsecs of distance." This simple statement had an enormous impact on cosmology: It suggested the idea that the expanding universe may have its origin in a huge explosion (the Big Bang theory). Estimates on the age of the universe range from 10 to 20 billion years.
Maschinenund Apparatebau Mfinchen
Schmidt, stimulated by previous works of the Frenchmen de Karavodin (1906-1909) and Marconnet (1909), develops his "Schmidt tube" [Schmidtrohr], a pulse jet engine, and obtains German and British patents. 375 Using a tube resonator with a length of about 3.6 m, a valve matrix at the entrance and a
104
P. Krehl Laval nozzle at the exit, it applies the reflected shock wave for periodical reignition (at about 50 Hz). 9 Further developed in Berlin by the Argus Motoren GmbH [Argus-Schmidtrohr], it served in World War II to power the V-l, the world's first cruise missile ( M - 0.47),
1931
Applied Mechanics Division, University of Michigan, Ann Arbor, MI
Donnel1376 studies longitudinal shock wave transmission in solid bodies when impacted and the dimensions are no longer very small compared to the velocities of such waves. He theoretically treats various cases of practical importance, such as the impact of thin bars with free or fixed ends, effects of a sudden change in the cross section or material of a bar, and waves due to a force applied at an intermediate section.
Television Laboratories Inc., San Francisco, CA
Farnsworth 377 suggests the continuous dynode "electron multiplier.". This simple but most effective and versatile device led in the 1960s to the development of the microchannel plate (MCP) at Bendix Research Laboratories. Its outstanding features, such as high intensification and fast shuttering, allowed CCD cameras to be used in the 1990s at ultrashort exposure times down to only a few nanoseconds (intensified CCD or ICCD), an important requirement for applications to shock wave recording in solids and hightemperature plasmas. In addition, MCPs provided trace intensity multiplication of about 1000 times, which allowed single-sweep viewing up to the oscilloscope's rise-time specification at bandwidths of as much as 1 GHz.
Harvard University, Cambridge, MA
Bridgman publishes his book The Physics of High Pressures, which will earn him the Nobel Prize in physics in 1946. Although exclusively dealing with static pressure phenomena, his book will stimulate following generations of solid-state shock physicists to study also the dynamic properties of materials beyond pressures not accessible by static compression techniques.
Imperial College, London
Chapman and Ferraro 378 suggest that the "sudden commencement" impulse coming from the sun is the front of a plasma cloud emitted from the sun and hitting the Earth's magnetic field. Their hypothesis will lead to the discovery of planetary shock waves by Gold 379 and planetary bow waves by Axford 38~ and Kellog. 3sl
School of Aeronautical Engineering, University of Rome
General Gaetano A. Crocco, 382 a former pioneer of Italian aviation and director of research of the Italian Air Force, coins the term aerothermodynamics, a combination of fluid mechanics and thermodynamics, to take account of aerodynamic heating at supersonic speeds that at this time are only
History of Shock Waves
105 reached by the propeller tips of high-speed aircraft. 9 This particular branch of super/hypersonic flow, later introduced and propagated by yon I~rman, 383 still is a serious problem in the design of high-speed air- and spacecraft, because one has to meet with the reduction of materials strength at elevated temperatures, which already begins at rather low Mach numbers. For the example of the Soviet SST Tu-144 (1968), it was observed that, when it flew for hours at Mach 2, air friction heated up the airframe over 300~ (150~ above the surrounding air with heat concentrations on the nose and leading edges of the wings. TM
1932
Zeiss-Ikon AG, Dresden
Joachim and Illgen 385 measure the gas pressure of rifles by using their "piezo-indicator," an instrument consisting of a piezo-crystal, an amplifier, and an oscilloscope with a cathode ray tube. Time sweeping occurs with a rotating drum covered with photo paper. Being the archetype of modern shock pressure recording techniques, it allows for the first time the recording of pressure-time profiles with a high temporal resolution.
Institut far Technische Physik, Berlin
Schardin 386 investigates theoretically the shock propagation in tubes and the condition for ignition of hydrogen-oxygen mixtures by incident and reflected shocks. His work, together with that of Kobes (-~ 1910) and Hildebrand (--~ 1927), form the basis of modern shock tube theory.
Milan, Italy
G. A. Crocco 387 reads a paper at the 20th Meeting of the Italian Association for the Advancement of Science that discusses future possibilities of superaviation, i.e., flight at very high altitudes (above 37,000 feet) up to about M = 3. He addresses the particular problems of high-speed flights in the stratosphere--such as lift and propulsion--and illustrates the economic efficiency.
Flugwissenschaftliches Institut, TH BerlinCharlottenburg
Wagner 388 studies the fundamental processes of percussion and gliding when a body at high speed hits the free surface of a fluid. 9 The water impact of a body under a small angle of incidence leads to a periodic bouncing along the surface, which can easily be demonstrated by throwing a small stone on a pond's surface under a low angle. This phenomenon, called ricocheting, had already attracted some early percussion pioneers, such as Marci (1639) who explained this effect by the law of reflection [De proportione motus, Propositio XXXX]. The skipping effect to which seaplanes are subject when they land on water is of great practical importance for the float construction and was tackled also in the United States 389 and the Soviet Union. 39~
106
1933
P. Krehl
Guggenheim Aeronautical Laboratory at CalTech, Pasadena, CA
Von K,4rm/m and Moore 391 perform a pioneering study on the resistance of slender, spindle-like bodies (such as projectiles) at supersonic speed. Their study takes into account a new type of drag (wave drag) that occurs when the body approaches the sound velocity. Later their study will be generally considered as the starting point of supersonic aerodynamics. ,, In the following year Taylor and Maccol1392 extended the study to the more general case of axisymmetric cones having any semivertical angle that is less than a certain critical angle under which the shock wave detaches from the body (Mach angle). For narrow cones both methods showed consistency.
Cavendish Laboratory, Cambridge
G. I. Taylor 393 calculates the forces on a thin biconvex airfoil moving at supersonic speed and compares his result with Stanton's drag data obtained in the wind tunnel at NPL, Teddington (Stanton-+ 1920s).
Langley Aeronautical Laboratory of NACA, Hampton, VA
Stack and Jacobs first photograph the transonic flow field over airfoils at speeds above the critical Mach number. They use the schlieren technique and correlate their flow analysis with detailed pressure measurements. 394 9 In 1951 Stack and his colleagues were awarded the prestigious Collier Trophy for their pioneering transonic wind tunnel work.
Institut ff~r Aerodynamik, ETH Zf~rich
Ackeret 395 operates the world's first continuous-flow supersonic wind tunnel with a closed loop using a kaval nozzle (M = 2; 40 x 40 cm2). It is anticipated not only for the testing of model aircraft but for use in ballistic research, and steam and gas turbine design as well. He designed the facility when he was still working at Brown-Boveri Company, which also built the device.
British Museum of Natural History, London
Spencer, 396 keeper of minerals, summarizes the available information on meteorite craters and cites five craters or crater clusters with associated meteoritic material: the Arizona Meteor Crater, the Odessa Crater of Texas, the Henbury craters of Australia, the Wahar craters of Arabia, and the Campo del Cielo craters of Argentina. 9 Since 1931 various authors have appealed to the impact and explosion of meteorites to account for the Ries and Steinheim basins of Germany, the Ashani Crater of the African Gold Coast, the K6fels Crater of the Tyrolian Alps, and the Pretoria Salt-Pan of South Africa. Most contemporary theoreticians, however, still favored some form of cryptovolcanic hypothesis, maintaining that the explosions were due to expansion of gases associated with ascending magmas.
University of Cincinnati, OH
Bucher 397 discovers in a large quarry, about two miles east of Kentland [Newton County, Indiana] curious striated cup-and-
History of Shock Waves
107 cone structures, so-called "shatter cones," with apical angles ranging from 75 to nearly 90 degrees, and as long as 2 meters in limestone and 12 meters in shale. Considering these unusual "cryptovolcanic structures" (Branca and Fraas 1905) as disturbances in deranged Paleozoic beds, he ascribes their origin to a deep-seated explosion of gases derived from an igneous intrusion.
1934
Institute of Chemical Physics of the Soviet Union, Leningrad
Semenov 398 publishes his monograph Chain Reactions, which contains the development of a theory of nonbranching chain reactions. It is the result of previous discoveries that he and his team made on the basis of the study of critical phenomenamsuch as the limit of ignition--during oxidation of vapors of phosphorus, hydrogen, carbon monoxide, and other compounds. He writes, "In 1927 and 1928 in Oxford, Leningrad and partly at Princeton the chain theory was applied to a study of the reactions leading to inflammation and explosion. What is particularly important, the theory has advanced here hand in hand with new experiments, which led to a discovery of new and the explanation of old, long ago forgotten, and quite unintelligible phenomena, and they have outlined the field of those reactions which are specific in the new conception. They have aroused a broad interest in this new reaction field and have brought to life in 1930, 1931, 1932, and 1933 a wave of new kinetical investigation . . . . It is hoped that the analysis given here will enable us to make some new generalization and thus to advance somewhat further in the question of the classification of reactions and of finding new laws common to wide classes of chemical change." J His thorough and continuous investigations earned him the 1956 Nobel Prize in chemistry which he shared with the British Hinshelwood.
Lehrstuhl fur Luftfahrttechnik, TH Aachen
The first German supersonic wind tunnel 399 is installed at Wieselsberger's institute under the leadership of R. Hermann. The Laval nozzle is covered with a layer of plaster of Paris, which ensures a sufficient surface smoothness and is easier to form than wood or metal. The ideal nozzle geometry was determined graphically using the method of characteristics. The 10 • 10 cm 2 wind tunnel can be operated up to M = 3 and will be used in 1936 to test models of the liquidpropellant rocket A-3 (short for Aggregat 3), the forerunner of the V-2 (Peenem~nde -~ 1942).
Mt. Wilson Observatory and CalTech, Pasadena, CA
Only 18 months after Chadwick's discovery of the neutron in England, Baade and Zwicky 4~176 connect supernova explosions to the formation of neutron stars, stating, "With all reserve, we advance the view that a supernova represents the transition
108
e Krehl of an ordinary star into a neutron star." 9 The first observational evidence was given 33 years later with the discovery of a rapidly rotating magnetic neutron star, a so-called "pulsar," in the center of the Crab Nebula, which is a remnant of an explosion seen by the Chinese (1054). Further evidence was given by the spectacular event 4~ of the Supernova 1987a.
1935
Institute of Physics, U.S.S.R. Academy of Sciences
(2erenkov, 4~ then a postgraduate student, observes that radiation of blue light is emitted when an energetic charged particle passes through a transparent nonconductive material at a velocity greater than the velocity of light within the m a t e r i a l . . Later Tamm and Frank (1937) theoretically treated this "Cerenkov effect" and concluded that velocity phenomena, similar to a head wave in supersonic aerodynamics, exist also in the micro cosmos when an energetic particle moves through a medium at a velocity greater than the phase velocity of light in this medium. For this unique discovery and interpretation, the three Soviet scientists earned together the 1958 Nobel Prize in physics. 4~
Services des Poudres, France
Michel-L~vy and Muraour 4~ study the interaction of a single shock wave generated by an explosive with a solid boundary or of two shock waves generated by two simultaneously fired explosives and observe in both cases an intense luminosity. This effect is also observed when a shock wave is reflected from a very light obstacle, such as cigarette paper. They give a correct interpretation that the luminosity is solely attributed to the shock wave itself and not to any phenomena due to the explosion processmfor example by the emission of burnt particle--thus rejecting a previous hypothesis (Siersch-+ 1896).
5th Volta Conference, Rome
G. A. Crocco is president of this international meeting with the topic High Velocity in Aviation. It is the first time that leading supersonic aerodynamic engineers from around the world discuss together the possibilities of supersonic flight. Shortly afterward, however, some nations, starting with Germany and Italy, will classify this topic because of its military relevance. Busemann 4~ extends the existing linear airfoil theory to include terms of higher orders. In his famous concept of "sweepback wings," he predicts that his "arrow wings," with a geometry such that they remain within the shock cone at supersonic speed, would have less drag than straight wings exposed to the head wave (a shock wave). However, since propeller-driven aircraft of the 1930s still lack the ability to enter supersonics, his idea cannot be realized immediately, but will influence most future high-speed aircraft designs. Prandtl 4~ reports on strange shock-like
History of Shock Waves
109 phenomena that he has observed in his supersonic nozzle. Wiese|sberger first suggests that this phenomenon, later named condensation shock, might be caused by condensed water vapor when using atmospheric (moist) air. 4~ Studies by Oswatitsch 4~ and Hermann 4~ will prove that this supposition is indeed correct. Ackeret 41~ discusses the supersonic wind tunnel he has just completed for the Italians at Guidonia. Von K , q r m a n 411 presents a new theory of supersonic flow from the viewpoint of drag.
1936
Princeton University, NJ
Wigner and Huntington 412 suggest that insulating diatomic molecular solid hydrogen, subjected to very high pressures, might transform into a metallic monatomic solid phase, and estimate a pressure of transition to be not less than 250 kbar. 9 Their hypothesis stimulated numerous static and dynamic high-pressure studies. An experimental evidence for the existence of "metallic hydrogen" would be important not only fundamentally in condensed matter physics and astrophysics, but also technologically for possibly producing a high-temperature superconductor. 413
School of Aeronautical Engineering, University of Rome
Luigi C r o c c o 414 publishes a fundamental theoretical study on the relative merits of different types of supersonic wind tunnels. Later von Karm~in referred to this review article as the "bible of supersonic wind tunnels."
Chair of Chemical Technology, Imperial College of Science & Technology, London
Bone, Fraser, and Wheeler 415 study "spin detonation" in a moist 2CO + 0 2 medium and use a high-speed rotating mirror camera to measure the flame speed of detonation phenomena in tubes. They observe a periodic structure in the detonation wave and come to the following important conclusion: "A new view of the detonation-wave in gaseous explosions is advanced. For it can no longer be regarded as simply a homogeneous 'shock wave,' in which an abrupt change in pressure in the vicinity of the wave-front is maintained by the adiabatic combustion of the explosive medium through which it is propagated; but it must now be viewed as a more or less stable association, or coalescence, of two separate and separable components, namely of an intensively radiating flame-front with an invisible shock wave immediately ahead of it; and whether persistent 'spin' is developed or not depends upon the stability or otherwise of their association . . . . " 9 Their observations stimulated other researchers who, although coming to a different explanation of the origin of the periodic phenomena, essentially confirmed the inhomogeneity of the detonation front.
110
1937
1938
P. Krehl
Supersonic Wind Tunnel Division, Institute of Aerodynamics, TH Aachen
Hermann, Wieselsberger's assistant at the supersonic wind tunnel facility (M -- 3.3, working section 10 x 10 cm2), performs aerodynamic tests on models of the A-3 rocket, the first large liquid-fuel rocket (length 6.74m, weight 740kg) and antecedent of the A-4, later renamed V-2 (Peenemfinde--~ 1942). 416 By increasing the length of the tail unit he verifies a stable flight even at high Mach numbers. 9 In the following year plans were worked out "to build an aerodynamic-ballistic research institute, capable of furnishing all aerodynamic, stability, aerodynamic control, and heat transfer data needed for the development of numerous projects, such as supersonic projectiles (fired from guns), rocket-powered vehicles without wings, stabilized by fins (called missiles) and rocket-powered supersonic vehicles with wings and fin-assemblies or with delta wings. ''417
Safety in Mines Research Board, Sheffield, U.K.
Payman and Shepherd 418 rediscover the shock tube (although this term was not yet coined) as a powerful tool to study combustion processes in air-methane mixtures and to clarify whether a shock wave alone could start an explosion in a firedamp/air atmosphere.
5&ool of Aeronautical Engineering, University of Rome
L. Crocco 419 investigates fluids in chemical equilibrium and, combining the entropy equation with the momentum equation, obtains a relation between flow velocity V, vorticity V x V, and thermodynamic properties. The "Crocco equation" contains the important result that a vortex-free flow behaves isentropically in the whole flowfield. 9 His equation was later extended by Vazsonyi 42~ to take into account fluid viscosity and became known as the "Crocco-Vazsonyi equation."
General Electric Company, Schenectady, NY
Tonks, 421 studying high-current-density phenomena in lowpressure arcs, coins the designation "pinch effect.". In the dynamic pinch, the radius of the plasma column decreases with time and the cylindrical current shell moves inward, thus acting like a magnetic piston and sweeping up all of the charged particles it encounters (the snowplow concept). The pinch effect is mostly used to compress gaseous matter. Bless, 422 however, using pinched hollow metal conductors, first demonstrated the suitability of the pinch method for shock compression of solid miniature specimens as well.
Luftkriegsakademie Berlin-Gatow
O. von Schmidt 423 treats wave propagation at the boundary between two media of different wave speeds. He observes that any wave that enters a material with a higher wave propagation velocity produces a "von Schmidt head wave" [von Schmidt'sche Kopfwelle] in the material with the lower propagation velocity that appears similar to a head wave produced
History of Shock Waves
111 by a supersonic bullet. Other wave types in solids, such as transversal and bending waves, can produce head waves under different angles as well. The von Schmidt head wave, however, is a pseudo-supersonic phenomenon, independent of the presence of any shock waves and observable also with sound wave. Nevertheless, it is of great importance for seismology. 424
Southern Methodist University, Dallas, TX
Boon and Albritton 425 show that geologic structures of the Kentland type ("shatter cones", Bucher--+1933) are the product of a meteorite impact. According to their theory, high-velocity impact~many times faster than the velocity of a shock wave in any type of rock--compressed the rocks elastically, rather than deforming them plastically, after which they were "backfired" into a damped-wave disturbance. They assumed that the shatter-cones, typically pointing toward the impinging body, were formed during the initial or compressional stage of such a meteorite i m p a c t . . Later Dietz 426 suggested shatter cones as useful field criteria ("index fossil") for shock-wave fracturing in the geological past, thus constituting presumptive evidence for astroblemes-ancient meteorite impact scars.
U.S.S.R.
Belajev 427 first applies an exploding wire to produce detonation in nitrogen chloride and nitroglycerine. Subsequently Johnston 428 in the United States found that the shock wave, generated by an exploding wire, could also produce detonation in PETN (pentaerythritol tetranitrate), a less sensitive explosive in which detonation cannot normally be effected by a heated wire. This important discovery allowed one to replace the pill of primary explosive in the conventional detonator (Nobel---, 1863) with a secondary explosive, thus substantially reducing the handling hazards. The "safety detonator" became a much-applied device in missile and space vehicle technology, where exploding bridge-wire detonatorsmsuch as in state separators, cable cutters, and explosive bolts--are widely used. Another great advantage is the reduction in the time delay from milliseconds to microseconds, which allows an exact synchronization.
S iemens- Werke Berlin; Research Laboratory, General Electric Company, Schenectady, NY
The flash X-ray technique--the generation of high intensity X-ray pulses of microsecond durationmis introduced. Steenbeck, 429 who invented the method the year before at the Siemens-werke Berlin, uses a capacitor discharge through a mercury-vapor-filled capillary discharge tube, which provides a small focus. He immediately recognizes flash radiography as an outstanding diagnostic tool to stop motion of projectiles in
112
p. Krehl flight, shock waves in optically nontransparent media, and self-luminous events such as detonation waves. In the same year this new method is applied to make flash radiographs of detonating hemisphere-shaped charges (Thomanek-+ 1938). Kingdon and Tanis 43~ in the United States independently generate flash X-rays by using a different diode type to study mutation effects in biological samples.
Luftkriegsakademie Berlin-Gatow
Thomanek 431 discovers the importance of the cavity liner and documents the "shaped charge lined cavity effect." He started his studies on Schardin's hypothesis that the cavity effect might be caused by the Mach effect (Mach and Wosyka--+ 1875). Thomanek's first liner material is the glass recipient used in experiments to evacuate the cavity. His colleague Thomer 432 visualizes for the first time the jet formation by using the recently developed flash radiography technique (Steenbeck -+ 1938). This method allows the study of the collapse of the liner without the interference of smoke and flame associated with the explosion. 9 After World War II, the Swiss Mohaupt 433 claimed in an article to have already discovered the lined cavity effect as early as 1935.
Institut fiir Aerodynamik, ETH Zfirich
Preiswerk 434 investigates the applicability and limitations of the analogy between a two-dimensional shock wave and a hydraulic jump (i.e., a horizontal water flow at low depths and with a free surface). With examples of a hydraulic jump propagating through a plane Laval nozzle or being reflected obliquely at a solid boundary, he notices that with increasing strength of the hydraulic jump the water flow measurements increasingly deviate from the gas dynamic solution. 9 At this time this analogy was of particular interest because it would have allowed the replacement of expensive highspeed diagnostics--such as are required in the case of shock wave diagnostics--by relatively simple water-table installations.
KaiserWilhelmInstitut far Chemie, Berlin
Hahn and Strassmann 435 perform the first artificial nuclear fission of uranium using neutrons. They cautiously annotate that their results (published Jan. 6, 1939) "are in opposition to all the phenomena observed up to the present in nuclear physics.",, In the following year, their colleague Fl~igge436 first estimated the released energy of the uranium fission process and stated that one cubic meter of uranium oxide (corresponding to approximately 4.2 tons of pitchblende) contains sufficient fission energy to cover the consumption of electric energy of central Germany for a period of 11 years. He also speculated on the huge quantity of explosive energy
History of Shock Waves
113 that could be released artificially by nuclear fission within milliseconds. In nature, however, this event is quite unlikely because the concentration of uranium, even in highly enriched deposits, is far too low to maintain a chain reaction. Shortly thereafter, he states 437 in a Berlin newspaper that the fission energy of about 4 tons of uranium oxide would be sufficient "to throw the water mass of Lake Wannsee [a renowned lake in the Southwest of Berlin with a length of about 3 km] into the stratosphere." Given in a popular-science manner, it is the earliest example of illustrating the huge amount of energy that could be released explosively by nuclear fission. Less than six years later it will be realized in the first American atomic bomb. It is interesting here to note that in the same year when FLOGGE published his estimations, Zeldovich and Khariton delivered a report on this topic at a seminar held at the Leningrad Physico-Technical Institute, in which they elucidated the conditions for a nuclear explosion and estimated its destructive force. 438 Later they both contributed mainly to the developmenht of the first Soviet atomic bomb (1949).
1939
Heeresversuchsanstalt Peenemfinde, Baltic Sea, Germany
The world's most advanced supersonic wind tunnel is installed under the leadership of R. Hermann. It is a blowdown-to-vacuum complex (M----4.4, later extended to 5.3; working section 4 0 • 4 0 c m 2) with a three-component balance for measuring drag, lift, and pitching moment. Its main task will be the aerodynamic optimization of the rockets A-4, A-5, and the guided missile Wasserfall. Aerodynamic characteristics of these models, such as drag and lift, can be measured using an electromagnetic balance. Operation above M = 5, however, reveals that condensation effects of the air become significant and impair visualization. This discovery eventually will led to the installation of the first dryer system to take moisture out of the air before it enters the nozzle. 9 After the war the famous Peenem(inde wind tunnel was confiscated by the U.S. Army, dismantled and shipped to the Naval Ordnance Laboratory (NOL) at White Oak, MD. Later the facility was operated by the Naval Surface Warfare Center (NSWC), now defunct.
Heinkel Factory, Warnemfinde
The first successful flight of the Heinkel He-178 occurs (max. velocity 700 kin/h), the world's first turbojet aircraft which leads to an aviation revolution. It was designed by Pabst von Ohain, a graduate from the University of GOttingen.
114
P. Krehl
Aerodynamische Versuchsanstalt GOttingen
The first measurements are carried out on sweepback wings in a high-speed tunnel (cross section 11 • 11 cm 2) at velocities close to the sound velocity, thus following suggestions made by Busemann (--+ 1935) and Betz. 439 These studies were not published until after World War II and caused considerable sensation among foreign aerodynamic experts. 44~
1940
New York City
The Manhattan Project--code name for the U.S. effort during World War II to produce the atomic bomb--is initiated. 441 The initially slow-growing project was named after the Manhattan Engineer District of the U.S. Army Corps of Engineers, because much of the research was done in New York City. 442 O n April 1943 Serber, one of Oppenheimer's assistants, defines the goal more specifically: "The object of the project is to produce a practical military weapon in the form of a bomb in which the energy is released by a fast neutron chain reaction in one or more of the materials to show nuclear fission.",, The project had an enormous impact on the further evolution of shock wave physics and detonics. It led to the installation of a large number of special laboratories and test sites, operated by both governmental agencies, private research organizations and universities.
1940
United States
Von Form~in and Dryden, both on a business trip to Washington DC, discuss shock phenomena in planes that occurred at transonic speeds. In his memoirs von Karman 443 writes, "We talked about the phenomenon and decided that if we invented a word, it had to be something between subsonic and supersonic to indicate that the body travels 'through' the speed of sound and back. We chose 'trans-sonic'. However, there was an argument as to whether to spell it with one s or with two s's. My choice was one s. Dr. Dryden favored two s's ..... We agreed on the illogical single s and thus it has remained. Incidentally, I used this new expression in a report to Wright Field. Although we just made up the word, nobody asked me what it meant. They just accepted transonic as if it had always belonged to the language . . . . " . The transonic regime was then of great practical importance because pilots of the Lockheed P-38 reported that around Mach 0.8 their aircraft was shaken wildly and lost equilibrium. At high flight speed, the air moved over certain parts of the wing and tail at a speed greater than the speed of the plane because of the curvature of these sections. This phenomenon created shock waves that, dancing forward and back, caused dangerous vibrations of the skin structure (shock stall). In 1941 a Lockheed test pilot died when shock waves from the plane's wings
History of Shock Waves
115 created turbulence that tore away the horizontal stabilizer, sending the plane into a fatal plunge.
1941
Safety in Mines Research Establishment, Sheffield, U.K.
Payman and Shepherd 444 continue their shock tube combustion studies (---,1937) and make schlieren pictures of the shock wave with different driver gases. They notice that hydrogen as a driver gas results in higher shock pressures in the test chamber.
Institute of Chemical Physics, Leningrad [now Petersburg]
Zeldovich 445 presents his steady detonation model, assuming that a nonreactive shock wave is the leading element in the detonation, followed by a reaction zone in which detonation is initiated and completed, thereafter followed by the nonreactive flow. 9 Shortly thereafter the same idea was worked out independently by von Neumann 446 in the United States and by D6ring 447 in Germany, and is known today as the
Messerschmitt AG, Augsburg
Zeldovich-von-Neumann-DOring (ZND) theory. The Messerschmitt Me-262 becomes the world's first sweepback jet fighter. It is the fastest aircraft of that time (870 k m / h at an altitude of 6100m). Prior to this, the usefulness of sweepback wings was first proven at the Messerschmitt Company by wind tunnel tests. 448 9 In the same year the Me-163, a rocket plane that already approached the deltawing geometry, made its maiden flight (M = 0.84). At the end of the war, the plane was built with a 45 ~ sweepback in its wings.
Services des Poudres, France
Michel-Levy, Muraour, and Vassy 449 study luminous phenomena in various gases behind the shock front. They generate strong shock waves by head-on collision of shock waves emitted by explosives, and in argon they observe an intense light emission that increases toward the ultraviolet.
Abteilung fur Technische Physik & Ballistik, Luftkriesakademie Berlin-Gatow
Schardin 45~ first suggests the possibility that phase transformations might be induced by shock waves (shock-induced freezing). He fired bullets into a tank filled with carbon tetrachloride and water at speeds varying from 800 to 1800m/s and photographed the process. He found the region surrounding the bullet to be opaque in tetrachloride at 1200m/s and in water at 1800m/s, whereas water remained transparent at 800m/s.
Cavendish Laboratory, Cambridge
G. I. Taylor TM assumes a high-intensity point explosion and calculates the propagation law of the blast wave. 9 His results, then of greatest military importance and top secret, were not published in the open literature until 1950. At the beginning of his paper he stated, "This paper was written early in 1941 and circulated to the Civil Defense Research Committee of the Ministry of Home Security in June of that year. The present
116
e. Krehl writer had been told that it might be possible to produce a bomb in which a very large amount of energy would be released by nuclear fission--the name atomic bomb had not been used---and the work here described represents his first attempt to form an idea of what mechanical effects might be expected if such an explosion could occur. In the thencommon explosive bomb mechanical effects were produced by the sudden generation of a large amount of gas at a high temperature in a confined space. The practical question which required an answer was: Would similar effects be produced if energy could be released in a highly concentrated form unaccompanied by the generation of gas? This paper has now been declassified, and though it has been superseded by more complete calculations, it seems appropriate to publish it as it was first written, without alterations . . . . " He found that only for a point explosion with an instantaneous energy release (the ideal case) the shock wave moves with a steady speed (D = const), analogous to the case of a plane detonation, but that the pressure p with increasing distance r very rapidly decreases (p "- 1/r3). Independently from Taylor, an analogous solution for a point explosion was also obtained in the Soviet Union by Sedov. 452 The corresponding cylindrical problem was solved by Lin, 453 (1954), he obtained similar results (D = const, but p "~ l/r2).
1941
David W. Taylor Model Basin, Carderock Division, MD
[now NSWCCD]; ChemischPhysikalische Versuchsanstalt der Kriegsmarine, Kiel
Studies in the United States 454 and Germany 455 are initiated to study the oscillation of the gas globe ("bubble") of an underwater explosion, beginning on a laboratory-scale using Edgerton-stroboscopy and high-speed cinematography, respectively. The experimental investigations stimulate theoretical studies in the United States and England on the bubble motion and shock wave generation, leading throughout the war to a wealth of new data on underwater shock wave propagation and interaction phenomena with boundary surfaces (published in 1950 as UNDEX Reports). Berthe and Kirkwood 456 will demonstrate that after reaching the rebound point in bubble dynamics, a shock wave is emitted into the surrounding liquid, thus essentially confirming the photography studies. 9 Early investigations of an explosion from a single charge had revealed that the main shock is followed by a second large pulse and further small ones (Blochmann --~ 1898). Probably not later than in submarine warfare of WWI this phenomenon was recognized as a particular threat to a submarine's hull. Campbell 457 appropriately wrote in his report, "For some time, submarine personnel have noticed that more than one impact results from a single nearby
History of Shock Waves
11 7 underwater explosion, such as a depth charge. Successive shocks were noted, and it was believed that the intensity and the time between blows decreased with each successive blow. Motion pictures of the action of floating models subjected to underwater explosions corroborated this impression."
1942
Heeresver-
suchsanstalt Peenemfinde, Germany
On October 3 the rocket A-4 (later named V-2) covers a range of 191 km and reaches a record height of 84.5 km, thus being the first man-made vehicle to penetrate into space. 458 The missile has a total length of almost 14 meters and was capable of transporting a payload of 750 kg with a velocity of up to Mach 4.
LASL, Los Alamos, NM
The Los Alamos Scientific Laboratory (LASL) is established by the U.S. government to centralize nuclear bomb research and development for the Manhattan Project (---~1940s), which had hitherto been performed at the Universities of Chicago, Cornell, Minnesota, Purdue, Stanford and Wisconsin, and the Carnegie Institution of Washington. 459
Institut far Gasdynamik, LFA, Braunschweig
Guderley, 46~ treating the implosion of cylindrical and spherical shock waves mathematically, predicts infinite shock strengths at the implosion center. Real gas effects, however, furnish a natural limit to these theoretical singularities.
Institute for Advanced Study, Princeton, NJ
Von Neumann speculates on a method to find an equivalent Huygens principle for waves of finite amplitude (shock waves). 461 9 Unfortunately, details of this interesting approach have not been passed onto us.
Dept. of Physics, Cornell University, Ithaca, NY
Bethe 462 calculates the stability of shock waves for an arbitrary equation of state and deals with the case when a phase transition is induced by the shock, at that time a phenomenon thought to be possible in strong underwater explosions. In his introduction he circumscribes the goal of his study: "The theory of shock waves thus far has been developed mainly for ideal gases. Even for these, the question of stability of shock waves has received little attention. Recently, the problem of shock waves in water has gained much practical importance. Therefore, it seems worthwhile to investigate the properties of shock waves under conditions as general as possible . . . . " He treats the Hugoniot curve H(v, s) in terms of volume per unit mass, v, and entropy per unit mass, s, and derives the following three stability conditions of a shock wave: (i) 32p(v,s)/3v2>O; (ii) v Op(v,e)/3e>-2; and (iii) 3p(v, e)/Ov < 0 He concludes that the transition from solid to liquid, from solid to gas, and from liquid to gas, as well as the reverse transitions, should not affect the stability of the shock, while in the solid-solid transitions the shock front
118
P. Krehl would split into successive shocks, the first one raising the medium to a metastable state and the second one transforming it into the new stable phase.
1943
Applied Mathematics Group, Institute for Advanced Study, Princeton, NJ
Von Neumann 463 develops a "two-shock theory" of regular reflection and a "three-shock theory" of Mach reflection. He also coins the term Mach effect to denote such a three-shock configuration. 9 The quantitative experimental evidence of his theory of oblique reflection of shock waves came mainly from the four following sources (all provided in 1943): (i) Aberdeen ballistic photographs; (ii) Princeton shock-wave tube photographs; (iii) Teddington supersonic wind-tunnel photographs; and (iv) Prof. Wood's model shock interaction experiments at Johns Hopkins University using Mach and Wosyka's method of shock generation and soot recording (--->1875). The Mach effect was actually used in the bombing of Hiroshima and Nagasaki (--+ 1945) to determine the position (height of burst, HOB) of the atomic bomb best suited for optimum damage. 464
The Johns Hopkins University, Baltimore, MD
Wood 465 repeats E. Mach and Wosyka's soot experiments (-+ 1875) and E. Mach's and L. Mach's schlieren photography of two interacting spark waves (-+ 1889). Wood's results fully confirm the existence of "Mach disk" (or "Mach bridge") formation.
Ballistic Research Laboratories (BRL), Aberdeen, MD
Charters 466 analyses Wood's soot experiments (--+1943) and discovers in E. Mach's "opposite V-gliding-spark" arrangement (--+ 1875) "lines of discontinuity" that separate areas of equal pressuremi.e., are not shock waves. However, they represent discontinuity lines for entropy, temperature and density, 467 = Later they were called contact discontinuity lines (Courant and Friedrichs 1948) or slipstreams (Bleakney and Taub, 1949).
Institut fftr Mechanik, TH Aachen
Schultz-Grunow 468 publishes a pioneering paper on shock wave propagation in ducts having an area change segment. He correctly treats the flow up- and downstream of the area change segment as an unsteady, one-dimensional flow using the theory of characteristics, but approximates the flow in the area change section as being quasi-one-dimensional and steady. 9 His method of approximation was an important achievement for many engineering applications in the precomputer era. For example, it allowed for the first time the determination of the exhaust flow of internal combustion engines, the flow in diffusers such as in shock tunnels and in converging/diverging nozzles used in rockets and jet propulsion engines. Today the flow through ducts with inserted area
History of Shock Waves
1 19 change segments can easily be handled as a truly two- (or three-) dimensional unsteady flow.469
1944
David W. Taylor Model Basin, Carderock Division, MD
Campbell, Spitzer, and Price 470 study interference effects of spherical shock waves resulting from two underwater detonations of small charges and first prove that Mach reflection also exists in water. However, they cannot detect any slipstreams, which is possibly caused by the fact that the pressure level of intersecting shock waves is too low.
Princeton University Station, NJ
Reynolds 471 uses a pressurized "pot" terminated by a diaphragm, which, when pierced, produces a steep pressure pulse with a rise time of only a few nanoseconds. His device, a kind of short shock tube, is very appropriate to calibrate piezoelectric gauges. 9 Since then the piezoelectric gauge has become the standard pressure gauge in most shock tube facilities and is used routinely at most head-on and side-on shock front positions.
Guggenheim Aeronautical Laboratory, Pasadena, CA
Von I~rm~in designs the first large modern American supersonic wind tunnel (working section 15 • 15 in.) for the BRL in Aberdeen, MD. The famous astronomer Hubble will temporarily act as a director of this facility.
BRL, Aberdeen Proving Ground, MD
Sachs 472 extends the Hopkinson scaling law (B. Hopkinson-+1915) to account for effects of altitude or other changes in ambient conditions on air blast waves ("Sachs scaling law").
Heel'esver-
Erdmann 473 modifies the nozzle of the supersonic wind tunnel and performs the first hypersonic wind tunnel tests at a Mach number close to 9. Shortly thereafter the evacuation of the Peenem~inde Supersonic Laboratory to Kochel begins. Plans are worked out to directly use the Walchensee Hydroelectric Plant for providing the required enormous power (about 60 MW) to operate a huge hypersonic wind tunnel (1 • 1 m 2, M = 10) in the future. 9 Hypersonic wind tunnel studies were not resumed until 1947 after the completion of the first American hypersonic facility by the NACA at Langley, VA. The tunnel had an l 1-in. 2 test section, capable of reaching hypersonic flow up to M -- 7.
suchsanstalt Peenemfmde, Kochel, Bavarian Alps
University of GOttingen
Oswatitsch 474 performs for the German Army Ordnance [Heereswaffenamt] the first theoretical and experimental studies to determine the factors influencing muzzle (or recoil) brake efficiency. These brakes recover momentum from the exhausting propellant gases by deflecting the flow away from the direction of fire. However, they also increase significantly the blast overpressure behind the gun in the vicinity of crew members. This problem will remain a
120
e. Krehl permanent challenge to postwar designers of large caliber cannons.
1945
BRL, Aberdeen Proving Ground, MD
Thomas 475 discusses Becker's theory of the shock front (---~1921). He shows that the shock front's thickness is always at least of the order of magnitude of a free path length and that the Boltzmann equation can be applied even for the most violent shocks.
Applied Mathematics Group, Institute for Advanced Study, Princeton, NJ
Von Neumann 476 proposes a new approach to the hydrodynamical shock problem that he applies to the collision of shock and rarefaction waves. His method, based on a simple pressure-density relationship as already proposed by Riemann ( ~ 1859), provides also a computational procedure and will be resumed in the following years by yon Mises and Geiringer (1948).
Palmer Physical Laboratory, Princeton University, NJ
Smith 477 uses a shock tube and photographs the oblique reflection of plane shocks in air, thus giving the first quantitative information about the validity of von Neumann's twoand three-shock solutions (von Neumann --~1943). He discovers that, contrary to the reflection of sound waves, a reflected shock reflects at a larger angle than the angle of incidence. At large shock strengths, the Mach reflection begins at nearly the angles at which the theory says regular reflection is not possible. For weak shocks, regular reflection continues to be seen at larger angles of incidence than where they are theoretically impossible. 478 His discrepancy will later be named the "von Neumann paradox." Smith also first observes complex Mach reflection. 9 In the same year YON NEUMANN,479 treating various shock wave interaction phenomena, termed in the case of Mach reflection the new shock wave--a mergence of the reflected shock with the incident shock in the vicinity of the reflecting wall---the
Mach stem. ISL, SaintLouis, Alsace, France
The Laboratoire de Recherches Balistiques et Ad.rodynamiques de Saint-Louis (LRSL), is founded. The first directors are Prof. H.
LASL, Los Alamos, NM
Goranson and coworkers initiate a program to determine equation-of-state data of shock-compressed materials, a subject of immediate interest for the design of nuclear weapons and their effects. It will stimulate also other laboratories in the United Statesmand shortly after also in the Soviet UnionNto initiate research in shock wave physics on
Schardin and Gen. R. Cassagnou. 9 In 1959 it was transformed into a joint French-German research institute to promote the scientific cooperation between France and Germany and renamed Institut Saint-Louis (ISL).
History of Shock Waves
121 a large scale.. At that time some theoretical studies on the behavior of shock waves in solids already existed, provided for example by early pioneers such as Christoffe148~ (1877), Hugoniot 481 (1889), Duhem 482 (1903), Hadamard 483 (1903), and Jouguet 484 (1920)] and some as well as on the theory of plastic waves (B. Hopkinson-+1905). Even so, the longplanned systematic campaign at Los Alamos and other national research laboratories and private research organizations can be regarded as the birth of modem solid state shock wave physics. 485
Trinity Site, Alamogordo, NM
On July 16, the first nuclear fission bomb is ignited at an altitude of 100 ft. The bomb is an implosion-type weapon that uses high explosive lenses to rapidly implode a hollow subcritical sphere of fissionable material into a solid supercritical sphere. Measurements are made by Fastax cameras (i) of the shock wave expansion by positioning cameras at halfmile stations; and (ii) of the mass velocity, using suspended Primacord and magnesium flash powder (upon analysis of the results, a total yield of 19,000 tons TNT equivalent was found). The peak pressure is recorded, using spring-loaded piston gauges. The excess shock velocity in relation to sound velocity is measured with a moving-coil loudspeaker pickup. Fermi 486 devised his own order-of-magnitude method of roughly determining the blast yield: "About 40 seconds after the explosion the air blast reached me. I tried to estimate the strength by dropping from about six feet small pieces of paper before, during, and after passage of the blast wave. Since, at that time, there was no wind, I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2~ meters, which, at the time, I estimated to correspond to the blast that would be produced by 10,000 tons of TNT . . . . " 9 Later investigations showed that, depending on the height of burst, about 45-55% of the fission energy appears as blast and shock. Since the positive duration of a blast wave from a nuclear explosion is longer than from a chemical one, damage effects to air blast loading are more severe than hitherto observed from conventional explosions.
Japan
On August 6 Hiroshima is bombed (Little Boy, uranium-guntype bomb, equivalent to 15,000 tons of TNT, HOB ~ 1900 ft) and on August 9 Nagasaki is bombed (Fat Man, plutonium implosion bomb, equivalent to 21,000 tons of TNT, HOB ~, 1850ft). Later the crew of the B-29 (Enola Gay), which dropped the bomb and witnessed the explosion from on board, contributed to the following official account 487 of
122
P. Krehl the first atomic air raid in history: "The flash after the explosion was deep purple, then reddish and reached to almost 8,000 feet; the cloud, shaped like a mushroom, was up to 20,000 feet in one minute, at which time the top part broke from the 'stem' and eventually reached 30,000. The stem of the mushroom-like column of smoke, looking now like a giant grave marker, stood one minute after the explosion upon the whole area of the city, excepting the southern dock area. This column was a thick white smoke, darker at the base, and interspersed with deep red. Though about fifteen miles (slant range) from the target when the explosion occurred, both escort aircraft, as well as the strike plane, reported feeling two shock waves jar the aircraft. Approximately 390 statute miles away from the target area, the column of smoke still could be seen piercing the morning sky." The second shock was caused by reflection of the primary shock at the ground. The precise yield of these two bomb explosions was difficult to state for that early type of weapon and remained a subject of later discussions and investigations. 488 In the subsequent long period of Cold War, the knowledge of yields was of particular interest in understanding the mechanism of observed damage on a wide spectrum of civilian targets and radiation effects on man, and in predicting damage scenarios in a possible future nuclear war.
1.9
NOTES
1. C. A. Truesdell. Euler's two letters to Langrange in October, 1759. In Leonardi Euleri Opera Omnia XIII [II]. Teubner, Leipzig (1926). See also Editor's introduction, pp. xxxvii-xli. 2. H. Cavendish. A measurer of explosions of inflammable air (laboratory note). In The scientific papers on the Honorable Henry Cavendish, F.R.S., (E. Thorpe, ed.) Chemical & Dynamical, vol. II, Univ. Press, Cambridge (1921), pp. 318-320. 3. J. Canton. Experiments to prove that water is not incompressible. Phil. Trans. Roy. Soc. London 52:640-643 (1762); Experiments and observations on the compressibility of water and some other fluids. Phil. Trans. Roy. Soc. London 54:261-262 (1764). 4. D. Bernoulli. Examen physico-mechanicum de motu mixto qui laminis elasticis a percussione simul imprimitur. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae XV: 2931 (1770). 5. G. Monge. M~moire sur la construction des fonctions arbitraires dans les int~grales des equations aux differences partielles. M~noires des mathCmatiques et de physique pr~sent~s l'AcadCmie...par divers sc~avans... 7 [l'ann~e 1773, 2e partie]:267-300, 305-327 (1773). Imprimerie Royale, Paris (1776); Sur la d~termination du fonctions arbitraires dans les int(.grales de quelques (.quations aux differences partielles. Miscellanea taurinensia (Torino) 5, 16--78 (1770-1773).
History of Shock Waves
123
6. R. Taton. ~oeuvre scientifique de Gaspard Monge. Presse Universitaire de France, Paris (1951); Gaspard Monge. Dict. Scient. Biogr. 9: 469-478. Scribner's Sons, New York (1973). 7. K. Oswatitsch. Uber die Charakteristikenverfahren der Hydrodynamik. ZAMM 25//27: 195208, 264-270 (1947). 8. R. Courant and K. O. Friedrichs. Supersonic flow and shock waves. Interscience Publ., New York (1948). 9. M. B. Abbott. An introduction to the method of characteristics. American Elsevier, New York (1966), pp. 35-44, 128-163. 10. E. Nairne. Electrical experiments by Mr. Edward Nairne, of London, mathematical instruJ mentJmaker, made with a machine of his own workmanship, a description of which is prefixed. Phil. Trans. Roy. Soc. London 6 4 : 7 9 - 8 9 (1774). 11. H. Mtiller. Gewehre, Pistolen, Revolver. Kohlhammer, Stuttgart (1979). See also: Windbfchsen und elektrische Zundung, pp. 154-156. 12. J. L. Lagrange: Memoire sur la th~orie du mouvement des fluides. Nouv. M~m. Acad. Roy. Sci. & Belles-Lettres Berlin (1781), pp. 151-198. 13. E R. Gilmore, M. S. Plesset, H. E. Crosley, Jr.: The analogy between hydraulic jumps in liquids and shockwaves in gases. J. Appl. Phys. 21:243-249 (1950). 14. C. Hutton. New experiments in gunnery, for determining the force of fired gunpowder, the initial velocity of cannon ball, the ranges of projectiles at different elevations, the resisatnce of the air to projectiles, the effect of different length of guns, and of different quantities of powder, &c, &c. In: Tracts on mathematical and philosophical subjects. T. Davidson, London (1812). See vol. 2, Tract XXXIV, pp. 306-384. 15. C. Hutton. Theory and practice of gunnery, as dependent on the resistance of the air. In: Tracts on mathematical and philosophical subjects. T. Davidson, London (1812). See vol. 3, Tract XXXVII, pp. 209-315. 16. H. Cavendish. Experiments on air. Phil. Trans. Roy. Soc. London 74: 481-502, 510-514 (1784), 7 5 : 1 5 - 2 2 (1785). 17. Count Morozzo. Account of a violent explosion which happened in a flour~warehouse, at Turin, December 14th, 1785; to which are added some observations on spontaneous inflammations. The Repertory of Arts and Manufactures 2:416-432 (1795). 18. J. L. Lagrange. Sur la percussion des fluides. Mcm. Acad. Roy. Sci. Turin I: 95-108 (17841785). 19. D. Bernoulli: Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Dulsecker, Strassburg (1738); translated into English by T. Carmody, H. Kobus: Hydrodynamics. Dover Publ., New York (1968). 20. J. Goodricke. A series of observations on, and a discovery of the period of the variations of light of the star marked ~ by Bayer, near the head of Cepheus. Phil. Trans. Roy. Soc. London 16: 48-61 (1786). 21. L. I. Sedov. Similarity and dimensional methods in mechanics. Infosearch Ltd., London (1959), pp. 305-353. 22. E yon Baader. Versuch einer Theorie der Sprengarbeit. Bergmannisches J. 1 (No. 3):193-212 (March 1792). 23. D. R. Kennedy. History of the shaped charge effect. The first 1O0 years. Company brochure prepared by D. R. Kennedy & Associates, Inc., Mountain View, CA (1983). 24. G. Pinet: Histoire de l'~cole polytechnique. Baudry, Paris (1887). 25. E. E E Chladni. ~ber den Ursprung der yon Pallas gefundenen und anderer ahnlicher Eisenmassen, und fiber einige damit in Verbindung stehende Naturerscheinungen. J. E Hartknoch, Riga (1794). 26. Journal der Physik 8, 20-21 (1794).
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27. I. I. Glass and J. P. Sislian. Nonstationary flows and shock waves. Clarendon Press, Oxford (1994), pp. 271-275. 28. J. M. Montgolfier: Note sur le belier hydraulique, et sur la maniere d'en calculer les effets. J. des Mines XIII, 42-51 (1802-1803). 29. Weinmann Sondermaschinen- und Steuerungsbau GmbH, 91217 Hersbruck, Germany. 30. J. B. Biot. Theorie mathematique de la propagation du son. J. Phys. 55:173-182 (1802). 31. J. B. Biot. Account of a fire-ball which fell in the neighborhood of EAigle: In a letter to the French Minister of the Interior. Phil. Mag. 16 [I]:224-228 (1803). 32. Brockhaus Konversationslexikon. Brockhaus, Leipzig (1908). See vol. 11, p. 815. 33. E L. Neher. Die Erfindung der Photographie. Kosmos, Stuttgart (1938), p. 36. 34. S. D. Poisson. Memoire sur la theorie du son. J. Ecole Polytech. (Paris) 7:319-392 (1808). 35. J. L. Lagrange. Sur une nouvelle methode de calcul integral pour les differentielles affectees d'un radical carre, sous lequel la variable ne passe pas le 4 e degre. Mem. Acad. Roy. Sci. Turin II: 218-290 (1784-1785). 36. J. L. Gay-Lussac and L. J. Thenard. De la nature et des proprietes de racide muriatique et de l'acide muriatique oxigene. M~n. Soc. Arcueil 2:339-358 (1809). 37. G.J. Singer and A. Crosse. An account of some electrical experiments by M. De Nelis. Phil. Mag. 46 [I]:161-166 (1815). 38. P. S. Laplace. Sur la vitesse du son dans l'air et dans l'eau. Ann. Chim. & Phys. 3:238-241 (1816). 39. H. Davy. (I) On the fire-damp of coal mines, and on methods of lighting the mines so as to prevent its explosion. Phil. Trans. Roy. Soc. London 106:1-22 (1816); (II) An account of an invention for giving light in explosive mixtures of fire-damp in coal mines by consuming the fire-damp. Phil. Trans. Roy. Soc. London 106:23-24 (1816). 40. Technisches Museum Wien, A-1140 Vienna, Austria. 41. L. M. H. Navier. Memoire sur les lois du mouvement des fluides. Mem. Acad. Roy. Sci. Paris 6: 389-440 (1823). 42. S. D. Poisson. Memoire sur les equations generales de l'equilibre et du mouvement des corps solides elastiques et des fluides. J. Ecole Polytech. (Paris) 13 (Cahier 20):1-174 (1831). 43. A. J. C. de Saint-Venant. Note sur l'ecoulement de l'air. C. R. Acad. Sci. Paris 21:366-369 (1845). 44. G. G. Stokes. On the theory of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc. 8:287-319 (1845). 45. J. C. Maxwell. On the dynamical theory of gases. Phil. Trans. Roy. Soc. London 157:49-88 (1867). 46. S. D. Poisson. Sur la vitesse du son. Ann. Chim. & Phys. 2 3 : 5 - 1 6 (1823). 47. S. D. Poisson. Sur la chaleur des gaz et des vapeurs. Ann. Chim. & Phys. 23:337-353 (1823). 48. W. E. Parry. Journal of the third voyage for the discovery of a North-West Passage from the
Atlantic to the Pacific; performed in the years 1824-25, in His Majesty's ships Hecla and Fury, under the orders of...W.E. Parry. J. Murray, London (1826). See Experiments to determine the rate at which sound travels at various temperatures and pressures of the atmosphere, Appendix, p. 86. 49. W. E. Parry. Journal of a second voyage for the discovery of a North-West Passage from the
Atlantic to the Pacific; performed in the years 1821-22-23, in His Majesty's ships Fury and Hecla, under the orders of...W.E. Parry. J. Murray, London (1824-25). See p. 140; and Abstract of experiments to determine the velocity of sound at low temperature, Appendix, pp. 237-239. 50. W. E. Parry and H. Foster. Reply to Mr. Galbraith's remarks on the experiments for ascertaining the velocity of sound at Port Bowen. Phil. Mag. 1 [II]:12-13 (1827).
History of Shock Waves
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51. D. Colladon and C. E Sturm. Ober die Zusammendruckbarkeit der Flussigkeiten. Ann. Phys. 12 [II]:161-197 (1828). 52. J. B. C. B~langer. Essai sur la solution num~rique de quelques problCmes relatifs au mouvement permanent des eaux courantes. Carilian-Gceury, Paris (1828). 53. E. Jouguet. Quelques probl~mes d'hydrodynamique g~n~rale. J. Math. Pures & Appl. 3 [VIII]:1-63 (1920). See p. 12. 54. D. E C. Arago. Uber die Explosionen der Dampfmaschinen. Ann. Phys. 18 [II]:287-314, 415436 (1830). 55. C. Wheatstone. An account of some experiments to measure the velocity of electric light. Proc. Roy. Soc. London 3:299-300 (1834). 56. E. Sabine. President's address. Proc. Roy. Soc. London 18:145-147 (1869). 57. L. Foucault. M~thode g~neral pour mesurer la vitesse de la lumi~re dans l'air et dans les milieux transparents. C. R. Acad. Sci. Paris 30:551-560 (1850). 58. B. W. Feddersen. Beitrage zur Kenntnis des elektrischen Funkens. Ann. Phys. 103 [II]:69-88 (1858). 59. P. W W. Fuller and J. T. Rendell. The development of high speed photography. In High speed photography and photonics, S. E Ray, ed. Focal Press, Oxford (1997), pp. 21-23; V. Parker and C. Roberts: Rotating mirror and drum cameras. Ibid. pp. 167-180. 60. D. E J. Arago. On thunder and lightning. Edinburgh New Phil. J. 26: 81-144, 275-291 (1839); Meteorological essays, with an introduction by A. yon Humboldt (translated by R. A. Sabine). Longman & Co, London (1855). 61. M. Faraday. On some supposed forms of lightning. Phil. Mag. 19 [III]:104-106 (1841). 62. G. K. Hubler. Fluff balls of fire. Nature 403:487-488 (2000). 63. A.J.C. de Saint-Venant and P. L. Wantzel. M4moire et exp4riences sur l'4coulement de l'air. J. Ecole Polytech. (Paris) 16:85-122 (1839). 64. C. Wheatstone. Description of the electromagnetic clock. Proc. Roy. Soc. London 4 : 2 4 9 - 2 7 8 (1840). 65. J. S. Russell. The wave of translation in the oceans of water, air and ether. Trubner & Co., London (1885), p. 315. 66. C. S. M. Pouillet. Note sur un moyen de mesurer des intervalles de temps extremement courts, comme la duree du choc des corps ~lastiques, cell du debondement des ressorts, de l'inflammation de la poudre etc.; et sur un moyen nouveau de comparer les intensit~s des courants ~lectriques, soit permanents, soit instantan~s. C. R. Acad. Sci. Paris 19:1384-1389 (1844). 67. M. Faraday and C. Lyell. On explosions in coal mines. Phil. Mag. 26 [III]:16-35 (1845). 68. G. B. Airy. Tides and waves. In Encyclopaedia Metropolitana. Fellowes, London (1845). 69. C. Haeussermann: Gedachtnisrede auf Christian Friedrich SCHONBEIN [Referat]. Z. f. d. gesamte Scheit~- und Sprengstoffwesen 4, 433-434 (1909). 70. J. Taylor. "Improvements in the manufacture of explosive compounds, communicated to me from a certain foreigner residing abroad." Brit. Patent No. 11,407 (Oct. 8, 1846). 71. E A. Abel. On the manufacture and composition of gun-cotton. Phil. Trans. Roy. Soc. London 156:269-308 (1866). 72. J. C. Doppler. Uber den Einflut~ der Bewegung des Fortpflanzungsmittels auf die Erscheinungen der ,~ther-, Luft- und Wasserwellen (presented in 1847). Abhandl. BOhm. Gesellsch. Wiss. Prag 5 [V]:293-306 (1848). 73. A. Sobrero. Sur plusieurs composes d~tonants produits avec l'acide nitrique et le sucre, la dextrine, la lactine, la marnite, et la glycerine. C. R. Acad. Sci. Paris 25:247-248 (1847). 74. G. B. Airy. The Astronomer Royal's remarks on Prof. Challis' theoretical determination of the velocity of sound. Phil. Mag. 32 [III]:339-343 (1848).
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J. Challis. On the velocity of sound. Phil. Mag. 32 [III]:494-499 (1848). G. G. Stokes. On a difficulty in the theory of sound. Phil. Mag. 33 [III]:349-356 (1848). G. B. Airy. On a difficulty in the problem of sound. Phil. Mag. 34 [III]:401-405 (1849). J. C. Maxwell. On the equilibrium of elastic solids. Trans. Roy. Soc. Edinburgh 20:87-120 (1850). 79. W. Goldsmith. Dynamic photoelasticity. In Kurzzeitphysik, K. Vollrath and G. Thomer, eds. Springer, Wien (1967), pp. 579-619. 80. W. H. E Talbot. On the production of instantaneous images. Phil. Mag. 3 [IV]:73-77 (1852). 81. W.J.M. Rankine. On Laplace's theory of sound. Phil. Mag. 1 [IV]:225-227 (1851). 82. S. Earnshaw. On the velocity of the sound of thunder. Phil. Mag. 20 [IV]:37-41 (1860). 83. G. G. Stokes. An examination of the possible effect of the radiation of heat on the propagation of sound. Phil. Mag. 1 [IV]:305-317 (1851). 84. J. Challis. Theoretical determination of the velocity of sound. Phil. Mag. 32 [III]:276-284 (1848); On the theory of the velocity of sound. Phil. Mag. 1 [IV]:405-408 (1851). 85. C. E. Munroe. Modern explosives. Scribner's Mag. 3:563-576 (1888). 86. J. Jamin. Description d'un nouvel appareil de recherches, fondr sur les interferences. C. R. Acad. Sci. Paris 42:482-485 (1856). 87. A. Toepler and L. Bohzmann. Uber eine neue optische Methode, die Schwingungen t6nender Luftsaulen zu analysieren. Ann. Phys. 141 [II]:321-352 (1870). 88. A. Kr6nig. Grundz~ge einer Theorie der Gase. Ann. Phys. 99 [II]:315-322 (1856). 89. R. Clausius. Uber die Art der Bewegung, welche wir Warme nennen. Ann. Phys. 100 [II]:353380 (1857). 90. J. C. Maxwell: Illustrations of the dynamical theory of gases. Part I: On the motion and collisions of perfectly elastic spheres. Phil. Mag. 19 [IV], 19-32 (1860); Part II: On the process of diffusion of two or more moving particles among one another. Ibid. 20 [IV], 21-37 (1860); On the dynamical theory of gases. Phil. Trans. Roy. Soc. London 157, 49-88 (1867). 91. L. Boltzmann. Weitere Studien tiber das Warmegleichgewicht unter Gasmolekulen. Sitzungsber. Akad. Wiss. Wien 66 [II]:275-370 (1872). 92. T. Koga: Introduction to kinetic theory stochastic processes in gaseous systems. Pergamon Press, Oxford etc. (1970). Cf. Appendix G: An historical sketch of kinetic theory, pp. 261-277. 93. J. P. Joule and W. Thomson. On the thermal effects of fluids in motion. (I) On the temperature of solids exposed to currents of air. Proc. Roy. Soc. 8:178-185 (1856-1857); (II) Temperature of a body moving through air. Proc. Roy. Soc. 8: 41-42, 556-567 (1856-1857). 94. T.J. Rodman. Reports of experiments on the properties of metals for cannon, and the qualities of cannon powder; with an account of the fabrication and trial of a 15-inch gun. Crosby, Boston, MA (1861); Mitteilungen fiber Gegenstande des Artillerie- und Geniewesens (Wien). Notizen S. 57 (1881); C. Cranz: Lehrbuch der Ballistik III. Springer, Berlin (1927). 95. A. Noble. On methods that have been adopted for measuring pressures in the bores of guns. Rept. Meet. Brit. Assoc. 64:523-540 (1894). 96. K. W. Knochenhauer. Uber den elektrischen Zustand der Nebenbatterie wahrend ihres Stromes. Sitzungsber. Akad. Wiss. Wien 33 [II]:163-204 (1858). 97. S. Earnshaw. On the mathematical theory of sound (communicated in Nov. 1858). Phil. Trans. Roy. Soc. London 150:133-148 (1860). 98. S. Earnshaw. "On the mathematical theory of sound." Rpt. Meet. Brit. Assoc. 28:34-35 (1858). 99. B. Riemann. Uber die Fortpflanzung ebener Luftwellen yon endlicher Schwingungsweite. Abhandl. KOnigl. Ges. Wiss. GOttingen 8 (Math. Physik. Kl.):243-265 (1860). 100. R. Carrington. Description of a singular appearance seen in the sun on Sept 1, 1859. Month. Not. Astron. Soc. 20:14-16 (1860); On two cases of solar spots in high latitudes, and on the surface currents indicated by the observation. Month. Not. Astron. Soc. 20:255-259 (1860). 75. 76. 77. 78.
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101. A. von Humbold. Kosmos. Entwurf einer physischen Weltbeschreibung, vol. III. Cotta, Stuttgart & Augsburg (1858), p. 127. 102. S. Earnshaw. On the velocity of the sound of thunder. Phil. Mag. 20: [IV] 37-41 (1860); On the triplicity of sound. Phil. Mag. 20 [IV]:186-192 (1860). 103. C. Montigny. Note sur la vitesse du bruit du tonnerre. Bull. Acad. Roy. Belg. 27 [II]:36-46 (1860); Observations sur l'acc~l~ration de la vitesse du bruit du tonnerre. Bull. Acad. Roy. Belg. 27 [II]:62-63 (1860); Fortschritte der Physik 16:165-167 (1860). 104. G. A. Hirn. Sur le bruit du tonnerre. Cosmos (Paris) 16:651-655 (1860). 105. E Raillard. Sur le bruit du tonnerre. Cosmos (Paris) 1 6 : 3 7 2 - 3 7 4 (1860); Du bruit du tonnerre, de ses variations ou de ses roulements, de sa vitesse &. Cosmos (Paris) 17: 166-172, 675-677 (1860). 106. A. B. Nobel. "Nitroglycerin" (manufacturing and fining). Brit. Patent No. 1813 (1864). 107. P. E. Le Bouleng~. M~moire sur un chronographe r (prCsentr 5 Dr 1863). MCm. Couronn. & M~'m. Savants Etrangers de l'Acad. Roy. (Bruxelles) 3 2 : 3 9 pages (18641865). 108. G. B. Airy. "Report on steam boiler explosions." Rpt. Meet. Brit. Assoc. 33:686-688 (1863). 109. R. Armstrong and J. Bourne. The modern practice of boiler engineering, containing observations on the construction of steam boilers. Spon, Leipzig (1856). See also chp. III: Explosions: an investigation into some of the causes producing them, and the deterioration of boilers generally. 110. Dampfkesselexplosionen. In Meyers Konversations-Lexikon, vol. IV Verlag Bibliograph. Inst., Leipzig (1875), pp. 954-956. 111. V. Regnault. On the velocity of the propagation of waves in gaseous media. Phil. Mag. 35 [IV]:161-171 (1868). 112. H.J. Rink. l]ber die Geschwindigkeit des Schalls nach Hrn. Regnault's Versuchen. Ann. Phys. 149 [II]:533-546 (1873). 113. A. Toepler. Beobachtungen nach einer neuen optischen Methode. Max Cohen & Sohn, Bonn (1864), p. 43. 114. H.W. Robinson and W. Adams (eds.). The diary ofR. Hooke. Taylor & Francis, London (1953). 115. L. Foucault. M~moire sur la construction des t~lescopes en verre argentS. Ann. de l'Observatoire Imperial de Paris 5:197-237 (1859). 116. A. Nobel. "Improvements in the manufacture of gunpowder and powder for blasting purposes." Brit. Patent No. 2359 (Sept 24, 1863). 117. E. Bergengren. Alfred Nobel. The man and his work. Nelson & Sons, London etc. (1962). 118. J. Le Conte. On the adequacy of Laplace's explanation to account for the discrepancy between the computed and the observed velocity of sound in air and gases. Phil. Mag. 27 [IV]: 1-32 (1864). The paper was written in 1861 but did not arrive in England until two years later because of the interruption of communication incident to the great revolutionary struggle. 119. H.W. Schr6der van der Kolk. On the velocity of sound. Phil. Mag. 30 [IV]:34-49 (1865); Note on the velocity of sound, and on the mechanical energy of chemical actions. Phil. Mag. 30 [IV] :391-392 (1865). 120. A. B. Nobel: Results of blasting experiments made with nitroglycerin at Vieille-Montagne mine. Phil. Mag. 30 [IV], 236-238 (1865). 121. R. D. Napier. On the velocity of steam and other gases, and the true principles of the discharge of fluids. Spon, London (1866). 122. O. Reynolds. On the flow of gases. Manchester Lit. & Phil. Soc. 2 5 : 5 5 - 7 1 (1885). 123. This letter, dated April 29, 1867, is now in the archives of the Technische UniversitFzt Dresden. 124. A. Toepler. Die vom elektrischen Funken in Luft erzeugte Welle. Ann. Phys. 131 [II]:180-215 (1867).
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424. O. von Schmidt. Zur Theorie der Erdbebenwellen. Die "wachsende" Reflexion der Seismik als Analogon zur "Kopfwelle" der Ballistik. Z. Geophysik 12:199-205 (1936). 425. J.D. Boon, C.C. Albritton, Jr.: Established and supposed examples of meteoritic craters and structures. Field & Laboratory (Dallas) 6:44-56 (1938); The impact of large meteorites. Ibid. 6, 57-64 (1938). 426. R. S. Dietz: Meteorite impact suggested by the orientation of shatter-cones at the Kentland, Indiana, disturbance. Science 105:42-43 (1947); Meteorite impact suggested by shatter cones in rock. 131:1781-1784 (1960). 427. A. E Belajev. The production of detonation in explosives under the action of a thermal pulse. Dokl. AN (SSSR) 18(No. 4-5):267-269 (1938). 428. H. Johnston. "Electric initiator with exploding bridge wire." U.S. Patent No. 3,040,660 (filed Nov. 8, 1944, applied June 26, 1962). 429. M. Steenbeck. Uber ein Verfahren zur Erzeugung intensiver ROntgenblitze. Wiss. VeriJff. Siemens 17:363-380 (1938). 430. K. H. Kingdon and H. E. Tanis. Experiments with a condenser discharge. Phys. Rev. 53: 128134 (1938). 431. H. Schardin. Uber die Entwicklung der Hohlladung. Wehrtech. Hefte 51 (No. 4):97-120 (1954). 432. A set of some of the first flash radiographs of the shaped charge were donated to Dr. C. Fauguignon, Adjoint Scientific Director of ISL, upon Dr. G. Thomer's retirement from ISL. See also: D. R. Kennedy. History of the shaped charge effect. The first 100 years. Company brochure prepared by D. R. Kennedy & Associates, Inc., Mountain View, CA (1983). 433. H. Mohaupt. Shaped charges and warheads. In Aero-space Ordnance Handbook, E B. Pollad and J. A. Arnold, eds. Prentice-Hall, Englewood Cliffs, NJ (1966), chp. 11. 434. E. Preiswerk. Anwendungen gasdynamischer Methoden auf Wasserstri~mungen mit freier Oberfliiche. Mitteilungen aus dem Institut ftir Aerodynamik, ETH Zurich (1938). 435. O. Hahn and E Strassmann. Uber den Nachweis und das Verhalten der bei der Bestrahlung des Urans mittels Neutronen entdtehenden Erdalkalimetalle. Die Naturwissenschaften 27: 1115 (1939); Uber die Bruchstiicke beim Zerplatzen des Urans. Ibid., pp. 89-95; Nachweis der Entstehung aktiver Bariumisotope aus Uran und Thorium durch Neutronenbestrahlung. Nachweis weiterer aktiver Bruchstucke bei der Uranspahung. Ibid. 163-164. 436. S. Flugge. Kann der Energieinhalt der Atomkerne technisch nutzbar gemacht werden? Die Naturwissenschaften 27:402-410 (1939). 437. S. Flligge. Die Ausnutzung der Atomenergie. Deutsche Allgemeine Zeitung 78 (No. 387) (Aug. 15, 1939). 438. Y. Khariton, Y. Smimov: The Khariton version. Bull. Atomic Scientists 49(No. 5):20-31 (1993). 439. A. Betz. "Flugzeug mit Geschwindigkeiten in der N/~he der Schallgeschwindigkeit." Top secret German Patent D.R.P. No. 732/42 (1939). 440. R. Smelt. A critical review of German research on high-speed airflow. J. Roy. Aeronaut. Soc. 50: 899-934 (1946). 441. E H. Shehon. Reflections of a nuclear weaponeer. Shelton Enterprise Inc., Colorado Springs (1988), chp. 1: The Manhattan Project. 442. The life and time of the Manhattan Project. See http://www.gis.net/carter/manhattan/project.html. 443. T. von Karman and k. Edson. The wind and beyond. Theodore yon Karman pioneer in aviation and pathfinder in space. Little, Brown and Co., Boston and Toronto (1967), p. 233. 444. W. Payman and W. C. E Shepherd. (VI) The disturbance produced by bursting diaphragms with compressed air. Proc. Roy. Soc. London A186:293-321 (1946). 445. Y. B. Zeldovich. On the theory of the propagation of detonation in gaseous systems. ZETP (SSSR) 10:542-568 (1940).
History of Shock Waves
141
446. J. von Neumann. "Theory of stationary detonation waves." Rpt. OSRD-549 (1942). 447. W. D0ring. Ober den Detonationsvorgang in Gasen. Ann. Phys. 43 [V]:421-436 (1943). 448. T. von Karman and L. Edson. The wind and beyond. Theodore von Kdrman, pioneer and pathfinder in space. Little, Brown and Co., Boston and Toronto (1967), p. 224. 449. A. Michel-L~vy, H. Muraour, and E. Vassy. Repartition spectrale ~nerg~tique dans la lumiere Cmise lors de la rencontre d'ondes de choc. Rev. Opt. Th~or. Instrum. 20:149-160 (1941). 450. H. Schardin. Experimentelle Arbeiten zum Problem der Detonation. Jb. dtsch. Akad. Luftfahrtforsch. (1940-1941), pp. 314-334. 451. G. I. Taylor. "The formation of a blast wave by a very intense explosion." Ministry of Home Security, R.C. 210, II-5-153 (1941) and Proc. Roy. Soc. London A201:159-186 (1950). 452. L. I. Sedov. Propagation of strong explosive waves. Prikl. Mat. Mekh. (SSSR) 10 (No. 2):241250 (1946). 453. S. C. Lin. Cylindrical shock waves produced by instantaneous energy release. J. Appl. Phys. 25:54-57 (1954). 454. "A photographic study of small-scale underwater explosions". David W. Taylor Model Basin, Confidential Test Report R-39 (Aug. 1941). 455. Weinert: "Unterwasserzeitlupenaufnahmen von Gasblasenschwingungen". Berichte der chemisch-physikalischen Versuchsanstalt6 der Kriegsmarine (Kiel) 245, Heft VI (1941). 9 This report which was cited by W. DOting and H. Schardin in a post-war review paper on detonations [In: Naturforschung und Medizin in Deutschland 1939-1946, (A. Betz, ed.) Verlag Chemie, Weinheim/Bergstr. (1953). Bd. 11: Hydro- und Aerodynamik pp. 97-125], could not be located in German archives. 456. H. A. Berthe and J. G. Kirkwood: "The pressure wave produced by an underwater explosion." NDRC Div. B, Progr. Rpt. OSRD-588 (1942). 457. D.C. Campbell. "Motions of a pulsating gas globe under waterma photographic study". David W. Taylor Model Basin Rpt. 512 (1943). 458. W. Dornberger. V2mDer Schuj~ ins Weltall. Bechtle, Esslingen (1952). 459. Los Alamos 1943-45; The beginning of an era. Brochure LASL-79-79, Los Alamos Scientific Laboratories, NM. 460. G. Guderley. Starke kugelige und zylindrische Verdichtungsstot~e in der Nahe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforsch. 19:302-312 (1942). 461. R.J. Seeger. On Mach's curiosity about shock waves. In Ernst Mach, physicist and philosopher, R. S. Cohen and R. J. Seeger, eds. Boston Studies in the Philosophy of Science 6:42-67 (1970). 462. H. A. Bethe. "On the theory of shock waves for an arbitrary equation of state." NDRC Div. B, Rpt. OSRD-545 (1942). 463. J. von Neumann. "Oblique reflection of shocks." Navy Dept., Bureau of Ordnance, Explosives Res. Rpt. No. 12, Washington, DC (1943). 464. J. von Neumann: The Mach effect and height of burst. In: (A.H. Taub, ed.): J. von Neumann collected works. Pergamon Press, Oxford etc. (1963), vol. VI, pp. 309-347. 465. R. W. Wood. "On the interaction of shock waves." Rpt. OSRD-1996 (1943). 466. R. J. Seeger. On Mach's curiosity about shock waves. In Ernst Mach, physicist and philosopher, R. S. Cohen and R. J. Seeger, eds. Boston Studies in the Philosophy of Science 6:42-67 (1970). 467. (A.H. Taub, ed.): J yon Neumann collected works. Pergamon Press, Oxford etc. (1963), vol. VI, pp. 309-347. 468. E Schuhz-Grunow. Nichtstatiomire, kugelsymmetrische Gasbewegung und nichtstatiomire Gasstromung in Dusen und Diffusoren. Ingenieur-Archiv 14:21-29 (1943). 469. O. Igra, L. Wang, and J. Falcovitz. Nonstationary compressible flow in ducts with varying cross-section. J. Aerospace Eng. 212 [Part G]:225-243 (1998).
142
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470. R. H. Cole. Underwater explosions. Dover Publ., New York (1948), pp. 257-258. 471. E H. Reynolds. "A preliminary study of plane shock waves formed by bursting diaphragms in a tube." Rpt. OSRD-1519 (1943). 472. R. G. Sachs."The dependence of blast on ambient pressure and temperature." BRL Rpt. 466, Aberdeen Proving Ground, MD (1944); Some properties of very intense shock waves. Phys. Rev. 69:514-515 (1946). 473. P. P Wegener. The Peenemunde wind tunnels. A memoir Yale Univ. Press, New Haven (1996), pp. 69-70. 474. K. Oswatitsch. "Flow research to improve the efficiency of muzzle brakes." Heereswaffenamt. Berlin Rpt. R1001 (1944). 475. L. H. Thomas. Note on Becker' theory of the shock front. J. Chem. Phys. 12:449-453 (1944). 476. J. von Neumann. "Proposal and analysis of a new numerical method for the treatment of hydrodynamical shock problems." Applied Mathematics Group, Institute for Advanced Study, Princeton Rpts. OSRD-3617 and NDRC-108 (1944). 477. L. G. Smith. "Photographic investigation of the reflection of plane shocks in air." Palmer Physical Laboratory, Princeton University Rpt. OSRD-6271 (1945); Phys. Rev. 69:678 (1946). 478. P. Colella and L. E Henderson. The von Neumann paradox for the diffraction of weak shock waves. In Proc. 9th Int. Mach Reflection Symp., H. Reichenbach, ed. Ernst Mach-Institut, Freiburg (June 1990). 479. J. von Neumann: Refraction, intersection and reflection of shock waves. In: John von Neumann. Collected Works. (A.H. Taub, ed.) Pergamon Press, Oxford etc. (1963), vol. VI, pp. 300-308. 480. E. B. Christoffel. lJber die Fortpflanzung yon St6gen durch elastische feste KOrper. Ann. di Mat. 8 [II]:193-243 (1877). 481. P. H. Hugoniot. M~moire sur la propagation du mouvement dans les corps et plus sp~cialement dans les gaz parfaits. 1e Partie. J. Ecole Polytech. (Paris) 5 7 : 3 - 9 7 (1887). 482. P. M. M. Duhem. Sur les thgor~mes d'Hugoniot, les lemmes de M. HADAMARD et la propagation des ondes dans les fluides visqueux. C. R. Acad. Sci. Paris 132:1163-1167 (1901); Sur les ondes longitudinales et transversales dans les fluides parfaits. C. R. Acad. Sci. Paris 132:1303-1306 (1901). 483. J. S. Hadamard. Lemon sur la propagation des ondes et les gquations de l'hydrodynamique. A. Hermann, Paris (1903). 484. J. C. E. Jouguet. Sur les ondes de choc dans les corps solides. C. R. Acad. Sci. Paris 171: 461464 (1920); Sur la cr162 des ondes dans les solides ~lastiques. C. R. Acad. Sci. Paris 171: 512-515 (1920); Sur la variation d'entropie dans les ondes de choc des solides r C. R. Acad. Sci. Paris 171:789-791 (1920). 485. C. E. Morris. Shock-wave equation-of-state studies at Los Alamos. Shock Waves 1:213-222 (1991). 486. E H. Shehon. Reflections of a nuclear weaponeer. Shelton Enterprise Inc., Colorado Springs (1988), p. 2:15. 487. From: Administrative History, History of 509th Composite Group, 313th Bombardment Wing, 20th U.S. Air Force, Activation to 15 August 1945. This information was kindly provided by the Smithsonian National Air and Space Museum, Washington, DC. 488. J. Malik. "The yields of the Hiroshima and Nagasaki nuclear explosions." Los Alamos National Laboratory, Rpt. LA-8819 (1985).
CHAPTER
2
General Laws for Propagation of Shock Waves Through Matter LEROYE HENDERSON Professor Emeritus, 8 Damour Avenue, East Lindfield, Sydney, New South Wales 2070, Australia
2.1 2.2 2.3 2.4
Introduction The Riemann Problem Length and Time Scales The Conservation Laws for a Single Shock 2.4.1 Laboratory Frame Coordinates 2.4.2 Shock Fixed Coordinates 2.5 The Hugoniot Adiabatic 2.5.1 The Hugoniot Equation 2.5.2 The Rayleigh Equations 2.5.3 Solution of a Simple Shock Riemann Problem 2.6 Thermodynamic Properties of Materials 2.7 Thermodynamic Constraints on the EOS 2.8 Nonthermodynamic Constraints on the EOS 2.8.1 Convexity 2.8.2 Shock Wave Stability Constraints 2.8.3 Monotonicity Constraints 2.9 The Bethe-Weyl (B-W) Theorem 2.10 Shock Wave Interactions 2.10.1 Dimensions of the Interactions 2.10.2 Two-Dimensional Shock Wave Interactions 2.10.3 Three-Dimensional Shock Wave Interactions 2.11 The Triple-Shock-Entropy and Related Theorems 2.11.1 The Theorems 2.11.2 Application to Shock Wave Interactions 2.12 Crocco's Theorem 2.13 The Refraction Law 2.14 Concluding Remarks References Handbook of Shock Waves, Volume 1 Copyright ~ 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
143
144 2.1
L. E Henderson
INTRODUCTION
The ideal objective for this chapter would be to present the theory that describes how shock waves propagate, and interact, as they pass through any material in any thermodynamic state. The conservation laws and some other laws and theorems meet this objective for equilibrium states, or other states that can be defined in some useful way. However it is useful to extend the discussion to theory that requires only mild constraints on the material's equation-of-state (EOS). Powerful results can then be applied to a broad class of materials; in particular to any material in a single phase state. An EOS so restricted has a profound effect on the nature of the phenomena that is observed. The seminal paper for multi-shock interactions in general classes of materials is by von Neumann (1963). Often length and time scales are insignificant for shock problems; these are called shock-Riemann problems (Sections 2.2, 2.3). The equations obtained from the conservation laws of mass, momentum, and energy are called the Rankine-Hugoniot (R-H) equations (Section 2.4). These equations contain the material velocities on both sides of a shock. If the equations are manipulated to eliminate the velocities, a single equation containing only state variables is obtained. It is the Hugoniot equation, and is the starting point for many studies. If two other equations, called the Rayleigh equations are appended to the Hugoniot equation, they comprise a set equivalent to the R-H equations (Section 2.5). The material properties that are required are defined in Section 2.6. Those that will be used most often are the adiabatic exponent ~, the Gruneisen coefficient F, and the fundamental derivative G. The EOS constraints required to ensure thermodynamic stability are presented in Section 2.7. Other EOS constraints presented in Sections 2.8 are needed to ensure the existence and uniqueness of shock-Riemann problems. They are also needed to control the monotonicity of state variables along the Hugoniot adiabatic, which is a plot of the Hugoniot in say the (v, p) plane, where v is the specific volume and p is the pressure. Furthermore they are needed to ensure that a shock does not become unstable by splitting, ot through ripple instability. The important admissability conditions are also presented in Section 2.8. They ensure that the shock is physically possilbe, in that it increases the entropy in the material as required by the second law, and that the flow of the material relative to the shock is supersonic on the upstream side and subsonic downstream, as required to prevent the splitting/ripple instabilities. The important Bethe-Weyl Theorem (B-W) is presented in Section 2.9. In its most general form, it guarantees the existence of a solution to the Hugoniot equation, but it is most powerful when the EOS is convex (Section 2.6). The
General Laws for Propagation of Shock Waves Through Matter
145
theorem shows that the solution is then unique; that the admissability conditions are satisfied; and that the entropy is monotonic along the Hugoniot adiabatic. The (l-D) and (2-D) interactions of shocks are discussed in Section 2.10, and the Triple-Shock-Entropy Theorem in Section 2.11. The theorem is useful for 2-D interactions. Next is Crocco's Theorem presented in Section 2.12; it is useful for curved shocks, and when flow gradients are present. Finally we present the Refraction Law in Section 2.13. It can be applied to 2-D shock interactions, especially when several shocks meet at a point; called a wave node."
2.2
THE RIEMANN
PROBLEM
"A Riemann problem is defined for a system of conservation laws such as mass, momentum and energy, as an initial value problem such that the initial data have no length or time scales, or in other words the data is constant along ray paths" (Courant and Friedrichs, 1948). The classic example is the shock tube problem studied by Riemann (1860). Many shock problems have this scale invariant character, but not all.
2.3 LENGTH
AND TIME SCALES
One length scale that is always present is the thickness of the shock wave. The simplest example is that of a monatomic gas. Its shock wave thickness is about four mean free paths, that is, it takes about four molecular collisions to adjust the equilibrium state upstream of the shock to downstream of it. The molecular processes inside the shock wave are not in equilibrium. A shock wave is thicker in polyatomic gases because molecular rotation and vibration require more collisions for equilibrium to be attained. For weak shock waves in the atmosphere the thickness may be of the order of one i km because of the large number of collisions required to attain vibrational equilibrium in nitrogen, especially when moisture is present (Johannesen and Hodgson, 1979). The shock wave thickness is also increased by chemical reactions as with detonations (Fickett and Davis, 1979), and by dissociation and ionization. More generally, the velocity and the thermal gradients inside the shock wave imply the importance of the material transport propertieswparticularly viscosity and heat conductivity (Zeldovich and Raizer; 1966; Thompson, 1972, p. 363). If the shock wave thickness length scale is too small to be of significance to the problem then it is sufficient to consider only the equilibrium states on both sides of the shock. One then has a shock Riemann problem.
146
L. E Henderson
Time scales are often important, but only two occurrences will be mentioned here. First, a time scale is present if the shock wave becomes unstable and splits into two waves moving in the same direction (Section 2.8.2). Suppose that an intense shock wave propagates into a metal, which is initially at atmospheric pressure and temperature, and suppose it also compresses the metal beyond its yield point. It is known that eventually the shock wave will split into two waves. The first is a precursor shock wave that compresses the metal to its yield point, and the second is a compressive plastic wave (Zeldovich and Raizer, 1966). Second, a shock wave may induce a change in phase of the material. A well-known example is the ~ ~ e (body-centeredcubic to hexagonal-close-packed) phase transformation in iron that takes place at 12.8Gpa, which can also cause splitting (Duvall and Graham, 1977). However, in many cases the time to attain equilibrium is orders of magnitude greater than the time for the shock wave to pass through the material. In this case there will be no phase change, and any equilibrium can only be metastable; there will then be no time scale. For example, if a shock wave
'....
VII
'
VIII 20
15
VI
Q
!0
II
"3--/
i I
-- I O0
.
.
.
.
.
Water
\ l
-- 50
,
0
_
I
50
_
_
i
O0
T~ FIGURE 2.1 Ice-water-phase diagram. (adapted with changes from Eisenberg and Kauzman 1969).
General Laws for Propagation of Shock Waves Through Matter
147
compressed water at atmospheric pressure to a pressure P > 104 atm (1000 MPa), and if thermodynamic equilibrium were attained, then ice (VII) would exist downstream of the shock wave (see Fig. 2.1). However, this does not usually happen because of the long time required for attaining equilibrium (Bethe, 1942). Instead, the water remains in the liquid phase but in metastable equilibrium. The time scale for thermodynamic equilibrium reduces rapidly, however, if the compressed state approaches a spinodal condition (Section 2.7).
2.4 THE CONSERVATION SINGLE
LAWS FOR A
SHOCK
2.4.1 LABORATORY FRAME C O O R D I N A T E S Suppose the material is contained in a cylinder of cross-sectional area A. One end of the cylinder is open, but the other is closed by a piston that is in contact with the material. Initially the system is at rest as shown in Fig. 2.2a. Suppose that at time t = 0, the piston impulsively acquires the finite velocity Up in the x-direction; it instantly begins to drive the material to the right at the same velocity Up. This is accomplished by a shock wave S that instantly appears on the face of the piston and propagates into the material with the finite velocity (/s > /alp (see Fig. 2.2b). As Us is finite (it must be less than the velocity of light!), the material to the left of the shock wave moves at the velocity Up, but the material to the right of it remains at rest. The equations from the conservation laws for mass, momentum, and energy can now be derived. It is assumed for simplicity that the system has adiabatic walls, that body forces such as gravity and electromagnetism are negligible and that there is no heat transfer by radiation across the shock.
Conservation of Mass After unit time the piston has moved a distance Up and the shock a distance Us, where Up, Us, are the scalar magnitudes (speeds) of the vectors/dp, (/s. During that time the shock compresses a mass of the material from its initial volume A U s to A ( U s - U p ) . The density therefore increases from P0 to p, so by conservation of mass PoUs = p ( U s - Up) = rh
(2.1)
where rh is the mass flux of material passing through the shock wave. Notice that it is strictly true that Us > Up, for if Us = Up then p = oo, which is physically impossible with the current state of knowledge.
148
L. E Henderson
I
.Po, Po o)
I 0 I
I .Po, Po
I
~
IS
P
jo, P, Up Up _~ '
Po,Po
Us- Up
r l
''
Us
b)
! _1
v !
,p ,P, Up- Us
c)
X
W
L. U -O, -U s
.,,
_ -Us
_____~_~, Po
t
FIGURE 2.2 Shock wave generated by the impulsive motion of a piston, a) Initial state at rest; b) state in unit time after the piston had acquired velocity/Jp impulsively; and c) motion in shock fixed coordinates (p is the piston and S is the shock wave).
Conservation of M o m e n t u m Suppose that P0 is the initial pressure of the material and p is the pressure of its compressed state. The piston applies a driving force ( p - po)A to the material, causing it to acquire a momentum per unit time of (PoUsA)Up - rhAUp. Then from conservation of momentum p - p0 - p0 U~Up
(2.2)
Conservation of Energy The compressive work that the piston does on the material in unit time is
PAUp. The energy gained by the material in unit time is the sum of the kinetic
149
General Laws for Propagation of Shock Waves Through Matter
!(PoUsA)Up2 and the internal energy (PoUsA)(e- %). Thus by conservation of energy pUp -- PoUs(1Up 4 - e - % )
(2.3)
The preceding equations are the conservation laws for a single shock wave. They are of fundamental importance.
2.4.2 SHOCK FIXED COORDINATES It is often convenient to transform the conservation laws into a coordinate system that is at rest with respect to the shock. This is easily accomplished by subtracting the shock wave speed Us from the (zero) particle speed ahead of the shock wave and also from the particle speed Up behind it. Then uo = - U s
(2.4)
u = Up - Us
(2.5)
and
where u 0 and u are the particle (material) speeds ahead of and behind the shock, respectively, and relative to it. The last of these equations can be written as
Up = u - u o
(2.6)
Substituting Eq. (2.4) into Eqs. (2.1), (2.2), (2.3) we acquire, after some algebra, the conservation laws in shock fixed coordinates; these are also called the Rankine-Hugoniot equations.
pu = PoUo p 4- pu 2 -- Po 4- PoU~ 1 u2 Po 1 P-4-e4- - - - 4 - % 4- u~ p 2 po
(2.7) (2.8) (2.9)
Equation (2.9) can also be written in terms of the enthalpy, h p/p + e, as: lu2
h 4.~
1
-- h 0 4.-~1/2 z
ht
which is called Bernoulli's equation. Here, ht is the total enthalpy.
(2.10)
150
2.5 2.5.1
L. E H e n d e r s o n
THE
HUGONIOT
ADIABATIC
THE HUGONIOT
EQUATION
If speeds U s and Up are eliminated from Eqs. (2.1)-(2.3), the conservation laws reduce to a single equation, which is a function only of the variables of state. It is called the Hugoniot equation, and it is fundamental to shock wave theory. 1 e - e o -- -~(p + po)(Vo - v)
(2.11)
Notice that a neater form of it is obtained if the densities P0 and p are replaced by the specific volumes v0 and v, respectively, where v = 1/p. In order to plot the Hugoniot curve in the (v, p)-plane it is necessary to know the initial state (v 0, P0) of the material and its EOS, or its equivalent such as a table of state properties.
2.5.2
THE RAYLEIGH EQUATIONS
If u o or u is eliminated from Eqs. (2.7) and (2.8), we obtain the Rayleigh equations 2
2
2 2
p0U~ - PoUo - p
21,12
Ap -
Av
(2.12)
where Ap = p - Po and Av = v - v0. If the pressure j u m p across a shock becomes vanishingly small, that is p --~ Po, then v ~ v0 and u ~ u0, and one also finds that the specific entropy is s ~ s o [Eq. (2.41) in Section 2.8.1]; then in the limit Eq. (2.12) becomes 22
P~176
[3P] ~ s
(2.13)
S
where a 0 is the speed of sound in the undisturbed material. It follows that - U s = u 0 --~ a 0, so that in the limit the shock wave propagates at the speed of sound, or in other words it is reduced to an acoustic wave. Note that -Vo[3p/3V]s, is the bulk modulus and that a 0 is called the longitudinal sound speed in solid mechanics and is appropriate for an unconstrained material. The sound speed in a thin bar is somewhat smaller (Kolsky, 1953).
151
General Laws for Propagation of Shock Waves Through Matter
Returning to Eqs. (2.1) and (2.2), and replacing Po and p by vo and v and eliminating Us the result, with the help of Eq. (2.6), is 1
1 Up -- (u - Uo)2 -- -~(p - po)(V 4- Vo)
(2.15)
which in laboratory frame coordinates is the gain in the kinetic energy per unit mass of the material by the passage of the shock wave. In shock fixed coordinates there is a loss of kinetic energy across the shock wave, because for a compression v < v0, and by Eq. (2.7) u < u 0, and so ~1/./2 < 89u~. From Eqs. (2.9) and (2.11) we get 1 (u02 _ u2) _ -~(p 1 - po)(V 4- v o) -~
2.5.3
(2.16)
S O L U T I O N OF A SIMPLE S H O C K R I E M A N N
PROBLEM The problem is illustrated in Fig. 2.2. Suppose the initial state (v0, P0) upstream of the shock is given, and also the downstream pressure p. It is required to find the compressed specific volume v and thus the downstream state (v, p). The problem can be solved in the (v, p)-plane when the EOS of the material is known p = p(e, v). The Hugoniot curve can be plotted by using Eq. (2.11) and the EOS, and v can then be found because p is given (see Fig. 2.3). The slope of
R
P
H
P I I I I I
I i I I I
I .... _
FIGURE 2.3
i ...... I
'A
ID
!
v
vo
Y
Hugoniot curve H and Rayleigh line R in the (v, p)-plane.
152
L. E
Henderson
the Rayleigh line A p / A v can now be calculated and it is a constant; this means that the Rayleigh line is straight in this plane. From Eq. (2.12), U2s/v 2 -- A p / A v , from which we find Us. By Eq. (2.11) the gain in the internal energy is represented by the trapezium ABCDA, while by Eq. (2.15) the gain in the kinetic energy per unit mass (laboratory frame) is represented by the triangle BECB. By Eq. (2.3) the total gain in energy per unit mass is represented by the rectangle AECDA.
2.6 THERMODYNAMIC MATERIALS
PROPERTIES
OF
It is important to notice that the conservation laws, the Hugoniot, and the Rayleigh equations are independent of any equation of state. Consequently, these laws and equations can be applied to any material. Nevertheless, the EOS has a decisive effect on the nature of the shock phenomena that appears in it. However, before these effects can be discussed it is necessary to define the thermodynamic properties that will be needed. The fundamental equation e = e(v, s)
(2.17)
contains all the thermodynamic information about the system (Callen, 1985). If by definition T--
~s v
(2.18)
3e]
(2.19)
and -p-
then by using Eqs. (2.18) and (2.19), in differential form, Eq. (2.17) becomes de - Tds - pdv
(2.20)
where T is the temperature. Equations (2.18) and (2.19) are the thermal and mechanical EOS, respectively, and they can also be written T -- T(v, s) and p - p(v, s)
(2.21)
It is often useful to define the EOS in terms of (v, e) rather than in (v, s). By Eqs. (2.18) and (2.19) this is always possible because e is a monotonically increasing function of s, as T > 0 and T - - 0 is unattainable, and so, T = T(v, e) and p = p(v, e)
(2.22)
153
General Laws for Propagation of Shock Waves Through Matter
The specific heats (2.23)
LOTIv and Cp - T -~ e
Cv -
The compressibilities Ks -- - -
(2.24)
and KT -- - V
V
S
T
The coefficient of thermal expansion l~V
(2.25)
p
Because of the thermodynamic relation Ks=
1
KT
fl2vT =
~
CpK T
Cp
CV
(2.26)
only three of these five properties are independent. It is conventional to choose these to be Cp, KT, and fl, as tables of them exist for many materials. In what follows, some of the properties obtained from the second and third derivatives of the energy are of special importance.
The adiabatic exponent
~' - p
Lav2Js
p~:s - p-~ = - p
~v s
where a is the speed of sound. For an ideal gas, 7 reduces to the ratio of the specific heats, 7 - Cp/Cv. Notice that ~ can often be found from Eq. (2.27) because 7 - aZ/P v. Some values of a, p, and p -- 1/v are given in Table 2.1.
The Grfineisen coefficient
r-
v 2e
TO~O~--T ~
] ~v v i i~i v
(228
This can be written to show that F determines the spacing of the isentropic curves in both of the (ln v, In p) and (v, p)-planes
F
l_avJ~ T
as v
pv T
as
ln v
(2.29)
154
L. E Henderson
Some Approximate Values of Shock and Material Properties. (Originally distributed at the 1989 Topical Conference on Shock Waves in Condensed Matter, Sponsored by the American Physical Society) TABLE 2.1
.
Material a
Water NaC1a KC1b
LiF Teflon PMMA Polyethylene Polystyrene
Brass A1-2024 Be Ca Cu
Fe b Pb U
.
.
.
/9
Cp
a
[kg/m3]
[kJ/kgK
[km/s]
1000 2160 1990 2640 2150 1190 920 1040 8450 2790 1850 1550 8930 7850 11350 18950
4.19 0.87 0.68 1.50 1.02 1.20 2.30 1.20 0.38 0.89 0.18 0.66 0.40 0.45 0.13 0.12
1.51 3.53 2.15 5.15 1.84 2.60 2.90 2.75 3.73 5.33 8.00 3.60 3.94 3.57 2.05 2.49
F
0.1 1.6 1.3 2.0 0.6 1.0 1.6 1.2 2.0 2.0 1.2 1.1 2.0 1.8 2.8 2.1
a Superscripts a and b refer to above and below phase transitions.
It follows at once that the isentropics cannot cross each in these planes when F > 0. By further manipulation and also by using Eq. (2.25) the following, useful relation between F and/~ is obtained: F=
vfl
(2.30)
CvKT For a thermodynamically stable system Cv > 0 and KT > 0 (see Menikoff and Plohr, 1989), and because v > 0, it follows that F and fl always have the same sign. When F is a constant then Eq. (2.30) becomes the famous Gruneisen EOS. For most materials, in most states, F and fl are positive. For the alloy Invar, they are almost zero at room temperature, but for water < 3.984 ~ and at 1 atm, both F and fl are negative. There are also many tetrahedrally bonded materials for which these quantities are negative for some domains of state (Table 2.2). For an ideal gas, F - 7 - 1 > 0. Some values of F are presented in Table 2.1, (Collins and White, 1964).
The reciprocal of the dimensionless specific heat pv -
g = T LO~s2Jv-
Cv~:
(2.31)
General Laws for Propagation of Shock Waves Through Matter
155
TABLE 2.2 Temperature Domains of Some Materials that have a Negative fl and F at a Pressure of I atma Material
Temperature domain
Water Diamond Vitreous silica ZnSe CdTe Ice I GaAs Ge InSb 0~-Sn
<3.384:'c <90K <289K <64K <72K <63K <55K <48K <55K <45K
,,
This quantity is strictly positive g > 0 for a system, which is thermodynamically stable (Section 2.7).
The fundamental derivative O2p 1
LS-JJs
1
2V (Oap~
(2.32)
by Eqs. (2.20), (2.27), G is a third derivative of e. Here it is given in nondimensional form f (Thompson, 1971)" G > 0 iff O2p ~v2j s > 0
(2.33)
The fundamental derivative is a measure of the curvature of the isentropics in the (v, p)-plane. If Eq. (2.33) is strictly true then any particular isentropic is convex, which means that it always lays above a tangent to any point on it (see Fig. 2.4). An isentropic is straight for any domain of states for which G = 0 and concave if G < 0. It will be shown in Sections 2.8.3 and 2.9 that the sign of G has profound physical consequences.
DEFINITION. An equation of state of a material is convex if its fundamental derivative is strictly positive (Bethe, 1942).
156
L. E Henderson
sO
0
u
FIGURE 2.4 Convex isentropic curves always lie above a tangent and have a negative slope, [32p/OV2]s> 0 and [Op/Ov]~ < o. This definition is of great importance for investigating the existence of a given shock wave phenomenon in large classes of materials. By differentiating Eq. (2.27) with respect to v, a relation between 7 and G is obtained:
G--~
,[
3,+ i - -
(2.34)
P2 F02P1 1 +--
(2.35)
or in terms of the density
G-
P~ L~Js
Values of G for some liquids are presented in Table 2.3 (Thompson, 1971).
157
General Laws for Propagation of Shock Waves Through Matter TABLE 2.3 Values of G for Some Liquids at 1 atm and 30 ~C a Liquid
Values of G
Water Acetone 1-Propanol Mercury Methanol n-Propanol Glycerine Ethanol n-Butanol
2.7 THERMODYNAMIC
3.60 6.0 6.2 4.94 5.81 6.36 6.1 6.28 6.36
CONSTRAINTS
ON
THE EOS The material EOS must satisfy the following constraints if the equilibrium states on both sides of the shock are to be thermodynamically stable (see Callen, 1985; Menikoff and Plohr 1989) 7 > 0; g > 0; ? g - F 2 > 0
(2.36)
If any of the equalities in Eq. (2.36) apply, the equilibrium will be only neutrally stable; instability can then be caused by vanishingly small fluctuations. Constraints that are equivalent to Eq. (2.36) are 1
>
Ks -
1
1
> 0, and
KT -
1
> ~
Cv -
> 0
(2.37)
Cp -
By Eqs. (2.23) and (2.24) these constraints imply that
ap] < 0
(2.38)
and -
T
>
0
(2.39)
v
that is, if a thermodynamically stable system is compressed isothermally its pressure will increase, which means that it is mechanically stable. If the system is heated at constant volume, its temperature will increase, which means that it is thermally stable.
158
L. E Henderson
T = const.
Meto-stoble region
Spinodal
M eta-stable region
FIGURE 2.5 Sketch of the spinodal (Wilson line) for a van der Waal's equation of state in the (v, p)-plane. Note the metastable regions SLP, saturated liquid line, SVP, saturated vapor line, CP, critical point.
For a fluid near the vapor-liquid phase transition it is often possible to produce metastable states, such that [ap/8v]T --+ O. For example a saturated liquid may be carefully expanded isothermally to a lower pressure in such a way that the constraint [Op/av]-r < 0 remains satisfied as lOp~Or]-r --+ O. The limit is called the spinodal or Wilson line (see Fig. 2.5) and is defined by
ap] _ 0
(2.40)
The super-expanded state is metastable, but if the limit is approached closely, it becomes inevitable that a fluctuation of sufficient magnitude will occur and cause Eq. (2.38) to be violated, after which explosive boiling will occur. This branch of the spinodal is the superheat limit. The other branch occurs on the vapor pressure side, and is also defined by Eq. (2.34); in this case it is associated with super-cooling (for more details, see Frost and Sturtevant, 1986; Shepherd and Sturtevant, 1982).
General Laws for Propagation of Shock Waves Through Matter 2.8
NONTHERMODYNAMIC
ON
THE
159
CONSTRAINTS
EOS
2.8.1 CONVEXITY Suppose that with Bethe, the Hugoniot equation (2.11) is expanded in a Taylor series; for a weak shock wave this gives to leading order
[~ LTviJs
1 As = s - s o -- - 12----T
VO)4
(2.41)
Now for a compressive shock wave, Av < 0, so if the entropy is to increase across a weak shock As > 0, then the EOS must be convex [Eq. (2.33)]. If on the other hand, the inequality equation (2.33) is reversed, so that the EOS is concave, then it is weak expansion shocks that entropically increase for an adiabatic system. For waves of strength, Bethe (1942) found that sufficient conditions for adiabatic compression shocks to be entropy increasing were that the EOS obeyed the convexity constraint as well as a constraint on the Gl~neisen coefficient
arbitrary
2p] > b-V~2j~
0 ==> G > 0
F>-2
(2.42) (2.43)
Bethe (1942) showed that all pure substances in a single-phase state obeyed Eq. (2.42) for practically all thermodynamic states. The final result of his m e t h o d is given in the Appendix at the end of this chapter. A s u m m a r y of convex (G > 0) materials is presented in Table 2.4. The constraint fails, (G < 0) for fluids of sufficiently high molecular weight (i.e., those containing at least seven atoms in their molecule) and when the fluid in the superheated vapor state nears its phase critical point (Bethe, 1942; Zeldovich and Razier, 1996). Many authors have used the van der Waal's EOS to find a locally
TABLE 2.4 Materialsthat have a Convex EOSa .....
9 Dissociating or ionizing gases 9 Single-phasevapors with <7 atoms in their molecules Single-phase solids at low and normal temperatures 9 Ideal gases with either constant or variable specific heats 9 Liquids at normal temperatures, including water ><3.984 ~ 9 Liquid-vapor phase transition; the convexity may be discontinuous 9 Some metal phase transitions; for example, g ~ e (BCC---~HCP) in iron 9
160 TABLE 2.5
L. E Henderson
Materials that have a Nonconvex EOS a
Metals at their yield point; elastic-plastic transition Transition between two condensed phases at one of the two boundaries between the pure and the phase mixture Single-phase vapors that have seven or more atoms in their molecules and in a state near their phase critical point
concave domain (Bethe 1942, Zeldovich and Razier 1966, Thompson 1971, Lambrakis and Thompson 1973, Cramer and Sen 1987, Cramer 1989). A sketch of a concave domain is presented in Fig. 2.6. Notice that the isentropics are concave near the critical point but convex elsewhere. Examples of other materials that may be concave are given in Table 2.5. As the inequality equation (2.42) is strict it implies that -[Op/OV]s may not be constant, and in particular it may not be zero. Thus by Eq. (2.27), 7 > 0 strictly, and this is also necessary for the thermodynamic stability equation (2.36). As ~ - 0 is forbidden, there are no stationary values for the pressure along an isentropic. Convexity also forbids the speed of sound being zero, such as occurs at phase triple points, for example ice/water/steam, for then by Eqs. (2.27) and (2.42), G - 0, and the material is neither convex nor concave. However when G > 0, then by Eq. (2.27)
p2a2=
-[~]
>0
(2.44)
s
thus an isentropic curve always has a negative slope in the (v, p)-plane when the EOS is convex (compare Figs. 2.4 and 2.6). The only material that Bethe found that did not satisfy Eq. (2.43) was melting ice at - 2 0 ~ which occurs at about 2500atm and then F , ~ - 2 . 1 . However, other examples are now known, such as vitreous silica, which has the remarkably low value of 1-" ,~ - 9 at about 25 K (Collins and White, 1964).
2.8.2
S H O C K W A V E STABILITY C O N S T R A I N T S
Bethe (1942) deduced constraints on the EOS that would be sufficient to prevent a shock wave from splitting into two waves that move in either the same direction, or else in opposite directions. Von Neumann (1943) gave an elegant discussion of the first type of splitting. He supposed the shock being divided into two parts. The first part joins the initial pressure P0 to an intermediate pressure p', and the second joins p' to the final pressure p. The velocity of each part is given by the Rayleigh equation (2.12). The shock wave
161
General Laws for Propagation of Shock Waves Through Matter
S=const.
g-O
9
:
k
\
"
>0
I
',\\
-
\.
..,~
.
.
.
.
.
.
.
.
V FIGURE 2.6 Sketch of nonconvex, G < 0, isentropics near the saturated vapor line in the (v, p,)plane. Note that meta-stable regions SLP, saturated liquid line; SVP, saturated vapor line; CP, critical point (after Menikoff and Plohr, 1989).
cannot split into two waves moving in the same direction if the speed U~ of the following wave is >_ speed U s of the leading wave U~ >_ U s
( P - P') > ( P ' - Po) (v'-
v) -
(2.45)
(Vo - v')
for all p' in p > p ' > P0- In Section 2.9 it is shown that this splitting is impossible with a convex EOS. In order to exclude a shock splitting into two waves moving in opposite directions, Bethe (1942) deduced that sufficient constraints on the EOS were convexity G > 0, and
F~-l[_~__]-p__( Y - F ) < O koY.] e
(2.46)
V
or equivalently F < ~,
(2.47)
162
L. E Henderson
The basis for Bethe's (1942) study of the materials that satisfied Eqs. (2.46) and (2.47) was the thermodynamic identity
o=g,
--
< 0
Bethe (1942) concluded that: 9 "Nearly all materials in a single phase obey this constraint, but that it breaks down for a few phase transformations. 9 This constraint seems to be more generally fulfilled than the convexity constraint. ~ If the constraint is to be fulfilled for phase transformations, it is required that, AeAs > 0, that is, the energy and the entropy must change in the same direction. This is fulfilled for practically all phase transformations, but some exceptions are ice I or ice III to ice V (see Fig. 2.1)." "Later D'Yakov (1956) and Erpenbeck (1962) used a linearized analysis to test shock stability against small 2-D, ripple perturbations. It was concluded that a shock was stable if the following inequalities were satisfied -I
General Laws for Propagation of Shock Waves Through Matter
163
have produced more restrictive upper limits. Fowles (1976) was able to replace (2.48) with the neat result - 1 < R < 1, but an even more restrictive limit was found by Kontorovich (1958) following an elaborate analysis of the impact of 2-D and 3-D disturbances on a shock. The result is 1 -- M 2 - (Vo/v)M 2 -1 < R < - 1 - M 2 q- (Vo/v)M 2
(2.49)
When this upper limit is violated, the shock should develop multidimensional ripple instabilities. They have been observed on shocks in dissociating and, or, in ionizing, gases (Griffith et al., 1975" Glass and Liu, 1978). Shock stability and materials properties It has already been noted that the lower limit cannot be violated if G > 0. Menikoff and Plohr (1989) have proposed F _< 7 - 1, to ensure that the upper limit is not violated; it is only sufficient, so it is possibly too restrictive. Alternatively if dv/dp[ h is obtained from Eq. (2.11), so R -- - ( v / p ) ( k p / k v ) [ 1 + -i1 F k v / v ] / [ 7 - 8 9 > 0, then with G > 0 => A p / k v < 0, so [1 + 89FAy~v] < 0, where the inequalities follow because R > 0 for this limit. But by Eq. (2.52) (Section 2.9), this is impossible if F < pv/e. It follows that these material property constraints on the EOS guarantee shock stability by virtue of Eq. (2.49). The admissibility constraints These are
as >__0
(2.50)
M0 > 1 > M
(2.51)
where M 0 is the flow Mach number upstream of, and relative to the shock. The first of them is necessary by the second law; the second is necessary for shock stability Eq. (2.45). Under plausible assumptions, Fowles (1975) has shown that (2.51) is a consequence of the second law for viscous, heat-conducting fluids with arbitrary EOS's. The convexity constraint is not needed in this case. By contrast, (2.51) is obtained from Bethe-Weyl Theorem (Section 2.9) for entropy increasing compression shocks As > 0, Av < 0, for any material with a convex EOS"
2.8.3
MONOTONICITY
CONSTRAINTS
Many researchers have helped formulate the EOS constraints described in this section. The key idea is to find the constraints needed to ensure the monotonicity of particular thermodynamic properties, the wave velocity Us and the particle velocities u 0 and u, along a Hugoniot adiabatic or isentropic. For
164
L. E Henderson
example the entropy is monotonic increasing along a Hugoniot adiabatic when G > 0; hence it may be used as a parameter (Section 2.9). The constraints presented here are necessary for the monotonicity of other properties and for the uniqueness of the solutions to 1-D shock interactions. Numerous theorems can be proved once the monotonicities are established. The strong constraint
pv F <m_+ --
r
1 (V-Vo) F >0
1+
2
(2.52)
V
When this constraint is satisfied, v is a monotonic decreasing quantity along a Hugoniot adiabatic. If also G > 0, then the curve is itself everywhere convex in both the (v, p)- and (u, p)-planes (see Fig. 2.7a). The ideal gas obeys Eq. (2.52) everywhere because from its EOS it is easy to obtain F = 7 - 1, G - 1(7 + 1) and pv/e - 7 - 1 - F, and so G > 0 and F - pv/e. All materials in a single phase obey Eqs. (2.33) and (2.52) for a large domain of states. The most notable exceptions are dissociating and ionizing gases, which violate Eq. (2.52) but still satisfy Eq. (2.33). In such circumstances the Hugoniot curve becomes locally concave in the (v, p)-plane, but still remains convex everywhere in the (u, p)-plane (see Fig. 2.7b). The medium constraint
F
i pv
(2.53)
The solution to a Riemann problem for 1-D interacting shock waves is unique when this important constraint is satisfied. The constraint also guarantees that e and u are monotonic increasing quantities along a Hugoniot adiabatic (Menikoff and Plohr, 1989). It is evidently weaker than Eq. (2.47), so it must be more generally satisfied than it is. The weak constraint F _< 27 --+ ? - -
1 P-Po F > 0 2 p
(2.54)
This ensures that pressure p and enthalpy h are monotonic increasing along a Hugoniot; all known materials obey it. Menikoff and Plohr (1989) show that when G > 0, then strong ~ medium ~ weak
(2.55)
so by Eqs. (2.52) to (2.54), with ? > 0 1 pv pv 27 >__7 - } - - ~ > ~ > F 2ee-
(2.56)
165
General Laws for Propagation of Shock Waves Through Matter
P
i
P
L
0 V
Oe-.s 'U
(a) Strong Condition P
~Vo .
0
'
0 e.... V
(b) Medium Condition p
!
~vo.
0 !
"
~
0 V
" '
U
(c) Weok Condition
~0 V
-U
(d) Violates Weak Condition FIGURE 2.7 Plohr, 1989).
Hugoniot locus, with G > 0, in the (p, v)- and (p, u)-planes (after Menikoff and
Figure 2.7 illustrates the effect on the Hugoniot curve when the strong, medium and weak constraints are successively violated while G > 0 remains satisfied. Henderson and Menikoff (1998) used constraint equations (2.33) and (2.54) to prove the following.
166
L. E Henderson
LEMMA. If G > 0 and F < 27 then the Hugoniot curve based on state (v 0, P0) contains a unique shock for any p > Po. Moreover, pressure-increasing shock waves are entropy increasing. If the shock wave strength is defined by A p - p applies to a shock of any strength.
2.9
THE
BETHE-WEYL
P0, then the lemma
(B-W) THEOREM
In its most general form, the theorem ensures that solutions to a single shock (Riemann) problem exist. If an EOS obeys the convexity constraint G > 0, then the theorem ensures much more, namely that the solution is unique; admissability conditions Eq. (2.50), and (2.51), are satisfied; and that M 0 > 1 > M, so that the lower stability limit in Eq. (2.49) is not violated. However even with G > 0 it is still possible for the upper limit to be violated, as for sufficiently strong shocks in dissociating or ionizing gasses (Section 2.8.3). The theorem can only be applied directly to a normal shock, that is one whose velocity vectors are perpendicular to it (Fig. 2.8a). If a velocity vector crt = cot0, which is parallel to the shock is added to h 0 and fi then a normal shock becomes an oblique shock with respect to the upstream and downstream flows (see Fig. 2.8b) - ( J s - (J0 - fi0 + ;r and U - u 4- v t
(2.57)
The theorem can then be extended to oblique shock waves by resolving the vectors U0 and (J to obtain the normal shock vectors fi0 and ft. Henderson and Menikoff (1998) state the theorem as
U
Uo ~
a)
,,
,,
,
,,
b)
S
FIGURE 2.8 Flow vectors for normal (a) and oblique (b) shock waves.
General Laws for Propagation of Shock Waves Through Matter
167
THEOREM (BETHE-WEYL). If G > 0 and F < 27 then the Hugoniot curve based on any state zero intersects every isentropic exactly once. Moreover f o r entropy increasing shocks s > s o , one has v < v o and u < a, while f o r entropy decreasing shocks s < s o , v > v o and u > a.
The first part of the theorem guarantees the existence of at least one solution to the Hugoniot equation. Bethe's proof depends on the asymptotic properties of the EOS and of the Hugoniot equation. An outline of a more elegant proof based on the same approach is now presented (Menikoff and Plohr, 1989). Define the Hugoniot function as 1 h(v, s) -- e(v, s) - eo + ~ [p(v, s) + po](V - Vo)
(2.58)
The function is now restricted to the isentropic s and designated hs(v ). Notice how the Hugoniot equation is recovered when hs(v ) - 0 . Next, suppose that the asymptotic properties of the EOS are such that p(v, s) --+ oo as v -+ 0 and that p(v) > 0. Furthermore, ? > 0, implies that [Op/Ov]s < 0, and from this and the previous assumptions it can be proved that e/p --+ 0 as v -+ 0. Now using these results, it follows from Eq. (2.58) that hs(v ) -+ - o o as v ~ 0. On the 1 (/) - v0), then with v > v0, hs(v ) --+ +oo other hand, because hs(v ) > - e 0 + ~p0 as v--, oo. By continuity, therefore, hs(v ) vanishes at least once, and at least one solution exists to the Hugoniot equation. In order to accommodate shocks of arbitrary strength, Bethe's proof of the second part required that G > 0, plus the sufficient (only) constraint lp > - 2 (Eq. 2.43). The proof presented by Henderson and Menikoff (1998) used the lemma in Section 2.8.3 and the alternative EOS constraints G > 0 with F < 27, where the weak constraint is again only sufficient. However, because 32 > 0, the two F constraints overlap and cover the entire real domain - o o < F < oo and this confirms the claim of Menikoff and Plohr (1989) that their proof is independent of F. Note that Henderson and Menikoff (1998) corrected an error in their proof. The theorem ensures that with G > 0, a unique solution exists. It also ensures that the entropy s is a monotonic increasing quantity along a Hugoniot; so s can be used as a Hugoniot parameter. Other quantities that can be shown to be monotonically increasing are the mass flux in through the shock wave, the shock velocity Us, and the negative slope of the Raleigh line, ( A p / A v ) . Using the implicit function theorem Henderson and Menikoff (1998) obtained, COROLLARY 2.1. If G > 0 and F < 232 the Hugoniot curve can be parameterized by the entropy s and consists of a single curve connected to the base state.
168
L. E Henderson
The B-W theorem asserts that u < a for an entropy-increasing compression shock and u > a for an entropy decreasing expansion shock. The inequalities are strict so sonic flow cannot occur downstream of either shock. The only exception is for an acoustic degeneracy, where p = P0 and u = a = a0 = u0, which is trivial. An important corollary can now be deduced. Notice that the conservation laws remain the same if the initial (v0, P0, e0, u0) and the final (v, p, e, u) states are interchanged. Similarly the Hugoniot and the Raleigh equations remain the same. The equations are also unaffected if the signs of the velocity vectors are reversed. Consequently, an entropy-increasing compression shock wave can be viewed mathematically as a reversed, entropy decreasing, expansion shock wave, which has supersonic flow on its lower density side. Hence, COROLLARY 2.2. If G > 0 and F < 2~ then for an entropy increasing shock the upstream state is supersonic u o > a o and the downstream state is subsonic u < a.
D o w n s t r e a m Sonic Flow The slope of the Hugoniot curve in the (p, s)-plane can be found by differentiating Eq. (2.11) and using Eq. (2.20)
T
=
Ay
h
[ -API (2.59) 7 -IF
7]
Note that the denominator is always positive when the weak constraint is satisfied. By using Eqs. (2.12) and (2.27) the numerator can be rewritten as
--P
-
~s+S-vv
-
p
v
-
where by the B-W Theorem the inequality in Eq.(2.60) is certainly valid when G > 0, that is for a compression shock, but it is also valid for an expansion shock with G > 0 (Thompson, 1971; Thompson and Lambrakis, 1973). Suppose that G has both positive and negative state domains for some material. If the initial state is on a particular isentropic in the G > 0 domain, then by geometry it is possible for the Rayleigh to be tangent to another isentropic in the G > 0 domain (Fig. 2.6). The Rayleigh and isentropic have the same slope at the tangent point Op/Ovl~ - A p / A v ; so there is sonic flow downstream of the shock u 2 = a 2 (Eq. 2.60). Furthermore by obtaining dp/dVlh from (2.11) it is
169
General Laws for Propagation of Shock Waves Through Matter
easily shown that the Rayleigh is also tangent to the Hugoniot Op/OVls = A p / A v - - d p / d v ] ~ . This condition corresponds to the lower shock splitting limit in Eq. (2.49). Hugoniot curves that cross from a G > 0 into a G < 0 domain also have a concave segment (Cowperthwaite, 1968; Thompson and Lambrakis, 1973; Cramer and Sen, 1987). Shock waves now become composite waves which can be compressions or expansions. A compression has the form shock/isentropic-compression/shock, while an expansion has the form isentropic-expansion/expansion-shock/isentropic-expansion. The shocks are called sonic shocks because there is sonic flow either upstream or downstream, or on both sides of them (see for example the discussion in Menikoff and Plohr, 1989). The G - 0 condition occurs not only at the separating boundary between G~0, but also at phase triple points where the speed of sound is zero. A concave segment of a Hugoniot, and an isentropic can contract to zero length to form a cusp, where formally G -- - o o ; this is well known at a phase change and at a metal yield point. The Rayleigh line can touch the cusp (Fig. 2.9), corresponding again to the lower limit of (2.49); shock splitting is observed at any higher shock pressures (Zeldovich and Raizer, 1966; Duvall and Graham, 1977; Thompson et al., 1986; McQueen, 1991). Finally notice from (Eqs. 2.59, 2.60) that when a Hugoniot has a sonic point then Os/Op] h = 0, and the entropy no longer increases monotonically along it." P
H
2_ 1
(
o
RI
,
9
~'~,~
' PO ) V
FIGURE 2.9 u - - c/.
Shock splitting instability when G < 0 locally at (/21, Pl), which is a sonic point
170
L. E Henderson
r2
cd
rI
I I I I 1
I..~. i
I
2 a)
12'
After
Before
cd I I ! t I I
13'
2
After
2
1 Before
b)
0
e
t
cd I I I ! I I
sl
1
FIGURE 2.10 The one-dimensional collision of two planar shock waves i I and i 2. a) Colliding shocks, b) Overtaking shocks, r-reflected shock wave, t-transmitted shock wave, e-expansion wave, and cd-contact discontinuity.
General Laws for Propagation of Shock Waves Through Matter
171
2.10 SHOCK WAVE INTERACTIONS 2.10.1
DIMENSIONS OF T H E I N T E R A C T I O N S
A normal shock wave is (1D) by definition (Fig. 2.8a). If two 1D shocks i I and i 2 are parallel and approach each other from opposite directions they will collide. After collision there will be two reflected shock waves r 1 and r2, and a contact discontinuity cd, (see Fig. 2.10a). There will also be a collision if i 1 and i 2 move in the same direction. This is an overtaking collision and there are two possible outcomes. If i 1 and i 2 are weak shock waves, then after collision there will be a transmitted shock wave t, a reflected shock wave r, and a contact discontinuity cd For stronger shocks, an expansion wave e (see Fig. 2.10b) will replace the reflected shock wave r. Such phenomena are defined to be 1D interactions because all the waves and contact discontinuities are parallel to each other.
2.10.2
TWO-DIMENSIONAL SHOCK WAVE
INTERACTIONS These (2D) interactions occur between nonparallel, that is, oblique shocks. A few examples are illustrated in Fig. 2.11, but there are many other possibilities. Classification of wave interactions is facilitated if a direction is assigned to every wave and contact in it (Landau and Lifshitz, 1959). Taking shock-fixed coordinates and resolving and /J0 and /J into perpendicular fi0 and fi, and parallel vt0 and zrt vectors, as in Fig. 2.8b, the direction of a shock is defined to be the same as ~r = zet- In a similar way, a direction can be assigned to any wavelet in a Prandtl-Meyer expansion. For a 2D contact discontinuity, the direction is the same as the particle path on either side of it. As illustrated in Fig. 2.11, 2D waves and contact discontinuities may either meet or emanate from a point in the flow called a node (Glimm et al., 1985). W h e n the direction of a wave points towards a node, we say that the wave arrives at the node or that it is an incoming wave. We say that it leaves the node or that it is an outgoing wave when it points away from the node. In this sense the wave direction is the same as that of the flow of information from flow disturbances. For example, an incident shock i always arrives at a regular reflection (RR) node while the reflected r shock always leaves it (see Fig. 2.11a). For a Mach reflection (MR), the i shock is incoming, while the r shock and Mach shock are outgoing, and so is the contact cd. It may be said that the i shock splits into the r and n shocks and the contact discontinuity (see Fig. 2.11c). By contrast, the very similar interaction (see Fig. 2.11d) has two in-coming i shocks, and one
172
L. E Henderson
cd
~RR
o)
2
i2
b)
,,_...~oMR
_
- ~ - - - D C ',
c)
OR
,
d)
t ~ cd
OR t ~
e)
f)
cd
OR
t~
~d
r
g)
FIGURE 2.11 Examples of two-dimensional shock wave interactions, a) Regular reflection node, RR. b) Cross node, CR. c) Mach reflection node, MR. d) Degenerate cross node, DC. e) to g) Various overtaking nodes, OR.
outgoing r shock and contact discontinuity. It can only exist when there is an extra boundary in the flow, which is needed to generate i 2, (Henderson and Menikoff, 1998).
2 . 1 0 . 3 THREE-DIMENSIONAL SHOCK WAVE INTERACTIONS Shock waves that have conical, cylindrical or spherical symmetry are simple examples of (3D) shock waves. The study of the interactions of these shock waves or those with more complicated shapes usually requires the help of
General
Laws for Propagation of Shock Waves Through Matter
173
computer graphics software in order to achieve success. For an example of a recent study, see Skews (1997).
2.11 THE TRIPLE-SHOCK-ENTROPY A N D RELATED THEOREMS 2.11.1
THE THEOREMS
The B-W theorem is directly applicable to either a 1D or a 2D shock wave, but not to the interactions of two or more of these shock waves. The triple-shockentropy (TSE) theorem gives information about shock interactions in materials whose EOSs satisfy G > 0 and F < 27. Application to 3D interactions is possible in regions where the radius of curvature of the shock waves is small compared to their thicknesses, and where also transverse flow gradients are small. THE TRIPLE-SHOCK-ENTROPY THEOREM. (Henderson and Menikoff, 1998)
Suppose that G > 0 and F < 27 are satisfied everywhere. Consider only the physically realistic, entropy increasing, shock waves. Then the entropy increase across a sequence of two shock waves is smaller than that across a single shock wave to the same final pressure. This is the key theorem; for its somewhat lengthy proof see the cited reference. Extension of it to an n shock sequence follows easily by mathematical induction; the result is, COROLLARY 2.3. A sequence of n shock waves has less entropy increase than a single shock wave to the same final pressure. Once the entropy inequality has been derived it easy to obtain inequalities for other state variables. The results are, COROLLARY 2.4. Consider the state downstream of a sequence of shock waves with the same final pressure as a single shock wave. Multiple shocks have a smaller enthalpy and temperature increases than the single shock. Moreover, if F < ? then multiple shock waves have a smaller specific energy increase, and if F > 0 then multiple shock waves have a smaller specific volume increase. A similar result can also be obtained for the particle velocity with F < 7, as follows.
174
L. E
THE TRIPLE-SHOCK-PARTICLE-BOUND THEOREM.
Henderson
(Henderson and Menikoff 1998)
Suppose that G > 0 and F < ? are satisfied everywhere. Consider a sequence of two, entropy increasing, shock waves from state 0 to state 1, and from state 1 to state 2, and a third shock from state 0 to state 2' with P2' - P 2 . Then (U 2 -- Ul) 2-4- (U 1 -- U0) 2 •
(2.61)
(U 2 -- U0) 2
2.11.2 APPLICATION TO SHOCK WAVE INTERACTIONS Consider the consequences of the inequality equation (2.61) to 1D shock interactions. Suppose that before collision there is an incoming left-facing shock wave (0-1, L) approaching an incoming right-facing shock wave (0-1, R). The Hugoniot curves in the (u, P)-plane and the wave diagram in the (x, t)plane are sketched in Fig. 2.12. We shall now prove that the outgoing waves produced after the collision can only be shock waves. We begin by assuming to the contrary that one of the outgoing waves, say (1-2, R) is an expansion wave. This means that the Hugoniot curve for (1-2, R) must cross the Hugoniot for (0-1, L), which implies that u 2 > u 2, (see Fig. 2.13). To prove that this is t
P
c
I I I
I
I
\._
_d
I I X
. . . .
(o)
(b)
FIGURE 2.12 The collision of two shock waves of opposite family a) Hugoniot loci in the (u, p)plane. The dashed lines are Hugoniot loci for the incoming shock waves and the solid lines are Hugoniot loci for the outgoing shock waves. The outgoing shock waves of the shock waves interaction correspond to the intersection point of the two solid curves, b) Wave diagram in the (x, t)-plane. The initial state is (0), the incoming shock waves are (1) and the outgoing shock waves are (2). The superscripts L and R denote left and right in the (x, t)-plane. The 2' denotes a single shock wave from the initial state with the same final pressure as the outgoing shock waves.
175
General Laws for Propagation of Shock Waves Through Matter
t
P
R
r
I
I
\ \
I 01"' I
(o)
(b)
FIGURE 2.13 Excluded situation in which the collision of two shock waves of the opposite family would result in a reflected rarefaction.
impossible, we must show that u 2 < u2,. (Here we have dropped the L and R subscripts for simplicity.) Now u 2 < 1/2' can be expressed as
11/2 - 1/11-I1/1 - 1/ol < l u 2 ' -
uol
(2.62)
But also because lu 2 - 1 / 1 1 - J i l l - 1/0[ < 11/2- 1/lj, then Eq. (2.61) implies that [u2 - U l ] < ]u2, -1/o], so Eq. (2.62) is satisfied. Hence, outgoing expansion waves are excluded and only outgoing shock waves are permitted. The wave diagram is quite different for overtaking shock waves (Fig. 2.14). In this case there may exist states where outgoing expansion waves are possible, that is, where
lU2 -- 1 / 1 1 + ]1/1 -~-uOI > lU2 ' -- 1~/01
(2.63)
As indicated in Fig. 2.14, this is possible if the shock waves are sufficiently strong, even though Eq. (2.61) remains satisfied. The result is an outgoing reflected expansion wave and a shock wave that move in opposite directions. Alternatively, both outgoing waves are shock waves when the incoming shock waves are sufficiently weak. Although the 2D interactions are more complex, the TSE theorem permits the immediate conclusion that a contact discontinuity must occur in the wave systems sketched in Figs. 2.11b, 2.11c, and 2.11d. By the extension to n shock waves the same conclusion follows for the systems sketched in Fig. 2.11e. Furthermore, inequalities for the thermodynamic states across the contact discontinuities are obtainable from Corollary 2.4, Section 2.12.1.
176
L. E Henderson
t
R
c
I
zR~I~ I i
,/
;
! I
B)~~-----~o t
'/ /
~'~:":" X -X
C)
(a)
(b)
FIGURE 2.14 The overtaking of two shock waves of the same family, a) Hugoniot loci in the (u, p)-plane, b) Wave diagram in the (x, t)-plane when the outgoing shock waves consist of a reflected rarefaction wave and a transmitted shock wave. c) Wave diagram in the (x, t)-plane when the outgoing shock waves consist of a reflected shock wave and a transmitted shock wave. The lines with arrowheads represent particle paths.
S FIGURE 2.15
A curved shock wave S standing off a blunt body B.
177
General Laws for Propagation of Shock Waves Through Matter 2.12
CROCCO'S
THEOREM
This theorem is useful for application to curved shock waves of variable strength, such as a bow shock wave standing off a blunt body (see Fig. 2.15). It relates the flow velocity U and the vorticity V x U vectors to the gradients of the entropy Vs and total enthalpy Vh t. It may be applied generally to shock wave problems because it is independent of any EOS. Given the Euler equation DU p ~ = -Vp Dt
or after expanding the left-hand side 3U p--~ + p(U. V ) U -
(2.64)
-Vp
Now rewriting the fundamental equation (2.20) in terms of the enthalpy Tds = d h - vdp, and because for 3D flow the differentials can be replaced by
the gradient operator, ]
TVs -- V h -
vVP- Vh-
(2.65)
_ Vp P
Eliminating Vp from Eqs. (2.64) and (2.65) 3U
(2.66)
TVs -- Vh +-a-:-+ ( U . V)U ot
But from Eq. (2.10), h - - ht - ~
lu2
1
2
(2.67)
+ (U. V)U
(2.68)
~ Vh=Vh t-~vU
Eliminating h between Eqs. (2.66) and (2.67) 1
3U
TVs -- Vh t - ~ VU 2 + ~
Finally, the vorticity vector can be introduced by means of the identity 1 u2 V1
( u . v ) u = u x (v x u)
so that Eq. (2.68) becomes Crocco's theorem (Crocco, 1937): 3U T V s - - V h t - U x (V x U)+-0---t-
(2.69)
178
2.13
L. E Henderson
THE REFRACTION
LAW
This is a kinematic equation and it is independent of any EOS. It is useful when a 2 or a 3D wave interaction has at least one node (see Fig. 2.11). Consider a reference frame at rest with respect to the undisturbed material. It is clear that if a wave system is to be stable against breaking up into some other system, it is necessary that all the waves meeting at a node must propagate at the same velocity along the node trajectory. An example of the node that occurs in regular reflection (RR) is shown in Fig. 2.16. This system appears for example when an incident shock i diffracts o v e r a r i g i d r a m p . If t h e w a v e v e c t o r s for t h e i a n d r s h o c k s are, respectively, U i,
/J~, if thecorresponding downstream particle (driving piston) velocity vectors are Upi, Upr, and if the angles of incidence and reflection are ~i and ~r, then by geometry the refraction law is
~_ ~U~ _ sin ~i
U; sin(n - ~ )
_-- U'r
(2.70)
sin ~
Ur Ur
(2 r
ai " I
I I
I
U
I I I I I
~ .
Upr \ \
ai i
,
I 7/'- ar Ll
I I
Ui FIGURE 2.16 Wave and particle velocity vectors in laboratory frame coordinates for the regular reflection RR of an incident shock wave i.
General Laws for Propagation of Shock Waves Through Matter
179
Because the incident shock wave compresses and accelerates the undisturbed material, the r shock propagates into a material that is both moving and compressed. If Ur is the velocity of r relative to the moving material that is upstream of it, then by geometry,
U~ -
[TJr -+- Upi c ~
(2.71)
0~r)
If the ramp surface is impervious, then one has the boundary condition
Op,
Opr
-
(2.72)
-Op
When the ramp angle (1 r t - cq) and the shock wave velocity ~Ji are given, the particle velocity Upi can be found from Eq. (2.3) and the EOS of the material. This leaves four unknowns 0~r, U~, Ur and Upr, and there are three equations, (2.70)-(2.72). A fourth equation can be obtained from Eq. (2.3) and the EOS to find the relation between Ur and Upr. Thus the RR problem can be solved when Ui, 0q and the EOS are given. The more complicated Mach reflection (MR) is presented as a second example in Fig. 2.17. In order to accommodate the Mach shock n, the
J
J
Ur ,.,.!
Ur
/
Upn/
~ _
Upi
I
--
Un
L'
Ui
FIGURE 2.17 Wave and particle velocity vectors in laboratory frame coordinates for the Mach reflection MR of an incident shock wave i.
180
L. E Henderson
refraction law is extended by adding another equation, which includes the corresponding wave and particle velocities Un and Upn, respectively ~!
~_
Ui = U_~ = sin a~
s i n o~r
Un
(2.73)
s i n o~n
The law provides a powerful means for finding when one wave system changes to another as a result of a continuous change in the system parameters. For example, consider the MR sketched in Fig. 2.11c, in which the Mach shock n, leaves the node. Suppose that 0~i decreases while the other independent parameters such as Ui are held constant. This causes 0~n to increase, and a condition can be reached where a n > rr/2. Hence, n now arrives at the node as in Fig. 2.11d, and an extra boundary c is needed to support the appearance of n = i2. If it is not present then the three shock wave system cannot appear, and an RR appears instead. The separating, or transition condition occurs when 0~n - 7 r / 2 , and Eq. (2.73) gives sin 0~N - sin 0~i = Un Ui
(2.74)
Equation (2.74) defines the mechanical equilibrium criterion for RR ~ MR transition, and is such that Mach shock n is a normal shock. Equations analogous to (2.71) and (2.72) require an extensive discussion and are omitted here (for more details, see Henderson, 1989).
2.14
CONCLUSIONS
9 The conservation laws (2.1)-(2.3) or (2.7)-(2.9), and the equivalent Hugoniot (2.11) and Rayleigh (2.12) equations. They can be applied to either a normal or an oblique shock wave in any material. 9 The Bethe-Weyl Theorem (Section 2.9). It can also be applied to either a normal or to an oblique shock wave in any material, but it is most powerful when the material EOS is convex G > 0 [Eq. (2.33)] and when the EOS also obeys the weak constraint F < 2? [Eq. (2.54)]. All materials in a single phase obey these EOS constraints for almost all thermodynamic constraints. Exceptions for a fluid in a single phase occur near a phase critical point, if the fluid has at least seven atoms in its molecule. Other exceptions exist at the yield point of a metal, and for some, but not all multiphase states. 9 Thermodynamic properties may be used to construct several constraints on a material's equation of state. They determine when a solution to a single-shock-Riemann problem exists and is unique (Section 2.9)" when
181
General Laws for Propagation of Shock Waves Through Matter
the solution to a 1-D shock-interaction problem is unique (Eq. 2.5.3); when state variables are monotonic along the Hugoniot adiabatic (Section 2.83, 2.9)" when a shock is stable against splitting or ripple instabilities Eq. (2.49)" also the necessary admissibility conditions for compression and expansion shocks to exist Eqs. (2.50, 2.51). Furthermore see Section 2.9 for other necessary existence conditions, for a compression shock (G > 0), and for an expansion shock (G < 0). 9 The triple-shock-entropy Theorem (Section 2.11). This theorem gives general properties for 1D and 2D shock wave interactions when the material EOS obeys the G > 0 and F _< 27 constraints. 9 Crocco's theorem (Eq. 2.69 in Section 2.12). This theorem is useful when there are entropy, total enthalpy, or velocity gradients upstream of the shock or when the shock is curved and with nonuniform strength. It is applicable to any material. 9 The refraction law (Eqs. 2.70 and 2.73 in Section 2.13). This theorem is useful when there are wave nodes present in 2D or 3D shock interactions. It provides a means for finding where one wave system changes into another under a continuous change of the system parameters. It can be applied to any material.
APPENDIX" THE C O N V E X I T Y OF AN EQUATION
OF STATE
Following Bethe (1942), the basis of his method for finding if [O2p/0V2]s > 0 for a given EOS is the equation
s
-
4-
C-~v ~-~ ~
v
L-~-vJT
1 - ~vv L-aYJvJ
REFERENCES American Physical Society (sponsor), 1989 Topical Conference on Shock Waves in Condensed Matter. Bethe, H. (1942). The theory of shock waves for an arbitrary equation of state, Clearing House for Federal Scientific and Technical Information, U.S. Dept. Commerce, Wash. DC Rept. PB-32189. Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatics, New York: John Wiley, (Second edition). Collins, J.G. and White, G.K. (1964). Thermal expansion of solids, in Progress in Low Temperature Physics, Vol. 4, C.J. Groter, ed., pp. 450-479.
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Courant, R. and Friedrichs, K.O. (1948). Supersonic Flow and Shock Waves, New York: Interscience Publishers Inc. Cowperthwaite, M. (1968). Properties of some Hugoniot curves associated with shock instability, J. Franklin Inst. 285: 275-284. Cramer, M.S. (1989). Shock splitting in single phase gases, J. Fluid Mech. 199: 281-296. Cramer, M.S. and Sen, R. (1987). Exact solutions for sonic shocks in van der Waals gases, Phys. Fluids 30: 377-385. Crocco, L. (1937). Eine neue Stromfunction fur die Erforschung der Bewegung der Gase mit Rotation. Z. Angew. Math. Mech., 17: 1-7. Duvall, G.E., and Graham, R.A. (1977). Phase transitions under shock-wave loading. Rev. Modern Phys. 49: 523-579. D'yakov, S. (1954). On the stability of shock waves, Zh. Eksp. Teor. Fiz. 27: 288. Eisenberg, D. and Kauzmann, W. (1969). The Structure and Properties of Water, Oxford Univ. Press. Erpenbeck, J.J. (1962). Stability of step shocks, Phys. Fluids. 5: 1181-1187. Fickett, W. and Davis, W. (1979). Detonation, Los Angeles: UC Berkeley Press. Fowles, G.R. (1975). Subsonic-supersonic condition for shocks, Phys. Fluids 18: 766-780. Fowles, G.R. (1976). Conditional stability of shocks - a criterion for detonation, Phys. Fluids. 19: 227-238. Frost, D. and Sturtevant, B. (1986). Effects of ambient pressure on the instability of a liquid boiling explosively at the superheat limit. TASME J. Heat Transfer, 108: 418-424. Gardner, C.S. (1963). Comment on "stability of step shocks". Phys. Fluids 6: 1366-1368. Gathers, G.R. (1994). Selected Topics in Shock Wave Physics and Equation of State Modeling. Singapore: World Scientific. Glass, I.I. and Liu, W.S. (1978). Effects of hydrogen impurities on shock structure and stability in ionizing monatomic gases. Part 1. Argon. J. Fluid Mech. 84: 55-77. Glimm, J., Klippenberger, C., McBryan, O., Plohr, B., Sharp, D., and Yaniv, S. (1985). Front tracking and two-dimensional Riemann problems. Adv. Appl. Math. 6: 259. Griffiths, R., Sandeman, J., and Hornung, H. (1975). The stability of shock waves in ionizing and dissociating gases. J. Phys. D8: 1681. Henderson, L.E (1989). On the refraction of shock waves. J. Fluid Mech. 198:365-386. Henderson, L.E and Menikoff, R. (1998). Triple-shock entropy theorem and its consequences. J. Fluid Mech. 366: 179-210. Johannesen, N.H., and Hodgson, J.P. (1979). The physics of weak waves in gases. Rept. Prog. Phys. 42: 629-676. Kolsky, H. (1953). Stress Waves in Solids, O.U.P. Kontorovich, V. (1957). On the stability of shock waves. Zh. Eksp. Teor. Fiz. 33:1525 (Soviet Phys. JETP 1957, 6: 1179). Kontorovich, V. (1958). Concerning the stability of shock waves. Zh. Eksp. Teor. Fiz. 33:1525 (Sov. Phys. JEPT 6: 1179). Lambrakis, K.C. and Thompson, P.A. (1972). Existence of real fluids with a negative fundamental derivative F. Physics of Fluids, 15: 933-935. Landau, L.D., and Lifshitz, E.M. (1959). Fluid Mechanics, Reading, MA: Addison Wesley. McQueen, R.G. (1991). Shock waves in condensed media: Their properties and the equation of state of materials derived from them, in High Pressure Equations of State: Theory and Applications, S. Eliezer, R. A. Ricci, eds. New York: North Holland. Menikoff, R. and Plohr, B.J. (1989). The Riemann problem for fluid flow of real materials, Rev. Modern Phys. 61(1): 75-130. Meyers, M.A. (1994). Dynamic Behavior of Materials, New York: John Wiley Neumann, J. yon (1963). Collected Works, Vol. 6, Pergamon.
General Laws for Propagation of Shock Waves Through Matter
183
Riemann, B. (1860). Uber die forpflanzung ebener luftwellen von endlicher schwingungsweite. In Collected Works of Bernard Riemann, 1953, H. Weber, ed., New York: Dover. Shepherd, J.E. and Sturtevant, B. (1982). Rapid evaporation at the superheat limit. J. Fluid Mech. 121: 379-402. Skews, B. (1997). Aspect ratio effects in wind tunnel studies of shock wave reflection transition. Shock Waves 7: 373-383. Thompson, PA. (1971). A fundamental derivative in gasdynamics. Phys. Fluids 14: 1843. Thompson, P.A. (1972). Compressible-fluid Dynamics, New York: McGraw-Hill. Thompson, P.A. and Lambrakis, K.C. (1973). Negative shock waves, J. Fluid Mech. 60: 187-208. Thompson, P.A., Carafano, G.C., and Kim, Y-G. (1986). Shock waves and phase changes a largeheat-capacity fluid emerging from a tube. J. Fluid Mech. 166: 57-92. yon Neumann, J. (1943). Oblique Reflection of Shocks, see Collected Works, vol. 6, New York: Pergamon. Wackerle, J. (1962). Shock-wave compression of quartz, J. App. Phys. 33: 922-937. Zeldovich, Ya., and Raizer, Ya. (1966). Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, New York: Academic Press.
CHAPTER
3 .1
Theory of Shock Waves 3.1
Shock Waves in Gases
GEORGE EMANUEL School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, Oklahoma, 73019, USA
3.1.1 Introduction 3.1.2 Jump Conditions 3.1.2.1 Steady Normal Shock Waves 3.1.2.2 Mach Number 3.1.2.3 Jump Direction 3.1.2.4 Unsteady Normal Shock Waves 3.1.2.5 Oblique Shock Waves 3.1.3 Shock Wave Configurations 3.1.3.1 Local versus Global Analysis 3.1.3.2 Single Shock System 3.1.3.3 Multiple Shock System 3.1.4 Interactions 3.1.4.1 Shock Impingement 3.1.4.2 Shock-Expansion and Expansion-Shock Interactions 3.1.4.3 Boundary-Layer Interaction 3.1.5 Real Gas Phenomena 3.1.5.1 Low-Temperature Phenomena 3.1.5.2 High-Temperature Phenomena 3.1.6 Perfect Gas Shock Waves 3.1.6.1 Steady Shock Waves 3.1.6.2 Unsteady Shock Waves 3.1.6.3 Characteristic Theory 3.1.6.4 Shock Formation 3.1.6.5 Steady, Two-Dimensional or Axisymmetric Shock Waves 3.1.6.6 General Theory References
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
185
186
3.1.1
G. Emanuel
INTRODUCTION
Explosive volcanic eruptions generate highly destructive shock waves; as evidenced by the Mount St. Helen's eruption in 1980. Shock waves, in fact, are as ancient as our planet. On a more benign level, lightning generates a shock wave whose attenuated form is heard as thunder. Aside from volcanic eruptions and thunder, shock waves are manmade. These are often generated by explosives, going as far back as Chinese firecrackers. Other early forms of manmade shocks stemmed from firearms and the explosion of steam boilers. As discussed in Chapter 1, the early theory and observation of shock waves goes back to the second half of the nineteenth century. At the time of the start of the Second World War only the basics of a shock wave in a perfect gas were known. Much of this chapter and much of this handbook is based on discoveries made during and since the war. Indeed, as will become evident when discussing real gas effects, shock waves are still an interesting and important research topic. The initial motivation for postwar research was the development of jet engines, rocketry, supersonic missiles and aircraft, and nuclear explosions. Gas-phase shock waves are usually viewed within the context of gas dynamics or compressible flows. In these inviscid flows, shocks represent by far the most distinctive single feature. Pedagogically speaking, the study of compressible flow is a natural way to introduce the subject. Nevertheless, shock wavesmtheoretically, computationally, and experimentally--represent a diverse science and technology in their own right; and one that goes well beyond the confines of the compressible flow of air. This chapter provides a general overview of the subject and leaves for later chapters a more detailed treatment. The one exception is the analysis of shock systems in a perfect gas. In Section 3.1.2 jump conditions associated with steady and unsteady shocks are outlined. Shock wave configurations are the subject of Section 3.1.3, while Section 3.1.4 surveys interaction processes. Real gas effects are discussed in Section 3.1.5. These include, for example, the effects of a phase change, chemical reactions, or radiative transport. Sections 3.1.2-3.1.5 are not comprehensive, but emphasize physical understanding with a minimal use of mathematics. Section 3.1.6 covers shock waves in a thermally and calorically perfect gas. It includes a standard treatment of normal and oblique shocks, and also covers characteristic theory, shock formation, and shock wave derivatives.
3.1
187
Shock Waves in Gases
3.1.2 JUMP CONDITIONS 3.1.2.1
STEADY NORMAL SHOCK WAVES
As sketched in Fig. 3.1.1, the simplest case of a fixed, or steady, normal shock is discussed. The upstream and downstream velocities are wl and ~2, respectively, and the velocities are normal to the shock. The change in state associated with the shock generally occurs over a few mean free paths. Under normal density conditions this means the shock can be viewed as a mathematical discontinuity in an otherwise smooth gas flow. For shocks in sea level air, this criterion is easily satisfied, and our discussion assumes the shock is a discontinuity. The conservation laws of fluid mechanics mandate that the fluxes of mass, momentum, and energy should be conserved across the wave, that is,
(
(PW)l = (Pw)2
(3.1.1a)
(/9 + pW2)l -~. (/9 -~- pw2)2
(3.1.1b)
1 ) ( 1 ) h+~w 2 h+~w 2 1
(3.1.1c) 2
In these relations, p, p, and h are the thermodynamic density, pressure, and enthalpy, respectively. The enthalpy appears in the energy equation, instead of the internal energy, because flow work is included. Conditions at the upstream state are considered to be known, while (p, p, h, w)2 are to be determined. With only three equations for four unknowns, a thermodynamic state equation of the form
f(p, p, h) - 0
(3.1.2)
is required. The form of this relation, or one of its many equivalents, is responsible for the wide range of physical phenomena that are encountered shock wave
f W2
w1
,,,..._
y
FIGURE 3.1.1
Steady normal shock wave.
188
G. Emanuel
throughout the handbook. The simplest, and still the most important case, is that of a thermally and calorically perfect gas (3.1.3a)
p - - pRT
7R h= ~ T 7-1
(3.1.3b)
where R is the gas constant, T is the absolute temperature, 7 is the constant ratio of specific heats, and where the specific heat at constant pressure Cp equals 7R/(7 - 1). As temperature is a new variable, two state equations instead of one are required. Most exceptions to the preceding equations are referred to as real gas effects. This includes temperature-dependent specific heats, dense gas flows where the pressure does not equal pRT, chemical reactions, etc.
3 . 1 . 2 . 2 MACH NUMBER It is convenient to characterize the strength of a shock wave with the ratio of two speeds; this ratio is called the shock Mach number M 1. For a steady, normal shock, it is the ratio of the upstream speed divided by the upstream speed of sound M1 --
(3.1.4)
Wl
al The speed of sound a represents the speed with which a weak acoustic disturbance would travel in the undisturbed fluid. Its thermodynamic definition is -
(3.1.5)
s
where s is the entropy, which is held fixed in the partial derivative. For a perfect gas, this definition simplifies to
( p) 1/2
a-
(TRT) 1/2 --
7
(3.1.6)
Of course, a Mach number M, can be defined as the ratio w / a at any point in a flow field. The following terminology is used to classify a steady flow: incompressible flow: subsonic flow: transonic flow: supersonic flow: hypersonic flow:
M 4< 1 M < 1 I M - 11 < 0.4 M > 1 M >> 1
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189
Shock Waves in Gases
As we shall see, shock waves can occur only in a steady flow if the upstream flow is supersonic, that is, M 1 exceeds unity for a normal shock. The strength of a steady, normal shock is also represented by the pressure ratio across it P 2 / P l . In air, this ratio is 18.5 when M 1 is 4, and this shock would be viewed as intense. As M1 increases, so does P z / P l . In the strong shock, or hypersonic limit, the pressure ratio becomes infinite as M 1 becomes infinite. On the other hand, the shock becomes weak, that is, an acoustic wave, as M1 approaches unity from above.
3.1.2.3 JUMP DIRECTION Engineers are often concerned about the change in state across a shock wave. This typically means the change in M, p, p, T, P0, P0, and To at state 2 as compared to that at state 1. Here, a zero subscript denotes a stagnation quantity. For a perfect gas, these are given by the isentropic relations P___O0= X7/(7-1),
/30 = X 1/(7-1),
p
p
TO = X T
(3.1.7)
where X-
1 + y - 1 M2 2
(3.1.8)
The change in state across a shock is adiabatic but irreversible. Although the shock is quite thin, there nevertheless is heat transfer and viscous stress inside the wave. Both processes are irreversible, and, by the second law of thermodynamics, s2 >_ Sx, where the equality sign holds when the shock has degenerated to a reversible acoustic wave. This is a fundamental result; it means there is a direction to a shock wave. "[he downstream state, state 2, must have an entropy value s2 that exceeds Sx. A homentropic flow has a constant entropy value, in contrast to an isentropic flow, where the entropy can vary from streamline to streamline. Homentropic flows are thermodynamically reversible and reversible in their flow direction as well. This is not the case for a flow with a shock wave; the direction of a flow that satisfies s 2 > s I cannot be reversed. This point is not especially obvious in view of the upstream/downstream symmetry of Eqs.
(3.1.1). For the flow in Fig. 3.1.1,the upstream and downstream Mach numbers are defined as M 1 ---- w--! ,
6/1
M z ---- w---~2
6/2
(3.1.9)
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and M 1 > 1 > M2
(3.1.10)
where M 2 has a finite lower bound, which is achieved when M 1 --+ oo. (For simplicity, the acoustic wave case is ignored.) For a perfect gas, the second law of thermodynamics helps yield P2 T2 P2
P02
..- 1;
-
Pl T1 Pl
-
P02
=
P01
To2
< 1,
P01
= 1
(3.1.11)
To1
The adiabatic requirement is reflected in the unity value for To2/Tol. Notice that P2/Pl and Po2/Pol have opposite trends with respect to M 1. The large decrease in Po2/Pol for a strong shock is a major system consideration in the design of a supersonic wind tunnel or an engine inlet. Inequalities (3.1.10) are general; they do not require a prefect gas assumption. A shock wave whose upstream and downstream states are in the vicinity of a coexistence curve, on the vapor side, may have P2, T2 , P2 < 1
(3.1.12)
Pl T1 Pl and still be in accord with the second law. Such a shock is called a rarefaction shock wave, but has yet to be experimentally demonstrated. This topic is further discussed in the real gas section.
3.1.2.4
UNSTEADY NORMAL SHOCK WAVES
A normal shock is considered that is moving with velocity w s-'' relative to a laboratory frame, see Fig. 3.1.2. A prime denotes the unsteady motion, and the three velocities sketched in the figure are parallel to each other and may be time dependent. The convention that a rightward (leftward) directed velocity is -"s may be positive (negative) is adopted. Thus, w 1-'' is always positive, but w positive or negative. The flow is analyzed by first introducing a velocity transformation that, at a given instant, fixes the shock's location, as in Fig. 3.1.1. We therefore write -" 1 W
-
-
-"1 W
-
-
W s' ,
-"2 - W
-"2 W
-
-
-"s W
(3.1.13)
This transformation leaves the thermodynamic variables, such as p, p, and h, unaltered. As a consequence, Eqs. (3.1.1) and (3.1.2) yield a solution for (p, p, h, w)2, after which the inverse transformation would yield w~. This procedure presumes that w's(t) is a known quantity.
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Shock Waves in Gases
/ ...#
....#
P
wI
M1
P
..~
Ws
h.__ v
FIGURE 3.1.2
Along with
/ s h o c k wave
t
WE
,,...._
r
Unsteady normal shock wave.
and M 2, we introduce !
M~
Wl
M~ -- w~
(3.1.14)
6/2
al
to obtain !
M 1 -- M~ - w-~s,
!
M 2 -- M~ - w--~s
6/1 !
(3.1.15)
6/2
!
With w S, al, and M 1 known, a 2 and M 1 c a n be evaluated. In turn, M 2 is determined by M1, and therefore M~ can be found. Note that the shock becomes weaker as w 'S increases; when M 1 = 1 it is an acoustic wave. (If w s increases further, the shock should be replaced with a rarefaction wave.) The shock moves into the upstream flow when w's is negative. In this case, the shock becomes stronger. !
3 . 1 . 2 . 5 OBLIQUESHOCK WAVES So far, only normal shocks have been discussed. Imagine a fixed oblique shock that has an angle fl relative to the upstream velocity Wl" see Fig. 3.1.3. The analysis is relatively straightforward. Decompose W1 into components that are tangential ~ and normal Ul tO the shock. A shock is possible only if ua/a 1 exceeds unity, and the strength of the shock depends on this Mach number. In other words, P2/Pl, P2/Pl, U2/6/2. . . . are determined by ul/al, which is the normal component M1,, of the wl/al Mach number. The downstream flow field is constructed by starting with a normal shock whose upstream and downstream velocities are fil and fi2, respectively. Simply add a uniform velocity ~ to both the upstream and downstream flows, as indicated in Fig. 3.1.3. This results in a shock wave that has an acute angle// with respect to the upstream velocity ~'1. The downstream velocity ~2 then has
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/
.2/
.
shock wave FIGURE 3.1.3
Oblique shock wave.
an acute angle 0, again relative to Wl- Oblique shocks are most easily pictured as stemming from a sharp turn in a wall, as pictured in Fig. 3.1.4. The upstream flow is supersonic and parallel to the adjacent wall. In order for the downstream flow to be parallel to its wall, a shock wave is required that compresses the gas. In view of Fig. 3.1.3, the velocity components are related by u I = w I sin 13 = v tan 13
(3.1.16a)
u 2 = w 2 sin(fl - O) = v tan(fl - O)
(3.1.16b)
With Eqs. (3.1.9), the normal Mach n u m b e r c o m p o n e n t s are
Mln =
U l - - M 1 sin al
13,
M2. = u-A2= M2 sin(fl - O)
(3.1.17)
a2
Further relations a m o n g these parameters are discussed in Section 3.1.6, where a multivalued solution is obtained.
shock wave
/
/
f
/
/
/
/
/
/
/
f
FIGURE 3.1.4
Supersonic flow over a ramp.
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Shock Waves in Gases
3.1.3 SHOCK WAVE CONFIGURATIONS 3.1.3.1
LOCAL VERSUS GLOBAL ANALYSIS
The sketches in Figs. 3.1.1-3.1.3 can be viewed as referring to a differential area on the surface of a shock. This local viewpoint is often analytically quite convenient. Shocks, however, are a global feature of any flow field, and their analysis is typically done with respect to a given wall configuration. Moreover, as will be discussed in Section 3.1.5.2, a global analysis is required if a second law violation is to be avoided. Here, we outline a handful of relatively simple cases; more complicated shock systems appear throughout the handbook. Noninteracting (single) shocks are first discussed, after which multiple shock systems are considered.
3.1.3.2
SINGLE SHOCK SYSTEM
Wedge Flow A steady, uniform, supersonic freestream encounters the two-dimensional wedge sketched in Fig. 3.1.5(a). The wedge angles 0 2 and 0 3 are sufficiently small, for a given upstream Mach number M 1, such that the shocks are attached. Generally, the flows in regions 2 and 3 are supersonic. For a given M 1 value, however, there is a very narrow range of wedge angles for which the downstream flow is subsonic. With attached shocks, the flows in regions 2 and 3 are independent of each other, and each of these can be viewed as the one sketched in Fig. 3.1.4. The two shocks are planar and the flows in regions 2 and 3 are uniform when supersonic. If, however, the flow in region 2, say, is /sonic
line
2 M1
02
MI 83 ...//
(a)
M>I
(b)
FIGURE 3.1.5 Supersonicflow over a wedge: (a) attached shocks; and (b) detached shock.
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G. Emanuel
subsonic, then the shock is no longer planar because of the upstream influence of the shoulder of the wedge, which is sketched in Fig. 3.1.5(b). With M 1 fixed, if either, or both, of the 0 angles become sufficiently large, then the shocks form a single, curved, detached, bow shock; see Fig. 3.1.5(b). The wedge is now followed by a supporting fiat plate. In the nose region, the flow is subsonic. Downstream of the sonic lines (really surfaces) that start at the shoulders, the flow is supersonic. Additional shocks may appear in these supersonic regions.
Wave Drag
Shock waves represent an irreversible, entropy producing process. For a supersonic or hypersonic vehicle, this shows up as drag, called wave drag. Wave drag, when present, is only part of the total drag. Nevertheless, it generally is the dominant form of drag for a high-speed vehicle, dominating, for example, viscous drag. A nominal drag coefficient is defined as d
2d
c a = ( 1 / 2 ) ( p w 2 ) o ~ A = ~(pM2)oo A
(3.1.18)
where A is a reference area, d is the drag, and an infinity subscript denotes freestream conditions. While c a may have roughly comparable subsonic and supersonic values for the same body, the M2 factor substantially increases the magnitude of the supersonic drag. The reason for this is partially apparent in Fig. 3.1.5(a), where the uniform surface pressures P2 and P3 can significantly exceed Pl when M 1 > 1. (Further increasing the wave drag is the reduced pressure, which is
Sweep The theory of shock systems is built on simple models that can be extended into configurations of greater complexity and utility. As an illustration, consider just the upper wedge, with the angle 02, in Fig. 3.1.5(a) with an attached shock (Emanuel, 1992a,b). Rotate the wedge in a plane containing the lower surface of the wedge, which remains aligned with the freestream velocity. Let the angle of rotation, or sweep, be A; see Fig. 3.1.6. In this three-
3.1 ShockWaves in Gases
195
shock
FIGURE 3.1.6 Wedgewith sweep. dimensional flow, the strength of the shock now depends on the component of that is in the sweep plane
M1
MI•
--
M 1 cos A
(3.1.19)
where the angles 0 2 and j82 are also in the sweep plane, and this plane is perpendicular to the leading edge of the wedge. In drawing Fig. 3.1.6, it has been assumed that MI• exceeds unity and the shock is still attached. This flow is analyzed by starting with the unswept wedge with the 02 angle in Fig. 3.1.5(a), but with a freestream Mach number MI• To this add a constant velocity, whose magnitude is w I sin A, that is normal to the plane of the unswept flow. This velocity is uniformly added to the flows both upstream and downstream of the shock. The procedure is thus similar to the way an oblique shock is generated.
Taylor-Maccoll Flow Another important flow, called Taylor-Maccoll flow, is that about a cone at zero incidence; see Fig. 3.1.7. This flow contains a conical shock whose axis, and that of the body, is aligned with the freestream velocity. This configuration is amline
Ml
FIGURE 3.1.7 Supersonicflow over a cone.
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G. Emanuel
particularly important for supersonic axially symmetric engine inlets. As with a wedge, if 0b becomes too large for a given M 1 value, then the shock becomes detached. The flow field between the attached shock and the surface of the cone is not uniform, but rather conical. This means that the pressure, Mach number, etc., are functions only of the angle q. In other words, these parameters are constants on a conical surface, or ray, whose apex coincides with that of the cone. A streamline, downstream of the shock, follows a gently curving path that ultimately becomes parallel to the cone surface. Along this streamline, the flow experiences a modest homentropic compression, that is, the pressure increases and the Mach number decreases. The slope of the streamline just downstream of the shock is slightly less than Oh; the solution is therefore singular at the apex. Generally, a conical flow field is supersonic. But, just as with a wedge, there is a narrow range of 0b values for a given M 1 value where it is subsonic. In fact, one of the rays may be a sonic line, in which case the flow adjacent to the body (shock) is subsonic (supersonic). If the cone axis is misaligned with the freestream velocity, the flow field is no longer conical and the shape of the attached shock deviates from that of a right-circular cone. Blunt-Body Flow Hypersonic flight of a missile generally requires a blunted (spherical) nose cone in order to reduce the high overall surface heat transfer rate. This surface shape produces a blunt body flow, as sketched in Fig. 3.1.8. A detached bow shock has a point where it is normal to the freestream. If the flow field is axisymmetric, the streamline through the normal part of the
M1 _
_
_
FIGURE 3.1.8 Blunt-bodyflow.
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197
S h o c k W a v e s in G a s e s
shock wets the body. The shock is strongest at this location, gradually weakening with increasing distance from this location. This weakening does not happen to the shocks in Figs. 3.1.5(a) and 3.1.7, providing the wedge and conical bodies continue to grow indefinitely. Of course, real bodies have a finite length, and in the process of termination generate expansion waves that interact with, and weaken, the bow shock. Thus, a bow shock becomes an acoustic wave far from the body that generated it. The weakening process, however, is quite gradual, as attested to by a ground level sonic boom that is caused by the shock system of a high-flying supersonic aircraft. At a relatively low supersonic Mach number, in the transonic range, a detached shock is well upstream of the body and rapidly spreads in the transverse direction, as roughly sketched in Fig. 3.1.5(b). As M 1 increases, the stand-off distance, which is the minimum distance between the shock and the body, decreases and the shock tends to wrap itself around the forward part of the body. In this circumstance, the flow field (viscous or inviscid) between the shock and the body is referred to as a shock layer.
Duct Flow A normal shock may propagate down a constant cross-sectional area duct into a quiescent gas, as pictured in Fig. 3.1.9(a). Conceptually, we imagine a piston, with speed w~, pushing on the gas and thereby generating the shock, which here represents a sharp demarcation between the moving and quiescent gases. This type of flow occurs in shock tubes. If the piston speed is constant, then the shock speed, which exceeds Wp, is also constant and the gas between the shock and piston surface has a uniform state. If the shock moves into a duct with an increasing, or decreasing, crosssectional area (see Fig. 3.1.9(b)), then the shock becomes nonplanar (Friedman, 1960). If there is an area increase (decrease), the overall strength of the shock decreases (increases). The moving shock induces a nonzero velocity downstream of it. If the flow is considered inviscid, this velocity is tangent to l
/ /
I I
....
I
~
/
'~
r -----1 I I
/
/
/
/
/
/
/
--.p
w,~ ~t
w; "'1
~
,v
/
I I I
1.,I . . . . . . . .
(a) F I G U R E 3.1.9
/
/
1 I
W s
-\
\ \
(b)
U n s t e a d y s h o c k p r o p a g a t i o n in: (a) c o n s t a n t cross-sectional area duct; a n d (b)
n o n c o n s t a n t area duct.
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G. Emanuel
~
tube
WlFe
(a) FIGURE 3.1.10
(b)
Exploding wire experiment: (a) before explosion; and (b) after explosion.
the wall, at the wall. Consequently, the shock is normal to the wall, at the wall. In a real flow, there is an unsteady, viscous boundary layer along the wall that starts at the foot of the shock. Near the wall, the shock wave is slightly perturbed from its normal position because of the displacement thickness of the boundary layer. Cylindrical and Spherical Flows Shock waves may have an unsteady cylindrical or spherical configuration. For example, consider an exploding wire experiment as shown in Fig. 3.1.10(a,b). Initially, a thin metal wire is stretched along the axis of a tube that is filled with an inert gas. The wire is rapidly vaporized by suddenly passing a high-voltage current through it (Jensen, 1976). A cylindrical shock wave is thereby generated that propagates through the surrounding quiescent gas. The shock reflects from the inner wall surface of the tube and becomes an imploding wave. This wave propagates into a radially nonuniform flow and the strength of the shock increases as its surface area decreases. Its strength becomes quite large as it reaches the axis of the tube. Similarly, the explosion of a nuclear device can be viewed as generating a spherical shock from a point source. The shock's strength rapidly decays with its increasing radius. It will remain spherical until it encounters some obstacle, or a variation in pressure and density with altitude.
3 . 1 . 3 . 3 MULTIPLE SHOCK SYSTEM Lambda Shock System A deceptively simple configuration occurs when two planar shocks coalesce to form a third shock, as sketched in Fig. 3.1.11. The attached wall shocks are produced by two sharp turns, and the pattern is called a 2-shock, or three-
3.1
Shock Waves in Gases
199
slipstream triplepoint
M1
9
/
/
l
i
/
/
~
/
]-5
c- j
3
z
FIGURE 3.1.11
Three-shock pattern.
shock system. Region 5 has a larger entropy than region 4 because the entropy jump (AS)l5 is larger than (AS)l2 + (As)23. There is an extremely weak pressure disturbance between regions 3 and 4, which is either an oblique shock or an expansion. This disturbance, although quite weak, is theoretically required (Johannesen, 1952; Emanuel, 1982, 1983). Because the disturbance is usually not optically detectable (Johannesen, 1952), it is generally ignored, as will be done in future discussion. Regions 4 and 5 are separated by a thin slipstream. The slipstream starts at the triple point, where the shocks merge. The shape of the slipstream is determined by requiring that the pressure be continuous across it and by a velocity tangency condition. These two conditions are also essential when performing a local analysis of the triple point. The slipstream is actually a (viscous) vortex sheet, because rotation is generated by the different flow speeds on the two sides of it. In drawing Fig. 3.1.11, it is assumed that the flow in region 2 is supersonic. Nevertheless, 2-shock patterns are common, as will become evident. Overexpanded Nozzle Flow Shocks form in the jet emanating from an overexpanded (as well as an underexpanded) nozzle (Chow and Chang, 197'2; Li and Ben-Dor, 1998). (In an overexpanded nozzle, the ambient pressure exceeds the static pressure in the exit plane of the nozzle.) To be specific, a uniform jet leaving an axisymmetric thrust nozzle is considered, see Fig. 3.1.12. For a modest amount of overexpansion, that is, the ambient pressure only slightly exceeds the nozzle exit pressure, the conical lip shock becomes a slightly weaker (approximately) conical shock downstream of the apex. This pattern is sketched in Fig. 3.1.12(a). The shock reflects as an expansion fan from the outer edge of the jet. For a greater level of overexpansion, see Fig. 3.1.12(b); the conical lip shock is relatively strong, and is unable to focus. Instead, a nearly normal shock,
200
G. Emanuel Mach disc
expansion fan
jet
lip shock (a)
(b)
FIGURE 3.1.12 Axisymmetric jet from a nozzle with: (a) slight overexpansion; and (b) significant overexpansion. called a Mach disk, develops. The 2-shock pattern now evident is similar to the one in Fig. 3.1.11. The triple line is circular and a cylindrical slipstream emanates from it. Directly downstream of the Mach disk the flow is subsonic while the flow is generally supersonic outside the slipstream.
Steady, Oblique Shock Wave Wall Reflection The reflection of a planar, oblique shock from a wall is discussed here, without the complicating feature of a boundary layer. Figure 3.1.13 shows two patterns; the one in (a) is called regular reflection (RR) and the other is called Mach reflection (MR). In both flows, a wedge generates an attached incident (I) shock, although the wedge angle 0 is larger in the MR case. The dashed line is a streamline; between the incident and reflected (R) waves, it is parallel to the lower surface of the wedge. In Fig. 3.1.13(b), the incident shock is too strong for RR; hence, we again have a 2-shock system. The nearly normal shock, labeled M, is now called a Mach stem. The flow immediately downstream of it is subsonic, while that between the slipstream and the reflected shock is usually supersonic. .....
O"
expansion
Mi
9
Ml-~
-
.
I
9
~)
.
r
R slipstream
I / / 1 1 / / / / / / / / 7 / / / / / / / / / / / /
s~'eamlinc (a) (b) FIGURE 3.1.13 (a) Regular reflection and (b) Mach reflection.
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201
Shock Waves in Gases
l
f
/
/
/
J
_f
J
f
R \
\
J
f
f
=Ms
I
M
\
FIGURE 3.1.14
Unsteady shock interaction with a ramp.
Unsteady Flow Generally, shock patterns in unsteady flows can be appreciably more complex than their steady counterpart. The unsteady counterpart to Fig. 3.1.13(b) is sketched in Fig. 3.1.14. This type of flow is produced in a shock tube. A planar incident shock propagates into a quiescent gas. Near the end of the tube, there is a ramp that causes part of the incident shock to reflect as a curved shock into the uniform, but moving, gas downstream of the incident shock. There is a triple point where the three shocks meet. Other patterns occur; the one that is shown is the simplest case. We may ask why an unsteady flow with shock waves is generally more complicated than an analogous steady flow. A steady inviscid flow is locally hyperbolic when the Mach number exceeds unity. Only hyperbolic flows admit waves, such as shocks and expansion fans. (In an unsteady flow, the expansion is referred to as a rarefaction wave.) Thus, a steady flow must have at least one region where the flow is supersonic if it is to contain a shock wave. On the other hand, an unsteady inviscid flow is everywhere hyperbolic regardless of the magnitude of the Mach number. Thus, all unsteady flows admit wave behavior. This fundamental difference between steady and unsteady flows influences the interaction process. Generally, an unsteady flow possesses a wider range of possible configurations.
