ADVANCES IN APPLIED MECHANICS VOLUME V
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ADVANCES IN APPLIED MECHANICS Editors
H. L. DRYDEN
TH. VON
KARMAN
Managing Editor
G . KUERTI Case Institute oJ Technologlj, Cleveland, Ohio
Associate Editors
F. H. VAN
DEN
DUNGEN
L. HOWARTH
J. PkREs
VOLUME V
1958 ACADEMIC PRESS INC., PUBLISHERS NEW YORK, N.Y.
COPYRIGHT @ 1958 BY
ACADEMIC PRESSINC. 111 FIFTHAVENUE NEW YORK3, N.Y.
ALL RIGHTSRESERVED N O PART O F THIS BOOK MAY B E REPRODUCED I N A N Y FORM, B Y PHOTOSTAT, MICROFILM, OR A N Y OTHER MEANS, WITHOUT WRITTEN
PERMISSION FROM T H E PUBLISHERS
LIBRARY OF CONGRESSCATALOG CARD NUMBER: 48-8.503
PRINTED IN
THE
UNITEDSTATESOF AMERICA
CONTRIBUTORSTO VOLUMEV
H. N. ABRAMSON. Southwest Research Institute, S a n Antonio, Texas
H. DERESIEWICZ, Columbia University, New Y o r k , New York
J. FABRI,Office National d’Etudes et de Recherches Akronautiques, Paris EDWARD A. FRIEMAN, Princeton University, Princeton, New Jersey RUSSELLM. KULSRUD,Princeton University, Princeton, New Jersey L. M. MACK, Jet Propulsion Laboratory, California Institzcte of Technology, Pasadena, California H. J. PLASS,The University of Texas, Austin, Texas E. A. RIPPERGER,The University of Texas, Austin, Texas CHARLESSALTZER, Case Institute of Technology, Cleveland, Ohio, and General Electric Compan y , Electronics Laboratory, Syracuse, New York
R. SIESTRUNCK, Office National d’Etudes et de Recherches Aironautiques, Paris A. I. VAN DE VOOREN,National Aeronautical Research Institute, Amsterdam
P. P. WEGENER, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California
V
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PREFACE This fifth volume of “Advances in Applied Mechanics” contains reviews and surveys of the current state of research in selected fields of applied mechanics and two brief papers which the Editors hope will be of interest to readers of the series. Contributions to the “Advances” are, in general, by invitation, but suggestions of topics for review and offers of special contributions are very welcome and will receive careful consideration. The Editors are convinced that summary articles by research workers are important contributions to the advance of knowledge, that they are rewarding to the authors in giving a broad assessment of where we stand, and that they stimulate both author and reader to further research. When, as occasionally happens, the summary article grows into a book, the Editors applaud the result, while regretting the loss of the article for Advances. We are thus pleased that Professor W. D. Hayes, invited by us to review hypersonic flow theor$ became so engrossed in his subject that he completed a book which will appear as a separate monograph. THEEDITORS December, 1957
* W. D. Hayes and R. Probstein. “Hypersonic Flow Theory.” Academic Press, New York, 1958, in preparation.
vii
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CONTENTS ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTRIBUTORS TO VOLUME V
PREFACE.
v vii
Supersonic Air Ejectors A N D R . SIESTRUNCK. Office National d’gtudes et de Recherches BY.J . FABRI A ironautiques. Paris I . Introduction . . . . . . . . . . . . . . . I1. Aerodynamic Flow Patterns in Jet Ejectors I11. Experimental Verification . . . . . . . . . IV. Optimum Jet Ejector Design . . . . . . .
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1 2 13 30
. . . . . . . . . . . . . . . . . . . . . . . . . .
Unsteady Airfoil Theory BY A . I . VAN DE VOOREN.National Aeronautical Research Institute. Amsterdam I. I1. I11. IV . V. VI . VII . VIII . I X.