3.1.4
INTERACTIONS
Shocks interact with any feature they encounter. The flows of the previous section can be viewed, for example, as wall-shock interactions. In this section, a few interaction processes are briefly discussed. An oblique shock may encounter a slipstream. The nature of this interaction must take into account the pressure and velocity tangency conditions across the slipstream as well as whether or not the flow on the side opposite to the
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G. Emanuel
incident shock is subsonic or supersonic. Shocks also interact with a contact surface. This type of surface separates two gases that initially have different thermodynamic states. The initially high- and low-pressure gases in a shock tube are separated by such a surface. The incident shock, after reflection from a shock tube end wall, interacts with the contact surface.
3.1.4.1
SHOCK IMPINGEMENT
Shock impingement, or interference, refers to a flow where an oblique shock impinges on the upstream portion of a blunt-body shock wave (Edney, 1968). The interaction is fundamentally different from that shown in Fig. 3.1.11. Figure 3.1.15 is a rough sketch that does not attempt to delineate the flow field downstream of the blunt-body shock. In fact, there are six different configurations for this interaction, where the configuration depends primarily on the location of the impingement point, or line. In one of these configurations, referred to as type VI, a small location on the blunted surface experiences a very high pressure and heat transfer rate. The heat transfer rate can exceed the noninterference rate at a stagnation point by more than an order-of-magnitude and result in significant structural damage for an aircraft or missile (Edney, 1968).
3.1.4.2
S H O C K - E X P A N S I O N AND EXPANSION-
SHOCK INTERACTIONS Figure 3.1.13(a) is a sketch of an expansion-shock interaction in which a centered Prandtl-Meyer expansion, from the shoulder of the wedge, interacts
blunt-body shock wave
oblique
M1 FIGURE 3.1.15 Obliqueshock impingement with a blunt-body shock.
3.1 ShockWaves in Gases
203
with the reflected wave (Li and Ben-Dor, 1996). Although not shown, the same interaction occurs in Fig. 3.1.13(b). Of course, in order for there to be an expansion, the flow upstream of it must be sonic or supersonic, and the expansion accelerates the flow that passes through it. An altered, noncentered expansion appears on the downstream side of the shock. In Fig. 3.1.13(b), this expansion can accelerate the subsonic flow between the wall and the slipstream to a supersonic state (Chow and Chang, 1972; Li and Ben-Dor, 1998). Overall, an interaction weakens both the shock and the expansion. This mutual weakening is a general feature that is not restricted to this particular configuration or to any special gas model. In Fig. 3.1.13, the expansion runs into the shock. Figure 3.1.16(a) also represents an expansion-shock interaction, but where the shock runs into the expansion, and, although attenuated, passes through it. In both sketches in the figure, wall B is drawn parallel to wall A, although this simplifying feature is not essential. In Fig. 3.1.16(a), the shock can interact with the expansion because 1~- 0 is larger than the Mach angle/11 of the leading edge of the expansion, where /t - sin- 1_1 M
(3.1.20)
A shock-expansion configuration is sketched in Fig. 3.1.16(b). Again, a planar shock interacts with a centered Prandtl-Meyer expansion. Regions 2 and 3 are supersonic, uniform flow regions. The border between these regions and the expansion are left-running characteristics, while between points b and d, and also from point c, they are right-running characteristics. (Characteristic theory is developed in Section 3.1.6.) The dashed lines from points b and c represent streamlines. (With surface B parallel to A, point c is actually at infinity.) In this case, the expansion does not pass through the shock but reflects from it as a vortical layer and as a family of right-running characteristics, such as bd. The reason for not passing through the shock, in contrast to _ shock
Prandtl-Meyer expansion M~_~
h
o
c
Prandtl-Meyer
~~vortical
e x p a n d _ _
k A ---
(a)
~ layer
B (b)
FIGURE 3.1.16 (a) Expansion-shockinteraction and (b) shock-expansioninteraction.
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G. Emanuel
Fig. 3.1.13(a), is the inability of a wave to have an upstream influence in a supersonic flow. The flows in regions 1, 2, 3, and in the expansion below the vortical layer are homentropic. The expansion causes a significant weakening of the shock outside of point b. Consequently, this portion of the shock is curved and the downstream flow, along streamlines, is isentropic, not homentropic. In turn, the entropy variation coincides with vorticity; hence, the vortical layer. This layer is analogous to a slipstream, except that it has a substantial thickness. There is a discontinuity in the curvature of the shock at points b and c. (This aspect is treated in Section 3.1.6.5.) On the other hand, the shock strength is continuous at these points; hence, the streamlines that border the vortical layer are not slipstreams. The wall pressure in region 3 is a constant. This region, however, terminates along the bd characteristic, and the wall pressure varies downstream of point d.
3.1.4.3
BOUNDARY-LAYER INTERACTION
The most important interaction process for a shock is with a boundary layer. If boundary-layer separation occurs, the structure of the flow field can be quite different from what it would be if separation did not occur. The separation process, in turn, alters the configuration of the shock wave system. If the impinging shock wave is relatively weak, it may not cause separation, and the sketch in Fig. 3.1.13(a) is then realistic. Separation depends on whether the boundary layer, just upstream of the incident shock, is laminar, transitional, or turbulent. Laminar boundary layers separate much more readily than turbulent layers; however, turbulent layers predominate in real flows. If the upstream boundary layer is laminar, the interaction, with or without separation, causes it to become turbulent. If the strength of the incident shock is not much greater than what is required for separation, boundary-layer reattachment quickly occurs, and the flow sketched in Fig. 3.1.13(a) may still suffice. As the incident shock increases in strength, however, the separated region increases in size and complexity.
3.1.5 REAL GAS PHENOMENA The perfect gas assumption greatly simplifies any fluid or aerodynamic analysis and is realistic for many applications, especially those involving a monatomic gas or room temperature air. Real gas effects, however, are often important and cannot be ignored. Thermodynamically speaking, they occur in two regions.
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Shock Waves in Gases
205
The first is in the general vicinity of the coexistence curve, the other region is at elevated temperatures.
3 . 1 . 5 . 1 LoW-TEMPERATURE PHENOMENA The critical temperature T c of most monatomic and diatomic species (He, Ne, Ar, H2, N2, 02, CO . . . . ) is well below room temperature. For instance, an average T c value of 133 K is used for air. For convenience in the subsequent discussion, coexistence curve effects are referred to as a low-temperature phenomena. This terminology is convenient even though a large polyatomic molecule has a T c value well above room temperature, and is usually a liquid or a solid at room temperature. It may be possible to approximate thermodynamic behavior with a thermally and calorically perfect gas model, for atoms and small molecules, when the temperature is well above Tc. For a monatomic gas, or a gas mixture of inert monatomic species, the approximation is quite accurate over a very large temperature range. It becomes inadequate at very low temperatures, where quantum effects or condensation may occur, or at a very high temperature, where electronic excitation occurs. Diatomics, such as those mentioned earlier, and an inert mixture of monatomic and diatomic species, are reasonably approximated by a perfect gas assumption over a more limited temperature range, but one that encompasses room temperature. In this range, the rotational modes of a diatomic molecule, with an internal energy of R T , are fully excited. Vibrational Modes At room temperature, the vibrational mode of most diatomic molecules is inactive. In other words, the molecule is in its vibrational ground state, and molecular collisions are energetically insufficient for populating the first excited vibrational state with more than a miniscule fraction of molecules. The key parameter is the characteristic vibrational temperature T v of the mode. A large T v value means a relatively stiff bond between the atoms, and a large energy spacing between adjacent vibrational levels of the mode. For instance, T v is 2219 K for 0 2 and 3352 K for N 2. The vibrational contribution starts to be significant when T / T v is about 0.2, which is well above room temperature for air. (When modeling air, an average T~, value of 3056 K can be used.) One diatomic exception is I2, where Tt, is only 309 K. For triatomics, and larger polyatomics, there are multiple vibrational modes whose characteristic temperatures may range from fairly low values, near 300 K, to large values that are in excess of 4000 K. A large polyatomic,
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especially one with heavy atoms, that is, a molecule with a relatively large molecular weight, will have a number of partially excited vibrational modes at room temperature. As a consequence, its constant volume specific heat c v has a pronounced temperature variation, even at room temperature. Triatomics and larger polyatomics are rarely represented near room temperature as a perfect gas with any degree of accuracy. This is because of a relatively large Tc value and one or more active vibrational modes. In this circumstance, Eqs. (3.1.3) are inadequate. We thus replace these equations with the general relations p -- p(T,
v),
c,:o _
cO(T)
(3.1.21a, b)
where Cvo is the part of c~ that accounts for the contribution of the translational and internal modes, and c~0 only depends on the temperature. For example, for a molecule without electronic excitation, we can write o
5 + ~R + R F gv
Cv = ~
,
(~
' sinh 0~
(3.1.22)
where gv is the degeneracy of a distinct vibrational mode, T~,
Ov = 2 T
(3.1.23)
and c~ -- - 2 , monatomic gas; = 0, linear polyatomic; and
(3.1.24)
= 1, nonlinear polyatomic The first term on the fight accounts for translational and rotational excitation, while the summation term is the harmonic oscillator model for the vibrational modes.
Reciprocity One of the Maxwell equations, called reciprocity, represents a constraint on the form for c v. This constraint stems from Eq. (3.1.21a), and is written as
I (o p
(3. 25)
The second-order partial derivative is performed with the specific volume held fixed, whereas the indefinite integral is performed with T held fixed inside the integrand. The integral term provides the contribution of the thermal state
3.1
207
Shock Waves in Gases
equation to cv. For a thermally perfect gas and a van der Waals fluid, this contribution is zero.
Fundamental Equation A fundamental equation is a thermodynamic potential function, with the proper independent variables, and which contains all of the equilibrium thermodynamic information about the system. This information is extracted by performing various partial derivatives of the potential function. For example, for a single component substance, the enthalpy is a potential function when written as
h = h(p, s)
(3.1.26)
whereas Eq. (3.1.2) is not a fundamental equation. The temperature and specific volume are defined as T --
,
v --
p
(3.1.27a,b) s
A thermodynamic system possesses many potential functions, for example, the Helmholtz free energy, when written as
f .= f ( T , v)
(3.1.28)
is a potential function. The various potential functions are equivalent to each other; the choice is a matter of convenience. One can show that Eqs. (3.1.21) are equivalent to a potential function. This means these two relations can analytically yield
s = s(T, v),
h = h(T, v),
a = a(T, v) . . . .
(3.1.29)
where a is the speed of sound. They can also produce the coexistence curve, the vapor pressure, and the specific heats. Thermodynamic consistency is ensured by starting with a fundamental equation or with Eqs. (3.1.21). Phase Change A shock wave may result in a phase change. This occurs, for example, when the wave passes through air containing fog, rain, or snow. The general equilibrium conditions between two phases, here written between a vapor with a g subscript and a liquid with an f subscript are
p(T, vf) = p(T, vg) v -~v vf
dv-O T
(3.1.30a) (3.1.30b)
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where T is held fixed in the integrand, and p in Eq. (3.1.30a) is the vapor pressure. The integral relation is equivalent to a chemical potential condition; it is called a Maxwell construction. For simplicity, consider a normal shock that propagates through an equilibrium air/fog mixture. For the upstream state, at least two new parameters are required. The first would be a mass fraction x of H20 (liquid plus vapor) in the air/fog mixture. The second, y, would represent the mass fraction of H20 that is in the form of liquid droplets. (The small amount of air dissolved in these droplets is disregarded.) Depending on the purpose of the analysis, still other parameters may be required, such as a mean droplet size or a droplet number density. Downstream of the shock, mass fraction x is unchanged from its upstream value. If equilibrium is assumed immediately downstream of the shock, then y changes across the shock. Equilibrium here means the droplets and gas have the same speed and temperature. As the strength of the shock parametrically varies from weak to strong, downstream of the shock y decreases from its preshock value until it becomes zero at some intermediate shock strength. The jump conditions, Eqs. (3.1.1), are unaltered, except that the thermodynamic parameters require mixture definitions. The density and enthalpy are now mass averaged values for the dry air, H20 vapor, and the droplets. The pressure is the sum of the air plus the vapor pressure. One could use Eq. (3.1.3a) for the partial pressure of the droplets, but because of their huge molecular weight, this pressure is negligible. Inside a droplet, the pressure is associated with the surface tension and is inversely proportional to the radius of the droplet. This pressure, which can exceed the vapor pressure, is also ignored.
Condensation Shock Waves Early, high Mach number, supersonic wind tunnels suffered from what is called a condensation shock wave (Wegener and Wu, 1977). These tunnels utilized ambient, often humid, air at their inlets. As the moist air isentropically cooled it would become supersaturated in the diverging section of the nozzle. At a sufficient level of supersaturation, the moisture would condense over a distance of about one centimeter with a detectable pressure change along the tunnel wall. The flow experiences heat addition in a condensation shock wave. If the flow consists of a pure vapor that becomes supersaturated, not all of the vapor can condense, and thermal choking may occur. Condensation shock waves in wind tunnels can be avoided by either drying the air before it enters the nozzle or by increasing the stagnation temperature.
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Shock Waves in Gases
209
Relaxation Process and the Downstream Equilibrium State An air/fog mixture may not be in equilibrium just downstream of a shock (Guha, 1992). Because of the extremely large difference in mass between a gas or vapor molecule and that of a liquid (or solid) droplet, the mass, momentum (or velocity), and energy (or temperature) of the droplets are unaltered when they pass through the gas-phase shock. For purposes of clarity, we again assume the shock is moving into a quiescent mixture consisting of a gas, vapor, and droplets. Downstream of the shock, the high-speed gas (plus vapor) flow accelerates the droplets, while the droplets provide a drag that decelerates the gas flow. Similarly, the high-temperature gas starts to vaporize the droplets, if they are liquid, while the droplets cool the gas. The preceding phenomenon, downstream of the shock, is called a relaxation process. When the upstream state is uniform and the shock strength is also uniform, a steady flow process can be modeled with a system of ordinary differential equations. Far downstream of the shock, the mixture achieves full thermodynamic and mechanical equilibrium. In this state, there is only one velocity and one temperature. The fluid may still involve droplets if the gas shock is not intense enough to fully vaporize them. The shock can be viewed two different ways. Our comment here is quite general; it is not restricted to a gas/liquid mixture. In the first case, there is a two-step process. The first step involves a discontinuous transition (i.e., a shock) that quickly equilibrates the translational and rotational modes of the gas (plus vapor). (In the droplet example, the vibrational modes would also be included, since the flow time for vibrational relaxation is generally much less than for a phase change.) The second step involves a relaxation process leading to a fully equilibrated state. Alternatively, the entire transition can be viewed as a single shock, which now may have a macroscopic thickness. Amazingly enough, the two final states are identical. Thus, one need only perform the relaxation process calculation if the process itself is the topic of interest. For instance, this is the way chemical rate coefficients are experimentally established with a shock tube. The overall equilibrium calculation is simplest, and should be used as a check on the two-step approach. Real gas behavior frequently involves some type of relaxation process just downstream of a discontinuous shock. This is especially true in the hightemperature regime. Nevertheless, real gas shock waves, without any relaxation process, do occur, as we now discuss.
Cryogenic Wind Tunnel A typical supersonic wind tunnel can properly simulate a freestream Mach number, but, because of the small size of the model, produces an unrealistically
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low characteristic Reynolds number. To increase the Reynolds number, thereby improving the viscous simulation, a cryogenic wind tunnel is sometimes utilized. Consequently, the state of the gas in the test section, usually N 2, just upstream of the model's bow shock is relatively close to the coexistence curve. The transition across the shock thus requires the use of Eqs. (3.1.21). Dense Gas Flow A different example is when a vapor possesses very large specific heat values near the critical temperature. In this circumstance, the molecule contains a large number of atoms and a large number of active vibrational modes. An example would be C16F26, whose critical point temperature is 701 K (Cramer, 1991). For these molecules, the critical temperature generally ranges from ~530-700 K. Two separate but related phenomena are involved. In the first, retrograde behavior produces a distorted shape for the coexistence curve in a T-s diagram: see Fig. 3.1.17. (For small molecules, the coexistence curve is bell shaped.) Condensation is now possible with an isentropic compression (Thompson and Lambrakis, 1973). States 1 and 2 indicate the possibility of a shock in which the upstream vapor is partially condensed downstream of the shock (Kobayashi et al., 1996). The parameter
r - 1 + - vT - (~-~~T) (Olna~ c~,
~,
(3.1.31)
Ov , I ;
is called the fundamental derivative of gas dynamics (Thompson and Lambrakis, 1973). For a perfect gas, F reduces to (7 + 1)/2. For a weak normal shock, the entropy jump across the shock As equals cF(Ap) 3, where c is a positive number and Ap is the pressure change. Thus, when F is positive, as is generally the case, the second law requires that Ap be positive, that is, the shock is compressive. On the other hand, a vapor with very large specific heats has a region adjacent to the coexistence curve where F is negative; see Fig. 3.1.18. This behavior raises the possibility of a single-phase rarefaction shock wave
/
critical point
71 existence curve
FIGURE 3.1.17 T-s diagram for a retrograde fluid.
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Shock Waves in Gases
P
critical point / / ~ -"-,\
FIGURE 3.1.18
F>O
p-v diagram showing the negative F region.
where Ap is negative. For further details, see the review articles by Cramer (1991) and Thompson (1991).
3 . 1 . 5 . 2 HIGH-TEMPERATURE PHENOMENA Electronic Excitation of Monatomic Species With increasing temperature, monatomic species first become electronically excited and then ionized. For example, the lowest excited electronic state, relative to the ground state, has a characteristic temperature of 2.766 x 10 4 K and 228.0 K for nitrogen and oxygen atoms, respectively. Thus, oxygen atoms are more easily excited electronically than are nitrogen atoms. These values, of course, are not applicable to N 2 and 02.
Vibrational Excitation For diatomic species, vibrational excitation is generally the first real gas effect to appear as the temperature increases. The effect, in terms of an increasing cv value, comes on gradually. For equilibrium air, for example, the ratio of specific heats 7 is 1.400 at 300, 1.376 at 600, and 1.344 at 900K. At higher temperatures, chemical reactions such as dissociation occur. At still higher temperatures, ionization occurs. In this circumstance, the gas is not necessarily a mixture of monatomic species. For instance, in high-temperature air NO is the first species to become ionized. Returning to the discussion of vibrational relaxation of a diatomic species, Eq. (3.1.3a) still holds, but cv is now temperature dependent. In this circumstance, set , 6 - 0, g v - 1, and c v - c ~ , 0, where c~,0 is given by Eq. (3.1.22). When Eq. (3.1.3a) applies and c~, = c v ( T ) the gas is referred to as ideal. All introductory thermodynamic texts have an extensive treatment of an ideal gas along with tabular results. The preceding discussion is based on equilibrium thermodynamics. As indicated earlier, when a shock passes through a gas there may be a non-
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equilibrium relaxation zone just downstream of the shock. For the case under discussion, the rotational and translational modes are equilibrated after only a few molecular collisions, whereas the vibrational mode may require many thousands of collisions. This stems from quantized vibrational states with a large energy gap between adjacent states. Only a very occasional collision has sufficient kinetic energy to overcome the barrier posed by the gap. Thus, there is a vibrational relaxation zone downstream of the shock that has a macroscopic thickness. The equilibrium vibration energy for a single harmonic oscillator mode is e~ib
RTv
eL/7-- 1
(3.1.32)
where T is the local translational and rotational temperature. A rate equation of the form
Devib = e*b - evib Dt z(T, p)
(3.1.33)
is then used to represent the relaxation process. In this relation, e~b is the actual (nonequilibrium) vibrational energy, D( )/Dt is the substantial derivative, and the relaxation time z has a Landau-Teller form (Vincenti and Kruger, 1965; Millikan and White, 1963). In a steady, one-dimensional flow, the substantial derivative reduces to wd( )/dx, where x is distance in the flow direction. In this case, Eq. (3.1.33) is an ordinary differential equation. More generally, it is a first-order partial differential equation. The relaxation zone terminates when the numerator on the right-side is effectively zero. This downstream state can be directly obtained by including the right-hand side of Eq. (3.1.32) in the enthalpy that is used in the (3.1.1) equations. Rayleigh Curve Theory A relaxation process may be endothermic or exothermic. Vibrational relaxation is endothermic, that is, it causes the gas temperature to decrease in the direction of flow. In either case, the flow in the relaxation zone may be viewed as a Rayleigh curve process (Emanuel, 1986). The principal assumptions are steady, inviscid flow in a constant area duct with, if necessary, averaged equilibrium inlet and exit conditions. Averaging is based on an evaluation of average mass, momentum, and energy fluxes. A perfect, or ideal, gas is not required, although often assumed. The length of the duct can be infinitesimal, when the relaxation process is assumed to be instantaneous, or it may be finite. The flow inside the duct may be three-dimensional and involve a complex, nonplanar shock system. It is a "black box" approach that connects steady, equilibrium end states.
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213
Shock Waves in Gases
Many flows with combustion, such as a detonation wave, satisfy the forementioned criteria. Because of its generality, a Rayleigh curve analysis represents a useful conceptual tool. For instance, it is a clear and simple way for evaluating thermal choking or if a second law violation has occurred. On occasion, Rayleigh curve considerations are utilized in the subsequent discussion. The Mach number just downstream of a steady normal shock is subsonic; hence, when the process is endothermic (exothermic) the Mach number decreases (increases) in the flow direction. For an oblique shock, use the Mach number component that is normal to the shock. If the process is sufficiently exothermic, as can happen with chemical reactions, the flow will thermally choke. This aspect is further discussed later.
Polyatomic Species With a triatomic, or larger polyatomic, the first real gas effect to be encountered, which may be present at room temperature, stems from the integral term in Eq. (3.1.25). We now have Eq. (3.1.21a) and cL, = q , ( T , v). These relations are to be used in conjunction with Eq. (3.1.1) for the jump conditions.
Chemical Reactions Chemical reactions can be triggered by a shock, thereby resulting in a downstream relaxation zone. The length of the zone depends on the strength of the shock, for example, whether it is normal or oblique. Once reactions start to occur, the composition of the gas becomes important. For fluid dynamics, this is most conveniently given in terms of moles of species i in a unit mass of fluid n i (Vincenti and Kruger, 1965). The molecular weight depends on this molemass ratio, and, consequently, the gas constant is no longer a constant. In addition, the enthalpy and entropy, per unit mass of fluid, are given by h = ~_. n i h i,
s = ~_. n i s i
i
i
(3.1.34a,b)
where the sum is over all species both reacting and inert in the mixture, and h i and s i are the molar enthalpy and entropy of pure species/. The h i and s i each must have an additive constant, evaluated at a common reference state, that accounts for the reactive energy of each species at that state. Equations (3.1.1) are unaltered, but the enthalpy is given by Eq. (3.1.34a). Equation (3.1.3a) often holds, but the gas constant now depends on n i. If the mole-mass ratios are constants in a flow, the flow is called chemically frozen. Ordinary aerodynamics treats air as a frozen flow. In an equilibrium flow, the n i change and are algebraic functions of T and p through the law of
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G. Emanuel
mass action. This circumstance prevails when the reactive process is much faster than fluid dynamic changes. For instance, chemical equilibrium conditions generally hold in the subsonic part of a nozzle, but become nonequilibrium at a location downstream of the throat. When the fluid dynamic change is rapid, nonequilibrium conditions prevail and the composition is governed by rate equations. This is the situation downstream of a shock wave, where the composition gradually relaxes to an equilibrium state. As previously mentioned, this equilibrium state can be evaluated by using Eqs. (3.1.1) and (3.1.3a) with R variable, Eq. (3.1.34a), and the law of mass action. The shock now equilibrates the translational mode, all internal modes (including the vibrational modes), and the composition. The result is an equilibrium shock wave. A very important distinction is whether the reactive system is endothermic (heat absorbing) or exothermic (heat producing). For example, the endothermic case corresponds to a shock that dissociates an otherwise inert gas, such as air. The dissociation occurs in a relaxation zone, just downstream of the shock. The shock must be relatively intense, as the temperature decreases in an endothermic relaxation zone. Dissociation preferentially occurs from the upper vibrational states, near the dissociation limit, and consequently vibrational relaxation may be coupled to the dissociation mechanism. In an exothermic relaxation zone, this generally does not occur (Asaba et al., 1965). A vibrational relaxation zone can still be used or, more simply, the vibrational modes can be equilibrated by the shock. In the nonequilibrium case, endothermic or exothermic, a series of rate or master equations are required that provide the rate of production, or depletion, of the individual species. Structurally, they are similar to Eq. (3.1.33), but must satisfy several conditions. The individual reactions, along with their forward and backward rate coefficients, must be realistic, that is, the process represented by the reaction should actually occur. Important chemical kinetic steps should not be overlooked. The ratio of the rate coefficients, for a given reaction, should equal the equilibrium constant for that reaction. Each reaction should be balanced, that is, the number of atoms of a given type on one side should equal the number of atoms of the same type on the other side. This condition ensures conservation of atomic type for a material volume. Finally, when the right-hand sides of all rate equations are set equal to zero, the resulting algebraic equations should uniquely yield the correct downstream equilibrium composition. For example, in a differential fluid particle the number of positive ions, weighted by the degree of ionization, should equal the corresponding number of negative ions, which may be largely electrons. This condition corresponds to electrostatic neutrality. Finally, if only the equilibrium state is of interest, then a linearly independent set of not necessarily kinetically realistic reactions is sufficient.
3.1 ShockWaves in Gases
215
Hereafter, the discussion focuses on exothermic combustion in a gaseous flow. If the process is only moderately exothermic, for example, because of a high percentage of diluent, then a flame, or deflagration wave, occurs. Across the wave, the pressure is nearly constant and the wave travels slowly at a subsonic speed governed by heat and mass diffusion. For sufficient exothermicity, a detonation wave develops. In simplest terms, this wave consists of a nonreacting shock that is driven by the chemical heat release just downstream of it. The shock and chemistry are closely coupled. A rather similar third process, shock-induced combustion is important, for example, for scramjet engines. In this circumstance, a premixed reactant mixture is raised to a temperature above its autoignition value by a shock and proceeds to react. Typically, the shock is generated by a ramp or wedge and not by the combustion process. In the detonation and shock-induced combustion cases, there may be a macroscopic induction length (or flow time), just downstream of the shock, before the highly exothermic reactions kick in (Eubank et al., 1981). The delay allows the density of the free radicals, such as OH, to accumulate. These are needed for the principal reactions to proceed with explosive speed. The rate of free radical production is temperature sensitive (Asaba et al., 1965). Hence, the length of the induction zone is sensitive to the post-shock temperature and, therefore, to the strength of the shock. The induction length may range from zero to many meters (Nicholls et al., 1963). The next three subsections treat planar detonation waves. The first two of these discuss normal detonation waves in terms of the two principal tools used for their studymnamely, the detonation tube and the shock tube. The third subsection deals with oblique waves. The fourth, and final, detonation wave subsection discusses nonplanar waves. This handbook considers shock waves in gases, liquids, and in solids. An interesting in-between topic combines both aspects. This is a detonation wave in a condensed phase, liquid or solid, explosive (Kapila et al., 1997). In this circumstance, a normal shock vaporizes the condensed phase. Just downstream of the shock, the density of the gas is quite large, that is, it is comparable to that of the condensed phase. The co-volume of the gas is now of major importance and must be included in the non-ideal thermal state equation.
Detonation Tube Flow This device consists of a rigid tube that is closed at one end and open at the other which is actually sealed with a mylar film. The tube is filled with a fuel/oxidizer mixture that is ignited at the closed end. Various measurements, such as density and pressure, are made near the open end (Kistiakowsky and Kydd, 1955, 1956).
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G. Emanuel
For purposes of simplicity, initial transients are ignored and the wave that propagates down the tube is assumed to be one-dimensional. When the wave is a deflagration, the gas between it and the closed end of the tube has zero speed. Consequently, the combustion zone acts as a piston on the upstream gas (Landau and Lifshitz, 1987). This piston action causes a weak normal shock to propagate into the unburned gas. In turn, the shock slightly preheats the gas, thereby accelerating the combustion process. Remember that the flow between the shock and deflagration is subsonic and thus they can influence each other. If the combustion, that is, heat addition, is sufficiently intense, then the deflagration will transition into a detonation wave (Lee, 1977). A detonation wave rapidly develops when the combustion is highly exothermic, as in a stoichiometric mixture of hydrogen and oxygen. As before, a normal detonation wave is analyzed in a fixed frame, as sketched in Fig. 3.1.19(a). States 1 and 2 are equilibrium states, and w 1 represents the propagation speed of the detonation into a quiescent gas. In contrast to the nonreactive case, where the w 1 speed is presumed known, the propagation speed of the detonation is unknown, unless it is experimentally established. From a theoretical viewpoint, an additional assumption is required in order to establish w 1. The first such assumption made over a century ago is that the Mach number, in a wave fixed frame and just downstream of the combustion zone, is unity, that is, w 2 -- a 2. The additional approximation of an instantaneous heat addition yields the Chapman-Jouguet (C-J) model. Despite its antiquity, it is still controversial (Takayama, 1992). The reason for this controversy will become evident in the subsequent discussion. It was noticed by Taylor (1949) that a C-J detonation was incompatible with a zero speed end wall condition in a detonation tube. This condition could only be met by inserting a rarefaction wave into the flow just downstream of the combustion zone [Fig. 3.1.19(b)]. The Mach number at the leading edge of the rarefaction is unity in the C-J model. Although the detonation wave in the figure is fixed and steady, the rarefaction, now called a Taylor wave, is unsteady with its width increasing linearly with time. shock ~ W1
combustion zone Wl
(a) FIGURE 3.1.19 wave.
,.~shock combustion zone
(b)
rarefaction wave
Schematic of a fixed detonation wave: (a) no Taylor wave; and (b) with a Taylor
3.1 ShockWaves in Gases
217
All detonation waves in a detonation tube require a Taylor wave. Its existence has been confirmed by Kistiakowsky and Kydd (1955). Moreover, their experiments show excellent agreement with the C-J, Taylor wave theory. The one exception to this occurs in the vicinity of the shock, where the macroscopic thickness of the combustion zone overlaps the leading edge region of the rarefaction. (If the overlap is ignored when modeling the wave, the C-J condition still applies.) The rarefaction is most intense in this region and overlap would tend to quench the more temperature dependent reactions. On the other hand, the reactions may be starting before the translational and internal modes are fully equilibrated by the shock (Kistiakowsky and Kidd, 1956), an effect that accelerates the kinetics. An approach in which the instantaneous heat release is replaced by a finite rate kinetics model is referred to as a Zel'dovich, von Neumann, DOring (ZND) model. The discontinuous shock, C-J condition, and Taylor wave remain unaltered. Under appropriate conditions, the reaction zone consists of an induction zone, often called an ignition delay region, followed by a heat release zone. The reaction zone is subsonic with the Mach number climbing to unity at the end of the heat release region. There is no induction zone in the Kistiakowsky and Kydd (1955) experiments. The initial mixture consisted of 20% argon, 40% acetylene, and 40% oxygen at 300 K and i atm. The estimated C-J temperature is 4350 K. (This system is more exothermic than an air/fuel mixture because of the nitrogen.) As previously discussed, the heat addition process whether instantaneous or over a macroscopic length can be viewed as a subsonic Rayleigh flow. This flow can only terminate at a thermally choked, M = 1 state because for a detonation tube, a Taylor wave is required and this wave must be sonic at its leading edge. Detonation waves are often referred to as underdriven or overdriven. In an underdriven (overdriven) steady detonation wave, the Mach number just downstream of the heat addition is supersonic (subsonic). The C-J condition represents the demarcation between these two modes of operation. In a shock fixed frame, the Mach number just downstream of the (nonreacting) normal shock is subsonic. Thus, underdriven operation, quite generally, is a violation of the second law. On the other hand, overdriven operation is not a violation and does occur, although not in a detonation tube. In this regard, experimental data need to be modeled to first fit the downstream part of the Taylor wave (Kistiakowasky and Kydd, 1955), because of the overlap of the kinetics region with the leading edge region of the Taylor wave. Shock Tube Flow Detonation waves are often studied with a shock tube (Lee, 1967). A perfect gas analysis can be performed that is similar to the shock tube discussion in Liepmann and Roshko (1957), but with instantaneous heat addition after the
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G. Emanuel
shock. When this is done for the incident flow (Ashford, 1994), it is found that the detonation wave may or may not require a Taylor wave (Yu et al., 1992). In either case, a global solution is required that encompasses the flows from region 1 to region 4, see Fig. 3.1.20(a), as is done in the nonreactive case. Heat addition occurs between states 2 and 3. In Fig. 3.1.20(a), M 3 < 1, the detonation wave is overdriven and there is no Taylor wave. Figure 3.1.20(b) sketches the C-J case with M 3 = 1 and a Taylor wave. Thus, the C-J condition does not always apply. When it does, however, we obtain 1
P3 --~(P2 + 1)
(3.1.35)
where Pi - Pi/Pl. In addition, T 2 / T 1 and T B / T 1 are constants. In other words, when only the initial pressure ratio P4 varies, the flow adjusts by changing the strength of the Taylor wave, with the strength of the detonation wave fixed. Whether or not a Taylor wave occurs depends on the choice for the highpressure gas, the pressure ratio P4, and the initial composition of the lowpressure gas. A critical value P4q can be determined, where
P4 > P4cJ, P4 < P4cJ,
overdriven detonation C-J condition with a Taylor wave
With T 1 -- T 4 -- 300 K and H2/O 2 as the low-pressure gas, Fig. 3.1.21 shows P4CJ v e r s u s the fuel/oxidizer equivalence ratio ~b (~b - 0 for pure O 2) for three different high-pressure gases. Suppose the high-pressure gas is He. When P4 is above the He curve, the detonation is overdriven. When P4 equals P4CJ, M3 -- 1 and the Taylor wave has zero strength. Finally, when P4 is below the curve, we have the C-J condition with a Taylor wave. With He and a pressure ratio P4 of about 100, the equivalence ratio has to exceed 0.2 in order to obtain a Taylor wave.
t rarefaction wave
contact I I
// //
combustion / zone
X (a)
FIGURE 3.1.20 Taylor wave.
/
/
'
wave
X (b)
Shock tube flow with a detonation wave with: (a) no Taylor wave; and (b) with a
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219
Shock Waves in Gases
105
. . . .
lg
N
Ar
10
2 10
10
1 -2 10 FIGURE 3.1.21 Critical initial gases (Ashford, 1994).
-1 10
P4/Pl ratio
1
3 10
vs the equivalence ratio for three high-pressure side
Experimentally observing a Taylor wave in a shock tube flow is not simple. At early times, because of the diaphragm rupture process, the relatively compact Taylor wave may not be discernible. At later times, when the detonation is close to an end wall, the gradients inside the spread-out Taylor wave are relatively weak, and the reflected rarefaction wave may have overtaken part of the wave. A shock tube can be used in conjunction with a wind tunnel in order to generate a high-enthalpy air flow in the test section of the wind tunnel. The device is called a shock tunnel. One particular device configuration (there are many other possibilities) utilizes a detonation wave to generate a highpressure, high-temperature gas in the high-pressure chamber of the shock tube (Lu et al., 1998). A relatively low-pressure hydrogen/oxygen mixture is initially contained in the driver section of the shock tube, while the driven section contains air as the test gas. Among several possibilities, spark ignition can be used to detonate the gas at either the closed end of the tube or the diaphragm end. In either case, the initial behavior is that of a detonation wave in a detonation tube. Consequently, the detonation wave is followed by a Taylor wave. An attached boundary layer develops, starting at the foot of the shock, as the incident shock travels down the tube. After the shock reflects from the end wall, there is an interaction between the reflected shock and the now upstream
220
G. Emanuel
boundary layer. For a nonreactive shock tunnel flow, the reflected shock causes the boundary layer to separate (Davies and Wilson, 1969). A 2-shock pattern with a triple point and slipstream is produced. This is an unsteady flow in that the triple point moves toward the centerline of the tube. This phenomenon imposes a severe run-time constraint on shock tunnel operation (Davies and Wilson, 1969). For some time, pseudosteady and unsteady shock wave reflection processes have been (optically) studied by placing a ramp or wedge near the end wall of the low-pressure section of a shock tube (Ben-Dor, 1992). The incident shock interacts with the wedge to form various patterns, such as a single Mach reflection pattern. The same type of experiment could be run with a detonable gas mixture in the low-pressure side of the tube. It would be interesting to compare the reflection process when the incident detonation is overdriven with when it is followed by a Taylor wave. This type of experiment would also be of value for the interpretation of transverse wave phenomena, as discussed in the nonplanar detonation wave subsection.
Plane, Oblique Detonation Wave As will be discussed in Section 3.1.6, the fl, 0 angles in Fig. 3.1.4 are related by an equation that also involves the ratio of specific heats and the upstream Mach number M 1. If the shock is replaced by a detonation wave, an additional parameter, the (instantaneous) heat release, just downstream of the shock, needs to be considered. Plots of fl versus 0, with a normalized heat addition q are commonplace (Powers and Stewart, 1992; Lasseigne and Hussaini, 1993). These plots are conceptually useful; for example, they indicate detachment and the different modes of operation. A typical sketch is shown in Fig. 3.1.22(a) (Ashford and Emanuel, 1994) in which the gas is perfect, 7 = 1.4, and M 1 = 5. The outermost curve, with no heat addition, shows the strong and weak (attached shock) solution branches, with detachment occurring when 0 is about 42 ~ With q > 0, the curve becomes a closed loop, where the upper branch is an overdriven detonation, and the weak solution branch has underdriven and overdriven segments. These are separated by the C-J condition, whose location is indicated by an asterisk. By using the component of the Mach number that is normal to the wave, we observe that the weak, underdriven solution violates the second law, in accordance with Rayleigh curve theory. (The actual Mach number downstream of the oblique shock is usually supersonic.) The resolution of this dilemma, first pointed out by Ashford and Emanuel (1994), is that a steady-flow counterpart to the unsteady Taylor wave is required. This is just a PrandtlMeyer expansion. In terms of Fig. 3.1.4, the detonation wave is immediately followed by a centered expansion. At the leading edge, the C-J condition
3.1 ShockWaves in Gases
221
~ u t i o n 6080f ~
q=lO/o~verW~rken }
,olu _ 0
(a)
0
10
20
30
e
40
50
80 60
1340 20 . . . .
0
i
. . . .
10
I
,
,
,
,
I
,
20 e 30
,
i
,
I
i
40
,
,
,
50
(b)
FIGURE 3.1.22 /~, 0 detonation plot for a perfect gas when V= 1.4 and M underdriven regime violates the second law; and (b) no violation occurs.
1 -"
5 when: (a) the
prevails for the normal component of the Mach number, and the expansion terminates when the velocity is parallel to the downstream wall. A revised fl, 0 plot is shown as Fig. 3.1.22(b). To the left of the asterisk, fl is constant as is the strength of the detonation. Any change in 0 is accommodated by a change in the angular spread of the Prandtl-Meyer expansion. At the asterisk, the expansion has zero strength, and it is finite to the left of the asterisk. The horizontal line actually continues into the region where 0 is negative. A Prandtl-Meyer expansion, as is the case for much of the discussion in these subsections, has important implications for scramjet engines that utilize an oblique detonation wave. The flow downstream of the wave must be supersonic; such engines are referred to as oblique detonation wave engines (ODWE) (Morrison, 1980; Menees et al., 1992). A propulsive force is generated by the high-pressure gas, downstream of the detonation, expanding against the wall of the vehicle inside a nozzle. Engine performance is, therefore, adversely affected if a Prandtl-Meyer expansion occurs first. An ODWE engine therefore, should avoid operation in the region to the left o the C-J state.
222
G. Emanuel
Cycle analysis indicates that any thermal engine operates most efficiently when the entropy production per unit mass of fluid is a minimum. For an ODWE, there is entropy production associated with the shock and with the combustion process, where these are additive. For complete combustion, the combustion entropy increase is roughly constant with wave angle. The shock wave entropy increase, however, is rapid with the wave angle. These considerations suggest that optimum performance of an ODWE is at the C-J state. Figure 3.1.23(a) shows an H2/air detonation plot when T 1 = 300 K and M 1 - - 5. The dashed line is for a perfect gas with 7 - 1.4, while the solid line, which is for a real gas, is based on an equilibrium code (Gordon and McBride, 1989) computation. The differences in the figure, between the two types of calculations, primarily stems from the temperature dependence of the specific heats rather then free radical (H, O, N, OH) formation (Ashford, 1994). For example, at a wave angle/3 of 66 ~ the percentage of molecules that are radicals is only 1.1% when the equivalence ratio is 0.4. For a given wedge angle, the difference in/~ and in the detachment value for 0 increases significantly with the equivalence ratio.
8O 6O 13 4o
o.o
I
2O
I !
Real gas i
i
i
i
,
2'o .....o
o
,
(a)
i
20'
60
80
,=
6O
o.ojj .o
13 40
20 Real gasgas -- .
0
.
.
.
I
20
,
,,
,
,
I
40
,
,
,
(b)
,
60
o FIGURE 3.1.23 ]3, 0 plot for H2/air detonation when T1 - 300 K under the perfect gas versus real gas assumptions: (a) M] = 5; and (b) M] -- 10 (Ashford, 1994).
3.1 ShockWaves in Gases
223
The corresponding M 1 -~ 10 case is shown in Fig. 3.1.23(b). The equilibrium real gas curves are now bunched together for/~ values in excess of about or ~ 35 ~ The dominant effect here stems from the formation of radicals. For instance, at a/~ value of 66 ~ the percentage of molecules that are radicals is 20% when the equivalence ratio is 0.5. Even with no combustion, the percentage is still 12%. For air combustion there is typically an induction zone, whether the shock is normal or oblique. In fact, because of the lower postshock temperature, an oblique shock has a longer induction zone. The induction zone is subsonic for a strong solution shock and generally supersonic for a weak solution shock. As noted earlier, when a shock is caused by a wedge or ramp, the process can be viewed as shock-induced combustion. If there is an induction zone where the flow is supersonic, then a detonation wave cannot form at the tip of the wedge (Li et al., 1994). Figure 3.1.24 indicates the combustion starting some distance downstream of the shock. In the induction zone, there is a gradual, nearly isothermal buildup of free radicals, such as OH. The heat addition in the deflagration region produces a compression wave that propagates from the wall along converging, left-running characteristics. (Note that the leading edge characteristic of the compression must intersect the shock.) The angular spread of the compression gradually decreases. When this wave runs into the shock, an oblique detonation, with a stronger shock and little or no induction zone, forms. The flow that passes through the deflagration wave is separated from that which passes through the detonation wave by a slipstream or by a vortical layer of finite thickness. It is worth noting that the figure and the Li et al. (1994) computation are for a weak, overdriven detonation, where there is no Prandtl-Meyer expansion just downstream of the oblique detonation wave. A calculation with an expansion would have to have conditions to the left of the C-J state, and would have to be global in the sense that the downstream wall tangency condition is imposed.
detonation
induction
deflagration ~ \ / zone ~ j 2
vortical layer ~ ~
shock ~ ~ ' - ' FIGURE 3.1.24 Formationof an oblique detonation wave when the flow is supersonic downstream of an attached oblique shock.
224
G. Emanuel
Nonplanar Detonation Waves Cylindrical and spherical detonation waves, both exploding and imploding, have been studied (Lee and Lee, 1965; Ahlborn and Hunt, 1969; Bach et al., 1969). When the shock is relatively strong, that is, its radius is small, the detonation is overdriven. As it weakens, the shock front and reaction front separate, there is an induction zone and good agreement is obtained with the C-J condition (Bach et al., 1969). The experiments of Ahlborn and Hunt (1969) and Lee and Lee (1965) are in good agreement with the ChesterChisnell-Whitham theory (Friedman, 1960). When a blunted projectile is fired into a combustible gas mixture, shockinduced combustion can occur. The strength of the curved detached bow shock varies, thereby resulting in a variation in the length of the induction zone. Depending on flow conditions, the flow just downstream of the shock may be steady or oscillatory (Lehr, 1972; Alpert and Toong, 1972; Matsuo and Fujiwara, 1993). There is another phenomenon of major importance associated with a nominally normal detonation wave with a finite length for the induction zone. In this circumstance, there is a subsonic region just downstream of the shock. Disturbances can propagate and amplify in this region (Clavin et al., 1997). The result is referred to as transverse waves (Strehlow and Fernandes, 1965) that propagate in a direction parallel to the shock. These waves were first observed by White (1961), who referred to their effect as turbulence. The shock wave has a dimpled appearance that is part of a cellular structure (Lee, 1984; Lefebvre et al., 1993). The nonplanar shock thus has a twodimensional, roughly periodic, structure that produces entropy and vorticity where its curvature is greatest. Moreover, the transverse waves are moving shock waves. A cell thus involves incident, reflected, and Mach stem shocks with triple points. For instance, Lee (1984) (courtesy of D.H. Edwards) shows an interferogram for a double Mach-reflection configuration. Lee also notes that the average velocity of the wave is close to the equilibrium C-J velocity, as would be expected from Rayleigh curve theory. It needs to be experimentally demonstrated that an oblique detonation wave with a supersonic induction zone does not experience these transverse waves. A detailed calculation for this type of flow by Li et al. (1994), does not in fact contain transverse waves. Plasma Shock Waves A gas that is partially or fully ionized is called a plasma. Generally, the gas is hot enough that all polyatomic molecules are fully dissociated. Thus, a plasma consists of a mixture of monatomic neutral species, their positive and negative ions, and free electrons. For a noble gas plasma, negative ions do not occur,
3.1
225
Shock Waves in Gases
and the plasma would consist of A, A +, and e, where A represents helium, neon, argon, xenon, or krypton. A fully ionized plasma would consist of A +, A ++ . . . . . and electrons. A relatively low-temperature plasma has a radiative field, but its intensity is too weak to alter the fluid dynamics. The book by Woods (1987) discusses magnetoplasma shock waves for tokamak application, including jump conditions, stability, and structure. A radiative field is not considered. Adamovich et al. (1998) discuss a normal shock wave that propagates into an already weakly ionized plasma. As they point out, the temperature is too low for the radiative intensity to have an impact on the fluid dynamics. A number of anomalous effects are discussed including shock acceleration, nonmonotonic variation of flow parameters downstream of the shock, shock weakening, and shock-wave splitting. The physics underlying this behavior is still under investigation. Above ~ 1 0 4 K, radiative effects are important. This type of plasma is typically studied downstream of the incident shock in a shock tube using a noble gas. Aside from fluid dynamics, this type of plasma physics involves quantum mechanics, radiative transport theory, and Maxwell's electromagnetic field theory. In supernova explosions and astrophysical jets, shock waves occur where the plasma also requires a relativistic treatment. For a plasma consisting of A, A +, and e, the ionization steps are A + A ~-A + + e 4- A
(3.1.36a)
A 4- e ~ A+ 4- 2e
(3.1.36b)
The first of the radiative processes, called Bremsstrahlung radiation, is A + 4- e --+ A + 4- e 4- hv
(3.1.37a)
A 4- e --+ A 4- e 4- hv
(3.1.37b)
where h and v are, respectively, Planck's constant and the frequency, and hv is the energy of the emitted photon. Neither reaction involves a quantized state; hence, the transitions are free-free, which produces continuum radiation. Freebound transitions, or radiative recombination, is given by A + 4- e ~ A(p) 4- hv
(3.1.38)
where A(p) represents an electronic ground or excited state of atom A. In this circumstance, the spectrum is continuous, with a sharp cutoff at the low frequency end. The A+ 4- e relative collisional energy must equal or exceed the energy required for the atom to be in the p-state. There may be several different p-states, each with its own cutoff frequency. Finally, bound-bound transitions are represented by A(p) --+ A(q) 4- hvpq
(3.1.39a)
A(p) 4- hvpq --~ A(q) 4- 2hvpq
(3.1.39b)
226
G. Emanuel
where p and q are different electronic states. The first process is spontaneous emission, while the second one is stimulated emission. The frequency of the emitted photon is Vpq,where the upper and lower states are quantized, and the line is broadened by natural (due to the uncertainty principle), collisional, and Stark processes. Reaction equations (3.1.37a)-(3.1.39b) are for emission; absorption is obtained by reversing the direction of the arrows. Early theoretical analysis assumed a steady, normal, high Mach number shock in a perfect gas. The temperature downstream of the shock is assumed to be high enough to cause a significant radiative heat flux, some of which crosses the shock. Ionization is not considered, and a gray gas approximation (the absorption coefficient is independent of frequency) is made, which greatly overestimates the absorption of the radiation by the gas upstream of the shock. The physical picture is that of a discontinuous translational shock at x -- 0, which is embedded in a thicker shock that extends from x - - o o to x - oo. The cold gas at x < 0 absorbs the radiation from the hot gas. The nature of the mathematical singularities at x -- 4-oo were investigated by Emanuel (1965). This type of flow is referred to as a radiation-resisted shock wave (Heaslet and Baldwin, 1963; Mitchner and Vinoker, 1963). If the upstream Mach number is not too large and the radiation is sufficiently intense, a translational shock need not occur, and the flow has a smooth transition from its upstream to its downstream state. Physically, this does not occur, since a modest, say, 4, Mach number shock does not produce a detectable radiative intensity. In the subsequent discussion of shock tube experiments, the radiative intensity is never sufficient to alter the flow upstream of the incident shock. Actually, the radiative intensity is large; it is the miniscule absorption properties of the cold, nonunionized gas, upstream of the shock, that prevents the interaction. A number of studies have been performed that involve the production and relaxation of an ionized plasma in a noble gas downstream of an incident shock in a shock tube flow (Petschek and Byron, 1957; Horn et al., 1967; Semerjian, 1972; Cambier, 1991). The fluid state is not uniform downstream of the ionizational relaxation region because of radiative cooling. The flow in a differential streamtube that is coincident with the centerline of the shock tube is compared with experimental observations. In a shock-fixed frame, this flow is steady and one-dimensional. The enthalpy for an A, A +, e mixture is used, where the internal energy of the electronically excited states is negligible and is generally ignored. The thermal state equation has the form p = (1 + a)pRT
(3.1.40)
where 0~is the degree of ionization, and, in local thermodynamic equilibrium, is provided by a Saha equation (Vincenti and Kruger, 1965). The formulation
3.1
227
Shock Waves in Gases
for the radiative cooling starts with a steady form of the radiative transfer equation OIv
l"Ji%- z - - P(Jv- ~%Iv)
(3.1.41)
where Iv is the radiative intensity at frequency v, lj are the direction cosines associated with the direction of radiative propagation, Jv is the mass emission coefficient, and ~cv is the mass absorption coefficient. The equations for mass and momentum are unaltered, while the energy equation is written as
d(1)
pw-~
h + ~ w2
- -Q
(3.1.42)
where Q is the radiant energy loss, per unit volume. This parameter stems from a solution of the radiative transfer equation and accounts for the geometry of the flow field and of the bounding walls. The loss term consists of optically thin continuum radiation, reactions (3.1.37) and (3.1.38), and optically thick line radiation. The line radiation tends to be self-absorbed in the plasma, although some reaches the wall. The optically thin radiation tends to propagate to the walls. Some of it passes through the cold gas that is upstream of the shock and also through the gas that is on the other side of the contact surface. The modeling is in good agreement with the measurements. On the other hand, the modeling of conditions behind the reflected shock are in poor agreement for the electron number density, temperature, and pressure with measurements (Bengtson et al., 1970; Phillips and Pugatshew, 1980). The reflected shock, of course, is now moving into a plasma. Moreover, the shock, boundary-layer interaction (Davies and Wilson, 1969), discussed in the shock tube subsection, may be occurring.
3.1.6
PERFECT
GAS SHOCK
WAVES
Steady and unsteady, normal and oblique, shock waves are the respective topics of the next two subsections. The flow across a differential surface area of a shock is considered; this is a local viewpoint. The third subsection presents the basic elements of characteristic theory as applied to a steady, two-dimensional or axisymmetric flow. On occasion, we have already had recourse to consequences of this theory. It will be applied in the fourth subsection, where steady and unsteady shock formation is discussed. Characteristic equations for unsteady, one-dimensional flow are provided in this subsection. The next subsection globally treats steady, two-dimensional or axisymmetric shocks.
228
G. Emanuel
Derivatives of flow parameters, such as the pressure and Mach number, are evaluated just downstream of the shock. In addition, a theory is developed for derivatives in the streamline and Mach line directions, for wave reflection from the downstream side of a curved shock, for shock produced vorticity, and for the singularity associated with a discontinuity in the curvature of the shock. The last subsection very briefly mentions several generalizations.
3.1.6.1
STEADY S H O C K WAVES
Normal Shock Waves This type of shock, in a perfect gas, is the simplest possible nontrivial case. Equations (3.1.1), (3.1.3), (3.1.4), and (3.1.6) apply. Conditions upstream and downstream of the shock are, respectively, denoted with subscripts 1 and 2, where M 1 is the shock Mach number. By straightforward algebraic manipulation, one can obtain the jump conditions across the shock in terms of the ratio of specific heats 7 and M 1. The contents in Table 3.1.1 are thereby obtained. In the table, X and Y are introduced for notational convenience. The subscript 0 denotes a stagnation quantity, see Eq, (3.1.7), while p, T, p, w, s and R, are, respectively, the pressure, temperature, density, flow speed, entropy, TABLE 3.1.1
Steady, N o r m a l S h o c k J u m p
C o n d i t i o n s for a Perfect G a s -- 1M2 ' X = 1 + - -7- ~
Y = 7M 2 - - 2 1/2
P2
2
Pl
7+1
-- = ~ Y 1
r-7 = P2
wl
Pl
w2
s2-sl
= -
7 + 1M2 2
X1
(Pf_22"} In ,,t, ol /
~'o2 rOl
----1
Pl
*1
3.1
229
Shock Waves in Gases
and gas constant. The downstream Mach number varies from unity, when M 1 -- 1, to [ ( 7 - 1)/(27)] 1/2, when M 1 --> oo. Thus, M 2 is subsonic but has a positive lower bound. The Mach number relation can be inverted to obtain the M 2 equation, whose similarity to the M 1 equation is obvious. The pressure, temperature and density ratios are unity when M 1 = 1. On the other hand, when M 1 --+ oo we have P2 Pl
27 M2 ' 7 4- 1
2T ,~ 27( 7 - 1) M2 '
P2 = 7 + 1
T1
Pl
(7 q- 1) 2
7-
(3.1.43)
1
While the pressure and temperature ratios become infinite with M1, the density ratio is finite. In fact, for air it is only 6. Conservation of energy dictates that TO is a constant across the shock. A shock, however, is an irreversible process, and therefore and s 2 > s 1 and Pol > P02- When M 1 -- 1, of course, P01 equals P02 and s 2 equals Sl. When M 1 ~ 00, we obtain (7 if-,!) (7+1)/(?-1) PO___2~, Pol
2 71/(~-1)(7--1)
(3.1.44)
Po2/Pol
and decreases as MF 5 when 7 = 1.4 and M 1 is large compared to unity. The last relation in the table is the Rayleigh-Pitot pressure formula. It is used for a Pitot tube that is aligned with a supersonic freestream.
Oblique Shock Waves Figures 3.1.3 and 3.1.4 are reconsidered. The normal velocity components and Mach numbers are provided by Eqs. (3.1.16) and (3.1.17) or by Table 3.1.2. Equation (3.1.16) in conjunction with u__!1= 7 + 1 u2
2
M2 sin2 fl
(3.1.45)
1 + 7 - 1 M2 sin2 fl ' 2
yields tan 0 -- cot fl
(+1
M~2 sin 2 fl _ 1
1+
)
(3.1.46)
7-------- sin 2fl M 2 2
which provides 0 in terms of 7, M1, and ft. Quite often it is more convenient to have fl in terms of 7, Ma, and 0. An explicit result (Emanuel, 2000, Appendix C) is obtained by first rewriting the preceding equation in terms of tan ft. This turns out to be a cubic equation,
230
G. Emanuel
TABLE 3.1.2 Steady, Oblique Shock Jump Conditions for a Perfect Gas
Mln "- M1sin 1~, tan 0 = cot ~
M2. = M2 sin(]~- 0)
MRsin2 1~-- 1 1 + ( ~ -1-- sin2~)M12
X=1+Y-1M2
Z=M 2 - 1
2
y+ 1 2 = [Z2 - 3X(l +----~M2) tan2 0] 1/2 A
Z3-9X X + - - ~ M 4 tan20
%= tan 1~=
23 Z + 22 cos[(41rg + cos-1 Z)/3] 3X tan0
= 0,
strong shock solution
= 1,
weak
shock solution
- 1 M2
[7+i( 1 0 d = tan-
I [[~+1 1_3 ? - M2 1/ MI~ 2 4 7~+4 _
1
--~+n
1/2
4 n)
)]1/2 M12-4+H 1 (2"/- 1)M12+ 4 - H
] ] 1/2
1/2
sin~ - O) cos ~ sin(~ - O) + - - ~ sin 0 P 0 2 (~~___~1)(?+I)/(y-1) P01
(M1 sin ]~)27/(?-1)
[1 q-~-~ (M1sin/J)2];'/()'-l)[),(M1sin/J)2 - ~-~11/(7-1)
with relatively simple coefficients, for which a closed form solution is available. This solution is provided in a computationally convenient form by the 2 . . . . . equations in Table 3.1.2. Figure 3.1.25 is a fl, 0 plot when 7 = 1.4. Observe that fl is double valued; on the left-hand side is the weak solution, on the right-hand side is the strong solution. These are separated by the curve labeled 0d, which stands for the
3.1
231
Shock Waves in Gases 50
Mz"3
2
1.5 !M2 =1
~I ~' i~
40 l
~
0.8
0161
./ ',~ , . .~,,: '
" I l/l/ttr ~
,'\~,~,'~,/ -I
:I
30
M,=IO~////
/
M,=2j~...!
':\~.~i [
20
I 10
:
, i~
:
,
ii :
7 /
:
:
20
30
40
:,-..,
.6, 1.4
,
",'~i!
"~'~,\.~-I,~
', \
"~,:xx~ ~ w ! .........
0
10
FIGURE 3.1.25
50
60
70
80
I
90
]7, ~ plot when 7 = 1.4.
detachment value of 0. Equations for 0 a and/7 a are listed in Table 3.1.2. Also shown in the table are equations for M 1 and M 2 in terms of 7,/7, and 0. An attached shock, such as the one in Fig. 3.1.4, is always a weak solution shock. A strong solution shock occurs in the blunted region of a detached shock (see Fig. 3.1.5(b)). This type of shock corresponds to a single M 1 curve in Fig. 3.1.25. At the location where the shock is normal to the freestream velocity, 17 is 90 ~ and 0 is 0 ~ As the M 1 curve is followed, a point is reached where the velocity turn angle is a m a x i m u m ~ n a m e l y 0 a. Shortly thereafter, M 2 is sonic. As we move farther along the shock, 17 decreases until is zero. At this condition, the shock is an accoustic wave and 17 is renamed the Mach angle, which is given by Eq. (3.1.20). From the figure, observe that is M 2 subsonic for a strong solution shock and that it is supersonic for a weak solution shock, except for a narrow region to the left of the detachment curve. When 0 < 0 a, the cubic equation for tan 17 has three real roots. Two of the roots correspond to the strong and weak solutions, the third is nonphysical. If the wall turn angle 0 in Fig. 3.1.4 increases, the shock will suddenly detach when 0 = 0 a and move upstream as a curved shock, which is normal to the wall, at the wall. Detachment also occurs if 0 is fixed, but M1 decreases. Physically, detachment happens when the velocity just downstream of an attached shock can no longer be parallel to the wall. A weak solution oblique shock causes a substantially lower loss in stagnation pressure (see Table 3.1.2) than its strong shock counterpart. It is for this reason that an oblique shock system is used in supersonic diffusers and inlets.
232
G. Emanuel
For instance, with ~ - 1.4 and M 1 - - 6 , a normal shock has Po2/Pol of 2.965 x 10 -2, whereas an oblique shock with, for example, 0 - 26.3 ~ has Po2/Pol equal to 2.415 x 1 0 -1" this is an order-of-magnitude difference.
3 . 1 . 6 . 2 UNSTEADY SHOCK WAVES Incident Shock Waves As sketched in Fig. 3.1.26, a constant speed normal piston generates a constant speed normal shock that propagates into a quiescent perfect gas. We readily obtain t
W 1 - - W s,
t
l
1
Wz -- Wp -- Ws -- W2
(3.1.47)
where a prime denotes an unsteady parameter. With Eqs. (3.1.9) and the Wl/W2 equation in Table 3.1.1, the connection between the piston speed and shock Mach n u m b e r is obtained !
!
Wp ____W___22__--~ ~ 2 Z 1 al al 7 q- 1 M 1
(3.1.48)
where Z equals M 2 - 1. Note that M1, the shock Mach number, can also be written as M's --(W's/a'l). The Mach numbers in the unsteady (or laboratory) frame are !
M] -- wl -- 0 al
Mt2_w'2_al w~_ --
t
(T~)1/2
a---7 - - a--~ a--7 piston
2
Z1
y@
1 M-1
(3.1.49a) (3.1.49b)
Z1
(X 1
y-~)1/2 1
W2
(1) (1)
FIGURE 3.1.26
fixed frame.
X
(a) (b) Constant speed piston that generates a shock: (a) x, t diagram; and (b) shock
3.1 ShockWaves in Gases
233
Because P01 = Pl, the pressures transform as P2
P2 --
P~
Pol - Pl P'2
2 =
~
Pl
(3.1.50a)
Y1
?q- 1
I
\2YI ]
-~I (? - 1)M2 +
2
(3.1.50b)
It is important to note that while P 2 - P~ and T2 --T~, P02 # P02 and To2 ~ T~2. This sharp difference between static and stagnation parameters is evident from the energy equation =
(3.1.51)
T1
T 1 q-
2
\ a 1/
which shows that T ' o 2 / T 1 is the sum of a static temperature increase plus an increase caused by the shock-induced flow speed w~. Another way of viewing this is to realize that the piston directly imparts energy and momentum to the flow, thereby increasing T'02 and P'o2. Table 3.1.3 summarizes the incident normal shock relations. Except for those equations that involve the piston speed, these are jump conditions that hold at an arbitrary instant of time. Therefore, they do not require a constant piston speed. The introduction of a constant speed for the piston is merely a simple way to motivate the TABLE 3.1.3 Incident Normal ShockJump Conditions for a Perfect Gas X = 1 + -Y-- - ~1M2'
y = ?M 2
2 ' w'p a1
M1 ~
7- 1
~ 4~,
1+
M,1 =0,
2
Z=M2-1
Z1
~ + 1M 1
1+
M,2 --
i~ +
1)w~j
Z1 (X1Y1) 1/2
P'2 P2 2 p'~ -- p~ -- y + 1 Y~
p;1
P0____2= (Y + 1~ 1/(~'-')[M12 (7- 1)M~+
Pl
Pl
~k-'~-I /
/ X1
rl Ti-
r;1 T1
-1,
1 5
M12
- ~ +----/ (~ -
1)M~ +
234
G. Emanuel
subsequent analysis. The M 1 equation is the inversion of Eq. (3.1.48). Equations for parameters that are not listed are easily derived from those that are tabulated. By way of illustration, suppose we are to estimate the piston speed for generating a Mach 2 shock in room temperature air. In this situation, y = 1.4, a 1 ~ 340 m/s, and Eq. (3.1.48) yields w pl - 425 m/s. This is faster than the maximum piston speed obtainable in a car engine. In a shock tube, however, much larger shock Mach numbers are generated, where the contact surface represents the constant speed piston. Nonconstant speed normal shocks are discussed in the last subsection, after characteristic theory is introduced.
Reflected Shock Waves A normal shock is moving to the left with a speed w 'r, which is negative (see Fig. 3.1.27). For clarity in the later discussion, it is convenient to use the notation as shown in the figure. For purposes of generality, the gas, both upstream and downstream of the shock, may be in motion. The connection between the steady and unsteady frames is 1~/2 - -
'
'
W2 -- Wr,
W 3 --
'
'
(3.1.52)
W 3 -- W r
where a caret is used to distinguish ~v2 from w 2. Introduce the Mach numbers !
Mr
w2 =_~, a2
l
p Wr M r-'-~, a2
!
f W2 M 2 =~, a2
t W3 M 3 :~, a3
M 3-
W3 a3
(3.1.53) where M r (not M'r) is the reflected shock Mach number, because the strength of the shock depends on M r . Equations (3.1.52) can be written as (3.1.54a)
M r -- M '2 + M 'r M 3 -- M 3 -I-
(3.1.54b)
M'r
A
w~ (3)
(2) (a)
wE
W 3
(3)
(2) (b)
FIGURE 3.1.27 Reflectedshock schematic when; (a) it is moving; and (b) fixed.
3.1
Shock Waves in Gases
235
Table 3.1.1 is used to eliminate M3 and T 2 / T 3, where M 1 in the table is replaced by M r. After M r is eliminated from Eq. (3.1.54b), the fundamental result is obtained 1 M~ =
-
7+ 1 7 - I(M ~ +M~r)2 2 M',(M~+ M ' r ) + 2
(3.1.55)
which provides the downstream Mach number, M~, in terms of M~ and M'r, which are presumed to be known. The preceding relation covers three special cases. In the first, the unsteady Mach number downstream of an incident shock is obtained, by setting M 'r --+ M 1,
M '2 - - O ,
M 3' --+ M '2
(3.1 56)
In the second case, the jump condition for a fixed normal shock is recovered by setting M~ --~ M 1,
M'r --0,
M~ --~ M2
(3.1.57)
The most interesting case corresponds to the reflected shock in a shock tube flow, where M~ --0. With the aid of Eq. (3.1.54a), this yields M2 r -
7+ 1 , ~ M 2 M
2
or
r -
1 -- 0
[ ( 211j2
M r _- 7 -+-1- ~ M 2, q- 1 +
7 +41 M ~
(3.1.58a)
(3.1.58b)
The flow upstream of the reflected shock corresponds to that downstream of the incident shock. Thus, w~ and M~ are the same for the incident and reflected flows. Consequently, Eq. (3.1.49b) can be used to eliminate M~, with the elegant result
(3.1.59 that directly provides M~ in terms of the incident shock Mach number. The reflected shock Mach number is bounded as 1 < M~ < -
-
- 2.646
(3.1.60)
7 - 1
where ? = 1.4 for the numerical value. One can show that M~ < M 1, where the disparity is especially large when M 1 is large. We also note that M~M2 = 1
236
G. Emanuel TABLE 3.1.4 Reflected Normal Shock Jump Conditions for a Perfect Gas
M~ = -~- ,
M,r = - - ,< -,
M,2 = -w~ -,
f12
a2
a2
M~ =
M~ -- _w3' , g/3
M3 - w~ ~/3
1 _ 7 +___~1M,r(M2 + M,.) + ~ - f (M~ + Mlr) 2 2 {[1 +7-1___~(M2, + M,r)2] [y(M~ + M,r)2 ~ J / 1 / 2
For a shock tube flow, where w~ = 0:
P3 P 0 3 Pl Pl T3
T~3
2 Y1 [ 3 7 - 1 M } _ ( 7 _ 1 ) ] 7 + 1X 1 2
( 2 )2 1 [
~][37-1M2-(y-1)]
where M2 is the Mach number downstream of the incident shock in an incident shock fixed frame. A few key equations are collected in Table 3.1.4.
Oblique Shock Waves A curved shock that is moving in an upstream direction is considered, as would be the case for the reflected shock in Fig. 3.1.14. Because the shock is devoid of mass, only its motion normal to itself is relevant. Moreover, as the shock jump conditions are algebraic, a knowledge of its past motion and of the unsteady flow are not required for the analysis at a given instant of time. As in the previous discussion, a differential element of shock surface area is utilized, and a frame fixed to the shock is used for its study. Thus, only the upstream and downstream relative velocities, at a given instant of time, are significant. As before, a velocity transformation is used that fixes the position of the shock at an arbitrary instant of time. It is analytically convenient to combine the moving and fixed frames in a single sketch; see Fig. 3.1.28. In the sketch, -" is the velocity of the shock. It is pictured as if w'~ < 0, actually, w's may be Ws -"1, which has an angle negative or positive. The upstream gas velocity is w -"2, which has an fl'relative to the shock. The unsteady downstream velocity is w -"1 that are normal and tangential to angle 0' relative to w -"1. The components of w the shock are u-"1 and 5' By adding -w~ to the velocity field, a steady frame is obtained for the shock that leaves all thermodynamic parameters unaltered. Because ws'' is normal to the shock, ~' = ~, but fl r fl' and 0 r 0'. The W1and w2 velocities are thus obtained, as shown in the figure.
3.1
237
Shock Waves in Gases
~'=~
wl
w2
-~'
~
s
~
fl
fl
_
v FIGURE 3.1.28
Schematic of the differential area of an unsteady, oblique shock wave.
The Mach n u m b e r s !
!
MS =-~,
Ws al
!
!
[
W1
I
M 1 = ~,
W2
W1
M2 --~,
al
M1 --~,
a2
al
M 2 ~-
W2 a2
(3.1.61)
' < 0) w h e n w 'S < 0 (w'S > 0). The shock paraare defined, where M'S > 0 (M S meters associated with its motion, orientation, and upstream state, ?, M'S, fl', and M~, are considered known. Our objective is to determine 0' and M~. It will first be necessary, however, to obtain fl and M 1. From the figure, observe that ]
!
tan fl - ul - w__________~ u
[
tan
fl' -- u-L1 /)
(3.1.62)
which yields t a n f l - - (1 + M - - - - ~ fl,) ~ tan fl' M~ sin Hence, fl >
(3.1.63)
fl' w h e n w'S is negative. From the relation w 'I sin fl' - w s = w 1 sin fl
(3.1.64)
MI" M1 = sin fl
(3.1.65a)
Mln -- M 1 sin fl' 4- M's
(3.1.65b)
we readily obtain
where
Thus, fl and M 1 are easily established; consequently, 0 and M2 are also available. The strength of the shock depends on Mln, and this parameter m u s t exceed unity if a shock is to exist.
238
G. Emanuel
From the relations l
l
l
tan(fl - O) -- u2 - w s,
tan(fl' - 0')
/)
u2
(3.1.66)
V
and
a__~l(T~) 1/2__y + l
M,f lsin ,/2 1/2
a2
2
(l+Y-lM~sin2fl)2
TM sin 2 fl (3.1.67)
we obtain sin(fi t -
O')[ tan(fl -
O)
Lta--~-~- o')
7+1 2 (
!
--1 M 2 M'sM1 sin fl
7 _ 1 M2sin2fl)l/2( 1+ 2
7M21sin2fl- ) 72- - 1
1/2
(3.1.68)
The only unknowns are M~ and 0', while t a n ( f l - 0)can be written as tan(fl - O) -
2 7 q- 1
1 + ~ - 1 M12sin2 fl 2 M12sin fl cos fl
(3.1.69)
A second equation involving M~ and 0' is required. It stems from Eq. (3.1.67), (3.1.70)
u 2 -- w~ sin(fl' - 0') - w's -- w 2 sin(fl - O) and
1 +7M2 sinZ(fl _ O) _
1M12sinzfl
2 7M12sin2ft. Y - 1
(3.1.71)
with the result
M~ sin(fl' - 0') M'~M1 sin fl
7+1
1/2
1+ x
2
M12sin2/3 2
+ 1
7M12sin2fl-
M~ sin 2 fl
M;M1sin fl
- 1
2 1
(3.1.72)
3.1
239
Shock Waves in Gases
Eliminate M~ from Eqs. (3.1.68) and (3.1.72), to obtain tan(fl' - 0') =
(3.1.73)
l + 7 - 1 M 2 s i n 2 fl 7 + 1 M'sM1 sin fl 2 2 7+ 1 M 2 sin/J cos
(3.1.74)
where 2
~_
Finally, we have
O' = f l ' - tan -1 ~
(3.1.75)
Once /3, M 1, and 0' are established, all other shock parameters can be evaluated. The foregoing analysis reduces to conventional, steady, oblique shock theory when M's is zero. It also has the proper limits when f l ' - 9 0 ~ (normal shock) and when Mln is unity.
3.1.6.3 C H A R A C T E R I S T I C T H E O R Y Only flow fields governed by hyperbolic partial differential equations can exhibit wave phenomena, such as expansion or compression waves. All unsteady, inviscid flows are hyperbolic, including a one-dimensional, subsonic flow. Steady, two- and three-dimensional, inviscid, supersonic flows are hyperbolic. Hyperbolic equations can be solved with the aid of characteristic theory. The computational application of the theory, which is not discussed, is called the method-of-characteristics (MOC). Many books treat characteristic theory, for example, works by Liepmann and Roshko (1957), Ernanuel (1986), and Landau and Lifschitz (1987). A brief overview is presented here that focuses on the basic concepts that underlie the theory. In an elliptic flow, a small disturbance steadily decays with distance in all directions from the source of the disturbance. On the other hand, in a hyperbolic flow, there are preferred directions along which the disturbance need not decay. Moreover, only the flow field downstream of these directions is altered by the disturbance; there is no effect on the upstream flow. In a twodimensional or axisymmetric flow, these directions are referred to as Mach lines or characteristics. In a three-dimensional flow, the Mach lines become Mach cones whose axis are streamlines. For purposes of simplicity and conciseness, the steady Euler equations for a two-dimensional (cr = 0) or axisymmetric (~r = 1) flow are considered. The development is somewhat different for an unsteady, one-dimensional flow (Rudinger, 1969). (Table 3.1.6, discussed later, provides the relevant characteristic equations.)
240
G. Emanuel
The key idea behind characteristic theory is to introduce a transformation of the independent variables such that the original partial differential equations become ordinary differential equations (ODEs) with respect to the new coordinates. These coordinates are called characteristics and they form a nonorthogonal grid. The corresponding ODEs are referred to as compatibility equations. In certain circumstances, the compatibility equations can be analytically integrated. Generally, the characteristic and compatibility equations form a coupled system of ODEs that is numerically solved using the MOC. The forementioned Euler equations can be written as
~(x~PWl-----~)+ O(x~Pw2~) = 0 ~X 1
(3.1.76a)
OX2
Owl Ow1 10p Wl ~ + w2 Tx 2 + ---pOx1 0
(3.1.76b)
=
Ow2
Ow2
1
1
Y P F- (w 2 + w 2 2 ) - h 0
(31.76d)
2
where x I and w I are associated with the freestream direction and x2 and w2 are in the transverse direction. When the flow is two-dimensional, Xl and x 2 are Cartesian coordinates. The unknown dependent variables are p, p, w 1, and w 2. A uniform supersonic freestream is assumed; thus, h 0 is a constant and the entire flow field is homenergetic. We assume the freestream encounters a shock wave. If the shock is planar or conical, the flow downstream of the shock is irrotational and homentropic. For a curved shock, the downstream flow is rotational and isentropic. The distinction between a planar or conical shock and a curved one is therefore important, and the analytical treatment is generally different. When writing Eqs. (3.1.76) in characteristic form, it is usually advantageous to also transform the dependent variables. (Over the years, many different choices for these variables have been utilized.) For instance, r, M, and P are used here, where P is In P0, and fl is the streamline angle with respect to x 1, as pictured in Fig. 3.1.29. The compatibility equation along the ~1 coordinate is
dO
dv
Z 1/2 dP
d~ 1
d~l
?M 2 d~ 1
o- sin 0
1 dx 1
cos 0 - sin 0x 2 d~ 1
--0
(3.1.77)
where v is the Prandtl-Meyer function; see Table 3.1.5. Observe that d{1 cancels, and that Z a/2 is complex if M < 1. Even after the d~l cancellation, Eq. (3.1.77) only holds along {2 = constant curves. The full set of characteristic equations is given in the table. The C+ compatibility equation is just Eq.
3.1
241
Shock Waves in Gases
X2
~
4:o, Co, streamline
/
-~ G , C_ ,right-running characteristic
X1 FIGURE 3.1.29
Characteristic coordinates.
(3.1.77) and holds along those characteristics whose dx2/dx I equation stems from
OX2/O~ 1
dX 2
OXl/O~l dXl ~2
-- t a n ( # + O)
(3.1.78a)
TABLE 3.1.5 Characteristic Equations for Steady, Two-Dimensional or Axisymmetric Flow of a Perfect Gas Z 1/2
o
sin 0
dx--L= 0 ]
d O - d r +-~-~dP + Z1/2 cosO _ sin 0 x2 dx__~2= Z 1/2 sin 0 + cos 0 dx 1 Z 1/2 cos 0 - sin 0
C+
I
P = constant |
dx2 _
dx--T -
tan
0
J Co
Z 1/2 a sin 0 dx---L= 0 ] dO + dv - - ~ dP - sin 0 + Z 1/2 cos 0 x 2 C_ dx 2 _ Z 1/2 sin 0 - cos 0 dx I
sin 0 + Z 1/2 cos 0
/
where P -- In P0 Z=M2-1
v +1 1J2 tan1 ( )1j2 zlj2] tanlZXJ2 Z1/2 dr=
MX
dM
242
G. Ernanuel
where # is the Mach angle. The C o characteristics are streamlines and a constant value for P means the flow is isentropic, not homentropic. The C+ equations are similar to the C ones, where the characteristic equation is based on
3x2/0~2 = dx---~2] - tan(p - O) OX1/a~ 2 dx I ~,
(3.1.78b)
If the flow is rotational, then P varies from streamline to streamline, and all six equations in Table 3.1.5 are required. This is the case even if the flow is two-dimensional and r~ = 0. When the flow is irrotational, P is the same constant for each streamline, and the Co equations are unnecessary. Moreover, if the flow is also two-dimensional, the C+ compatibility equations integrate to 0 - v -- J+,
C+
(3.1.79a)
0 + v - J_,
C_
(3.1.79b)
where the constants of integration J+ are called the Riemann invariants. For a rotational flow, P is a Riemann invariant that is constant along streamline characteristics. Now, J+ are constants of integration for the C+ compatibility equations in Table 3.1.5. The P and J+ invariants can change from one characteristic to the next. If one of the invariants, say J+, originates in a uniform flow region, then the J+ invariants from this region have the same value.
3.1.6.4
SHOCK FORMATION
This subsection briefly addresses why shocks form and how they strengthen. Formation occurs when nonstreamline characteristics of the same family attempt to overlap. Overlap would result in a multivalued solution, which is physically unrealistic. Instead, a very weak shock starts to form along the characteristic that is farthest upstream. It then gradually strengthens as downstream characteristics merge with it. In a steady flow, the merging process results in a curved shock. Downstream of this shock, the flow is isentropic, not homentropic, and vortical. This is the way shocks form and develop internally in the jet from an underexpanded nozzle. For an unsteady, normal shock, the downstream flow is isentropic, that is, different particle paths have different entropy values. This discussion is not appropriate to a blunt body shock. In the nose region, downstream of the shock, the flow is elliptic, not hyperbolic. In this case, the shock owes its existence to the velocity tangency condition on the body.
3.1
243
Shock Waves in Gases
Steady Flow Infinitesimal disturbances propagate outward C+ along characteristics starting at the wall, as sketched in Fig. 3.1.30(a). Here, the shock is caused by a sharp wall turn. As the supersonic flows in regions 1 and 2 are uniform, the constant slopes of the C+ characteristics are -- tan # 1 ,
--
1
tan(p 2 + 0)
(3.1.80)
2
With M 1 > M 2 > 1 and 0 > 0, it can be seen that/*2 + 0 > fl > Pl- Thus, the C+ characteristics, on both sides of the shock, run into it. Sharp wall turns, however, do not occur because of the smoothing effect of an attached, or detached, boundary layer. In effect, the wall compression is gradual, as sketched in Fig. 3.1.30(b). A compression wave propagates outward along the C+ characteristics. As indicated by these equations, the slope of the characteristics gradually increase along the wall in the downstream direction. The crossing of characteristics is avoided by having them merge along the shock.
Unsteady Flow The piston in Fig. 3.1.26 accelerates impulsively to a constant speed. A more realistic situation is sketched in Fig. 3.1.31, where the piston gradually accelerates from a zero speed. At the face of the piston, the gas and piston speeds are the same. Region 1 is a quiescent gas that is separated from a nonuniform flow region by a gradually strengthening shock wave. The governing characteristic equations (Rudinger, 1969), are listed in Table 3.1.6. A particle path is shown in Fig. 3.1.31 as a C O characteristic. The entropy in region 1 is a constant, but varies from particle path to particle path in region 2. shock
x 2
shock ,
l l
l
l
r
l
l
l
l
X1
l
(a) FIGURE 3.1.30
t ..... r~, (b)
Steady shock formation: (a) at the wall; and (b) gradual.
244
G. Emanuel TABLE 3.1.6 Characteristic Equations for Unsteady, One-Dimensional Flow of a Perfect Gas
dt
a+ w
=a---d-~
dx dt
C+
--=w+a
s = constant | dx J Co
&
=
w
a_(2 dt
- 1
dx
&
C_
at = w-a
where d+ 0 0 dt - Ot + (w + a) d_ 0 O
Thus, C O and C+ are characteristics. Conditions just downstream of the shock affect the flow at the face of the piston, along C_ characteristics. In other words, the infinitesimal increase in shock strength is transmitted to the gas at the face of the piston along a C_ characteristic, where, for example, it results in an infinitesimal increase in the speed of sound. On the other hand, the piston acceleration strengthens the shock along the C+ characteristics. For an arbitrary piston speed W p ( t ) a numerical solution for region 2 is required, which utilizes the MOC in conjunction with algebraic shock jump
piston
shock
.J
/
4y" ~~/ I
(1)
X
c FIGURE 3.1.31
Shock generated by a variable speed piston.
3.1
245
Shock Waves in Gases
conditions. Approximate analytical treatments can be found in Friedrichs (1948) (see also Lax, 1948 and Sharma et al., 1987).
3.1.6.5
STEADY,T w o - D I M E N S I O N A L
OR
AXISYMMETRIC S H O C K WAVES Aside from a perfect gas, a steady flow, without swirl, and with a uniform, supersonic freestream is assumed. The shape of the shock, which is prescribed, y =f(x)
(3.1.81)
is either two-dimensional (or = 0) or axisymmetric (or = 1). The coordinate system is sketched in Fig. 3.1.32, where y is the radial coordinate when the shock is axisymmetric. Arc length coordinates along the C+, C o, and C_ characteristics are denoted as ~+, ~o, and ~_, respectively. The velocity components u and v are tangential and normal to the shock, and g is the Mach angle. The velocity itself is tangent to (o and has an angle 0 with respect to the x-axis. In contrast to a more general analysis, the foregoing assumptions allow explicit, closed form results to be obtained. These include the derivatives of flow parameters that are normal and tangential to the shock. With these derivatives available, derivatives in any downstream direction can be found. Much of the material in the following subsections is based on Emanuel and Liu (1988) and Emanuel (2000).
,
u
/ / / ~ (
/r
o
, /s~ c ~t r _____-~/ ~
I 'r
eamline
"~-
Y
l FIGURE 3.1.32
_.shock
X
Characteristic directions downstream of a shock.
246
G. Emanuel
Jump Conditions The wave angle is
fl--tan-l(-~)
(3.1.82)
and its derivative with respect to arc length along the shock s is
fl,
dfl
= d s = c~
(3 1.83)
d2f
fl -dx -~
The longitudinal curvature of the shock then is ~cl
= -fl'
(3.1.84)
For a convex-shaped shock, as sketched in Fig. 3.1.32, fit is negative and the curvature is positive. The j u m p conditions initially reduce to cos fl
(3.1.85a)
W2 - - W1 COS(fl -- O)
tan fl P2 - Pl tan(fl - O)
(3.1.85b)
sin fl sin 0 P2
-
(3.1.85c)
Pl + (pW2)l c o s ( 3 - O)
1 w21sin 0 sin(2fl - O) h2 - hi + ~
(3.1.85d)
cos2(fl - O)
For notational convenience we introduce m
--
M21,
w
--
(M 1 sin fl)2
X '
--
~-1
1 4- 7 - 1 w 2
Y - -
'
?w
2 (3.1.86)
where w should not be confused with wl or w2. The tangential and normal velocity components are related to wl and w 2 with ul = u2 = wl cos 3 = w2 c o s ( 3 - 0)
v 1 = w 1 sin fl,
v2 = w 2 sin(fl - 0)
(3.1.87a) (3.1.87b)
Table 3.1.7 summarizes the jump conditions in a standard, computationally convenient format. For later use, several M 2 relations are provided as are s i n ( / / - 0) and c o s ( f l - 0). Note that cr does not appear, even though the equations hold for axisymmetric shocks. The M22 equation is readily obtained from Eq. (3.1.71). Other stagnation parameter jump conditions can be obtained by remembering that T02 = T01.
3.1
247
S h o c k W a v e s in Gases
TABLE 3.1.7
J u m p C o n d i t i o n s for a T w o - D i m e n s i o n a l or A x i s y m m e t r i c S h o c k Wave m----M12,
w=msin
y_?w
2fl,
- -7--~1,
X=I+Y-1
w,
[~,.pw2jl = YPl m
/.l2 "-- W1 COS fl 2
X
V 2 --" 7 - - - ~
P2
Wl ~
m sin fl
2 Y ~'(7 + 1) "rW2"l m ?+1
M ~- - 1 = ~' + 1 (~ + 1)row 2
+ 2 + ( , / - 3)w - 2,/w2
4
XY
2
PO,2 --
cos fl - O)
---=--7 + 1
= z
(
]~(]~ _}_ 1) (/014/2)1 \ tan 0 --
Tangential
W
1
7 - 1 ~ 2 ~ ;,/(;.-1) y 1 + ----~'"2] m w-1
tan fl (~ + 1)m/2 + 1 - w
7 + 1 m sin fl cos
2
x
x
Derivatives
The arc lengths s and n, along and normal to the shock are introduced. The coordinate n is zero at the shock, is positive downstream of it, and is welldefined only at the shock. As in boundary-layer theory, this causes no difficulty. The tangential derivatives are obtained by differentiating the equations in Table 3.1.7 with respect to s. The results are listed in Table 3.1.8. Note that all the derivatives, including dO/ds, are proportional to fit.
Normal
Derivatives
These derivatives require the steady Euler equations in scalar form, where one coordinate is along the shock and the other is normal to it. These equations can
248 TABLE 3.1.8
G. Emanuel Tangential Derivatives for a Two-Dimensional or Axisymmetric Shock Wave
Ou) 2-- --W1fl, sin fl -~S -
2 ( 7-1 ?+lWl 1---~w
_
_
_
)fl'c~ w
2
(~s) = ~-~ (Pw2)lfl'sin fl c~ fl 2
3~'~--(? +I)pl
fl'm sin fl cos fl X2
2
(3M2~ _
(7+1) 2 ( 17+ - 1 ) m (14-?w 2) fl'msinflcosfl
\--s
(Opo)
-~s 2=
--X-
(
-y + 1
(pw2), 1 +
7+i ~m(1
dO ds
2
2
(xY?
"~--1 )'/(?-l)fl'(W--1)2 -~
M2
mX tanfl
+ w) + 1 - 2 w - 7w2
2
fl,
w+l-2w-Tw 2 1 (OM2~ 2M2(Mi - 1) 1/2 k, 3s ,]2
(7+1) l + - - - - ~ m
(0~/) ;
2-----
be written as (see Emanuel, 2000, Chapter 6)
O(.u)
3---~ + 3(pv) + fl'pv - rr___p_p (u sin fl - v cos fl) = 0 On 3u
u
y
+
3v u
u~4-v
3u
3v +
13p
+ fl'uv + -;
= o
10p - fl'u + - o,, = o
(3.1.88a) (3.1.88b) (3.1.88c)
P
7-1p+~(u 24-v2) - 0
(3.1.88d)
For notational simplicity, a subscript 2 is suppressed, although these equations are valid only just downstream of the shock. (This is evident from the appearance of ,6', which is defined only at the shock.) Remember that u and v are the velocity components tangential and normal to the shock. Values for u, v, p, and p are known from Table 3.1.7, while their s derivatives are known from Table 3.1.8. For instance, for the 3(pu)/3s term in continuity,
3.1
249
Shock Waves in Gases TABLE 3.1.9
Normal Derivatives for a Two-Dimensional or Axisymmetric Shock Wave 3u)
gift' cosfl
2= w l g 2 - - y - - ~ m ( l + 3 w )
(__~) =
1
m(w--1)sinfl-
(2)2
(mg5 --]-g6) #,
2-- Pl(mg3 + g4)
2-
~
crY w my l ~tan ]3
o'Ycos#
--FPl y-----~
xy2
L 4x2( w - 1)
(--~)2 = (pw2)i ( 1 + @M22)7'/(}'-1)[ ( w - xl) c~
2/J'
The gi(?, w) functions are: 1
m = v - ~ [-(~ + 5) + (3 - ~)~1 2
g2 = (y 4- 1]2 [(y -- 1) -- 2(? -- 1)w + (5y + 3)w 2] J
2+1 g3= ~ [ 2 ( y + 2 ) + ( 3 - 7 ) w + 3 ( y - 1 ) w
2]
1
g4 - - ~ [ 2 - - ( 2 2 + ~ + 6 ) w + ( 7 2 - 4 2 + 1 ) w 2 - ( 7 -
1)(27+1)w 3]
g5 = -(? - 1) + (? + 5)w + 2(2 7 - 1)w 2 2
-- I) + 2(27 -- l ) w + (~2 _ 72 - 2 ) w 2 - (372 - l ) w 3]
g6 -- ~ [ - - ( 7
y+l g7 = - - ~ [ 4 -- (y -- 1)(~ + 3)w +
g8 =
()~2 jr_ 18~ --
3)w 2 -- 4y(2 -- ~)w 31
--2(7 - i) + 2(2 - 1)(3 - 7)w + (972 -- 1 4 ~ Jr- 1)w 2 + (73 -- 1722 -- y + 1)w3 + 7(--322 + 4~ + 3)w 4
we use
o(p.)
ou
op
& + i
o--7-= P~ + " G - k 2 x
Pl
x)
( - w 1 sinfl) + (w 1 cosfl)
[(? + l)p I fl'm sinX 2fl cos fl]]
= (2 + 1)(PW)l = (? + l)(PW)l
fl'sinfl(
1
)
fl'sinfl( 3 ?-1 X2 m - ~ w - --~
) w2
X2
--~ wX -4- m cos 2/3
(3.1.89)
250
G. Emanuel
Consequently, Eqs. (3.1.88) are four, linear, inhomogeneous, algebraic equations for &t/0n, Or~on, 3p/on, and 3p/on. The solution, determined with Cramer's rule, is given in Table 3.1.9. Each term on the fight-hand side is proportional to either fl' (i.e., the shock curvature) or rr/y, which stems from continuity. The gi a r e functions of ~ and w and are listed at the bottom of the table. For an axisymmetric, blunt-body shock, both cos fl and y are zero where the shock is normal to the freestream. The indeterminant ratio cos fl/y, which appears in the table, equals the curvature of the normal part of the shock in the limit of y going to zero. In the w = 1 limit, the shock becomes a Mach wave. In this limit, some of the terms in the table are indeterminate, since fl' and w - 1 go to zero. A more interesting limit is the hypersonic one. If the shock wave is normal, or nearly normal, to the upstream flow, the rightmost terms in the gi dominate, and the X, Y, and w - 1 factors simplify in an obvious manner. Another hypersonic limit is for slender bodies, when w is of order unity, and the gi, X, Y, and w - 1 factors do not simplify. Nevertheless, the equations do simplify because of the presence of m, which approaches infinity.
Derivatives Along Streamlines and Mach Lines The differential operators along streamlines and Mach lines are often useful, for example, in checking MOC codes. In view of Fig. 3.1.32, we have
(3~(o)--cos(fl-O), 2
(3-~o) = sin(fl - O)
(3.1.90)
where f l - 0 is the acute angle between a streamline and the shock wave. Hence, the streamline derivative is
O 2--
05 ~-'554-0T0~0 Chl 0 2
--
COS(fl-O ) ~ 24- sin(fl -
0)
2
(3.1.91)
where the sine and cosine coefficients are provided in Table 3.1.7. Hereafter, subscript 2 is understood but may not be shown. The sines and cosines of p + fl - 0 and p - fl + 0 are needed for the Mach line directions. These are the angles that the ~_ and ~+ characteristics have
3.1 ShockWavesin Gases
251
with respect to the shock. As an example, one of the sines is evaluated as sin(/z + fl - 0) -- sin(fl - 0) cos/z + cos(fl - 0) sin tt --
[
1+
32+ l m s i n f l c ~
)
2
x I'(M22-1)1/2M 2 q- TYq-lrnsinflc~ (M22 -- 1)1/2 q- ? +
2
/vX 1/2
1 m sin fl cos fl X
(M22 _ 1,1/2) q Y + 1 msin fl cos fl
--~X)
2
l+(?+lmsinflc~ 2 2X
X
(3.1.92a)
where it is not convenient to eliminate (M 2 - 1) 1/2. In a similar manner, the following relations are obtained:
COS(#--]-fl- O)- (X) 1/2
? + 1 m sin fl cos fl gM-2 ~ 2 -- 1~1/2, 2 X
sin(/2- fl + 0)= ( y ) 1/2
2+lmsinflcosfl_, 2 X
1
l+(?+lmsinflc~ 2 2X
(3.1.92b)
e~m2-1)1/2 (3.1.92c)
1 4 - ( 7 + l2m s i n f2i c ~ X
COS(fl- fl--~-O)- (y) 1/2
? + 1 rosin fl cos fl tM2 ~, 2 - 1~1/2j + 1 2 X
1+(2 -k2 l m2sinXfic~ fl)
(3.1.92d)
For the right-running characteristic direction, just downstream of the shock, we now have as
= cos(/, + fl - 0),
On
= sin(/, + fl - 0)
(3.1.93)
252
G. Emanuel
and as 3
On 3
7+1 m sin fl cos fl)2] -1
( Y ) 1/2 I --
lq
X
2
X
.7 + 1 m sin fl cos fl (M 2 2 X
+[(M22
1)1/2
1/2
7 + l m s i n Xfl c o s fl -~n3}
(3.1.94a)
For the left-running characteristic direction, we similarly obtain O Os O On O = t O~+ O~+ as O~+ On 8 3 = - cos(~ - fl + 0 ) ~ + sin(~ - fl + 0)~n ---(Y)
1+
2 +[(M2_1)1/2
7+21msinflc~
X
3s 7 + l m s i n Xfl c o s fi -~n3}
(3.1.94b)
While Eqs. (3.1.91) and (3.1.94) are not simple, they nevertheless provide explicit relations for the derivatives of interest, and are readily evaluated with a computer.
Wave Reflection From a Shock Wave Left-running (C+) Mach lines reflect from the downstream side of a shock as a wave consisting of right-running ( C ) Mach lines. This is the case in the upper half-plane sketched in Fig. 3.1.32; for the lower half-plane, the two families are reversed. The reflected wave is an expansion wave if the Mach lines diverge from each other when they are traced in the downstream flow direction. If the lines converge, the wave is compressive. Moreover, converging lines that start to overlap must be replaced with a developing shock wave. Thus, an internal shock may form in a supersonic flow containing converging Mach lines of the same family. This happens in an underexpanded jet, where flow conditions just upstream of the internal shock are nonuniform.
3.1 ShockWaves in Gases
253
It is also useful to know whether or not the reflected wave is compressive or expansive. This wave impinges on the body and alters the boundary layer. For instance, if the wave is compressive, it may cause the boundary layer to thicken and perhaps separate. On the other hand, if the wave is expansive, it would cause the flow in a laminar boundary layer to accelerate and thereby delay transition. The slope of the C_ Mach lines, just downstream of the shock, i s / z - 0 relative to the x-axis. As we travel outward along the shock, these Mach lines tend to converge when/z - 0 increases. Thus, the reflected wave is compressive (expansive) if the positive angle ~ - 0 increases (decreases). The C_ wave is a compression if a ( ~ - 0)
ds
> 0
or
a ( ~ - 0)
d]3
< 0
(3.1.95)
where the second form is analytically convenient. It is hopelessly complicated to attempt to derive an equation for d(lz - O)/d[3 without the assistance of the foregoing theory. Start with d# 1 = dM M(M 2 - 1) 1/2
(3.1.96)
and write
dla dM dO d ( # - O) dM ds ds d-------~ d__~ ds
1[ 1 dM dO] fl' M ( M 2 - 1) 1/2 ds F dss
(3.1.97)
where again a subscript 2 is suppressed. With the aid of Tables 3.1.7 and 3.1.8, we have a ( , - 0)
(7 4- 1)3/2(1 4- 7- 1 m)(1 4- 7w2)m sin ~ cos fl \ 2
[ t
2XS/2Y 1/2 14-
7 42- 1 m sin X fl cos
_
[(7 4- 1)mw + 2 + (7 - 3)w - 27w2] 1/2
7 +____11m(1 + w) + 1 - 2w - 7w 2 2 (7 4. 1 ) ( l q - ?~+ 1m ) w + l - 2 w - y w 4
(3.1.98) 2
This result does not depend on the nature of the incoming C+ wave, which may be expansive or compressive. Moreover, it is independent of whether or
254
G. Emanuel
not the flow is two-dimensional or axisymmetric, and is also independent of the local shock wave curvature -fl'. Although complicated, the right-hand side depends only on 7, M1, and fl; hence, the influence of the incident wave is limited to its effect on the wave angle ft. For a blunt-body shock (see Fig. 3.1.8), the foregoing analysis holds only when fl* > fl, where fl* is the wave angle when M 2 is sonic. A relation for fl* is obtained by setting M 2 - 1 equal to zero in Table 3.1.7. This yields (7 + 1)row* + 2 + (7 - 3)w* - 27(w*) 2 - 0
(3.1.99)
which also means that d(p- fl)/dfl is positively infinite at the sonic point. Thus, near the sonic point the reflected wave is expansive. The preceding relation becomes 1 7(M1 sin ft.)4 _ : [ y _ 3 4- (y + 1)M21](M1 sin ft.)2 _ 1 -- 0
(3.1.100a)
which produces sin fl* -- ( 47M21 1 {7 - 3 + (7 + 1) M2 X
1/2
+ [1772 - 67 + 9 + 2(7 - 3)(7 + 1)M12 + (7 + 1)2M~111/2}) (3.1.100b) With 7 - 1.4 and 1.5 _< M 1 _< 8, fl* is in the 61~ ~ range. As is generally the case, let the incoming wave be an expansion, thereby weakening the shock. Detailed calculations with ~ - 1.4 show that the reflected wave is expansive for all fl values when M 1 < 1.59. For larger M 1 values, there is a range of fl values
Pl < f i < -
39 ~
M 1 -- 1.59
38~ 36 ~ 35 ~
2<M 1<4 M 1 -- 6 M 1 -- 8
(3.1.101)
for which the reflected wave is compressive. The compressive fl region starts at M 1 -- 1.59, where #(1.59) - 39 ~ For still larger fl values, the reflected wave is expansive. Hence, for freestream Mach numbers in excess of 1.59, both types of reflection processes may be present. The expansive reflection starts at the sonic point, while the compressive reflection would occur farther downstream, where the shock is relatively weak. The hypersonic small disturbance theory limit is defined by M 1 --> oo,
K~ -- M 1 sin fl -- O(1)
(3.1.102)
3.1 ShockWaves in Gases
255
With
X--l+ ~' - 1K~,
?-1
Y -- ?'K~
2
(3.1.1o3)
Eq. (3.1.98) becomes
d(p-O) 7-1 dfl
1+7K} ? q- 1K2flX1/2y 1/2
2 l+K} ? q- 1
K2fl
(3.1.104)
to leading order. For instance, when Kfl is unity, this yields d(~ - o)
2(3 - ~)
dfl
?+ 1
(3.1.105)
and the reflected wave is compressive.
Curvature
Singularity
Let us examine more closely the flow in the neighborhood of point b in Fig. 3.1.16(b). Between points a and b, the curvature of the shock is zero, whereas between points b and c, it is finite, and the curvature has a jump discontinuity at point b. This is a general phenomenon whenever a dispersed wave starts or stops interacting with an upstream shock. For instance, there is another discontinuity in shock curvature at point c. In the subsequent analysis, an exact formula is derived for the curvature just above point b. A result such as this is useful when an MOC code has to deal with this type of singularity. Let point 2 t in Fig. 3.1.33 be a point on the shock above point 2. (For the subsequent analysis it is convenient to alter the notation.) Our objective is to determine the (negative of the) curvature, fl~ of the 2 - 2' (circular) arc in the limit of point 2' approaching point 2, which is fixed. In this limit, the reflected wave from the shock has no influence on fl~. Hereafter, subscript 2 refers to the curved part of the shock. The radius of curvature R, which equals _(fl~)-l, is perpendicular to the 2 - 2' arc. As fl is continuous, the arc is tangent to the planar shock at point 2. Simple trigonometry is used to evaluate a number of angles. For the limit of interest, set
co --~ -d~l,
0 ~ dO
(3.1.106)
The law of sines for the 0-1-2 triangle establishes l 1 --
sin(fl I - Ow) cos 0 w s i n ( p / - flI + Ow)
(3.1.107)
256
G. Emanuel
/a , - P , + Ow
9ff - l~ + fl~ - Ow- O+ ~
2
90o+/~_fl~+O_co
shock
l~
Y
0 [ ~ "~" -/P, I~'~ _X
F I G U R E 3.1.33
q
Shock wave curvature singularity schematic.
where x 1 = 1 is used. The law of sines for the co triangle is ll
12
sin co
sinOr/2 + P , -
(3.1.108a) fl, + Ow - co)
w h i c h becomes sin co sin(fl z - Ow) l2 - -
cos Ow sin(~i - / 3 i + Ow)cos(#i - flI + Ow - co)
(3.1.108b)
The law of sines for the 0 triangle yields R
R-I
s i n ( n / 2 + PI - r1 + O. - co)
2
sin(rt/2 - PI + r1 - 0 , - 0 + co)
(3.1.109a)
w h i c h becomes cos(#1 - / ~
+ 0 . - co) - cos(~i - / ~ sin co s i n ( f l / - Ow)
cos 0 w s i n ( p / - -
+ 0w + 0 - co)
1
(3.1.109b)
flI + Ow) R
Equations (3.1.107) and (3.1.108b) are thus used to eliminate l I and l 2. By expanding the left-hand side and using Eqs. (3.1.106), (3.1.109b) becomes s i n ( # t - r1 + Ow)dO - - -
dr/sin(fl - Ow)
1
cos 0 w sin(/~ I - flI 4- 0w) R
(3.1.109c)
3.1
257
S h o c k W a v e s in G a s e s
or finally
ds) _
Ow)
sin(fir -
2 - - - - COS 0 w s i n 2 ( p /
--
(3.1.110)
flI q- Ow)
because R dO = ds. This equation relates the differential arc length along the shock, just above point 2, to the differential angle in the Prandtl-Meyer expansion, at the leading edge (LE) of the expansion. Note that dr/is negative. Along the shock, Table 3.1.8 provides
@)_ 2 7 4- 1 P~176176176 sin 2fit 2~
~
,
(3.1.111)
In conjunction with Eq. (3.1.110), this yields f12 --
Y + 1 cos 0w sinZ(pi - - fit -}- O w ) l ( d p ) 2 sin 2ill sin(fir - 0~) PoeM2 ~
2
(3.1.112)
where the pressure derivative is next evaluated at the LE of the expansion. Remember that this derivative is constant along the LE and therefore holds at point 2. The pressure variation, inside the expansion, can be written as (Emanuel,
2000)
)7/(7_1) p__ =
PI
2
7q- 1
X1 cos 2 z
(3.1.113)
where X - 1 + 7 + 1 M2 2
'
z-
yI +
+
- r/
(3.1.114)
and v is the Prandtl-Meyer function. By differentiation, we obtain
dp __ 27 ( 2 )7/(7-1) d r / - (72 - 1) 1/2pI 7 + 1Xt sinz(c~
(3.1.115)
The value of z at the LE, where r / = PI, can be shown to equal z2 - ZLE -- tan-1
-- 1 (M/2 _ 1) +1
(3.1.116)
With this Eq. (3.1.115) can be shown to simplify to
d p ) _ 27 ~/2 2 7-+1 PI(M2-1)
(3.1.117)
258
G. Emanuel
The final result is obtained by combining this relation with Eq. (3.1.112) 27 7M2 sin2 flI
v+l
ML
7-1 2 (M/2 _ 1)1/2 c~ 0w sin2(#1 - flI + Ow) sin 2ill sin(fix - Ow) (3.1.118)
Table 3.1.7 can be used to evaluate parameters such as ( M ~ - 1) 1/2 and sin(fli- Ow). Note that MI, ill, and 121 refer to the supersonic, uniform flow region just downstream of the planar shock. We also observe that fl~ is negative, as expected. Although a global analysis is used to derive the preceding equation, the result is actually a local one, providing the wall slope at point 1 is discontinuous. To see this, examine Fig. 3.1.34, where 0w ~ 0,
fl~fl,
#I--~#,
MI ~ M
(3.1.119)
The planar shock need only be of infinitesimal length, and 0, fi . . . . refer to flow conditions between the planar shock and LE of the expansion immediately adjacent to point 2. This result stems from the fact that fl~ does not depend on the overall strength of the expansion, that is, the magnitude of the expansive wall turn angle at point 1 in Fig. 3.1.33. The magnitude of fl~ in Fig. 3.1.34 depends on 7, M~, 0, and the fact that the LE of an expansion intersects the shock at point 2. Other parameters, such as fl, M, and # that appear in Eq. (3.1.118), are derived quantities.
shock
_s
.ml=o
/Q)~ +-
/ ~ LE of expansion
unif.orm flow region
FIGURE 3.1.34 Localview of curvature singularity.
3.1 Shock Waves in Gases
259
Vorticity From Emanuel (2000, Problem 6.3), the magnitude of the vorticity, just downstream of the shock, is
02 = -
-~s 2+
2+ fl'u2
(3.1.120)
With the aid of Tables 3.1.7-3.1.9, this becomes 2 0.)2 -- ~ w 1 7+1
( w - 1)2 cos fl fl,
wX
(3 1 121) . .
This relation holds for two-dimensional and axisymmetric flows, and ~2 is normal to the plane of the flow. Because the freestream is uniform, this is shock-generated vorticity. It is zero for a planar or conical shock, where fl' -- O. It is also zero when the wave is acoustic and w -- 1, and when it is a normal shock and fl = 90 ~ Hence, there is a location on every blunt body shock where leo21 is a maximum, or where dcoz/ds is zero.
3 . 1 . 6 . 6 GENERAL THEORY Thomas (1948) relates the streamline curvature to the shock curvature for a perfect gas with a uniform freestream. Brown (1950) extends this work by assuming the upstream flow can be rotational. Lazarev et al. (1998) apply the theory of singular surfaces to shock propagation. (See these references for additional references.) Again, the gas is perfect, uniform, and quiescent upstream of the shock. Emanuel (2000, Chapter 6) presents an analysis where the gas need not be perfect, the upstream flow may be nonuniform and unsteady, and the unsteady shock surface is three-dimensional.
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260
G. Emanuel
Ashford, S.A. and Emanuel, G. (1994). Wave angle for oblique detonation waves. Shock Waves 3: 327-329. Bach, G.G., Knystautas, R., and Lee, J.H. (1969). Direct ignition of spherical detonations in gaseous explosives, in Proc. of Twelfth Sym. (Inter.) on Combustion, 853-864. Ben-Dor, G. (1992). Shock Wave Reflection Phenomena, New York: Springer-Verlag. Bengtson, R.D., Miller, M.H., Koopman, D.W, and Wilkerson, T.D. (1970). Comparison of measured and predicted conditions behind a reflected shock. Phys. Fluids 13: 372-377. Brown, W.E (1950). The general consistency relations for shock waves. J. Math. and Phys. 29: 252262. Cambier, J.-L. (1991). Numerical simulations of a nonequilibrium argon plasma in a shock-tube experiment. A/AA 91-1464. Chow, W.L. and Chang, I.S. (1972). Mach reflection from overexpanded nozzle flows, A/AAJ. 10: 1261-1263. Clavin, P., He, L., and Williams, F.A. (1997). Multidimensional stability analysis of overdriven gaseous detonations. Phys. Fluids 9: 3764-3785. Cramer, M.S. (1991). Nonclassical dynamics of classical gases, in Nonlinear Waves in Real Fluids, A. Kluwick, ed., New York: Springer-Verlag, 91-145. Davies, L. and Wilson, J.L. (1969). Influence of reflected shock and boundary-layer interaction on shock tube flows. Phys. Fluids 12: Part II, May 1969, 1-37-I-43. Edney, B.E. (1968). Effects of shock impingement on the heat transfer around blunt bodies. AIAAJ. 6: 15-21. Emanuel, G. (1965). Structure of radiation-resisted shock waves with vibrational nonequilibrium. Phys. Fluids 8: 626-635. Emanuel, G. (1982). Near-field analysis of a compressive supersonic ramp. Phys. of Fluids 25: 1127-1133. Emanuel, G. (1983). Numerical method and results for inviscid supersonic flow over a compressive ramp. Computers and Fluids 11: 367-377. Emanuel, G. (1986). Gasdynamics: Theory and Applications, AIAA Education Series. New York, American Institute of Aerodynamics and Astronautics. Emanuel, G. (1992a). Oblique shock wave with sweep. Shock Waves 2: 13-18. Emanuel, G. (192b). Oblique shock wave with sweep II. Shock Waves 2: 273-275. Emanuel, G. (2000). Analytical Fluid Dynamics, 2nd ed., Boca Raton, FL: CRC Press. Emanuel, G. and Liu, M.-S. (1988). Shock wave derivatives. Phys. Fluids 31: 3625-3633. Eubank, C.S., Rabinowitz, M.J., Gardiner, Jr., W.C., and Zellner, R.E. (1981). Shock-initiated ignition of natural gas-air mixtures, Eighteenth Sym. (Int.) on Combustion, The Combustion Inst., 1767-1774. Friedman, M.P. (1960). An improved perturbation theory for shock waves propagating through non-uniform regions. J. Fluid Mech. 8: 193-209. Friedrichs, K.O. (1948). Formation and decay of shock waves. Comm. on Appl. Math. 1: 211-245. Gordon, S. and McBride, B.J. (1989). Computer program for calculation of complex chemical equilibrium compositions, rocket performance, incident and reflected shocks, and ChapmanJouget detonations. NASA SP-278. Guha, A. (1992). Structure of partly dispersed normal shock waves in vapor-droplet flows. Phys. Fluids A4: 1566-1578. Heaslet, M.A. and Baldwin, B.S. (1963). Predictions of the structure of radiation-resisted shock waves. Phys. Fluids 6:781-791. Horn, K.P., Wong, H., and Bershader, D. (1967). Radiative behavior of a shock-heated argon plasma flow, J. Plasma Phys. 1: 157-170. Jensen, R.J. (1976). Metal-atom oxidation lasers, in Handbook of Chemical Lasers, R.W.E Gross and J.E Bott, eds., New York: John Wiley, 703-732.
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Johannesen, N.H. (1952). Experiments on two-dimensional supersonic flow in corners and over concave surfaces. Phil. Mag. 43: 568-580. Kapila, A.K., Son, S.E, Bdzil, J.B., Menikoff, R., and Stewart, D.S. (1997). Two-phase modeling of DDT: Structure of the velocity-relaxation zone. Phys. Fluids 9: 3885-3897. Kistiakowsky, G.B. and Kydd, PH. (1955). Gaseous detonations. VI. The rarefaction wave. J. Chem. Phys. 23: 271-274. Kistiakowsky, G.B. and Kydd, P.H. (1956). Gaseous detonations. IX. A study of the reaction zone by gas density measurements. J. Chem. Phys. 25: 824-835. Kobayashi, Y., Watanabe, T., and Nagai, N. (1996). Vapor condensation behind a shock wave propagating through a large molecular-mass medium. Shock Waves 5: 287-292. Landau, L.D. and Lifshitz, E.M. (1987). Fluid Mechanics, 2nd ed., New York: Pergamon Press, chapter 14. Lasseigne, D.G. and Hussaini, M.Y. (1993). Interaction of disturbances with an oblique detonation wave attached to a wedge. Phys. Fluids A5: 1047-1058. Lax, A. (1948). Decaying shocks. Comm. Appl. Math. 1: 247-257. Lazarev, M.P., Ravindran, R., and Prasad, P (1998). Shock propagation in gas dynamics: explicit form of higher order compatibility conditions, Acta Mechanica 126: 139-151. Lee, B.H.K. (1967). Detonation-driven shocks in a shock tube. A/AA J. 5: 791-792. Lee, J.H.S. (1977). Initiation of gaseous detonation. Ann. Rev. Phys. Chem. 28: 75-104. Lee, J.H.S. (1984). Dynamic parameters of gaseous detonations. Ann. Rev. Fluid Mech. 16: 311-336. Lee, J.H. and Lee, B.H.K. (1965). Cylindrical imploding shock waves. Phys. Fluids 8: 2148-2152. Lefebvre, M.H., Oran, E.S., Kailasanath, K., and Van Tiggelen, P.J. (1993). Simulation of cellular structure in a detonation wave, in Dynamic Aspects of Detonations, Prog. in Astronautics and Aeronautics, 64-77. Lehr, H.E (1972). Experiments on shock-induced combustion. Astronautica Acta 17: 589-597. Li, C., Kailasanth, K., and Oran, E.S. (1994). Detonation structures behind oblique shocks. Phys. Fluids: 6:1600-1611. Li, H. and Ben-Dor, G. (1996). Oblique-shock/expansion-fan interaction-analytical solution. AIAA J. 34: 418-421. Li, H. and Ben-Dor, G. (1998). Mach reflection wave configuration in two-dimensional supersonic jets of overexpanded nozzles. AIAA J. 36: 488-491. Liepmann, H.W. and Roshko, A. (1957). Elements of Gasdynamics, New York: John Wiley, 79-83. Lu, EK., Stuessy, WS., Wilson, D.R., Bakos, R.J., and Erdos, J.I. (1998). Recent advances in detonation techniques for high-enthalpy facilities. AIAA, 98-0550. Matsuo, A. and Fujiwara, T. (1993). Numerical investigation of oscillatory instability in shockinduced combustion around a blunt body. AIAA J. 31: 1835-1841. Menees, G.P, Adelman, H.G., Cambier, J-C, and Bowles, J.V. (1992). Wave combustors for transatmospheric vehicles. J. Prop. and Power 8: 709-713. Millikan, R.C. and White, D.R. (1963). Systematics of vibrational relaxation. J. Chem. Phys. 39: 3209-3213. Mitchner, M. and Vinoker, M. (1963). Radiation smoothing of shocks with and without a magnetic field. Phys. Fluids 6: 1682-1692. Morrison, R.B. (1980). Oblique detonation wave ramjets, NASA-CR-159192. Nicholls, J.A., Adamson, Jr., T.C., and Morrison, R.B. (1963). Ignition time delay of hydrogenoxygen-diluent mixtures at high temperatures, AIAA J. 1: 2253-2257. Petschek, H. and Byron, S. (1957). Approach to equilibrium ionization behind strong shock waves in argon. Ann. Phys. 1: 270-315. Phillips, M.G.R. and Pugatschew, A.A. (1980). An investigation of the structure of plasma produced by reflected shock waves. Phys. Fluids 23: 34-37.
262
G. Emanuel
Powers, J.M. and Stewart, D.S. (1992). Approximate solutions for oblique detonations in the hypersonic limit. A/AAJ. 30: 726-736. Rudinger, G. (1969). Nonsteady Duct Flow, New York: Dover Publications, Chapter 3. Semerjian, H. (1972). Spectroscopic study of the behavior of xenon behind a shock wave, Project Squid, NR-098-038, Purdue University, Lafayette, Indiana. Sharma, V.D., Ram, R., and Sachdev, EL. (1987). Uniformly valid analytical solution to the problem of a decaying shock wave. J. Fluid Mech. 185: 153-170. Strehlow, R.A. and Femandes, ED. (1965). Transverse waves in detonations. Comb. and Flames 9: 109-119. Takayama, K. (1992). Report on the first international ram accelerator workshop. Shock Waves 2: 61-62. Taylor, G. (1949). The dynamic of the combustion products behind plane and spherical detonation fronts in explosives. Proc. Roy. Soc. (Lond.) A200: 235-247. Thomas, T.Y. (1948). Calculation of the curvatures of attached shock waves. J. Math. and Phys. 27: 279-297. Thompson, PA. (1991). Liquid-vapor adiabatic phase changes and related phenomena, in Nonlinear Waves in Real Fluids, A. Kluwick, ed., New York: Springer-Verlag, 147-213. Thompson, P.A. and Lambrakis, K.C. (1973). Negative shock waves. J. Fluid Mech. 60: 187-208. Yu, H-R, Esser, B., Lenartz, M., and Gr6nig, H. (1992). Gaseous detonation driver for a shock tunnel. Shock Waves 2: 245-254. Vincenti, W.G. and Kruger, Jr., C.H. (1965). Introduction to Physical Gas Dynamics, New York: John Wiley. Wegener, PP. and Wu., B.J.C. (1977). Gasdynamics and homogeneous nucleation, in Advances in Colloid and Interface Science, vol. 7, Elsevier Scientific Publishing Co., Amsterdam, The Netherlands: 325-417. White, D.R. (1961). Turbulent structure of gaseous detonation. Phys. Fluids 4: 465-480. Woods, L.C. (1987). Principles of Magnetoplasma Dynamics, Oxford: Clarendon Press, Chapter 5.
CHAPTER
3
~
2
Theory of Shock Waves 3.2
Shock Waves in Liquids
SHIGERU ITOH Shock Wave and Condensed Matter Research Center, Kumamoto University, Kumamoto, Japan
3.2.1 Fundamental Properties of Liquid 3.2.1.1 Density of Liquid 3.2.1.2 Compressibility of Liquid 3.2.1.3 Viscosity of Liquid 3.2.2 Wave Motion in Liquids and Equation of State 3.2.2.1 Pressure Wave in Liquids 3.2.2.2 Equation of State for Liquids 3.2.2.3 Plane Shock Relation for Water 3.2.3 Shock Waves in Water Due to Underwater Explosion of High Explosives 3.2.3.1 Observational Investigation 3.2.3.2 Numerical Procedure 3.2.3.3 Experiments of Underwater Shock Waves 3.2.4 Von Neumann Reflection of Underwater Shock Wave 3.2.4.1 Introduction 3.2.4.2 Experimental Method 3.2.4.3 Wave Configuration of Oblique Interaction of Underwater Shock Waves 3.2.5 Application of Underwater Shock Waves 3.2.5.1 Shock Compaction of Powders 3.2.5.2 Exposive Forming by Underwater Shock Waves References
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
263
264
s. Itoh
3.2.1 F U N D A M E N T A L PROPERTIES OF LIQUID
3.2.1.1
DENSITY OF
LIQUID
The density of liquid, similar to the density of any other matter, is defined as the mass per unit volume. The specific volume is the reciprocal of the density. The density of water is exactly 1 gm/cm 3 at atmospheric pressure (0.1 MPa) and a temperature of 3.98 ~ Thus the density depends strongly on both pressure and temperature. The dependence of the density of water on temperature at atmospheric pressure is shown in Fig. 3.2.1. Dorsey (1948) presented a detailed analysis of the density of water. The dependences of the densities of mercury and a few oils on temperature at atmospheric pressure are shown in Figs. 3.2.2 and 3.2.3, respectively. 3.2.1.2
C O M P R E S S I B I L I T Y OF L I Q U I D
In machine design, for example, pumps, water turbines, etc., liquids are usually treated as incompressible fluids. However, as is the case with all matter, liquids are compressible, and similar to gases, the compressibility of the liquid becomes very important when the fluid velocity exceeds the speed of sound. The flow Mach number of a fluid is the ratio between the fluid velocity and its speed of sound. When a pressure of 0.1 MPa is imposed on iron its volume is contracted by about 6 x 10 -7. Under the same pressure water is contracted 120 to 200 times more than iron. Thus liquids are not too difficult to compress. When a pressure P2 is imposed on a liquid having initially a specific volume t~ and a pressure/91,
0.99
09s 5 0.97
0.96 0.95
\
o.
I"
t'O |
0 10 20 30 40 50 60 70 BO 90 100 FIGURE 3.2.1 Density of water at atmosphere pressure (Kell, 1968).
3.2
Shock Waves in Liquids
265
!
i
I
I
,,:l',/,
!
lq~ 13.6
x
ib-/mmmmmm imJ,-immmmm llmm,,,~mmmm NILIINI III/I[.T//I I//IIW51/I Illl/Bb~ll I/lllllkLil
13.41--
!mmllI~Immm,~
mll
ml
-20 FIGURE 3.2.2
0
20
40
80
80
100
Density of Mercury at atmospheric pressure (Stoff and Bigg, 1928).
1.0
xlO 3 , , m
m
m
m
m
m
m
m
m
m
m
~~nmnummmn 0.95
~mnnnnw~un n w;gunnnnh~ mmmm/~qmm| ichino OIF
0.90 e~
E o
~L
-",
v
0.85
DieselON
.Light
~
0-800
k~
iio,r
T I1
!
20
FIGURE 3.2.3
~-
~~-
40
1
60
Density of oil (Itaya, 1966).
80
266
S. Itoh
its specific volume changes as follows" -Av
= flv(p 2 -
Equation (3.2.1a) can be rewritten as Au -
-
-
/)
flAp
=
Pl)
(3.2.1a)
(3.2.1a)
where A p - P2 - P l and fl is the modulus of the compressibility of liquid. Similar to the density and the specific volume, the compressibility also depends on the pressure and the temperature. The compressibility for an isothermal compressing process, that is, the isothermal compressibility, is & - -7
Similarly, the compressibility for an isentropic (adiabatic and reversible) compressing process, that is, the isentropic compressibility, is 1(3~)
(3.2.3)
10 5
1
2 3456810
2 34568102
2 34568103 2 34568104
FIGURE 3.2.4 Compressibility of water for isothermal values obtained from speed of sound data (Vedam and Holton, 1968).
3.2
267
Shock Waves in Liquids
where s is the entropy. The isothermal compressibility can be obtained with the aid of PVT (P: pressure; V: volume; T: temperature) isobars, and the isentropic compressibility is usually obtained directly from the acoustic propagation of the liquid using the following relation:
fls =
1
2 PoCs
(3.2.4)
where cs is the isentropic speed of sound of the fluid and P0 is its initial density. Values of fl; for water as obtained from speed of sound data are shown in Fig. 3.2.4 (see Bridgeman, 1949). Values of fl; for other liquids are given in Table 3.2.1. The inverse values of fl, that is, K - 1/fl is the bulk modulus. This parameter is also very important for understanding shock waves in liquid.
3.2.1.3
VISCOSITY OF
LIQUID
It is well known, that in many fluids (for example, water) that flow in a laminar flow, z o~ du/dy, where z is and the shear stress and du/dy is the fluid velocity gradient. The proportional constant is defined as the dynamic viscosity /z Fluids obeying this relation are known as Newtonian fluids. However, many TABLE 3.2.1
Isothermal Compressibility of Various Liquids
Liquid
T[~
Acetone Acetone Ethyl alcohol Ethyl alcohol Ether Ether Glycerin Glycerin Chloroform Mercury Mercury Toluene Carbon dioxide Benzene Water Water Water Water Methyl alcohol Methyl alcohol
20 20 20 20 20 20 20 20 20 21.9 21.9 20 20 20 20 20 20 60 20 20
P[atm]
flT[GPa -1 ]
1 5000 1 5000 1 5000 1 5000 1 1000 10000 1 1 1 1 1000 5000 10000 1 5000
1.26 0.21 1.11 0.22 1.87 0.22 0.21 0.12 1.01 0.0388 0.03 0.91 0.93 0.95 0.45 0.36 0.18 0.12 1.23 0.21
268
s. Itoh 0.018 0.016 _ \
0.014
\
\i
0,012 _
.
: -.._,
|
9
.
~,~
,X
0.010 0.008
\
0.006
I
Water (1at)
I
_m_
"o., r
0.oo4 E
.....
-U-
0.002- ~,-
1
i [
o.ooo 0
FIGURE 3.2.5
I
I
t
I
I
I
Temperature (~C)
li
l
10 20 30 40
50 60
70
I
I
1
80
I
I
1
I
1
1
~
T [
='= t
90 100 110 120 130 140 150
Kinematic coefficients of viscosity of liquid (Korosi and Fabuss, 1968).
liquids, such as large polymers, colloid liquids involving solid powders, oil paintings, etc., show marked deviations from this behavior. For this reason they are called non-Newtonian fluids. Values of the dynamic viscosity of polymers can be found in Ferry (1970). The kinematic viscosity defined by v = ,u/p is also an important parameter in fluid dynamics. The dependence of the kinematic viscosity of water on the temperature is shown in Fig. 3.2.5. 3 . 2 . 2 W A V E M O T I O N IN L I Q U I D S A N D EQUATION OF STATE 3.2.2.1
PRESSURE WAVE IN LIQUIDS
The propagation of a one-dimensional (1D) disturbance (e.g., a pressure wave) in the x-direction is given by Euler's equation of motion ~w = ~t
1~
(3.2.5)
p ~x
where w is the velocity component in the x-direction, p is the density, and p is the pressure. The continuity equation for 1D unsteady and uniform flows is
~p
~
~(pw) ~p
+ ~
~p
= ~t + w ~
~w
+ ~~
= 0
(3.2.6)
3.2
269
Shock Waves in Liquids
The last equation can be rearranged to read 3w _ 3x--
1 3p p 3t
w 3p p 3x
(3.2.7)
The second term on the fight-hand side of Eq. (3.2.7) is negligibly small compared to the other terms, and therefore, the continuity equation becomes 3w 3x
1 3p
=
(3.2.8)
p 3t
In the case of a gaseous phase such as air, if the changes are isentropic then pv ~ - const. Here, o is the specific volume and ~c is the ratio of the specific heat capacities. Differentiating this relation results in the following: 0K -- tct)~C-Xdt)- 0
(3.2.9a)
which can be rearranged to read d~= v
@ Kp
(3.2.9b)
The specific gravity ? can be expressed as ? - Pg - g/o. Differentiating this expression results in the following: dv -- = v
d? = ?
dp p
(3.2.10)
Substituting Eq. (3.2.10) into Eqs. (3.2.8) and (3.2.9b) results in 0w= 3x
1 0p Kp 3t
(3.2.11)
Solving for w by differentiation and elimination, one obtains the following wave equation: 32w = ~cp 32_~w= a2 32w 3t 2 p 3x 2 3x 2
(3.2.12)
This 1D form of the wave equation is satisfied by any function of the form f ( t + x/a) and f ( t - x/a). The minus or plus signs indicate that the wave is advancing either in the positive or in the negative x-direction. In the case of the liquid phase, a similar procedure can be followed while using the bulk modulus K instead of the specific heat capacities ratio to. This yields d__p_p= dp p K
(3.2.13)
270
s. Itoh
Similarly to the case of a gaseous phase, one gets for the case of a liquid phase the following wave equation: 02W __ K 02_~W - - a 2 02w Ot 2
p Ox 2
(3.2.14)
Ox 2
From which one obtains a- ~
(3.2.15)
Using this relation, the speed of sound of water can easily be calculated. At room temperature the bulk modulus and the density of water are K = 2.03 GPa and p = 1000 kg/m 3, respectively; consequently, the speed of sound of the water is 1425 m/s.
3.2.2.2
EQUATION OF STATE fOR LIQUIDS
In general, the Mie-Gl~neisen equation of state EOS is used for solids or condensed matter. The pressure change under a cold compression when the temperature is changed from 0 to T K in a constant volume process is given by
op)
p-pc(v)--
~
3~
T -- -c T
(3.2.16)
U
where ~ is the coefficient of linear expansion and ( is the isothermal compression rate. By introducing the parameter F, which is defined as 3or
F -~-~v
(3.2.17)
Eq. (3.2.16) becomes F
p - - p~(v) - t - - - C v T
v
(3.2.18)
where C v is the specific heat capacity at constant volume. Assuming that e - ec = CvT results in F
p -- pc(v) + - - ( e - ec) t)
Replacing both p~ and
ec
by
PH
and
e H,
(3.2.19)
respectively, yields: F
p - pn(V) +--(e ~)
-- ell)
(3.2.20)
3.2
271
Shock Waves in Liquids
Equation (3.2.19) or (3.2.20) is known as the Mie-Gruneisen equation of state (Meyers, 1994). The parameter F is known as the Gruneisen constant and it is defined in thermodynamics as
r - v(~P)~
(3.2.21)
The following relation is often used, because it is very convenient for real calculations: ~ or
(3.2.22)
_F_
t)
DO
Rice, McQueen and Walsh (1957) introduced F ~ s - 1, where the parameter s is obtained from fitting the results of shock Hugoniot experiments to the line (3.2.23)
U s = a o 4- SUp
where Us is the shock wave velocity, Up is the shock-induced flow (particle) velocity, and a 0 is the speed of sound ahead of the shock wave. Rice and Walsh (1957) carried out experiments to get Hugoniot data for water up to 250 kbar. Using their results the relation between the pressure p and the density p can be obtained. The results are shown in Fig. 3.2.6. The bold line in Fig. 3.2.6 is the results obtained by Tait (see Heukroth and Glass, 1968). Tait's equation reads as follows:
Po + A w
25 '
"
I
i 9
0
.... ~ IoL .
-~,
9
.,,
i l l
!
/
/
!\ \
! i ,Z [// I//
.....
\ 2SO~
r
A ,,
i
"
FIGURE 3.2.6
v /-o~,----
\ l /" 4 /
,.
0 500
/
! !
I/ !1
,
/
!
.,, .
10
h
iii
I
IIXi
Tait m ~
~
/
i /x
. . . .
15-
I
i
/./
~ 1000
/J
f J
Oer,sity (kl/,,J,) .....
I
,
,
L
......
1500
2000
2500
Relations between pressure and density up to 25 GPa.
272
s. Itoh
where A w is a constant and the subscript 0 indicates the initial condition (P0 -- 0.1 MPa at room temperature). For p < 2500 MPa, Aw = 296.3 MPa and n -- 7.415. Marsh (1980) provides Hugoniot data for a variety of solids and liquids. The values of the Gruneisen constant for various materials are listed in Table 3.2.2. Mader's (1979) data regarding the EOS of water are shown in Table 3.2.3. While studying underwater detonations of pentolite cylinders, Sternberg (1987) used a power law to relate the pressure p and the density p and obtained good numerical results. The relation between Us and Up for liquids is given in Table 3.2.4, which is taken from Marsh (1980). TABLE 3.2.2
Grfineisen Constant and s for Various Materials a
Standard PMMA Magnesium AZ31B Copper Iron SUS304 Uranium-3wt% Molybdenum Gold Platinum ,
p [g/cm 3]
C O [km/s]
S
F
1.18 1.775 8.93 7.85 7.896 18.45 19.24 21.44
2.26 4.516 3.940 3.574 4.569 2.565 3.056 3.633
1.816 1.256 1.489 1.920 1.49 1.531 1.572 1.472
0.75 1.43 1.96 1.69 2.17 2.03 2.97 2.40
,
a Marsh (1980).
TABLE 3.2.3
Speed of sound c 1483m/s
Hugoniot Data and Mie-Gr/ineisen Constant of Water Specific heat at constant volume C,.
Line expansion
constant s
Mie-Griineisen constant F
2.0
1.87
1.0
00001 m/K
Us - U p
TABLE 3.2.4
The Relation Between Us and up for Liquids
Acetone Ethyl alcohol Ammonia liquid Bromoethane Cyclohexadiene Ether Ethylene glycol Mercury Oxygen liquid
Us ~ Us ~ Us ~ U~ ~ Us ~
Us ~ Us ~ Us ~ U~ ~
1.94 4- 1.38up 1.73 4- 1.75Up 2.0 4- 1.51Up 1.58 4- 1.36Up 1.87 4- 1.33Up 1.70 4- 1.46up 2.15 4- 1.55up 1.75 4- 1.72up 1.88 4- 1.34Up
3.2
273
Shock Waves in Liquids
3.2.2.3
PLANE S H O C K RELATION FOR W A T E R
While shock wave theory (generation and propagation) both in gaseous and solid media is well understood, the theory of shock waves in liquids is much less well known. In order to simplify the presentation of the theory of shock waves in liquids, 1D water flow will be considered. In a dissipative flow, when waves of finite amplitude propagate along the flow direction they develop a steep front, known as a shock wave, which is balanced by the dissipative processes of the medium. The shock wavefront, is very narrow, and is of the order of a few angstroms. When dissipation is neglected the steep shock wavefront can be considered as a sharp front across which the flow properties such as pressure, density and energy change discontinuously. Working separately, Rankine and Hugoniot (see Lamb, 1932) developed the equations relating the flow properties across such a discontinuity. These relations are known as the Rankine-Hugoniot relations. If an observer moves with the velocity Us of the shock front into a region where the particle velocity is u 0 and density is P0, the incoming relative velocity with respect to the observer is Us - %. If the pressure, density and particle velocity behind the sharp front are p, p and Up, respectively, the following relations can easily be derived from the conservation laws of mass, momentum and energy. The conservation of mass is ,Oo(Us - Uo) = p ( u ~ - up)
(3.2.25)
The conservation of linear momentum is ,oo(U~ -
Uo)(Up -
Uo) = p - p o
(3.2.26)
The conservation of energy is pup -
pouo -- P(U s -
%)
e -
e0 +.~
where e0 is the internal energy per unit mass ahead of the sharp front. If the flow in the region ahead of the shock front is quiescent, then u0 = 0. In addition, usually the pressure ahead of the shock front P0 is much smaller than the post-shock pressure p, thus P0 can be neglected. Under these assumptions the forementioned relations become p(Us - Up) = PoUs
(3.2.28a)
p = poUsup
(3.2.28b)
e - eo -- ~p
-
(3.2.28c)
Results obtained using the EOS given in Table 3.2.3, are shown in Fig. 3.2.7. It is evident that the speed of sound behind an underwater shock wave is
274
s. Itoh "'i
A
!
I'
'
I
I
r 0
r
s= J -
j
r
o .=J
,.
S
s
s
s
I
I
I
I
"'
Sound s ' * " ss ~ s ~ ~
~
-
~Shock
J
,p"
PRESSURE (GPa) FIGURE 3.2.7
Shock relation for water calculated using the EOS proposed by C. Mader (1979).
higher than the velocity of the shock wave at the same pressure. This means disturbances that occur in the water overtake the preceding shock and attenuate it. This is opposite to the situation in a gaseous medium, where the shock wave is faster than the speed of sound behind it, and as a result for short durations the shock wave keeps its original strength. It should be noted here that the most significant difference in the RankineHugoniot relation between water and a gaseous medium is in the fact that the speed of sound of water is higher than the shock wave velocity. This mean that the shock velocity is more easily attenuated as it propagates in water than in gases. This fact is very important for understanding and treating shock waves in liquids.
3.2.3 SHOCK WAVES IN WATER DUE TO UNDERWATER EXPLOSION OF HIGH EXPLOSIVES 3.2.3.1
OBSERVATIONAL I N V E S T I G A T I O N
Because the compressibility of liquids is smaller than that of gases, it is very difficult to obtain shock waves in liquid using the ordinary shock tube (see Chapter 4.1 by M. Nishida). One technique of obtaining underwater shock
3.2 ShockWaves in Liquids
275
waves is by underwater explosions of high explosives. Shock waves generated by underwater explosions are usually observed via high-speed photography together with standard shadowgraph systems. In the following, a didactic description of the experimental procedure of underwater shock wave research conducted at the Shock Wave and Condensed Matter Research Center in Kumamoto University in Japan is given. A layout of the observational facilities is shown in Fig. 3.2.8. The experiments are carried out in an explosion chamber (8 m x 8 m and a height of 5 m). The photographs are taken by an image converter camera (IMACON 790, made by Hadland Photonics Co., with a maximum flaming speed of 20 x 106 ffames/s and a maximum streak capacity of 1 ns/mm). The image converter camera is coupled with a xenon flashlight source (HL 20/50 type Flashlight, also made by Hadland Photonics Co., with an output of 500J and a flash time of 50 gs) using the standard shadowgraph system. In addition, in order to investigate the velocity of the shock front, streak photographs are taken by inserting a slit in a direction along the shock front surface. The initiation time of the explosives and the starting time of the xenon flashlight source are controlled by three delay generators type JH-3CDG (Hadland Photonics Co.). In some cases a detonation cord is used in order to get a longer delay time. 3.2.3.2
NUMERICAL PROCEDURE
The simplified arbitrary Lagrangian-Eulerian (SALE) method is used to simulate the experiments (for details, see Itoh et al., 1996). The calculation
FIGURE 3.2.8 Observationsystem for high-speed photography.
276
s. Itoh
using the SALE method is divided into three phases: Phase 1. An explicit Lagrangian calculation, in which the velocity field is updated by the effects of all the forces acting on the flow field. Phase 2. A Newton-Raphson iteration that provides the time-advanced pressures and velocities. Phase 3. Performance of all the advective flux of the mass calculations and the mesh rezoning. All the materials are treated as compressible fluid media. The 2D governing equations are as follows. The conservation of mass a(pu) 4 8(pv) = 0
ap
at + ax
(3.2.29)
---q-
The conservation of linear momentum in the x- and y-directions O(pu2) +
~)(puv) +
O(p + q) +
-
o
Ot Ox Oy Ox ~O(pv) _ ~- ~(puv) + ~(pv 2) + ~c)(p 4- q) = o at
Ox
Oy
(3.2.30) (3.2.31)
Oy
The conservation of internal energy ~O(pe) _ O(pev) 8t ~- O(peu) ox + Oy + (p + q)D -- 0
(3.2.32)
where D is the velocity divergence Ou
Ov
Ox
Oy
D = -- + --
(3.2.33)
In the foregoing equations q is the artificial viscous pressure. For the computation of shock waves q = 2oPAD min(0, D)
(3.2.34)
u and v are the velocity components in the x- and y-directions, A is the computational cell area, and ~-0 is a proportionality constant. For water and polymethylmethacrylate (PMMA) the following Mie-Gruneisen equation of state is used: (f --- ~-~)2 1 -
+ FPoe
(3.2.35)
The various parameters in Eq. (3.2.35) are given in Tables 3.2.2 and 3.2.3.
3.2
Shock Waves in Liquids
2"/7
Explosive
TABLE 3.2.5
JWL Parameters of a SEP
A [Gpa]
B [Gpa]
R1
R2
co
eo [J/kg]
364.99
2.3097
4.30
1.00
0.28
2.16 • 106
The JWL equation of state is used for the detonation products (for details, see Minota, Nishida and Lee, 1996), and it is as follows: P -- A 1
R1~7 e x p ( - R 1 V ) + B 1
exp(-R 2V) q - - - ~
where the JWL parameters A, B, R1, R2 and o) are determined by the experiment, and V is the ratio of the density of the initial explosive to the density of the detonation products (Pe/P). The experimentally determined JWL parameters of a SEP explosive are listed in Table 3.2.5.
3.2.3.3 EXPERIMENTS OF U N D E R W A T E R S H O C K WAVES 3.2.3.3.1 Spherical Explosion-Generated Shock W a v e s In order to understand the characteristics of shock waves in liquids, underwater explosions of spheres of high explosives are presented. Two types of high explosives have been used in experiments. One type, known as SEP explosive, is based PETN (65% wt PETN and 35% wt paraffin). Its detonation velocity is about 7km/s. The other type, known as HMX explosive, is based on PBX explosives and its detonation velocity is about 8.4 km/s. The diameters of the explosive spheres were 20 mm, 38 mm and 48 mm, and they were placed in a water chamber as shown in Fig. 3.2.9. The explosives were detonated using a No. 6 electric detonator (made by Asahi Chem. Co.). The propagation process was recorded using framing photographs. Streak photographs were also taken by setting a slit as shown in Fig. 3.2.9b. The length scale on the streak photographs was calibrated by a block gauge and the time scale by a known pulse wave. A series of framing photographs is shown in Fig. 3.2.10 in which SW denotes the shock wavefront. The time interval between each two consecutive photographs was 2 Its. It is evident from Fig. 3.2.10 that the front of the shock wave propagates in a self-similar manner. Streak photographs of four experiments are shown in Fig. 3.2.11. The type of the explosive and the diameter of the explosive charge are given in each photograph; SW indicates the shock
278
s. Itoh
FIGURE 3.2.9 Experimental assemblies of explosive spherical charge. (a) The spherical charge is suspended by a line. A detonation cord is used to obtain the optimum delay time in order to take the framing photography. (b) The spherical charge is placed on a PMMA support in order to enable application of streak photography. A detonating cord is also used to obtain the optimum delay time.
wavefront, EX shows the explosive, and the white line d e n o t e d by SL indicates the self-illumination of the explosives. A c o m p u t e r i z e d evaluation of the e x p e r i m e n t a l data provides a detailed x - t wave relation for the u n d e r w a t e r s h o c k waves. Using this x - t wave p h o t o g r a p h y and a curve-fitting m e t h o d (see
FIGURE 3.2.10
A series of the framing photographs obtained for a SEP spherical charge.
rar~ O
~~ ~~ ~~ ga,
FIGURE 3.2.11
Streak photographs for a SEP spherical charge. t,~ "-,1
280
s. Itoh
Bervington, 1969), the following time-dependence position of the spherical charge is obtained: r -- A I [ 1 - e x p ( - B
it)] + A2[1 -
exp(-B2t)] +
A3[1 - e x p ( - B 3 t ) ]
+ cot
(3.2.37) Differentiating by t results in the shock wave velocity
dy
dt = A1B1 exp(-Blt) + A2B2 exp(-B2t) 4- A3B 3 exp(-B3t) + Co
(3.2.38)
Note that as t --, oo the shock wave velocity approaches co as indeed should be the case. Using the impedance matching method (see, e.g., Kinslow,1967) the shock-induced flow velocity Up, that is, the velocity of the interface between the water and the explosives, is obtained. For the SEP explosive, Up = 2240 m/s and for the PBX explosive, Up -- 3090 m/s. Using the value of Up, the initial shock velocity at the boundary of the spherical charge u r can be obtained. The initial shock wave velocities for the SEP and the PBX explosives are Ur = 5480 m/s and u~ = 7010 m/s, respectively. The shock wave velocities for the SEP and the PBX explosives as a function of the propagation distance are shown in Fig. 3.2.12. For both explosives, the shock waves attenuate very quickly along a very short distance after which they reach asymptotic velocities. Once the shock wave velocity is known, the shock-induced pressure is obtained using the following equation: P - Po U~~U~ - Co + Po
(3.2.39)
S
where P0 and P0 are the ambient pressure and density ahead of the shock wave, respectively.
llO
7000 - - - SEP PBX
"~ 6000
~
5000 4000
- 3000 2000 t v) 1000 J~
0
0
I
!
I
20
40
60
80
Distance r (mm) FIGURE 3.2.12
An underwater shock wave velocity versus the propagation distance.
3.2
281
Shock Waves in Liquids
1# 10 10 4 9,
103
10 10 1
10
100
1000
(R+r)/R FIG. 3.2.13 The pressure ratio across the shock wave versus the nondimensional distance from the charge center for a PBX spherical charge.
Figure 3.2.13 illustrates the pressure ratio P/Po across the shock wave as a function of the nondimensional distance from the charge center (R 4- r)/R for a PBX charge (R is the charge radius) in a logarithmic coordinate system. The circles are obtained from the empirical equation (3.2.39), which is based on experimental results, and the triangles are the pressure measurement results as obtained in the experimental investigation of Itoh et al. (1996). The solid line represents the numerical calculations. It is evident from the figure that the various results coincide with each other. This implies that the method of deducing the shock strength from the streak photograph is valid. Similar results but for a SEP charge are shown in Fig. 3.2.14. The circles, triangles and squares P/Po
105
.
.
_~.
..j , ~. i
10 4
~ .1
1 03
- -
'
mLj
, ~-,,, I
"
r.L,
. i
102~
L 1
.....
~ i
.i.
i: ~
,i
~"-:
L . . . . .
.
_
-
,
A o ---- ---
Experiment Experiment Calculation Calculation Calculation
18) (R=29) ( R - 1 O) (R= 18) (R=29)
: 10 (R+r)/R
FIGURE 3.2.14 The pressure ratio across the shock wave versus the nondimensional distance from the charge center for a SEP spherical charge.
282
s. Itoh
are obtained from the empirical equation (3.2.39), which is based on experimental results, for the following charge radii R = 10, 18 and 29 mm, respectively, and the solid lines represent the numerical calculations. The agreement between the experimental and the numerical results seems to be very good. With the aid of Figs. 3.2.13 and 3.2.14 one can get a relation between the pressure ratio across the spherical shock wave P/Po and the nondimensional propagation distance from the center of the explosive charge:
Po For a PBX charge, A = 2 . 2 x A = 1.1 x 10 5 and a = 1.8.
l0 s and a = 1 . 8
and for a SEP charge
3.2.3.3.2 Planar E x p l o s i o n - G e n e r a t e d Shock Waves In the following the case of a plate explosive will be presented. The plate, shown in Fig. 3.2.15, consists of an explosive lens and a main explosive (A). The explosive lens is a combination of two high explosives (B and C), which have different detonation velocities. The length, the width and the thickness of the main explosive are L, H and T, respectively. An electric detonator is used to detonate the explosive plate after it is immersed in the water. Once the detonator is set off a detonation wave propagates from the electric detonator at a constant velocity, that is, the detonation velocity. The experimental setup for the investigation with an explosive lens is shown in Fig. 3.2.16. A high explosive HABW is used as the explosive with the
FIGURE 3.2.15 An experimental setup for the SEP plate explosive. The main explosive is A. The explosive lens is a combination of two explosives B and C.
3.2 ShockWaves in Liquids
283
FIGURE 3.2.16 An experimental setup for taking a framing photograph to investigate the performance of the explosive's lens. low detonation velocity and a SEP explosive is used as the explosive with the high detonation velocity. The detonation velocity of the HABW is about 5000m/s. As shown in Fig. 3.2.16, the explosive lens assembly is immersed in a water tank made of PMMA, in an orientation perpendicular to the light source. Using an image converter camera, a self-illuminated picture such as shown in Fig. 3.2.17 is obtained. It can be seen that as the detonation wave propagates, its shape (indicated by the white lines), becomes fiat. Hence, the explosives lens indeed produces planar detonation waves. When this detonation wave collides with the water, an underwater shock wave is transmitted into the water. The experimental setup for recording this process is shown in Fig. 3.2.18. The forementioned observational system has been used. A series of the typical framing photographs of the detonation wave
FIGURE 3.2.17 Self-illuminationsof the main explosive. The white lines indicate the front of the detonation wave.
284
s. ltoh
FIGURE 3.2.18 An experimental assembly of the SEP plate for underwater explosions. The slit indicates the region in which a streak photograph is recorded. propagation in a SEP plate charge is shown in Fig. 3.2.19. The numerical simulation of the process shown in the framing photographs is also shown in Fig. 3.2.19, where SWW indicates the underwater shock front, W is water, EX is the explosive, PP is the PMMA plate, which holds the explosive, and SWP denotes the shock wave emerging from the PMMA plate. As is evident from Fig. 3.2.19 the experimental results are very well simulated numerically. A streak photograph of the process described in the preceding is shown in Fig. 3.2.20. Using the forementioned computerized evaluation technique, the relation between the distance propagated by the shock wave and the propagation time can be obtained. The results are shown in Fig. 3.2.21, where the solid line indicates the numerical results; HP31 and HP55 denote the results of SEP explosives, having thicknesses of 3 m m and 5 ram, respectively. Figure 3.2.21 indicates that the relation between the distance and the time of propagation coincides for all the cases. The pressure ratio across the underwater shock wave can easily be calculated from these data. The results are shown in Fig.
FIGURE 3.2.19 A series of framing photographs of the detonation propagation in a SEP plate charge (upper part); a series of the numerical simulations of the process of the process recorded in the framing photographs (lower part).
3.2 Shock Waves in Liquids
285
FIGURE 3.2.20 A typical streak photograph of a SEP plate charge. The horizontal scaled length corresponds to 2 ~ts and the vertical scaled length to a distance of 50 mm.
3.2.22 (in a logarithmic scale), where the circles indicate the experimental results and the solid line is the numerical simulation. The horizontal axis is the nondimensional distance from the boundary of the explosive to the water. The linear dependence (Fig. 3.2.22) between the pressure ratio and the n o n d i m e n sional distance from the b o u n d a r y of the explosive to the water implies that
P----= P0
A(T+rt-a
(3.2.41)
T
where T is the thickness of the plate charge. The constants A and a depend on the supported material. While the values of A for both PMMA and SUS 304 were found to be the same, that is, A = 6.6 • 104, the values of a were found to be different. They were found to be a - 0.69 and a -- 0.48, for PMMA and SUS 304, respectively.
60 50 t
HP31 9 DI):7Omm ._.'S 40~" : DD-'-5Omm o
HP55 A D0:5 a D~p-3 o Dp'I
~ 2010 2
4 6 Time t (use<:)
8
10
FIGURE 3.2.21 X-t wave diagrams obtained by the streak photographs for SEP plate charge.
286
s. Itoh -o
Numerical calculation S t r e a k photograph : :::~,'i:: 9 '"' ......
lOS
:
: ~ "' i::": I I ,",'ij':,, IIII
I I I IIIII
I r~llll
0 Q.
%1
I
"
' ''"'III
,
"9
9
Il _
I
!!!,!!!L! i l i'~'':'" I,I I'"'""
....
::',:~:i; I I l IIIII ]
I II IIIII
III
i
ILlll
IIIII
I
I I II IIII
I I I IIIII
i
I l l[lllt
I II llill
,oI l llil[llII[IIIII I
10
100
Reduced distance (T+x)/T FIGURE 3.2.22 The relation between the pressure ratio across an underwater shock wave and the nondimensional distance of propagation for a SEP plate charge. The experimental setup for generating underwater shock waves by a short cylindrical SEP explosive is shown in Fig. 3.2.23. An explosive lens was used in order to produce plane detonation waves in the explosive. Framing photographs and the numerically calculated shadowgraph (see Minota, Nishida and Lee, 1996) are shown in Fig. 3.2.24. The computational shadowgraphs were obtained by solving
02P +
~p
02p
+ ~
(3.2.42)
FIGURE 3.2.23 An experimental setup for generating underwater shock waves by a short cylindrical SEP explosive.
3.2 Shock Waves in Liquids
287
FIGURE 3.2.24 A series of framing photographs and computational shadowgraphs of underwater shock waves generated by short-length cylindrical SEP explosives. As can be seen from Fig. 3.2.24, the underwater shock wave that first emerges as a planar wave (see its top part) assumes a curved shape very quickly as it propagates outwards. Thus the disturbances affect the shock waves m u c h more in liquids than in gases. The experimental setup for generating underwater shock waves by a long cylindrical SEP explosive is shown in Fig. 3.2.25. The diameter of the cylindrical SEP explosive is 20 m m and its length is 250 mm.
FIGURE 3.2.25 An experimental setup for generating underwater shock waves by a long cylindrical SEP explosives.
288
S. Itoh
FIGURE 3.2.26 A series of framing photographs of the SEP cylindrical explosive charge.
Figure 3.2.26 shows framing photographs of the explosion of a cylindrical charge in water; EX denotes the undetonated explosive; SW indicates the underwater shock wave trajectory along the slit direction; SL denotes the selfluminous light; and W is the water. The underwater shock wave and the detonation wave in the explosive are seen to move in the same direction. Because the shape of the front of the underwater shock wave is seen to be unchanged, it is obvious that the shock wave velocity is constant in the direction of the detonation. Streak photography is applied in order to deduce the shock velocity along a direction perpendicular to the propagation direction of the detonation wave. A typical streak photograph is shown in Fig. 3.2.27. The vertical axis represents the distance traveled by the shock wave along the streak slit. The horizontal axis indicates the traveling time. Here again EX denotes the undetonated explosive, SL denotes the self-luminous light, and SW indicates the underwater shock wave trajectory along the slit direction. It is evident from the photograph that the slope of the underwater shock wave trajectory slowly decreases with time. This implies that the velocity of the underwater shock wave decreases gradually with time or position.
FIGURE 3.2.27 A typical streak photograph of the SEP cylindrical explosive charge.
3.2 ShockWaves in Liquids
289
70 60
10
/
, 5
~
10
15 20 25 30 Time (IJS) FIGURE 3.2.28 A propagation-distance versus propagation-time wave diagram of the SEP cylindrical explosive charge. A computer image-processing technique was applied on the streak photograph in order to obtain quantitative information regarding the underwater shock wave propagation parameters. The obtained propagation-distance versus propagation-time data are presented in Fig. 3.2.28, where, in addition to the experimental data the results of the numerical simulation are also shown. The agreement between the experimental results and the numerical simulations is seen to be very good. By differentiating the propagation-distance versus propagation-time data shown in Fig. 3.2.28, the underwater shock wave velocity can be obtained. Once the underwater shock wave velocity is known the pressure ratio across it can be obtained. The pressure ratio across the underwater shock wave P/Poas a function of a nondimensional propagation-distance (R + r)/R (R is the radius of the charge) is shown in Fig. 3.2.29. Both experimental results (symbols) and calculated data (lines) also are shown in the figure. The solid line corresponds to calculations with a charge diameter of 10 ram, the dashed line corresponds to calculations with a charge diameter of 30ram, and the dotted line corresponds to calculations with a charge diameter of 50ram. The results indicate that, in logarithmic scale, the dependence of the pressure ratio across the underwater shock wave on the propagation distance is very close to being linear. Using a relation similar to Eq. (3.2.41) the data shown in Fig. 3.2.29 can be described by the following similar expression:
P---=A(R+r)-a p0
(3.2.43)
R
Based on the data shown in Fig. 3.2.29 the constants A and a of Eq. (3.2.43) are A = 6.6 x 104 and a - 1.12.
290
s. Itoh
o ...... ......
Expedment ( ~ 1 O) Calo .latlon ( ~ 1 O) Cal9 ulation (d) 30) Calc ubtlon ( ~ 50)
105~ ~-a
; ~"I =";';;;; 1 I II I":,, IX~t .! ! I 1 III!
l kJ -~ ,-
I I l II!i
llJi
~ ( ~'~ ili
10 4 ~
I
II . I' -'~'' l"t~t~
!
I
!.i IIIIrV',
!
~ I 1 I I rill
I
I
I I 11~
,o 1 I ll]lllil 1
10
;L, ;I '.;~I I I ] 1
1
l l
I
! I
I !! i
I
l~_
;I
illll
Ill
11111
lllll
!11i i iiill[llll I
I
I
1 .I,I
I
I
I
I I
I
I I
! If
I[11
I Illl
II1Ti
i1111
Illll
Ill Jllll lOO
(R+ r ) / R
FIGURE 3.2.29 Pressurechange versus nondimensional distance for the SEP cylinder.
3.2.4 VON N E U M A N N R E F L E C T I O N OF UNDERWATER S H O C K WAVE 3.2.4.1
INTRODUCTION
It has well known that when a shock wave moving in a gaseous phase interacts with a wedge, either a regular reflection RR or the an irregular reflection IR will occur (Ben-Dor (1991); Sasoh and Takayama (1994)). The irregular reflections can be further divided into two categories: 9 Mach reflection MR, which could be subdivided to a single-Mach reflection SMR, a transitional-Mach reflection TMR (see section 8.1.4.5), and a double Mach reflection DMR (see section 8.1.4.6). 9 Von N e u m a n n reflection vNR (see section 8.1.4.7). All the MR wave configurations have sharp triplepoint at which the incident, the reflected and the Mach stem shock waves meet. In the vNR wave configuration the reflected shock wave is replaced by a band of compression waves. As a result a triple point does not exist in a vNR, and the incident and the Mach stem shock waves are connected together
3.2
Shock Waves in Liquids
291
smoothly. The occurrence of an MR or a vNR depends upon both the incident shock wave Mach number and the reflecting wedge angle. Generally speaking, vNR is typical for small shock wave Mach numbers and small reflecting wedge angles. As the analytical model for solving an MR, that is, the threeshock theory (originally proposed by von Neumann, 1943), failed to predict vNR-related experimental results, before the vNR was recognized as a reflection-different form MR, the occurrence of vNR was considered as a "von Neumann paradox". The forementioned difference between experimental observations and theoretical explanations was first reported by Birkhoff (1950). The vNR has been intensively studied both experimentally and numerically in recent years by a number of investigators (e.g., Sakurai et al. (1989), and Colella and Henderson (1994)). Particularly, Colella and Henderson (1994) proposed the transition process and existence criterion of the vNR after investigating both experimentally and numerically the diffraction of weak shock waves over wedges. Ben-Dor (1991) pointed out that if the velocity of the reflected shock wave in an MR becomes equal to the local speed of sound, the onset of the vNR occurs. Sasoh, Takayama and Saito (1992) reported their research results on the occurrence conditions of the vNR, as well as the MR <-+ vNR transition process by employing Whitham's shock-shock theory (1957). As described subsequently, vNR wave configurations are very abundant in the reflection of underwater shock waves. The present author has been devoting considerable effort to exploring a variety of metalworking methods by utilizing underwater shock waves generated by underwater explosion of high explosives. For example, in the course of the study of shock compaction of powders, an assembly for generating underwater shock waves was developed with the aim of using the extremely high pressures behind converging underwater shock waves to consolidate powders. The numerical simulation computation of this compaction process indicated that the underwater shock wave reflection pattern was not an ordinary MR, but a reflection with an extremely curved Mach stem. In addition, Adadurov, Dremin and Kanyel (1969) obtained optical pictures of the wave configuration following the interaction of shock waves generated by implosive explosions of high explosives with PMMA cylinders. A very curved Mach stem was evident in the wave configurations recorded by them. Carton, Stuivinga and Verbeek (1996) who conducted experimental studies of the shock compaction of powders in cylindrical geometry, also presented recently flash X-ray photographs of shock wave configurations with an extremely curved Mach stem. All the preceding results indicate that when shock waves obliquely intersect or reflect off a rigid wall in condensed matter (e.g., liquids, solids or powders), the resulting wave configurations consist of extremely curved Mach stems. The
292
s. Itoh
reason for the appearance of wave configurations with very curved Mach stems is attributed to the relatively high speeds of sound of condensed matters. The high speed of sound implies low shock wave Mach numbers, that is, weak shock waves, which in turn increase the possibility of obtaining vNR rather than MR. It should be noted here that the reflection process of shock waves in condensed matters is yet to be understood. Krehl, Hornemann and Heilig (1977) conducted experimental studies of the interaction of shock waves with carbon tetrachloride (CC14) liquid by flash radiography. Unfortunately, their attention was focused mainly on MR and not on vNR. In the following section the experimental results of optical observations of vNR resulting from the reflection of underwater shock waves, which are generated by detonating highly efficient explosives in water, together with corresponding numerical simulations on the reflection phenomenon are presented.
3.2.4.2
EXPERIMENTAL M E T H O D
The apparatus for generating an oblique interaction of underwater shock waves is illustrated schematically in Fig. 3.2.30. The SEP explosive was used to
FIGURE 3.2.30 An experimental setup for the underwater shock reflection.
3.2
Shock Waves in Liquids
293
FIGURE 3.2.31 Series of framing photographs of the wave configurations, which result from the collision of two plane underwater shock waves.
K~ ot~
e~
~.
FIGURE 3.2.32
Series of computational shadowgraphs of the framing photographs shown in Fig. 3.2.31.
296
s. Itoh
generate the underwater shock waves by detonation. Two SEP sticks, 110 m m long, 5 0 m m wide and 5 mm thick, were placed in a V-shape at an opening angle of 20. A PMMA plate supported the two sticks. In the experiments, the opening angle was varied from 30 ~ to 115 ~ in steps of 15 ~. An explosive lens was mounted at the top end of the two SEP sticks to achieve a planar initiation. The apparatus was placed in a PMMA-made aquarium filled with water. After simultaneously detonating the two SEP sticks at the tip of the V-shape, two underwater shock waves were generated. The two shock waves both approached gradually the central line (subsequently referred to as the reflection surface), where, finally, they obliquely collided with each other to produce a reflection configuration. The photographic system for recording this phenomenon and the numerical method for simulating it were described earlier.
3.2.4.3
W A V E C O N F I G U R A T I O N OF O B L I Q U E
I N T E R A C T I O N OF U N D E R W A T E R S H O C K W A V E S Figures 3.2.3 la to 3.2.3 if show a series of flaming photographs of the oblique collision of underwater shock waves at different angles of collision. The opening angles for Figs. 3.2.31 a, b, c, d, e and f, are 0 = 15 ~ 22.5 ~ 30 ~ 37.5 ~, 45 ~ and 52.5 ~, respectively. The photographs for all the cases were taken in 4-gs intervals. One can see that in the case of 0 > 45 ~ a greatly curved Mach stem appeared during the intersection of the two underwater shock waves. This means that von Neumann reflection took place. For 0 = 15 ~ for which the Mach stem is seen to be almost straight, a typical Mach reflection is evident. Figures 3.2.32a and 3.2.32f present the corresponding computed shadowgraphs of the previously described experimentally obtained configurations (see Figs. 3.2.31a to 3.2.31f) of the oblique collision of the underwater shock waves at different collision angles. The dark lines in the figure represent shock waves SW, RS and MS denote the incident, the reflected and the Mach stem underwater shock waves. In the case of 0 = 15 ~ an obvious reflected shock wave and a straight Mach stem can be clearly identified in the reflection configuration. Consequently, the wave configuration is a Mach reflection. Unlike this wave configuration, in the case of 0 = 45 ~ the Mach stem is seen to have a very large curvature and there is no evidence of a reflected shock. Consequently, the wave configuration is a v o n Neumann reflection. The foregoing results indicate that with large opening angles, the interaction angles between the underwater shock waves are large enough to result in von Neumann reflections.
3.2 ShockWaves in Liquids 3.2.5
APPLICATION
297 OF UNDERWATER
SHOCK
WAVES
3.2.5.1
S H O C K C O M P A C T I O N OF P O W D E R S
3.2.5.1.1 Introduction Shock consolidation is one of the techniques to produce bulk materials from powders. It is pointed out that a crack-free bulk of hard material can be obtained by shock consolidation (see Gourdin, 1986). Shock compaction of powders using cylinders was achieved by using explosives in direct contact with the powder container. While this technique is known to give good consolidation results in the case of soft powders, for example, pure copper and mild steel, it is less successful in the case of hard powders. The disadvantage of this technique arises from the fact that the tensile strength imposed by the rarefaction wave that is generated when the shock wave is reflected off the free end could generate cracks. This rarefaction wave causes a rapid drop in the temperature of the bulk while it propagates through the bulk. This excessive temperature change results in thermal stresses, which finally cause many cracks. In addition, as pointed out by Meyers and Wang (1988), Mach or von Neumann reflections could occur at the center of the powder cylinder in the cylindrical method. This shock wave reflection could cause extremely high temperatures at the center part, and the high temperatures, could in turn, blow out the powder. For these reasons, another consolidation technique for consolidating some difficult-toconsolidate powders using a converged underwater shock wave was proposed by Chiba et al. (1992). Their technique produced longer pressure durations in the powder than other techniques. Crack-free and central-hole-free bulks of Ti-A1 metallic compound were obtained using this technique, which is presented in the following section as an illustration of the industrial application of underwater shock waves.
3.2.5.1.2 Experimental Procedures Figure 3.2.33 shows a schematic illustration of the consolidation apparatus using the underwater shock waves. The apparatus consists of three parts: an explosive container; a water tank; and a powder container. Detonating an explosive by an electric detonator generates a detonation wave. The detonation wave propagates from the top to the bottom of the explosive chamber. When the detonation wave impinges on the water, an underwater shock wave in generated in the water chamber. As the underwater shock wave travels through the water chamber, it interacts with the chamber wall with a cone angle of ~.
298
FIGURE 3.2.33
s. Itoh
Experimental setup for powder consolidation using underwater shock waves.
When it interacts with the wall, it reflects from it either as a Mach or a v o n Neumann reflection, which in turn results in a high pressure. This increased shock-induced pressure and shock duration facilitate the conditions required for the consolidation of the powder. By adjusting the mass and the detonation velocity of the explosive one can easily control the shock pressure and the shock duration in powder. In order to better understand the mechanism of the converging process of an underwater shock wave, high-speed photographs were taken by a standard shadowgraph system and the previously mentioned facilities. For convenience of photography, the experimental setup for the observation was not axisymmetrical but planar. Similar to the axisymmetrical setup, it consisted of an explosive lens, an explosive chamber, and a water chamber. The explosive chamber was 25 mm high and 30 mm wide. The water chamber consisted of two parts: the conical water chamber and the powder container part as shown in Fig. 3.2.33. The conical part had an angle of 20~ SUS 304 steel was used for the water chamber. The whole setup was submerged in a water tank made of PMMA plates for taking the framing photographs. Numerical simulations were also carried out.
3.2
Shock Waves in Liquids
299
A direct pressure measurement using a manganese piezoresistance gauge was also carried out in order to investigate the effects of the converging shock wave on the water chamber. The gauge was 7 m m long, 0.5 m m wide and 6 ~m thick. It was confirmed by means of a calibration using an inclined-mirror method (see Nakamura and Mashimo, 1993), that the gauge had a good accuracy up to 40 GPa. A schematic diagram of the assembly for the pressure measurement is shown in Fig. 3.2.34a. It consists of four parts: a 30-mm
FIGURE 3.2.34 (a) Schematic illustration of the experimental setup and the recording setup and the recording layout for pressure measurement by a manganese piezoresistance gauge. (b) Schematic illustration of the manganese pressure piezoresistance sensor.
300
s. Itoh
diameter and a 70-mm high explosive chamber; a varying cross-sectional water chamber having a conical angle of 20~ a measurement part on which the manganese piezoresistance gauge was placed; and a circuit for signal recording. In order to keep a 5A constant current flow for a long duration, a high-voltage pulse source was used. A DS-8122 dual-beam oscilloscope (100MHz maximum frequency) and an HP 54522 digital oscilloscope (0.Sns minimum sampling time) were used for recording the results. A 2-mm thick circular PMMA plate and a 10-mm thick rectangular PMMA plate were used for sandwiching the gauge (see Fig. 3.234b) in order to insulate it from the water. Consequently, the measured pressure was the pressure in the PMMA 2 m m from the bottom of the water chamber. Therefore, a correction was needed in order to get the real pressure on the bottom of the water chamber.
3.2.5.1.3 Results and Discussion
Framing photographs of the converging process of the underwater shock wave are shown in Fig. 3.2.35. The photographs were taken in 2-gs time intervals. As can be seen, the detonation wave propagated downward in the explosive chamber, and the detonation-produced gas gradually expanded outward (seen in black at the top of each photograph). At a time of 2 gs, the underwater shock wave interacted with the wall and reflected as a curved underwater shock wave. At a time of 4 ~ts an almost plane shock wave can be seen to travel downward. Finally, the largely deformed underwater shock wave entered the powder chamber. This may be the effect of a new underwater shock wave caused by the preceding shock wave in the SUS 304 steel. Therefore, in practice, it is very important that these preceding shock effects be prevented from occurring by the shock consolidation assembly. As seen in Fig. 3.2.33, this was done by separating the powder chamber from the water chamber. A comparison of the pressures induced by the converging shock, as measured by the manganese piezoresistance gauge, and results of the numerical simulation are shown in Fig. 3.2.36. Because the measured pressure is averaged over the manganese foil area, the correspondingly computed pressure is also averaged by means of a numerical integration. The bold line indicates the pressure history as recorded by the manganese piezoresistance gauge. The peak pressure is 17 Gpa, but quickly decreases to 5 GPa after only 1 gs This indicates that a pressure of more than 10GPa is maintained for at least 500 ns, which is very effective for utilizing underwater shock waves. The fine line indicates the numerical calculation results. The maximum calculated pressure of 16.2 GPa, is about 5% smaller than the measured one. The experimentally recorded and numerically computed pressure profiles are very similar. Even though the measured pressure profile is only 0.5 gs long, it can be concluded
r O t~
e~
~~ ~~
FIGURE 3.2.35
A series of f r a m i n g p h o t o g r a p h s of a traveling u n d e r w a t e r s h o c k w a v e in the w a t e r c h a m b e r .
302
S. Itoh
FIGURE 3.2.36 Comparison between the pressure measured by the manganese piezoresistance gauge and the numerical calculation.
that the n u m e r i c a l code well s i m u l a t e s the p r e s s u r e of the u n d e r w a t e r s h o c k wave. F i g u r e 3.2.37 s h o w s the p r e s s u r e c o n t o u r m a p s o b t a i n e d by the n u m e r i c a l calculation at the i n s t a n t w h e n the u n d e r w a t e r s h o c k wave enters the p o w d e r
FIGURE 3.2.37 Numerical calculated pressure contours obtained at the instant when the underwater shock wave enters into the powder chamber.