. . . . 36 Introduction . . . . . . . . . . . . . . . . . . . . . . . . The Fundamental Equations . . . . . . . . . . . . . . . . . . . . 31 The Oscillating Airfoil in Two-dimensional Subsonic Flow . . . . . . . 44 The Oscillating Airfoil in Three-dimensional Subsonic Flow . . . . . . 52 The Oscillating Airfoil in Supersonic Flow (Supersonic Edges) . . . . . 59 The Oscillating Airfoil in Supersonic Flow (Subsonic Edges) . . . . . . 65 Non-linear Approximations . . . . . . . . . . . . . . . . . . . . . 72 . . . . 77 Indicia1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 84 References . . . . . . . . . . . . . . . . . . . . . . . . .
The Theory of Distributions BY CHARLES SALTZER Case Institute of Technology. Cleveland. Ohio. and General Electric Company. Electronics Laboratory. Syracuse New York
.
I. I1. I11. IV . V. VI . VII . VIII . I X. X.
.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Theory of Distributions . . . . . . . . . . . . . . . . . . . . The Singularity Functions and the Finite Part of An Integral . . . . A Heuristic Approach . . . . . . . . . . . . . . . . . . . . . . . Fourier Series and the Poisson Transformation . . . . . . . . . . . Ordinary Differential Equations . . . . . . . . . . . . . . . . . . Applications to Fourier Transforms . . . . . . . . . . . . . . . . Fourier Transforms of Distributions . . . . . . . . . . . . . . . . Generalized Harmonic Analysis and Stochastic Processes . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
91 92 95 98 99 102 104
104 107 109 110
Stress Wave Propagation in Rods and Beams BY H . N . ABRAMSON. H . J . PLASS. A N D E . A. RIPPERGER. Southwest Research Institute. San Antonio. Texas. and The University of Texas. Austin. Texas I . Introduction . . . . I1. Longitudinal Waves . I11. Flexural Waves . . References . . . . .
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ix
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111 113 151 188
CONTENTS
X
Problems in Hydromagnetics
BY EDWARD A. FRIEMAN A N D RUSSELL M. KULSRUD. Princeton University. Princeton. N e w Jersey I. I1. I11. IV . V.
Introduction . . . . . . . . . . . . . . . Fundamental Equations . . . . . . . . . . General Processes . . . . . . . . . . . . Stability of Hydromagnetic Equilibria . . Hydromagnetic Waves . . . . . . . . . . References . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 195 . . . . . . . . . . . . . 196 . . . . . . . . . . . . . 198 . . . . . . . . . . . . . . 204 . . . . . . . . . . . . . 216 . . . . . . . . . . . . . 231
Mechanics of Granular Matter BY H . DERESIEWICZ. Columbia University. N e w Y o r k . N e w Y o r k 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Geometry of a Granular Mass . . . . . . . . . . . . . . . . . . . . I11. Some Recent Results of Contact Theory . . . . . . . . . . . . . . . IV . Mechanical Response of Granular Assemblages . . . . . . . . . . . . V. Suggestions for Further Research . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
233 236 251 267 300 303
Condensation in Supersonic and Hypersonic Wind Tunnels BY P. P. WEGENERA N D L . M . MACK. Jet Propulsion Laboratory. California Institute of Technology. Pasadena. California I. I1. I11. IV . V. VI .
Equilibrium Condensation Limits . . . . . . . . . . . . . . . . . . 307 Condensation of Water Vapor in Supersonic Nozzles . . . . . . . . . 320 Condensation in Steam and Hypersonic Nozzles . . . . . . . . . . . . 343 Diabatic Flows and Thermodynamics of Condensation . . . . . . . . . 365 Kinetics of Condensation . . . . . . . . . . . . . . . . . . . . . . 403 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . 433 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
Author Index
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449
Subject Index
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456
Supersonic Air Ejectors BY J. FABRI
AND
R. SIESTRUNCK
Office National d ' l h d e s et de Recherches A kronautiques. Paris Page
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . 11. Aerodynamic Flow Patterns in Jet Ejectors . . . . . . . . 1. Supersonic Flow Patterns . . . . . . . . . . . . . . . 2. Mixed Flow Patterns . . . . . . . . . . . . . . . . . 3. Mixed Flow Patterns with Primary Separation . . . . . 4. Theoretical Performance Curves . . . . . . . . . . . . 111. Experimental Verification . . . . . . . . . . . . . . . . 1. Experimental Set-up . . . . . . . . . . . . . . . . . 2. General Performance Curves . . . . . . . . . . . . . . 3. Operation without Induced Flow . . . . . . . . . . . . 4. Study of the Main Parameters . . . . . . . . . . . . . 6. Pressure Distribution along the Walls of the Mixing Tube IV. Optimum Jet Ejector Design . . . . . . . . . . . . . . .