3.2
303
Shock Waves in Liquids
chamber for two wall angles 0 = 20 ~ and 0 = 30 ~ A 10-mm diameter powder chamber was used in the numerical calculation. In this figure, r indicates the time measured from the moment the explosives were detonated by the electric detonator. The boundary of the dark region corresponds to the head of the shock wave. It is clearly seen that the shock wave in the case of 0 = 30 ~ is more deformed than that for 0 = 20 ~ The pressure histories obtained for both cases are shown in Fig. 3.2.38. The solid lines, indicated by 0.0mm, show the pressure history at the center of the assembly. The dotted lines, indicated by 2.5 mm, show the pressure history 2.5 m m from the center of the powder chamber. Finally, the bold lines, indicated by 5.0mm, show the pressure history at the wall boundary of the powder chamber. Due to the effects of the deformation of the shock wave the pressure difference between the center and the wall boundary is larger for 0 = 30 ~ than for 0 = 20 ~ The maximum pressure, however, is about 1.5 times that obtained for 0 = 20 ~ In order to get higher pressures and smaller pressure differences, the corner of the entrance to the powder chamber was rounded. Figure 3.2.39 shows the numerical contours for the case of 0 = 30 ~ The radii of the rounded corner Rs are 10, 20, and 30 mm, respectively. It is evident from Fig. 3.2.39 that the shock wave becomes flatter as Rs increases. The pressure histories for the formentioned three values of Rs are shown in Fig. 3.2.40. No pressure difference is obtained when Rs is equal to 30 mm. The time duration when the pressure is over 10GPa, is about 1 gs. This time duration and the zero pressure difference are sufficient to successfully consolidate a Ti-A1 metallic compound by the shock wave without any cracks.
40.0 O.Omm 30.0
8=30"
o
....
2 5ram
20.0 ,it.
10.0
0.0 loo Tlme (u seo) FIGURE 3.2.38 Pressure histories obtained at the center, middle position and the wall boundary for each value of 0.
304
S. Itoh
FIGURE 3.2.39 Numerical calculated pressure contours obtained at the instant when the underwater shock wave enters the powder chamber. The Rs indicates the radius of the rounded entrance corner of the powder chamber.
3.2 ShockWaves in Liquids
305
40.0
................. O.Omm 30.0 . . . . 2.5mm 20.0 50mm 8 =30 . ",, ,,,, L~Rs-2Omm -30 o=ov" "~ 8 .
~-" 10.0,, % , ,
,,,,.
Rs=l O m m ~ ,
,
. . . . _-:
10.0 Time
(.
120 sec)
....
'
14.0
FIGURE 3.2.40 Pressurehistories obtained at the center, middle position and the wall boundary for each value of Rs.
3.2.5.2
E X P L O S I V E FORMING BY UNDERWATER
SHOCK WAVES 3.2.5.2.1 I n t r o d u c t i o n Explosive forming is a kind of metalworking technique that employs the shock wave as well as the energy stored in high-pressure gaseous products that result from the detonation of explosives. Recently, it has become popular to use, in addition to high-pressure gaseous products, the high pressure resulting from detonation for metalworking. The detonation-induced pressures are high enough to damage thin plates, sheets and piping elements in direct forming. For this reason the explosives are usually placed at a distance from the workpiece in a suitable energy-transferring medium such as water. The detonation wave propagates into the water and generates an underwater shock wave (see, e.g., Ezra, 1973). Then the underwater shock wave propagates in the energy-transferring medium towards the objective workpiece. It is very easy to control the pressure that is eventually imposed on the workpiece by changing the distance from the explosive to the workpiece. At the same time, the energy-transferring medium, for example, the water, also acts as an insulation from the explosion-generated heat. Explosive engraving is one of the excellent applications of explosive forming. Monroe (1888), whose work is considered the earliest record of this technique, engraved apple and maple tree leaves on an iron plate by explosive-generated pressures. Unfortunately, his technique was not developed further. In 1970, a young researcher, Prof. Fujita 1, started to study Monroe's 1Professor Fujita retired in 1999 from KumamotoUniversity. During his last ten years there he served as Director of the High Energy Rate Laboratory. The name of the laboratory was changed recently to the Shock Wave and Condensed Matter Research Center.
306
s. Itoh
technique. He and his coworkers attempted to engrave a Chinese character on an iron plate using a metal die. Their experimental setup is shown in Fig. 3.2.41. The metal die was attached to the base metal by a double-faced tape so that it would remain in position when hit by the underwater shock wave. Using this procedure, Prof. Fujita and his coworkers obtained a very fine engraved Chinese character, which, unfortunately, was contaminated by a clear trace of the thin double-faced tape. This accidental experience led him to begin investigating intensively underwater forming using soft dies (see Fujita et al., 1993). This technique is popularly known as the Fujita Method. In recent years, paper, cloth, leather and bamboo leaves have been used as the die. A better pressure control was required for the forming technique using soft dies than that used for metal dies. This is the major reason for developing easier means of generating better-controlled underwater shock waves. A brief illustration of underwater explosive forming will be given in the following. The detonation of the explosive placed above the plate to be formed produces a shock wave in the surrounding water that spreads gradually outward from the explosion center. The detonation products, at the first stage, drastically expand due to the initial high pressure and energy. Then, as the pressure in the water becomes higher than that in the detonation products, the contraction of the detonation products begins to increase the pressure in the products, finally resulting in a second pressure wave in the water. This type of motion is repeated several times. The pulsation from the explosion center is governed by the restricting conditions of the surrounding water. When the first shock wave reaches the plate, the plate is accelerated rapidly to a high speed by the very high but short-duration pressure. Because the pressure rapidly drops, the water on the plate is not able to follow the deformation of the plate, thus a cavitation is produced between the water and the plate. However, from the contraction of the detonation products, a second pressure wave appears, which pushes water through the cavitation space, thereby further increasing the speed of the plate.
FIGURE 3.2.41 Underwaterexplosive forming of a copper plate using a metal die. The metal die is attached to the base metal by a double-faced tape.
3.2 ShockWaves in Liquids
307
In this method, the deformation of the plate consists of two phases, one due to initial shock wave, and the other due to the following pressure waves. This deforming process causes the plate to strike the model die a few times, which causes the high pressure in the model die to incur plastic flow and work hardening. As a result, the dimensions of the die are greatly changed, thus affecting the final forming quality. Initially, if sufficient energy is imparted, it will cause a deformation that will force the plate and the model die to be in complete contact. Hence, the surface of the model die is duplicated on the plate and the forming is completed. Consequently, compared with conventional static forming, the explositive forming, as it is a dynamic deformation process, has a different deformation mechanism. Underwater explosive forming is usually done in a water tank or a closed vessel, but it can also be conducted in a water pool or in the sea. In such cases, there is no limitation on the geometry of the forming part and a large-sized part can be formed in practice. However, there are some shortcomings, for example, the working places are limited because of the explosion noise and scattering matter that accompanies the process, and the control of the pressure becomes difficult. In the case of a closed vessel, the forming is conducted in a closed container so that the energy can be better utilized and the noise and the scattering problems may be prevented. Moreover, corresponding to the changes of the forming container and the setting of the explosive, the pressure can be controlled and highly precise forming parts may be obtained. This, however, is somewhat troublesome and time-consuming.
3.2.5.2.2 Outlines of Underwater Explosive Forming Underwater explosive forming generally consists of the following operating procedures. 9 Step 1. Designing the pattern for a model and making the pattern on a cutting sheet. 9 Step 2. Putting the sheet on a steel plate as the forming model. 9 Step 3. Holding the model and a copper plate in place in the forming assembly, practicing the forming operation. 9 Step 4. Cutting the formed copper plate to the desirable shape, etching it with an acid solution, treating the surface with a special substance in order to improve its protectiveness. In Step 1, a pattern is designed and a soft die is made. At this stage, a suitable scanning machine scans the design pattern. The method of engraving the Chinese character "yume," shown in Fig. 3.2.42 and which means "the dream," is described in the following.
308
S. Itoh
FIGURE 3.2.42 The Chinese character "yume," which means "the dream." It was drawn by Prof. Hou Zengshou, Taiwan University of Technology, China. After getting the image data, the data are reversed to b e c o m e a mirror image according to w h i c h the die will be made. T h e n a cutting plotter cuts the reversed paper die. In Step 2, the paper die is pasted onto a steel plate. Figure 3.2.43a shows a n e w technique in w h i c h a triple-layer paper die was made. The b o u n d a r y of the cut-off zone is s h o w n in Fig. 3.2.43b. A paper of only 0.24 m m in thickness was used as the die. (It s h o u l d be n o t e d that paper 0.01 m m thick could be successfully used as the paper die.)
FIGURE 3.2.43 (a) The overall view of the paper die of the Chinese character "yume" pasted on the steel plate (comparison to Fig. 3.2.42 indicates that the character is reversed); (b) An enlarged revised die. (The triple layers can be seen at the boundary.)
3.2 ShockWaves in Liquids
309
FIGURE 3.2.44 The closed vessel. The thin copper plate is seen on the paper die.
At Step 3 the paper die, pasted to the steel plate, is placed into the closed vessel. Then the thin plate to be formed (e.g., a 3-mm thick copper plate) is placed on the die as shown in Fig. 3.2.44. Figure 3.2.45 shows a schematic diagram of the pressure chamber used together with the closed vessel and the arrangement of the detonation cord. A variety of available explosives have been investigated for their explosive characteristics. It was confirmed that the detonating cord was the most suitable
FIGURE 3.2.45 Schematic illustrations of the pressure chamber and the arrangement of the detonation cord: (a) a circular pattern; (b) a swirling pattern.
310
s. Itoh
for underwater explosive forming because it was both cheap and very easy to handle. Figure 3.2.46 shows a typical pressure profile at a point 50 mm below the detonating cord. This pressure value is high enough to deform the thin copper plate. In practice, changing the distance between the explosive and the copper plate controls the strength of the underwater shock wave. All of the chamber is submerged in the water tank prior to firing the electric detonator. Finally, in Step 4, the formed copper plate is cut to the desired shape, etched with an acid solution, and then treated with a special substance to better protect it. The engraving of a Chinese character is shown in Fig. 3.2.47. The very sharp triple-edged contours of the Chinese letter are clearly seen on the copper plate. Figure 3.2.48 shows the finished engravings of a memorial plate presented to the Ben Gurion University of the Negev, Israel on the occasion of signing an academic agreement and friendship cooperation with Kumamoto University. Similar memorial plates were presented to the following universities with which Kumamoto University has established close collaboration: Chinese Academy of Science, China; Texas Tech. University, USA; Kasetsart University,
200
150
100 A
~g r
=z a,,
50
10
-50
Time(/~ s)
FIGURE 3.2.46 The underwater detonation-induced pressure profiles as obtained by a detonating cord. The detonation velocity is about 6300m/s.
3.2
Shock Waves in Liquids
311
FIGURE 3.2.47 A photograph of the engraving of the Chinese character '~mme" (which means the dream). (After surface treatment).
FIGURE 3.2.48 A photograph of the memorial plate, which was presented to the Ben Gurion University of the Negev. An annealed copper plate (365 mm x 365 mm x 0.3 mm) was used. The forming quality seems to be good.
312
s. Itoh
FIGURE 3.2.49
A photograph of the formed Chinese letter koo (heaven).
FIGURE 3.2.50
A photograph of the formed image of an owl.
3.2
Shock Waves in Liquids
313
Thailand; North Carolina State University, USA; Georgia Institute of Technology, USA; University of Alberta, Canada; and Bai chai University of Korea. Two samples of Chinese characters "koo," which means "a Heaven" and Owl are shown in Figs. 3.2.49 and 3.2.50, respectively. Although the Fujita Method has received excellent achievement in fine arts forming, the author wishes to further conduct the research on this aspect.
REFERENCES Adadurov, G.A., Dremin, A.N., and Kanyel G.I. (1969). Zhurn. Prikl. Mekh. Tekhn. Phys. 2: 126127. Ben-Dor, G. (1991). Shock Wave Reflection Phenomena, New York: Springer-Verlag. Ben-Dor, G. and Takayama, K. (1992). The phenomena of shock reflectionna review of unsolved problems and future research needs. Shock Waves 2: 211-223. Bervington, P.R. (1969). Data Reduction and Error Analysis for the Physical Science, Ch.1, New York: McGraw-Hill. Birkhoff, B. (1950). Hydrodynamics, A study in Logic, Fact and Similitude, Princeton, NJ: Princeton Univ. Press. Bridgeman, P.W. (1949). The Physics of High Pressure, London: Bell & Sons, Ltd., 169. Carton, E.P., Stuivinga, M., and Verbeek, R. (1996). A new arrangement for dynamic compaction in the cylindrical configuration. Explomet'95, 29-36. Chiba, A., Itoh, S., Tomoshige, R., Miyazaki, K., and Fujita, M. (1992). Pressure analysis on powder compaction process using converging underwater shock wave assembly. Proc. Intl. Syrup. Intense Dynamic Loading & Its Effects, 768-771. Colella, P. and Henderson, L.E (1994). The yon Neumann paradox for the diffraction of weak shock waves. J. Fluid Mech. 213: 71-94. Dorsey, N.E. (1948). Properties of Ordinary Water Substance, New York: Reinhold Book Co. Ezra, A.A. (1973). Principles and practice of explosive metalworking. An Indust. Newspaper Publ., 1-15. Ferry, J.D. (1970). Viscoelastic Properties of Polymers, 2nd ed., New York: John Wiley & Sons, Inc. Fujita, M., Ishigori, Y.,Nagano, S., Kimura, N., and Itoh, S. (1993). Explosive precision of fine arts using regulated underwater shock waves, in Advances in Engineering Plasticity and Its Application, Amsterdam Elsevier Sci. Publ., B.V. Gourdin, W.H. (1986). Dynamic consolidation of metal powders. Prog. Mat. Sci. 30: 39-80. Heukroth, L.E. and Glass, I.I. (1968). Low-energy underwater explosions. Phys Fluids II 10: 20952107. Itaya, M. (1966). Hydrodynamics (Japanese Ed.), Japan: Asakura Book Co., 16-17. Itoh, S., You, N., Kira, A., Nagoano, S., Fujita, M., and Honda, T. (1996). Fundamental characteristics of underwater shock wave due to underwater explosion of high explosives. JSME J. 62(601): 50-55. Kell, G.S. (1968). Density, thermal expansivity and compressibility of liquid water from 0 ~ to 150 ~ J. Chem. & Eng. Data, 20(1): 97-105. Kinslow, R. (1967). Stress waves in laminated materials. AIAA Paper 67: 140. Korosi, A. and Fabuss, B.M. (1968). Viscosity of liquid water from 2 5 C to 150~ measurements in pressurized glass capillary viscometer. Anal. Chem. 40: 157. Krehl, P., Hornemann, U., and Heilig, W. (1977). Flash radiograph), of unsteady regular and Mach reflection in a liquid, in Shock Tube and Shock Wave Research, B. Ahlborn et al. eds. 303-312.
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Lamb, H. (1932). Hydrogynamics, 6th ed., England: Cambridge Univ. Press. Mader, C.L. (1979). Numerical Modeling of Detonations, Berkeley: UC Berkeley Press, 278-285. Marsh, S.P. (1980). Laser Shock Hugoniot Data, Berkeley: UC Berkeley Press. Meyers, M.A. (1994). Dynamic Behavior of Materials, New York: Wiley-Interscience, John Wiley & Sons, Inc., 126-140. Meyers, M.A. and Wang, S.L. (1988). An improved method for shock consolidation of powders. Acta Metal. 36(4): 925-936. Minota, T., Nishida, M., and Lee, M.G. (1996). Vortex-ring impingement on a wall with shock formation. Shock Waves 1,545-550. Monroe, C.E. (1888). Modern explosives, Scribner Magazine 3: 563. Nakamura, A. and Mashimo, T. (1993). Calibration experiments of a thin manganin gauge for shock wave measurement in solids: Measurements of shock-stress history in alumina. Japanese J. Appl. Phys. 32(10): 4785-4790. Rice, M.H. and Walsh, J.M. (1957). Equation of state of water to 250 kilobars. J. Chem. Phys. 26(4). Rice, M.H., McQueen, R.G. and Walsh, J.M. (1957). Compression of Solids by Strong Shock Waves. Solid State Physics. 6, E Seitz and D. Turnbull, eds, New York: Academic Press. Sakurai, A., Henderson, L. E, Takayama, K., Walenta, Z., and Collera, P. (1989). On the yon Neumann paradox of weak Mach reflection. Fluid Dyn. Res. 333-345. Sasoh, A. and Takayama, K. (1994). Characterization of disturbance propagation in weak shockwave reflections, J. Fluid Mech. 277: 331-345. Sasoh, A., Takayama, K., and Saito, T. (1992). A weak shock wave reflection over wedges. Shock Waves 2: 277-281. Sternberg, H.M. (1987). Underwater detonation of pentolite cylinders. Phys. Fluids 30(3): 761-769. Stott, V. and Bigg, P.H. (1928). International Critical Tables, vol. 2, p. 457, New York: McGraw-Hill Book Co. Vedam, R., and Holton, G. (1968). Specific volumes of water at high pressures, obtained from ultrasonic propagation measurements. J. Acoust. Soc. Am. 43: 108. yon Neumann, J. (1943). Collected Works, 6: 238-308. Whitham, G.B. (1957). A new approach to problems of shock dynamics. Part I: Two-dimensional problems. J. Fluid Mech. 2: 145-171.
CHAPTER
3.3
Theory of Shock Waves 3.3
Shock Waves in Solids
KUNIHITO NAGAYAMA Department of Aeronautics and Astronautics, Kyushu University, Fukuoka 812-8581, Japan
3.3.1 Introduction 3.3.2 Basics 3.3.2.1 Shock Jump Conditions 3.3.2.2 Weak Shock Formulas 3.3.3 Experimental Method 3.3.3.1 Procedure of Shock Wave Generation 3.3.3.2 Measurement Methods 3.3.4 Shock Hugoniot Curve and High-Pressure Equation of State for Solids 3.3.4.1 Empirical Linear Relation 3.3.4.2 Reflection and Transmission of Shock Waves at the Material Interface 3.3.5 Shock Thermodynamics 3.3.5.1 Grflneisen Equation of State for Condensed Media 3.3.5.2 Irreversibility of Shock Compression Process 3.3.5.3 Temperature Calculation 3.3.6 Topics of Applications 3.3.6.1 Elastic-Plastic Shock Waves 3.3.6.2 Wave Splitting by Elastic-Plastic Transition or High-Pressure Phase Transition References
3.3.1 I N T R O D U C T I O N "Solid" is sometimes used to represent the concept of idealized substances whose dynamical properties are the opposite of "fluid." Unlike this concept of Handbook of Shock Waves, Volume 1 Copyright ~ 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0m12-086431-2/$35.00
315
316
K. Nagayama
solids, all solid materials have a very large but finite rigidity. In other words, all solid materials are compressible, although their bulk modulus ranges from tens of Giga Pascal (GPa) to thousands of kilobars. Furthermore, as explained in what follows, solid materials can flow under some conditions, and their dynamics should be described by fluid equations. The very large but finite value of the bulk modulus of solid materials indicates that even solid materials can be compressed appreciably when an instantaneous stress or pressure is exerted at an extremely fast rate. In exactly the same sense as gases, compressible shock waves can be induced in any solid materials. The most impressive difference of the shock waves in solids from those of gases is the very high pressure necessary to produce shock waves in solids. The concept of high compression of solids by high-pressure shock waves was introduced in the 1940s. The most comprehensive study of the theory of shock waves in media with an arbitrary equation of state was published by Bethe (1942). Because solid materials have quite a different equation of state from that of gases, the introduction of shock waves in solids adds another wide dimension to shock wave science and technology (Zel'dovich and Raizer, 1967). However, in some sense, the range of related fields associated with shock wave physics and engineering in condensed media can be considered to be much wider than those in gases. Because the major part of shock wave studies on solid materials has been pursued from the viewpoint of the science of materials and not in the wave behavior itself, most of the shock wave studies on condensed media have been on materials science (Davidson and Shahinpoor, 1997). Although material properties and various macroscopic and microscopic properties of materials can be understood by microscopic terms, theoretical prediction of the properties based on information only on the constituent atoms is not always possible. Thus a variety of the material properties should in principle be studied experimentally. More than a thousand solid materials have been studied so far to determine their shock wave properties. Most of the shock Hugoniot data have been compiled in databases, which can be accessed or purchased in a book or from an electronic file on a disk (Marsh, 1981; van Thiel, 1966; etc.) As explained already, shock waves in solids are always associated with very high pressure states of solids. Therefore, shock wave science has a close relationship with high-pressure physics (Al'tshuler and Bakanova, 1969; Eliezer, Ghatak, and Hora, 1986). Other areas related to shock wave science include the following: the problem of terrestrial or planetary interiors (Whipple, 1947); material synthesis by shock wave loading (Decarli and Jamieson, 1961); shock compaction (Prummer, 1987); explosive welding of metal plates (Cowan and Holtzman, 1963); space debris (Christiansen, 1995); elasticplastic behavior of materials (Mashimo, 1993); high-pressure structural phase transitions (Duvall and Graham, 1977); pulse laser-induced waves
3.3
317
Shock Waves in Solids
(Dlott, 1995); medical applications (Noack and Vogel, 1998; Nakahara and Nagayama, 1999); nuclear fusion (Brush, 1967; Eliezer, Ghatak, and Hora, 1986); etc. Due to the wide variety of materials involved in the science of shock waves in solids, it is difficult to introduce all the aspects of this science by a single author and in limited space. Recently, however, a series of monographs on shock wave phenomena in condensed media has been published, and should be a great help to readers interested in specific problems (Davidson and Shahinpoor, 1997). Progress in the field was reported in a series of Proceedings of the Binenial Conference on Shock Compression of Condensed Matter held in the United States. Hence, instead of explaining or attempting to discuss all the related topics, this chapter will concentrate on several fundamental topics and give an appropriate introduction. Figure 3.3.1 shows the correlation of shock wave science to other areas of sciences. Although shock wave science in solids seems to be an interdisciplinary science, the shock wave propagation process itself is a welldefined fundamental phenomenon. It is often encountered in various situations involving very high-energy density states caused by the very fast deposition of any form of energy, that is, explosion, high-speed impact, laser pulse, etc. In this chapter, elementary concepts of shock compression of solids are explained with special emphasis on the difference from those of gases. Later, how some of the general concepts established in gas shock theory lose their general character will be shown, and several new concepts (new only to gas dynamicist) will be introduced. Among the shock-related fields, fundamental discussion of shock wave phenomena by thermodynamics, microscopic theories, and hydrodynamics will be presented. The vast field of shockpressure-induced phase transitions, material strength-related behaviors (elastic-plastic behavior), and rate-dependent phenomena is omitted due to space limitations. Various areas that are omited in this chapter are mentioned and discussed in the works cited earlier. As a science of waves, shock waves in condensed media contain many interesting topics and problems left to be solved.
FIGURE 3.3.1
Shock wave science is an interdisciplinary science.
318 3.3.2 3.3.2.1
K. Nagayama
BASICS SHOCK JUMP CONDITIONS
Shock wave propagation is governed by flow equations together with equations of state for the substance studied. At the shock front, two states across the front are related by the relations PoUs = p [ u s - up]
(3.3.1)
axx = Po + PoUsUp
(3.3.2)
PoUs s H - s o + - ~ u
- - ax~U p
(3.3.3)
where us, %, p, axx, and e denote the shock velocity, the particle velocity, the density, the xx-component of the stress tensor and the specific internal energy, respectively; 0 and H denote the value at the initial state and the compressed state, respectively. As the shock-compressed state of the condensed substance is a very high-pressure state, the initial pressure P0 can be neglected by comparing the value with that of the compressed state. This is not the case, however, for the reflected shock wave whose initial state itself is a highpressure state. These conservation equations should be derived from the general conservation laws in continuum mechanics. We therefore adopt the xx-component of stress in the momentum equation (3.3.2). The foregoing expressions, however, do not contain the heat flux term, which is included in the generalized energy conservation equation. The shock-compressed state in solids is assumed here to be in a thermally and dynamically equilibrium state. The stress tensor is not, however, hydrodynamic pressure contains some deviatoric stress component due to the finite value of the yield strength even at the high-pressure state. Notice from the discussion so far that the stress ratio or the pressure ratio a x ~ / p o may not be a good parameter to describe shock wave propagation in solid materials. This is in sharp contrast to that for ideal gases. In fact, the pressure ratio is entirely useless in the case of condensed materials. Rather, physically, there is no change in the initial conditions between the atmospheric conditions and the evacuated conditions (vacuum). Thus, generalization of the shock wave theory of gases in terms of the pressure ratio fails in this sense. Hence, we need another common concept of a nondimensional parameter to represent shock strength in universal terms, although such work on this topic has not been completed yet (Nagayama, 1994a). One of the major objectives of the solid shock experiments is to explore the high-pressure equation of state for the material, because the attained high-
3.3 ShockWaves in Solids
319
pressure state by shock compression, that is, the Hugoniot state, is well defined by Eqs. (3.3.1) to (3.3.3). By measuring some of the shock wave parameters arising in the jump conditions, it is possible to specify the attained highpressure states in terms of the state variables arising in the shock jump conditions. It is therefore possible to obtain information on the form of the equation of state of the material under the high pressure in the form
F(p, v, ~) = 0
(3.3.4)
This combination of variables for the equation of state is not coincidental, but has clear physical meaning. They are all mechanical variables, although the internal energy has a thermal aspect as well. In this sense, no explicit information can be obtained on the thermal variables such as temperature or entropy by the measurement of the characteristics of shock wave propagation. This is the consequence of the fact that the shock wave is driven by a break in the mechanical variables. Even so, thermal aspects of materials have a sufficiently large influence on the propagation behavior. Thus it is of fundamental importance to discuss thermal contributions to the equation of state for solids at very high pressures and temperatures reached by strong shock compression.
3 . 3 . 2 . 2 WEAK SHOCK FORMULAS It is very important to develop weak shock theory in order to elucidate the basic ideas and properties of shock waves in condensed media. However, the shock waves realized in the experiments are sometimes too strong to be described by weak shock theory. In a later formulation, the xx-component of the stress tensor is assumed to be equal to hydrostatic pressure. Most of the formulation is quite general for arbitrary fluids, and such a formulation was introduced by Bethe (1942). The increase in pressure ApH, that in internal energy Ae H, and those in other variables are assumed to be expressed by the Taylor series expansion in terms of the increase in volume Av. In weak shock approximation, the change in entropy is calculated to be the third order of the volume change by
T~
--
1 (~2p~ (Av) 3
1-2 \ ~ , J So
(3.3.5)
320
K. Nagayama
where To and AS denote the initial temperature and the change in entropy, respectively. Up to the second order of volume change, the pressure and energy change can be expressed as
Ap- ( Ae-
SoAv q- -~1(02P~\~v2~ ] So(Av)2 ~0e s0AV+ ~ \0V2]s0 (Av)2 -
(3.3.6)
-p~
- -~ -~v So
These equations all are valid for any fluids with an arbitrary equation of state. For the compression shock wave to be stable, the change in entropy across the shock front should be positive, because the shock transition is both an adiabatic process and a spontaneous process. The necessary and sufficient condition of this is
(
02P~ > 0
(3.3.8)
which is derived from Eq. (3.3.5). This result is satisfied for almost any fluids except for the region of phase transition. Rarefaction shock possibility has been discussed for the liquid-gas phase transition. This type of anomalous behavior can also be realized by the structural phase transformation from the highpressure phase to the lower-pressure phase of various solid materials.
3.3.3 EXPERIMENTAL
METHOD
3 . 3 . 3 . 1 PROCEDURE OF SHOCK WAVE GENERATION As explained in the Introduction, very high-energy density has been realized to produce high-pressure shock waves in condensed media. Methods of generating shock waves in condensed media can be divided into two procedures. One is the direct contact of some media in a high-energy density state with the specimen. The high-energy density media can be a high-temperature plasma, gaseous products of high explosive charges, etc. (Keeler and Royce, 1971). The other procedure involves the impact of a high-speed projectile accelerated by some means. High-speed impact is a process whereby at least half of the kinetic energy of the projectile is converted into the internal energy of the collided material, which brings it to a high-energy density state. Energy sources that can generate high-energy density flux of gas flow or electric power or light beam (Dlott, 1995) or particle beam, and nuclear explosion are candidates for shock wave generation. In other words, the explosion of high explosive charges,
3.3 ShockWaves in Solids
321
FIGURE 3.3.2 Schematicillustration of a compressed gas gun. electrical explosion of foils, pulse lasers or particle-beam generators can be used to induce shock waves in condensed media. If it is used to accelerate a kind of projectile to high velocity, it is also possible to realize the second type of shock generation. A propellant can be used to accelerate a projectile to high velocity. This is one of the very common and established means of shock generation, although a propellant cannot realize the high-energy density flux of gases compared with other energy sources. However, a volume of compressed gas can accelerate projectiles (Kinslow, 1970; Nagayama, 1993). A device designed to accelerate a projectile and to perform the impact experiment is called a gun. A compressed gas gun, a powder gun, and a two-stage gas gun are established experimental devices for generating plane shock waves (Fowles et al., 1970). Figure 3.3.2 shows the schematic of a gas gun used to accelerate a projectile for impact shock studies (Nagayama, 1993). To generate a plane wave of high-pressure shock waves in solids and to measure the shock propagation, a relatively thin plate should be used. Figure 3.3.3 shows the schematic of the impact shock experimental assembly. A flyer plate is the nose part of the projectile to be collided with the target assembly. The target assembly consists of the driver plate, specimen plate, etc. A precisely controlled head-on collision of the flyer with the driver plate induces a highpressure plane shock wave within the driver plate and flyer plate. All of the components of the experimental assembly should be plates, and not rods. This is to realize the strict 1D strain condition. Compared to the case of shock tube experiments for gases, no rigid wall exists for any solid materials. The 1D strain
FIGURE 3.3.3 Conceptof shock wave experiments by high-speed impact.
322
K. Nagayama
condition to ensure the plane shock wave condition can only be satisfied by finalizing the measurement before the rarefaction wave from the free surface of the side peripheries of these plates reaches the measuring area (central area) of the assembly. The normal thickness of the specimen is several millimeters, which indicates that the shock generation time is a microsecond or less. To obtain high-precision shock wave parameters, high temporal and spatial precision is required.
3.3.3.2
MEASUREMENT METHODS
One of the major objectives of shock compression experiments is to obtain information on the high-pressure state attained by shock compression. The simplest way to do this is to measure the two shock wave parameters in Eqs. (3.3.1) to (3.3.3) in one shot. In many cases, the shock velocity and particle velocity are measured simultaneously. As explained later, the time history of some physical variables is also measured. This kind of measurement is beginning to be a must for shock compression measurements, as profiles of the quantities contain information on the dynamic processes occurring inside the shock rise. Such information includes, for example, elastic-plastic transition, structural phase transition at high pressure, relaxation structure due to the stress relaxation, etc. As explained in the previous subsection, the duration of the phenomena is about a microsecond or less, and accurate measurement of the phenomena requires a high time resolution of nanoseconds or higher. In laser shock experiments, whose duration is in the range of tens of nanoseconds or shorter, the time resolution should be picoseconds or shorter. Many experimental methods have been developed to measure shock and particle velocity (Cheret, 1992). Among these, several are still being used in shock compression experiments. They are divided roughly into optical measurement methods, and electric signal measurements. Several kinds of in-material gauges were developed to measure the stress profile with time (Graham, 1992). Most of the gauge materials were developed to minimize the temperature effect, and a major part of the gauge response to shock waves is the stress-induced strain of the gauge material. As these gauges are embedded within the specimen plates, the stress history along a particle is recorded. Figure 3.3.4 shows a typical example of the signal of a Polyvinylidene Fluoride (PVDF) gauge for shock waves in polyethylene specimen (Mori and Nagayama, 1998). In this case, polymer material exhibits a stress relaxation behavior, which resulted in the very fast rise at the shock front followed by a slowly approaching relaxation structure (Barker and Hollenbach, 1972).
3.3 ShockWaves in Solids
323 0.4
.
t
+
!
9
~'
v .- i v - - --
0.3 =
0.2
.....
(1.1
' 0
-
-.-1.
,
,
,
i
i
,
,
|
1
,
.....
J ....
2
3
4
Time [~sf FIGURE 3.3.4 Recordedstress profile for a relaxing polymer by a PVDF in-material gauge.
The most established and popular method of monitoring the particle velocity history is the VISAR method introduced by Barker and Hollenbach (1972). VISAR is an acronym for velocity Interferometer System for Any Reflector; its fundamental experimental layout is shown in Fig. 3.3.5. Physical principle of the method is a wide-angle Michelson Interferometer with delay leg. Reflected light by the surface is divided into two paths, one of which is delayed through a delay leg. Interference of these two beams provide information on the change in the Doppler shift due to the moving velocity of the surface. VISAR can measure the acceleration-sensitive signals from which very precise velocity history of the surface can be determined. One of the conventional methods of registrating free surface motion is the inclined mirror method (Keeler and Royce, 1971). Figure 3.3.6 shows a modified system of a similar technique (Mori and Nagayama, 1998). In this
FIGURE 3.3.5 Illustrationof velocity interferometersystem for any reflector.
324
K. Nagayama
FIGURE 3.3.6 Concept of a prism technique to record shock wave parameters.
case, the pulsed light source (in this case, a long-pulsed laser) illuminates the optical prisms, at the bottom of which light is reflected totally. Reflected light beams are recorded by a high-speed camera. Light intensity changes by the arrival of a shock wavefront at the prism bottom face, and reflectivity changes due to the totally reflected light to the partially reflected light. Figure 3.3.7 shows a typical example of the recorded streak photograph. In this experiment, there are three prisms, two of which are placed on the driver plate, and a large inclined prism is set above the specimen. The flyer, driver and specimen material are all polyethylene. From Fig. 3.3.7, one can see an instantaneous
FIGURE 3.3.7 Recorded shock arrivals and free surface displacmeent history by the prism method.
3.3 ShockWaves in Solids
325
intensity change by the shock arrival and a slightly curved free surface displacement, indicating the velocity relaxation behavior. Because most of the solid materials are opaque, one of the simplest ways of observing the shock front is by flash x-ray apparatus (Jamet and Thomer, 1976). Such a picture is quite intuitive, but does not give high-precision data for the shock wave parameters. Many other efforts have been made to measure variables other than shock wave parameters. A very special x-ray apparatus was used to record the flash xray diffraction from the compressed lattice, which shows the changing lattice configurations under the shock wave compression process. Although temperature is very important among various thermodynamic variables, the value at the shock-compressed state of solid materials is not easy to measure. Moreover, we have no reliable experimental method of determining shock temperature applicable to any shock conditions. In some restricted experimental conditions, radiation measurement from shocked solids are found to give reasonable temperature estimates.
3.3.4 SHOCK HUGONIOT CURVE AND HIGHPRESSURE EQUATION OF STATE FOR SOLIDS 3.3.4.1
EMPIRICAL LINEAR RELATION
For many solid substances and over wide ranges of pressures, the following empirical relationship is known (Zharkov and Kalinin, 1971; Kinslow, 1970): (3.3.9)
u~ = A 4- Bup
where A and B are material parameters. This relationship is for plastic shock waves, but it is also valid in relatively weak compressions; the parameter A is expected to play the role of sound velocity. Experimental values of A, however, does not coincide with those of longitudinal sound velocity. Further, A is related to the hydrostatic compressibility through the relation K
K4-4G
p-
.
4 G
4 2
(3.3.10)
5
which is a square of the velocity of a virtual longitudinal sound wave due to hydrostatic compression. We call this the bulk sound velocity cb whose value can be estimated by Eq. (3.3.10) or from the data of hydrostatic compression of the material. Parameter A is found to be in good agreement with the bulk sound velocity in many materials. Parameter B represents the nonlinearity of the dynamical properties of materials. The empirical formula, Eq. (3.3.9), is
326
K. Nagayama
then considered to be a kind of expansion approximation in terms of particle velocity under the weak shock approximation. The empirical formula is known to hold well over the convergence limit of the weak shock approximation. Various attempts have been made to understand this universal behavior (Rodean, 1977). However, no sufficient physical explanation of the empirical formula has yet been reported. By combining Eq. (3.3.9) with the shock jump conditions, Eq. (3.3.1) to (3.3.3), the shock Hugoniot compression curve can be given by --
PH
(3.3 11)
p ~
(1 - Br/)2 v
17= 1
(3.3.12)
Vo
In this expression, it is assumed that the shock-compressed state is in hydrostatic compression, and the xx-component of the stress tensor is set to the hydrostatic pressure. This assumption is valid for the case where the shock pressure is much larger than that of the yield strength of the material at that pressure. Figure 3.3.8 shows the periodicity of the parameters A and B for elements. Parameter A, the bulk sound velocity, shows clear periodicity, while parameter B has the value in a very narrow range. Figure 3.3.9 shows the pressure-density Hugoniot compression curve for selected materials. All of these curves are calculated based on the measured value of A and B together with the initial density of materials. All Hugoniot curves are plotted up to the pressure value,
10
.
.
.
.
.
.
.
.
|
Us=A+Bup
_
.............
e
Zn . . . . . . . . .
,
,
i
20
,
,
.
.
........................
~ ..................................
i i
......
,
.
.
]
h i ........ , ......................................
~ ~
0
.
i
..................
o
.
,
Sb i ~ La i
'.9 . . . . . . . . . . . .
J
40
~ .......................
,
.............
,
i
~ ~...4_ ......................................
,
......... --?-~u..
~-.~
,
,
60
................................
, i "Pb
,~i
........
,
..... ~~
J
80
,
,
...........
~
100
Atomic number Z FIGURE 3.3.8 Periodicity of the material parameters in the empirical linear relation between shock and particle velocity.
3.3
327
Shock Waves in Solids 300
......
i
250 I
200
150
iiiiii
100
,,~
50
0 1.0
1.2
1.4
1.6
1.8
2.0
P/P0 FIGURE 3.3.9
Pressure-density Hugoniot compression curve for selected materials.
where the empirical linear relation cannot apply due to the lack of data or to phase transitions. As is seen from Fig. 3.3.9, for the same value of density ratio, higher shock pressure is generated for heavy materials, or materials of high shock impedance. The concept of shock impedance will be defined in the following section.
3.3.4.2 R E F L E C T I O N AND T R A N S M I S S I O N OF S H O C K WAVES AT THE MATERIAL INTERFACE In sharp contrast to the case of gases, solid materials have a sharp material boundary, and it is a very easy task to put several materials together with a well-defined interface. A shock wave induced in one of the materials can propagate into another adjacent material through the interface. This phenomenon is used in several of the experimental methods as explained in the previous section. Here, several concepts concerning the wave reflection and transmission are explained. Reflection and transmission of shock waves at the material interface can be understood almost by the reflection and transmission of sound waves. One of the key parameters of a sound wave in a medium is acoustic impedance, which is defined as the product of density and sound velocity. At the interface of two materials with the same acoustic impedance value, the sound wave transmits with no reflected waves. This is called
328
K. Nagayama
impedance matching. Extending the concept of acoustic impedance, shock impedance Z is defined as
Z-
(3.3.13)
poUs
As in the case of sound waves, the shock wave transmits into the other materials without reflection if the shock impedance of materials is equal. Except for the instant of shock arrival only at the interface, the x.x-component of the stress tensor and particle velocity normal to the interface should have the same value. Change of state by the wave interaction at the material interfaces can be considered by drawing the possible wave reflections to occur at the interface. This should be done in the plane of the particle velocity versus stress. Figure 3.3.10 shows the pressure-particle velocity Hugoniot compression curve for the same materials as those plotted in Fig. 3.3.9. This plot is useful in estimating the shock pressure produced by the high-speed impact of two materials, whose Hugoniot curve is shown in this plot. It is also possible to choose the impactor material whose Hugoniot curve should be replotted starting at the point of the impact velocity up = u o and p - 0, and by plotting the curve to the velocity-decreasing direction. The crossing point should be the realized compressed state. Such a plot is invaluable to the experimenter in designing the experiment.
300
250
I
....
- ..................................................... W.. i .......................... ~. ....................................................
200
i 150
~b
i
i
...............................i ...........................i ........................ i ............................~l ...................
lOO
................................i........................ i ................................ ~...........................i........~~ ...........
%
,.~
50
......................... J
o 0.0
~.o
]
.......................... ......................... i...............
2.0
3.0
~ ................. At..
4.0
~.o
Particle velocity - km/s FIGURE 3.3.10
Pressure-velocity Hugoniot curve for selected materials.
3.3
329
Shock Waves in Solids
3.3.5
SHOCK THERMODYNAMICS
3 . 3 . 5 . 1 G R U N E I S E N E Q U A T I O N OF STATE FOR CONDENSED MEDIA Besides shock wave parameters arising in the shock jump conditions, other thermodynamic variables are often required to determine their values. These are temperature, entropy, thermal internal energy, specific heat, etc. However, the most important parameter in the equation of state for condensed media at high pressure and temperatures is the so-called Gl~neisen parameter, which is defined by the following thermodynamic equation (Gr(lneisen, 1926):
/;
Although various efforts have been made to measure the value of the Gruneisen parameter at the shocked state, the precision of the data is not very good. The volume dependence of the eigenfrequencies of lattice vibration is by definition a quantity that is equivalent to the Gruneisen parameter, and various microscopic theories and calculations have been published to evaluate the parameter at arbitrary high-pressure conditions. Because the parameter can be expressed by other measurable thermodynamic quantities such as the bulk modulus, the specific heat, and the volume expansion coefficient, the temperature dependence of the Gl~neisen parameter was calculated at least at atmospheric pressure conditions. Based on thermodynamic measurements, the Gruneisen parameter is almost independent of the temperature 7 -- y(v)
(3.3.15)
to a high temperature near the melting point of the material (Steinberg, 1981). Based on this empirical law, together with the definition, Eq. (3.3.14), the pressure-volume-energy relation can be written as p = pc(v) +
~(v)
[~ - ~c(v)]
(3.3.16)
where pc(v) and ec(v ) are cold pressure and cold internal energy, respectively, and which are the values of pressure and internal energy at zero temperature Kelvin. In this equation, we have three material functions p~(v), ec(v), and 7(v), which we must know. Functions of cold pressure and energy are the isotherms at zero Kelvin, and they are also isentropes at S = O. Therefore, we have another relation p~(v) -
dec(v) dv
(3.3.17)
330
K. Nagayama
By using this relation, we need to know two functions ec(v ) and 7(v). If the Gl~neisen parameter is given by a function of volume, the whole state surface can be determined by the measurement of only one of the thermodynamic paths, that is, an isotherm, an isentrope, or a Hugoniot. If we have a measured Hugoniot curve, we get
pH(v) -- pc(v) + 7(v) [eu(V) _ ~c(v)] /)
(3.3.18)
which is obtained by applying Eq. (3.3.16) to the Hugoniot state. In this formulation of the equation of state for solids at high pressures, the Gn)neisen parameter is a key to determine the entire functional relationship of Eq. (3.3.18). The Grfineisen parameter is defined in microscopic terms as d In vj
=
(3.3.19)
dlnv
where vj denotes the lattice vibration frequency of the jth mode (Eliezer, Ghatak, and Hora, 1986). Although this depends on the mode itself, it is proved to be an identical definition as Eq. (3.3.14) under the assumption of Eq. (3.3.15). Equation (3.3.19) states that the value of the parameter changes according to the change in volume. As the lattice vibration frequency should be given by the change in the form of lattice potential function, function 7(v) should be determined by the behavior of ec(v). Several model theories have been proposed for the behavior of 7(v) as a function of the lattice potential function: d2(pcv 2m/3) "~(v) . . . .
d(13r
,
(3.3.20)
dv where m = 0, 1, 2, which corresponds to the Slater model (Slater, 1939), the Dugdale-MacDonald model (Dugdale-MacDonald, 1953), and the VaschenkoZubarev model (Vaschenko and Zubarev, 1963), respectively. If the shock wave data are available, functional relations pH(v), ell(v) are known. Because the relationship between cold components is given by Eq. (3.3.17), the unknown functions in Eq. (3.3.18) are 7(v) and ~c(v). By combining Eq. (3.3.17) and (3.3.18) with one of the model theories for 7(v), it is possible to calculate the volume behavior of ~,(v). From such calculations, 7(v) is found to decrease with compression. Because experimental determination of function y(v) was found to be quite difficult, theoretical determination of 7(v) is still very important.
3.3 ShockWaves in Solids
331
For a given set of state variables for a high-pressure state, say, (p, v), internal energy at the state can be calculated by
-- eH(V) + p -- PH(v-----~) PY
(3.3.21)
This calculation is applied to the states off the Hugoniot. As the equation of state contains only p, v, and e, only these variables can be calculated by this method. We need additional assumptions to determine other variables such as temperature or entropy. For very high-pressure shock waves in solids, an appreciable amount of electronic excitation will make an increasing contribution to the thermal energy. At extreme pressures, it will be the dominant part of the thermal energy. In this case, the so-called three-term equation of state is assumed:
p = pc(V) + pT(V, T) + Pe(V, T)
(3.3.22)
e = ec(v ) + ~,r(v, T) + ~,e(V, T)
(3.3.23)
where Pr and er are the second term of Eq. (3.3.16) and the quantity inside the parenthesis, respectively, and are called thermal pressure and internal energy due to lattice vibration. The third terms of Eq. (3.3.22) and (3.3.23), that is, p~ and ee, are the pressure and internal energy due to the electronic excitation. The electronic Gruneisen parameter is also assumed as
pe(v, T) --7e ee(V ' T) v
(3.3.24)
Finally, the shocked state at extreme pressures is described by the quantum statistical atom model.
3 . 3 . 5 . 2 IRREVERSIBILITY OF S H O C K COMPRESSION PROCESS Shock compression of materials leads to the irreversible production of entropy, which is the cause of shock heating of substances. Basically, volume compression lowers the freedom of spatial distribution of constituent particles, resulting in the decrease of entropy. Even so, entropy increases due to shock compression. This means that the increase in the specific internal energy by shock compression should have a larger effect on the increase in entropy than the decrease in volume. For solid materials, lattice configurational energy increases by volume compression, which is not the thermal energy. This increase in energy is neither associated with the increase in entropy, nor is it associated with the increase in temperature.
332
K. Nagayama
Entropy increase by shock compression along the Hugoniot curve is described by the following equation: 2 dig s
THdSH _ up u s dup dup
(3.3.25)
which is derived only by using the conservation equations and thermodynamics. Combining this equation with the empirical linear relation, Eq. (3.3.9), and integrating over particle velocity, we obtain
TdS -- ~
[M2 - 4M 4- 3 4- 2 In M]
(3.3.26)
where we define the so-called shock Mach number M -- us A
(3.3.27)
by using the bulk sound velocity. This parameter is found to work well as a reasonable parameter for defining the strength of shock waves in condensed media, although the sound velocity used in the definition is the one where there are no waves that do not propagate. Simple evaluation of the irreversibility of shock compression is possible by dividing the quantity fi-i TdS by the total intemal energy increase A~H:
flq Td___~S_-- M 2 - 4M 4- 3 4- 2 In M AeF/
(3.3.28)
( M - 1) 2
Although the quantity fH TdS is not equal to the irreversible heating of materials, it is found to be close to it. It is noticeable that this approximate formula contains only the Mach number and no material dependence. Although the shock pressures in the condensed substance realized in shock experiments are very large, ranging from several GPas to hundreds of GPas, the shock Mach number has a relatively low range. Most experiments are categorized and limited to those shock pressures whose shock Mach number is less than 2. This still corresponds to a very strong shock wave in condensed media.
3.3.5.3
TEMPERATURE CALCULATION
Because temperature at the shocked state, that is, the shock temperature, is very difficult to measure, theoretical estimation of temperature at the shocked state is still a reliable method of obtaining information about the thermal properties of materials under shock compression. Nevertheless, we need to know several other quantities except for the shock Hugoniot curve. One is the
3.3 ShockWaves in Solids
333
Gr~neisen parameter at high pressure 7(0, and the other is the specific heat at constant volume C v and its volume dependence. Temperature is calculated by integrating the thermodynamic relationship along the shock Hugoniot curve, as the Hugoniot curve is the most reliable thermodynamic path of the high-pressure region known for most materials. For these theoretical calculations to have sufficient accuracy, the Gruneisen parameter and the specific heat should be known or measured under high pressure. Normally, however, such data are quite limited. Rather, as previously stated, the Gr~neisen parameter itself is found to be one of the most difficult quantities to measure at high pressures. In this sense, high-pressure behavior of these quantities are assumed theoretically. For the specific heat at constant volume, the Debye model is commonly adopted. This model is successful in predicting the temperature behavior of specific heat for various materials. This model cannot explain the measured lattice vibration frequency spectrum of materials, but the success of the model may suggest that the specific heat value is quite insensitive to the detailed frequency mode spectrum. It is assumed that this situation holds even at high shock pressures. Another assumption is the volume behavior of the Gruneisen parameter. The Gruneisen parameter is known to decrease by compression. In many calculations of shock temperature, the very simple assumption P? -- PoTo = const.
(3.3.29)
is adopted instead of complicated model theories described by Eq. (3.3.20). In any case, one cannot derive an analytical expression for shock temperature even if Eq. (3.3.29) is assumed. Temperature is calculated by using the thermodynamic relationship dT T
--
p~dv
dS +
--
Cv
dO --
~
dS + - -
0
(3.3.30)
Cv
where | denotes the Debye temperature. Nagayama introduced new thermal variables in the framework of the Gruneisen equation of state that lead to a new simple algorithm of calculating shock temperature and the Grfmeisen parameter (Nagayama, 1994b; Nagayama and Mori, 1994; Nagayama, 1997). This calculation of shock temperature contains only the contribution from the lattice vibration. For very high-pressure shock waves in solids, an appreciable amount of electronic excitation will have an increasing effect on the thermal energy. At extreme pressures, it will be the dominant part of the thermal energy. In this case, we have to use Eqs. (3.3.22) and (3.3.23) instead of Eq. (3.3.16) as an equation of state.
334
3.3.6
3.3.6.1
K. Nagayama
TOPICS
OF APPLICATIONS
ELASTIC-PLASTIC SHOCK WAVES
Impact shock wave experiments could realize precisely controlled plane shock compression. As explained before, the produced shock wave compresses the material with the condition of ideal 1D strain. This condition is exactly the same as that realized by the shock tube instrument for gas shock waves. In this sense, shock-compression data are a very unique and reliable source of information on the mechanical behavior of materials under purely uniaxial compression condition. Shock compression, however, is special with regard to the following: (i) the strain rate is very fast; and (ii) the stress-strain curve is complicated due to the effect of compression. Above some limit of the value of the deviatoric stress, the material yields, and the stress-strain relation curves downward. The point of deflection of the Hugoniot curve is called the Hugoniot elastic limit. The value of stress at the point should be a material parameter to specify the elastic-to-plastic transition. Figure 3.3.11 shows a schematic illustration of the elastic-plastic Hugoniot compression curve. In this case, the elastic wave and the plastic wave propagate at different velocity. Apparently, the propagation velocity of the elastic wave should be equal to the longitudinal sound velocity, which is always larger than plastic-wave velocity. For the elastic shock wave, stress and strain at the shock front can be given by
cr~ - (,~ + 2~)e~ Cryy -- Crzz = 2e~x
%
(3.3.31)
hydrostatic isotherm goniot elastic Limit
.....
0 FIGURE 3.3.11
Vo
~V
Elastic-plastic Hugoniot curve and the point of Hugoniot Elastic Limit (HEL).
3.3
335
Shock Waves in Solids
where cr,o,, ~yy, ~rzz denotes components of the stress tensor, exx denotes the 1D strain component, and 2, # denotes the Lame constant. If yielding occurs behind the elastic precursor wave, the shock yield stress Y can be given by Y = 2/~exx
(3.3.32)
This result is obtained by assuming either the maximum shear stress or von Mises criterion for yielding due to the condition of 1D strain. Hydrostatic pressure at the wavefront is defined as P - ~ (Crx~ + ffyy "+- fiZZ) --
~ "~--3 ].l gxx
(3.3.33)
Departure between the Hugoniot curve and the hydrostatic compression curve is calculated by using Eqs. (3.3.31) and (3.3.33) as ~
4 2 -- p + ~ ~e~ - p 4- ~ Y
(3.3.34)
Yield strength has a rate-dependent parameter and a pressure dependence as well. It is found that Y is an increasing function of hydrostatic pressure. The shock-compression process is characterised by the very large strain rate. Thus the propagation of the elastic-plastic wave is a very complicated phenomenon strongly dependent on material (Mashimo, 1993).
3.3.6.2 W A V E S P L I T T I N G BY E L A S T I C - P L A S T I C T R A N S I T I O N OR H I G H - P R E S S U R E PHASE TRANSITION At high pressures, structural phase transition will take place in various materials. Lattice structure changes to a more stable configuration for the pressure and temperature realized by strong shock compression. In such cases, the shock Hugoniot compression curve can be seen as in Fig. 3.3.12. A similar situation is realized by the elastic-plastic transition as well. According to thermodynamics, the isotherm around first-order phase transition in the pressure-volume plane has a horizontal line segment at the mixed-phase region. Shock Hugoniot in this region should have a negative slope due to the temperature and entropy increase caused by decreasing volume. From the conservation equation of continuum mechanics, change of state at the shock-front rise occurs along the so-called Rayleigh line. The value of the shock velocity is proportional to the square root of the slope of the Rayleigh line. If the Hugoniot curve has a shape such as shown in Fig. 3.3.11, the shock wave splits into two waves, with the weaker wave propagating with the velocity that is determined by the slope between the initial point and the deflection
336
K. Nagayama
PH 9
high-pressure phase
_
D
~j
transitionpoint B
mixedregion
V
O FIGURE 3.3.12 formation.
Vo
Hugoniot curve for a material undergoing a high-pressure polymorphic trans-
point, and the succeeding wave propagating with the velocity determined by the slope between the deflection point and the final compressed state. In Fig. 3.3.12, the final state is assumed to be state C. The propagation velocity of the fast (and first) wave UsB is given by
l~sB ~
VO
V/DO PB ~ UB
(3.3.35)
and the propagation velocity of the second wave Usc then given by (3.3.36) where the term on the right-hand side denotes the particle velocity of the fast wave. The shock velocity of Eq. (3.3.36) is the one viewed from outside the process. Particle velocity is the flow velocity in which the second wave propagates. Such a splitting takes place when the condition
~/ PB > PV~I-- PB i) 0 ~
1) B
~
(3.3.37)
i) C
is valid. This condition holds for high-pressure phase transition or elasticplastic transition. This double-wave structure disappears when the final pressure reaches the point C in Fig. 3.3.12, where the slope from point A to B and that of B to C are equal. The double-wave structure is observed in the shock pressure smaller than Pp.
3.3
Shock Waves in Solids
337
ACKNOWLEDGMENT T h e a u t h o r w i s h e s to t h a n k Mr. M o r i for useful d i s c u s s i o n s a n d o t h e r assistance in c o m p l e t i n g this b o o k . He also w i s h e s to t h a n k the e d i t o r for p r o v i d i n g the o p p o r t u n i t y to w r i t e this chapter.
REFERENCES Artshuler, L.V. and Bakanova, A.A. (1969). Electronic structure and compressibility of metals at high pressures. Sov. Phys. Uspekhi [English Translation] 11: 678-689. Barker, L.M. and Hollenbach, R.E. (1972). Laser interferometer for measuring high velocities of any reflecting surface. J. Appl. Phys. 43: 4669. Bethe, H. (1942). Theory of shock waves in a medium with arbitrary equation of state, original paper in Report republished in Classic Papers on Shock Compression Science, J.N. Johnson and R. Cheret, eds., London: Springer, (1998), pp. 421-492. Brush, S.G. (1967). Progress in High-Temperature Physics and Chemistry, vol. 1, C.A. Rouse, ed., Oxford: Pergamon Press, p. 1. Cheret, R. (1992). Detonation of Condensed Explosive, New York: Springer-Verlag. Christiansen, E.L. (1995). Hypervelocity impact testing above 10 km/s of advanced orbital debris shields, Proc. APS Conf. Shock Compression of Condensed Matter, p. 13. Cowan, G.R. and Holtzman, A.H. (1963). Flow configuration in colliding plates: Explosive bonding. J. Appl. Phys. 34: 928. Davidson, L. and Shahinpoor, M. (eds.). (1997). High-Pressure Shock Compression of Solids I-IV, New York: Springer. Decarli, P.S. and Jamieson, J.C. (1961). Formation of diamond by explosive shock. Science 133: 1821. Dlott, D.D. (1995). Picosecond dynamics behind shock front. J. de Physique IV, C4; Suppl. J. de Physique III, 5: C4-337. Dugdale, J.S. and MacDonald, D.K.C. (1953). The thermal expansion of solids. Phys. Rev. 89: 832. Duvall, G.E. and Graham, R.A. (1977). Phase transitions under shock wave loading. Rev. Mod. Phys. 43: 523-579. Eliezer, S., Ghatak, A., and Hora, H. (1986). An Introduction to Equation of State: Theory and Applications, Cambridge: Cambridge Univ. Press. Fowles, G.R., Duvall, G.E., Asay, J., Bellamy, J.P., Feistmann, E, Grady, D., Michaels, T., and Mitchell, R. (1970). Gas gun for impact studies. Rev. Sci. Instrum. 41: 984. Graham, R.A. (1992). Solids Under High-Pressure Shock Compression, New York: Springer-Verlag. Gnmeisen, E. (1926). Handbuch der Physik, H. Greiger and K. Scheel, eds., Berlin: Springer, 10, pp. 1-59. Gustov, V.W. (1994). High Pressure Physics and Chemistry of Polymers, A i . Kovarskii, ed., Boca Raton: CRC Press Inc., Chap. 8. Jamet, E and Thomer, G. (1976). Flash Radiography, Amsterdam: Elsevier Pub. Co. Keeler, R.N. and Royce, E.B. (1971). Physics of High Energy Density, P. Cardirola and H. Knoepfel, eds., New York/London: Academic Press. Kinslow, R. (1970). High-Velocity Impact Phenomena, New York/London: Academic Press. Marsh, S.P. (1981). Los Alamos Shock Hugoniot Data, Berkeley: University of Califomia Press. Mashimo, T. (1993). Shock Waves in Materials Science, A. Sawaoka, ed., Tokyo: Springer-Verlag, p. 113.
338
K. Nagayama
Mori, Y. and Nagayama, K. (1998). Sensitive optical detection of the shock front and fast moving surface for shock study in condensed media in the 1 GPa stress region. Rev. Sci. Instrum. 69: 1730-1734. Nagayama, K. (1993). Shock Waves in Materials Science, A. Sawaoka, ed., Tokyo: Springer-Verlag, p. 195. Nagayama, K. (1994a). New method of calculating shock temperature and entropy of solids based on the Hugoniot data. J. Phys. Soc. Japan 63: 3737. Nagayama, K. (1994b). New thermal variables of condensed matter at high pressures and temperatures. J. Phys. Soc. Japan 63: 3729. Nagayama, K. and Mori, Y. (1994). Simple method of calculating Gruneisen parameters based on the shock Hugoniot data for solids. J. Phys. Soc. Japan 63: 4070. Nagayama, K. (1997). Cold potential energy function for solids based on the theoretical models for Gr~neisen parameter. J. Phys. Chem. Solids 58: 271. Nakahara, M. and Nagayama, K. (1999). Water shock wave emanated from the roughened end surface of an optical fiber by pulse laser input. J. Mat. Proc. Tech. 85: 30. Noack, J. and Vogel, A. (1998). Single-shot spatially resolved characterization of laser-induced shock waves in water. Appl. Opt. 37: 4092. Prfimmer, R. (1987). Explosiveverdichtung Pulvriger Substanzen, Berlin: Springer-Verlag. Rodean, H.C. (1977). J. Appl. Phys. 48: 2384. Slater, J.C. (1939). Introduction to Chemical Physics, New York: McGraw-Hill, Chap. XIII. Steinberg, D.J. (1981). The temperature independence of Gruneisen's gamma at high temperature. J. Appl. Phys. 52: 6415-17. van Thiel, M. (1966). Compendium of Shock Wave Data, Livermore, CA: Univ. Calif. Press. Vaschenko, V. Ya. and Zubarev, V.N. (1963). Conceming the Grtineisen constant. Sov. Phys. Solid State 5: 653. Whipple, E.L. (1947). Meteorites and space travel. Astronomical J. 52: 137. Zel'dovich, Ya. B. and Raizer, Yu. P. (1967). Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (English Translation), vol. 2, New York/London: Academic Press, p. 685. Zharkov, V.N. and Kalinin, V.A. (1971). Equation of State for Solids at High Pressures and Temperatures, New York/London: Consultants Bureau, p. 29.
CHAPTER
3.4
Theory of Shock Waves 3.4
Rarefaction
Shocks
ALFRED KLUWICK Institute of Fluid Mechanics and Heat Transfer, Vienna, University of Technology, Vienna, Austria
3.4.1 Introduction
3.4.2 Shock Adiabat 3.4.3 Shock Admissibility 3.4.4 Shock Structure 3.4.5 Weak Shocks 3.4.6 Shock Dynamics 3.4.7 Concluding Remarks References
3.4.1
INTRODUCTION
One of the main issues in the theory of gasdynamic shocks is the question of the type of shocks, that is, compression or rarefaction shocks, which can be sustained by a given fluid. The possibility of shock formation was envisaged first by Stokes (1848) but significant progress as to this question was not achieved until more than 50 years later through the milestone papers by Jouguet (1901), Zempl6n (1905), and Duhem (1909) (translated into English by Gendron, 1989). In these studies it was recognized that the second law of thermodynamics provides a powerful tool to eliminate shocks that are physically impossible. For example, when applied to the special case of perfect gases it was found that rarefaction shocks are impossible as they lead to a decrease of entropy. By contrast, compression shocks are compatible with the requirement of the second law that the entropy must not decrease in adiabatic processes. In addition, Duhem (1909) was the first to demonstrate that the first and second derivatives of the entropy s with respect to the density p evaluated at the upstream state p = Pb vanish while
(d3s
_
1 H,
H-- (2dP
d2p
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
(3.4.1)
339
340
A. Kluwick
Here, p and T denote the pressure and the absolute temperature. From this relationship he concluded that the region downstream of a (weak) shock is where the fluid is more dense or less dense if the quantity H is positive or negative. "In the first case, we say that the shock wave propagates a condensation; in the second case, it propagates an expansion." The entropy increase across weak shocks was calculated independently by Becker (1922) and Bethe (1942). Their results combined with Eq. (3.4.1) reveal that the quantity H represents a measure for the curvature of isentropes in the p, v-diagram, which most conveniently is expressed in nondimensional form (Hayes, 1958) /)3 02pl
H
F - - 2 a 2 0 v 2 1 s = 2a2
(3.4.2)
where a = ~/Op/Opl~is the equilibrium speed of sound and v = 1/p denotes the specific volume. In a general fluid, shocks that are sufficiently weak will therefore propagate a condensation if the isentropes are curved up (as in the case of perfect gases) and propagate an expansion if the isentropes are curved down. As it turns out, the curvature of isentropes is not only important as far as the theory of shocks is concerned, but also plays a significant role in other areas of gasdynamic flows. Following Thompson (1971) we shall therefore refer to F as fundamental derivative. For example, the importance of F in describing nonlinear acoustic waves follows from the observation that an equivalent definition of F is F - 1_ Op___aIa
a Op
(3.4.3)
Well-known relationships for planar unidirectional waves then show that F determines the variation of the convected sound speed a with the density perturbations carried by a wave do = 4- a F
dp
p
(3.4.4)
Thompson and kambrakis (1973). Here, the upper and lower sign correspond to right- and left-moving waves. Accordingly, compression waves will steepen if F > 0 and flatten out if F < 0. Similarly, rarefaction waves will steepen if F < 0 and flatten out if F > 0. This ties in nicely with the conclusions drawn from Eqs. (3.4.1) and (3.4.2). The double role played by the curvature of isentropes, first to select shocks that satisfy the second law of thermodynamics, and second to explain how these shocks can form naturally from smooth initial conditions was recognized first by Becker (1922, pp. 332 and 357) "In a given medium only compression shocks or rarefaction shocks are possible both mechanically and thermodynamically if 02p/OV2ls is positive or negative."
3.4 RarefactionShocks
341
At present, two mechanisms are known that suggest the existence of fluids having embedded regions of negative F, thus opening the possibility for the formation of rarefaction shocks. The first of these is relevant to vapors consisting of complex molecules. Due to the large number of internal degrees of freedom and the associated large heat capacities, isentropic compression or expansion of such vapors causes relatively small temperature changes. This suggests that with increasing molecular complexity the differences between isentropes and isotherms will decrease. As a consequence, the vapor behaves as a retrograde rather than as a regular fluid, that is, isentropic expansion may lead to condensation. Estimates based on corresponding states calculations indicate that fluids will exhibit retrograde behavior if cCoo> 11.2R, where cCoo denotes the isochoric ideal gas heat capacity at the critical temperature, and R is the universal gas constant (Lambrakis, 1972). Furthermore, it is well known that the curvature of isotherms in the p, v-diagram changes twice in the general neighborhood of the thermodynamic critical point. For sufficiently large values of cvoo c one expects isentropes to behave similarly, which in turn causes F to be negative over a finite range of pressures and densities. Because F characterizes the nonlinear dependence of the convected sound speed on the density disturbances caused by a wave, fluids of this type are said to exhibit negative or mixed nonlinearity. It has been shown by Bethe (1942) and independently by Zel'dovich (1946) that retrograde van der Waals fluids exhibit embedded negative F regions if cCoo takes on values that are larger than about 17.5R. Using more sophisticated and accurate equations of state, several candidates for real negative F fluids, which include hydrocarbons and fluorocarbons of moderate complexity, have been identified by Thompson (1971), Lambrakis and Thompson (1972), and Thompson and Lambrakis (1973). Because of the significance of each of these studies such fluids are now commonly referred to as Bethe-Zel'dovich-Thompson (BZT) fluids (following a proposal by Cramer, 1991). More recently, the properties of fluorocarbons have been investigated also by Cramer (1989a). Table 3.4.1 summarizes the properties of commercially available fluorocarbons that were found to have a negative F region in the single-phase region. In nine cases this region is seen to be large enough to include the critical isotherm. The following considerations will focus on single-phase equilibrium flows. Nevertheless, it is interesting to note that the slope of equilibrium isentropes at the saturation boundary is discontinuous k(T) =
OplOvl~ -
OplOvl~ m
' > 0
(3.4.5)
which has important consequences as far as the properties of shocks with phase transition are concerned. Here, the subscript m denotes the equilibrium derivative on the mixture side of the coexistence curve. Geometrically, this so-
342
A. Kluwick
TABLE 3.4.1 Negative F Fluids. Fmin(Tc) Denotes the Minimum Value of F on the Critical Isotherm Based on Calculations Using the Martin-Hou Equation (3.4.27) (Cramer, 1989a) Fluid CloF22 CloF18 ( P P 5 ) CllF2o ( P P 9 ) C13F22 (PP10) C14F24 (PP11) C16F26 (PP24) C17F3o (PP25) C12F27N (FC-43)
C15F33N (FC-70) C18F39N (FC-71) CllF23HO3 C14F29HO4 C17F35HO5
pf-decane pf-decalin pf-methyldecalin pf-perhydrofluorene pf-perhydrophenanthrene pf-fluoranthene pf-benzyltetralin pf-tributylamine pf-tripentylamine pf-trihexylamine fluorinated ether E3 fluorinated ether E4 fluorinated ether E5
(K)
(bar)
(kg/m3)
Pc
cC~
]-'min(Tc )
578 565 587 632 650 701 687 567 608 646 536 568 595
13.1 17.5 16.6 16.2 14.6 15.3 11.0 11.4 10.3 9.42 10.8 8.41 7.70
574 661 669 626 626 623 626 621 622 619 592 570 619
74.8 64.5 72.8 78.4 97.3 112 123 93.0 119 145 82.9 109 136
0.04 0.11 0.05 -0.08 -0.15 -0.36 -0.22 -0.03 -0.17 -0.29 0.10 -0.02 -0.11
Tc
Pc
R
,,
called "kink" can be interpreted as a region of negative F, which has contracted onto a single curve, yielding F = - o o at the saturation line. This suggests that flows of retrograde fluids with phase changes may experience the effects of mixed nonlinearity and thus lead to the occurrence of phenomena that are qualitatively similar to those in single-phase BZT fluids. Theoretical as well as experimental studies have shown that this is indeed the case (Thompson and Sullivan, 1975; Dettlev, Thompson, Meier and Speckmann, 1979; Thompson and Kim, 1983; see also Thompson, 1991). Probably the most spectacular example of this correspondence is provided by the evaporation shock that forms during isentropic expansion of a retrograde liquid or liquid/vapor mixture (Chaves, Lang, Meier and Speckmann, 1985; Thompson, Carafano and Kim, 1986; Thompson, Chaves, Meier, Kim and Speckmann, 1987; and SimOes-Moreia and Shepherd, 1999). As the pressure decreases across such a discontinuity it represents the two-phase analog to the single-phase rarefaction shock. In closing this short digression to flows with phase change we note that this analogy can be made precise if the deviations from thermodynamic equilibrium are negligible (Menikoff and Plohr, 1989; Kluwick, 1998). As pointed out earlier, it is the large heat capacity cCo~ that is responsible for the formation of negative F regions in the p, v-diagrams of fluids with moderate and high molecular complexity. However, as is well known, the isochoric heat capacity of general fluids tends to infinity as the thermodynamic critical point is approached (for example, Landau and Lifschitz, 1987). It has been suggested that the critical point singularity of Cv provides a second mechanism that may cause the occurrence of negative and mixed nonlinearity
3.4 RarefactionShocks
343
in an even broader class of single-phase fluids (Borisov, Borisov, Kutateladze and Nakoryakov, 1983; Kutateladze, Nakoryakov and Borisov, 1987; Gulen, Thompson and Cho, 1990; see also Kluwick, 1995). Calculations using renormalization group theory carried out more recently by Emanuel (1996) do not seem to support this conjecture or the results obtained by Gulen et al. (1990). On the other hand, it has been claimed by Novikov (1949) that ordinary steam exhibits negative nonlinearity in the near critical portion of the two-phase regime. Novikov's conclusions (based on an unreferenced steam table) were confirmed by Kahl and Mylin (1969) for a region between P/Pc ~ 0.92 and 1. Given the universality of (the nonanalytic part of) the thermodynamic relations in the neighborhood of the critical point as expressed by the hypothesis of scale invariance (e.g., Widom, 1965; Schofield, 1969) one expects negative nonlinearity in the two-phase regime to be possible also for some wider class of fluids. Clearly, however, further efforts will be necessary to obtain a full picture of the behavior of F in the limit p - - , Pc, v ~ v c. This behavior is of importance also in connection with the shock tube experiments of Borisov et al. (1983) who generated rarefaction waves of constant form in Freon-13 at near critical conditions that were interpreted as single-phase rarefaction shocks. Taking into account, however, that Freon-13 is not a BZT fluid (Cramer, 1989a; Brown & Argrow, 1998), and the foregoing discussion of negative nonlinearity near the critical point, the possibility that the propagating rarefaction front caused the fluid to enter the two-phase regime must also be considered (Thompson, 1991; Cramer, 1991). Further evidence for this point of view is provided by recent theoretical results of Piechor (1997), which show that pressure oscillations downstream of the rarefaction front similar to those recorded by Borisov et al. (1983) are indicative of the presence of capillary forces. In summary, it appears that fluids having large specific heats are the best candidates for fluids that admit single-phase rarefaction shocks. The present review, therefore, will concentrate on shock formation in the dense gas regime of BZT fluids. Although the main emphasis is on rarefaction shocks the properties of compression shocks will also be addressed, the reason being twofold. First, it is found that embedded regions of negative F that are a necessary ingredient for the existence of rarefaction shocks strongly affect the behavior of compression shocks. Second, when dealing with general initial and/or boundary value problems, rarefaction and compression shocks may be generated simultaneously and even interact. This chapter is structured as follows. In section 3.4.2 the properties of BZT fluids and the role of the second law in ruling out impossible shocks are treated in more detail. Section 3.4.3 considers further restrictions resulting from the so-called stability condition that requires that the wave speeds immediately upstream/downstream of the shock front must not be larger/smaller than the
344
A. Kiuwick
shock speed. This is seen to result in a number of nonclassical phenomena including sonic shocks, double sonic shocks, and split shocks. Although plausible, the stability criterion cannot be derived from the jump relations. Rather it represents an additional requirement that in a sense serves to mimic the effects of physical mechanisms such as internal friction and heat conduction neglected in the derivation of these relations. This is confirmed by the treatment of the shock layer problem in Section 3.4.4, which shows that the stability condition is a necessary but not sufficient condition for the existence of admissible shocks, that is, shocks that have a thermoviscous structure. In the case of shocks of finite strength the shock layer problem has to be solved numerically. However, if the shock strength is sufficiently small, then analysis is possible providing additional physical insight. Furthermore, the results summarized in Section 3.4.5 are found to be in excellent qualitative agreement with the numerical solutions of the full shock layer equations. While Sections 3.4.2 to 3.4.5 deal with the local properties of shock discontinuities an attempt is made in Section 3.4.6 to investigate how the unusual properties of sonic shocks, split shocks, etc., affect the global behavior of flows. To this end again the assumption of weak nonlinearity is adopted. Finally, Section 3.4.6 briefly summarizes recent developments in steady gasdynamic flows not covered herein and indicates other areas where effects of mixed nonlinearity are of importance. 3.4.2
SHOCK
ADIABAT
In the context of classical gasdynamics shock waves represent discontinuous solutions of the governing equations in which thermoviscous effects are neglected. Due to their vanishing thickness shocks do not carry mass and as a consequence the Rankine-Hugoniot jump conditions that connect the upstream and the downstream states are not affected by inertia. This in turn means that one can always choose a coordinate system such that relationships expressing the conservation of mass, momentum and energy locally assume the form [pu] = 0, (3.4.6) [p] -- -m2[v]
(3.4.7)
[e] -- -[v] pb + Pa 2
(3.4.8)
holding in the case of one-dimensional (1D) steady flow (Hayes, 1958). It is therefore sufficient to restrict the following considerations to normal discontinuities, that is, discontinuities having vanishing tangential velocities; see, Fig. 3.4.1.
3.4 RarefactionShocks
345
Pb, flb~
eb~
Pa, Pa, ea~ ...
...
v
llb
Ua
FIGURE 3.4.1 Normal shock. In Eqs. (3.4.6) and (3.4.7) and the Hugoniot equation (3.4.8), the subscripts b and a refer to flow properties before and after the shock, and the square brackets characterize the jump in any quantity Q: [Q] - Qa - Qb. Furthermore, p, v-- l/p, p, e, u and m - pbub = PaUa are the density, the specific volume, the pressure, the specific internal energy, the velocity and the mass flux, respectively. An additional condition stems from the second law of thermodynamics, which requires that the jump of the specific entropy s across a shock discontinuity must not be negative: [s] >_ 0 (3.4.9) Equation (3.4.8) contains thermodynamic quantities only. As a consequence, this relationship is invariant under the transformations used to simplify the jump conditions and thus applies to general shock discontinuities including moving and oblique shocks as well. Furthermore, Eq. (3.4.8) is sufficient to determine the locus of thermodynamic states that can be connected by shock discontinuities. Following common practice we shall refer to this locus and in particular its graph in the p, v-diagram as the shock adiabat. The evaluation of Eq. (3.4.8) requires further information concerning the constitutive properties of the medium under consideration. A convenient choice is to specify the thermal equation of state and the ideal gas heat capacity at constant volume: p - p(v, T) (3.4.10)
cvoo - cvoo(T) -- ~!irnooc~(T, v)
(3.4.11)
Here T denotes the absolute temperature. The standard thermodynamic relationships ae-
( c~o~(7) -
7
ioo -~2p ~ (v', T)dv' t u
d7 +
T2 ~ tP(TT!tdv
(3.4.12)
346
A. Kluwick
can then be used to express e and s, which enter Eqs. (3.4.8) and (3.4.9) in terms of v and T. With p(v, T), e(v, T) and s(v, T) given, the shock adiabat and the entropy inequality then form a closed set of equations that fully determine the thermodynamic properties of shock discontinuities. Before tuming to a more detailed discussion of these properties and in particular the conditions that must be satisfied to allow for the occurrence of rarefaction shocks, it is convenient to summarize some results that will be useful later. Combining the Gibbs' relation (3.4.14)
Tds = de + pdv
with Eqs. (3.4.8) and (3.4.9) yields immediately
Jvbl)aTds -- I vbvapdv -- Iv] Pb -[--2 Pa = AH -
AR
(3.4.15)
The terms on the right-hand side represent, respectively, the areas AH and A R under the shock adiabat and the straight line (Rayleigh line) that connects the upstream and downstream states in the p, v-diagram (Fig. 3.4.2). It thus follows that compression/rarefaction shocks are admissible, that is, satisfy the entropy inequality (Eq. (3.4.9)), only if the Rayleigh lines lie (at least partially) above/below the shock adiabat. The investigation of thermoviscous effects in Section 3.4.3 will show that the Rayleigh lines must lie entirely above/below to allow the construction of a shock layer solution, but this conclusion cannot be drawn from the jump relations alone. However, even the weaker requirement resulting from these relations suffices to infer that a necessary condition for the formation of rarefaction shock waves is the existence of a range of pressures and specific volumes including Pb and vb for which the shock adiabat is curved down rather than curved up as in the case for a perfect gas. The expansion of [e] in Eq. (3.4.8) in terms of [s] and [v] for weak shocks [s] -~ 0, [v] -~ 0 assuming F b ~ 0 immediately yields the celebrated result
[s]=
I'ba~ [v]B +... 6Tbvb3
(3.4.16)
first derived by Duhem (1909) and independently by Bethe (1942). In agreement with the preceding discussion, Eq. (3.4.16) shows that the type of (weak) shock possible depends on the curvature of the shock adiabat in the p, v-diagram. Second, Eq. (3.4.16) expresses the well-known fact that the slope and curvature of the shock adiabat at v b, Pb agrees with the slope and curvature of the isentrope passing through this point. This in turn means that the shape of shock adiabats and isentropes is qualitatively similar locally, that is, in the
3.4
347
Rarefaction Shocks
Pb
~
Rayleigh line
Pa
x~~.~ ~ /
shock adiabat
ml Vb FIGURE 3.4.2
Va
V
Hugoniot diagram shock adiabat, Rayleigh line and areas AH (111),AR (=).
vicinity of the upstream state. If the considerations are limited to weak shocks, which will be treated in fuller detail in Section 3.4.5, then the search for rarefaction shocks reduces to the search for media having embedded regions of negative F. (However, the considerations in Section 3.4.5 will show that this is also a necessary requirement for the formation of rarefaction shocks of finite strength.) As pointed out in the Introduction, van der Waals gases with large values of the specific heats belong to this group of media (Bethe, 1942; Zel'dovich, 1946). For fluids to exhibit negative F behavior the requirement of large specific heats is a crucial one while the specific form of the thermal equation of state is less important. This can be seen directly from Eqs. (3.4.12) and (3.3.14). In fact these equations imply that e(v, T) and s(v, T) for a general fluid can always be decomposed in the form e = O(T) 4- ~f(v, T)
(3.4.17)
s = g/(T) 4- 6g(v, T)
(3.4.18)
where O(T) and W(T) depend on the ideal gas heat capacity Cvoo(T) only and R
c~= ~ c~oo
(3.4.19)
is a nondimensional parameter that characterizes the relative magnitude of the universal gas constant R and the value of cvoo evaluated at the critical point
348
A. Kluwick
temperature Tc. For perfect gases and other media of low molecular complexity is of order one and the dependency of e and s on T and v is equally important. However, media of moderate and high molecular complexity have ~ << 1 and both e and s depend mainly on T with small corrections due to variations in v accounted for by the functions f(v, T) and g(v, T) that reflect the specific properties of the thermal equation of state. Substitution of Eq. (3.4.17) into Eq. (3.4.18) and assuming c~ << 1 yields the result that the temperature changes caused by shock discontinuities are small: [T]/T b = O(~), c~ << 1. As a consequence, shock adiabats, isentropes, and isotherms differ only slightly in the p, v-diagram. This implies F
v3 ~2pl
v3 c~2p
2a 2 a v 2I
2a 2 av2
+0(6)
(3.4.20)
T
which represents a variant of a relationship obtained first by Bethe (1942). Even more important, however, one concludes that the form of shock adiabats and isentropes for fluids with moderate and high molecular complexity (c~ < ( 1 ) is qualitatively similar not only locally, that is, in the vicinity of the upstream state, but also globally. Furthermore, this form can be inferred from that of isothermes that are known to exhibit regions of negative curvature in the general neighborhood of the thermodynamic critical point. This is confirmed by more detailed calculations based on specific equations of state. In this connection it is illustrative to consider van der Waals gases with constant isochoric heat capacity:
RT
P-v-b
o~
v2'
cv - c o n s t .
(3.4.21)
which has the advantage that isentropes, isotherms, and shock adiabats can be expressed in closed form. Despite its simplicity this gas model (in which the positive constants ~ and b account for intermolecular forces and molecular size) predicts the correct qualitative thermodynamic behavior in the range of pressures and densities, which is of interest here, and therefore is frequently used to investigate basic properties of dense gas flows. For the purpose of presentation it is convenient to introduce the nondimensional quantities
--E P
Pc'
~_k Pc'
i--r
s
Tc' -g- c~,
(3.4.22)
where as before the subscript c denotes the values at the thermodynamic critical point. Substitution of the constitutive relationships into the Hugoniot
3.4 RarefactionShocks
349
equation (3.4.8) then yields the shock adiabat in the following form (Thompson and Lambrakis, 1973; Cramer and Sen, 1987)" 2[~] ( 3(ub + va + 3(c~- 1)~,~a) ) [P] = - 3 ( 2 + a)~a - 3a~b -- 2 ,,3pb(1 + a) + (~t,) 2 / (3.4.23) Furthermore, isotherms and isentropes are found to be given by
-
P-Pb--(5~-i)
(
pb((3Db -- i)l+a -- (3D -- i)l+a) + - Pb --
+
3Pb+
(~b-fl
J
3(V2(3Vb- 1)l+a- V~(3V(DDb)2
(3.4.24)
i)l+a)
( 3 ~ - 1) l+a (3.4.25)
It is easily shown that Eqs. (3.4.23), (3.4.24), and (3.4.25) coincide in the limit c~ -+ 0 and differ by terms of order c~only if the gas has large specific heats c5 << 1. Representative results obtained by evaluation of these equations are plotted in Figs. 3.4.3 and 3.4.4. In addition, Figs. 3.4.3 and 3.4.4 display the F - 0 locus calculated from the explicit relationship (Cramer and Sen, 1986),
3 4T(2 + 3~ + ~2) _/3(3 - p)3 F -- 2 (3 - fi)(4T(1 + a) - fi(3 - fi)2)
(3.4.26)
In agreement with the earlier considerations isotherms, isentropes, and shock adiabats are seen to approach each other in the limit of high specific heats c~ --+ 0. For c~sufficiently small, therefore, the shock adiabats corresponding to upstream states in the dense gas regime exhibit two inflexion points located in the vicinity of the F - 0 locus. Between these inflexion points the shock adiabats are curved down rather than up as in fluids with low molecular complexity, thus creating the possibility for the formation of rarefaction shock waves. In addition, the behavior of isentropes and shock adiabats displayed in Figs. 3.3.4 and 3.4.4 suggests that shocks in the dense gas regime of BZT fluids may cause unusually small entropy increases. This will be confirmed in Section 3.4.4. Computations of shock adiabats using more sophisticated equations of state yield results that exhibit the same general trends as those plotted in Figs. 3.4.3 and 3.4.4. Detailed numerical calculations have been performed by Cramer and Crickenberger (1991) who chose the Martin-Hou equation of state (Martin and Hou, 1955) RT p -- ~
v- b
4 + ~
i-1
dF~(v) Qi(T)~
dv
(3.4.27)
350
A. Kluwick 1.2
I.I
0.9
0.8
0.7
0.6
0.5 0.5
FIGURE 3.4.3 ~ with ~ = 0.02;
1
1.5
2
2.5
3
Isentropes,--- shock adiabats and ...... isotherms for a van der Waals gas saturationline, - . - F = 0 locus.
which has the advantage of a strong analytical basis and therefore requires a m i n i m u m number of input data. Furthermore, the studies of Thompson and Lambrakis (1973) and Cramer (1989a) suggest that it is conservative in the sense that it underpredicts rather than overpredicts nonclassical gas behavior (see also Leidner, 1996). In Eq. (3.4.27) the positive constant b again accounts for molecular size. The functions Fi(v ) comprise inverse integral powers of v - b 1
Fi(v) -- - b -
)
(3.4.28)
while the functions Qi(T) are taken to be of the form Q~(~)
-
, ,, ,,, qi+x "4- qi+i T 4- qi+l exp(-5.475T/Tc)
(3.4.29)
3.4 RarefactionShocks
351
1.2
of'-'~ I.I
0.9
0.8
\
0.7
0.6
0.5
u_
0.5.
1
1.5
2
2.5
3 V
FIGURE 3.4.4
with c~= 0.001"
Isentropes,--- shock adiabats and ...... isotherms for a van der Waals gas saturation line, - . - l" - 0 locus.
The constants entering these relationships depend on the molecular weight, the values of the pressure, the temperature and the specific volume at the critical point, and the normal boiling temperature. To complete the description of the thermodynamic properties of the gas it is necessary to specify the ideal gas heat capacity Cvoo(T). In the investigations of Cramer and Crickenberger (1991) Cvoo(T)was approximated by the power law
(3.4.30)
where n is a constant and c~oo is the ideal gas specific heat evaluated at the reference temperature T ~'. As pointed out by Cramer (1989a), both n and Cvoo * can be estimated by the group contribution method of Rihani and Doraiswany (1965).
352
A. Kluwick
Representative s h o c k adiabats calculated from Eqs. (3.4.8), (3.4.12), (3.4.27), (3.4.28), (3.4.29), and (3.4.30) for the BZT-fluid P P l l of Table 3.4.1 are plotted in Fig. 3.4.5. For this fluid c5 = 0.01 and as in the case of the c o r r e s p o n d i n g calculations based on the van der Waals equation of state the s h o c k adiabats exhibit regions of negative curvature if T is sufficiently close to T c. O n e thus concludes that s h o c k adiabats that are curved d o w n over a finite range of pressures and densities in the dense gas regime c a n n o t be excluded on t h e r m o d y n a m i c g r o u n d s b u t in fact represent a characteristic feature of real fluids having large specific heats. According to Eq. (3.4.16) 1.5 1.4
1.3
1.2 m
T - 1.05 I.i
1.04
1.0
~- 1.03 " 1.02 ~" 1.01
0.9
"990.98
saturated-vapor line
0.8
0.7
0.6
0.5 0.5
1.0
! .5
2.0
2.5
3.0
3.5
ly FIGURE 3.4.5 Shock adiabats for PP11. The adiabats for values ]" < 1 intersect the saturated vapor line at ~-. The adiabats for ~- > 1 go though the point ~ - 1.35, ~-. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press.
3.4
353
Rarefaction Shocks
fluids of this type are characterized by the presence of an embedded negative F region in the single-phase region of the p, v-diagram. Examples of such fluids that are frequently used in engineering applications are summarized in Table 3.4.1.
3.4.3
SHOCK
ADMISSIBILITY
In addition to the jump conditions (Eqs. (3.4.6)-(3.4.8)) and the entropy inequality (Eq. (3.4.9)) admissible shocks must satisfy the condition Mb > 1 > Ma
(3.4.31)
where M i--u-!,
i-b,a
(3.4.32)
ai
denotes the values of the Mach number immediately before and after the shock front (lax, 1957; Oleinik, 1959). Equation (3.4.31) is usually referred to as a stability condition, which is taken to be necessary for existence and persistence of shocks. Equation (3.4.31) is equivalent to the speed-ordering relation, which requires that the shock speed must be larger (or at least equal) to the convected sound speed upstream of the front and smaller (or at most equal) to the convected sound speed in the region downstream of the front. Following Cramer and Kluwick (1984) shocks having Mb = 1 or M a = 1 will be referred to as sonic shocks. Similarly, shocks with Mb = Ma = 1 will be termed double sonic shocks. Taking into account Eq. (3.4.16), which states that the variation of the entropy along the shock adiabat vanishes at v = vb, one obtains
~b
b
Equation (3.4.7) [ p ] / [ v ] - - m 2 - --pbU~ combined with Eq. (3.4.33) then shows that the first inequality of Eq. (3.4.31) is equivalent to the requirement dp) > [p]
(3.4.34)
A second result that relates the slope of the shock adiabat at v - v a to the slope of the Rayleigh line is obtained by combining Eq. (3.4.8) with the Gibbs' relation equation (3.4.14) to yield
(3.4.35)
354
A.
Kluwick
Expressing the pressure variation at the shock adiabat in terms of the variation of the entropy and specific volume using the thermodynamic relationships v
~
-&e v d S - - ~ d v
(3.4.36)
one obtains the expression
d m[p]= ~
aa(M2a-1)
Iv] Ga) [v] Va2(1 + ~Va
(3.4.37)
which is equivalent to a result derived in Cramer (1989b). Here,
c=v v
= fla2
(3.4.38)
Cp
denotes the so-called Grtineisen parameter and ,6 is the coefficient of thermal expansion
#-
1 3p] #~r
(3.4.39)
The denominator of the term on the right-hand side of Eq. (3.4.37) vanishes if 2va + [vlG a = 0
(3.4.40)
This is the usual singularity that limits the maximum possible density increase across compression shocks (if the coefficient of thermal expansion is positive as will be assumed here throughout). In the case of perfect gases G can be expressed in terms of the ratio of the specific heats y: G = y - 1 and one recovers the well-known result V~aI max
_ Pal y__+ l , Pb m a x ~ - 1
~CP ? = cv
(3.4.41)
Subsequent discussion will be restricted to the portion of the shock adiabat to the right of the singularity. Equation (3.4.37) then implies that the second inequality in Eq. (3.4.31) will be satisfied if a-<~'[v]
(3.4.42)
3.4 RarefactionShocks
355
Equation (3.4.35) may also be written as
H
o-N/
o
(3.4.43)
which together with the relationship equations (3.4.42) and (3.4.15) shows that sonic points on the shock adiabat, that is, points where the Rayleigh line coincides with the tangent, are local extrema in entropy: ~-]--
~
,[
d--vv , - 0 .
(3.4.44)
In summary, the stability condition equation (3.4.31) can be expressed in the equivalent form (3.4.45) o
which has the advantage that it contains only quantities that can be inferred from the shape of the shock adiabat in the p, v-plane and therefore can be checked visually. The requirement that the Rayleigh line must not cut the shock adiabat at an interior point can be interpreted graphically also and together with Eqs. (3.4.45), (3.4.9), and (3.4.15) this is sufficient to isolate physically acceptable, that is, admissible, shock discontinuities. For example, application of these results shows that the discontinuity leading from 1 to 2 in Fig. 3.4.6 represents an admissible rarefaction shock. Similarly, the discontinuity leading from 1 to 2 in Fig. 3.4.7 is seen to represent an admissible compression shock. If the amplitude of these shocks is increased by decreasing/increasing Pa one eventually obtains the limiting cases 1 --+ 3 where the Rayleigh lines are parallel to the tangents to the shock adiabat in 3. According to Eq. (3.4.37), shocks of this type have critical downstream states M a -----1 and therefore represent sonic shocks. Further increase of the shock strength leads to configurations where the Rayleigh line cuts the shock adiabat in an interior point. As a result the stability criterion (Eq. (3.4.45)) is violated. Jumps of amplitude i --> 4 therefore are inadmissible, that is, cannot propagate as stable shock discontinuities. It should be noted however, that the entropy inequality equation (3.4.9) is satisfied in both cases if the points 4 and 3 are sufficiently close. One therefore concludes that the stability criterion equation (3.4.45) is more restrictive than the requirement [s] > 0 if the curvature of the shock adiabat changes sign. By contrast, Eqs. (3.4.9) and (3.4.45) are equivalent if the shock adiabat is strictly convex. Although pressures below P3 in Fig. 3.4.6 cannot result from a single rarefaction shock under steady flow conditions, they can of course be generated by (unsteady) isentropic expansion downstream of a rarefaction
356
A. Kluwick
H
pb
Vb
FIGURE 3.4.6 isentrope.
V
Admissible and nonadmissible rarefaction shocks. H...shockadiabat, I . . .
P
H
i
Pb
I
.....
Vb FIGURE 3.4.7 isentrope.
Admissible and inadmissible compression shocks. H . . . s h o c k adiabat, I . . .
3.4
357
Rarefaction Shocks
shock of maximum strength, that is, starting with the sonic state 3. According to Eq. (3.4.44), the entropy is stationary at sonic points of the shock adiabat. As a consequence, state 3 corresponds to a local maximum in entropy. Furthermore, as s increases with p at v --= const., (3.4.46)
Os I - pcpa2
the isentrope I passing through 3 is found to lie above the shock adiabat. Any pressure jump that leads from state 1 to a state on I, say, 1 ---> 4', then will immediately disintegrate into a sonic rarefaction shock and a downstream rarefaction wave fan formed by left-moving wavefronts (characteristics) that propagate with the convected sound speed a -- u - a
(3.4.47)
This phenomenon, referred to as shock splitting, is shown in Fig. 3.4.8 where x and t denote the distance normal to the shock and the time. Similar results are obtained for the case of compression shocks considered in Fig. 3.4.7. Positive pressure jumps 1---> 4', which are larger than those associated with the sonic shock 1 ---> 3, split into a sonic shock 1 ---> 3 and a downstream compression fan. As the strength of this compression fan increases further, however, a new phenomenon arises. To this end we first note that a satisfies the relationship equation (3.4.4) with the minus sign that follows immediately from the definition of the fundamental derivative equation (3.4.3) and the compatibility condition on right-moving characteristics u 4-
ll ~a(P ) dp' .
pl
- const.
(3.4.48)
Second, differentiation of Eq. (3.4.43) with respect to v yields d2s T dv----i :
[v] 82p] 2
8v 2 s =
[v] 2a2F v
(3.4.49)
v2
at sonic points of the shock adiabat. In combination with the results obtained earlier that sonic points are local extrema in entropy this shows that the sign of F alternates at successive sonic points. In Fig. 3.4.9 the transition curve F = 0 therefore crosses the shock adiabat in two points A, B (located in the vicinity of the inflexion points). Furthermore, let v i denote the value of the specific volume at the inflexion point of I, that is, where I crosses the F - 0 lOCUS.
According to Eq. (3.4.4) the convected sound speed equation (3.4.47) increases monotonically with increasing density for values of v~ < v < v2 indicating that the sonic shock compression fan combination represents a
OO
unsteady wave fan
X
1-+2
X
1-+3
X
1-+4'
FIGURE 3.4.8 Space-time-diagrams corresponding to the various pressure jumps included in Figs. 3.4.6 and 3.4.7; characteristic.
shock,
3.4
359
Rarefaction Shocks
p
I ~7
M
R2 F=0 Pi "[....
B
R1
A
//1
Pb
vi
~..'~
1
Vb
V
FIGURE 3.4.9 Compression split shocks: H1,H2...shock adiabats; R1, R2...Rayleigh lines; M... curve formed by the endpoints of all possible shock adiabats/-/2; I isentrope.
single-valued, that is, admissible solution. For values v /)i on I inside the negative F region thus initially is curved down. Because H 2 is tangent to I in 3 the second compression shock that terminates the fan has an upstream sonic state (Fig. 3.4.10). With increasing downstream pressure the points 3 and 5, which characterize, respectively, the states before and after the second shock, move to larger and smaller values of v. This in t u m results in a weaker fan and a stronger terminating shock. Finally, in the limit as v approaches the value v6 the fan vanishes and both shocks coincide in the x, t-plane. As a result, the pressure increase Pl - + P6 can be viewed as generated by either the combined action of two sonic shocks 1 -+ 2, 2 -+ 6 or by a single shock discontinuity. However, larger pressure jumps, say, Pl -+ P7, always give rise to the formation of single compression shocks.
360
A. Kluwick
I~2
2--+3
3--+5
x FIGURE 3.4.10 Disintegration of an imposed pressure jump (1 ~ 5 of Fig. 3.4.9) into a shockfan-shock combination; shock, ~ characteristic.
pT P=O
//2 2 \
M 1.
1-r
R1 1~ --+ 2 H1
-
Vb
~)i
-
V
FIGURE 3.4.11 Rarefaction split shocks: H1, H2... shock adiabats; R1, R2... Rayleigh lines; M . . . curve formed by endpoints of all possible shock adiabats H 2" I . . . isentrope.
3.4 RarefactionShocks
361
Compression shocks leading from state 1 to a state between the two sonic points 2 and 4 of the shock adiabat H I violate the stability condition equation (3.4.45) and therefore have been ruled out. It should be noted, however, that larger pressure jumps P4 < Pa < P6 satisfy the wavespeed-ordering relationship. Moreover, if the downstream pressure is sufficiently close to P6 the entropy criterion equation (3.4.9) also will be satisfied. Nevertheless, the preceding discussion of the phenomenon of shock splitting, that is, the formation of shock fan combinations as Pa increases beyond the value Pi, strongly suggests that compression shocks of this type whose Rayleigh lines cross the shock adiabat in two interior points are inadmissible even if they satisfy the second law and the stability condition. This will be confirmed in the following section dealing with the internal dissipative shock structure. Before addressing this problem in more detail let us return to the case of rarefaction shocks. As pointed out previously, the amplitude of such shocks having an upstream state 1 in Fig. 3.4.6 is limited by the occurrence of a downstream sonic state. Closer inspection of the general shape of the shock adiabat, however, shows that the stability condition equation (3.4.45) may not only limit the maximum possible shock strength, but also the minimum possible shock strength. As depicted in Fig. 3.4.11 this occurs if the upstream state 1 is located to the left of the inflexion point A of the shock adiabat. The rarefaction shock of minimum strength then has an upstream sonic state and is represented by the Rayleigh line R1, which is tangent to H I in 1 and causes a pressure drop Pl ~ P3" Very weak pressure drops starting with Pl therefore will be achieved isentropically and result in a rarefaction wave fan. This solution is correct as long as v < v i the specific volume corresponding to the inflexion point on I where the fundamental derivative changes sign. For v > v i the fundamental derivative is negative and the expansion fan therefore curls over to become triple valued. Elimination of the region of muhivaluedness results in a rarefaction shock that terminates the wave fan. With increasing values of v the rarefaction wave fan weakens and vanishes in the limit v ~ v3 in which the terminating shock coincides with the rarefaction shock of minimum strength. So far, we have shown that the occurrence of inflexion points on the shock adiabat may lead to a limitation of the maximum or minimum possible shock strength. Shocks of limiting strength are characterized by the presence of upstream or downstream sonic states and represent an essential ingredient of the shock-splitting phenomenon. In this connection it is interesting to note that rarefaction shocks (in contrast to compression shocks) can also arise in the form of double sonic shocks, that is, shocks that have both upstream and downstream sonic states. An example is depicted in Fig. 3.4.12, which in addition indicates how a double sonic rarefaction shock may form naturally inside an expansion wave fan leading from state 1' to state 2'. Here, I 1 and 12 denote the isentropes passing through the points 1 and 2 characterizing the states upstream and downstream of the shock front.
362
A. Kluwick
H
t
1'-+1 1-
2
2--+2'
/2 .,,
x
FIGURE 3.4.12
Double sonic rarefaction shock: H . . . shock adiabat;/1, I2... isentropes.
The qualitative picture of nonclassical shock behavior in the dense gas region of fluids with high specific heats that has emerged from the foregoing considerations is fully supported by quantitative predictions based on specific equations of states. Detailed calculations using the van der Waals gas model and adopting the assumption cvo~(T) -- const, have been carried out by Cramer and Sen (1987), and Cramer (1989b). As pointed out earlier, these constitutive relationships lead to an analytic expression for the shock adiabat. Moreover, it is possible to derive exact solutions for sonic shocks that considerably simplify the treatment of shock splitting. Representative results showing the evolution of the density distribution are summarized in Figs. 3.4.13 and 3.4.14. As before the coordinate system is assumed to move with the shock speed. Furthermore, ~c - x/L and t = t ~ / L denote a nondimensional distance and time that are defined in terms of an artifical length L and the parameters a, b entering the equation of state (3.4.21).
3.4.4 SHOCK STRUCTURE In the preceding sections shock waves have been treated as discontinuities propagating in an inviscid and thermally nonconducting fluid. In reality,
3.4
363
Rarefaction Shocks
(a) ............................. Pb = 0.776, Pb = 1.013, Mb = 1.059
0.75
0.5
t=O
J
.............................
N.
Pa = 0 . 5 1 7 ,
Pa = 0 . 8 6 3 ,
Ma = 1
I \ \ 5 10 15
0.25
|
0
,,
1
(b) t-
15 ~-.
0
10 5 \
I
/ .......................... Pb = 0.952, Pb = 1.059, Mb = 1 1.0
0.75
o
-1
...... Pa = 0.737, Pa = 0.999, Ma = 0.953
0.5
t
0
1
FIGURE 3.4.I3 Splitting of an initially imposed density jump into: (a) shock-fan; (b) fan-shock combination. Van der Waals gas with 6 - 0.02. Results from Cramer and Sen (1987).
however, thermoviscous processes neglected so far lead to a continuous transition between the upstream and downstream states. To this author's knowledge, this was demonstrated first in independent studies by Rankine (1870) and Prandtl (1906) in which the effect of heat transfer was included, leading to an estimate of shock-layer thickness. The mechanisms of both heat conduction and internal dissipation were accounted for later by Taylor (1910) whose celebrated weak shock solution beautifully displays their combined effect on the shock structure. Further analytical progress was reported in the investigation by Becker (1922) that included the important observation (attributed in fact to Prandtl) that the inclusion of heat transfer alone may not be sufficient to obtain completely smooth shock profiles. Similar to the
t 1.25 ~= 15
-1.0 ...................
Pb = 0.994, Pb = 1.069,
Mb = 1
0.75
0.5 ............................. Pa = 0.476, Pa =0.826, M a = l
I
!
-2
-1
0.25 ~~~:. 0
1
2
FIGURE 3.4.14 Splitting of an initially imposed density j u m p into fan-shock-fan combination. Van der Waals gas with J = 0.02. Results from Cramer and Sen (1987).
3.4 RarefactionShocks
365
work by Rankine, Prandtl and Taylor the investigations carried out by Becker concentrated mainly on perfect gases. Real fluid effects on the structure of shock waves have been investigated by Weyl (1949) and in more detail by Gilbarg (1951), who proved the existence and uniqueness of the shock-layer solutions for the class of fluids considered by Weyl (1949) with the property that F and G are strictly positive (as pointed out by Menikoff and Plohr (1989) the assumption G > 0 is for convenience but is not crucial). The shock-layer problem for fluids with embedded negative F regions has been treated by Cramer and Crickenberger (1991). In the context of dense gas dynamics the investigation of the shock-layer problem is important for at least two reasons. First, it is important to establish admissibility criteria excluding nonphysical solutions of the simplified governing equations obtained in the limit of zero viscosity and heat conductivity. In the field of classical gas dynamics dealing with positive F fluids, simple rules such as the stability criterion equation (3.4.31) or the requirement that entropy must not decrease are sufficient to select physically acceptable shock solutions. However, as pointed out in the preceding section neither rule is any longer equivalent if F is allowed to change sign. In such situations the stability criterion (Eq. (3.4.31)) typically is found to be more restrictive than the entropy criterion equation (3.4.9). However, even Eq. (3.4.31) was seen to be insufficient to rule out discontinuities whose inadmissibility is strongly suggested by the analysis of the shock-splitting phenomenon. In this connection it should be noted that no general procedure is known at present that allows the strict derivation of appropriate admissibility conditions on the basis of the simplified set of governing equations alone. In fact, it has recently been shown that the standard argument leading to the stability criterion (Eq. (3.4.31)), for example, by Germain (1972) contains a serious loophole (see Kluwick et al., 2000). It is tacitly assumed that the states upstream and downstream of a shock discontinuity can be varied independently. However, this assumption may be violated if dispersive effects enter the shock-layer problem as is the case, for example, for internal waves in two-layer fluids (Cox et al., 2000) and concentration waves in suspensions of particles in fluids (Kluwick, 1991b; Kluwick et al., 2000). In both cases (which share the property of mixed nonlinearity with the type of flow considered here) the stability criterion equation (3.4.31) then is found to be both too weak and overly restrictive. It is too weak because it admits the formation of sonic shocks that may be inadmissible, and overly restrictive because it excludes the occurrence of nonclassical shocks (,Jacobs et al., 1995; Hayes and Le Floch, 1997; Le Floch, 1999), which emanate rather than absorb characteristics. It thus appears that the investigation of the shock-layer problem is the only safe way to derive admissibility criteria that properly mimic all the physical effects neglected in the reduced set of governing equations.
366
A. Kluwick
A second reason why the study of the thermoviscous shock structure is felt to be important is in connection with the shock-splitting phenomenon. This is found to occur if the strength of a compression shock that bridges the whole negative F region (for example, 1 -~ 7 in Fig. 3.4.9) is continuously reduced so that the Rayleigh line eventually touches the shock adiabat. According to the theory of shock discontinuities discussed in Section 3.4.3, the wave propagates as a single shock as long as the Rayleigh line is (even only slightly) above the shock adiabat, but immediately splits into two shocks if it falls (however little) below the sonic point 2. Taking into account, however, that thermoviscous effects cause information to spread over finite distances, it is reasonable to expect that the shock structure will in some sense be able to anticipate the impending splitting phenomenon before the Rayleigh line even becomes tangent to the shock adiabat. Following Cramer (1989b) and Cramer and Crickenberger (1991) the investigation of the shock structure problem is restricted to single-phase Navier-Stokes fluids. As before the flow is taken to be 1D and steady. The balance equations for mass, momentum, and energy then assume the form dpu
-- 0
(3.4.50)
dx
du
pu-~
(2 4- 2p)
dp 4-
- - ~ ( p u - (2 4- 2/~)u
(3.4.51)
4- N
(3.4.52)
Here, #(p, T), 2(p, T) and k(p, T) denote the shear viscosity, second viscosity, and thermal conductivity, which are assumed to satisfy the inequalities __>0, 3 2 + 2 p > 0 ,
k>0
(3.4.53)
The first integral of Eqs. (3.4.50)-(3.4.52) may be written as // -=
/)
m
d/)
m(2 + 2/~)-77~- -- Fv(V, T) = p - Pi + m2( v - vi)
(J.~.~',
(3.4.55)
61,X
1
k dT _ FT(v ' T) =_ (e - ei) - -~ m2(v2 - I)2) + (Pi 4- m2vi)(v - vi) mdx
(3.4.56)
3.4 RarefactionShocks
367
where the subscript i refers to either the upstream ( i - b) or downstream ( i - a) state. The boundary conditions associated with Eqs. (3.4.55) and (3.4.56) are (T,v)~T~,v~)
as
( T, v ) ---> ( T a , Va )
x~-oo
as
x--->oo
(3.4.57)
They imply the conditions Fr(v ~, Ti) = 0
Fv(v i, Ti) = O,
(3.4.58)
which, in combination with the requirement m = const., are easily shown to be equivalent with the jump relationships (Eqs. (3.4.6)-(3.4.8)). According to Eq. (3.4.58) the pairs %, T b and v a, T a represent singular points of the shock-layer equations (3.4.54)-(3.4.56). It may be shown that a subsonic critical point is a saddle point while a supersonic critical point is a repulsive node (Weyl, 1949). Furthermore, using standard thermodynamic relations the following expressions may be shown to hold (Gilbarg, 1951; Menikoff and Plohr, 1989): OF v
> 0,
aT dT
p
dT
> / < --d-vv
dv F~
OF~
-~
> 0
(3.4.59)
l)
supersonic/subsonic critical point
(3.4.60)
Fr
If the fundamental derivative F is strictly positive, the Rayleigh line intersects the shock adiabat in two points only. As a result, v b, T b and v a, T a are the only roots of Fv(v, T ) = 0 and Fr(v, T ) = 0. As shown by Gilbarg (1951), Eqs. (3.4.59) and (3.4.60) together with the classification of these singular points mentioned in the foregoing are then sufficient to establish existence and uniqueness of solutions to the shock-layer problem. The corresponding phase plane is sketched in Fig. 3.4.15. Its basic features are easily inferred from Eqs. (3.4.59) and (3.4.60), which require the curve F v = 0 to lie above the curve F r -- 0 and by noting that dT dv
>0
for
Fv > 0 , F T > 0
or
Fv < 0 , Fr < 0
(3.4.61)
that is, for every v, T-pair not lying on the F v = 0, F r ----0 curve while dT
d-v < 0
for
Fv < 0 , Fr > 0
(3.4.62)
As pointed out by Cramer (1989b), Gilbarg's analysis can be extended without difficulty to cases where F is strictly negative so that %, T b and v a, T~ are again the only roots of Fv(V, T ) = 0 and Fr(v, T) = 0. Moreover, as Eqs.
368
A. K l u w i c k
T FT, Fv > O
,, T.
.
.
.
.
.
.
.
> O, F,, < 0
Tb
FT, Fv < 0
Va FIGURE
3.4.15
P h a s e p l a n e for s t r i c t l y p o s i t i v e F ; - . -
',! '',.,
Vb F L. -- 0, - - -
V F r = O, ~
trajectories.
(3.4.59) and (3.4.60) are valid for arbitrary media they can readily be used to work out the general form of the phase plane for the shock-layer problems associated with fluids having embedded regions of negative F (Cramer, 1989b; Menikoff and Plohr, 1989). A typical example is depicted in Fig. 3.4.16 where it has been assumed that the Rayleigh line intersects the shock adiabat in four rather than two pairs (v i, Ti). Inspection of this graph shows that, as before, there exist trajectories that connect neighboring roots (v~, Ti). However, it is clearly not possible to construct solutions that describe the transition between states represented by nonneighboring pairs (v i, Ti). Consequently, one concludes that the Rayleigh line of admissible shocks, that is, shocks having a thermoviscous structure, must not cut the shock adiabat in interior points and therefore must lie entirely above or entirely below the shock adiabat. This requirement is seen to be equivalent to the Oleinik criterion (Oleinik, 1959) and to Lax's generalized entropy condition (Lax, 1971). According to the entropy inequality equation (3.4.9) and Eq. (3.4.15) shocks of the first type will be compression shocks while shocks of the second type will be rarefaction shocks. Numerical solutions to the shock-layer equations (3.4.54)-(3.4.57) have been obtained by Cramer and Crickenberger (1991) who adopted the MartinHou equation of state. The methods developed by Chung et al. (1984), Chung et al. (1988) were used to calculate the shear viscosity and thermal conductivity of the BZT fluids PP10, PP11 and FC- 71. Due to lack of data the bulk
3.4
369
Rarefaction Shocks
T
T
.........
-:.-:.-.
............ "'",. ',
FT, F,, > 0
T4
:
i
T3 i
i
i
.
i
:
,, it
T2
T~
........ F T , F v < O
U4 FIGURE 3.4.16
x
........i .................... i
U3
i .....
U2
Phase plane for F ~ 0 ; - . -
U1
F,: = 0, - - -
F~ = 0, - -
trajectories.
viscosity #b = 2 + 2p/3 was taken to be proportional to p. Furthermore, several numerical tests indicated that different choices of the ratio Pb/# do not affect the basic qualitative features of the dissipative shock structure that are of primary interest. Results for three rarefaction shocks in PP10 are plotted in Figs. 3.4.17, 3.4.18, and 3.4.19, where the distance x is nondimensionalized with the length scale L - #,x lye~pc and p, denotes the value of p at the reference state, TABLE 3.4.2 N u m e r i c a l Data for the E x p a n s i o n Shocks of Figs. 3.4.17-3.4.19. Fluid is PP10 w i t h vb = 1.75Vc, T b = 0.99Tc, Pb = 0.9194pc, l'b = --0.298, and ~ b / ~ -- 0.5. R e p r i n t e d w i t h P e r m i s s i o n from C r a m e r and Crickenberger: The Dissipative Structure of Shock Waves in Dense Gases. J. Fluid Mech. 223. C o p y r i g h t 1991, C a m b r i d g e University Press Shock
valv c
PalPc
TalTc
Fa
Mb
Ma
sa - sb ( x 10 - s )
1 2 3
1.95 2.15 2.35
0.8896 0.8589 0.8279
0.9876 0.9855 0.9836
-- 0.1633 -- 0.0222 0.1002
1.014 1.023 1.027
0.989 0.988 0.995
0.72 3.93 7.63
R
370
A. Kluwick
J
o.9 --
t~a
--
t/b Vb
2/~~/~ 1
U 0.7
l(
0.6
0.$
O.4
02
&l
0 -~
-2.1) - I J
- 1.0 -O.S
0
0.$
1.0
I.S
~(• X
2.0
2.$
-31
FIGURE 3.4.17 Variation of specific volume versus x for three rarefaction shocks in PP10. Numerical data are given in Table 3.4.2. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press.
typically the critical temperature and one atmosphere. All three shocks have the same upstream state given in Table 3.4.2, which also lists the downstream conditions and the Mach numbers. According to Table 3.4.2 the shocks i and 2 of smallest and intermediate strength leave the medium in the negative F region while l"a > 0 downstream of the strongest shock 3. In agreement with the considerations of Section 3.4.3 the downstream value Ma of the Mach number increases with increasing shock strength. Shock 3 is only slightly weaker than the shock of maximum possible strength having a sonic downstream state. The salient features of Figs. 3.4.17 to 3.4.19 can be summarized as follows: (i)
(ii)
rarefaction shocks may exhibit pronounced regions inside which the entropy is smaller than the value sb far upstream (of course s - sb always becomes positive as the downstream state is approached to satisfy the entropy inequality); the shock thickness does not necessarily decrease with increasing shock strength. In Fig. 3.4.17 the shock thickness is seen to decrease initially. However, this trend is reversed as the shock approaches its maximum strength characterized by Ma = 1; and
3.4
371
Rarefaction Shocks IO [
8
-
8a
8b
-
-
-
0
Sb
.
.
.
.
.
.
' "
9
I
~
~
//
/
-30
-40
-60
-2.s-i.o-;.~-i.o-;.~
o
0.'5
,'.o J~s 2'.o z~ x
Z (x10-31
FIGURE 3.4.18 Variation of entropy versus x for three rarefaction shocks in PP10. Numerical data are given in Table 3.4.2. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press.
~,,
M - Ma
Mb- Ma
~.~. i ,,L~ l
9,
! \\
~i\," 2 3 $
-2~
-2.0 - I J
- I . 0 -G$
0
0.5
1.0
IJ
x
2.0
2.5
( x lo -3)
FIGURE 3.4.19 Variation of Mach number versus x for three rarefaction shocks in PP10. Numerical data are given in Table 3.4.2. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press.
372
A. Kluwick
TABLE 3.4.3 Numerical data for the compression shocks of Figs. 3.4.20-3.4.22. Fluid is FC-71 with vb =2.5Vc, T b = T c , Pb =0.8523pc, l-'b = 0 . 2 1 6 0 , and ~b/[.I,=0. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press Shock 1 2 3 4 5 6 7
va/vc
Pa/Pc
Ta/Tc
Fa
Mb
Ma
sa -R sb ( x 10-5)
1.3000 1.2875 1.2750 1.2625 1.2500 1.2375 1.2250
1.0528 1.0551 1.0575 1.0599 1.0624 1.0649 1.0675
1.0088 1.0089 1.0091 1.0092 1.0094 1.0096 1.0097
0.8330 0.8979 0.9644 1.0322 1.1011 1.1707 1.2408
1.017 1.017 1.018 1.019 1.020 1.021 1.022
0.949 0.942 0.934 0.926 0.917 0.907 0.897
13.94 18.04 22.82 28.36 34.76 42.10 50.50
1.0 -
v -- Vb
.:
[
6 ~tl I/.--/--3
0.9
7 ...i I
t/a - - Vb
r.--'2
/ /
0.8
i / 'i / /
0.7
/ 0.6
0.$ 0.4 0.3 0.2
0.1 O -3
~ -2
-I
0
I
2
3
4
$
X
~(xlO -3) FIGURE 3.4.20 Variation of specific volume versus x for seven compression shocks in FC-71. Numerical data are given in Table 3.4.3. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press.
3.4
373
Rarefaction Shocks
(iii) the distribution of the Mach number inside the shock layer may not be monotonic. Although M(x) decreases monotonically for shocks 1 and 2 it exhibits a local minimum for shock 3, which causes F to change sign.
In order to shed further light on these phenomena the properties of weak shocks will be considered in more detail in Section 3.4.5. Before doing so it is instructive to consider representative examples of numerical shock-layer solutions for compression shocks (Figs. 3.4.20, 3.4.21, and 3.4.22). The upstream conditions (which are the same in all cases considered), the downstream conditions, and the Mach numbers are summarized in Table 3.4.3. All shocks (which are numbered according to their strength) take the fluid across the negative 1-" region. As a result, they cannot become arbitrarily weak but, as pointed out in Section 3.4.3, must be stronger than the shock of limiting strength whose Rayleigh line touches the shock adiabat in an intermediate point where F < 0 (Fig. 3.4.9). Similar to the results for rarefaction shocks
2o!
. . . . . . .
"
"4"
" " '
"1"~"
,5
' "" II ill ,,, i, i i ~ ,l~,l '1 I i
II
,o
.'.~1,,I II I ~F~,!,i ,! I
tlll l , !
I
'
I
I
/ I I /I
o
2x (x10-3) Variation of entropy versus x for seven compression shocks in FC-71. Numerical data are given in Table 3.4.3. Reprinted "with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases.J. Fluid Mech. 223. Copyright 1991, Cambridge University Press. FIGURE 3.4.21
374
A.
Kluwick
already discussed a variety of nonclassical phenomena is observed and summarized in what follows.
(i)
(ii)
As in the classical case of fluids with strictly positive F the shock thickness is seen to increase with decreasing shock strength. Initially, this thickening does not affect the qualitative form of the density distribution inside the shock layer that exhibits a single inflexion point. Eventually, however, two additional inflexion points form, which in turn cause a dramatic increase of the shock thickness. It thus appears that the internal shock structure anticipates the inviscid shock-splitting process. The admissibility criterion (Eq. (3.4.31)) requires the Mach number to decrease from a supercritical to a subcritical value. For all shocks considered here this transition is not monotonic but rather is characterized by the presence of a local Mach number minimum and Mach number maximum. Furthermore, the local maximum is seen to exceed the critical value 1 in each case. One thus concludes that 9
w|
I
9
Iii
9
I
9
9
1
9
9
I
9
9
9
I v
9
6 1.0
M - M ~
0.$
1 iii ,
-3
".-'a
'2,
"
;
'
;
"
i
"
;
"
..]
,
s
x ~(xIo -3) FIGURE 3.4.22 Variation of Mach number versus x for seven compression shocks in FC-71. Numerical data are given in Table 3.4.3. Reprinted with permission from Cramer and Crickenberger: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223. Copyright 1991, Cambridge University Press.
3.4 RarefactionShocks
(iii)
375
compression shocks that bridge the negative F region can have an internal layer of supersonic flow. The entropy is larger than the upstream value throughout the shocklayer. Furthermore, the entropy distribution exhibits a single maximum if the shock strength is sufficiently large. As to be expected, however, a second maximum occurs in shocks that are so weak that the shock layer, anticipating the inviscid shock-splitting process, develops a double layer structure.
Finally, it should be noted that the shocks listed in Tables 3.4.2 and 3.4.3 cause extremely small entropy jumps. This agrees with the suggestion made in Section 3.4.2 and may be of importance as far as practical applications of BZT fluids are concerned (see, for example, works by Cramer, 1991 and Kluwick, 1998).
3.4.5 WEAK SHOCKS The shock-layer solutions obtained by Cramer and Crickenberger (1991) have revealed a variety of phenomena not encountered in the field of classical gasdynamics. Some of these effects are plausible in the sense that they are (as in the case of impending shock splitting) suggested by simple physical considerations. However, these considerations as well as purely numerical computations are insufficient to determine the exact conditions under which these phenomena will occur. Further insight into the relevant physical processes is possible if the considerations are restricted to shocks of small amplitude. In addition, such a strategy has the advantage that it not only produces interesting analytical results but also allows the study of the global behavior of flow fields with embedded shocks. To simplify the analysis the flow under consideration is assumed to be 1D. Furthermore, the unperturbed state is taken to be uniform and at rest. It is then convenient to introduce the nondimensional quantities ~c _ x
-s
-t _ a o t
_
u
a
P -- Po
ao
ao
Poa~
P '
Po
T -~oo '
s-s
o
Cvo
(3.4.63) Here, x, t and L are, respectively, the propagation distance, the time, and a characteristic wavelength. The subscript 0 denotes quantities evaluated at the
376
A. Kluwick
undisturbed state. If the thermoviscous effects are negligibly small, which will be investigated first, the governing equations then can be written as dfi
dp
3-~
t5-~4-h--~---~--~
d2
on
d~
-~-a4-h,
~-0
on
d~
-~--h
(3.4.64)
where the nondimensional function fi is given by -- Cvo flTpa 2 po a2
(3.4.65)
cp
Furthermore, let e << 1 denote the small but finite wave amplitude. The various field quantities are then expanded in the form fi "-- 8/11 + 82R 2 - 4 - ' ' ' '
;f = 1 + e T 1 + 82T2 999
t 5 - 1 + 8Pl So --
--
-+- 8 2 p 2
"~- 8 n s 1
(3.4.66)
~- " " "
-'t- " " "
CvO
As in the study by Cramer and Kluwick (1984) the considerations will be restricted to right-moving shocks, that is, shocks that propagate in the positive x-direction. The various perturbation terms then depend on the wave coordinate = .~- t
(3.4.67)
and the slow time z
(3.4.68)
= g~t
The values of the parameters n, m depend on the position of the point P in the p, v-diagram that characterizes the unperturbed state relative to the transition line (Fig. 3.4.23). If P is located sufficiently far from the transition line F = 0, then F = O(1) and small pressure and density disturbances of 0(5) cannot cause the fundamental derivative to change sign. Hence, the leading order nonlinear correction to the convected sound speed is 0(5) indicating that significant distortions of the wave profile occur over propagation distances and times of O(1/5) as in the case of classical nonlinear acoustics, which has been studied intensively in the past (see, for example, Crighton, 1979). The proper choices then are m = 1, n - 3 and one obtains the evolution equation 3P13z+ ~'pl ~
-
O,
F -
F0
(3.4.69)
3.4
377
Rarefaction Shocks
P
I - - C.On 8 tq
//
v FIGURE 3.4.23 Regions where F exhibits different orders of magnitude: I, I': F - O(1); II, ir: F = O(g); III: F = O(s2). - . - transition line F = 0, saturation line.
and recovers Eq. (3.4.16):
~4
[S1] - - 6~voTo [])1] 3
(3.4-.70)
Here, F 0 denotes the value of F in the unperturbed state. Depending on the sign of Fo ><0, Eq. (3.4.69) causes waves to steepen forward or backward leading in turn to the formation of compression or rarefaction shocks, which cannot be generated simultaneously. This is possible, however, if the unperturbed state has F 0 -- O(e) as in regions II, ir of Fig. 3.4.23, so that the pressure disturbances carried by the wave may lead to a sign change of F. It may then be inferred from Eq. (3.4.4) that the perturbations of the convected sound speed are O(s 2) rather than O(s) as before. As shown in Cramer and Kluwick (1984), this implies m -~ 2, n -- 4 and the wave evolution is governed by
~. ~ f~p, +-~p~
- o,
_ ro ~'
~, - A0 - (ar(~, ~)) (3.4.71) \ ah o
378
A. Kluwick
Furthermore, the entropy jumps caused by shocks satisfy a~ [pl] 3 f, + [Sl]- 6cv0T0
(Pl~ + Plb)
(3.4.72)
Not only F but also its first derivative with respect to p may change sign if the unperturbed state is in region III of Fig. 3.4.23, that is, in the vicinity of the point where the isentrope is tangent to the transition line F = 0. Hence, F 0 and A 0 are O(e 2) and O(e), respectively. This in turn leads to disturbances of the convected sound speed of O(e3). In addition, inspection of Bethe's relationship (Eq. (3.4.16)) suggests that the entropy changes associated with weak shocks will be of O(eS). One thus concludes that m = 3, n = 5 and as demonstrated by Cramer and Crickenberger (1991), the form of the evolution equation consistent with these scalings is
0p,+ 1~0 -- F~
+ z~
-7'
A~
=-T'
=o, N -- ( 027(/3' s))
\
(3.4.73)
o
Finally, it may be shown that the proper generalization of Eq. (3.4.70) is given by (Kluwick, 1993):
IS1]
"
-
-
agTo [oil 3 F + 6Cvo
2- (Pla + Fib) + ~-~(3P~a + 4PlaPll, + 3P12~) (3.4.74)
It is useful to recast the evolution equations (3.4.69), (3.4.71), and (3.4.73) in a more compact form by introducing the perturbation mass flux J(Pl) = ~- Pl2 + ~ p3 + 2-4 p4
(3.4.75)
where the definitions of the parameters I~, z~, and/~/depend on the unperturbed state, that is, z~ = N = 0 for states in region I, I ~and N -- 0 for states in II, II' including the high-pressure and low-pressure branch of the transition line. One then finally obtains the kinematic wave equations Opl ~ dj OPI -- OPl ~ OJ(Pl) -
(3.4.76)
which have been investigated in detail in the past (see, for example, Whitham, 1974).
3.4
379
Rarefaction Shocks
First, it follows immediately from the first equation in (3.4.76) that the density perturbation Pl is constant on characteristic curves propagating with the (convected) sound speed dj v w = dp 1
(3.4.77)
Second, evaluation of the second equation in (3.4.76) for discontinuous solutions yields the relationship [J] v~ = [p~]
(3.4.78)
expressing the shock speed in terms of the jumps in density and perturbation mass flux. It may be verified that this result is fully consistent with the leadingorder approximations to the jump conditions (3.4.6)-(3.4.8) (Cramer and Kluwick, 1984; Cramer and Crickenberger, 1991). The relationships (Eqs. (3.4.77) and (3.4.78)) can be interpreted graphically by plotting the curve j--Jl(Pl)" According to Eq. (3.4.77) the slope of this curve is directly related to the characteristic speed corresponding to the value 191 considered. Similarly, it follows from Eq. (3.4.78) that the slope of the straight line connecting two points on this curve equals the speed of a shock having the density jump [Pl]- One thus concludes that, just as in the classical theory of weakly nonlinear acoustic waves, the graph J(Pl) plays a double role. It relates isentropic changes of the field quantities but can be interpreted also as the leading-order approximation of the shock adiabat in the j, pl-plane. In the following, straight lines that connect the upstream and downstream states of shocks will therefore again be referred to as^Rayleigh lines. Finally, inspection of the definitions of F, A and/~ shows that the local value of the fundamental derivative is given by I ~ --
F~m
d2J dp 2
(3.4.79)
with values m = 0, 1 and 2 for cases where the unperturbed state is in regions I, II, and III of Fig. 3.4.23. This is seen to be consistent with the general theory of finite amplitude unidirectional planar waves that are governed by the evolution equation (Thompson, 1971) 0t -t-
~ F(~)d~ - - - 0 0 P Ox
(3.4.80)
Numerical solutions to this equation, which in contrast to their weakly nonlinear versions requires the density distribution to be continuous, have been obtained by Cramer and Sen (1986).
A. Kluwick
380
Within the framework of classical weakly nonlinear acoustic waves thermoviscous effects were studied first by Lighthill (1956). Specifically, he showed that a self-consistent description of these effects involves a linear term being proportional to the second derivative of t91 with respect to ~, which is added to the transport equation (3.4.69) for nondissipative flows. This result carries over unchanged to waves exhibiting mixed nonlinearity (Cramer and Kluwick, 1984; Cramer and Crickenberger, 1991). In summary, then, the dissipative form of Eq. (3.4.76) may then be written as OPl+ OJ(Pl)
0~
O~
(3.4.81)
~ ~2fll
-
2
O~2
Here, ~5 - ~
1 (
20
2+--+ ~o
70-1) Pr
'
Poao L Re = ~ , l~o
Pr --
#oCpo
(3.4.82)
ho
denote, respectively, the acoustic diffusivity of a general fluid, the Reynolds number (based on a characteristic wave length L), and the Prandtl number. Furthermore, l = 1, 2 or 3 for F 0 = O(1), O(e) or O(G2). While the expansions for ~, T and ~ in Eq. (3.4.66) remain unchanged the expansion for ~ now has to take into account that the entropy disturbances inside a shock layer are much larger than the overall entropy jump: -- ~SO "-F gn-1 S1 "~ O(gn-1) Cv0
(3.4.83)
as can be seen from the energy equation in nondimensional form, which upon using the first-order results T1
/'ll - -
Pl
=
Go
(3.4.84)
yields S1 __
~0G0 Pr
~ Ofll 2 + 20//10 + (70 - 1 ) / P r 3~
(3.4.85)
Equation (3.4.81) describes the evolution of weakly nonlinear, weakly dissipative acoustic waves in the dense gas regime of BZT fluids. In the limit A -- N = 0 it reduces to the classical Burgers equation, which can be linearized by the Cole-Hopf transformation (see, for example, Lighthill, 1956). By contrast, powerful analytical methods to calculate closed-form solutions of the modified Burgers equation (3.4.81) with /~ g: 0 and/or /r # 0 are not known at present. (For example, as proved by Nimmo and Crighton (1982), no linearizing Backlund transformations can exist that would yield analytical solutions to the cubic Burgers equation f ~ - / q / - 0.) In the absence of such
3.4 RarefactionShocks
381
exact methods one therefore has to resort to numerical computations (Cramer, Kluwick, Watson and Pelz, 1986). In this section the main interest in Eq. (3.4.81) is in connection with the thermoviscous structure of weak shocks. In order to determine this structure one seeks traveling-wave solutions of the form Pl = Pl(r/),
rI = ~ - vs~
(3.4.86)
where vs denotes the shock speed. By imposing the boundary conditions Pl --+ Plb as r / ~ oo and Pl --+ Pla as r / ~ - o o one recovers the result, Eq. (3.4.78), which relates vs to the shock strength. Furthermore, it is found that the density distribution inside the shock layer is governed by (Kluwick, 1998): dp 1 = J(Pl) -- J(Plb) -- Vs(Pl -- Plb)" 2 drI
(3.4.87)
Due to the fact that the perturbation mass flux J(Pl) is given by polynomials of the form of Eq. (3.4.75), the shock-layer equation (3.4.87) can be solved in closed form (Taylor, 1910; Cramer and Kluwick, 1984; Cramer, 1987; Cramer and Crickenberger, 1991). However, by noting that the right-hand side of Eq. (3.4.87) represents the distance between the curve j - - J ( P l ) and the Rayleigh line in the j, pl-plane, some interesting qualitative features of shocks can be obtained also by simple geometric considerations. In this connection it is useful to consider the most general case where terms up to p4 have to be included in the expression for the perturbation mass flux (Fig. 3.4.24). If the strength of a shock that bridges the whole negative F region is sufficiently large the right-hand side of Eq. (3.4.87) exhibits a single maximum as Pl increases from fllb to 191a. Hence, the resulting shock profile will have a single inflexion point as in the case of a perfect gas (Fig. 3.4.24(a)). If the upstream state of the shock is kept fixed while Pla decreases, the slope of the Rayleigh line is seen to become equal eventually with the slope of the tangent in the inflexion point I 1 of the j, pl-graph. For even weaker shocks therefore the expression on the right-hand side of Eq. (3.4.87) exhibits two maxima and one minimum in the interval Plb <--Pl <--Pla, which in tum leads to shock profiles with three inflexion points (Fig. 3.4.24(b)). This is the phenomenon of impending shock splitting discussed in the preceding section. Similar to the solutions of the full Navier-Stokes equations also the dissipative structure of weak shocks govemed by Eq. (3.4.87) anticipates the shock-splitting phenomenon, which according to inviscid theory was found to occur where the Rayleigh line becomes tangent to the j, pl-graph. As the Rayleigh line approaches this limiting position the slope of the shock at the middle inflexion point tends to zero and the overall density increase across the shock is then achieved essentially by two individual weaker shocks that are separated by a
impending shock splitting :" /
~O O0 bO
...""
ff..""".."'"'"" ia) Pl .....
. . . . . . . . . . . . . . . .
Ill
r>0 ~
J (c)
(b)
.
~b.__
F I G U R E 3.4.24 Thcrmoviscous shock structure. Admissible profiles with (a) one and (b) three inflcxion point(s) and inadmissible profile (c).
383
3.4 Rarefaction Shocks
pronounced plateau region. Further reduction of the shock strength causes the Rayleigh line to cut the j, pl-graph in intermediate points. These represent singular points of the shock-layer equation (3.4.87) and as a result a continuous transition from Plb t O Pla i s no longer possible (Fig. 3.4.24(c)). This is seen to be equivalent to the shock admissibility criterion derived from the full Navier-Stokes equations that the Rayleigh line of admissible shocks must not cut intervening branches of the shock adiabat. However, due to the fact that the weakly nonlinear approximation to the shock adiabat in the j, Pldiagram is given by relatively simple polynomials of the form Eq. (3.4.75) this condition as well as the stability criterion (3.4.88)
vwa > v s > vwb
can now be evaluated analytically (Cramer and Kluwick, 1984; Kluwick and Scheichl, 1996). The results are displayed in Fig. 3.4.25 for the two cases where the undisturbed state of the fluid is in the vicinity of the high- or low-pressure branch of the transition line or in the neighborhood of the point where this line is tangent to the isentrope (regions II, II', and III of Fig. 3.4.23). In the latter case the condition for the possible formation of rarefaction shocks that the curvature of the shock adiabat changes sign requires that the value of the parameter
C = ~2
(3.4.89)
be less than one-half. Density pairs violating the condition placed on the location of the Rayleigh line are located within the hatched regions. In order to meet the stability criterion equation (3.4.88) regions representing admissible combinations are labeled according to the sign of A. Density pairs on solid curves respresent sonic shocks. The points D and D' in Fig. 3.4.25(b), therefore, define double sonic (rarefaction) shocks. Finally, pairs located on the dotted curve correspond to split shocks whose associated Rayleigh lines contact the shock adiabat in an interior point. Additional information conceming the general properties of shock profiles may be gained by recognizing that the distance between the Rayleigh line in the j, pl-plane and the graphj = J(Pl) close to the intersection points/91 - - Pll, l -- a and/or b varies linearly with fll - - fill if Vs and the convected sound speed Vwl are different J(Pl) --J(Plb)
-- Vs(Pl
-- Plb)
~" (Vwl -
Vs)(Pl --
Pll)
(3.4.90)
384
A. Kluwick
PlbF $
(a) ~,
.,.,""""
h
k>o
.,.,.""
~.>o
-s
P~"T
~i./,i-
i
Ple
"
!
j
1>o
j,
//"
./"
)-
,,,
(b)
-2 -4 -6 -8
-1o
- !o
-8
-6
-4
-2
o
2
Plb
4
6
~7
FIGURE 3.4.25 Admissible and inadmissible shock waves in the Pla, Plb-Plane" Density pairs within the hatched region violate the requirement that the Rayleigh line must not cut intervening branches of the j, pl-diagram. In order to be able to meet the stability criterion equation (3.4.88) regions representing admissible shocks are labeled according to the sign of A. (a) F 0 = O(e). Reprinted with permission from Cramer and Kluwick: On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142. Copyright 1984, Cambridge University Press. (b) F 0 = O(~2). Reprinted with permission from Kluwick and Scheichl: Unsteady transonic nozzle flow of dense gases. J. Fluid Mech. 310. Copyright 1996, Cambridge University Press.
3.4
385
Rarefaction Shocks
Investigation of the shock-layer equation then immediately yields the result known from classical gasdynamics that the density profile approaches the upstream and/or downstream asymptote exponentially Pl -- Pll ~ e x p ( 2 ( v ~ t f
v~) (rl + Cl)),
,rl[ -+ oo
(3.4.91)
where CI is an arbitrary constant. However, if the shock under consideration has a sonic upstream or/and sonic downstream state then dv w (Pll)(Pl J(Pl) -J(Plb) - Vs(Pl -- Pll) ~" ~1 dp 1
-
(3.4.92)
Pll) 2
and consequently the approach of Pl to the upstream or/and downstream value is algebraic rather than exponential
Pl - Pll "" - d v ~ (Plz)0/+ CI) dpl
,
I~/I--~ oo
(3.4.93)
Representative examples of shock profiles for the case that the unperturbed state is in the neighborhood of the high-pressure or low-pressure branch of the transition line, that is, N = 0, are depicted in Fig. 3.4.26. The quantities G, B, and ~ are identical to those introduced by Cramer and Kluwick (1984) and are given by 2 6 -- ~
Pla + Plb ' 1+
2
B -r ,
6F
a[p 1]
125 n
14 ; r
" 2
'
(3.4.94)
It is easily shown that shocks with upstream and downstream states such that B < 1 violate the admissibility criteria discussed earlier, and therefore will not have physically acceptable profiles. The value B -- 1^corresponds to shocks having sonic upstream (A > 0) or sonic downstream (A < 0) conditions. The slow algebraic decay of sonic shocks for @--+ oo compared to the much faster exponential decay of admissible nonsonic shocks B > 1 is clearly visible in Fig. 3.4.26. For large values of B" 1~//~ >> 1, Pl = O(1) and the effects of mixed nonlinearity are small. Consequently, the shock speed and the wavespeeds before and after the shock (approximately) satisfy the bisector rule v s = (Vwa + Vwb)/2 , which in turn implies that the upstream and downstream decay rates in Eq. (3.4.91) are equal. The shock profiles then are almost antisymmetrical with respect to { = 0 as in the case of a classical Taylor solution that is approached in the limit B --+ oo.
386
A. Kluwick I
l l i ~
'
I
~'~.gL",dkl'~k~
I
B=I~ ~
G
"
I
"
I
I
5
J
--l--
'
t
.
-4
I
....
I
-2
l
. . . . .
0
,
2
~
t
4
FIGURE 3.4.26 Shock structure. B = 1 corresponds to a shock having sonic conditions as ~ ~ e~. Reprinted with permission from Cramer and Kluwick: On the propagation of waves exhibiting both positive and negative nonlinearity.J. Fluid Mech. 142. Copyright1984, Cambridge University Press. If the unperturbed state of the medium and the upstream/downstream state of the shock is fixed while the shock strength is increased, specific values of correspond to increasingly smaller values of the original coordinate r/. In general, this trend is not significantly affected by the associated changes of the parameter B and the shock thickness therefore is found to decrease with increasing shock strength as in the case of perfect gases. However, if this process eventually leads to a sonic shock this tendency may be inverted by the transition from the exponential to the algebraic approach to the downstream/upstream state, which in the G, ~-plane is associated with a rapid increase of the shock thickness. In fact, it is this mechanism that is responsible for the nonmonotonic dependency of the shock thickness on shock strength evident in Figs. 3.4.17 to 3.4.19. Detailed calculations within the framework of the weakly nonlinear theory for unperturbed states that are in the neighborhood of the high- or low-pressure branch of the transition line have been performed by Cramer (1987). In this case the (dimensional) shock thickness A measured by the maximum slope criterion can be expressed in the form A --=Ld
2F 2 /~2[Pll2Gma x
,
La=
12I/~I~L 132
(3.4.95t
where L a is a nonlinear diffusion length. Results obtained for the two cases where either the upstream or the downstream state is the unpertubed state are plotted in Fig. 3.4.27. In addition, Fig. 3.4.28 displays the Rayleigh lines of
3.4 RarefactionShocks
387
A 2Ld
H
{ 2~>0, ^
pl.=O
. A
P~b=O
I
IH
{ A>O, p~b=O ^ A < O, Pla = 0
/~- > O, Plb = 0 ?, < O, Pl,, = 0 0.1-
-S0
-4D
-S~)
-2.0
-I.0
~
tO
2.0
"--0
4.0
5D
-= pla or
r
F
Plb
FIGURE 3.4.27 Plot of scaled shock thickness versus strength. From Cramer (1987), modified. admissible shocks in the j, pl-plane assuming 13 > 0, /~ > 0 for definitness. Shocks characterized by points on curve I in Fig. 3.4.27 then represent compression shocks with unperturbed upstream state. They can never assume a sonic downstream state and as a result the shock thickness decreases with increasing shock strength. By contrast, curve II characterizes compression shocks with unperturbed downstream state. As seen from Fig. 3.4.28, their amplitude cannot exceed that of the shock having a sonic upstream state. In agreement with the earlier qualitative considerations, the shock thickness therefore initially decreases with increasing magnitude of the density jump but eventually increases again as the decay rate in Eq. (3.4.91) becomes sufficiently small and finally vanishes to give rise to the algebraic behavior exhibited in Eq. (3.4.93). Points on curve HI in Fig. 3.4.27 represent rarefaction shocks with unperturbed upstream conditions. The shock of minimum amplitude has a sonic upstream state but all stronger shocks are nonsonic. As a result, the shock thickness is found to decrease with increasing magnitude of
388
A. Kluwick
plA F
-1 --,
..
!
j~2 v
F >0 II:pl,, = 0
_
FIGURE 3.4.28
f,a
III: Plb = 0
j, pl-Diagram w i t h / ( / = 0, terminology is for f" > 0,/~ > 0.
[/91]. Similar considerations for other sign combinations of 1~,/~ reveal that the nonmonotonic variation of the shock thickness with shock strength will occur in rarefaction/compression shocks if f" < / > 0. For shocks that are sufficiently weak so that mixed nonlinearity does not play an important role, B >> 1, Eq. (3.4.95) may be used to derive the estimate Aex .... it~ ( 20 7 - 1 ) aoFo[p ] 2 + -/~o - + Pr
(3.4.96)
Inspection of this result reveals two mechanisms that can lead to a significant increase of the shock thickness: 1) small values of F 0, that is, if the unperturbed state is in the vicinity of the transition line; and 2) large values of 20/~0 as found, for example, in common substances such as CO2 and H2 (Emanuel, 1990; Emanuel and Argrow, 1994) and in situations where the unperturbed state is near critical (Borisov et al., 1983). An additional interesting conclusion based on the analysis of weak shocks can be drawn from Eq. (3.4.85). It shows that compression/rarefaction shocks exhibit a significant entropy excess/deficit over most of the shock profile. This
3.4 RarefactionShocks
389
is seen to be in complete qualitative agreement with the fully nonlinear solutions of the Navier-Stokes equations depicted in Figs. 3.4.18 and 3.4.21. In addition, Eq. (3.4.85) predicts that the entropy distribution of compression shocks that bridge the whole negative Fregion will have two maxima and a minimum if the density profile exhibits three inflexion points. Again, this behavior has been found to occur in the results for shocks of finite strength presented in Fig. 3.4.21. To leading order, the local Mach number M in a frame moving with the shock is given by M - 1 ~ e~(v~- v~)
(3.4.97)
where r = 1, 2 or 3 for F 0 = O(1), F 0 = O(0 or F 0 = O(e2). Differentation of the structure equation (3.4.87) with respect to r/ shows that v s - v w can be expressed in terms of PI which yields the relationship (Cramer and Crickenberger, 1991) M -
1 ~, _ ~ r _~ d 2 p l / d r l 2 2 dpl/dr I
(3.4.98)
One thus concludes that the flow is sonic at inflexion points of the density profile. In general, the Pl versus r/ curves exhibit one inflexion point that separates regions of supersonic and subsonic flow. However, in the case of impending shock splitting there will be three sonic points. Moreover, because of the inequalities Mb > 1 and Ma < 1 Eq. (3.4.98) then predicts the formation of a supersonic pocket between the second and third sonic point. Indeed, just such an embedded region was found in the numerical results depicted in Fig. 3.4.22.
3.4.6
SHOCK
DYNAMICS
So far, the discussion of shocks in the dense gas regime of BZT fluids has focused on their local properties. It is the aim of this section to outline briefly how these affect the global behavior of flow fields. Most work in this area has been carried out for weakly nonlinear waves, which therefore will be the main topic in the following. Waves of finite amplitude have recently been studied numerically by Argrow (1996), Brown and Argrow (1997, 1998), and Bates and Montgomery (1999). It is useful to start with the investigation of 1D planar waves for which the evolution equation has been derived in the preceding section. As already pointed out, Eqs. (3.4.77) and (3.4.78) can immediately be used to obtain graphical solutions to boundary or initial value problems. According to the formal solution of the kinematic wave equation Pl is constant on character-
390
A. Kluwick
istics which therefore, are represented by straight lines in the ~, r-diagram. Moreover, inspection of the result for the wavespeed shows that the slope of characteristics in this diagram is given by the slope of the tangent to the j versus Pl plot in the corresponding point. If characteristics intersect, the solution has to be supplemented with a shock front that is parallel to the Rayleigh line connecting the upstream and downstream states in the j,/91plane. As an example we consider two cases of square pulses defined by the initial conditions
z--O:
for
~-f < 1
Pl F -0
(3.4.99)
elsewhere
adopting the assumption that the j, pl-relationship is of the form Eq. (3.4.75) with N - 0, Fig. 3.4.28. For definiteness F and /~ are again taken to be positive. It should be noted, however, that the results plotted in Figs. 3.4.283.4.30 are valid for all possible sign combinations if the notation compression and rarefaction shock F > 0 and F < 0 are modified appropriately (Cramer and Kluwick, 1984). In the first case considered, Fig. 3.4.29, the parameter/3 o is negative but the amplitude of the wave is so small that the fluid remains in the positive F region. Consequently, the discontinuous density drop at ~ - ~ 2 / ~ imposed initially, spreads into a wave fan while a compression shock forms at -- - F 2 / A . As in classical nonlinear acoustics the trailing shock is weakened by its interaction with the wave fan and for large time its strength is proportional to 1/~/z. If the amplitude of the negative pulse is increased the fluid inside the pulse eventually enters the negative F region. As shown in Fig. 3.4.30 the resulting wave pattern then is completely different from the classical result. First, the discontinuity at ~ = 1~2//~ disintegrates only partly into a wave fan, which is followed by a sonic rarefaction shock. Second, the amplitude of the initially prescribed discontinuity at ~ = - l ~ 2 / / k is too large to lead to an admissible compression shock. It thus splits into a compression fan that is terminated by a sonic compression shock. The sonic rarefaction shock is weakened by its interaction with the compression wave fan. In order to remain sonic the rarefaction shock emanates characteristics that leave the front in the tangential direction. In this way a precursor region is formed. Points inside this region are no longer directly connected with the initial data but the information propagates along characteristics that cross the sonic rarefaction shock. Furthermore, since the compression and rarefaction shock propagate with different speeds they eventually collide at z - z c. This gives rise to another interesting
3.4
391
Rarefaction Shocks
T
\,
I' compression shock
II
II
I
m~ -2
r=
20
f_
0
-
"2"
10
.iT
v
0
'__/
FIGURE 3.4.29 Wave evolution for /30 = - 0 . 9 . Reprinted with permission from Cramer and Kluwick: On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142. Copyright 1984, Cambridge University Press.
phenomenon as can be seen from the density profiles corresponding to different values of ~. While according to the classical theory of weakly nonlinear acoustic waves two merging shocks result in a single shock of larger amplitude, the collision of a rarefaction and a compression shock is found to be associated with the formation of a sharp spike in the density distribution as z - z c -+ O- and a sudden reduction of the shock strength at z = z~. For values of z that are large compared to the collision time z~ the waveprofile approaches a linear sawtooth and the shock strength decays as 1 / ~ again as in the case of a classical fluid. The reader interested in additional information concerning the behavior of 1D planar waves resulting from
392
A. Kluwick
localized disturbances is referred to the original papers by Cramer and Kluwick (1984), and Cramer et al. (1986). In the latter study also thermoviscous effects that are governed by Eq. (3.4.81) are taken into account. The evolution of inviscid periodic disturbances have been investigated by Cramer and Sen (1986) and Kluwick and Koller (1988); see also Kluwick (1991a). In all these references the assumption is made that the unperturbed state is in the vicinity of the high- or low-pressure branch of the transition line. Unsteady planar flows, where the relationship J(Pl) has to be approximated by a quartic, have not been treated in detail so far. The theory of strictly 1D planar waves can be extended in various ways. An obvious and important generalization is to include three-dimensional (3D) effects into the analysis (Kluwick and Czemetschka, 1990; Kluwick, 1991a;
I
single compression shock
4O
I
rarefaction shock
30
compression shock
20
compression fan
-19
-IS
i0
-11
-7
-3
0
3
t -45
40
30
FIGURE 3.4.30 Wave evolution for P 0 - - 2 . 2 . Reprinted with permission from Cramer and Kluwick: On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142. Copyright 1984, Cambridge University Press.
3.4 RarefactionShocks
393
Cramer and Webb, 1998; Kluwick and Cox, 1998). A representative example of these types of flows is given by the propagation of quasi-one-dimensional waves in a channel whose cross-sectional area varies slowly with propagation distance. As before the considerations will be restricted to unperturbed states where the leading-order approximation to the perturbation mass flux is given by a cubic polynomial. Expansion of the field quantities in the form of Eq. (3.4.66) with independent variables - t - ~,
~ - E2~
(3.4.100)
then yields the evolution equation (Kluwick and Czemetschka, 1990)
OPl
F -+-/~' P ) Pl OPl + ~Pl -dA - =0
(3.4.101)
Here A(~) denotes the cross-sectional area nondimensional with a suitable chosen reference value. Introducing the transformed quantities -- p x ~ ,
/3 -- ~ P l ,
~-5{
(3.4.102)
and recasting Eq. (3.4.101) into divergence form one obtains ^
O~ O)_O, 3--~-F 3--~
~2
)--
2~/-A
~3
-
(3.4.103)
~SA
As in the treatment of strictly 1D waves the formal solution to this kinematic wave equation is given by -- const,
on
d~
~j
----_--= d.~ O~
~
~2
~r~
2A
(3.4.104)
In general, the solution to a given initial or boundary value problem will contain shock discontinuities. Equations (3.4.104) then have to be combined with appropriate jump conditions that have already been discussed in the treatment of 1D planar waves. Although these conditions hold unchanged, the variation of the density disturbances caused by the changes of the crosssectional area A(~) may nevertheless lead to the occurrence of a significant new phenomenon. This is seen most easily by following the evolution of a single shock initiated by the boundary conditions:
P- {
forf~ o
(3.4.1o5)
394
A. Kluwick
Furthermore, it will be assumed that A(~) increases with propagation distance ~ and A(~) ~ oo, ~ ~ oo as, for example, in the case of outgoing cylindrical (l = 1) and spherical (l = 2) waves d l ~ l n A -- x-
(3.4.106)
Finally, the following discussion will again be for f~ > 0, /~ > 0. The modifications required to cover other sign combinations are easily inferred from the definitions Eq. (3.4.102) of the transformed quantities/3, ~. If/30 > 0, the boundary conditions Eq. (3.4.105) cause the formation of a compression shock (Fig. 3.4.31(a)). Due to the requirement that the crosssectional area grows with increasing distance it follows from Eqs. (3.4.102) and (3.4.104) that the scaled density disturbances /3 downstream of the shock decay. Still, however, the shock strength vanishes only asymptotically in the limit ~ ~ oo. This well-known result of classical nonlinear acoustics has been expressed very well by Hayes (1972) who coined the phrase: "old shocks never die, they are like soldiers who only fade away." For values of/30 between - 3 and 0 the density jump imposed at k = x0 fully (Fig. 3.4.31(b)) or partly (Fig. 3.4.31(c)) disintegrates into a wave fan. Finally, a single rarefaction shock is generated if/30 < - 3 (Fig. 3.4.31(d)). It is easily seen that the evolution of this latter shock differs significantly from that of the compression shock discussed earlier. Again, the shock strength is found to decrease with increasing propagation distance. This implies that the flow regimes (d), (c), and (b) indicated in Fig. 3.4.31 correspond to subsequent stages of the wave evolution. As a consequence, the conditions for a sonic shock will be reached at some distance ~ = k s. For ~ > xs, a sonic rarefaction shock forming a precursor region emerges from the initially generated single shock front. Moreover, this sonic shock must vanish at a finite distance xst > xs where the scaled density perturbation immediately downstream of the shock front assumes the value 13st - - 1 corresponding to the inflection point I of the )/~/A versus/3 graph that separates regions (c) and (b). One thus concludes that shocks in the dense gas regime of BZT fluids may experience sudden death. Quantitative results that exhibit this nonclassical phenomenon are depicted in Fig. 3.4.32 for the case of a spherical wave A ( ~ ) - ~2 initiated by the boundary conditions =x0=2:
/3=
-4 0
for 0 < ~ < 4 elsewhere
(3.4.107)
As pointed out previously, the density jump imposed at k - 2, ~ = 0 then is large enough to generate a single rarefaction shock. The increase of the crosssectional area slows the shock that propagates with the unperturbed speed of
3.4
Rarefaction Shocks
II
v
r
:
v
r
395
03
U3
O
~~
396
A. Kluwick
10
35 o o~ oo-s ooo 9 s o s~176 J 9 ss~ 9 oo sos
............................
.~ PToI
9 'j
0
oo
. . . . . ~176176 . . . . . . ~176176176
o~"
FIGURE 3.4.32
~ ,E'~~
~
o ~
I0
I0
~
-.-'':i'i 20
30
6 3 ~
""
-4
-2
0
Evolution of a spherical square pulse for P0 = - 4 .
sound when the density perturbations downstream of the front assume the value = - 3 . At larger values of ~ > xs ~ 3 its strength and speed are determined by the requirement that the upstream state is a sonic state. The sonic shock terminates at x s t - 8 and the corresponding smooth density distribution exhibits a plateau region generated by characteristics carrying constant values r0 - r(x0, ~)- However, the plateau region vanishes when the characteristic leaving the endpoint of the sonic shock merges with the trailing shock. At large distances ~ ~ oo the density distribution inside the precursor approaches a linear sawtooth. In Fig. 3.4.32 the length of the pulse was chosen such that the sonic rarefaction shock terminates before it collides with the trailing compression shock. However, both shocks will collide if the length/amplitude of the initial disturbance is sufficiently small/large, which then leads to a flow pattern similar to that depicted in Fig. 3.4.30. In all cases the trailing shock remains intact up to ~ - - o o , its strength decreasing proportional to ~:-l(ln~:) -1/2 asymptotically as in classical nonlinear acoustics provided 1~ # 0. In this connection the following nonclassical property of the trailing shock is important. Due to the fact that the positive density jump prescribed at = x0, ( - 4 is too strong for an admissible compression shock it splits into a compression wave fan and a compression shock with a sonic upstream state, again similar to the case of the planar square pulse shown in Fig. 3.4.30. In contrast to this case, however, the density disturbances carried by the characteristics forming the wave fan decay with increasing propagation distance. This in turn causes the characteristics to curve and to merge with
plfit/f"
3.4 RarefactionShocks
397
the trailing shock, which is clearly visible in Fig. 3.4.32. As a result the trailing shock has nonsonic upstream conditions for all ~ > x0. This brief discussion of quasi-one-dimensional waves has shown that shocks experiencing mixed nonlinearity may terminate at finite distance from their origin due to the geometrical spreading of disturbances. This is possible also in strictly planar waves if the unperturbed state is not constant but varies in the propagation direction (Kluwick and Cox, 1992). If these variations are sufficiently slow the obvious generalization of Eq. (3.4.71) is OPl
(
F(~;) + A(0) p ) 0pp l
-- O,
--5-
z-
t-
iIi
dcr
J
(3.4.108)
Here, a0(~) denotes the speed of sound in the unperturbed medium nondimensional with its value at ~ - 0:a0(0 ) - 1. Figure 3.4.33 shows the evolution of a density jump imposed at ~ - x0
/~(0)_ ~: -- X0"
/9 -- t91 l~(0)
Po
for ~ - f~2(0)% > 0
0
elsewhere
(3.4.109)
assuming that 1~ varies linearly with
f~(})- f~(}o)(l+ K(}- Xo))
(3.4.110)
If F(~) does not change sign, K > 0, the resulting wave pattern in the ~, ~plane qualitatively resembles that of the I~-constant case: the imposed density jump disintegrates into a fan for 1~ ~ 0 and P0 X 0 while a shock forms for F ~ 0 and P0 ~ 0. If, however, K < 0, the fundamental derivative changes sign at ~,-~c o -1/K, which leads to a completely different behavior. Results for F(~o) ~ 0 and P0 ~ 0 are plotted in Fig. 3.4.33(a). The imposed density jump initially causes the formation of a wave fan. Once I~ has changed sign the characteristic curves generating this fan start to converge and they eventually focus at the point ~ - 0, ~ - xs - 2~, - x0-As a consequence, the shape of the density distribution imposed at ~ - x0 is recovered at this value of the propagation distance. Due to the changed sign of 1~ the associated density jumpnow produces a stable shock front that propagates into the region ~ > ~s. If F(~0) ~ 0 and P0 <>0, then as seen in Fig. 3.4.33(b), a shock discontinuity is formed at ~ = x0. Because [1~[ decreases with increasing values of ~ the speed of this shock decreases also despite the fact that the shock strength remains unchanged up to ~ - ~, where sonic conditions are reached. For larger values of ~ the resulting flow pattern therefore will be similar to that obtained for 1~ ~ 0, P0 ~ 0 and K > 0. Consequently, the sonic shock front at ~ - ~, disintegrates into a wave fan of finite strength.
6 9~ -
2.~,-:~o
~:- .~o
.~o
--,
! .~,
-
~o
o2
~(~o) -l
~o+6
0+2
~(~o)
_-
~*
d
i
2
3~
ko + 4
2:~,, ~o
(a)
(b)
FIGURE 3.4.33 Evolution of a single density jump: (a) l~(0)X0, ~o<~0, (b) l~(0)~0, ~o~0 (I/~o/l~(0)[--i, K---0.5). Reprinted with permission from Cox and Kluwick: Propagation of weakly nonlinear waves in stratifiedmedia having mixed nonlinearity.J. Fluid Mech. 244. Copyright 1992, Cambridge University Press.
3.4 RarefactionShocks
399
The examples discussed so far indicate that the unusual form of the admissibility criteria for shocks in fluids having mixed nonlinearity strongly affects the global flow behavior that exhibits a number of new and interesting phenomena. Unfortunately, however, difficulties associated among others with the required high-pressure and high-temperature levels have prevented so far the direct experimental observation of sonic shocks, split shocks, double sonic shocks and terminating shocks in single-phase BZT fluids. Even the experimental generation of a single rarefaction shock of constant strength represents a task that still has to be carried out. In principle, such a shock can be generated in at least two different ways, that is, using shock tubes or resonance tubes. Shock tube experiments in the dense gas regime, which have been simulated numerically by Argrow (1996) and Brown and Argrow (1997), require highly sophisticated test rigs and measuring techniques. They produce a discontinuous set of data but have the advantage of a relatively simple theoretical basis. Experiments in resonance tubes are expected to be easier. Moreover, by controlling the piston frequency it should be possible to vary the shock strength continuously over the interesting range of thermodynamic states. The theoretical interpretation of the results requires the investigation of interacting oppositely traveling waves, which is significantly more difficult than the treatment of unidirectional waves considered so far, even if the wave amplitude is small. As a first step towards a better understanding of the relevant physical processes, Cox and Kluwick (1996) investigated the longtime evolution of the signal in closed resonance tubes applying a small amplitude, small rate approximation and assuming a periodic piston displacement of the form #h(co~ for operating frequencies in the neighborhood of the linear resonance frequencies com = m/2, m = 1, 2 . . . . :
co = tom(1 + ~2A),
A = O(1)
(3.4.111)
As before, ~ ~ 1 characterizes the magnitude of the density perturbations. The signal evolves on the long time scale ~ - ~2cot but is periodic in the fast scale r = c~t. Formal analysis then yields the amplitude of the piston (nondimensional with the tube length L) # ~ 1 necessary to sustain density oscillations of order E. However, this result can be derived also from an energy balance that requires that the energy dissipated by a shock per unit time P0T0[~] has to be supplied by the work of the piston per unit time poa~h. Due to the assumption F = O(~) adopted in Cox and Kluwick (1996) [~] R O(~4), which together with ~ = O(~) and fi = O(#) immediately yields ~~3.
A. Kluwick
400
Taking h(cot-) - 1 sin(2zrcot) and m - 1 one then finds that the signal on the closed-end boundary of the tube is given by the evolution equation 3g
OT
Fg q-
/~kg2 --2-
A q-
/~ [r+l 2---r
gR(z)dz
)
3g
g-
rt
~cos(2gr)
~ ~)2g -- 2 Or2
(3.4.112)
where g denotes the Riemann invariant on right-running characteristics and Pl = 2 g . In the limit of steady-state inviscid flow, which is of primary interest, Eq. (3.4.112) can be simplified by introducing the transformed quantities G -v-g, F
y - r sgn A
(3.4.113)
which gives + G - 2 ~y + A cos(2rty) -- 0
(3.4.1141
Here, the quantity 2 represents the effective detuning parameter that combines the effects of frequency detuning and detuning caused by mutual wave interaction while A has the meaning of an effective forcing parameter: /k
7~/~k2
1 Ii +1 G2(r)d r (3.4.115)
A - - ~ - ~ sgnA Similar to the case of perfect gas oscillations (Betchov, 1958; Chester, 1964; Jimenez, 1973) physically acceptable solutions of Eq. (3.4.114) are required to be single valued and to satisfy a zero mean condition. The resulting solutions then are found to contain shock discontinuities for a certain range of the controlling parameters 2 and A that forms the resonance band (see Fig. 3.4.34). Furthermore, it may be shown that inside this resonance band four regions related to different qualitative flow behavior have to be distinguished. Points in region (1) correspond to solutions that exhibit qualitative similar features to perfect gas oscillations and acceptable solutions are constructed from separatrices that connect saddle points (Fig. 3.4.35(a)). For the purpose of discussion discontinuities that increase with y will be referred to as compression shocks. Again, it should be noted, however, that the actual physical classification depends on the signs of F and A. Region (2) is distinguished from region (1) by a qualitative change of the integral curves. Nevertheless, acceptable solutions can still be obtained from saddle point integral curves, Fig. 3.4.35(b). Region (3) is characterized by the occurrence of a sonic compression and a
3.4
401
Rarefaction Shocks
0,80
0.00
1
2
3
4
5
A FIGURE 3.4.34
Resonant shock band.
rarefaction shock (see Fig. 3.4.35(c)). Finally, points in region (4) correspond to solutions that contain both a sonic compression and a sonic rarefaction shock (see Fig. 3.4.35(d)). Using these results it is possible to simulate resonance tube experiments. As an example consider the case where the unperturbed thermodynamic state of the fluid is kept constant while the piston frequency varies. In terms of the transformed variables this corresponds to solutions with fixed effective forcing parameter and varying effective detuning parameter. Results for A = 5 are depicted in Fig. 3.4.36. If 2 is sufficiently large, the corresponding point in the A, ),-plane is outside the resonance band and the solution is continuous. A transition from one shock to two shocks is observed as 2 decreases through the resonant frequency band. This corresponds to a passage through regions (1), (3), and (4) of Fig. 3.4.35. The transition occurs through the single compression shock becoming sonic at a point on the boundary between regions (1) and (3). The rarefaction shock generated as 2 continues to decrease eventually becomes sonic on entering region (4). The amplitudes of both shocks decay as 2 is further detuned away from resonance. After leaving the resonance band continuous solutions can again be constructed. These results indicate that resonance tubes are indeed a very effective means to study experimentally the distinguishing features of BZT fluids in the dense gas regime, that is, the formation of rarefaction shocks, sonic shocks and the simultaneous occurrence of compression and rarefaction shocks.
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3.4.7 CONCLUDING
REMARKS
The present chapter focused mainly on the conditions under which rarefaction shocks may form in single-phase fluids and on the local properties of such discontinuities. A very limited number of examples of how the occurrence of rarefaction shocks affects the global flow behavior have been discussed briefly in Section 3.4.6. A more detailed description of these and related phenomena would be the subject of a volume, as yet unwritten, on dense gas dynamics. It
404
A. Kluwick
should be noted, however, that considerable progress in the understanding of general dense gas flow has been achieved during the past 15 years. Some of these developments have been reviewed by Cramer (1991), Kluwick (1991a, 1998) and Thompson (1991). These include steady internal flows, which within the framework of a quasi-one-dimensional approximation, have been studied by Chandrasekar and Prasad (1991), Cramer and Best (1991), Cramer and Fry (1993), and Kluwick (1993). Results for two-dimensional (2D) nozzle flows have been obtained by Schnerr and Leidner (1993a), Schnerr and Molokov (1994), and Aldo and Argrow (1995). In addition, unsteady quasi-one-dimensional nozzle flows have been calculated by Kluwick and Scheichl (1996). Investigations of flows past single airfoils and through cascades have concentrated so far mostly on transonic flows that appear to be of significant importance as far as practical applications in turbomachines are concerned; see Cramer and Tarkenton (1992), Tarkenton and Cramer (1993), Schnerr and Leidner (1993b), Leidner (1996), and Rusak and Wang (1997, 2000). In the case of single airfoils it is found that dense gas effects may lead to a substantial increase of the lower critical Mach number (Cramer et al., 1992). This indicates the possibility of shock-free flows up to almost Mach number one even without applying elaborate techniques to optimize the airfoil shape. Furthermore, the possibility of constructing shock-free supersonic flows in cascade configurations has been demonstrated by Monaco et al. (1997). A start has been made also to include viscous effects. Investigations deal both with Fanno-type flows (Cramer et al., 1994) and boundary-layer-type flows (Cramer and Whitlock, 1993; Zieher, 1993; Kluwick, 1994; Cramer et al., 1997; Kluwick, 1998; Cramer and Park, 1999; Kluwick, 2000). Effects of mixed nonlinearity are not only important in gases where they may lead to the occurrence of rarefaction shocks, but also in various other areas of fluid mechanics. In fact, negative shocks that cause a sudden decrease in density seem to have been investigated first both theoretically and experimentally in the context of suspensions of particles in liquids (Kynch, 1952; Shannon and Tory, 1965). More recent studies dealing with the propagation of small-amplitude disturbances in homogeneous suspensions indicate additional interesting features of concentration shocks. Due to the presence of dispersive effects shocks experiencing mixed nonlinearity may violate the stability criterion equation (3.4.88) and yet be admissible (Kluwick, 1991b; Kluwick et al., 2000). The occurrence of negative nonlinearity in resuspension flows has been demonstrated by Loimer and Schaflinger (1998). As mentioned already in the Introduction, liquid vapor flow with phase transition, which has been investigated intensively at the Max Planck Institut fur StrOmungsforschung in GOttingen and the Rensselaer Polytechnic Institute in Troy, New York, represent a rich source of mixed nonlinearity. Further theoretical studies
3.4 RarefactionShocks
405
based on the thermocapillarity equations of liquid vapor systems have been carried out among others by Slemrod (1983, 1984a,b), Shearer (1988), Piechor (1994, 1995, 1996, 1997), and Kluwick (1995). In addition, also two-layer fluid flows are known to exhibit negative and mixed nonlinearity, which manifests itself among others in the formation of negative hydraulic jumps (Kakutani and Yamasaki, 1978; Helfrich, 1984; Melville and Helfrich, 1987; Gavrilov and Liapidevskii, 1996; Liapidevskii, 1996; Choi and Shen, 1997; Chen and Liu, 1998; Cox et al., 2000). However, as pointed out by Bertozzi et al. (1999), related phenomena are possible also in single-layer flows with temperature-dependent surface tension. Further examples of flows that may give rise to negative and mixed nonlinearity include first, second, and fourth sound waves in superfluid helium (Osborne, 1951; Temperley, 1951; Dessler and Fairbank, 1956; Garrett, 1981; Turner, 1981; Atkin and Fox, 1984; Torcynski et al., 1984; Torcynski, 1985; Cramer and Sen, 1990; Cramer and Kluwick, 1993; Braun et al., 1996), waves in two fluid plasmas (Bezzerides et al., 1978; Mitrovich, 1981) and waves in anisotropic plasmas (Zakharov, 1989; Zakharov and Shikin, 1990). Finally, it should be noted that the occurrence of negative and mixed nonlinearity is by no means restricted to fluids. Illustrative examples taken from the field of solid mechanics are listed in Cramer (1991); see also Carman and Cramer (1993) and Tarkenton and Cramer (1994).
ACKNOWLEDGMENTS The author wishes to thank Dr. Braun, Dr. Cox, Prof. Cramer, Prof. Meier, and Prof. Thompson for numerous stimulating discussions on dense gas dynamics. He also is deeply indebted to Mrs. Hubert, Dr. Braun and Dr. Cox for their assistance during the preparation of this manuscript. This chapter is dedicated to Prof. Dr. PA. Thompson.