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1 2 4 8 11 12 13 14 16 19 23 29 30
I. INTRODUCTION The industrial use of jet ejectors, though already very old, has been confined mainly to very particular cases of operation. The experimental results obtained with such ejectors were therefore insufficient to allow a safe extrapolation of the performance curves to more general cases. Large-sized modern jet ejectors, driven by high powered air-compressors and designed for large ranges of operating conditions, cannot be based on these earlier results, if one wishes to be sure of the final outcome. For this reason systematic experimental and theoretical researches have been resumed in many countries ([l], [2], [3]) on air/air jet ejectors with high $yesswe ratios, in which the primary air flow is supersonic. The tangential action and the turbulent mixing of air jets, made even more complicated by the proximity of the walls of the mixing tube, is one of the most difficult problems of gas dynamics. No correct solution has yet been given for this problem, and the performance of jet ejector operations cannot be predicted from such studies. The theories of jet ejectors, and in particular the one we proposed earlier [4] in an attempt to give theoretical interpretations of experimental results are all essentially aevodynamical. In such theories, the exhaust 1
2
J. FABRI A N D R. SIESTRUNCK
performance of jet ejectors is determined by the compatibility conditions of two air streams flowing in the same duct, called the mixing tube. The aerodynamic conditions of compatibility are written for a simplified geometric representation of the experimental set-up and do not take into account viscosity and diffusion effects. The success of the simple aerodynamic theories stems from the fact that viscosity and diffusion effects are essential in the building-up of the flow system, but, in an established movement, play a very minor part in maintaining the flow under nearly perfect-gas conditions. Of course, these aerodynamic conditions are incomplete. They do not show the effect of such important parameters as the minimum mixing-tube length necessary to induce secondary flows, or the best relative position of primary and secondary nozzles to give optimum ejector performance. However, the predicted rates of induced mass flows are very close to the experimental results obtained with actual ejector set-ups, if the mixing tube is long enough and the geometric configuration of the set-up is similar to the theoretical one. It is, however, necessary to take the friction losses into account, but this can be done by means of the classical results obtained for friction losses in smooth ducts. The resulting aerodynamic theory can be considered as giving very correct overall descriptions of the actual operation of jet ejectors. As the theory shows the existence of some similarity rules, the results of ejector performance calculations can be condensed into a small number of diagrams. By means of these diagrams the performance of any jet ejector with fixed air-supply conditions can be rapidly computed, and jet ejectors can be rapidly designed for any preassigned performance.
11. AERODYNAMICFLOW PATTERNS IN
JET
EJECTORS
Fundamentally, a jet ejector is designed to draw a given mass flow rate M" of induced secondary flow from a reservoir at a given stagnation pressure pirf into a vessel of higher pressure p , this latter pressure being generally the atmospheric pressure. Such a compression is obtained by means of a high pressure primary flow (mass flow rate M') which is expanded from the primary stagnation pressure pi' to the same pressure 9. A convenient representation of such an operation is obtained on the nondimensional p, &-diagram, where p = M"/M' is the ratio of the two mass flow rates and & = pi"/p, the secondary reservoir pressure divided by the exhaust pressure, is the inverse of the compression ratio. The characteristic performance curves ( p vs. &) are given for fixed primary conditions. Figure 1 shows a schematic representation of a jet ejector similar to those used in our experimental studies of ejector performance. Actually such an ejector differs from the industrial set-ups only in minor points.