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Schnerr, G.H. and Leidner, P. (1993b). Numerical investigation of axial cascades for dense gases. PICAST1-Pacific International Conference on Aerospace Science and Technology, pp. 1-8. Schnerr, G.H. and Molokov, S. (1994). Exact solutions for transonic flows of dense gases in twodimensional and axisymmetric nozzles. Phys. Fluids 6: 3465-3472. Schofield, P. (1969). Parametric representation of the equation of state near a critical point. Physical Review Letters 22: 606-608. Shannon, P.T. and Tory, E.M. (1965). Settling of slurries. Ind. Eng. Chem. 57: 18-25. Shearer, M. (1988). Dynamic phase transitions in a van der Waals gas. Quarterly of Appl. Math. 46: 631-636. Sim6es-Moreira, J.R. and Shepherd, J.E. (1999). Evaporation waves in superheated dodecane. J. Fluid Mech. 382: 63-86. Slemrod, M. (1984a). Dynamics of first order phase transitions, in: Phase Transformations and Material Instabilities in Solids, M. Gurtin, ed., New York: Academic Press, pp. 163-203. Slemrod, M. (1984b). Dynamic phase transitions in a van der Waals fluid. J. Differential Equations 52: 1-23. Slemrod, M. (1983). Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81: 301-315. Stokes, G.G. (1848). On a Difficulty in the Theory of Sound. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, pp. 349-356. Tarkenton, G.M. and Cramer, M.S. (1994). Nonlinear second sound in solids. Physical Review B 49: 11794-11798. Tarkenton, G.M. and Cramer, M.S. (1993). Transonic Flows of Dense Gases. ASME paper 93-FE-9, 1-10. Taylor, G.I. (1910). The conditions necessary for discontinuous motion in gases. Proc. Roy. Soc. A 84: 371-377. Temperley, H.N.V. (1951). The theory of propagation in liquid helium II of "temperature waves" of finite amplitude. Proc. Phys. Soc. Lond. A 64: 105-114. Thompson, P.A. (1991). Liquid-vapor adiabatic phase changes and related phenomena, in: Nonlinear Waves in Real Fluids, A. Kluwick, ed., Berlin: Springer, pp. 147-214. Thompson, P.A. (1971). A fundamental derivative in gasdynamics. The Physics of Fluids 14: 18431849. Thompson, P.A. and Lambrakis, K.C. (1973). Negative shock waves. J. Fluid Mech. 60: 187-208. Thompson, P.A. and Sullivan, D.A. (1975). On the possibility of complete condensation shocks in retrograde fluids. J. Fluid Mech. 70: 639-649. Thompson, P.A. and Kim, Y.-G. (1983). Direct observation of shock splitting in a vapor-liquid system. Phys. Fluids 26: 3211-3215. Thompson, P.A., Carofano, G.C., and Kim, Y.-G. (1986). Shock waves and phase changes in a largeheat-capacity fluid emerging from a tube. J. Fluid Mech. 166: 57-92. Thompson, RA., Chaves, H., Meier, G.E.A., Kim, Y.-G., and Speckmann, H.D. (1987). Wave splitting in a fluid of large heat capacity. J. Fluid Mech. 185: 385-414. Torczynski, J.R. (1985). Nonlinear fourth sound. Wave Motion 7: 487-501. Torczynski, J.R., Gerthsen, D., and Roesgen, T. (1984). Schlieren photography of second-sound shock waves in superfluid helium. Phys. Fluids 27: 2418-2423. Tumer, T.N. (1981). New experimental results obtained with second-sound shock waves. Physica 107B: 701-702. Weyl, H. (1949). Shock waves in arbitrary fluids. Comm. Pure Appl. Math. 2: 103-122. Whitham, G.B. (1974). Linear and Nonlinear Waves, New York: John Wiley and Sons. Widom, B. (1965). Equation of State in the Neighborhood of the Critical Point. The Journal of Chemical Physics 43: 3898-3905.
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Rarefaction Shocks
411
Zakharov, V.Yu. (1989). Possibility of rarefaction shocks in anisotropic plasma. Fluid Dynamics 24: 625-628. Zakharov, V.Yu. and Shikin, I.S. (1990). Longitudinal simple waves in a plasma with anisotropic pressure. Fluid Dynamics 25: 959-962. Zerdovich, Ya.B. (1946). On the possibility of rarefaction shock waves. Zh. Eksp. Teor. Fiz. 4: 363364. ZemplCn, G. (1905). On the possibility of negative shock waves in gas. Comptes Rendus de l'Acad~nie des Sciences 141: 710-713. Zieher, F. (1993). Kompressible Grenzschichtstr6mungen bei Zust~nden in der N~ihe des kritischen Punktes. Masters Thesis, Vienna University of Technology, 1-144.
CHAPTER3
o5
Theory of Shock Waves 3.5
Stability of Shock Waves
NIKOLAI M. KUZNETSOV N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, 117977, Russia
3.5.1 Introduction 3.5.2 Hydrodynamic Conditions of Shock-Wave Stability 3.5.2.1 'One-Dimensionar Conditions of ShockWave Instability 3.5.2.2 Corrugation Stability of Shock-Waves 3.5.2.3 Nonuniqueness of Shock Front Representation 3.5.2.4 Regions where a Shock-Wave Discontinuity is Unstable and where its Representation is Nonunique 3.5.2.5 On the Physical Meaning of the Solutions with Steady-State Corrugation Perturbations of a Shock Wave and with Acoustic Waves Emanating by the Shock Front 3.5.2.6 Resonance Reflection of a Sound Wave and Shock-Wave Stability 3.5.2.7 General Characteristics and a Simple Example of Relation Between Instability and Nonuniqueness of Steady-State Regimes 3.5.2.8 Stability of Shock Waves Pertaining to the Lower and Upper Branches of the Z-Shaped Segment of the Shock-Wave Hugoniot Curve. Splitting of an Unstable Shock Wave 3.5.2.9 Simple Interpretation of the Instability Mechanisms and Criteria for Instability
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
413
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N.M. Kuznetsov
3.5.2.10 Feasibility of Experimental Observation of Hydrodynamic Instability of Shock Waves 3.5.2.11 Stability of Shock Wave Supported by a Piston 3.5.3 Stability of the Structure of Shock and Detonation Waves 3.5.3.1 Experimental Data on Structural Instability of Shock Waves 3.5.3.2 The Structure of Shock Waves and Stability of Viscous Compression Discontinuities 3.5.3.3 On the Hydrodynamic Approach to Flows with Structurally Unstable Shock-Waves 3.5.3.4 On the Mechanisms of Structural Instability of Shock and Detonation Waves 3.5..3.5 Two-Fronts Model of a Shock (or Detonation) Wave with Instantaneous Heat Release 3.5.3.6 Two-Fronts Model of Shock and Detonation Waves with NonInstantaneous Relaxation References
3.5.1 I N T R O D U C T I O N The following problems of shock-wave stability are considered: 9 hydrodynamic stability of shock waves, that is, stability of a shock discontinuity itself, regardless of the processes taking place within the front and of its structure; and 9 stability of the steady shock-wave structure (i.e., of the transition region separating the material ahead of and behind the shock wavefront and which is often assumed to be a discontinuity in hydrodynamics). In Section 3.5.2 we consider a simple approach to determine the criteria for hydrodynamic instability based on the linear analysis of small perturbations (D'yakov, 1954; Kontorovich, 1957a; Iordanskii, 1957; Erpenbeck, 1962). The relative positions of the segment of the shock-wave Hugoniot curve satisfying the conditions of nonunique shock wavefront representation and the instability
415
3.5 Stability of Shock Waves
criteria are discussed. In this connection, the analogy between hydrodynamic shock-wave stability with other phenomena that may be either stable or unstable is considered. The mathematical and physical meaning of steady corrugating perturbations that appear in the theory of lipear shock-wave stability is discussed. The relation between these perturbations and shockwave stability is thoroughly analyzed. In Section 3.5.3 some experimental data on structural instability and the results of theoretical analysis of the conditions under which the shock-wave structure becomes unstable are presented. In both parts the feasibility of experimental observation of the shock-wave instability are briefly discussed (see, e.g., the reviews by Kuznetsov, 1989, 1995).
3.5.2 HYDRODYNAMIC CONDITIONS SHOCK-WAVE STABILITY
OF
The history of theoretical studies of shock-wave stability is a very interesting and demonstrative example of treating complicated physical phenomena. The results obtained in the course of solution of the problem not only provided the answers to particular questions but raised new more complicated problems. In tum, their solution revealed the relations between various aspects of the problem and provided explanation of the instability mechanisms.
3 . 5 . 2 . 1 " O N E - D I M E N S I O N A l 2 ' C O N D I T I O N S OF SHOCK-WAVE INSTABILITY The general relations for propagating gasdynamic perturbations derived from the gasdynamic differential equations together with typical thermodynamic properties of the materials (except for some thermodynamic anomalies) satisfy "one-dimensional" conditions of shock-wave stability, that is, conditions not associated with possible violation of the flow homogeneity in the direction aligned with the shock wavefront surface. The one-dimensional conditions for shock-wave stability comprise a condition for an increase in the entropy behind the shock wavefront and inequalities for the Mach number: M1 -----/)1 r
>
1,
M2 -- -/)2 -
<
1,
(3.5.1)
r
where v is the material velocity with respect to the shock wavefront and c is the speed of sound. Subscripts 1 and 2 denote the flow regions ahead of and behind the shock wavefront. The continuity conditions for the mass, momentum, and energy flows (Landau and Lifshits, 1987) satisfied at the shock-wave
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N.M. Kuznetsov
discontinuity (the Rankine-Hugoniot conditions) are symmetrical with respect to the direction of shock-wave propagation. In other words, these conditions allow the existence of both compression and rarefaction shock waves. However, as a shock wave reverses its propagation direction, the entropy increment changes its sign. The entropy increases in a compression shockwave (and decreases in a rarefaction shock wave), provided the second isentropic derivative of the specific volume v with respect to pressure g is positive at each point of the shock-wave Hugoniot curve (Landau and Lifshits, 1987):
(
O2v) > 0 ~p2J s
(3.5.2)
If the foregoing inequality equation (3.5.2) reverses within some pressure range, depending on the position of the initial point at the shock-wave Hugoniot curve and on the position and the size of the region where inequality equation (3.5.2) is violated, rarefaction shock waves become possible (Zeldovich, 1946). Inequality (3.5.2) is violated very rarely and only within a restricted region of the thermodynamic parameters. In particular, this is true in gases with a large specific heat near the curve of phase equilibrium with the liquid and in the vicinity of the critical point (Zeldovich, 1946). The feasibility of rarefaction shock waves theoretically predicted for this case was later confirmed experimentally (Kutateladze et al., 1980). Rarefaction shock waves are also possible when the shock Hugoniot has a kink as shown in Fig. 3.5.1. The derivative (~2V/~2)5 a t the kink point does not exist but, as far as the sign and the value of the entropy change across the shock wavefront are concerned, this kink is equivalent to a smooth bend of the shock Hugoniot, which causes violation of inequality (3.5.2) in the vicinity of the kink (Zeldovich and Raizer, 1967). Kinks at the shock-wave Hugoniot curves arise when solids undergo plastic deformations or phase transitions (polymorphic transformations and melting). Compared to the situations where inequality equation (3.5.2) is violated, kinks on shock-wave Hugoniot curves are encountered more frequently ~. Interestingly, the feasibility of generating rarefaction shock waves in materials with kinks at the shock-wave Hugoniot curves remained unnoticed for a long time although rarefaction shock waves in materials and states with a negative derivative (02v/Op2)s were sought for. The number of materials capable of producing rarefaction shock-waves and the practical importance of the phenomenon suggest that materials with kinks at the shock-wave Hugoniot curves are of major interest. This was realized by Drummond in 1957 following experiments performed by Bancroft et al., in 1956 in which shock compression accompanied by a plastic flow and/or phase transitions caused formation of two or more shock waves instead of one. Under certain conditions rarefaction shock waves are formed in the course of unloading (Ivanov and Novikov, 1961).
3.5
417
Stability of Shock Waves
B .
.
2 B
.
.
!
I
/ 3 sl,f S'F
_
"
1
0
v
0
u
FIGURE 3.5.1 Shock Hugoniot with a kink or a smooth bend (the dashed line in Fig. 3.5.1a) at point s-. Condition L < - 1 is met along the s- - s + segment. In the p, u coordinates (Fig. 3.5.1b), the dashed lines show schematically the isentropes (for downward transitions) or shock Hugoniot curves (for upward transitions); A and B are the boundaries of nonunique representations of the shock-wave discontinuity; H is the shock-wave Hugoniot curve with the initial point 3; and S at the bottom is the isentrope with the initial point at the top (see the shock-wave with pressure P2 and isentropic rarefaction wave in Fig. 3.5.4d,e).
Condition equation 3.5.1 implies that the velocity of a shock wave is supersonic with respect to the unperturbed material and is subsonic with respect to the material behind the shock front. The necessity of these conditions for shock-wave stability is obvious--due to the first of the inequalities (Eq. (3.5.1)), perturbations generated by the shock wave do not run away ahead of the front and therefore the shock discontinuity is not smeared like some unsteady transition layers. This includes, for example, a centered rarefaction wave with a layer with variable temperature (or concentration of a diffusing component) in heat conduction (diffusion), with an initial temperature (concentration) discontinuity, and with initial homogeneous distributions of the temperature (concentration) at each side of the discontinuity in an infinite medium. The second inequality in (3.5.1) provides continuous communication of the shock wavefront with the region behind the shock wave. Without such connection the shock-wave amplitude (if the shock wave is assumed to be an infinitely thin layer behind the discontinuity) would drop instantaneously to zero because there would be no energy required for heating and accelerating the material passing through the discontinuity. In mathematical terms, conditions (Eq. (3.5.1)) show that the solution of the problem of interaction of an arbitrary small-amplitude gasdynamic pertur-
418
N . M . Kuznetsov
bation and a shock-wave is unique. A small perturbation results in a unique change in the shock-wave amplitude upon its reflection at or transmission through the discontinuity when three boundary conditions, namely, continuity of the mass, momentum, and energy fluxes through the discontinuity are satisfied in a unique manner. Therefore, the number of independent secondary perturbations, that is, the change in the shock-wave amplitude and oneparametric acoustic and entropy perturbations originating from the shock wavefront into which the initial perturbation can be decomposed, must be equal to three and, hence, there must be only two outcoming perturbations. If inequalities (3.5.1) are met, the number of outgoing perturbations is exactly two, namely, one acoustic and one entropy (contact discontinuity) perturbation. These perturbations propagate downstream of the shock wavefront through the material behind it (Landau and Lifshits, 1987).
3 . 5 . 2 . 2 CORRUGATION STABILITY OF SHOCK WAVES Conditions (Eq. (3.5.1)) are necessary but insufficient conditions for shockwave stability because they do not rule out the instability caused by the interaction between the neighboring elements at the discontinuity surface (and the related stream tubes). This interaction could cause a growth of random local inhomogeneities (distortions) at the shock discontinuity surface. This instability is called corrugation instability. It was analyzed by D'yakov in 1954 (see also the subsequent studies by Kontorovich, 1957a; Iordanskii, 1957; and Erpenbeck, 1962) who applied the linear stability theory (LST). They solved the dispersion equation for the complex frequencies of acoustic perturbations of the exp [i(kr- wt)] type and analyzed the sign of Im co. The conditions for corrugation instability were determined by D'yakov (1954) in the form of the following inequalities: L<-landL>
l+2M 2
(3.5.3)
where
J is the density of the material flux through the shock wavefront, (Ov/Op)iqis the derivative of the specific volume with respect to pressure taken along the shock-wave Hugoniot at point p = P2, and D is the shock-wave velocity. Examples of shock-wave Hugoniot curves satisfying the first of inequalities equation 3.5.1 at segments s- - s+ and s~ - s+ are shown in Figs. 3.5.1 and 3.5.2. Segment s- - s+ of the shock-wave Hugoniot curve shown in Fig. 3.5.3
3.5
Stability of Shock Waves
419
s~ oo
L v
FIGURE 3.5.2
A shock Hugoniot version with segment sG - s+ within which L < -1.
satisfies the s e c o n d i n e q u a l i t y e q u a t i o n (3.5.3). F o r the v a l u e s of p a r a m e t e r L in t h e r a n g e 2
1 - OM 2 - M22
- 1 < L < L o, where 0
=
L~
(3.5.5)
2-M 2
Vl/V 2 is t h e c o m p r e s s i o n ratio at t h e s h o c k w a v e f r o n t , c o r r u g a t i n g
p e r t u r b a t i o n s d e c a y w i t h t i m e a c c o r d i n g to t h e p o w e r law. I n t h e r e m a i n i n g r a n g e of t h e p a r a m e t e r L, t h a t is, at L 0 > L > 1 + 2M 2
(3.5.6)
2 In D'yakov's work (1954), the expression for L0 was written erroneously because when finding the angular boundary between the waves arriving at the shock discontinuity and emanating from it, the flow of the material behind the discontinuity was not taken into account. As a result, the following expression was derived: L0 = (1 - 0M2 + 2M2)/(1 4- 2M~). The correct equation (3.5.5) for L0 written here was derived by Kontorovich (1957a) and Iordanskii (1957).
420
N.M. Kuznetsov P -" .
i
s"4.
.
.
3\ \
H s- S
~z v
0
0
o
u b
FIGURE 3.5.3 Shock Hugoniot with a segment s - - s + where L > 14-2M 2. In the p,u coordinates (b), the lower dashed line is the isentrope tangent to the shock Hugoniot at point s+ and the upper dashed line is the shock Hugoniot curve drawn from point s-; A and B are the boundaries of the regions where the shock discontinuity is nonunique; S is the isentrope starting at point 2; H is the shock Hugoniot curve emanating from point 2 (see the isentropic rarefaction wave and shock-wave with pressure P3 respectively, in Fig. 3.5.4b,c).
very peculiar solutions in the form of acoustic waves emanating from the shock discontinuity at a certain angle depending on L were obtained using the LST (D'yakov, 1954; Kontorovich, 1957a; Iordanskii, 1957; Erpenbeck, 1962). In this solution, corrugating perturbations travel along the discontinuity surface, being neither attenuated nor amplified, that is, they are stationary in a frame of reference attached to the discontinuity surface. It looks as if the shock-wave generates sound waves. The physical meaning of these solutions and the related problem of shock-wave stability at values of parameter L in the range (Eq. (3.5.6)) are analyzed in Section 3.5.2.5.
3 . 5 . 2 . 3 N O N U N I Q U E N E S S OF SHOCK F R O N T REPRESENTATION After the results of the investigation by D'yakov (1954) concerning the instability regions were repeatedly confirmed using different approaches (Kontorovich, 1957a; Iordanskii, 1957; Erpenbeck, 1962), Gardner (1963) reported in a short communication that he noticed that when the second inequality equation (3.5.3) holds, the shock wavefront can be represented by a configuration of two waves---a shock-wave of a lower amplitude and a rarefaction wave propagating backwards (with a contact surface between them;
3.5
421
Stability of Shock Waves P
Y
o
P3
"•
I
-.3 P2
P3
I
Ic I
I I
Y b P
Pz o Pl
~
'
I
cii
h
" d
c} 7-
Pl ~o
i
e
FIGURE 3.5.4 Nonunique representation of a shock-wave discontinuity--solitary wave (a) and alternative configurations with waves traveling in different directions (b, c) and in one direction (d, e); C is the contact surface.
see Fig. 3.5.4b). Further analysis of the condition under which the shock discontinuity admits nonunique representations was performed by Kuznetsov (1985) and Fowles (1981) and demonstrated the following: 1. Nonunique shock wavefront representation exists also within the instability region with L < - 1 . Thus, in all the situations where the shock discontinuity is unstable (i.e., when any of conditions (3.5.3) is met) the front, like the arbitrary discontinuity (Riemann problem), can be decomposed into three gasdynamic discontinuities, two waves and a contact surface. 2. This decomposition is nonunique because in each particular case an unstable shock-wave discontinuity can be represented by two different wave configurations. For example, when the second of inequalities equations (3.5.3) holds, the shock-wave discontinuity can be represented not only by the configuration suggested by Gardner (1963) (Fig. 3.5.4b) but by a configuration composed of two shock-waves propagating in opposite directions as well (Fig. 3.5.4c). 3. The solution of the problem of the unstable shock-wave decomposition into elementary waves yields condition equation 3.5.3 for the instability boundaries.
422
N.M. Kuznetsov
4. The fact that unstable shock-wave discontinuities can be decomposed into other waves (i.e., nonuniqueness of the front representation) does not necessarily mean that the opposite is true. The nonuniqueness of the shock wavefront representation does not imply its instability. Such properties of shock-wave discontinuities as instability and nonunique representation of the front are related, but they are not equivalent. The forementioned claims, points (1) to (4), are proven in Sections 3.5.2.4 and 3.5.2.7. The instability boundaries 3.5.3 can be found more easily by the shock discontinuity decomposition into elementary waves (called hereinafter the discontinuity decomposition or Riemann solution) than by the LST methods. Of course, since the instability boundaries (Eq. (3.5.3)) were initially derived by the LST method the benefits indicated in the preceding text regarding the Riemann solution method are insignificant as applied to these boundaries. However, the discontinuity decomposition method enables us not only to determine the instability boundaries but also to find the two versions of the wave configurations resulting from an unstable (i.e., actually nonexistent) shock-wave discontinuity. However, it still remains unclear which of the two possible configurations is realized in experiments (see also Section 3.5.2.10).
3.5.2.4
R E G I O N S W H E R E A SHOCK-WAVE
D I S C O N T I N U I T Y IS UNSTABLE AND W H E R E ITS REPRESENTATION IS N O N U N I Q U E Nonunique representations of shock discontinuities as well as the Riemann problem can be conveniently studied in the p-u coordinates (the pressure-material velocity in the laboratory frame of reference). Indeed, as these values are continuous at the contact surface, the set of three equations for three parameters is separated into a set of two equations for p and u and an independent equation for the density elevation at the contact surface (or for entropy and temperature increments related uniquely with the density change). Shock-wave representation (Fig. 3.5.4a) is called nonunique if the wave (i.e., a discontinuity satisfying the Rankine-Hugoniot relations at its front) can be represented by other gasdynamic discontinuities--two shock-waves 3 (Fig. 3.5.4c,d) or a shock-wave and a rarefaction wave (Fig. 3.5.4b,e). Hereafter, all the waves constituting these configurations except for the shock-wave propagating forward through the initially unperturbed material will be called secondary. There is also a contact surface in addition to the secondary 3Each secondarywave is assumed to be unique. Otherwise, the number of waves into which the initial shock-wave can be split will be larger than two and the number of contact discontinuities will exceed one.
3.5 Stabilityof Shock Waves
423
waves. Configurations with the waves propagating in one direction are physically realizable only if the second necessary condition for shock-wave stability (Eq. (3.5.1)) is violated. Otherwise, these configurations are only formal solutions that should also be taken into account in order to elucidate the relation between the regions of shock-wave instability and the nonuniqueness of its representation. Graphically, nonuniqueness of shock wavefront representation pertaining to an arbitrarily chosen point 2 at the initial shock Hugoniot curve (with pressure P2 and velocity u 2) corresponds to the nonunique intersection at the (p, u)plane of the initial shock-wave Hugoniot curve with the Hugoniot curve of the secondary shock wave or isentrope of the secondary rarefaction wave. One of the intersections must be at point 2 and the second must be at another point (we denote it as in Fig. 3.5.3 by point 3 with pressure P3 and velocity u3). These multiple intersections occur when the initial shock-wave Hugoniot curve has Z-shaped segments. Figures 3.5.1 and 35.3 display two qualitatively different versions of these shock Hugoniot curves. In Fig. 3.5.3b the initial shock Hugoniot curve crosses: (1) at the lower point 3, the isentrope S emerging from point 2, which serves as the initial point of the rarefaction wave; and (2) at the upper point 3, with the shock-wave Hugoniot curve H emerging from point 2 as an initial point of the secondary shock wave. It is important to note which of the points 2 or 3 is the initial point of the isentrope because these points are separated by a contact discontinuity. The first of the forementioned solutions corresponds to the configuration shown in Fig. 3.5.4b and the second, to the configuration displayed in Fig. 3.5.4c. Thus, for an initial shock-wave (Fig. 3.5.4a) corresponding to any point in the middle of the Z-shaped segment of the curve confined between the points where this shock Hugoniot and isentrope 4 are tangent (points s- and s+ in Fig. 3.5.3b) one can suggest two more representations in the form of the configurations shown in Figs. 3.5.4b,c. Hereafter, these three branches of the Z-shaped segment are called Z-branches. In materials with the other possible orientation of the Z-shaped Hugoniot curve segment (Fig. 3.5.1b), intersections of the initial shock-wave Hugoniot curve with the Hugoniot curves of the secondary shock waves are also nonunique. The qualitative difference between this case and that considered earlier is as follows. The initial shock-wave Hugoniot curve in the latter case crosses the isentropes and the shock-wave Hugoniot curves of the other family (as compared to those shown in Fig. 3.5.3), characterized by the other (positive) sign of the change in u caused by variation of p, which corresponds 4At point s- the initial shock-waveHugoniot is tangent to the shock-waveHugoniot curve H. However the amplitude of the secondaryshockwavevanishes at this point (Fig. 3.5.4c, at P3 ----P2); therefore the Hugoniot curve H coincides with the isentrope.
424
N.M. Kuznetsov
to the propagation of the secondary waves and the initial shock wave in the same direction (see Fig. 3.5.4d,e). Under these conditions, upon decomposition of a shock-wave discontinuity corresponding to any point 2 at the middle Z-branch (Fig. 3.5.1) into the elementary gasdynamic waves, points 3 are initial for the Hugoniot curves of the secondary waves rather than final as in the first case. The final point is point 2 (see the shock-wave Hugoniot curve 3-2 with the lower point 3 and isentrope 3-2 with the upper point 3 in Fig. 3.5.1b). Otherwise the topology of the solution is identical in both cases. The condition for tangency of the initial shock-wave Hugoniot curve and adiabates of the isentropic waves is expressed by the following equation (3u/3p)n = (3u/3p) s
(3.5.7)
where subscript H denotes that the derivative is calculated along the shockwave Hugoniot curve. The derivatives (3u/3p) satisfy the relations ( o u / o p ) H = (1 - IO/2J,
(ou/ap)s = + v / c
(3.5.8)
Here the minus sign pertains to the isentropes in Fig. 3.5.3b and to the configuration shown in Fig. 3.5.4b, and the plus sign corresponds to the isentropes shown in Fig. 3.5.1b and the configuration displayed in Fig. 3.5.4e. Substitution of Eq. (3.5.7) into Eq. (3.5.8) yields L = 1 :F 2M 2
(3.5.9)
When the sign in Eq. (3.5.9) is minus, that is, the sign in Eq. (3.5.8) is plus, M 2 = 1 at the point where the isentrope and the shock Hugoniot curves are tangent (see, e.g., Landau and Lifshits, 1987). Hence, in this case L ----- 1
(3.5.10)
Thus, the tangency points, that is, the boundaries of segments s- - s + of the initial shock-wave Hugoniot curves plotted in Figs. 3.5.3 and 3.5.1 satisfy Eq. (3.5.9) with the plus sign and Eq. (3.5.10), respectively. It can be easily shown that within these segments L > 1 4- 2M 2 (Fig. 3.5.3) or L < - 1 (Fig. 3.5.1), that is, conditions equation 3.5.3 for the shock-wave instability are satisfied. The reverse inequalities hold outside these segments at the initial shock-wave Hugoniot curve. Thus, according to the LST, only the middle Z-branch of the shock-wave Hugoniot curve corresponds to the unstable shock waves. As will be shown in Sections 3.5.2.7-3.5.2.9 this is not accidental. In the foregoing discussion the nonuniqueness of the shock wavefront representation corresponding to the middle Z-branch located between the tangency points (Figs. 3.5.1b and 3.5.3b) was considered. Now we can readily prove that the front representation is nonunique at the segments belonging to the lower and the upper Z-branches of the shock-wave Hugoniot curve adjacent
3.5
Stability of Shock Waves
425
to the tangency points s- and s+ from below and above, respectively. Each tangency point is one of the boundaries of these segments. Their second (outer) boundaries are not determined by the local properties of the shock-wave Hugoniot curve and, therefore, cannot be written in the form of simple relations for parameter L. To find the outer boundary, one should solve a boundary problem, that is, find a point at the initial shock-wave Hugoniot curve (beyond the middle Z-branch) which is the initial point for the Hugoniot curve of the secondary wave. This secondary Hugoniot curve has another common point with the initial shock-wave Hugoniot curve, namely, the tangency point. Locations of these points (A and B) are schematically shown in Figs. 3.5.1 and 3.5.3. Note that these tangency points do not generally coincide with the boundaries of the middle Z-branch because in this case the Hugoniot curves of the secondary waves are different, that is they have different initial points. Thus, for a shock-wave Hugoniot curve as displayed in Fig. 3.5.3, for example, the Hugoniot curves of the secondary waves begin from the tangency points while in the considered case they start at points A and B. Obviously, a similar pattern of the location of the Hugoniot segments satisfying the condition of nonuniqueness of shock discontinuity representation can also be observed in materials with the other shock Hugoniot curves (Fig. 3.5.1). We do not consider here this alterative version of the shock-wave Hugoniot curve. However, it is important to note that although the shock wavefront representation is nonunique in this case as well on the lower Zbranch, this nonuniqueness is only formal because both wave configurations obtained after decomposition of the initial shock-wave discontinuity into the elementary gasdynamic waves cannot be realized (they are both shown in Fig. 3.5.4e but have different pressures P3). In this connection see the comment on these formal solutions at the beginning of this section. The analysis of the full pattern of these nonunique representations and the related wave configurations for all three Z-branches and for each type of shock-wave Hugoniot curve (Figs. 3.5.1 and 3.5.3) can be found in a review by Kuznetsov (1989).
3.5.2.5. On THE PHYSICAL M E A N I N G OF THE S O L U T I O N S W I T H STEADY-STATE C O R R U G A T I O N PERTURBATIONS OF A S H O C K W A V E AND W I T H A C O U S T I C WAVES E M A N A T I N G FROM THE SHOCK FRONT The range of values of parameter L mentioned in the Introduction (3.5.1) and defined by inequalities equation (3.5.6) seems to be somewhat puzzling within the linear stability theory. The authors who studied shock waves by the LST
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N.M. Kuznetsov
methods did not rule out the possibility of shock-wave instability in this range (D'yakov, 1954, 1958; Kontorovich, 1957a), indicating that this problem could be solved only beyond the LST approach (D'yakov, 1954). Moreover, even leaving aside the shock wavefront stability problem, we must recognize that without considering nonlinear effects the physical meaning of the solutions with stationary acoustic waves leaving the shock wavefront cannot properly be understood (Kuznetsov, 1986a, 1986b, 1987). Perturbations of all the parameters in these solutions are proportional to exp[i(kx
4- ly -
wt)]
(3.5.11)
with real values of k, l, and co. (The x- and y-axes are directed along the shock wavefront and normally to it.) The ratio k/l determining the orientation of the acoustic wave with respect to the shock wave depends on parameter L and in the range determined by inequalities equation (3.5.6) if it corresponds to the outgoing waves. The formulas specifying this dependence can be found in Kuznetsov (1986a, 1989). Solutions (3.5.11) within the range (Eq. (3.5.6)) are peculiar because they imply a supersonic phase velocity v t of the perturbations propagation along the shock-wave discontinuity surface. The velocity of the acoustic waves propagating along the shock wavefront in a frame of reference attached to this front is c s = c ( 1 - M2) 1/2. The velocity v t equals cs only at L = L0 and rises monotonically with L tending to infinity at L = 1 4- 2M 2 (Kuznetsov, 1986a, 1989). Acoustic waves in the system do not exhibit frequency dispersion. Hence, any acoustic wave packet retains its shape and the group velocity is identical with the phase velocity. Thus, according to this solution, an acoustic signal will propagate along the shock wavefront at a supersonic velocity, which is impossible and contradicts the causality principle. In this connection, the following questions arise. 1. What is the origin of these noncausal solutions? 2. Do they have physical meaning? 3. How they are related to the shock-wave stability problem? Before resorting to answering these questions, consider first another result inferred from the LST approach (D'yakov, 1954; Kontorovich, 1957a; Iordanskii, 1957) that is important for further analysis. Solutions with steady-state corrugating perturbations of a shock-wave exist within a range of parameter L, which is wider than the range defined by inequalities equation (3.5.6): L* < L < 1 4- 2M2
(3.5.12)
where L* is some value of the parameter L within the interval - 1 < L* < L0. However, unlike the interval defined by inequality (3.5.6), the acoustic waves of type (3.5.11), which are adjacent to the shock wavefront, are incident and not outgoing waves.
3.5
427
Stability of Shock Waves
2
0 l / /
C// /
3
FIGURE 3.5.5 Triple-wave configuration with an incident weak wave (3); (1) and (2), the shockwave prior to and after the perturbation, respectively; C is the contact surface. The arrows indicate the directions of the streamlines in a frame of reference attached to point 0.
The simplest way of answering the questions raised here is based on the theory of wave intersections. In the linear approximation, corrugating perturbations of a shock-wave with emitted or incident acoustic waves are equivalent to triple-wave configurations, that is, they consist of the unperturbed and perturbed shockwaves and an acoustic wave (Kuznetsov, 1985). Figures 3.5.5 and 3.5.6 show these configurations with incident and emitted acoustic waves, respectively. By definition, the possible values of angles 7 for incident and emitted waves are determined by the following inequalities, respectively: 0 < ? < 7o
(3.5.13)
70 < ? < rc
(3.5.14)
where ?0 = c o s - 1 M2- Only one such triple configuration with a specific orientation of the acoustic wave with respect to the shock wavefront corresponds to each value of the parameter L from the interval equation (3.5.14), similarly to the solutions obtained by the LST methods and already discussed here. 2
I
0
I
X
\
FIGURE 3.5.6 Triple-wave configuration with an outgoing weak wave (3); (1) and (2), the shock-wave prior to and after the perturbation, respectively; C is the contact surface; (I) the region ahead of the shock front. In the flow region behind the shock wavefront only acoustic perturbations from sector II* can arrive at a point 0; this sector is bounded by lines 1 and 7 = 70 (the dashdotted line). The arrows indicate the directions of the streamlines in the frame of reference attached to point 0.
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N.M. Kuznetsov
0
1
\ FIGURE 3.5.7 Four-waveconfiguration: 1 and 2, shock-waveprior to and after the perturbation, respectively; 3f and 3r are incident and reflected weak waves; C is the contact surface. The arrows indicate the directions of the streamlines in a frame of reference attached to point 0. Generally, interactions between shock and acoustic waves at an arbitrary incidence angle independent of the parameter L yield a 4-wave configuration that comprises unperturbed and perturbed shock waves and incident and reflected (or refracted, if the original wave is incident from the region ahead of the shock front) acoustic waves (Fig. 3.5.7). Therefore the triple-wave configuration with an incident acoustic wave implies that the reflected wave is missing. The triple configuration with an emitted acoustic wave corresponds to a hypothetical reflection without incidence. The same is true for the steady-state LST solutions already considered here. The results of the theoretical analysis of acoustic-wave reflection from a shock-wave (D'yakov, 1954, 1958; Kontorovich, 1959; Fowles, 1981; Kuznetsov, 1986a) are in full agreement with this degeneration of the 4 wave configuration. According to these results, the reflection coefficient vanishes when the preceding solutions yield configurations with the incident waves (and to the related triple configurations, Fig. 3.5.5) and tends to infinity (resonance, according to the terminology of D'yakov, 1958 and Kontorovich, 1959) when the acoustic waves are emitted--z = Pr/Pf, where pf and Pr, are the pressures in the incident and reflected waves. In some publications these resonances are identified, although without appropriate justification, with the shock-wave instability (see, e.g., Fowles, 1981). To answer the three questions formulated here, the near-resonance and resonance reflection of acoustic waves was analyzed in a nonlinear (quadratic) approximation (Kuznetsov, 1986a). It turned out that there are two solutions for the reflection coefficient: Z -----P_Z-- --4/(?r) 4- (Det) 1/2
pf Det =
(3.5.15)
2apf r
)apf - 4~(~f )apf
(3.5.16)
3.5 Stabilityof Shock Waves
429
@(Tf) < 0 is in the range determined by inequalities (3.5.6). Here 7f and 7r are the angles 7 for the incident and reflected waves, respectively, a is a coefficient depending on the incidence angle, the parameter L, and some other parameters characterizing the Hugoniot curve and amplitude of the shock-wave (Kuznetsov, 1986a), and ~ is a smooth function of the angle. For further analysis it is important only that at the resonance point, that is, at 7 = ~res, the function @(7~) will be equal to zero and can be written near this point as follows (3.5.17)
~t(7~) -- const(7~- 7res)"
Note that in the range defined by inequalities (3.5.13), whereby the reflection is nonresonant, the function ~ also is equals to zero but only at a certain incidence angle 7f; @(Tf) = 0. At this angle g = 0. Solution (3.5.15) can conditionally be called weak (with the plus sign of the determinant) and strong by analogy with the known results for wave reflection from a rigid wall (Landau and Lifshits, 1987; Courant and Friedrichs, 1977). In the weak solution, the reflected shock-wave pressure Pr near the angle Yres 1/2 tends to zero as pf . When 7 deviates from Yr~s, the weak solution approaches the appropriate result obtained in the linear reflection theory. The reflection coefficient as a function of 7 r - 7res is shown schematically in Fig. 3.5.8 for linear and nonlinear approximations. Equation (3.5.15) shows that at any arbitrarily small, but finite, pressure in the incident wave and at apf < 0 (i.e., at @(Tf)apf > 0) there is a small interval ATr of near-resonance values of 7~, where the determinant (3.5.16) is negative, and, hence, the problem has no solution. As pf ~ 0, ATr ~ 0. Depending on the sign of the parameter a, the determinant becomes negative either in the compression phase or in the rarefaction phase of the incident wave. Thus either the phenomenon cannot be described using the 4-wave configuration (i.e., the
\ X
mo)
v
o
\
\
~- ~'~,,~
--Xmo x
FIGURE 3.5.8 Ratio Z = P,./Pf as a function of the reflection angle 7r, in the vicinity of the resonance point at apy > 0. The dashed lines represent the linear approximation.
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N.M. Kuznetsov
2
1
FIGURE 3.5.9 Mach reflection of a weak wave 3; 1 and 2 shock wave prior to and after the perturbation, respectively; 4--reflected wave; 5--Mach stem.
regular reflection is impossible) or the quadratic approximation is insufficient to calculate the configuration. In any case, amplitudes of the waves arising upon reflection could be finite when pf ~ O. For example, when the regular reflection changes to Mach reflection (Fig. 3.5.9), the Mach stem must travel over the front of an unperturbed shock wave with a velocity not lower than that at which point 0 will propagate provided the incident wave with the same slope is reflected regularly (Fig. 3.5.7). However, at any slope of the incident wave, except for the slope angle 70, point 0 travels at a supersonic velocity that grows when the angle Vf decreases, that is, the greater is the difference ~0 - YfHence, the Mach stem must be a shock wave with a finite amplitude independent of the small amplitude pf of the incident acoustic wave. Recall that only the range of angles ~ - ~res within which the regular reflection is impossible depends on the value of pf. The precediing square-root dependence of Pr on pf for near-resonance reflection implies, in particular, that infinitely small perturbations reflected from the shock wavefront do not cause a finite change in the shock-wave amplitude. The amplitude of the reflected shock-wave remains infinitesimally small and only the order of smallness changes. This result enables us to attribute physical meaning to the steady-state solutions in the linear stability theory and to the appropriate solutions for triple configurations (Fig. 3.5.6). In the linear approximation in terms of Pr, the main cause for perturbing the shock-wave can be an acoustic wave incident at the resonance angle. Its amplitude is proportional to pr2. Therefore, this wave as well as other nonlinear effects are not taken into account in the linear theory. Thus, the causality of this phenomenon cannot be analyzed in the linear approximation but it is required for the existence of triple configurations with outgoing acoustic waves. In the strong solution, Pr is independent of pf as it tends to zero. This solution does not restore the result of the linear approximation at incidence
3.5 Stabilityof ShockWaves
431
angles far from the resonance angles. At p f - 0 it corresponds to a triple configuration of the waves with a weak emitted wave with a finite amplitude whose value affects the angle ~ (Fig. 3.5.6). The resulting configuration does not satisfy the causality principle and, hence, cannot exist without being supported extemally. Examples of this extemal influence (a hypothetical experiment with a kind of piston in the form of a thin plate, which looks like an infinite thin needle in the plane of Fig. 3.5.6), that slides freely along the tangential discontinuity with the end at point 0) are discussed by Kuznetsov (1986b). We present also another well-known example relevant to the discussed problem that shows how a solution of the hydrodynamic equations with a shock discontinuity is correct (i.e., physically meaningful) only in the presence of an extemal influence that is not present in the equations and the boundary conditions. We refer to the solution for the so-called underdriven detonation (Landau and Lifshits, 1987; Zeldovich and Kompaneets, 1960; Shchelkin and Troshin, 1963). As commonly recognized, a self-sustaining detonation propagates at the Chapman-Jouguet (CJ) velocity D ~ Dq but the solution of the hydrodynamic equations with the appropriate continuity relations at the shock wavefront also allows other velocities D > Dc2, with the same boundary conditions at infinity. The solution with D > Dcj acquires physical meaning and describes a real phenomenon if the material is "ignited" by some initiator other than a shock-wave, for example, by an electric discharge or a focused light beam propagating through the material at a preset velocity D > Dcj. The energy and momentum introduced by these initiators are negligible and are disregarded in the hydrodynamic equations. The initiator prescribes only the propagation velocity of the wave, which cannot propagate without external initiation. Violation of the causality principle indicates only that the solution is not correct but does not allow for the reason to be determined. It seems interesting and helpful to elucidate this facet of the problem. Solutions with an emitted acoustic wave obtained with the appropriate boundary conditions, for example, those given by Eq. 3.5.11 and those appropriate to 3-wave configuration (Fig. 3.5.6), with the prescribed velocity of the intersection of all discontinuities (zero points in Fig. 3.5.6) prove to be incorrect. The incorrecmess in the case of the triple configuration relates only to a single point 0 and it is associated with the fact that the second in the necessary conditions for shock-wave stability (Eq. (3.5.1)) that is, the condition M2n < 1, is violated at this point (although in deriving these solutions this condition was supposed to be met). Here subscript n denotes the normal component of v in Eq. (3.5.1). The existence of this incorrectness was proved by Kuznetsov (1986a, 1987) and in what follows it is presented using a slightly different approach.
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N . M . Kuznetsov
At all angles 7 in the triple configurations with an emitted acoustic wave, that is, within the range given by Eq. (3.5.14), the causal relation between the shock wavefront and the flow behind it is violated. Signals from the perturbed region in the flow behind the shock front (which should be related causally with a point 0) do not reach point 0. Obviously only these signals determine the velocity at which this point travels along the unperturbed shock wavefront. The second condition (Eq. (3.5.1)) is met only formally at point 0 in compliance with hydrodynamics laws. Indeed, perturbations from the region behind the shock front arrive at point 0 only from a sector bounded by the angle 70, that is, from sector II* between the front 2 and the line 7 = 7o (Fig. 3.5.6), which is the boundary of the angles of the incident and outgoing acoustic waves. However, for outgoing waves when the angles satisfy inequalities (3.5.14), the entire region of the perturbed flow--sector III in Fig. 3.5.6m is located beyond sector II '~. Therefore perturbations from sector III do not reach point 0 and, hence, the motion of this point and the state of material at this location are not causally related to the flow parameters in sector III. Because of the violation of the causal relation between point 0 and the perturbed flow behind the shock wavefront, the wave configuration, which is initially prescribed as shown in Fig. 3.5.6 (or in the form of solution equation (3.5.11) bounded by the corrugated shock wave), immediately splits into gasdynamic waves; a rarefaction wave propagates from point 0 into sector III, wavefronts 2 and 3 will be curved, and incident and reflected, from the shock wavefront, waves are formed. The latter wave pattern can be determined by stability analysis of solutions (Eq. (3.5.11)) when the acoustic waves are emitted or using equivalent solutions with a triple-wave configurations (Fig. 3.5.6). Kuznetsov (1987) showed that the local perturbations arising in the vicinity of point 0 of a triple-wave configuration with outgoing waves, that is, when the angles are determined by inequalities equation (3.5.14), occupy the region growing in time and the angles between the fronts strongly deviate from those in the initial triple configuration. A version of local perturbations and their development are illustrated in Fig. 3.5.10. Notably, the amplitude of the perturbation does not grow unboundedly with time. If an initially plane front 3 and, hence, the plane section of a wavefront 2 have a finite width, the rarefaction waves propagating from right to left from the region of the flow behind the shock wavefront unperturbed by the acoustic wave 3 cause a complete attenuation of wave 3, that is, the triple configuration that distorted the plane shock wavefront vanishes. The preceding reasoning is based on the results of nonlinear analysis and provides the answers to the first two questions raised in the beginning of this section, that is, how the solutions contradicting the causality principle are obtained and how one can assign a physical meaning to them. In addition, we demonstrated why the solutions admitting supersonic propagation of small
3.5 Stabilityof Shock Waves
433 I
o~ o2 o3o~
o'2
!
03
--
1
~o//~ / // //
3a/ FIGURE 3.5.10 Propagationof a weak emitted wave 3 perturbed in the vicinity of point 0; (I) flow region ahead of the shock wavefront, (1) shock wavefront (plane within a weak kink at point 0). Subscripts 1 to 3 denote times t l , t] 4- At, and t 1 4- 2At, respectively. The dashed lines are the continuation of the plane shock wavefront 3. perturbations are not correct, what exactly is incorrect in solutions allowing supersonic velocity of small perturbations, and how a perturbation of a shock wave determined by this solution attenuates.
3.5.2.6
R E S O N A N C E R E F L E C T I O N OF A S O U N D
W A V E AND S H O C K - W A V E STABILITY Consider shock-wave stability within the range of the parameter L given by Eq. (3.5.6). As already noted here, steady-state solutions with a corrugated shock wavefront and acoustic waves emitted from the front cannot be properly explained within LST. This suggests that shock waves with values of the parameter L within the range specified by inequalities (3.5.6) could be unstable. Thus these solutions are not correct and the flows that they describe exist only in the presence of external mechanisms. However, this does not imply that the solutions are not relevant to the shock-wave instability and that the shock waves that satisfy condition (Eq. (3.5.6)) are necessarily stable. Indeed, lowamplitude random perturbations, for example, flow fluctuations ahead of the shock wavefront or behind it, may act as such external factors. If these perturbations arrive at the shock wavefront at the resonance angles, what occurs for the L values determined by relation (3.5.6), and the order of the smallness of their amplitude reduces upon their refraction or reflection at the shock front. Therefore, one has to examine whether this suffices for shock-wave instability to grow. For this end consider the interaction between a shock wave and random perturbations or a fluctuation of thermodynamic nature arriving at the shock-wave discontinuity from regions ahead of (acoustic and entropy perturbations) and behind (only acoustic perturbations). The acoustic fluctuations have various wave vectors and can be represented by an integral super-
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N.M. Kuznetsov
position of plane waves with different orientations (Fourier integral with respect to the wave vector k). The pressure amplitude of the waves with the absolute value of the wave vectors within the interval dk is proportional to dk (i.e., to the differential of the incidence angles d~/f,) and, hence, it is infinitesimally small. Equation (3.5.15) implies that the amplitude of the wave reflected from the shock wavefront is of the same order of smallness as the incident wave amplitude, except for the resonance point, ?r = 7res, and can be written as
dpr = [-~(Tf )/~(},r)]dpf
(3.5.18)
The pole-type singularity in Eq. (3.5.18) does not result in the divergence of the integral pressure Pr of the reflected waves due to their interference. Hence, in calculating Pr we must use the principal value of the integral:
,/,(~fl dp.f p~ = _ i ~,(~)d~jd~s
(3.5 ~9)
Thus, a random acoustic fluctuation does not change the order of its smallness upon reflection from a shock wave. Therefore the reflected perturbations do not substantially differ from other fluctuations departing from the shock front and cannot cause its instability. Entropy fluctuations (perturbations of the density and, consequently, temperature) at a constant pressure generate acoustic perturbations upon passing through the shock-wave discontinuity. It can be shown that qualitatively these perturbations do not differ from the reflected acoustic waves considered here. In the vicinity of the resonance angle they interfere and in the relative units (p,./p) they are of the same order of magnitude as the initial density perturbation (Ap/p). As already mentioned in relation to Eq. (3.5.15), small perturbations incident at a shock wavefront at the angle corresponding to the near-resonance regular reflection remain small, but the order of their smallness reduces (Pr ~Pf..I/2xJ. Therefore, when analyzing shock-wave stability, one must also answer the following question. Small acoustic near-resonance perturbations with a finite amplitude, random or produced at various instants by an external source, after reflection from the shock wavefront and changing their order of 1/2 smallness from pf to pf , catch up in the course of their downstream propagation with the entropy perturbations generated by the previous reflections of near-resonance acoustic perturbations. These entropy perturbations can be shown to be of the same order of smallness, that is, pff2. The interaction of acoustic and entropy waves produces an incident wave of the higher order of smallness (~. pf). It is of interest whether these repeated reflections would lead to perturbations with an infinitely growing amplitude. The answer to this question is negative. The point is that the waves reflected from the entropy perturbations arrive again at the shock wavefront at angles 7, which differ from
3.5 Stabilityof Shock Waves
435
the resonance angle. The angles ~ are governed by the laws of specular reflection from a nearly plane surface of the entropy perturbation and it would be extremely unlikely that angle ~ equals 7res- Reflections of such waves from a shock wavefront do not change the order of smallness of the perturbation and hence these repeated reflections of the perturbation cause their damping. If the extemal source generates incident small perturbations with a finite amplitude with orientations and phases (amplitude sign) such that the regular reflection is either impossible at all or cannot be described within the quadratic approximation (Eq. (3.5.15)), the shock wavefront perturbations may be of the order of the initial shock-wave amplitude. A weak incident wave with a finite amplitude and an appropriate angle and phase causes a total change of the shock wave, that is, it splits into a shock wave propagating forward but with an essentially different amplitude, and other finite-amplitude waves. Thus, the shock wave in the range (Eq. (3.5.6)) is unstable with respect to small perturbations of a special type, namely, finite-amplitude waves with a certain sign incident at a nearly resonance angle. However, this instability is rather special because after the source of the perturbations is turned off, the initial shock wave may recover. In any case, as we have already shown here, this shock-wave is stable with respect to small perturbations of a thermodynamic fluctuating origin. In this connection, it must be noted that (see Egorushkin, 1984) the incident waves resulting from the nonlinear interactions of the emitted waves may lead, at a certain shape of the shock-wave Hugoniot curve with values of the parameter L in the range (Eq. (3.5.6)), to the unbounded amplification of perturbations of the shock wave. 5 However, this result applies only to monoharmonic finite-amplitude waves incident at the resonance angle but not to fluctuation white noise.
3.5.2.7
G E N E R A L C H A R A C T E R I S T I C S AND A
S I M P L E E X A M P L E OF R E L A T I O N B E T W E E N INSTABILITY AND N O N U N I Q U E N E S S
OF
STEADY-STATE R E G I M E S The relation between the criteria of shock-wave instability and Z-shaped segments at the shock Hugoniot curve already discussed here is typical, in 5Egorushkin (1984) supposed that the primary perturbation is a wave departing at the resonance angle. Self-sustained propagation of such a wave does not comply with the causality principle. The physically meaningful formulation of the problem corresponds to prescribing the primary perturbation in the form of the incident waves. However, in the mathematicalanalysis of the nature of the feedbackbetween the shock wave and the flow behind it the direction of the initial perturbation propagation (toward the shock wavefront or away from it) is not significant.
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N.M. Kuznetsov
its mathematical basis and geometrical interpretation, to a very wide class of stationary processes (not necessarily of the same physical nature) in which, depending upon the parameters of the problem, the nonuniqueness of the relations between the governing parameters either appears or vanishes. Electric discharge is one of the simplest examples of such processes. Voltage-current characteristics of discharges have a Z-shaped segment and look similar to the plots in Fig. 3.5.11. In the range of voltages from U1 to U2 the Ohm's equation I U/R has three solutions, that is, there are three values of electric-current (and, hence, three values of the resistance R) for the same voltage. Two currents pertaining to the upper and lower branches of the Z-shaped curve are stable to small perturbations of the voltage or current, whereas the third value (for the middle branch) is unstable. The boundaries of the middle branch corresponding to the unstable regimes are the tangency points (s+) of the voltage-current characteristics and the lines of constant voltage. The intersections of these lines with the voltage-current characteristics determine the outer boundaries A and B of the segment where three current values correspond to each selected voltage. All these properties of the voltage-current characteristic are trivial and we dwell on them only because this example offers a simple interpretation and a way for memorizing the essentially similar (but more complicated for perception) patterns of the shock-wave Hugoniot curve segments pertaining to the nonunique representation and instability of shock discontinuities. The analogy becomes obvious if we take into account that the voltage-current characteristic and constant-voltage lines are similar to the shock-wave Hugoniot curve and Hugoniot curves of the secondary waves, respectively (see Fig. 3.5.11). The three solutions for the current are similar to the three solutions for the shock discontinuity (solitary shock wave and two alternative wave configurations).
B
I I
f 0
i
l
i
UI
U2
U
FIGURE 3.5.11 Voltage-current characteristic with a Z-shaped segment AB whereby three current values correspond to each voltage value.
3.5 Stability of Shock Waves
437
Correlations between potentials and currents of the Ohm's law type are encountered quite often. They describe heat conduction, diffusion, viscous flows, including motion of solid particles under the action of an external force (Stokes' formula), mobility of ions in electrolytes, and the relation between the generalized forces and flows in thermodynamics of irreversible processes (in biology this is the dependence of population size on food resources and in economics it is the dependence of the demand for goods on their "reverse" price, etc.). Certainly, the domain with three nonunique solutions does not occur in all cases. This depends on whether the proportionality coefficient between the "force" and the "current" changes abruptly or not as the "flow" increases up to some critical value. If this is the case and a Z-shaped segment arises at the curve of the current, the properties of the steady-state processes of very diverse nature have much in common. 3.5.2.8 STABILITY OF S H O C K WAVES P E R T A I N I N G TO THE L O W E R AND U P P E R BRANCHES OF THE Z - S H A P E D S E G M E N T OF THE S H O C K W A V E H U G O N I O T CURVE. S P L I T T I N G OF AN UNSTABLE S H O C K W A V E Shock waves pertaining to the upper and lower Z-branches bounded by points A and B (Figs. 3.5.1 and 3.5.3) are stable to small corrugating perturbations, but admit, as was previously shown here, decomposition into two different wave configurations. One of these configurations contains an absolutely unstable shock wave corresponding to the middle Z-branch, and is not of interest. However, the waves of the second configuration are stable with respect to small perturbations. If these waves propagate in opposite directions or in one direction but so that the second wave lags behind the first, decomposition of the original solitary shock wave into this configuration is irreversible. This means that shock waves of such strength are stable with respect to small perturbation but may change to the alternative wave configuration at sufficiently high-amplitude perturbations. After reflection of a certain perturbation (an incident wave (shock or rarefaction wave) with a given amplitude), one of the following situations must be realizedmeither the original shock wave or the alternative wave configuration. As the incident wave is characterized by a single parameter, namely, its strength, it may be expected that there exists a limiting amplitude of the incident wave separating between perturbations to which the original shock wave is stable and those which cause its instability, that is, transition to the alternative wave configuration. This applies to perturbations arriving from both sides of the shock discontinuity (of course, with different amplitude boundaries). Apart from the compression and rarefac-
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N.M. Kuznetsov
tion waves, contact surfaces of an arbitrary strength (by the absolute value and sign) with the appropriate threshold for the shock-wave stability can arrive as a perturbation from the flow region ahead of the shock wavefront. The shockcapturing schemes that employ solution of a Riemann problem cannot be applied for the analysis of shock-wave stability to relatively large perturbations in the form of stepwise functions. Indeed, the nonuniqueness of the solution of a Riemann problem, because of the Z-shaped shock-wave Hugoniot curves, will produce erroneous numerical solutions. In implicit schemes, in order to avoid nonphysical results, instead of smearing the discontinuities by formally introducing an artificial viscosity or nonlinear filtering, the real kinetics of relaxation inside the shock wavefront must be taken into account. For strong shock waves the problem cannot be solved within the hydrodynamic approach and requires an analysis of the shock wavefront by molecular-dynamics methods. To the best of our knowledge, the problems of determining amplitude boundaries for these perturbations were not analyzed either quantitatively or qualitatively. A solitary shock wave was not observed experimentally within the s_ - B segment including the s+ - B segment where, according to the LST criteria, the wave is stable to small perturbations (see Section 3.5.2.10) in the case of the shock Hugoniot curves shown in Fig. 3.5.1, (Bancroft et al., 1956; Duvall, 1977; Al'tshuler, 1978). Instead of this wave, a wave configuration consisting of two shock waves propagating in the same direction is formed (Fig. 3.5.4d). Several attempts were undertaken to prove that a solitary shock wave does not exist in this case but they were based on rather simplified models of the shock wavefront structure that are not applicable to strong shock waves. In order to prove instability of the shock discontinuity at the s+ - B segment (Fig. 3.5.1a), the following arguments (Al'tshuler, 1978) are presented (see also Duvall, 1977). The Michelson-Rayleigh line connecting point 1 with arbitrary point 2 at the s+ - B segment crosses the shock-wave Hugoniot curve at two other points. These intersections formally allow interpretation of the initial shock-wave discontinuity 1 - 2 as a sequence of three shock waves separating states 1 - 2', 2 ' - 2 " , and 2 ' - 2 (Fig. 3.5.1a) and propagating at the same velocities. When wave 1 - 2 is weak and nearly isentropic, it is not difficult to demonstrate (Kuznetsov, 1966) that the entropy in the wave 2 ' - 2 " reduces, that is, such a compression shock-wave cannot exist. Thus it is possible to infer from this that shockmwave discontinuity i - 2 is unstable. However, this proof is explicitly or implicitly based on an assumption that the entire phase trajectory of the transition from state 1 to state 2 is projected in the p, V plane into the continuous motion along the straight line 1 - 2. The latter is valid only for weak, nearly isentropic, waves and only when the shear viscosity is insignificant. Analysis of the existence and uniqueness of shock-wave transition based on qualitative examination of steady-state solutions of the differential gasdynamic
3.5 Stabilityof Shock Waves
439
equations describing the flow of a viscous and heat-conducting gas with subsequent tending of the viscosity and thermal conductivity to zero and, therefore, tending of the finite thickness of the transition layer to the shock discontinuity (see Rozhdestvenskii and Yanenko, 1983, and references there) is also applicable only to weak waves. As it is well known, the equations of state, gasdynamic equations, and the linear relations of thermodynamics of irreversible processes appearing in these equations which have been used in the above analysis are inapplicable to any description of the structure of strong shockwaves. Temperature cannot be defined by the analysis of the structure of these discontinuities even for the translational degrees of freedom of the molecules. Nevertheless, the results of this analysis are interesting at least for modeling. Within such a model, it was proven that the shock transition from point 1 to point 2 is possible only if the entire interval 1 - 2 of the shock-wave Hugoniot curve between these points is located on the p, V-plane to the left of the straight line drawn between these points. Hence, intersection of this straight line with the shock-wave Hugoniot curve segment 1 - 2, except for its boundaries, implies that the steady shock-wave transition 1 - 2, that is, the transition via a solitary shock wave, is impossible (unstable), at least within the considered model.
3.5.2.9 SIMPLE INTERPRETATION OF THE INSTABILITY M E C H A N I S M S AND CRITERIA FOR INSTABILITY The positive feedback between the "voltage" and "current" is the fundamental feature of the instability mechanisms. At least in most cases, if not always, this feedback can be detected quite clearly, and shock waves are not an exception in this respect. The factors causing the loss of stability of a shock discontinuity proper (i.e., without its corrugation) can be determined without invoking a sophisticated mathematical analysis and they were discussed at the beginning of Section 3.5.2.1 in relation to the one-dimensional stability conditions (3.5.1). We will show now that each of the two criteria Eq. (3.5.3) corresponds to a different mechanism of corrugating instability and will provide a simple explanation of these mechanisms. Case with L < - 1 We will show that in this case the shock-wave velocity reduces with increase in pressure P2. Differentiating J2 with respect to P2, we obtain from Eq. (3.5.4)
2JdJ/dp2 = (1 +
L ) l ( v ~ - v2)
(3.5.20)
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N.M. Kuznetsov
Since J and (U1 --U2) are positive, Eq. (3.5.20) yields that is,
dD/dp2 < 0
dJ/dp2 < 0 at L < - I , (3.5.21)
The latter also can be easily seen from Figs. 3.5.1a and 3.5.2--in the s- - s + (Fig. 3.5.1a) and s~ - s+ (Fig. 3.5.2) intervals, the absolute value of the slope of the Michelson-Rayleigh line, which is proportional to D 2, reduces with increasing pressure behind the shock front. This means that the initial plane shock discontinuity becomes convex (cylindrical for two-dimensional and spherical for three-dimensional perturbations) after the pressure behind the shock wavefront is perturbed randomly and locally. The convex front area increases with time as t or t 2 for a cylinder and a sphere, respectively. The latter, as in the case of any diverging shock wave, causes a decrease of the pressure behind the shock wavefront compared to the pressure behind a plane shockwave with identical parameters of the flow far away from the discontinuity. The pressure decrease for the wave with the amplitude corresponding to inequality L < - 1 increases further the wave velocity and reduces further the pressure and so on. Similarly, it can be shown that at the concave areas at the shock wavefront (converging shock wave) the pressure increases with time and the wave propagation velocity decreases. This implies the excitation of the corrugating instability of a shock-wave in this range of the parameter L. It can be shown that within the s - - s + interval of the shock-wave Hugoniot curve (Fig. 3.5.1 where L < - 1 , the "one-dimensional" condition for shockwave stability equation (3.5.1) for the Mach number M 2 is not satisfied. Thus the wave cannot exist without corrugation, because its strength drops instantaneously to a value at which the flow behind the shock wavefront becomes subsonic. This is the reason why in studying shock-wave stability within the linear theory (D'yakov, 1954) the shock-wave Hugoniot curves, similar to those shown in Fig. 3.5.1 were not analyzed. However, for a shock-wave Hugoniot curve of another shape (Fig. 3.5.2) (D'yakov, 1954), which also meets inequality L < - 1 along the sgo - s + segment, the other stability conditions, that is, conditions Eq. (3.5.1) is satisfied. The boundaries of the shock-wave Hugoniot curve segment that satisfy inequality L < - 1 are found from the condition dJ/dp2 = 0 or dJ/dp2 = - o o under which, as it is seen from Eq. (3.5.20), L = - 1 or L = - o o . The region of corrugating instability L < - 1 is fully consistent with the LST results (D'yakov, 1954). The preceding simple proof of shock-wave instability may serve as an illustrative interpretation of its mechanism at L < - 1 . Case with L > 1 4-2M2 The corrugating instability of a shock wave is associated in this case with the reflection of weak perturbations from the shock wave. The interpretation of the
3.5 Stabilityof Shock Waves
441
shock-wave instability mechanism discussed in what follows for this case is not based on a rigorous mathematical analysis and therefore it should be considered as qualitative. (The corrugating instability criteria (Eq. (3.5.3)) were interpreted in connection with an analysis of the shock-wave propagation through a tube with a variable cross section (Landau and Lifshits, 1987)). The coefficient of the reflection of an acoustic perturbation from a shock front (for normal incidence) can be written as follows (Kuznetsov, 1984): )(, = - [ L
-
(1 -
2M2]/[L
- (1 4- 2M2)]
(3.5.22)
Inspection of Eq. (3.5.22) shows that the reflection coefficient is negative for L > 1 + 2M 2 and its absolute value exceeds unity. This suggests that after the reflection the phase changes and the amplitude of a weak perturbation increases. This interaction of a shock wave with perturbations arriving at its surface from the flow region behind the shock wavefront results in corrugating instability of the wave. Suppose that a local perturbation in the form of a weak rarefaction wave (-I~p/I) arrives at the shock wavefront from the flow behind the wavefront. In the considered range of the parameter L the pressure at the shock wavefront becomes P2 + 6 P r in the course of reflection, exceeding the initial pressure P2 by a value of C~pr > . Because of the finite size of the local perturbation its reflection lasts for a finite time, during which the perturbed area of the shock discontinuity moves at a velocity that is higher than the velocity of neighboring front areas. As a result, the discontinuity surface becomes convex. The local region with the elevated pressure persisting throughout the reflection time behind the perturbed area of the discontinuity propagates as compression waves incident on the neighboring areas of the shock front retain their strength espy. Reflection of these compression waves produces rarefaction waves of a higher amplitude 6 and reduces the strength and the propagation velocity of the reflecting areas of the shock wavefront. As a result, concave areas are formed on the shock-wave discontinuity surface near the convex area, and so on. The latter process continues and because the waves are amplified upon reflection, eventually it excites the corrugating instability. The condition for realization of the forementioned instability is specified by the second of inequalities (3.5.3), that is, it is identical with the LST result.
16pfl
6It can be shown that at L > 1 + 2M2 both, the amplitude rise and the phase reversal similar to Eq. (3.5.22) occur when the perturbation incident on the shock wavefront is oblique (Kuznetsov, 1986a).
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3.5.2.10
N.M. Kuznetsov
FEASIBILITY OF EXPERIMENTAL
OBSERVATION OF HYDRODYNAMIC INSTABILITY OF S H O C K WAVES So far only a few results in the theory of the hydrodynamics of shock-wave stability were verified or can be verified experimentally. The criteria (Eq. (3.5.3)) for corrugating of shock-wave instability derived in 1954 still remain only theoretical. Although the shock-wave Hugoniot curves (Fig. 5.2.3) satisfying these criteria (Figs. 5.1-3), do not contradict the thermodynamic inequalities ( ~ / a v ) ~ < 0 and c v ~ O, they are nevertheless anomalous from the thermodynamic point of view. Such materials with equations of state that result in the required behavior of the shock-wave Hugoniot curves have still not been found. Shock wave Hugoniot curves with a kink or a smooth bend (Fig. 3.5.1a) are frequently encountered and are associated with phase transitions or the beginning of plastic flows. We will discuss later in this section the dynamic experiments with materials that have shock-wave Hugoniot curves of this type. As mentioned here these results are not of interest for verifying the LST results. Indeed, the one-dimensional condition of the shock-wave stability (3.5.1), which is not related to the front corrugation, is violated at these shock-wave Hugoniot curves when the inequality L < - 1 is satisfied. The criterion L -~ - 1 , is met, although formally, in materials with realistic equations of state with shock-wave Hugoniot curves similar to that shown in Fig. 3.5.1 while materials where the second of the criteria (Eq. (3.5.3)) was satisfied have not been found yet. Shock-wave Hugoniot curves of a dissociating and ionized gas (see, e.g., (Kuznetsov, 1965)) and solids with a large initial porosity have a segment with a positive pressure derivative over the specific volume. However, the absolute value of this derivative, in all known cases, is not large enough for the second criterion (Eq. (3.5.3)) to be met; the same applies even to the instability criterion for the shock-wave interaction with a piston (Eq. (3.5.23)) (see section 3.5.2.11). Notably, even smaller values of the parameter L satisfying criterion 3.5.6 with a certain angle of the resonance acoustic wave reflection corresponding to each value of L were observed only in exceptional cases, namely, within some region of states of two-phase liquid-vapor systems (copper (Bushman, 1976) and water (Kuznetsov and Davydova, 1988)). The common thermodynamic properties of two-phase liquid-vapor systems (see Kuznetsov, 1982; Kuznetsov and Timofeev, 1986) suggest that the region (Eq. (3.5.6)) exists also in the shock-wave Hugoniot curves of other two-phase systems, for example, freons. However, the feasibility of observing the resonance reflection and other phenomena associated with shock-waves interaction with low-amplitude perturbations, incident at nearly-resonance angles, is restricted due to the
3.5 Stability of Shock Waves
443
gravitation-induced convection (buoyancy of vapor bubbles) and long relaxation times (heat and mass transfer). Thus, materials with shock-wave Hugoniot curves similar to that shown in Fig. 3.5.1 offer virtually the only possibility of observing what is formed instead of an unstable solitary shock-wave. Only by analyzing systems with such shock-wave Hugoniot curves can one hope to answer experimentally the question as to whether a solitary shock wave can be observed at the upper and lower Z-branches where the criteria of stability to small perturbations are met but the altemative representation of the shock discontinuity is possible. As already mentioned in Section 3.5.2.8, solitary shock waves were not observed in experiments at the middle branch and at the upper Z-branch. However, this does not necessarily rule out the possibility that these waves cannot be found at the upper Z-branch. It is conceivable to suggest that solitary shock waves were not observed in experiments because the solitary shock wave has been obtained in experiments by decomposition of a discontinuity (e.g., decomposition of a discontinuity formed by impact of a striker or a shock wave on a target made of the tested material) and not because they inevitably decompose into another 2-wave configuration. Furthermore, generating strong steady shock waves and measuring their parameters in materials with phase transitions is very complicated because of the finite phase transition time, which is not always large compared with the experimental characteristic time scale. These difficulties can be eliminated or substantially reduced under the following experimental conditions. First, a shock wave of a high amplitude is generated that transfers the material to some point at the shock-wave Hugoniot curve above point B (Fig. 3.5.1). Then the shock-wave velocity is measured when its amplitude is gradually reduced. In the P, v-plane this corresponds to the downwards translation of point P2 along the shock-wave Hugoniot curve. The purpose of this experiment is to find out whether the shock wave splits into two waves after passing through point B (Fig. 3.5.1). Probably when the upper Z-branch corresponds to strong nonisentropic shock waves such a solitary shock-wave can be observed below the point B. Unfortunately, experiments of this type are extremely difficult to perform, and it is doubtful that they will be performed unless they are motivated by a particular applied problem. On the contrary, at the lower Z-branch between points A and s_ only solitary shock waves are observed in the experiments. The 2-wave configuration is formed only for values of P2 that are higher than the kink point at the shock-wave Hugoniot curve s-. In this case the problem of alternative wave configurations is solved unambiguously. Both such configurations correspond to the same Fig. 3.5.4e and differ only by the first wave pressure (P3)-As mentioned in Section 3.5.2.4, these configurations correspond only to formal solutions and are unrealistic because the second wave catches up
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N . M . Kuznetsov
with the first one. Moreover, in one of the two configurations the shock wave with pressure P3 belongs to the middle Zobranch, that is, if it is absolutely unstable.
3.5.2.11 STABILITY OF A S H O C K W A V E S U P P O R T E D BY A P I S T O N A shock wave stable by itself can, together with the flow behind the front, form an unstable flow pattern due to the special boundary conditions for the flow behind the shock wavefront far away from it. This possibility may be realized when the boundary conditions are set at a finite distance from the shock wavefront rather than at infinity (y--4-oo). Strictly speaking, this is an independent problem that concerns stability of flows with shock waves. However, we can distinguish at least one particular case in this p r o b l e m ~ this is an analysis of the propagation of a shock wave supported by a piston. The latter case is of direct relevance to the theory of shock-wave stability. In many cases, the shock wave is produced by a piston that starts moving at some initial time t - - 0 . For example, in shock tubes the gas in the highpressure chamber acts as a piston. When the wave is produced by impact of the tested material on a target with a high acoustic impedance, the target plays the role of the piston. A projectile moving in a medium at a supersonic velocity can be considered as a piston of a finite size supporting steady propagation of the detached (bow) shock wave. Some time after shock-wave formation by a moving piston (depending on the space and time scales of the particular problem, e.g., this time is very large in the case of a detached shock wave formed by a projectile) because the distance between the shock wavefront and the piston is finite a problem of stability of the flow in which perturbations are repeatedly reflected from the shock wavefront and the piston must be addressed. The problem can be formulated as follows. Plane piston moving at a constant velocity u produces a shock wave propagating with a certain velocity. Let the velocity of the piston change instantaneously by a small value c~u. The produced weak plane perturbation catches up with the shock front, reflects from the front, then reflects from the piston, and so on. Will this sequence of repeated reflections converge, that is, will it result in a new steady flow in which the shock-wave velocity differs from the initial velocity also by a small value? This problem was solved analytically (Kuznetsov, 1984) and graphically (Fowles and Swan, 1973). The answer to this question follows also from an analysis of the repeated reflections from the parallel fronts of two shock waves
3.5 Stabilityof Shock Waves
445
propagating in opposite directions (Galin, 1975). The solution can be written in the form of an inequality imposed on parameter L. The sequence of repeated reflections converges if L < 1 and diverges if L > 1
(3.5.23)
that is, the interaction between the wave and the piston is stable in the first case and is unstable in the second. 7
3.5.3
STABILITY
SHOCK
AND
OF THE
DETONATION
STRUCTURE
OF
WAVES
Generally, a steady one-dimensional flow (depending on the coordinate aligned with the streamline) in the transitional layer separating the material ahead of the shock or detonation front behind it can be either stable or unstable. In hydrodynamic calculations of flows with shock-waves using explicit introduction of discontinuities, the flow stability analysis in these transitional layers is redundant because the layers are replaced by discontinuity surfaces. If a wave is unstable as a hydrodynamic discontinuity, it is unstable in general (i.e., it does not exist). However, if the flow in the transitional layer is unstable, this does not necessarily imply that the shock wave is unstable, but only that the structure of this layer becomes more complicated, multidimensional, and turbulent.
3.5.3.1
EXPERIMENTAL DATA ON
S T R U C T U R A L I N S T A B I L I T Y OF S H O C K W A V E S Due to the vast experimental evidence on shock-wave stability and because the criteria for hydrodynamic shock-wave instability are very rarely met in practice, the problem of instability of shock wavefront structure was considered for a long time as a problem of mostly academic interest. However, new experimental data on the propagation of strong shock waves in various gases 7At point L = 1 the derivative of the mass velocity over the pressure (0u/0p)vi at the shockwave Hugoniot curve equals zero. At L > 1 the inequality (0u/0p)H < 0 holds. This, together with the fact that the region where the shock Hugoniot curve has an anomalous shape (the region with L > 1, i.e., with (0u/0p)H < 0) is always bounded, suggests that at least three values of P2 correspond to any value of u from the shock Hugoniot curve segment where L > 1 (Zababakhin and Simonenko, 1967). Two of these values (minimum and maximum) pertain to a stable flow, and the third, to an unstable flow. It is this latter value of P2 that belongs to the shock Hugoniot curve interval where L > 1. This is another example of the relation between instability and nonuniqueness of steady regimes.
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N.M. Kuznetsov
with complex and high-rate energy-relaxation processes provided evidence of some deviations from the simple steady one-dimensional structure of shock waves whereby a homogeneous flow in the directions parallel to the shock wavefront with monotonic (or in some special very rare cases weakly nonmonotonic) density and pressure changes along the streamlines. The types of deviations are diverse, varying from a sequence of one-dimensional density maxima and minima along the streamlines (Hilton, 1952; Ryazin, 1980), to radical change of the wave structure (Ryazin, 1980; Tsykulin and Popov, 1977; Gordeev, 1978; Griffith et al., 1975; Mishin et al., 1981; and Savrov, 1985) and its flow turbulization (chaotic distribution of dark and light spots detected in direct observations of very strong shock waves through the window at the shock tube end in gases such as argon and xenon (Savrov, 1985)). The full list of experimental publications is much larger than the studies discussed in this chapter. Some original experimental data and references relevant to the problem can be found in Mishin et al. (1981). Clearly, the phenomena detected in these observations are different. Therefore, it is of great importance to verify experimentally when the shock-wave instability is observed and when various boundary effects or interaction of an ionizing shock wave with external fields (e.g., with electric discharge (Gorshkov et al., 1987)) cause flow instability and are not related directly to shock-wave stability. Because of the technical difficulties, usually no such analysis was performed. However, there is some experimental evidence of these boundary effects (see Vasil'eva et al., 1985; Davydov et al., 1983; Shuropov et al., 1985; Gvozdeva et al., 1985, Bashenova et al., 1989; and the review by Kuznetsov, 1995). The lack of experimental data on the structural instability of shock waves warrants comprehensive theoretical analysis to solve both fundamental and applied problems.
3 . 5 . 3 . 2 THE STRUCTURE OF SHOCK WAVES AND STABILITY OF VISCOUS COMPRESSION DISCONTINUITIES The transitional layer in a shock wave consists of the so-called viscous compression layer (viscous jump) where the initial thermodynamic state changes to a new state with equilibrated translational and rotational degrees of freedom of the molecules. The compression layer is followed by a considerably wider zone of slow relaxation (relaxation zone) within which the vibrational, chemical and ionic composition, etc. equilibrium is attained. The effect of these relaxation processes on the shock wavefront structure in weak shock waves is less significant as compared to that of the processes taking place
3.5 Stability of Shock Waves
447
in the viscous compression jump (e.g., the changes in the vibrational energy and the gas composition are rather small), but in strong shock waves it is manifested very strongly. The available experimental data and theoretical estimates (see review by Kuznetsov, 1989 and the references therein) indicate that the structural instability is associated with the processes within the relaxation zone of strong shock waves rather than inside the compression jump. Very short compression jump zone in strong shock-waves (in gases they are of the order of the mean free path of the molecules) allow us to replace this zone by a discontinuity surface when studying the structural instability of the wave. When the flow in the relaxation zone is unstable, the compression jump also becomes inhomogeneous on the scale of turbulent fluctuations. However, if the jump itself is hydrodynamically stable and does not resonantly reflect incident perturbations (these two conditions are always met, at least in gases) s , the front inhomogeneity is only a normal response to the incident perturbations which are not amplified upon their reflection.
3.5.3.3 O N THE H Y D R O D Y N A M I C A P P R O A C H TO F L O W S W I T H STRUCTURALLY UNSTABLE S H O C K WAVES On the scale exceeding the characteristic size of turbulent fluctuations the averaged flow is one-dimensional. At long distances from the compression jump prior to or during transition of each elementary material volume to the local thermodynamic equilibrium the turbulent fluctuations are damped, and their energy and pressure become negligible due to viscosity. At these distances the common one-dimensional continuity relations for the mass, momentum and energy fluxes through the shock waves are valid. Therefore, replacement of the entire transition layer separating the flow regions ahead of the shock front and at long distances behind it by the discontinuity surface, as has been done in the hydrodynamic approximation, is equally applicable to the shock wave with unstable transitional layer. Therefore, all the results of the analysis of the shock-wave stability obtained by the hydrodynamic methods are valid, irrespective of how the wave structure is formed. However, any questions concerning the wave structure should be considered only when the wave is hydrodynamically stable. 8Within the viscous compressionjump, the intemal degrees of freedom of the gas molecules are virtually unexcited. The shock-wave Hugoniot curve describing gas compression in the jump has no anomalies and the parameter L does not satisfy conditions equations (3.5.1), (3.5.3), and (3.5.6).
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N.M. Kuznetsov
3 . 5 . 3 . 4 O N THE MECHANISMS OF STRUCTURAL INSTABILITY OF SHOCK AND DETONATION WAVES The problem of flow stability in the transitional layer of a shock wave (structural stability) attracts attention predominantly in studies of flows on scales comparable with the transitional layer width. Investigations into the mechanisms and measurements of the rate constants for relaxation processes with the aid of shock and detonation waves belong to the same class of problems. The detonation wave structure is unstable because of the strong positive feedback between the perturbations, particularly entropy waves, emitted by the shock wavefront, and the response to the acoustic perturbations arriving from the reaction zone. The stronger the temperature dependence of the reaction rate the higher is the amplitude of the response perturbations. Numerous experimental data (see, e.g., the monographs of Shchelkin and Troshin, 1963, and Dremin et al., 1970) and the results of calculations with approximate qualitative criteria by the linear stability theory based on various models (Shchelkin, 1959; Aslanov, 1966; Dremin, 1983; Pukhnachev, 1963; Erpenbeck, 1964; Zaidel and Zeldovich, 1963; Aslanov and Volkov, 1992; Kuznetsov, 1994) suggest the existence of the structural instability of detonation waves caused by this mechanism. The following result is common for all these investigations, even though they differed in the approximations and modeling approaches used--if the reaction rate depends strongly on shock-wave amplitude, the detonation wave structure is unstable.
3 . 5 . 3 . 5 T w O - F R O N T S MODEL OF A SHOCK (OR DETONATION) WAVE WITH INSTANTANEOUS HEAT RELEASE Perturbations propagating between the shock wavefront and the relaxation zone were analyzed using the two-fronts, one-dimensional model of a wave structure consisting of a shock wavefront followed by a zone of the reaction induction period and then by a front of instantaneous energy release (Chernyi, 1969; Levin et al., 1974). This model always predicts instability to highfrequency perturbations at any activation energy in the Arrhenius kinetics. Unfortunately, it is precisely at high frequencies that the model is unsatisfactory from the physical point of view; the heat release front cannot be assumed infinitely thin (i.e., the heat release cannot be assumed instantaneous) with respect to short waves. In spite of this serious drawback of the model, it is quite
3.5 Stabilityof Shock Waves
449
helpful and can be used to derive approximate estimates of the instability conditions for finite heat release times. Although (Chernyi, 1969; Levin et al., 1974) the two-fronts model was applied to detonation waves, it is valid for shock waves as well. In the latter case the relaxation with heat absorption rather than heat release occurs instantaneously at the second front. Note that the assumption of an infinitely long initial perturbation in a plane )2, parallel to the shock wavefront, does not restrict the applicability of the model. Indeed, due to repeated reflections an initial perturbation of a finite length in the plane Y. becomes arbitrarily long along )2 as compared to the relaxation zone width.
3.5.3.6
TwO-FRONTS MODEL OF SHOCK
AND D E T O N A T I O N WAVES W I T H N O N I N S T A N T A N E O U S RELAXATION The conditions for structural instability of shock and detonation waves derived using the two-fronts model with a finite heat release (relaxation) time z were analyzed by Kuznetsov (1994). Hereafter, the term heat release denotes release or absorption of heat depending on the sign of the reaction heat (Q). The preceding discussion on response of the two-fronts model to small perturbations suggests that in reality (when the relaxation zone width following the induction period zone is finite) the amplitude coefficient of the reflection of the perturbations from this zone will grow as the zone width reduces (or the characteristic relaxation time 1: following the induction period z i, decreases). As mentioned here, a strong dependence of the induction period on the temperature (or on the shock-wave strength) promotes excitation of the instability. However, this dependence (as well as the dependence of cross sections of various inelastic molecular collisions on the energy of their relative motion) usually occurs when the reaction rate (cross section of inelastic collisions) is rather low. Thus, the induction period depends highly on the temperature when it is large. The preceding two factors facilitating the excitation of the instability, that is, small ~ and large zi, can be combined to formulate a simple qualitative criterion for structural instability:
~/~i << 1
(3.5.24)
For chemical reactions in a self-sustained or slightly overdriven detonation typical wave structures are the structures with two or more subsequent stages whereby the first stage is relatively slow and only weakly influences the
450
N.M. Kuznetsov
gasdynamic parameters (the induction period). It is followed by the exothermic reactions leading to a significant pressure and density reduction within a narrow subregion of the relaxation zone. Hence, condition (3.5.24) is met, although not as a strong inequality: "t/'c i '~
0.1 - 0.5
(3.5.25)
Contrary to detonation waves, relaxation processes in shock waves are often not self-accelerating. Relaxation starts immediately at the shock wavefront while the Maxwell distribution sets in or immediately after this. Because in the Maxwellian-distribution the temperature decreases in the course of the relaxation due to the excitation of the new degrees of freedom (molecular vibrations and breaking of the chemical bonds), relaxation does not accelerate and is even depleted. The induction period for dissociation and monomolecular reactions is a time required in order to excite the molecular vibrational levels and to establish a quasisteady population of the vibrational levels. Dissociation proceeds predominantly from the upper vibrational levels after this induction period. However, it is known that this period is much shorter than the characteristic dissociation time. Generally, the two relaxation stages, where one is slow and changes only insignificantly the pressure and the temperature (i.e., leaving the flow behind the compression jump unchanged) and the other (which is faster and more energetic) is accompanied by a considerable rise in the density and reduction of the temperature in the steady shock-wave structure, are not typical of shockwaves. However, there are exceptions and one can probably find a number of such exceptions. However, there is an example that is represented not by some rare relaxation process but by the well-known process of ionization of a monatomic gas behind a strong shock wave with an appreciable (of the order of tens of percent) ionization degree in the gas at equilibrium behind the shock wavefront where the slow initial build-up of free electrons and the subsequent fast avalanche ionization accompanied by heating of the electrons due to electron-ion collisions (see Fig. 3.5.12) are well separated in time. The following inequality IKfK~[ > 1
(3.5.26)
may serve as a criterion for structural shock-wave instability in the onedimensional approximation, that is, when the flow (stable or unstable) is assumed to be one-dimensional. Here Kf and K r are the coefficients of perturbation reflection from the wavefront and the relaxation zone, respectively. It must be emphasized that it suffices that inequality equation (3.5.26) is met for the absolute values. Indeed, the absolute value of a local perturbation grows unboundedly as a result of subsequent repeated reflections from the
3.5
451
Stability of Shock Waves
ne/ne
Y FIGURE 3.5.12 Qualitative dependence of the electron number density n e on the coordinate y (as measured downstream from the compression jump) in the ionization structure of a strong shock wave in a rare gas; h e i s the equilibrium electron concentration.
front and the relaxation zone both at KfKr < 1 and KfKr < - 1 . Using the twofronts model with instantaneous heat release, coefficients Kr for entropy and acoustic perturbations were calculated by Chernyi (1969) and Levin et al., (1974). When the heat is released within a finite time, Kf was calculated by Kuznetsov (1994) where K~ was also estimated approximately. According to Kuznetsov (1994) in a shock wave ionizing monatomic gas
[KfK,.I "~ 0.041:/E--L "c kT e
(3.5.27)
where E 1 is the energy of the first excited atomic level (e.g., in argon and xenon E 1 = 11.6 eV and 8.3 eV, respectively) and Te is the temperature of free electrons. For some values of shock-wave amplitude and gas density the inequality (3.5.26) is possible. Thus one of the qualitative instability criteria for the structure of shock waves and detonation waves is the small ratio of the characteristic relaxation time to the induction period. Excitation of the instability does not require the presence of the von Neumann spike in the steady-state structure but only a sharp pressure change in the relaxation zone (of any sign). The latter statement is not very significant for detonation waves because the von Neumann spike normally occurs there but it is essential for shock waves. Certainly the preceding criteria for the stability of the shock-wave structure are only approximate estimates. Numerical calculations for various model structures of shock and detonation waves are required in order to refine these results.
452
N . M . Kuznetsov
REFERENCES Artshuler, L.V. (1978). Phase transitions in shock waves (review). J. Appl. Mech. Tech. Phys., 19 (4): 496-505. Aslanov, S.K. (1966). Criterion for one-dimensional instability of gases detonation. Doklady Akademii Nauk Ukrainskoj SSR, 7:871-874 (in Russian). Aslanov, S.K., and Volkov, V.E. (1992). About stability and structure of waves in plane channels. Matematicheskoje Molelerovanie, 4:18-20 (in Russian). Bancroft, D., Peterson, E.L., and Minshall, S. (1956). Polymorphism of iron at high pressure. J. Appl. Phys., 27 (3): 291-298. Bazhenova, T.V., Gvozdeva, L.G., Lagutov, Yu.P., Lakhov, V.N., Faresov, Yu. V., and Fokeev, V.P. (1989). Nonsteady Interactions of Shock and Detonation Waves in Gases, New York: Hemisphere. Bushman, A.V. (1976). Tezisy Dokladov Vtorogo Vsesoyuznogo Simpoziuma po Impulsnym Davleniyam. Abstracts (2nd USSR Symp. on Pulse Pressures). VNIIFTRI, Moscow, 39 (in Russian). Chernyi, G.G. (1969). Initiation of oscillations during attenuation of detonation waves. App. Math. Mech., 33: 451-461. Courant, R., and Friedrichs, K. (1977). Supersonic Flow and Shock Waves, New York: SpringerVerlag. Davydov, A.N., Lebedev, E.E, and Shurupov, A.V. (1983). Rayleigh-Taylor instability in a cylindrical explosive flow. Soviet Tech. Phys. Lett., 9 (4): 185-186. Dremin, A.N. (1983). Pulsating detonation front. Combust. Expl. Shock Waves, 19 (4): 521-530. Dremin, A.N., Savrov, S.D., Trofimov, V.S., and Shvedov, K.K. (1970). Detonatsionnye volny v kondensirovannykh sredakh (Detonation Waves in Condensed Media), Nauka, Moscow (in Russian). Drummond, W.E. (1957). Multiple shock production. J. Appl. Phys., 28 (9): 998-1001. Duvall, G.E. (1977). Phase transitions under shock-wave loading. Rev. Modern Phys., 49 (3): 523579. D'yakov, S.P. (1954). On the stability of shock waves. Zh. Eksper. Teoret.Fiz., 27, 3(9): 288-295 (in Russian). D'yakov, S.P. (1958). Interaction of shock waves with small perturbations. Sov. Phys. JETP, 6: 729742. Egorushkin, S.A. (1984). Nonlinear instability of a spontaneously radiating shock wave. Fluid Dyn., 19 (3): 436-443. Erpenbeck, J.J. (1962). Stability of step shocks. Phys. of Fluids, 5 (10): 1181-1187. Erpenbeck, J.J. (1964). Stability of idealized one-reaction detonations. Phys. Fluids, 7 (5): 684-696. Fowles, G.R. (1981). Stimulated and spontaneous emission of acoustic waves from shock fronts. Phys. Fluids, 24 (2): 220-227. Fowles, G.R., and Swan, G.W. (1973). Stability of plane shock waves. Phys. Rev. Lett., 30 (21): 1023-1025. Galin, G.Ya. (1975). Interaction of small perturbations with regular and combined shock waves. Izv. Akademii Nauk SSR, Mekhanika Zhidkosti i Gaza, 3:164-167 (in Russian). Gardner, C.S. (1963). Comment on stability of step shocks. Phys. Fluids, 6 (9): 1366-1368. Gordeev, V.E. (1978). Oscillating detonation and instability of shock waves in gases with relaxation. Doklady Akademii Nauk SSSR. Fizika, Khimija, 239:117-119 (in Russian). Gorshkov, V.A., Klimov, A.I., Mishin, G.I., Fedotov, A.B., and Yavor, I.P. (1987). Behaviour of electron density in a weakly ionized nonequilibrium plasma with a propagating shock wave. Sov. Phys. Tech. Phys., 32 (10): 1138-1141. Gvozdeva, L.G., Lagutov, Yu. P., and Fokeev, V.P. (1985). Influence of the physicochemical properties of gases on shock interaction with convex cylindrical surfaces. J. Eng. Phys., 48 (2): 171-174.
3.5
Stability of Shock Waves
453
Griffith, R.W., Sandeman, R.J., and Hornung, H.G. (1975). The stability of shock waves in ionizing and dissociating gases. J. Phys. D. (Appl. Phys.), 9 (12): 1681-1691. Hilton, W.E (1952). High-speed aerodynamics, London: Longmans. Iordanskii, S.V. (1957). About stability of a plane shock wave. Prikl. Mat. Mekh., 321: 465-471. Ivanov, A.G., and Novikov, S.A. (1961). Rarefaction shock waves in iron and steel. Sov. Phys. JETP, 13 (6): 1321-1322. Kontorovich, V.M. (1957a). Concerning the stability of shock waves. Sov. Phys. JETP, 6 (6): 11791180. Kontorovich, V.M. (1957b). Reflection and refraction of sound by a shock wave. Sov. Phys. JETP, 6 (6): 1180-1181. Kontorovich, VM. (1959). Reflection and refraction of sound by a shock waves. Akust. Zh. [Sov. Phys. Acoust.], 5 (3): 314-323 (in Russian). Kutateladze, S.S., Borisov, A1.A., Borisov, A.A., and Nakoryakov, V.E. (1980). Experimental detection of a rarefaction shock wave near a liquidwapor critical point. Sov. Phys.--Doklady, 25 (5): 392-398. Kuznetsov, N.M. (1965). Termodinamicheskie Funktsii i Udarnye Adiabaty Vozdukha pri Vysokikh Temperalurakh (Thermodynamic Functions and Shock Hugoniots of Air at High Temperatures), Moscow: Mashinostroenie (in Russian). Kuznetsov, N.M. (1966). A feature of shock compressibility with disappearance of the two-wave configuration. J. Appl. Mech. Tech. Phys., 7 (1): 76. Kuznetsov, N.M. (1982). Equation of state and the phase-equilibrium curve of liquid-vapor systems. Sov. Phys.mDoklady, 27 (9): 742-744. Kuznetsov, N.M. (1984). Criterion for instability of a shock wave maintained by a piston. Sov. Phys.--Doklady, 29 (7): 532-534. Kuznetsov, N.M. (1985). Contribution to shock-wave stability theory. Sov. Phys.--JETP, 61 (2): 275-284. Kuznetsov, N.M. (1986a). Reflection of weak perturbations by a shockwave front. Nonlinear analysis. Sov. Phys. JETP, 63 (2): 433-438. Kuznetsov, N.M. (1986b). Physical meaning of solutions with outgoing sound waves in the theory of shock-wave stability. Sov. Phys.--JETP, 64 (4): 781-783. Kuznetsov, N.M. (1989). Stability of shock waves. Soviet Physics~Uspekhi, 32 (11): 994-1012. Kuznetsov, N.M. (1994). Stability of shock and detonation wave structures. Sov. J. Chem. Phys., 12 (3): 391-405. Kuznetsov, N.M. (1995). Hydrodynamic and structural stability of shock and detonation waves. Chem. Phys. Rept., 14 (9): 1307-1340. Kuznetsov, N.M., and Timofeev, E.I. (1986). Shock-wave compression of a broad range of twophase liquid-vapor systems. High Temperature, 24 (1): 94-99. Kuznetsov, N.M. (1987). Theory of the stability of shock waves; physical meaning of steady-state solutions for a perturbed shock front with weak outgoing waves. Sov. Phys.~JETP, 66 (6): 1132-1135. Kuznetsov, N.M., and Davydova, O.N. (1988). Regions of resonance reflection of sound by a shock front in a two-phase water-vapor system. High Temperature, 26 (3): 426-429. Landau, L.D., and Lifshits, E.M. (1987). Fluid Mechanics, Oxford: Pergamon. Levin, V.A., Solomakha, B.P., and Chikova, S.P. (1974). About stability of a plane detonation wave. Nauchnyje Trudy Instituta Mekhaniki MGU (Proc.Inst. Mech., Moscow State Univ.), 32:44-59 (in Russian). Mishin, G.I., Bedin, A.P., Yushchenkova, N.I., Skvortsov, G.E., and Ryazin, A.P. (1981). Anomalous relaxation and instability of shock waves in gases. Sov. Phys.~Tech. Phys., 26 (11): 1363-1368. Pukhnachev, V.V. (1963). About stability of ChapmanoJouguet detonation. Zh. Prikl. Mekh. Tekh. Fiz., 6:66-73 (in Russian).
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Rozhdestvenskii, B.L., and Yanenko, N.N. (1983). Systems of Quasi-Linear Equations and Their Applications to Gas Dynamics, Providence, Rhode Island: American Math. Soc. Ryazin, A.P. (1980). Ionization instability of a shock wave in xenon. Sov. Tech. Phys. Lett., 6 (5): 222-223. Savrov, S.D. (1985). Dynamics of excitation of radiation shock waves in inert gases. High Temperature, 23 (1): 23-27. Shchelkin, K.I. (1959). Two cases of unstable combustion. Sov. Phys.mJETP, 9: 416-422. Shchelkin, K.I., and Troshin, Ya.K. (1963). Gazodinamika goreniya (Gasdynamics of Combustion), Moscow: Akad. Nauk SSSR (in Russian). Shurupov, A.V., Gal'burt, V.A., and Lebedev, E.F. (1985). Effect of Rayleigh-Taylor instability on the structure of explosive cylindrical flow. Sov. Tech. Phys. Lett., 11 (5): 230-232. Tsykulin, M.A., and Popov, E.G. (1977). Izluchatel'nye svoistva udarnykh voln v gazakh (Radiative Properties of Shock Waves), Moscow: Nauka (in Russian). Vasil'eva, R.V., Zuev, A.D., Moshkov, V.L., Tkhorik, L.G., and Shingarkina, V.A. (1985). Turbulent mixing of driver and driven gases in shock tube channel. J. Appl. Mech. Tech. Phys., 26 (2): 271277. Zababakhin, E.I., and Simonenko, V.A. (1967). Discontinuities of shock adiabats and the nonuniqueness of some shock compression. Sov. Phys.--JETP, 25 (5): 876-877. Zaidel, R.M., and Zeldovich, Ya.B. (1963). One-dimensional stability and attenuation of detonation. Zh. Prikl. Mekh. Tekh. Fiz., 6:59-65 (in Russian). Zeldovich, Ya.B. (1946). The possibility of the rarefaction shock waves. J. Phys. (USSR), 10 (4): 325-326. Zeldovich, Ya.B., and Kompaneets, A.S. (1960). Theory of Detonation, New York: Academic Press. Zeldovich, Ya.B., and Raizer, Yu.P. (1967). Physics of Shock Waves and of High-Temperature Hydrodynamics, in 2 volumes, New York: Academic Press.
CHAPTER
3.6
Theory of Shock Waves 3.6
Shock Waves in Space
MICHAEL GEDALIN Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
3.6.1 3.6.2 3.6.3 3.6.4 3.6.5
Introduction Magnetohydrdodynamic (MHD) shocks Shock Morphology Bow Shock Observations Collisionless Shock Theory 3.6.5.1 Field Structure 3.6.5.2 Nonlinear Waves and Ramp Width 3.6.5.3 Noncoplanar Magnetic Field 3.6.5.4 Ion Motion 3.6.5.5 Electron Heating 3.6.6 Shock Particle Acceleration 3.6.6.1 Shock Drift Acceleration 3.6.6.2 Diffusive Acceleration 3.6.6.3 Electron Acceleration 3.6.7 Conclusions References
3.6.1 I N T R O D U C T I O N Gasdynamic shocks form each time a high velocity supersonic flow is stopped by an obstacle. One of the most important distinctions between space shocks and ordinary shocks in gases is that the medium space shocks form in is a plasma-ionized gas. Another very significant property of these shocks is that the plasma almost always is embedded in an ambient magnetic field, which is compressed by the shock. Therefore, these shocks form whenever the plasma flow velocity exceeds the corresponding signal velocity in the magnetized plasma (de Hoffman and Teller, 1950; Tidman and Krall 1971). Shocks are ubiquitous phenomena in space. Many different kinds of shocks have been observed in the solar system. Planetary bow shocks (Russell, 1985) are formed when the solar wind encounters a planetary magnetosphere, Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
455
456
M. Gedalin
whether natural (as on Earth, Jupiter, and Saturn, which have their own magnetic field) or induced (as on Mars and Venus). Cometary shocks are produced by the interaction of the solar wind with charged particles of cometary origin. Interplanetary shocks appear whenever fast solar wind overtakes slow wind. Since the solar wind is super-magnetosonic, and the solar system velocity relative to the interstellar plasma is also super-magnetosonic, both the solar wind and interstellar flows should be decelerated to subsonic velocities at the heliopause that separates the two plasmas. This is achieved at the termination shock and heliospheric bow shock, respectively, which are formed on both sides of the heliopause. Shocks, believed to be produced by supernova explosions, play an important role in cosmic ray generation, that is, acceleration of charged particles to high energies. Another feature of these shocks that distinguishes them from gasdynarnic shocks is that the mean free path for Coulomb collisions in the system is much larger than the system size itself. For example, at the Earth orbit the electron Coulomb mean free path is larger than the distance to the Sun. This means that these shocks are essentially collisionless. As a result, the shock features (scales, particle energization, and dissipation) are quite different from those of shocks where collisions are substantial. In this chapter we will discuss the modern theory of these shocks and observations. In Section 3.6.2 we present the magnetohydrodynamic (MHD) description of the shock transition. In Section 3.6.3 we briefly discuss shock classification. Section 3.6.4 is devoted to in situ observations of the Earth bow shock. Theoretical developments related to the shock structure and particle motion in the shock front are discussed in Section 3.6.5. Shock particle acceleration is described in Section 3.6.6. For the reader's convenience throughout we prefer to cite more recent books and review papers, which contain a number of further references, rather than priority papers.
3.6.2 MAGNETOHYDRODYNAMIC (MHD) SHOCKS The magnetohydrodynamic (MHD) approach is the basis for shock theory. In MHD, a shock is a one-dimensional (1D) static discontinuity in an MHD fluid (de Hoffman and Teller, 1950; Tidman and Krall, 1971) that is described by hydrodynamic equations for average fluid variables (density p, velocity V, and pressure p, which we assume here to be scalar) and magnetic field B:
Otp+ V. (pV) -- 0 p[o~ + ( v . v)]v = - v (B2) p+ ~ + I(B.V)B 8tB = V x (V • B)
(3.6.1) (3.6.2) (3.6.3)
3.6
457
Shock Waves in Space
where we should add the state equation p = p(p), and no dissipation is included. We shall take this state equation in the polytropic form p (x pY. The shock is assumed to be stationary 0t = 0 and 1D, that is, all variables depend only on a single coordinate (we choose x) along the shock normal. In this case one has
Jx = pVx = const
(3.6.4)
B2
Px -- PV2x + P + 8--n -- const
(3.6.5)
P x = P V x V • - BxB-l~ = const
(3.6.6)
E t- = VxB • - B x V • = const
(3.6.7)
4n
where 2_ refers to the shock normal direction. Equation (3.6.4) describes the mass conservation across the shock. Equations (3.6.5) and (3.6.6) describe the momentum conservation across the shock front, while (3.6.7) is the constancy of the tangential electric field component, as Ohm's law in this case reads E + ( V / c ) x B = 0. Of course, B x is constant throughout the shock. These relations must be valid for any x dependence provided the system is stationary and 1D, which means that they can be applied to the infinitesimally thin shock transition from the asymptotically homogeneous state upstream (subscript u hereafter) to the asymptotically homogeneous state downstream (subscript d). In this case the preceding expressions provide the connection between upstream and downstream values p , V ~ = paVxa, etc. It can be shown (de Hoffman and Teller, 1950; Landau and Lifshitz, 1960) that it is possible to choose a reference frame so that the upstream fluid velocity (V~,, V zu ), upstream magnetic field (B x, B zu), the corresponding downstream values, and the shock normal ~: are in the same plane (so called coplanarity plane), which is convenient to choose as ( x , z ) plane. For this choice B, = (Bx, 0, Bzu ) = B,(cos 0, 0, sin 0), where 0 is the angle between the shock normal and upstream magnetic field, while E• = (Ey, 0). There are two basic choices of reference frame used in the shock studies--the normal incidence frame (NIF) where the upstream plasma velocity is along the shock normal V u = Vu~ and E y - - "CuBu sin O/c, and the de Hoffman-Teller frame (HT) where the upstream plasma velocity is along the upstream magnetic field and Ey = 0. Both frames are equivalent and can be used on one's convenience. The HT frame moves relative to NIF with the velocity V~h = Vu tan 0 along the z-axis. It is obvious that the HT frame exists only when Vsh < c, that is, tan 0 < c/Vsh, so that the HT frame exists only for sufficiently oblique shocks. In the perpendicular case, the 0 = 90 ~ HT frame does not exist.
458
M. Gedalin
In this section we shall work in NIF, so that Eqs. (3.6.4)-(3.6.7) take the following form: p . V u = paVxa,
(3.6.8)
Pu V2 + Pu + - ~ -- PuV~ p---~"+ Pu Pa BxBzu 4re = Pu Vu Vz't
+-8re
BxBza 4~
V,,Bz~ -- V, Bz a P___u__~ BxVz a Pa
(3.6.9) (3.6.10)
(3.6.11)
where we have used Eq. (3.6.8) in the subsequent equations. Theseare the socalled Rankine-Hugoniot relations, which allow us to find the downstream values of V,~, Vzd, Bzd, and Pa from the upstream parameters Pu, Vu, B,, and 0 (a reminder: B x = B, cos 0 and Bz~ = Bu sin 0). In what follows it is convenient to express everything in terms of the density compression ratio Y = Pa/Pu and the magnetic compression ratio Z - Bza/Bz~. Note also that VA = B , / v ' T ~ p is the upstream Alfv4n velocity and v~ -- v/Vpu/p, is the upstream sound velocity. Equations (3.6.8)-(3.6.11) immediately give Z =
V2 -- U~COS2 0
Y
(3.6.12)
fl (1 - yr) + 1 q-sin 20
(3.6.13)
v2 - v2 cos~ 0Y
2V2 Z---- v2sin20
1-
where fl = 8rcpu/B 2. Equating Eqs. (3.6.12) and (3.6.13) we arrive at the following relation: v =
(v~ - v~ cos 2 0) 2 y2 _ 1 ( v 2 _ v2 cos2 oy) 2
-
-
sin2-----v2 ~
1 --
-- sin 20 (1 -- Y~) -----0
(3.6.14)
which should be solved for Y > 1 (the flow is decelerated at the shock). Analysis of the behavior of the function F(Y) is straightforward. The solution Y = 1 is always a solution of Eq. (3.6.14). For a second solution to exist, the condition dF/dY < 0 should be satisfied in Y = 1, which means that either Vst < Vu < v1 or vv < V,, where (Kennel et al., 1985): Vr,st. =
v2 + vs2 -t-
(v 2 + v2) 2 - 4 v l v 2
(3.6.15)
3.6 ShockWaves in Space
459
are the fast and slow velocities, respectively, and v I -~ VA[ cos 0] is the intermediate velocity. It is accepted to define the Alfven Mach number M A = V u / v A and magnetosonic (fast) Mach number Mms -- V u / v F. The rest of this chapter is devoted to fast shocks Mms > 1 which constitute the vast majority of shocks found in space. These shocks are compressional--pa/p u > 1 and Bzcl/Bzu > 1. The foregoing derivation is adapted from Kennel (1988). It should be emphasized that the main role of the shock is to decelerate the high-velocity plasma flow and transfer the energy of the directed flow of incident upstream ions into the thermal energy of downstream ions and electrons, energy of compressed downstream magnetic field, and the energy of accelerated particles (see Section 3.6.6). The main task of the collisionless shock theory is that of explaining the mechanism of this transfer and shock maintenance. The preceding MHD theory describes the connection between the upstream and downstream shock parameters but it is unable to describe the transition itself unless some dissipation is included (e.g., resistive or viscous).
3.6.3
SHOCK MORPHOLOGY
Because Earth's bow shock is the closest space shock to us, as well as the one that is most significant to us, the question regarding position and shape of planetary bow shocks becomes rather important. The position and shape of a bow shock is determined by the interaction of the solar wind with the planetary magnetosphere (Spreiter et al., 1966; Fairfield, 1971; Formisano, 1979; Spreiter and Stahara, 1985; Cairns and Grabbe; 1994; Farris and Russell, 1994; Spreiter and Stahara, 1995; Petrinec and Russell, 1997). Roughly speaking, the solar wind is stopped where its ram pressure nmi V2 is balanced by the planetary magnetic field pressure B2/87r. Schematically planetary bow shock of a magnetized planet is shown in Fig. 3.6.1. The bow shock stands ahead of the magnetopause (marked r 0) that separates the planet magnetosphere (where particles are essentially trapped and "belong" to the planetary environment) from the solar wind with solar origin plasma. The upstream plasma comes from the Sun with the intrinsic magnetic field, as shown in Fig. 3.6.2. At the bow shock, the direction of the magnetic field lines and of streamlines changes abruptly. At each point the bow shock can be considered to be planar because its curvature can be neglected (at least in the local approximation). As the magnetic field is the most important component of the interaction it is usually chosen as the main characteristic of the shock transition. The angle ~)Bn between the upstream magnetic field and the shock normal is different in different bow shock locations. The angle 0Bn is one of the main shock
460
M. Gedalin / Bow Shock ~ / ,
Super-
/
sonic t-..-."""~
' ~ S . , : .............
Sonic line--~ /
t/
Subsonic
Magnetopause
f
/s.,
. - . . . . . . . . .
'" "" "" "'" i__ 1 ;___Z J _ _ _ J_'L ro rp
""
"'" J- . . . . . . . .
Magnetic obstacles Planet Mercury Earth Mars (?) Jupiter Saturn
rp,km
ro/rm
2400 6400 3400 71000 60000
1.5 10.0 1.1 70.0 22.0
FIGURE 3.6.1 Planetary bow shock position. Reproduced from Spreiter and Stahara (1985), copyright by the American Geophysical Union (AGU). The planetary radius rp is shown. The magnetopause separates the solar wind particles (flowing around the magnetopause surface) from those that belong to the planetary magnetosphere (i.e., trapped in the planetary magnetic field).
parameters according to which shocks are classified as quasiperpendicular OBn > 45 ~ and quasiparallel OB, <_ 45 ~ Another parameter used for shock classification is the kinetic-to-magnetic pressure ratio /Y-8zrnT/B 2. The most important shock parameter is its Mach number, either Alfven M A - - V u / v A (where V u is the NIF upstream plasma velocity and /)A is the Alfven velocity), or magnetosonic Mms = Vu/v ~ (where vF is the fast magnetosonic velocity). Theory predicts that the shock structure should change drastically when the downstream plasma velocity exceeds the downstream sound velocity (Kennel et al., 1985 Kennel, 1988). This happens at some critical M c. Accordingly, shocks with Mms < M c are called subcritical and those with Mms > M c are called supercritical.
3.6.4 BOW SHOCK OBSERVATIONS The fortuitous existence of a shock in the vicinity of the Earth has allowed us to perform extensive in situ measurements of the shock parameters and structure both from the many spacecraft launched in the past and those launched recently and even today. These measurements include almost all
3.6 ShockWaves in Space
461 Bow shock Magnetic field lines Streamlines of solar wind plasma flow Ionospheric pressure
Solar wind pressure
FIGURE 3.6.2 Solarwind flow lines and magnetic field lines at the magnetized planet. variables at the shock frontmmagnetic field vector, thermal ion and electron distributions, high-energy particle distributions and electric and magnetic field Fourier spectra. Some of these measurements are shown in Fig. 3.6.3. The electric field is the worst measurement variable. Early experiments (Formisano, 1982; Wygant et al., 1987) allowed measurements of only one component of an electric field with low time resolution (one measurement per 3 s on International Sun-Earth Explorer (ISEE) 1 and 2). Measurements on POLAR changed this tradition, providing three components of the electric field with a resolution of 8 vectors/s. Magnetic field measurements are best and have a time resolution of 16 vectors/s at best (Russell and Greenstadt, 1979) (at ISEE) or even 32 vectors/s (at Active Magnetospheric Particle Tracer Explorer (AMPTE)). Particle measurements are slower and more difficult. Cold beams are hardly identified and only one distribution is usually obtained during 3 s of measurements (at best), which involves some averaging and this is sometimes difficult to consider. Under observation, shocks are usually classified by low Mach number(subcritical, marginally critical, and weakly supercritical) and high Mach
462
M. Gedalin
100
ISEE 1 7 INBOUND 7 NOV 1977 ~,,ili~i~i'iinw,lii~,,,,~-~ ,~i,J,i i~ltlii~i,;i~,.i~w,~ ,~u~,~,~r,'~,~iI"
'
I
,
"
'
'
"
!.
NE
!
10 ~__ . . . . . . . . . . . . . . . . . . . . . . . . . . .
1~ 0.1 ..................
Tp 106
=,,
1 7
-
-
.....
LI,
,,
I ~
___ .
300 100
.........
i ~ : 1 .....
T E 105 r
Vp 200
i
.
.
I
.
-_
_
II
_
_
~
-......
....
I .
l
1t
l
,
I
I
,.~
i
;
. . . . .
-P E
0.4~
I
0,2
t
!
in
30 _-. 20 =" 10 ------
B
!
I
I t
!
I
Ii-
I
135 0B
90
.
..
45 UT
22:49
FIGURE 3.6.3 Example Reproduced from Sckopke N E, ion density NI, proton PE, and the total magnetic
50
51
52
53
54
22:55
of magnetic field and plasma measurements at the shock front. et al. (1983), copyright by AGU. From top to bottom: electron density Tp and electron TE temperatures, plasma velocity Vp, electron pressure field B.
number shocks (hereafter, we consider only quasiperpendicular shocks because quasiparallel shocks do not possess clear structure and are much more complicated). Figure 3.6.4 shows magnetic field profiles of a low Mach number- and high Mach number bow shock as measured by ISEE 1 (Russell et al. 1982). The low Mach number shock (Mms -- 2.1) is a quasiperpendicular (0Bn = 78~ lOW ,/7 = 0.1 shock with a rather smooth profile (Greenstadt et al., 1975), low level of magnetic field fluctuations both upstream and downstream. The transition from the upstream to the downstream state (ramp) is rather smooth and monotonic, and a low overshoot of the magnetic field just behind the ramp does not spoil the picture of a quiet simple front. By constrast, the high Mach number, Mms - - 4.3, shock with comparable moderate ,/7 = 0.3 and
3.6
463
Shock Waves in Space
Aug.2 ',"1978
.
.
.
.
.
.
.
l[ ,~
20 . . . . . . .
Mms = 43 [3= 01
OI
.
8BN = 78 ~ 1
1
,
2005 UT
I.
,~
I
i
2006
|
..
I
2007
|
.i
:
2008
(a)
Dec.2, 1977 40
20
J
Mms= 43
"_~. -~-----
[3= 03
.......... 0
'
2021 UT
eBN = 68 ~ l
..
//
,,,,
,
~
.._
2023
~
~
i
~
....
j.........
2025
(b) FIGURE 3.6.4 Magnetic field measured at the Earth bow shock: a) low Mach number shock observed on Aug. 27, 1978, 2007 UT; and b) high Mach number shock observed on Dec. 2, 1977, 2023 UT. Reproduced with permission from Nature, Russell et al. (1982), Copyright (1982), Macmillan Magazines Limited.
0B, = 68 ~ shock exhibits a rather well-structured front with rapid increase of the magnetic field followed by a high overshoot before the magnetic field goes down to the downstream value. It was suggested phenomenologically (Scudder et al., 1986a), that a typical shock front may be considered as a stationary 1D structure consisting of a gradual increase of the magnetic field (foot) followed by a sharp magnetic field jump (ramp) with the subsequent overshoot of the magnetic field and more or less pronounced large amplitude oscillations of the magnetic field downstream (see Fig. 3.6.5). This phenomenological shock structure was taken for granted and was used in almost all theoretical studies until recently, when it was shown that there is no single typical profile of high Mach number shocks and the dependence of the fine shock structure on the shock parameters is rather sensitive and complicated (Newbury et al., 1998). Rather obviously, observational analysis of the magnetic profile of the shock concentrated on several distinctive features, including ramp width (Morse and Greenstadt, 1976; Mellott and Greenstadt, 1984; Farris et al., 1993) (foot width
464
M. Gedalin
overshoot
ramp
v
foot
downstream
upstream z (arbitrary units) FIGURE 3.6.5
Typical high Mach number shock profile as suggested by Scudder et al. (1986a).
is believed to be satisfactorily predicted by theory, see Section 3.6.5), overshoot height (Mellott and Livesey, 1987), and noncoplanar magnetic field (Thomsen et al., 1987b; Gosling et al., 1988). The last one is absent in MHD shock theory and is directly related to the finite width of the shock front. Figure 3.6.6 shows the three components of the magnetic field of a low Mach number shock. This magnetic field is already rotated into the normal shock coordinates. The coplanarity theorem states that the upstream and downstream magnetic field vectors lie in a plane L - N (coplanarity plane), N being the direction of the shock normal. The component BN fluctuates, which means the presence of nonstationary fields. The noncoplanar component BM is naturally explained within the two-fluid hydrodynamics theory (see in what follows). The oscillations of B/_ ahead of the ramp are the phasestanding whistler precursor. The ramp width (marked with two vertical lines in the figure) is of most interest, since this is the shock part on which the main changes in the shock parameters occur. In statistical analyses the ramp width of low Mach number shocks was compared with several model length parameters that have been proposed to explain ramp width (Morse and Greenstadt, 1976; Mellott and Greenstadt, 1984; Farris et al., 1993): marginal stability length, ion acoustic length, whistler precursor wavelength 2rcccosOB,/(M 2 - 1)l/2cOpi, and ion inertial length c/O~pi. It was found that the best correlation is with whistler
3.6
465
Shock Waves in Space
40
"-"~,-'r
F
-20
!
4~ ,~
p"
- -,-- --'--' ~ -~-------~
'
o L .............................
' i.
~
t .................... J
40
i-9. 2 0 1"13
9
0
00:07 8/28/78
......
: : ...........: ...................." . . . . . . . . . . . . . . . . .
00:08
,
00:09
,,
,,,
i
I
00:10
FIGURE 3.6.6 Magnetic field vector of a low Mach number shock rotated into the normal shock coordinates LNM (N is along the shock normal and L - N is the coplanarity plane). Reproduced from Farris et al. (1993), copyright by AGU.
precursor length (even when such a precursor is not present), and that the width is always of the order of the ion inertial length. Similar analysis of high Mach number shocks encounters difficulties due to separating of spatial and temporal variations and because of the structured shock profile (Newbury et a|., 1998). In the only comprehensive study of a high Mach number shock profile (Scudder et al., 1986a,b,c) it was found that the ramp width is of the order of the whistler precursor wavelength. It was suggested the the shock analyzed by Scudder eta|. (1986a,b,c) is "typical." However, observations show (Newbury et al., 1998) that there is a rich variety of shock profiles and even shocks with apparently similar parameters may look quite different. Observationally, it can be concluded that the main transition
466
M. Gedalin
occurs at the scale of the ion inertial length C//(_Opibut in a number of shocks small-scale structure is observed with the typical scale of (0.1 - 0.2)c/copi. It is not clear at this time whether this small-scale structure is always quasistationary or time-dependent. Particle observations concentrated on the ion (Sckopke et al., 1983; Thomsen et al., 1985; Sckopke et al., 1990) and the electron (Feldman et al., 1982; Feldman, 1985; Thomsen et al., 1985; Thomsen et al., 1987; Schwartz et al., 1988) energization. Strong ion heating that starts before the ramp is shown in Fig. 3.6.3. The ion temperature increases drastically and the downstream ]~i '~ 1 even if the upstream ]~i << 1. The ion heating is related to the gyration and reflection at the shock front. Figure 3.6.7 shows the evolution of the ion distribution across the shock front. The high-energy ions appearing ahead of the ramp are ions which are reflected off the shock transition layer. These ions are again brought to the ramp by the magnetic field and motional electric field and cross it to produce the high-energy population of the downstream ion distribution, thus contributing significantly to ion heating (Sckopke et al., 1983, 1990). Electron heating is typically substantially weaker and occurs at the ramp itself. Figure 3.6.8 shows the cuts through the downstream electron distribution measured at different angles with respect to the local magnetic field. The upstream electron distribution is nearly Maxwellian. The distribution of heated electrons is fiat-topped, a typical feature for electrons heated in strong shocks (Feldman et al., 1982; Feldman, 1985). Statistical analysis of the electron heating and shock parameters has shown that heating correlates only with the energy of the incident plasma flow (Fig. 3.6.9) (Thomsen et al., 1987; Schwartz et al., 1988).
7 NOV 1977
ISEE 2 ORBIT 7 INBOUND
I oe
,) I i I
22.50.96
22.51.04
22.51.07
22.51.13
922.51.25
22.51.31
22.51.37
22.51.43
22.51.19
. . . . . . . . . . 22.51.48
i
FIGURE 3.6.7 Evolutionof ion distribution across the high-Machnumber shock front. Reproduced from Sckopkeet al. (1983), copyrightAGU.
3.6
467
Shock Waves in Space
ISEE 2
~ = -60 ~
13 Dec, 1977 150 eV
150 eV lo s _
,.
I
I
SW
MS
104
.~ 103
~5 ,-
iTi
<> 17.34 = 08.9
I
lOO
I -5
-10
I
I 0
510
I
"X"
I 5
I
I 10
I
Electron Speed (xl08 cm s-1) FIGURE 3.6.8 Shown are cuts across the downstream electron distribution at different angles with respect to the magnetic field, from solar wind (SW) to the downstream region (mangetosheath-MS). Reproduced from Feldman (1985), copyright by AGU.
25 -
Daliy Averages Range shown by Bars
2O v o
15
I-I
10-
B
5-
o_~ q-.~Q 9
0;
.z
I
1
"
t
J
i
2
J
3
i
4
V2u- V 2 (105 km2/s 2) FIGURE 3.6.9 Dependence of electron heating on the difference between the upstream and downstream plasma flow energy. Reproduced from Thomsen et al. (1987), copyright by AGU.
468
M. Gedalin
3.6.5
COLLISIONLESS
SHOCK
THEORY
The best approach to the study of shocks is one that is based on the kinetic equation. However, the mathematical difficulties are enormous, and the theory has been developing within two-fluid hydrodynamics (field structure) and single particle motion analysis (ion dynamics). Only in certain cases of electron motion is it possible to apply collisionless Liouville mapping.
3.6.5.1
FIELD S T R U C T U R E
Shock field structure is usually studied within two-fluid hydrodynamics assuming stationarity a/at - 0 and one-dimensionality 0/0y -- O/az -- 0. The corresponding equations read: n V x -- const.
(3.6.16)
d V x _ q_~SEx + qs [Vj_,s x B• x - ldPs,xx Vx dx - m s msc n dx
(3.6.17)
dV•
Wx ~
:
dx
q--LEx m s
+ q-----L-s[Vx(5;x B_L) 4- Bx(V_L x ,~)] msc dB_L
47me
--~ = -7-(V• Bx -- const,
Ey -- const,
1 dP_~,s n
(3.6.18)
dx
V_Le)
(3.6.19) E z -- 0
(3.6.20)
where s = i, e, qi = - - q e = e, ~c is the unity vector in the shock normal (x) direction, B• = 0, V l 9~----0, and P_~--(Pyx, Pzx) are the off-diagonal components of the pressure tensor. Quasineitrality n e = n i = n is explicitly assumed, which means also that Vx, e = Vx, i = V x. Further analysis of Eqs. (3.6.16)-(3.6.20) requires invoking additional assumptions.
3.6.5.2
N O N L I N E A R WAVES AND RAMP W I D T H
In the perpendicular case B = (0, 0, B(x)) of a cold plasma Pe - Pi = 0 the system equations (3.6.16)-(3.6.20) reduces to the following equation for the
3.6 ShockWaves in Space
469
magnetic field (Sagdeyer, 1966): c2
b2-1~~___b]_(1
b2-1"~ d
b2-1~
where MA --Vu/v A is the Alfven Mach number, COpe- (4nne2/me) 1/2 is the electron plasma frequency, and b = BIB o is the normalized wave amplitude. This equation has soliton solutions for MA = Vu/v A > 1. It was suggested by Sagdeev in 1966 that inclusion of weak dissipation can transform this solution profile into a shock-like profile. Unfortunately, this scenario is impossible in the oblique case, where there are no super-magnetosonic solitons (see Gedalin, 1998, and references therein). Instead, periodic large-amplitude nonlinear waves are present and are described by the following parametric representation: V2 bz - s i n O 4 - 2 # s i n O ( l _ X ) +
( 1 ) 1-~-~
cos20 ( 1- )1
sin 0(1 - Z)
4-
Ffl -2psin0(l_x)(F_l)
3cos20 2~ sin 0(1 - Z) V2
(N r -
(NF_I _ 1)
1)
(3.6.22)
and by - +(2V2(1 - l/N) + (1 - N c) + sin 2 0 - b2z)1/2
3.6.23
where lw = c cos Ovo/V~2 u, ~u = eBu/mic, and Z = cos20/V~ Obviously, the solution is limited by the conditions N > 0 and b~ > 0, which gives ]bzl _< [2V2(1 - 1/N) + fl(1 - Nr)] 1/2
(3.6.24)
It is natural to expect that the ramp width (at least of low Mach number shocks) is related to the wavelength of this large amplitude nonlinear wave. It is easy to see that in the low and moderate fl ~ 1 cases the typical scale is the whistler scale lw - c cos O/Mcopi. Numerical analysis of Eqs. 3.6.22 and 3.6.23 has shown that the typical wavelength is several whistler lengths. The nondissipative hydrodynamics (Eqs. (3.6.16)-(3.6.20)), however, cannot describe a shock profile with different asymptotical magnetic field at x --~ 4-oo. It is expected that inclusion of small dissipation would result in a shock-like profile but to date no detailed analysis has been done.
3.6.5.3
NONCOPLANAR MAGNETIC FIELD
The MHD theory of discontinuities predicts (Tidman and Krall, 1971) that the magnetic field of a shock remains in the same plane. Real shocks are not infinitesimally thin and inside the ramp there is a magnetic field excursion out
470
M. G e d a l i n
of the coplanarity plane. The appearance of this component is naturally described within two-fluid hydrodynamics (Jones and Ellison, 1987; Thomsen et al., 1987b; Gosling et al., 1988; Jones and Ellison 1991; Gedalin, 1996a; Newbury et al., 1997a). If we do not assume isotropy of ion pressure (see Section (3.6.5.4) a straightforward algebra gives the following approximate relation for the noncoplanar magnetic field in a quasiperpendicular shock front: d
By = lw - ~ B z +
Bx n m i v2
P~
(3.6.25)
The first term in this expression corresponds to the rotation of the magnetic field in the oblique whistler wave (according to the scale the shock front is a nonlinear whistler). The second term describes the contribution of the nongyrotropic ion distribution formed within the shock, and is responsible (in the stationary 1D case) for the observed deviation of the integral f Bydx from the expected value IwAB z (Jones and Ellison, 1987, 1991).
3.6.5.4
ION M O T I O N
Ion motion in the shock front can be analytically studied only within single particle analysis. The overall width of the shock transition layer is ~
~ = vy,
e
e
bx = ~ E x + ~ VyB z mi mic e
r
mi
mic
by = ~ Ey - ~
vxB z
(3.6.26) (3.6.27) (3.6.28)
are not integrable. However, one can integrate them across the thin shock transition, substituting d/dt = vx(d/dx ) to get V2x = V2xo- e o / m i + 2 1 VyD(x)dx
(3.6.29)
vy - vy o - I ( Vunu - D ( x ) ) d x \ Vx
(3.6.30)
where E x = - d o / & , D ( x ) = eBz(x)/mic, and Ey = VuBu/c. The effect of the magnetic field is to deflect the ion in the y direction and to produce deceleration in addition to the electrostatic one. These effects are proportional to the width of the layer and can be neglected in the lowest-order approxima-
3.6 ShockWaves in Space
471
tion for the ramp with width
l -~y-y--y-,g-y-y~
X
i X
FIGURE 3.6.10 From left to right: downstream gyration of an insufficiently decelerated transmitted ion, specular reflection of an ion at the ramp, and nonspecular reflection due to gyration behind the ramp.
472
M.
Gedalin
.... ..... :.,-- :.,,.'. -......... ....- ...- ....- -( ".... _
. .
....... . . . . . . . .-...:'::..:::..:'..~::: "..>f i. . . . . . . . ..i
i
i
FIGURE 3.6.11 Trajectories of initially Maxwellian distributed ions. Transmitted and nonspecularly reflected ions are clearly seen.
ahead of the ramp. The corresponding ion current is responsible for the formation of the foot--gradual increase of the magnetic field preceding the steep ramp. The foot length is easily estimated in the specular reflection model with an almost cold beam (Woods, 1971; Leroy, 1983; Gosling and Robson, 1985) in the simplest perpendicular geometry. Indeed, let the incident ion with velocity V = ( V u, 0) (the motion is in x - y plane, B is along z axis) is specularly reflected at x = 0. Immediately after the reflection the ion velocity is (-V,, 0). Equation of motion immediately gives x - - Vut - -
2V. ~sin(f~ut)
(3.6.31)
where f~u is the ion gyrofrequency in the constant upstream magnetic field at x < 0, and t is time since the reflection. The ion turns back to the shock where Vx -dx/dtO, that is, cos(f~,t)- 1. The corresponding turning distance [xt[ ~ 0.68(V,/f~u). This value has been used widely for determination of shock velocity with single spacecraft measurements. More sophisticated analyses provided the foot length for the general oblique geometry (Gosling
3.6
473
Shock Waves in Space
and Thomsen, 1985). It was shown recently that this length may be fi dependent (see Gedalin, 1997a for review and references).
3.6.5.5
ELECTRON HEATING
The electron mass is much smaller than the ion mass, which means that electrons are much more sensitive to the rapid spatial and temporal variations of the electric and magnetic field inside the shock front. However, the prevailing view is that the electron motion in the shock front is also governed by the quasistationary fields (Feldman et al., 1982; Goodrich and Scudder, 1984; Feldman, 1985; Schwartz et al., 1988; Scudder, 1995). If this is the case and the electron motion is adiabatic, v2_t_/B-- const, the collisionless Liouville mapping is straightforward in the de Hoffman-Teller frame, where energy conservation depends only on the coordinate along the shock normal. These two conservation laws give
v•2 a -- V2o,u(Bd/Bu) V•2
-Jr- V~,d ~ 122•
(3.6.32)
-+- 122ll,u q- 2eq~/me
(3.6.33)
where subscripts u and d refer to upstream and downstream, respectively, and it is assumed that the field inhomogeneity in shock front (overshoot) does not change Eqs. (3.6.32) (3.6.33). The parallel velocity vLi- v. B/[B[, and v• = x/v 2 - v~. In this case, if the upstream electron distribution is shifted Maxw~llian f,(Vll, u, v•
n,
[
2-~v2rexp -
(v
LI,,
- V~h)2 + v2~,l
(3.6.34)
2v2
where V~h- Vu/cos0B, is the upstream plasma velocity along the magnetic field in the de Hoffman-Teller frame, application of Eqs. (3.6.32) and (3.6.33) gives the Liouville mapping fa(Vll,a, V• v• where
V_L,u -- V•
1/2,
Vz, u -- x//-Q sign v•
- vll,d + v• a 1 --
-
me
(3.6.35) (3.6.36)
provided Q > 0. In this scenario, the electrons are accelerated along the magnetic field by the cross-shock electric field, while the perpendicular degree of freedom gets energy due to the adiabatic magnetic compression. Figure 3.6.12 shows the downstream electron distribution (in the v , - v• plane), which corresponds to the upstream Maxwellian. The electrons with v, > 0 cross the shock from upstream to downstream, while those with vii < 0
474
M.
Gedalin
~
o-
"".
%-..
~
I
0 vii
FIGURE 3.6.12
Downstream distribution corresponding to the upstream Maxwellian.
cross the shock from downstream to upstream. There is a large hole in the downstream electron distribution, corresponding to Q < 0. This region is inaccessible to the upstream electrons. It cannot be mapped to the upstream distribution and, therefore, should be filled due to other processes or by a preexisting downstream population of heated electrons. The downstream temperature (by an order of magnitude) would be determined by the energy of accelerated electrons, that is, cross-shock potential, although the precise value of the temperature would depend on the details of the hole filling process. Recently it was suggested (Balikhin et al., 1993; Balikhin and Gedalin, 1994; Gedalin et al., 1995) that in thin or strongly inhomogeneous shocks electrons become demagnetized in a part of the ramp. This scenario can be realized in extremely thin shocks (Newbury and Russell, 1996). As a result, these electrons are accelerated across the magnetic field and get more energy directly in their perpendicular degree of freedom. Figure 3.6.13 shows schematically adiabatic and nonadiabatic behavior. In the adiabatic cases the gyration velocity increases as ~/B. In the nonadiabatic cases the previously non-gyrating electron
3.6
475
Shock Waves in Space
X
X
FIGURE 3.6.13 Adiabatic (left) and nonadiabatic (right) electron motion in the model shock front. For simplicity perpendicular geometry is shown. The figures are not in the same scale. Units are arbitrary.
gyrates strongly behind the ramp. Although the details of the heating process differ in the adiabatic and nonadiabatic cases, difficulties with the gap filling persist and apparently cannot be resolved within the collisionless model and simple Liouville mapping.
3.6.6
SHOCK PARTICLE ACCELERATION
One of the most important roles space shocks play is that of charged particle accelerators. High Mach number supernovae shocks are believed to be responsible for producing high-energy cosmic rays. Two mechanisms are proposed for particle acceleration in shocks--shock drift acceleration, based on energizing of particles trapped in the vicinity of the shock front, and diffusive acceleration, based on the multiple scattering of particles moving long distances from the shock front.
3.6.6.1
SHOCK DRIFT ACCELERATION
Shock drift acceleration can occur in nearly perpendicular and perpendicular shocks. Energization occurs in the following way. An ion (electron acceleration is briefly discussed here) that enters the shock is stopped by the cross-shock potential and returned to the upstream region. There the ion is again returned to the shock by the motional electric field. During this cycle the ion is drifting along the shock front and acquires substantial energy due to the potential drop along the shock front caused by the motional electric field. If the normal
476
M.
Gedalin
component of the ion velocity at the entry to the shock front is sufficiently small, the ion can be reflected again and acquires additional energy so that the velocity along the shock front increases. The ion may be trapped for some time in the shock front so that its drift along the shock front is substantial, which leads to strong energization of the ion. Eventually all ions are transmitted downstream in the perpendicular shock. In the oblique shock geometry some high-energy ions penetrate downstream but a fraction of these ions escape upstream along the magnetic field lines. This type of shock drift acceleration (surfing mechanism) (Lee et al., 1996) is efficient for a small part of the upstream ion distribution and, if this distribution is sufficiently hot, in order to be reflected and trapped an ion has to have a very low initial velocity. For usual upstream Maxwellian distribution centered at the normal velocity Vu = MAY A the thermal velocity v~i should be of the order of Vu, which requires fli ~ M2A and is very high for high Mach number shocks. For low temperatures there is no low-energy ion injection in the surfing, and the mechanism is inefficient. One possibility is that another ion population with high effective thermal velocities is present. Such ions appear upstream of the cometary bow shocks, where the solar wind ionizes neutrals. These neutrals initially stand in the shock frame, so that the newborn ions move with the velocity - V u relative to the upstream plasma, thus having rather low normal velocities when they are dragged to the shock front and encounter it. A more traditional shock drift acceleration scenario (Armstrong et al., 1985) assumes multiple crossing of the shock front with the accompanying drift along the shock front. This can occur when the initial gyration velocity of the ion is sufficiently high, so that the cross-shock potential has almost no effect on the ion motion and shock crossings, as shown in Fig. 3.6.14. Although analysis shows that this process may result in ion energization, the mechanism is not especially efficient because the ion leaves the acceleration region too early. It has been shown (Decker, 1990) that the efficiency can be increased if the magnetic field downstream of the shock is no longer 1D (magnetic loops), so that additional ion reflection occurs behind the shock. Thus, the ion can spend substantially more time in the shock front, drifting in the direction of the motional electric field and acquiring energy more efficiently. If this non-ID magnetic field pattern is nonstationary (natural generalization), the shock drift acceleration mechanism becomes somewhat similar to the diffusive acceleration (see in what follows).
3 . 6 . 6 . 2 DIFFUSIVE ACCELERATION The diffusive acceleration mechanism (Axford, 1981; Forman and Webb, 1985; Scholer, 1985; Blandford and Eichler, 1987; Draine and McKee, 1993) is based
3.6
477
Shock Waves in Space
UPSTREAM
YO-
-1-
I..IJ ec I.O3 Z
0ICI
q=2
-13-
OBn =84.3 Vp - IOV B -22-
a ._1
-31 -
u..I m 1.1_ 0 re !0 I..IJ _..I
I.U
0 -45 '1
1.'4
4.s'
718
1'1 x
FIGURE 3.6.14 The x - y projection of the motion of an ion in the shock of strength (magnetic compression) 2 and 0Bn--84.3 ~ Length in upstream convective gyroradii V,/I2,. The ion was overtaken and reflected by the shock. Reproduced from Armstrong et al. (1985), copyright by
AGU. on the Fermi-acceleration (Fermi, 1949) principle, in which a particle bouncing off two wails acquires energy. In the case of a shock the upstream turbulence, drifting toward the shock with approximately plasma flow velocity, plays the role of one of the walls, while the shock front (or downstream irregularities that essentially stand in the shock frame) plays the role of the other wall. Since the return of the particle to the shock front occurs after multiple scattering with the turbulence, the process should be described by some kind of diffusion. The diffusive-kinetic equation for the particle 1D distribution function f(p) reads: of u v o of P v . v Of at~ + ~ ~cijxj - 3 ~ -- Q
(3.6.37)
where it is assumed that the particle velocity v >> V. Here p is the particle momentum, ~cijis the (momentum dependent, in general) diffusion tensor, and
478
M. Gedalin
Q is the source term, that is, rate at which particles are injected. The usual approach is to solve (Eq. (3.6.37) separately in the asymptotically homogeneous upstream and downstream regions and to take into account the shock as matching boundary where V. V ~-0 and Q r The particles most freely move along the shock normal in a parallel shock, and the formalism is the simplest in this case. In the steady state one has
0
of
of
x 3xx + v 3xx - 0
(3.6.38)
where x is along the shock normal. The boundary condition at the shock is obtained by integration of Eq. (3.6.37) over an infinitesimal region that encloses the shock, assuming that 3f/Ox and 3f/Op are continuous through the shock transition layer. Then one has: ofx - - ~p (k I -- k2)~x
_ v2) ~Of = Q
(3.6.38)
For a simple monoenergetic source function Q o( ( p - P0), Eqs. (3.6.38) and (3.6.39) allow power law solutions f o( E -~ at the shock for p > P0, where 7 = 3V1/2(V1 - - V2). Spatial dependence of the distribution function is determined by the diffusion coefficient ~c which, in general, is momentum dependent, so that the power law distribution would be observed only in the vicinity of the shock and in the downstream region. One of the problems of the described approach is that the diffusion-kinetic equation (3.6.37) can be written in this simple form only for particles whose energy is already well above the thermal energy, thus requiring preceding injection of particles from the initial thermal distribution and preliminary acceleration to energies when the diffusive mechanisms start to work. Such pre-acceleration can occur at quasiperpendicular shocks due to the shock drift acceleration mechanism.
3.6.6.3
ELECTRON ACCELERATION
The shock drift acceleration and diffusive acceleration are well developed for ions. Electrons are not accelerated equally efficiently because of the small mass. They cannot be trapped in the shock front as ions can since they easily escape along the magnetic field lines. Diffusive acceleration of electrons also has problems because Alfv~n turbulence, which is efficient in scattering ions, is inefficient in scattering electrons. It is possible that whistler turbulence develops, which can scatter electrons. Another mechanism was proposed for electron acceleration at nearly perpendicular shocks. The mechanism (Leroy and Mangeney, 1984; Wu, 1984) is based on the electron reflection off the
3.6 ShockWaves in Space
479
magnetic field jump because of the magnetic moment conservation. Since this reflection occurs in the de Hoffman-Teller frame, the reflected electron would have a velocity of ~ 2 V u tan 0B, in the normal incidence frame. For small OBn this velocity may greatly exceed the electron thermal velocity. The reflection occurs at the tail of the electron distribution equation (3.6.34), for v 2 ,u ( B a / B u - 1) > v2II,, 4- 2ec~/m e, and the reflected electron flux is small, but the energies are high. In the opposite regime of substantially oblique shocks and small V u tan O/v~e the reflected electron flux is much higher, but the energies are barely suprathermal.
3.6.7
CONCLUSIONS
Some aspects of the collisionless shock physics are not covered in this chapter. These include developments of nonstationary shock theory (see, e.g., Krasnoselskikh, 1985), wave measurements and wave features at collisionless shocks (Gurnett, 1985), and instabilities and their relation to the shock structures and ion and electron heating (Papadopoulos, 1985). We have not touched quasiparallel shocks (Greenstadt, 1985) and numerical simulations (Winske and Omidi, 1996), and nothing was said about shocks in relativistic plasmas (like a shock in a relativistic pair plasma). Physics of collisionless shocks in plasmas in general and of space shocks in particular is a very wide field. Research in this field includes studies of particle motion in various configurations of electric and magnetic fields, analysis of linear and nonlinear wave evolution, consideration of instabilities, and the interpretation of a large amount of observational data.
REFERENCES Armstrong, T.P., Pesses, M.E., and Decker, R.B. (1985). Shock drift acceleration, in Collisionless Shocks in the Heliosphere: Review of Current Research, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 35, pp. 271-285, Washington, DC: AGU. Axford, WI. (1981). Acceleration of cosmic rays by shock waves, in Proc. Int. Conf. Cosmic Rays 17th. 12: 155-203. Balikhin, M., Gedalin, M., and Petrukovich, A. (1993). New mechanism for electron heating in shocks. Phys. Rev. Lett. 70: 1259-1262. Bahkhin, M., and Gedalin, M. (1994). Kinematic mechanism for shock electron heating: comparison of theoretical results with experimental data. Geophys. Res. Let., 21: 841-844. Blandford, R., and Eichler, D. (1987). Particle acceleration at astrophysical shocks: A theory of cosmic ray origin. Phys. Rep. 154: 1-75. Burgess, D., Wilkinson, WP., and Schwartz, S.J. (1989). Ion distributions and thermalization at perpendicular and quasi-perpendicular supercritical collisionless shocks. J. Geophys. Res. 94: 8783-8792.
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Cairns, I.H., and Grabbe, C.L. (1994). Towards an MHD theory for the standoff distance of Earth's bow shock. Geophys. Res. Lett. 21: 2781-2784. de Hoffman, E, and Teller, E. (1950). Magneto-hydrodynamic shocks. Phys. Rev. 80: 692-703. Decker, R.B. (1990). Particle acceleration at shocks with surface ripples. J. Geophys. Res. 95: 11,993-12,003. Draine, B.T., and McKee, C.E (1993). Theory of interstellar shocks, Ann. Rev. Astron. Astrophys. 31: 373-432. Fairfield, D.H. (1971). Average and unusual locations of the Earth's magnetopause and bow shock. J. Geophys. Res. 76: 6700-6716. Farris, M.H., and Russell, C.T. (1994). Determining the standoff distance of the bow shock: Mach number dependence and use of models. J. Geophys. Res. 99: 17681-17689. Farris, M.H., Russell, C.T., and Thomsen, M.F (1993).Magnetic structure of the low beta, quasiperpendicular shock. J. Geophys. Res. 98: 15,285-15,294. Feldman, W.C. (1985). Electron velocity distributions near collisionless shocks, in Collisionless Shocks in the Heliosphere: Reviews of Current Research, R.G. Stone, and B.T. Tsurutani eds., Geophys. Monogr. Ser., vol. 35, pp. 195-205, Washington, DC: AGU. Feldman, W.C., Bame, S.J., Gary, S.P., Gosling, J.T., McComas, D., Thomsen, M.F, Paschmann, G., Sckopke, N., Hoppe, M.M., and Russell, C.T. (1982). Electron heating within the earth's bow shock. Phys. Rev. Lett. 49: 199-201. Fermi, E. (1949). On the origin of the cosmic radiation. Phys. Rev. 75: 1169-1174. Fromisano, V. (1979). Orientations and shape of the Earth's bow shock in three dimensions. Planet, Sp. Sci. 27: 1151-1161. Formisano, V. (1982). Measurement of the potential drop across the earth's collisionless bow shock. Geophys. Res. Lett. 9: 1033-1036. Forman, M.A., and Webb, G.M. (1985). Acceleration of energetic particles, in Collisionless Shocks in the Heliosphere: A Tutorial Review, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 34, pp. 91-114, Washington, DC:AGU. Gedalin, M. (1996a). Noncoplanar magnetic field in the collisionless shock front. J. Geophys. Res. 101: 11,153-11,156. Gedalin, M. (1996b). Transmitted ions and ion heating in nearly-perpendicular low-Mach number shocks. J. Geophys. Res. 101: 15,569-15,578. Gedalin, M. (1996c). Ion reflection at the shock front revisited. J. Geophys. Res. 101, 4871-4878. Gedalin, M. (1997a). Ion distributions at the quasiperpendicular collisionless shock front. Surveys in Geophysics, 18: 541-566. Gedalin, M. (1997b). Ion heating in oblique low-Mach number shocks. Geophys. Res. Lett. 24: 2511-2514. Gedalin, M. (1998). Low-frequency nonlinear stationary waves and fast shocks: Hydrodynamical description. Phys. Plasmas, 5: 127-132. Gedalin, M., Gedalin, K., Balikhan, M., and Krasnoselskikh, V.V. (1995). Demagnetization of electrons in the electromagnetic field structure, typical for oblique collisionless shock front. J. Geophys. Res. 100: 9481-9488. Goodrich, C.C., and Scudder, J.D. (1984). The adiabatic energy change of plasma electrons and the frame dependence of the cross shock potential at collisionless magnetosonic shock waves. J. Geophys. Res. 89: 6654-6662. Gosling, J.T., and Robson, A.E. (1985). Ion reflection, gyration, and dissipation at supercritical shocks, in Collisionless Shocks in the Hemisphere: Reviews of Current Research, R.G. Stone, and B.T. Tsurutani eds., Geophys. Monogr. Ser., vol. 35, pp. 141-152, Washington, DC: AGU. Gosling, J.T., and Thomsen, M.E (1985). Specularly reflected ions, shock foot thicknesses, and shock velocity determinations in space. J. Geophys. Res. 90: 9893-9896.
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Gosling, J.T., Winske, D., and Thomsen, M.E (1988). Noncoplanar magnetic fields at collisionless shocks: A test of a new approach. J. Geophys. Res. 93: 2735-2738. Greenstadt, E.W (1985). Oblique, parallel, and quasi-parallel morphology of collisionless shocks, in Collisiontess Shocks in the Hemisphere: Reviews of Current Research, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 35, pp. 169-184, Washington, DC: AGU. Greenstadt, E.W., Russell, C.T., Scarf, EL., Formisano, V., and Neugebauer, M. (1975). Structure of the quasi-perpendicular laminar bow shock, J. Geophys. Res. 80: 502-514. Gumett, D.A. (1985). Plasma waves and instabilities, in Collisionless Shocks in the Hemisphere: Reviews of Current Research, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Set., vol. 35, pp. 207-224, Washington, DC: AGU. Jones, EC., and Ellison, D.C. (1987). Noncoplanar magnetic fields, shock potentials, and ion deflection. J. Geophys. Res. 92: 11,205-11,207. Jones, EC., and Ellison, D.C. (1991). The plasma physics of shock acceleration. Space Sci. Rev. 58: 259-346. Kennel, C.E (1988). Shock structure in classical magnetohydrodynamics. J. Geophys. Res. 93: 8545-8557. Kennel, C.E, Edmiston, J.P., and Hada, T. (1985). A quarter century of collisionless shock research in Collisionless Shocks in the Heliosphere: Review of Current Research, R.G.Stone and B.T. Tsurutani, eds., Geophys. Mongr. Ser., vol. 35, pp. 1-36, Washington, DC: AGU. Krasnoselskikh, V. (1985). Nonlinear motions of a plasma across a magnetic field. Sov. Phys. JETP, Engl. Transl. 62: 282-288. Landau, L.D., and Lifshitz, E.M. (1960). Electrodynamics of Continuous Media, Oxford: Pergamon Press. Lee, M.A., Shapiro, V.D., and Sagdeyev, R.Z. (1996). Pickup ion energization by shock surfing. J. Geophys. Res. 101: 4777-4789. Leroy, M.M. (1983). Structure of perpendicular shocks in collisionless plasma. Phys. Fluids, 26: 2742-2753. Leroy, M.M., and Mangeney, A. (1984). A theory of energization of solar wind electrons by the Earth's bow shock. Ann. Geophys. 2: 449-456. Mellott, M.M., and Greenstadt, E.W. (1984). The structure of oblique subcritical bow shocks: ISEE 1 and 2 observations. J. Geophys. Res. 89: 2151-2161. Mellott, M.M., and Livesey, WA. (1987). Shock overshoots revisited. J. Geophys. Res. 92: 13,66113,665. Morse, D.L., and Greenstadt, E.W. (1976). Thickness of magnetic structures associated with the earth's bow shock. J. Geophys. Res. 81: 1791-1793. Newbury, J.A., and Russell, C.T. (1996). Observations of a very thin collisionless shock. Geophys. Res. Lett. 23: 781-784. Newbury, J.A., Russell, C.T., and Gedalin, M. (1997a). The determination of shock ramp width using the noncoplanar magnetic field component. Geophys. Res. Lett. 24: 1975-1978. Newbury, J.A., Russell, C.T., and Gedalin, M. (1998). The ramp widths of high-Mach-number, quasi-perpendicular collisionless shocks. J. Geophys. Res. 103: 29,581-29,593. Papadopoulos, K. (1985). Microinstabilities and anomalous transport, in Collisionless Shocks in the Heliosphere: A Tutorial Review, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 34, pp. 59-90, Washington, DC: AGU. Petrinec, S.M., and Russell, C.T. (1997). Hydrodynamic and MHD equations across the bow shock and along the surfaces of planetary obstacles. Sp. Sci. Rev. 79: 757-791. Russell, C.T. (1985). Planetary bow shocks, in Collisionless Shocks in the Heliosphere: Reviews of Current Research, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 35, pp. 109130, Washington, DC: AGU.
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Russell, C.T., and Greenstadt, E.W (1979). Initial ISEE magnetometer results: Shock observations. Space Sci. Rev. 23: 3-37. Russell, C.T., Hoppe, M.M., Liversey, WA., Gosling, J.T., and Bame, S.J. (1982). ISEE-1 and -2 observations of laminar bow shocks: Velocity and thickness. Geophys. Res. Lett. 9: 11711174. Sagdeev, R.Z. (1966). Cooperative phenomena and shock waves in collisionless plasmas. Rev. Plasma Phys. 4: 23-45. Scholer, M. (1985). Diffusive acceleration, in Collisionless Shocks in the Heliosphere: Review of Current Research, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 35, pp. 289301, Washington, DC: AGU. Schwartz, S.J., Thomsen, M.E, Bame, S.J., and Stansbury, J. (1988). Electron heating and the potential jump across fast mode shocks. J. Geophys. Res. 93: 12,923-12,931. Sckopke, N., Paschmann, G., Bame, S.J., Gosling, J.T., and Russell, C.T. (1983). Evolution of ion distributions across the nearly perpendicular bow shock: Specularly and nonspecularly reflected-gyrating ions. J. Geophys. Res. 88: 6121-6136. Sckopke, N., Paschmann, G., Brinca, A.L., Carlson, C.W, and L~ihr, H. (1990). Ion thermalization in quasi-perpendicular shocks involving reflected ions. J. Geophys. Res. 95: 6337-6352. Scudder, J.D. (1995). A review of the physics of electron heating at collisionless shocks, Adv. Space Res. 15(8/9): 181-223. Scudder, J.D., Mangeney, A., Lancombe, C., Harvey, C.C., Aggson, T.L., Anderson, R.R., Gosling, J.T., Paschmann, G., and Russell, C.T. (1986a). The resolved layer of a collisionless, high ]~, supercritical, quasi-perpendicular shock wave, 1, Rankine-Hugoniot geometry, currents, and stationarity. J. Geophys. Res. 91: 11,019-11,052. Scudder, J.D., Mangeney, A., Lacombe, C., Harvey, C.C., and Aggson, T.L. (1986b). The resolved layer of a collisionless, high ]~, supercritical, quasi-perpendicular shock wave, 2, Dissipative fluid electrodynamics, and stationarity. J. Geophys. Res. 91: 11,053-11,073. Scudder, J.D., Mangeney, A., Lacombe, C., Harvey, C.C., Wu, C.S., and Anderson, R.R. (1986c). The resolved layer of a collisionless, high ]~, supercritical, quasi-perpendicular shock wave, 3, Vlasov electrodynamics. J. Geophys. Res. 91: 11,075-11,097. Spreiter, J.R., and Stahara, S.S. (1985). Magnetohydrodynamic and gasdynamic theories for planetary bow waves, in Collisionless Shocks in the Heliosphere: Reviews of Current Research, R.G. Stone, and B.T. Tsurutani, eds., Geophys. Monogr. Ser., vol. 35, pp. 85-107, Washington, DC: AGU. Spreiter, J.R., and Stahara, S.S. (1995). The location of planetary bow shocks: A critical overview of theory and observations. Adv. Sp. Res. 15(8/9): 433-449. Spreiter, J.R., Summers, A.L., and Alksne, A.Y. (1966). Hydromagnetic flows around the magnetosphere. Planet. Sp. Sci. 14: 223-253. Thomsen, M.E, Gosling, J.T., Bame, S.J., and Mellott, M.M. (1985). Ion and electron heating at collisionless shocks near the critical Mach number. J. Geophys. Res. 90: 137-148. Thomsen, M.E, Gosling, J.T., Bame, S.J., Quest, K.B., Winske, D., Livesey, W.A., and Russell, C.T. (1987b). On the noncoplanarity of the magnetic field within a fast collisionless shock. J. Geophys. Res. 92: 2305-2314. Thomsen, M.E, Mellott, M.M., Stansbury, J.A., Bame, S.J., Gosling, J.T., and Russell, C.T. (1987). Strong electron heating at the Earth's bow shock. J. Geophys. Res. 92: 10,119-10,124. Tidman, D.A., and Krall, N.A. (1971). Shock Waves in Collisionless Plasma, New York: Wiley Interscience. Wilkinson, W.P., and Schwartz, S.J. (1990). Parametric dependence of the density of specularly reflected ions at quasiperpendicular collisionless shocks. Planet. Sp. Sci. 38: 419-435. Winske, D., and Omidi, N. (1996). A nonspecialist's guide to kinetic simulations of space plasmas. J. Geophys. Res. 101: 17,287-17,303.
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Shock Waves in Space
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Woods, L.C. (1971). On double structured, perpendicular, magneto-plasma shock waves. J. Plasma Phys. 13: 289-302. Wu, C.S. (1984). A fast Fermi process: Energetic electrons accelerated by a nearly perpendicular bow shocks. J. Geophys. Res. 89: 8857-8862. Wygant, J.R., Bensadoun, M., and Mozer, EC. (1987). Electric field measurements at subcritical, oblique bow shock crossings. J. Geophys. Res. 92: 11,109-11,121.
CHAPTER
3
.7
Theory of Shock Waves 3.7
Geometrical Shock Dynamics
ZHAO-YUAN HAN AND XIE-ZHEN YIN Department of Modern Mechanics, University of Science and Technology of China
3.7.1 Shock Wave Propagation through Quiescent Gases--Fundamental Concepts and Theoretical Basis 3.7.1.1 Chester-Chisnell-Whitham Relation 3.7.1.2 Free-Propagation Assumption 3.7.1.30rthogonal Curvilinear Coordinate System 3.7.1.4 Two-Dimensional Shock Diffraction 3.7.1.5 Three-Dimensional Shock Wave Diffraction 3.7.1.6 Diffraction of Shock Waves Propagating into Nonuniform Quiescent Gases 3.7.2 Shock Waves Propagation through Moving Gases 3.7.2.1 Shock Waves Propagation through Uniform Flow Fields 3.7.2.2 Shock Wave Propagation through Nonuniform Flow Fields References Geometrical shock dynamics is a special method that can be used for calculating and analyzing shock wave-related phenomena, such as diffraction, propagation, reflection, refraction, and interaction of shock waves. The key problem for the method is diffraction of shock waves. On the basis of gasdynamic fundamental equations, geometrical shock dynamics establishes its own governing equations and theoretical system, which are different from gasdynamics. The main difference between shock dynamics (for convenience the term "geometrical" is omitted here) and gasdynamics is that for the former, the objective is to concentrate on the shock front (shock surface), while for the latter, the objective is the whole flow field (of course, this includes the shock waves). Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All fights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
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z-Y Han and X-Z Yin
Since shock dynamics studies only shock waves, that is, strengths, orientations and shapes of shock waves, it is a simple, useful, and fast method for solving shock wave problems. In addition, by using the concept of disturbance propagating along the shock surface, it is possible to analyze some shock phenomenamfor example, what is the reason for the formation of a curved shock and which part of a shock is stronger, etc.? Shock dynamics may be divided into two parts. 9 Part 1 explains a shock wave propagating through a quiescent gas, including a uniform gas and a non-uniform gas. 9 Part 2 explains a shock wave propagating through a moving gas, including a uniform flow and a nonuniform flow.
3.7.1 SHOCK WAVE PROPAGATION THROUGH QUIESCENT G A S E S - FUNDAMENTAL CONCEPTS AND THEORETICAL BASIS 3 . 7 . 1 . 1 CHESTER-CHISNELL-WHITHAM RELATION When a shock wave propagates through a varying cross-sectional area tube, the strength of the shock wave will change with the cross-sectional area of the tube. Chester (1954), Chisnell (1957), and Whitham (1958) obtained the same relation between the shock wave Mach number (or pressure ratio across it) and the cross-sectional area of the tube under the condition of a uniform quiescent gas ahead of the shock wave by using three different methods. The relation is:
2MdM (M 2 - 1)K(M)
dA
-F~=0 A
(3.7.1)
where M is the shock wave Mach number, A is the cross-sectional area of the tube, and K(M) is a slowly varying function, which is expressed given by:
K(M)---- 2(2p 4- 1 4-M-2)-1 1 -~ 7 4- 1 and
[
(~ - 1)M2 + 2 ]
p
3.7 GeometricalShock Dynamics
487
For strong shock waves (M ---> oo) and K(M) = 0.3941(7 = 1.4), and for weak shock waves (M --> 1) and K(M) - 0.5. Equation (3.7.1), which is known as the C.C.W. relation, is the basis of shock dynamics. Milton (1975) and others provided modified or simplified relations. Although Chester, Chisnell, and Whitham used different methods to obtain the same relation, their starting point for the derivation was based on the same assumption, that is, the assumption of "free-propagation."
3 . 7 . 1 . 2 FREE-PROPAGATION ASSUMPTION We know that when a moving shock wave propagates through a tube of varying cross section, the shock wave itself and the flow field in the region behind it are disturbed by the wall surface of the tube. The change in the strength of the shock wave will generate some disturbances along a direction opposite that of the shock wave propagation. These disturbances propagate into the nonuniform region behind the shock wave. Such kinds of disturbances include pressure waves, and contact surfaces and their interactions, which generate new disturbance waves. While such kinds of disturbance travel through the nonuniform flow field of a tube of varying cross section, some reflected disturbances in the nonuniform region will be generated. The disturbances, which are moving in the same direction as the moving shock, will overtake the moving shock and thereby change its strength. This is the original shock/flow field phenomenon. There are three types of disturbances: 9 First, the pressure waves, which propagate through the nonuniform region behind the shock, generate transmitted and reflected pressure waves. The transmitted pressure waves move in the same direction as the shock and may overtake it. 9 Second, the contact surfaces, which travel through the nonuniform region behind the shock wave, also generate forward and backward pressure waves. The forward pressure waves may overtake the shock wave. 9 Third, the interaction of the pressure waves with the contact surface generates the transmitted and reflected pressure waves. The transmitted wave is the third type of disturbance that may overtake the shock. Some calculations show that the three types of disturbances may eliminate one another to some extent, that is, the combined effect of the three types of disturbance waves on the moving shock may be diminished. It is therefore assumed that we may neglect the interaction of the three types of disturbances
488
z-Y Han and X-Z Yin
with the shock, this makes the calculation of the shock diffraction and propagation simpler. This assumption is the basis of C.C.W. relation.
3.7.1.30RTHOGONAL CURVILINEAR C O O R D I N A T E SYSTEM As already mentioned, the C.C.W. relation (Eq. (3.7.1)) is only suitable for a moving shock propagating through a tube with varying cross section. Obviously, this is a one-dimensional (1D) problem of concern only for the change in the shock wave strength along a tube with solid walls. The relation is not of concern regarding the shape and orientation of a curved shock. Thus, the treatment and analysis of shock dynamic phenomena are limited by using only relation equation (3.7.1). In order to extend the forementioned idea to calculate and analyze two(2D) and three-dimensional (3D) shock wave diffraction and propagation phenomena, it is necessary to establish 2D and 3D shock dynamic equations. How are these equations to be set up? First, we need to generate many "tubes" in the 2D and 3D flow field and then establish the shock dynamic equations relating the "shock wave surface" and the "tube." Whitham (1957) established 2D shock dynamic equations for a shock wave propagating into a quiescent gas and obtained the relations for disturbances propagating along shock wave surfaces. The 2D equations are established on the basis of an orthogonal curvilinear coordinate system. The orthogonal curvilinear coordinate system consists of rays and positions occupied by a moving shock at successive times as shown in Fig. 3.7.1. The rays are regarded as the orthogonal trajectories of the curved shock at successive times (a rigorous definition of rays will be given in Section 3.7.2.1). The thick lines denote the positions of a shock wave, and the thin lines denote the rays. The curvilinear coordinates (~, ~) are introduced in the following. The shock positions are represented by the curves a = constant, and the rays are represented by the curves fl--constant. Since the positions of the shock wave depend on the time, ~ and t have a definite relation; thus the following relation can be taken: = t
(3.7.2)
= a 1t
(3.7.3)
or
a 1 is the speed of sound in the uniform quiescent region ahead of the shock wave.
where
3.7 GeometricalShock Dynamics
489
FIGURE 3.7.1 Positionsoccupied by a shock wave at successive times and corresponding rays. The rays and the positions occupied by the shock wave at different times construct the network. The line elements for the networks can be expressed as Mdo~ and Adfl. The expression Mdo~ represents the distance along a ray between the shock wave positions denoted by ~ and ~ 4- d~ and Adfl represents the distance along a curved shock wave between two rays denoted by /~ and /~ 4- d/~. The M and A are the coefficients for the correspondent line elements Md~ and Ad,8. If the relation equation (3.7.3) is used; Mdo~ can be written as
Mdo~ = Ma l dt = W~dt
(3.7.4)
Because Mdo~ is the distance through which the shock propagates, W s must be the shock speed and M must be the shock Mach number. The two neighboring rays construct a tube or a channel, which is called a ray tube. The C.C.W. relation equation (3.7.1) can be used for the ray tubes, so as to extend the method already mentioned here to 2D flows. The width or the area of a ray tube is given as Ad,8. Since d/~--constant along a ray tube, A must be proportional to the width or the area of ray tube.
3 . 7 . 1 . 4 Two-DIMENSIONAL SHOCK DIFFRACTION 3.7.1.4.1 Two-Dimensional Shock Dynamics Equations in a Curvilinear C o o r d i n a t e System On the basis of orthogonal curvilinear coordinate system, a set of 2D shock dynamic equations can be established. The equations consist of the so-called geometrical relations and the area relation. The geometrical relations may be
490
z-Y Han and X-Z Yin
regarded as the kinematics relations and the area relation, which in fact is the C.C.W. relation and is suitable for any ray tube in 2D flow fields, can be regarded as the kinetic relation.
3.7.1.4.1.1 Geometrical Relations The network of rays and positions occupied by the moving shock wave at the time interval between t and t + dt is shown in Fig. 3.7.2. From the network we can obtain
Q R - PS
0 -- ~
8~
1 ~A
= ~
PQ
M~
= ----3fl M fi~
(3.7.5)
or 1 0.4
O0 =
(3.7.6)
M
where 0 is called the wave angle, that is, the angle between the normal to the shock wave surface and the x-axis. Obviously, 0 is equal to the angle between the ray direction and the x-axis. Similarly, one can obtain another relation from the foregoing described network.
80 1 8M = - - : - 8 "~Ap O--~
P
S
Shock front R
(3.7.7)
P 0
"
Ray tube
FIGURE 3.7.2 Schematicillustration of the network consisting of rays and positions occupied by the shock wave at the time interval fit.
3.7 GeometricalShock Dynamics
491
3.7.1.4.1.2 Area Relation for Ray Tubes Under the condition of 2D flows, the C.C.W. Eq. (3.7.1) is valid for any ray tube. In this case, the area relation assumes the following form
1 8A
2M
A O~
OM
(3.7.8)
(M2 - 1)K(M) 8~
The partial differentials of A and M with respect to a represent in Eq. (3.7.8) the change in A and M along a ray tube ( f l - constant). Summarizing the preceding relations, a set of 2D shock dynamic equations can be written as follows: 00
1 8A
Off
M O~
O0 Oe
10M A Off
(3.7.9)
2M OM (M2 - 1)K(M) O~
1 0A A O~
3.7.1.4.2 Disturbances Propagation on the Shock Wave Surface In the analysis of geometrical shock dynamics, the change in the strength, orientation and shape of a shock wave that moves forward can be regarded as a result of the disturbance waves that propagate along it. Substituting the third of Eqs. (3.7.9) into the first one results in
O0 1 dA 8M = 0
Off
M dM O~
(3.7.10)
00 1 0M ~ + ~ ~ =0
8~
A(M) Off
A concept of speed of disturbance c(M) is induced into Eq. (3.7.10). Rearranging the equation yields
(00
8_~~)
A'(M) 8M
(80
8~)
A'(M)8M
-~-~- c
l OM
--0
l OM --0 + c----~ O~ ~ A Off
(3.7.11)
492
z-Y Han and X-Z Yin
As A and c are a function of M, dM/Ac is a total differential, and the following relations can be obtained
0 IdM _ 10M 0-~ Ac--A---~O--~
(3.7.12)
I ~dM _- 1 0M
(3.7.13)
and
i9~ Ac
Ac O~
Substituting Eqs. (3.7.12) and (3.7.13) into Eq. (3.7.11) results in 4- c
4- ~ 4- c
-~c = 0
(3.7.14a)
+ ~-c
~=0
(3.7.14b)
and
~-c or
(3.7.15a) and
to
(3.7.15b)
Equations (3.7.15a) and (3.7.15b) can also be written in the following form 0+
~
iaM
= constant along
~a~
c(M)
(3.7.16a)
dfl = -c(M)
(3.7.16b)
and 0-
IdM ~=
constant along
da
where c(M) = ~/-M/A~(M)A is the speed of the disturbance. The speed of disturbance can be rewritten as
c(M) - ~,
(M2 - 1)/((w)
(3.7.17)
3.7
493
Geometrical Shock Dynamics
In fact, the true speed of the disturbance propagating along the shock front is W a, which is expressed as
Ad~ ~/~ Wcl -- dt - a l A C - al (M2 - 1)K(M)
(3.7.18)
It follows from Eqs. (3.7.17) and (3.7.18) that the stronger the moving shock wave, the faster the disturbance wave on the shock front propagates.
3.7.1.4.3 2-D Shock Dynamics Equations in a Rectangular Coordinate System
3.7.1.4.3.1 Transformation of Curvilinear Orthogonal Coordinates to Rectangular Coordinates The functional relations between the curvilinear orthogonal coordinates (~, ]~) and the rectangular coordinates (x, y) can be expressed as ~ = ~(x,y)
(3.7.19)
]~ -- ]~(x, y)
(3.7.20)
and
which can also be expressed in the following differential form 0~ 0~ de- ~dx+~dy
(3.7.21)
d~ - -~ dx + -~ dy
(3.7.22)
and
There is a point P in two-dimensional flow fields, which can be expressed by curvilinear and rectangular coordinate systems as shown in Fig. 3.7.3. In the vicinity of point P on the figure, we have
Mdo~ = dx cos 0 + dy sin 0
(3.7.23)
Ad~ = - d x sin 0 + dy cos 0
(3.7.24)
d~ = cos 0 dx + sin 0 dy
(3.7.25)
and
Ol"
--if-
494
Z-Y Han and X-Z Yin
Y
I
,B ........
dx
p
Direction for ray
0 FIGURE 3.7.3
x
The relationship between the two coordinate systems in the vicinity of point P.
and O cos 0 d]~ = - sin_____:dx + 9dy A A
(3.7.26)
Comparing Eqs. (3.7.25) and (3.7.26) with Eqs. (3.7.21) and (3.7.22), respectively, we can obtain the differential relations between the two coordinate systems ~0~
COS 0
0x
M
Oct
sin 0
~y
M
(3.7.27)
and 3/~ 0x
sin 0 A
~/~
cos0
0y
A
(3.7.28)
Inversely, we can obtain 0x 8x
= M cos 0 (3.7.29) = - A sin 0
3.7 GeometricalShock Dynamics
495
and 8y = MsinO 8~ Oy =
(3.7.30)
A cos 0
3.7.1.4.3.2 Description of Disturbance Propagation in Rectangular Coordinate System The disturbance propagation in the curvilinear coordinate system already mentioned here is expressed by
d_._~= +c(M) d0c
(3.7.31)
which is also an equation for the characteristic curves. In order to express the relation for the disturbance propagation along the shock front in a rectangular coordinate system, Eqs. (3.7.19) and (3.7.20) can be written in an inverse form x = x(o~, 13)
(3.7.32)
y = y(ce, 13)
(3.7.33)
and
Whereas 0~ and ]3 in the preceding equation are independent, the relations given represent the relation equation between the two surfaces. If one wants to establish the relation between the curves in the two coordinate systems, Eqs. (3.7.32) and (3.7.33) should be written in the following form x = x[oc,/~(o0]
(3.7.34)
y = y[~,/~(o0]
(3.7.35)
and
which means that 0c and ]~ are not independent. If 13 = ]3(o0 is a family of characteristic curves, its differential equation is
cl~/cl~ = c(M), or cl~/cl~ = - c ( M ) . The differential relation for the characteristic curves in rectangular coordinate system may be written in the following form dy/dx = f(M). The derivation of this relation is given by
dy = dyldo~ a,,
_
Oylao~ + (OylO,6)(d,61doO ox/O +
(3.7.36)
496
z-r Han and X-Z Yin
Substituting Eqs. (3.7.36) results in
(3.7.29)
and
and dfl/do~ = +c(M) into Eq.