SUPERSONIC AIR EJECTORS
3
A high pressure primary flow (stagnation pressure fi%‘, stagnation temperature T*’)is supplied to the primary nozzle. The supersonic primary flow leaves the primary nozzle and enters the mixing tube of larger cross section, on the axis of which this nozzle is placed. The induced secondary flow (stagnation pressure fi,”, stagnation temperature T,”) comes tangentially into contact with the primary flow, throughout the length of the mixing tube. At the exit of the mixing tube both flows are ejected into the surrounding atmosphere at pressure fi. Sometimes a diffuser is used a t the mixing tube exit, and the pressure condition fi is imposed only a t the exit of the diffuser. In this schematic description of a jet ejector, the induced secondary air is taken from the surrounding atmosphere through a duct equipped with a flowmeter and a control valve. As it passes through the control valve, the secondary flow suffers a pressure drop depending on the valve setting. The secondary stagnation pressure becomes subatmospheric, and as the induced flow expands from the secondary chamber to the mixing-tube entrance, its pressure decreases occasionally to very low values. In the industrial jet ejector facilities for aeronautical applications, the useful part of the secondary stream is between the control valve and the mixing tube (jet-engine combustion chamber for low-pressure combustion tests). It is not always necessary to have a secondary chamber slowing down the flow to stagnation conditions, and in the induced-flow supersonic wind tunnels the high speed secondary flow is drawn along directly by the primary stream. In some cases, jet pumps may also be used as injectors, and the useful flow is then the high pressure stream a t the mixing tube exit. The mixing tubes considered in this study are cylindrical. Their length L is measured from the exit section of the primary nozzle. For the sake of simplicity this primary exit section S is taken as unit section. The mixing tube section is then designated by AS, and A a S designates the exit section of the downstream diffuser. The throat section of the primary nozzle is ASIA, ; A* then represents the ratio of the mixing-tube section to the primary throat section. The dimensionless speed M * determines the local aerodynamical conditions. M * is the ratio of the local mean velocity of each stream to its respective critical sound velocity a * . A single prime (’) refers to the primary flow conditions and a double prime (“) to the secondary flow, if these two streams are still distinct; “no prime” means a homogeneous mixing of both streams. Subscripts 1, 2 and 3 designate the three successive sections shown in Fig. 1. Subscript 0 is reserved for a section of possible separation of the primary flow, when the primary stagnation pressure is too low for the primary flow to fill the whole primary diffuser. Subscript i refers, as already stated, to stagnation conditions. Specific heats of both gases are supposed to be independent of the temperature and keep the same values before and after
4
J . FABRI A N D R. SIESTRUNCK
mixing. The expression m2=(y+ l)/(y- 1) defines a fundamental combination of the ratio y of the specific heats a t constant pressure and constant volume. SECONDARY STAGNATION PRESSURE NDARY CHAMBER LINES TO MANOMETERS
MIXING
TUBE
STAGNATION COMPRESSED AIR
SECONDARY FLOWMETER
FIG. 1. Schematic representation of a jet ejector.
1. Supersonic Flow Patterns
For high values of the ratio pi’/$, the primary flow entering the mixing tube is supersonic and remains supersonic throughout the mixing tube (Fig. 2a)* and the jet ejector operates with a supersonic flow pattern. In this case, the downstream pressure conditions have no limiting effect on the mixing-tube flow and the induced secondary flow rate i s the m a x i m u m rate compatible with the coexistence, in the mixing tube, of the primary and the secondary streams. In this case the two streams remain distinct well after section 1. The subsonic secondary flow may be described sufficiently well by means of the
* The shadowgrams of Fig. 2 show a two-dimensional primary nozzle of average M’*l = 1.78 (opening angle of the diverging nozzle 10’). The two-dimensional mixing tube has a section ratio of 1 = 2.61; the four shadowgrams correspond to the same setting of the control valve.
SUPERSONIC AIR EJECTORS
5
classical quasi-one-dimensional assumption, but the primary expansion should be studied by the method of characteristics. However, when in section (1) the secondary pressure PI’’i s lower than the @%nary pressure # , I ,
FIG. 2 . a : b: c: d:
supersonic flow pattern, pj‘ = 69, p = 0.222, saturated supersonic flow pattern, pi‘ = 5 p , p = 0.266, mixed flow pattern, pi’ = 4 9 , p = 0.300, mixed flow pattern with separation, pi’ = 3 9 , p = 0.445.