(3.7.30)
dy M sin 0 + A cos 0 . c sin 0 + cos 0 . (Ac/M) -- = = dx M cos 0 - A sin 0. c cos 0 - sin 0 . (Ac/M)
(3.7.37)
Letting tan v - Ac/M Eq. (3.7.37) can be written as
dy dx
=
sin 0 + cos 0. tan v cos 0 - sin 0 9 tan v
= tan(O + v)
(3.7.38)
Similarly, we can obtain the relation corresponding to dfl/da- -c(M) as
dv
- J = t a n ( 0 - v) dx
(3.7.39)
3.7.1.4.3.3 Two-dimensional Shock Dynamics Equations in a Rectangular Coordinate System The relation between the two coordinate systems can be rewritten as follows x -- x(~x, fl)
(3.7.40)
y -- y(o~, ,8)
(3.7.41)
and
and the corresponding differential relations as
0~
0Ox
0Oy
OOx
OOy
---- = ax0~ 0y0~
(3.7.42)
and
o
3--fi = 3xx3---fi+ ~ 3--fi
(3.7.43)
Substituting Eqs. (3.7.29) and (3.7.30) into Eqs. (3.7.42) and (3.7.43) yields 3 3 3 Oa = M cos 0 3xx + M sin 0 -x-oy
(3.7.44)
3 3 3 O--fi= - A sin 0 3xx + A cos 0
(3.7.45)
and
3.7 GeometricalShock Dynamics
497
Substituting Eqs. (3.7.44) and (3.7.45) into Eq. (3.7.10) results in the following geometrical equations in a rectangular coordinate system
~x~(~A ~) ~ ~ (~~0) _ o (3.7.46)
~x~(~~-~)~~ (~0) 0 The area relation along a ray tube in a rectangular coordinate system is then derived. Equations (3.7.40) and (3.7.41) under the condition of f l - c o n s t a n t (along a ray tube), can be expressed as x -- x(oO
(3.7.47)
y --y(~)
(3.7.48)
and
with the corresponding differential relations as
~
=
] O x d ~ ~yd~
(3.7.49)
Substituting Eq. (3.7.49) into Eq. (3.7.8), we get
l (OA dx OAdy~ _ A -~x d--~4- ~d-~] Dividing Eq. (3.7.50) by
A
2M
(OM dx SM dy)
(M 2 - 1)K(M) ~-~-xd-~4- -~-d-~
(3.7.50)
(dx/dcz) yields (M2 - 1)K(M)
(3.7.51)
Obviously, dy/dx in Eq. (3.7.51) for the ray tube is equal to tan0 and therefore the area relation can be written as
A -~x 4- tan 0
-- (M 2 - 1)K(M) -~x 4- tan 0 ~
(3.7.52)
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Z-Y Han and X-Z Yin
Summarizing the preceding geometrical equations and the area equation, one gets
ts~ 0) = o (s~ 0) - ~ 0 (c 0) = o +~
a
-~x + tan 0 - -
= (M2 _ 1)K(M)
(3.7.53)
-~x + tan 0
In the characteristic Eqs. (3.7.16a) and (3.7.16b) the expressions for the characteristic Eqs. (3.7.38) and (3.7.39) are used instead of the expression in the curvilinear coordinate system equation (3.7.31) and one can obtain the characteristic relations in the rectangular coordinate system 0+
JdM -~c -- constant along
~dy=
tan(O+ v)
(3.7.54)
0-
IdM ~cc = constant along
dy= ~
t a n ( 0 - v)
(3.7.55)
and
We can also derive the forementioned characteristic equations directly from Eqs. (3.7.53).
3.7.1.4.4 Two-Dimensional Shock Diffraction When a plane shock wave propagates through a channel having curved walls as shown in Fig. 3.7.4, the plane shock wave is disturbed by a series of
FIGURE 3.7.4 A plane shock disturbed by the disturbance wave generated on upper and lower surfaces.
3.7
499
Geometrical Shock Dynamics
disturbance waves that are generated on the upper and lower surfaces of a curved wall. This phenomenon is called shock wave diffraction. It can be classified as shock wave diffraction along a convex and a concave wall. This phenomenon is discussed here from the shock dynamic viewpoint. The disturbance waves, which are generated continuously on the wall surface, are called shock expansion in the case of the diffraction of a plane shock wave along a convex wall. They propagate on the shock front, and decrease its strength. They also change both the orientation and shape of the shock wave. The disturbance waves, which are generated continuously on the wall surface, are called shock compression in the case of the diffraction of a plane shock wave along a concave wall. They propagate on the shock front, and increase its strength. It should be noted that the shock-compression propagating along the shock front does not last for very long, because its behavior will change when a disturbance wave is overtaken by a following disturbance wave. In this case, the characteristic relations equations (3.7.54) and (3.7.55) cannot be used. A new type of disturbance wave, a shock-shock, which is a disturbance wave like a shock discontinuity propagating along the shock front, is generated. When the shock-shock propagates to some location on the shock front, the strength and orientation of the disturbed part of the shock change suddenly. The forementioned disturbances are discussed in detail in the following material. 3.7.1.4.4.1 Shock Expansion and Shock Compression
In the general case, a plane shock traveling through a 2D channel or tube is disturbed by the disturbance waves that propagate both upwards and downwards. In order to discuss the behavior of the undisturbed and the disturbed shock waves in more detail, we consider only the case of a simple wave. In the case of a simple wave, either upward or downward waves, but not both, are present on the shock front. Considering the case of an upward simple wave, Eqs. (3.7.54) and (3.7.55) can be written in the following simple form: 0+
I
M -dM ~ c - constant
along dy _ tan(O+ v)
(3.7.56)
1
0-
i
~adM ~ = constant everywhere
(3.7.57)
1
Combining Eqs. (3.7.56) and (3.7.57) results in 0 = constant and M - constant along ~dy-
tan(0 + v)
(3.7.58)
500
Z-Y Han and X-Z Yin
We know from Eqs. (3.7.37) and (3.7.38) that tan v = 1 ~ l ( M 2 _ 1)K(M)
(3.7.59)
From Eq. (3.7.58) and (3.7.59), we have v = constant along ~dy-
tan(O + v)
(3.7.60)
therefore dy =tan(O + v) - constant
(3.7.61)
Y - Yw = (x - xw) tan(O + v)
(3.7.62)
dx
or
where the subscript w is the value of the wall surface. It follows from Eqs. (3.7.61) and (3.7.62) that the family of characteristics for upward waves is a set of straight lines. It should be noted that the relation between the strength (M) and the orientation (0) along the disturbed shock front (across upward simple disturbance waves) is not Eq. (3.7.56) but Eq. (3.7.57). According to the initial and boundary conditions as shown in Fig. 3.7.5, M = M 0 and O0 = 0 for 0c=0 and 0 < f l < o o " and O - O w for / / - - 0 and 0<~
FIGURE 3.7.5
I
-
-
1 Ac
= 00 -
-
-
Ac
A plane shock disturbed by a simple wave.
(3.7.63)
3.7 Geometrical Shock Dynamics
501
or I S dM
0=
(3.7.64)
Ac
0
Summarizing Eqs. (3.7.62) and (3.7.64), we have
O-
0o -
o
M 2 _ 1)K(M)
(3.7.65)
Y - Yw = ( x - Xw) tan(O + v)
It follows from Fig. 3.7.5 and Eq. (3.7.58) that strength (M) and orientation (0) at points a s, b s, cs, d s . . . on the shock surface are equal to those at points a w, bw, cw, dw... on the wall surface, respectively. Therefore, the shape of the disturbed shock is similar, qualitatively, to the shape of the wall surface. In the case of the diffraction of a plane shock around a convex sharp c o m e r as shown in Fig. 3.7.6, we have the following relations
tan v, - -~
(M~ - 1)K(Mi)
(3.7.66)
#
~ = tan coi - t a n ( 0 / + vi) The strength and orientation of the plane shock are given as M o and 0 o = 0 and the deflected angle of the wall surface is given as 0 w.
Plane moving shock
I-M~ ,, Oo=O
y
l
Mo s
, ,~ First sturbance ~ line
/
Last
\~-~~,.~sturbance 9 line FIGURE 3.7.6 Diffraction of a plane shock around a convex sharp comer.
502
z-Y Han and X-Z Yin
Substituting M 0 and 00 = 0 into the second and third expressions of Eq. (3.7.66) yields
_
dy _ tan coo =
1 [1 (Mg - 1)K(Mo)
]1/2
(3.7.67)
This is the first disturbance line for the moving shock. By substituting 0 w into the first expression of Eqs. (3.7.66), one can obtain the strength of the shock on the surface of the deflected wall M w and then obtain the following relation from the second and the third expressions of Eqs. (3.7.66) dv - " = tan cow -- tan(0 w + Vw) dx
(3.7.68)
where tanvw = ~-s
(M2~- 1)K(Mw)
This is the last disturbance line for the moving shock. At any point on the shock surface, if coi is given, one gets 1/2
_ i)K(M)] l/2dM d- arctan[~ (M2i -1)K(Mi)]
= COi
From the preceding relation, one can obtain Mi a t point i on the shock surface, then get 0 i from the first expression of Eq. (3.7.66). If K(M) is regarded as a constant in some cases, the Eqs. (3.7.66) can be further simplified. The shape of the curved part of the shock wave can be expressed by the following equation,
Mi) -
L
(3.7.69) 0,)
where r i is the distance between point 0 and point i. If M0, L and coi are given, M i and 0 i are obtained by using Eq. (3.7.66) and can be obtained. 3.7.1.4.4.2
ri
Shock-Shock
As already mentioned, the diffraction of a plane shock along a concave wall is different from that of a plane shock along a convex wall. In the diffraction along a convex wall, the disturbance, which was initiated earlier, could not be overtaken by the disturbance generated later, so the simple wave in shock-
3.7
503
Geometrical Shock Dynamics
expansion form can be kept. In the diffraction along a concave wall, however, the disturbance generated later on the wall surface is faster than that of the disturbance generated earlier, and thus the disturbance generated later will overtake the disturbance generated earlier, and the simple wave cannot be kept. Particularly, in the case of the diffraction of a plane shock along a concave sharp comer (or diffraction by a wedge), shock compression disappears and another type of disturbance, that is, shock-shock, occurs (see Fig. 3.7.7). The shock-shock changes the strength and orientation of a plane shock suddenly. In Figure 3.7.7, a plane shock, whose shock Mach number is M0, is disturbed by a shock-shock generated by the wedge. The first deflected angle of the wall surface is 00, and the second deflected angle of the wall surface is 01. The angle between the shock-shock locus and the second surface of wall, Z can be expressed as Z = Zo - 01
(3.7.70)
where Z0 is the angle between the shock-shock locus and the x-axis. The shock-shock relations are derived in the following. The basic ideas of the derivation follow. First, both the undisturbed and the disturbed shocks are connected at point P. The connecting point travels along a shock-shock locus, which means that both the undisturbed and the disturbed shocks have the same velocities along the shock-shock locus, so that M~176 M1 a~ ve = cos(zo - 0o) = cos(zo - 01)
(3.7.71)
where a0 is the speed of sound in the field ahead of both the undisturbed and the disturbed shocks.
Mo
Moving shock
11~176176
..... t ...........................................
Oo FIGURE 3.7.7
/
I
.Mo
7 sOo:SosOO
x,~..o,i.
zo ol
A disturbance, shock-shock, propagating along a plane shock.
504
z-Y Han and X-Z Yin
Equation (3.7.71) can also be written as cos(Zo - 00)
M0
cos(Zo - 01)
=
(3.7.72)
M1
Second, it can be seen in Fig. 3.7.7 that there is a relation between the widths of the ray tubes for undisturbed and disturbed shocks. This relation can be expressed as: L --
A~
=
sin(z o - 00)
A~Afl
(3.7.73)
sin(z0 - 01)
where Aft along the ray tubes for undisturbed and disturbed shocks is kept constant. Therefore, Eq. (3.7.73) can be written as sin(z o - 0o)
=
sin(Zo - -
a0
01)
(3.7.74)
a 1
Third, the area relation for the ray tube can be expressed as 1 dA
A d~
=
2M (M 2 -
dM
1)K(M) da
(3.7.75)
where f l - constant. Or -2M d(ln A) = (M2 _ 1)K(M)
(3.7.76)
Integrating Eq. (3.7.76) gives (3.7.77) along the ray tube for undisturbed shock, and A l = k , e x p [ - I ( M 2 2i~K(M) ]
(3.7.78)
along the ray tube for disturbed shock. Assuming that ko = kl, one has A1 f(M1) A0 --f(M o) where
(3.7.79)
3.7 GeometricalShock Dynamics
505
Summarizing the preceding equations, one has the following set of shockshock relations"
cos(Zo -
0o)
cos(Zo
-
01)
Mo
M1
sin(z0 - 00)
sin(z0 - 01)
A0
A1 A1
f(M1)
Ao
f(Mo)
(3.7.80)
If 0o - 0 one has 01 - - 0 w and X - Zo - 0wRelation (3.7.80) can be written as cos(Zo +
Ow) cos Z M1
Mo sin(z
-
0w)
A1
Ao
Eliminating A0 and
A1
sin Z
A1
f(M1)
Ao
f(Mo)
(3.7.81)
results in cos(X+0w) cos Z sin(;( + 0w) sin Z
Mo M1
(3.7.82)
f(M o) f(M1)
It follows from Eq. (3.7.82) that when a plane shock whose Mach number is M0 diffracts over a wedge whose angle is 0w, the solution can be obtained by using two equations for the two unknown variables M 1 and Z. The 2D shock dynamic equations in a curvilinear orthogonal coordinate system and a rectangular coordinate system, and the following disturbances: shock expansion, shock compression and shock-shock, which propagate on shock fronts, have been discussed so far. The Mach reflection of a weak shock over a wedge with a small angle is analyzed by using the shock dynamic equations and the disturbance propagating along a shock front, and a curved Mach stem can be obtained (Li et al., 1994a). Three-dimensional diffraction of a shock wave propagating into a uniform quiescent gas will be discussed in what follows.
506
z-Y Han and X-Z Yin
3.7.1.5 THREE-DIMENSIONAL SHOCK WAVE DIFFRACTION
3.7.1.5.1 Three-Dimensional Shock Dynamic Equations The position of a 3D shock in the flow field can be expressed as (3.7.83)
~x(x, y, z) - a l t where a 1 is the speed of s o u n d in the region ahead of the shock. Equation (3.7.83) can also be written as
(3.7.84)
S(x, y, z) = ~(x, y, Z) - a lt = 0
Differentiating Eq. (3.7.84) with time t results in ..r
V0~. W s - a 1 -- 0 where Va is the normal to the shock surface and W s is the velocity of the moving shock. Consequently, the shock wave Mach n u m b e r M can be expressed as M = ~
1
(3.7.85)
For the case of a uniform quiescent gas ahead of the shock, the ray is always normal to the shock surface, so the unit vector i for the ray can be written as -. V0~ i = ~
(3.7.86)
Volume V, which is s u r r o u n d e d by the side surface of ray tube S and two end surfaces which is a shock surface cut out by the ray tube at two different times, S 1 and S 2, is shown in Fig. 3.7.8.
f
$2
Side surface of ray tube Shock surface cut out by ray
n2 _. i2
-. .......... i
fit
FIGURE 3.7.8 A volume surrounded by the side surface and two end surfaces of ray tube.
3.7
507
Geometrical Shock Dynamics
By applying the divergence theorem to the volume one gets
f V. v
dV -
i. nds +s~+s2 A
(3.7.87)
where fi is the outward normal of the ray tube or the normal to shock surface and A is the cross-sectional area of the robe. It follows from Fig. 3.7.8 that on $1, il "fil -- --1; on S2, i2" fi2 -- +1; on S, i - f i -- O. Therefore, one has V. As the diameter of the ray tube diminishes to zero, the two integrals on the right-hand side of Eq. (3.7.88) tend to the same value, hence
or
0
89,
Summarizing Eqs. (3.7.89), (3.7.85), (3.7.86), and (3.7.79) results in the set of the shock dynamic equations
M=
1
(3.7.90)
IWl
A Ao
f(M) f(Mo)
3.7.1.5.2 Different Forms of the Equations In a rectangular coordinates system fi -
cos
Ox~x +
cos
Oy~y +
cos
Oz~z
(3.7.91)
and
f i - i - MVo~ - M-~x~ x + M-~ ~y + M-~ze z
(3.7.92)
508
Z-Y Han and X-Z Yin
where cos 0 x, cos Oy, and cos 0 z are the directional cosines and ex, ey and ez are the unit vectors in the directions of the x-, y- and z-axis, respectively. Comparing Eqs. (3.7.91) and (3.7.92) yields M~xx = cos0 x,
M~-
cos 0y,
M~Zz -- cos0 z
(3.7.93)
Substituting Eq. (3.7.93) into the first equation of the set given by Eq. (3.7.90) and noting that
Ox
=~
Oxx
and
~
7zz = ~
7x
results in the following equations of shock dynamics in a rectangular coordinates system,
~ (co:0~)+ o(~o:oy)+ 0 (~)o ~ (co;0y) ~0 (co~0x) 0x
o (co~Oz), o (co~O/) ~x
(3.7.94)
~z
cos 2 0 x
cos 20y
+
+
cos z 0z = 1
A = A(M) By using a similar method, we can get the shock dynamic equations in the cylindrical and spherical coordinates. For the cylindrical coordinates, we have
(
0 rcos0~ Or A
) + ~(co;0)q, ~(~cos0x)- - 0 ~
O (~o~Ox)_ a -g
Vx
+ Oxx
(~o~O,)
0 (rcosO~) o (C~r) o-7
M
=
cos 2 Or + cos 2 0q~ + cos 2 0 x a = A(M)
-- 1
A
(3.7.95)
3.7 Geometrical Shock Dynamics
509
For a spherical coordinate system one has 1 a (co:0~) + ~ 1 O ( r c o s O n s i n ~ l ) + ~2 c o s Or 0 ( r cos Or) 4 =0 Or A r sin t/&p r sin r/Or/ A
O(r cosOn)
O (COMOr)
~ (FCM0tP) -- 1 ~ (cOMOr) Or sin r/&p
cos 2 Or + cos 2 0~ + cos 2 0~ -- 1
A = A(M)
(3.7.96)
3.7.1.5.3 T h r e e - D i m e n s i o n a l Shock-Shock Relations Figure 3.7.9 shows a ray tube that intersects a small portion of the surface of a shock-shock. Because the value ~ of the shock at any given time is the same even across the shock-shock surface, the tangential derivative of 0c on the two sides of the surface of the shock-shock must be equal, that is, fiss x (V~)o -fis~ x (V~)I
(3.7.97)
where fiss denotes a unit vector of the shock-shock surface, V~ is the normal to the shock surface, and the subscripts 0 and 1 denote values on the two sides of the surface of the shock-shock, respectively. Substituting Eqs. (3.7.85) and (3.7.86) into Eq. (3.7.97) results in nss x 10 ass x 11 = ~ M0 M1
Area of shock cut out by ray tube
(3.7.98)
AI
Area of shock-shock surface cut out by ray tube FIGURE 3.7.9 An elemental ray tube that intersects a shock wave surface and a shock-shock surface.
510
z-Y Han and X-Z Yin
One can also get the following relations from Fig. (3.7.9) Asshss "i0
-- A0
Assfiss "11
--
A1
where Ass is the area of the shock-shock surface cut by the ray tube and A0 and A 1 are the surfaces of the undisturbed and the disturbed shocks cut out by the ray tubes on both sides of the shock-shock surface, respectively. From the preceding equations one can get nss 9 10 nss 9 11 = ~ A0 A1
(3.7.99)
In summary the 3D shock-shock relations are x 10
nss
x 11
nss
M0
M1
nss .10
Ilss .11
Ao
A1
AI Ao
(3.7.100)
f(M1) f(Mo)
3.7.1.5.4 Diffraction of a Plane Moving Shock Wave over a Cone at Zero Angle of Attack Whitham (1959) studied the diffraction of a cone by using the theory of shock dynamics. Figure 3.7.10 shows the diffraction of a shock over a cone at a zero angle of attack. Under the condition of Mach reflection, from the viewpoint of shock dynamics, the triple-point trajectory is the shock-shock locus, and the curved Mach stem can be calculated using the shock dynamic method. For the case here, the phenomenon is axisymmetric and self-similar and thus the shock dynamic equations in a cylindrical coordinate system equation (3.7.95) can be simplified to an axisymmetric form,
8 (rsinO) Or
A
o(c:o)
A - A(M)
8 (rcosfl)_O +~ A (3.7.101)
3.7 Geometrical Shock Dynamics
511
M'0"........
Y
Movingshock
Shock-shock locus ....
..............
...............
M'0 "
cone . x >
FIGURE 3.7.10 Diffractionof a plane moving shock over a cone.
where 0 is the angle between the normal to shock and the cone axis. Introducing a similarity variable, r / = a r c t a n ( r / x ) , Eqs. (3.7.101) can be transferred to the following set of ordinary differential equations
dO dr/
tan 0 c o t ( r / -
O)(M2 -
1)K(M)
sin r/cos r/(1 + tan r/tan 0)[2M 2 t a n ( r / - 0) - (M 2 - 1)K(M) c o t ( r / - 0)]
1 dM _ t a n ( r / - 0) dO M dr/ dr/ (3.7.103)
Equation (3.7.103) can be applied to calculate the shape of the curved Mach stem with the following boundary conditions: at the shock-shock, r / = %, 0 = 01, and M : M 1 tan 01 = (M2 - M~)1/211 -
(f(M1)/f(M~
[M 0 -4-~(M1)/f(Mo))M1)]
--f(M1) tan
~
f(Mo)
1_(Mo/M1)2 ,]1/2 1 --
(3.7.104)
q(Ml)/f(Mo))2j
and on the surface of the cone, r/-- O~
and
O-
Ow
(3.7.105)
Solving Eqs. (3.7.103) using boundary conditions equations (3.7.104) and (3.7.105), one can find the angle of shock-shock locus Z0 and the changes in the strength and orientation of the Mach stem with the similarity variable r/. For the strong shock, use the following A - M relation
A where n -- 5.0743 .
(_~)n
(3.7.106)
512
z - r H a n a n d X~ Yin
The shock dynamic equations and the boundary conditions can be simplified as follows, dO
tan 0
dr/
sin/7 cos r/(1 + tan r/tan O)[n tan2(r/- O) - 1]
1 dR R dr/
= t a n ( r / - O)
(3.7.107)
dO dr/
(R2 - 1)1/2(1 - R12.)1/2 tan/91
"--
1 + R i -n /7 -- %0
tan )~0 --
(3.7.108)
~-~n
0-
0w
r / = 0w
(3.7.109)
where R = M / M o. Yu and Groenig (1984) studied the diffraction of plane shocks by cones further.
3.7.1.5.5 Diffraction of a Plane Moving Shock Wave over a Circular C y l i n d e r or Sphere When a plane shock wave moves through a cylinder or a sphere, regular reflection always occurs at the nose. The theory of shock dynamics can hardly handle a regular reflection, in which the disturbance wave has not caught up with the incident shock wave. In order to find the solution at the nose, the regular reflection at the nose may be regarded as a Mach reflection with an extremely small Mach stem, and the small Mach stem is straight and normal to the surface (Bryson and Gross, 1961). Figure 3.7.11 shows the diffraction of a plane moving shock over a cylinder, where the radius of the cylinder is normalized to unity. For the plane shock wave, which travels through distance x at time t with Mach number M 0, one has x
o~ -- a l t - - - M0
(3.7.110)
and the distance x can be expressed as x -- 1 - (1 4- b) cos (p
(3.7.111)
3.7 Geometrical Shock Dynamics
513
FIGURE 3.7.11 Diffractionof a plane moving shock over a circular cylinder.
Substituting Eq. (3.7.111) into Eq. (3.7.110) gives --
[1 - (1 4- b) cos ~o]
(3.7.112)
M0 We k n o w that for the Mach stem, ~ has the same value as the plane shock at any given time, that is, Eq. (3.7.112) can also be used for the Mach stem. For a straight Mach stem, which has the same direction with the radial, one can get 1
--IVy[-
M 1
0
rO~
--
1 (1 4-
d~
b/2) dq~
(3.7.113)
where M 1 is the Mach n u m b e r of the Mach stem. Considering that the two areas along the ray tube are A 0 -- (1 4- b)sin ~0 A1 -b one has b f ( M 1) -(1 4- b) sin (p f(Mo)
(3.7.114)
For strong shock, Eq.(3.7.114) can be written as
(1 4- b)sin ~0 where n -
5.0743.
(3.7.115)
514
z-Y Han and X-Z Yin
Substituting Eqs. (3.7.112) and (3.7.115) into Eq. (3.7.113) and rearranging it, we get the following ordinary differential equation d--b-b dq9
(1 + b) tan cp -
(3.7.116) cos cp
(1 + b) sin
From Fig. 3.7.11, the relation of the shock-shock locus can be written as y = ( 1 + b) sin ~o (3.7.117) x = 1 - (1 + b) cos q~ The slope of the tangent to the shock-shock locus is tan ;~o =
= ~
=
(b + 1) cos q~ 4- sin r.p(db/dqg) (b + 1) sin q9 - cos q~(db/dqg)
(3.7.118)
Equations (3.7.116), (3.7.117), and (3.7.118) can be used to find the Mach number of the Mach stem and the shock-shock locus. For the range of noses with small angle cp, one can obtain the following simplified solution b
-- sin n+l r
y = (1 + sin n+l q~)sin ~o (3.7.119) x = 1 - (1 + sin "+1 ~o) cos ~o tan Zo = cot q~ The diffraction of a plane moving shock over a sphere is shown in Fig. 3.7.12. For the same reason, and noting that the ray tube areas are different from the case of a cylinder, one can get the following equation: 2(1 + b/2)b 11In d&o b = (1 + b) tan ~o - (1 cos + b/2) ~o (1 + b-)2s~n ~oJ
(3.7.120)
The equations describing the shock-shock locus are the same as Eqs. (3.7.117) and (3.7.118). For small angle q~, one has I sinn+l
(3 7.121)
Comparing Eq. (3.7.121) with Eq. (3.7.119) for the case of a cylinder, we find that the Mach stem for the sphere is shorter than the Mach stem of the cylinder due to the 3D effect.
3.7 Geometrical Shock Dynamics
515
Moving shock l y Shock-shock surface
X
FIGURE 3.7.12 Diffractionof a plane moving shock over a sphere.
3 . 7 . 1 . 5 . 6 D i f f r a c t i o n of a P l a n e M o v i n g S h o c k W a v e o v e r a C o n e a n d an Elliptic C o n e at A n g l e of A t t a c k W h e n a plane moving shock wave diffracts over a cone with an angle of attack, the p h e n o m e n o n is 3D. If Mach reflection occurs, the shock-shock surface is a curved surface. In Fig. 3.7.13, the angle between the incident shock wave and the vertical plane to the axis of the cone is denoted by 2, 6c is the half apex angle of the cone, tic is the unit vector of the cone surface, M s is the incident shock wave Mach number, fis is a unit vector normal to the plane incident shock, fiss is the unit vector of the shock-shock surface and Z is the angle between the axis of the cone and the shock-shock locus, which is a function of the circumferential angle of cone q~.
Shock-shock surface
Elliptical cone X ,-.
~)
FIGURE 3.7.13 Diffractionof a plane shock over a cone and elliptic cone at an angle of attack.
516
z-Y Han and X-Z Yin
In spherical coordinates (r, r/, ~0), the plane ~ p - 0 corresponds to the symmetric meridian plane and the forementioned unit vectors can be written as
fi s - (cos 2 cos ~ / - sin 2 cos ~0)~r -
(cos 2 sin r / + sin 2 cos r/cos ~0)~r + sin 2 sin ~0. ~ 1
fiss v/Z2+ %a (0. er + %" er - Z'" e~)
(3.7.122)
.. _ [cos ~c/sin2 ~cer + sin @ cos @(1/b 2 - 1/a2)e.~] n c - ~/cos2 ~0/a2(1 + 1 / a 2) + sin 2 ~p/b2(1 + 1/b 2) where Zt - dz/dq~, 7~r, er, and 7~ are the unit vectors of the spherical coordinate system. For an elliptic cone, a and b denote the two axes of the elliptic cross section (for a cone, a - b and tic - er). As the p h e n o m e n o n is self-similar (0/Or - 0), Eq. (3.7.96) can be simplified to read
3M
3M
30r
3M G+
3M
30r
30~
+
3M
30~
- Co
(3.7.123)
30r
COS2 0 r -~- COS2 0 r -F" COS 2 0~o = 1
where E 1 =
M2e cos 0~0, E 2 -- M2e cos 0 r sin 1/, E 6 -- - M 2 sin 0 r sin ~/,
E7 =
-M2e sin 0~, E o -- - M 2 ( 2 cos 0 r sin r / + cos 0 r cos r/),
G 1 --
cos
0 r,
G 8 = - M 2 sin 0~ sin 1/, G 2 - - cos 0~ sin r/, G 5 - M sin 0 r, H 2 -- cos Or, G o = - M cos 0 r cos 0r H 4 -- M sin Or, H 0 -- - M cos 0 r, e - - (M 2
2M
1)K(M)'
[(Y-1)M2 =
-
K(M)= 2 [ ( 2# + 1 - ~1 ) (
1+
2 1-#2)1-1 7+ 1 #
+ ~)] -
T h e 3D shock-shock equations (3.7.100) are used to find the strength and the orientation of the Mach stem just behind the shock-shock surface, which
3.7
517
Geometrical Shock Dynamics
can serve as the boundary condition of Eq. (3.7.123). In the other boundary condition, the Mach stem must be normal to the surface of the cone, that is, fi.fic--0
at
~/-~c
(3.7.124)
Yin et al. (2000) solved the set of equations (3.7.123), by using the method of line and found the shapes of the shock-shock surface and the strength and orientation of the Mach stem.
3.7.1.6
D I F F R A C T I O N OF S H O C K W A V E S
PROPAGATING INTO N O N U N I F O R M QUIESCENT GASES 3.7.1.6.1 Area Relation for the Case of Shock Waves Propagation Through Nonuniform Quiescent Gases When a shock wave propagates through a varying cross-sectional area tube with a nonuniform quiescent gas, the strength of the shock wave depends not only on the area of the tube, but also on the parameters in the region ahead of the shock. Catherasoo and Sturtevant (1983) derived the relation for nonuniform quiescent gas ahead as follows edM +fd7 4- g
da I ax
+ h
dpl Pl
--
dA A
(3.7.125)
where 2M (M2 - 1)K(M) (~ - ~) f -- ~(~ + 1-------g(M, ~ 7) g = g(M, ~) =
2#(M 2 - 1) + [(7 - 1)M2 + 21 ( y - 1)M2 if- 2
1 { (7 + 1) 2# M2 / h - 2y(M2 _ 1) 2(M2 - 1) + #[27M 2 - (7 - 1)] - (} -~ iyM-i $ 2 ] In Eq. (3.7.125) M denotes the shock wave Mach number, A is the tube area, 7 is the ratio of the specific heats, a I is the speed of sound, and Pl is the pressure in the region ahead of the shock wave.
518
z-Y Han and X-Z Yin
3.7.1.6.2 Shock Dynamics Equations for Shock Waves Propagation Through Nonuniform Quiescent Gases For the case of a nonuniform quiescent gas ahead of a shock, the following relation is used: cz=t By using the same method as for the case of uniform quiescent gas, we obtain the following shock dynamic equations. The equations in the orthogonal curvilinear coordinates (a, fl) are: 30 0--~=
1 O A Off (Mat)
30
1 3A
Off
Ma 1 3o~
OM
~
(3.7.126) h Dpl
13A
Pl 00~
A 3~
g Oa1
The equations in 3D form are:
~. (~v~)_o 1 Mal :
(3.7.127)
IV~l
edM 4-fd7 4- g
@1
dal
4- h ~ = al Pl
dA A
The equations in a rectangular coordinates system:
~ (co:o~)+~~(co:o~)+~~(co: o~):0 lcoso,~
~ (coso~_o
(coso:~
o (coso,,~
~ M,.. / - ~ ,,-~--~-,./ -
(3.1.128)
o
cos 20 x + c o s 20y+cos 20 z - 1 edM 4-fd7 4- g
da I al
dpl 4- h ~ = Pl
dA A
3.7
519
Geometrical Shock Dynamics
3.7.1.6.3 Characteristic Relations for Shock Waves Propagation T h r o u g h N o n u n i f o r m Quiescent Gases As in the uniform gas case, we can similarly obtain characteristic relations from the shock dynamic equations. In an orthogonal curvilinear coordinate system, one gets
+~o(M) ~
do (1)
()u~dO(2)-c~
d/3 along ~ - +c
- -(V + c6) (1)
~
(3.7.129)
(2)- -(V -
d/3 along~-~a = - c
c6)
where subscripts (1) and (2) represent two characteristics, and
al c-~-
M2 -
)K(M)
M 0a 1
A 0/~ 6-~ ~+-,~~+f F
~
~
~
The speed of the disturbance wave propagating along the shock surface is
(3.7.130)
Compared with the speed of the disturbance wave in a uniform gas (Eq. 3.7.18), one finds that they have the same form, but it should note that a 1 in Eq. (3.7.130) is a variable and a 1 in Eq. (3.7.18) is a constant. The characteristic relations in a rectangular coordinate system are
+ co(M) (1)
- - F(1 )
(d_~O) (d_~~) -- co(M)
(2)
along
(1)
-- F(2) (2)
(1)
along
(dY)_tan(O_v) (2)
(3.7.131)
520
z-Y Han and X-Z Yin
where subscripts (1) and (2) represent two characteristics, respectively, and tan v -F(1)--
1
4~
( 1 _ tanOtanv 1 )[( t a n O - g t a n v ) a1l ~0 a 1 -
- tan v - -
pl \ ~X
F(2) --
- tan v . f
( 1 + tan10 tan v)[(tanO+gtanv)l~al a--i-~ 4- tan v Pl~
3.7.2
+ tan 0
SHOCK
THROUGH
-~X +
tan 0
4- tan v . f
10a
I
(g tan 0 tan v + 1) - -
~x + tan 0
10a
I
+ (g tan 0 tan v - 1) ~al ~y Oxx 4- tan 0
WAVES PROPAGATION
MOVING
GASES
The propagation of a shock wave through a moving gas will be discussed here. This is a new area, of course, and it is much more important for geometrical shock dynamics. Phenomena involving shock waves propagating into moving gases are more widespread. Some examples of this kind of shock propagation and diffraction are given in the following. In Fig. 3.7.14A, the reflected shock is propagating into a uniform flow field, which is induced by a plane incident shock. In Fig. 3.7.14B, the transmitted shocks and the Mach stems propagate into a nonuniform region and a conical flow field is formed by the supersonic flow passing through the cone. This kind of shock propagation and diffraction can be divided into two cases, that is, shock waves propagation into a uniform flow field and a nonuniform flow field. Chester (1960) extended Whitham's method to the case of a nonuniform flow ahead of the shock, but the results are only suitable for one-dimensional (1D) flows, in which the normal to shock surface coincides with the direction of the flow ahead of the shock wave.
3.7.2.1
S H O C K WAVES P R O P A G A T I O N T H R O U G H
UNIFORM F L O W FIELDS In the case of a shock wave propagating into a moving gas, in general, the rays are not normal to the shock surface, that is, the rays and the positions occupied
3.7
Geometrical Shock Dynamics
FIGURE 3.7.14
521
Shock waves propagating through uniform and nonuniform flows.
by a curved shock at different times do not construct an orthogonal curvilinear coordinate system as shown in Fig. 3.7.15B. In order to establish the geometrical relations and the area relation for this case, a method that differs from the one shown in Section 3.7.1 will be used. The basic idea is to make a transformation of coordinates. By means of this transformation, the nonorthogonal coordinate system is transferred into an orthogonal one, and then the original method mentioned in Section 3.7.1 can be used. Whitham (1968) presented this method. In Fig. 3.7.15A, a moving curved shock, which is reflected on the surface of a wedge, propagates through a uniform flow field induced by a plane shock.
522
z-Y Han and X-Z Yin
A
B
FIGURE 3.7.15 The moving and the stationary frames of reference; x, y and t are the coordinates of the stationary frame and x', y', and t' are the coordinates of the moving frame.
This is the case to be discussed. In the general case, the shock surface and the rays can be expressed in Fig. 3.7.15B. Let us introduce two frames of reference, one a stationary frame and the other a moving frame. The moving frame is attached to the uniform flow in front of the moving shock and the stationary frame is attached to the body over which the moving shock diffracts. In the moving frame of reference, the flow ahead of the moving shock is at rest and the rays are normal to the shock surfaces; therefore an orthogonal network that consists of rays and shock positions is formed. The equations for a quiescent gas ahead of a shock wave, which were given in Section 3.7.1, can be used in the case of a moving frame of reference. The purpose here, however, is to establish a set of equations for a uniform flow ahead of a shock wave in the stationary frame. Thus a transformation of coordinates from the moving frame of reference to the stationary frame of reference is necessary. In the general case of a uniform flow ahead of the shock wave, the relations between the coordinate systems of the moving and the stationary frame of references can be expressed as x = x t + Ucos~ 9tt y = yr + U sin ~. t t=t
I
(3.7.132)
3.7
523
Geometrical Shock Dynamics
where U is the flow velocity and e is the angle between the direction of the flow ahead of the shock and the x-axis. Under the condition that the direction of the flow ahead of the shock coincides with the x-axis, Eqs. (3.7.132) can be simplified to read x = x ~+ U( y =y'
(3.7.133)
t=t'
The moving shock can be described by using shock surface functions, or czt(x~,yt) in the stationary and the moving frames of reference, respectively. It is evident that functions ~ and e' have different forms. In the stationary frame of reference, the equation of shock for the surface can be expressed as cz(x,y)
cz(x, y) = a I 9t
(3.7.134)
where a 1 is the speed of sound in the region ahead of the moving shock. In the moving frame of reference, the equation of shock for the surface can be expressed as o~'(x', y ' ) = a I 9 t'
(3.7.135)
Although the functions of the moving shock surface in the two different frames of reference are different, from the third expression of Eq. (3.7.132), one can obtain the following relation od(x', y ' ) = o~(x, y )
(3.7.136)
Equation (3.7.134) can also be written as m. o~(x,y) = m . a 1 9t -- U . t
(3.7.137)
where m is the flow Mach number in the region ahead of the moving shock wave.
Substituting Eq. (3.7.137) into Eq. (3.7.133) results in x = x' + mcx(x, y ) y=y'
(3.7.138)
t=t'
The differential relations between the two coordinate systems will be derived in the following.
524
z-Y H a n a n d X-Z Yin
Taking the derivatives of the first and second expressions of Eq. (3.7.138) with respect to x' and yt, and considering 8 / / a x / - 0, 0x~/~y' = 0, ax'/ax' = 1, ~ y ' / 8 / = 1, one gets 0
1
0
---7 ax = 1 - m. 0~,,axx a
m.O~y
By'
(3.7.139a)
O
0
1 - m . O~xaX
ay
(3.7.139b)
Considering Eq. (3.7.136), one obtains a0~t
0x'
--
a0~
--
0x'
0~x
1 - m.0c x
0~' _ O0~ -~Y ~' By' 1 - m . 0 ~ x
(3.7140a)
(3.7140b)
Next, the shock dynamic equations for a uniform flow ahead of a shock wave are derived. In the moving frame of reference mentioned earlier, the following equations can be obtained (Whitham, 1968)" 9 VIo(
M! ~
= 0
1
(3.7.141)
IV'~'I
A' = A'(M')
In Eqs. (3.7.141), M' is the shock wave Mach number in the moving frame of reference. It is evident that the shock strength, which can be expressed by the shock wave Mach number or the pressure ratio across the shock wave, is kept unchanged, when the transformation from the moving frame of reference to the stationary frame of reference is made, that is, M'--M
(3.7.142)
Thus, the set of equations (3.7.141) can be written as
v,
v, )0 1
IV'~'i A' - A(M)
(3.7.143)
525
3.7 Geometrical Shock Dynamics
where A' is the cross-sectional area of the ray tube in the moving frame of reference. It should be noted that A' cannot be transferred to A, which is the crosssectional area of the ray tube in the stationary frame, because the ray does not coincide with the particle path behind the moving shock wave in the case of the stationary frame. This problem will be discussed later. In the 2D case, the equations in the moving frame of reference can be expressed as
ax,
5-2x,/+~
Oy,/- o (3.7.144)
M -- (~2 ..1._~ ) - 1 / 2
A ' = A'(M) By substituting Eqs. (3.7.139) and (3.7.140) into Eq. (3.7.144), the shock dynamic equations in the stationary frame of reference are obtained (Whitham 1968),
Ox LA,(~x ~+
M=
0~2)1/2] -~-~ LA#-~#+ ~-)-~?~J - o (3.7.145)
1 - mo~x
A' = A'(M)
The set of equations (3.7.145) can also be expressed by the shock wave Mach number M and shock wave angle, 0. As already mentioned, the shock strength is kept unchanged in the transformation of coordinates from the moving frame of reference to the stationary one. Similarly, the shock wave angle 0 is also kept unchanged in the transformation process of the coordinate systems, that is, M=M',
0
-
-
O'
(3.7.146)
The unit vector of the normal to the shock surface can be written as fi = cos 0~.x 4 - s i n O~.y
(3.7.147)
Considering the second expression of Eqs. (3.7.145), one has
w
M(~x~-x+ %~y)
[V~]
1 - m~ x
(3.7.148)
526
z-Y Han and X-Z Yin
A comparison of Eq. (3.7.147) with Eq. (3.7.148) results in O~ ~x Oa
cos 0 M + mcos0 sin 0
Oy
M + tacos0
(3.7.149)
Substituting Eq. (3.7.149) into the Eqs. (3.7.145) results in the following equations, which are expressed by M, 0, m, A' (Han and Yin 1989, 1993)
L(~u ~ cosO)A'J + -~ (M + m cos O)A' = 0 __OO(M x +
m co~ o)1 - ~
M + m cos o
= o
A' = A'(M) The area relation, the third expression of the set given by Eq. (3.7.150) can be expressed as 1 ~A /
0M
---~ -- e ~ A' ao~' oo~,
(3.7.151)
according to
x'= x'(o~')
and
y ' = y'(~')
(3.7.152)
Equation (3.7.151) can be expressed as
10.4'
___xaa'=
A-S~x---7 + tan 0 A' Oy'
(OM
-e ~
+ tan 0
ig_~~p)
(3.7.153)
Substituting Eq. (3.7.139) into Eq. (3.7.153) yields
XOA'
xaa'
e(aM
~)
a--;--~- + G a---;-~- = - \--~x + G~___
(3.7.154)
where G = M sin O/(Mcos 0 + m). Finally, the shock dynamic equations for a uniform flow ahead of the shock wave can be expressed as 0
McosO+m
]
O (M
O)A' =0 +,.cos a [M c~ 0 ] = 0
-~ (M + ,,,-~ ~yA'J+-~ O[
sin0 ]_ (M 4- m cos O) ~
4- m cos
13A'IOA' (OM~) a--;~ + Ga--;~ = -e ~ + G
(3.7.155)
3.7 GeometricalShock Dynamics
527
The third expression of the set given by Eq. (3.7.155) can also be written as follows
A-7-~x 4.
A ' 3y - - e
--~x 4-
~
(3.7.156)
By comparing Eq. (3.7.156) with (3.7.154), one obtains
dy _ M sin fl dx M cos fl 4. m
(3.7.157)
Equation (3.7.157) is the differential equation for the ray in the stationary frame of reference and dy/dx - M sin fl/(M cos fl 4- m) is also the slope of the ray in the stationary frame of reference. This problem will be further discussed when Eqs. (3.7.165) and (3.7.167) are presented. The first expression of equations (3.7.150) can also be expressed in the following form, which is similar to the case presented in Section (3.7.1), V.(A ) =0
(3.7.158)
where i is the unit vector of the ray in the stationary frame of reference and A is the cross-sectional area of the ray tube in the stationary frame of reference. One can prove that under the condition of a shock wave propagating into a uniform flow, Eq. (3.7.158) is still valid (Whitham 1968). The derivation of Eq.(3.7.158) is omitted here. To find the expression for i, the unit vector i can be expressed in the following form,
i = fl (x, y)ex 4- f2(x, Y)ey
(3.7.159)
Substituting Eq. (3.7.159) into Eq. (3.7.158) results in 3
Y) 4-
= 0
(3.7.160)
By comparing Eq. (3.7.160) with the first expression of Eqs. (3.7.150) yields
fl(x,y)
Mcos04- m
A f2(x, y) A
(M 4- m cos O)A' M sin 0 (M 4- m cos O)A'
(3.7.161)
where fl(x, y) and f2(x, y) are the components of the unit vector i and thus the following condition should be satisfied
[f21(x, y) + f~(x, y)]~/2 _ 1
(3.7.162)
528
z-Y Han and X-Z Yin
or
+L(M7
-- 1
(3.7.163)
Using Eq. (3.7.163) one can obtain the following expression for A'/A: A ~ (M 2 + 2Mm cos 0 + m2) 1/2 A- -(M + m cos O)
(3.7.164)
By substituting Eqs. (3.7.161) and (3.7.164) into Eq. (3.7.159) one gets -~ (M cos 0 + m)f~x + (M sin O)~y i -(M 2 + 2Mm cos 0 + m2) 1/2
(3.7.165)
From the foregoing discussion, one finds that under the condition of a uniform flow ahead of the shock waves, the unit vector of the normal to the moving shock surface fi is different from the unit vector of the ray in the stationary frame i. It follows from Eqs. (3.7.165) and (3.7.147) that -, _~ (M + m cos 0) 1. n -- ( M2 + 2Mm cos 0 + m2) 1/2
(3.7.166)
Equation (3.7.166) means that i is not normal to the shock surface in the general case and that it depends on the strength and orientation of the moving shock, and the flow direction and the flow Mach number in the region ahead of the shock waves. Expression (3.7.165) can be rewritten in the following form -.
1 --
M+,~
(3.7.167)
16t + ~1
or
, -where
~I -- M ~ ,
Cn -- m ~ f , M
-- W s / a
(3.7.168) I,m
-- U 1 / a
I , ~ --
cos OE x -6 sin O~.y and
Iff ~ e x-
It follows from Eq:~(3.7.168) that the direction of i coincides with that of the resultant vector of (W s + U 1) or (M + ~ ) , as shown in Fig. 3.7.16. It should be noted that in the stationary frame of reference, the direction of the ray is different from that of the flow behind the moving shock, so one cannot use the area relation along the ray tube in the stationary frame of reference. In Eqs. (3.7.150) and (3.7.155), the cross-sectional area of the ray tube in the moving frame A' is used instead of A; this is the case because for the
3.7
529
Geometrical Shock Dynamics
Mfi
Mfi + mrr,f
Moving shock
...
m'r, f
Direction of the flow ahead of the moving shock FIGURE 3.7.16
The unit vector ]' coincides with the vector/Vl + ~.
moving flame of reference, the ray coincides with the particle path immediately behind the moving shock wave. Therefore, the C.C.W. equation can be used along the ray tube in the moving frame. The concept of the disturbance propagation along the shock front can also be suitable for shock waves, which propagate into a uniform flow field. From the set of equations (3.7.155), one can obtain the following characteristic relations
(-~)+
co(M)(~-)=0 along (-~) -- tan(O+vl)
1
(~)+
1
co(M)(-~--)- 0 along 2
2
(-~) -- tan(0 + v2) 2
where
c o ( M ) - - [ ( M 2 - 2)K(M).] 1/2 [(1/2)(M 2 - 1)K(M)]1/2 -- rosin 0 M + m cos 0 [(1/2)(M 2 - 1)K(M)] 1/2 4- rosin 0 tan/)2 m_ M + mcosO tan v1 =
(3.7.169a)
1
(3.7.169b)
530
zoY H a n a n d X-Z Yin
The preceding relations can be expressed as 0+
I
co(M)dM = constant along
= tan(O +
Vl)
(3.7.170a)
co(M)dM -- constant along
= t a n ( 0 - v2)
(3.7.170b)
1
01
From the characteristic equations (3.7.170) it can be seen that the first equation represents an upward disturbance wave and the second represents a downward disturbance wave. As an example, one can consider the regular reflection of a moving shock wave on a wedge (Skews, 1971; Han, 1991), as shown in Figure 3.7.17. In the case of regular reflection, the reflected shock is disturbed only by a simple wave, which is generated at the comer o. The disturbance wave propagating along the reflected shock is a downward simple wave, which can be described by the following relations: 0~ +
(M 2 1
0 r
--
i)K(M
dM = constant everywhere
(3.7.171a)
~
M2- i)K(M)
dM -- constant along ~ -(3.7.171b)
where tan Or - -
[ 1 / 2 ( M ~ - 1)K(Mr)] 1/2 + msinO~ M~ + m cos Or
FIGURE 3.7.17 Regularreflection of a moving shock on the surface of a wedge.
3.7 GeometricalShock Dynamics
531
The downward disturbance wave propagates from point Q to point P along the reflected shock, and reduces its strength from the first disturbance point to the last disturbance point. A regular reflection of a plane shock, which interacts with an expansive corner, was investigated using the shock dynamic theory by Li et al. (1994b). In the case of a Mach reflection of a moving shock over a wedge, there are double disturbance waves, that is, upward wave and downward wave, which propagate along the reflected shock wave. The strength and orientation of the reflected shock depend on the double disturbance waves.
3.7.2.2
S H O C K WAVE P R O P A G A T I O N T H R O U G H
N O N U N I F O R M F L O W FIELDS 3.7.2.2.1 2-D Shock Dynamic Equations for Shock Wave Propagation through Nonuniform Flow Fields 3.7.2.2.1.1 T r a n s f o r m a t i o n of Coordinate System
As mentioned earlier, in order to establish the shock dynamic equations for the case of a uniform flow ahead of the shock wave, it is necessary to use a moving frame of reference, which is attached to the flow field ahead of a shock wave so as to construct an orthogonal curvilinear coordinate system. Then by using the transformation of the coordinates from the moving frame of reference to the stationary one, we can obtain a set of shock dynamic equations for a uniform flow ahead of the shock wave. The foregoing description does not apply to the case of a shock wave propagating into a nonuniform flow field. If one still uses a transformation of coordinates similar to that in the case of a shock wave propagating into a uniform flow, then one must find a way to establish the shock dynamic equations, that is, how to establish the moving frame system for the case of a shock wave propagating into a nonuniform flow field? The basic idea for solving the forementioned problem and deriving the shock dynamic equations for a nonuniform flow ahead of a shock are discussed here (Han and Yin, 1993). First, the nonuniform flow field immediately ahead of the shock surface is divided into many small fluid elements (regions). Each region is regarded as a uniform flow field, that is, the parameters, such as flow Mach number, flow angle, speed of sound, and so on in the region, are assumed to be constant, but change from one region to another. Second, many moving frames of reference are attached to the small fluid elements (regions). In this case, each region has its own moving frame of
532
z-Y Han
and X-Z Yin
reference, that is, there are many moving frames of reference for a nonuniform flow field, but only one stationary frame of reference. This is because the shock dynamic equations for the description of the shock wave propagation through a nonuniform flow field should be expressed by only one stationary coordinate system. Third, the shock dynamic equations in the moving frame of reference in any small region can be established and then by using a transformation of coordinates from the moving frame of reference to the stationary frame of reference, one can obtain the shock dynamic equations in the stationary frame of reference in any small region. Obviously, the equations obtained for any small region have the same form; if the flow Mach number and flow angle in the equations are regarded as variables, the equations, which are suitable for a small region, can be extended to the entire flow field. Many small regions in the nonuniform flow ahead of the shock, the corresponding moving frames, and stationary frame are illustrated in Fig. 3.7.18. The relations between the moving and the stationary frames of reference are x = x' q- U cos e. t' 4- A y = y ' + U s i n e . t' + B t--
(3.7.172)
t~
where x, y,and t are the coordinates in the stationary frame of reference, x', y', and tt are the coordinates in the moving frame of reference, and A and B are constant for each region, but they have different values for different regions,
FIGURE 3.7.18 Shownare many moving frames of reference in the field immediatelyahead of the shock waveand only one stationaryframe of reference attached to the body over which a shock wave diffracts.
3.7
533
Geometrical Shock Dynamics
which represent the positions of the moving frames of reference at the initial time t = 0. In the moving frames of reference, the equations of the shock surface can be written as (3.7.173)
~'(xt, y') = t'
In the stationary frame of reference, the equations of the shock surface can be written as a(x, y) = t Although a t ( x ' , / ) and
(3.7.174)
are different functions, they are equal because
a(x, y)
tt=t.
a'(x', y') = 0~(x,y)
(3.7.175)
Thus Eqs. (3.7.172) can be written as x = x ' 4- m a .
cos ~. 0~(x,y) + A
y = y' + ma.
sin s. a(x, y) 4- B
t--
(3.7.176)
tI
where m, ~ and a are the flow Mach number, the angle between the direction of the flow ahead of the shock wave and the x-axis, and the speed of sound in the region ahead of the shock wave, respectively. The corresponding differential relations for the two coordinate systems are
~x
~y
~x,
~xax'
~yax"
~y'
~x~y'
~y~y'
~ ax
~ 8y
(3.7.177)
The derivatives with respect to x', y', x, and y in Eq. (3.7.177) are unknown, so if it is necessary to take derivatives of the first and second equations of (3.7.176) with respect to x' and y' respectively, one has
~x
~
a x ' - - ~x' + m a .
~y
~y
. o~x 9~x--; + m a .
cos s. o~y 9 ~x--;
~y'
8x--i = 8x---i + m a .
~x
ax cos ~
~
ay sin
~ . o~x 9~
+ ma .
sin e. 0Cy9 8x--;
~x
~y
~y, - - ~y, 4- m a .
cos ~. 0cx 9~-7 4- m a . cos ~. 0Cy. ~y--;
~,
sin ~. a x 9~-7
- - ~ , 4- m a .
4- m a .
sin ~.
0Cy. ~y--;
(3.7.178)
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z-Y Han and X-Z Yin
In the moving flame of reference, x' and y' are independent variables, so one has ~y'
0x--7
~x' o, o~
=
=
ax'
0,
0x--7
1 =
,~
= 1
(3.7.179)
Substituting Eqs. (3.7.179) into Eqs. (3.7.178), solving for a x / a x ' , ay/ax',ax/Sy', and ay/sy' and then substituting the results into Eqs. (3.7.177), one gets
a
(1 -
Ox'
1 -
0 _
_
ay'
sin e , . O ~ y ) a / a x
ma.
ma. =
=
ma.
cos e, .
sin 8"~x" O/Oy
+ ma.
o~x -
ma .
cos 8-~y-alOx + (1 -
sin 8.0~y (3.7.180)
cos 8-czx)alo~
ma.
_
1 -
ma.
cos 8 .
gx -
ma.
sin e .0~y
Considering Eq. (3.7.175), one obtains the relations for the derivatives of 0~ and 0( with respect to x', yt, x, and y, respectively, as follows cgCX p Ox'
O~x 1 -
ma .
cos e
9 o~x -
ma .
sin e - O~y (3.7.181)
Oy ~
3.7.2.2.1.2
1 -- ma.
cose.0t x
Two-Dimensional
-- m a .
Equations
sinS.~y
Denoted
by
the
Function
aO~,r) In a moving frame of reference, which is attached to the flow field ahead of the shock wave in a small region, the rays are normal to the shock surface. According to Section 3.7.1.4, one can obtain the following equations
Ma =
I
(3.7.182)
IV'~'l
A' = A'(M, p, a)
where A' = A ' ( M , p , a ) is a simple expression, which will be derived later. In the case of 2D flows, the first and the second equations of (3.7.182) can be written as o
0x,
TOx,/+~-F~/-o (3.7.183)
Ma =
\Ox'J
+
\ay'/
3.7 GeometricalShock Dynamics
535
Substituting Eq. (3.7.181) into Eq. (3.7.183) results in the geometric shock dynamic equations
(3.7.184) Ma--
1 - m a . cos ~ 9 o~x -- m a . sin e 9~y
+
where J - ~x" cos e - % . sin e. The preceding equations are the geometric shock dynamic equations for any small region in the stationary frame of reference. By using a method similar to the one presented in Section 3.7.2.1, from the first expression of Eq. (3.7.184), one can obtain the unit vector for the ray in the stationary frame of reference -,
i --
ma. O~x)~.y [(~2 4- o~2)(1 4- m2a2j2)]1/2
(3.7.185)
A' -- (1 4- m2a2j2)1/2A
(3.7.186)
(ocx + m a . afl)e-x + ( % -
The unit vector of the normal to the shock surface, fi, can be expressed as VO~
_.
n
--
O~xe.x 4- O~ye.y
(3.7.187)
M(Tl(O~xe.x + %F.y)
]Vo~--~= (O~x 2 4- o~2)1/2 = 1 - ma. cos
~'~x
-
ma.
sin ~ .OCy
and -- cos O. ex 4- sin O. ~y
(3.7.188)
Comparing Eqs. (3.7.187) and (3.7.188), one can obtain the following relation between 0~(x,y) and M, m, 0, e, and a COS 0
oc~
a[M 4- m cos(O- e)]
(3.7.189)
sin 0 % = a[M 4- m cos(0 - ~)] The area equation in the stationary frame of reference is derived in the following. In the moving frame of reference, according to Section 3.7.1.4, one has dM
e~
h dp
g da
1
1 dA'
4- p~7~t 4- a ~-~7~4- ~7 4- A--/d0c---7= 0
(3.7.190)
The slope of the tangent to the ray in the moving frame of reference can be written as ! -
-
dx'
= tan 0
(3.7.191)
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z-Y Han and X-Z Yin
The differential relation between ~' and x' and y' along a ray tube in the moving frame can be expressed as
d 3dx' ady' = } d~' Ox' d~' 3y' d~x'
(3.7.192)
Substituting Eq. (3.7.192) into Eq. (3.7.190), and making a transformation of coordinate from the moving frame of reference to the stationary one by using the relations (3.7.181) and (3.7.189), one gets
lax l aA' A----~x4-GA'Oy--~---
[( aM ,.h--~xgaa) ( aM h~ g3__~)] e--3x4-~
4--a-~x 4-G e ~
4-p
4--a
(3.7.193) where G=
M sin 0 + m sin e M cos 0 + m cos e
(3.7.193)
This equation is the area equation for a small region in the stationary frame of reference. It should be noted that in Eq. (3.7.193) we still use the crosssectional area of the ray tube in the moving frame of reference A'.
3.7.2.2.1.3 Two-Dimensional Equations Denoted by M and Substituting Eqs. (3.7.189) into the geometric equations (3.7.184), and combining with the area equation (3.7.193), one gets the following set of shock dynamic equations for any small region in the stationary frame of reference:
1
1
co o+ cos _M__sin0+msine um 2 ~-))A'J + ~ L(M + m c~-s(O - ~))A'J -
sinO
cosO
3x a(M 4- rncos(O - e)) - - ~
0
]
a(M 4- m cos(0 - ~)) -- 0
(3.7.194)
1 0,4' 1 aA' - - ~ 4- GA-7-~- = - ( E x + GEy) A' ax
where aM
hap
gaa
Ex -- e-~x 4- F-~x 4- ---'~Ox
aM hap gaa Ey -- e--~y 4- - - y- 4- a ~y
It should be noted that while deriving Eqs. (3.7.180) and (3.7.181), which are used for the transformation of the coordinates from the moving frame of reference to the stationary frame of reference, m, e, and a are regarded as constants, which means that the results are suitable only for a small fluid
537
3.7 Geometrical Shock Dynamics
element (a small region), but not for the entire nonuniform flow field ahead of the shock wave. However, the Eqs. (3.7.194) have the same form, even if m, ~, and a are different in different small regions. Therefore, provided m, g and a are regarded as variables, Eqs. (3.7.194) are valid for the entire flow field and so the preceding equations are a set of 2D shock dynamic equations for the case of a nonuniform flow ahead of a shock. Substituting the third expression of Eq. (3.7.194) into the first one and eliminating A' results in a set of partial differential equations, in which the unknown variables are M and/9. The parameters m, ~, p and a are given. As an example, one can calculate the interaction of a plane shock wave with a vortex by using the foregoing equations (Han and Yin, 1993; Yin, 1995; Yang et al., 1996). Equation (3.7.185) for the unit vector for the ray tube in the stationary frame of reference can also be written in the following form by using Eqs. (3.7.189): -. (M cos 0 4- m cos e)ex 4- (M sin 0 4- m sin/3)ey i= [M2 4- 2Mm cos(O - e) 4- m2] 1/2
(3.7.195)
or
i--
M(cos O. ex 4- sin O. ~y) 4- re(cos ~-~x 4- sin s. ~y) [M2 4- 2Mm cos(O - e) 4- rrl2]1/2
(3.7.196)
Equation (3.7.196) can be expressed in the following form
.
i --
Ma + m§ IMa + m fl
(3.7.197)
or
-.,_ W~fi 4- UI'~f IWs~ 4- U17tfl
(3.7.198)
where W s = Ma I is the shock speed relative to the flow ahead of the shock wave, U 1 = ma 1 is the flow velocity in the region ahead of the shock wave, and '~f - cos e. ex + sin e- F.y is the unit vector of the flow ahead of the shock wave. It follows from Eqs. (3.7.197) and (3.7.198) that the results for the case of a nonuniform flow ahead of the shock wave are the same as for the case of a uniform flow ahead of the shock wave, with the direction of i coinciding with that of the resultant vector of (Wsfi + Ullf) or (Mfi 4-m~rf), as shown in Fig. 3.7.19. Equation (3.7.198) means that the ray direction is the direction of the locus of any point on the surface of shock. In the case of a quiescent gas ahead of the shock wave, the ray is normal to the shock surface, but in the case of a
538
z-Y Han and X-Z Yin
M-fi~.
M-fi+ m~f m ~j,
n
Moving shock _
{f
x
>
stationary frame FIGURE 3.7.19 The ray direction in the stationary frame of reference is the direction of the locus of any point on the shock surface.
moving gas ahead of a shock, in general, the ray is not normal to the shock surface. From the preceding equations, one can obtain the characteristic relations, which can be used to describe the disturbances propagation along the shock surface. However, as the flow ahead of the shock wave is nonuniform, equations compatible with the characteristic ones are complicated. Therefore, characteristic equations are omitted here (Han and Yin, 1993).
3.7.2.2.2 Three-Dimensional Shock Dynamic Equations for Shock Wave
Propagation Through Nonuniform Flow Fields
3.7.2.2.2.1 Shock Waves Propagation through Three-Dimensional Nonuniform Flow Fields When a shock wave propagates through a 3D nonuniform flow field, the shock surface and the flow field ahead of the shock wave are as illustrated in Fig. 3.7.20. The unit vector for the normal to the shock surface can be expressed as fi = cos 01 9ex 4- cos 02 9~.y 4- cos 03 9e-z
(3.7.199)
where 01 , 02 and 03 are the angles between the normal to the shock surface and the x-, y-,and z-axes, respectively and ex, ey, and ez are the unit vectors in the x, y and z directions. The unit vector in the direction of the flow ahead of the shock can be expressed as ;rf - cos el" ex 4- cos ~2" ey 4- cos ~3" ez
(3.7.200)
3.7
539
Geometrical Shock Dynamics n
Y
..Ray tube in the stationary~
9
9o ~ ooOO
~.,
9
,...Oo
frame 0
FIGURE 3.7.20
Shock wave propagation through a 3D nonuniform flow.
where gl, g2 and •3 are the angles between the direction of the flow ahead of the shock and the x-, y-, and z-axes, respectively. Obviously, the angles in Eqs. (3.7.199) and (3.7.200), are not independent and they can be expressed as
COS2 01 -[- COS2 02 -~- COS2 03 -- 1
(3.7.201)
COS2 gl -~" COS2 ~2 -[- COS2 83 -- 1
(3.7.202)
In the following the relation for the unit vector of the ray in the stationary frame of reference i, (Eq. (3.7.197)), which applies to 2D flows, will be extended to 3D flows. The relation is
Ma + ,.§ 1 --
[M~ + m;rf[
(3.7.203)
From Eqs. (3.7.203), (3.7.199), and (3.7.200) one has w a + m§ - (M co~ ox + m co~ ~ ) ~ + (M co~ O~ + m co~ e~)~
+ (M cos 03 -[- m cos g3)e.z
(3.7.204)
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z-Y Han and X-Z Yin
IMfi + m'~fl -- [M 2 -4- m 2 q- 2Mm(cos 0 1 COS/31 "[- COS 0 2 COS/32 "~- COS 0 3 COS/33)]1/2 (3.7.205) --, 1 i -- ~ [ ( M c o s 01 + m cos ~1)~x 4- (M cos 02 4- m cos r + (M cos 03 + m cos s
(3.7.206)
where N -- [M 2 + m 2 + 2Mm(cos/91 cos/31 -~- cos 0 2 cos/32 -~- cos 0 3 cos/33)] 1/2 From Eqs. (3.7.206) and (3.7.199) one has 7 -* L 1 " n = -- = cos 7 N
(3.7.207)
where from Fig. 3.7.20 one can obtain A = A' cos 7
(3.7.208)
AN = A'L
(3.7.209)
and
3.7.2.2.2.2 Three-Dimensional Equations for Shock Wave
Propagation Through Nonuniform Flow Fields in a Rectangular Coordinate System On the basis of the 2D equations derived earlier, the 3D shock dynamic equations for the case of a nonuniform flow ahead of the shock wave are derived in the following using a method that is different from that used to derive 2D equations. The basic idea is given here. First, Eq. (3.7.158) is still valid for the case of a 3D nonuniform flow ahead of the shock wave. It can be rewritten as
(AI 0 By substituting Eqs. (3.7.206) and (3.7.209) into Eq. (3.7.210), one obtains the following first geometric shock dynamic equation for 3D ~ ( M cos /91-'}- m cos /31)
8x
NL
~ ( M cos 02 --[-m cos /32)
4- -~
AIL 0 (Mc~176 + -~Z AfL
(3.7.211)
3.7
541
Geometrical Shock D y n a m i c s
Second, the other geometric equation is derived by using the same method as that in Section 3.7.1.3. The equation of the shock surface in the 3D case can be expressed as follows: S ( x , y , z, t) -
co(x, y , z ) - t -
0
(3.7.212)
Differentiating Eq. (3.7.212) with t results in igS
--+ 8t
VS.
V~ -
(3.7.213)
0
where V s is the velocity of any point on the shock surface and V S is the normal to the shock surface. The relation can also be written in the following form, 1 - Vow. ( W s + U 1 ) - - 0
(3.7.214)
where Ws is the shock speed relative to the flow ahead of the shock wave, the direction of the shock speed coincides with the normal to the shock surface, and U 1 is the velocity of the flow ahead of the shock wave. Equation (3.7.214) can also be written as 1 - IWI" I(M + l~)al cos y -- 0
(3.7.215)
or
-. ](M + rh)la cos 2, =
1
IWl
(3.7.216)
1
[M + m(fi. 7rf)]a - - IV~I
(3.7.217)
Equation (3.7.217) is the second geometric equation. By eliminating 0~ one can obtain the equations denoted by M, m, 0 and e. The procedure of the derivation is given by Vet a - Iva---[= [M + m(a. ;rf)]aVa - L a V a
(3.7.218)
Comparing Eqs. (3.7.218) and (3.7.199) yields COS 01 O~x - -
La COS 0 2
OCy---
La COS 0 3
% =
La
(3.7.219)
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z-v Han and X-Z Yin
From Eqs. (3.7.219) one can obtain the geometric equations corresponding to Eq. (3.7.217) as follows
~ \ La / - - ~
=o
(3.7.220)
-~ \ La ]--fiZZ
--0
(3.7.221)
Summarizing the derivation of the preceding equations, one finds that Eqs. (3.7.211), (3.7.220), (3.7.221), and (3.7.201) are the geometric equations for the case of a nonuniform flow ahead of the shock wave in a rectangular coordinate system. The area equation for 3D, nonuniform flows is derived by the following. The area relation for a ray tube in the case of 2D, nonuniform flows can be expressed as
dM h dp g da 1 1 dN e d-~t 4- P ~ , 4- a ~ 4- ~-; 4- ATdo---7 = 0
(3.7.222)
Obviously, this equation also suits 3D, nonuniform flows, but in this case, 0( is only a parameter and there are no other curvilinear coordinates. The change in cz' means movement along a ray tube. Consequently, the relation between the rectangular coordinates and 0( can be expressed as
x' = x'(~'),
y' = y'(cz')
z' = z'(cz')
(3.7.223)
Expanding Eq. (3.7.222) by using the functional Eq. (3.7.223) results in
\
~+~
+ ~ +~ o~,/+ . ~ + ~ + ~ +~ Oy,/~, ( aM h Op s ~4--~4~oz' ,,Oz'
4- e ~ 4 -
\ Oz'
1 OA'
~ oz'/-~
--0
(3.7.224)
where
dy/
COS 0 2
d.x'
COS 01
dz' dx'
cos 03 cos 01
By using a transformation of the coordinates from a moving frame of reference, which is attached to the flow ahead of the shock wave, one can obtain the area relation for a stationary frame of reference. The method used here is similar to that used in the case of 2D flows.
3.7
543
Geometrical Shock Dynamics
The relations between the coordinates in the moving frame of reference and the stationary frame of reference can be written as X ~ XA "-[- U c o s e I 9t / ~ x 2 -~- ma cos/~1 " ~(X, y, z) y = y' + U cos e2 " f = Y' + ma cos e2" ~(x, y, z) z = z' + U cos e3" t' = z' + ma cos e3" ~(x, y, z)
(3.7.225)
From Eqs. (3.7.225) one can find some differential expressions relating x, y, and z and x', y', and z' that are similar to Eqs. (3.7.180) and (3.7.181). Then by substituting these differential expressions into Eq. (3.7.224) one can obtain the following area equation in the stationary frame of reference
e---~ + p-~x + a-~x + A' ax ] + G1 e---~ + p-~ + a-~ + -~ (aM
hop
g aa
l aA'~
+G~ ~ - z - + : ~ + : ~ - + : , =0 (3.7.226) \ oz p oz a oz l-t' az ] where G1
--
M cos 02 + m cos e2 , M cos 01 + m cos el
G2 --
M cos 03 + m cos e3 M cos 01 + m cos el
Equation (3.7.226) can be rewritten as 1 (3A' q- G I -3A' 3A'~ = _(Ex + GIEy + G2Ez ) (3.7.227) A--;\Ox ~ --q- G2 Oz,]
where aM
hOp
gaa
aM
hOp
gaa
aM
hOp
gOa
Summarizing the foregoing discussion, one can obtain the following set of shock dynamic equations for 3D, nonuniform flows in a stationary frame of reference a
COS 01 "~- m c o s e 1
-~
) 0 ( cos0 -~-
A'L
-~
A'L
+ -~Z
A'L
a(cos01) -~ \ La ] - - ~ \ La =0 a (cos 02~
(co;o3) - ~ \
0y
La / = 0
cos 201 + cos 2 02 + cos 2 03 = 1 1 (3A' 3A' 3A"I -- -(Ex + GIEy + G2Ez) A--;\ax + G I - ~ --+G2 az,]
(3.7.228)
Z-Y Han
544
and
X-Z Yin
From the first and the fifth expressions, A' can be eliminated, and then considering the fourth expression, one can obtain three equations with three unknown variables M, 01, and 02 (or 03), which means that for problems of 3D unsteady shock diffractions, only three unknown functions are solved by using shock dynamic equations. 3.7.2.2.2.3
Three-Dimensional Equations for Shock Wave Propagation Through Nonuniform Flow Fields in a Cylindrical Coordinate System
The cylindrical coordinate system is illustrated in Fig. 3.7.21, and the unit vector for the normal to the shock wave and the flow ahead of the shock wave can be expressed as
fi -- COSOrP.r 4- COS Otpecp 4- cos Oxe.x
(3.7.229)
~f -- COSf,rer 4- COSetpeq~ 4- COS~,xe.x
(3.7.230)
Neglecting the detailed derivation of the 3D equations, one can directly write the following equations in cylindrical coordinates as
0 ((Mc~176 3r A'L
0 (COS0r~
O (Mc~176 4- -ff-~q~ A'L 0 ((McosOxWmc~ 4- -Ox A'L
0
0 (COS0x~
-~x \ La ,I--Or\ La / - - 0 0 (rc~
La
(3.7.231)
0 (COS0r~
-G\
/=0
COS2 Or 4- COS2 O~ 4- COS2 0 x -- 1
I(OA'
,0,4'
A---;\ Or 4- G1 ~
,OA'~__(Er4-
4- G20x J
,
G1E~~
4-G2Ex) '
where
OM h ~r g 3a Er = e--~r-r4- p 4 - aOr ---,
3M h Op g 3a Er = e ~ 4- p-~-~ 4 - arOq~ -~ 3M
G~ = M cos 0~0 + m cos er M cos
Or 4- m cos er
h Op g 3a G~ = M cos 0x + m cos ex M cos Or 4- m cos ~r
3.7
545
Geometrical Shock Dynamics
Y
x
A: FIGURE 3.7.21
The unit vectors in a cylindrical coordinate system.
For axisymmetric flows, that is, 3 / O q ) - O, and without a circumferential flow, that is, 0~o - g / 2 and qo - r~/2 letting 0 x - O, ex - e and Or -- re~2 -- O, e.r -- r~/2 -- e equations (3.7.231) can be simplified to read O [ (MsinO + m s i n e ) r ] 3 [ (McosO + m c o s ) r ] Or ( M + m ~o s (0 - E~ ' J + ~ L(~ u ~ ~-o-~0 - ~ ' J - o ax
sin0
] 0[
M + m c o s ( O - e))a
--~r
cos0
(M+mcos(O-O)a
]
--0
(3.7.232)
1 3.4' G' 1 3,4' _(Er + G,Ex ) A -73---~ + A'ax =
where
Er -
aM h Op g Oa e-~F "~--~ ar 4----,aOr L
aM
Ex - ~ ~ GI =
h Op
g 3a
+ -p~ + --~ax
M sin 0 + m sin e Mcos0 + mcose
3.7.2.2.2.4 S h o c k - S h o c k R e l a t i o n s for S h o c k W a v e Propagation T h r o u g h M o v i n g Gases Two-dimensional and 3D shock-shock relations for the case of a quiescent gas ahead of the shock wave were presented in Sections 3.7.1.2 and 3.7.1.3. For a shock wave propagating into a uniform and a nonuniform moving gas, one can also establish the shock-shock relations by using the same ideas. To obtain
546
z-Y Han and X-Z Yin
intuitive expressions, one can consider only 2D flows. As mentioned earlier, the shock-shock relations include both tangential and normal relations, The tangential relation means that the tangential derivative of 0~ for undisturbed and disturbed shock waves along the shock-shock surface (in the case of 2D, along shock-shock locus) in the stationary frame of reference are still equal, that is,
rlss X ( V o 0 0 - Ilss X (VO01
(3.7.233)
where fiss is the unit vector of the normal to the shock-shock surface, and (V000 and (V00x refer to the undisturbed and the disturbed shock waves, respectively. Under the condition of a moving gas ahead of a shock wave, the unit vectors of the normal to the undisturbed shock and the disturbed shock can be expressed as rio ---- [Mo 4- m(fi o 9;rf)]a(V~) o
(3.7.234)
nl = [M1 '[- m(nl" ~f)]a(V~
(3.7.235)
One can get from Eqs. (3.7.233), (3.7.234), and (3.7.235) nss x n o
M0 + ,.(ao. §
nss • n I
(3.7.236)
M 1 -'[- m(n I 9;rf)
In the case of 2D flows, as shown in Fig. 3.7.22, Eq. (3.7.236) can be written as cos(Zo - 0o) cos(Zo - 01) = M 0 + m cos(O 0 - e) M 1 + m c o s ( O 1 - g)
(3.7.237)
It follows from the preceding equation relation that the velocities along the shock-shock locus for undisturbed and disturbed shock waves are the same.
FIGURE 3.7.22 A shock-shock propagating along the shock wave in the case of a 2D, uniform flow ahead of it.
3.7 GeometricalShock Dynamics
547
The relation for the normal derivative of 0~ for both undisturbed and disturbed shock waves along the shock-shock surface is derived in the following. However, as already mentioned, for ray tubes for the undisturbed and the disturbed shock waves across the shock-shock it is necessary to use a moving frame of reference for the rays normal to shock surface. Hence, a moving frame of reference attached to the flow immediately ahead of the intersection point of the undisturbed and the disturbed shock waves is used. In this moving frame of reference, one can obtain the following two equations, which are the same as those given in Section 3.7.1.2 except for Z', sin(;~ - 0o)
A~
=
sin(;~ - 01)
A~
(3.7.238)
At the same time, one has cos(z~ - 0o)
M0
=
cos(z~ - 0~)
(3.7.239)
M1
Eliminating Z' from Eqs. (3.7.238) and (3.7.239) results in tan(01 _ Oo) = (M~ - M~)I/2(A'o2 - A~:))1/2 A'1M 1 - A'oM o
(3.7.240)
It follows from Eq. (3.7.240) that the angle between the normals to the undisturbed and the disturbed shock waves is independent of the parameters in the flow ahead of the shock waves, which means that the angle (01 - 00) is kept unchanged, when the transformation from the moving frame of reference to the stationary one is made. By combining Eqs. (3.7.237) and (3.7.240) one obtains tan(Zo - 0o) -- B1 (A'~ + Ai M~ - B~ + A'~176 Bo(A~2- " 1a,2al/2t~2 1/2 , ~'"1 - M ~ )
(3.7.241)
where B 1 -" M 1 + m c o s ( 0 1 - 8), B 0 - - M o + m cos(0 o - e). Rearranging this relation results in , tan(Zo - 00) = A~
2 - M~)I/2 - m sin(00 - e)(A~2 - A~2)1/2 [M0 - m cos(00 - e)](A~2 - A~2)1/2
(3.7.242)
The area relation of the rays for the undisturbed and the disturbed shock waves across the shock-shock in the case of a moving frame of reference is derived in the following
dM g da h dp e--d~' + a--d-~' + p-~&x' =
1 dA' A' d~z'
(3.7.243)
548
z-Y Han and X-Z Yin
It should be noted that the changes in p and a relative to those in M and 0 across the shock-shock can be neglected and so Eq. (3.3.2.113) can be simplified to read dM e~ = d~'
1 dA'
(3.7.244)
A' da'
or
A_~ -f(M1)A~ f(Mo)
(3.7.245)
Summarizing Eqs. (3.7.240), (3.7.242), and (3.7.245) gives a set of shockshock relations for the case of a moving gas for both uniform and nonuniform flows ahead of the shock wave, which also suits axisymmetric shock diffractions of a moving gas ahead of a shock wave, and as follows (Han and Yin, 1989; Li and Ben-Dor, 1995), , 2 - M2)1/2 - m sin(00 - s)(A{)2 - A~2)1/2 tan(z 0 - 00) = Ao(M1 [M0 - m cos(00 - s)](A;2 - Ai2)1/2 tan(01 _ Oo) -- (M2 - M2)1/2(A'o2 - A12)1/2 A I, M 1 _ AoM ' o
(3.7.246)
Ai f ( M 1 ) Xo
f(Mo)
As shown by Li and Ben-Dor (1995), Eqs. (3.7.246) can also be expressed in the following form: cos(Zo - 0o)
cos(Zo - 01)
B0
B1
M0 sin(z0 -/90) + m sin(z0 - s)
M1 sin(Zo - 01) + m sin(Zo - s)
BoXo
BIAI
A~
f(M1)
Xo
f(Mo)
(3.7.142)
3.7.2.2.2.5 I n t e r a c t i o n s and D i f f r a c t i o n s o f S h o c k W a v e Propagation T h r o u g h M o v i n g Gases Let us consider the diffraction of a moving shock around a cone, flying at a supersonic speed when its axis coincides with the normal to the shock wave as shown in Fig. 3.7.23. The parameters, such as M, m, 0, s, and a are self-similar,
3.7
549
Geometrical Shock Dynamics
[ Moving shock > Transmitted shock Mach stem
~ ~ ~
..........Shock-shock locus
FIGURE 3.7.23 The transmitted shock and Mach stem waves in the case of head-on interaction of a moving shock with the bow shock attached to a cone.
that is, the parameters are functions of the angle 1/only, which can be expressed as
r / = arctan(~)
3 3x
O 3r
(3.7.248)
r d x 2 4- r 2 drl
(3.7.249)
x d x 2 4- r 2 drl
By substituting Eq. (3.7.249) into the axisymmetric Eqs (3.7.232), one can obtain the following set of self-similar equations,
( cos0+ os ) + d ( M s i n O 4 - m s i n e ) cot r/9 ~ A'L dr? A'L (s,nO o) - 0 tanr/.~--~ \ - - ~ - j 4-~--~ a~
--
MsinO4-msina A'L sin 2 r/
l dA' ( dM h dp g da) A' d~? - - e--d-~ 4- p-d-~ 4- -a-d-~ (3.7.250) where L - M 4- m cos(O - e).
z-Y Han and X-Z Yin
550
If the flow ahead of the shock wave is homoentropic, for example, it is a conical flow, p and a are functions of the flow Mach n u m b e r m as shown in the followings 1 da a dr/ ldp p dr/
7-
1
m
dm
[1 + ( ( y - 1)/2)m 2] dr/ 7m
dm
(3.7.251)
[1 + ( ( y - 1)/2)m 2] dr/
The purpose is to calculate the changes in the strength and the orientation of the transmitted and the Mach stem shock waves. The procedure for this calculation is given in the following First, the conical flow field around the cone before interaction is obtained by using the conditions given. Second, the calculation of the interaction of a plane shock with the bow shock attached to the cone is made, and the transmitted shock wave at the intersection point can be obtained. Finally, the transmitted and the Mach stem shock waves in the conical flow field can be calculated by using the Eqs. (3.7.250), the shock-shock Eqs. (3.7.246), and the boundary conditions on the surface of the cone (Hart et al., 1990; Li et al., 1996). Li and Ben-Dor (1997) investigated the case of an oblique interaction of a moving shock wave with the bow shock wave attached to a wedge, including regular and irregular shock-on-shock intersections, by using the three-shock theory and geometrical shock dynamics.
REFERENCES Bryson, A.E. and Gross, R.W.E (1961). Diffraction of strong shocks by cones, cylinders and spheres. J. Fluid Mech. 10: 1-6. Cathersoo, C.J. and Sturtevant, B. (1983). Shock dynamics in a non-uniform media.J. Fluid Mech. 127: 539-561. Chester, W. (1954). The quasi-cylindrical shock tube. Phil. Mag. 45: 1293-1301. Chester, W. (1960). The propagation of shock wave along ducts. Adv. in Appl. Math. 6: 119-152. Chisnell, R.E (1955). The normal motion of a shock wave through a non-uniform one dimensional medium. Proc. Roy. Soc. Lond. A232: 350-370. Chisnell, R.E (1957). The motion of shock wave in a channel, with applications to cylindrical and spherical shock waves. J. Fluid Mech. 2: 286-298. Chisnell, R. F., (1965). A note on Whitham's rule. J Fluid Mech. 22: 103-104. Chisnell, R.E and Yousaf, M. (1982). The effect of the overtaking disturbance on a shock wave moving in a non-uniform medium. J. Fluid Mech. 120: 523-533. Collins, R. and Chen, H.T. (1970). Propagation of a shock wave of arbitrary strength in two half planes containing a free surface.J. Comp. Phys. 5: 415-422.
3.7
Geometrical Shock Dynamics
551
Collins, R. and Chen, H. T. (1971). Motion of a shock wave through a non-uniform fluid. Proc. 2nd Intl. Conf. Num. Meth. Fluid Dyn. Han, Z.Y. (1991). Shock dynamics description of reflected shock waves. Proc. 18th Intl. Symp. Shock Waves, Sendai, Japan. Han, Z.Y. and Yin, X.Z. (1989). Shock Dynamics USTC, Heifei, China: Univ. Sci. and Tech. China, (Teaching material for graduate students). Han, Z.Y. and Yin, X.Z. (1992). Shock-shock relation for moving gas ahead of shocks. Sci. China A7: 725-733. Han, Z.Y. and Yin, X.Z. (1993). Shock Dynamics, Dordrecht Boston/London: Kluwer Academic Publishers, Sci. Press. Han, Z.Y. and Yin, X.Z. (1989). Two dimensional equations of shock dynamics for moving gas ahead of shock wave. Sci. China A32: 1333-1344. Han, Z.Y., Feng, J.Q., and Yin, X.Z. (1990). A new method for calculating head-on interaction of a moving shock with a cone flying at supersonic speed. J. Chinese Soc. Astro. 4: 1-6. Henshaw, L.E, Smyth, N.E, and Schwendmann, D.W (1986). Numerical shock propagation using geometrical shock dynamics. J. Fluid Mech. 171: 519-545. Li, D.Y., Yin, X.Z., and Han, Z.Y. (1996). Calculation of shock dynamics for oblique interaction of plane moving shock with bow shock of a cone. J. USTC, 26, Univ. Sci. & Tech. China, Heifei, China. Li, H and Ben-Dor, G. (1997). Analytical investigation of two-dimensional unsteady shock-onshock interactions. J. Fluid Mech. 340: 101-128. Li, H., Ben-Dor, G., and Han, Z.Y. (1994a). Modification of Whitham's ray shock theory for analyzing the reflection of weak shock waves over small wedge angles. Shock Waves 4: 41-45. Li, H., Ben-Dor, G., and Han, Z.Y. (1994b). Analytical prediction of the reflected-diffracted shock wave shape in the interaction of a regular reflection with an expansive comer. Fluid Dyn. Res. 14: 229-239. Li, H and Ben-Dor, G. (1995). Reconsideration of the shock-shock relations for the case of a nonquiescent gas ahead of the shock and verification with experiments. Proc. Roy. Soc. Lond. 451: 383-397. Milton, B.E. (1975). Simplified ray-shock theory parameters for use with diatomic gases. UNSW/ FMT/2, Univ. New South Wales, Sydney, Australia. Schwendmann, D.W. (1988). Numerical shock propagating in non-uniform media. J. Fluid Mech. 188: 383-410. Skews, B.W. (1971). The shape of a shock in regular reflection from a wedge. CASI Trans. 4: 16-19. Whitham, G.B. (1957). A new approach to problem of shock dynamics. Part 1: Two-dimensional problems. J. Fluid Mech. 2:146-171. Whitham, G.B. (1958). On the propagation of shock waves through regions of non-uniform area or flow. J. Fluid Mech. 4: 337-360. Whitham, G.B. (1959). A new approach to problem of shock dynamics. Part 2: Three-dimensional problems. J. Fluid Mech. 5: 369-386. Whitham, G.B. (1968). A note on shock dynamics relative to a moving frame. J. Fluid Mech. 31: 449-453. Whitham, G.B. (1973). Linear and Nonlinear Waves, New York: Wiley-Interscience. Whitham, G.B. (1987). On shock dynamics. Proc. Indian Acad. Sci. (Math. Sci.) 96: 71-73. Yang, H.J., Yin, X.Z., and Han, Z.Y. (1996). Differential method of shock dynamics in moving media. J. USTC 23, Univ. Sci. and Tech. China, Heifei, China. Yin, X.Z. (1995). Characteristic method of shock dynamics used in shock-vortex interaction. Trans. Japan Soc. Aero and Space Sci. 38(120): 151-160. Yin, X.Z., Wang, Y., and Luo, X.S. (2000). Estimate on 3-D triple point trajectory for reflection of shock over cone. to be published.
5 52
Z-Y Han and X-Z Yin
Yin, X.Z., Yang, H.J., and Han, Z.Y. (1993). Differential method of shock dynamics in quiescent media. J. USTC, 23, Univ. Sci. and Tech. China, Heifei, Chaina. Yu, S. and Groenig, H. (1984). A simple approximation for axially symmetric diffraction of plane shocks by cones. Z. Naturforsch. 39a: 320-324.
CHAPTER
4.1
Shock Tubes and Tunnels: Facilities, Instrumentation, and Techniques 4.1
Shock Tubes
MICHIO NISHIDA Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Japan
4.1.1 Introduction 4.1.2 Shock Jump Relation 4.1.3 One-Dimensional Propagation of a Small Disturbance 4.1.4 Shock Tube Theory 4.1.4.1 General Description of a Shock Tube 4.1.4.2 Relations between Regions (1) and (2) 4.1.4.3 Relations between Regions (2) and (3) 4.1.4.4 Relations between Regions (3) and (4) 4.1.4.5 Reflection of a Shock Wave from the Shock Tube End Wall 4.1.4.6 Interaction between the Reflected Shock Wave and the Contact Surface 4.1.5 Techniques for Shock Tube Operation 4.1.5.1 Diaphragm 4.1.5.2 Variable Cross-Section Shock Tube 4.1.5.3 Shock Tube for Generating Strong Shock Waves References
4.1.1 I N T R O D U C T I O N Shock tubes have been employed by a number of shock wave researchers as the most reliable facility for investigating the structure of a shock wave, shock
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
553
554
M. Nishida
~"
f high-pressure I section diaphragm
I
~
shock wave
low-pressure section
FIGURE 4.1.1
Shock tube.
wave propagation and reflection, and so on, because their operation principle is surprisingly very simple. Vieille (1899, 1970) described shock tube experiments in 1899, and his work today is recognized as the first published paper on shock tube flows. In his experiments he divided a cylindrical tube into two parts by a diaphragm and obtained a discontinuity (shock wave) with a speed over 600m/s by rupturing the diaphragm with a high-pressure gas that was supplied into one side. This type of facility is still used at present as a fundamental type of a shock tube (Fig. 4.1.1). The effect of the diaphragm thickness on the discontinuity speed was investigated by varying the rupturing pressure using colloidion diaphragms of 0.29 and 0.11 mm in thickness and a paper diaphragm. Further, Vieille attempted to increase the propagation speed of the discontinuity by using a driver section with a larger volume but keeping the diaphragm thickness the same. The result showed that the propagation speed was independent of the volume of the driver section, which today is explained by shock tube theory. In memory of his pioneering work, an invited paper titled "Paul Vieille Lecture" is being presented at the meetings of the international symposia on shock tubes and waves.
4.1.2
SHOCK JUMP RELATION
In the following a brief description of the jump relations across a normal shock wave is given for easier understanding of a shock tube flow and the wave propagation in it. For details see Chapter 3.1. As mentioned there, when a perfect gas flowing supersonically with pressure Pl, density Pl, temperature T 1 and velocity u 1 encounters a discontinuity, then the pressure jumps to P2, the density to P2, the temperature to T2 and the velocity to u 2 behind the discontinuity. Such a discontinuity is called a shock wave. Figure 4.1.2 shows the flow-property jumps across a shock wave in a fixed shock coordinate. The steady-state flow across a shock wave is governed by the following fundamental conservation equations: Mass /91l/1 "--/921,/2
(4.1.1)
4.1
Shock Tubes
555 Shock Wave
It 2
/t 1
P2, P~, T2
pl, Pl. T1
U2
l pl
Tpl
FIGURE 4.1.2
Shock wave.
Momentum
Plu~i + Pl = p2u~ + P2
(4.1.2)
CpT 1-]-
(4.1.3)
Energy -- CpT 24 u~ ~
where Cp is the specific heat at constant pressure. The pressure ratio across the shock wave P2/Pl can be easily obtained from Eqs. (4.1.1) to (4.1.3) and
556
M. Nishida
expressed as a function of a density ratio across the shock w a v e /92//01 as follows:
7 + 1 P2 i P_! = 7 - 1 fll Pl 7 + 1 P2 7-1 Pl
(4.1.4)
where ? is the ratio of specific heats. An alternative form of Eq. (4.1.4) is lq P__~2= Ul = Pl U2
7+1
P2
7 - 1 Pl 7 + 1 P2 i 7 - 1 Pl
(4.1.5)
Equations (4.1.4) and (4.1.5) are called the Rankine-Hugoniot relations. The pressure jump across the shock wave P2/Pl is plotted vs the density ratio P2/Pl in Fig. 4.1.3. It is apparent from Eq. (4.1.5) that P2/Pl ~ (7 + 1)/(7 - 1) for P2/Pl ~ oo. This limit is also shown in Fig. 4.1.3. Figure 4.1.3 also shows the line for the isentropic change given by pp-~ - const. The difference between shock relation and isentropic relation increases with increasing P2/Pl. The density, velocity, pressure and temperature ratios, and the velocity change across a shock wave can be expressed as a function of the pre-shock flow Mach number M 1 (= ul/al) as follows: P_! = Ul = (7 + 1)M12 Pl u2 2 4- (7 - 1)M12 2y P_! = 1 4(M12 - 1) Pl 7+I
(4.1.6)
T_! = 1 4 - 2 ( 7 - 1______+______~1 2) 72 M 1(M _ 1) T1 (7 4- 1) 2 M12
(4.1.8)
u 2 - u 1= - 2 al 7+1
M1-
(4.1.7)
(4.1.9)
where a is the speed of sound. The entropy change across the shock wave s 2 - Sl is given by s 2 - S l = In R
1+
(M12 - 1) 7+ 1
(7
(-7 + 1)M12 1)M12q- 2
(4.1 10) "
where R is the specific gas constant. The total pressure ratio across the shock wave is expressed as P 0___~2= P01
[ 2,,, 1+ ~
(M12 - 1)
]-l,,u-,[ (.,,,+I)M1 ],,,,u-, (7 - i)M-~~- 2
(4.1.11)
4.1
557
Shock Tubes
50 -
.o
40 -
.~
30-
,5
g
P2
TI 20
10
3/=1.4
1
5z_~ 3~-1
I
I
10
15
/O2
/91
P2/Pl vs P2/Pl (Rankine-Hugoniot
FIGURE 4.1.3
relation).
7 6 5 /92 "~1
4 3 2 1
1
2
3
4
5
6
7
8
Mt FIGURE 4.1.4
P2/Pl vs
M 1.
9
10
l
I
l
0
I
~'~
4.1
559
Shock Tubes
9
I
I
I
I
I
I
I
I
m
76-
=7/5
U2-U 1 5 -
al
=5/3
4 3
i
2 1 071
1
2
I
I
t
I
I
I
i
3
4
5
6
7
8
9
10
MI FIGURE 4.1.7
7|
I
I
- ( u 2 - l~l)/al vs M 1.
I
I
'"
I
I
I
I
/
-I
6
$2-S 1
1
2
3
4
5
6
7
8
Ml FIGURE 4.1.8
(s2 - s])/R vs M 1.
9
10
560
M. Nishida
1
0.8 0.6 Po2 Po~ 0.4 0.2 o
1
2
3
4
5
6
7
8
9
10
MI FIGURE 4.1.9
Po2/Po, vs M 1.
Equations (4.1.6) to (4.1.11) are plotted for a monatomic gas (7 = 5/3) and a diatomic gas (7 = 7/5) in Figs. 4.1.4 to 4.1.9, respectively. The shock jump relations are expressed by the pressure ratio P21 -- P2/Pl for the convenience of the application to a shock tube flow: P_! -- u_[ = 1 q- 0r Pl t12 ~ -4- P21
(4.1.12)
T2 ~ + P21 T-~ = P21 1 + ~P21
(4.1.13)
u_! = M1 =./~P_21 .q-_i al V 1+=
(4.1.14)
1,/2 --1/1 . _ _
al
( 0 ~ - 1)(p21 -- 1)
(4.1.15)
V/(1 + a)(1 + ~P21)
where a = (7 + 1)/(7 - 1).
4.1.3 ONE-DIMENSIONAL A SMALL DISTURBANCE
PROPAGATION
OF
The continuity equation for a nonsteady one-dimensional (1D) flow is 1 Dp 8u - ~4---= 0 (4.1.16) p Dt ax
4.1
561
Shock Tubes
where D/Dt = O~Ot + uO/Ox, and t and x are the time and distance, respectively. Assuming an isentropic flow,
Dp(dplsDpl Dp
(4.1.17)
where ( )s denotes isentropic condition. From equations (4.1.16) and (4.1.17) one obtains the following equation: 1 Dp
D--t-4-
a20U
~xx -- 0
(4.1.18)
Multiplying the equation of motion Du/Dt + (llp)3p/Ox = 0 by a, Du
a 3p
Dt
p 3x
a~4--
----0
(4.1.19)
Subtracting Eq. (4.1.19) from Eq. (4.1.18), and adding Eqs. (4.1.18) and (4.1.19), results in the following two equations:
aN+a
N+a
=o
(4.1.20)
-0
(4.1.21)
Let us consider the meaning of D/Dt 4- aO/Ox. Lines given by dx/dt -- u :F a in the (x, t)-plane indicate the wavefronts of small disturbance traveling upstream and downstream, respectively (see Fig. 4.1.10). These wavefronts are called left-facing Mach lines for dx/dt = u - a and right-facing Mach lines for d x / d t - - u 4- a. The left- and fight-facing Mach lines are expressed by F - const, and f2 - const., respectively, where F and f2 are a function of x and t: -- u - a,
-- u + a
F
(4.1.22)
f2
Next, one considers the change of G, which is an arbitrary function of x and t, along the left- and right-facing Mach lines:
(-~t)
3G
-K a-K+
3G (3x)
-at a
0G
3G
DG
3G
Dt+a
It can be understood from these two equations that D / D t - a O l O x and D/Dt 4-aO/Sx represent the time derivative along the left-facing and right-
562
M. Nishida
facing Mach lines, respectively. Hence, Eqs. (4.1.20) and (4.1.21) become, respectively, --
+
=
0
(4.1.23)
+
= 0
(4.1.24)
Equations (4.1.23) and (4.1.24) can be rewritten as 1 7:du + - - dp = 0 pa
(4.1.25)
where the signs 7: mean, respectively, along the left- and right-facing Mach lines. From the first and second laws of thermodynamics, l dp P
dh
Tds
(4.1.26)
where h is the enthalpy and s is the entropy. The flow across a small disturbance is isentropic, so that setting ds -- 0 in Eq. (4.1.26) and using the resultant equation in Eq. (4.1.25), one obtains 7:aclu + dh - 0
(4.1.27)
Equation (4.1.27) is rewritten for ideal gas as follows: 2 du 7: ~ y-1
da - 0
(4.1.28)
Integration of Eq. (4.1.28) yields 2 ?-1
a -- Q = const.
(4.1.29)
on the left-facing Mach line (dx/dt = u - a) and 2 u 4-~a y-1
= P = const.
(4.1.30)
on the right-facing Mach line (dx/dt - u + a), where 2 u7:~a ~-1
is called the ~Riemann invariant. The right- and left-facing waves are called P-waves and Q-waves, respectively.
4.1
563
Shock Tubes
dx
u _ a
dt
X FIGURE 4.1.10
4.1.4
Small disturbance wave propagation.
SHOCK TUBE THEORY
4.1.4.1 TUBE
GENERAL DESCRIPTION OF A S H O C K
To date, various types of shock tubes have been designed and developed. Of these, the operation principle of the simplest shock tube with a constant cross section will be explained here. As illustrated in Fig. 4.1.11, this type of a shock tube is composed of high-pressure and low-pressure sections, which are initially separated by a diaphragm. The rupture of the diaphragm generates a shock wave, which travels into the low-pressure section. The shock wave is reflected from the end wall of the shock tube and encounters an opposing contact surface that is the interface initially separating the high-pressure and low-pressure gases. The reflected shock wave partly penetrates the contact surface and partly is reflected from it as a shock wave or an expansion fan. Detailed description of 1D wave interaction is given in Chapter 7. Thereafter, multiple wave reflections occur between the end wall and contact surface. The high-pressure gas is expanded into the low-pressure section of the shock tube immediately after the diaphragm rupture. This generates an expansion wave traveling at the speed of sound from the initial position of the diaphragm to left, into the high-pressure section and carrying the information of the expansion to the undisturbed portion of the high-pressure gas. This is an expansion wavefront (or a rarefaction wavefront). The pressure and temperature distributions at time t = t 1 after the diaphragm rupture are shown in Fig. 4.1.11. In the x - t diagram shown in Fig. 4.1.11, regions (1) and (4) are the initial states of the low-pressure and high-pressure sections, respectively. If the
564
M. Nishida
t ~ .,,.,..oo~176176176 ,,
',,
"
o:,,,"
:.
i
..
o~ ....
.
k
i
-..:.,. .... "L" \
i
',, expansion
(3)
,..-" ".... ".,,, ',,,,.~vo
-....... ~ , ( ",.."-" _ __ _(_4)___
II
~
,
I
I
I
I
, ~:~.t ..... "~ (~) _~
~o'~:.~ ............... ~
~
! ',~I
~,~,~,,ossu, o ',i section
....
~
~
~'*'''*''~
(1)
_
l
I
I~
I
,o,,,,.,:,,o~,,,o so~,~o,,
diaphragm
P
P4
__
L
r~
r,
LI FIGURE 4.1.11 ture.
Wave propagation in a shock tube, and distributions of pressure and tempera-
4.1 ShockTubes
565
initial conditions in regions (1) and (4) are specified, the entire flow behavior, after the diaphragm rupture, could be predicted. This process is outlined later.
4.1.4.2 RELATIONS BETWEEN REGIONS AND (2)
(1)
The shock jump relations are employed to determine the state of region (2) behind the moving shock. As shown in Fig. 4.1.12(a), one can consider a shock wave traveling into a quiescent gas (from left to right) at a speed Usl. The shock wave induces a velocity u 2 (from left to right) at a speed Usl. The shock wave induces a velocity u 2 (from left to right) on the post-shock gas. In order to apply the shock jump relations between regions (1) and (2), one needs to stop the moving shock. This is done by imposing a constant velocity equal to the shock wave velocity but in the opposite direction Usl o n the entire flow field. The result is a stationary shock wave as shown in Fig. 4.1.12(b). Note that hereafter u and v denote the flow velocity in a moving-shock coordinate system and that in a fixed-shock coordinate system, respectively. Using Eq. (4.1.6), one obtains P_! = Vl = ~ U s l _-- (Yl + 1)M21 Pl V2 Usl - u 2 2 + (71 - 1)Ms21
(4.1.3 1)
where Ms1 = U s l / a I is the shock wave Mach number. From Eq. (4.1.31)
u2 = ~ 714-1
Ms1-
(4.1.32)
The shock jump relations for pressure and temperature can be obtained by replacing M 1 by Msl in Eqs. (4.1.7) and (4.1.8), respectively. They are written along with Eq. (4.1.31) as follows:
F,_~=
( ~ + 1)M~
P-~ = 1 + 2 ~ (M~ - 1) Pl 71 + 1
~--~= T1
1 + 2 ( ~ - 1) ~ M ~ + I (Ms5 - 1) (71 + 1) 2 M21
(4.1.33) (4.1.34) (4.1.35)
566
M. Nishida
(2)
(1) Us,
U2
U l --
V2 - - - U s I
0
(a) moving shock FIGURE 4.1.12
"q- U2
U1 --- - - U s 1
(b) fixed shock
Moving and fixed reference frames ((a) and (b)) for a shock wave.
The shock jump relations expressed by P21, which has been given by Eqs. (4.1.12) to (4.1.15), can be applied to the shock tube flow as follows: P_! = vl = 1 + 0~lP21 t91
V2
(4.1.36)
0~1 "~- P21
T2 0~1 "+- P21 T--~= P21 1 + 0~1P21 V2
2
a-~l= Ms1
v2 - Vl
--
u2
--
=
(4.1.37)
0~1P21+ 1 1 ~ ~1
(gl- 1)(/921- 1) ~/(1 + 0tl)(1 + 0tiP21)
(4.1.38)
a1
(4.1.39)
where 0~1 - (Yl + 1 ) / ( Y l -- 1).
4.1.4.3 R E L A T I O N S BETWEEN R E G I O N S AND (3)
(2)
The region behind the contact surface is referred to as region (3). Although the temperature and the density could be different across the contact surface, the velocities and pressures at both sides of the contact surface must be equal, that is:
u3 = u 2
(4.1.40)
P3 -- P2
(4.1.41)
4.1
567
Shock Tubes
4 . 1 . 4 . 4 RELATIONS BETWEEN REGIONS AND (4)
(3)
The space between regions (3) and (4) is characterized by the expansion waves. Changes across the expansion wave are governed by an isentropic process, hence, the isentropic relationship can be employed for evaluating the flow from region (3) to region (4). Therefore, the following isentropic relation is applied: 274
P4
(a4~ ~-7--~
P3
k,a 3 /
(4.1.42)
The expansion wavefront, traveling upstream from the initial diaphragm position is a Q-wave. This wave is reflected from the end wall as a P-wave and then travels downstream towards region (3). As the value of P is constant on a P-wave, P4 = P3
(4.1.43)
Using Eq. (4.1.30) in Eq. (4.1.43), 2 114 3-
"
?4-
1
2
C/4 -- U3 3-
?3- 1
a3
(4.1.44)
Because the gas in region (4) is at rest, ll4 = 0. Hence, Eq. (4.1.44) is reduced to
?4
2 -
2 I a4 - - 113 3-
?3-
1
a3
(4.1.45)
Combining Eqs. (4.1.32), (4.1.34), (4.1.40) to (4.1.42), and (4.1.45) yields
P4_[ 271 2 ]1 Pll 1 3- 713.... 1 (Ms1- 1)
1
?4 713-1
1 aI
(4.1.46) Ms1_
a4
where 74 ~ " 73 has been employed. Thus, the pressure ratio P4/Pl c a n be determined by specifying Msl , ?1, ?4, al/a4. Inversely, specifying the conditions in the high- and low-pressure sections, the shock Mach number Msl is uniquely determined. In Fig. 4.1.13 the shock Mach number Msl is plotted vs the initial pressure ratio P4/Pl for T4 = T 1. Using the same gas in both the
568
M. Nishida
high- and low-pressure sections of the shock tube and setting T4 -- T 1, it is apparent from Eq. (4.1.46) that for P4/Pl ~ oo: Msl ~ 4.24
for monatomic gas
Ms1 ~ 6.16
for diatomic gas
(7 = 5/3) (7 - 7/5)
For simulating shock waves experienced by spacecraft during reentry into the earth atmosphere, much higher shock wave Mach numbers are needed. For P4/Pl -+ oo, Eq. (4.1.46) may be approximated by Ms 1 ~ ~1 + 1 a 4 ~ 4 - 1 a]
(4.1.47)
It is seen from the Eq. (4.1.47) that for generating strong shock waves in a shock tube, the value _
a4 = 7 / ' ml T4 al V~I m4 T1
(4.1.48)
must be as large as possible, where m is the gas molecular weight. Hence, it is desired to employ high-temperature gas with low molecular weight at the highpressure section (driver gas). Figure 4.1.14 shows the effect of the driver gas molecular weight on the obtained strength of the shock wave when the driven gas is nitrogen. It is apparent from Fig. 4.1.14 that using He as the driver results in a much higher shock wave Mach number than N2. The shock wave Mach number vs P4/Pl is also shown in Figs. 4.1.15 and 4.1.16 where the |
|
|
!
T
7
]
1
MS1
1
10~
101
102
103 104 P4/Pl
lOs
106
FIGURE 4.1.13 ShockMach number Msl vs P4/Pl for monatomic and diatomic gases.
4.1
569
Shock Tubes
20 18 16 14 12 M,i 10 8
He/N2 -
6 4 2 0~
I
I,
I
I
t
101
102
10 3
10 4
10 5
10 6
Pa/Pl FIGURE 4.1.14
Shock Mach number Ms1 vs P4/Pl for shock tubes with He/Nz and N2/Nz.
driven gas is N2 or Ar, respectively. The effect of the driver gas temperature on the obtained shock wave strength is shown in Fig. 4.1.17. It should be noted here that the results shown in Figs. 4.1.13 to 4.1.17 are applicable to perfect gas only. Once the temperature in the flow field inside the shock tube is high enough to excite internal degrees of freedom, the assumption of a perfect gas behavior is no longer applicable.
20
I
I
I
I
!
,6f
14 '
12
M,1 10 8 6 4
i
2 010 ~
1 01
1 02
103
1 04
105
1 06
Pa/Pl FIGURE 4.1.15
Shock Mach number Ms1 vs P4/Pl for shock tubes with H2/N2 and and He/N2.
570
M. Nishida
2O ! 18
Hf
16 14 12
He/Ar
M,1 10 8
6 4 2 00
101
102
103
104
105
t
106
P4/Pl FIGURE 4.1.16
Shock Mach number
30
Msl vs P4/Pl for
i
!
shock tubes with argon as driven gas.
i
I
He/N2
Ms, 15 10
1
10~
101
102
-
|
!
|
103
104
10s
10 6
P41Pl FIGURE 4.1.17 Effectof driver gas temperature on shock Mach number Ms].
4.1.4.5
R E F L E C T I O N OF A S H O C K W A V E FROM
THE S H O C K T U B E E N D W A L L A shock wave is reflected from the end wall of the shock tube and then travels upstream (see Fig. 4.1.18). Then it is desirable to know the conditions in region (5) behind the reflected shock wave. The gas in this region is at rest,
4.1 ShockTubes
571
thus u s = 0. The reflected shock wave travels into the oncoming flow whose velocity is u 2. As shown in Fig. 4.1.19(b), the movement of the reflected shock wave is stopped by imposing a velocity Us2 on the flow field shown in Fig. 4.1.19(a). The shock jump relations (4.1.36) to (4.1.39) are applied to the relations across the reflected shock wave by replacing the subscripts I and 2 by 2 and 5, respectively. Then one obtains P__~5= 0r
P2
V5
-
-
(4.1.49)
4- 1
~162 P52
/'5 _ 0~1 4- P52 T2 --P5214- ~1P52
(4.1.50)
v2___Z= /~LP524- 1 a2 V 1 4- o~I
(4.1.51)
V2r =
U2=
a2
a2
--(0~ 1 -- 1)(p52 - 1) v/(1 4- Oel)(1 4- o~1P52)
(4.1.52)
where P52 = Ps/P2, and V2r denotes the flow velocity in region (2) relative to the reflected shock wave as shown in Fig. 4.1.19(b), so that V2r is not equal to v2 employed in the preceding section. Because v5 - V2r = - u 2 as shown in Fig. 4.1.19(b) and also v2 - v I = u 2 in Eq. (4.1.39), (0~1 -- 1)(/952- 1)
u2 - ,/-(1 + =6(1 +
a 2 --
(0~1 -- 1)(p21 - 1)
,/(1 + =1)(1 + =1p 1)
a1
(4.1.53)
Because a2/a I = (T2/T1) 1/2, one can obtain the following equation from Eq. (4.1.37): a2 _ / ~ , ~14.P21 al ~/~'211 + 0r
(4.1.54)
Using Eq. (4.1.54) in Eq. (4.1.53), then P__55_~ P21(~14. 2 ) - 1 P2 0~14. P21
(4.1.55)
The combination of Eqs. (4.1.52), (4.1.54) and (4.1.55) yields (1 + 0~1)P21aI 4. 0r
(4.1.56)
V2r -- X/(1 4- 0r
Because u 2 -- V2r + Us2, the speed of the reflected shock wave U~2 is given by Us2 -- --
2p21 - ~ - ~ 1 - 1 a1 V/(1 + al)(1 4- alP21)
(4.1.57)
572
M. Nishida
Using Eq. (4.1.34) in Eq. (4.1.55), the pressure ratio across the reflected shock Ps/P2 can be expressed in term of the incident shock wave Mach number Msl as follows: P___~5= -2(71 - 1) 4- Ms21(371 - 1) P2 2 4- M21(71 - 1)
(4.1.58)
Figure 4.1.20 shows Ps/P2 vs Msl. Using Eq. (4.1.58) in Eqs. (4.1.49) and (4.1.50) allows plotting Ps/P2 and Ts/T 2 vs Msl , as is shown in Figs. 4.1.21 and 4.1.22, respectively. Let us express Ps/Pl and Ts/T 1 as a function of Ms1. The following two equations are derived from the combination of Eqs. (4.1.34) and (4.1.58), and that of Eqs. (4.1.35), (4.1.50) and (4.1.58):
P__~5= [271 M2 .-_ _(71 Pl
L
%
[2(~ 1
T1
[-2(71 - 1) + M21(371
71-t- 1 --1)] -
-
]2q-~c/sTl(-~l--ii--1)]
1)Ms214- 3 - 71][(371 - 1)Ms21 - 2(~ 1 2 (71 + I)2Ms1
-
-
1)]
(4.1.59)
(4.1.60)
and one has P___55__ P5 T1 Pl Pl T5
(4.1.61)
I ""=~ !'i' _1 Z
I I
~//////////////~
,_I2) ~, 115)~
~//////////////~ FIGURE 4.1.18 Shockwave reflection from the end wall of a shock tube.
4.1
573
Shock Tubes
(2)
(5)
Us2 us
-
-
--Us2
(b) fixed s h o c k
(a) moving shock FIGURE 4.1.19 (b)).
?)5
vsr = - U s 2 + us
u5 = 0
Moving and fixed reference frames for a reflected shock wave (see both (a) and
From Eqs. (4.1.59) to (4.1.61) and Eq. (4.1.46), one can show Ps/Pl, Ps/Pl and Ts/T 1 vs an initial driver to driven gas pressure ratio P4/Pl. These are shown in Figs. 4.1.23 to 4.1.25. In Figs. 4.1.20 to 4.1.25 the conditions of the same gas and equal temperature for the driver and driven gases are employed.
4.1.4.6
I N T E R A C T I O N BETWEEN THE R E F L E C T E D
S H O C K WAVE AND THE C O N T A C T SURFACE Consider the interaction of the reflected shock wave with the opposing contact surface. The two different interaction features that are possible are illustrated
I
I
I ....
I
-
I
I
I
I
i
4
~ =7/5
6 P_A P2
5 4
1
2
i
I
I
I
I
I
i
3
4
5
6
7
8
9
Ms~ FIGURE 4.1.20 P5/P2vs M~.
10
574
M. Nishida 4
3 P5
2
1
1
2
3
4
5
6
7
8
9
10
Msl
Ps/P2 vs Msl.
FIGURE 4.1.21
in Figs. 4.1.26 and 4.1.27, respectively. One is a case where the interaction causes the contact surface to travel back to the left, and, consequently, as shown in Fig. 4.1.26, expansion waves that propagate towards the end wall are generated. In the other case, after the interaction, the contact surface is still approaching the end wall as shown in Fig. 4.1.27. In the latter case, the
2.5
:T5
,,/=5/3
m
2
1.5
2
3
4
5
6
7
8
Ms, FIGURE 4.1.22
Ts/T 2 vs Ms1.
10
--
~%%%%%%%%%%%
'~,,
%%%
~',
~i,,,,
I
i~
~'
'
'
I
'
'
",
'
'
I
,~., ,'~-,~
\~
%%%%%
%%
'
"",.t
576
M. N i s h i d a
10
I
'
I'
I
.
.~
S S~SS
jS SS
T1
0
10 ~
I
I
I
101
10 2
10 3
10 4
P4 Pl FIGURE 4.1.25
Ts/T 1 vs P4/Pl.
reflected shock wave is partly transmitted across the contact surface and partly reflected from it. Let us determine the properties in regions (7) and (8) for these two cases. Equation (4.1.52) can be applied to the relationship between regions (3) and (8) by considering ~1 = ~2 and by replacing the subscripts as 2 ~ 3 and 5 --~ 8, (0~4 -- 1)(P83 - - 1) v8 -- v3 -- u 8 -- u 3 -- --a3 v/(1 + o~4)(1 --[-o4P83 )
-- N7 -- U2
(4.1.62)
where us = 1~7 and u 3 = u2 have been used because the flow velocities at both sides of the contact surface m u s t be equal. The gas in region (3) was initially in the driver section, thus ~3 = ~4 has also been employed in Eq. (4.1.62). Substituting u2 given in Eq. (4.1.39) into Eq. (4.1.62), the following equation can be obtained:
(~1- 1)(P21- 1) u 7 -- a I v/(1 + ~1)(1 + ~zlp21)
(0~4 - 1)(/983 - - 1) a3
~/(1 +
0~4)(1 -b 0~4P83)
(4.1.63)
4.1 Shock Tubes
577
If the wave traveling into region (5) is a rarefaction wave fan as shown in Fig. 4.1.26, then P7 < P5
(4.1.64)
Therefore, the relation across the rarefaction waves can be applied in this case, so that Q7 - Qs- Because u 5 -- 0, 2 U7 -- ~ a
)'1- 1
2 7 = -- ~ a
71- 1
5
(4.1.65)
The isentropic relation between regions (5) and (7) is given by P7__ (a55) a7 271/()'1--1) P5
(4.1.66)
From Eqs. (4.1.65) and (4.1.66), P7__
,~ 2.fl/(),1_1) 1 --1 7 1 - 1 117
(4.1.67)
Hence, one obtains ii 7 < 0. When the wave traveling into region (5) is a shock wave, then (4.1.68)
P7 > P5 t
\ (3 ~
(8) ' - , . ( 7 ) x x f )
stun fan
9"17~"~'~'(~'~e.x~- (2) ....... co,-," ~.....-~....._..-.----'~~r ~ a'4e (1)
end wall FIGURE 4.1.26 Back-facingcontact surface.
578
M. Nishida
t
~
shock ~a,4e (1)
end wall FIGURE 4.1.27
Forward-facing contact surface.
and hence Eq. (4.1.39) is applied in this case by changing as 1 -~ 5 and 2 --~ 7 and by considering/)7-/)5 --/~7: (oc 1 -
1)(p7 ~ -
1)
a5
(4.1.69)
u7 -- ~/(i + o~1)(i+ o~lP7s) which leads to u 7 > 0. One can summarize as follows: u7 < 0
for rarefaction waves, P7 < P5
u7 > 0
for a shock wave, P7 > P5
If u 7 = 0 could be realized, no wave would reflect at the contact surface. This particular case is said to be t a i l o r e d . More details of the tailored conditions will be given later. There is no analytical method for obtaining the properties in regions (7) and (8), thus one has to entrust this task to numerical calculations as explained in what follows. One can start initially by guessing the value of P83, and then u 7 can be temporarily determined from Eq. (4.1.63). 1. If u 7 < 0, one can proceed in the following way: (a) P75 can be determined by Ps3 --P75P52, as P52 is known from Eq. (4.1.55) or Eq. (4.1.58). (b) On the other hand, P75 is also obtained by employing the value of u7 in Eq. (4.1.67). (c) Compare these two values of P75 obtained in Steps (a) and (b). Vary
4.1
579
Shock Tubes
the value of P83 and iterate the forementioned process until the two values of P75 coincide with each other. Once P83 is finally determined, the states in regions (7) and (8) will become clear as described in what follows. (d) The speed of the transmitted shock wave Us3 can be determined by using the value of Ps3 in
3v = U u 3 -s a3 (e)
3 = foc4P83 + 1
a3
(4.1.70)
V 1 + o~4
a 7 is determined by using u 7 in the following equation, obtained from Eq. (4.1.65);
a7--a5(l
q 7-1u-~5)2
(4.1.71)
(f) The speeds of the rarefaction wave head and tail are given by, respectively, 1,t5 -b 6/5 = a 5 //7 "+- a7
(g) a s is obtained from the following equation derived from Eq. (4.1.50). a8 _ ~ p ~ + Ps3 a--3 83 1 + 0~4Ps3
(4.1.72)
2. If u 7 0, one can proceed in the following way: (a) u 7 can be also obtained from Eq. (4.1.69). (b) Compare these two values of u 7. Vary the value of Ps3 and iterate the forementioned process until the two values of u 7 coincide with each other. (c) Us3 can be determined by using the value of Ps3 in the following relation derived from Eq. (4.1.38): >
v_.2 = u3 - Us3 =
# 4P83 + 1
a3
V 1 + 0~4
a3
(4.1.73)
determined from the following equation which is derived from Eq. (4.1.37):
( d ) a 7 is
a7 a5
__
~p
0~1 -+- P75 75 1 + 0~1P75
(4.1.74)
(e) The speed of the shock wave reflected at the contact surface Us5 is
580
M. Nishida
calculated by
/0~1P75 + 1 Us5 - a s v 1 + 0~1
(4.1.75)
which is derived from Eq. (4.1.38). (f) as is obtained from
a83 -
W~ ~4 4- p83 83 1 --t-~4p83
(4.1.76)
which is derived from Eq. (4.1.37). Thus, not only can the properties in regions (7) and (8) be determined, but also the speeds of the transmitted shock wave, the shock wave reflected at the contact surface, and the rarefaction waves. Region (5) with higher temperature and higher pressure than in region (2) is convenient for the measurements of high-temperature gas properties. However, the maximum duration time of region (5) measured at the end wall depends on the arrival there of the wave reflected at the contact surface. If there is no such reflected wave, the contact surface will stand in front of the end wall as shown in Fig. 4.1.28, which leads to the longer test time. This is the tailored state mentioned earlier. Our next step is to find the tailored conditions.
(8) (3) ~
(5)[
.,,o.,..~176176 ......... o.o.o.,.(2"i" end wall FIGURE 4.1.28 Tailoredwave pattern.
4.1 ShockTubes
581
In the tailored case, neither wave reflects at the contact surface, so that u 7 = 0. Therefore, P8 = P7 = P5 and P3 = P2 leads to P83 = P52, and hence the following expression can be derived from Eq. (4.1.62): ( ~ 4 - - 1)(P52 - 1) u 2 -- a 3 V/(1 + ~4)(1 + 0~4P52)
(4.1.77)
If P52, Pl, T1 and T4 are specified, five unknowns u 2, a 3, P2, P4 and Msl will be determined from Eqs. (4.1.34), (4.1.39), (4.1.42), (4.1.46)and (4.1.77), and thus the tailored conditions are determined. However, the method of solution must be done by iteration as described in what follows. 1. T 1 and T4 are specified and hence a I and a 4 are known. Furthermore, P52 and Pl are specified. 2. P2 is obtained from Eq. (4.1.34) for the guessed value of Msl and hence the value of P21 (-~ P 2 / P l ) will be clear. 3. Substituting the value of P21 obtained at Step (2) into Eq. (4.1.39), then u 2 is determined. 4. The value of a 3 is determined from Eq. (4.1.77) for the known value of P525. By using the value of M~I in Eq. (4.1.46), one obtains P46. P4 is determined from Eq. (4.1.42) with P3 -- P2 for known value of a3 obtained at Step (4). 7. Vary the guessed value of M~I and iterate Steps (2) to (6) until the two values of P4 obtained at Steps (5) and (6) are coincident.
4.1.5 TECHNIQUES FOR SHOCK TUBE OPERATION 4.1.5.1
DIAPHRAGM
The diaphragm separating two sections, the driver and driven sections, is ruptured to generate a shock wave. This is done in two ways: 1) by pressure difference between the two sections; and 2) by pricking the diaphragm with a needle. When a metal diaphragm, usually made of aluminum or iron, is employed, it is recommended that the diaphragm be cross-scratched to reproduce spontaneous rupturing (see Fig. 4.1.29). With this, the rupture pressure is within 1% reproducibility (Bradley, 1962). When the diaphragm without the scratch is employed in a circular-type shock tube, a rough measure of the spontaneous rupture is given by P4 _ 4 d f
Pl
r
(4.1.78)
582
M. Nishida
FIGURE 4.1.29
Cross-scratched diaphragm.
where r is the diameter of the diaphragm, d is its thickness, and f is the stress at which breakage occurs in tension (for more details, see Bradley, 1962). For high degree of reproducibility and rapid turnaround time for the experiment, a diaphragmless shock tube using a fast-action valve has been proposed (e.g., Abe et al., 1992; Teshima, 1995; Kim, 1995). According to Kim (1995), a shot-to-shot reproducibility of better than 0.2% scatter in shock speed was achieved using air for both the driver and driven gas over the shock Mach number range of 1.1 to 2.5.
4.1.5.2
VARIABLE C R O S S - S E C T I O N S H O C K TUBE
A driver section with a larger diameter than a driven section was considered, with a strong shock wave being expected. Alpher and White (1958) made a theoretical estimation of a shock wave Mach number generated in the driven section for various combinations of driver/driven gases. Their results are shown for four cases of different cross section of the driver and driven cross sections in Table 4.1.1. According to Alpher and White (1958), the shock wave Mach number is increased by approximately up to 1.1 times compared to the
4.1
583
Shock Tubes
TABLE 4.1.1
T4/T 1 = T4/T 1 = T4/T 1 -T4/T 1 = T4/T 1 = T4/T 1 = T4/T 1 = T4/T ~ = T4/T 1 = T4/T 1 =
M a x i m u m Values of Shock Mach N u m b e r M s for P 4 / P l ~
oo
Driver-driven gas combination
A4 / A 1 = 1
1.51
2.25
oo
1 1 1 1
N2/N2 N2/A He/N2 He/A
6.18 7.48 10.9 12.8
6.34 7.69 11.3 13.3
6.47 7.84 11.6 13.7
6.76 8.20 12.5 14.7
1 1 2 2
H2/N 2 H2/A He/N2 He/A
22.6 27.5 14.8 18.0
23.2 28.3 15.4 18.8
23.6 28.8 15.9 19.3
24.7 30.1 17.1 20.8
2 2
H2/N 2 H2/A
31.9 38.9
32.8 40.0
33.4 40.7
34.9 42.6
case of equal cross section of the driver and driven sections even for A 4 / A 1 = 00.
4.1.5.3 S H O C K TUBE FOR G E N E R A T I N G S T R O N G S H O C K WAVES Reentry to the earth atmosphere of a spacecraft at high altitudes generates a very strong shock wave ahead of the vehicle nose. For example, a typical reentry flight Mach number is approximately 25 at the altitude of 80 km. Such a strong shock wave causes the shock layer gas to be vibrationally excited, dissociated and ionized, whereby the flow is in thermochemical nonequilibrium. Detailed experiments are required to reveal the structure of such strong shock waves. According to Eqs. (4.1.47) and (4.1.48), the combination of hightemperature lighter driver gas and heavier driven gas can generate a strong shock wave. The idea of a free-piston shock tube was based on this principle (for details, see Stalker, 1964, 1966; Greif and Bryson, 1965). The schematic of the freepiston shock tube is shown in Fig. 4.1.30. Initially light gas, for example, helium gas, is put into the compression tube and test gas is inserted into the low-pressure section. Reservoir gas (high-pressure gas), for which any gas may be used, releases a heavy piston that travels towards the right and compresses helium gas in the compression tube. Because a heavy piston is employed, the compression process is close to being adiabatic. Thus compressed and heated helium gas ruptures a diaphragm, thereby generating a strong shock wave in the low-pressure section. This type of shock tube is often called a Stalker Tube.
584
M. Nishida
reservoir compression tube
low-pressure section
i
ii:iii , I piston FIGURE 4.1.30 TABLE 4.1.2
diaphragm Free-piston shock tube.
Maximum Requirements and Major Dimension of HIEST
Maximum requirements stagnation enthalpy test time at maximum enthalpy model length stagnation pressure driver gas
Major dimension shock tube compression tube piston mass nozzle exit diameter
composition compression ratio pressure
length diameter length diameter
25 MJ/kg 2 ms 500 mm 50 ~ 150 MPa He 50 50 "~ 150 MPa
17 m 180 mm 42 m 600 mm 300 ~ 560 kg 1.2m
Free-piston shock tubes have been extensively developed to investigate the structure of a strong shock wave followed by ionization (for more details, see Nishimura et al., 1973; Takano et al., 1980; and Honma and Yoshida, 1980). Recently, the free-piston shock tube has been further extended to a largesized free-piston shock tunnel to experimentally simulate a high-enthalpy flow around a reentry vehicle. Of these, Itoh et al. (1997) reported on the largest free-piston driven shock tunnel (High Enthalpy Shock Tunnel HIEST), which was recently constructed at National Aerospace Laboratory--Kakuda. The maximum requirements and major dimension of HIEST are given in Table 4.1.2.
REFERENCES Abe, T., Funabiki, K., and Oguchi, H. (1992). A combined facility of ballistic range and shock tunnel using a fast action valve, in Shock Waves, Proc. 18th Int. Symp. Shock Waves, Sendai 1991, K. Takayama, ed., pp. 1025-1030, Berlin: Springer.
4.1
Shock Tubes
585
Alpher, R.A. and White, D.R. (1958). Flow in shock tubes with area change at the diaphragm section. J. Fluid Mech. 3: 457-470. Bradley, I.N. (1962). Shock Waves in Chemistry and Physics, London: Methuen and Co. Ltd.; New York: John Wiley & Sons Inc. Greif, R. and Bryson, A.E. (1965). Measurements in a free piston shock tube. A/AAJ. 3: 183-184. Honma, H. and Yoshida, H. (1980). Ionizing shock structure for a strong shock wave, in Shock Tubes and Waves, Proc. 12th Int. Symp. Shock Tubes and Waves, Jerusalem 1979, A. Lifshitz and J. Rom, eds., Israel: The Magnes Press, The Hebrew Univ., pp. 215-221. Itoh, K., Ueda, S., Komuro, T., Sato, K., Takahashi, M., Miyajima, H., and Koga, K. (1997). Design and construction of HIEST (High Enthalpy Shock Tunnel), in: Proc. Int. Conf. Fluid Eng., vol. I, pp. 353-358. The Japan Soc. Mech. Eng. Kim, Y.W. (1995). A new diaphragmless shock tube facility for interface instability and Mach reflection studies, in Shock Waves @ Marseille I, Proc. 19th Int. Syrup. Shock Waves, Marseille 1993, R. Brun and L.Z. Dumitrescu, eds., Berlin: Springer, pp. 227-232. Nishimura, M., Teshima, K., and Kamimoto, G. (1973). Multi-step ionization relaxation of argon behind a shock wave, in Recent Developments in Shock Tube Research, Proc. 9th Int. Shock Tube Symp., Stanford 1973, D. Bershader and W Griffith, eds., Stanford: Stanford Univ. Press, pp. 294-305. Stalker, R.J. (1964). Area change with a free-piston shock tube. AIAA J. 2: 396-397. Stalker, R.J. (1966). The free-piston shock tube. The Aeronautical Quarterly 17: 351-370. Takano, Y., Miyoshi, S., and Akamatsu, T. (1980). Reflection processes of ionizing shocks in argon on an end wall of a shock tube, in Shock Tubes and Waves, Proc. 12th Int. Symp. Shock Tubes and Waves, Jerusalem 1979, A. Lifshitz and J. Rom, eds., Israel: The Magnes Press, The Hebrew Univ., pp. 187-196. Teshima, K. (1995). High-frequency generation of high-pressure pulses using a diaphragmless shock tube, in: Shock Waves @ Marseille I, Proc. 19th Int. Symp. Shock Waves, Marseille 1993, R. Brun and L.Z. Dumitrescu, eds., Berlin: Springer, pp. 227-232. Vieille, M.P. (1899). Sur les discontinuites par la d~tente brusque de gaz comprim~s. Comptes Rendus de l'Acadonie des Sciences 129: 1228-1230. Vieille, M.P. (1970). On the discontinuities produced by the sudden release of compressed gases (translation of Vieille (1899)), in Shock Tubes, Proc. 7th Int. Shock Tube Symp. Toronto, Canada 1969, I.I Glass, ed., Toronto: Univ. Toronto Press, pp. 6-8.
CHAPTER
4 .2
Shock Tubes and Tunnels: Facilities, Instrumentation, and Techniques 4.2 Free Piston-Driven Reflected Shock Tunnels RICHARD MORGAN The Center of Hypersonics, The University of Queensland, Brisbane, Australia
4.2.1 Introduction 4.2.2 Driver-Heating Mechanism 4.2.3 Conclusions References
4.2.1 INTRODUCTION The starting point for high-performance shock tunnels is in the driver section, and driver design has lead the quest for high-flight speed-simulation capability A schematic of a generic driver-driven tube configuration is shown in Fig. 4.2.1. This covers the main variations and all free piston arrangements, and can be used with an idealized one-dimensional (1D) analysis to explain the primary operating principles and limitations. This chapter presents a basic analysis of the reflected shock tunnel, from which it is possible to gain an understanding of some of the primary operating characteristics. Reference is made to sources for more detailed analysis. Different geometrical configurations of the driver to driven tube interface have a strong influence on the nature of the process by which the driver gas expands to the driven gas velocity. The traditional shock tube maintains a constant sectional area between adjacent tubes, which only allows for expansion in the longitudinal direction by means of unsteady expansion fans. In an idealized situation, the diaphragm rupture is instantaneous and a centered fan
Handbook of Shock Waves, Volume I Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
587
588
R. Morgan Steady expansion region Unsteady , expansion , .
l ! 1 ' ,, / , i ; 11,' / /' / ,/
Driver
t
~t~ng
4'
vA
unsteady .... expansion,-.. "-..
I,///]
.......ill .
.
.
.
t
:
t':///
/
~
i / / / //"
"
,i
. . ..~~. 3
,//,::.:.:/
Test
/
/"
, ,, ,,
/ : / / /,,: ] ,////./
/
i t e
Nozzle-starting process
s
..lntetla~
.....; .,
X
Transition section starting process
i pressure a i r reservoir
compression tube
t
'J ,/'"
Test gas shock tube
dump tank
I
FIGURE 4.2.1 Schematicillustration of a generic reflected shock tube with a corresponding x-t diagram.
forms. A feature of the unsteady expansion fan is that total temperature and pressure are reduced at subsonic speeds but increased at supersonic speeds. To make best advantage of this, shock tubes can be arranged with an area reduction at the inlet to the driven section. This can create a sonic throat separating region of steady (or quasisteady) subsonic expanding flow from an unsteady supersonic expanding region. On primary diaphragm rupture, a package of unsteady expansion waves propagates into the driver tube. If an area contraction is present (from driver to driven tubes), and if the driver is long enough to delay reflections of the starting waves from the end wall, then a region of steady flow forms between the start of the contraction and the throat. In this instance there will be two sets of unsteady expansion waves (one subsonic and one supersonic), with a subsonic steady expansion sandwiched between them. For even moderately small area ratios of 4 and over, due to the strong coupling of the area ratio with the Mach number, the total pressure and temperature drop across the subsonic starting expansion becomes vanishingly small, and the driver behaves much like the ideal configuration described previously. In this instance the pressure ratio between the driven and the driver gases is given by:
--{
P3 _ P4
* (~/1 -- l)} 2~'/~'~-1
F 1 - M3
2
(4.2.1)
4.2
589
Free Piston-Driven Reflected Shock Tunnels
where F 1 is a term that relates to the geometry of the transition from driver-todriven tubes, through the Mach number M ll at which the change from steady to unsteady expansion occurs. Setting M l l to zero gives the straight-through constant area configuration; setting it to unity gives the ideal for optimum pressure recovery (steady expansion in the subsonic region, followed by supersonic unsteady expansion); and supersonic values of M l l allow for a steady expansion into the driven tube after a sonic throat. Values of M l l between zero and unity are unreal because the unsteady expansion traveling at sonic speed would propagate into the upstream region of steady expansion. The value of F 1 is given by:
The velocity in regions 2 and 3 (Fig. 4.2.1) is expressed most conveniently as a Mach number M~ based on the unperturbed driver speed of sound in region 4, and termed the "driver equivalent Mach number" (4.2.3)
M; -- U3 a4
By expressing it in this nondimensional way, and considering flow velocity rather than shock velocity, the pressure ratios become independent of the driven gas and therefore have universal applicability. In the ideal gas form presented, the only specific gas-dependent parameter is the driver-gas specific heat ratio (Yl)- Driver-gas temperature, speed of sound, and density are decoupled. From Fig. 4.2.2, it can be seen that the value of F 1 is always a number fairly close to unity for all configurations. However, because the coupling of F1 with -------gamma=1.67 - - - gamma=1.4 1.4
j
1.2
o.8 u_
0.6 0.4 0.2 0
,
0
FIGURE 4.2.2
2
,
Mll
4
Plot of the driver function F 1 versus Mll (Eq. (4.2.2)).
590
R.
Morgan
- - - - - Ml1=.6 ------0.8 1
1.0EK)0
1.2
-
,r 1.0E-01 (1. 1.0E-02 n 1.0E-03
- - 1.4 - 1.6
------- 1.8 -2 -------2.2
1.0E-04
.... 0
0.5
1
1.5
2
2.5
3
2.4
. . . . . . 2.6 ....
Ml1"
2.8
.-.-----3
FIGURE 4.2.3
Nose-to-tail pressure ratios of the driver gas (~, = 5/3).
pressure ratio is strong (to the seventh power for ~1 - - 1.4), and the leverage of the driver function on the pressure ratio acts on the difference between the two expressions in Eq. (4.2.1), the influence of the driver configuration on the pressure recovery is very great. The data are summarized in Figs. 4.2.3 and 4.2.4 for values of ~)1 of 1.67 and 1.4, respectively. Conditions where the expanded driver gas is traveling faster than the test gas (i.e., approximately when M l l is greater than M]) have been removed from the figure because they generate a reverse shock propagating back into the driver gas and a revised analysis is required. They are generally not advantageous conditions to target, as they remove the enthalpy-adding supersonic section of the unsteady expansion.
Ml1=.6
0.8 1
1.0E+00 ~ . . ~ , _ ~1" 1.0E-01
~
1.2
~
-
'
~
~ ,
-.----1.4
~_.....,,
1.6
'~
1.0E-02
riP)
~
~
- "
"
1.8
-
.....
1.0E-03 ..-----.--2.2 1.0E-04 0
0.5
1
1.5 M11"
2
2.5
....
2.4
......
2.6
....
2.8
.
FIGURE 4.2.4
.
.
.
Nose-to-tail pressure ratios of the driver gas (7 = 7/5).
4.2
591
Free Piston-Driven Reflected Shock Tunnels
It is seen that the benefits of optimal design increase significantly with nondimensional speed (M~), and are more evident for gammas of 1.67 (representing the monatomic gases) than 1.4 (characteristic of cold hydrogen). At nondimensional velocities of 2.2, the high-performance driver configuration (Mll = 1) has an order of magnitude improvement in pressure ratio over the constant area tube. Equation (4.2.1) can also be used to indicate the maximum possible velocity that can be delivered by a given configuration by equating the pressure ratio to zero. This gives:
M~ --
2F 1
(4.2.4)
71--1
This function is plotted in Fig. 4.2.5. It is seen that there is a relatively small influence of configuration on ultimate speed, but a significant influence of specific heat ratio. These ultimate speeds are not achieved in practice, and the benefit of the configurations with small pressure ratios is evident in developing higher speeds through reduction of viscous limitations. The forementioned nondimensional relations are test-gas independent, within the limits of a perfect gas approximation for the driver gas, except for the dependency on driver-specific heat ratio (71)- Real gas effects in the driver will significantly change the results. However, for the commonly used monatomic helium the discrepancy is small. Cold hydrogen drivers also agree well, although dissociation and vibrational excitation will cause departures from the simple theory for heated hydrogen. Real gas effects in the test gas do not change the performance insofar as the post-shock pressure (P3) is concerned.
6 5 4 ~~ ,~r~
3
........
-
84
-
-
gamma 1.4
,gamma
2
0
.2, . . . . .
1.67
I
I
I
1
2
3
Mll FIGURE 4.2.5
592
R. M o r g a n
However, they seriously alter the filling pressure (P1) required to achieve the required flow speed. The pressure developed in the reflected shock region (Ps) cannot be decoupled from the test-gas properties in the same way as for the primary shock. However, using the strong shock approximation and postulating perfect gas behavior in the test gas with a tailored interface, a useful analytical expression for the reflected shock pressure can be obtained. The speed of sound and temperature in region 2 are approximated by a2 =
1)72]0.5 and
U2[(7 2 -
T2 = 2C~2
(4.2.5)
and the Mach number of the reflected shock (Mr) is approximated by
Mr__72q-1 . / . 2 ( 7 2 + 1 ) 2 + V 1 6 ( 7 2 - 1)72
- - ~
4- 1
(4.2.6)
Combining this with the expression for P3/P4 (Eq. (4.2.1)), and using the expression for the pressure ratio across a normal shock the following is obtained:
P--25=P3P-25=P--2(M 22~'2 ~'2- ~I P4
P4 P3
P4
~/2 + 1
(4.2.7)
Y2 +
The results of this calculation are shown in Fig. 4.2.6. Again, the great advantage of using an area contraction between the driver and the driven tubes ( M l l = 1) is evident at high shock speeds, rising to an order of magnitude at M; -
2.
The trend of decreasing stagnation pressure with flow velocity is evident. This effect may be quantified nondimensionally with total enthalpy by again using the strong shock approximation, which for a perfect gas decouples all of
.,.oE.oo 1.0E+01
1,
,., ,,,,,
,,, ,,, ,,,
"
~. 1.0E-01
. . . . . .
Ml1-=0
1
" " 1.0E-02 1.0E-03, 0
,.
,
!
1
2
3
M3* F I G U R E 4.2.6
1
.... ----
.4
1.8 .--
Ml1-=2
Reflected shock pressure (3'1 -- 5 / 3 , ?2 = 7/5).
4.2
593
Free Piston-Driven Reflected Shock Tunnels 10
:0 ~0.1 n
r
-----Ml1= ~
0.01
0.001
0 FIGURE 4.2.7
2
4 2 Hs/a4
6
8
[
The stagnation enthalpy versus the reflected shock pressure (~1 = 5/3, 72 = 7/5).
the driven gas properties except ~1" This is done by coupling the temperature ratio across the reverse shock with the known temperature T2, giving H5 72(M2 272 a--~ -- 2 71 + 1
72 - }) (?2 - ])M2 q-2 72 -Jr (?1 + 1)M2
(4.2.8)
The results of this calculation are shown in Fig. 4.2.7 for a driver gamma of 1.67 and a test gas 71 = 1.4. Again, as evident from this figure is the importance of high driver gas speed of sound and good configuration (i.e., M l l in the region of 1) for developing high enthalpy flows at reasonable pressures.
4.2.2
DRIVER-HEATING
MECHANISM
The coupling of the pressure ratio with the speed is given by the nondimensional relationship shown in Eq. (4.2.1), and displayed in Fig. 4.2.7. However, when considering real applications, the true dimensional properties are of great importance, and high driver speed of sound is required to develop hypervelocity conditions. This may be achieved through a combination of light driver gas and high driver gas temperature. Since the time when shock tubes were first developed, many techniques have been used to increase the unperturbed driver speed of sound a 4. These have included electrical resistance and arc heating (for instance, Warren et al., 1962; Dannenberg, 1978; Park, 1991; Sharma and Park, 1990a,
594
R. Morgan
1990b; Holden et al., 1997) and combustion drivers (Bakos et al., 1996; Erdos et al., 1997; Nagamatsu et al., 1957). Also, shock-heated drivers have been employed (Henshall, 1958), which use an additional shock-heated section to prepare the driver gas. These facilities have been shown to be capable of producing high-enthalpy hypervelocity flows in the reflected shock mode, but have a number of operational disadvantages. First, gases requiring pre-heating impose difficult design constraints due to structural heating loads, and the use of combustion produces heavy gas products, which partially offset the advantages of increased temperatures. In addition, the length of the driver tube required to delay the arrival of the reflected expansion wave from the driver end wall increases the length of the tube, which has to be designed to withstand the high temperatures and pressures of the driver gas. Structural limitations when pre-heated driver gas is used limit pre-heat temperatures to the order of 600 K, whereas those using transient heating such as arc discharge or combustion can use much hotter gas. As can be seen from Fig. 4.2.7, such facilities pay a significant penalty in terms of pressure recovery. For these reasons, the free-piston compression process was developed by Stalker (1961, 1966, 1967, 1972, and 1987) to provide a versatile highperformance shock-tube driver. The operation and use of these facilities were fully reviewed by Gai (1992). The basic arrangement is shown in Fig. 4.2.1. The compression tube is filled with driver gas normally consisting of helium or helium/argon mixtures initially at ambient temperature and a low pressure. The driver gas is compressed by means of a heavy piston, driven by expansion from a high-pressure room-temperature reservoir. Volumetric compression ratios typically of the order 20 to 120 heat the driver gas to temperatures in the range of 1500 to 7000 K, with rupture pressures in the range of tens to (low) hundreds of MPa. The process approximates isentropic compression, but heat losses reduce the achieved temperature somewhat (see, e.g., Jacobs, 1994b). Several researchers have attempted to improve the efficiency of the compression by introducing entropy-raising processes. Knoos (1969) used "by-pass piston" concept, which employed a combination of shock heating, throttling and direct piston compression. This concept was successfully proved, but there has been no further development since. Bogdanoff (1990) and Kendall et al. (1997) investigated the introduction of an entropy-raising expansion into the compression tube bv means of an intermediate diaphragm bursting into an evacuated section. However, they found heat losses severely limited the practical advantages of the process. Large amounts of energy are added to the piston (tens of megajoules for the larger facilities), thus it is of considerable importance that this energy be transferred to the driver gas before the piston reaches the end of the tube, otherwise serious structural damage may be caused. Stalker (1967) solved the basic dynamics of the piston motion with an analysis that has been used ever
4.2 Free Piston-DrivenReflected Shock Tunnels
595
since to calculate "tuned" operation of the facility. Subsequent studies (Hornung, 1988a; Tanno et al., 1999) have confirmed the approach numerically, and hence it will not be discussed in detail here. The primary requirement is to use matching values of the reservoir pressure, the driver-gas fill pressure, and the piston mass, which are compatible with the facility geometry. To meet this condition a substantial area reduction between the driver and the driven tubes is required. Typical values are between 9 and 20. The area reduction also serves to create a region of steady subsonic flow up to the throat which increases performance as seen in Fig. 4.2.7. A driver requirement for achieving good test conditions is to have an adequate axial length to delay the reflection of the unsteady expansion fan from the end wall until after the flow of test gas ceases. For a free-piston compressor, this requirement is eased due to the fact that piston motion may be used to cancel wave reflections. This effectively occurs if the piston velocity is equal to the subsonic velocity of the gas entering the converging section leading to the choked throat. At this condition, the swept volume of the piston matches the mass flow of driver gas through the ruptured diaphragm, and the driver pressure remains approximately constant for a small holding time after rupture. This is achieved by giving the piston slightly more speed than is necessary to match the flow rate at rupture, which gives a pressure overshoot before the decay is apparent. This typically occurs at piston speeds of between 100 and 300m/s, depending on conditions. The value of the holding time required depends on the application. High enthalpy nonreflected shock tunnels (Stalker and Mudford, 1973; Stalker, 1980) have very short test times of the order of tens of microseconds and will not be adversely affected by a fairly rapid decay in pressure after rupture. These facilities may best be served by using a fairly high compression ratio (values up to 120 have been tried) with the consequent small slug of compressed driver gas. Low enthalpy conditions, which may have several milliseconds of test time, need the post-rupture pressure to be maintained for a substantial duration, and may use lower compression ratios and lower speed of sound mixture of gas 0enkins et al., 1991). Some facilities have found it necessary to use several different piston masses to give tuned operation over the full range of conditions (Tanno, 1999; McIntyre et al., 1995). The first demonstration of the free-piston concept for driving shock tubes was by Stalker (1961), at the National Research Laboratories, Ottawa, Canada. Shock speeds up to Mach 26 were demonstrated. (Professor Stalker is shown in Fig. 4.2.12). This work was continued by Stalker at the Australian National University, and led to the commissioning of the "T" series of shock tunnels (for more details see Stalker, 1966, 1967, 1972) which are now widely referred to as "Stalker tubes." The largest of the early facilities was T3, which has been used extensively for a wide range of aerodynamic studies of high-enthalpy real gas
596
R. Morgan
CALSPAN, B u f f a l o vehicle
~~ f t i r ~ n .,T4 B r i s b a n e
i0 "3
T3
heated facilities
c-
i0 -4
_
I0
FIGURE 4.2.8
0
.
,
t
2
1 .
4
.
.
.
.
I
6
A
8
..
tO
Simulation limits of various facilities (from Hornung, 1988a).
effects, (for example, Hornung, 1976; Hornung and Smith, 1979; Stalker and Stollery, 1975). In 1931, T3 was used for the first time for supersonic combustion experiments (see, e.g., Stalker and Morgan 1982). These experiments gave useful insights into the operation of scramjet combustors (Stalker, Morgan and Netterfield, 1998). but also revealed the need to develop larger facilities with greater scaling capacity. In Fig. 4.2.8, which is taken from Hornung (1988b), the binary scaling capabilities of various impulse facilities are shown. The operational characteristics of T3 are adequate to achieve supersonic combustion in a scramjet duct, but not to evaluate hypersonic combustion or to scale supersonic combustion at low altitudes. To achieve these objectives, higher values of the binary scaling parameter (see Gai, 1992) are required. This parameter is given by the product of the density with a characteristic length scale, and is therefore directly proportional to the stagnation pressure (Ps) developed in the shock tunnel. In practice, freepiston shock tunnels have been found to deliver less-reflected shock pressure than the theoretical values shown in Fig. 4.2.7, and to develop higherperformance facilities this issue needed to be addressed. Page and Stalker (1983) correlated the pressure recovery against aspect ratio for many freepiston-driven facilities, and showed that the best performance was consistently given by those facilities with a large compression tube aspect ratio L/d. The results of this study are shown in Fig. 4.2.9. Although the cause of this correlation is not fully understood, it gave a clear indication that longer drivers would be required for improved facilities. Subsequent analysis by Jacobs
4.2
Free Piston-Driven Reflected Shock Tunnels
597
0.2
'4~ O O 2
-0.2
0
0.2
I
0
O~
i.,
i
-
d''++"
-
X X
- O- +,, -
I
O.t.__ ~
tog(~--;)
9
4. -0.6
-
-O.E
K
J~
-1.0 --
FIGURE 4.2.9
Pressure recovery in free piston shock tunnels (from Page and Stalker, 1983).
(1994a) indicated that the cause was due to a coupling of the driver dimensions, the piston dynamics, and the heat transfer. In 1987, the T4 shock tunnel was commissioned (Fig. 4.2.10). This facility has a compression tube aspect ratio of 109, and is designed for a maximum rupture pressure of 200 MPa, although throat erosion problems require very high maintenance above 100MPa, (Itoh et al., 1998). The approximate performance envelope is shown in Fig. 4.2.8, although drive>contamination
FIGURE 4.2.10
The T4 reflected shock tunnel of the University of Queensland.
598
R. Morgan
problems severely restrict test time at enthalpies above 15 MJ/kg (see Skinner and Stalker, 1996). This facility has been used extensively for scramjet research, covering both fundamental studies and engines designed to produce more thrust than drag. The development of the stress waveforce balance has enabled net thrust to be measured on scramjet modules (see Paull, Stalker and Mee, 1995). Due to the high temperatures experienced in the stagnation region of the shock with a different tube, reflected shock tunnels produce test gas having a different composition than that experienced in flight. The primary discrepancy is due to excess dissociation in the free stream caused by chemical freezing in the nozzle. This changes the similarity between laboratory testing and free flight, and has been the subject of considerable investigation. However, for flows behind strong shocks (such as reentry bodies and hypersonic lifting surfaces at large angle of attack), and when the primary chemical process behind the shock is binary (such as a dissociation reaction), then matching the total enthalpy and the binary scaling parameter gives a good simulation of flight condition (see, e.g., Hornung, 1988a, 1988b; Stalker, 1989; Stalker and Krek, 1992). For a more complicated process, such as hydrogen combustion, it is not evident that the pressure-length scaling rule prescribed by binary processes will apply. Despite this, useful information from scaled combustor tests has been obtained (see Morgan and Stalker, 1987). with developing interest in scramjet propulsion systems, it became evident that even larger facilities than T4 were needed to reduce the uncertainty involved in scale testing of combustion processes. Towards this end, several large free-piston facilities have been built around the world for the purposes of scramjet testing and hypervelocity aerodynamics at high Reynolds numbers. The leading dimensions of some of these facilities are shown in Table 4.2.1. The largest of all these facilities, the HIEST tunnel TABLE4.2.1 GeometricData of Some Large Free-Piston Shock Tunnels Compression Compression ShockNozzle tube tube Comp Shock- tube Shock Piston exit length diameter tube tube diameter tube mass diameter Facility (m) (m) 1/d (m) length (mm) 1/d (kg) (mm) T3 T4 T5 HEG HIEST
6 25 30 22 42
0.3 0.228 0.3 0.55 0.6
20 109 100 60 70
6 10 12 17 17
76 75 90 150 180
79 133 133 113 94
90 300 90 388 150 314 760 880 220-780 1200
4.2
Free Piston-Driven Reflected Shock Tunnels
FIGURE 4.2.11
FIGURE 4.2.12
599
HIEST the world's largest free-piston reflected shock tunnel.
Professor Ray Stalker--the inventor of the free-piston shock tunnel.
at Kakada, Japan, is shown in Fig. 4.2.11 (for more details see Itoh et al., 1997).
4.2.3
CONCLUSIONS
Free-piston shock tunnels, or Stalker tubes, have played a major role in the experimental study of hypersonic aerodynamics since the first prototype
600
R. Morgan
compressor was tested in 1959. They represent a cheap and effective means of simulating high-enthalpy real gas effects in the laboratory. Recent development of much larger and higher-pressure facilities offers the potential for more extensive (and not so cheap) evaluation of models with a much closer matching of flight conditions and length scale.
REFERENCES Bakos, R.J., Calleja, J.E, and Erdos, J.I. (1996). An experimental and computational study leading to new test capabilities for the HYPULSE facility with a detonation driver. AIAA Paper 96-2193 19th AIAA Adv. Measurement and Ground Testing Tech. Conf., New Orleans, U.S.A. Bogdanoff, D.W. (1990). Improvement of pump tubes for gas guns and shock tube drivers. AIAA J. 28(3): 483-491. Dannenberg, R.E. (1978). A new look at performance capabilities of arc-driven shock tubes, in Shock Tube and Shock Wave Research, B. Ahlborn, A. Hertzberg and D. Russell, eds., Seattle, WA: University Press, 416-431. Erdos, J.I., Bakos, R.J, and Castrogiovanni, A. (1997). Dual mode shock-expansion/reflected shock tunnel. AIAA Paper 97-0560, 35th Aerospace Sci. Meet. and Exh., Reno, Nevada, U.S.A.. Gai, S i . (1992). Free piston shock tunnels: developments and capabilities. Prog. Aero. Sci. 29: 141. Henshall, B.D. (1958). The theoretical performances of shock tubes designed to produce high shock speeds, Aeron. Res. Council Current Papers, CP 407. Holden, M., Harvey, J., Boyd, I. George, J. and Horvath, T. (1997). Experimental and computational studies of the flow over a sting mounted planetary probe configuration. AIAA Paper 97-0768, 35th Aero. Sci. Meet. and Exh., Reno, Nevada, U.S.A. Hornung, H.G. (1976). Non-equilibrium ideal gas dissociation after a curved shock wave, J. Fluid Mech. 53: 149-176. Homung, H.G. and Smith, G.H. (1979). The influence of relaxation on shock detachment. J. Fluid Mech. 93: 225-239. Hornung, H.G. (1988a). The piston motion tubes and tunnels. GALCIT Rept. FM-88-1. Hornung, H.G. (1988b). Experimental real gas hypersonics. AIAA J. 92: 379-389. Itoh, K., Ueda, S., Komuro, T., Sato, K., Takahashi, M., and Miyajima, H. (1997). Design and construction of HIEST (High Enthalpy Shock Tunnel). JSME Centennial Grand Cong. Itoh, K., Ueda, S., Komuro, T., Sato, K., Takahashi, M., Miyajima H., Itoh, H.K., Tanno, H. and Muramoto, M. (1998). Improvement of a free piston driver for a high enthalpy shock tunnel. Shock Waves 8. Jacobs, P.A. (1994a). Numerical simulation of transient hypervelocity flow in an expansion tube. Computers & Fluids 23(1): 77-101. Jacobs, P.A. (1994b). Quasi-one-dimensional modeling of a free-piston shock tunnel. AIAA. J. 32(1): 137-145. Jenkins, D.M., Stalker, R.J., and Morrison, W.R.B. (1991). Performance considerations in the operation of free piston hypersonic test facilities, in Shock Waves K. Takayama, ed., Sendal, Japan, Springer-Verlag. Knoos, S. (1969). Bypass piston tube, a new device for generating high temperatures and pressures in gases. AIAA J. 9(11): 2119-2127. Kendall, M.A., Morgan, R.G., and Jacobs, P.A. (1997) A compact shock assisted free piston driver for impulse facilities. Shock Waves 7(4): 219-230.
4.2
Free Piston-Driven Reflected Shock Tunnels
601
Mclntyre, TJ., Maus, J.R., Laster, M.L., and Ettelberg, G. (1995). Comparison of the flow in the high-enthalpy shock tunnel in Gottingen with numerical simulations, in Shock Waves @ Marseille, 1, Berlin-Heidelberg, Germany: Springer-Verlag, 251-256. Morgan, R.G. and Stalker, R.J. (1987). Pressure scaling effects in a scramjet combustion chamber. AIAA Paper No. 87-7080, 8th Int. Symp. Air Breathing Engines, Cincinati, Ohio, U.S.A. Nagamatsu, H.T., Geiger, R.E., and Sheer, R.E. (1957). Hypersonic shock tunnel. ARS J. 29: 332340. Page, N.W. and Stalker, R.J. (1983). Pressure losses in free piston driven shock tubes. Proc. 14th Int. Symp. Shock Waves and Shock Tubes, R.D. Archer and B.E. Milton, eds., Sydney, Australia. Park, C. (1991). An overview of Ames facilities, in Shock Waves, K. Takayama, ed., Sendai, Japan: Springer-Verlag. Paull, A., Stalker, R.J., and Mee, D.J. (1995). Experiments on supersonic combustion ramjet propulsion in a shock tunnel. J. Fluid Mech. 296: 159-133. Sharma, S.P. and Park, C. (1990a). Operating characteristics of a 60-cm and 10-cm electric arcdriven shock tube--Part 1: The driver. J. Thermophys. & Heat Transfer 4: 259-265. Sharma, S.P. and Park, C. (1990b). Operating characteristics of a 60-cm and 10-cm electric arcdriven shock tube--Part 2: The driven section. J. Thermophys. & Heat Transfer 4: 266-272. Skinner, K.A. and Stalker, R.J. (1996). Mass spectrometer measurements of test gas composition in a shock tunnel. AIAA J. 34: 203-205. Stalker, R.J. (1961). An investigation of free piston compression of shock tube driver gas. Div. Mech. Eng., Nat. Res. Council, Canada, Rept. MT-44. Stalker, R.J. (1966). The free piston shock tube. Aero. Quart. 17: 351-370. Stalker, R.J. (1967). A study of the free piston shock tunnel. AIAA J. 5: 2160-2165. Stalker, R.J. (1972). Development of a hypervelocity wind tunnel. J. Roy. Soc. Lond. 76: 376-384. Stalker, R.J. (1980). Shock tunnel measurement of ionization rates in hydrogen. AIAA J. 18: 478. Stalker, R.J. (1987). Shock tunnel for real gas hypersonics, AGARD CP 428. Stalker, R.J. (1989). Hypervelocity aerodynamics with chemical nonequilibrium. Ann. Rev. Fluid Mech. 21: 37-60. Stalker, R.J. and Krek, R.M. (1992). Experiments on space shuttle orbiter models in a free piston shock tunnel, Aero. J. Roy. Aero. Soc. 96: 249-259. Stalker, R.J. and Morgan, R.G. (1982). Parallel hydrogen injection into a constant area, high enthalpy supersonic airflow. A/AAJ. 20(10). Stalker, R.J., Morgan, R.G., and Netterfield, M.P. (1998). Wave processes in scramjet thrust generation. Combust. & Flame 71: 63-77. Stalker, R.J. and Mudford, N.R. (1973). Starting process in the nozzle of a nonreflected shock tunnel. A/AAJ. 11: 265-266. Stalker, R.J. and Stollery, J.L. (1975). The use of a Stalker tube for studying the high enthalpy nonequilibrium air flow over delta wings. Proc. l Oth Int. Shock Tube Symp. Kyoto, Japan 55-56. Tanno, H., Itoh, K., Komuro, T., and Sato, K. (1999). Experimental study on the tuned operation of a free piston driver. Shock Waves 9. Warren, D.A., Rogers, D.A., and Harris, C.J. (1962). The development of an electrically heated shock driven test facility. GE Space Sci. Lab., USA, Rept. R62D37.
CHAPTER
4.3
Shock Tubes and Tunnels: Facilities, Instrumentation, and Techniques 4.3
Free-Piston Driven Expansion Tubes
RICHARD MORGAN The Center of Hypersonics, The University of Queensland, Brisbane, Australia
4.3.1 Introduction 4.3.2 Application of a Free-Piston driver 4.3.2.1 Super-Orbital Configurations 4.3.2.2 Super-Orbital Applications 4.3.2.3 Super-Orbital Expansion Tube Operation 4.3.3 Development of Larger Facilities 4.3.4 Test Times 4.3.5 Conclusions References
4.3.1 INTRODUCTION The development of flight vehicles has always required a combination of theoretical understanding of the subject matter, analytical and computational skills, and reliable experimental data. As flight has progressed to higher and higher speeds, the difficulty, danger, and importance of getting good experimental data have continually increased. In the twentieth century, the wind tunnel has been the primary means for obtaining experimental data. However, at speeds >2000m/s, continuous tunnels are not viable due to excessive power and heat dissipation requirements. To simulate these higher speeds in the laboratory, short-duration "pulsed" facilities have been developed. Since shock tunnels were first used for the study of high-speed flow phenomena, the attempts to enhance performance have concentrated on the driver design. Many driver techniques have been tried with varying degrees of Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
603
604
R. Morgan
success. In recent years tile free piston driver technique, developed in Australia by Stalker (1972), has been the preferred option for most high enthalpy facilities, see Chapter 4.2. However, both reflected and nonreflected shock tunnels are limited, independently of driver considerations, in the total enthalpy, which they can simulate because all the energy is added across a shock. This causes a radiation limit for reflected shock tunnels, and a highly dissociated plasma for nonreflected tunnels at very high shock speeds. For practical purposes of aerodynamic testing, such facilities are limited to earth orbital velocities and below, although very useful fundamental studies of plasma behind shock waves have been performed at much higher shock speeds (Sharma and Park, 1990a,b). Arc jets have been used to simulate superorbital flows at very high speeds, up to 30km/s, and have been used in experiments associated with Jupiter entry (see, e.g., Shepard, 1972). Combined radiation and convective heating have been simulated by focusing argon arc lamps on the stagnation region (Peterson et al., 1971). Arc jets produce hypervelocity flows with a low supersonic Mach number, like nonreflected shock tunnels, and do not produce good high-quality flow suitable for aerodynamic testing. They are primarily useful for creating realistic blunt body stagnation point heat transfer levels for materials testing. The high enthalpy simulation capability of free piston reflected shock tunnels is discussed in Chapter 4.2. They are seen to cover a wide range of the suborbital flight envelope, as shown in Fig. 4.2.8. However, at the higher enthalpy end they are severely limited in test time due to driver gas contamination (Skinner and Stalker 1996), and have reduced pressure recovery, see Fig. 4.2.7. Additionally, in the extended stagnation region (region 5, Fig. 4.2.1.), extensive dissociation and ionization may occur, which may not fully recombine in the steady expansion through the nozzle to the test gas state (Harris, 1966). Many practical flight applications occur at super-orbital speeds, such as the return of the Apollo capsules from the moon, the entry of space probes into the atmospheres of the gas giants, and the reentry of sample return missions to Earth from space. The heat shields for these missions have been designed with access to scarce experimental data, which has led to the use of conservative design techniques, and safety factors much larger than normal in the weight watching aerospace world (Gnoffo et al., 1999). The expansion tube concept first proposed by Ressler and Bloxham (1952) offers the potential to increase both the total enthalpy and the total pressure of shock tunnel flows, and also to reduce the levels of free stream dissociation and ionization. A schematic of an expansion tube is shown in Fig. 4.3.1. Only part of the energy is added to the flow through a shock wave, and acceleration to hypervelocity conditions is achieved by means of an unsteady expansion fan.
4.3
605
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> x
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i
Schematic illustration of a generic expansion tube with a corresponding
x-t
In this way energy and total pressure are added to the flow, at the expense of test time, without the flow dissociation, which would occur if all the energy were to be added across a single shock wave. The perfect gas analysis used in Chapter 4.2 can be extended to explain some of the operating characteristics of expansion tubes. In this analysis, onedimensional (1D) perfect gas flows are assumed, and flow velocities are normalized by the unperturbed driver sound speed (a4). This model is useful for giving an understanding of the overall operating principles of expansion tubes, but does not give a good indication of the temperature and the density of the flow, see Fig. 4.3.2. In Fig. 4.3.3, the normalized total pressures for reflected and nonreflected shock tunnels are shown, Ps and P02, respectively, using notation from Fig. 4.2.1. Clearly evident is the significant drop in the total pressure for flow speeds corresponding to values of M] > 2. In Fig. 4.3.4 the total enthalpy and the pressure gains across an unsteady expansion in region 2, Fig. 4.3.1, are shown. Using the strong shock approximation for the primary shock (so the Mach number in region 2 approaches a constant value of M 2 -- x/2/(y2 - 1)Y2), then the flow conditions
606
R. Morgan
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FIGURE 4.3.2 Schematic illustration of a generic superorbital expansion tube with a corresponding x - t diagram.
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The total pressures for reflected and nonreflected shock tubes (71 = 5/3,
4.3 Free-PistonDriven Expansion Tubes 40
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are found purely by the absolute Mach number in region 7, the total enthalpy ration:
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~
+
M 7.
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1 + (Y2- 1/2)M7
(4.3.1)
and for the total pressure ratio: 2Y2/Y2--1
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(12) 1+
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It is seen that flow Mach numbers of the order of 10 give increases in the total pressure by a factor of ~ 25, and in the total enthalpy by ~ 2.5. This effect potentially allows flows at higher enthalpies than the reflected shock tubes to be created with reasonable levels of total pressure, or alternatively, by lowering the value of M] for a given enthalpy, the total pressure is increased over a reflected shock tunnel condition. The overall performance of the expansion tube can be found from multiplying the pressure ratios of Figs. 4.3.3 and 4.3.4. It is evident that for a given total enthalpy, optimum total pressure is achieved by having a large acceleration across the unsteady expansion (High M7) , and a low value of the intermediate shock speed (M~) However, as shown in what follows, values of M] must be kept sufficiently high to ensure clean flows
608
R. Morgan
(Paull and Stalker, 1992). Values of the order of 1.3 and above are characteristic of clean flows. Note that the gain in the total pressure is typically an order of magnitude higher than the gain in the total enthalpy. The total enthalpy of the flow can be estimated from the multiplier in Fig. 4.3.4 and the approximate expression: H02 -- (1/2)Tz+la~(M~) 2. The mechanism for increasing the total enthalpy and the pressure can be seen from inspection of Fig. 4.3.5, which is taken from Morgan (1997a). Consider a Lagrangian slug of a test gas convecting through the unsteady expansion. The leading edge of the packet does work on the gas ahead of it, in proportion to the pressure-length integral. Similarly, when the trailing edge comes through, it has work done on it by the gas behind. However, due to the growth of the unsteady expansion with time, this section of the gas has a larger pressure-length integral than the leading section, and a net gain of the total enthalpy in the expanded gas occurs. As this process relies on a forwards cascade of energy from the unexpanded to the expanded gas, not all of the test gas can be processed this way, and a significant reduction in the test time occurs. Typical test times for expansion tubes are in the range of tens to several of hundreds of microseconds. The mechanism by which the test time is terminated is by a system of waves rather than arrival of the test gas-driver gas interface, and Paull et al. (1988) determined guidelines for estimating the test time and for selecting tube lengths to maximize it. Expansion tubes, driven by cold helium, were successfully developed and operated in the 1960s and 1970s (see, e.g., Trimpi, 1966; Norfleet et al., 1975; Miller, 1977). However, their suitability as a research tool was restricted, because very few usable operating conditions could be established, major disturbances to the test flow were found through Pitot measurements on the center line for many conditions, which rendered the flow unusable for testing purposes. Nevertheless, the useful performance potential of the facility type was demonstrated for a range of test gases by Miller and Jones (1983). / Time l
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4.3
Free-Piston Driven Expansion Tubes
4.3.2 APPLICATION DRIVER
609
OF A FREE-PISTON
In the late 1980s, a free-piston driver was attached to an expansion by Paull et al. (1988), and a wide range of operating conditions, both good and bad, were achieved. Theoretical work by Paull and Stalker (1992) revealed that the unsteady expansion could in some circumstances act to focus free stream disturbances to I single disruptive frequency, which precluded the establishment of steady flow conditions. Analyzing the propagation of acoustic waves in the expansion tube, Paull and Stalker (1992) predicted instabilities that correlated closely to those seen by previous workers (Norfleet, et al., 1975), and which could also be reproduced in the pilot free piston facility TQ at the University of Queensland. Extending this approach, guidelines were developed for achieving usable test flow, and the facility was then used for a range of good test conditions in air and argon up to 10 km/s (Neely et al., 1992). In addition, pressure scaling of established conditions with the free piston driver enabled parametric variation of Reynolds number to be obtained at approximately constant total enthalpy. This indicated versatile future uses for expansion tubes if the flow instability was avoided. This work was followed by an international revival of interest in expansion tubes, and to the recommissioning of the Langley tube (Miller, 1977) as HYPULSE (Tamagno et al., 1990; Erdos et al., 1997) at GASL Inc., for supersonic combustion studies. Paramount to the use of expansion tubes for scramjet work was the fact that lower levels of free stream freezing can be achieved at a given enthalpy than with a reflected shock tunnel. The HYPULSE facility was originally recommissioned using Langley configuration and established test conditions with a cold helium driver. It was ten successfully used for scramjet combustor simulations by attaching a 5.46:1 area ratio diffuser to create adequate pressures for supersonic combustion (Bakos et al., 1992). A free piston driver attachment was then designed for the facility (WBM Stalker Pry. Ltd., Brisbane, Australia) to increase performance, but was never built. Instead a detonation driver was installed, and dual-mode, reflected shock tunnel/expansion tube operating capability was established. This has proven to be a very successful modification, and has enabled scramjet simulations to be performed at high pressures and Mach numbers, and fairly large length scales (Bakos et al., 1996). The mechanism identified by Paull and Stalker (1992) for eliminating flow instabilities in expansion tubes was to run the primary shock wave highly overtailored, so that the speed of sound ratio across the driver/driven gas interface (a3/a 2, termed f in the following) was substantially < 1. This created an acoustic buffer between the driver gas and the unsteady expansion, which prevented drive-induced noise front focusing into major disturbances and
610
R. Morgan
propagating through the unsteady expansion and into the test gas. The value of f required to filter the noise adequately is not clearly defined, but is likely to depend on the tube diameter. Values of the order of 0.8 have been found to be effective. All the previously documented conditions, which had unusable test flow, were found to have higher values of f. Extending the forementioned gas analysis to calculate flow conditions that satisfy the sound buffer requirement, it is seen that the primary shock speed is the determining factor, with gas properties and driver configuration (as expressed in the "driver function" defined in Chapter 4.2.) having a second order effect. This is shown in Eq. (4.3.3), and the resulting values of M~ are plotted in Fig. 4.3.6 M~ = F1
1 f~/~2(~2 - 1)/2 ~- ~)1 -- 1/2
(4.3.3)
Taking 0.8 to be a realistic value for f (Paull and Stalker 1992), then minimum values of M] of ~ 1.4 will be required. This has serious implications for expansion tube testing of medium to low enthalpy flow conditions. Values of M] at this level do not attract a prohibitive pressure penalty (Fig. 4.3.3), but the pressures will be significantly reduced from the levels that might otherwise have been realized. It may also mean that the driver sound speeds may have to be artificially reduced. For instance, a condition previously used on HYPULSE for combustion studies has a primary shock speed of 2500 m/s, for which a maximum driver sound speed of about 1500 m/s would be needed. This may be done either by running the driver with a low compression ratio (which means doing more piston work for a given rupture pressure P4) o r using a heavier gas than helium (argon/helium mixtures are often used). Noting that the main feature of the free piston driver is high sound speed, (values of c/4 with helium are commonly around 4000m/s), low enthalpy expansion tube runs have to be run at far from ideal conditions. Despite this, the free piston driver is well suited for the application, as it is easy to fine tune the operating conditions. As a study for a very high-performance suborbital facility, Stewart et al. (2000), investigated the feasibility of commissioning the RHYFL reflected shock tunnel (designed for the Rocketdyne Company by WBM Stalker Pty. 2
~,~
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0.5
,
0.6
0.7
.