6
J. FABRI AND R. SIESTRUNCK
the primary jet expands into the mixing tube, and a t the section of maximum expansion (subscript e) this flow is nearly uniform. Actually both flows may be described accurately by one-dimensional isentropic expansions between sections 1 and 2, although such an assumption does not imply that the mean pressures p,' and p," are equal. The wake issuing from the primary-nozzle trailing edge, very apparent on the shadowgrams of supersonic flow patterns (Fig. 2a and b) cannot be neglected in small-size set-ups, such as the one used in our experiments. This wake represents a certain fraction 1 - A' (Fig. 3) of the primary flow.
\ \\\\\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\,
\ \ \
, \
FIG. 3. Experimental arrangement for supersonic flow pattern (schematic).
It can be assumed quite safely that between sections 1 and e this wake is fed by neither flow, as the mixing process has hardly begun within such a short distance. Let v represent the maximum width of the expanded primary flow (section e) ; the mass conservation equations of both flows yield
The corresponding pressures are those of an isentropic expansion
When writing the momentum conservation of both flows between sections 1 and e, one notices that the pressure integrals on the wake interfaces
SUPERSONIC AIR EJECTORS
7
between primary and secondary flows vanish, since the local pressure is the same on both sides. Therefore one obtains by adding the two momentum equations
Finally, the condition of maximum induced secondary mass flow is expressed by the equation M Y , = 1. This completes the set of aerodynamic equations which can be solved for given values of the geometric and aerodynamic parameters of the ejector (A’,M i l ) and for given primary conditions (pj’, Ti’). For each value of pl“ one obtains the auxiliary set of values of v, p l , p,”, M i , , and the speed M i l of the secondary flow. Then, by means of p1” and M i l , the corresponding point of the (p, &) diagram is obtained:
-
Pl“
Saturated supersonic flow patterns. As pif‘increases and M Y l attains the critical value MYl = 1, the aerodynamic choking of the secondary duct in its minimum section (1’ - 1 ) limits the maximum induced flow. Thus the optimum operational condition becomes M t l = 1, and for the corresponding saturated supersonic flow pattern the p, &-performance curve is linear :
This straight line corresponding to the saturated flow pattern is tangential to the performance curve of the pure supersonic pattern, represented by (2.4). The slope of the lines (2.5) is inversely proportional to the primary stagnation pressure. I t is clear, however, that the secondary mass flow rate induced by a jet ejector operating with a saturated flow pattern is independent of pi’, for a given value of &. Such a result is not specific to parallelstream jet mixing; it occurs whenever a well-defined geometric throat limits the secondary flow. Similarity Laws of Supersonic Flow Patterns. Supersonic flow patterns of jet ejectors obey very simple similarity laws. The stagnation temperatures of both flows appear only in the ratiop of the two mass flow rates. Therefore,
8
J . FABRI A N D R. SIESTRUNCK
if the primary stagnation pressure is high enough to induce a supersonic flow pattern, the reduced performance curve [,u(Ti”/Ti’)1’2vs. G] is a thermal invariant of the set-up. The solution of either equation (2.4) or (2.5) depends only on the ratio pj”/p,’ of the secondary and primary stagnation pressures. The reduced performance curve [,u(fii‘/fii”)vs. &] is then the pressure invariant of the set-up. If these two similarity laws are correctly combined, the numerical calculation of jet ejectors with supersonic flow patterns is greatly simplified, a t least as long as the ratio y of the specific heats can be assumed to be constant. I t is interesting to note, that the operation of the jet ejector is described by the laws of perfect gases, when the primary conditions allow the establishment of a supersonic flow pattern. Viscosity and turbulence of the two air flows as well as the downstream geometric shape of the ejector do not interfere with the exhaust performance. Their only effect would be to advance or to retard the establishment of these supersonic flow patterns. 2. Mixed Flow Patterns
When the primary pressure decreases, the supersonic primary jet eventually breaks down, the downstream shock waves moving towards the primary nozzle exit (Fig. 2c). The subsonic secondary flow is then induced by a subsonic primary flow, and the induced mass flow rate is limited by the exhaust capacity of the mixing tube, whether or not it is followed by a diffuser, since the final exit pressure of the mixed and homogenized flow is the pressure of the surrounding atmosphere. In order to define the overall mixing of the two streams, the geometrical and mechanical characteristics of the set-up must be taken into account. The geometric characteristics have already been defined. The only interesting mechanical characteristic is the turbulent wall friction coefficient, f. The mean pressure drop between sections 1 and 2 is given by the classical one-dimensional representation 2
where p designates the density, 5 the ratio of the area of lateral surfaces to that of the transverse section AS, and M , represents a mean value of the speed in any intermediate section. The approximate overall expression
9
SUPERSONIC AIR E J E C T O R S
where 5 = 4 LID represents now the non-dimensional length of the mixing tube of diameter D ,gives a quite accurate estimate of that pressure drop. In (2.7), M e 2 is the mean value of M , in section 2 where uniform pressure and velocity are assumed. In the parallel-stream mixing considered here the momentum equation takes the form
m2 - M,,
= AP2
m 2 - M *2 2
If the stagnation temperatures are replaced by the corresponding critical sound velocities, a*‘, a*‘‘ and a,, the mass conservation equation Pl‘
+ (A’- 1 )
$1”
(2.9)
a
and the equation of total enthalpy conservation
take very similar forms, the last one replacing the condition (2.2) of isentropic flow. The ejector performance, directly derived from the secondary flow characteristics (pit', M i can now be calculated from (2.8), (2.9j, and (2.10), if the aerodynamic data (PI‘, M i of the primary flow, the thermodynamic data (a*’, a,”) of both flows and the mixing tube exit conditions (p2, M*2) are given. The values of the latter quantities are computed from the diffuser exit data (p3, M,3) by means of the one-dimensional flow equations. The mass conservation equation between sections 2 and 3, (2.11)
g=-
P2 P3
M*2(m2 - M2,3) M*3(m2- M2,2) ’
is independent of friction. The equation of motion takes into account the friction losses estimated as in equation (2.6) and is written in differential form : (2.12)
10
By elimination of obtaines
J. FABRI AND R. SIESTRUNCK
p
between (2.12) and the differential form of (2.11) one
In the particular case of a conical diffuser with apex angle u, Eq. (2.13) can be integrated [ 8 ] , and the numbers M*8 and M,, a t the entrance and the exit of the diffuser are related by (2.14)
where 5 = 2 tan a/f y. At the diffuser exit the subsonic flow pressure is necessarily the pressure of the surrounding atmosphere: (2.15)
P3
= P,
and M,, must be chosen so as to cover the whole mixed flow performance curve, where 0 and p are still defined by (2.4). Of course, a correct choice of f has to be made. The use of either von Kkrman's universal relation [9] (2.16)
f - 1/2 = 4 log ( Rf'") - 0.4
or one of its usual approximate representations yields sufficiently accurate estimates off, as the flow is always turbulent and the walls generally smooth. In (2.16), &! represents the transverse Reynolds number of the mixing-tube flow. Approximate Similarity Laws. Equations (2.8), (2.9), and (2.10) show that, using the same primary nozzle ( M , l ) for different mixing tubes (A) with similar geometry (c, u, and a) and in the same thermodynamic conditions (a,', a,"), one obtains nearly the same induced flow coefficient p, if the values of $,"(A - l ) / A and #,'/A can be kept the same. This approximate similarity law is valid only if the variation of the friction coefficient with the Reynolds number and the geometric imperfections of the actual set-up, i.e. the difference A - A', can be neglected. The influence of the stagnation temperatures cannot be represented in such a simple way. However, for small values of the induced flow coefficient p, the non-dimensional combination ,u(T,"/T,')''2 can be considered as an approximate thermal invariant of the ejector.