0.8
f Target sound speed ratio
FIGURE 4.3.6
0.9
1
(a3/a2)
The requirement for filtering noise from the test gas (71 = 53, ~2 -- 7/5).
4.3 Free-PistonDriven Expansion Tubes
611
Ltd., Brisbane, Australia, but mothballed before commissioning) as an expansion tube for high Reynolds number scramjet testing. This facility promises to have unique simulation capabilities, but without the materials problems which arise for reflected free piston facilities at very high enthalpies and pressures.
4.3.2.1 SUPER-ORBITAL CONFIGURATIONS From inspection of Fig. 4.3.3, it is seen that the pressure recovery drops off dramatically for values of M~ much beyond 2, with the possibility of approximately doubling the enthalpy through the unsteady expansion, giving a maximum practical total enthalpy equivalent to about 3 x the original driver speed of sound. To achieve higher speeds, an extra process has to be introduced. Inspecting Fig. 4.3.1, and noting that the shock is overtailored, the speed of sound in the driven gas (region 2) is higher than in the expanded driver gas (region 3). if the driven gas in region 1 were replaced by a cold driver gas instead of the test gas, then the shock-heated driver gas would make a more effective driver than the expanded driver gas (region 3). If this shock-heated gas were subsequently used to drive the test gas tube, then the resulting transmitted shock would be faster than that produced by direct coupling of the driver to the driven tube. A configuration for achieving this is shown in Fig. 4.3.2. In order to pursue the concept further, Morgan and Stalker (1987) added a compound driver to the expansion tube, and used helium as the accelerator gas. This enabled true super-orbital aerodynamic flows to be created as energy input across both the shock and the unsteady expansion was maximized. A free piston helium driver was used to shock heat an intermediate helium driver tube, which was run with overtailored shock speed to create a more effective driver than the original isentropically compressed gas. Theoretical analysis using equilibrium theory gave good agreement with the measured experimental parameters of static and Pitot pressures, and with fiat plate heat transfer in test flows of air and carbon dioxide at velocities up to 13 km/s (for more details, see Neely and Morgan, 1994). Flow visualization using laser holography (see Wegener et al., 1996), on oblique wedges and blunt bodies has demonstrated the utility of the facility for simulating super-orbital flows, and agreed well with numerical simulations. Further development has led to larger facilities (Doolan and Morgan, 1999; Morgan, 1997b) and a major facility 65 m long has been built at the University of Queensland (see Fig. 4.3.7 and 4.3.8). 4.3.2.2
SUPER-ORBITAL APPLICATIONS
The missions for which such velocities might be required include rapid, nonminimum energy transfers between Barth and other planets, aeroassisted swing by maneuvers, and the aerocapture of planetary probes.
612
R. Morgan
FIGURE 4.3.7
The primary diaphragm station of the X3 superorbital expansion tube.
FIGURE 4.3.8
The launch tube of the X3 superorbital expansion tube.
4.3 Free-PistonDriven Expansion Tubes
613
The benefits of such maneuvers may be quite dramatic in terms of weight reduction, increased payload, extended mission objectives. For example, the recent Magellan probe operation using multiple glancing entries into the atmosphere of Venus produced velocity changes that would been totally impossible otherwise. To perform the orbit change using impulsive rocket burns, 800 kg of fuel would have been required, far exceeding the total fuel supply of 94 kg. (This particular operation was by a spacecraft not designed for atmospheric flight, and moving in the rarefied gas regime). Such flows cannot be produced at this time in the facilities around which this study is focused. It is however relevant as the first aeroassisted swing by, and as an example of the potential benefits of such techniques. It was documented in the January 1994 issue of Aerospace America. Analysis by Macrossan (1996) indicates that rarefied flows may be studied at superorbital speeds, by the addition of an expansion nozzle to the superorbital flows. Waverider studies by Anderson et al. (1991) indicated that reductions in fuel requirements by orders of magnitudes might result from using aeroassisted gravity swing bys. Some of the atmospheres of interest include air, carbon dioxide (Mars), and nitrogenmethane mixtures (Titan). The superorbital expansion tube facilities can simulate these flows by substituting the appropriate gas into the shock tube section, and has been used for air (Neely and Morgan 1994), carbon dioxide, and helium. Neon mixtures simulate the Jovian atmosphere at super-orbital speeds. A feature of all these flows is the strong influence real gas effects will have on the thermodynamics and fluid dynamics of the flow field surrounding the entry body. The energy transfer from kinetic to chemical, thermal and radiative forms as the flow passes around the body is so high that dramatic changes to the gas properties occur. This includes nonequilibrium chemistry, nonequilibrium thermal storage modes, nonequilibrium radiation, and severe ionization. The interaction of these processes with the flow field is complex, and not fully understood at present. The problem of combining all these effects into one numerical package is complex, and is compounded by the fact that even the basic relaxation rates for the various processes involved are not precisely defined. A further complication is provided by the fact that ionization levels will be much higher than experienced in "normal" entry flows, and will act over a larger region of the body than for suborbital flows, that is, ionization levels which occur behind a normal shock in sub-orbital flows will be found behind oblique shocks at higher speeds. Associated with the ionization is an increase in electron density and radiation levels. Nonequilibrium radiation is known to be most intense in the high-temperature chemical relaxation zone behind the shock, and can substantially lower the shock layer temperature. The high diffusivity of electrons enables the influence of the shock layer to be
614
R. Morgan
felt further from the body than is the case in nonionized reacting flows. In light of these considerations, the usefulness of a facility, which can directly induce the real gas phenomena in the laboratory, is evident. The influence of these complicated and incompletely understood processes is quite fundamental from an engineering point of view in order to fly atmospheric craft successfully at these velocities. For example, the entry probe for Titan was designed with a large safety factor on the heat shield because of the unknown heat transfer rates on the ablative surface and as a consequence some of the payload had to be sacrificed, thereby reducing the benefits and increasing the cost of the mission. Basic flight parameters, such as lift, drag and center of pressure, may change dramatically. In an analogous example, the Gemini and Apollo capsules overshot the target landing spots by up to 200 miles on their first landings, which has been attributed to real gas effects shifting the center of pressure. The US space shuffle also experienced shifts in the center of pressure, which pushed the aerodynamic control flaps to the limits of their design. Similar effects we to be expected on superorbital missions due to some of the processes mentioned here. Leeward heat transfer was very large on the recent Galileo Jovian entry probe with an entry speed of ~ 45 km/s, and it is unclear how important it will be for missions at lower velocities, which can potentially be simulated in expansion tubes. The separated flow region behind the capsule will take longer to start than the windward flow. Recent studies in America by Hollis et al. (1995) showed that leeward flows behind blunt bodies can be usefully studied in expansion tubes at suborbital speeds, and summarized the criteria for generating steady flow. The possibility of studying such flows at superorbital speeds is currently being investigated. Extensive use of the superorbital aerodynamic maneuvers already discussed here will only be possible if the processes involved can be better understood and quantified. This cannot happen through theory and computation alone, because experimental testing and verification are necessary. The superorbital facilities offer the means to start this process in a higher flight regime than was previously possible. Nonreflected shock tunnels are capable of producing shocks that create flows of the appropriate enthalpy for this sort of study. However, with all the energy added across the primary shock, the produced flow is an ionized plasma, with rapid loss of enthalpy through radiation from the shock heated core. The hypervelocity flows created are unsuitable for aerodynamic testing. In the expansion tube concept, the static enthalpy component is minimized as a major part of the energy is added across an unsteady expansion. This leads to higher Mach number flows than in a nonreflected tunnel, lower static temperature, and hence less ionization and dissociation of the test gas.
4.3 Free-PistonDriven Expansion Tubes
615
4.3.2.3 SUPER-ORBITAL E X P A N S I O N T U B E OPERATION In Fig. 4.3.2 an expansion tube configuration tube suitable for generating superorbital, hypersonic flows is shown. The notation is from Neely and Morgan (1994). The use of a shock-heated, secondary driver has been tried before to achieve higher shock speeds (see, e.g., Henshall, 1958; Stalker and Plumb, 1968). Morgan and Stalker (1991) document the first application of this technique to expansion tubes. The primary factor that determines the advantage--in nose-to-tail pressure ratio--of using an extra driver section is the flow speed behind the secondary shock, normalized by the primary driver speed of sound. For large values of this parameter, termed "driver equivalent flow Mach number" in Chapter 4.2, substantial advantages are evident, see Fig. 4.3.9. The data in Fig. 4.3.9 are plotted in terms of the velocity and the pressure behind the shock, rather than the more usual shock speed and the preshock ambient pressure, to decouple the influence of the test gas and of the viscous effects between the shock wave and the contact surface. It was calculated for helium/helium/air, in a constant area shock tube, but within the limits of the strong shock approximation it also applies to any combination of monatomic driver gases, and any driven gas. For the compound driver configurations, the shock strength through the secondary driver has been optimized to give maximum pressure in the shock-heated test gas for each condition. The velocity plotted is that behind the secondary shock, rather than the final test velocity after the unsteady expansion. This still may be used to give an
1.00E+00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1.00E-01 1.00E-02
It,
~11.00E-03 ~t 1.00E-tM 1.00E-05
. i
[ 1.00E-06 t 1.00E-07
~
9 Compound Combustion . Single
1.00E-08 " Flow equivalent Math number M* FIGURE 4.3.9 Comparisonof driver options.
616
R. Morgan
indication of possible final flow speed, as typically, the unsteady expansion doubles the flow velocity (Lucasiewicz, 1972). It can be seen in Fig. 4.3.9 that the compound driver has significant advantages at equivalent Mach numbers > 2, and is an order of magnitude better at 2.5. The single stage driver cannot drive flows at equivalent Mach numbers > 3 (at any pressure), and the compound driver is the only option for the higher relative speeds. The compound driver could in theory accelerate gas up to 9x the original speed of sound, ignoring viscous effects. However, the gas density would be so low that viscous effects would prevent the high speeds from being achieved. Taking reasonable engineering limits of ~ 108 Pa for the reservoir pressure, "~ 105 Pa after the incident shock passes the test gas and before the unsteady expansion (giving ~ 106 Pa reflected shock pressure for opening the tertiary diaphragm), pressure ratios (P6/P4) of the order of 10 -3 will be required. Thus it can be seen from Fig. 4.3.3 that flow equivalent Mach numbers of ~ 4 in region 6 will probably represent an upper limit. The useful operating range of compound drivers will therefore be between equivalent Mach numbers of ~ 2 and 4. Also shown on fig. 4.3.9 for comparison is the estimated performance of a combustion-heated driver using a mixture of 8He + 3H 2 + 02, and driving a simple shock tube. The speed of sound for the mixture was taken from Lucasiewicz (1972). The universal applicability of this figure does not apply to the combustion curve, as the speed of sound of the combustion products is not an independent variable as it is for free piston compression. For comparison with the hot helium options, the normalizing speed of sound is that for a typical free piston compression ratio of 60, that is, 4000 m/s. It is seen that for the high speeds targeted in this study, the combustion driver is not a viable option. An incidental benefit of using the compound driver is that because overtailored operation is fundamental, a speed of sound increase always occurs across the interface between the expanded driver gas and the shock-heated driver gas. This acts as an acoustic buffet to noise generated in the driver and upon primary diaphragm rupture, and helps maintain good quality flow (Paull and Stalker, 1992).
4.3.3
DEVELOPMENT
OF LARGER FACILITIES
The first superorbital expansion tube set up at the University of Queensland, X1, had a 38-ram bore and used a fairly low-performance free piston driver with an area ratio 7.1:1 between the driver and the driven tubes. It was considered to be a pilot study to confirm the effectiveness of the concept, and to investigate the practicality of building larger-scale superorbital facilities. It
4.3
Free-Piston Driven Expansion Tubes
617
has been used to produce a range of superorbital conditions in air and carbon dioxide flows, and has established steady flows on both blunt and slender bodies. Flow conditions were estimated assuming equilibrium conditions in the unsteady expansion, and a simple inertial model to include the effect of the rupture of the tertiary diaphragm. The reverse shock formed when the flow hits the secondary diaphragm affects the whole of the test gas, and must be considered. The predicted pressure levels and test times are seen to be fairly close to the experimental ones. when a fiat, plate model at zero incidence was placed in the test section, the measured heat transfer levels were seen to be very close to laminar values predicted by the Eckert reference correlations (Eckert, 155), assuming equilibrium conditions, see Fig. 4.3.12. The test condition was obtained with a flow driver equivalent Mach number of ~2.6, before the unsteady expansion to 13 km/s, and with a shock tube ambient fill pressure of 800 Pa and an acceleration tube fill pressure of 10 Pa. At this speed the advantage of the compound driver was a factor of ~, 1000 over the single stage, and it was clear that such flow conditions would not have been possible without the compound driver. Although the experimental results agreed well with the equilibrium predictions for the conditions presented here, there is a need for more detailed understanding of the chemical composition of the test gas, and the details of the rupturing process. Other conditions on the larger X2 facility (Doolan and Morgan, 1999) showed better agreement with a frozen expansion calculation. Numerical analysis by Jacobs (1994a,b) and Stewart et al. (2000) gave good agreement with experimental data. Shock reflection off the secondary (or tertiary if the configuration of Fig. 4.3.2 was used) diaphragm had been found by many workers to affect the flow conditions developed in the test section. Morgan et al. (1991) developed a 1D inertial rupture model, which correlated with the measured wall pressures, and Wilson (1992) used a numerical holding time model that allowed for instantaneous rupture of a massless diaphragm and a predetermined time after shock reflection. In 1994, a project to build a very large expansion tube, designated X3, was started. The targeted dimensions were approximately 65 m total length, with a bore of 182.6 mm. It was required to be capable of both sub- and superorbital operating modes. In order to build the large-scale facility, a larger driver was needed. To save costs, an almost constant area driver to driven tube configuration was chosen. As seen in Chapter 4.2, this leads to a reduction in total pressure recovery in the test gas. Using the free piston compression driver, with compression ratios of the order of 100, this would create a prohibitively long driver section. To get around this problem, a two-stage driver process was developed by (Doolan and Morgan, 1999). A schematic of the arrangement is shown in Figs. 4.3.10 and 4.3.11. Large swept volumes and low pressures
00
Launchtube for first stage compression .
'High-pressure ( ~ air reservoir
~ ,,~H~..~'o..~'~ ,. . . . I r,,~" / ~ / / / / / . Lightweightouter piston [2"
Plastic buffer
\
\ _ / / / / / / / / .z , " / ~
~ FIGURE 4.3.10
~'x'~'~x~'~NNx'x~" ~oooo~yoom~os~,o~ tube ...... ~x.~.~'~"~
Testgas shocktube DI
! Accelerationtube low-press.tiregas D2
Schematic illustration of a two-stage driver tube.
0
4.3 Free-PistonDriven Expansion Tubes
619
Scale (mm) , , ,
0 100
300
500
700
Z
FIGURE 4.3.11 Schematicillustration of the actual piston assembly for the X3 expansion tube.
characterize the first stage of the compression, while the second stage has low swept volume and high pressures. This is achieved by using two pistons as is shown in Fig. 4.3.11. Energy is added to the pistons by expanding the reservoir gas as they travel down the launch tube, most of which is contained in the inner piston, which contains the bulk of the mass (300 kg). On reaching the compression tube, the outer lightweight piston (approximate mass 100 kg) is stopped by impact with a polyurethane buffer, and the internal piston finishes the compression process in the compression tube. Primary diaphragm rupture pressures are designed to be up to 100 MPa.
4 . 3 . 4 TEST TIMES The useful test time in expansion tubes is typically much less than in a reflected shock tunnel of the same length, and the addition of an extra driver section compounds this problem. The high enthalpy condition described by Neely and Morgan (1994) has a test time of only 20 kts, but this was shown to be adequate to start super-orbital flows around small bodies. A simple analysis based on wave tracking, and interface locations as given by Mirels (1963), is seen to give a reasonably good estimate of the usable test duration (see Fig. 4.3.12). Doolan et al. (1994) have extended this analysis to larger facilities, and test times of the order of 100s of microseconds appear to be possible for facilities of manageable size. An improved numerical analysis by Doolan and Jacobs (1996), has been found useful in predicting the operation of superorbital facilities.
620
R. Morgan 2e-3 -
1
st/x-O.s
(mm-o.5)
t
-_--- ........
a
__._
A ]1_ .
.
.
.
.
.at . . . . . . .
A
le-3 .......
0
0
ltO
210
310
I
40
,
laminar
12a
laminar
121a
A
vft:507
m 9
vft508 vftS09
I
50
60
70
x, Distance from leading edge (ram) FIGURE 4.3.12 A plot of St normalized by x ~ versus the distance from the leading edge for 108 MJ/kg (from Neely and Morgan, 1994).
4.3.5
CONCLUSIONS
The free piston driver has been shown to be suitable for generating both suband superorbital flows in expansion tubes, and with appropriate tailoring of operating conditions, good quality flows suitable for aerodynamic testing can be obtained. The superorbital conditions are obtained by using an extra shockheated driver to increase the strength of the primary shock waves, which can be produced. Extra enthalpy is produced at the expense of the test time, and for this reason a long facility of this type has been developed at The University of Queensland. The facilities have the potential for useful experimental simulation of nonequilibrium ionizing, radiating and dissociating flows characteristic of superorbital travel around all the atmospheres of the solar system.
REFERENCES Allen, M.G. (1998). Diode laser absorption sensors for gasdynamic and combustion flows. Meas. Sci. TechnoI. 9: 545-562. Anderson, J.D., Lewis, M.J., and Kothari, A.P. (1991). Hypersonic waveriders for planetary atmospheres. AIAA J. Spacecraft & Rockets 28(4). Doolan, C.J. and Morgan, R.G. (1999). A two stage free piston driver. Shock Waves 9: 239-243. Doolan, C.J. and Morgan, R.G. (1994). Hypervelocity simulation in a new large scale experimental facility. 18th AIAA Aerospace Ground Testing Corf., Colorado Springs, U.S.A. Doolan, C.J. and Jacobs, P.A. (1996). Modelling mass entrainment in a quasi one dimensional shock tube code. J. AIAA 34(6). Eckert, E.R.G. (1955). Engineering relations for friction and heat transfer to surfaces in high velocity flow. J. Aero. Sci. 585-587. Gnoffo, RA., Weilmuenster, K.J., Hamilton, I.I., H.H. Olynick, D.R., and Ventatapathy, E. (1999). Computational aerothermodynamic design issues for hypersonic vehicles. AIAA J. Spacecraft & Rockets 36(1): 21-43.
4.3
Free-Piston Driven Expansion Tubes
621
Harris, C. (1966). Comment on the non-equilibrium effects of high enthalpy expansions of air. AIAA J. 4(6): 1148. Henshall, B.D. (1958). The theoretical performances of shock tubes designed to produce high shock speeds. Aero. Res. Council Current Papers, CP 407. Hollis et al. (1995). Hypervelocity aero-heating measurements in wake of Mars mission entry vehicle. 26th AIAA Fluid Dyn. Conf. Jacobs, P.A. (1994a). Numerical simulation of transient hypervelocity flow in an expansion tube. Computers & Fluids 23(1): 77-101. Jacobs, P.A. (1994b). Quasi-one-dimensional modeling of a free-piston shock tunnel. AIAAJ. 32(1): 137-145. Lucasiewicz, J. (1972). Experimental Methods of Hypersonics, New York: Marcel Dekker Inc. Macrossan, D.N. (1996). DSMC calculation of high densit), high speed nozzle flow from an expansion tube. Private communication. Miller, C.G. (1977). Operational experience with the Langley expansion tube with various test gases. NASA TM-78637. Miller, C.G. and Jones J.J. (1983). Development and performance of the NASA Langley Research Center expansion tube/tunnel. A hypersonic-hypervelocity real-gas facility. AIAA 14th Int. Syrup. Shock Tubes & Waves, R.D. Archer and B.E. Milton, eds., Sydney, Australia. Mirels, H. (1963). Test time in low pressure shock tubes, Phys. Fluids 6(9): 1201-1214. Morgan, R.G. (1997a). A review of the use of expansion tubes for creating super-orbital flows. AIAA 35th Aero. Sci. Meet. & Ex., Reno, Nev. U.S.A. Morgan, R.G. (1997b). Super-orbital expansion tubes. 21st Int. Syrup. Shock Waves, Great Keppel Island, Australia. Morgan, R.G. and Stalker, R.J. (1987). Pressure scaling effects in a scramjet combustion chamber. 8th Int. Syrup. Air Breathing Engines, Cincinnati, Ohio, U.S.A. Morgan, R.G., Bakos, R.J., Tamagno, J., Stalker, R.J., and Eedos, J.I. (1991). Scramjet testingground facility comparisons. ISABE Conf, Manchester, England. Neely, A.J. and Morgan, R.G. (1994). The super-orbital expansion concept, experiment and analysis. The Aero. J. Roy Aero. Soc., 97-105. Neely, A.J., Stalker, R.J., and Paull, A. (1992). Enthalpy, h)qgervelocity flows expansion tube. Aero. J. 175-186. Norfleet, G.D., Lacey, Jr., J.J., and Whitehead, J.D. (1975). Results of an experimental investigation of the performance of an expansion tube. Proc. 4th Hypervelocity Techniques Syrup., Arnold AF Station, 49-110. Paull, A. and Stalker, R.J. (1992). Test flow disturbances in an expansion tube. J. Fluid Mech. 245: 493-521. Paull, A., Stalker, R.J., and Stringer, I.A. (1988). Experiments on an expansion tube with a free piston driver, AIAA 15th Aero. Testing Conf. San Diego, Ca., U.S.A. Peterson, D.L., Gowen, EE., and Richardson, C. (1971). Design and performance of a combined radiative-convective heating facility. AIAA Paper 71-255. Ressler, E.L. and Bloxham, D.E. (1952). Very high Mach number flows by unsteady flow principle, Cornell Univ. Grad School Aero. Eng., limited distribution monograph. Sharma, S.E and Park, C. (1990a). Operating characteristics of a 60-cm and 10-cm electric arcdriven shock tube--Part 1: The driver. J. Thermophys & Heat Transfer 4: 259-265. Sharma, S.P and Park, C. (1990b). Operating characteristics of a 60-cm and 10-cm electric arc-driven shock tubemPart 2: The driven section. J. Thermophys. & Heat Transfer 4: 266-272. Shepard, C.E. (1972). Advanced high power arc heaters for simulating entries into the outer planets. AIAAJ. 10(2): 117-118. Skinner, K.A. and Stalker, R.J. (1996). Mass spectrometer measurements of test gas composition in a shock tunnel. AIAA J. 34: 203-205. Stalker, R. J. (1972) Development of a hyqgervelocity wind tunnel. J. Roy. Soc Lond. 76:376-384.
622
a. Morgan
Stewart, B.A, Morgan, R.G., Jacobs, P.A., and Jenkins, D.M. (2000). The RHYFL facility as a high performance expansion tube for scramjet testing. 21st AIAA Adv. Measurement Tech. & Ground Testing Conf., Denver, CO., U.S.A. Trimpi, R.L. (1966). A theoretical investigation of simulation in expansion tubes and tunnels. NASA TR R 243.
Wegener, M.J., Bishop, A.I., McIntyre, T.J., Rubinsztein-Dunlop, H., Stalker, R.J., and Morgan, R.G. (1996). Visualization and analysis of bow shocks in a super-orbital expansion tube. AIAA J. 34(10): 2200--2202.
CHAPTER
4.4
Shock Tubes and Tunnels: Facilities, Instrumentation, and Techniques 4.4
Blast Tubes
ROBERT ROBEY Los Alamos National Laboratory, Los Alamos, USA
4.4.1 General Description 4.4.2 Experimental Design Specification 4.4.3 Experimental Design Configurations 4.4.3.1 Centered Explosive Design 4.4.3.2 Explosive at the End of the Blast Tube 4.4.3.3 Explosive Outside the Blast Tube 4.4.4 Driver Design 4.4.5 Detonable Gas Driver 4.4.6 Simulation Scaling 4.4.7 Simulation Envelope 4.4.8 Instrumentation 4.4.9 Applications 4.4.9.1 Non-ideal Blast Wave Simulations 4.4.9.1.1 Dust-Laden Blast Waves 4.4.9.1.2 Wall Jets 4.4.9.2 Model Studies 4.4.9.3 Civil Defense Studies 4.4.9.4 Detonation Studies 4.4.10 Conclusion References
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
623
624 4.4.1
R. Robey
GENERAL
DESCRIPTION
A blast tube facility is more completely described as an "explosively driven shock tube." The shorter description "blast tube" will be used for simplicity. A blast tube is composed of two parts: 1. a long tube; and 2. an explosive energy source.
The simplest and most common blast tube is a circular steel tube open at both ends. The explosive may be placed anywhere within the length of the tube. Operating a blast tube facility requires experience with explosive handling and storage. This at first might seem to make it a more difficult facility to operate than the compressed gas-driven shock tube. However, the blast tube requires no mechanical pumps, compressed gas storage or diaphragms, thereby eliminating many of the common safety and maintenance concerns regarding the operation of shock tubes. In addition, the blast tube undergoes very little thrust loading, which simplifies the tube design. As a general rule, the shock tube is the simpler facility for smaller size experiments and the blast tube is more appealing for larger facilities. Still, those with no experience with explosives should not undertake the operation of a blast tube without suitable training and preparation. A practical consideration for a blast tube is that of being able to enter the tube to install the explosives. Smaller blast tubes have used explosives encased in foam that are slid into the driver chamber, but this produces dirty detonation products full of particulates. Typical sizes of blast tubes range from 1-6 m in diameter and hundreds of meters long. Some of the larger facilities include the CERF 6-m diameter shock tube at Kirtland Air Force Base near Albuquerque, New Mexico, the nearby Sandia National Laboratories Thunderpipe facility, and the New Mexico Tech 6-m diameter shock tube located in Soccoro, New Mexico. The locations of these facilities point out another important requirement for an explosively driven shock tubemlots of space. The long-range propagation of low pressures is highly variable and dependent on local conditions. Space is required to provide an adequate buffer to prevent civilian property damage.
EXPERIMENTAL SPECIFICATION
4.4.2
DESIGN
Blast tubes are large facilities most commonly used for blast effects studies. Shock tubes are smaller and generally used for shock phenomenology studies.
4.4
625
Blast Tubes
The design specifications used here will focus on those for blast effects studies. Design specifications for other types of studies can also be used in blast tubes, but the design configuration to achieve the specified environment may be different from that presented here. For blast effects studies, the most common design specification is an ideal blast wave from an open-air explosion. The typical waveform is an exponentially decaying pressure time history. The peak static pressure is usually specified or alternatively a distance from an explosion of a specified strength. From the specifications, a static pressure time history can be determined (for details see Baker, 1973). To be complete, a dynamic pressure history should also be determined, though this step is not always done in practice. Once the blast wave is specified, the goal is to match the specifications as closely as possible for the positive phase of the blast wave. The rationale for using a blast tube for blast effects studies is simple. As shown in Fig. 4.4.1, the blast tube is analogous to placing a tube from the explosive to the test station on an open-air explosion. The tube does not allow any energy to cross the boundary of the tube, thus conserving energy from the explosive to the test station. The explosive source can be much smaller than that required for an open-air explosive test while producing a similar blast wave. A more correct blast tube for simulation of an open-air explosion might be a conically shaped tube. A few blast tube facilities have been built based on a conical section, most notably the DASACON facility at Dahlgren, Virginia. However, conically shaped blast tubes are impractical due to the high fabrication cost and thus not very common. The blast tunnel at Foulness, England is an example of the compromises that have to be made for larger blast facilities. Its design was based on a conical shape, but due to fabrication costs, the tunnel has a stairstep shape.
( \
FIGURE 4.4.1
..
"
.
.
.
Test Station
.
_
] Superimposed Blast Tube
Blast tube as a cylindrical section of an open-air explosion.
626
R. Robey
Blast runnels such as the Foulness blast tunnel are larger facilities than the blast tube, and specifically designed to test full-scale targets. A blast tube built for this purpose would have to be 2 to 3 km long. Blast tunnels use various techniques to minimize both the tunnel length required and fabrication costs. The driver sections are typically smaller in diameter than the test section and special techniques are used to limit the effect of the rarefaction wave from the end of the tunnel.
4.4.3 EXPERIMENTAL CONFIGURATIONS
DESIGN
Because there is no fixed location at which the high-pressure gas and diaphragm must be located in the blast tube, the designer has a lot of freedom in choosing an experimental configuration most appropriate for the task at hand. The explosive driver can be located anywhere along the length of the blast tube or even outside of it. Multiple drivers can be used of varying strengths to custom tailor a blast environment. Gases or cross-sectional area changes can be used to modify the blast environment. To keep the design cases simple, only a constant cross-section tube with one explosive driver will be considered.
4.4.3.1
CENTERED EXPLOSIVE DESIGN
A good understanding of the wave-shaping capabilities of a blast robe facility can be gained by first looking at a design with the explosive placed in the center of the blast tube as shown in Fig. 4.4.2. The centered explosive design is interesting because it directly correlates to the traditional closed-end shock tube design. Because the tube configuration is perfectly symmetrical around the center of the blast tube, it can be modeled as half the problem with a rigid wall at the center of the length of the blast tube. This makes it identical to a shock tube with a pressurized gas driver and a single diaphragm to release the high-pressure gas. The explosive section usually consists of multiple strands of explosives equally spaced in the cross section of the tube with the length of the explosive running down the axis of the robe to create the length of the explosive section shown in the diagrams. For experimental design purposes, the explosive section is treated as if it were uniform pressure and temperature. Though there are small effects from the detonation direction and radial blast, they can
4.4 BlastTubes
627 Explosive
Test Station ~:3
2:~
Explosive
Test Station
FIGURE 4.4.2 Centeredexplosive driver configuration.
be safely ignored for most simulations. With these simplifications, the design is a one-dimensional (1D) problem. For the centered explosive design, the wave system is composed of the following components: 1. a fiat-topped shock wave emerging from the high-pressure explosive section; 2. a rarefaction wave traveling back into the explosive section; 3. a rarefaction formed when the shock wave exits the shock tube and travels back up towards the test station; and 4. the material interface of the detonation products traveling towards the test station. The sequence of events is as follows. First the fiat-topped shock wave starts down the blast tube from the explosive section towards the test station. Then rarefaction waves enter the explosive section from both ends of the section. The rarefactions cross at the middle of the explosive section with one of the rarefactions heading towards the test station. In the full blast tube model, a rarefaction enters from the opposite side of the explosive section from the test station, passes the other rarefaction at the tube centerline, and then continues on towards the test station. In the half-problem, the rarefaction enters the explosive from the test station side of the explosive section, reflects off the rigid wall, and then travels back towards the test station. In actuality, these two scenarios are identical in all respects. The rarefaction will catch up with the initial shock wave at about 2--4 driver lengths down the blast tube, turning it into an exponentially decaying blast wave. A rarefaction from the end of the blast tube is formed as the shock wave exits the tube. It travels into the blast tube and passes over the test station.
1~. Robey
628
The detonation products travel down to the test station. This starts at the time the explosive goes off, but the material interface travels more slowly than the shock and rarefaction waves. During the travel time, the gases behind the material interface have cooled off due to the expansion of the detonation gases. However, if the density of the detonation products is significantly different than the density of the compressed air just ahead of the material interface, they can change the dynamic pressure at the test station. The following calculations can help roughly determine some of the key parameters of the design. The first key parameter is the pressure of the fiattopped shock wave. The Riemann equation can be used to solve for either the driver pressure or the pressure of the fiat-topped shock wave. The full derivation of this equation and some of the following equations can be obtained from many textbooks (e.g., Currie, 1974) that deal with fluid and/or gas dynamics. Usually the equation must be solved iteratively. -- 2~dri .... / (~driver - -
1)
()~driver- 1)Cam~b( psh~ 1) Pdriver __ Pshoc.____kk 1 -Cdriver \ Pamb . PareD Pamb ~ [( (Pshock)1 2~/air ?air -- 1) + (~air + 1) ~ 1 \ Pamb where p is the pressure, c is the speed of sound and 7 is the ratio of the specific heat capacities. The speed of sound c can be calculated by the following equation:
The blast tube is normally filled with atmospheric air at ambient conditions. For initial designs at elevations close to sea level, the standard atmospheric conditions can be used (Handbookof Chemistryand Physics,1975-1976). They are:
Pair = 101,325 Pa Cair = 340.29205 m/s Pair = 1.2250140 kg/m Tair = 288.16 K where T is the temperature and p is the density. The actual conditions at the site should be used for the final experimental design or for post-shot analysis. Detailed calculations for the driver conditions will be discussed in Section 4.4.4, which deals with the driver design. But for estimation of the shock tube design, the specific heat capacities ratio 7 of the
629
4.4 Blast Tubes
detonation products will usually be ~ 1.3 and the driver pressure obviously must be less than that which the steel of the tube can survive. This usually limits the driver pressures to < 2 MPa for strong blast tubes. After selecting a driver pressure that is below the maximum pressure, the temperature can be estimated by assuming a perfect gas, T2 _ P2 T1 Pl The shock speed of the fiat-topped shock wave can be calculated with the following equation, Vshock -- MshockCai r where !])air- 1)-~-(Tair + l)(Psh~
Msh~ --
27air
k, Pair J
The arrival of the rarefaction wave from the muzzle end of the blast tube can be roughly estimated by assuming a wave traveling at the speed of sound of the ambient air. In reality, the shock wave will be traveling faster than the speed of sound as it travels to the end of the blast tube. Then the rarefaction wave will be traveling slower than the ambient speed of sound on the return journey as it fights its way back up against the flow. These effects roughly cancel out. The usual practice is to take the positive phase duration that is desired and calculate how far from the end of the blast tube the test station should be placed. Ltest station to tube exit
Cairtpositive phase
The velocity of the material interface can be calculated with the following equation: 1/2 l/interfac e ~ Cair
[
XF'shock
7air (~air -- 1) + (Tair -~- 1)
(?air)] x,/ashock/ /
The distance from the driver to the test station can be estimated to keep the detonation products from arriving during the positive phase of the blast wave.
630
R. Robey
This is calculated by assuming the material interface travels at its initial velocity the entire time, yielding the following equation, Ldriver to test station - - Uinterfacetpositive phase
This estimated length is high because the velocity of the material interface slows down substantially after the rarefaction wave catches up to the initial shock wave. Further analytical calculations can be done, but in practice a 1D compressible fluid dynamics modeling program is much more practical for fine tuning the design. Experience with the centered explosive configuration has shown that the rarefaction entering into the explosive is not strong enough to produce the desired static pressure waveform. The waveform at the test station typically has too long a tail to simulate the corresponding waveform from an open-air explosion. An example of a static pressure waveform produced by an explosive driver located in the interior portion of the blast tube is shown in Fig. 4.4.3. Note that the pressure decay is exponentially decaying towards ambient pressure rather than below ambient pressure. A simplified explanation is that the expansion driving the rarefaction is only 1D in the shock tube while the expansion in the open-air explosion is three-dimensional (3D). Based on this simple explanation, it is clear that a configuration with the explosive at the end of the blast tube will better simulate the exponential decaying pressure waveform from an open-air explosion.
4.4.3.2 EXPLOSIVE AT THE END OF THE BLAST TUBE The blast tube configuration with the explosive section at the end of the blast tube is shown in Fig. 4.4.4. The blast tube allows the blast energy to be conserved from the explosive section to the test station so that a smaller explosive charge can simulate the effects of an equivalent open-air explosion. The energy flux through the test section can be determined from the blast waveform specification. Multiplying this energy flux by the area of the blast tube gives the energy that must be sent down the blast tube by the explosive driver. Assuming half of the energy of the driver goes towards the test station and half goes towards the breech end of the blast tube, a good estimate of the required driver energy can be calculated. The four components of the wave system are shown in Fig. 4.4.4. These are the same as the components in the centered explosive configuration except that the rarefaction entering the explosive section is combined with rarefaction
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from the tube exit. This combined rarefaction travels at the same velocity as that of the centered-explosive design, because the head of the rarefaction can only travel at the speed of sound in the fluid through which it is traveling. However, the rarefaction wave is stronger and sharper, with the tail of the rarefaction traveling faster than the rarefaction tail in the centered-explosive design. This is due to the greater expansion area for the blast energy at the end of the blast tube. As a result, this design comes closer to matching the blast wave specification from an open-air explosion. The same equations from the centered driver can be used to estimate the key parameters for this configuration. To fine tune the design, a 1D compressible fluid dynamics modeling program should be used. To properly model the rarefaction from the end of the blast tube, an area change should be incorporated into the model. For the purposes of modeling the flow inside the blast tube, it is sufficient to use a smooth expansion of the tube diameter with a linear or quadratic function rather than an abrupt change to a tube of infinite area. The explosive driver section inside the blast tube produces a radial pulse, which is seen in the downstream pressure records at the characteristic period of the shock tube. This characteristic period is t --
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This is the tube diameter divided by the local sound-speed at the gauge location. Any radial pressure pulse will reflect this periodic time interval and will be apparent in the experimental gauge data. The radial pulses from the driver slowly decrease in amplitude to a smooth waveform several driver lengths down the blast tube.
4.4.3.3
EXPLOSIVE O U T S I D E THE BLAST TUBE
This driver configuration places the explosive just outside the end of the blast tube. In many ways, this is just an extension of the concept that was used to justify the placement of the driver section at the end of the blast tube. The
4.4 BlastTubes
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rarefaction wave enters almost immediately instead of having to pass through the driver to catch up with the shock wave. In addition, the radial pulse from the explosive is not reflected by the tube wall, greatly reducing the radial pressure waves down the tube. The explosive charge is no longer constrained by the strength of the steel thus permitting the length of the explosive to be shortened. Experimental tests of this configuration used a thin sheet of explosive on either a wood or foam backing just outside the end of the tube. Multiple-point ignition is required for the sheet explosive to have the best effect. With only a single-point ignition, a curved blast wave will be produced at the entrance to the blast tube and the stagnation pressure will not be uniform across the blast tube. With the explosive outside the blast tube the waveform produced close to the driver is much smoother because the radial pressure pulse is nearly eliminated. An example of the static pressure waveform produced is shown in Fig. 4.4.5. The driver outside the blast tube is preferred when high-fidelity (smooth) environments close to the driver are required.
4.4.4 DRIVER DESIGN The explosive driver is modeled as if the explosive energy is added instantaneously to the driver section to form detonation products at a uniform pressure 40
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and temperature. Even with this simplification, calculating the driver conditions is complex because the normal treatment for calculating explosive energy is not sufficient. The presence of plentiful oxygen in the air surrounding the explosive strands creates additional chemical reactions that can account for half the energy produced by the explosive driver. The source of fuel for the chemical reactions with the oxygen is the jacket material of the explosive. Some of the explosives used in the blast tube have had jacket masses greater than the explosive mass. The explosive jacket material is usually a plastic material. Reasonable results have been achieved by assuming this additional material is polyethylene and modeling it as an ethylene reaction with twothirds of the energy released and the other one-third used in decomposing the polyethylene into ethylene. Each explosive used in the blast tube has to have a similar model constructed and calibrated to predict the driver pressures and temperatures accurately. The steps used in calculating the driver pressure and energy start with the calculation of the mass of the explosive, the jacket material, and the air in the driver section. A small length of explosive is taken apart and the components weighed to determine the fraction of jacket material. The second step is to determine the standard composition of the detonation products from a suitable reference (e.g., Mader, 1979). The products are usually given in moles of each constituent produced and have to be converted from mole fractions to mass fractions. Then the moles of oxygen in the driver section are calculated from the air and in the detonation products. Assuming the ethylene reaction given in what follows, calculate the amount of jacket mass that reacts and the mass quantities of burn products created, C2H4 if- 302 ~ 2CO 2 + 2H20 Add together the masses of each gas constituent from the detonation products, burn products, and the air originally in the driver section. Remember to subtract out the oxygen consumed in the chemical reaction. Then calculate the specific heat of the mixture using a mass average of the specific heats of each of the gas constituents. The ratio of the specific heats 7 can then be calculated, Cpdriver ~driver -- Cvdriver
The energy for the driver is calculated by adding both the energy produced by the explosive and released by the chemical reaction to the original energy in the air in the driver section. The density of the driver is determined by adding the explosive mass and air mass and dividing by the driver section volume.
635
4.4 Blast Tubes
The pressure, temperature, and speed of sound of the driver section can be determined using the perfect gas equation of state, Pd~iver -- ednver(~driver -- 1)Pd~iver Mdriver Tdriver -- edriver(Ydnver -- 1) Cdriver
__ ~/TdriverPdriver V
where Mdriver is the molecular weight of the driver gas and ~ is the universal gas constant.
4.4.5
DETONABLE
GAS DRIVER
An alternative to using strands of explosive for the explosive driver source is to use a detonable gas. Some of the detonable gas drivers that have been tried are methane/air, MAPP gas/air, and hydrogen/oxygen. A thin plastic sheet is used to seal off the driver section and all the cracks and leaks are plugged with caulking. To ensure detonation, the amount of gases put into the driver must be measured so that it is near the stoichiometric mixture. Then it must be mixed well using a disposable fan in the driver section or a pump extemal to the blast tube, but in either case it must be a sparkless device. A small explosive initiator or an ignition spark is then used to initiate the detonation. All operations must be done remotely for the gas fill, mix, and initiation to ensure safety. This may still be preferable over the solid explosive handling procedures for many experimenters. Others will find the messy gas handling operations to be less desirable than the solid explosives. The waveforms produced are not substantially different than those produced by the solid strands of explosives if the same number of initiation points is used in the gas driver as the number of solid strands of explosive.
4.4.6
SIMULATION
SCALING
The length of the shock tube and the maximum driver pressure provide hard limits to how large a blast simulation can be achieved. A common technique to avoid these limitations is to conduct a scaled simulation. Various forms of scaling laws have been developed (for more details, see Baker, 1973). The most common blast wave scaling is Hopkinson scaling. In Hopkinson scaling, the length dimensions are scaled linearly by the scale factor and the cube of the scaling factor is used to scale the yield. For a simulation of a 1-kiloton
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explosion at 500m from the explosion, a 1/10th-scale simulation would require only a ton of explosive with the test station at 50 m. The test object would be a 1/10th-scale model in every dimension. Although all the blast parameters scale very nicely according to this scaling law, boundary layers, viscosity, and structural properties do not scale the same way. To apply a scaling factor to a blast tube simulation, adjust the blast wave specification by dividing the time axis by the scaling factor. This reduces the positive phase duration by the same scaling factor as well as the length of the blast tube required to perform the simulation.
4.4.7
SIMULATION
ENVELOPE
The simulation envelope for a facility can be determined by the physical constraints of the maximum driver pressure and the tube length. An example of a simulation envelope is shown in Fig. 4.4.6 for a hypothetical blast tube 500 m long with a maximum driver pressure of 1800 kPa. The maximum driver pressure of the blast tube limits the line on the fight of the simulation
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envelope. The maximum driver pressure limits the maximum fiat-topped shock that can be propagated down the blast tube, assuming that the temperature and 7 of the detonation products cannot be greatly varied for a given detonation pressure. For the hypothetical blast tube, a driver pressure of 1800 kPa will generate a 500-kPa fiat-topped shock, making this the maximum simulation pressure. The length of the blast tube determines the curved line on the left-hand side of the simulation envelope. The positive phase duration of the blast wave is limited by the length of the blast tube and thereby limits the size of the blast that can be simulated. Since lower pressures have longer time durations for an explosion of a given size, the envelope is smaller at lower pressures.
4.4.8
INSTRUMENTATION
Pressure transducers are the most common forms of instrumentation used in blast tubes. The typical transducer must be small in size with high frequency response to resolve the shock wavefronts as they cross the sensing element. Several commercial suppliers such as Kulite, Endevco, PCB, Keller, and Kistler exist for pressure transducers with different models suitable for different applications (For the reader's convenience the websites of the companies producing these pressure transducers are given in the References). The blast environment creates special difficulties for pressure transducers with its explosive flash, heat, and small particle debris. Sensing elements in close proximity and direct line of sight of the explosives must have a protective filter to increase the gauge survivability and avoid anomalous data effects. Filters that have commonly been used include screens, small pieces of electrical tape, and coatings such as silicon, zinc oxide, or General Electric RTV (for more details see Doerr, 1993). These filters must be selectively applied to gauges that require protection as the loss of frequency response should be avoided when the protection is unnecessary. Some gauge models have built-in protection, which may be preferred over that applied by the experimenter. Static pressure measurements are the most common type of measurement in the blast tube. Mounting the pressure gauges for static pressure measurements requires special considerations. Strong pressure and acceleration waves will be present in the blast tube walls and even in the nearby surroundings. The gauges should be mounted in an acceleration damping material such as teflon, delrin, or phenolic mounts. The most common practice is to machine threads on the inside and outside of a hollow cylinder of the mounting material. The gauge is then inserted into the mounting device, which is then mounted in the steel tube wall. Flush mounting of the gauge for static pressure measurements
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is very important to obtain the proper pressure data. Instrumentation wiring must also be done with care. The strong vibration environment can cause loose connections to vibrate and cause sharp spikes in the data. Connections should be firm and protected as much as possible from vibration. Stagnation pressure measurements are the second most common measurement. Static pressure gauges are not sufficient to fully determine the flow properties at a particular point in the blast tube. Either stagnation or dynamic pressure data is required and stagnation pressures are easier to measure. To obtain stagnation pressure measurements, a mounting structure is necessary to enable the gauge to be placed facing upstream. It is best to have more than one pressure gauge at a tube cross section to detect variations in the stagnation pressure across the tube. One of the mounting wings that has been used for collecting stagnation pressure measurements is shown in Fig. 4.4.7. Note that the gauges are inserted into the front of the long probes extending into the flow. The purpose of the probes is to get the gauge in front of the bow shock that forms in front of the mounting wing. The gauges are slightly recessed in the probe and protected with a small screen inserted at the front of the probe. Static pressure measurements can be obtained from the wing-mounted gauges by closing the front facing hole with a conical point and placing a set of holes around the diameter of the probe in front of the gauge. Dynamic pressure measurements can be made with differential pressure gauges with a stagnation configuration in the front and a static pressure configuration behind the gauge. Examples of these probe configurations are shown in Fig. 4.4.8. Calibration of the gauges is performed in a laboratory prior to installing them in the blast robe. Once they are installed in the blast robe, it can be difficult and time-consuming to remove them and calibrate the gauges for
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every test. A reasonable solution is to test the installed gauges with an in situ device to ensure proper functioning. The calibration constant used for data reduction remains the laboratory calibration, but the in situ test can detect gauge problems in the installed configuration and greatly improve data quality. An in situ test device can be a simple attachment to a pressure bottle that is placed over the gauge position. The gauge output is recorded and checked to see if it is reasonable. More elaborate tests can be performed with a pressure pulse instead of a constant pressure, but the additional complexity is not usually necessary. If there is a problem with the gauge, it is usually obvious with a simple test. Digital data recording has all but replaced the older magnetic tape recording systems. The driving factors are the cost of the data recording equipment and the labor required to manually extract the data from the magnetic tapes. Digital systems produce higher quality data, but they are less forgiving for errors in the start and end time estimates and the maximum data values to be observed. Data recording should be done as close as practical to the experimental facility to minimize the frequency loss from long data lines.
4.4.9
APPLICATIONS
The applications for a blast effects facility include ideal and nonideal blast effects studies, survivability studies with scaled models, civil defense structures, accidental explosion scenarios, terrorist threats, and explosive properties. Real world explosions generate blast waves that can be substantially different than an ideal blast wave. Blast waves that differ from the ideal blast wave are generally referred to as nonideal blast waves. Nonideal blast waves include the effects that have been observed for real explosions including multiple-blast sources, dusty flow conditions (Newell, 1989), shock reflections, and nonuniform shock speeds in layers of gas with different speed of sounds
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(Ganong and Stockham, 1989). These environments are highly variable as they are dependent on localized conditions. The blast tube provides a way to study the behavior of these complex environments and determine hardware or structural survivability under these real conditions. More detailed discussion on nonideal blast environments simulations are covered in a later section. The size of a typical blast tube only permits scaled model studies of survivability. Full-scale survivability tests require a larger cross section and longer positive phase duration. These types of environments are only available in blast tunnel facilities. However, small-scaled model studies can be useful for parameter studies to look at the effects of different design options on increased survivability of a test system or to determine sensitivity of the design to different types of blast environments. Civil defense structures are designed to survive specified blast environments. Critical components such as the air supply system must have valves that survive these environments. Testing of these critical components is often performed in blast tubes. Accidental explosions and terrorist threats have been an increasing concern in recent years. Tests of potential scenarios and the effectiveness of countermeasures have been conducted in blast tubes. Even natural events such as explosive volcanism have been studied in a vertical blast tube (Sturtevant et al., 1991). The properties of solid explosives, liquid sprays, and gas mixtures continue to be studied in blast tubes and other experimental facilities. These properties can be used to improve our understanding of the complex physical and chemical processes that are occurring as well as to determine critical parameters to be used for numerically modeling these events. Material properties of other solids, liquids, and gases are often tested in shock tubes. The purpose of these material tests may be either to understand the behavior of shock waves passing through the material or to determine the fundamental properties of the material for use in other applications.
4.4.9.1
N O N - I D E A L BLAST WAVE S I M U L A T I O N S
4.4.9.1.1 Dust-Laden Blast Waves The introduction of dust into the blast tube produces a more realistic blast environment. It also produces more severe conditions for instrumentation and accurate measurements. The proper amount of dust, its particle size, and distribution will be highly variable and dependent on local conditions of a blast site. A reasonable dusty flow condition should be selected and the studies conducted. Once the potential effects of the dust-laden blast wave are
641
4.4 Blast Tubes
determined, speculation can be made about variations from the design conditions. Accurate and repeatable simulation of dusty flows in a blast tube is also very difficult. The first challenge is to loft the proper amount of dust into the flow. Spreading dust out on the floor of the blast tube produces relatively low amounts of dust in the flow, due to the reduced length and time scales in a scaled simulation. Other experimental techniques are needed to produce heavier dust concentrations. Further, a lot of variation in the dusty flow occurs across the blast tube within very short distances. The introduction of dust into the flow creates the problem of trying to measure it. The measurement techniques for dust flow are based on work originally developed by Bannister (1959). He designed gauges that he called greg and snob gauges. The terms were shorthand for gregarious and snobbish, which describes the way the gauges were designed to behave towards the dust in the flow. The snob gauges were designed to ignore the dust and the greg gauges were to register the effects of the dust. In a more technical description of their behavior, it is the amount of dust momentum flux they register, or a dust registry coefficient that ranges from zero to one. A snob gauge should have a dust registry coefficient close to zero and a greg gauge should be close to one. The actual dust registry coefficients can be far from this ideal and are strongly dependent on dust size and the Mach number of the flow. Shown in Figs. 4.4.9 and 4.4.10 are examples of a greg and a snob gauge. The greg gauge is blunt shaped, with the sensor recording either the stagnation effects of the particles or the particle impacts. The snob gauge stagnates the flow by choking it. The dust particle should pass through with minimal effect on the pressure measurement at the front of the gauge. The amount of dust loading is determined by subtracting the snob gauge from the greg gauge. In practice, reasonable results are hard to obtain because
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the gauges are not located at the same point in the flow and any instrumentation problem such as baseline shift or anomalous data spikes can dominate the differences between the gauges.
4.4.9.1.2 Wall Jets The presence of air layers next to the ground with different temperatures or densities than the surrounding air can significantly modify blast wave behavior. The effects of fast and slow sound-speed layers have been studied by Needham and Schneider (1993) Ben-Dor and Rayevsky (1992), and Rayevsky and BenDor (1994). The high sound-speed layer produces a shock precursor that runs ahead of the primary shock wave. This produces a complex shock structure with a precursor shock, a triple point, and a Mach stem as shown in Fig. 4.4.11. Behind the shock waves a vortex is formed, forcing a cold gas jet along the surface. The effect is to enhance the dynamic pressure conditions while reducing the peak static pressure. The high sound-speed layer can be simulated in the blast tube with a layer of helium gas. A mylar sheet is stretched out above the floor and at the front and back of the helium layer. The helium is introduced into the blast tube prior to the experiment. The concentration of the helium is confirmed through sound-speed measurements. The explosive is set off and the blast wave propagates down the blast tube. When the blast wave encounters the helium layer, a precursor shock or compression wave will be propagated down the blast tube ahead of the main shock wave. The mylar essentially shatters under the suddenly imposed loading. The complex shock structure will then form
4.4
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Blast Tubes
FIGURE 4.4.11
Precursor shock structure caused by a high sound-speed layer.
and can be observed at various points along the blast tube. Shown in Fig. 4.4.11 is a frame from a high-speed camera of the shock structure that occurs in the experiment. The shock in the helium is running out ahead of the main shock and a diagonal shock connects the two shocks, forming a triple point. The dark rolled-up structure is the mylar from the experiment. Shown in Fig. 4.4.12 is a numerical calculation of the high sound-speed layer. The pressure contours show the same shock structure as in the experiment. The structure behind the shock waves is not as complex as in the experiment because many of these structures are density changes, which do not show up in the pressure contour plot. A low sound-speed layer causes the main shock wave to run ahead of the shock wave in the layer of gas near the surface. A complex shock structure develops with either a Mach reflection or a regular reflection. Shown in Fig. 4.4.13 are results from an analytical and numerical study done by Ben-Dor and Rayevsky (1993). The main shock wave in the air can be seen ahead of the complex shock structure in the dense gas layer. The diagonal shock from the main shock layer to the floor can be either a Mach reflection or a regular reflection depending on the conditions of the flow. The static pressure environment at the surface can be greatly enhanced by the shock reflections. Behind the shock waves, the flow rolls up in a similar manner as that seen in
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FIGURE 4.4.12
Numerical calculation of a high sound-speed layer.
the high sound-speed layer case but in an upward direction rather than a downward one. The low sound-speed layer can be simulated in a similar manner as the high sound-speed layer but with a denser gas. Freon-12 has been used in small shock tube experiments (for more details, see Kuhl et al., 1991). The gas layer was injected through a porous floor just prior to the experiment. A plastic sheet was not used to constrain the gas. The Freon-12 gas is heavier than air and so it tends to stay on the floor of the shock tube. This experiment looks at the turbulent structure that develops behind the shock waves rather than the shock structure itself. The rich shock and flow structures that occur in both the high- and low sound-speed layers provide a rich source of experimental and numerical work for the future. Higher fidelity simulations of the wall layer will be needed for more detailed studies of these shock wave phenomena.
4.4.9.2
MODEL STUDIES
Scaled models can be tested in blast tubes to better understand the blast effects on test objects. Once a model is placed into the flow, the inviscid flow assumptions used for the blast environment must be reexamined. For the diffraction phase loading, which is the loading caused by the shock initially passing over the object, the inviscid flow assumptions work well. For the drag phase (late time) loading of the test article, two viscous effects are important. The first is the boundary layer from the walls and floor of the blast tube. The second is the flow separation point on the object. To properly match the viscous flow effects in the incompressible part of the flow, the Reynolds number must match between the simulation and the real event. For the compressible parts of the flow, the Prandtl number should also be matched.
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Just matching the Reynolds number would probably be considered sufficient because the effects are at later time in the blast wave and consequently at lower pressure. Matching the Reynolds number in an experiment is difficult, as it requires using a different gas. This is usually not practical in a blast tube experiment. The impact of not correctly modeling the viscous flow effects is to limit the scale factor that experimenters can use. Scaled model tests in blast tubes can be used to determine the loading environment under different blast conditions. These loads can be used in designing or modifying the full-scale versions of the test article. Evaluating model prototypes early in design studies can be very effective in evaluating different design concepts. This is similar to the use of wind tunnels to evaluate design prototypes prior to the construction of full-scale prototypes. Another use of scaled model tests is to validate numerical response models for different types of test objects. Rouquand and Tournemine (1989) provide an excellent example of this kind of work in developing an enhanced model for overturning semi-truck trailers in the SORROCO blast tunnel. In their work, a handful of experimental tests were used to validate a response model composed of a collection of springs, shock absorbers and sliding contacts.
4.4.9.3
CIVIL DEFENSE STUDIES
Civil defense studies in blast tubes usually look at the propagation of blast waves into civil defense structures in some manner. The propagation of blast waves in tunnels with various branching, turns, and various chambers has been a constant source of experimental work in blast tubes, shock tubes, and open-air experiments. Blast valve survivability studies have looked at the survivability of blast valves and their effectiveness at reducing blast wave propagation into structures. Both active and passive blast valves have been the subjects of these studies. Various environments have been used, including side-on, direct, and those environments caused by reflections from nearby walls. Blast tube and open-air experiments have also examined the survivability of various types of construction such as brick walls. The results of these studies can be used for the establishment of building codes for civil defense structures.
4.4.9.4
DETONATION STUDIES
A close relative to the blast tube is the detonation tube used in studies of detonation behavior for gases, liquid sprays, and dust. These facilities are usually called detonation tubes, even though they may be exactly the same
4.4 Blast Tubes
647
facility as the blast tube already described here. Other possible configurations include a circular tube closed at one end, a square or rectangular tube, or a rectangular tube with a removable top wall. The facility must use a large tube so that the diameter is larger than the critical diameter for detonating the gas being studied. If the tube is smaller than the critical diameter, the gas cannot be detonated. The critical diameter is a property of each fuel and is dependent on fuel composition and its pressure. At lean gas mixtures, the critical tube size can be the size of a small blast tube. Detonation and deflagration studies have increased steadily in the last two decades due to high profile accidents. These accidents have included the hydrogen bubble that occurred in the Three Mile Island incident (PA) (Berman, 1981), several petrochemical accidents, railway accidents with LPG, airline disasters, and a series of grain dust explosions (Lee, 1981). With every accident comes a flurry of experimental activity as researchers try to understand the physical processes that occurred and how to prevent such incidents in the future. 4.4.9.4.1 Transition from Deflagration to Detonation A fuel that is initiated will often start as a burn or a deflagration. The deflagration will under certain conditions transition to a detonation with much greater propagation speeds and pressures. The damage caused by a detonation can be far greater than that produced by deflagrations, making this situation of great concern in the petrochemical industry. Understanding the conditions that increase the likelihood of a transition to a detonation can help in the design of safer gas handling facilities at ports, pipelines, and petrochemical complexes. The experimental design is a circular tube with one end closed. A combustible gas mixture is introduced into the tube. An initiation with a spark or small explosive starts the combustible gas burning at the closed end of the detonation tube. The gas initially starts to react as a deflagration and measurements are taken to see if the deflagration transitions to a detonation and how long it takes. The shock speed and pressure increase dramatically when it becomes a detonation. Both pressure and time-of-arrival gauges can be used to determine when this occurs (for more details, see Ohyagi et al., 1991). Changing the conditions of the tube by adding obstacles can greatly accelerate the transition to a detonation. The obstacles are added to observe the sensitivity of the fuel mixture to these conditions. Inhibitors of the reaction can be introduced to determine their effectiveness at preventing the transition to detonation or perhaps even quenching the deflagration. Studying the conditions of detonations in gases is also important in understanding the detonation processes in solids and other materials. The length
R. Robey
648
scales in gases are far larger than in a solid explosive, allowing better experimental analysis. The length scales have been lengthened further by techniques such as detonations at lean limits or in the presence of an inert gas. 4.4.9.4.2 Turbulent Flame Acceleration Greatly increased flame speeds and pressures have been observed in accidents where only a deflagration is believed to have taken place. This has stimulated research into deflagrations in highly turbulent flow that causes flame propagation speed and pressure to increase over the laminar deflagrations. To study these enhanced deflagrations, a flame acceleration tube is used. It is a circular or rectangular tube with orifice plates or obstacles placed in the flow. The deflagration accelerates with the turbulent flow through the orifices. Strangely enough, the flame may also be quenched by the orifice for mixtures near the rich and lean limits. 4.4.9.4.3 Hot Jet Ignition The ability for a deflagration to pass through a small opening into another chamber or out into the open air and ignite a combustible mixture is of serious concern. Large experimental facilities have been constructed to study the ability of the so-called "hot-jet" to ignite combustible mixtures. A detonation or flame acceleration tube with orifice plates is configured to exit into a larger tube or open air. This produces a hot turbulent jet at the expansion into the larger tube, which has been observed to produce pressures comparable to detonation pressures (for more details, see Hjertager et al., 1981). 4.4.9.4.4 Dust Detonations Dust detonations have been studied in both horizontal and vertical shock tube configurations. The detonation tube is filled with a dust/air mixture and then an explosive initiator is fired from one of the ends. The propagation of the initiator into the dust is observed to see if it decays or proceeds at the Chapman-Jouget (CJ) velocity (for more details, see Kauffman and Nicholls, 1981). Deflagrations are also studied for many of the same phenomenology as seen for gas detonations.
4.4.10
CONCLUSION
Blast tubes have played an important role in the continuum of experimental test facilities between the small laboratory-scale shock tubes and full-scale
4.4 Blast Tubes
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testing in open-air tests and blast tunnels. This role will continue into the future as experimental techniques strive to keep up with the rapid numerical advances of the last decade. Improvement in experimental techniques is vitally necessary not only for improved experimental results, but also as the fuel for the further advancement of the numerical modeling techniques. Higher fidelity experiments are needed as well as better measurement techniques to detect when the theoretical models are insufficient to describe the observed behavior. Higher fidelity blast waves will be one of the necessary components to improve the experimental methods. Use of sheet explosive drivers or even explosive lenses outside the end of the tube will be necessary to produce more uniform and smoother decaying waveforms. Coupled with the improved blast waves will be better experimental measurement techniques to resolve the structure of detonations and chemical reactions that occur over very short physical and temporal intervals. The ability to measure and describe turbulent processes accurately will be necessary to drive the scientific understanding of turbulence. Many of the processes that still remain to be understood are extremely complex. The ability to conduct experiments with a two- or a three-dimensional structure will be needed to address many of these problems. One of the lessons learned from many years of experimental and numerical studies is that little is learned from either experiments or numerical studies alone. It is when these two tools are combined and contrasted that leaps of understanding occur. It will be vitally necessary in future work to closely couple these tools together to produce new breakthroughs in scientific theory.
REFERENCES Baker, W.E. (1973). Explosions in Air, San Antonio: Wilfred Baker Engineering. Published in cooperation with the Southwest Research Institute. Bannister, J.H. (1959). Particulate dynamics research at Sandia Laboratory, Proc. 6th Midwestern Conf. Fluid Mech., Austin: Univ. Texas. Ben-Dor, G. and Rayevsky,D. (1993). Numerical and analytical investigation of the interaction of a blast wave with a high density layer, 13th Int. Symp. Military Appl. Blast Simulation, The Hague, The Netherlands. Ben-Dor, G. and Rayevsky, D. (1994). Shock wave interaction with a high density step like layer. Fluid Dyn. Res. 13(5): 261-279. Berman, M. (1981). Hydrogen combustion research at Sandia, Proc. Int. Conf. Fuel-Air Explosions, Montreal, Canada: Univ. Waterloo Press. Currie, I.G. (1974). Fundamental Mechanics of Fluids, McGraw Hill, Inc. Doerr, S. (1993). Selection and evaluation of a pressure transducer for blast measurements, 13th Int. Symp. Military Appl. Blast Simulation, The Hague, The Netherlands. Endevco Catalog at http://www.endevco.com Handbook of Chemistry and Physics 1975-1976, 56th ed., Cleveland, OH: CRC Press.
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Ganong, G.P. and Stockham, L.W (1989). Problems associated with production of wall jets in shock tubes, l lth Int. Symp. Military Appl. Blast Simulation, Albuquerque, NM. Glowacki, WJ., Kuhl, A.L., Glaz, H.M., and Ferguson, R.E. (1985). Shock wave interaction with high sound speed layers, Proc. 15th Int. Symp. Shock Waves and Shock Tubes, Berkeley, CA. Hjertager B.H., Fuhre K., and Eckhoff R.K. (1981). Large-scale experiments on turbulent flame and pressure development, Proc. Int. Conf. Fuel-Air Explosions, Montreal, Canada: Univ. Waterloo Press. Kauffman, C.W and Nicholls, J.A. (1981). Dust explosion research at the University of Michigan, Proc. Int. Conf. Fuel-Air Explosions, Montreal, Canada: Univ. Waterloo Press. Keller Catalog at http://www,keller-druck.ch Kistler Catalog at http://www.kistler.co.uk Kuhl, A.L., Reichenbach, H., and Ferguson, R.E. (1991). Shock interactions with a dense-gas wall layer, Proc. 18th Int. Symp. Shock Waves, Sendai, Japan. Kulite Catalog at http://www.kulite.com Lee, J.H.S. (1981). Explosion research of the shock wave physics group at McGill, Proc. Int. Conf. Fuel-Air Explosions, Montreal, Canada: Univ. Waterloo Press. Mader, C.L. (1979). Numerical modeling of detonations, Berkeley, CA: Univ. California Press. Needham, C. and Schneider, K. (1993). Numerical predictions to assist in the design of a precursor experiment, 13th Int. Syrup. Military Appl. Blast Simulation, The Hague, The Netherlands. Newell, R.T. (1989). Dust laden blast environments in shock tubes, 1 lth Int. Symp. Military Appl. Blast Simulation, Albuquerque, NM. Ohyagi, S., Ochiai, T., Yoshihashi T., and Harigaya, Yi. (1991). Growth of cell widths in gaseous detonations, Proc. 18th Int. Syrup. Shock Waves, Sendai, Japan. PCB Catalog at http://www.pcb.com Rayevsky, D. and Ben-Dor, G. (1992). Shock wave interaction with a thermal layer. A/AAJ. 30(4): 1135-1139. Rouquand, A. and Tournemine, D. (1989). TRUCK2V, a modified version of the truck computer code for overturning calculation of semi-trailers exposed to blast waves, 11 th Int. Conf. Military Appl. Blast Simulations, Albuquerque, NM. Sturtevant, B., Glicken, H., Hill, L., and Anilleumar A.V. (1991). Explosive volcanism in Japan and the United States: Gaining an understanding by shock tube experiments, Proc. 18th Int. Symp. Shock Waves, Sendai, Japan.
CHAPTER
4.5
Shock Tubes and Tunnels: Facilities, Instrumentation, and Techniques 4.5 Supersonic and Hypersonic Wind Tunnels BRUNO CHANETZ* Head of Unit "Experimental Simulation", Head of Hypersonic Hyperenthalpic Project, ONERA, 8 rue des Vertugadins 92190 Meudon, France
AMER CHPOUNt t Laboratoire d'Adrothermique du CNRS, 4 ter, Route des Gardes 92190 Meudon, France
4.5.1 4.5.2 4.5.3 4.5.4 4.5.5
Introduction The Nozzle The Diffuser Start-Up Process Supersonic and Hypersonic Continuous Wind Tunnels 4.5.5.1 Return-Circuit Continuous Wind Tunnels 4.5.5.2 Open-Circuit Continuous Wind Tunnels 4.5.6 Blow-Down Wind Tunnels 4.5.6.1 Preliminary Remarks 4.5.6.2 Description of a Classical Cold Blow-Down Wind Tunnel 4.5.6.3 Induction Blow-Down Wind Tunnel 4.5.6.4 Description of a Hot Blow-Down Wind Tunnel 4.5.7 Experimental Techniques 4.5.7.1 Pitot Probe Technique 4.5.7.2 Multihole Pressure Probes
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
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4.5.7.3 Electron Beam Fluorescence Technique (EBFT) 4.5.7.4 Heat Flux Measurement by Surface Measurement Techniques 4.5.7.5 Infrared Thermography Technique 4.5.7.6 Laser Doppler Velocimetry (LDV) 4.5.8 Summary References
4.5.1
INTRODUCTION
Wind tunnels are used for simulating aerospace systems flight conditions in the laboratory. In principle, these facilities include a gas supplying system (source), a nozzle that transforms the gas from reservoir conditions to test conditions, the test section, and finally a diffuser and an exhaust system. Several features characterize a given supersonic wind tunnel depending on its operating aspects: 9 supersonic, hypersonic or hypervelocity wind tunnels (depending on the Mach number and temperature required in the test section); 9 low- or high-enthalpy wind tunnels (depending on the inlet gas stagnation enthalpy); 9 blow-down or continuous operating wind tunnels (regarding running time); 9 open- or closed-circuit wind tunnels (continuous wind tunnels); 9 free or closed jet wind tunnels; 9 atmospheric inlet or high-pressure inlet wind tunnels (blow-down wind tunnels); 9 atmospheric or vacuum exhaust wind tunnels (blow-down wind tunnels); 9 planar or axisymmetric nozzle wind tunnels (depending on the nozzle exit section geometry); 9 single or multinozzle wind tunnels (depending on the number of nozzles); 9 fixed geometry or variable geometry nozzle wind tunnels; and 9 rarefied gas flow or continuous gas flow regime wind tunnels (depending on the gas density in the test section). Any wind tunnel can be characterized by a combination of these listed features. In general, the Mach number 5 is considered as the limit between supersonic and hypersonic wind tunnels. However, this limit is a matter of controversy. Hypervelocity wind tunnels operate at considerably higher pressure and temperature supply than do hypersonic facilities. Trisonic (also
4.5 Supersonicand Hypersonic Wind Tunnels
653
known as polysonic or multisonic) wind tunnels operate over three speed ranges, that is, subsonic, transonic and supersonic regimes. The available time for testing a model in a wind tunnel, called the running time, depends on the source of the gas and of the type of exhaust system. Blowdown wind tunnels are characterized by a short running time, whereas the running time for a continuous flow wind tunnel is virtually infinite. For a blow-down tunnel, the gas supply can be either the atmosphere or a highpressure gas storage tank. For the test gas two options of discharge are available, discharge into a vacuum tank or into the atmosphere. In a blowdown wind tunnel, the testing time depends on the capacity of the gas storage tank. If the discharge is into a vacuum tank, the volume of the exhaust tank determines the testing time. Long testing time can be obtained when a compressor or the atmosphere supplies the gas and the exhaust condition is achieved by vacuum pumps. The corresponding facility is a continuous flow wind tunnel. The limiting factor for a continuous facility is the continuous power supply for driving the compressor. In the case of a supersonic flow wind tunnel with a large test section, the power required can be several hundreds megawatts. For this reason, large test section wind tunnels are usually of the blow-down type, where energy can be stored for a long period and then released for a short time. In the case of continuous flow tunnels, if the gas is supplied to compressor from the atmosphere, the facility operates in an open circuit. A closed-circuit wind tunnel operates with a single machine that is both a vacuum pump and a compressor. In such a case, in order to avoid a continuous rise in gas temperature due to continuous energy release from the compressor, heat must be extracted from the gas by means of a heat exchanger. To simulate high-altitude rarefied flows, low-density wind tunnels are used. These facilities operate with very small mass flow rates and with large capacity vacuum pumps. Specific measurement techniques are required in these facilities. The tunnel nozzle and diffuser may be located in a big vacuum chamber (called Eiffel chamber). In addition, if a duct does not connect the nozzle and the diffuser, a supersonic free flow surrounded by vacuum can be achieved. In this case the facility is called a flee-jet wind tunnel. To simulate some important aspects of high-velocity flow, for example, real gas effects, and chemically reacting gas, a supply of high specific enthalpy gas is required. Variation of the flow Mach number during the test time can be achieved by varying the ratio between the nozzle exit area and its throat area, that is, by using variable geometry nozzles. Such a flow Mach number variation can be achieved only for planar nozzles. For an atmospheric inlet wind tunnel in which the flow Mach number in the test section is larger than 5, the flow expansion in the nozzle can lead to
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very low static temperatures in the test section. To avoid liquefaction, the gas must be heated prior to entering the nozzle. This can be achieved by using, for example, a graphite heater or electrical resistance. The nozzle is an essential element of the wind tunnel. The theoretical working conditions of a supersonic nozzle are given in the next section.
4.5.2
THE NOZZLE
Figure 4.5.1 shows a supersonic converging-diverging nozzle. For a fixed geometry, the nozzle cross-sectional area variation is a known function. In addition, we assume quasi-one-dimensional (1D) flow throughout the nozzle. The flow properties at each section of the nozzle can be calculated from classical gas-dynamics relations. For example, Figs. 4.5.2 and 4.5.3 show, respectively, the evolutions of the pressure (normalized by the reservoir pressure) and the flow Mach number at each section along the nozzle axis. Depending on the nozzle exit pressure, four flow regimes in the nozzle can be identified from these figures. In the first regime (curves a and b), the flow remains subsonic through the nozzle. The nozzle throat cross-section A is greater than the critical crosssection A*. The pressure evolution presents a minimum at the throat. The flow accelerates through the converging part of the nozzle and decelerates through the diverging part of the nozzle. Curve c presents the limit case where the downstream ambient pressure is sufficiently low, so that the pressure at the throat reaches the critical value. The flow reaches sonic condition at the throat. The mass flow rate rh is now fixed to its critical value: th
-
(4.5.1)
Wp~tA*
Entry plane
Exit plane
FIGURE 4.5.1
Supersonic nozzle.
4.5
655
Supersonic and Hypersonic Wind Tunnels
v/vi ~
ab
O,/v ~ . Shock
Entry section FIGURE 4.5.2
Throat
Exit section
Flow regimes in a supersonic nozzle: pressure distributions.
where tp is a function of the gas specific heats ratio ~, and Pst and Tst are the stagnation pressure and temperature, respectively. After the throat the flow decelerates through the diverging part and remains subsonic. The second regime (curve d) corresponds to the situation where the exit pressure is sufficiently low, so that a normal shock wave is formed in the diverging part of the nozzle. The flow is supersonic between the throat and the shock wave. Downstream of the shock the flow decelerates within subsonic
M
Shock
Entry section FIGURE 4.5.3
Throat
Exit section
Flow regimes in a supersonic nozzle: Mach number distributions.
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conditions, the pressure increases, and the flow Mach number decreases. The shock stands at a position where the local Mach number provides a downstream pressure, which matches the exit pressure after the compression process. The shock moves downstream when the exit pressure decreases but it no longer affects the upstream supersonic flow. Curve e corresponds to a limit situation where the normal shock wave stands exactly at the nozzle exit cross section. Curves f and g characterize the third regime. The flow in the diverging part of the nozzle is entirely isentropic and reaches the ambient pressure through a series of external oblique shocks. Curve h presents a unique situation in which the ambient pressure is identical to the actual isentropic exit pressure Pis- The flow downstream of the nozzle forms a uniform isentropic jet. This situation is called adapted regime. For ambient pressures below that of the exit isentropic pressure (curve k), the exhausted jet reaches the ambient pressure through a series of expansion waves (fourth regime).
4.5.3
THE DIFFUSER
Figure 4.5.4 illustrates schematically an ideal open-cycle supersonic wind tunnel that is composed of a nozzle (N), a test section, and a diffuser (D). The gas accelerates through the nozzle and achieves supersonic conditions at the test section. After passing the test section, the gas reaches the diffuser, which acts as an inverse nozzle. The flow decelerates through the nozzle and must reach the same sonic conditions at the diffuser throat as for the nozzle. However, the flow situation is unstable due to the positive pressure gradient existing downstream of the diffuser throat. For this reason, the diffuser is designed to achieve subsonic flow conditions, by means of a normal shock somewhere in the diverging part of the diffuser. The mass flow rate of the gas
Nozzle
Test section
Diffuser
FIGURE 4.5.4 Ideal supersonic wind tunnel composed of a nozzle (N), a test section, and a diffuser (D).
4.5
657
Supersonic and Hypersonic Wind Tunnels
flowing in the test section can be expressed as a function of either the nozzle throat or the diffuser throat as D fit - - W P + t ( A
.D
N
.N
) = WPstCA )
(4.5.21
As the flow is adiabatic, that is, there are no thermal losses, (T~ = T~). Then . D N .)N ) - Pst(A . On the other hand, due to friction and the possible shock waves associated with the model placed in the test section one expects a drop in the flow stagnation pressure. That is, p~ < p~N t, and therefore D
Pst(A
CA*)'::' _
(A.)N -- pD > i
(4..5.3)
Because the cross-sectional area of the nozzle throat is the flow critical area, that is, (At) N --(A*) m and (At) t) > (A*)t) one gets (A0 D
(A*) D
CAt)N > (At) N > 1
(4.5.4)
which yields, (At)D> (At)N. This relation shows that the diffuser throat must be larger than the nozzle throat. If the diffuser throat is too small, a supersonic flow cannot be achieved in the test section. On the other hand, if the diffuser throat is too large, a normal shock wave is expected to stand in the diverging section of the diffuser, causing a high overall drop in the stagnation pressure.
4.5.4
START-UP PROCESS
During the start-up process, for example for a closed-circuit wind tunnel, the compressor simultaneously lowers the diffuser exit pressure and raises the nozzle inlet pressure. The resulting flow process is sufficiently slow, so that during start-up the flow in the wind tunnel can be considered quasisteady. Figure 4.5.5 shows the static pressure (normalized by the nozzle inlet pressure) distribution along the wind tunnel at different stages during the start-up process. One assumes that the diffuser throat section is large enough so that the flow does not choke at the diffuser throat. As the pressure ratio is decreased, a normal shock wave is formed at the nozzle exit section as illustrated by curve d in Fig. 4.5.5. In this situation, the flow Mach number in ahead of the shock wave is the nozzle exit Mach number, which is the largest
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---
d
n
9
Shock
FIGURE 4.5.5
Pressure distributions during the start-up of an ideal supersonic wind tunnel.
flow Mach number in the wind tunnel circuit. Thus in this situation the drop in the stagnation pressure (which must be compensated by the compressor) attains its largest value. As the pressure ratio is decreased, the shock wave moves rapidly downstream (through the test section) and stands in the diffuser divergent part, at an area that is slightly larger than the test section one. Curve e in Fig. 4.5.5 illustrates this particular situation. The drop in the stagnation pressure in this case remains high. In order to reduce losses in the stagnation pressure, the pressure ratio can be increased either by adjusting the compressor pressure ratio or by increasing the diffuser throat area (variable geometry wind tunnel). In such a case, the normal shock moves upstream and positions itself near the diffuser throat. The corresponding stagnation pressure drop reaches a minimal value at this condition.
4.5.5 SUPERSONIC AND HYPERSONIC CONTINUOUS
WIND
TUNNELS
4.5.5.1 RETURN-CIRCUIT CONTINUOUS W I N D TUNNELS In general, continuous operating facilities need a very big amount of power supply, which can reach several megawatts for a suitable cross-sectional size
4.5
659
Supersonic and Hypersonic Wind Tunnels
and in the cases of high-density flows. For this reason, such facilities operate at moderate hypersonic and supersonic (M < 5) conditions and only a few such facilities are available in research laboratories. The power supply can be reduced by closing the gas circuit and, therefore, recovering partly the exhaust gas energy. However, the continuous rising of the circuit mean temperature can delay significantly the beginning of tests. Figure 4.5.6 illustrates a typical closed-circuit supersonic wind tunnel. In this case a single machine (compressor) lowers the pressure at the exit of the diffuser and raises the pressure at the entry to the nozzle. To avoid liquefaction during the flow expansion in the nozzle, the closed circuit is filled with dry air. Before entering the tunnel, the gas is dried by an air-drying device and is stocked in a reservoir by means of a small compressor. Due to the continuous release of mechanical energy that is partially transformed into heat by friction, head losses and other dissipative processes, for example, shock waves, the gas temperature rises quickly in the circuit. Consequently, heat must be withdrawn by a heat exchanger in order to lower the circuit mean temperature. The VKI-S1 (von Karman Institute for fluid dynamics) supersonic wind tunnel is an example of such a facility. This facility is located in Belgium and operates at flow Mach numbers in the range of 0.4 < M < 2.25. The test section size is 40 x 40 (cm). The flow is driven in a closed circuit by a 615-kW axial compressor. The LRBA-C4 (Laboratoire de Recherche Balistiques et A6rodynamiques) supersonic wind tunnel, which is located in France, is an example of a closedcircuit, continuous-flow type. The flow Mach number is in the range of 0 . 1 5 < M <4.29. Two compressors powered by 13.5-MW electric motors drive the C4. The test section size is 40 x 40 (cm). The ONERA (Office National d'Etudes et de Recherches A6rospatia!es) S2MA continuous-flow, closed-circuit supersonic facility is located at Modane-
Compressor Motor
,
.-
Heat exchanger
Nozzle
Test section
Diffuser
.
Air dryer Dry
FIGURE 4.5.6
air
A continuous operating, closed-circuit supersonic wind tunnel.
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Avrieux, France. A 57-MW compressor powered by four hydraulic Pelton turbines drives the flow. The test section has a size of 175 x 193 (cm) and the flow Mach number is in the range of 1.5 < M < 3.1. The S5Ch wind tunnel is another ONERA supersonic closed-circuit continuous-flow facility. It runs at flow Mach numbers of 1.2 and 1.45-3.15. The test section size is 30 x 30 (cm). The Aircraft Research Association (ARA) supersonic wind tunnel is located in Bedford, United Kingdom. This facility is a continuous-flow, closed-circuit type wind tunnel. The flow Mach number of this wind tunnel is in the range 1.4 < M < 3. The test section size is ~0.80 x 0.80 m. The Royal Aeronautical Establishment (RAE) has two supersonic continuous-flow, return-circuit wind tunnels operating in Bedford, United Kingdom. Both facilities are equipped with flexible nozzles to allow operation at any flow Mach number in the range 2 . 5 < M < 5 for the 0.91 x 1.22m and 1.35 < M < 2.5 for the 2.44 x 2.44m RAE wind tunnels, respectively. The 0.91 x 1.22m RAE facility has two 18-stage axial-flow and two 8-stage centrifugal compressors that can be run either in parallel or in series. The driving power reaches 66MW for the 0.91 • 1.22m and 68MW for the 2.44 x 2.44 m RAE wind tunnels, respectively. The DLR (Deutsche Forschungsanstalt for Luft und Raumfahrt) TWG (2.5 x 2.5 m) wind tunnel located in Goettingen, Germany, is a continuousflow, return-circuit trisonic facility. The maximum flow Mach number during supersonic operations is 2. An 8-stage 12-MW axial blower provides the airflow. The DLR PIK, P2K and P3K hypersonic wind tunnels are located at KOlnPorz, Germany. These facilities operate in continuous-flow and closed-circuit conditions in a wide range of flow Mach numbers (5 < M < 20) using conical nozzles. The flowing gas (dry air, nitrogen, argon, or helium) is heated up to 6000 K using electric arc heaters. The nozzle exit section diameters are 110 mm for P IK and 250mm for P2K and P3K.
4.5.5.2
OPEN-CIRCUIT CONTINUOUS WIND
TUNNELS Open-circuit wind tunnels offer remarkably stable flow conditions within a short time. However, due to high energy costs a few examples of these wind tunnels exist worldwide. To illustrate this kind of facility, Fig. 4.5.7 shows a typical supersonic, continuous-flow open-circuit wind tunnel from CNRS running at flow Mach 4.96 located at Meudon, France. This is a free jet facility. One advantage of such a facility is its almost unlimited running time.
o
Heater
Com pressor Motor '
Pump
Q~
f
L
Oil separator
ffuser ,-
...... pq
L~f_l Cooler
Test cham ber Air dryer
/,~ Filter Atmosphere
Atm osphere FIGURE 4.5.7
Motor
A continuous operating, open-circuit, free jet supersonic wind tunnel.
o
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B. Chanetz and A. Chpoun
The long flow duration can overcome instrumentation response time and allow good stabilization of flow conditions to be obtained prior to experiments. Twostage volumetric compressors (MPR-RC300) supply air from the atmosphere, after passing through a filter at a jet stagnation pressure of 850 kPa. To lubricate the blades, oil is furnished continuously to the compressor. Consequently, an oil-separating device is placed at the exit from the compressor. In addition, the atmospheric air is dried before entering the nozzle. The nozzle exit diameter is 127 mm. The area ratio between the exit and the throat sections is 25. Consequently, the test-section-flow Mach number, based on inviscid theory, was designed to be exactly 5. However, due to viscous effects the actual flow Mach number (4.96) is based on stagnation pressure measurements in the supersonic jet and behind the normal part of the bow shock wave formed ahead of a Pitot gauge. This value is maintained within a variation of <1% along the 200-mm diameter jet. To avoid liquefaction during its expansion in the nozzle the flow is heated up to a stagnation temperature of 453 K. The mass flow rate is ~.0.76 kg/s. The corresponding supersonic jet velocity is 870m/s. The temperature and the pressure are 76.5 K and 1683 Pa, respectively. The Reynolds number per unit length, based on these conditions is 1.28 x 10 7 m -1. This high Reynolds number minimizes viscous effects and, as a consequence, the experimental results can be considered suitable for comparison with inviscid theories. The supersonic flow emerges from the nozzle into the test chamber. This type of open-jet test section, surrounded by a large vacuum chamber, makes integration of various diagnostic equipment easier. A group of three parallel pumps (MPR-1000) evacuates the air to the atmosphere after passing through a conical diffuser. An air-water heat exchanger is used to lower the air temperature before entering the vacuum pumps. The overall electric power required for running this facility is about 1MW.
Figure 4.5.8 provides a schematic view of the CNRS-SR3 (located in Meudon, France) continuous open-circuit facility, which simulates rarefied flow conditions. In such a facility, a relatively low gas mass-flow-rate is supplied to the nozzle by a small compressor. An alternative way is to supply the necessary gas from a compressed bottle or by using the local gas network. On the other hand, a large volume flow rate pump must be installed. For the particular facility shown in Fig. 4.5.8 the pumping group is composed of three stages, including six rotative vacuum pumps, two Root pumps, and two oil diffusion pumps. The last pumping group can withstand volume flow rates of about 40m3/s under pumping pressures of up to about 10 - 4 a t m . The DLR VIG, V2G, and V3G open-circuit wind tunnels located in Goettingen, Germany, operate in a continuous manner. The nozzle exit section diameters are 25, 40, and 130 cm for VIG, V2G, and V3G, respectively. These
i.a s= 0-t
o
T[ST II[AT[R NOZ/L[ CHAHO[R
OIFFUSER
VACUUHGAY[
TO HIGII HASSFLOW PUHPIN5 SYSI[H VACUUHGAll
TEST caAm3zn
o
NITROG[N AIR
Oo
l:::t r l=t cgl
ilil i
LOWO[NSITY
PUHPS
TUNN[t CONTR.OLAND IXSTRUH[NTATION FIGURE 4.5.8
ROTARY PUHP
ROOTS _~OIl VAPORBOOST[RPUHP
View of a rarefied flow regime supersonic wind tunnel.
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facilities simulate high-flow Mach numbers (up to 25) and high-altitude, lowdensity flows. The wind tunnels use high-vacuum diffusion pumps.
4.5.6 BLOW-DOWN WIND TUNNELS 4.5.6.1
PRELIMINARY REMARKS
Before describing the operation of a blow-down wind tunnel it should be noted that this type of wind tunnel is the one most frequently found in hypersonic laboratories. These are also supersonic blow-down facilities and hypersonic continuous facilities. As in a conventional supersonic/hypersonic wind tunnel, in blow-down wind tunnels it is also necessary to dry the air used in the facility in order to prevent the condensation of the water vapor contained in normal air. As far as hypersonic facilities are concerned, the pressure ratio ~z between the upstream and the downstream parts of the facility, which is required for flow initiation, increases very rapidly with increasing flow Mach number M. Figure 4.5.9 provides the evolution of this ratio versus M. The second curve 7~a = f ( M ) overestimates the previous ratio (7~ a = 1.4~z), giving an increase generally adopted to take into account the particular case of a free jet test chamber and the existence of a model at a moderate incidence inside the testing chamber (Rebuffet, 1966). These ratios are for "cold" wind tunnels in which no real-gas effects are present. Furthermore, for constant pressure ratio lr the absolute level of the stagnation pressure must increase as the flow Mach number increases, so that the Reynolds number does not become too small. Hypersonic flow is usually considered to begin at above M = 5. Thus, as far as hypersonic facilities are concerned, this limitation can be important. In addition to obtaining a free-stream flow Mach number of >5, it is absolutely necessary to heat the reservoir air to avoid its liquefaction during the expansion through the nozzle. The minimum stagnation temperature Tst needed for preventing liquefaction is given for a perfect gas (7 = 1.4) by the following formula: 380 (l+-~)Tst
--4.45 + 3 . 3 5 1 o g ] 0 ( 1 + - ~ ) -
1Ogl0Ps t
(4.5.5)
where Pst is the stagnation pressure and M is the flow Mach number. To illustrate this formula, consider first a stagnation pressure of 0.1 atm and a flow Mach number of 5 for which the required minimum stagnation temperature is 279 K.
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Supersonic and Hypersonic Wind Tunnels
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For a larger stagnation pressure, it is impossible to avoid liquefaction without heating the air reservoir. For example, at M = 5 and Pst = 10atm, the required minimum stagnation temperature is 369 K and for M = 5 and Pst = 100 atm the required minimum stagnation temperature rises to 440 K. For supersonic facilities (and even more so for hypersonic ones) the pressure ratio ga required to start the nozzle can be quite large. To obtain large values of ga, it is necessary to place a vacuum tank with a very low pressure downstream of the diffuser. Furthermore, there are two types of hypersonic wind tunnels. 9 "Cold" wind tunnels are those in which the air is heated just enough to avoid its liquefaction during the expansion in the nozzle. The temperature
666
B. Chanetz and A. Chpoun of the air at the nozzle exit is very low. Such a wind tunnel is not sufficient to simulate all of the conditions experienced during a space vehicle reentry. "Hot" wind tunnels are those in which the air is heated to a temperature high enough so that the air at the end of the expansion remains hot enough to simulate correctly atmospheric reentry conditions.
4.5.6.2
D E S C R I P T I O N OF A CLASSICAL C O L D
BLOW-DOWN WIND TUNNEL Figure 4.5.10 shows the complete aerodynamic system of a blow-down wind tunnel having the same characteristics as the wind tunnel R3CH located at ONERA~ Chalais-Meudon center in France. This wind tunnel operates at M - 10 with a stagnation pressure in the range of 12.5 < Pst < 125 bars and a stagnation temperature T s t - 1100 K, which corresponds to unit Reynolds numbers in the range 8.4 x 105 < Re < 8.4 • 106. Similar wind tunnels exist elsewhere in the world. For example another ONERA one, but located at Modane-Avrieux, France, is a large hypersonic wind tunnel called S4MA. The nozzle exit diameters are in the range 7001000 mm. The energy is obtained by burning propane with air and it is stored in an accumulation heater. The stagnation pressure is 40 bars at M - - 6 and 150 bars at M = 10 and M = 12, which leads to unit Reynolds numbers in the range 3 x 106 < Re < 2.7 x 107. At the Center of Aerodynamic and Thermal Studies (CEAT) in Poitiers, France, the hypersonic blow-down wind tunnel H210 operates at M = 7 and M = 8. The stagnation pressure is in the range 22 < Pst < 100 bars, giving a unit Reynolds number in the range 1.3 • 106 < Re < 9.2 x 106 at M - 7 and 1.5 x 106 < Re < 4.2 x 106 at M - 8. At the National Aerospace Laboratory (NAL) in Chofu-shi near Tokyo, Japan, there is a 50-cm hypersonic wind tunnel, which produces hypersonic flows at M---5, 7, 9, and 11. The stagnation pressure is in the range 10 < Pst < 100bars, which leads to a unit Reynolds number in the range 3 x 105 < Re < 7.2 x 106. At the Aircraft Research Association (ARA) in Bedford, United Kingdom, the M7T blow-down wind tunnel produces hypersonic flows at M = 6, 7, and 8. The stagnation pressure is in the range 100 < Pst < 200 bars, which leads to a unit Reynolds number in the range 3 x 106 < Re < 4.5 x 106. At NASA Langley, Virginia, USA, the 31-in Mach 10 air tunnel has been calibrated for operation at stagnation pressures in the range 8.5 < Pst < 10 bars
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with a reservoir temperature of 1000 K, which produces a freestream unit Reynolds number in the range 7.5 x 104 < Re < 6.3 • 105. In all these facilities upstream compressed gas (air or another gas) is heated before expanding through the nozzle. The gas then goes through the test chamber and is compressed by the diffuser before reaching the vacuum tank. Each element of the facility is described in details here. 4.5.6.2.1 The Air Heater The air upstream of the nozzle is heated to a stagnation temperature Tst by a Joule effect heater (electrical heating). As it is a cold hypersonic wind tunnel, the stagnation temperature is just sufficient to prevent the air from liquefaction through the nozzle. As an example, for a flow Mach number of 10 at the nozzle exit plane, the temperature ratio between the reservoir (stagnation) temperature Tst and the static temperature T measured at the nozzle exit is equal to 21. This value is found from the isentropic relation: Ts--2t- 1 + 7-- 1M2
T-
-T-
(4.5.6)
Thus, for a stagnation temperature of 1050 K, the static temperature is very low, only 50 K. This explains the term "cold" used for this type of wind tunnel. Usually the heated air is not kept in the tunnel but ejected to the atmosphere via a 3-way valve. When the desired temperature (Tst) is reached, a fast-acting 3-way valve, located at the heater outlet, discharges the flow into the nozzle within about one-tenth of a second. 4.5.6.2.2 The Nozzle Design The nozzle design is a particularly difficult task at high flow Mach numbers, since an improper design, especially in the vicinity of the throat section, could generate perturbations that might coalesce around the nozzle axis. The best way to avoid such problems is to use an axisymmetric profiled nozzle designed to produce a uniform flow at its outlet plane under nominal conditions. Most parts of the nozzles in current use were designed before the advent of modern computers and the current widespread use of Navier-Stokes solvers. In the first step, a calculation is performed using the method of characteristics, with a specified flow Mach number along the nozzle longitudinal axis, to obtain the desired isentropic nozzle contour. The second step consists of calculating the boundary layer that would develop on this shape. One deduces from this calculation the displacement thickness C~l(X) (here x is the longitudinal abscissa). This displacement thickness is then added to the isentropic
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Supersonic and Hypersonic Wind Tunnels
core (increasing the radius) to provide a first approximation to the real nozzle shape. On this new shape, another boundary layer calculation is performed. The new displacement thickness thus obtained is subtracted from this shape, which gives a contour very close to the initial isentropic shape. The process is repeated until the difference in the radii becomes <0.01 mm. The process usually converges in three iterations. At the end of this process, one obtains dimensions that define the nozzle shape. To eliminate small irregularities, smoothing algorithms are used. While the part near the throat has a positive curvature (concave away from the flow), the last part of the nozzle has a negative curvature (concave toward the flow). The continuity of the profile in the neighborhood of the inflection point is ensured by a sinusoidal interpolation function. For a hypersonic nozzle, the throat radius is very small compared with the exit diameter. For example, in the R5Ch wind tunnel already mentioned here, the throat radius is equal to r c = 5.279 x 10-3m for a nozzle exit of D o = 0.355 m. One also notes that the ratio ~t/r c (~t is the meridian contour radius, and r c is the throat radius) is often large enough to result in a very gradual expansion in the first part of the throat and to avoid the formation of shock waves along the nozzle (in the RSCh facility this ratio is equal to 80). The aforementioned procedure gives excellent results as has been recently proven by Navier-Stokes calculations performed, a posteriori, on the R5Ch nozzle (Chanetz and Coet, 1993).
4.5.6.2.3 The Nozzle Qualifications The flow produced by the nozzle must be qualified experimentally before it is used for testing models. The most convenient method is to use a Pitot probe to obtain the radial distributions of flow Mach number in different sections between the nozzle outlet plane and the diffuser inlet plane. A Pitot tube measures the stagnation pressure ptst behind a normal shock, which is formed in front of the Pitot tube. The Mach number is deduced from the measurements by one of two methods according to the probed flow region. 9 In the isentropic region (undisturbed flow), these values are calculated directly from the ratio between the total pressure Ptst given by the Pitot tube and the reservoir stagnation pressure Pst measured upstream of the nozzle throat. The flow Mach number is obtained by numerical inversion of the following classical formula giving the ratio of stagnation pressures across a normal shock:
Pst-
(7 - 1)M2 4- 2
[.27Mi -- ( 7 - 1)
(4.5.7)
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B. Chanetz and A. Chpoun
9 In the dissipative region (boundary layer and mixing layer) the flow Mach number is calculated by assuming that the static pressure p is constant in the radial direction and equal to the value determined from the flow Mach number given at the edge of the boundary of the dissipative region. For the supersonic part of the boundary layer, the flow Mach number is then calculated from the following formula: p --
12yM ~ ~ ( 7 - 1)
(4.5.8)
9 For measurements performed in the subsonic part of the boundary layer the following formula is employed: P'st- Pst = [ 7-1 P P 1 4- 2 M2
]?/7-1
(4.5.9)
The obtained flow Mach number distribution clearly indicates the undisturbed part of the'flow, where the Mach number is constant and the mixing layer surrounds the inviscid fluid core. 4.5.6.2.4 The Test Chamber The nozzle discharges into a test chamber. The test section shown in Fig. 4.5.11 is of the common free-jet type (ONERA Mach 10 R3CH wind tunnel). For a properly contoured nozzle, the flow Mach number in the test chamber is uniform. For conical nozzles, the flow Mach number in the test chamber continues to increase with axial distance away from the nozzle exit. 4.5.6.2.5 The Diffuser
The diffuser assembly includes a converging recovery section followed by a cylindrical mixer and it ends with a diverging section connected to a line leading to the vacuum sphere. The design of the diffuser is of great importance, since the duration of the run is a function of the compression ratio. The design of the diffuser should follow very precise laws in order to optimize flow compression. Monnerie (1966) performed a detailed study of a family of diffusers suitable for hypersonic wind tunnels with low Reynolds numbers. It stresses the advantages that could be drawn from natural recompression of the flow in the diffuser. In effect, the longer the flow in the diffuser preserves a pressure above the pressure in the downstream tank, which increases as it fills, the longer it takes for unstarting to occur, which in turn extends the useful duration of the run. The following parameters were considered: 9 the angle 0er of the conical converging element;
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diameter D r of the inlet converging element; length L m o f the cylindrical mixer; diameter D m o f this mixer; and angle ~a of the diverging element connected to the vacuum tank.
From this study it was concluded that most of the pressure recovery is achieved in the mixer. The length of the mixer must be large enough to establish a subsonic regime at its extremity. The mixer efficiency depends essentially on diameter Din, with higher efficiency obtained for smaller values o f D m. However, D m should not be smaller than D0/1.29, where D O is the nozzle exit diameter. This limitation is due to conditions necessary for starting the flow. It has also been shown that the conical converging element should adhere to the following values: D r = ~/3D o
and
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These conditions are particularly important when the mixer diameter D m is smaller than the nozzle exit diameter D O. The compression ratio pe/p delivered by the entire unit (converging element plus mixer) can be calculated in terms of: 9 9 9 9
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nozzle exit diameter Do; displacement thickness C~l(X); exit flow Mach number Mo; and mixer diameter Dm; and
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(4.5.10)
where p is the static pressure at the beginning of the converging element (the same pressure as that prevailing at the nozzle exit) and Pe is the pressure at the end of the diffuser.
4.5.6.2.6 The Vacuum Tank A diffuser in which the flow is compressed is usually connected to a vacuum tank in which the pressure is low. For the aerodynamic circuit shown in Fig. 4.5.10, the vacuum tank is a large-capacity sphere. The ONERA R5Ch wind tunnel has such a sphere. Its volume is 500 m 3. In this case, the pressure inside the sphere at the beginning of a run is only a few pascal. Such a low pressure can be obtained by a succession of pumps.
4.5 Supersonicand Hypersonic Wind Tunnels 4.5.6.3
673
INDUCTION BLOW-DOWN WIND
TUNNEL In most of the blow-down wind tunnels, a vacuum tank lowers the pressure downstream of the diffuser. An alternative method to obtain gas suction in the diffuser is by blowing air, vapor water, or hot water at the diffuser exit. Such a water steam generator is used in the SIGMA 4 wind tunnel of the Institut A~rotechnique de Saint-Cyr, which is located in Saint-Cyr l't~cole near Paris, France (Brocard, 1962). This original hot-shot ejector device has been conceived by SNECMA-FRENZL. The water, constituting the motor fluid, is heated to 558 K at a pressure equal to 65 bars. A momentum exchange between the test gas creates a powerful suction, which ensures the flow in the test section. The mixing chamber downstream of the diffuser is long enough to ensure good flow recompression conditions. A gravity separator recovers onehalf of the water.
4.5.6.4
DESCRIPTION OF A H O T B L O W - D O W N
WIND TUNNEL During their return flight to Earth, space vehicles are exposed to high aerothermal loads due to the conversion of kinetic energy into thermal energy (air braking due to drag). The very large vehicle velocity can give rise to real-gas effects due to the excitation and dissociation of molecules throughout the very hot shock layer. These effects can be generated in a hotshot wind tunnel. In such a wind tunnel, the large energy required to heat the flow is obtained via an arc-heater chamber and then the gas expands through the nozzle, passes through the test chamber, and reaches a vacuum tank as in a classical cold blow-down wind tunnel. The run duration is extremely short, a few hundred milliseconds. The various components of such a hot-shot wind tunnel are schematically described in Fig. 4.5.12. The description is based on the working principle of the ONERA F4 high-enthalpy wind tunnel (Sagnier et al., 1999; Chanetz et al., 1992). Another wind tunnel that works according to this principle is the Impulse wind tunnel IT-2 at TsAGI, in the Central Aerohydrodynamics Institute, Zhukovsky, Russia. The IT-2 wind tunnel works with a stagnation temperature in the range of 1500-5000K and a stagnation pressure in the range 1201500 bars. The run duration is around 0.1 s. In such wind tunnels, high flow stagnation parameters are achieved due to electric discharge in a closed spacefilled gas.
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B. Chanetz and A. Chpoun .
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FIGURE 4.5.12 Hot-shot wind tunnel.
4.5.6.4.1 The Impulse Generator The impulse generator produces the high energy level required for heating the gas in the arc chamber. In the F4 facility the impulse generator consists of a momentum wheel of 15,000 kg coupled to a 150-MW ahemator. The ahemator works as a motor at variable frequencies to run the wheel up to 6000 rpm, which represents an energy of 400 MJ. The electric motor circuit is then disconnected and the wheel runs the rotor of the altemator to produce a continuous electric current. This current is then ignited between the two electrodes by a fusible electric wire. (In the IT-2 wind tunnel at TsAGI a powerful bank of capacitors generates the electric discharge.)
4.5.6.4.2 The Arc Chamber The reservoir arc chamber is filled with cold gas at 10-100 bars. This chamber is typically small in size--a few liters. In the F4 wind tunnel, the chamber has an adjustable volume of between 10 and 151. The electric current produced by the impulse generator is initiated between the two electrodes by a fusible electric wire. The spiral electrodes produce a rotating magnetic field, which keeps the arc impact point moving, thus reducing erosion. The chamber has a liner to protect it. This liner can be made of copper, but this was shown to induce contamination in the flow. Contamination is avoided by using a carbon-carbon or graphite liner. For the same reason, electrodes made of copper are coated with graphite. These protections improve the flow quality but leads to a decrease of the maximum reservoir enthalpy. For obtaining the required reservoir conditions, the F4 facility needs a power of up to 100200 MW during several tens of milliseconds.
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4.5.6.4.3 The Nozzle and the Test Chamber Once the prescribed reservoir conditions are reached, the electric supply is shut down. Firing a pyrotechnic plug, after a few milliseconds, opens the nozzle throat. At the end of the run a pyrotechnic valve located in the arc chamber is fired to allow the remaining gas to be quickly evacuated into a dump tank. In the F4 wind tunnel, maximum run duration of 400 ms can be obtained. During the run, the reservoir pressure and the reservoir enthalpy decrease continuously due to heat and mass losses of the test gas in the arc chamber. However, the process is generally slow enough (a decay of 1% per ms for both the pressure and the enthalpy in the F4 wind tunnel) to perform measurements corresponding to mean values during a sufficiently long period. 4.5.6.4.4 Difficulties in Evaluating the Flow Conditions in a Hotshot Wind Tunnel In high-enthalpy wind tunnels, it is not easy to know accurately the flow conditions needed for initialization of the computations because of the large number of parameters whose knowledge is needed simultaneously (e.g., freestream density, velocity, chemical composition, and vibrational temperatures). When all these parameters cannot be measured simultaneously (which is generally the case), the missing parameters must be deduced by making some assumptions about the thermochemical state of the wind tunnel flow. Sagnier and VCrant (1998) solved the problems encountered in the ONERA high-enthalpy F4 wind tunnel and have done a complete study of this problem. The reservoir enthalpy in the F4 wind tunnel is calculated by using a semiempirical correlation, involving heat transfer rates (measured in the stagnation region of spherical catalytic probes) and Pitot pressures. Based on the results obtained from 60 Navier-Stokes computations for stagnation point heat fluxes, assuming nonequilibrium (catalytic wall) and perfect gas flows, Sagnier and V~rant (1998) proposed the following enthalpy-heat flux relationship:
Qstv/Rn~
(4.5.11)
ihst - hw/RTa ] 1.069---- 23.787
This formula is given at the stagnation point with an accuracy equal to 4-12%. The reservoir enthalpy and the wall enthalpy are hst and hw, respectively. Using this formula, the reservoir enthalpy hst is inferred from the stagnation pressure, P's~ is measured by a Pitot probe located in the test chamber and the stagnation heat flux Qst is measured by a hemispherical probe of radius Rnose. All variables are in SI units (the gas constant R -288.2 u SI).
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B. Chanetz and A. Chpoun
Without going into details, the relevance of the methodology used to determine the reservoir enthalpy has been demonstrated by recent comparisons with velocity measurements using different optical techniques. In calibration of the F4 wind tunnel (Sagnier and Verant, 1998), another problem appeared when the nozzle wall pressure measurements were compared with computations made for high-enthalpy conditions. The experimental pressures were found to be significantly larger than the pressures obtained from "conventional" nonequilibrium predictions at high-enthalpy although the experimental/theoretical agreement was good for low-enthalpy runs. With the help of pseudo-one-dimensional Euler solver coupled with boundary layer models, the nozzle flow charts were designed for a wide range of reservoir conditions and for extreme limits of thermochemical assumptions. The models used for the charts were equilibrium, thermochemical nonequilibrium, or frozen core flows coupled with either laminar or turbulent (from the nozzle throat) boundary layers.
4.5.7 EXPERIMENTAL TECHNIQUES 4.5.7.1 PITOT PROBE TECHNIQUE In a supersonic wind tunnel one technique currently employed for measuring the flow Mach number is the Pitot tube technique. The Pitot probe (see Fig. 4.5.13), which is formed of a hollow small tube, is placed parallel to the flow direction. In supersonic flows, a detached bow shock wave is formed ahead of the probe. The tube is connected to a pressure transducer, which measures the stagnation pressure behind the bow shock. The free flow streamline parallel to the tube axis crosses the normal part of the bow shock. Thus in this case the Rankine-Hugoniot relations for a normal shock wave apply. For example, the following equation gives the ratio of stagnation pressure upstream and down-
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4.5
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Supersonic and Hypersonic Wind Tunnels
stream of the normal shock wave (Pst~ and Pst2, respectively) as a function of upstream flow Mach number, that is, M 1. Thus, measuring Psq and Pst2 yields to the upstream flow-Mach number. Psq
4.5.7.2
+ 1 + (~, + i)M 2
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M U L T I H O L E PRESSURE PROBES
Measuring a three-dimensional (3D) flow available in wind tunnels calls for more and more discrete instruments with better and better performance. Although muhihole pressure probes are an intrusive instruments, they are commonly used to probe velocity fields in complex flows. A great effort has been made to miniaturize them and to enable their operation over a wide envelope of incidence angles, velocities, and pressures. These miniaturized probes are difficult to design and construct, as they are to operate through broad parameter domains. Often they have complex responses requiring the use of modern computer facilities. The probes themselves must be calibrated first and the final accuracy of the measurements depends on the accuracy of this calibration (including position measurements) and also on the fineness of the grid and methods of numerical analysis (Gaillard, 1990). Muhihole probes are used for measuring velocity vector directions and amplitudes, as well as the local stagnation pressures in supersonic flows. Different types of multihole probes are used. To measure the velocity component in a plane, a flattened three-hole probe are used. The central hole of a three-hole probe acts as a Pitot probe and the two other holes are located on the right-hand side and on the left-hand side of the central hole. The three holes are in the same plane. For measuring 3D flows, at least three unaligned holes are needed, but one generally uses five-hole probes, with two holes aligned with the central hole in the same plane and the two others aligned with the central hole in the perpendicular plane. Miniaturization (diameter of 1 or 1.5 mm) is desired for these probes in order to reduce the measurement control volume and the choking effect. They are suitable for averaged measurements (instantaneous values can be measured using hot-wire probes or more easily by LDV measurements).
4.5.7.3
E L E C T R O N BEAM F L U O R E S C E N C E
TECHNIQUE (EBFT) To measure local flow density, the electron gun technique is employed in rarefied supersonic and hypersonic wind tunnels (Fig. 4.5.14). The electron
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B. Chanetz and A. Chpoun
filter
tomultiplier
mirror
lens
slot
beam receiver
turbomolecular pump
nozzle
n gun
FIGURE 4.5.14
Density measurements: the electron gun technique.
gun axis is perpendicular to nozzle flow direction. The electrons, accelerated by a high-voltage electrical field, emerge from the gun through a small orifice; then they pass through the flow and are collected by a beam receiver. A turbomolecular pump, together with a primary pump, keeps the gas pressure at a very low level (10 -3 Pa) inside the gun. For local density measurement, an optical system associated with a photomultiplier allows measuring the induced fluorescence of a small control volume of the analyzed gas. At higher flow densities, an X-ray detector is used to record the X-ray emission of the control volume. Then, in both cases, the local density is deduced from fluorescence or X-ray measurements after appropriate calibration.
4.5.7.4 H E A T F L U X M E A S U R E M E N T BY SURFACE MEASUREMENT TECHNIQUES To measure surface temperatures and heat fluxes, the thin wall method is still one of the most efficient techniques. The surface temperature of a model is measured directly, using thermocouples embedded in a thin wall. The thermocouple wires have a small cross section in order to ensure a negligibly small heat loss by conduction along the wires. When the model is subjected to a convective flux, its wall is heated up, and under some conditions the
4.5 Supersonicand Hypersonic Wind Tunnels
679
temperature time derivative is proportional to the transmitted heat flux. Consequently, for a given local heat capacity of the thin wall, it is possible to deduce the local convective heat flux from the recorded time derivative of the wall temperature. This method, also called the calorimetric method, requires a very thin skin (thickness of between 0.1 and 0.5 mm), so that it is not relevant for complex models. Another method is then applied using a thick wall. Two techniques are available. In one of them, thermocouples made of the same material as the model are embedded in the thick skin. In the other a thermometer element, such as a platinum film, is applied to an insulating support, such as a Macor ceramic. In this case, the whole model can be made of ceramic or only the part upholding platinum films. In the two techniques, the heat fluxes are determined from the surface temperature rise recorded during a run.
4.5.7.5
INFRARED THERMOGRAPHY T E C H N I Q U E
An infrared camera is also used for measuring wall temperatures and convective heat fluxes. In general, a thermovision camera is equipped with two rotating mirrors, for vertical and horizontal scanning and with an optical detector cooled by liquid nitrogen (see Fig. 4.5.15). The main difficulty in measuring surface temperatures by infrared thermography is to quantify precisely the emissivity of the model surface. In general, models are painted with graphite-charged paint and the paint emissivity is obtained after calibration. The philosophy of measurement is similar to the thin wall technique. Successive thermograms are recorded at a frequency of several thermograms per second, and, for a known local heat capacity of the wall, the analysis of successive thermograms yields the heat flux on the investigated surface. Generally, only the convective heat flux is measured, while the effects of
lens~ ( h
rotatingmirror orizontal sweeping)
.
~
control
display investigated plane
rotating~firror
cooleddetector
(verticalsweeping)
FIGURE 4.5.15 Heat flux measurements: infrared thermography device.
680
B. Chanetz and A. Chpoun
radiation and thermal conduction are neglected. These conditions are satisfied if the thermograms are recorded just at the starting time when the model begins to heat up, so that the surface temperature does not rise more than a few degrees during the test. For angles of inclination >60 ~ between the axis of the camera and the normal to the viewed surface, the real and apparent emissivities of the model surface may be quite different. This complicates the interpretation of thermograms for models of complex shapes, especially near round leading edges.
4.5.7.6
LASER DOPPLER VELOCIMETRY
(LDV)
The advent of Laser Doppler Velocimetry (LDV) has offered an incomparable experimental tool to analyze complex aerodynamic flows whose exploration by material probes (hot wires, multihole pressure probes) was complicated, hazardous, or impossible. Indeed it is now possible to investigate in great detail 3D, separated, highly fluctuating flows in a velocity range extending from low subsonic conditions to high supersonic flow Mach numbers. However, it is impossible to probe hypersonic velocity fields by this technique. Indeed, the LDV technique requires particles seeded in the flow, which are detected in the interference fringes. In hypersonic flowfields, particle drag is an important phenomenon and the particle velocity is not the same as that of the gas velocity. Other laser techniques are available in this domain, such as Coherent Anti-Stokes Raman Scattering (CARS). For more details regarding this technique see Chanetz et al. (1999). In the subsonic and supersonic regime, LDV is a very powerful measuring technique, which can provide a volume of data comparable in size to that produced by advanced computational fluid dynamics codes. The LDV technique measures the components of the mean velocity and all terms of the Reynolds stress tensor. This nonintrusive technique is now being used more and more. In the 3D version, three pairs of beams converge at the measurement point, producing three distinct fringe patterns. The signals from the three counters are validated by a check of simultaneity and are then routed to the acquisition system. Bragg cells are provided on each of the three channels to suppress the ambiguity on the velocity sign in the region of separation. Thus in research studies, a profitable use of the large amount of results obtained for the local flow field mean and turbulent properties calls for development of sophisticated processing methods capable of adequately revealing and displaying the complex flow structure. This capability is vital not only to arrive at a clear physical understanding of the flow but also to permit comparisons with theoretical models (Chanetz et al., 1987).
4.5 Supersonic and Hypersonic Wind Tunnels
681
4 . 5 . 8 SUMMARY Supersonic and hypersonic wind tunnels of various types are currently used in research laboratories and in aerospace related industries. Most of them were built in the late 1950s, the 1960s, and the early 1970s. These facilities w o r k on the basis of the classical gas d y n a m i c theory. The w i n d tunnels test crosssection sizes that can vary from a few centimeters up to an order of m a g n i t u d e of a meter. W i n d tunnel types differ by s o m e of their operating aspects. C o n t i n u o u s operating w i n d tunnels offer long d u r a t i o n flow b u t need a high p o w e r supply in high-density flow cases. In general, low-density flow facilities are of c o n t i n u o u s type. O n the other hand, b l o w - d o w n w i n d tunnels are characterized by short r u n n i n g time but can p r o d u c e high Mach n u m b e r and h i g h - e n t h a l p y flows. Blow-down w i n d tunnels are the m o s t used facilities, with w i n d tunnels e q u i p p e d with specific i n s t r u m e n t a t i o n d e p e n d i n g on their flow regimes.
REFERENCES Allegre,J. (1992). The SR3 low density wind tunnel. Facility capabilities and research development. AIAA Paper 92-3973. Anon (1990). Technical Date and Information on Foreign Test Facilities, Aerospace Tech. US General Accounting Off. Brocard, J. (1962). Une soufflerie transsonique et supersonique de conception moderne, la soufflerie Sigma 4 de l'Institut A~rotechnique de Saint-Cyr. ATMA. Brun, E.A., Mamartinot-Lagarde, A., and Mathieu, J. (1968). M~.caniquedes Fluides-1, Bordas-Paris: Dunod-Universit~. Candel, S. (1990). Mdcanique des fluides, Bordas-Paris: Dunod-Universit~. Charwat, A.E (1965). A generalized analysis of the performance characteristics and operating ranges of hypersonic-flow test facilities, R-432-PR USAF Project Rand. Chanetz, B. and Coet, M.-C. (1993). Shock wave boundary layer interaction analyzed in the R5Ch laminar hypersonic wind tunnel. Aerospace Res. 1993-5. Chanetz, B., Bur, R., Dussillols, L., Joly, V., Larigaldie, S., Lefebvre, M., Marmignon, C., Mohamed, A., Oswald, J., Perraud, J., Pigache, D., Pot, T., Sagnier, P., V~rant, J.-L., and William, J. (1999). Hyperenthalpic hypersonic project at ONERA, AAAF Symp., Arcachon, France (ONERA Reprint 1999-43). Chanetz, B., Coet, M.-C., Nicout, D., Pot, T., Broussaud, P., Francois, G., Masson, A., and Vennemann, D. (1992). New hypersonic experiment means developed at ONERA: the R5Ch and F4 facilities: Theoretical and experimental methods in hypersonic flows. Proc. AGARD Conf., 514. Chanetz, B., Molton, P., Pagan, D., and Pot, T. (1987). Analysis of complex three-dimensional flows by laser Doppler velocimetry data proccessing and display. ONERA Reprint 1987-64. Gaillard, R. (1990). Development of a calibration bench for small anemoclinometer probes, Symp. Aerodynamic Measuring Techniques Transonic and Supersonic Flow in Cascades and Turbomachines, VKI. Krill, A.M. (1962). Advances in Hypervelocity Techniques, New York: Plenum Press.
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B. Chanetz and A. Chpoun
Le Sant, Y. and Bouvier, F (1992). A new adaptative test section at ONERA/Chalais Meudon. ONERA Reprint 1992-117. Monnerie, B. (1966). Study of a family of diffusers for low Reynolds number wind tunnels. Aerospace Res. 114. Panhkurst, R.C. and Holder, D.W. (1952). Wind Tunnel Technique, London: Pitman Press. Rebuffet, P. (1966). A~.rodynamique Exp~rimentale, vols. 1 and 2, Paris: Librairie Polytechnique B6ranger, Dunod. Sagnier, P. and V6rant, J.-L. (1998). On the validation of high enthalpy wind tunnel simulations. Aerospace Sci. Tech. 7: 425. Sagnier, P., Ledy, J.-P., and Chanetz, B. (1999). ONERA wind tunnel facilities for re-entry vehicle applications, AAAF Symp., Arcachon, France (ONERA Reprint 1999-106).
Measurement Techniques and Diagnostics 5.1
Flow Visualization
HARALD KLEINE Department of Aeronautics and Space Engineering, Graduate School of Engineering, Tohoku University, Sendai, 980-8579 Japan
5.1.1 Density-Sensitive Flow Visualization 5.1.2 The Shadow Technique 5.1.3 Schlieren Methods 5.1.4 Color Schlieren Techniques 5.1.5 Direction-Indicating Color Schlieren Method 5.1.6 Interferometry 5.1.7 Shearing Interferometry 5.1.8 Holographic Interferometry 5.1.9 Light Sources and Recording Materials 5.1.10 Time-Resolved Visualization and Animation References
In his much acclaimed 1982 Album of Fluid Motion Milton van Dyke began his Introduction with the remark that "we who work in fluid mechanics are fortunate [ . . . ] that our subject is easily visualized." This is particularly true in the area of shock waves and associated flows, although one may occasionally argue whether this visualization is, in fact, an "easy" task given the complexity that some visualization setups have attained. A great number of techniques have been developed since flow visualization was recognized as an eminently important discipline to study fluid mechanics, with its capability to yield both qualitative and quantitative information regarding the flows under investigation. In fact, from the days when shock wave research was still in its infancy until now, the development and application of some visualization methods have been crucial in helping to Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
683
684
H. Kleine
detect and understand fundamental mechanisms of shock wave propagation and interaction. The importance of flow visualization as a diagnostic tool has even increased throughout this century, with visualization techniques often being the sole, or at least the main, suppliers of information about a flow field. The optical techniques described here are also relevant for another important reason: They are nonintrusive, allowing one to take measurements within a flow field without disturbing or influencing it. Sensors that would be used for a quantitative assessment of the physical properties of the flow, such as its temperature or its pressure at a certain location, can only be placed along natural physical boundaries, such as the walls of a test chamber or a solid test object within the fluid. Otherwise they would generate a significant disturbance that would alter the flow and its properties, especially in compressible, high-speed flows. Measuring these flow properties along such a natural boundary of a flow avoids these disturbances. However, the amount of information to be gathered from this is limited, in particular if data about the inner flow field are to be extracted. Knowledge about the structure of a flow and the processes within it would also simplify the task of interpreting measurements obtained with sensors; a flow picture would, for instance, possibly enable one to detect and identify the reason for peculiar measurement traces. In many cases, in particular in the area of shock wave research, flow visualization represents practically the only experimental method to investigate a flow field and to obtain a qualitative/phenomenological description of the flow structure and the mechanisms that establish it (e.g., Takayama, 1983). Refined visualization techniques can, to a limited extent, also provide quantitative information, in particular, if combined with the results of numerical flow simulation (e.g., Kleine et al., 1995). One aspect of shock wave research, which without flow visualization would have been difficult to develop beyond purely theoretical treatment, is, for instance, the classification of the various forms of shock wave reflection. This is outlined here in Chapter 8.1 by Ben-Dor. The methods to be discussed in this chapter have been used for other applications as well; some of them may not have been originally intended for use in compressible fluid mechanics, but their suitability for this research field soon became evident. Consequently, their further development was intimately linked to the study of compressible flows, to an extent that some techniques are nowadays exclusively associated with this research area. The inherently fascinating aspect of the methods described in the following sections is that they enable one to "see the invisible." The motion of gases (like air) can be felt and its effects can be seen (if for instance one observes leaves and tree branches moving and rustling in the wind), but the moving object itself--the gas, being a fully transparent medium--remains invisible to the naked eye.
5.1
685
Flow Visualization
One obvious way to visualize such a fluid flow would be to add foreign and visible particles and then observe their motion. This principle is indeed being followed in a vast number of applications involving a great variety of so-called tracer materials. The main concern is, however, whether or not a fluid particle and such a tracer particle behave in identical fashion; the tracer particle may have dynamics of its own and thus not be able to follow the fluid motion faithfully. Indeed, this restriction limits the application of methods involving foreign objects to low-speed flows. Occasionally, variants of this visualization principle have been used in flows involving shock waves; however, overall they do not play a very important role in this field and are thus not discussed here. Nevertheless, in some research areas, such as in the study of large-scale blast waves, tracer particle methods become relevant as other techniques are less readily applicable (see Chapter 13.1 by Dewey).
5.1.1 DENSITY-SENSITIVE FLOW VISUALIZATION The most commonly used techniques to visualize compressible fluid flows utilize a fundamental physical principle--the fact that the speed of light depends on the density of the (transparent) medium it traverses. Light that passes through an area of varying density will undergo certain modifications. The recording of these modifications essentially then results in a picture of the flow. Strictly speaking, one does not take a picture of the flow itself but of the changes and disturbances it induces in an array of light beams. This effect can be observed frequently in everyday life, mostly when air masses of different temperature meet. The hot air rising from a stove, for instance, casts a shadow against a wall if adequately illuminated by sunlight. This shadow creates a picture of the flow; in fact, this example almost ideally represents one of the visualization methods that will be discussed in the subsequent sections. In gases, a simple linear relation exists between the speed of light and the density of the fluid: Co
- - - - n -- 1 + K p
(5.1.1)
c
where co denotes the speed of light in vacuum, c the speed of light in the medium under investigation, p the density of this medium, and K the so-called Gladstone-Dale constant. The ratio of co and c is defined as the refractive index, n, of the medium; K is a function of certain characteristics of the medium and also dependent on the wavelength of the light. However, for the applications described here, the latter dependency can be neglected. For air at a temperature
686
I-I. Kleine
T = 288 K, for instance, the value of K varies between 0.225 and 0.23 cm3/g over the range of visible wavelengths (Merzkirch, 1987). Because of the direct relation Eq. (5.1.1) between n and p, both properties will be used synonymously with respect to their role in influencing a traversing light beam. This influence manifests itself in the following way: A light beam is displaced, deflected, and retarded or accelerated when it traverses an area of varying density. In other words, it will reach a recording plane behind this area at a different location, under a different impingement angle, and at a different time. Each of these modifications can be recordedmin fact, each modification corresponds to a particular set of techniques. Furthermore, it can be shown (Merzkirch, 1987) that each modification yields a different derivative of the density: The displacement is proportional to the second derivative of the density, the angular deflection to the density gradient, that is, the first derivative, while the retardation or acceleration is related to the density p itself. All relevant methods to record these modifications are listed in Table 5.1.1. Displacement and deflection angle are quantities that have a vector character, that is, for a full description one would have to specify both their magnitude and their direction. The techniques to record these quantities can, however, only deliver either direction or magnitude, while a hybrid combination would be necessary to measure both simultaneously. Furthermore, some techniques are characterized by a directional sensitivity, which means that they can only detect modifications of the light beam that occur in a certain preset preferred direction. All methods are integral, line-of-sight techniques, which implies that the measured modification is the sum of all local modifications encountered along the optical path. Mathematically, one might express this as measured change -
Q ds
with
ds - (dx 2 + dy 2 + dz2) 1/2 (5.1.2)
where s is the arc length along the (usually curved) light ray between the limits ~1 and ~2, and Q denotes the physical property that generates the change. From the initial comments it is already clear that Q will be some function of the density p. The integral in Eq. (5.1.2) is considerably simplified if one can introduce the following assumptions: 9 Q attains nonzero values only within a given test section depth L; the light rays are not influenced outside of this test section. 9 curvature and displacement of the light rays within the test section are negligibly small, so that the actual path of the light ray can be approximated by the depth of the test section L. The arc length coordinate
TABLE 5.1.1
Overview of Density-sensitive Visualization Techniques
Method
Optical component in focal plane of first spherical mirror
Optical component in focal plane of second spherical mirror
Recorded quantity
Shadow
slit or pinhole
Schlieren
slit or pinhole
knife edge
Color schlieren (conventional)
slit or pinhole
color filter assembly
~
Op ~x
or
Color schlieren (dissection technique)
rectangular color filter source mask
slit
~
Op Ox
or
Direction-indicating color schlieren
annular color filter source mask
pinhole
Shearing interferometry
polarizer and Wollaston prism
Wollaston prism and polarizer
Reference beam interferometry (Mach- Zehnder interferometry, holographic interferometry)
(slit or pinhole)
~-7+
p
ap Ox
or
no
0; 0y ~p
0y ~p
0y
Op
oB Op Ox
~
p(x, y)
ap or
Preferred direction
0y
(magnitude and sign)
yes
(magnitude and sign)
yes
(magnitude and sign)
yes
(direction)
no
(magnitude and sign)
yes no
O~ O0 -.4
688
H. Kleine s then coincides with one of the Cartesian coordinates (in commonly used notation, the light ray path is described by the coordinate z).
In the following sections, which will describe in greater detail the various physical properties that the symbol Q comprises, Eq. (5.1.2) will thus have the simpler form measured change -
Q dz
(5.1.3)
The light beam that traverses the test section can in general attain arbitrary shapes, but both the theoretical evaluation as well as the experimental realization are simplified if a parallel light beam is used. Another advantage of this concept is that the test object may be placed anywhere along the beam. It has to be emphasized that even in the simplified form, Eq. (5.1.3), the measured change remains the result of an integration along the optical path. Evidently, if the contributions to the measured effect vary strongly along this path, that is, if Q - Q(z), it may become difficult to determine the local value of Q(~) for any location ~ between 0 and L. Only in the case of a plane flow, that is Q :# Q(z), or in the case of an a priori known functional relation Q(z) can the measured change directly be related to a numerical value of Q(~). Strictly speaking, plane flows do not exist in nature; however, a multitude of flows exists, which are essentially plane, because three-dimensional (3D) effects are negligible and/or restricted to a very narrow portion of the test section so that their contribution to the measured change can be neglected. For flows where 3D effects have to be considered, the visualization methods are still very useful to obtain a qualitative description of the phenomena. The characteristics of the visualization techniques described in what follows can be illustrated in the test problem schematically shown in Fig. 5.1.1. A shock wave (M s = 1.45 in N2) is initially diffracted at a sharp 90 ~ comer, which represents a sudden change in cross-sectional area. The upper part of the wave, which has not yet been affected by this diffraction, hits an obstacle in the form of a square cylinder. In the flow pattern that develops in the process of this interaction, all flow features encountered in shock wave research appear: shock waves, vortices, expansion and compression zones, slipstreams, boundary layers, and regions of separated flow. Figure 5.1.2(a)-(h) demonstrates how this rather complicated flow field is visualized by the subsequently described visualization techniques. At the 90 ~ corner, a diffraction pattern is formed (Skews, 1967), whose main elements consist of the diffracted shock that propagates downward into the duct, an expansion wave that propagates back upstream (at the instant shown here, this wave has already left the visible test section), and a slipstream rolling up to a vortex, which remains attached to the corner.
5.1
689
Flow Visualization
FIGURE 5.1.1
Schematic of the test problem.
When the shock wave interacts with the square cylinder a reflected wave is generated, which initially moves back towards the left side of the test section. Its lower portion is, however, almost immediately influenced by the wave system that was created during the diffraction process. This portion eventually follows the leading diffracted shock into the duct, after passing through the vortex that was formed during the diffraction. As a result of the latter interaction, the vortex has been "cut off" at the corner, where two smaller vortex subsystems develop. The part of the incident wave that was diffracted by the square cylinder has generated two regions of separated flow, starting at the front corners of the top and bottom cylinder surfaces and ending by a vortex. These regions develop because the stagnation flow at the frontal surface of the cylinder cannot negotiate the sharp corners. At the rear end of the square cylinder (top and bottom corners), the incident wave is diffracted once more, which leads to the generation of two more expansion waves and vortices. Subsequently, the diffracted waves collide and form a reflection pattern, which initially is regular, but develops into a Mach reflection later on. A line, which upon closer inspection represents a small V-shaped funnel, starts immediately behind the intersection of the two waves, where a Mach stem has already become visible. These two elements indicate that the onset of this reflection occurred several microseconds before the instant of photography. The diffracted wave at the bottom side of the square cylinder has not propagated as far as its counterpart on the top side, which is an indication of the influence of the interaction with the flow system generated at the 90 ~ corner. A further analysis and description of this flow would go beyond the scope of this text, but the given explanation should be sufficient to compare the different visualization results.
FIGURE 5.1.2 Visualization of the test problem (Fig. 5.1.1) (from Kleine, 1994). (a) Shadow method; (b) monochrome schlieren (vertical knife edge); (c) holographic interferometry; (d) classical color schlieren; (e) color schlieren, dissection technique, symmetrically placed vertical cutoff; (f) color schlieren, dissection technique; top: detail of (e); bottom: with shifted sensitivity range (asymmetrically placed cutoff); (g) direction-indicating color schlieren (cutoff device: pinhole); and (h) direction-indicating color schlieren (cutoff device: cylindrical plate). (See Color Plate 1).
(p~nu!~uoD) ~ ' I ' g 3~IFl9Irl
.o 14
~z
FIGURE 5.1.2
(Continued)
5.1 FlowVisualization
693
A more extensive treatment of some of the visualization techniques discussed in the following sections as well as the description of other methods can be found in corresponding monographs and textbooks (e.g.. Merzkirch, 1987; Oertel and Oertel, 1989; Ray, 1997). The text of this chapter is largely a translation of the corresponding sections in the author's thesis (1994), which is also the source of all figures and photos presented here. Further information on special aspects and instrumentation is available in the proceedings of conferences dedicated to the subject of flow visualization (for example, The International Congress on High-Speed Photography and Photonics or the Flow Visualization Symposium).
5.1.2
THE SHADOW
TECHNIQUE
The shadow technique represents the simplest and most straightforward method of density-sensitive flow visualization techniques as it can be realized with a minimum of optical components. The recorded quantity is the displacement of a light beam that has traversed a region of varying refractive index (i.e., the test section). As only the overall magnitude of the displacement Ar is recorded (Ar = ( A x 2 --t- Ay2)l/2), there exists no directional dependence. Therefore, the technique is equally sensitive in both the x- and the y-direction. It can be shown (see, e.g., Merzkirch, 1987) that in a recording plane this beam displacement leads to variations of light intensity AI/I, which are dependent on the second derivative of the density p. Here it is assumed that the beam displacement is not compensated by an additional optical system in between the test section and the recording plane. AI_ l I
O-~ + o
ln(1 + Kp) dz ~ IK
~
+
(p) dz
(5.~.4)
o
The last part of Eq. (5.1.4) was derived considering that almost always Kp << 1, which allows one to expand the logarithm. The dependency on the second derivative of p already indicates that a shadowgraph--a picture taken with a shadow visualization system--can only depict variations of the density gradient; in other words, discontinuities such as shock waves, slipstreams, or zones of high turbulence such as in the wake of a projectile. Phenomena that have a more gradual or spatially expanded nature, such as expansion fans and compression waves, cannot be easily detected on a shadowgraph. In order to obtain a shadow effect, the recording (or shadow) plane has to be placed at a certain distance l away from the test section. In the simplest realization of this method, this plane is defined by placing a screen or a film at a usually small distance behind the test section (Fig. 5.1.3(a)). This so-called
694
U. Kleine
(a)
(b) test section i
film illumination intensity
--normal
! I
^
'
---l l o r m a l
-dark X
--normal
imaging lens (position f, c, or b )
test section
A
,,
film '
_.~__/~
Z
II
I
I
v f
I <0
l=O
~ >0
(f)
(~)
(b)
"
v c
b
shadow plane FIGURE 5.1.3 system.
Principle of the shadow technique. (a) Direct method; and (b) with an imaging
"direct" shadow technique has the advantage of avoiding all the possible distortions that an optical imaging system might introduce. However, it becomes more and more impractical for larger test sections and in setups where the film or the screen cannot be protected from (unwanted) light in the environment. In these cases, an optical imaging system has to be introduced (Fig. 5.1.3(b)), which, however, does not image the test section but the shadow plane. As a result of this, the contours of objects in the test section are not perfectly in focus; this effect is however only of minor consequence as the optical systems usually found in flow visualization apparatuses provide a sufficiently large depth of field. The introduction of the optical imaging system also allows one to put the shadow plane at arbitrary locations, even in front of the test section. Such a shifting of the shadow plane (from position f to b or vice versa in Fig. 5.1.3(b)) leads to a change of sign in the intensity distribution on the recording plane, as illustrated in Fig. 5.1.4. If the test section itself is imaged, the shadow effect is reduced to a minimum. In case of a very narrow test section with negligible extension in the direction of beam propagation, the optical system would completely compensate the beam displacements caused by the varying refractive index in the test section; hence, no intensity variations would be recorded and no shadow effect would be observed. Discontinuities of the density or the refractive index, respectively, are shown on a shadowgraph as a combination of one bright and one dark line each (for a density increase, the sequence is dark/bright for the shadow plane located behind the test section, and bright/dark for the shadow plane in front of the test section). The respective second line of this combination is usually less pronounced than the first one. The front of the discontinuity corresponds to the front of the first of the two observed lines.
FIGURE 5.1.4
Detail of test problem (Fig. 5.1.1), with the shadow plane before (a) and behind (b) the test section (from Kleine, 1994).
696
n. Kleine
The sensitivity of the method is largely governed by the distance l between test section and shadow plane. This distance should not however be chosen too large as the blurring of the test section contours increases with increasing value o f / (Fig. 5.1.3(a)). Furthermore, displaced beams should not cross before reaching the shadow plane as this would prevent one from localizing exactly the discontinuity fronts. For most applications, l should not exceed 75% of the test section depth. Large light deflections in the vicinity of objects or walls may lead to a distortion of the contour in the image of the test section. Light beams close to the edges of an object can either be deflected into regions usually covered by the shadow of the object, or they may be deflected away from the contour, letting the object appear larger than it is in reality. Shadowgraphs are very suitable to capture and measure the geometry of a system of discontinuities. They may also be used for a qualitative description of a flow field, although some phenomena (as mentioned earlier) cannot or can only hardly be made visible. Further data that might theoretically be obtained from processing the recorded shadowgraphmfor example, a twofold integration of the intensity distribution, Eq. (5.1.4), in order to derive the density distribution p(x, y)--is usually very inaccurate. As a result, the application of shadowgraphs for a quantitative investigation of a flow field is somewhat limited. On the other hand, shadowgraphs often provide the clearest and best structured flow pictures, but at the expense of smaller details that they suppress. This can easily be seen when comparing the corresponding pictures in Fig. 5.1.2.
5.1.3
SCHLIEREN
METHODS
A Schlieren system detects and records the change of direction that a light beam experiences on its path through a medium of varying refractive index. Please note that "schliere" is taken from the German language, in which all nouns are capitalized; in the current English adaptation the capital "S" has been replaced by a small "s" in order to avoid creating the impression that the technique is named after a person of this name. In order to comply with this notation, the capital "S" will not be used any more in this text from now on. A schliere is generally defined as a small portion of a transparent object that changes the direction of a traversing light beam. Spatially extended objects such as a transparent wedge, which introduce a constant light beam deflection over a larger area, are not considered to be schlieren (Schardin, 1942; Wolter, 1956). The principle of the schlieren technique is to separate deflected from undeflected (i.e., undisturbed) light beams, for example, by blocking off
5.1 Flow Visualization
697
light s o u r c e
pinhole or slit
undisturbed ray disturbance I deflected (schliere) I ray
. . i m a g i n g lens 0
lens F2
image plane ( s c r e e n or film)
lens FI test section
k n i f e edge
FIGURE 5.1.5 Toeplerconfiguration of a schlieren setup.
deflected beams so that they do not reach a recording screen onto which the test section is imaged. Unlike the shadow apparatus that may be realized without an imaging unit, optical imaging is imperative for the schlieren method. If the deflected beam is intercepted, the corresponding point in the test section, where the light deflection occurred, appears darker. Similarly, one can reverse this system by blocking off undisturbed light beams and only admitting deflected ones to the screen, where corresponding image points of the test section would now appear brighter. The most c o m m o n realization of a schlieren apparatus is the so-called Toepler system (Toepler, 1864; Schardin, 1942; Holder and North, 1956). In an often-found version of this setup, a light source is placed in the focal point of a lens F1, thus generating a parallel light beam, which then traverses the test section (Fig. 5.1.5) 1. If individual pencils of rays are considered (Liepmann and Roshko, 1957) one can show that each point in the test section receives light (or, in a more general description, information) from all portions of the light source, which in this case may be of extended size. If the parallel light beam is focused again by means of the schlieren head F 2, an image of the light source is generated in the focal point of F 2, where each point in the source image receives light from every portion of the test section. If in this focal
In the original system built by Toepler, the lens F l was omitted. The object, which exhibited a comparatively small spatial extension in the direction of ray propagation, was located immediately behind the second lens F2, called the "schlieren head" (Toepler, 1864). Introducing F 1 allows one to generate a parallel light beam and renders the system applicable to spatially extended objects, but does not change its characteristics.
698
H. Kleine
plane a portion of the light is blocked (e.g., by introducing a cutoff device like a knife edge), all ray pencils are equally affected. In this case, the image of the test section, usually created by another lens O (Fig. 5.1.5) would be uniformly darkened. This uniform change of illumination of the test section image is the indication that the knife edge has indeed been placed in the focal point of F 2. For any location of the knife edge other than the focal point, the test section image would darken or brighten on one side earlier than on the other. However, if at one point of the test section a schliere changes the direction of one of the light rays in, say, the y-direction, the ray pencil associated with this test section point is shifted by an amount Ah at the location of the knife edge (i.e., the focal plane of F2), where Ah is given as Ah =f2~y
(5.1.5)
with f2 being the focal length of the lens F 2 and ~y the deflection angle in the y-direction. Upon exiting (but still within) the test section, the deflection angle of the light ray attains the value (see Fig. 5.1.5 for definition of coordinates):
, l i( n(x, 1y, z)
~Y --
dz
(5.1.6)
where L is the depth of the test section. For a plane (two-dimensional (2D)) flow (i.e., n ~ n(z)), applying Eq. (5.1.1) and Snell's law leads to --~
~ LK
(5.1.7)
/l 0
for the deflection angle outside of the test section. Here, n o is the refractive index of the medium surrounding the test section. Usually, one has n o ~ 1, which was introduced into the last part of Eq. (5.1.7). Thus, the deflection is a linear function of the density gradient with the light ray being deflected in the direction of increasing density. The displacement Ah, Eq. (5.1.5), then leads to a proportional change of illumination intensity at the corresponding point of the image of the test section, if the forementioned cutoff device has been placed at the focal plane of F2 in the fashion described earlier on. Only displacements vertical to the edge can be recognized that way, while a horizontal shift Ahhor (Fig. 5.1.6) does not lead to a change of illumination in the recording plane. Thus, this visualization method yields the density gradient in a predetermined, preferred direction, that is, vertical to the edge of the cutoff device. This means that only a component of the density gradient in the test section is represented, which would necessitate a second visualization with a rotated knife edge (preferably, with a rotation angle of 90~ In shock wave studies, this require-
5.1 FlowVisualization
699
initial light source i m a g e ~
deflected
light source i m a g e
I...... : :.
!~, '
|h- i h r
1,LAh, FIGURE 5.1.6 Deflectionof the light source image at the location of the cutoff device.
ment is usually equivalent to having to reproduce an experiment, which presents certain difficulties. In order to overcome this potential problem, systems have been devised that split the beam shortly before being focused by F 2, which enables one to use two knife edges simultaneously (Barry and Edelman, 1948). This adds, however, to the complexity of the apparatus and may also lead to problems with the instrumentation, in particular with respect to the available brightness of the light source. The relative intensity variation AI/I in the recording plane, also defined as contrast, can be formulated as:
where h denotes the portion of the light source image that passes the knife edge in the absence of any gradients in the test section (Fig. 5.1.6). The last part in Eq. (5.1.8) is only valid for a plane flow under the assumptions stated for Eq. (5.1.7). The sensitivity S of a schlieren apparatus is given by
S ~
Ah/h
f2
"~--
(5.1.9)
h
Both characteristic values of the setup, contrast and sensitivity, are thus largely determined by design parameters of the apparatus, namely, the size of the light source image and the focal length of the second lens or schlieren head F 2. Apart from the sensitivity, the sensitivity range of an apparatus is an essential design criterion for a schlieren setup. The maximum values of detectable deflection angles are determined by a complete deflection of the light source image over or under the knife edgemany further deflection will not lead to a change of illumination on the recording plane. For a total height
700
H. Kleine
hat of the light source image (Fig. 5.1.6), the largest deflection angle to be registered towards the knife edge is given by
h (5.1.10)
~ymax ----f-22
while for deflections away from the knife edge, angles in excess of ~ymin "--
hT-h
(5.1.11)
f2
will not lead to a further brightening of the corresponding point in the test section image. Equal sensitivity for positive and negative deflections requires h 7 ----2h
(5.1.12)
If a schlieren system is designed with, say, h - 1 mm and f2 = 2000 mm (these values can be considered as fairly typical), one would then obtain ~ymax -- -~-0.0005 rad -- 1.72' =,
(~9_~~)
kg ~ 4-44 m4
(5.1. lOa)
max
The latter part of Eq. (5.1.10(a)) follows with Eq. (5.1.7) for a (typical) test section depth L = 5 0 m m and air as test gas (Gladstone-Dale constant K = 0.227 cm3/g). This sensitivity range is comparatively small; the calculated maximum value may easily be exceeded in a complicated flow field, as the test problem (Fig. 5.1.2(b)) indicates. If one tries to increase the range by increasing either h or hr, the sensitivity of the apparatus is negatively affected. For constant f2, the sensitivity may, in turn, be raised by decreasing h, but this is limited by (so far not considered) diffraction effects at the knife edge and the necessity of a minimum background illumination. Furthermore, a reduction of h by blocking off a larger portion of the light source image would lead to a onesided shift of the sensitivity range as already small deflections would lead to a complete darkening of the corresponding image point of the test section. If h is reduced by selecting a smaller source size hr, the measurement range is reduced correspondingly. The minimum density gradient that can be detected by the eye in such an apparatus is usually associated with a 10% change of illumination. For the values of h, f2, L, and K given in the preceding, this would correspond to
(~)min~,~ 4-4.4 m4kg
(5.1.11a)
The values given in Eqs. (5.1.10(a)) and (5.1.11(a)) apply to the system used for Fig. 5.1.2(b).
5.1
Flow Visualization
701
Schlieren setups with very long focal lengths for the schlieren head F 2 can visualize down to 1% of this value, which corresponds roughly to a density gradient related to a temperature change of 1 K over 1 m in ambient air. If the test section reaches a certain size, it becomes increasingly difficult and expensive to obtain lenses of adequate optical quality. Usually, the lenses are replaced by spherical or parabolic mirrors, which first generate and later focus the parallel light beam (Fig. 5.1.7). For the focal length/diameter ratios most likely to be encountered in schlieren systems, the difference between a spherical and a parabolic mirror is usually negligible. For this reason, these mirrors will be referred to from now on as spherical mirrors, which represent the more frequently found type. The inevitable off-axis arrangement generates two additional image distortions, which however can be eliminated or minimized by an appropriate layout of the system: Coma is eliminated in a symmetrical setup (equal focal lengths of both mirrors, equal angles to the main optical axis through the test section on opposing sides) (C7erny and Turner, 1930), while astigmatism can be minimized by introducing a correcting lens. As a consequence of the astigmatism of the system, two focal points exist for the spherical mirror, one each in the meridional and in the sagittal plane. The distance between the two, the so-called astigmatic difference, can amount to several millimeters, depending on the off-axis angle and the focal length (Hecht and Zajak, 1974). A vertical knife edge is to be placed in the focal point of the meridional plane, while a horizontal knife edge would have to be located in the sagittal one; only there one would obtain the uniform change of illumination of the test section image. In cases where one works only with one of these knife edge options, a correction is not mandatory, but it becomes imperative when knife edges are introduced of an orientation that is not compatible with the orientation of the main optical planes. In these cases, the forementioned uniform change of illumination of the test section image cannot be achieved without an astigmatism correction. The latter can be achieved, for example, by introducing a cylindrical lens shortly behind the focal point of the first spherical mirror (see Fig. 5.1.7), which helps to reduce the astigmatic difference to negligible values (Prescott and Gayhart, 1951). Spherical aberration, which could also occur for in-line setups realized with lenses rather than the off-axis mirror arrangement, is mostly negligible, if the ratio of focal length and diameter of the optical component is sufficiently large (f/D > 5). Other knife edges, with shapes that differ substantially from a straight edge (such as slits, circular orifices, gratings, etc.) create generally stronger diffraction effects, which lead to a significant decrease in image quality. All schlieren setups described here including the shearing interferometer (Sections 5.1.2-5.1.7) can be realized with a more or less identical setup, that
polarizer and Wollaston-prism
shearing interferometer
SM
9 spherical mirror with focal length f CL : cylindrical lens
color filter source
mask
color schlieren m e t h o d (dissection-technique) CL
SM
1
slit or
light .
.
.
source
monochrome schlieren methods/ classical color schlieren
.
monochrome schlieren m e t h o d s / classical color schlieren color filter
SM
2
OlOr s c h l i e r e n m e t h o d issection-technique) s l i t or pinhole
/ shearing interferometer Wollaston-prism and polarizer
FIGURE 5.1.7
Schematic layout of the setups used for schlieren and shearing interferometry flow visualization (from Kleine and Gr6nig, 1993).
b.J
5.1 FlowVisualization
703
is, the Toepler Z-configuration as it is schematically shown in Fig. 5.1.7. Switching from one method to the other requires only the replacement or the realignment of a few optical components and can be achieved without major effort. The visualization of the test problem (Fig. 5.1.2(b)) clearly shows that the amount of information contained in a schlieren picture exceeds by far the one of a shadowgraph. Expansion and compression zones become visible and magnitude and sign of occurring density gradients can be estimated or determined, respectively. On the other hand, it may become more difficult to interpret the flow field, in particular, because the measurement range of the system has obviously been exceeded several times and this leads to larger zones of either total darkness or brightness. Furthermore, some flow features remain invisible, namely those which generate a density gradient parallel to the orientation of the knife edge (such as the boundary layers along the wails of the test section). In the shown flow picture, the knife edge was oriented vertically.
5.1.4 COLOR SCHLIEREN TECHNIQUES As the human eye is generally more sensitive to detecting changes of hue rather than changes of illumination intensity, it has been tried almost since the beginning to introduce color to schlieren systems. A monochrome schlieren apparatus, as outlined in the previous section, may easily be modified to a color schlieren setup by replacing the knife edge with a set of colored filters (Rheinberg, 1896; Schardin, 1942; Holder and North, 1952b). According to the experienced deflection, the light source image would pass through different filter segments, thus creating changes of hue in the recording plane. The sensitivity of the system is governed by the ratio of the width of the light source image and the width of a filter segment. However, even for comparatively wide filter segments (and thus a low sensitivity of the system) one observes a strongly reduced image quality in the recording plane, caused by diffraction effects at the filter segments. Diffraction appears at all imperfections and nonuniformities of the filter system, in particular at the boundaries between segments. For an acceptable image quality, the achievable sensitivity of the system is about one order of magnitude lower than for an identical but monochrome system with a knife edge. The color system can be improved by using continuously changing color filters such as an interference graded filter (Meyer-Arendt and Appelt, 1978), or a photographic transparency of a white light spectrum (Howes, 1984). When compared to a black-and-white system, however, the sensitivity of such a color setup remains inferior. This should be adequately illustrated by
704
H. Kleine
comparing Figs. 5.1.2(b) and (d). In the case of Fig. 5.1.2(d), the knife edge had been replaced by a slide of the spectrum of a Xe-lamp. If one uses a prism to achieve the separation of (usually white) light into different colors (Schardin, 1942; Holder and North, 1952a) and a slit to replace the knife edge, essentially the same problems are encountered: High sensitivity requires small widths of the slit, which in turn causes an increase of diffraction. Cords (1968) developed an empirical relation between the required slit width w and the demanded optical resolution d in the recording plane, which is formally similar to the Rayleigh criterion: w - 0.65 2g d
(5.1.13)
where 2 is the wavelength of the used light, and g is the distance between the test section and the schlieren head. For the setup used to perform the visualizations represented in this text, one obtains Wmin " - 5 . 5 m m for g = 2800mm, 2 = 550nm and d = 0 . 2 m m . Choosing a smaller value for w while leaving the rest of the setup unchanged will reduce the attainable resolution d. For a schlieren setup designed in the conventional way as described earlier on, the criterion, Eq. (5.1.13), implies that only low sensitivity setups may be realized. In order to overcome this problem, Cords (1968) suggested performing the separation into different colors already in the focal plane of the first spherical mirror, in order to provide a sufficiently large spatial separation between the individual color sectors (dissection technique). This latter aspect presents the main difference between Cords's approach and the previously mentioned prism technique. The color sectors may be separated by a distance large enough so that the slit in the cutoff plane can be adjusted to a width compliant with Eq. (5.1.13). This approach utilizes the fact that a spherical mirror can generate a largely parallel beam even if the source in its focal point exhibits a certain spatial extension rather than being an ideal point source. This spatial extension may be as large as 1% of the focal length of the spherical mirror. Consequently, in schlieren setups with lenses, it is more difficult to apply the dissection technique because of the usually smaller dimensions of the lenses, which would require correspondingly smaller color sectors. This latter aspect, however, may be considered primarily as a manufacturing issue as the preparation of the color sector arrangement becomes more challenging for smaller dimensions. One technique that has produced very agreeable results consists of applying cut-out gelatin filter segments to a black/transparent template that can easily be produced with high-contrast black-and-white negative film (Cords, 1968; Stong, 1971). The whole assembly is generally referred to as a color filter source mask. The color segments, which are illuminated from the back by an adequately strong white-light source, provide an ensemble of approximate point-like
5.1
Flow Visualization
705
sources for the spherical mirror, which generates, accordingly, a set of parallel light beams, one for each segment in its respective color. In order to assure that each of these light beams completely fills the spherical mirror (thus ascertaining that the whole information of the light source is received by each point in the test section), it has to be enlarged and spread by means of a diffusing screen. This screen is placed immediately in front of the filter system, facing the light source. Ground glass plates with a surface roughness of 4-10 lam are most useful for this purpose (screens that diffuse the light more strongly increase the light losses, while screens with less pronounced diffusion characteristics lead to a nonuniform illumination of the test section). The generated parallel beams overlapman observer sees a white light beam (possibly with thin colored fringes at the edges) traversing the test section. The second spherical mirror "undoes" the overlapped beams and generates an image of the filter system in its focal plane (for a setup with identical mirrors this image is inverted but of equal size to the filter system illuminated by the light source). There, a slit arrangement serves as the knife edge. In the initial setting (no disturbance in the test section), usually only light from one segment passes this slit and reaches the recording plane, thus providing the background color. When the refractive index in the test section changes, the light of other segments can pass the slit and thus generate other hues in the recording plane. It is characteristic of this method that the colors usually become more and more "watery" and faded the higher the amount of deflection, as light from more and more filter segments reaches the film, without blocking the light from the previously admitted segments. Therefore, strong deflections are generally represented by a form of tinted white (Kleine and Gronig, 1993). By choosing the proper selection and arrangement of filter segments, a given density gradient range may be represented by as many as eleven hues (see Fig. 5.1.8(b)). Only two "pure" colors can be generated--the ones of the innermost segments on each side--while all other hues are mixtures of various colors. The width of each segment determines the sensitivity of the system. Widths < 0.2 mm, however, can generate diffraction effects when the color filter source mask is illuminated. A second criterion for a minimum width of the segments is that the amount of transmitted light has to be sufficient to expose the film (or to be detected by the eye). Considering that the intensity of the applicable light sources is limited, this requirement leads to a minimum illuminated surface area of the segments. As their length is limited (1% of the focal length of the spherical mirror, see in the preceding material), this translates into a minimum width for the segments. The final dimensions of a color filter mask (Fig. 5.1.8(a)) have to be determined considering the optical properties of each setup and can usually only be found in an empirical way. Another criterion that should be considered in the design of a color source mask is, of course, the already mentioned ability to manufacture the system.
706
H. Kleine
(
(b)
!)
bluish
~'
green yellow blue red light blue violet blue grey~ ight violet ddish white
~thte
~ :.
I I
+,
.
i
:
-ioo green
~
blue
red
~
yellow
'
:
_:
:
:
9
o 8x
FIGURE 5.1.8 Color schlieren (dissection technique). (a) Color filter source mask; (b) diagram for correlation between color and gradient magnitude.
The template/gelatin filter arrangement may be replaced by a photographically obtained color transparency, but mostly at the expense of clarity and good color definition. The design of the color source masks makes it obvious that a color schlieren system based on the dissection technique is very inefficient in terms of using light, as already a significant portion of the light provided by the source is not used and blocked before reaching the test section. This feature presents a severe limitation for these techniques, at least in short-duration applications such as shock wave studies. If the slit in the cutoff plane is positioned roughly symmetrically to the image of the color filter mask, the ranges of negative and positive density gradients are represented by approximately the same number of hues. However, by shifting the slit to one side it is possible to increase the resolution into different colors for one particular zone of the gradient range, while the remainder may be represented by but one color. Similarly, one could design a color filter mask in a correspondingly asymmetrical fashion. This corresponds to a one-sided shift of the sensitivity range in a monochrome system by changing the size of h (Fig. 5.1.6). An example for this shift of the sensitivity range is given in Fig. 5.1.2(f), where the same flow field is shown twice, using both a symmetrical and an asymmetrical color coding. The calibration of a schlieren system may be performed in a simple fashion, if an object of known refractive index distribution is placed inside the test section (Schardin, 1942). A suitable calibration object is for instance a piano-
5.1 FlowVisualization
707
convex lens of long focal length (the light rays deflected by the lens are not to cross before reaching the knife edge). In monochrome setups with a high sensitivity, slightly curved glass plates may be more useful, as they represent a range of smaller deflection angles and as they also are simpler to manufacture than lenses (Scheitle and Wagner, 1992). The deflection angle caused by an object such as a piano-convex lens is | ~ tan | - yA
(5.1.14)
where Yz signifies the distance of the ray to the center of the lens (0 < Yz < lens radius) and fcz the focal length of the calibration lens. Equating this angle to the deflection angle caused by a planar flow field with refractive index variations, Eq. (5.1.7), one obtains the following expression for the density gradient:
(8AP~ -
Yz LKfd
(5.1.15)
The coordinate y for the gradient is perpendicular both to the direction of light propagation and the knife edge--and thus parallel to Yz (Kleine and GrOnig,
1991). If light absorption by the calibration lens is negligible, Eqs. (5.1.15) and (5.1.8) can easily be combined to relate deflection angles to variations of light intensity in the recording plane. A photo of the test section with the calibration lens immediately yields the sensitivity range and the sensitivity itself, which in the case of a color schlieren setup is determined by the width of each color zone. The dissection technique system that was used to obtain Figs. 5.1.2(e), (f) and 5.1.10(b), is characterized by a range of (~p/0Y)max = • kg/m 4 in air. The smallest density gradient that causes a shift of colors with respect to the initial setting amounts to (OP/~Y)min ~-~ 4-8kg/m4 in air. The classical color schlieren method (Fig. 5.1.2(d)) yields (8p/~Y)max ~,~ 55 kg/m 4 and (Sp/c~y)min ~, 18 kg/m 4. As can be seen from these values, color schlieren techniques are generally less sensitive than the monochrome method but they provide a greater sensitivity range. The dissection technique offers a very good compromise and proves to be superior to the classical system. In similar fashion as for the classical black-and-white technique, it is not mandatory to provide an astigmatism correction. Figure 5.1.2(e) and (f) demonstrate how the test problem may be visualized with the described color schlieren technique. A direct comparison to the classical techniques (Figs. 5.1.2(b) and (d)) illustrates clearly the gain of information about the flow field, in particular in the zones where vortices are predominant. By taking two pictures, with a turn of both the color mask and the slit by 90 ~ the gradient components in both x- and y-direction can be made
708
H. Kleine
visible, which then may be combined to determine the overall gradient, similar to the procedure in a monochrome setup. 5.1.5 DIRECTION-INDICATING SCHLIEREN METHOD
COLOR
The color schlieren technique described in the last section is essentially equivalent to a monochrome method with respect to its main characteristics. Both systems record the same quantities and differ only in the way they code and display the information. As mentioned before, color shifts are generally easier to detect by the human eye than changes of illumination, and the sensitivity range of a color schlieren apparatus is, for the most part, greater than that of a corresponding monochrome system. These features by themselves, however, would mostly not be sufficient to justify the increase in complexity for bringing color into a schlieren system. Color may, however, be used as the carrier of information that would be difficult if not impossible to display by means of just changes of illumination, namely, the direction of a density gradient, as opposed to its magnitude, which is recorded by the techniques described so far. This may be realized using a color mask as shown in Fig. 5.1.9(a). Again, such a mask is conveniently assembled by placing gelatin filter segments behind a black/transparent template, with this system being located behind a diffusing screen. In the original version of this technique, which was described in 1970 by Settles, the mask was composed of four straight segments arranged in the form of a square, displaying the colors green, red, blue, and yellow. However, since
(c)
yello
~,,,~'~N~ _
blue
~
direction of a positive density gradient
FIGURE 5.1.9 Direction-indicating color schlieren. (a) Color filter source mask; (b) color filter source mask, corrected for astigmatism; (c) diagram for correlation between color and gradient direction.
5.1 FlowVisualization
709
in additive color mixing 2 there are only three primary colors (green, red, blue--yellow is obtained by mixing red and green), this system led to a partial ambiguity in relating hues and directions. This problem is overcome by using a mask of circular shape as shown in Fig. 5.1.9(a), which yields six different color sectors of largely equal size (Fig. 5.1.9(c)). In this system, the knife edge is replaced by a circular cutoff (such as an iris diaphragm), where the opening corresponds to the inner diameter of the color mask (Settles, 1985). The color coding correlation represented in Fig. 5.1.9(c) can be obtained experimentally in the described fashion of taking a picture of an object that would create uniform ray deflections in all directions, such as a piano-convex lens. Each of the three primary and three secondary colors can be chosen as a background color by a corresponding shift of the cutoff device towards the selected segment(s). It is advantageous to slightly enlarge the inner portion of the segment(s) with the chosen background color on the color source mask. This would already admit a certain amount of light to the film in the absence of any deflections in the test section, while allowing one to align the cutoff and the image of the source mask concentrically. The color correlation diagram of Fig. 5.1.9(c) applies to Figs. 5.1.2(g) and (h), while the correlation diagram for Fig. 5.1.10(a) is given with the photo. Requiring the same sensitivity in all directions as well as a uniform color distribution makes it imperative that the source mask image be focused exactly in the cutoff plane. In order to achieve this, the astigmatic difference has to be reduced to negligible values (see Section 5.1.3). If this is accomplished by introducing a cylindrical lens, one has to consider that this additional element distorts the source mask image; a source mask of circular shape is now imaged as an ellipse in the cutoff plane. In order to maintain the use of a circular cutoff, the contour of the source mask has to be designed to compensate this distortion. Therefore, the source mask is to be laid out as an ellipse, where the ratio of the main axes corresponds to the distortion factor introduced by the cylindrical lens (Fig. 5.1.9(b)). The correlation diagram (Fig. 5.1.9(c)) remains unchanged. Since all gradients normal to the direction of light ray propagation are detected by this technique, in a plane flow all phenomena that alter the local fluid density are visualized. Vortices are usually represented by a color pattern corresponding to the inverted color correlation diagram such as the one shown in Fig. 5.1.9(c). The rolling-up of the slipstream into a vortex is shown more clearly by this technique than in any other schlieren method (Fig. 5.1.10(a)). Here, the slipstream is visible as a dark line, as the corresponding light rays are deflected to the outside, that is, away from the edges of the circular cutoff. The slipstream alone can be made visible in a similar fashion by a shadowgraph, which, however, would not show other flow elements. It is often less clearly 2In the additive formation of colors, spectrum colors can be generated by mixing corresponding proportions of the primary colors green, blue, and red. If light of these colors illuminates a screen, the region where all beams overlap appears white (full reconstruction of white light). This should not be confused with the subtractive formation of colors or the combining of different pigments in painting. The primary colors in the latter are red, yellow, and blue.
FIGURE 5.1.10 Visualization of the diffraction of a shock wave (M s = 1.32 in N2) at a vertical edge (from Kleine, 1994). (a) Direction-indicating color schlieren with diagram for correlation between color and gradient direction; (b) magnitude-indicating color schlieren (dissection technique); (c) reconstructed holographic interferogram; (d) detail of (c). (See Color Plate 2).
o
J ~ ~ N
~~ o
FIGURE 5.1.10
(Continued)
712
H.
Kleine
recognizable in a magnitude-indicating schlieren system, which depicts a vortex by two almost semicircular zones representing the density decrease towards the vortex core and the density increase from the core towards the outer flow, respectively. Consequently, the two neighboring zones (which meet at the vortex core) are shown in colors or shades of gray of opposite sides of the calibration range (Fig. 5.1.10(b)). In a flow visualization that yields the density distribution itself, that is, an interferogram (see Section 5.1.8), the slipstream is normally hardly visible (Figs. 5.1.10(c),(d)). The photos obtained with the direction-indicating color schlieren technique are generally very clearly structured and thus most suitable for phenomenological interpretation, even though a potential symmetry of the flow field is normally not reflected in a symmetrical color coding. If for instance the horizontal center line of the test section corresponds to the symmetry axis of the flow, processes that occur in equivalent (but mirror-image) fashion on both sides of this axis are usually depicted by different colors. However, with the help of the correlation diagram, the interpretation of the flow picture, for the most part, presents no problem. Particular attention is required when it comes to interpreting color changes in the immediate vicinity of solid body contours. For the most part, these changes can be identified as zones of strong gradients within boundary layers, where generally the density gradient caused by a temperature change becomes visible. In Fig. 5.1.2(g) the green and violet borders, respectively, along the horizontal test section walls indicate an increase of density towards the wall, that is, from the (hot) flow towards the (cold) wall. However, if the density profile exhibits an inflexion point in the vicinity of the contour of a model or a test section wall (e.g., as a consequence of an extended entropy layer, as it may exist, for instance, in hypersonic flows), two corresponding lines are created, with colors of opposing sectors in the correlation diagram. The zone in immediate vicinity to the wall, which usually represents the density increase towards the wall, is often shown only as a faint line. The direction-indicating technique may also be used with a different sort of schlieren cutoff: The circular cutoff (iris diaphragm) may be replaced by a circular plate whose diameter corresponds to the outer diameter of the color source mask. The slip lines in vortices now appear as bright zones and lines as the outward deflection of the light rays leads to an increased exposure of the film. An example of the results to be obtained with this method is given in Fig. 5.1.2(h), for which the correlation diagram of Fig. 5.1.9(c) also applies. On flow pictures of this sort, all discontinuities are shown in high contrast. The method is, however, prone to "overshoot" and generates large, bright zones in which no details may be seen, if flow fields with strong gradients are observed. In a direction-indicating color schlieren system the magnitude of a gradient can, at best, be coarsely determined, as it is represented by the usually not easily quantifiable color saturation. For equal directions (identified by the
5.1 FlowVisualization
713
angle fl), gradient magnitudes up to (~p/~r)fl=const < 100kg/m 4, where r - (x2+ y2)1/2 are mostly represented equally, that is, within this range, a change of the actual gradient magnitude does not lead to a clearly visible change of color saturation on the flow picture. An additional coding of gradient magnitude could be achieved by means of a grid-like cutoff, which would introduce variations of brightness superimposed on the colors. The medium of color is conveniently used for the information concerning the gradient direction, while gradient magnitudes would be indicated by changes of brightness. However, such a system would essentially eliminate all the advantages of the dissection technique and yield flow pictures substantially degraded by optical diffraction effects (Scheitle and Wagner, 1990). Similar quality losses and, in addition, an ambiguous correlation between colors, gradient magnitudes, and directions are also encountered in an extended conventional color schlieren method, where the knife edge is replaced by an array of filter segments (Wuest, 1967). Displaying simultaneously gradient magnitude and direction and thus increasing the amount of information contained in a flow picture is definitely useful for the observation of simple flows, but will most likely make the interpretation of more complex flow fields more difficult and sometimes even impossible. The reader may try to imagine how the visualized test problem would appear with a superimposed bright/dark pattern indicating gradient magnitude in addition to the color that signifies gradient direction. In the dissection technique, this additional information can only be introduced by changing the source mask layout, while the cutoff has to remain largely unmodified if diffraction effects are to be avoided. Showing gradient magnitude and direction simultaneously just by using different colors, that is, without the previously mentioned brightness modulation, is usually rendered impossible by the limited number of clearly distinguishable hues. A corresponding modification of the filter source masks of Fig. 5.1.8(a) and Fig. 5.1.9(a) would therefore create considerable problems establishing an unambiguous correlation between color and gradient value and/or direction.
5.1.6
INTERFEROMETRY
Interferometric measurement methods allow one to determine the difference between the optical paths 3 and thus the "travel time" of two or more light waves, of which at least one has traversed the phase object under investigation. 3In order to avoid confusion it should be noted that the optical path is defined as o.p. = f n(s) ds, with s as geometricalcoordinate and n as refractiveindex, whereas the geometrical path is given by g.p. = f d s
714
H.
Kleine
If the individual waves fulfill the conditions for interference, their combination yields an interference pattern typical of the experienced optical path difference. Interferometers can be operated in two settings. If in the absence of a phase object no interference fringes are visible, the system is set in the so-called infinite fringe configuration, which means that the difference in optical path between object and reference beam is identical and constant for all points of the test section. If a phase object is introduced, the occurring fringes represent lines of constant phase shift, which in the case of gases correspond to lines of constant density (so-called isopycnics). However, with a deliberate misalignment of one component of the setup, a range of phase shifts throughout the test section can be introduced in the absence of a phase object. In this case, a set of interference fringes becomes visible before the test object is introduced. In this finite fringe setting, a test object will cause a deformation of the existing fringes rather than the generation of new ones. While a quantitative evaluation of the flow field is possibly simplified by choosing the finite fringe setting, this is usually achieved at the expense of the sensitivity of the system. In a so-called reference beam interferometer the beam that traverses the test object (hence called object beam) and the second, so-called reference beam, are split and separated in a way that the latter passes through an area of usually known and constant properties, independent of the test object. Later, both beams are recombined and interferometrically compared (Liepmann and Roshko, 1957). This type of interferometry is represented by the MachZehnder interferometer (for a comprehensive review of this technique, see e.g. Merzkirch 1987) and the setups that are described in the section on holographic interferometry. In a shearing interferometer, on the other hand, the reference beam also traverses the test section, normally only separated from the object beam by a small distance. Here, it would also be influenced by the test object, so that the resulting comparison would yield the differences in optical path between two a priori unknown states.
5.1.7
SHEARING
INTERFEROMETRY
In a shearing interferometer, both beams that later interfere with each other pass through the test object, that is, the flow field, normally at a very small distance e, the so-called beam separation. The subsequent interference thus yields a measure for the differences between two neighboring locations in the test object. An analysis of the recorded quantities indicates that the formation of interference fringes in an infinite fringe setting or their displacement in a
5.1 FlowVisualization
715
finite fringe configuration is dependent on a density gradient within the flow field (Merzkirch, 1974): Ke
Ii
igp(x,ayy, z) dz - N2,
N = 0, 4-1 4-2 . . . .