11
SUPERSONIC AIR E J E C T O R S
3. Mixed Flow Patterns with Primary Separation For primary pressures that are still lower, the exhausting capacity of the primary flow depends on the geometric shape of the primary nozzle. Actually the one-dimensional theory of jet ejectors yields a low-pressure limitation for the mixed flow pattern: this limit corresponds to the case where the local secondary pressure pl" a t the primary nozzle exit equals the pressure which would be obtained if, for the primary exit conditions (pl', M i a normal shock were to occur in the exit section 1, i.e.
pl"
-- Pl'
m2Mi21- 1 m2- M,, '2 ,
This would lead to a classical subsonic ejector. Such a flow pattern may appear in well-designed nozzles in which a quasi-normal shock in the divergent nozzle changes the supersonic flow into a nearly uniform subsonic flow. However, in the more usual conical nozzles such shock waves do not appear. Instead of emerging from the primary nozzle lips, the primary jet separates from the nozzle walls along a parallel situated inside of the divergent part of the nozzle (Fig. 2d). The corresponding section is designated by the subscript 0. It seems reasonable to assume that the subsonic secondary pressure pl" prevails in the whole part of the primary nozzle which is not filled with the primary flow, i.e. between sections 0 and 1. Then (2.9) and (2.10) still hold, but (2.8) takes the form a 0 Po'
(2.17)
+
a - (a' - 1: m2
"1
=
a P2
of course, the isentropic expansions -a* ao=7-( 1
(2.18)
A
M*o
Po'
m2 - M,2 2
4
m2 - 1
l/(Y- 1),
m2--*o
m2 - M , ,
Pi'
hold in the supersonic flow. According to a generally accepted rule [lo] the upstream pressure of the primary flow in the separation section represents always the same fraction
12
J. F A B R I A N D R. SIESTRUNCK
of the quasi-uniform downstream pressure. Such an assumption means that the oblique shock waves that separate the primary flow from the nozzle walls and deflect the jet toward the nozzle axis have a nearly constant pressure ratio. For the conical nozzles used in rocket motor tests* the experimental value of this compression ratio is close to 2.5. This value of the critical compression ratio agrees quite well with the theoretical predictions of flow separation due to shock wave-boundary layer interaction in the Mach number range considered [ l l ] . In the absence of a more accurate rule, the condition (2.19)
Pl"
= 2.5 p,l
represents the complementary equation. The number of unknown quantities being therefore maintained, the performance of the jet ejector in the low pressure range can now be calculated. This mixed flow pattern with primary separation disappears when (2.20)
plr'< 2.5 pl'. 4. Theoretical Performance Curves
It appears from Sections 1, 2, and 3 that for given primary conditions (pi', M i l ) and a given value of the secondary compression ratio (or a given value of the secondary pressure coefficient GI) each set of equations gives a value of the induced flow coefficient, and the actual value of p has to be chosen between these values. This choice depends of course on the downstream exhaust possibilities. We shall assume that the supersonic flow pattern appears whenever the mixed-flow assumption leads to a too optimistic value of p. That is to say, one chooses between supersonic or mixed flow patterns the one which, for a given secondary pressure, induces a smaller amount of secondary flow. Thus when the primary stagnation pressure is higher than a given critical value &', the supersonic flow pattern appears even for very small amounts of induced secondary flows, and the saturated pattern appears when p increases (Fig. 4a). When pi'< 3i'r the small values of p correspond to the mixed flow pattern; as p increases, the expansion of the primary flow decreases, and eventually the supersonic pattern may appear (Fig. 4b). In some cases, the flow pattern passes directly from mixed flow to saturated supersonic flow (Fig. 4c). * Generally these nozzles are designed for expansion ratios pi'/p,' of 15 to 30 (1.6 < M'*, < 1.9). Such a rule does not apply to nearly transonic flows (for M'*, = 1.37 the compression ratio 2.5 is obtained by means of a novmal shock).
13
SUPERSONIC A I R E J E C T O R S
When pif is still lower, this transition becomes impossible in the usual range of operation (& < 1) although the primary throat remains sonic.
0
ml
0
a
b
C
d
0 FIG. 4. Typical exhaust performance curves for diminishing primary pressures. S supersonic pattern, SS M Ms
saturated flow, mixed flow, mixed flow with separation.