(5.1.16)
This equation is valid for a beam separation e in y-direction. For a plane flow this can be simplified to Op KeL-~
= N2,
N -- 0, 4-1, + 2 , . . .
(5.1.17)
In this configuration, the fringes that become visible signify lines of constant density gradient (infinite fringe setting). The similarity between shearing interferometry and a schlieren technique is obvious. Both methods measure and yield the density gradient, albeit by detecting different properties of the light beams that have been influenced by it. Thus, a density gradient is represented by both a deflection of the beam as well as a phase shift between two neighboring points. The close relation of shearing interferometry to a schlieren technique is also seen in the experimental realization of the method: In the simplest version, one has only to replace the knife edge by a system of two polarizers and a Wollaston prism (Merzkirch, 1965). If the center of the prism is located exactly in the focal plane of the schlieren head, the interferometer is adjusted for infinite fringe distance. Axial deviations from this position (i.e., in the direction of the light beam propagation) generate a system of background interference fringes (finite fringe setting) whose order can be changed by moving the prism normally to the direction of the light beams. In such a configuration, a Wollaston prism is the optical element responsible for beam separation. Other optical components may yield the same effect, but are less frequently used. A Wollaston prism consists of two prisms made of birefringent materials, such as calcite or quartz. A birefringent material exhibits an anisotropy in its optical characteristics as it displays two different indices of refraction, depending on the polarization direction of the light that passes through them. Consequently, an incident ray, which generally exhibits a random polarization direction, is refracted and separated into two differently polarized components. In the case of the Wollaston prism, the prism components are arranged in such a way that the rays leaving the prism are polarized perpendicular to each other (Fig. 5.1.11(a)) and diverge at an angle e. On the other hand, the process may be reversed inasmuch as two rays entering the prism at exactly this angle ~ are recombined by such a prism. Each of the rays is split into two perpendicularly polarized components as a result of the birefringence. One component of each pair exits the prism on an identical path, but as they are differently polarized, they can only interfere with each
716
H.
(a)
Kleine
polarizer
(b) I
I
|
polarization I directions
\
FIGURE 5.1.11 Ray propagation through a Wollaston prism. (a) Ray separation; (b) ray recombination.
other once the subsequent polarizer has removed this polarization difference (Fig. 5.1.11(b)). In order to have interfering ray components of equal intensity, both incident rays have to be polarized in such a way that their polarization exhibits equal parts in the two principal axes of the prismnthus, they will be split evenly. This is achieved by placing another polarizer ahead of the prism. From the foregoing it follows that the principal axes of both polarizers have to be at an angle of 45 ~ to the polarization axes of the prism. The polarizers can be arranged with their principal axes parallel or perpendicular to each other; in an experiment the latter setting is easier to detect as it corresponds to the scenario in which the polarizers block the light completely. In the described configuration, the distance e between the two beams inside the test section can be determined as e =f2 ~
(5.1.18)
where f2 is the focal length of the schlieren head. Following Eq. (5.1.16), this beam separation e is the determining factor for the sensitivity of the system. According to Eq. (5.1.18), the focal length f2 appears, similarly to the results for a schlieren setup, as the essential optical parameter, if the divergence angle e of the prism is fixed. When the system is run with a white light source, the outlined simple realization of the shearing interferometer delivers a comparatively dark image of the test section with low fringe contrast. This is caused by the fact that the light source has to be point-like in order to fulfil the condition of spatial coherence, necessitating the use of a pinhole and thus blocking a relevant portion of the emitted light. However, with a second, identical Wollaston prism placed in the focal plane of the first spherical mirror, the coherence properties
5.1
717
Flow Visualization
are greatly improved, which allows one to work with an extended light source. Consequently, the brightness of the image is much enhanced. Such a system, which includes placing the first polarizer in front of the first Wollaston prism, represents the usual setup for a polychromatic light source (see Fig. 5.1.7). For a low beam separation and an infinite fringe setting, shearing interferograms are very similar to corresponding schlieren pictures. Analogous to the schlieren techniques described in Sections 5.1.3 and 5.1.4, shearing interferometry can only detect changes (i.e., density gradients) in a certain preferred direction, which corresponds here to the direction of beam separation. If white light is used, the density gradients are represented by fringes of different color, depending on magnitude and sign of the gradient. The correlation between color and gradient magnitude, however, is fixed and cannot be influenced (Oertel and Oertel, 1989). There may exist an ambiguity with respect to the sign of a gradient because of the symmetrical occurrence of fringes around the zeroth-order interference fringe (Fig. 5.1.12). For a largely unambiguous correlation, one has to choose a fringe of higher order as
-400
yellow range of calibration lens for 6 = I' (center at approx. 665 n m )
- -~te_ i
i
I
--2{}0
I s
f
grey
\ \ \
-
\
black
\
k k
grey [nm]
k
\ \
200
-white
\ \
yellow
\ \ \
400 B
orange red
Zviolet blue m
800
green yellow orange
l i f l i 1 / / / / / /
FIGURE 5.1.12
Phase shifts and interference fringe colors.
718
H. Kleine
reference (and thus as background). Calibration of the system may be conducted in the same way as for a color schlieren apparatus. A deflection angle ~, coded with a color or a variation of image brightness by a schlieren system, corresponds to a color-coded phase shift A~. The measurement range of a shearing interferometer is determined by the divergence angle e if the focal length f2 of the schlieren head is given. In all presented examples, it is about 40% higher than the range provided by the color schlieren technique described in Section 5.1.4. In these examples, the calibration can only cover a range of (Op/Oy) - +100 k g / m 4 (corresponding to a phase shift of 4-870 nm), but it is also possible to represent stronger gradients in a corresponding flow picture through fringes that represent higher phase shifts. The maximum values that would still be possible to detect amount to a p p r o x i m a t e l y (0fl/~Y)max = 4-140 kg/m 4. Figure 5.1.12 also indicates the chosen measurement range for the interferogram shown in Fig. 5.1.13(a). For a larger beam separation (and thus a higher sensitivity) the fringe pattern on a calibration picture is more densely packed. The measurement range is reduced as the contrast diminishes with increasing fringe order. For phase shifts > 8 0 0 n m the interference colors become increasingly faded
FIGURE 5.1.13 Visualization of the test problem (detail) with shearing interferometry (from Kleine, 1994). (a) Low beam separation (divergence angle e - 1'); (b) high beam separation (divergence angle e = 5'). (See Color Plate 3).
5.1 FlowVisualization
719
mixtures of basic colors, which also begin to repeat themselves (Oertel and Oertel, 1989). For a divergence angle of 51 (0.00145 rad), for instance, the range of the forementioned setup decreases to • 60 kg/m 4. For a finite fringe setting, the use of white light (and thus color) has only a few advantages over a monochrome method. In contrast to a reference beam interferometer, a shearing interferometer does not yield a permanent displacement of fringes so that the fringe returns to its original position if the density gradient is reduced to zero. A significant difference between shearing interferometry and a schlieren system becomes obvious in the representation of solid body contours and discontinuities. Unless they are parallel to the direction of beam separation, these boundary surfaces appear as double images (i.e., with a half-shade border of width e) on a shearing interferometry picture. The actual position of the contour or discontinuity is in the center of this border. The interpretation of a flow picture obtained with this technique is rendered difficult by this effect; for large beam separations, the images of flow fields with discontinuities become very intricate and hardly suitable for a quantitative evaluation (Fig. 5.1.13(b)), which, however, is possible for small beam separations (Kramer 1965). For flow fields that are discontinuity-free, this technique yields readily a graphic representation of the encountered gradients (as seen in Fig. 5.1.13(b), for the stagnation flow left of the square cylinder).
5.1.8 HOLOGRAPHIC
INTERFEROMETRY
In the methods contained under the term "holographic interferometry" two (or more) light waves are interferometrically compared, of which at least one is generated from a holographic reconstruction. This latter aspect represents the main difference between these techniques and other interferometric methods. It is therefore useful to quickly recapitulate the characteristics of holography. Normal photography generates, with the help of optical components, an image of an object. In contrast to this, holography is capable of recording and storing the actual light waves emanating from this object, thus maintaining all the information with respect to amplitude and phase of the wave. After recording these waves and after adequate processing of the recording material, the stored waves can be "released" or reconstructed at any arbitrary later point in time. In this reconstruction, a copy is created, which has essentially all the properties of the original waves. Photographic film or other optical detectors, including the human eye, can only register the intensity of the radiation generated by light waves, but all information concerning the phase is lost. If this is to be prevented, the information about the phase distribution is to be converted into an equivalent
720
H. Kleine
intensity distribution. This conversion is achieved by letting the light wave emanating from the object (henceforth called "object wave") interfere with a coherent reference wave. The interference generates a pattern whose intensity distribution is a function of both wave amplitude and phase of each wave. Thus the essential information about the object wave is stored in this interference pattern. If one illuminates the photographic plate (the so-called hologram) after exposing and processing it with the reference wave again (which in this part of the process is called "reconstruction wave"), the recorded interference pattern acts like a diffraction grating (Fig. 5.1.14(b)). In addition to a modulated transmitted portion of the reconstruction wave, two diffracted waves are generated, one of which is essentially an exact copy of the original object wave. When viewing an illuminated hologram, a virtual image of the object at its initial position is formed, that is, behind the hologram. The second diffracted wave is the conjugated object wave, which creates a real image of the object at the same distance in front of the hologram. Corresponding points of both images have the same distance to the hologram plate. As the images are created on different sides of the hologram, the real image appears in a pseudoscopic mode, that is, with inverted depth coordinate. This means that the observer perceives the image of an object point of the rear end to be in front of the image of a point of the object's front portion. Because of the difficulties thus encountered during the observation and evaluation of the real image, it is mostly the virtual image that is used for any further analysis. Although virtual and real image are located on different sides of the hologram and thus spatially separated, they can only be detected individually, if reference and object wave meet at a sufficiently large angle | as shown in Fig. 5.1.14(a). Such a system, realized at first by Leith and Upatnieks (1964) is called an off-axis configuration, while the principle of holography was initially investigated and experi-
viewing angle for virtual image
(b)
(a)
photographic plate reference
wave:
,~ ~ _ ~
hologram
~, ~ reconstructiOnwave ~ , ~ ~
fiii~i:i;ii;il;i
objec
virtual .... ~ g image
~
halo transmittedwave.
i ~;~i!iii real image
..... ~ viewing angle for real image FIGURE 5.1.14 Holography.(a) Hologram recording; (b) hologram reconstruction.
5.1
721
F l o w Visualization
mentally verified in the so-called in-line arrangement by Gabor (1948). In this system, object and reference wave coincide so that the generated images can only be perceived simultaneously. Although all holograms share certain characteristics, there exists a multitude of different types and categories, which can be distinguished according to the properties of the object and the reference wave, the used optical components, or the way the hologram is processed (Ostrovsky et al., 1980). All holograms presented here are phase transmission holograms, where object and reference wave impinge on the photographic film from the same side under an angle | with respect to each other. The interference fringes, which attain a mostly normal position with respect to the film surface, are separated by a distance b given by the so-called Bragg condition (Bragg, 1913) for a known wavelength 2, b=
2 2 sin(|
(5.1.19)
After processing, this interference fringe pattern can be converted into a phase grid by means of a bleaching process, thus modulating not the amplitude but the phase of the reconstruction wave. The main purpose of the conversion into a phase hologram is to increase the diffraction efficiency for the reconstruction wave at the grid, expressed through the diffraction efficiency coefficient (Vest, 1979) Iobj ~diff =
(5.1.20)
ir w
where Iobj is the average light-wave intensity emanating from the object, and Irw is the light-wave intensity of the reconstruction wave. For a hologram that is generated without any further objects between the object and the holographic plate as sketched in Fig. 5.1.14(a), the so-called speckle pattern presents a limit for the attainable resolution. This effect is always observed when an object is illuminated with coherent light; it represents the random interference of light waves diffusely reflected and scattered from a surface, which for optical standards is to be considered as rough. As a consequence of this effect, the surface of the object image in the hologram appears grainy. The typical diameter dsp of a speckle can be estimated by considering the statistical intensity distribution of a speckle pattern (Vest, 1979) dsp ~ 1.22 2g D
(5.1.21)
where 2 is the wavelength of the light, g the distance between object and recording plane (hologram), and D the object size.
722
H. Kleine
This relation is formally analogous to the Rayleigh criterion, which describes the resolution capability of an optical component (e.g., Hecht and Zajak, 1974). In order to keep the losses in resolution caused by these speckles to a minimum, the distance g between object and hologram should be as small as possible (see Fig. 5.1.14(a)). Several other methods to reduce the speckleeffect have been described in the literature, but they are in general characterized by limitations in their application and success (e.g., Hariharan, 1984). For so-called imaging holograms, which will be described later on, the restrictions for the optical setup caused by the speckle effect are not relevant. If the reconstruction wave is not identical to the reference wave used for generating the hologram (i.e., if there are differences with respect to the wavelength or the shape of the wavefronts), aberrations and distortions will be present in the reconstructed hologram. For the ideal case of Gaussian optics and a surface hologram (where the thickness of the emulsion is negligibly small compared to the wavelength), these distortions can be theoretically calculated as a function of the wavelength ratio/~ = 2rec/2ref and the coordinates of corresponding points of reference and reconstruction wave. These aberrations only exist if there are differences between the imaging and the reconstructing optical system, but in addition further aberrations inherent to each hologram occur. These aberrations can be described and classified in similar fashion as the aberrations of a lensmspherical aberration, coma, astigmatism, Petzval field curvature, and distortion (Hariharan, 1984)--but they are only of higher order and thus of mostly negligible consequence for the applications described in this chapter. However, if holograms are to be quantitatively evaluated more intensively than indicated here, these aberrations are to be considered. Even though the basic assumptions made for an ideal surface hologram are not fulfilled in a real holographic process (in fact, the recordings are volume holograms, whose mathematical treatment is significantly more complicated) and although the real optical system will show some deviations from Gaussian optics, the derived relations for this ideal case are also generally applicable to a real hologram. Further details about principles and applications of holographic recording are explained in greater detail in the corresponding literature (among others, Smith, 1975; Caulfield, 1979; Hariharan, 1984). In holographic interferometry the principles of holography are applied to record an object wave at an instant to, where the state of the object is generally known. The reconstruction of this wave is then superimposed on another object wave emanated at a later instant tl, when the state of the object is generally unknown. Both superimposed waves interfere, if the necessary conditions are met, with the interference leading to a (macroscopic) fringe pattern that is caused by the changes the object has undergone in the time interval A t - t 1 - t 0. There will be no fringe pattern if the object does not
5.1 FlowVisualization
723
change its state during At or if the changes within At exceed the measurement range of the system. The number and shape of these fringes allow one to draw conclusions concerning the changes and possibly their causes. In conventional interferometers, the light waves to be compared are separated spatially so that they have traversed different paths at the same time when they are recombined again. The interference fringes register the phase shift, that is, the difference in optical path length A~, which is a result of the separated (but generally geometrically equally long) optical paths" zX~(x, y) - K
[p(x, y, z) - 0r~f] dz - N,t,
N -- 0, + 1 , + 2 . . . .
(5.1.22)
Here, p is the density along the path of the light beam traversing the object, while Pref signifies the (constant) density along the path of the reference beam. In this equation, the Gladstone-Dale relation, Eq. (5.1.1), has already been incorporated. For a holographic interferometer, the light waves are separated with respect to time; the waves traverse approximately the same physical path, but at different instants. Again, the recorded quantity is a phase shift or optical path difference, which however in this case results from changes along one light path within the time interval A t - t 1 - t o : A~(x, y) -
[n(x, y, z, tl) - n(x, y, z, to)] dz
= K
= N2,
[p(x, y, z, q ) - O(x, y, z, t0)l dz
(5.1.23)
N -- 0,-4-1, 4-2 . . . .
From this it follows that in contrast to conventional interferometers no restrictions apply for a holographic interferometer with respect to the shape of the wavefronts to be compared. In holographic interferometry, two object waves are compared with each other as opposed to an object wave and another, usually object-independent reference wave, as in a conventional interferometer. Distortions that both waves encounter in equal fashion at both instants to and tl, do not appear in the final comparison as only changes within At are recorded. Thus the requirements for the quality of the optical components of the setup are drastically reduced without compromising the quality of the resulting flow picture. It is for instance possible to use test section windows made of a material that would not meet higher optical standards, for example, standard glass or highly light transmitting plastics such as Plexiglas. This also enables one to investigate larger objects, for which a high-quality optical system, mandatory for a conventional interferometer, would not be realizable or affordable.
724
H. Kleine
Storing both the amplitude and the phase of the object wave also enables one to view the object from different angles. The range of viewing angles provided by a hologram is essentially governed by its size and its position relative to the object, but also by the object itself and by its "optical accessibility" to illuminate it from different angles. The case of visualizing the flow of a transparent medium yields a couple of special aspects: The holographic interferogram creates a three-dimensional (3D) image of the test section, in which a fringe system becomes visible that may exhibit a certain spatial distribution. The interpretation and localization of this fringe system is, however, only easily feasible for uncomplicated flow patterns. It should be emphasized that one does not get a 3D image of the flow (the flow itself remains invisible for a transparent medium) but a virtually infinite number of two-dimensional (2D) projected, line-of-sight views. The recorded light waves have traversed the flow field and carry the information and phase-shift that has been accumulated and integrated along their optical path. This is in contrast to the holographic recording of an opaque object, where waves that were reflected from an object are stored with the phase information of the corresponding object point. On the other hand, elements of a flow that generate a significant (overall) change of the refractive index and thus the phase along the line-of-sight (AgO > 22) appear indeed as if they were 3D. A shock wave, for instance, or any other sufficiently large density discontinuity, appears as a plane when viewed from the side (i.e., when looking through the test section from an oblique angle). This plane is composed of a succession of interference fringes (see Fig. 5.1.15(a)). An interference fringe is generated when, under the current corresponding viewing angle ~, the elongation of the optical path attains a multiple of the wavelength 2. If the shock wave is, in a simplified description, considered as a jump in density and refractive index, the phase shift changes linearly when the discontinuity is viewed under an angle ~. For the geometrical conditions shown in Fig. 5.1.15(b), one obtains: / . f_ Aq~ - l [n(~, y, f:, tl) - n(~, y, f:, to)] d~ (5.1.24) J0 where coordinates marked by a tilde are defined as ~--
J
J
cos~,
for
j--x,z,L
For n ~ n(y) and the simplified step-function distribution of the refractive index schematically indicated in Fig. 5.1.15(b), Eq. (5.1.24) can be expressed as
AgO -
n'~ + L(ntl -
nto )
where
n' -
-
ntl -
nt~
sin ~ cos
(5.1.25)
5.1
725
Flow Visualization
FIGURE 5.1.15 Viewing a shock wave under an angle ~. (a) Reconstructed holographic interferogram showing a shock wave passing over a cone-shaped obstacle (M s = 2.92 in N2; initial pressure in test section Pl = 100kPa; cone angle 50 ~) (from Kleine, 1994). (b) Localization of interference fringes: coordinate definition.
Thus, the condition to be met for the generation of interference fringes is n'~fr + L ( n t l - nto ) - N2,
N -- 0, 4-1,-t-2 . . . .
(5.1.26)
which results in a sequence of equidistant, parallel fringes at locations given by ~fr =
N2
- L(ntl glr
-
nto )
(5.1.27)
Counting the number of fringes yields the magnitude of the discontinuity with sufficient accuracy, if the "flow background" is largely constant so that it does not contribute to the optical path change. Moving from one interference fringe to the next corresponds to a density increment determined by the apparatus, see Eq. (5.1.29), if no discontinuity is located in between. In Fig. 5.1.15(a) the undisturbed shock wave is composed of six fringes, which corresponds to a density jump of 0.34 k g / m 3, while the theoretical shock relations predict a change of P2 - Px = 0.32 k g / m 3. For discontinuity systems that are composed of several elements, one can in some cases determine the different density jumps across each element of the system. In a Mach reflection pattern, for instance, one can obtain the different density jumps across the incident shock and the Mach stem for a shock wave propagating over a wedge-shaped obstacle. Other flow elements, for which the change of the refractive index along the line-of-sight is not sufficiently large (A~ < 2), are not perceived as 3D. This is, for instance, observed for a vortex, which when seen from a frontal view (~ = 0 ~ appears as a sequence of concentric tings. The change of refractive index is distributed over a larger area, so that along the line of sight there is at
726
H. Kleine
most a phase shift of 2. Thus, a vortex does not appear as a tube, as one might have expected, but the circular structures become ellipses when changing the viewing angle. The corresponding fringes are generated at different locations within the test section, so that the ellipses appear to be inclined, but overall the observer does not get a marked 3D impression (see Fig. 5.1.16). A single holographic interferogram contains the information of a large number of conventional interferograms, each of which would yield the corresponding projected view of the test section seen from a fixed angle ~. It is in principle possible to determine the 3D density field by analyzing a number of these projections (optical tomography), but this generally requires a huge computing capacity (Sweeney and Vest, 1973; Herman, 1980). Furthermore, the range of viewing angles covered by a single holographic interferogram is insufficient to reconstruct a general, fully 3D density field. For flow fields with axial symmetry the evaluation is greatly simplified; for a symmetry axis that is appropriately located with respect to the viewing direction, often a single projected view can be sufficient to determine the whole flow field (Schardin, 1942; Pearce, 1958). In order to evaluate a holographic recording as well as to present the results without having to set up a reconstruction apparatus, it becomes necessary to photograph the virtual image. For small distances between hologram and camera, the flow field can be recorded under viewing angles that give a 3D perception of the object (see Fig. 5.1.19(a)). However, the photograph generally tends to do far less justice to this 3D effect than the human eye itself, which is why photographic representations of reconstructed holograms
FIGURE 5.1.16 Reconstructedholographic interferogram showing the vortex generated during the diffraction of a shock wave at a sharp 90~ corner ( M 5 -- 1.58 in N2); different views of a s i n g l e interferogram (from Kleine, 1994).
5.1 FlowVisualization
727
appear to be considerably less impressive than what is experienced in a direct view of the virtual image. If one uses a camera with a long focal length and views the hologram from a correspondingly larger distance, the parallax is suppressed, which renders the recording very similar to a conventional interferogram. For plane flows such visualizations are sufficient for an evaluation and an interpretation. However, the light intensity of the virtual image is generally low, which limits the amount of possible photographic magnification of flow details. For such cases, so-called imaging holograms (which will be described later on) are much more appropriate. The quality of a photographic reproduction of a reconstructed hologram is also strongly influenced by the already introduced speckle effect. Similar to what was observed for the recording of a hologram, Eq. (5.1.21), the size of a speckle pattern in the reconstruction process, dsprec, can be estimated to be dsprec "" 1.22 2r~cf D
(5.1.28)
if the hologram is viewed by means of a lens with focal length f and diameter D (Vest, 1979). The ratio f/D represents the f-number of the viewing optical system. As Eq. (5.1.28) indicates, the price for a high degree of depth of field is an enlargement of the speckle structures. Imaging holograms, as indicated earlier on, are not affected by this effect. The type of holograms described so far is very suitable for a qualitative/ phenomenological description of a flow field, for checking whether a flow is mostly plane or not, and for comparatively simple quantitative evaluations like the determination of shock strength, but also for estimating the errors that are introduced through a projected 2D view of a mostly 3D flow field. In all applications where the possibility to view a flow field under various angles would yield only an insignificant amount of new information, the socalled imaging hologram provides a more useful means of flow visualization. This type of hologram differs quite substantially from the previously described holographic recordings. Analogous to regular photography, the object wave itself is not recorded, but rather an image of the object, generated by an optical system. While in a "normal" hologram the information of one object point is spread over the whole recording area (for this reason, a virtual image of the entire object can be reconstructed with just a fragment of the holographic plate used for recording) there is now, as in photography, an unambiguous relation between object and image point. This means that each part of the hologram contains only information of one particular portion of the object, which eliminates the forementioned possibility of reconstructing the whole object from a part of the recording. If the light waves pass the phase object only in one direction (as with a parallel light beam), the information about the depth of the phase object is lost, which renders the viewing of the virtual image from
728
H. Kleine
different angles impossible--in other words, the 3D effect no longer exists. The resulting flow picture corresponds to a conventional interferogram (e.g., from a Mach-Zehnder interferometer) or the parallax-free reconstruction of a "normal" hologram. The main advantages of this interferogram type compared to the conventional versions are the drastically lower requirements concerning the optical quality of the apparatus components as well as a higher flexibility with respect to test section shapes. It is, for instance, possible to visualize flows in test sections that are only optically accessible from one side (see, for example, Takayama et al. 1985). The size of a flow field that can be visualized is only limited by the dimensions of the beam-expanding optics and the power of the light source. In an imaging holographic interferogram, the fringe pattern is localized on the holographic plate, which enables one to view and reconstruct it with white light. During reconstruction, enlargements of details can be made to an almost arbitrary e x t e n t , limited only by the resolution capability of the holographic film material and the power of the light source used for reconstruction. As mentioned before, the principle of holographic interferometry is to record the changes that a phase object undergoes in a time interval At = t 1 - to by comparing the light waves that were sent through the object at these two instants t 1 and t0. Theoretically, the time interval At may attain arbitrarily high values, if an ideal light source with respect to its coherence properties is assumed. The changes of the phase object must, however, remain sufficiently small so that the conditions for interference of the two light waves are not violated. Usually, the light wave emanated at time to is holographically recorded. In principle, one can proceed in exactly the same fashion to record the wave at instant t 1. Both recorded waves can then be jointly reconstructed at an arbitrary later point in time. It is, however, also possible to superimpose the waves of time t 1 > to directly, that is, without recording them holographically, on the reconstructed wave of time to (so-called holographic real-time interferometry). This method requires that the hologram of instant t o is placed exactly at its original position after processing it (with an accuracy of fractions of the wavelength) in order to generate an exact copy of the recorded wave. The superposition, that is, the interference, is then either observed visually or recorded photographically. The required positioning accuracy leads to severe problems concerning the physical stability of the setup, usually rendering it very complex and expensive. For the observation of highly transient phenomena there are additional stringent requirements concerning the temporal resolution of the recording apparatus, which often cannot be met. For these cases, the first option (holographic recording of both light waves) is more useful. This version of the technique is commonly referred to as holographic double pulse interferometry. The holograms are generated within a short time
5.1
Flow Visualization
729
interval (usually < i ms), mostly on the same holographic material, and are subsequently jointly reconstructed. The mechanical stability requirements for such a setup are rather low if a movement of the setup within this short time interval can be avoided. One has, however, to find a suitable light source that can generate light pulses of sufficient intensity with the ability to be initiated with adequate temporal accuracy. The forementioned interferometric comparison of the light waves at instants t 1 and to yields a system of interference fringes. In the case of viewing a phase object, these fringes correspond to lines of constant refractive index (for gas flows, they are identical to lines of constant density, the isopycnics), if the change of the phase object represents the only change within the whole system during At (this restriction is essential because interference fringes can also be generated by modifications of the optical components without changing the phase object). For this configuration, which corresponds to the "infinite fringe" setting in a conventional interferometer, these isopycnics yield a projection of the entire density field. In the case of a plane flow, the density increment between two lines can be determined from Eq. (5.1.23) as 2 Ap -- ~ (5.1.29) KL In order to evaluate the number of interference fringes obtainable in an interferogram, the dimensionless density increment Ap/p(to) is more suitable (p(to) is the initial density in the test section) because it can directly be compared to the maximum density ratio within the flow. In order to achieve a large number of fringes and thus a good resolution, this dimensionless increment should attain small values. Therefore, the initial density p(to) should be as high as possible, as the other factors in Eq. (5.1.29) are mostly fixed or limited in their range. Increasing L, for example, by traversing the test section several times, leads to decreasing the density increment and thus to a higher resolution, but this procedure is likely to create problems with respect to a quantitative evaluation, caused by the more complex route of the rays. Similar to a Mach-Zehnder interferometer, but in contrast to shearing interferometry and to schlieren setups, the optical system itself (lenses, spherical mirrors) is only used to generate the parallel light beam, but influences neither the sensitivity of the apparatus nor its measurement range. It is only possible to assign a numerical value to these interference fringes, if for at least one point (P) of the flow field the actual density pp is known (from an independent measurement or through calculations), and if no discontinuity separates this point and the fringes to which the values are to be assigned. Along the fringe next to this point (P), the density assumes the value flfr -- tiP -[" Aft
(5.1.30)
730
H. Kleine
where A~ < Ap, with Ap from Eq. (5.1.29). The value of At5 can be determined through a densitometer measurement of the intensity distribution across an interference fringe. The errors associated with this measurement and the generally not exactly known starting value pp determine the accuracy of the subsequent evaluation of the density field. Counting the fringes and using Eq. (5.1.29) yields the density at other locations in the flow field, provided that no discontinuity renders an unambiguous fringe counting impossible. There is, however, in every case, an ambiguity with respect to the sign of the density change, as an increase of n and p, respectively, is represented in exactly the same way as a decrease of identical magnitude. The sign of the density change has to be determined either from physically plausible considerations or from an additional flow picture that yields the density gradient, for example, a schlieren picture. As mentioned in the preceding, the described configuration corresponds to the so-called infinite fringe setting, where fringes appear only if the phase object changes. The ambiguity with respect to the sign of a density change can be avoided, if one of the light waves is modulated with an appropriate carrier frequency. This modulation leads to the generation of a fringe pattern (generally a system of parallel fringes), even for an unchanged test object. Subsequent changes of the test object then result in distortions of this fringe pattern, which can be measured and determined with respect to magnitude, sign, and direction. Such an additional fringe pattern is created if between the two exposures (i.e., within At = t 1 - t0) one of the optical components either in the object or in the reference beam is slightly changed. The simplest implementation of this idea can be achieved by moving a component of the reference beam (Takayama, 1983), while it could be shown that the best results are obtained by introducing a wedged plate immediately in front of the phase object (Ostrovsky et al., 1980). An example for an interferogram with finite fringe setting is given in Fig. 5.1.17(a). To obtain the modulation, a lens in the reference beam was moved slightly during At. The corresponding infinite fringe interferogram is shown in Fig. 5.1.17(b). Further details for this recording technique are discussed in the corresponding literature on this subject (Vest, 1979; Ostrovsky et al., 1980). The setups sketched in Fig. 5.1.18 represent typical arrangements for a holographic interferometer. A double pulse ruby laser is used as light source, which emits two pulses of equal intensity. Each pulse is separated into object and reference beam by means of a beamsplitter. The geometrical lengths of both object and reference beam should, if possible, be equal within the accuracy of measurement (i.e., within a few millimeters to centimeters). Adequate individual settings (pulse intensity, beamsplitter ratio) are to be chosen to obtain a high contrast of the generated interference fringes (Vest, 1979).
5.1
Flow Visualization
731
FIGURE 5.1.17 Interaction of a shock wave with a cylinder (M s = 1.33 in N2) (from Kleine, 1994). (a) Reconstruction of a finite fringe imaging hologram; (b) reconstruction of an infinite fringe imaging hologram, recorded in a separate test approximately 20 ps later than (a).
The high number of different directions of ray propagation, which allow one in a "normal" hologram to view the object from a range of angles and thus create the 3D effect of the virtual image, are obtained with the help of a diffusing screen placed immediately in front of the test section. The holographic plate is put right behind the test section so that most of the available recording surface can be used (Fig. 5.1.18(b)).
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H. Kleine
(a) P Bs SM E C Hol DS
9 9 9 9 9 9 9
plane m i r r o r beamsplitter spherical m i r r o r b i c o n c a v e lens collimating lens hologram diffusing screen
double
pulse
laser
continuous
alignment
laser
/
reference beam
PI
objec
tlb
P' ~ ' ~
earn
/p
. . . . .
test section
Hol
I
(b) double
pulse
continuous
reference
laser
alignment
laser
Bs
SM
DS I
t ]
1- - - j ~ [
test section
FIGURE 5.1.18 Setups for holographic interferometers to record: (a) Imaging holograms; and (b) diffusely illuminated transmission holograms.
During the experiment, the recording of the reference condition at time t o is typically performed shortly before the shock wave arrives in the test section; the second pulse is then initiated at an instant t 1 = to + At, which usually has to be set prior to firing. The increment At is chosen in a way that the desired
5.1 FlowVisualization
733
flow field has established itself in the test section. In the most common infinite fringe setting, both pulses are automatically triggered in sequence with an interval At usually of between 1 and 1000 las. The triggering signal can be generated, for example, by a pressure transducer located ahead of the test section. For finite fringe setting, two independent single pulses are fired with a time interval of several seconds, which would allow one to perform the necessary changes for the modulation of one of the beams. The holograms are reconstructed with the setups sketched in Fig. 5.1.19. The reconstruction light source may be either a laser or also, for imaging holograms, a white light source. The reconstruction beam follows a path according to the geometrical conditions in the recording setup (Fig. 5.1.18). For a quantitative evaluation of the holograms, the wavelength ratio ~/r --" '~rec/'~ref has to be considered. Regular black-and-white film can be used to r e c o r d ~ i n full or with enlargements of details~imaging holograms or the virtual image of a "normal" hologram. Figure 5.1.2(c) shows a reconstructed holographic interferogram of the test problem. The high amount of information on this flow picture renders the qualitative interpretation and phenomenological description of the process slightly more difficult than it was in the case for the corresponding schlieren pictures. While some elements of the flow are clearly visible and interpretable
(a)
hologram
ra~]
_
=80...135mm
E
C
continuous laser
spatial filter
came
(b)
hologram
E
continuous laser or spectral lamp
spatial filter (only with laser)
film
C: collimating lens E: biconcave lens
FIGURE 5.1.19 Setups for the reconstruction of: (a) diffusely illuminated transmission holograms; and (b) imaging holograms.
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H. Kleine
(e.g., the vortex systems), other phenomena are depicted less graphically: The expansion wave, for instance, which reflects from the upper boundary of the test section (this wave was generated when the leading shock wave diffracted at the rear surface of the square cylinder) only appears as a distortion of the fringe pattern, while it is more clearly visible on the schlieren pictures. Similarly, the interaction of the reflected wave in the lower channel with the vortex, that is, the penetration of the wave into the vortex core, is only hinted at. Since at no point of the flow field is the density known or the first-order fringe identifiable, the density field cannot be numerically determined without further information.
5.1.9 LIGHT SOURCES AND RECORDING MATERIALS In order to achieve instant "snapshots" of highly transient flow phenomena, suitable light sources have to be used that can deliver a sufficiently intense light pulse of extremely short duration (pulsewidth < i ps). For monochrome setups, pulsewidths of ~ 2 0 n s can be realized by using pulsed lasers or adequately designed spark sources (Miyashiro and Gr6nig, 1985; Miyashiro et al., 1992). The overall amount of light has to be sufficient to expose the recording material. This is, for the most part, not a problem for pulsed lasers, but in the case of spark sources the requirements of high intensity and short duration are generally contradictory, so that a compromise solution has to be found. In most cases, the required amount of light is obtained by enlarging the pulsewidth. This conflict becomes very obvious when it comes to designing an adequate light source for a color schlieren apparatus. More than 80% of the emitted light is lost at the color source mask and, therefore, a very intense pulse is required to expose ordinary film sufficiently. In addition, it is required to have a mostly uniform spectral distribution within the emitted light, as the spectral range defined by the filters in the source mask has to be completely covered. These conditions can only be met by a spark generated in a Xe- or Ar-environment, where Xe is the preferable choice due to its higher light emission efficiency. A discharge in Xe will usually reach the desired light energy levels and the uniform spectral distribution, but at the price of an increased pulse duration. In particular, one observes an afterglow caused by a very slow decay of the light emission. Adding other gases like N2 and/or H2 to the Xe filling can accelerate this decay without a major influence on the peak emission value. While each of these gases alone would produce the desired change in the temporal emission characteristics, it would also lead to a gas-specific shift in the spectral distribution so that it is usually necessary to add both components. Even when the forementioned "stopping gases" are added, the decay of the light emission can still last for several microseconds, which may lead to a "long
5.1 FlowVisualization
735
time exposure" for flows photographed with this source. Visualized by the light that is emitted during the afterglow, phenomena of a less transient nature (such as the quasisteady slipstream in a Mach reflection), which do not change significantly in the decay period, may appear in the flow picture even if they have not existed at the actual instant of photography. In Figs. 5.1.2(d)-(h), for instance, the lines that become visible ahead of the shock front, can be identified as slipstreams that were generated within 5 las after the instant of exposure. Highly transient objects such as the shock front tend to appear slightly blurred as a result of this "long" exposure, which begins to degrade the image quality for shock Mach numbers higher than ~ Ms = 2. It should be possible to reduce this effect by appropriate design changes in the light source circuit. Regular slide film can be used as recording material for color schlieren and shearing interferometry photos. The sensitivity depends on the actual optical layout and the established image size. Typically, ASA 200 to ASA 400 color reversal film can be used for color schlieren visualization, while the more light efficient sheafing interferometry can be conducted with ASA 50 or ASA 64 color reversal film, if the same light source is used (see, e.g., Kleine and GrOnig, 1993). For holographic interferometry, the coherence requirements as well as the demanded temporal characteristics make a pulsed laser light source the only viable option. Commonly used are pulsed ruby lasers (wavelength 2 - - 6 9 4 n m ) which can emit pulses of high energy (several Joules) and short duration (typically 20ns) within a time interval of characteristically up to I ms. Other sources like pulsed Ar-, Kr- or Nd :YAG-lasers have also been used with various degrees of success. High-quality holograms have to be taken on corresponding holographic film material, which is characterized by a very high resolution (and consequently a low sensitivity, which in tum requires a high light output). Regular, lowsensitivity, black-and-white film may also be used, but at the expense of image quality. Best results can be expected for holograms recorded on holographic glass plates, which, however, also represent the most expensive film material. A large variety of processing recipes exist, which can either be requested from the holographic film supplier or found in the corresponding literature.
5.1.10 TIME-RESOLVED AND ANIMATION
VISUALIZATION
In the standard version of the setups described in the previous sections, one single photo is obtained per trial. For some flow processes, it may be possible to reconstruct the entire event by a sequence of pictures, taken in several trials with an adequately delayed instant of exposure. In all flows, however, where a random element governs to some extent the process (e.g., the onset of instability or chemical reactions), this approach may not lead to a true
736
H. Kleine
representation of the observed phenomena. In these cases, a time-resolved visualization is preferable. All described visualization techniques can, in principle, be used in a time-resolved fashion, but the technical challenges are drastically increased. For this reason, it has to be weighed carefully if the increased amount of gained information indeed justifies the substantially higher degree of complexity that the apparatus will require. In this context it should be considered that, generally, the image quality of a single picture is recognizably higher than that in a frame of a time-resolved sequence. The temporal resolution is achieved either on the light emitting or on the light receiving side, that is, by combining either a stroboscopic light source and a continuously recording camera or a continuously emitting light source and a framing camera. Various systems have been constructed, some of which are commercially available, while others have been specially developed for particular applications and exist only in prototype versions. The biannual conference of High Speed Photography and Photonics is probably one of the best forums to obtain updated information on the latest developments on this sector. Historically, the first option (i.e., using a stroboscopic light source) has been preferred in the framework of shock wave research. The first apparatus, which is commonly referred to as a Cranz-Schardin system (Cranz and Schardin, 1929) is composed of an array of spark sources, which are triggered in sequence. In the usual setup, all spark sources are located in the focal plane of the first spherical mirror; however, the light emitted from each of these sources traverses the test section at a slightly different angle, which results in a certain degree of parallax between the individual beams. Each light source generates its own image of the test section on the recording plane (usually a large photographic plate). If the dimensions of the source array are small in comparison with the diameter of the spherical mirror, the influence of the parallax becomes negligible. Such a system can usually deliver up to 24 frames, with a temporal separation of one to several microseconds. In refined versions, each source can be triggered individually so that slow motion and time-lapse effects can be achieved. Another system, which avoids the parallax problem, was designed and utilized at an early stage in laser development (Oppenheim et al., 1966) and subsequently refined (Kleinschnitger, 1978). It consists of a ruby laser stroboscope that is capable of emitting a train of pulses of about 20 ns each, separated by a preset time interval of one to several microseconds. Combined with a rotating mirror camera, an apparatus of this kind can yield photo sequences of sufficiently high spatial and temporal resolution with an overall number of frames that may exceed several hundred. Realistic framing frequencies may go up to 500 kHz (one frame every 2/.ts), which is sufficient for most applications in shock wave research.
5.1 FlowVisualization
737
Using a system like the forementioned ruby laser stroboscope and the rotating mirror camera, shadowgraphs as well as monochrome schlieren and shearing interferometry pictures can be taken in a time-resolved fashion. The system becomes more complex as a result of introducing these new elements, but the overall apparatus remains unchanged. The previously mentioned Cranz-Schardin source is realistically only useful in time-resolved shadowgraphy and to a lesser amount in a monochrome schlieren setup. Unlike the laser stroboscope, the Cranz-Schardin system requires optical components of sufficient size in order to utilize the light of all its array elements. Theoretically, it is also possible for all methods that involve color to expand a regular single frame setup to a time-resolved technique. Because of the spectral requirements that prescribe a white light source, it appears more promising to perform the temporal separation by means of a framing camera on the receiving side. A sufficiently strong Xe-flash with a flash duration of several milliseconds can be used as the light source. In practice, however, only visualizations with low framing rates have been achieved. In holographic interferometry, the combination of a high-speed camera and a real-time holographic interferometer would offer the desired time-resolved visualization, but in general the achievable time resolution is far below 10 kHz, which may be too low for shock wave flows. Since the real-time interferometer requires very stringent tolerances with respect to positioning and movement of the hologram, the effort in designing a setup to meet these requirements is tremendous (Hintze, 1987). Other systems involve a sequence of electrooptical shutters and variable beamsplitters that yield a repeatedly split system of object and reference beams (Racca and Dewey, 1990; Ehrlich et al., 1993); however, the image quality usually suffers significantly as a consequence of adding these components. If a sequence of flow pictures has been obtained, either by applying one of the indicated time-resolved recording techniques or by a number of single-shot experiments (the latter assumes a very high degree of reproducibility of the experiment), the frames can be combined to yield an animated movie of the event. The result of such an animation is very useful and instructive for a qualitative understanding of the process, but for a more detailed evaluation, the researcher will most likely have to refer to individual frames again. As for time-resolved visualization in general, the necessary effort (which can become very significant) has to be weighed against the expected gain in information (Yang et al., 1996).
ACKNOWLEDGMENT The work that eventually led to the composition of this chapter was performed at the Stot~wellenlabor, RWTH Aachen, Germany, under the supervision of
738
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Prof. Dr. rer. nat. H. Gr6nig, whose guidance and advice are most gratefully acknowledged. The author is also indebted to all members of the Stogwellenlabor, who assisted him in various ways during his years at this institute, and to Dr. M. P. Butr6n Guillen and Mr. J. Nerenberg, who provided useful discussions and help in the preparation of this manuscript.
REFERENCES Barry, EW and Edelman, G.M. (1948). An improved schlieren apparatus. J. Aer. Sci. 15: 364-365. Bragg, WL. (1913). The diffraction of short electromagnetic waves by a crystal. Proc. Cambridge Phil. Soc. 17: 43-57. Caulfield, H.J. (ed.) (1979). Handbook of Optical Holography, New York: Academic Press. Cords, P.H. (1968). A high resolution, high sensitivity color schlieren method. SPIE J. 6: 85-88. Cranz, C. and Schardin, H. (1929). Kinematographie auf ruhendem Film und mit extrem hoher Bildfrequenz. Zeitsch. f. Physik 56: 147-183. Czerny, M. and Turner, A. (1930). Uber den Astigmatismus bei Spiegelspektrometern. Zeitsch. f. Physik 61: 792-797. Ehrlich, M.J., Steckenrider, J.S., and Wagner, J.W (1993). New system for high-speed time-resolved holography of transient events. SPIE 1801. Proc. 20th Intl. Congr. High Speed Photography & Photonics, Victoria~Canada 1992, J. Dewey and R.G. Racca, eds., SPIE, 372-379. Gabor, D. (1948). A new microscope principle. Nature 161: 777-778. Hariharan, P. (1984). Optical Holography, Cambridge: Cambridge University Press. Hecht, E. and Zajak, A. (1974). Optics, Reading, MA: Addison-Wesley. Herman, G.T. (1980). Image Reconstruction from Projections, New York: Academic Press. Hintze, W (1987). Echtzeitholographische Untersuchungen einer oszillierenden Str6mung yon feuchter Luft in einer Lavald~ise. Dissertation, RWTH Aachen, Institut fur Allgemeine Mechanik, Germany. Holder, D.W. and North, R.J. (1952a). Color in the wind tunnel. The Aeroplane 82: 16-19. Holder, D.W and North, R.J. (1952b). A schlieren apparatus giving an image in color. Nature 169: 466. Holder, D.W and North, R.J. (1956). Optical methods for examining the flow in high-speed wind tunnels, Part I: Schlieren methods. AGARDograph AGARD-AG-23. Howes, WL. (1984). Rainbow schlieren and its applications. Appl. Optics 23: 2449-2460. Kleine, H. (1994). Verbesserung optischer Methoden fur die Gasdynamik. Dissertation, RWTH Aachen, Stogwellenlabor, Germany. Kleine, H. and Gr6nig, H. (1991). Color schlieren methods in shock wave research. Shock Waves 1: 51-63. Kleine, H. and Gr6nig, H. (1993). Visualization of transient flow phenomena by means of color schlieren and shearing interferometry. SPIE 1801. Proc. 20th Intl. Congr. High Speed Photography & Photonics, Victoria~Canada 1992, J. Dewey and R.G. Racca, eds., SPIE, 400-409. Kleine, H., Ritzerfeld, E., and GrOnig, H. (1995). Shock wave diffraction--new aspects of an old problem, in Proc. 19th Intl. Symp. on Shock Waves, Vol. IV, Marseille/France, R. Brun and L.Z. Dumitrescu, eds., Berlin: Springer, pp. 117-122. Kleinschnitger, K. (1978). Entwicklung einer stroboskopischen Laser-Lichtquelle im Frequenz-bereich von I kHz bis 1MHz. EMI Freiburg Bericht E2/78. Kramer, C. (1965). Die Differentialinterferometrie als Mel~verfahren der gasdynamischen Forschung. Abhandlungen aus dem Aerodynamischen Institut Aachen, Heft 18, pp. 37-43.
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Leith, E.N. and Upatnieks, J. (1964). Wavefront reconstruction with diffused illumination and three-dimensional objects. J. Opt. Soc. America 54: 1295-1301. Liepmann, H.W and Roshko, A. (1957). Elements of Gasdynamics, New York: John Wiley and Sons. Merzkirch, W. (1965). A simple schlieren interferometer system. AIAA J. 3: 1974-1976. Merzkirch, W. (1974). Generalized analysis of shearing interferometers as applied for gasdynamic studies. Appl. Opt. 13: 409-413. Merzkirch, W. (1987). Flow Visualization, 2nd ed., New York: Academic Press. Meyer-Arendt, J.R. and Appelt, H. (1978). Microscopy using a new type of color schlieren technique. Microscopia Acta 80:111-114. Miyashiro, S. and GrOnig, H. (1985). Low-jitter reliable nanosecond source for optical shortduration measurements. Exp. Fluids 3: 71-75. Miyashiro, S., Kleine, H., and Gronig, H. (1992). Novel nanosecond spark source for optical measurements in shock wave research, in: Proc. 18th Intl. Symp. on Shock Waves, Sendai/Japan, K. Takayama, ed., Berlin: Springer, pp. 973-978. Oertel, H., Sr. and Oertel, H., Jr. (1989). Optische StrOmungsmef~technik, Karlsruhe, G. Braun. Oppenheim, A.K., Urtiew, P.A., and Weinberg, EJ. (1966). On the use of laser light sources in schlieren-interferometer systems. Proc R. Soc. Lond. A 291: 279-290. Ostrovsky, Y.I., Butusov, M.M., and Ostrovskaya G.V (1980). Interferometry by Holography, Berlin: Springer. Pearce, W.J. (1958). Calculation of the radical distribution of photon emitters in symmetric sources. in Conference on Extremely High Temperatures, H. Fischer and L.C. Mansur, eds., New York: John Wiley and Sons. Prescott, R. and Gayhart E.L. (1951). A method of correction of astigmatism in Schlieren systems. J. Aer. Sci. 18: 69. Racca, R.G. and Dewey, J.M. (1990). High speed time-resolved holographic interferometer using solid-state shutters. Optics and Laser Technology 22: 199-204. Ray, S.E (ed.). (1997). High Speed Photography and Photonics, Oxford: The British Association for High Speed Photography, Focal. Rheinberg, J. (1896). On an addition to the methods of microscopical research, by a new way of optically producing color-contrast between an object and its background, or between definite parts of the object itself. J. Royal Microscop. Soc. 373-388. Schardin, H. (1942). Die Schlierenverfahren und ihre Anwendungen. Ergeb. Exakten Naturwiss. 20: 303-439. Scheitle, H. and Wagner, S. (1990). A four colors line grid schlieren method for quantitative flow measurement. Exp. Fluids 9: 333-336. Scheitle, H. and Wagner, S. (1992). A simple lens-like method of determining small light ray deflections. Exp. Fluids 12: 208-209. Settles, G.S. (1970). A direction-indicating color schlieren system. AIAA J. 8: 2282-2284. Settles, G.S. (1985). Color-coding schlieren techniques for the optical study of heat and fluid flow. Intl. J. Heat & Fluid Flow 6: 3-15. Smith, H.M. (1975). Principles of holography, 2nd ed., New York: John Wiley & Sons. Stong, C.L. (1971). Schlieren photography is used to study the flow of air around small objects. Scientific American 224: 118-124. Sweeney, D.W. and Vest, C.M. (1973). Reconstruction of three-dimensional refractive index fields from muhidirectional interferometric data. Appl. Opt. 12: 2649-2664. Takayama, K. (1983). Application of holographic interferometry to shock wave research, in Industrial Applications of Laser Technology, W.E Fagan, ed., Proc. SPIE 398: 174-181. Takayama, K., Kleine, H., and Gronig, H. (1987). An experimental investigation of the stability of converging cylindrical shock waves in air. Exp. Fluids 5: 315-322. Toepler, A. (1864). Beobachtungen nach einer neuen optischen Methode, Bonn: Max Cohen & Sohn.
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Van Dyke, M. (1982). An Album of Fluid Motion, Stanford: Parabolic. Vest, C.M. (1979). Holographic Interferometry, New York: John Wiley and Sons. Wolter, H. (1956). Schlieren-, Phasenkontrast- und Lichtschnittverfahren. Grundlagen der Optik, Handbuch der Physik, vol. 24, Berlin: Springer, pp. 555-645. Wuest, W. (1967). Sichtbarmachung yon Str6mungen IV. Das Schlierenverfahren. Archiv fur Techn. Messen V144-5: 81-86. Yang, J.M., Babinsky, H., Takayama, K., Saito, T., and Sislian, J.P. (1996). Holographic interferometric animation of shock diffraction and reflection using a highly repeatable shock tube. in Proc 20th Intl. Symp. on Shock Waves, Pasadena~USA, B. Sturtevant, J.E. Shepherd, and H. Hornung, eds., Singapore: World Scientific, pp. 141-146.
FIGURE 5.1.2 Visualization of the test problem (Fig. 5.1.1) (from Kleine, 1994). (d) classical color schlieren; (e) color schlieren, dissection technique, symmetrically placed vertical cutoff; (f) color schlieren, dissection technique; top: detail of (e); bottom: with shifted sensitivity range (asymmetrically placed cutoff); (g) direction-indicating color schlieren (cutoff device: pinhole); and (h) direction-indicating color schlieren (cutoff device: cylindrical plate).
FIGURE 5.1.2
(continued)
FIGURE 5.1.10 Visualization of the diffraction of a shock wave (M s = 1.32 in N2) at a vertical edge (from Kleine, 1994). (a) Direction-indicating color schlieren with diagram for correlation between color and gradient direction; (b) magnitude-indicating color schlieren (dissection technique).
FIGURE 5.1.13 Visualization of the test problem (detail) with shearing interferometry (from Kleine, 1994). (a) Low beam separation (divergence angle ~ - - I t ) 9 (b) high beam separation (divergence angle ~ -- 5t).
FIGURE 5.2.9
OH PLIF concentration imaging. For details see text.
CHAPTER
5
.2
Measurement Techniques and Diagnostics 5.2
Spectroscopic Diagnostics
DAVID E DAVIDSON AND RONALD K. HANSON Mechanical Engineering Department, Stanford University, Stanford, California, 94305, USA
5.2.1 Introduction 5.2.2 Absorption Theory and Line Shapes 5.2.3 Ultraviolet and Visible Laser Absorption Techniques 5.2.3.1 Visible and Near-Ultraviolet Transitions Available Without Frequency Doubling: CN, Sill, CH, NCO, C 2, Sill 2, NH 2, TiN 5.2.3.2 Ultraviolet Transitions Available with Frequency Doubling: OH, NH 5.2.3.3 Ultraviolet Transitions Available Using BBO Frequency Doubling: CH 3, NO, 02, HO2 5.2.3.4 Lamp Absorption: Working Without Lasers 5.2.4 Frequency Modulation Methods 5.2.4.1 Theory and Experiment 5.2.4.2 NH 2 and 1CH2 5.2.5 Infrared Laser Absorption and Emission Techniques 5.2.5.1 Room-Temperature Diodes 5.2.5.2 Pb Salt Diode Lasers 5.2.5.3 CO Discharge Lasers 5.2.5.4 Emission Methods 5.2.6 Atomic Resonance Absorption Spectroscopy 5.2.6.1 Experimental Method 5.2.6.2 Calibrations and Applications 5.2.6.3 Shock Tube Impurities 5.2.7 Planar Laser-Induced Fluorescence 5.2.7.1 Theory 5.2.7.2 Measurement Strategies 5.2.7.2.1 Species Imaging
Handbook of Shock Waves, Volume 1 Copyright :~ 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
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D. E Davidson and R. K. Hanson 5.2.7.2.2 Temperature Imaging
5.2.7.2.3 Velocity Imaging References
5.2.1
INTRODUCTION
There are a wide variety of spectroscopic methods available to the researcher who wishes to investigate the properties of a shock tube flow field and the distribution and concentration of individual species, both stable and transient, in that field. These methods take advantage of the known spectroscopic properties of electronic and ro-vibrational transitions of individual target species, and, in general, can provide quantitative and nonintrusive measurement results. Both absorption and emission methods have been used in shock tubes to monitor chemical composition and gasdynamic parameters. Although classical absorption and emission methods based on older technologies for light sources and detectors continue to be used, many aspects of spectroscopic investigation have been improved by the use of lasers, and these are the methods that most modern investigators now consider. Laser methods have permitted improvements in every area of concern to experimentalists: signal-to-noise ratio (SNR) and sensitivity, species and transition specificity, and temporal and spatial resolution. Advancements in the developments of lasers have not slowed down in recent years, and every new generation of lasers offers the potential to investigate new species or improve existing performance. In particular, progress continues to be made in three areas of lasers diagnostic methods: cw narrow-line width ring dye laser absorption, used in line-of-sight measurements; planar laser-induced fluorescence (PLIF) with pulsed lasers, used to spatially resolve complex flows; and semiconductor diode lasers, used in inexpensive, compact and rugged absorption diagnostic systems. The discussion in this chapter is divided into an introduction and six sections. In the first discussion section, Section 5.2.2, on absorption theory and line shapes, we introduce the basic equation describing absorption. We also describe the present state of experimental knowledge of high-temperature collision-broadening and -shift parameters that are needed in the quantitative application of this method. In Section 5.2.3, laser absorption techniques, we describe the application of ultraviolet and visible, narrow-line width, ring dye laser absorption to the detection of transient and stable species in shock tubes. The wide range of species investigated reflects the active and diverse research being performed in this area. In Section 5.2.4, frequency modulation (FM)
5.2 SpectroscopicDiagnostics
743
methods, we describe how the application of FM techniques to classical laser absorption methods can significantly improve diagnostic sensitivity. In Section 5.2.5, infrared laser absorption and emission techniques, we describe recent developments in the application of semiconductor diode lasers to shock tube measurements, as well as review classical infrared emission methods. In Section 5.2.6, atomic resonance absorption spectroscopy (ARAS), we introduce the use of low-pressure microwave discharges as sources of far ultraviolet radiation and their use in the detection of atomic species. A more extensive discussion of the ARAS method, including its use in combination with Excimer laser photolysis methods, is given by Michael in Sec. 16.3. In Section 5.2.7, planar laser-induced fluorescence, we touch on some theoretical considerations of this method, and describe some current PLIF imaging schemes that take advantage of recent developments in high-power pulse lasers and high-speed, solid-state cameras. Because of the extent of this field, many areas of spectroscopy are not discussed. These include, among others, Mie scattering methods, nonlinear methods such as coherent anti-Stokes Raman spectroscopy (CARS) and degenerate four-wave mixing (DFM), and a fuller discussion on emission techniques. However, for the areas we do discuss, we have attempted to include a broad selection of references. All the work referenced comes from the years since 1970, and 70% of it was published in the last 10 years. We could not hope to survey the literature thoroughly, and we apologize to those whose important work was inadvertently omitted in this process.
5.2.2 ABSORPTION SHAPES
THEORY AND LINE
The narrow and tunable line width of cw laser radiation permits a simple, theoretical description of the line-of-sight absorption for the uniform concentration profile generally found in shock tubes. The transmitted beam obeys the Beer-Lambert law for the fractional transmission of light at a single wavelength: I / I o -- e x p ( - k a P t o ~ l X i L )
(5.2.1)
where I and I0 are the transmitted and incident intensities (watts-cm-2), or power (watts), at wavelength 2, P is the total pressure (atm), Xi is the mole fraction of the absorbing species i, L is the shock tube diameter (cm), and ka is the absorption coefficient (atm -1 cm -1) which is a function of species i, wavelength 2, shocked gas temperature T, and pressure P and composition {M} of the composition of the bulk carrier gas. The absorption coefficient k~ is the product of the integrated line strength S (atm -1 cm -2) and the line shape function ~(v - %) (cm), the first of which is a function of temperature and the
744
D. E Davidson and R. K. Hanson
specific transition, and the second, which is a function of the pressure, temperature, composition and diagnostic wavelength. The integrated line strength is directly related to the oscillator strength or Einstein coefficients for the transition and is usually determined from other direct experiments, such as radiative life time measurements, or is derived from calibration experiments that use known concentrations of absorbers and known, or assumed, line shapes (Herzberg, 1950; Mitchell and Zemansky, 1971; Schatz and Ratner, 1993). The Voigt profile is normally used to describe the line shape function and has proved very successful. A Voigt profile represents the convolution of a Gaussian line shape function resulting from Doppler broadening with that of a Lorentzian line shape function resulting from collisional processes. Given that the Doppler-broadening component of the Voigt profile is well understood and simple to calculate, the collision broadening contribution is usually determined from experimentally determined values for the collision-widths and -shifts. These values are derived from experiments where the absorption feature is scanned over a wavelength range, which include its peak and wings. Surveys of the results of these measurements performed in shock tubes for a variety of absorption features are given in Tables 5.2.1 and 5.2.2. For the shocked gas conditions normally found in shock tubes, the collision-broadening and-shift coefficients have been found to vary linearly with pressure, and are, in general, sufficiently well described by a single inverse power law dependence in temperature with a coefficient usually falling between 0.5 and 1.0. (Breene, 1961) kv(width)
= Av o
P (To~T) n
ka(shift) = ka o e (To~T) m
(5.2.2) (5.2.3)
Here Av0 is the collision-broadened line width (full-width at half maximum (FWHM)) per unit pressure (cm -1 atm -1) and P is the pressure (atm). As evident from the data in the tables, there is a paucity of collision-shift measurement data. In actual practice this is not always a disadvantage, especially for work at pressures near 1 atm. Centering of the diagnostic laser frequency on the desired peak of the line shape can often be done simultaneous to the shock tube experiment in an accompanying atmospheric pressure flame. A variety of sources contributes to the overall uncertainty in the concentration determination from narrow-line width laser absorption experiments. These include path length uncertainties based on boundary layer thickness, oscillator strength uncertainties based on lifetime or calibration experiments, higher-order terms in the Honl-kondon factor of the integrated line strength, signal to noise contributions, shocked gas state variable uncertainty, Voigt profile uncertainties, and interference from other species. These sources all
o
r~,,J.
TABLE 5.2.1
Collision-Broadening and Collision-Shift Parameters for Visible and Ultraviolet Transitions Measured in Shock Tube Experiments
~d o
Species, Transition
2 (nm)
{M}
To (K)
OH (A2y~+-X21-I) Rl(5)
306.5
OH (A2Z+-X2FI) R1(7-11) R21(7-11) NH (A3IIi-XSZ -) (0,0) R branch CN (B2E+-X2E +) P1 + P2(4-14) ** -- {0.0560 exp(--0.201N") + 0.11} NO (A2Z-X2H) (0,0) Q2 q- R12(13) NO (A2G-X2H) (0,0) P21 + QI(26)
306.5 336.1 388
Ar N2 Ar Ar Ar
2000 2000 300 2000 298
0.032 0.042 0.140 0.0320 **
0.77 0.90 0.75 0.94 0.80
225.4 225.4 225.4 225.4 597.3
02 H20 Ar N2 Ar
295 295 295 295 2000
0.53 0.79 0.505 0.585 0.0175
0.66 0.79 0.65 0.75 0.5
NH2 (A2AI-X2B1) (090 *-- 000)Y~PQ1,N7
Avo (cm -1 atm -1)
n
A60 (cm -a atm -1)
m
Reference
-0.305
0.45
(Rea et al., 1987) (Rea et al., 1987) (Davidson et al., 1996) (Chang and Hanson, 1989) (Wooldridge et al., 1995b)
-0.16 -0.21 - 0.180 - 0.180
0.52 0.71 0.58 0.56
(DiRosa and Hanson, 1994) (DiRosa and Hanson, 1994) (Chang et al., 1992) (Chang et al., 1992) (Kohse-Hoinghaus et al., 1989)
--,1 k.a
--q
TABLE 5.2.2
Collision-Broadening and Collision-Shift Parameters for Infrared Transitions Measured in Shock Tube Experiments
Species, Transition
2 {M} (nm)
H20 (v 1 -F v3 and 2v 1)
1405 Ar
300
0.033
0.53
--0.023
1405 N2
296
0.083
0.59
--0.017
H20 296 CO2 296 CO2 300 N2 300 02 300 H2 300 Ar 300 Ar 300 HCN 1000 N2 300 Ar 300 1300 Ar * 300
0.40 0.093 0.212 0.117 0.070 0.093 0.050 0.114 0.676 0.102 0.065 0.027 0.141
0.84 0.43 0.70 0.76 1.12 0.87 0.61 0.632 1.19 0.62 0.57 0.90 0.82-0.60
1405 1405 H20 1383 1383 1383 1383 CO2 (00~ ,-- (00~ R(48, 50, 52) 4200 HCN v1 (10~176 P(10) 3000 3000 NO 1/2,0,R(21/2) and 3/2,0,R(21/2) 5228 5300 H20 Y2 123,10 ~ 112.9 5258 CO (1 ~---0) (2 *-- 1) (3 ~ 2) 4740 9 CH4/Air Flame Products
To (K)
Av0 n (cm -1 atm -1)
Ago m (cm -1 atm -1)
Reference
0.85 (Nagali et al., 2000) (Nagali et al., 1999) 1.0 (Nagali et al., 1997) (Nagali et al., 1999) (Nagali et al., 1997) (Nagali et al., 1997) (Langlois et al., 1994) (Langlois et al., 1994) (Langlois et al., 1994) (Langlois et al., 1994) (Wooldridge, 1995) (Chang and Hanson, 1985) (Chang and Hanson, 1985) (Falcone et al., 1983) (Falcone et al., 1983; Hanson et al., 1976) (Salimian and Hanson, 1983) (Varghese and Hanson, 1981)
5.2
747
Spectroscopic Diagnostics
conspire to limit the best overall uncertainty in a concentration measurement using narrow-line width laser absorption to about :k5 %.
5.2.3 ULTRAVIOLET AND VISIBLE LASER ABSORPTION TECHNIQUES A variety of species can be probed with cw laser light in the visible and the ultraviolet regions. These are summarized with references, in Table 5.2.3 for the wavelength range 216-614nm. This list includes species that can be investigated with either fixed-frequency or tunable-frequency lasers. Fixed
TABLE 5.2.3
Ultraviolet and Visible Laser Absorption Diagnostics
Wavelength Species Method (nm)
Reference
216.6
CH3
BBO Freq.-Doubling. Ring Dye Laser
225.3
NO
226.1
HO2
(Chang et al., 1991; Davidson et al., 1993a; Davidson et al., 1995a; Davidson et al., 1995b; Davidson et al., 1992b; Davidson et al., 1995c; Davidson et al., 1993b; Duet al., 1996) (Chang et al., 1992; Davidson et al., 1992a; DiRosa et al., 1991; DiRosa and Hanson 1994) (Moser, 1993)
227.4
02
257.3 303.4
03 NH
304.7
NCO
306.5
OH
BBO Freq.-Doubling Ring Dye Laser BBO Freq.-Doubling Ring Dye Laser (Davidson et al., 1992a) BBO Freq.-Doubling Ring Dye Laser Freq.-Doubled Ar+ Laser (Thoma and Hindelang, 1995) AD*A Freq.-Doubling (Szekely et al., 1984b) Ring Dye Laser AD*A Freq.-Doubling (Louge et al., 1984) Ring Dye Laser (Braun-Unkhoff et al., 1995; Davidson et al., 1988; AD*A Freq.-Doubling Davidson et al., 1989; Davidson et al., 1991a; Ring Dye Laser Davidson et al., 1996; Dean et al., 1991b; Hanson et al., 1988; Hanson et al., 1983; Markus 1995; Masten et al., 1990; Mertens et al., 1995; Petersen et al., 1996a,b; Petersen et al., 1999; Rea et al., 1987; Rea and Hanson 1983; Rea and Hanson 1988; Ryu et al., 1995; Schading and Roth 1997; Shin et al., 1989; Szekely et al., 1984a; Wooldridge et al., 1994a; Wooldridge et al., 1994b; Wooldridge et al., 1995a; Wooldridge et al., 1995c; Wooldridge et al., 1995d; Yu et al., 1995) (continued)
748
D. E Davidson and R. K. Hanson
TABLE 5.2.3
(continued)
Wavelength Species Method (nm)
Reference
336.1
NH
LiIO3 Freq.-Doubling Ring Dye Laser
388.4
CN
Ring Dye Laser
413.6
Sill
Ring Dye Laser
431.1
CH
Ring Dye Laser
440.5
NCO
Ring Dye Laser
467.4 472.7 516.6 579.3 590.7 597.4
C2 NO2 C2 Sill2 1CH2 NH2
Ring Dye Laser Argon Ion Laser Ring Dye Laser RingDye Laser FM Spectroscopy RingDye Laser
597.4
NH2
FM Spectroscopy
614.0
TiN
Ring Dye Laser
(Chang and Hanson, 1989; Davidson et al., 1990c; Klatt et al., 1995; Mertens et al., 1989; Mertens et al., 1991a; Mertens et al., 1992a; ROhrig et al., 1994; R6hrig and Wagner 1994) (Davidson et al., 1991b; Wooldridge et al., 1993; Wooldridge et al., 1995b; Wooldridge et al., 1995c; Wooldridge et al., 1996b) (Markus 1995; Markus and Roth, 1996; Mick et al., 1995) (Dean and Hanson, 1989; Dean and Hanson, 1992; Dean et al., 1990; Dean et al., 199lb; Louge and Hanson, 1986; Markus 1995; Markus and Roth, 1992; Markus and Roth, 1995; Markus et al., 1996; Markus et al., 1994; Markus et al., 1992; Rohrig et al., 1997b; Woiki et al., 1998) (Louge and Hanson, 1984a; Louge and Hanson, 1984c; Louge et al., 1984; Mertens et al., 1992b) (Kruse and Roth, 1995) (Bates et al., 1998; Rohrig et al., 1997a) (Kruse and Roth, 1995; Kruse and Roth, 1997) (Markus, 1995) (Deppe et al., 1998) (Davidson et al., 1990; Kohse-HOinghaus et al., 1989; Mertens et al., 1991b) (Deppe et al., 1998; Naumann et al., 1997; Votsmeier et al., 1999a; Votsmeier et al., 1999b; Votsmeier et al., 1999c) (Deppe et al., 1999) (Herzler et al., 1998)
frequency lasers, such as the Ar + gas discharge laser, can be used for species, such as NO2 and 03, which either have a multitude of transitions where the chance of an overlap between the fixed frequency laser and a strong species transition is great, or have very broad absorption spectra where the exact fixed frequency of the laser is not critical. Tunable ring dye laser systems offer many important experimental advantages over fixed-frequency lasers, a wide frequency tuning range with a single laser system, which permits access a wide variety of species transitions, and a narrow-line width for improved tuning on and off selected transitions. Both types of lasers offer excellent spatial locating and pointing ability, and the potential to achieve very large SNR.
5.2 SpectroscopicDiagnostics
749
A schematic of a typical ring-dye laser absorption system is given in Fig. 5.2.1. Radiation at the fundamental wavelength (e.g., 597 nm for a typical NH 2 transition or 613 nm for an OH transition) is generated in a ring-dye cavity, which is pumped by an Ar § laser (typically near 475 nm) or a solid state laser (532nm). Mode quality is monitored using a scanning interferometer, and wavelength is monitored using either a wavemeter or by centering the laser on an absorption feature in a static cell or flame. If required (as in the case of the OH transition at 306.5 nm), doubled radiation is generated either intracavity, or extracavity, by passing the fundamental radiation through a nonlinear crystal, such as LiIO 3, ADA*, or BBO. The output radiation is split into a reference beam (I0), which carries information about the amplitude variations, and a probe beam (I), which is sent through a train of optics to focus it through the test gas in the shock tube where it is absorbed by the target species. The reference beam and the transmitted probe are collected on photodetectors, and electronically, or digitally, divided to produce a common-mode rejected (I/I o) signal which is recorded and can be directly analyzed using the Beer-Lambert law. Calibration of a narrow-line width laser absorption diagnostic can be accomplished several ways. The simplest method is to generate known
FIGURE 5.2.1 Ring dye laser experimental schematic. The wavelengths indicated, 306 and 612 nm, are those used for OH absorption.
750
D. E Davidson and R. K. Hanson
quantities of the target species and measure the absorption in the desired temperature and pressure range. This can easily be accomplished if the target species is relatively stable at high temperatures such as NO. But it can also be accomplished by the rapid decomposition of known quantities of an appropriate precursor, such as in the case of the target species CH 3, which can be derived from the rapid high-temperature decomposition of azomethane or ethane. Known quantities of target species can also be generated from thermodynamic equilibrium, such as in the case of forming OH plateaus at long times in the high-temperature combustion of H2/O 2 mixtures. An alternative to direct calibration is to calculate the absorption coefficient from fundamental spectroscopic parameters. This can be done, as in the case of CH and NH, from independently derived values for the oscillator strength,
100000 .
10000CN "T
E
"T,
C2 1000-
S
E
i
H
~
NH
v
SiH
,4.,,.,
[: ._<2 (p o O to c~ I... o ~0 .o <[
100
CH3
OH
NO H
10
CO
~
.
NH 2
f 0.1
'
I
'
I
'
I
'
I
'
I
'
I
'
I
'
500 1000 1500 2000 2500 3000 3500 4000 Temperature
FIGURE 5.2.2
(K)
Selected absorption coefficients for ring dye laser diagnostics.
5.2
751
Spectroscopic Diagnostics
assumed or measured values for the collision broadening and shifts, and an understanding of the statistical mechanics of the species transition. We have summarized the performance of these laser systems in Figs. 5.2.2 and 5.2.3 where we show absorption coefficients for available wavelengths for selected species and minimum detectivities for typical operation of ring dye laser absorption systems in shock tubes. The absorption coefficients are based on Voigt line shapes calculated with i atm Ar as the primary collision partner. These values are for a particular transition, and though they may be the most commonly used, they may not necessarily be the strongest absorbing features of a particular band or species. The minimum detectivities are based on S / N - 1 for bandwidths of 1 MHz, pathlengths of 10 cm, with a minimum detectable absorption of 0.1% for wavelengths <300nm, and 0.3 % for
1000
,
i
,
w
,
w
,
t \ ~ l
,
I
,'
i
02 NH NO 2
100 Q.
E
o O
g
HO 2
lO
NCO NH 2 FM
@ E e--
OH .
Sill
~
=> o
~
0.1
NH
D
E
CN
E -~ 0.01
C2 1E-3
I
500
'
I
'
I
'
I
'
I
'
I
'
i
'
1000 1500 2000 2500 3000 3500 4000
Temperature (K) FIGURE 5.2.3
Selected minimum detectivities for ring dye laser diagnostics.
752
D. E Davidson and R. K. Hanson
wavelengths > 250nm. This increase in minimum detectable absorption for the shorter wavelengths reflects the current experimental difficulty in maintaining very low noise in BBO frequency-doubled systems in the far ultraviolet. The minimum detectable absorption for NH 2 using FM methods is 0.01%. The absorption coefficients can be divided into four groups. The largest values are found for the diatomics with transitions between like systems: CN (a Z - E transition) and C 2 ( H - II). These give minimum detectivities of < 10ppb. The hydrides, CH ( A - 1-I), NH ( H - E), Sill ( A - H), and OH ( E - H) which is considered to be the standard for comparison, form the group with the next largest absorption coefficients. They typically have minimum detectivities of 0.1-1 ppm. Triatomics and larger molecules have lower absorption coefficients, in part because of the larger statistical manifold in which the molecule can exist. Their minimum detectivities fall in the range of 1-100ppm. Finally NO and 0 2 both have small absorption coefficients, for 02 ( E - E) this is so because the lower state of the transition that can be accessed with BBO frequency doubling is v = 5 and for NO ( H - E) this occurs because it is not a diagonal system. Some details of 14 of these diagnostic systems are given here. The diagnostics are grouped into three categories: visible and near-ultraviolet wavelengths available without the need for frequency doubling, the ultraviolet transitions which require frequency-doubling systems, and the ultraviolet transitions accessible using BBO frequency doubling.
5 . 2 . 3 . 1 VISIBLE AND NEAR-ULTRAVIOLET TRANSITIONS AVAILABLE W I T H O U T FREQUENCY DOUBLING"
CN, Sill, CH, NCO,
C 2, S i l l 2, N H 2,
TiN In this wavelength region and others, the selection of diagnostic wavelengths is dependent on several factors. First, a strong absorption transition from the electronic ground state of the target species is preferred. Second, predictable temperature and pressure dependence is needed if a wide range of conditions is to be investigated. Between these two factors, one must often chose either a larger absorption coefficient, which occurs with the overlap of many lines at the band head and hence has a complicated temperature and pressure dependence, or a weaker single absorption line with a simple predictable line shape and temperature and pressure dependencies. Third, the diagnostic wavelength should be away from interfering absorption from other species found in the chemical kinetic system being investigated. This can sometimes
5.2 SpectroscopicDiagnostics
753
be dealt with by just moving to an adjoining line, or it may require moving to a completely different band or system. The experimentalist should be aware that hot bands, transitions that occur for vibrationally excited species, might also become a problem in shock tube test-gas environments. Fourth, reliable operation of the laser system should be possible at the diagnostic wavelength. Within the constraints of these four factors, a wide variety of transient species have been studied at visible and near-ultraviolet wavelengths and these are described in what follows. These include CN at 388 nm, Sill at 413 nm, CH at 431 rim, NCO at 440 nm, C 2 at 467 nm, Sill 2 at 579 nm, NH 2 at 597 rim, and TiN at 614 nm. For CN, NH 2 and Sill2, calibration is relatively direct; however, for the other species experimental calibration is difficult or not possible. There are no simple quantitative Sill, CH, NCO, TiN or C 2 sources; the absorption coefficient of these species must be calculated from the spectral parameters or fit to kinetic models, often with the associated uncertainties based on the oscillator strength and collision parameters or kinetic rate coefficient uncertainties. The CN B2~ + - X2~ + (0,0) transition is very strong, and the peak sensitivity of this diagnostic is similar to that found in the very sensitive HARAS diagnostic. The CN can be produced quantitatively from the decomposition of C2N 2. The CN diagnostic, in conjunction with NCO and NH 2, has played an important role in elucidating the fuel-nitrogen submechanism in hydrocarbon and propellant chemistry (Davidson et al., 1991b; Wooldridge et al., 1993; Wooldridge et al., 1995b; Wooldridge et al., 1995c; Wooldridge et al., 1996b). Another strong transition, C2, has been investigated by Roth and coworkers, as part of a study of C2H 2 decomposition, at 467.440nm in the C 2 d3I-I - a3I-I (1,0) transition using the overlapping rotational lines R1(37 ), R2(36 ), and R3(35 ) (Kruse and Roth, 1996). The Sill and Sill 2 absorption diagnostics have been used in conjunction with the Si-ARAS diagnostic to form a relatively complete Si diagnostic system. The A2A - X2FI (0,0) transition of Sill is at 413.5 nm. (Markus 1995; Markus and Roth 1996; Mick et al., 1996). The A1B1 - X~A1 (0,2,0)-(0,0,0) transition of Sill 2 is at 579.3 nm (Markus 1995). The Sill 2 can be produced quantitatively by pyrolysis of Si2H6. Dean and Hanson found the CH ARA- X2[-[ (0,0) Qld+2c(7) line pair is more suitable than transitions in the C-X system for CH absorption measurements because of the improved accuracy in the oscillator strength and line positions available for the A-X transition (Dean and Hanson, 1989; Dean and Hanson, 1992; Dean et al., 1990; Dean et al., 1991b; Louge and Hanson, 1986; Markus, 1995; Markus and Roth, 1992; Markus and Roth 1995; Markus et al., 1996; Markus et al., 1994; Markus et al., 1992; ROhrig et al., 1997b; Woiki et al., 1998).
754
D. E Davidson and R. K. Hanson
NCO was monitored by several investigators at 440.479nm, the A (00~ - X (0010) P2 4-PQ12 bandhead. Though it is not possible to directly calibrate this diagnostic, improvements in the understanding of the HNCO chemical kinetics mechanism has permitted the overall uncertainty to be reduced to approximately +25 % (Louge and Hanson, 1984a,b; Louge et al., 1984; Mertens et al., 1992b). A wide variety of possible lines are available for the detection of NH 2. Several workers have found the A2A1 - X2B1 (090-000) ~PQ1,N(7) doublet at 597.375nm suitable for quantitative analysis, the NH 2 can be produced quantitatively by the pyrolysis of ammonia NH3, hydrazine H 2 N - NH 2, or monomethylamine CH3NH2. The spectroscopy of the NH 2 molecule is complicated and an unequivocal determination of some transitions is difficult; an experimental determination of the absorption coefficient is preferred for this diagnostic (Davidson et al., 1990c; Kohse-HOinghaus et al., 1989; Mertens et al., 1991b). The sensitivity of this diagnostic has been improved by the use of frequency modulation techniques that are described in Section 5.2.4 (Deppe et al., 1998; Deppe et al., 1999; Votsmeier et al., 1998a,b,c). Recently, Herzler et al. (1998) have detected TiN qualitatively in a shock tube using the A2I-[- X2]~ (0,0) system of 15 overlapping rotational lines R12(18-24)+Ql(19-26) at 620.125nm, as well with 19 overlapping lines R21(5) 4- R2(30-37) + Q21(30-39) at 614.036 nm.
5.2.3.2
ULTRAVIOLET TRANSITIONS AVAILABLE WITH FREQUENCY DOUBLING: OH, N H Many important combustion species have strong absorption transitions at wavelengths shorter than 390 nm, the approximate lower limit of ring dye lasers. These wavelengths can be reached by using frequency doubling. In this method, laser power at the fundamental frequency is circulated, either intracavity or extracavity, through nonlinear crystals that have a small coherent output at twice the input frequency. A variety of crystals and tuning methods are available and these are further described in the references. Frequency doubling near 220nm using BBO is discussed separately in the next subsection. Perhaps the most important of these species is OH. It is a very clear and strong indicator of hydrocarbon oxidation, and in common combustion systems, its measurement does not suffer from strong interferences. The OH A2~ + - X3II (0,0) band near 306.5 nm is easily accessible using temperaturetuned ADA* frequency doubling. Developments in this method by Rea and others include rapid scanning, twoqine thermometry, and fundamental studies
5.2
Spectroscopic Diagnostics
755
on collisional line-shape parameters (Rea et al., 1987; Rea and Hanson 1988; Rea et al., 1984). The first rapid-scanned spectrally resolved absorption determination of temperature behind reflected shock waves was done using OH (Chang et al., 1987). The OH can be simply generated in known quantities from the thermodynamic equilibrium of combustion products in H2/O 2 ignition, or by modeling the time dependent concentration. The kinetic mechanism describing this system is well-established and very accurate at pressures near 1 atm. Several groups in chemical kinetic rate coefficient studies have used this diagnostic and it has also been particularly useful in product distribution studies where it has been used in combination with other diagnostics (Davidson et al., 1988, 1989, 1991a, 1996; Dean et al., 1991b; Hanson et al., 1988; Masten et al., 1990; Reaet al., 1987; Rea and Hanson, 1988; Wooldridge et al., 1994a,b; 1995a,c,d). The NH can be measured at the nearby wavelength 336.100nm and this is currently done using frequency doubling with LilO 3 crystals. The NH A31-Ii- X3•-(0, 0) band has both discrete lines, for example the Q2(9) line, and a strong overlapping structure, the Q1 bandhead, which have been used as diagnostic features. The NH is generally found in systems that contain nitrogen (in the fuel or as an additive), and often kinetically as a product of NH 2 reactions. Predicting NH concentrations in these complicated systems usually requires accurate NH 2 concentration predictions or measurements. These NH 2 measurements can be performed using laser absorption at 597.375 nm. NH can be produced quantitatively by the pyrolysis of HN 3 or the photolysis of HNCO (Chang and Hanson, 1989; Davidson et al., 1990c; Klatt et al., 1995; Mertens et al., 1989, 1991a, 1992a; R6hrig et al., 1994; Rohrig and Wagner, 1994).
5 . 2 . 3 . 3 ULTRAVIOLET TRANSITIONS AVAILABLE USING B B O FREQUENCY DOUBLING: C H 3 , NO, 02, HO2 An important frequency doubling technology is based on fl-BaB204 (BBO) crystals. Doubled output at wavelengths as short as 205 nm is possible with this crystal. One of the interesting and important applications of this technology, and one near its blue limit, is the generation of light to probe the CH 3 B(2A~)- X(2A~) transition at 216nm [called the ill(0-0) band by Herzberg (1950)]. This band is composed of two prominent and overlapping features, the P + Q and R band heads. Known quantities of methyl radicals can be produced from a variety of sources. At temperatures > 2000 K, the simplest source is the decomposition of ethane. At lower temperatures azomethane cleanly breaks into two methyl
756
D. E Davidson and R. K. Hanson
radicals with near 100 % efficiency, and methyl iodide has also been used with some success. Some organometallics are potential sources, but are very toxic and cannot be recommended. Broad weak interference features from C2H 2 and C2H4 exists at the CH 3 wavelength, but can usually be accounted for by kinetic modeling. Interference from the lines of the Schumann-Runge 02 band is possible and must be avoided in the hot gases of oxidative systems. This diagnostic has been used at present for several sets of experiments including studies of methane and ethane decomposition and methyl-methyl reaction rate coefficients, and oxidative systems (Chang et al., 1991; Davidson et al., 1992b, 1993a,b, 1995a,b,c; Duet al., 1996. Between 225 and 230nm, there are several other important species that have been accessed using narrow-line width laser absorption. Of these, NO so far has been the best characterized. The NO A-X (0,0) band is rich with multiple and single line configurations which are suitable for concentration or flow parameter measurement. For example, the overlapped P21Ql(26) lines at 225.357nm have been used for concentration measurements; the line pair R2(24) + Q2(30) at 225.31 nm has been used for flow parameter determinations. A discussion of NO spectroscopy as it applies to high-temperature shock tube measurements, including collision-broadening and -shift parameters can be found in DiRosa and Hanson (1994). Interference from 02 can be minimized by selecting NO absorption lines that lie in the relatively structureless minima of the Schumann-Runge bands. These interferences increase with temperature. The NO bands further to the blue, for example the (1,0) at 215nm, though still accessible with this method, are more susceptible to interference from larger molecules. The NO diagnostic has been used in flow experiments, as well as kinetic studies. By rapidly scanning and fully resolving NO line pairs, the gas velocity, temperature, pressure, density, and mass flux have been determined in shock tube flow fields. Velocity is determined by measuring the Doppler shift of line spectrum, which is taken from an absorption trace that traverses the flow at an angle relative to one that traverses perpendicular to the flow. Temperature is derived from the temperature-dependent relative peak heights of line pair members. Different line pairs can be chosen to improve sensitivity in different temperature regimes. Pressure is derived from the fractional absorption in flows of known concentration or from the collision-broadening of the line. Density and mass flux are determined from these three variables (P,T,V) (Chang et al., 1992; Davidson et al., 1992a; DiRosa et al., 1991; DiRosa and Hanson, 1994). 02 can be measured at higher temperatures, above about 2000 K, where the Schumann-Runge bands, particular the B-X(5,5)P1,2,3(39) transition at 227.354 nm, is accessible using the same BBO crystal cut and laser system as used in the forementioned NO experiments. With an absorption coefficient of
5.2 SpectroscopicDiagnostics
757
near 1 atm -1 cm -1, this transition is relatively weak, but offers a quantitative method to measure 0 2 concentration. This diagnostic has been applied to the measurement of product formation in the N20 decomposition system (Davidson et al., 1992a). HO 2 absorption has been attempted at a wavelength of 226.05 nm, again using the same experimental setup as for the NO experiments. Strong HO 2 signals were observed in lean H2/O2/Argon mixtures, permitting a preliminary determination of the H + 02 recombination rate coefficient (Moser, 1993).
5 . 2 . 3 . 4 LAMP ABSORPTION: W O R K I N G W I T H O U T LASERS Many absorbing species have spectral regions where there is substantial overlap of discrete transition lines. This produces broad absorption features that can be probed with broad band lamps. The trade-off is that although these lamp systems are simpler and less expensive than laser absorption diagnostics, the spectral intensities and hence the SNR for lamps is usually much poorer. This poorer performance requires larger target species concentrations to maintain suitable SNR, and these larger concentrations inherently complicate the interpretation of the kinetic mechanisms. To improve signal quality in lamp systems, broadband absorption features can be probed with wider diagnostic wavelength intervals. This reduces the species selectivity and sensitivity when compared to laser diagnostics, and complicates the theoretical calculation of the absorption coefficient, requiring a direct calibration in most cases. A survey of some applications of this method, which includes examples in the wavelength region 151-450nm, is given in Table 5.2.4. A wide variety of different lamps are used and details for these can be found in the individual references.
5.2.4 FREQUENCY
MODULATION
METHODS
Improvements in the sensitivity of conventional absorption methods are possible using frequency modulation techniques. A more sensitive diagnostic permits measurement of reaction rates with lower initial concentrations and therefore possibly without significant interference from secondary reactions. This sensitivity exists with ARAS methods, but for diatomic and molecular species, the sensitivity limit has not been reached. With state-of-the-art laser noise cancellation methods, the major component of the remaining noise in conventional absorption measurements is caused by the passage of the
758
D. E Davidson and R. K. Hanson
TABLE 5.2.4 Lamp Absorption Diagnostics Wavelength Species (nm)
Method
~twave Discharge Lamp ~twave Discharge Lamp Iodine Lamp Zinc Lamp Xe-Hg Lamp HO 2 Lamp HO2 Lamp N20 Xe Lamp C1NO Xe Lamp 02 Xe Lamp 03 Xenon Lamp NF2 Deuterium Lamp Benzyl Xe-Hg Lamp 2-Methylbenzyl Xe-Hg Lamp NH Hg Lamp ~twave Discharge Lamp OH
151.0 174.4 206.2 213.9 216.6 226.1 230.0 230.0 250.0 250.0 254.0 260.0 260.0 265.0 303.4 306.5
CO C2H4 HN3 CH3 CH3
339.0 388.4
BrO CN
Xe Lamp Hg Arc Lamp
405.0 415.0 421.6 450.0
NO2 Br2 CN NO2
Xe Lamp Xe Lamp Xe Lamp Tungsten Lamp
Reference
(Frank et al., 1986; Markus, 1995) (Saito et al., 1995; Zelson et al., 1994) (Kajimoto et al., 1979) (Baeck et al., 1995; Hwang et al., 1993) (Hwang et al., 1988; Moiler et al., 1986) (Kijewski and Troe, 1971) (Hippler et al., 1990b) (Endo et al., 1979) (Endo et al., 1979) (Endo et al., 1979) (Takahashi et al., 1995a) (Koudriavtsev et al., 1995) (Hippler et al., 1990a) (Hippler et al., 1994) (Szekely et al., 1984b) (Bott and Cohen, 1984; Bott and Cohen, 1989; Hidaka et al., 1982) (Takahashi et al., 1995a) (Louge and Hanson, 1984b; Szekely et al., 1983a; Szekely et al., 1983b; Szekely et at., 1984a; Szekely et al., 1984b; Szekely et al., 1984c; Szekely et al., 1984d) (Endo et al., 1979) (Takahashi et al., 1995a) (Fueno et al., 1973) (Fifer and Holmes, 1982)
transmitted radiation through the shock tube and test gas itself--scattering, beam steering, and birefringence. Improvements in detection electronics do not yield improvements in the minimum detectable fractional absorption. Frequency modulation spectroscopy is a differential absorption scheme using two closely spaced side bands of the main frequency. This scheme is less sensitive to shock tube and flow generated noise and thus allows for significant improvement in the SNR and minimum detectivity. Early work (Bjorklund, 1980) in the detection of stable species and other workers in the detection of transient species (Whittaker, Pokrowsky et al., 1983; Whittaker, Wendt et al., 1984; Chang and Sears, 1996) has led to the application of this method to the detection of transient species in shock tube experiments (Deppe et al., 1999; Deppe et al., 1998; Votsmeier et al., 1999a,c).
5.2 SpectroscopicDiagnostics 5.2.4.1
759
THEORY AND EXPERIMENT
A brief description of the experimental setup should help in explaining the application of frequency modulation to the ring dye laser absorption diagnostic scheme (see Fig. 5.2.4). As in conventional ring dye laser systems, the fundamental radiation frequency (for example 5.2 x 1014 Hz or 597nm for NH 2) is generated in a cavity by pumping the dye with an Ar + or solid state pump laser. The beam is passed through an electrooptical modulator (EOM) driven by an amplifier at the modulation frequency (600MHz), which generates side bands with typically 20 % of the carrier intensity. The modulated beam is passed through the windows and test gas volume, and is collected with high-speed detectors that have a bandwidth at least as great as the modulation frequency. The detector output signal is filtered, amplified, and demodulated with a double balanced mixer. The local oscillator of the mixer is driven by the same amplifier as the EOM but with an appropriate phase adjustment so that only the absorption part of the FM signal was detected. The output of the mixer in the preceding configuration increases with differential absorption between the two side bands. The output can be related to absorption at the central frequency or species concentration several waysmdirect calibration with known quantities of the target species (NH2), direct comparison with classical absorption signals, or by the use of a scanning etalon with known finesse. Current systems can achieve ~0.5 ppm NH 2 minimum detectivity at 1500 K, i atm in a single pass with a bandwidth of 1 MHz. This is equivalent to 0.01% absorption. Improvements in this value can occur at longer path length (double passing) or with smaller bandwidth. The RF modulation CORFof the refractive index of the EOM crystal results in the addition of side bands coo + coRF and coo - coRF tO the central frequency coo. In a symmetric system such as a balanced mixer, these two beat signals are 180 ~ out-of-phase and cancel perfectly, resulting in no RF amplitude modulation observed. When the beam is propagated through an absorbing medium, differential absorption or phase shift occurs, resulting in imperfect cancellation, and an amplitude modulated RF signal is generated. In the limit of small absorption, dispersion and modulation index, the output signal at the modulation frequency is equal to IFM --"
MIo{(~+ - c~_)cos 0 4- (~_ - 2~ c 4- ~+) sin O}
(5.2.4)
Here M is the modulation index, I0 is the carrier beam intensity, 6+,_ are the amplitude attenuations of the higher and lower side band frequencies, respectively, and ~b+,_ ,c are the optical phase shifts in the upper, lower, and carrier signals, respectively.
760
D. E Davidson and R. K. Hanson
FIGURE 5.2.4
FM spectroscopy experimental schematic.
The phase 0 can be adjusted so that if zero, the signal reflects the absorption differences only. If the modulation frequency is small compared to the width of the probe spectral line, the phase-modulated signal is proportional to the derivative of the absorption feature. If the modulation frequency is larger compared to the width of the probe spectral line, the phase-modulated signal resembles the absorption profile.
5.2.4.2
N H 2 AND 1 C H 2
FM methods have recently been applied to two important chemical kinetic problems in combustion. In the first application, Deppe et al. and Votsmeier et al. have measured ppm and sub-ppm NH 2 concentrations in NH2/NO shock tube experiments. Both groups attempted to resolve persistent questions about the magnitude of the branching ratio and overall rate of the NH 2 + NO products reaction at elevated temperatures. Earlier measurements have had difficulties with interfering reactions because of the larger NH 2 concentrations needed with conventional absorption methods. With the increased sensitivity allowed with FM methods, these experiments required only 5-ppm initial concentration of NH 2 and were not complicated by interfering reactions (Deppe et al., 1999; Deppe et al., 1998; Votsmeier et al., 1999a,c).
5.2 Spectroscopic Diagnostics
761
In the second application, Deppe et al. have demonstrated the detection of ppm levels of 1CH2 generated in the photolysis of ketene. Until this study, there had been no successful absorption measurements of CH 2 in shock tubes. The detection of CH 2 takes advantage of the fact that excited singlet methylene 1CH2 does have a weak laser-accessible transition near 590nm, and that at combustion temperatures this state is populated. In the Deppe et al. experiments, the authors probed the biB1(0, 14, 0)404 - alAl(0, 0, 0)414 transition at 590.707 nm and could detect 50ppm of 1CH2 with a SNR of approximately 10 with a 1-MHz bandwidth (Deppe et al., 1998).
5.2.5 INFRARED LASER ABSORPTION AND EMISSION TECHNIQUES Probing IR absorption transitions of target species offers a different trade-off between performance and cost from that found in visible transitions. While the absorption coefficients are typically lower for the IR transitions, the low cost and ease of use of individual laser systems makes these methods very attractive. Recent technical advances have extended the wavelength range available in semiconductor diode lasers beyond that of the earlier CO gas discharge and Pb salt diode lasers, and these new lasers are beginning to find application in shock tube studies. A survey of infrared laser diagnostic systems is given in Table 5.2.5. Representative absorption coefficients and detectivities are presented in Figs. 5.2.5 and 5.2.6. A full review of the currently available diode lasers is given in Allen (1998).
5.2.5.1
ROOM-TEMPERATURE DIODES
As described by Allen, the shortest wavelength semiconductor lasers are made of InGaA1P to 630nm, followed by A1GaAs lasers in the 780-900nm range. Midrange lasers are made using InGaAs and InGaAsP on a variety of substrates, and wavelengths out to 2100 nm can be reached using lasers made of InGaAsP with InGaAs/InAsP substrates. Longer wavelengths (to 2300nm) can be reached with the addition of antimony to these compounds. A variety of species transitions are accessible with these diodes. These include H20 with an absorption band near 1340nm; CO which has a second overtone at 1550 nm; CO 2 which can be detected at 1575 nm and 2010 nm; OH which can be measured in flames in the 1500-2000 nm region; CH4 which has an absorption feature near 1300nm; and NO at 1800nm. The H20 and C2H2
762
D. E Davidson and R. K. H a n s o n
100
E
|
~
10
C
O
24200nm
"7,
E o
~
"7
E
CO 4 8 8 0 n m
v
c-.mQ2
o ,.i,-
o o t. ~O e~ o e~
~ ~ ~
HCN 3330n
1
NO 5300nm H20 5258nm NO 5166nm
01
1392nm 0.01
, 0
I 500
,
I 1000
i
I 1500
,
i 2000
,
I
,
2500
i 3000
,
I 3500
. 4000
Temperature (K) FIGURE 5.2.5
Selected absorption coefficients for infrared laser diagnostics.
can also be weakly detected in the visible and NO 2 has an extensive spectrum in the visible, which is accessible with 635- and 670-nm lasers. Of these species, only a few have been successfully measured in a shock tube. The H20 is the most important of these, and Nagali and others have performed a full study of several absorption lines at 1390nm to determine collisional parameters and calibrations (Langlois et al., 1994; Nagali et al., 1997; Nagali et al., 1998, 1999; Nagali and Hanson, 1997). This band has also been used to measure temperature, pressure, and velocity simultaneously (Arroys et al., 1994a,b). The A band ( b l X - X3•) of 0 2 at 763 nm has been used by Philippe and Hanson (1992, 1993) with wavelength modulation spectroscopy and second harmonic absorption to measure supersonic velocities and gasdynamic properties, including mass flux, behind incident shock waves. Although the absorp-
5.2
763
Spectroscopic Diagnostics
10000
I
'
I
'
I'
'
I
'
I
'
I
'
..c .r
I
'
1392nm
c)_
E co 1000 O
NO 5166nm
H20 5258nm
@
E o. 100
HCN
r-
3330n
~
~
NO 5300nm
~
>
CO 4 8 8 0 n m
co a
10
CO 2 4200nm
E E r-
|
o
I
500
,
I
1000
i
I
1500
,
I
i
2000
Temperature FIGURE 5.2.6
I
'
2500
I
3000
'
I
3500
'
4000
(K)
Selected minimum detectivities for infrared laser diagnostics.
tion coefficient of this transition is very small, excellent SNR was achieved using wavelength modulation methods. The atomic oxygen transition line (3s5S~ -3pSP3) at 777.2nm and the atomic nitrogen transition line (3s4p~ 3p4p~ at 821.6nm have been scanned in shock tube flows to determine the temperature where the conditions are extreme enough to dissociate atoms from the parent molecule (Chang et al., 1994, 1995).
5.2.5.2
P b SALT D I O D E LASERS
Lead salt diode lasers have been available for many years and can be used at wavelengths between about 3000 and 30,000nm. Although they are of low
764 TABLE 5.2.5
D. E Davidson and R. K. Hanson Infrared and Near-Infrared Laser Diagnostics
Wavelength Species (nm)
Method
760.0 777.2 821.6 1390
02 O*-atoms N*-atoms H20
Semiconductor Semiconductor Semiconductor Semiconductor
3000 3390
HCN CH4
4200 4764
CO2 CO
4880
CO
5166
NO
5228 5258
NO H20
Reference
(Philippe and Hanson 1993) (Chang et al., 1995) (Chang et al., 1994) (Arroyo et al., 1994a; Arroyo et al., 1994b; Langlois et al., 1994; Nagali and Hanson, 1997) Pb Salt Diode Laser (Chang and Hanson, 1985) He-Ne Laser (Heffington et al., 1976; Olson et al., 1978) Pb Salt Diode Laser (Wooldridge et al., 1996a) Pb Salt Diode Laser, CO Laser (Hanson, 1977; He et al., 1995; Hsu et al., 1983; Thaxton et al., 1997) Pb Salt Diode Laser (Varghese and Hanson, 1981; Yu et al., 1995) CO Laser, Pb Salt Diode Laser (Hanson et al., 1975; Hanson et al., 1976; He et al., 1995; Klenk et al., 1997; Roose et al., 1981; Salimian et al., 1984; Thaxton et al., 1997) CO Laser (Falcone et al., 1983) CO Laser (Salimian and Hanson, 1983; Salimian et al., 1984; Thaxton et al., 1997) Diode Diode Diode Diode
power (~ 100~tW), require cryogenic operation and detection, and have output at multiple wavelengths, they did provide an early demonstration of laser absorption techniques. They continue to be useful in wavelength regions where room-temperature semiconductor diodes are not available. The first application of laser-resolved absorption in shock tubes was the scanning of CO lines at 4600nm to determine concentrations and temperatures (Hanson, 1977). Line-shape measurements have also been made in NO by Hanson and coworkers at 5166nm (Falcone et al., 1983; Hanson et al., 1976), and HCN at 3000nm (Chang and Hanson 1985).
5 . 2 . 5 . 3 CO DISCHARGE LASERS The CO gas discharge lasers have output at a wide variety of wavelengths in the infrared, some which are coincident transitions of important combustion species. Water (H20) at 5258nm and NO at 5166nm have both been measured using this laser system (Hanson et al. (1975, 1976)" Salimian and
5.2
765
Spectroscopic Diagnostics
Hanson, 1983). M. C. Lin and co-workers have used this laser for many years to measure CO concentrations (He et al., 1995; Hsu et al., 1983" Thaxton et al., 1997).
5 . 2 . 5 . 4 EMISSION METHODS Infrared emission has historically been an important diagnostic for the investigation of stable and transient species at high temperatures in shock tubes. There exists an extensive literature surrounding this method, a small sample of which is given in Table 5.2.6. For most poly-atomic, and to some degree diatomic species, the vibrational relaxation times found in shock-heated gas mixtures are sufficiently short that the emission from their transitions can be assumed to be in vibrational equilibrium with the bath gas translational temperature. Then, knowing the temperature and pressure of the shocked gas, one can determine the concentration of the infrared-active species from the emission. A different situation exists in the visible and ultraviolet, where electronically excited states of some species may be created in excess by certain reactions, particularly during times of increased radical pool growth. For example, OH emission peaks seen in
TABLE 5.2.6
Emission Diagnostics
Wavelength Species Method (nm)
Reference
306.5 336.1 388.4 538.0 3330 3000 3400 3500 4300 4500
OH NH CN NH2 HCN NH3 CH4 CH20 CO2 N20
UV Emission UV Emission UV Emission Visible Emission IR Emission IR Emission IR Emission IR Emission IR Emission IR Emission
4764 5050 5300
CO CN NO
IR Emission IR Emission IR Emission
6300 10500
H20 NH3
IR Emission IR Emission
(Hidaka et al., 1982) (Roose et aI., 1980; Roose et al., 1981) (Slack, 1976) (Roose et al., 1980; Roose et al., 1981) (Chang and Hanson, 1985) (Roose et al., 1980) (Petersen et al., 1998) (Dean et al., 1980) (Jachimowski 1977) (Dean, 1976; Monat et al., 1977; Roose et al., 1977; Roose et al., 1981; Salimian et al., 1984) (Hsu et al., 1983) (Brupbacher and Kern 1973; Jachimowski, 1977) (Flower et al., 1977; Hanson et al., 1975; Monat et al., 1977; Roose et al., 1977; Roose et al., 1981) (Flower et al., 1977) (Roose et al., 1977; Roose et al., 1980; Roose et al., 1981; Salimian et al., 1984)
766
D. E Davidson and R. K. Hanson
ignition experiments reflect small excited OH* population which may not be simply proportional to the larger ground state population (Hidaka et al., 1982). Consider the fundamental (Av = 1) band in a species which is in thermal and vibrational equilibrium with its bath gas, such as 10.5-1am band of NH 3. In the linear or optically thin limit, the total infrared radiation from a vibrationalrotational band is proportional to the energy stored in that vibrational band. For a detector with a uniform spectral response across the entire extent of the fundamental band, this emission is proportional to the product of the molar concentration of the emitter and the average vibrational energy in the observed mode per mole. I -- Cr [X] R Ov/(exp(~ v - 1)
(5.2.5)
Here I is the emission signal, C c is a calibration constant which reflects the collection efficiency of the detector system,[X] is the concentration of the emitter molecule, R is the universal gas constant, and 0v is the characteristic vibrational temperature for the observed vibration-rotation band. Knowing the shocked gas temperature and the emission signal, the emitter concentration can then be determined. For diatomic species with long vibrational relaxation times, it is possible to determine the vibrational temperature or monitor the relaxation to equilibrium using this method as well.
5.2.6
ATOMIC
RESONANCE
ABSORPTION
SPECTROSCOPY The atomic species most relevant to hydrocarbon combustion, H- and Oatoms, and others such as D-, C-, N-, S-, Cl-, Br, I and Si-atoms, are best measured quantitatively in a shock tube using atomic resonance absorption spectrometry (ARAS). In practice, resonant radiation is generated in a lowpressure microwave discharge cavity placed adjacent to the entrance window of the shock tube, and transmitted radiation that has passed through the absorbing line-of-sight test gas medium is collected at the exit window. Quantitative measurements are made either by using known quantities of absorbers or by theoretical modeling of the process. Large oscillator strengths make these diagnostics sensitive to sub-ppm levels of atoms, and in fact, limit their usefulness to experiments that generate a maximum of several ppm of atoms in the measurable test time. A survey of ARAS diagnostic systems is given in Table 5.2.7. Selected absorption coefficients and typical minimum detectivities are shown in Figs. 5.2.7 and 5.2.8. In these figures minimum detectable absorptions of 2 % are used for O-, H- and C1-ARAS, and 5 % for other atomic species. These values are based on typical published experimental data.
5.2 SpectroscopicDiagnostics 10 6
'
E O
~,
E
I
767 '
I
'
I
'
I
'
I
'
I
'
I
'
H 121.6nm ~ ~
10s
GI
N 1 199 9nm
S 147.4nm
t(D
.w
O o,.. (D O
(,.) cO .t=,,
"10
4
CI r 1137"9 ..~...
~
30.4nm
0
<
B
103
57.7n,,,
I
'
I
~....~
'
I
'
Si251.6nm
I
'
I
'
I
'
I
'
500 1000 1500 2000 2500 3000 3500 4000 Temperature (K) FIGURE 5.2.7 Selectedabsorption coefficients for ARASdiagnostics.
5.2.6.1
EXPERIMENTAL M E T H O D
Several workers have described this method. A typical ARAS system has two sides--the emission side and the detection side. To generate atomic resonance emission, a low-pressure discharge is typically formed in a 2.45-GHz EvanstonBroida cavity with a 1-cm dia glass tube passing through it. Care must be taken to protect workers from scattered microwave and ultraviolet radiation. Different discharge gases are used to generate different emission lines. The 1% mixtures of the parent gas in helium have been used successfully, H 2 for Hatoms, N 2 for N-atoms, 02 O-atoms, CO or CO 2 for C-atoms, and H2S for Satoms. The H-atom radiation can also be generated using commercially pure helium as the discharge gas. These discharges usually have sufficient hydrogen
768
D. E Davidson and R. K. Hanson 10
m
m m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
r-
E o
.t..,*
Br / /
@
S N
E r
C
I
>
(D
cJ
0.1
a E E
O
! i
..,..
t-
H 0.01 0
500 1000 1500 2000 2500 3000 3500 4000 Temperature (K)
FIGURE 5.2.8
Selected minimum detectivities for ARAS diagnostics.
impurities to produce Ha emission, but with such low H-atom concentration to preclude self-absorption. Without the complication of self-absorption, it is possible to calculate the absorption coefficient theoretically from simple basic spectral constants. The emission from these lamps passes into and out of the shock tube and then into the detection side of the system through MgF 2 windows (or LiF 2 if N-atoms are being measured). The radiation at the exit side of the shock tube is collimated into a vacuum monochromator. Line filters have also been used successfully instead of monochromators. Detection is done with solar-blind PMTs with CsI cathodes. The following references are examples of a variety of N-ARAS systems, (Davidson and Hanson (1990a,b), Fujii et al. (1991),
769
5.2 SpectroscopicDiagnostics TABLE 5.2.7 ARASDiagnostics Wavelength Species (nm)
Method Reference
119.9
N-atoms ARAS
121.5 121.6 130.4 137.9
D-atoms H-atoms O-atoms Cl-atoms
147.4 156.1
S-atoms ARAS C-atoms ARAS
157.7 183.0
Br-atoms ARAS I-atoms ARAS
251.6
Si-atoms ARAS
ARAS ARAS ARAS ARAS
(Davidson and Hanson, 1990a; Davidson and Hanson, 1990b; Fujii et al., 1991; Natarajan and Roth 1991; Roth et al., 1986; Roth and Thielen, 1985; Thielen and Roth, 1987; Thielen and Roth 1996) See Section 16.3 See Section 16.3 See Section 16.3 (Kumaran et al., 1994; Kunz and Roth, 1998; Lim and Michael, 1993; Lim and Michael, 1994; Michael et al., 1993; Saito et al., 1995) (Markus, 1995; Woiki and Roth 1995; Woiki and Roth, 1996) (Dean et al., 1991a; Dean and Hanson, 1992; Kruse and Roth, 1995; Markus, 1995) (Takahashi et al., 1995b) (Takahashi et al., 1997; Takahashi et al., 1995b; Wintergest and Frank, 1995; Yamauchi et al., 1997) (Kunz and Roth, 1998; Mick et al., 1995; Mick and Roth, 1995; Mick et al., 1993; Woiki et al., 1997)
Natarajan and Roth (1992), Roth et al., (1986), Roth and Thielen (1986), Thielen and Roth (1987, 1996)).
5.2.6.2.
CALIBRATIONS AND APPLICATIONS
As described in Section 16.3, Michael et al. have calculated the absorption coefficient of a non-self-absorbed He transition and have found good agreement with the predicted absorption and measured values. For other selfabsorbed emission systems, experimental calibration is necessary. As in the case of laser absorption, several methods are possible. Known quantities of Hatoms can be produced by shock heating mixtures of N 2 0 / H 2 / A r . The simple reaction scheme of N 2 0 q- M = N 2 + O + M and O + H 2 = OH + H with its well-known rate coefficients can be used to predict the H-atom concentration with approximately 25 % uncertainty. The well-known kinetic behavior of N 2 0 decomposition can be used for O-atom calibration. The high-temperature equilibrium partition of N2/N has been used to generate known quantities of N-atoms; C-atoms can be generated from the complete decomposition of CH 4. In systems with complicated H-atom chemistry, some isolation of individual pathways can be achieved through the use of deuterated compounds and the
770
D. E Davidson and R. K. Hanson
measurement of D-atoms. The isotopic shift puts this transition near to the H~ transition so that nearly identical detection systems can be used, but sufficiently far away from the H~ transition as to avoid H-atom interferences. Ground state (3p) and electronically excited (1D) S-atoms have been detected and the concentration of the (1D) state S-atoms can be determined by equilibrium experiments with COS (Markus, 1995; Woiki and Roth, 1995, 1996). Measurements of Si-atoms have been performed as part of studies of Sill and Sill 2. The Si-atom calibrations can be determined from modeling the decomposition of silane (Kunz and Roth, 1998; Mick et al., 1995; Mick and Roth, 1995; Mick et al., 1996; Woiki et al., 1997). With the advent of environmental concerns about halogenated compounds, there has been a renewed interest in the measurement of halogen atoms CI-, Br-, and I-atoms and their reactions (Kunz and Roth, 1998; Lim and Michael, 1993; Lira and Michael, 1994; Michael et al., 1993; Saito et al., 1995). Another area that takes advantage of the ability to measure I-atoms quantitatively is the investigation of the decomposition of iodoethane, C2H5I. Upon decomposition of C2H5I, the subsequent rapid decomposition of the product C2H5 generates quantities of H-atoms at approximately 9 0 % yield. Understanding of I chemistry has helped confirm the yield of this process, and permitted the use of iodoethane as a relatively clean source for H-atoms. Two other areas of concern in the application of ARAS diagnostics are nonresonant radiation and pressure effects. Nonresonant radiation can contribute to a non-Beer's law behavior at large absorptions, and typically accounts for 5-15 % of the emission signal in some H-ARAS systems. Its contribution to the absorption signal is usually handled empirically. On the other hand, pressure dependence of the absorption coefficient is a topic of current interest and the effects of pressure shifts and collision broadening of absorbing atomic lines have not yet been thoroughly investigated in shock tube systems.
5.2.6.3
S H O C K TUBE IMPURITIES
One of the primary results of H-ARAS measurements is the confirmation that all shock tubes have impurity levels of H-atoms in the shock-heated test medium. The high sensitivity of the H-atom diagnostic, combined with the wide possible sources of H-atoms in machining fluids, cleaning solvents, and reactant impurities, ensures that measurable amounts of H-atoms can be seen in most shock tubes. This situation is accentuated at higher temperatures. The practical result of this observation is that processes that are sensitive to small initial quantities of impurities may have scattered results. A classic example of this is the scatter seen in ignition times of H2/O 2 mixtures. This problem is
5.2
Spectroscopic Diagnostics
771
usually unsolvable, though with care some shock tube workers have reduced the measured H-atom concentration at lower temperatures to an almost unmeasurable level.
5.2.7 PLANAR LASER-INDUCED FLUORESCENCE Two-dimensional (2D) visualization of fluorescence-based signals is a powerful imaging technique that has been made possible with the availability of highpowered pulsed lasers, image-intensified solid-state cameras, and computerbased image processing. The use of planar laser-induced fluorescence (PLIF) was first reported in 1982 by several workers (see references that follow) and was first applied to shock tube and shock tunnel flows in 1989 by McMillin et al. (1989, 1992). In the intervening years, it has become widely used as an imaging tool, especially in propulsion and combustion systems. We present here a short overview of its theory and applications to imaging in shock tube and tunnel flows. Extended overviews of PLIE which include fuller theoretical discussions, are given by Hanson and coworkers (see Hanson, 1986; Hanson et al. 1990; Seitzman and Hanson, 1993).
5.2.7.1
THEORY
Fluorescence is generally defined as radiation emitted by a molecule or atom when it decays by spontaneous emission of a photon in an optically allowed transition from a higher to a lower energy state. This de-excitation process occurs in parallel with other processes that de-excite the molecule, including collisional energy transfer to other molecular states. The population of the upper state can occur by chemical reaction, molecular collisions, and radiative interactions. In laser-induced fluorescence, the upper state is populated by absorption of laser radiation at an allowed resonant frequency between the discrete lower state and the excited upper state. After excitation the laserpopulated upper state may undergo any of several different competing processes. 1) The molecule could return to the ground state by laser-induced stimulated emission 2) The molecule could absorb another photon and get still further excited. 3) The rotational or vibrational internal energy distribution of the molecule could be rearranged by inelastic collisions with other molecules. 4) Quenching or electronic energy transfer by inelastic collisions could Occur.
772
D. E Davidson and R. K. Hanson
5) "Internal" collisions could cause dissociation of the molecule, which would include predissociation, the change from a stable to repulsive electronic arrangement. 6) Or finally, The original state and its energetically nearby neighbors, populated by processes like that in 3), can emit spontaneously, or fluoresce.
The linear fluorescence equation, which describes the region where fluorescence is proportional to laser intensity, is used to model PLIF measurements in practice. Rp-
rl 1 B12
Iv
[Azl/(A21 + Q21)]
(5.2.6)
Where Rp (photons-cm-3s -1) is the steady state fluorescence rate, n 1 (cm -3) is the initial population of the lower absorbing state, B12 (s -1 W -1 c m 2 Hz) is Einstein B coefficient, Iv (Wcm -2 Hz -1) is the laser spectral intensity, Azl (s -1) is the Einstein A coefficient for spontaneous emission (i.e., fluorescence), and Q21 (s-l) is collisional transfer coefficient, which can be extended to include other loss rates. From kinetic theory, the collisional loss rate Q21 -norm (v), where n (cm -3) is the total number density, a m (cm 2) is the mixture-averaged quenching crosssection, and (v) (cms -1) is the appropriate mean molecular speed. From this equation we can see that the fluorescence signal is proportional to the lower state population and to a factor [(Azl/(A21 + Qzl)] called the SternVollmer factor or fluorescence quantum yield. The PLIF signal is then a measure of the light absorbed at each flow field point modified by the local fluorescence yield. This yield is affected by the collisional quenching rate. Current PLIF research is investigating strategies that avoid this problem. These include taking the ratio of two PLIF images to determine temperature, and monitoring velocity by Doppler-shifted absorption. By using signal differences and ratios, local collisional quenching effects can be effectively minimized.
5 . 2 . 7 . 2 MEASUREMENT STRATEGIES We will discuss three applications of PLIE species, temperature, and velocity imaging. These three methods typify the wide range of technical options available to the experimentalist. 5.2.7.2.1 Species Imaging The primary use of PLIF imaging has been for species measurement, with these measurements generally qualitative rather than quantitative. Thus, although
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the image data has not been converted to absolute concentration, it can still be used for locating distinct regions in the combustion structure such as flame fronts, recirculation zones and regions of burnt gases. The PLIF image collected for species measurement can be interpreted as being dependent on a two-step process, absorption of laser light and subsequent emission or fluorescence as shown in the linear fluorescence equation. The first step, the absorption process, is proportional to the number density of target molecules in the absorbing state (a function of Boltzmann statistics) times the number of photons incident on the measurement volume (a function of laser system performance). The second step is characterized by the fluorescence yield or the Stern-Vollmer factor (a function of the collisional environment in which the target molecule exists). With proper accounting of these last two factors, the signal is directly related to species concentration. Of the three plates shown here, Plate 1 shows the OH fluorescence that occurs during the auto ignition of H 2, which is traversely injected into a supersonic 02 crossflow. The conditions are similar to those found in a Scramjet engine during flight at Mach 13. The OH-fluorescence intensity can be qualitatively linked to OH mole fraction. At the combustion pressures obtained in this work, the fluorescence signal is proportional to the mole fraction and the Boltzmann fraction of OH molecules in the absorbing state achieved by the Ql(7) transition of the Az2E+ - X21-I (1,0) band. Temperature plays a relatively minor role in interpreting the signal in the regions observed to contain OH, and the fluorescence intensity can be considered as a direct indicator of OH mole fraction. 5.2.7.2.2 Temperature Imaging Temperature can be quantitatively imaged using two-line PLIE This strategy uses the sequential excitation of two absorption transitions of a single tracer, with similar quenching properties, to infer the local temperature. The simplest variation of this two-line strategy takes advantage of the fact that the point-bypoint ratio of the two fluorescence images is proportional to the relative populations of the absorbing lines, and is a function of temperature through Boltzmann statistics. The ratio would be proportional to exp{-AEab/kT}where AEab is the difference in energy between the two absorbing states. This approach is useful for flows that are compressible and of variable composition, because the dependence on number density, absorber mole fraction, and collisional quenching is effectively removed. The experimental setup is shown in Plate 2 and an example image is shown in Plate 3 (Palmer and Hanson, 1996). A two-wavelength OH PLIF system used in a shock tunnel facility to determine the flow field temperature is shown in Plate 2. Frequency-doubled
774
FIGURE 5.2.9
D. E Davidson and R. K. Hanson
OH PLIF concentration imaging. For details see text. (See Color Plate 4).
FIGURE 5.2.10 PLIF experimental setup for two-wavelength OH temperature measurement in a shock tunnel facility. For details see text. (See Color Plate 5).
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FIGURE 5.2.11
775
OH PLIF temperature imaging. For details see text. (See Color Plate 6).
photons from a tunable, XeCl-excimer-pumped dye laser with Coumarin 540 dye were used to excite transitions in the A-X (1,0) band of OH near 282 nm. The laser sheets, each ~ 50 mm wide and 0.35 mm thick with ~ 0.25 mJ before entering the test section, were formed with cylindrical lenses, and directed through the mid-plane of the flow field at - 1 2 1 and 59 ~ with respect to the jet axis using planar, broadband UV mirrors. One sheet was delayed by 230ns with respect to the other following three round-trip passes through a 12-m long reflector cavity. The resulting fluorescence was imaged onto two 578 x 384 pixel, cooled, intensified charge-coupled device (ICCD) arrays with 230-ns gating by Nikkor ultraviolet lenses with f/4.5. Resonant fluorescence and laser scattering were blocked by 2-mm, WG-305 Schott glass filters placed in front of the cameras, along with 2-mm, UG-5 filters used to block visible radiation. Approximately 60 % of the fluorescence from the A-X (1-1) and (0,0) bands of OH at 312 and 306 nm, respectively, were collected. The rotational temperature field was inferred from the single-shot P1(1.5) and Q1(5-5) images and a frame-averaged field was obtained by evaluating the temperature from 20 individual image ratios and averaging the results. They may be compared with the Method of Characteristics (MOC) prediction of the temperature field, as shown in what follows. A thresholding operation was performed on the raw PLIF images to eliminate the small OH signal levels in the ambient region of the flow field, and the measured signal ratio expected at
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x/D = on the jet axis, before conversion to temperature. The symmetry of the single-shot temperature field, expected because the flow field is axisymmetric, demonstrates the similarity of the laser pulses used in acquiring the single-shot PLIF images. Blurring of the shock structure caused by differences in its position in the single-shot images is evident. The agreement between the measured and calculated fields is generally good throughout the free jet flow field. 5.2.7.2.3 Velocity Imaging The primary strategy for velocity imaging in PLIF is based on the Doppler effect. The velocity component parallel to the incident laser pulse direction has the effect of shifting the absorption transition frequency that will affect the fluorescence intensity. This frequency shift has a value o f AVdoppler = --Uo/]C, where u 0 is the velocity component in the direction of the incident light, and 2 is the wavelength of the incident light. In the case of pulse lasers, the laser excitation line shape is relatively broadband, but of a known or predictable form. In this case, it is preferred to have the target molecule absorption feature Doppler shift along the broadband laser line shape. This results in effectively stronger or weaker laser intensity. With the absorption feature properly positioned in frequency space on the laser excitation feature, more Doppler shift or greater velocity is equivalent to stronger fluorescent signal.
ACKNOWLEDGMENTS In our laboratory, the work in this area has been supported for many years by several agencies including the U. S. Department of Energy, the U. S. Department of Defense through the Army Research Office, the Air Force Office of Scientific Research and Office of Naval Research, and the Gas Research Institute. We would also like to thank the long series of graduate students and postgraduate researchers who have contributed to these studies.
REFERENCES Allen, M.G. (1998). Diode laser absorption sensors for gasdynamic and combustion flows. Meas. Sci. Technol. 9:545-562 Arroyo, M.P., Birbeck, T.P., Baer, D.S., and Hanson, R.K. (1994a). Dual diode-laser fiber-optic diagnostic for water-vapor measurements. Opt. Lett. 19: 1091-1093. Arroyo, M.P., Langlois, S., and Hanson, R.K. (1994b). Diode-laser absorption technique for simultaneous measurements of multiple gasdynamic parameters in high-speed flows containing water vapor. Appl. Opt. 33: 3296-3307. Baeck, H.J., Shin, K.S., Yang, H., Qin, Z., Lissianski, V., and Gardiner, W.C. (1995). Shock tube study of the reaction between CH 3 and H 2. J. Phys. Chem. 99: 15925-15929.
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Bates, R.W., Hanson, R.K., Bowman, C.T., and Golden, D.M. (1998). Measurement of the thirdbody efficiency of water for the H + 02 = M = HO 2 + H20 reaction at 35 atm and 1200 K. 27th Syrup (Int.) on Combust., Combust. Inst., Work-In-Progress Poster W4B14. Bjorklund, G.C. (1980). Frequency modulation spectroscopy: A new method for measuring weak absorptions and dispersions. Opt. Lett. 5: 15-17. Bott, J.E and Cohen, N. (1984). A shock tube study of the reaction of the hydroxyl radical with propane. Int. J. Chem. Kinetics 16: 1557-1566. Bott, J.E and Cohen, N. (1989). A shock tube study of the reaction of the hydroxyl radical with H 2, CH 4, c-CsH10, and i-C4H10. Int. J. Chem. Kinetics 21: 485-498. Braun-Unkhoff, M., Naumann, C., and Frank, P. (1995). A shock tube study of the reaction CH 3 + 0 2. 19th Int. Syrup. Shock Waves, Vol. 2, 203-208. Breene, R.G. (1961). The Shift and Shape of Spectral Lines, Oxford: Pergamon Press. Brupbacher, J.M. and Kern, R.D. (1973). Reaction of cyanogen and hydrogen behind reflected shock waves. J. Phys. Chem. 77: 1329-1335. Chang, A.Y. and Hanson, R.K. (1985). Shock-tube study of HCN self-broadening and broadening by argon for the line P(10) of the v1 band at 3 microns J. Quant. Spectrosc. Radiat. Transfer. 33: 213-217. Chang, A.Y. and Hanson, R.K. (1989). Measurements of absorption lineshapes in the A37zi - X3y~-(0, 0) band of NH in the presence of Ar broadening. J. Quant. Spectrosc. Radiat. Transfer 42:207-217. Chang, A.Y., Rea, E.C., and Hanson, R.K. (1987). Temperature measurements in shock tubes using a laser-based absorption technique. Appl. Opt. 26, 885-891. Chang, A.Y., DiRosa, M.D., Davidson, D.E and Hanson, R.K. (1991). Rapid tuning CW laser technique for measurement of gas velocity, temperature, pressure, density, and mass flux using NO. Appl. Opt. 30:3011-3022. Chang, A.Y. DiRosa, M.D. and Hanson, R.K. (1992). Temperature dependence of collision broadening and shift of NO A-X (0,0) band in the presence of argon and nitrogen. J. Quant. Spectrosc. Radiat. Transfer 47:375-390. Chang, B-C. and Sears, T.J. (1996). Transient frequency modulation absorption spectroscopy of free radicals in supersonic free jet expansion. Chem. Phys. Lett. 256: 288-292. Chang, H.A., Baer, D.S. and Hanson, R.K. (1994). Semiconductor laser absorption diagnostics of atomic nitrogen for hypersonic flow field measurements. AIAA Paper 94-0385. Chang, H.A., Baer, D.S. and Hanson, R.K. (1995). Semiconductor laser diagnostics of kinetic and population temperatures in high-enthalpy flows. 19th Int. Syrup. Shock Waves, Vol 2, 33-36. Davidson, D.E, Chang, A.Y., and Hanson, R.K. (1988). Laser photolysis shock tube for combustion kinetics studies. 22th Syrup. Ont.) Combust., Combust. Inst., 1877-1885. Davidson, D.E, Chang, A.Y., Kohse-Hoinghaus, K., and Hanson, R.K. (1989). High temperature absorption coefficients of 0 2, NH 3, and H20 for broadband ArF excimer laser radiation. J. Quant. Spectrosc. Radiat. Transfer 42: 267-278. Davidson, D E, Chang, A.Y., DiRosa, M.D., and Hanson, R.K. (1991a). Continuous wave laser absorption techniques for gasdynamic measurements in supersonic flows. Appl. Opt. 30: 2598--2608. Davidson, D.E, Chang, A.Y., DiRosa, M.D., and Hanson, R.K. (1993a). A CW laser absorption diagnostic for methyl radicals. J. Quant. Spectrosc. Radiat. Transfer. 49: 559-571. Davidson, D.E, Dean, A.J., DiRosa, M.D., and Hanson, R.K. (199 lb). Shock tube measurements of the reactions of CN with O and 0 2. Int. J. Chem. Kinetics 23: 1035-1050. Davidson, D.E, DiRosa, M.D., Chang, E.J., and Hanson, R.K. (1995a). An improved determination of the 216.615 nm absorption coefficient for methyl radicals. J. Quant. Spectrosc. Radiat. Transfer 53: 581-583. Davidson, D.F and Hanson, R.K. (1990a). High temperature reaction rate coefficients derived from N-Atom ARAS measurements and excimer photolysis of NO. Int. J. Chem. Kinetics 22: 843-861.
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Davidson, D.E and Hanson, R.K. (1990b). Shock tube measurements of the rate coefficient for N + CH 3 -- H2CN + H using N-Atom ARAS and excimer photolysis of NO. 23th Symp. (Int.) Combust., Combust. Inst., 267-273. Davidson, D.E, Hanson, R.K., and Bowman, C.T. (1995b). Communication: Revised values for the rate coefficients of ethane and methane decomposition. Int. J. Chem. Kinetics 27: 305-308. Davidson, D.E, Kohse-Hoinghaus, K., Chang, A.Y., and Hanson, R.K. (1990c). A pyrolysis mechanism for ammonia. Int. J. Chem. Kinetics 22: 513-535. Davidson, D.E, Roehrig, M., Petersen, E i . , DiRosa, M.D., and Hanson, R.K. (1996). Measurement of the OH A-X (0,0) 306nm absorption bandhead at 60atm and 1735K. J. Quant. Spectrosc. Radiat. Transfer 55: 755-762. Davidson, D.E, DiRosa, M.D., Chang, A.Y., and Hanson, R.K. (1992a). Shock tube measurements of the major product channels of N20 + O. 18th Int. Symp. Shock Waves, Sendal Japan, 813-818. Davidson, D.E, DiRosa, M.D., Chang, A.Y., and Hanson, R.K. (1992b). A shock tube study of methane decomposition using laser absorption by CH 3. 24th Symp. (Int.) Combust., Combust. Inst., 589-596. Davidson, D.E, DiRosa, M.D., Chang, E.J., Hanson, R.K., and Bowman, C.T. (1995c). A shock tube study of methyl-methyl reactions between 1200 and 2400 K. Int. J. C hem. Kinetics 27:1179-1196. Davidson, D.E, DiRosa, M.D., Hanson, R.K., and Bowman, C.T. (1993b). A study of decomposition ethane in a shock tube using laser absorption of CH 2. Int. J. Chem. Kinetics 25: 969-982. Dean, A.J., Davidson, D.E, and Hanson, R.K. (1991a). A shock tube study of reactions of C atoms with H 2 and 02 using excimer photolysis of C302 and C atom atomic resonance absorption spectroscopy. J. Phys. Chem. 95: 183-191. Dean, A.J. and Hanson, R.K. (1989). Development of a laser absorption diagnostic for shock tube studies of CH. J. Quant. Spectrosc. Radiat. Transfer. 42: 375-384. Dean, A.J. and Hanson, R.K. (1992). CH and C-atom time histories in dilute hydrocarbon pyrolysis: Measurements and kinetics calculations. Int. J. Chem. Kinetics 24: 517-532. Dean, A.J., Hanson, R.K., and Bowman, C.T. (1990). High temperature shock tube study of reactions of CH and C-atoms with N 2. 23th Symp. (Int.) Combust., Combust. Inst., 259-265. Dean, A.J., Hanson, R.K., and Bowman, C.T. (1991b). A shock tube study of reactions of C atoms and CH with NO including product channel measurements. J. Phys. Chem. 95: 3180-3189. Dean, A.M. (1976). Shock tube studies of the N20/Ar and N20/H2/Ar systems. Int. J. Chem. Kinetics 8: 459-474. Dean, A.M., Johnson, R. L., and Steiner, D.C. (1980). Shock-tube studies of formaldehyde oxidation. Combust. & Flame 37: 41-62. Deppe, J., Friedrichs, G., and Wagner, H.G. (1998) FM spectroscopy behind shock waves. 27th Syrup. (Int.) Combust., Combust. Inst., Work-in-Progress Poster W4F05. Deppe, J., Friedrichs, G., Romming, H.-J., and Wagner, H.G. (1999). A kinetic study of the reaction of NH 2 with NO in the temperature range 1400--2800 K. Phys. Chem. Chem. Phys. 1: 427-435. DiRosa, M.D., Chang, A.Y., Davidson, D.E, and Hanson, R.K. (1991). CW laser strategies for multiparameter measurements of high speed flows containing either NO or 02. 29th Aerospace Sci. Meet., Reno, NV, Paper 91-0359. DiRosa, M.D. and Hanson, R.K. (1994). Collision broadening and shift of NO g(0,0) absorption lines by 02 and H20 at high temperatures. J. Quant. Spectrosc. Radiat. Transfer 52: 515-529. Du, H., Hessler, J.P., and Ogren, P.J. (1996). Recombination of methyl radicals. 1. New data between 1175 and 1750 K in the falloff region. J. Phys. Chem. 100: 974-983. Endo, H., Glanzer, K., and Troe, J. (1979). Shock wave study of collisional energy transfer in the dissociation of N2, CINO, Q3 and N20. J. Phys. Chem. 83: 2083-2090. Falcone, P.K., Hanson, R.K., and Kruger, C.H. (1983). Tunable diode laser measurements of the band strength and collision halfwidths of nitric oxide. J. Quant. Spectrosc. Radiat. Transfer 29: 205-221.
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Filer, R.A. and Holmes, H.A (1982). Kinetics of the HCN + NO 2 reaction behind shock waves. J. Phys. Chem. 86: 2935-2944. Flower, W.L., Hanson, R.K., and Kruger, C.H. (1977). Experimental study of nitric oxide decomposition by reaction with hydrogen. Combust. 5ci. & Techn. 5: 115-128. Frank, P., Bhaskaran, K.A., and Just, T. (1986). High-temperature reactions of triplet methylene and ketene with radicals. J. Phys. Chem. 90: 2226-2231. Fueno, T., Tabayashi, K., and Kajimoto O. (1973). Bimolecular dissociation of cyanogen behind incident shock waves. J. Phys. Chem. 77: 575-581. Fujii, N., Murayama, S., Kobayashi, T., and Nosaka, Y. (1992). High temperature reaction of NO and H2 behind shock waves. 18th Int. Syrup. Shock Waves, 741-746. Hanson, R.K. (1978). High-resolution spectroscopy of shock-heated gases using a tunable infrared diode laser. 11th Int. Symp. Shock Waves, 432-438. Hanson, R.K. (1977). Shock tube spectroscopy: Advanced instrumentation with a tunable diode laser. Appl. Opt. 16: 1479-1481. Hanson, R.K. (1986). Combustion diagnostics: Planar flow field imaging. 21st Syrup. (Int.) Combust., Combust. Inst., 1677-1691. Hanson, R.K., Chang, A.Y., and Davidson, D.E (1988) Modern shock tube methods for chemical studies in high temperature gases. Thermophysics Plasmadynamics and Lasers Conference, San Antonio TX, paper 88-2712. Hanson, R.K., Monat, J.P., Flower, W.L., and Kruger, C.H. (1975). Decomposition of NO studied by infrared emission and CO laser absorption. 10th Int. Symp. Shock Waves, 536-543. Hanson, R.K., Monat, J.P., and Kruger, C.H. (1976). Absorption of CO laser radiation by NO. J. Quant. Spectrosc. Radiat. Transfer 16: 705-713. Hanson, R.K., Salimian, S., Kychakoff, G., and Booman, R.A. (1983). Shock tube absorption measurements of OH using a remotely located dye laser. Appl. Opt. 22: 641-643. Hanson, R.K., Seitmian, J.M., and Paul, PH. (1990). Planar laser fluorescence imaging in combustion gases. Appl. Phys. B50: 441-445. He, Y., Wu, C.H., Lin, M.C., and Melius, C.E (1994). The reaction of CN with NO at high temperatures in shock waves. 19th Int. Syrup. Shock Waves, 89-94. Heffington, W.M., Parks, G.E., Sulzmann, K.G.P., and Penner, S.S. (1976). High-temperature absorption coefficient of methane at 3.392 microns. J. Quant. Spectrosc. Radiat. Transfer 16: 839-841. Herzberg, G. (1950). Spectra of Diatomic Molecules, Princeton, Van Nostrand. Herzler, J., Leiberch, R., Mick, H.J., and Roth, P. (1998). Shock tube study of the formation of TiN molecules and particles. Nano Structured Materials Vol 10, 1161-1171. Hidaka, Y., Takahashi, S., Kawano, H., and Suga, M. (1982). Shock-tube measurement of the rate constant for excited OH(A) formation in the hydrogen-oxygen reaction. J. Phys. Chem. 86: 1429-1433. Hippler, H., Reihs, C., and Troe, J. (1994). Shock tube absorption study of the oxidation of benzyl radicals. 25th Symp. (Int.) Combust., Combust. Inst., 37-43. Hippler, H., Seisel, S., and Troe, J. (1994). Pyrolysis of p-xylene and of 4-methyl benzyl radicals. 25th Symp. (Int.) Combust., Combust. Inst., 875-882. Hippler, H., Troe, J., and Willner, J. (1990b). Shock wave study of the reaction HO 2 + HO 2 = H20 2 + 02 confirmation of a rate constant minimum near 700K. J. Chem. Phys. 93: 1755-1760. Hsu, D.S.Y., Shaub, W.M., Creamer, T., Gutman, D., and Lin, M.C. (1983). Kinetic modeling of CO production from the reaction of CH 3 and 02 in shock waves. Bet. Bunsenges. Phys. Chem. 87: 909-919. Hwang, S.M., Rabinowitz, M.J., and Gardiner, W.C. (1993). Recombination of methyl radicals at high temperature. Chem. Phys. Lett. 205: 157-162.
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Hwang, S.M., Wagner, H.G., and Wolfe, T. (1988). Recombination of CH3 radicals at elevated pressures and temperatures. 23rd Symp. (Int.) Combust., Combust. Inst., 99-105. Jachimowski, C.J. (1977). An experimental and analytical study of acetylene and ethylene oxidation behind shock waves. Combust. & Flame 29: 55-66. Kajimoto, O., Yamamoto T., and Fueno, T. (1979). Kinetic studies of the thermal decomposition of hydrazoic acid in shock waves. J. Phys. Chem. 83: 429--435. Kijewski, H. and Troe, J. (1971). Study of the pyrolysis of H202 in the presence of He and CO by the use of UV absorption of HO 2. Int. J. Chem. Kinetics 3: 223-235. Klatt, M., Spindler, B., and Wagner, H.G. (1995). Minor decomposition channels of CH3NH 2 at high temperatures. Zeit. fur Phys. Chemie 191: 241-249. Klenk, W, Stuhler, H., and Frohm, A. (1997). Experimental investigation of the excitation of internal degrees of freedom of NO behind shock waves. 21st Int. Symp. Shock Waves, 207-212. Kohse-H6inghaus, K., Davidson, D.E, Chang, A.Y., and Hanson, R.K. (1989). Quantitative NH 2 concentration determination in shock tube laser-absorption experiments. J. Quant. Spectrosc. Radiat. Transfer 42: 1-17. Koudriavtsev, N.N., Sukhov, A.M., and Shamshev, D.P. (1995). Pressure influence on the rate of the NF 3 decomposition behind shock waves. 19th Symp. Shock Waves, Vol. 2, 43-46. Kruse, T. and Roth, P. (1996). Kinetics of C2-radical reactions during shock induced pyrolysis of acetylene. 21st Int. Syrup. Shock Waves, 851-856. Kruse, T. and Roth, P. (1997). C 2 reactions with 02, H2, and N 2 studied by perturbation of shock induced pyrolysis of acetylene. 21st Int. Symp. Shock Waves, 213-217. Lim, K.P and Michael, J.V. (1994). Thermal rate constants for the C1 + H 2 and C1 + D2 reactions between 296 and 3000 K. J. Chem. Phys. 101: 9487-9498. Kunz, A. and Roth, P. (1998). Dissociation of SiCl 4 based on C1- and Si-concentration measurements. 27th Symp. (Intl.) Combust., Combust. Inst., 261-267. Langlois, S., Birbeck, T.P., and Hanson, R.K. (1994). Temperature-dependent collision-broadening parameters of H20 lines in the 1.4 micron region using diode lasers absorption spectroscopy. J. Molecular Spectroscopy 167: 272-281. Lira, K.P. and Michael, J.V. (1993). The thermal decomposition of CH3C1 using the Cl-atom absorption method and the bimolecular rate constant for O + CH 3 (1609-2002 K) with a pyrolysis photolysis-shock tube technique. J. Chem. Phys. 98: 3919-3928. Lim, K.P. and Michael, J.V. (1994). Thermal decomposition of VH2C1z. 25th Symp. (Int.) Combust., Combust. Inst., 809-816. Louge, M.Y. and Hanson, R.K. (1984a). High temperature kinetics of NCO. Combust & Flame 58: 291-300. Louge, M.Y. and Hanson, R.K. (1984b). Shock tube study of cyanogen oxidation kinetics. Int. J. Chem. Kinetics 16: 231-250. Louge, M.Y. and Hanson, R.K. (1984c). Shock tube study of NCO kinetics. 20th Symp. (Int.) Combust., 665-672. Louge, M.Y. and Hanson, R.K. (1986). Shock tube study of high temperature absorption spectroscopy of CH at 431 rim. 15th Int. Symp. Shock Waves, Berkeley, California, 827-833. Louge, M.Y., Hanson, R.K., Rea, E.C., and Booman, R.A. (1984). Quantitative high temperature absorption spectroscopy of NCO at 305 and 440nm. J. Quant. Spectrosc. Radiat. Transfer. 32: 353-362. Markus, M. (1995). Laser-Anwedungen in Der Hochternperaturkinetik: Farbstoff-Ringlaser-Spektroskopie und UV-Laser-Photolyse, Doktor-lngenieur Thesis, Univ. Duisburg. Markus, M.W and Roth, P. (1992). On the reaction of CH with CO studied in high temperature CH4/Ar + CO mixtures. Int. J. Chemical Kinetics 24: 433-445. Markus, M.W and Roth, P. (1995). Shock tube study of the reaction CH + NO -- products using a perturbation method. 19th Symp. Shock Waves, 95-100.
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Markus, M.W. and Roth, P. (1996). Development of a quantitative ring dye laser absorption diagnostic for free Sill radicals. J. Quant. 5pectrosc. Radiat. Transfer 56: 489-499. Markus, M.W., Roth, P., and Just, T. (1996). A shock tube study of the reactions of CH with CO 2 and 02. Int J. Chem. Kinetics 28: 171-179. Markus, M.W., Roth, P., and Tereza, A.M. (1994). Thermal decomposition of CH 2 verified by product concentration measurements of C, H and CH. 25th 5ymp. (Int.) Combust., Combust. Inst., 705-712. Markus, M.W., Woiki, D., and Roth, P. (1992). Two channel thermal decomposition of CH 3. 24th 5ymp. (Int.) Combust., Combust. Inst., 581-588. Masten, D.A., Hanson, R.K., and Bowman, C.T. (1990). Shock tube study of the reaction H + 02 = OH + O using OH laser absorption. J. Phys. Chem. 94: 7119-7128. McMillin, B.K., Lee, M.P., Palmer, J.L., and Hanson, R.K. (1992). Two-dimensional imaging of shock tube flows using planar laser induced fluorescence. 18th Int. 5ymp. Shock Waves, 819-824. McMillin, B.K., Lee, M.P., Paul, P.H., and Hanson, R.K. (1989). Planar laser induced fluorescence imaging in a shock tube. AIAA Paper 89-2566. Mertens, J.D., Chang, A.Y., Hanson, R.K., and Bowman, C.T. (1989). Reaction kinetics of NH in the shock tube pyrolysis of HNCO. Int. J. Chem. Kinetics 21: 1049-1067. Mertens, J.D., Chang, A.Y., Hanson, R.K., and Bowman, C.T. (1991a). A shock tube study of reactions of NH with NO, 02 and O. Int. J. Chem. Kinetics 23: 173-196. Mertens, J.D., Chang, A.Y., Hanson, R.K., and Bowman, C.T. (1992a). A shock tube study of reactions of atomic oxygen with isocyanic acid. Int. J. Chem. Kinetics 24: 279-295. Mertens, J.D., Dean, A.J., Hanson, R.K., and Bowman, C.T. (1992b). A shock tube study of reactions of NCO with O and NO using NCO laser absorption. 24th Syrup. (Int.) Combust., 701710. Mertens, J.D., Kohse-Hoinghaus, K., Hanson, R.K., and Bowman, C.T. (1991b). A shock tube study of H + HNCO = NH 2 + CO. Int. J. Chem. Kinetics 23: 655-668. Mertens, J.D., Wooldridge, M.S., and Hanson, R.K. (1995). A laser photolysis shock tube study of the reaction of OH with NH 3. 19th Int. Symp. Shock Waves, 37-42. Michael, J.V., Kim, K.P., Kumaran, S.S., and Kiefer, J.H. (1993). Thermal decomposition of carbon tetrachloride. J. Phys. Chem. 97: 1914-1919. Mick, H.J., Kruse, T., and Roth, P. (1996). A shock tube study of the reaction Si + H 2 -- Sill + H. 20th Symp. Shock Waves, 875-880. Mick, H.J. and Roth, P. (1995). A shock tube study of the oxidation of silicon atoms by NO and CO 2. 19th Int. Symp. Shock Waves, Vol 2, 47-52. Mick, H.J., Smirnov, V.N., and Roth, P. (1993). ARAS measurements on the thermal decomposition of silane. Ber. 13unsenges. Phys. Chem. 97: 793-798. Mitchell, A.C.G. and Zemansky, M.W. (1971). Resonance Radiation and Excited Atoms, London: Cambridge Univ. Press. Moller, W., Mozzhukhin, E., and Wagner, H.G. (1986). High temperature reactions of CH3. 1. The reaction of CH 3 4- H 2 = CH 4 4- H. Ber. Bunsenges. Phys. Chem. 90: 854. Monat, J.P., Hanson, R.K., and Kruger, C.H. (1977). Kinetics of nitrous oxide decomposition. Combust. Sci. & Techn. 16: 21-28. Moser, L. (1993). Anwendunge hochentwickelter Lasermethoden im Maschinenbau (The Moser Report), Stanford Univ. Nagali, V., Chou, S.I., Baer, D.S., and Hanson, R.K. (1997). Diode-measurements of temperaturedependent half-widths of H20 transitions in the 1.4 micron region. J. Quant. Spectrosc. Radiat. Transfer 57: 795-809. Nagali, V., Davidson, D.E, and Hanson, R.K. (2000). Measurements of temperature-dependent argon-broadened half-widths of H20 transitions in the 7117 cm -1 region. J. Quant. Spectrosc. Radiat. Transfer Vol 64, pp. 651-655.
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Nagali, V. and Hanson, R.K. (1997). Design of a diode-laser sensor to monitor water vapor in highpressure combustion gases. Appl. Opt. 36: 9518--9527. Nagali, V., Herbon, J., Homing; D., Bates, R., Davidson, D.E, and Hanson, R.K. (1999). Diode-laser based diagnostic to monitor water-vapor in high-pressure environments. AIAA Paper 99-0942. Natarajan, K. and Roth, P. (1992). High temperature rate coefficient of the reaction: CN + N = C + N 2 obtained by resonance absorption of N atoms. 18th Int. Syrup. Shock Waves, 787-792. Naumann, C., Braun-Unkhoff, M., and Frank, P. (1997). A shock tube study on ammonia decomposition and its reaction products. 21st Int. Syrup. Shock Waves, 133-138. Olson, D.B., Mallard, W.G., and Gardiner, W.C. (1978). High temperature absorption of the 3.39 micron He-Ne laser line by small hydrocarbons. Applied Spectroscopy 32: 489-493. Palmer, J.L. and Hanson, R.K. (1996). "Temperature imaging in a supersonic free jet of combustion gases with two-line OH fluorescence." Applied Optics, Vol. 35,485-499. Petersen, E.L., Davidson, D.E and Hanson, R.K. (1990). Kinetics modeling of shock-induced ignition in low-dilution CH4/O 2 mixtures at high pressures and intermediate temperatures. Combust. & Flame. Vol. 117, pp. 272-290. Petersen, E.L., Davidson, D.E, Rohrig, M., and Hanson, R.K. (1996a). High-pressure shock tube measurements of ignition times in stoichiometric H2/O2/Ar mixtures. 20th Int. Syrup. Shock Waves, 941-946. Petersen, E.L., ROhrig, M., Davidson, D.E, Hanson, R.K., and Bowman, C.T. (1996b). High-pressure methane oxidation behind reflected shock waves. 26th Syrup. (Int.) Combust., Combust. Inst., 799-806. Philippe, L.C. and Hanson, R.K. (1992). Laser-absorption mass flux sensor for high-speed air flows. Opt. Lett. 16: 2002-2004. Philippe, L.C. and Hanson, R.K. (1993). Laser diode wavelength modulation spectroscopy for simultaneous measurement of temperature, pressure, and velocity in shock-heated oxygen flows. AppI. Opt. 32: 6090-6103. Rea, E.C., Chang, A.Y., and Hanson, R.K. (1987). Shock tube study of pressure broadening of the A2y~+ - X2rc (0,0) Band of OH by Ar and N2. J. Quant. Spectrosc. Radiat. Transfer. 37: 117-127. Rea, E.C., and Hanson, R.K. (1983). Rapid extended range tuning of single-mode ring dye lasers. Appl. Opt. 22: 518-520. Rea, E.C. and Hanson, R.K. (1988). Rapid laser-wavelength modulation spectroscopy used as a fast temperature measurement technique in hydrocarbon combustion. Appl. Opt. 27: 4454-4464. Rea, E.C., Salimian, S., and Hanson, R.K. (1984). Rapid tuning frequency doubled ring dye laser for high resolution absorption spectroscopy in shock-heated gases. Appl. Opt. 23: 1691-1694. ROhrig, M., Petersen, E.L., Davidson, D.E, and Hanson, R.K. (1997a). A shock tube study of the pyrolysis of NO 2. Int. J. Chem. Kinetics 29: 483-493. R6hrig, M., Petersen, E.L., Davidson, D.F., Hanson, R.K., and Bowman, C.T. (1997b). Measurement of the rate coefficient of the reaction CH 4- O2 = products in the temperature range 2200 to 2600K. Int. J. Chem. Kinetics 29: 781-789. R6hrig, M., Romming, H.-J., and Wagner, H.G. (1994). A direct measurement of the reaction NH 3 4- NH = 2NH 2. Bet Bunsenges. Phys. Chem. 98: 1332-1334. Rohrig, M. and Wagner, H.G. (1994). The reaction of NH(X) with the water gas components CO 2, H20 and H 2. 25th Symp. (Int.) Combust., Combust. Inst., 975-981. Roose, T.R., Hanson, R.K., and Kruger, C.H. (1978). Decomposition of NO in the presence of NH 3. llth Int. Symp. Shock Waves, 245-253. Roose, T.R., Hanson, R.K., and Kruger, C.H. (1980). Thermal decomposition of NH 3 in shock waves. 12th Int. Syrup. Shock Waves, 476-485. Roose, T.R., Hanson, R.K., and Kruger, C.H. (1980). A shock tube study of the decomposition of NO in the presence of NH 3. 18th Symp. (Int.) Combust., Combust. Inst., 853-862.
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Spectroscopic Diagnostics
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Roth, P., Louge, M.Y., and Hanson, R.K. (1986). O- and N-atom measurements in high temperature C2H2 4-O kinetics. Combust. & Flame 64: 167-176. Roth, P. and Thielin, K. (1986). Measurement of N atom concentrations in dissociation of N 2 by shock waves. 15th Int. Symp. Shock Waves, 245-252. Ryu, S.-O., Hwang, S.M., and Rabinowitz, M.J. (1995). Shock tube and modeling study of the H + 0 2 -- OH + O reaction over a wide range of composition, pressure and temperature. J. Phys. Chem. 99: 13984-13991. Saito, K., Oda, A., and Tokinaga, K. (1995). High-temperature unimolecular decomposition of ethyl chloroformate: Comparison of the secondary competing steps with ethyl formate. 19th Int. Symp. Shock Waves, Vol. 2, 119-124. Salimian, S. and Hanson, R.K. (1983). Absorption measurements of H20 at high temperatures using a CO laser. J. Quant. Spectrosc. Radiat. Transfer 30: 1-7. Salimian, S., Hanson, R.K., and Kruger, C.H. (1984). Ammonia oxidation shock-heated NH3-NO2-Ar mixtures. Combust. & Flame 56: 83-95. Schading, G.N. and Roth, P. (1997). Shock-wave induced sublimation and dissociation of cyanuric acid aerosols. 21st Int. Symp. Shock Waves, Vol. 1,219-223. Schatz, G.C. and Ratner, M.A. (1993). Quantum Mechanics in Chemistry, Englewood Cliffs: PrenticeHall. Seitzman, J.M. and Hanson, R.K. (1993). Planar fluorescence imaging in gases, in Experimental Methods for Flows with Combustion, A. Taylor, ed., London: Academic Press, Chapter 6. Shin, K.S., Fujii, N., and Gardiner, W.C. (1989). Rate constant for O + H 2 - - O H + H by laser absorption spectroscopy of OH in shock-heated H2-O2-Ar mixtures. Chem. Phys. Lett. 161: 219. Slack, M.W. (1976). Kinetics and thermodynamics of the CN molecule. III. Shock tube measurement of CN dissociation rates. J. Chem. Phys. 64: 228-236. Szekely, A., Hanson, R.K., and Bowman, C.T. (1983a). High-temperature determination of the rate coefficient for the reaction H 2 + CH -- H + HCN. Int. J. Chem. Kinetics 15: 915-923. Szekely, A., Hanson, R.K., and Bowman, C.T. (1983b). Shock tube determination of the rate coefficient for the reaction CH + HCN -- C2N 2 + H. Int. J. Chem. Kinetics 15: 1237-1241. Szekely, A., Hanson, R.K., and Bowman, C.T. (1984a). High temperature determination of the rate coefficient for the reaction H20 + CN -- HCN + OH. Int. J. Chem. Kinetics 16: 1609-1621. Szekely, A., Hanson, R.K., and Bowman, C.T. (1984b). Shock tube study of the reaction between hydrogen cyanide and atomic oxygen. 20th Symp. (Int.) Combust., Combust. Inst., 647-654. Szekely, A., Hanson, R.K., and Bowman, C.T. (1984c). Shock tube study of the thermal decomposition of cyanogen. J. Chem. Phys. 80: 4982-4985. Szekely, A., Hanson, R.K., and Bowman, C.T. (1984d). Thermal decomposition of hydrogen cyanide behind incident shock waves. J. Phys. Chem. 88: 666-668. Takahashi, K., Honda, J., Inomata, T., and Jinno, H. (1994). High-temperature reactions of ozone with bromine behind shock waves. Rate constant for the reaction BrO + BrO -- 2Br + 02. 19th Symp. Shock Waves, Marseille, vol. 2, 65-70. Takahashi, K., Inomata, T., Abe, T., and Fukaya, H. (1997). Flame inhibition mechanism of HFC23: Kinetic studies on the reactions of CHF 3 4- H, CF 3 + O, and CHF 2 + O. 21st Int. Symp. Shock Waves, vol. 1, 163-168. Takahashi, K., Inoue, A., and Inomata, T. (1996). Direct measurements of rate coefficients for thermal decomposition of methyl halides using shock tube ARAS technique. 20th Int. Syrup. Shock Waves, vol. 2, 959-964. Thaxton, A.G., Lin, M.C., Lin, C.-Y., and Melius, C.E (1997). Thermal oxidation of HCN by NO 2 at high temperatures. 21st Int. Symp. Shock Waves, vol. 1,245-249. Thielin, K. and Roth, P (1987). Resonance absorption measurements of N, O, and H atoms in shock heated HCN/O2/Ar mixtures. Combust. & Flame 69: 141-154.
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Thielin, K. and Roth, P (1996). N atom measurements in high-temperature N 2 dissociation kinetics. AIAAJ. 24: 1102-1105. Thoma, M. and Hindelang, EJ. (1995). Experiments on shock-heated ozone dissociation in oxygen/air using UV laser spectroscopy. 19th Intl. Syrup. Shock Waves, Vol. 2. Marseilles, 59-64. Varghese, PL. and Hanson, R.K. (1981). Collision width measurements of CO in combustion gases using a tunable diode laser. J. Quant. Spectrosc. Radiat. Transfer 26: 339-347. Votsmeier, M., Song, S., Davidson, D.F., and Hanson, R.K. (1999a). Sensitive detection of NH 2 in shock tube experiments using frequency modulation spectroscopy. Int. J. Chem. Kinetics, Vol. 31, pp. 445--453. Votsmeier, M., Song, S., Davidson, D.E, and Hanson, R.K. (1999b). Shock tube study of monomethylamine thermal decomposition and NH 2 high temperature absorption coefficient. Int. J. Chem. Kinetics, Vol. 31, pp. 323-330. Votsmeier, M., Song, S., Hanson, R.K., and Bowman, C.T. (1999c). A shock tube study of the product branching ratio for the reaction NH 2 + NO using frequency modulation detection of NH 2. J.PC.A Vol 103, pp. 1566-1571. Whittaker, E.A., Pokrowsky, P. et al. (1983). Improved laser technique for high sensitivity atomic absorption spectroscopy in flames. J. Quant. Spectrosc. Radiat. Transfer 30: 289-296. Whittaker, E.A., Wendt, H.R. et al. (1984). Laser FM spectroscopy with photochemical modulation. Appl. Phys. B 35: 105-111. Wintergest, K. and Frank, P (1995). Direct measurement of the reaction H + CO 2 at elevated temperatures. 19th Int. Syrup. Shock Waves, Vol 2. 77-82. Woiki, D., Kunz, A., and Roth, P. (1997). Laser flash photolysis of SiN4 and Si2H6 behind shock waves. 21st Int. Syrup. Shock Waves, 183-187. Woiki, D. and Roth, P (1995). On the formation of S(3P) and S(1D) during the thermal decomposition of COS behind shock waves. 19th International Shock Tube Symposium, Marseilles, Vol 2. 53-58. Woiki, D. and Roth, P (1996). Bimolecular S-atom reactions studied in pyrolysis and photolysis shock-wave experiments. 20th Int. Syrup. Shock Waves, 845-850. Woiki, D., Votsmeier, M., Davidson, D.E, Hanson, R.K., and Bowman, C.T. (1998). CH-radical concentration measurements in fuel-rich CH4/O2/Ar and CH4/O2/NO/Ar mixtures behind shock waves. Combust & Flame 113: 624-626. Wooldridge, M.S. (1995a). Shock tube studies of elementary hydroxyl-radical reactions important in combustion systems. PhD thesis. HTGL Rept. No. T-324, Stanford Univ. Wooldridge, M.S., Hanson, R.K., and Bowman, C.T. (1994a). A shock tube study of the CO + OH = CO 2 + H reaction. 25th Syrnp. (Int) Combust., Combust. Inst., 741-748. Wooldridge, M.S., Hanson, R.K., and Bowman, C.T. (1994b). A shock tube study of the O + OH -- H20 4- H reaction. Int. J. Chem. Kinetics 26: 389-401. Wooldridge, M.S., Hanson, R.K., and Bowman, C.T. (1995d). A shock tube study of nitric acid decomposition. 19th Int. Syrup. Shock Waves, Vol. II. Marseilles, France, 83-88. Wooldridge, M.S., Hanson, R.K., and Bowman, C.T. (1996a). A shock tube study of CO 4- OH = CO 2 4- H and HNCO + OH = products via simultaneous laser absorption measurements of OH and CO 2. Int. J. Chem. Kinetics 28: 361-372. Wooldridge, S.T., Hanson, R.K., and Bowman, C.T. (1993). Development of a CW laser absorption diagnostic for measurement of CN in shock tube experiments. J. Quant. Spectrosc. Radiat. Transfer 50: 19-34. Wooldridge, S.T., Hanson, R.K., and Bowman, C.T. (1995b). Measurements of argon collision broadening in the CN B2y~+ - X2Y~+ (0,0) spectrum.J. Quant. Spectrosc. Radiat. Transfer 53: 481-492. Wooldridge, S.T., Hanson, R.K., and Bowman, C.T. (1995c). Simultaneous laser absorption measurements of CN and OH in a shock tube study of HCN + OH = products. Int. J. Chem. Kinetics 27: 1075-1087.
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Spectroscopic Diagnostics
785
Wooldridge, S.T., Hanson, R.K., and Bowman, C.T. (1996b). A shock tube study of reactions of CN with HCN, OH, and H 2 using CN and OH laser absorption. Int. J. Chem. Kinetics 28: 245-258. Wooldridge, S.T., Mertens, J.D., Hanson, R.K., and Bowman, C.T. (1994). A shock tube study of the reactions of CN and NCO with NO 2. 25th Symp. (Int.) Combust., 983-991. Yamauchi, N., Kosaka, K., Miyoshi, A., Koshi, M., and Matsui, H. (1997). Thermal decomposition of alkyl radicals. 21st Int. Symp. Shock Waves, 127-132. Yu, C.-L., Wang, C., and Frenklach, M. (1995). Chemical kinetics of methyl oxidation by molecular oxygen. J. Phys. Chem. 99: 14377-14387. Zelson, L.S., Davidson, D.E, and Hanson, R.K. (1994). VUV absorption diagnostic for shock tube kinetics studies of C2H4. J. Quant. Spectrosc. Radiat. Transfer 52:31-43.
FIGURE 5.1.13 Visualization of the test problem (detail) with shearing interferometry (from Kleine, 1994). (a) Low beam separation (divergence angle ~ - - I t ) 9 (b) high beam separation (divergence angle ~ -- 5t).
FIGURE 5.2.9
OH PLIF concentration imaging. For details see text.
FIGURE 5.2.10 PLIF experimental setup for two-wavelength OH temperature measurement in a shock tunnel facility. For details see text.
FIGURE 5.2.11
OH PLIF temperature imaging. For details see text.
CHAPTER
6
Shock Capturing PHILIP ROE W. M. Keck Foundation Laboratory for Computational Fluid Dynamics, Department of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan, 48109-2118, USA
6.1 6.2
6.3
6.4
6.5 6.6 6.7 6.8 6.9
Introduction Analytical Background 6.2.1 Conservation 6.2.2 Weak Solutions 6.2.3 Physical Solutions: Entropy Conditions 6.2.4 Quasilinear Form, Jacobians 6.2.5 Wavespeeds, Hyperbolicity, Nonlinearity, and Convexity 6.2.6 Characteristic Variables, Centered Waves 6.2.7 Riemann Problems Numerical Background 6.3.1 Finite-Volume Methods: The Lax-Wendroff Theorem 6.3.2 Error and Accuracy 6.3.3 The Simplest Hyperbolic Problem 6.3.4 Time-Stepping, Flux Integration, SemiDiscretization One-Dimensional Methods 6.4.1 The Godunov Scheme 6.4.2 A Linearized Riemann Solver 6.4.3 The Entropy Fix 6.4.4 Positivity 6.4.5 High-Resolution Schemes 6.4.6 Essentially Nonoscillatory (ENO) Schemes 6.4.7 Avoiding the Riemann Problem Source Terms Multidimensional Application 6.6.1 Flux Calculation Grid Generation and Adaptivity Anomalous Solutions "Genuinely" Multidimensional Methods
Handbook of Shock Waves, Volume 1 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086431-2/$35.00
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6.10 Further Reading 6.11 An Example 6.12 Concluding Remarks References
6.1
INTRODUCTION
Faced with the task of writing a computer code to solve some flow problem expected to contain shockwaves, the simplest decision would be simply to ignore any complexities that they might cause, and to write the code as though the flow were continuous everywhere. The most likely outcome would seem to be that the shocks would appear as more-or-less well-defined transition regions. We might speak of the shocks as being "captured" by the computation, much as the spirit of a personality can be captured by an artist in a quick sketch. But there could very well be some doubts about how valid these solutions actually were. Would these shockwaves actually be in the right place, and would the transitions across them really resemble the true jumps? The shock-capturing techniques that are nowadays widely used for many important calculations have arisen from systematic efforts to allay these fears. Despite the fact that no special measures are taken to detect the presence of shocks, care is taken with other aspects of the method so that rather precise statements can be made about the results. In particular, as the size of the computational mesh is systematically reduced, and hence the width of the transition regions is also reduced, the jumps across them will indeed obey the Rankine-Hugoniot conditions, both as regards the strength of the jump and the speed of its propagation. To achieve this much is actually quite easy, but there are other desirable properties that are more elusive. 1. The quality of the transition should be crisp; neither excessively smeared out nor accompanied by meaningless oscillations. 2. Only shocks that are physically valid should be produced; for example, rarefaction shocks should never appear. 3. The method should be computationally robust; the code should not require that any free parameters must be adjusted to maintain good behavior over a wide range of problems. 4. It should also be physically robust; it should not be possible to give meaningful data that cause the code to break down. In addition, the code should be economical to run. Despite at least 40 years of very energetic research and some outstanding predictive successes, there is still no consensus as to the best practical methods of achieving these objectives, and a beginner would find the literature
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bewildering. Therefore, this chapter will not attempt to be comprehensive. I will focus on the approaches that I personally know best, but will mention and refer to viable alternatives. I will not conceal the remaining uncertainties, but will attempt to orient the reader for his or her further explorations. I hope that the descriptions of several different approaches will not be found confusing or intimidating. Shocked flows of simple substances under conditions that are not too extreme can be computed accurately and reliably using a basic "toolkit" of nonlinear reconstruction and Riemann solver that is very easy to master. Alternatives are sometimes required if the flow conditions are extreme, or if computational cost or high accuracy are of major concern. I have however included a short section (6.9) on the recent trend toward "genuinely multidimensional" methods. The approach is neccessarily mathematical (in language if not in rigor) but concentrates as much as possible on concepts rather than formulas. Detailed algebra is sometimes given to support conclusions that can be made precise; a less formal style seemed appropriate elsewhere. Throughout, the well-explored Euler equations will be used to illustrate the ideas, but frequent remarks concerning other sets of equations will appear. I hope that these remarks, supplemented by the bibliography, will make this review useful to a diverse range of readers.
6.2 ANALYTICAL BACKGROUND 6.2.1
CONSERVATION
Conservation is the bedrock of all successful shock-capturing methods. A quantity q(x, y, z, t) is said to be conserved if we find, having drawn some arbitrary control volume f2, that changes in the amount of q contained within f~ depend only on events at the boundaries. For example, mass is a conserved quantity that changes only because it flows through the boundary. Energy is also counted as a conserved quantity because it changes due to flow through the boundary and also because of work done at the boundary. Commonly, the densities of mass, momentum, and energy are spoken of as the "conservative variables", but in some contexts other quantities may also be conserved. More abstractly, a set of conservation laws is a set of statements about some vector of length m containing unknowns u; that for any control volume f2 with boundary 0f2 (See Fig. 6.1)
a l udV 4- I
0t
F.ndS-0
(6.1)
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FIGURE 6.1 A control volume f2, its boundary 3fl, and a normal n to that boundary (left-hand side); the control volume divided into two parts by a surface of discontinuity Z (right-hand side).
Here the first term is the rate of change of the conserved quantities within the control volume. In the second term F is a set of d v e c t o r s F 1 . . . Fd, where d is the number of space dimensions, with each vector of length m. These are called the flux vectors. An element of 3f2 has an outward normal defined by d n - d n 1. . . . . dn d and the inner product F2dn 1 + - - . + Fadn a is the flux through that element (accounting as explained above for all surface forces as well as the flow rates). The system is closed if the fluxes F i can be expressed as known functions of the conserved quantities u. For inviscid gasdynamics 1, the appropriate definitions are
u --
P
PVl
pv2
PV3
pv I
pv21 + P
pVl V2
PVl V3
pv 2
, F 1 --
pv iv 2
, F 2 --
pv 2 + p
, F3 - -
Pv2v 3
flY3
flVl V3
pV2V3
pv 2 + P
E
vl(E + r)
vz(E + p)
v3(E + p)
(6.2)
where p is the mass density, v is the velocity, E is the density of total (i.e. internal + kinetic 2 energy, and p is the static pressure, which can be obtained from the conserved variables via an equation of state if the flow is in 1Allowing for viscosity and heat conduction does not destroy conservation, but the fluxes now contain viscous stresses and heat flow terms that depend on the derivatives of the solution. 2In more general contexts, the total energy is the sum of whatever energies are expected to participate in the flow. Depending on context these may include potential energy, magnetic field energy, chemical energy, or rest mass energy.
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Shock Capturing
thermodynamic equilibrium. For example, if the fluid is an ideal gas P - (7 - 1)pe and for more general fluids p - p(p, e) where E 1 e .... [[vll2 p 2 +
l/1
2
+
U21
"
The combination E + p appears frequently enough to deserve its own name of total enthalpy; h = (E + p)/p is called the specific total enthalpy. Any problem that can be expressed as such a set of integral conservation laws can also be expressed as a set of partial differential equations, because Gauss's Theorem can be used to rewrite Eq. (6.1) as I (0tU+ d i F)dV v -
0
~2
which can only be true for an arbitrary control volume if 0tu + div F -- 0
(6.3)
The equations are then said to be in divergence form. However, we have derived this by implicitly assuming that the derivatives exist. Bearing in mind that we wish to deal with the possibility of shock waves, there may be places where the solution does not posess derivatives, and so cannot be described by a differential equation. However, the integral form is always valid, and it is the realization of that simple fact which makes the capturing strategy respectable.
6.2.2 WEAK SOLUTIONS A useful theoretical construct that combines the differential and integral viewpoints is the concept of a weak solution of Eq. (6.1) (Lax, 1954). This is some function u(x, t) such that I ~b(x, t)(0tu + div F)dV - 0
(6.4)
for every test function ~(x, t). Clearly this condition is met if the solution is smooth (differentiable) and satisfies Eq. (6.3) everywhere. However, if there is a surface E across which the solution jumps discontinuously, as shown on the right-hand side of Fig. 6-1, the notion of a weak solution can also be justified. If it is assumed that we can integrate Eq. (6.4) by parts, then we obtain d ~2
~b(x, t)(udx - F d t ) - [
(nOt(/) + F - g r a d
Fdp)dV
(6.5)
d~2
where (dx, dt) is the outward normal to an element of 0D. in the (d + 1)dimensional space (x, t). Now consider a control volume ~2 that crosses Z and
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is divided by E into two parts, f~l and ~22. If we apply Eq. (6.4) to f~ (by taking ~b(x, t) to equal unity inside ~ and zero outside) and then subtract from this the result of applying Eq. (6.4) to f]l, f]2 separately, the outcome is
Jz* ~ ( xt){(U2 , -
ul)dx - (F2 - F1)dt} - 0
(6.6)
where the integral is over the portion E* of g cut off by ~. This is only possible for an arbitrary test function ~b(x, t) if, over the whole of E, the jump conditions (U2 -- Ul)dX - (F 2 -
F1)dt =
0
(6.7)
are satisfied. This is made more understandable by noting that the d-vector dx/dt = Vr~ is the speed with which the discontinuity surface moves normal to itself, so that (U 2 - - U l ) V E - - F 2 -
F 1
(6.8)
These are the Rankine-Hugoniot conditions. Remarkably, the notion of weak solution incorporates both the differential equations and the jump conditions. In the theory of computational conservation laws one of the main objectives is to prove that the solutions computed by a particular method will converge, as the mesh is refined, to a weak solution of the problem under consideration.
6.2.3 PHYSICAL S O L U T I O N S : ENTROPY CONDITIONS However, it is not always guaranteed that the solution to which the code converges is physically correct. It is well known that although the jump conditions (Eq. (6.8)) are reversible, in the sense that if they are satisfied by the pair of states (ul, u2) they will also be satisfied by the pair of states (u2, Ul), it is not always the case that both transitions are physically realizable. For example, in gasdynamics, shocks that increase the density of the fluid are allowed, but shocks that decrease the density are not. On the other hand, contact discontinuities that separate parallel flows at different temperature, or vortex sheets across which only the tangential component of velocity changes, have no such "forbidden directions." If we are to have confidence in the output from a computer code, it is essential that the scheme employed respects these conditions whenever they apply. In other words, it is not enough that the captured discontinuities satisfy the Rankine-Hugoniot conditions, but that the jump takes place in an admissible direction.
Shock Capturing
793
There are at least three essentially independent tests for the validity of a jump in the solution. The evolutionary condition (Lax, 1957) is that a valid shock must be able to form out of a smooth compression wave. This can also be interpreted that the shock is associated with waves of a particular family, which enter the shock from both sides, as in Fig. 6.2(a). There is also the entropy condition that some scalar quantity, which is a convex function of the conserved variables and which is itself conserved in smooth flows, always increases across the jump (Lax 1971, Liu, 1976). For gasdynamics this is the thermodynamic entropy. For more general problems it may not always be apparent that a function with these mathematical properties exists. 3 Finally there is the viscosity condition that the jump is a mathematical limit of some more complete model (such as the Navier-Stokes equations) as the parameters involved (viscosity, conductivity, or other dissipations) tend to zero from positive values (Liu, 1981). The terms selection criteria and admissibility conditions are used to describe these tests. Sometimes they are all referred to collectively as entropy conditions. The discontinuities that they select are said to be admissible, entropy satisfying, or entropic. In ideal gasdynamics, the validity of any proposed jump can be settled by appeal to any of these criteria, and they will all agree. However, for systems of equations more complex than gasdynamics, the validity of a jump can be a controversial issue. For example, the intermediate shocks in magnetohydrodynamics (MHD) are still being debated (De Sterck, 1999; Falle and Komissarov, 1999; Wu, 1995). Therefore one cannot guarantee the validity of computed results, since even the mathematical model is unclear. Nevertheless, appeal is often made to these ideas in the design of numerical methods. Sometimes it is possible to prove that a discrete version of the thermodynamic entropy condition is satisfied. Usually, however, proof is only possible for rather simple methods that may not be very accurate. The viscosity condition may be used as semiempirical justification for including in the code some artificial terms that do not precisely represent the physical dissipation but that have a dissipative effect. Generally, the inclusion of such artificial dissipation makes the code more robust but less accurate. A more sophisticated practice is to employ the evolutionary condition to increase dissipation only where it is needed. The local use of increased dissipation is sometimes called an entropy fix (Section 6.4.3).
3The question is connected with whether the equations can be written in symmetric form (see Harten (1993)) which in turn links with well-posedness of the initial-value problem (Friedrichs (1958)). These issues are of practical importance, because attempting to write code for sets of equations that lack a proper mathematical underpinning can be a frustrating business.
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6.2.4 QUASILINEARFORM,JACOBIANS An alternative way to write the conservation laws is to introduce the Jacobian matrices 4 ~gF i
Ai = - ~ ,
i-- 1,2,3
Then the divergence form of Eqs. (6.3) becomes ~u 0 = -~ + divF ~
~ ~OF t 1 l ~F 2
~u =
(6.9)
~u +
~F3 au
+
(6 10) ~u
+
The last form is called the quasilinearform of the equations, if, as is usually the case, the matrices Ai depend on u. If the matrices are constant, the equations are linear. Equations in quasilinear form do not always derive from conservation laws. The matrices A~ may not be the derivatives of any flux (and in such a case it is not correct to call them Jacobian matrices). We then sometimes speak of the equations as being in nonconservative form. This may happen either because we have deliberately chosen not to use the conserved variables as the unknowns, or else because we are considering a problem that cannot be expressed purely as a set of conservation laws. This is the situation for many formulations of two-phase flow, and it is an open question in such problems how to systematically obtain jump relationships. Shock capturing may be attempted for problems in nonconservation form, but loses much of its theoretical justification. The results are often found satisfactory for weak discontinuities, especially if any conservation property that does exist is preserved. For example, in a bubbly liquid, momentum is not conserved either for the liquid or the bubbles, but should be conserved for the mixture (Wallis, 1967). In magnetohydrodynamics, mass, momentum, and energy are conserved (with magnetic field intensity contributing to the energy) and the magnetic field components obey equations that can be put into conservation form, but there turn out to be advantages in not treating them as conserved quantities (Powell et al., 1999). Some general observations about the possibility of splitting equations into conservative and advected subsets are made by 4Given two vectors p, q such that q depends on p, a Jacobian matrix A contains the partial derivatives of q with respect to the elements of p; thus ai.j = Oqi/~p) and a small change dp will change q by dq = Ad p.
795
Shock Capturing
(Fedkin et al., 2000), who also exploit the distinction made between "genuinely nonlinear" and "linearly degenerate" waves in Section 6.2.5. We now give the Jacobian of the flux vector F 1 in the Euler equations: 0 7-1
A1 =
Ilvll 2 - Vl2
(3 -- ~)V 1
--V1V 2
V2
--VIV 3
V3
2
2 - ? 2
1
V1 ]]v]]
vl a2
a2
~- 1
7- 1
3(2 - ~)Vl2 + 1 ilvl12 2 "2
0 -(7-
0 1)v2
-(?-
0 1)v3
7- 1
v1
0
0
0
v1
0
--(7- I)VlV2
--(~--I)VlV3
(6.12)
~Vl _
Usually the coefficient matrices can be substantially simplified by choosing nonconsevative variables. If we choose variables p - - ( p , v 1, v2, v3, p) (usually called the primitive variables), the matrix A 1 is much simpler:
A1 =
-V 1
/9
0
0
0
0
v1
0
0
1/p
0
0
3/1
0
0
0
0
0
V1
0
0
pa 2
0
0
v1
(6.13)
For this reason, it is usually easier to state analytical properties in terms of nonconservative variables. However, a shock-capturing code needs to use the conserved variables whenever they are appropriate. In either case, the eigenstructure of the matrices is very significant. The eigenvalues will be shown in the next section to be the speeds of plane waves; they will be found to be the same whatever variables are used. The right eigenvectors give the directions in state space along which a plane wave changes the solution. The left eigenvectors will appear in Section 6.2.6.
796 6.2.5
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W A V E S P E E D S , HYPERBOLICITY,
N O N L I N E A R I T Y , AND C O N V E X I T Y Any surface of discontinuity behaves essentially like a one-dimensional jump in the normal direction. Without loss of generality we can orient the coordinates so that this normal direction coincides with the x-axis. The normal speed is then S such that 5 S[u] = [Fx] (see Eq. (6.8)). If the wave is weak, the jumps across it will be such that Ax[u ] = [Fx], where A x is the Jacobian matrix containing the derivatives of F x with respect to u. Comparing these two conditions shows that the propagation speed of a weak wave must be an eigenvalue of A x, and the change produced by the wave must be along the corresponding right eigenvector. If Ax is m x m, representing m equations in m unknowns, and all m of its eigenvalues are real, and all m of its eigenvectors are distinct, the equations are said to be hyperbolic. Every solution of such equations (in one dimension) is locally the superposition of some or all of m independent wavelike motions. In higher dimensions a problem is said to be hyperbolic if it is hyperbolic in the one-dimensional sense for every possible orientation of the waves. If two or more eigenvalues coincide, there is no problem provided that the corresponding eigenvectors are distinct; this just means that several different kinds of wave happen to travel with the same speed. But if there is a missing eigenvector (Jordan block) there are solutions that are not wavelike, and that generally grow linearly with time (Bouchut and James, 1999). If some of the eigenvalues are complex, there are nonwave-like solutions that grow exponentially with time. Both of these situations are commonly found in mathematical models representing two-phase flows (such as those of liquids containing numerous small bubbles (Coquel et al. 1997; Stewart and Wendroff, 1984; Wallis, et al., 1967)). There is controversy over whether such models might be valid (or are simply erroneous) and over how they should be handled numerically if they are valid. For gasdynamics, the five eigenvalues (of Eqs. (6.12) or (6.13) are v 1 - a , v 1, v 1, v 1, v 1 4-a. The waves with speed v 1 -4-a are the acoustic waves. The right eigenvector corresponding to such a wave in conserved variables is (Godlewski and Raviant, 1996; Hirsch, 1988; Leveque, 1992)
du c -
dp
1
dm I
v I -]- gl
dm 2
(x
V2
dm3
v3
dE
h 4- v i a
(6.14)
5We follow the common practice of denoting the jump of some quantity q across a discontinuity by [ql
Shock Capturing
797
In primitive variables we have dp
1
dVl dv p -
dv 2
:koa
(6.15)
cx
dv 3 dp
0 a2
If a weak acoustic wave passes through a uniform region and slightly raises the density, a second acoustic wave that follows it can be shown to travel slightly faster (Courant and Friedrichs, 1976). A sequence of such waves would eventually overtake each other and coalesce into a shockwave. On the other hand, if each wave reduces the density, then sucessive waves move slightly slower, and the disturbances fan out. These typical phenomena are illustrated in Fig. 6.2(a,b). Waves that affect the propagation speed of subsequent waves of the same kind are called genuinely nonlinear. The gasdynamic waves that propagate ,with speed u are contact surfaces and shear waves or vortex sheets. Across either kind of wave the pressure does not jump, and neither does the normal component of velocity. Across a contact surface there is a jump in the thermodynamic properties other than pressure. Across a vortex sheet all thermodynamic properties are constant, but the tangential components of velocity change. (Because there are two such components the algebra counts the waves twice.) The slip surface that flows downsteam from a shock intersection is the superposition of a contact surface and a vortex sheet. None of these waves affect the subsequent propagation of similar waves (although a contact surface does change the speed of an acoustic wave). Waves having this property are said to be linearly degenerate (Godlewski and Raviart, 1996; Leveque, 1992). To determine mathematically which is the case, note that the change du through a weak wave is the eigenvector r h corresponding to the eigenvalue 2 h of the Jacobian matrix A x. Now 2 h is a function of the variables u, with respect to which it has a gradient
(a)
(b)
li,.!II (c)
(O)
FIGURE 6.2 (a) Successive acoustic waves that raise the density overtake each other and coalesce into a shock. (b) Acoustic waves that reduce the density spread out to form a fan. (c) Waves that merely elevate the temperature do neither of those things. (d) Magnetoacoustic waves that progressively change the transverse field, causing it to pass through zero, first coalesce and then diverge, resulting in a "compound wave."
798
P. Roe
vector Vu2 k. If the wave is linearly degenerate, we will have Vu2 k 9r = 0. For the Euler equations, eigenvectors for the linearly degenerate waves expressed in conserved variables are (Godlewski and Raviart, 1996; Hirsch, 1988; Leveque, 1992)
du c -
dp
1
0
0
dm 1
V1
0
0
dm 2
oc
,
V2
dm 3
V2
89
0
0
V3
dE
,
(6.16)
v3
2
and in primitive variables dp dv 1 dtl p --
dv 2
Of
1
0
0
0
0
,
V2
0 0 ,
0
dv 3
0
0
V3
ae
0
0
0
(6.17)
The term "shock-capturing" is normally used to indicate a method in which all types of wave will be computed automatically, without specific treatment,
but the name might give an impression that shock discontinuities are the ones on which attention must be concentrated. In fact each type of wave gives rise to distinctive difficulties, with shocks in some ways the easiest. For every type of wave, truncation error has a dissipative effect tending to spread the wave out, but for shocks the compression due to the merging of characteristics competes with this. This competition brings about a transition having finite width measured in terms of a few cells, just as the physical processes in a real flow bring about a finite width of a few molecular collision distances. The solution computed within the transition usually has no physical meaning; it is an artificial internal structure that depends on the numerical scheme employed. For linearly degenerate waves, the numerical dissipation still exists but there is no compression due to wave focusing that provides competition. Linearly degenerate waves therefore tend to smear themselves out indefinitely, unless special numerical measures are employed. When examining the sample results displayed by an author offering some new method, it is well to give special attention to the quality of weak shocks and linearly degenerate waves. If results are only presented for strong shocks, some scepticism is in order. Rarefaction waves also have particular problems. They may not spread out quickly enough, giving solutions that resemble rarefaction shocks. Further, strong rarefactions may generate near-vacuum conditions, and the numerical solution may generate negative density and/or pressure. Usually, the code will
Shock
Capturing
799
then crash because the sound speed has become imaginary. A scheme that always predicts physically realistic states is said to be a positive scheme. A further classification of wave behavior is called convexity. Imagine a situation in which waves of the same family continually arrive, and that the changes due to them accumulate. In other words, the material is taken farther and farther from its initial state, never returning to an earlier state. The condition that the speed of these waves changes monotonically is that Vu2 k 9r does not vanish but remains of one sign, and it is then said that that family of waves is genuinely nonlinear and convex (Godlewski and Raviart, 1996). For the commonest examples, such as an ideal gas, or for shallow water, the waves are indeed convex, and so all shocks can be thought of as arising from the progressive steepening of compression waves. For fluids with certain nonideal equations of state, (even ideal) MHD (Brio and Wu, 1988; Roe and Balsara, 1996), solid materials with certain nonlinearities (Menikoff and Plohr, 1989), and some two-phase flow problems such as the displacement of oil by water (Tveito and Winther, 1991, 1995), nonconvex behavior is found. Shocks can be found that satisfy, for example, the entropy and viscosity conditions but not the evolutionary condition, as illustrated in Fig. 6.2(d). In these cases, the structure and admissibility of shockwaves contain open questions at the analytical level, and progress in their computation will have to involve close cooperation between mathematicians, physicists, and numerical analysts.
6.2.6 CHARACTERISTIC VARIABLES, CENTERED WAVES For an equation in the quasilinear form, 0u 0u & 4- A~xx -- 0
(6.18)
the left eigenvectors of A also play important roles. In fact, multiplying (6.18) through by some left eigenvector lk gives 0u 0u 0 - lk -ff[ 4-
lhA Ox
0u 0u - lh ~ + 2klh ax
= lk. ~ +,~h ~
(u)
showing that the ordinary differential equation lh 9du = 0 holds along characteristic lines or characteristics defined by dx/dt = 2k(u } Generally it is not
800
P. Roe
possible to integrate these equations and find quantities that are constant along the characteristics unless A is either constant or 2 x 2. (Lax, 1972). When this is possible the quantities are called Riemann invariants or characteristic variables. This nomenclature is often used rather loosely in the computational literature to refer to the quantities 1. u, which are almost constant along the characteristics over short distances. We can find self-similar solutions to Eq. (6.18) of the form u(x, t) -- u(x/t) = u(~)
(6.18)
which are easily shown to satisfy (Courant and Friedrichs, 1976): du - o ar
(A- ~)~
(6.19)
so that the changes in u take place only along some right eigenvector. If this is the kth right eigenvector then we speak of a centered k-wave. Such waves, together with linear and nonlinear discontinuities, are the building blocks for solutions of the Riemann problem, as discused in the next section.
6.2.7 RIEMANN PROBLEMS The initial-value problem for some given set of conservation laws, in which the data consists of two adjacent constant states, is called a Riemann problem (Riemann, 1860). The most familiar example is the shock tube, in which two gases at different pressures (and perhaps different temperatures) are kept apart by a diaphragm. The experiment is initiated by bursting the diaphragm. On the time scale associated with the experiment the two gases remain separate, but their interface moves into the low-pressure region, preceded by a shockwave. At the same time a rarefaction wave moves back into the low-pressure gas. These events are represented in an (x, t) diagram in Fig. 6.3 (left-hand side). This is a special case of the Riemann problem, obeying the practical constraint that both gases are initially at rest. Without that constraint, there are many more possibilities. In the one-dimensional case, the wave traveling to the right may either be a shock or a rarefaction, or absent. The same is true for the wave traveling to the left. The center wave may be a contact or absent. This gives 18 possible types of behavior, one of which is shown in Fig. 6.3 (right-hand side). For gasdynamics and other relatively simple systems a graphical solution can be given (Courant and Friedrichs, 1976, Fig. 6.4) that can be translated via the secant or tangent methods into a fairly fast iteration. Because the pressure and velocity do not change across the contact discontinuity, it simplifies the problem to use them as variables; we only have to find how they change across the two outer waves. Plot the right state as a point in a u, p diagram. If this state
Shock Capturing
801
X
X
FIGURE 6.3 Wave structure corresponding to shocktube problem (left-hand side). Wave structure corresponding to two initial states that strongly diverge from each other (right-hand side).
P
Hu~iot
./ "'-...,.
I sentr0.p~_ .......... :"~: 's ........ --
...-" UR JLsentrope -.
FIGURE 6.4 Graphical procedure for solving a Riemann problem in gasdynamics or related systems. The curves passing through the given states ULR show how those states might be changed by, respectively, left and right-going waves.
e n c o u n t e r s a r i g h t - m o v i n g s h o c k its p r e s s u r e a n d velocity will b o t h be increased, a n d it will m o v e to s o m e state o n the c u r v e labeled H u g o n i o t . If it e n c o u n t e r s a r i g h t - m o v i n g rarefaction it will m o v e to s o m e state on the c u r v e labeled Is e n t ro p e . T h e s e two c u r v e s t o g e t h e r f o r m a m o n o t o n e - i n c r e a s i n g curve t h a t defines all p o s s i b l e o u t c o m e s for the fate of the r i g h t state. Similarly t h e r e is a m o n o t o n e - d e c r e a s i n g c u r v e that defines all possi bl e o u t c o m e s for the fate of the left state. W h e r e v e r the initial p o i n t s uL, uR are p l a c e d these two curves h a v e a u n i q u e 6 i n t e r s e c t i o n u* r e p r e s e n t i n g the state b e t w e e n the waves. By this time w e also k n o w w h e t h e r each w a v e is a s h o c k or a rarefaction; w e can therefore find the n e w d e n s i t y of each state.
6The only case in which this procedure does not give a unique intersection is if the velocity difference u R - uL is SO large that the isentropes reach the axis p = 0 before intersecting. This means that a vacuum forms between two powerful rarefaction waves, and within this vacuum region the velocity is undefined. This would not invalidate the numerical use of the solution, because the flux is defined (it is zero).
802
P. Roe
The density is usually discontinuous across the contact. Quickly convergent numerical procedures have been given by, for example, Van Leer (1979). We show in Section 6.4.1 that the flux across the interface between two finite volumes can be found by solving a Riemann problem, and that method turns out to have some nice features but is often too expensive (even for ideal gasdynamics, it is about twice as costly as the main alternatives). We present in Section 6.4.2.1 a discussion of linearized Riemann solvers, but many other methods have been proposed to solve the Riemann problem approximately. One of these is based on the fact that the Hugoniot and isentrope curves are rather similar (they always have the same slope and curvature at their common point). So one of them can be used to approximate the other, and since the Hugoniot usually has a simpler equation than the isentrope, the so-called twoshock approximation has been used.
6.3 NUMERICAL B A C K G R O U N D 6.3.1
FINITE-VOLUME METHODS:
THE L A X - W E N D R O F F T H E O R E M The clearest way to mimic conservation properties in a computer code is to adopt the "finite-volume" strategy in which space is divided into a set of contiguous nonoverlapping regions (the finite volumes, or "cells") and each volume is supposed to contain certain amounts of mass, momentum, and energy. Now suppose that each face of every volume is either an interface between two volumes or else located on the boundary of the computational domain. Then the "flux" through each face either serves to transfer a conserved quantity from one cell to another, or else it introduces conserved quantities into the system. This is immediately obvious in the case of mass. The mass flux across an interface removes mass from one cell and adds exactly the same amount to its neighbor. But also any force acting on an interface will add to the momentum in one cell and remove it from the other, and the work done by that force on material that crosses the interface will add to the energy in one cell and subtract from the other. Although the cells in Fig. 6.5 have been shown as (irregular) hexahedra, there is no need for the volumes to be of any particular shape, and arbitrary mixtures of shapes are quite acceptable. This helps in dividing up awkwardly shaped regions.
803
Shock Capturing
FIGURE 6.5
Two adjacent control volumes, their common interface, and its normal.
In a typical application, the solution is supposed known at some moment of time t" and advanced to the next m o m e n t t n§ (the n, n 4- 1 are superscript labels, not powers) by a formula such as r:"+l/2 -nj,k Vju)" - Vju~*I - At E "j,k
(6.20)
k
Here j refers to a particular cell, and k to its neighbors, with { j, k} labeling the interface between two cells. The volume of the jth cell is Vj, and nj, k is the outward normal to the interface (not a unit normal, but scaled proportional to the area). There is some geometry involved. If the interface is a (skewed) quadrilateral, the scaled normal is just half the cross product of its two diagonals (giving due care to the sign) but calculating the volume is surprisingly nontrivial (Davies and Salmond, 1985). It is, however, important to calculate all geometrical terms exactly and without approximation; otherwise conservation is lost. The fluxes have been given a superscript n + 1/2 to indicate that they are supposed to be averages over the time interval t" <_ t <_ t n+l. In fact, if it were possible to compute those averages exactly, Eq. (6.20) would be an exact result. However, we will actually have to estimate them somehow, usually from a knowledge of the data at time t". The precise way in which that estimate is made distinguishes one finite-volume method from another. For finite-volume schemes Lax and Wendroff (1980) succeeded in proving convergence to weak solutions of the given conservation laws provided the scheme is stable (it must converge to something as the mesh is refined) and that it is consistent. Consistency is defined in terms of the way that the flux on the interface is calculated from the states of two or more neighboring cells: F* -- F*(ul, u 2 . . . u , )
804
R Roe
A scheme is said to be consistent if, whenever all of these states are identical, the flux corresponding to this state is returned, that is to say F*(u, u . . . u ) = F(u) The consistency of a proposed scheme is usually self-evident; its stability usually is not. Often stability is assumed if it can be proved in some simple special cases, such as a one-dimensional linear problem on a uniform grid. In fact, this is usually more satisfactory than it sounds. A scheme that satisfies both conditions will, if it is applied to a problem that should contain a shockwave, capture that shockwave in the sense that as the mesh is made finer, and the transition region therefore becomes narrower, the discrete solution will converge to a weak solution of the continuum problem. A further condition must be satisfied if the method is to converge reliably to a steady state. The flux function must be Lipshitz continuous, meaning that a change of order c in either of the given states must only produce a change of order E in the computed flux. In empirically derived methods there is the possibility that this will not be satisfied if the small change causes a wavespeed to change sign.
6.3.2
E R R O R AND A C C U R A C Y
Before it can be said that some numerical method produces a solution that converges, on a fine enough grid, to the exact solution of the problem, we must discuss the notion of numerical error. Intuitively this is the difference between the numerical solution and the exact solution, but even when an exact solution is available, that difference is not easy to measure, especially if the true solution happens to be discontinuous. Technically the difficulty is that the exact solution contains an infinite amount of information whereas the computed solution naturally contains only finite information. The trick is to boil down the exact solution to give only the information that the numerical solution attempts to compute, for example the solution values at some finite number of grid points, or the average states in some finite number of cells. This is called projecting the exact solution into the discrete solution space. Comparing the finite sets of information from the exact and numerical solutions allows a precise measure of the error. However, comparing the solutions at some fixed set of points is not, in fact, appropriate for discontinuous flows. If the discontinuity has been smeared out over a finite region then there must be large errors at points within that region. The better measure is to compare average values of the solutions over the finite volumes into which the space has been divided. The error measured in this way will usually be smaller, and corresponds better to the judgments that one would
Shock Capturing
805
make visually. Defining the error carefully is important because one often wants to verify that a method achieves its predicted order of accuracy, perhaps to be reassured that the coding is correct. In fact the output of a finite volume code is firmly intended to be some set of estimates for the average states in the volumes. If the errors in these average states become arbitrarily small as the grid is sytematically refined 7, then the method employed is said to converge in the sense of finite volumes (for more, useful, detail see Rezqui (1998)). If the size of a typical computational cell is h, and if the error is found to be proportional to hp as h is reduced, the method is said to be p-th order accurate. Actually, it is hard to predict what the order of accuracy of a given method will be, if it is applied to a discontinuous flow. The usual idea of truncation error for a finite-difference scheme can be used to predict that a certain order of accuracy should obtain for smooth flows (perhaps also requiring that the grid is in some sense well-behaved), but it is unlikely that the predicted accuracy will be realized if the flow contains discontinuities. For one thing, there must be, even in the sense of finite volumes, fairly large errors near the captured discontinuities, and it is also possible that these will contaminate the solution elsewhere. Typically, shockcapturing practitioners use methods that are nominally of second-order accuracy, but practical tests on discontinuous problems usually reveal (by plotting the error versus h on a log-log graph) that the errors behave roughly like hp with p in the range 1.6-1.8, and this is said to be the experimental order
of accuracy. There is no doubt that as a strategy shock capturing does entail some inevitable error as a result of portraying the discontinuities in a rather literally "broad-brush" manner, and there is little theory to suggest how big this is. Based on numerical experiments, Casper and Carpenter (1995) find that all shock-capturing methods of the finite-volume kind are only first-order accurate on extremely fine grids, although methods that are nominally higher order are usually much preferable on the size of grid that is affordable in practice. The order of accuracy that is worth striving for seems to depend on the context, although first-order methods are seldom satisfactory. If the interest in the problem lies largely in the behavior of strongly nonlinear shockwaves, a method that is nominally second-order will usually be adequate. If the interest lies more in the behavior of vortices and contact surfaces, or the propagation of weakly nonlinear waves over large distances, higher-order methods often prove worthwhile, despite their complexity. The reader is recommended to consult the literature for problems similar to their own, to see what the experience has been. 7Small in some appropriate norm such as L 1 or L2; the Loc norm makes no sense for shockcapturing methods because the maximum error (close to a discontinuity) is always (_9(1).
806 6.3.3
e Roe
THE SIMPLEST HYPERBOLIC PROBLEM
The linear advection equation
Otu + aOxU = 0
(6.21)
is analytically trivial, being satisfied by any function f ( x - at). The solution is classical if f is differentiable, and weak otherwise. There is a single unknown u that might, for example, be the density of some impurity that is carried along a pipe by fluid moving with constant speed a, if no diffusion takes place. The flux of this impurity is f ( u ) = au. Despite this simplicity, effective numerical solution procedures are rather difficult to find but, once found, can be generalized to much more complicated and realistic problems. This simple problem is therefore a good place to start. The unknowns will be the cell averages
(j+l/2)Ax
- = u~
u(x, tn)dx JO-1/2)Ax
which are estimates of the mean density in cell j at time level n, and the generic finite-volume method is
AX~ = A x ~ +1 - At[fj+l/2 n+l/2 - Jion+l/2] j-l/2
(6.22)
where
fn+l/2 Jj+1~2
1 [(n+l)At
=
--
~ dnAt
f(xj+l/2, t)dt
a I (n+l)At
R(Xj+1/2, t)dt
nAt
6.3.3.1 Flux Estimation If it were possible to evaluate this flux exactly, the cell averages would be updated exactly. The reason we cannot do this is that we only have available the cell averages, not the detailed distribution of u(x) within the cells. This is available, if at all, only on the first timestep. But if it were available, we could use the fact that u is constant along the characteristics to rewrite the integral for the flux as an integral in x (see Fig. 6.6); n+l/2 _ 1 ['~j+l/2 u(x, tn)dx +1/2 -- At axj+l/2_aAt
(6.23)
807
Shock Capturing
t
x-at
Xj_I/2
X
FIGURE 6.6 Transferring the integral from the t-axis to the x-axis using characteristics; in the case shown the wavespeed a is positive.
The success of any particular finite-volume scheme turns in part on its ability to estimate this integral accurately. To begin with, note that the limits of the integral lie to the left of the interface if a is positive and to the right if a is negative. This reflects the fact that all wave propagation has a sense of direction associated with it, and the most successful shock-capturing methods exploit this, even though this is not actually essential. A general strategy begins by trying to reconstruct the function u(x) from the given cell averages, perhaps as a polynomial. Suppose that we select some set of p cells in the neighborhood of the interface j + 1 and find the polynomial of degree p whose average value in each cell reproduces the given average. Examples of such polynomials are illustrated in Fig. 6.7. Then we insert that polynomial in Eq. (6.23). The resulting flux formulas are shown in Table 6.1. In these formulas there appears the important parameter (6.24)
v = aAt/Ax
which is called the Courant number or CFL number (Courant et al., 1928). It measures the relative propagation speeds of the exact solution a and the numerical solution Ax/At; it also measures the proportion of the cell TABLE 6.1 p
Flux Functions for Some Simple Finite Volume Schemes
Cells
Flux formula
1
j
afij
2
j,j+l
~[(1 + v)aj+ (i -
2
j--l,j
a[(v -
Name First-order Upwind, Donor Cell
Stable if 0 < v< 1
a
v)fi)+l]
1)fij_1 + (2 - v)fi7]
Lax-Wendroff
-1 < v < 1
Second-Order Upwind
0 < v< 2
Third-order
0 < v< 1
a
3
j-l,j,j+l
gt(v 2 - l)fij_1 + (1 + v)(5- 2vlfi) + (1 - v ) ( 2 - v)fi)+l]
808
P Roe
j-1
j m
i
_ ......
j+l ! ! ~ ~
X
Ax _aAt j-1
j
U
j-1
j
j+l
. . . . . . . Ax
gat j+l
.. j-1
j
j+l
---i--= [ ,I _
N Ax aAt
X
Ax aAt
X
FIGURE 6.7 Calculating the flux of an advected impurity. Its density is known in each of the three cells shown. The heavy dashed line represents a polynomial function (constant, linear, quadratic) that matches the given mean value in certain cells. The integral under this line (the shaded region) is the amount of impurity that will cross the heavily drawn interface in time At.
contents that will cross the interface in one timestep. Whenever this quantity is an integer, the complete contents of any cell at time level n appear in some other cell at level n + 1 and the finite volume method can become exact. Check the values taken by these flux formulas for small integral v to see how this works. Of course there is no such property in more general problems, because there is usually more than one propagation speed, and these are not usually constants. Nevertheless, many methods work best if the timestep is chosen so that the most significant waves travel roughly one cell width per timestep. One important aspect of these formulas is whether their properties remain the same if the sense of propagation (sign of a) is changed. This is only true of the Lax-Wendroff scheme, which takes information from the two cells j, j + 1 either side of the interface. If a is changed to - a , then v is changed to - v , and the weights given to the left and right neighbors is simply reversed. The scheme transforms to its own mirror image and we speak of a centered or central scheme. Such schemes, having been derived for Eq. 6.2.1, can be extended to linear systems of equations (6.18) simply by reinterpreting a (and hence v) as matrices. For the other schemes displayed this is not the case, because they take information asymmetrically from either side of the interface. The first-order upwind scheme, if a is made negative, becomes unstable because it is no longer based on relevant information. To keep a scheme with similar properties we need to change the cell supplying the information from j to j + 1, so that
Shock Capturing
809
fij+l/2 n+l/2 -- au7+1 This is also true for the other two methods. Whenever the set of cells used to determine a flux depends on the propagation direction we speak scheme 8. of an upwind scheme or an uvwind-biased, 6.3.3.2 Some Numerical E x p e r i m e n t s A good overall impression of the performance of a method for hyperbolic problems can be gained by applying them to the simple problem (Eq. (6.21)). This has been done in Figs. 6.8-6.10. The initial data consist of four "typical" signals, a square wave, a raised cosine, a narrow Gaussian, and a half-ellipse. This is a revealing set proposed by Zalesak (1987) where more examples can be found. The grid is composed of 100 uniformly spaced points, with 20 devoted to each signal and 5 in between. This grid is really not fine enough to resolve these waves, but here and later a deliberately coarse grid brings out the differences between various schemes. The boundary conditions are periodic (u(1, t) = u(0, t)) so that the solution should be transported to the fight and reenter on the left. By taking 400 timesteps with v = 0.25, or 200 timesteps with v = 0.5, or 133 timesteps with v = 0.75, the solution should travel around the domain once and sit exactly where the initial data was. It is quite clear that the First-Order Upwind Scheme loses all semblance of the solution. An adequate computation would require many more grid points. About ten times the density used would be needed. The Lax-Wendroff scheme is better, but suffers seriously from overshoots that trail behind the waves. Again more points are needed, perhaps four times as many. The third-order scheme does a much better job, although there is still clearly some loss of definition.
6.3.3.3 von N e u m a n n Analysis It is important to know whether these schemes will be stable if operated for large numbers of timesteps. The principal tool for investigating this is von Neumann analysis, suggested by John yon Neumann when consulting for the Manhattan project. The details can be found in many books on numerical analysis, for example, see works by Tannehill et al. (1997), Morton and Myers (1994), Gustafsson et al. (1995), and Vichnevetsky and Bowles (1982). One considers initial data that is sinusoidal in x; u~ - exp i(jO), with 0 < fl < ~z, 0 being the phase change from one cell to the next. We need only consider fl in the stated range because 0 = n is the Nyquist or folding frequency, which is the 8If this procedure is applied to each characteristic variable in a set of equations, such as the acoustic equations, to which solutions remain solutions under mirror reflection, the final scheme will appear symmetrical. Probably one should still speak of upwind schemes, if they were derived using "upwind arguments."
810
P. R o e
1
-0.50
"',
0.00
,
',
-',
0.20
', .
.
0.40
.
.
.
.
,
0.60
'
0.80
o. o
'
o.;o
1.00
u - 0.25 1.50
0.50 -
-0.50
'
,
o.oo
,
"
-r
0.20
u 1.50
.
.
J
o.: o
.
.
'
1.00
w"
1.t
0.50
-
.
.
.
.
.
. 0 ..6 0
.
.
.
0.50-
-0.50 0.00
, . . . . 0.~0
r 0 94 0 .
"
u
-
.
'
0.80
0.75
FIGURE 5.8 Numericalresults for the First-Order Upwind Scheme.
highest frequency that can be resolved on the mesh. Marching forward one timestep produces a solution u~ - g ( O ) e x p i ( j O ) where g(O) is a complex number called the amplification factor. The modulus of g must be less than unity; otherwise the solution will grow without limit, but values less than unity will eventually cause the solution to be damped out. It can be shown that for small 0 Ig(0)l -- 1 -
7(v)Op+2 (p even)
[g(0)[ -- 1 --
7(v)Op+I (p odd)
The quantity 7(v) (which is different for each scheme) must be nonnegative for stability. In addition to determining stability the analysis provides information
811
Shock Capturing
1.5oj
~o
050 --0.50 , 0.00
~%_ :/
~ ~
I
' 0.~10
0.20
~
,
% /
0.60
/ 0.80
, 1.00
u = 0.25 .
1.50
.
.
.
0.50
-0.50__ 0.00
,
,
,
,
0.20
I
0.40
,
,
0.60
,
I
0.80
1.00
u = 0.50 1.50-
A
0.50 -
--0.50 0.00
t
t
0.20
l .
.
.
A A .
.
0.~10
w
'
0.60
'i'
0.80
'
1.00
u = 0.75 FIGURE 6.9 Numerical results for the Lax-Wendroff Scheme. The results for the Second-Order Upwind Scheme are similar, except that the oscillations lead instead of trailing.
on the error committed. For wave propagation it is important to recognize two different varieties of error. A positive value of ?(v) represents the damping error or dissipation. The argument of g represents the phase change at some cell position in one time step. Ideally this should be -vO. The quantity
P(O) - arg g(O)
812
e. Roe 1.50
'
0.50
--0.50 0.00
",
'
,
0.20
'
"
0.~10
',
'
,
I
0.60
i
0.80
1.00
u - 0.25 1.50
0.50 -
-0.50
I
I
0.00
I
0.20
I
I
I
0.40
I
I
0.60
l
I
0.80
1.00
u - 0.50 1.50
0.50
-0.50
0.00
I
I
0.20
"
-
FIGURE 6.10
I'
0.~10
"!
0.;0
....
I
0.;0
1.00
0.75
Numerical results for the Third-Order Scheme.
is called the phase error and at any future time will give the relative displacement of the wave from its true position. It can be shown that for small 0, P(O) - a(v)O p (p even)
P(O) --
0~(v)0p+I (p odd)
Schemes that have an even order of accuracy suffer from a curious disadvantage. Their damping error is much smaller than their phase error. This has the consequence that short waves will travel with the wrong speed but will not be damped out, and this creates trains of unwanted oscillations. These can be seen in Fig. 6.9. Because of this phenomenon, in numerical wave propagation some
Shock Capturing
813
amount of dissipation is usually seen to be desirable. A method is said to be dissipative in the sense of Kreiss if 7(v) is positive at all non-zero frequencies (Kreiss, 1964). The Lax-Wendroff Scheme fails this test at 0 = ft. Nevertheless, excessive dissipation is ruinous to accuracy. The design of successful methods for wave propagation really amounts to creating dissipation with the lightest possible touch. 6.3.3.4 Godunov's T h e o r e m The oscillations are reduced but not eliminated by employing the third-order scheme; it is natural to ask if they are inevitable. The answer to this question is contained in Godunov's Theorem 9 (Godunov, 1959; in translation 1960). Define a monotonicity-preserving scheme to be one such that monotone data (u-ff.+l -
always of one sign) alway yields a monotone solution. Then all monotonicitypreserving schemes of the form fjn+l/2 +1/2 -- E ~k(V)tlT~-k k
are only first-order accurate. The class of schemes considered includes all schemes in which the flux is found by employing any fixed interpolation and integration strategy at every interface. When these schemes are generalized to handle nonlinear systems of equations, the resulting oscillations are often disastrous. Not only is the solution polluted by meaningless waves, but the density may be driven negative, which causes the code to crash. Our alternatives are therefore either to live with the dissipation errors inherent in first-order schemes, or else to develop interpolation strategies that circumvent Godunov's Theorem. Although firstorder schemes are seldom satisfactory, they have a role to play as "fallback" schemes when the higher-order methods fail. Usually this is in the vicinity of discontinuities. All successful schemes for shock capturing are based on somehow detecting discontinuities, and taking special action in their vicinity.
6.3.4 T I M E - S T E P P I N G , F L U X I N T E G R A T I O N , SEMI-DISCRETIZATION The schemes just outlined are of the type called fully discrete. We make the update by integrating the flux across the interface with respect to time. For a first-order method it is enough to take the flux as constant over the timestep, but for higher-order methods the variation with time must be accounted for. There are several ways to do this. For the simple problem previously 9The two principal mathematical results of shock-capturing theory are the Lax-Wendroff Theorem (which is encouraging) and Godunov's Theorem, which is discouraging.
814
R Roe
considered, constancy of the solution along characteristics allows the integral to be performed with respect to x rather than t. This is not always possible for nonlinear problems. Instead, the solution along the interface can be represented as a Taylor series u(t) -- u(O) 4- tatu(O ) 4- } t2()ttu(O) 4- 1 t3atttu(O) where the time derivatives (for a one-dimensional problem) can be related to the space derivatives by am = -A0~u OttU =
(6.25)
(6.26)
~x(A0xF)
atttu = -~x((Au~xF)axF 4- Aax(AaxF))
(6.27)
Taking account of p terms in the series will give a method of order p + 1. Methods constructed in this way often have the nice property that the results get better as the timestep gets larger. For example, all of the schemes in Table 6.1 are exact when the CFL number is 1.0. However, it can be seen that the expressions for the higher derivatives become complicated and expensive (in Eq. (6.27) Au is an m x m x m tensor). One attempt to simplify is not to integrate the flux directly, but to evaluate it at certain moments within the timestep (the Gauss points of the interval (t", t"+l)). For example, to evaluate the integral with second-order accuracy it is enough to know the instantaneous flux halfway through the timestep. To obtain fourth-order accuracy we need only the fluxes at t = t"(1 q---~~3)+ t n + l ( i T -~~3) However, this is really not all that useful, as there is no inexpensive way to find even these values. Therefore the majority of higher-order 1~ methods are based on the idea of semi-discretization, sometimes called the method of lines in the older literature. The idea is to evaluate, at the instant t ~, merely the time derivative of the solution. Since we are dealing with the conservation law ~tu 4- 0xF = 0
we have only to integrate this over the interval xj_l/2 < x
<
Xj+I/2 tO obtain
Zkx~tfij(t-- t") 4- F~_1/2 - F7_1/2 = 0
(6.28)
If the interface fluxes can be found at the beginning of the timestep, as some function of the states in the surrounding cells, we have, in effect, a large system of ordinary differential equations to solve. It is not normally a good idea to turn 1~ this context, higher-order generally means better than second-order.
815
Shock Capturing
these over to an ODE solver from a software library, because that solver was probably written to deal with awkward issues of stiffness that do not usually arise in the present context. Instead, rather simple Runge-Kutta methods are preferred. Specialized versions have been developed for use when stability is more important than accuracy, for example if only the final steady state is of interest (Tai, 1990, Lynn, 1995), or to maintain monotonicity properties (Swanson and Turkel, 1997), (Shu and Osher, 1988). For a promising alternative approach to high-order discretization see Yee (1997).
6.4 ONE-DIMENSIONAL METHODS This will be a long section, not because one-dimensional problems are that important, but because most of the issues can be displayed in this simple context. As a patient reader will discover, there have been a great many methods proposed. I have not attempted to be comprehensive. My selection criterion has been either that a particular method has been found sufficiently promising that it has been put to practical use by significant numbers of people, or in a few cases because it raises an important theoretical issue. The level of discussion will be found uneven, and partly reflects some subjective judgments about how important the various methods are. However, some methods are given only a brief discussion because they do not fit well into my chosen framework, which focuses on finite-volume methods. Whenever the main feature of a method is how it represents the physics, the discussion will be in terms of how it applies to one-dimensional systems of conservation laws atu + 0xF ----0 (6.29) also writable in integral form as ~(udx
F
dt) -- 0
(6.30)
when they are solved by the finite volume method
a~+ ~ - ~ ' -
At [l:zn+l/2 __ Fn+l/2 ~-j+~/2 j-~/2 ]
(6.31)
That is to say, it will be concerned mostly with various prescriptions for ~n+l/2 estimating the fluxes "j+l/2. If the main interest is in the mathematics of interpolation I will usually illustrate this with the linear scalar example of Section 6.3.3. To a large extent these two problems are complementary; the two main questions to be asked when building a code are: how shall we interpolate? and how shall we represent the physics? (and in higher dimensions, how shall we build a grid?). However, I have not succeeded in organizing my material quite so neatly.
816
P. Roe
Most of the current methods for multidimensional problems are in fact fairly straightforward extensions of their one-dimensional counterparts, but some discussion will be given in Section 6.9 of "genuinely" multidimensional methods.
6.4.1
THE GODUNOV SCHEME
Among all monotonicity-preserving schemes for Eq. (6.21), the one with smallest dissipation (7(v) in Section 6.3.3.3) turns out to be the First-order Upwind Scheme for which the flux function is simply the average flux in the cell that is "upwind" from the interface (sometimes called the donor cell). This is therefore the scheme that offers the best compromise between reliability and accuracy, and it will be interesting to generalize it. Generalization to a more complex situation is of course not usually unique, but one interpretation of the First-order Upwind Scheme is rather natural and suggestive. The Riemann problem for the linear advection equation has the simple solution u(x/t) = ULH(a -- x/t) -F URH(X/t - a)
where H ( x ) = (1 + s g n x ) / 2 is Heaviside's unit function. The First-Order Upwind Scheme amounts to taking the flux between two cells as the solution at the interface of the Riemann problem with those two cells as data. Godunov proposed to take that as a general prescription for the flux, for any set of conservation laws. That is called the Godunov flux, FG(UL, UR) , evaluated at x = 0 from the exact solution of the Riemann problem with data uL, uR. A nice physical interpretation of this strategy comes from noting that for small enough time-steps (such that waves from adjacent interfaces do not intersect) we are solving exactly the initial-value problem with the cell-average states, taken constant within the cells, as initial data. In principle, what is done next in every cell is to average the solution that has been found from the two Riemann problems but there is never any need to consider the complex details. The formulas (Eq. (6.31)) and the use of the Godunov flux do this averaging automatically. Unlike most numerical solution methods, the Godunov method does not approximate the evolution of the data, but the data itself. (See Fig. 6.11.) This interpretation can be used to show that the Godunov method is very robust, and will always produce solutions having positive density and pressure. This property was defined as positive conservation in Einfeldt et al. (1991). The simple proof is worth giving.
Shock Capturing
817 _
n+l
uj
n+l
j-1
j
j+l
FIGURE 6.11 Illustrating Godunov's method. The new average state in the cell is found by integrating around the control volume displayed, taking the interface fluxes to be those obtained by solving the Riemann problem. The timestep must be such that waves from one interface do not reach the neighboring interfaces. They may intersect the neighboring waves.
Think of the conserved variables (p, m, E) as defining a three-dimensional phase space. Within this space, the physically meaningful region P (positive density and pressure) is defined by E ttl2 P:p > 0 and -p- ~ p2 > 0 It is easy to show that P is a convex region11, meaning that if any two points both lie in P, so does the segment of a straight line that joins them. Now suppose that the data for the Godunov method consists of a collection of cell average states, each lying in P. Solving the Riemann problems exactly gives solutions that remain in P (a proof is given in Smoller (1983)) and averaging the solution within each cell produces new averages that also lie in P (because of convexity). There are, however, sets of equations for which the convexity property does not hold, and possibly the Godunov method might fail for these, at least for some data. In its original form, Godunov's method is not very commonly used, because even for the Euler equations the flux is quite expensive to compute, and considerable attention has been given to simplifying it. The simplified versions may lose certain properties of the original, but these can often be regained by "fixes" that need only be done locally. Methods that replace the exact evolution with some physically motivated approximation, but that still stress careful approximation of the data, are loosely referred to as Godunov-type methods. An autobiographical article that describes the original development of Godunov's scheme has both human and technical interest (Godunov, 1999).
liThe convexity property discussed here is unrelated to the convexity property discussed in Section 6.2.5. Convexityof the MHD equations is established by Janhunen (2000).
818 6.4.2
P. Roe
A LINEARIZED
RIEMANN
SOLVER
Suppose the problem to be solved Otu + AOxu = 0
is linear, so that A is a constant matrix. The solution to the Riemann problem is then immediate. Each wave is a j u m p that takes place across a line x = 2 i t , where 2 k is an eigenvalue of A, and the change in state across that line is proportional to the right eigenvector r k, so that [U]k = ~krk, where [u]k is the j u m p in state across the kth wave and 0~k is a coefficient measuring the strength of the wave. As the sum of these changes must account completely for the difference in the given states, we have uR - uL = ~
o~krk
(6.32)
k Upon multiplying this equation by one of the left eigenvectors, and exploiting the orthonormality of left and right eigenvectors, we obtain 0oK - IK(UR -- UL)
(6.33)
We are interested in the flux at the interface. As the flux is F = Au and the eigenvectors satisfy A r k = 2kr k, we have that the change in flux across the kth wave is 0~k2krk. There are now several ways to evaluate F w h e n x = 0. First, we can start with the flux in the left cell and add the changes that occur across each left-running wave in Fig. 6.12. F(0) = FL +
~ 0~/2krk k:2k<0
(6.34)
FIGURE 6.12 Structure of a linearized Riemann problem. The solution is constant between the waves, all of which are discontinuous.
819
Shock Capturing
Alternatively we can start with the flux in the fight cell and subtract the changes that occur across each right-running wave F(0)-- F R -
~
~h2krh
(6.35)
k:2k>0
As there is no change in flux across a stationary wave there is no need to account for any wave that may have 2 = 0. Finally, we can take the average of these two equivalent statements to get the forms that are usually given: F(0) -- I[F L + FR] -- 89 ~ czkl2klrk
(6.36)
k
--2--![FL + FR] -- 89 ~ lhlAhlrkCuR -- UL)
(6.37)
k
+ FR] -- i IAl(uR - u[)
(6.38)
where [A[ is the so-called absolute value of the matrix A. For coding, the form (6.37) is often used, but is not efficient for large systems of equations. The most efficient form is to choose whichever of Eqs. (6.34) or (6.35) involves summing over the fewest waves. See works by Fedkiw et al. (1998, 2000) and Roe (1998). This trick, although elementary, is not as well known as it should be, so that the expense of upwind schemes is frequently exaggerated. In the Appendix a simple example of a one-dimensional first-order Euler code is given that employs this device.
6.4.2.1 Choice of a Linearization In analytical work, one often linearizes around some assumedly constant background state, but in a computation this would defeat the object of solving the full nonlinear equations. To compute just the flux across a particular interface, however, we may solve a problem that is linearized around some state representative of conditions near that interface. Most commonly, one solves a problem (6.39)
0tU +/~(UL, UR)OxU "-- 0
where ,4(u L, UR) is some constant matrix obtained by considering the two input states u L, u R. For consistency with the given conservation laws we need to have ~,(u, u) -- ACu)
(6.40)
Simple possibilities are A(u,,
= 1 [A(u,) +
or
uR)-
(u, +
820
P. Roe
and these are quite adequate if only weak discontinuities are expected. A slightly more elaborate form of averaging is worthwhile for more severe problems, and is often taken to be the default, since the additional expense is not great. In 1981 the author proposed to choose the average matrix .~i(uL, uR) such that (6.41)
A(u L, uR)(u R - u/_) - F R - Ft_
which would not be true for either of the forementioned choices. Imposing this property means that Riemann problems whose solution is a single discontinuity (shock or contact) are computed exactly. This can be proved by noting that in such cases the two input states must satisfy the Rankine-Hugoniot condition 5(uR - u~) = FR - FL
and so in these circumstances any A with the property of Eq. (6.41) will have S as an eigenvalue and (uR - ut) as an eigenvector. Therefore, the wave decomposition of (u R - u t ) will contain only a single term, which will be the exact solution. There are many possible ways to construct such a matrix, but at least for the Euler equations only one of the form A(ut_, uR) = A(fi)
(6.42)
fi = fi(uL, UR)
(6.43)
That is to say, there is a unique average state fi(u t, uR) such that linearizing around that state enforces the property of Eq. (6.41). By definition of the matrix A we must have that
[pv] -[pv]
(6.44)
-- ~ (7 - 3)~2[p] - (? - 3)~[pv] + (? - 1) ~ -P
{1
[v(E + p)] - ~ ~ (7 - 2)~ 2
+
?-1
(6.45)
a ~ } [p]
?-1
, (y -
2
i +~2
}
3)~ 2 [pv] + ~[~]
The first of these equations is trivial; the second reduces to
~2[p] _ 2~[pv] + [pv 2] - o
(6.46)
821
Shock Capturing
Solving this quadratic, selecting the root that makes sense when u L = u R, and simplifying, yields the average velocity as 12 -- ~/~vz + ~/-fiTvR
(6.47)
The third equation may now be solved for h; the result is
where
h2 -- ( 2 - 1)[h _ 1521
(6.48)
_ ~/-fiThL + ~/~hR
(6.49)
,/g/+,/N
The eigenvalues and eigenvectors of the average matrix are simply obtained from those of the full Euler equations by inserting the averaged quantities fi, h, h. Then Eqs. (6.34), (6.35), or (6.36), gives the interface flux. A code that uses the flux function defined by Eqs. (6.34), (6.35), or (6.36) with the choices of Eqs. (6.47) and (6.49) has the property of returning the exact solution after any number of timesteps if the initial data is either a stationary shockwave or a stationary contact. This latter property is valuable because when these methods are extended to the Navier-Stokes equations, the "Euler part" of the code "sees" the boundary layer as a series of contact discontinuities (Van Leer et al., 1987). Generalizations of this procedure to a nonideal gas are given in Toumi (1992), Mottura et al. (1997) and earlier work cited there. Generalization to ideal MHD and other systems can be found in Cargo and Gallice (1997).
6.4.2.2 Failings of Linearized Solvers It would be surprising if the drastic simplification involved even in a local linearization did not involve some loss of the special properties of the Godunov scheme. Perhaps the surprise is that so many are retained. For example, if the exact Riemann problem or the linearization in Section 6.4.2.1 are each used to form a numerical flux, and the results are compared for a simple shock-tube problem, the results are quite different after one timestep, but after about 30 timesteps differ only in the fourth significant figure. Because capturing methods inevitably blur out the details to some extent, no great benefit in accuracy comes from working excessively hard to resolve that detail in the flux function. This is particularly true because most computation takes place in smooth regions where the choice of a flux function is unimportant. The crucial behavior is close to discontinuities, which tend to s~eparate out, so that accuracy for isolated waves will be sufficient. Regions where waves collide 12We write the result in a form that stresses symmetry; a small reduction in computation comes from dividing the numerator and denominator either by v/-p-7 or v/-p-7. This will save one square root evaluation every time the flux subroutine is called.
822
P. Roe
are not possible to resolve precisely using capturing methods, but the correct transmitted waves emerge quite promptly. Using an approximate Riemann solver can, however, cause a loss in robustness. It is no longer possible to give any proof that the solution will not predict negative densities or pressures. Indeed, Einfeldt et al. (1991) showed that a flux based on any consistent linearization (A(uL, uR) -- A(fi) with fi defined in any manner at all) would produce negative pressure for some data. The problem is that zero does not have any special significance to a linear solver. Nor is it possible to prove that the weak solutions to which the method converges are entropy-satisfying. In fact, if a stationary rarefaction shock is offered as initial data to the code described in Section 6.4.2.1 it will be exactly preserved, just as a stationary contact or shockwave would be. Linear solvers do not recognize the behaviour of genuinely nonlinear waves. A third failing will not be catalogued here because it is shared by the exact solver and is therefore not associated with the linearization. It will be described in Section 6.8. At the time of writing, there does not seem to be any perfectly satisfactory flux function. The standard practice is to take a cheap, reasonably good one, analyze why it fails, and add fixes if they are needed for a particular application. The fixes must not add to the computational cost and so should involve only simple formulas. 13
6.4.3
T H E ENTROPY F I X
The typical situation that causes the entropy condition to be violated occurs when the interaction between two cells should involve a rarefaction wave that spans the interface, and would therefore affect both cells. A linear approximation will replace the rarefaction fan by a single jump that is either left-going or right-going, and which therefore affects only one cell. Any kind of fix that forces information to travel both ways, for example by adding artificial dissipation, will alleviate this problem, but a more delicate approach would be preferable. Figure 6.13 shows a typical situation where the linearized solver has predicted one wave to be almost stationary with speed 2. If this is an acoustic wave it should spread if the jump in velocity [v] > 0, and then the difference A2 in the speeds of its rightmost and leftmost rays should be, for an ideal gas, (7 + 1)[[v]1/2. The wave should span the interface if !A2 > 12l 2 13Reliable calculations can be made using very simple first-order schemes on extremely fine grids, for example, the Lax-Friedrichs scheme noted in Section 6.4.7.1. Usually this is prohibitively expensive.
823
Shock Capturing
E*
t
~=k | t
v*
x
x
FIGURE 6.13 An entropy fix may be needed for a linearized Riemann solution if the mean position of a rarefaction wave lies close to the axis (left-hand side). The wave is 'opened out' on the right-hand side.
A simple correction for this effect can be obtained by assuming that the linearized solver has correctly computed the jumps across the waves, and so in particular has set this wave in the correct environment. The fluxes F~_,Rand the states u~_,R to either side of it will therefore be accepted, but instead of an abrupt jump we assume that the state u changes linearly with ~ = x/t. Now, within a rarefaction fan dF/du : ). = ~, (see Eq. (6.2.6)) so with du/d~ constant, dF
de
= r
:=~F-F(0)--~
du
(6.50)
d~
1 ~2du d--~
(6.51)
If this formula is to be consistent, then for some F(0) it must predict both F E and F~. Therefore we must have 1 F~ - F~_ -- ~[(r
_ . . _ _ du _ (~/~)2] de
= ~1 [(r 1
= ~(r
_ (r
+ ~)(u;
This condition is guaranteed by taking 1 ~(~ + ~)-
u~ ~ - uL'
- u~)
824
P. Roe
thereby simply spreading the rarefaction equally either side of its location in the linearized solution. Therefore we can write the interface flux as 1
F(0)--F{~-2(~L F/,.
.)2 u ; - u{
~--~--~*L
_(1~.)2L -2"
F~-F~
1/2({; +
~*~* r,v R -~*t)
= F{ - ( ~ ) 2 _ (~;)2 [F~ - F{]
= F~
-
(~/~)2 (~/~)2 __ (~{)2
[F R -
F{]
We can regard this as a modification of the flux formula (Eq. 16.34)). The sum should now be taken over the waves for which 2 - ~1 (A,~)k < 0, with the factor (r _ ( ~ ) 2 ) 6 [0, 1] applied to the last term. If Eqs. (6.35) or (6.36) are used instead there is a similar modification. This flux formula was derived originally in a paper by Van Leer et al. (1989), but is given here with an original demonstration. With this entropy fix results are obtained for rarefaction waves that are very close to those obtained by using the exact (Godunov) flux. When embedded in the kind of high-resolution methods to be described in Section 6.4.5, it proves quite satisfactory. With first-order methods the modification needed is more drastic; it can be shown that changing the factor to ~E/(r - ~ ) is equivalent to a method derived for scalar problems in Roe's work (1985). Methods that are sufficiently diffusive do not need entropy fixes. The point of using the fixes is to be selective as to where dissipation is added.
6.4.4 POSITIVITY We have already shown that the Godunov flux leads to a scheme that keeps the density and pressure positive for all time and mentioned that no linearization can give a Riemann solver with that property. For the pair of rarefactions shown in Fig. 6.3 (right-hand side) the exact solver predicts a vacuum if (see Smoller, 1983): 2 vR - vL > ~
(aL + aR)
7-1 By contrast, a linearized solver attaches no special significance to the value zero, and as the velocity difference is increased the central pressure and densities fall proportionally and eventually become negative. The proof appears in Einfeldt et al. (1991). It is not correct to describe the problem as an
825
Shock Capturing
instability, although the effects may be similar. It cannot usually be cured by reducing the timestep; when this is tried the solution fails at about the same elapsed time; the solution exits the region P in a thoroughly stable manner. An empirical response is to reset the density and pressure, whenever they become negative, to small user-defined values. This is occasionally satisfactory, but errors incurred in this way tend to accumulate. There are various Riemann solvers, other than the exact one, that can be proved to preserve positivity. This can take the form of a "fix" to the linearized solver, or can derive from an independent approach. The simplest fix is based on the same observation of convexity that allows proof of positivity for the Godunov scheme. If the solutions of the approximate Riemann problem contain no states not in P, then neither can their averages. Assume that the approximate Riemann solution is conservative (the one in Section 6.4.2.1 is). Then in a situation like the double rarefaction (Fig. 6.3, right-hand side) the density will only go negative if the wavespeeds are underestimated. Figure 6.14 shows a variety of approximate solutions to a double rarefaction problem. In (a) the exact density distribution is compared with a linearized solution in which the density becomes negative. Because of conservation the area underneath the two graphs is the same. In (b) we see that applying an entropy fix does not improve the situation. If the linear solution is modified so that the two
_
_
!
I
I
l
(a)
X
p . . . .
." "
I
. . . .
I -.
I
I
I
"..I
"
I
l
-
l i
t.'"
t I'. I -.
-t
I"
i i
I I
l
-
t
I I
I
I
i
-'1 9 I
"
9
!
~
I.__.__ ~ . . . . . . . . . . . . . . . . . . . . . . . .
I
i X
(b) FIGURE 6.14
l
I
i
T'-"
i
i
(c)
The exact solution to a strong double rarefaction problem and various ways to modify the linearized solution: (a) comparison of exact and linearized solution; (b) comparison of linearized solutions with and without entropy fix; and (c) comparison of linear and nonlinear approximations.
826
e. Roe
waves move faster, keeping the same area, the density will be increased. This is the basis for the modified flux function proposed by Einfeldt et al. (1991). An important result is that if a certain flux function preserves positivity in one space dimension, it also does so in more than one (Jiang and Shu, 1996; Linde and Roe 1996). Preservation of positivity for MHD flows is discussed by Balsara and Spicer (1999a) and by Janhunen (2000). 6.4.4.1 Dubrocca's Proposal A very interesting proposal that guarantees positivity was made by Dubrocca (1998). To simplify the algebra it will be explained here just for the flow of an isothermal gas, for which the conserved variables, flux vector, and Jacobian are given by
()
(
P ' F = pv
v-
)
pa 2 pv +pv 2 ' A -
[ 0
a2 - v2
iI
2v
where a is a constant sound speed (p = a2p). Dubrocca writes Ap = a2Ap + flApAvA(pv) - flApAvA(pv)
and so, formally [ -
0
1 ] ( 6 . 5 3 2O+flApAv
a2_o2_flAvA(pv)
)
which still meets the condition ,4[u] = [F] under the choice (Eq. 6.47)) (so that isolated shocks are still recognized). However, the wavespeeds are now the roots 2-, 2 + of 22 - (2~ + flApAv)2 + ~2 + flA(pv)Av - a 2 = 0 (and are always real if fl >_ 0). It can be shown that the density in the central region is given by p, _ (2+ - ~)PR + (~' - 2-)PL - ~Av -
(6.54)
2+_2_
-
1
= ~(p, + pR)
(fl(Ap)2 + 2~/PLPR)AV
....
2~(/~Ap) 2 + 4fl pC'~p-~)(Av)2 + 4a 2
which decreases monotonically (if fl > 0) with Av toward a finite limit 1 P * - - ~(PL + PR)
_
fl(ap) 2 + 2 ~ / p L p a 2V/(flAp) 2 + 4 f l ~ / p L p -R
This limit will be zero if fl satisfies (the algebra is omitted) fl2(Ap)2 + 4 f l V r ~ -
1 - 0
(6.55)
Shock Capturing
827
the positive root of which is /3
-
1 ( 4 7 [ + 4-F~) 2
(6.56)
With this choice the method predicts a vacuum state for infinitely strong rarefactions. It is still not clear to me whether there is a physical interpretation to this proposal, or if it is merely an exercise in curve-fitting, but it does also remarkably simplify the expressions for the wavespeeds; we obtain 2+ _ 1 (v/_ + vR)+ ~/a 2 + 1 (Av)2
(6.57)
in which we see specifically the required speeding-up of waves. A version for the ideal-gas Euler equations requires two parameters, since positive density no longer guarantees positive pressure (Dubrocca, 1998). More analysis and practical experience are needed to evaluate this suggestion.
6.4.4.2 Kinetic Schemes The Riemann solvers discussed in Sections 6.41 to 6.4.4.1 make the same continuum assumption as the equations being solved. We could also derive the flux across an interface by appealing to a molecular picture. It would not be appropriate to use a very precise picture based, for example, on the full Boltzmann equations, because the expense would not be justifiable. But simple pictures inspired by kinetic models may be useful, especially to enforce positivity. If we imagine that all ceils are populated at time t = t" by some set of particles, each of which carries positive mass and internal energy, and some of which pass into the neighboring cell, then it is certain that the new populations at t = tn+l also have positive mass and internal energy. When these populations are averaged at the new time level, a kind of instantaneous collision takes place that restores thermodynamic equilibrium. The problem with such kinetic schemes, however, is that they resolve contact discontinuities rather poorly. Imagine two adjacent cells that differ only in temperature. The molecules in each have the same mean velocity but different random velocities. Unless collisions are accounted for, the two sets of molecules stream through each other, diffusing the temperature. However, if collisions are allowed for, the flux evaluation becomes expensive. Perhaps the most complete development of these ideas is to be found in the work of Xu et al. (1995), who populated each cell with a Gaussian velocity distribution corresponding to its initial (equilibrium) state, and then solved (analytically, in terms of error functions) at each interface a linear version of the Boltzmann equation with BGK relaxation terms. The relaxation time is left
828
e. Roe
as a free parameter. In the smooth regions of the flow it may be given a fairly arbitrary, and possibly realistic, value. Near shocks it is increased substantially to promote the "streaming through"; this gives stability at some cost to accuracy. The dissipation provided by this flux function is more or less that of the Navier-Stokes equations, but at an artificially low Reynolds number (at least near shocks). This BGK kinetic scheme is provably positive, and Xu claims that it does not admit the "carbuncle phenomenon" (see Section 6.8). Other interesting kinetic schemes are due to Fey (1998a,b) (in rather general form, usefully modified by Noelle (1999)), Croisille (1995) and Xu (1999) (for MHD), and Mieussens (1998 and to appear) for rarefied gases.
6.4.5
H I G H - R E S O L U T I O N SCHEMES
We now need to confront the consequences of Godunov's Theorem (Section 6.3.3.4) that schemes employing the same interpolation strategy everywhere are either oscillatory or only first-order accurate. Independently, several people have found how to overcome this, and all methods involve some form of feedback from the data to the algorithm. The generic term high-resolution schemes has been coined to denote any scheme that employs such feedback to produce well-resolved discontinuities.
6.4.5.1 MUSCL-type schemes The common element to most of these is some form of limited reconstruction. We return to the scalar advection problem (Eq. (6.21)). Recall the strategy for reconstructing the data near an interface from the nearby cell averages (Fig. 6.7). The flux is then calculated by assuming that it is the reconstructed polynomial that has to be advected. This works well if the data is well represented by a polynomial (if it is "smooth") but can work badly if not. We therefore constrain the reconstruction so that it cannot cause overshoots, and do this by specifying that the reconstructed polynomial does not introduce values outside the range of those present in the data. This is most easily explained when the "polynomial" is a straight line. The reconstruction in cell j will be of the form
ujO0 =
+
jO,-
(6.58)
Shock Capturing
829
where sj is a slope that remains to be determined, and ~j is the centroid of the cell. Previously we took for sj one of the two choices
(2nd-order Upwind),
z~ sj -- sJ~ =
(6.59) (Lax-Wendroff)
Ax
We could have chosen a symmetrical formula
(6.60)
In Fig. 6.15(a) we do just that; there is a light sloping line that represents the RHS and a heavy dashed line parallel to it that defines the cell reconstruction. For smoothly varying data this creates no problem. However, with the more rapidly varying data in Fig. 6.15(b) this strategy (thin sloping line within the central cell) gives a value at the right-hand edge of cell j greater than the average in cell j + 1. To avoid any risk that this overshoot could advect into the next cell we reset the gradient sj so that this situation is just averted. The value that we need is s j - 2s~. In Fig. 6.15(c) the only choice that will avoid
z ~X
(a)
X
~X
aA&
(b)
(c)
~
Ax
aAt=
x
aAt
FIGURE 6.15 Limitedlinear reconstruction of data near an interface. The heavy dashed line gives a reconstruction of the data in the central cell. The reconstruction reproduces the given value and is given a slope that will not cause overshoots.
830
P. Roe
introducing new values is sj - 0 and this will be the case any time ~j is a local extremum. A formula that covers all cases and can be coded efficiently is sj -- H(s}s~)sgn(sC)min(isC[, 2Is}l, 2{s~I)
(6.61)
The first factor (H is again the Heaviside function) sets the reconstructed slope to zero at extrema. The second factor gives it the correct sign and the third factor the correct magnitude. There are other recipes that will also avoid overshoots. For example, the harmonic mean 2H(s~-s~)
(6.62)
is less expensive and has similar properties. The "minmod" average, usually giving inferior results, is
sj = H(s~sR)sgn(sC)min([s~[, [s;[)
(6.63)
More generally we may write sj -- M(sL, sR)
(6.64)
where M(a, b) is a generalized mean such that
1. 2. 3. 4.
M(ka, kb) = kM(a, b) (independence of measurement units), M(a, b) = 0 if ab < 0 (switch-off at extrema), ]M(a, b)] < 2max(]a], ]hi) (no overshoots), and M(a, a 4- E) -- a 4- (9(E) (true average in smooth regions).
Once a reconstruction strategy has been decided on, the flux follows by integration as fj+l/2 = at 1 (1 + a)hj + 89(1 -- a)~j+ 1 + ~l a(1 -[v[)Axsj+ 1/2(,-~)1
(6.65)
where a = sgn(a). This procedure was introduced by van Leer (1979), in the context of a scheme to which he gave the acronym MUSCL, standing for Monotone Scheme for Conservation Laws. Simpler alternatives for the overall scheme have been devised, but the limited reconstruction remains an important element in many methods. It is exceptionally robust and has become highly popular but is, nevertheless, slightly heavy-handed. For one thing we could have allowed some overshoot at the edge of the cell, provided that the average advected across the interface did not exceed the neighboring value. This requires us to build the CFL number into the limiting process and since the CFL number will be different for each wave, care is needed when treating a system of equations (Section 6.4.5.4).
831
Shock Capturing 1.50
0.50
-
-0.50
, 0.00
.~ 0.20
,
9 0.40
,
, 0.60
0.80
,
o.~o
'
1.00
u - 0.25 1.50
,
0.50-
-0.50-
o.oo
,
o.~o
'
o.~o
'
o.~o
'
,.oo
u - 0.50 1"60 i
0.00
0.20
0.40
u FIGURE 6.16
-
0.60
0.80
1.00
0.75
Numerical results for the minmod limiter.
Another objection is that the method does not distinguish between spurious oscillations and genuine extrema. Actually that is rather a difficult distinction to make, and there is really no way to do it based on information about just three cells. Any method capable of overcoming this problem would have to use more information than this, with quite severe consequent increase in computing time. Nevertheless, there will be applications where this pays off (see Section 6.4.6)
832
P. Roe
6.4.5.2 M o r e E x p e r i m e n t s Analogously with Section 6.3.3.2, we conduct the same numerical tests to show the difference between the various limiter functions (see Figs 6.16 to 6.18). First, a comparison with the "parent" Lax-Wendroff scheme (Fig. 6.9) is rather dramatic. Not only are all the overshoots removed, but the phase errors have become remarkably small. Second, the relationship between the quality of the results and the use of the limiter turns out to have an intuitive quality. The rather cautious minmod limiter is rather diffusive and loses information about the extrema. The "aggressive" Superbee limiter (see Section 6.4.5.4) avoids
1.50
0.50-
-0.50
r
9
o.oo
,
~
o.~o
o.~o
'
o.~o
'
o.~o
. . . . .
i
o.~o
~.oo
u - 0.25 1.50
-0.50
~
o.oo
~
o.~o
'
o.~o
j
o.~o
1.oo
v - 0.50 1.50
0.50-
-0.50
o.oo
~
i
0.20
,
o.~o
i
o.~o
'
0.~08
i
v - 0.75 F I G U R E 5.17
N u m e r i c a l results for the h a r m o n i c m e a n limiter.
1.00
833
Shock Capturing 1.50
0.50
-0.50
I
o.oo
o.~o
J
I
o.~o
o;o
,
'
o.;o
J
1.oo
u - 0.25 1.50
0.50
-0.50
0.00
.
i
0.20
o.~o
I
u
-
~
o.~o
J
o.;o
o;o
'
0.80 ~
1.00
0.50
1.50
0.50- t ~
-0.50 0.00
o.~o
'
o.~o
u FIGURE 6.18
-
'
'
1.00
0.75
Numerical results for the Superbee limiter.
losing extreme values, but only by building false "plateaus." The moderate "harmonic mean" compromises between these extremes (see Fig. 6.17).
6.4.5.3 Hancock's Scheme There is more than one way to extend the MUSCL scheme from linear advection to systems of conservation laws. One of these is based on a neat geometrical/physical interpretation of what happens when the reconstruction is linear. The integral giving the flux through the interface (see Fig. 6.19) is just aAt times the mean height of the shaded region, which is also the recon-
-
-- n+l/2
n+I/2 9
~A~ .
t ~
.
n+|t2
t
t
n+l/2
t n
.
. !
[
x
Ax
aAt/2
Ax
x
aAt/2
FIGURE 6.19 Geometric interpretation of the flux calculation for a linear reconstruction. On the left-hand side is the case a > 0; on the right-hand side a < 0.
835
Shock Capturing
structed value that will pass through the interface at t - t" 4- 1 At. Therefore, let the solution inside each cell evolve for 1 At according to
a, uj(x) = -asj This simply shifts the initial data by aAt/2. If a > 0 this will bring to the right interface the state that it should have. If a < 0 this will bring to the left interface the state that it should have. So every interface has the correct state either on its left or on its right. Solving a Riemann problem with these neighboring states as data will give the correct flux. Remarkably, this actually works for systems of equations! The algorithm can be summarized thus. To avoid clutter the time superscripts are mostly omitted. We begin with the data at time level n. 1. Find the gradients of the conserved variables u = {u(m)}, where/,/(m) is the mth conserved variable, between all pairs of consecutive cells, . (m)
. (m)
.(m) uj+ 1 -- uj ~j+1/2 Ax 2. Find the limited gradients of the conserved variables in each cell, _
[ (m)
~(m)
3. Reconstruct the conserved variables within each cell according to +
4. Allow each cell to evolve "internally", that is, without interacting with its neighbors, for half a timestep, according to Uj(X, t n+l/2) -- flj + [(x -- ~))I - 89 AtA(fij)](s))
This need only be evaluated at the endpoints of the cell, Xj4_1/2, 5. Find the flux through the interface by solving the Riemann problem with the states immediately to each side, that is
Uj(Xj+I/2, tn+l/2), Uj+I(Xj+I/2, t n+l/2) 6. Update the solution to the new time level, using Eq. (6.31). The fact that this procedure automatically collects, even for systems of equations in higher dimensions, all of the effects needed both to ensure second-order accuracy in smooth regions and to preclude oscillations near discontinuities is not self-evident, and will not be proved here. It was observed by Steve Hancock while working on the design of nuclear reactor safety codes, and communicated privately to Bram van Leer, who made use of it (van Leer
836
P. Roe
1982). It is a fairly economical process to encode. It is natural to consider the calculation of the flux as a subroutine that calls two other subroutines. One of these selects a particular nonlinear average M ( a , b) and one selects a Riemann solver. Whatever choices are made the method is nominally second-order accurate, although the experimental order of accuracy will probably be less if shocks are involved. One way to think about the choices is that they are analogous to selecting options on a motor vehicle. There may be a standard model available that performs adequately under "normal" conditions, but operation under extreme conditions may require some extra features such as four-wheel drive or air-conditioning. et al.
6.4.5.4 Flux-Limiting, F l u c t u a t i o n - S p l i t t i n g As already noted, there are various ways to take a nonlinear average of two gradients. They can be analyzed in terms of their effect on the propagation of a linear discontinuity in (Eq. (6.21). Constant-coefficient second-order schemes will spread such a discontinuity, giving it a width proportional to the cube root of time (Hedstrom, 1975). Experimentally (Roe and Baines, 1984) it is found that most of the schemes described here do the same. It is possible to pursue the averaging rather more agressively; even biasing the average toward the larger of the two slopes if it is permissible to do so. In practical problems this can help to improve the resolution of contact surfaces and vortex sheets, but it may oversharpen shockwaves, leading to instabilities. It may therefore be useful to have a method that allows different limiters to be applied to different waves. Retum to the linear advection problem. The flux for the MUSCL scheme can be written as follows, where the absolute value Iv[ and the logical value ~r = sgn(a) take care of the different cases a ~ 0, and where superscripts are omitted to avoid clutter, fj+l/2 = a[l( 1 + cr)fij + 1(1 - ~)fij+l + 1~( 1 - I v l ) A x s j + l / 2 ( l _ ~ ) ] = a[89 - cr)hj 4-1(1 - a)aj+l + 1or(1 -[v[)AxM(Sj+l/2, Sj+l/2_~)] = a[89(1 - 6)a) + ~1 (1 - crlaj. 1 4 - l a ( 1 -[v[)M((hj+ 1 - hj), (hj+l_ ~ - fij_~))] = a[ 1 (1 4- cr)fij 4- 21 (1 - a)fij. 1 1 +~a(1
(
-[v[)(aj+ 1 - aj)M 1 (hj+l_~ '
(aj+
-
-)]
uj_~)
aj)
= a[ 1 (1 4- a)fij 4- 89(1 - a)aj. 1 4- i a(1 - Ivll(hj. 1 - hi)M(1, r;.1/2) ] = a[l(hj 4- hi. 1) - l a ( 1
- (1 -[v[)dp(r[.~/2))(hj+ ~ - hj)]
(6.66)
837
Shock Capturing
where (6.67) and we have defined the limiter function or flux limiter ~b(r) - M(1, r)
(6.68)
In the manipulation, various properties of the generalized mean have been exploited. In particular the property M(ka, kb) - kM(a, b) has been used in the form M(a, b) = aM(l, b/a) to show that the scheme really depends on a single "smoothness parameter," which is just the ratio of two consecutive gradients. In fact any symmetrically defined generalized mean can be equivalently represented, through eq. (6.68), by a scalar function ~b(r) such that 4)(0 - rd?(1/r). The proof is
The conditions already given on M translate into the conditions ~b(r)<2,
r
<2
~b(l+E)--l+(fi(E)
~b(r)=0ifr<0
A diagram of ~b(r) versus r is called a limiter diagram; limiters that have broadly similar visual appearance in this diagram produce similar results. The shaded region is the region within which the function must lie according to a simple analysis that assumes ~b(r)- 0, r < 0. Limiters that lie toward the lower boundary give more diffusive results. Those toward the upper boundary give sharper but sometimes more erratic results. We can think of the different limiters as representing "cautious" or "agressive" policies toward the estimation of gradients. An agressive policy could be one that biases the gradient toward the larger of the two alternatives if this will not cause an overshoot. This has the remarkable effect of halting the otherwise inevitable numerical diffusion of linear discontinuities (Roe and Baines, 1984), so that they do not broaden beyond some finite width. One then speaks of "compressive limiters" for example, the upper bound of the diagram defines the "Superbee" limiter. ~b - max[0, min (2r, 1), min (r, 2)] (6.69) To ensure that no overshoots appear in the solution we may impose a condition 14 //;+1 _ / / ~
0 <
< 1
(6.70)
14In many texts reference is made to the Total Variation Diminishing or TVD condition (Harten, rl 1983) that requires the scheme to be of the form ~ + l _ ~ __ Cj+I/2 (~ nj+l - u~) - Dj+l/2(u~ m mU)-l) with Cj+I/2 > 0, Dj+I/2 > O, Cj+I/2 + Dj+I/2 _< 1. In practice, the analysis is almost always made on the simpler form (Eq. (6.70)).
P. Roe
838
which expresses the condition that the new value of uj must lie in the range defined by u~_1, u~. This condition is appropriate for a > 0 and becomes, for a finite volume update,
( n+l/2 ~Ln+l/2 0 < Atdj+l/2
j-l~2
< 1
7-uT_
-
For the flux defined by Eq. (6.66) this becomes
O
:=~
=r
% + 1/2(1 - v)qb(rj+l/2)(%+l - %) - %-1 - 1 / 2 ( 1 - v)dp(rj_l/2))(fi j - ~ - 1 )
1 r 1 0 < 1 + ~(1 - v ) ~ - - ( 1 -rj+l/2 2
1
~)(r'j+1/2)
-- 1 <--(1 - 2
1 - v)~b(rj_l/2) < - v 1
-- V ) ~ - - - ( 1 ./+1/2 2
2
V)ffi(rj_l/2) <
-
~
V --
-
1-v
-
~(rj+~/2) <
1 --
<1
2 4~(r;_~/2)
rj+l/2
<
-
--
v
This is the "limiter inequality." As it stands there is no way to simplify it because it deals with two independent values of the limiter; to make progress we have to make "worst-case" assumptions. For example, if we assume that neither ~b nor dp/r will ever be negative, then the worst case for the left inequality is that ~(rj+l/2) = 0. Similarly the worst case for the right inequality is that ~b(rj_l/2)= 0. So sufficient conditions are that 2 O<~b(r)<-, v
O<
~b(r) r
2 < -1-v
~b(l+e)-l+C(r
qS(r)-Oifr
The case a < 0 is dealt with just by adding modulus bars to the v. These bounds are substantially more generous than those in Eq. (6.69), although those turn out to be the most generous that can be set independently of v. One way to exploit the dependence on v is to pick the limiter so that the default scheme in smooth regions of the flow is the third-order scheme contained in Table 6.1. Comparing that flux with Eq. 6.66, one finds that they are identical if ~b - 1[(2 - v) 4- (1 q- v)r]
(6.72)
839
Shock Capturing
In smooth regions of the flow we have third-order accuracy, and the limiter inequality is satisfied if we do this for v 3+v-
4-v -1-v
and use the upper limits elsewhere. The resulting limiter function is displayed in Fig. 6.20 (right-hand side), (see Arora and Roe (1997)). A further extension of the limiter concept derives from noting that the limiter inequality can be satisfied if (see Spekreijse (1987)) - m < ~b(r) < 2 - m,
-m <
qb(r)
< 2- m
(6.73)
F
This admits v a n A l b a d a ' s l i m i t e r (van Albada et al. (1982)) r+r ok(r) - -
2
l+r 2
whose use makes the flux function a differentiable function of its arguments, a useful property for implicit methods.
6.4.5.5 Application to Linear and Nonlinear Systems To apply this method to a system of linear equations, we can consider separately the flux due to each characteristic field. We write the identity u(0) = ~ (l h 9u(0))r h
(6.74)
h
Now (s " u = R h ) is a Riemann invariant carried with speed 2 k, so that we have s
u(0) - l h 9 [ l ( f i ; + fij+l) - l a ( 1 - (1 -Ivhl)c~(~+l/2))(fi j - fij+l)]
r (r) /
~,~,, -
F,7~
!
1 ~/~
-
0 (r)
r/ O-
1
2
3
r "~
1
f
1
2
3
r
4
5
6
FIGURE 6.20 Limiter diagram. On the left-hand side no allowance is made for the CFL number. On the right-hand side, allowance is made and the case v -- ~ is shown.
840
p. Roe
and therefore u(O)
-
-
~ lh 9[l(fij + fi)+l) - 89 k
{7 k
-
- (1 -]vkI)~(rj+l/2))(u j
I -- i(fij + U)+l) - ~ I Y~ [crh(1 - (1 -[vhl)r k
ak
h 9 (fi
_
U)+l)]r h
-
- U)+l)]r h
(6.75) Hence Fj+I/2 = Au(0) 1
--2--l(~j + Fj+I ) --2
12kl(1- (1 -Ivl)c~(r;+l/2))lk.(~t j
-
-
l~j+l) r h (6.76)
Compare this expression with the flux obtained by solving the linearized Riemann problem equation (6.36). The only difference is in the factors ( 1 - ( 1 - ]v])dp(r~+l/2) ) which modify the strengths of each wave. With the choice ~ - - 0 we recover precisely Eq. (6.36). With the choice 0 ~ - 1 (LaxWendroff) the factor becomes I vkl , showing that the wave strengths are reduced, and this is in fact the modification that achieves second-order accuracy. If these formulas are applied to an appropriate local linearization (Section 6.4.2.1) then they can be used in the nonlinear case.
6.4.6 ESSENTIALLY N O N - O S C I L L A T O R Y
(ENO)
SCHEMES
The schemes just discussed compute the flux from a knowledge of the state in at most three cells. There is a limit to how much can be accomplished with that much information. In particular it is impossible to know whether a value of r very different from unity has arisen because we are close to a discontinuity or to a local extremum (see Fig. 6.21). Lacking the ability to make this distinction is a handicap. In the case where we are close to a discontinuity some form of limiting is usually essential. However, close to an extremum, it leads to an unwelcome clipping phenomenon. Suppose that at some time we happen to sample values that just bracket the extreme value, as in Fig. 6.21 (right-hand side). At the next timestep, these become the new limits on the allowable values, and the true extremum can never reappear. The previously described methods are therefore unsuitable for the long-time 15 propagation of, for 15A reasonable definition of "long-time propagation" would be a situation in which waves are required to travel a distance more than, say, 10 times their wavelength.
841
Shock Capturing
U ,.~
~////)A//
" ~FI:/F~
z//////
9 "
-_*_-_*_-_-_-_
.////////~?'/////"
1/11111 ~'//////.
,11/11/
N
'
I
I
~<
~///..~ ]
FIGURE 6.21 Illustrating the difficulty of distinguishing jumps from extrema. The three central cells have the same values in each of these two situations, but on the left they form part of a discontinuity whereas on the right they form part of a local maximum.
example, acoustic waves with short wavelengths. To improve the situation, more information is required, for example, second derivatives. There is a general prescription for reconstructing a function to arbitrarily high order while avoiding unwanted oscillations. The idea will be explained in terms of interpolating in a table of point values uj = u(xj) and it is assumed that we want to reconstruct the function within some interval xj < x < Xj+l. Firstorder reconstruction is trivial; we just take the straight line joining (xj, uj) to (xj+ 1, Uj+l). To obtain a second-order interpolant we consider adding to the pair of points {J, J + 1} either the point J - 1 or the point J 4- 2. In either case we can find a quadratic function that passes through all three points, and therefore still passes through the points J,J 4- 1. We select whichever of the two quadratics has the smallest absolute value of the second derivative (in other words the smoothest). To obtain a cubic interpolant, we start with the quadratic already chosen, and consider extending its stencil either by one point on the left or by one point on the right. With either choice we can find a cubic interpolant that passes through some set of four data points, always including J, J 4- 1. Out of these two cubics we select the one with the smallest absolute value of the third derivative. And so on. Finding an interpolant of order p requires that we consider p - 1 points on either side of the interval under consideration; thus 2p points in all. The elegant result is that the interpolant, defined over the subset of p 4- 1 points actually employed, contains in this range of x no more local extrema than were to be found in that list of p 4- 1 values. The interpolant therefore introduces no new exrema; it is however allowed to introduce new extreme values, and would adequately resolve the situation shown in Fig. 6.21. There are many variations on this basic idea. One is that in a finite volume scheme we actually have cell averages as data rather than point values, and require the reconstruction in each cell. This is neatly dealt with by forming partial sums of the cell averages: j+1/2 ~ l~j,
sj+3/ = % +
= aj + aj+ + aj+,,
tc.
842
P. Roe
3
2
5
6 o
0
O
3
5
6
o
1
6
0
FIGURE 6.22 Illustrating the Essentially Nonoscillatory technique for interpolation in the interval 3-4. We start with linear interpolation and at each stage consider interpolating a polynomial of one order higher on two competing extensions of the stencil. We always choose the smoothest alternative.
The values of S are the values of the integral of u at the interfaces. We make a monotone interpolant of S and then differentiate it. The original ENO concept as expounded by Harten and Osher (1987) and Harten et al. (1987) has been found to be slightly unreliable, perhaps because it puts too much stress on the mathematical nature of interpolation and not enough on the physics. For example, the smoothest interpolant will always be chosen, even if the values defining it are all downwind of the point j. This seems to cause a slow buildup of error, despite the nominal accuracy. Therefore, more sophisticated procedures have been developed, among them the Weighted ENO, or WENO method (Jiang and Shu, 1996), that biases the stencil toward the physically relevant one. Timestepping for ENO and WENO schemes is not usually done by integrating the interpolant, because in practical situations involving nonlinear equations the work of obtaining all the necessary terms in Eq. (6.27) is prohibitive. Instead, semidiscrete methods are employed (see Section 6.3.4). The flux at each stage is found by solving a Riemann problem for the reconstructed states either side of a given interface. An authoritative survey of ENO methods is given by Shu in the book edited by Barth and Deconinck (1999).
843
Shock Capturing
6.4.7
A V O I D I N G THE R I E M A N N P R O B L E M
Solving the Riemann problem, with whatever degree of approximation, appears complicated, although it need not be expensive with efficient coding. For the Euler equations at most one of the waves has to be accounted for, as in Eqs. (6.34) and (6.35). More valid reasons for avoiding the solution of Riemann problems would be that for some particular problem the structure of the equations makes the Riemann solution complicated, or that a code needs to be written that will explore numerous different governing equations. 6.4.7.1 Lax-Friedrichs In its original form, the Lax-Friedrichs method (Lax, 1954) stands at the extreme of robustness and simplicity. The basic idea is to use a staggered grid, like the pattern of bricks in a wall. One way to index such a grid is to continue to use integer indices for the cells but with the understanding that j + n is even. The interfaces can then be labeled with the odd combinations. Suppose that at time tn the data is constant in each cell. Suppose also that the timestep is small enough that waves from the old interfaces do not reach the new interfaces. Then the fluxes along the new interfaces are constant at their old values and the new average value in cell j can be simply computed by integrating f(u dx - F dt) = 0 around the control volume shown in Eq. (6.22): 2Axu~ +1 -- Ax[ll~_ 1 -~- 11~+1]- At[F~+ 1 - F~_I]
(6.77)
This new solution is in fact the average over the staggered grid of the exact solution to the piecewise constant data, just as the Godunov method gave the average over the unstaggered grid. The same argument can therefore be used to show that the Lax-Friedrichs scheme is also positive in the sense of Section 6.4.4. It is sometimes said that use of staggered storage eliminates any need for solving Riemann problems because the waves are "hidden" and never affect the interfaces. Although this is true, it is also true, for the same reason, that there is no possibility of deriving any benefit from the information that Riemann solutions provide. Indeed, although for linear advection the first-order upwind scheme is the least diffusive of the monotone schemes, Lax-Friedrichs is the most diffusive. In this simple form, the method has very little practical use, but does form the basis for some useful methods. 6.4.7.2 N e s s y a h u - T a d m o r Suppose that in Fig. 6.23 the data at time t n are not piecewise constant but piecewise linear within each cell. This will change Eq. (6.77) to read 2Axu~ +1 -- Ax 11~_1 -F- g
r
- ~
- Jt,, [Fj+~ - Fj_~]dt
844
P. Roe
j
j-2
/+2 n+l
Ax At
n-1 j-3
j-I
j+l
j+3
FIGURE 6.23 The staggered grid used for the kax-Friedrichs and Nessyahu-Tadmormethods.
Now the integrands are no longer constant but we can estimate their first derivatives as being 0r - - A i } t u - -A2s, and hence fiT+l = ~1 _[u)_, . + tip+,] - ~At[ F p + 1 -F.",_,] - ~1 Ax (~
A 2 A t2 "~
n -
Ix 2
(6.78) In the Nessyahu-Tadmor (1990) method, the slope evaluated as (and see Eq. (6.63)).
Sj + l ,
for example, is
(6.79) Good results are obtained for a variety of shock problems. Disadvantages of the method are that the CFL number is restricted to 0.5, and that the results deteriorate at small values of the CFL number. This may make the method less suitable as a foundation for viscous problems, although some of these issues are addressed in the recent paper by Kurganov and Tadmor (2000). There is an ingenious version of the method for unstructured grids (Armingjon et al. 1994), although it becomes rather intricate geometrically. 6.4.7.3 HLL, HLLE, HLLC ... Another scheme that "hides the physics," not quite so secretively, is the Harten, Lax, van Leer scheme. We draw lines moving from the origin of the interface with speeds b/, bR, (bE < 0 < bR), as in Fig. 6.24, and choose bL,R big enough to enclose the fan of waves in the Riemann solution. We can find the average state over the interval A C by a similar integration to Eq. (6.77), over the control volume A B C D E F . fiAc(bR -- bL) -- bR6R -- bLfiL -- [F R - FL]
(6.80)
845
Shock Capturing
i-~ approximate domain exact domain
c:.-i
At Ax D
E
:-,::3
F
~ -,::3
bL At
r
bRAt
FIGURE 6.24 Flux functions of the HLL family are derived by supposing that the domain of influence is a priori known to be EAC.
Next we assume (with no justification except that the resulting scheme may have interesting properties) that the solution over AC is actually constant at this mean state. Since this is again an average of states that occur in the exact solution, we have again a positive scheme. Then we calculate the change that has been brought about in one of the cells, say the fight one, and find it to be given by Z ~ ( u ~ +1 - u ~ ) - -
bRAt(fiAt - faR)
FR- - F L]
V bLbR -- At LbR -
bE
(fir
-
6L)
--
bg b R --
bL
= At[F* - FR] where in the last line F* is supposed to be a flux on the interface responsible for producing this change. Solving for F* one obtains F* -- bRFL - bLFR bR - bL
+
bLbR (u R - UL) bR -- b L
(6.81)
The formula does indeed have some interesting properties. Note that if all the information travels in one direction (bL = 0 or ba - 0) we recover an upwind scheme; F * - - F R or F L. If the estimates b/_,R are accurate then in some circumstances the flux is even exact. For example, if the solution is an isolated shock and the estimates for bL,R are the slowest and fastest wavespeeds, h 4- h in the linear Riemann solver of Section 6.4.2.1. However, these estimates do
846
P. R o e
not always cover the exact fan and the scheme is not guaranteed to be positive. To obtain a positive scheme the estimates bt_ = min (v/_ - a/, ~ - h)
bR
--
min (v R + ae, ~' + a)
were suggested in Einfeldt et al. (1991). This is the HLLE scheme. Other choices of bL,R are possible. The choice be = -bL = A x / A t recovers the Lax-Friedrichs scheme, and taking bR = - b L to be the maximum in absolute value of the eigenvalues of AL and AR gives the venerable Rusanov scheme. However, taking bL + bR ~: 0 gives a useful upwind bias. In an important case, however, the HLL and HLLE schemes are highly inaccurate. This is the case when the exact solution to the Riemann problem contains a strong contact discontinuity. The change that should take place at x / t - u is spread over the whole region AC. Toro et al. (1994) present a modification that restores the contact discontinuity (the HLLC scheme) while also preserving positivity if the wavespeeds are properly chosen (Batten et al., 1997). A similar device was used by Linde (1998) to provide an economical flux for the MHD equations, on the assumption that only the fast waves and contact discontinuities needed to be resolved. It makes sense to think of the schemes in Sections 6.4.7.1 to 6.4.7.3 as a systematic hierarchy. The Lax-Friedrichs scheme could be derived by making the grossly simplified assumption that the solution to the Riemann Problem is constant within the region - A x / A t <~ x / t <~ A x / A t and then finding the flux consistent with this assumption. The Rusanov scheme improves on this by narrowing down the domain of the disturbance to -[2[ma ~ ~< x / t ~ [/~[max and the HLL scheme improves again by accounting for the asymmetry of the domain. HLLC accounts for the fact that the disturbance does not create a uniform state. By making the assumptions more numerous and more realistic we could, if we wished, end up by solving the Riemann problem exactly.
6.4.7.4 Flux-Vector Splitting, CUSP, A U S M . . . All of the preceding methods (Sections 6.4.1, 6.4.2, 6.4.4.1, 6.4.7.2, 6.4.7.3) for computing a flux are variations offlux-difference splitting. Given two states uL,R we estimate, with greater or lesser sophistication, the interaction between those states and so compute a flux through their common interface. The essence is to find the flux difference AF = F(un) - F(uL) and then to decompose it into left and right-moving effects: AF = AF + AF
(6.82)
847
Shock Capturing
leading to a scheme of the form At > u; +1 -- u; - ~ [ A F ) _ I / 2 4- AF)+I/2]
(6.83)
An alternative is to carry out the decomposition before the differencing, by splitting ,.=
-)
F -- F 4- F
(6.84)
and then creating an interface flux Fj+I/2 -- Fj 4- Fj+ 1
(6.85)
This is called flux splitting or flux-vector splitting. Although the two procedures are equivalent for linear problems, flux-vector splitting is not so physically motivated. Clearly, given just one state, we can say nothing about how it may be changing. A mathematical justification is to consider whether the flux formula (Eq. (6.85)) would lead to a stable scheme. Given some arbitrary decomposition of the form of Eq. (6.84), the condition for stability is that the matrices A, A, defined to be the Jacobians ~- OF A--~,
~ OF A--~
are respectively nonpositive and nonnegative (Steger and Warming, 1981, van Leer, 1982). This definition allows decompositions to be defined that give rise to workable methods, but in fact all share the defect noted in conjunction with kinetic methods, of which flux-splitting schemes are a generalization. That is, that a stationary contact discontinuity is diffused in proportion to the difference of the densities (Dubois, 1999, and Gressier et al. 1999). This non-physical effect stems from the weakly motivated procedure. A number of attempts have been made to devise hybrid methods that combine the accuracy of flux-difference splitting with the economy of fluxsplitting. Several of these are developed for the Euler equations on the supposition that there is some significance to the representation of the flux as an "advected part" plus a "pressure part."
F -- VxU 4- pw where w - (0, 1, 0, 0, Vx) ~. The idea is that the first term corresponds to the organized velocities in a gas-kinetic viewpoint, and the second term to the random velocities. Although this is not strictly correct, the reader will have realized by now that the devising of a flux function need not be a rigorous business. The usual approach is to take the "convected" term from the "upwind" cell (j 4- 89 with various definitions of the terms
848
P. Roe
involved, and to take a pressure term that is a combination of the pressures in the two cells. Examples include Liou's AUSM flux (Lion and Steffen, 1993), Jameson's CUSP, and the modified AUSM scheme described in Tatsumi et al. (1994), Radespeil and Kroll (1995). 6.4.7.5 Flux-Corrected Transport Flux-Corrected Transport (FCT) was historically the first method to treat the problem of discontinuity propagation correctly by introducing nonlinear feedback. In FCT this is done by calculating the solution twice, once by a low-order method known not to cause overshoots, and also by a high-order method that may 16. An extremum in the solution will be accepted only if predicted by both methods. Otherwise, a correction is made to the fluxes between some pairs of cells. Because one flux is used consistently for updating both neighbors to a given interface, conservation is guaranteed. In FCT, both the low- and high-order methods are central schemes, as defined in Section 6.7. In the low-order scheme, the data are advected, but also undesirably diffused. In the high-order scheme, they are diffused less. The difference between the high- and low-order methods can be thought of as an antidiffusive effect that must be kept under control. For this strategy to be effective, the advection from the low-order scheme should be as accurate as possible. Within the class of monotonicity-preserving central schemes for Eq. (6.21) it can be shown that the scheme with the smallest phase error (~(v) in Section 6.3.3.3) is the Low Phase Error (LPE) Scheme for which the flux is given by fj+I/2,LPE
-
-
a [ 1 + 3V6v+ 2v2 uj - 1 - 3v6v+ 2v 2 uj+x 1
(6.86)
As the high-order method we will take the Lax-Wendroff Scheme (Table 6.1). fj+l/2,Lw = a
1 +v 2
uj +
1-v 2
uj+l
]
(6.87)
The difference between these two is called the antidiffusive flux,
fj+I/2,AD --fj+I/2,LW
1 ~
V2
--fj+I/2,LPE = a ~ ( 1 / j + l
-- l/j)
(6.88)
The flux that will actually be taken is fj+l/2 = fj+I/2,LPE -1- cofj+I/2AD
(6.89)
where co 6 [0, 1] plays here a similar role to a limiter function. ~6In actual implementation two complete evaluations are not neccessary, but this is a convenient way to think about it.
849
Shock Capturing
The calculation is conveniently performed in two steps. A provisional solution u* is calculated using the LPE method with u" as data
~; = ~' _ ~At[~n+~/~,P~ - ~ _ ~ / ~ . , ~ ]
(6.90)
and then the diffusion is removed, conservatively, using the provisional solution as data,
At[
u7 +1 -- uy - - ~
OOj+I/2f~/+I/2,AD -- ~ j - 1 / 2 f j*I/2,AD
]
(6.91)
Now we have to develop a strategy for limiting the antidiffusion. If the provisional solution u* contains a local extremum ~ , this will be enhanced by the antidiffusive step. This is permitted, since the LPE method has already authorized an extremum 17. Thus we only have to consider the case where the sequence u~_ 1, u~, U~+l is monotone. At this point the argument becomes an algebraic exercise. To prevent the antidiffusive step from creating a new extremum we may insist that either (A) .
0<
R•+I -u~
0<
R;+I -u~
<1.0
or that (B) .
-
u;§
<1.0
7 -
These express the fact that u~+1 is required to lie in one of the intervals uy], [u;, uj+{~] and reduce to
0 < -
[Uj_I,
1- 2 E 6
coj_l/2 - c 0 j + 1 / 2 ~ ~ < 1.0 Aj_l/2u* j -
or
0 < --
6
v2[ ,lj2u (_Oj_l/2 . . . . Aj+I/2R*
(_Oj+l/2
1
< 1.0. -
17The method has been called "Phoenical SHASTR' where SHASTA means Sharp and Smooth Transport Algorithm, and the adjective describes how mutilated extrema arise from the ashes of the first-order scheme.
850
P. Roe
Bearing in mind that the ratio of differences must be positive for monotone data, and that the values of co are also positive, it can be seen that a sufficient condition to enforce one of (A,B) is that
6 ~Aj_I/2H* Aj+3/2H*~ coj+I/2 --< V' 1 -------~ min ~ ~
(6.92)
If co = i is not allowed by this, then we take the largest value that is allowed. The apparatus works only so long as the provisional solution can be guaranteeed free of spurious extrema. Since monotonicity of the low-order scheme can be shown to require that v < 0.5, this is also the practical limit of operation of FCT schemes. Often the scheme is simplified by noting that over the useful range of Courant numbers the coefficient C - (1 - v2)/6 varies only within the range [1/6, 1/8]. Replacing C everywhere that it appears in the preceding analysis by its maximum value of 1/6 gives a valid scheme that is somewhat simpler, especially for application to systems of equations. Taking the minimum value of 1/8 instead carries no guarantee, but often works well in practice. Either choice implies some loss of the full second-order accuracy. In Fig. 6.25 the results from the FCT method are shown, including the case v - 0.75 for which monotonicity is not expected. It is fair to say that the appearance of FCT in 1973 revolutionized the computation of strongly discontinuous flows, but also that in recent years it has lost out somewhat to the methods based on upwinding and characteristic analysis. Although the results in Fig. 6.25 for simple advection are extremely good, the main problem for FCT is that of deciding, for systems of equations, which variables are to be examined when testing for extrema. In one-dimensional gas dynamics, for example, the pressure may develop new extrema, but the Riemann invariants may not. Further practical details of FCT can be found in work by Oran and Boris (1987) and a dramatic application in Baum et al. (1995).
6.4.7.6 Jameson's Method This is another method of both historical and practical significance. The paper by Jameson et al. (1981) brought about a complete reevaluation of the Euler equations as a means of computing transonic aerodynamic flows. Previously, aerodynamicists had concentrated most of their efforts on simpler mathematical models, such as potential flow, believing that this would be adequate for their needs, since strong shocks are usually of no interest in aircraft design, which concentrates on efficient (low-drag) shapes for which the shocks are
Shock Capturing
851
1.50
0.50
-0.50 , 0.00
,,
,,,
,
,
0.20
,,
0.40 u -
,
,
0.60
,
,
0.80
1.00
0.25,
1.50
0.50 -
-0.50
o.oo
'
o.~o
'
....
o.~o u -
I
o.;o
I
''
!
o.;o
1.{
0.50
1.50
0.50-
-0.50 0.00
,
., 0.20
,
o~o
' u
'
o~o
-.
o~o
'
1 00
0.75
-
FIGURE 6.25 Numericalresults for the Flux-Corrected Transport Scheme.
weak. It was also believed that Euler computations would be expensive, and that the expense would not be justifiable. Not only did Jameson succeed in reducing the cost of Euler computations far below previous levels, but it turned out that even for weak shocks the potential methods proved unreliable, and were quickly supplanted by Euler methods. Jameson's m e t h o d relies on the simplest central scheme
+1/2
--
2
852
P Roe
together with a cleverly chosen artificial viscosity. This is taken to be a mixture of first and third differences, so that n+l~2
+1/2
__ -
-
![Fj .3r_Fj+I ] 2 + ~-7 [q+l/2tuj+~ -
u~]- ej+l/2[u~+2 -
3u~+1 + 3 g - U~_l]J
These lead, respectively, to second-order and fourth-order dissipation when the fluxes are differenced. The coefficients d 2), E(4), which must be dimensionless, are chosen to give a nonlinear response to strongly-varying data. The central scheme by itself would be unstable. To preserve accuracy, d 2) is chosen to be (.o(kx2), specifically
c+1/2• 2-- )max(Vj, Vj+I) where Vj senses the local variation of the pressure through Vj - / 4 2) IP;+I - 2pj + P;-ll Pj+ l q- 2pj + Pj-1 and is (_9(kac2) except near shocks. The second-order term by itself is found to permit small oscillations in the smooth flow and these are suppressed by the fourth-order term, which must however be switched off near shocks. A simple way to do this is to take ej(4) (2) +1/2 _ max ( 0 , / 4 4) - c)+a/2) The constants K (2), K(4) are determined empirically. In general, the method serves admirably its intended purpose of computing weak shocks in mildly nonlinear flows, but is sometimes found to lack robustness for strong shocks or rapidly transient flows. 6.4.7.7 Chang's Method An original set of ideas has been put forward by Chang et al. (1999) and references therein), originally on the basis of attempting a symmetrical treatment of space and time coordinates. Although this idea has some merit for linear problems, the irreversible nature of shockwaves enforces some asymmetry, which is incorporated into the later versions. The method uses the same staggered mesh as the Lax-Friedrichs scheme, but keeps solution gradients as independent unknowns. Formally the method is only secondorder accurate, but is claimed to perform very well. Like other staggered mesh schemes it becomes geometrically intricate in higher dimensions, but it deserves further analysis.
853
Shock Capturing
6.5
SOURCE
TERMS
In many applications, conservation is modified by "source terms." For example, mass might be created, from the viewpoint of fluid dynamics, if solid fuel burns to give a gaseous product. Then a certain amount of new mass is available to participate in the flow. Or there might be several gaseous species present, which exchange their identity by chemical reactions. In this case, there is a continuity equation for each species, of the form OtPi q- div(PiV) -- Pi - Di
(6.93)
Here Pi is the density of species number i, P~ is the rate per unit volume at which species i is produced, and D~ is the rate at which it is destroyed. In the divergence term it has been assumed that all species move with the same speed v. Summation with respect to i will yield the regular equation for conservation of mass with ~-~i Pi = ]9. The structure of such equations is relatively simple, because all of the additional equations can be put into characteristic form as ~)t(Pi/P) + v . V ( P i / p )
= (Pi - D , ) / p
All of these characteristic equations, for the mass fractions ~zi = (Pi/P), have a characteristic speed equal to the flow velocity, and so do not add new terms to an approximate Riemann solver. Of the versions (Eqs. (6.34) and (6.35)), we can always select the one that does not need the corresponding term. It is probably not neccessary to solve each component of Eq. (6.93) in conservation form, since only their sum is needed to describe a shock transition. This is true on the assumption that the physical shocks are so thin that any special chemistry taking place within them can be neglected. A more complicated situation arises if the different species do not move with the same speed. For example, if tiny bubbles are distributed through a fluid, they might be described computationally by the fraction of mass or volume that they represent, but they will not in general move precisely with the fluid. In this case there are separate momentum equations for each species, of the form ~t(PiVi) + div(PiV i • v i -~ pI) = ~ Fi, k k
(6.94)
where Fi, k might be a rather complicated expression for the force exerted by species k on species i. Another common situation arises in relaxing flows, for example at low densities it may be necessary to consider a fluid as not being in thermodynamic equilibrium, but possessing several temperatures, each of which is attempting to regain the value of the thermodynamic temperature, but possibly being driven out of equilibrium by events such as shockwaves.
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P. Roe
The representation of these complicated situations by sets of differential equations is intricate and sometimes controversial, but the effect is always that the conservation laws are modified to read
Otu + E AjO,,ju = S(u, x)
(6.95) J There is not a great deal of general analytical theory on systems of this kind (although progress is being made (kiu, 1987; Natalini, 1999)) and correspondingly little by way of soundly based numerical procedures. In practice a very common way to proceed is with operator splitting, also known as the method of fractional steps. One solves in alternating order the homogeneous problem
Otu "It-E Aj~xj U
(6.96)
-- 0
J and the system of ordinary differential equations Otu = S(u, x)
(6.97)
Symbolically this may be represented by defining evolution operators
Eh(At)u(t )
(for (6.96))
u(t + At) -- Es(At)u(t )
(for (6.97))
u(t 4- At) =
Then evolution of the full problem over one timestep is given approximately by any of the following operators: u(t + At) --
Eh(At)Es(At)u(t )
or
u(t + At) -
Es(At)Eh(At)u(t )
or
u(t + At) - Es( 89At)Eh(At)Es( 89At)u(t) u(t + At) --
Eh(89At)Es(At)E h(89At)u(t) y
FIGURE 6.26
or
f
Part of a quadrilateral grid, with the local coordinate at an interface.
Shock Capturing
855
In theory the first pair of operators are each first-order accurate and the second pair (known as Strang splitting after Strang (1968)) are each second-order accurate. In practice, very little difference is observed. The m e t h o d is satisfactory provided the interaction between the two subproblems is weak. It is perfect if the operators commute, or if they are orthogonal, but these conditions are rare in practice. In Roe (1991) and Arora and Roe (1993) examples are given where the subproblems are solved exactly but the combined m e t h o d is poor. Operator-splitting is a useful technique for dealing with weak source terms, but w h e n the source terms are strong 18 it is less good. There is no consensus on the proper way to proceed, although insightful analysis may be found in works by Bereux and Sansaulieu (1997), Caflisch et al. (1997), Jin and Levermore (1999), Leveque (1998), and Pember (1993).
6.6 MULTIDIMENSIONAL APPLICATION
6.6.1 FLUX CALCULATION There are rather straightforward mechanisms for extending the forementioned methodology of Section 4 to higher dimensions. See, for example, Hirsch (1998), Godlewski and Raviart (1996) or Toro (1999). Basically, any subroutine that can compute the interface flux given a pair of neighboring states can be adapted to two or three dimensions using the fact that the Euler equations (and most other conservation laws) are rotationally invariant. Suppose that the equations are written in rectangular coordinates x - (x, y, x) with velocities v - (v x, Vy, Vz) and fluxes F = Fx, Fy, F z. Then in some rotated coordinate system x f - 1-Ix the velocities are v t - 1-Iv and the fluxes are F ' - - l i E . We always assume that there is some rectangular Cartesian coordinate system that underlies our computation, and it is in those coordinates, for the sake of consistency, that we measure m o m e n t u m . We will call these the reference coordinates. However, to figure out how, say, x - m o m e n t u m is transferred between cells it is convenient to employ local coordinates oriented with each interface. 18It is usual to compare typical timescales for the two subproblems (Eqs. (6.96) and (6.97)). Typically the computationally relevant time scale for Eq. (6.96) is the time for a wave to cross one computational cell, and the time scale for Eq. (6.97) is a reaction or relaxation time. If the latter is small compared with the former, the calculation cannot proceed as fast as that of the homogeneous calculation, and the problem is said to be stiff. This is slightly different terminology from the ODE case, where stiffness means a large or small ratio between two timescales in the continuous problem.
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To compute the flow through a given interface separating states uL, UR and having a normal vector n, the first step is to compute the components of momentum along and normal to n. This amounts to multiplying the momentum components in the reference system by a rotation matrix H. Now compute the flux in the direction of n by any one-dimensional method, that is, F__' - - F * ( H u t , 1-IUR)
This gives a valid estimate for the flux of mass and energy through the interface, but the flux of momentum relates to momentum in the local coordinates. However, we actually need the flux of the Cartesian momentum, so we compute F -- I-I-1F*(FIuL, HUR)
(6.98)
A common strategy for achieving second-order accuracy is to compute a gradient within each cell by applying Gauss's Theorem to the centroids of neighboring cells:
lj
grad Ui, j --- SN(i,J)
u.n
(6.99)
NO,j)
SN(iO)
where Ni, j a r e some set of neighbors (Fig. 6.27) and is the area of the polygon defined by their centroids. By analogy with the one-dimensional treatment, this symmetric reconstruction would lead to overshoots, so a test is made to see whether the reconstructed distribution would go outside the range defined by Ui, j and its immediate neighbors ui0+l, Ui::kl, j. There is a question here which set of variables should be chosen to perform the test. The characteristic variables lose much of their significance in more than one
ii
~ ,~.
~-~-~"
FIGURE 6.27 The gradient inside a cell can be estimated by applying Gauss's Theorem to some set of surrounding cells.
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Shock Capturing
dimension, and the primitive variables are usually chosen. So if p denotes an arbitrary primitive variable we reconstruct p within the cell as pi,j(x) -- Pi,j 4- ki,j(xi+l/2,j - ~i,j) grad Pi,j where k is a scalar factor playing the role of a limiter. To find a suitable k we begin by computing, for each immediate neighbor of cell i, j, a set of four numbers kiq_l,j, ki,j+ 1 such that, for example,
Pi,j + ki+l,j(xi+l/2,j - :~i,j) grad Pi,) where Xi+1/2,j is the midpoint of the edge between cells (i,j), (i + 1,j) This gives us a scalar multiplier for the gradient such that the reconstruction will exactly recover the neighboring average at the midpoint of the interface. If this multiplier turns out to be greater than 1.0in every case then the original gradient was safe. If any of the multipliers turn out to be negative, we choose ki, j = 0. Otherwise, there will be one or more multipliers in the range [0, 1], and we choose the smallest of these. We have now completed a limited reconstruction within every cell, and in particular we have a left- and rightreconstruction at the midpoint of every interface. Solving the Riemann problem, in the sense of Eq. (6.98), for these states, gives a flux that is a second-order away from discontinuities. This can be used to give a semidiscrete scheme, or the Hancock procedure can be employed (Section 6.4.5.3). Extensions to this procedure giving higher-order accuracy by means of higher-order polynomial reconstruction, and to cells that may be triangular, are given in Barth and Fredrickson's paper (1990). Irregular (triangular, tetrahedral) grids can either be used to store average values within the triangles themselves, or on the dual grids centered at vertices. A useful survey of ideas and methods for unstructured grids can be found in the AGARD report (1992).
6.7
GRID GENERATION
AND ADAPTIVITY
The generation of suitable grids is a vast topic in itself, and I will make no attempt to deal with it in general. However, a topic that is particularly relevant to shock capturing is that of grid adaptivity. This is the efficient use of computer resources by adopting a finer grid in active regions of the flow than elsewhere. Without such a strategy some calculations are quite impossible. The main alternatives are structured (logically rectangular) multiblock (structured ceils within unstructured blocks) and unstructured grids (in principle with no restrictions, but in practice usually an assembly of triangles or tetrahedra). For a long time it was thought that adaptive grids were most compatible with the unstructured approach, because refinement in a structured
858
P. Roe
context tends to yield contiguous cells of markedly different size. With many schemes, waves tend to reflect off the boundary between coarse and fine grids. However, beginning with the landmark paper by Berger and Colella it has been realized that upwind schemes contain the information needed to prevent this, and the combination of Cartesian adaptive grids with upwind schemes have proved remarkably robust. In work by Gombosi et al. (1994) the largest and smallest cells differ by a factor of 222 -~ 4.2 x 106. No numerical difficulties were encountered. Remarkable images of very complex flows can be found in Baum et al. (1995), Quirk and Karni (1996) and Powell et al. (1999) as well as the works already cited. A further example appears in Figs. 6.29 and 6.30. The choice between structured and unstructured adaptation turns on several practical issues. Unstructured grids are easier to fit into awkward geometry, and probably rather fewer cells are required to define the solution with a given quality, but certain essential information (the storage locations of the neighbors of a given cell) has to be expensively stored and accessed. Moreover, for dynamic calculations where the active regions are in continual movement, the grid information needs to be updated almost every timestep. On a purely Cartesian grid, the difficulty is that general boundaries intersect the grid, producing irregularly shaped (and perhaps very small) cells that require special treatment. Any implementation of Adaptive Mesh Refinement (AMR) has to be based on some refinement criterion, a method for signaling which cells are to be refined (and, for reasonable efficiency, subsequently derefined). Ideally these would be based on some kind of error estimate. There is a well-developed theory of error estimators for elliptic problems, but not for hyperbolic problems. The main difficulty is that once they have been committed, errors tend to propagate along the characteristics lines of the solution. Thus errors are found in places other than where they were created. There is active research in this area at the time of writing, but for practical purposes most mesh refinement algorithms employ empirical measures of the flow activity rather than attempt to estimate the error. This involves inspecting cells in pairs and computing differences of the variables, or perhaps in small clusters and computing measures such as the curl or divergence of the velocity. In any case, cells that are found to participate in some measure of activity that is unusually high are flagged for refinement. This can be done cell-by-cell, leading to a tree structure for connectivity (Charlton and Powell, 1997) or the flagged cells can be clustered into structured blocks (Berger and Colella, 1989; Quirk, 1996; Stout et al. 1997). In the latter case each block will usually contain some cells that did not need to be refined, so there is some redundancy in storage, compensated by more efficient coding, especially on parallel machines. With either strategy, typically only a few percent of the run time is spent in "managing" the data.
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859
6.8 ANOMALOUS SOLUTIONS There is no general proof that nonlinear hyperbolic sytems have unique solutions for prescribed boundary conditions. Indeed, one can find clear-cut examples of nonuniqueness. The simplest is shown in Fig. 6.28 (left-hand side) where two solutions of the Euler equations are shown for two-dimensional supersonic flow past a fiat-faced plate. The upper picture is the usual solution, with a smoothly curved bow shock and an embedded subsonic region. However, the lower picture is also a solution, if the shaded region is supposed to contain a region of stagnant air at the pressure that would be generated by flow over a wedge of the appropriate angle. All of the discontinuities involved are entropic, and in the absence of viscosity or heat conduction this is a valid solution. It is even a physically realizable solution, as can be verified from the experimental photograph given as Plate 272 in a book by van Dyke (1982). In the experiment, a thin splitter plate is placed ahead of the body. The boundary layer that forms on this plate creates vorticity that diffuses outward to create a circulation around the stagnant region. A similar effect is possible in the flow past a sphere. Figure 6.28 (right-hand side) shows the regular solution at the top, and at bottom a complicated flow that can can be induced experimentally by placing a needle along the axis of symmetry as shown in the experiments of Maull (1960). There was even a proposal to
FIGURE 6.28 Two solutions for the flowpast a flat step. Two solutions for the flowpast a sphere (right-hand side).
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induce such flows during the reentry of blunt space vehicles to reduce the heating rates (Holden, 1966) but the mechanism (the aerospike) was eventually found insufficiently repeatable for engineering applications. There has been some concern over the fact that solutions like those on the right-hand side are sometimes spontaneously produced by Euler codes. The first systematic investigations were made by Quirk (1994a) and more recent work is summarised in Robinet et al. (to appear) where a connection with genuine physical instabilities is also made. It has usually been supposed that they were a purely numerical artifact, by authors not aware of the nonuniqueness results. It has been claimed on an experimental basis that certain methods will only yield the regular solutions. However, Pandolfi and d'Ambrosio (2000) have found that many of these methods do produce the anomalous solutions on fine enough grids. The schemes most likely to exhibit the anomalies on coarse grids are those that have been designed to have the smallest numerical dissipation, for example the flux formulas of Godunov (Section 6.4.1) and Roe (Section 6.4.2). Generally the anomalies appear only on structured grids and only if the features of the flow are aligned almost perfectly with the grid lines, which tends to keep the dissipation particularly small. To remove them, substantial numerical dissipation can be needed, which goes quite against the spirit of high-resolution schemes. There is some hope that the "genuinely multidimensional" methods discussed in the next section will not suffer this problem. The difficulty is not sufficiently universal that it should discourage the adoption of high-resolution methods but a potential user of these methods needs to be aware of it. 6.9 "GENUINELY" METHODS
MULTIDIMENSIONAL
For one-dimensional problems it is possible to develop a theory that is fairly respectable. The gap between what can be proved and what needs to be done is mostly, due to nonlinearity, but in practice things work rather nicely. The extension to higher dimensions in Section 6.6 is essentially empirical. It stems quite naturally from the representation of the solution as piecewise polynomial within cells and generally discontinuous across interfaces. If, in the spirit of Godunov's original concept (Section 6.4.1, Fig. 6.11), we ask how such approximate data would physically evolve, certainly one-dimensional waves would be exchanged across those interfaces. But this is simply a consequence of the crude data representation and has nothing to do with real physical behavior. Notwithstanding the huge investment in these methods and the numerous predictive triumphs, it is not unreasonable to seek something even better. (For a critique of current methods see Roe, 1993).
Shock Capturing
861
Many authors have sought to develop "genuinely multidimensional" methods, but I am inclined to recognize as such only those that eliminate the "gridoriented waves." This more or less compels the adoption of a continuous representation of the solution, most simply by linear triangular or tetrahedral elements. In turn, the nonlinearity required to circumvent Godunov's Theorem cannot come from any reconstruction procedure of the kind described in Section 6.4.5.1. In a slow progress, several researchers have attempted to rebuild shock-capturing methods along completely new lines. One class of methods initiated by Roe (1987) is based on evaluating the residual over each element, and if this is not zero, distributing some pattern of changes over the nodes of that element. Considerable detail is given in the book edited by Deconinck and Koren (1997) although the methodology is still developing steadily. The requisite nonlinearity can be inserted into the distribution scheme. An attractive programming feature is that the method can be coded as a loop over the elements, with no access to other data. These methods, at least for steady flows, are capable of everything that conventional highresolution methods can achieve; see for example the impressive thesis by van den Weide (1998) and also van der Weide et al. (1999). They also seem to be less sensitive to the "quality" of the mesh (Abgrall to appear). However, the aim is to create methods actually superior to the current ones, and this involves considering what sort of physics the code should imitate. One aspect of this involves realizing that many practical problems exhibit a mixture of elliptic and hyperbolic behavior and that the machinery of upwinding and limiting is needed only for the hyperbolic part. Making this separation cleanly at the discrete level is one way to construct compressible Euler codes that accurately approach the incompressible limit (Mesaros and Roe, 1995; Rad and Roe, 1999; Sidilkover, 1999).
6.10
FURTHER
READING
Many aspects of shock capturing have been curtailed in this account to a quick sketch of the main ideas. There are fuller treatments available. Very comprehensive treatment, both of the numerical schemes and the underlying mathematical theory, is given by Godlewski and Raviart (1996). More basic coverage of the essentials can be found in Leveque (1992) and Morton and Myers (1994). A readable, practical introduction is the book by Toro (1999). Also comprehensive, with an emphasis on aerospace applications, is the two-volume set by Hirsch (1988). A book that covers the basics of hyperbolic problems and extends them to include viscous effects is Tannehill et al. (1997). Some of the historical material has been collected by Hussaini et al. (1997). A volume giving particular emphasis to high-order schemes is the proceedings of a NATO
862
R Roe
Short Course held in September 1998 (Barth and Deconinck, published in 1999). A previous Short Course, sponsored by AGARD, dealt with aspects of unstructured grid methods (AGARD, 1992). The majority of references in this Chapter have been cited in respect of some computational technique. It may be worth collecting some of them here that relate to specific types of flow. Additionally some material is cited here that did not find a place in the body of the text. The large amount of recent work on ideal MHD has been represented by Balsara and Spicer (1999a, b), Brio and Wu (1988), Cargo and Gallice (1997), Croisille et al. (1995), Dai and Woodward (1998), de Sterck (1999), Gombosi et al. (1994), Linde (1998), Janhunen (2000), Myong and Roe (1998), Powell et al. (1999), Toth (2000), Xu (1999) and Zachary et al. (1994). The modification of MHD to include anisotropic ion pressure appears in Wegmann (1997). For relativisitic flows the reader may consult Bona et al (1995, 1997), Donat et al (1998), Eulderink and Mellema (1995), Font (2000), Komissarov (1999) and Yamada (1997). Two-phase flow is treated in Bereaux and Sansalieu (1997), Coquel et al. (1997), Sainsaulieu (1995), Stewart and Wendroff (1984) and Toumi (1992). For flow of non-ideal gases consult Liou et al. (1990), Vinokur and Montagn~ (1990), Toumi (1992) and Mottura et al (1997). For reactive flows consult Bihari and Schwendeman (1999), Bourlioux and Majda (1992), Oran and Boris (1987), and Quirk (1994b). Multi-material flows are to be found in Karni (1996), Saurel and Abgrall (2000), Tang and Sotiropoulos (1999), and Ton (1996). An application to turbulence closures can be found in Brun et al. (1999).
6.11 AN EXAMPLE There are many impressive examples of shock-capturing calculations in the literature, but this review would not be complete without something to whet the reader's appetite. Figure 6.29 shows computed pressure contours in the flow around a rocket-propelled missile shortly after it has left its silo. A adaptive tree-structured mesh is used to capture the fine details of the flow, and the adaptation strategy is designed to cope with changing boundary topology. The geometry here is two-dimensional, and the length scales are indicated in meters in the accompanying figures. A tree-structured adaptive Cartesian mesh is used to capture the fine details of the flow, as discussed in Section 6.7. The number of cells starts off at around 20,000, and increases to about 500,000. The flux function was the linearized Pdemann solver described in Section 6.4.2 with the Superbee limiter function (6.69). The calculation is intended to demonstrate a capability, rather than to solve the problem with realistic parameters. At time t = 0, the missile is in the silo, and the initial temperature and pressure in the "propulsion tank" are each 50 times the
Shock Capturing
863
FIG. 6.29 Shaded contours of Mach number in the flow around a missile leaving its silo. The flow features are explained in the text. (See Color Plate 7).
ambient (atmospheric) values. These are representative thermodynamic conditions for rocket propulsion. However, the mass of the missile was set to the unrealistically low value of 15 kg, in order to enable the missile to travel a reasonable distance in a few thousand time steps. In reality, the mass of a missile of the given size would be three or four orders of magnitude greater than the chosen value, and the missile would certainly not reach a supersonic speed by the time it starts to emerge from the silo. The trajectory of the missile was computed by treating it as a rigid body. The total force and moment were computed by integrating the pressure force and the pressure moment around the surface of the missile, and the equations of motion were time-integrated using a forward-Euler scheme. The most prominent features of the flow are the strong bow shock ahead of the missile that extends all the way around the missile to the ground, the strong shock generated by the blast of the high pressure gas from the silo and its reflection in a Mach stem on the ground, and the various interacting shock, shear, and contact waves around the opening of the silo and around the nozzle. Eventually, the blast shock overtakes the bow shock.
864
e. Roe
The grid on which the computation was performed is shown in Fig. 6.30. It has evolved dynamically by clustering around the rapidly-varying features of the flow, not all of which are prominent in the plot of Mach number, but which are visible in other visualizations. It clearly supports the contention in Section 6.7 that upwind schemes permit the use of grids whose size can change very suddenly.
6.12
CONCLUDING
REMARKS
Shock capturing seems to be a simple idea. The Lax-Wendroff Theorem guarantees that we can capture authentic weak solutions. The complications arise from Godunov's Theorem, which tells us it will not be easy to achieve high accuracy. Therefore, various ingenious mechanisms are put forward to reduce numerical dissipation, but this carries with it the risk that we may also capture discontinuities that violate entropy. The attempt to juggle the conflicting requirements of accuracy and robustness has led to a profusion of rival methods. There are no clear winners. The only thing that all successful
FIG. 6.30 The grid used to compute Fig. 6.28. The coarsest and finest cells displayed differ by a factor of 64 in linear dimension.
865
Shock Capturing
methods have in common is some form of nonlinear feedback to evade Godunov's Theorem. Those who write the codes make personal choices dependent on their tastes and priorities. I have had to wield some rather arbitrary knives to keep this review within manageable proportions. Although I was rather horrified to see how long the bibliography had grown, it could very easily have been two or three times longer. Faced with the choice between offending my friends and overwhelming my readers I have opted for the former and beg their forgiveness.
ACKNOWLEDGMENTS I am very grateful to two former doctoral students at the W. M. Keck Foundation Laboratory for Computational Fluid Dynamics. Dr. Sami Bayyuk of the CFD Corporation performed the simulation shown in Fig. 6.29, and Dr. Timur Linde of the University of Chicago ASCI Flash Center wrote the Euler code that appears in Appendix. I would also like to thank the students of class Aero 623 in the University of Michigan, who during the Winter Term of 2000 cheerfully allowed me to inflict drafts of this chapter on them in lieu of course notes. Their splendid response verified both the viability and the learnability of the material.
APPENDIX: A SIMPLE CODE FOR ONE-DIMENSIONAL GASDYNAMICS The following is a simple code written in C to solve the Euler equations in one dimension using a first-order scheme based on the ideas in Sections 6.4.2.1 and 6.4.3. /***** EULERID - Simple o n e - d i m e n s i o n a l Euler code *****/ #include <stdlib.h> #include <stdio.h> #include <math.h> main () { int n; int i,j; double dx; double time; double tmax;
/* Number of grid points */ /*Conter variables for loops "i/ /* Cell size */ /* Time variable */ /* M a x i m u m time * /
866 double double double double double double double double double double double double
R Roe
s; d; u; c; h; ul,ur; cl,cr; pl,pr; alpha; speed; gamma; fd, fm, fe;
/* /* /* /* /* /* /* /* /* /* /* /*
Temp variable */ Roe- averaged density "i/ Roe- averaged v e l o c i t y */ R o e - a v e r a g e d sound speed */ Roe- averaged specific enthalpy *j/ Left and right velocities */ Left and right sound speeds */ Left and right pressures */ Acoustic wave weight for Roe solver */ M a x i m u m characteristic speed */ Adiabatic constant */ Interface ~luxes */
double* U[3] ; double* R [3] ;
/* Array of conservative variables */ /* Array of residuals */
FILE* fptr;
/* Pointer to output file */
p r i n t f ( " N u m b e r of points: ") ; scanf ("%d", &n) ; /* Allocate work arrays with 2 ghost points */ for (i=0; i<3 ; i++) { U[i] = (double*)calloc((n+2),sizeof(double)) ; R[i] = (double*)calloc((n+2) ,sizeof(double)) ;
} p r i n t f ( " A d i a b a t i c constant: scanf ("%if", &gamma) ;
") ;
/* Read left state and store it in the left ghost point */ printf("Left state (primitive) : ") ; scanf("%if %if %If",&U[0] [0],&U[I] [0],&U[2] [0]); U[2] [0] = 0.5*U[0] [0]*U[I] [0]*U[I] [0]+U[2] [0]/(gamma-l.) ;
U[l] [0] *=u[0] [0]; /* Read right state and store it in the right ghost point */ printf("Right state (primitive) : ") ; scanf("%if %if %if",&U[0] [n+l] ,&U[I] [n+l] ,&U[2] [n+l]); U[2] [n+l] = 0.5*U[0] [n+l]*U[l] [n+l]*U[l] [n+l]+U[2] [n+l]/ (gamma-1. ) ; U[I] [n+l] *= U[0] [n+l] ; /* Set up Riemann p r o b l e m initial conditions */
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/* C o m p u t e
left w a v e w e i g h t
*/
a l p h a *= ( p r - p l - d * c * ( u r - u l ) ) / ( 2 . * c * c ) /* C o m p u t e
interface
;
fluxes */
fd = U[0] [ i ] * u l + a l p h a ; fm = U[0] [ i ] * u l * u l + p l + a l p h a * s ; fe = u l * ( U [ 2 ]
[i]+pl)+alpha*(h-u*c)
;
} else
{ /* O n l y r i g h t - g o i n g
w a v e is n e e d e d
*/
/* A p p l y r i g h t w a v e e n t r o p y fix */ alpha = 4.*(ur-ul+cl-cr)
;
a l p h a = a l p h a > 0. ? a l p h a alpha = fabs(s=u+c) (s > 0. ? s : 0.) /* C o m p u t e
: 0.;
> 0.5*alpha
?
: 0.5*(s*s/alpha+s+0.25*alpha)
right wave weight
*/
a l p h a *= ( p r - p l + d * c * ( u r - u l ) ) / ( 2 . * c * c ) /* C o m p u t e
interface
;
fluxes */
fd = U[0] [ i + l ] * u r - a l p h a ; fm = U[0] [ i + l ] * u r * u r + p r - a l p h a * s ; fe = u r * ( U [ 2 ]
[i+l]+pr) - a l p h a * ( h + u * c )
;
} /* U p d a t e m a x i m u m
characteristic
s p e e d */
s p e e d = s p e e d > (s=fabs (u) +c) ? s p e e d /* D i s t r i b u t e
residuals
-= fd;
R[I] [i] += fm; R[I] [i+l]
-= fm;
R[2] [i] += fe; R[2] [i+l]
-= fe;
} /* U p d a t e
solution
for(j=0;j<3;j++)
*/ {
for (i=l ; i<=n; i++)
{
U[j] [i] -= R[j] [i]/speed; R[j] [i] = 0.;
} R[j] [ 0 ] = 0.; R[j] [n+l] : 0.;
}
*/
R[0] [i] += fd; R[0] [i+l]
: s;
;
869
Shock Capturing
time += dx/speed;
/* U p d a t e timer */
/* O p t p u t s o l u t i o n in p r i m i t i v e form "i/ fptr = fopen ("eulerld.dat", "w") ; for (i=l ; i<=n; i++) f p r i n t f ( f p t r , " % i f %if %if %if\n", (i-0.5)*dx,U[0] [i],U[l] [i]/U[0] [i], (gamma-l.)*(U[2] [i] -0.5*U[I] [i]*U[l] [i]/U[0] [i])) ; fclose(fptr) ; /* Free a l l o c a t e d m e m o r y */ for (i=0 ; i<3 ; i++) { f r e e ( U [i] ) ; free(R[i] ) ; }
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876
P. Roe
Yamada, S. (1997) An implicit Lagrangian code for spherically symmetric general relativistic hydrodynamics with an approximate Riemann solver, Astrophys. J. 475: p 720. Yee, H.C. (1997) Explicit and implicit multidimensional compact high-resolution shock-capturing methods: formulation, J. Comput. Phys. 131: p 216. Zachary, A., Malagoli, A., Colella, P. (1994). A higher-order Godunov method for multidimensional ideal magnetohydrodynamics. SIAM J. Sci. Comp. 15: 263. Zalasak, S.T. (1987). A preliminary comparison of modern and shock-capturing schemes: linear advection, in Advances in Computer Methods for Partial Differential Equations VI, R. Vichnevetsky, and R.S. Stepleman, eds., IMACS Rutgers Univ., New Jersey
FIGURE 6.29 Shaded contours of Mach number in the flow around a missile leaving its silo. The flow features are explained in the text.
INDEX ,,
A Abbe, C., 6 Abbot, H. L., 13 Abel, E A., 17 Absorption theory and line shapes, 743-747 atomic resonance absorption spectroscopy, 766-771 infrared laser absorption and emission techniques, 761-766 lamp, 757 ultraviolet and visible laser absorption techniques, 747-757 Acceleration, space particle diffusive, 476-478 drift, 475-476 electron, 478, 479 Ackeret, J., 14 Acoustic waves, 796-797 Adaptive Mesh Refinement (AMR), 857-858 Adiabatic exponent, 153 Adiabat shock, 344-353 Admissibility, 353-362, 793 Aerothermodynamics, 97 Airy, G. B., 12 Album of Fluid Motion (van Dyke), 683 Al'tshuler, L. V., 16 Amplification factor, 810 Antidiffusive flux, 848 Arago, D. E, 10 Argand, 13 Atomic resonance absorption spectroscopy (ARAS), 766
calibrations and applications, 769-770 experimental method, 767-769 shock tube impurities, 770-771 AUSM, 846
B Bacon, R., 7 Bairstow number, 102 Ballistic galvanometer, 39 Ballistic impact studies, early, 4-5 Bancroft, D., 15 Barnaby, S. W., 14 Becker, R., 340, 363 Beer-Lambert law, 743 Bernoulli, D., 5 Bernoulli's equation, 149 Berthelot, M., 17, 18, 19 Berthelot wave, 68 Bethe, H. A., 7, 159-162, 340, 341 Bethe-Weyl (B-W) theorem, 144, 166, 167 Bethe-Zel'dovich-Thompson (BZT) fluids, 341, 342 adiabat shock, 344-353 admissibility, 353-362 dynamics, 389-403 structure, 362-374 weak, 374-389 BGK kinetic scheme, 828 Big Bang Theory, 7 Biot, J. B., 6
Handbook of Shock Waves, Volume 1
Copyright9 2001by AcademicPress. All rightsof reproductionin any formreserved. ISBN:0-12-086431-2/$35.00
878 Blast tubes. See Shock tubes and tunnels, blast (explosive) Bleakney, W., 11 Blochmann, R., 13 Blow-down wind tunnels air heater, 668 cold, 666-672 diffuser, 670-672 general description, 664-666 nozzle design, 668, 669 nozzle qualification, 669, 670 test chamber, 670 vacuum tank, 672 Blow-down wind tunnels, hot, 673 arch chamber, 674 evaluating flow conditions, 675, 676 impulse generator, 674 nozzle and test chamber, 675 Blow-down wind tunnels, induction, 673 Blunt-body flow, 196, 197 Bodenstein, M., 19 Bone, W. A., 19 Boon, J. D., 16 Boundary-layer interaction, 204 Bow shock observations, 460-467 Boys, C. V., 20 Bragg condition, 721 Bremsstrahlung radiation, 225 Bridgman, P. W., 15 Bunsen, R. W., 18 Bunsen waves, 72 Burton, C. V., 12
C Canton, J., 12 C2, visible and near-ultraviolet transitions, 752-754 Cavendish, H., 18 Cavitation damage, 14 Centered/central scheme, 808 Centered waves, 800 CH, visible and near-ultraviolet transitions, 752-754 Chain Reactions (Semenov), 99 Challis, J., 12 Chang method, 852 Chapman, D. L., 19 Chapman-Jouguet theory/state, 19, 83, 216, 217, 220, 221,632
Index Characteristic theory, 239-242 Characteristic variables, 799, 800 Charge-coupled device (CCD), 21 Chester, W., 486, 487, 520 Chester-Chisnell-Whitham relation, 486, 487 Chisnell, R. F., 486, 487 Chlorine-hydrogen explosion, 19, 30 Christoffel, E. B., 12, 15 1CH2, frequency modulation methods, 760, 761 CH3, ultraviolet transitions, 755, 757 CN, visible and near-ultraviolet transitions, 752-754 CO discharge lasers, 764, 765 Coefficient of thermal expansion, 153 Coherent anti-Stokes Raman scattering (CARS), 680, 743 Cole-Hopf transformation, 380 Collisionless Liouville theory electron heating, 473-475 field structure, 468 ion motion, 470-473 noncoplanar magnetic field, 469, 470 nonlinear waves and ramp width, 468, 469 Compressibilities, 153 Compression discontinuities, stability and viscous, 446, 447 Compression process and solids, irreversibility of, 331,332 Compression shock, 64 Condensation shock waves, 208 Configurations, of shock waves blunt-body flow, 196, 197 cylindrical and spherical flows, 198 duct flow, 197, 198 Lambda shock system, 198, 199 local versus global, 193 in multiple shock systems, 198-201 overexpanded nozzle flow, 199-200 in single shock system, 193-198 steady, oblique wall reflections, 200 sweep, 194, 195 Taylor-Maccoll flow, 195, 196 unsteady flow, 201 wave drag, 194 wedge flow, 193, 194 Conservation, shock capturing and, 789, 791 Conservation laws, 259, 276 conservation of energy, 148, 149, 539 conservation of mass, 147, 148, 538 conservation of momentum, 147, 539
Index Jacobian matrices, 794, 795 laboratory frame coordinates, 147-149 MUSCL (monotone scheme for conservation laws), 828-831 shock fixed coordinates, 149 Continuity equation, 560, 561 Continuous wind tunnels open-circuit, 660-664 return-circuit, 658-660 Convexity, 159, 160, 181 shock capturing and, 799 Cook, S. S., 14 Coplanarity plane, 457 Corrugation instability, 414-416 steady-state perturbations, 425-433 Courant, R., 145 Courant (CFL) number, 807, 808 Cramer's rule, 250 Cranz, C., 20 Cranz-Schardin system, 736, 737 Crocco's theorem, 145, 177 Crocco-Vazsonyi equation, 102 Crosse, A., 14 Cryogenic wind tunnels, 209, 210 Curvature singularity, 255-260 CUSP, 846 Cylindrical flow, 198
D Damping error, 811 Dautriche, H., 84, 85 Davy, H., 18 DeCarli, P. S., 16 Deflagrations, 647, 648 Degenerate four-wave mixing, (DFM), 743 de Hoffman-Teller frame (HT), 457, 473, 479 De Laval, G. P., 10 Density-sensitive flow visualization, 685-693 De Saint-Venant, A. J. C., 5, 10, 16 Descartes, R., 4 Detonation, use of term, 17 Detonation physics, evolution of, 17-19 Detonation studies, 646-648 See also Shock tubes and tunnels, blast (explosive) Detonation tube flow, 215-217 Detonation wave, 18 nonplanar, 224 plane, oblique, 220-223
879 Diffraction, in nonuniform quiescent gases, 517-521 Diffraction, three-dimensional equations for, 506-509 over a circular cylinder of sphere, 512-515 over a cone and an elliptic cone at angle of attack, 515-517 over a cone at zero angle of attack, 510-512 shock-shock relations, 509, 510 Diffraction, two-dimensional area relations for ray tubes, 491 description of, 498-505 equations in a curvilinear coordinate system, 489-491 equations in a rectangular coordinate system, 493-498 expansion and compression, 499-502 geometrical relations, 490 propagation of disturbance waves, 491-493, 495, 496 shock-shock relations, 502-505 Diffuser, 653, 656, 657 Diffusive acceleration, 476-478 Discontinuity decomposition, 422-425 Dissipation, 811,813 Dixon, H., 19 Donnell, L. H., 16 Donor cell, 816 Doppler effect, 40, 61 DOring, W., 19 Downstream equilibrium state, 209 Drift acceleration, 475,476 Dubrocca's proposal, 826, 827 Duct flow, 197, 198 Dugdale-MacDonald model, 330 Duhem, P. M. M., 12, 15 Dust explosions, 18 Dust-laden blast waves, 640-642 Dvor~ik, C., 20 Dynamic adiabate, law of, 34 Dynamite, 17, 53, 59
E Earnshaw, S., 6, 12 Ecole Centrale des Travaux Publics, 30 Ecole Polytechnique, 30 Einstein, A., 12 Einstein coefficients, 744, 772 Elastic-plastic shock waves, 334-336
880 Elastic-plastic transition wave splitting by, 335, 336 Electrohydraulic effect, 14 Electron acceleration, 478, 479 Electron beam fluorescence technique (EBFT), 677, 678 Electron heating, 473-475 Electrostatic generator, 7 Emden brothers, 10 Emission techniques, 765, 766 Energy equation, 227, 233 Enthalph-heat flux relationship, 675 Entropy, 331,332 condition, 793 fix, 793,822-824 near-resonance and resonance reflection, 428-430, 433-435 shock capturing and, 792, 793 Envelope simulation, 636, 637 Equation of state (EOS), 144 convexity, 159, 160, 181 for liquids, 270-272 nonthermodynamic constraints on, 159-164 for solids, 325-326 thermodynamic constraints on, 157, 158 Essentially nonoscillatory (ENO) schemes, 840-842 Euler, L., 16 Euler's equation, 168, 240, 247, 248, 779 of motion, 268 Evolutionary condition, 793 Explosion, use of term, 17 Explosive engraving, 305, 306 Explosive forming, underwater, 305-313 Explosive tubes. See Shock tubes and tunnels, blast (explosive) Explosive wave, 18, 68
F Faraday, M., 18 Farnsworth, P T., 21 Finite-volume methods, 802-804 Firedamp, 18, 33, 60, 62, 74 First-Order Upwind Scheme, 816 Flash radiography, 21 Flow visualization background information, 683-685 density-sensitive, 685-693 interferometry, 713, 714
Index interferometry, holographic, 719-734 interferometry, shearing, 714-719 light sources and recording materials, 734, 735 schlieren method, direction-indicating color, 708-713 schlieren methods, 696-703 schlieren techniques, color, 703-708 shadow technique, 693-696 time-resolved visualization and animation, 735-737 Fluctuation splitting, 836-839 Fluorescence quantum yield, 772, 773 Flux calculation, 855-857 Flux-corrected transport (FCT), 847-850 Flux-difference splitting, 846 Flux integration, 813-815 Flux limiting, 836-839 Flux vectors, 790 Jacobian of, 795 splitting, 846, 847 Folding frequency, 809, 810 Fourier's method, 34 Fox-Talbot, W. H., 20 Free-propagation assumption, 487, 488 Frequency modulation methods, 757-761 Friedrichs, K. O., 145 Fundamental equation, 207
G Galilei/Galileo 4 Gases, shock waves in configuration of shock waves, 193-201 history of, 8-12 interactions, 201-204 jump conditions, 187-192 perfect gas shock waves, 227-259 real gas phenomena, 204-227 Gas phenomena, real chemical reactions, 213-215 condensation, 208 cryogenic wind tunnels, 209, 210 dense gas flow, 210, 211 detonation tube flow, 215-217 excitation of monatomic species, 211 fundamental equation, 207 high-temperature, 211-227 low-temperature, 205-211 nonplanar detonation waves, 224
Index phase change, 207, 208 plane, oblique detonation wave, 220-223 plasma shock waves, 224-227 in polyatomic species, 213 Rayleigh curve theory, 212, 213 reciprocity, 206, 207 relaxation process and downstream equilibrium state, 209 shock tube flow, 217-220 vibrational excitation, 211,212 vibrational modes, 205, 206 Gas shock waves, perfect, 227 characteristic theory, 239-242 general theory, 259 incident waves, 232-234 normal waves, 228, 229 oblique waves, 229-232, 236-239 reflected waves, 234-236 shock formation, 242-245 steady, 228-232 steady two-dimensional or axisymmetric, 245-259 unsteady, 232-239 Gauss's theorem, 791,856 Gay-Lussac, J. L., 19 Geometrical shock dynamics applications, 485 Chester-Chisnell-Whitham relation, 486,487 difference between gasdynamics and, 485 diffraction, in nonuniform quiescent gases, 517-520 diffraction, three-dimensional, 506-517 diffraction, two-dimensional, 489-505 free-propagation assumption, 487, 488 orthogonal curvilinear coordinate system, 488, 489 wave propagation through moving gases, 520-550 wave propagation through nonuniform flow fields, 531-550 wave propagation through quiescent gases, 486-520 wave propagation through uniform flow fields, 520-531 Gibbs' relation, 346, 353 Gladstone-Dale constant, 685 Godunov flux, 816 Godunov scheme, 813, 816, 817 Godunov-type methods, 817 Goldberg, V., 14
881 Grid generation and adaptivity, 857, 858 Gruneisen, E., 15 Gn~neisen coefficient, 153, 159 G~neisen constant, 271,272 Gruneisen equation for condensed media, 329331 Guncotton, 17 Gunpowder, 7, 17
H Hadamard, J., 12, 15 Hancock's scheme, 833-836 Head wave, 10 Heat flux measurements, 678, 679 Heaviside's unit function, 816, 830 Helmholtz free energy, 207 Hertz, H., 5 HLL, HLLE, HLLC .... 844-846 High-pressure phase transition, wave splitting by, 335,336 High-resolution schemes application to linear and nonlinear systems, 839, 840 experiments, 832, 833 flux-limiting, fluctuation-splitting, 820-829 Hancock's, 833-836 MUSCL-type, 828-831 High-speed diagnostics, milestones in, 19-21 Hildebrand, F., 11 Him, G. A., 6 HMX explosive, 277 Hollow cavity effect, 66 Hollow charge effect, 66 Holographic interferometry, 719-734 Honl-London factor, 744 Hooke's law of deformation, 5 Hopkinson, B., 15, 16 Hopkinson, J., 15 Hopkinson law, 95 Hopkinson scahng, 635, 636 Hot jet ignition, 648 HO2, ultraviolet transitions, 755-757 Hugoniot, P. H., 8, 12, 15 Hugoniot curve, 77, 164 kinks in, 416 for solids, 325-328 stability of z-shaped part of, 437-439 Hugoniot elastic limit, 334 Hugoniot equation, 144, 150
882 Hugoniot relation, 70, 71, 84 Huygens, C., 4, 5 Hydraulic jump, 36 Hydraulic ram, 31 Hydrodynamica (Bernoulli), 5 Hydrodynamics, 10 Hydrodynamic stability. See Stability, hydrodynamic Hyperbolicity, shock capturing and, 796-799 Hyperbolic problem, shock capturing and, 806813 flux estimation, 806-809 Godunov's theorem, 813 numerical experiments, 809 yon Neumann analysis, 809-813 Hypersonic small disturbance theory limit, 254, 255 Hypersonic wind tunnels. See Shock tubes and tunnels, supersonic and hypersonic
I Impedance matching, 14 Impingement, 202 Inclined mirror method, 323, 324 Indurite, 74, 76 Infrared laser absorption CO discharge lasers, 764, 765 emission techniques, 765, 766 lead salt diode lasers, 763, 764 room-temperature diodes, 761-763 Infrared thermography technique, 679, 680 Instability. See Stability/instability, hydrodynamic; Stability/instability, structure Intensified charge-coupled devices (ICCDs), 21 Interferometry, 43 description of, 713, 714 finite fringe configuration, 714 holographic, 719-734 Mach-Zehnder, 9, 20, 70, 714 infinite fringe configuration, 714 reference beam, 714 shearing, 714-719 International Treatise of Petersburg, 51 Ion motion, 470-473 Isothermal equation of state, 9
J Jacobian matrices, 794, 795 Jameson's method, 851,852
Index Jamieson, J. C., 16 Jordan block, 796 Jouguet, E., 8, 12, 14, 15, 339 Jump conditions jump direction, 189, 190 liquids AND, 246, 247 Mach number, 188, 189 oblique shock waves, 191, 192 solids AND, 318, 319 steady normal shock waves, 187, 188 unsteady normal shock waves, 190, 191 validity tests for, 793 Jump relations, 554-560 JWL equation, 277
K Kinetic schemes, 827, 828 Kingdon, K. H., 21 Kobes, K., 11 Kolsky, H., 16 Krakatao, 66-64
L Lagrangian-Eulerian method, 275-277 EAigle Fall, 6 Lambda shock system, 198, 199 Landau-Teller form, 212 Laplace, P-S., 11 Laser absorption techniques, ultraviolet and visible, 747-757 Laser Doppler velocimetry (LDV), 680 Lattice potential function, 330 Laval nozzle, 10, 11, 107 Law of sines, 255-257 Lax-Friedrichs method, 843 Lax-Wendroff theorem, 802-804, 808, 809, 813, 848 Lead salt diode lasers, 763, 764 Le Chatelier, H., 18 Leiden jar, 7, 20 Length scales, 145, 146 Limiter function, 837 Linear advection equation, 806 Riemann problem for, 816 Linearized Riemann solver, 818-822 Linear stability theory (LST), 422, 425-433 Line shapes, 743-747 Lipshitz continuous, 804 Liquids, properties of
Index compressibility, 264-267 density, 254 viscosity, 267, 268 Liquids, shock waves in See also Underwater shock waves equation of state, 270-272 history of, 12-14 one-dimensional water flow, 273, 274 pressure waves, 268-270 wave motion, 268-274 Longitudinal sound speed, 150 Los Alamos Scientific Laboratory (LASL), 117 Low phase error (LPE), 849 Lummer, O. R., 7, 12 Lyell, C., 18
M Mach, E., 6, 8, 10, 19, 20 Mach, L., 10, 11, 20 Mach cone, 69, 70 Mach effect, 10 Mach lines, derivatives along, 240-242 Mach number, 62, 82, 94, 95, 188, 189 Mach reflection, 171, 179, 200 Mach wave, 70 Mach-Zehnder interferometry, 9, 20, 74, 714 Magnetohydrodynamics (MHD), 456-459, 793, 794 Mallard, E., 18 Manhattan Project, 16, 103, 114, 117, 809 Marsh, S. P., 16 Martin-Hou equation, 349, 350 Maxwelrs theory of elasticity, 15, 42 McQueen, R. G., 16 Measurement techniques. See Flow visualization; Spectroscopic diagnostics Mercury fulminate, 17 Meteor showers, 6 Method-of-characteristics (MOC), 239, 250, 775 Method of fractional steps, 853 Method of lines, 814 Meyer, T., 11 Michelson-Rayleigh line, 438, 440 Microchannel plate (MCP), 21, 104 Mie, G., 15 Mie-Gr~neisen equation, 270, 271,276 Mie scattering methods, 743 Mikhelson, V. A., 19 Moisson, A., 15
883 Montgolfier, J. M., 13 Montigny, C., 6 Multihole pressure probes, 677 Multiple Spark Camera, 20, 21 Munroe effect, 71 MUSCL (monotone scheme for conservation laws), 828-831
N Napier, R. D., 10 Napoleon, 30, 31 Navier, L. M. H., 5 Navier-Stokes equations, 31, 32, 793, 821,828 NCO, visible and near-ultraviolet transitions, 752-754 Near-resonance reflection, 428-438 Nernst, W., 19 Nessyahu-Tadmor method, 843, 844 Neumann, E, 5, 16 Newton, I., 4, 5, 11 Newtonian fluids, 267 Newton-Raphson iteration, 276 Newton's cradle, 5 NH, ultraviolet transitions, 754, 755 NH2 frequency modulation methods, 760, 761 visible and near-ultraviolet transitions, 752-754 Niepce brothers, 31 Nitroglycerine, 17, 48 NO, ultraviolet transitions, 755-757 Nobel, A., 17 Node, 171, 172 Nonconservative form, 794 Noncoplanar magnetic field, 469, 470 Nonlinearity, shock capturing and, 796-799 Non-Newtonian fluids, 270 Nonplanar detonation waves, 224 Nonthermodynamic constraints on EOS, 159-164 Normal derivatives, 247-250 Normal incidence frame (NIF), 457, 458 Nozzle flow, overexpanded, 199, 200 Nozzles, 653-656 Nuclear fission bomb, 121, 122 Nyquist frequency, 809, 810
O Oblique detonation wave engines (ODWE), 221,222
884 Oblique shock waves, 191,192, 229-232, 236-239, 296 OH mole fraction, 773 ultraviolet transitions, 754, 755 Ohm's law, 437 One-dimensional propagation of a small disturbance, 560-562
On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance (Rankine), 15 Operator splitting, 854, 855 Ordinary differential equations (ODEs), 240 Orthogonal curvilinear coordinate system, 488, 489 Oswatitsch, K., 26, 100, 110 02 ultraviolet transitions, 755-757 Oxyhydrogen, 18, 26, 67, 68
Index Positivity, 824-828 Prandtl, L., 10, 11, 14, 363 Prandtl-Glauert rule, 101 Prandt|-Meyer function, 90, 171,202, 203,221, 223, 257 Preiswerk, E., 12 Pressure measurements stagnation, 638 static, 637, 638 Pressure waves, in liquids, 268-270 Principal Hugoniot, 70 Principia (Newton), 5, 11, 33 P-waves, 562
Q Quasilinear form, 794, 795 Q-waves, 562
P Papin's safety valve, 10, 35 Parry, W. E., 8 Parsons, C., 15 Payman, W., 20 Perbuatan volcano, 66, 67 Percussion research, 4-6 Percussion wave, 68 Phase change, 207, 208 Phase error, 812 Photodisruption effect, 14 Photography, single-shot, 20, 21 Physics of High Pressures, The (Bridgman), 104 Picric acid, 17 Piston-driven tubes and tunnels. See Shock tubes and tunnels, free piston-driven expansion; Shock tubes and tunnels, free piston-driven reflected Piston support, stability and, 444, 445 Pitot probe technique, 676, 677 Pitot tube, 229 Planar laser-induced fluorescence (PLIF) measurement strategies, 772-776 species imaging, 772, 773 temperature imaging, 773-776 theory, 771,772 velocity imaging, 776 Plasma shock waves, 224-227 Poisson, S. D., 11, 12 Poisson isentrope, 34, 35 Polyvinylidene Fluoride (PVDF), 322
R Radiative transfer equation, 227 Raillard, E, 6 Rakhmatulin, K. A., 15 Ramsauer, C., 16 Rankine, W. J. M., 12, 15, 18, 363 Rankine-Hugoniot equations/relation, 70, 90, 144, 149, 273, 274, 458, 556, 788, 792, 820 Rarefaction shock waves, 190, 798 adiabat, 344-353 admissibility, 353-362 background information, 339-344 dynamics, 389-403 structure, 362-374 weak, 374-389 Rayleigh, Lord, 12 Rayleigh curve theory, 212, 213 Rayleigh equations, 144, 150, 151 Rayleigh line, 78 Rayleigh-Pitot pressure formula, 229 Reciprocity, 206, 207 Reference beam interferometer, 714 Reflection near-resonance and resonance, 428-430, 433-435 from a shock wave, 252-255 of a shock wave from shock tube end wall, 570-573
Index and transmission at material interface in solids, 327, 328 Refraction law, 145, 178, 179 Regnault, V., 9, 10 Regular reflection, 200 Relaxation process, 209 Relaxing flows, 853 Resonance reflection, 428-430, 433-435 Reynolds, O., 10 Rice, M. H., 16 Riemann, B., 8, 12 Riemann invariants, 242, 562, 800 Riemann problem/solution, 145, 422-425 for linear advection equation, 816 linearized, 818-822 shock capturing and, 800-802 solution of a simple shock, 151, 152 z-shaped part of Hugoniot curve and nonuniqueness of, 438 Riemann problem, methods for avoiding Chang, 852 flux-corrected transport, 848-850 flux-vector splitting, 846, 847 HLL, HLLE, HLLC .... 844-846 Jameson's, 850-852 Lax-Friedrichs, 843 Nessyahu-Tadmor, 843, 844 Room-temperature diodes, 761-763 Ross, J. C., 8
S Saha equation, 226 Salcher, P., 6, 10, 11 Scaling simulation, 635, 636 Schardin, H., 11, 20, 696, 697, 736, 737 Schlieren method, 9, 20 color, 703-708 description of, 696-703 direction-indicating color, 708-713 Schuster, A., 19 Sedov, L. I., 14 Selection criteria, 793 Semenov, N. N., 19 Semi-discretization, 813-8 15 SEP explosive, 277, 292, 296 Shadowgraphy, 9, 20, 63, 693-696 Shearing interferometry, 714-719 Shear waves, 797 Shock adiabat, 344-353
885 admissibility, 353-362 consolidation and compaction of powders, 297-304 diamonds, 73 dynamics, 389-403 formation of, 242-245 splitting, 357 structure, 362-375 Shock capturing anomalous solutions, 859, 860 background information, 788, 789 characteristic variables, 799, 800 conclusions, 864, 865 conservation and, 789-791 entropy conditions, 792, 793 errors and accuracy, 804, 805 example, 862, 863 finite-volume methods, 802-804 flux calculation, 855-857 flux integration and semi-discretization, 813-815 grid generation and adaptivity, 857, 858 hyperbolic problem, 806-813 Jacobian matrices, 794, 795 multidimensional methods, 860, 861 Riemann invariants, 800 Riemann problems, 800-806 source terms, 853-855 wavespeeds, hyperbolicity, nonlinearity, and convexity, 796-799 weak solutions, 791,792 Shock capturing, one-dimensional methods, 815 entropy fix, 793, 822-824 essentially nonoscillatory (ENO) schemes, 840-842 Godunov scheme, 813, 816, 817 high-resolution schemes, 828-840 linearized Riemann solver, 818-822 positivity, 824-828 Riemann problem, avoiding, 843-852 Shock dynamic equations, three~dimensional, 506-509 for wave propagation through nonuniform flow fields, 538-545 for wave propagation through nonuniform quiescent gases, 518 Shock dynamic equations, two-dimensional in curvilinear coordinate system, 489-491
886 Shock dynamic equations (continued) in rectangular coordinate system, 493-498 for wave propagation through nonuniform flow fields, 531-538 for wave propagation through uniform flow fields, 520-528 Shock-shock relations three-dimensional diffraction, 509, 510 through moving gases, 545-548 two-dimensional diffraction, 502-505 Shock thermodynamics Gruneisen equation for condensed media, 329-331 irreversibility of compression process, 331, 332 temperature calculation, 332, 333 Shock tubes and tunnels diaphragm, 581,582 flow, 217-220 general description, 563-565 for generating strong waves, 583, 584 interaction between reflected wave and contact surface, 573-581 invention of, 11 isentropic relation, 567 jump relations, 554-560 operating techniques, 581-584 reflection from end wall, 570-573 relations between regions 1 and 2, 565, 566 relations between regions 2 and 3, 566 relations between regions 3 and 4, 567-570 role of, 553, 554 tailored case, 578, 580, 581 variable cross-section, 582, 583 Shock tubes and tunnels, blast (explosive) applications, 639-648 center placement of explosive, 626-630 characteristic period, 632 civil defense studies, 646 conclusions, 648, 649 design configurations, 626-633 design specification, 624-626 detonation studies, 646-648 driver, detonable gas, 635 driver design, 633-635 dust-laden blast waves, 640--642 end placement of explosive, 630-632 envelope simulation, 636, 637 general description, 624 instrumentation, 637-639 model studies, 644-646
Index nonideal blast wave simulations, 640-644 outside placement of explosive, 632, 633 scaled simulation, 635, 636 wall jets, 642-644 Shock tubes and tunnels, free piston-driven expansion applications of driver, 609-6 16 background information, 603-608 conclusions, 620 development of larger facilities, 616-619 super-orbital applications, 611-6 14 super-orbital configurations, 611 super-orbital operation, 615, 616 test times, 619 Shock tubes and tunnels, free piston-driven reflected background information, 587-593 conclusions, 599, 600 driver-heating mechanism, 593-599 Shock tubes and tunnels, supersonic and hypersonic blow-down, 664-672 blow-down, hot, 673-676 blow-down, induction, 673 continuous, 658-664 diffuser, 653, 656, 657 electron beam fluorescence technique, 677, 678 features of, 652-654 heat flux measurements, 678, 679 infrared thermography technique, 679, 680 laser Doppler velocimetry (LDV), 680 multihole pressure probes, 677 nozzle, 653-656 Pitot probe technique, 676, 677 running time, 653 start-up process, 657, 658 Shock wave interactions boundary-layer, 204 contact surface and, 573-581 dimensions of, 171-173 expansion-shock, 202, 203 impingement, 202 shock-expansion, 203, 204 three-dimensional, 172 triple-shock-entropy (TSE) theorem and, 173-176 two-dimensional, 171, 172 Shock waves defined, 2, 3, 68
Index interactions, 171, 172, 201-204 percussion research, 4-6 properties of, 3 related fields, 3, 4 Shock waves, evolution of chronology of, 25-122 early speculations and natural, 6-8 in gases, 8-12 in liquids, 12-14 in solids, 14-16 Sill, visible and near-ultraviolet transitions, 752-754 Sill 2, visible and near-ultraviolet transitions, 752-754 Silver fulminate, 17 Simulation envelope, 636, 637 scaled, 635, 636 Singer, G. J., 14 Single shock system, configurations in, 193-198 Slater model, 330 Snell's law, 698 Sobrero, A., 17 Solids, shock waves in applications, 316, 317, 334-336 elastic-plastic, 334-336 empirical linear relation, 325-327 generating, 320-322 Gr~neisen equation for condensed media, 329-331 high compression, background of, 315-317 history of, 14-16 Hugoniot curve and equation of state for, 325-328 irreversibility of compression process, 331,332 jump conditions, 320, 321 measurement methods, 322-325 reflection and transmission at material interface, 327-328 temperature calculation, 332-333 wave splitting, 335, 336 weak formulas, 319, 320 Sommer, J., 10 Source terms, 853-855 Southwell, R., 8 Space, shock waves in bow shock observations, 460-467 collisionless Liouville theory, 468-475
887 magnetohydrodynamic (MHD) approach, 456-459 morphology, 459, 460 particle acceleration, 475-479 types of, 1, 2 Spark wave, 68 Species imaging, 772, 773 Specific total enthalpy, 791 Spectroscopic diagnostics absorption theory and line shapes, 743-747 atomic resonance absorption spectroscopy, 766--771 background information, 742, 743 frequency modulation methods, 757-761 infrared laser absorption and emission techniques, 761-766 planar laser-induced fluorescence, 771-776 ultraviolet and visible laser absorption techniques, 747-757 Spherical flow, 198 Spinodal, 158 Stability/instability, hydrodynamic corrugation instability, 414-416 criteria for instability, 439-441 discontinuity decomposition/Riemann solution, nonunique representations and, 422-425 linear stability theory (LST), 422, 425-433 near-resonance and resonance reflection, 428-430, 433-435 nonunique representations, 420-422 one-dimensional instability, 415-418 piston support and, 444, 445 relationship between instability and nonuniqueness of steady-state regimes, 435-437 steady-state corrugation perturbations, 425-433 triple-wave configuration, 424-428, 431-432 underdriven detonation, 431 verification of, 442-444 z-shaped part of Hugoniot curve and, 437-439 Stability/instability, structure experimental data on structural instability, 444, 445 mechanisms of instability, 448 two-fronts model with instantaneous heat release, 448, 449
888 Stability/instability, structure (continued) two-fronts model with noninstantaneous relaxation, 449-451 viscous compression discontinuities and, 446, 447 Stability constraints, 160-163 Stalker tubes. See Shock tubes and tunnels, free piston-driven reflected Stanton, T. E., 11 Static adiabate, 35 Steady normal shock waves, 187, 188 Steenbeck, M., 21 Stellar explosions, 7 Stern-Vollmer factor, 772, 773 Stodola, A. B., 10 Stokes, G. G., 12, 339 Strang splitting, 855 Streamlines, derivatives along, 250-252 Supersonic wind tunnels. See Shock tubes and tunnels, supersonic and hypersonic Sweep, 194, 195
T Tailored case, 578, 580, 581 Tait's percussion machine, 6 Tangential derivatives, 247 Tanis, H. E., 21 Taylor, G. I., 12, 15, 16, 363 Taylor-Maccoll flow, 195, 196 Temperature calculation, 332, 333 Temperature imaging, 757-760, 773-776 Th~nard, L. J., 19 Theory of Sound, The (Rayleigh), 60, 62, 77, 85, 92 Thermal state equation, 226 Thermodynamic constraints on EOS, 157, 158 Thermodynamic properties of materials, 152-157 Thompson, P A., 340, 341 Thornycroft, J., 14 Three-dimensional shock dynamic equations See also Diffraction, three-dimensional for wave propagation through nonuniform flow fields, 538-545 Three-dimensional shock wave interactions, 172 Thunder, 6, 38, 48 Time scales, 146, 147
Index TiN, visible and near-ultraviolet transitions, 752-754 TNT, 74 Toepler, A., 20 system, 697 Total enthalpy, 791 Transducers, pressure, 637 Transmission of shock waves at material interface, 327, 328 Treatise on Natural Philosophy (Tait and Kelvin), 71 Triple-shock-entropy (TSE) theorem, 144, 145, 173, 174 shock wave interactions and, 174-176 Triple-shock-particle-bound theorem, 174 Tsunamis, 79 Tumlirz, O., 12 Two-dimensional or axisymmetric shock waves, 245-259 curvature singularity, 255-259 derivatives along streamlines and Mach lines, 250-252 jump conditions, 246, 247 normal derivatives, 247-250 tangential derivatives, 247 vorticity, 259 wave reflection, 252-255 Two-dimensional shock dynamic equations, 488, 489 See also Diffraction, two-dimensional for wave propagation through nonuniform flow fields, 531-538 Two-dimensional shock wave interactions, 171, 172 Two-fronts model with instantaneous heat release, 448, 449 with noninstantaneous relaxation, 449-451 Two-shock approximation, 802
U Ultraviolet and visible laser absorption techniques, 747-757 Underwater shock waves applications, 297-313 compaction of powders, 297-304 experiments, 277-290 explosive forming, 305-313 numerical procedure, 275-277 observational investigation, 274, 275
Index planar explosion-generated, 282-290 spherical explosion-generated, 277-282 yon Neumann reflection of, 290-296 Unsteady flow, 201 Unsteady normal shock waves, 190, 191 Upwind (-biased) scheme, 809 first-order, 816
W van Albada's limiter, 839 van Dyke, M., 683 Vaschenko-Zubarev model, 330 Velocity imaging, 776 Vibrational excitation, 211,212 Vibrational modes, 205, 206 Vieille, P., 11, 18, 554 VISAR (Velocity Interferometry System for Any Reflector), 323 Viscosity condition, 793 Viscous compression discontinuities, stability and, 446, 447 Voigt profile, 744 von Karm~m, T., 9, 14, 15 von Lilienthal, O., 10 von Neumann, J., 10, 19, 160, 809 analysis, 809-813 reflection of underwater shock waves, 290-296 Vortex sheets, 797 Vorticity, 259
889 Water ricochets, 13, 14 Wave drag, use of term, 9, 194 Wave reflection. See Reflection Waves centered, 800 genuinely nonlinear, 797 linearly degenerate, 797, 798 shear, 797 Wavespeeds, shock capturing and, 796-799 Wave splitting, 335, 336 Weak shock formulas, 319, 320, 375-389 Weak solutions, 791,792 Weber, H., 12 Wedge flow, 193, 194 Weighted essentially nonoscillatory (WENO) schemes, 842 Wehrubsky, J., 20 Wentzel, J., 20 Wheatstone, C., 20 Whip cracking, 7 Whitehead, J., 11 Whitham, G. B., 486-488, 521 Wilson line, 158 Wind tunnels See also Shock tubes and tunnels, supersonic and hypersonic cryogenic, 209, 210 supersonic, 11, 67, 107, 113, 119 Wosyka, J., 10
X X-ray technique, flash, 103 W Wagner, H., 14 Wallis, D., 4 Wall jets, 642-644 Walsh, J. M., 16 Wantzel, P. L., 10, 28 Washington Meteor, 6, 61, 77 Water hammer, 13, 80
Z Zeldovich, Y. B., 19, 341 Zeldovich-von-Neumann-DOring (ZND) theory, 19, 115, 217 Z6mpl6n, G., 12, 339 Zhukovsky, N. E., 13