It is usual to observe flow separation in the primary nozzle for such values of the primary pressure. The rule given above still applies though the mixed flow gives a higher mass flow rate with separation than without, since in this case the choice is to be made between two similar flow patterns each of which corresponds to a different primary pressure range (Fig. 4d). 111. EXPERIMENTAL VERIFICATION
In the schematic representation accessible to theoretical calculations, the exit section of the primary nozzle extends into the mixing tube. Actually such a geometric configuration is not the only one that can be described
14
J. FABRI A N D R. SIESTRUNCK
theoretically. The equation of momentum conservation, on which the theoretical description of the mixed flow pattern is based, has equally simple overall expressions in the case of a constant section mixing process and in the case of a constant pressure mixing process (i.e. with constant pressure along the wall of the mixing tube). Although the second process may show better theoretical performance, [2], it does not seem to have any physical significance in the supersonic flow patterns. I t is also very hard to verify whether such a process actually takes place, and it is difficult to build the various necessary geometric configurations. For these reasons all our experimental investigations were confined to constant-section mixing, the exhaust performances of which are interesting enough. 1. Experimental Set-up
The experimental results were obtained with a small-scale apparatus, schematically shown in Fig. 1. The whole ejector is built of metal parts, all of them rotationally symmetric and screwed together end to end. The easy exchange of parts allowed rapid transition from one geometric configuration to another. The primary nozzles generally consisted of two cones with common base (nozzles B , C, D,E , and F of Fig. 5). Although the flow in the exit section of such nozzles (section 1) is nearly conical, their easy and accurate manufacturing made them preferable to nozzles of the type E , (Fig. 5), which is a better design but difficult to manufacture. Two mixing tubes of different diameters (Dl = 11.7 mm and D, = 15 mm) were designed for this set-up. As all primary nozzles have throats of nearly the same cross section, the mixing tubelthroat ratio A* has only two different values. The actual enlargement ratios I' are easily computed from the data shown in Fig. 5. The stagnation pressures are measured by means of pressure taps placed in the primary and secondary chamber walls. These two chambers have large sections compared to the flow sections, and the flow conditions are therefore close to the actual stagnation conditions. The primary mass flow rate is deduced from the primary stagnation pressure after calibration of the primary nozzles. The pressure drop in the primary supply lin$& which is nearly independent of the mass flow rate as long as the p r i i a r y nozzle remains supersonic, is estimated to be four percent of the primary pressure in our experiments. Thus the actual stagnation pressure of the primary flow is 96 percent of the measured value. In all the numerical calculations made hereafter this correction was taken into account, in order to keep as basis of comparison the value of the stagnation pressure measured by the primary manometers.
15
SUPERSONIC AIR EJECTORS
As the secondary chamber is close to the mixing-tube entrance, there is no such problem for the secondary stagnation pressure, and the compression ratio of the jet ejector may be derived directly from the secondary-chamber pressure measurements. 8-C-D-E-F
El
int. ext.
10.95
11.26 FIG. 5. Description of the primary nozzles.
All our experiments were performed with air as the primary and the secondary fluid. The stagnation temperatures of both flows are the same and the mean friction coefficient f is nearly independent of the induced mass flow rate. The value f = 0.0053 derived from (2.16) gives a good correlation between the theoretical performance curves and the experimental measurements.
16
J. F A B R I A N D R. SIESTRUNCK
In order to show the correctness of these theoretical calculations, all the diagrams appearing hereafter were computed independent1y of the experiments by means of the equations of Part 11, and all the curves shown in the following figures are theoretical curves. The corresponding experimental points are marked on the same diagrams and generally agree quite well with the calculated predictions.
FIG.6. Comparison of theoretical and experimental performances for various primary pressures (Nozzle D, 1, = 5.45; 6 = 61.5; u = 1).
2 . General Performance Curves
Figures 6 and 7, though seemingly very intricate, represent the performance curves of two similar jet ejectors for various primary pressures. The first corresponds to the simple case of a mixing tube ejecting into the surrounding atmosphere, without a downstream diffuser (a = 1). I n Fig. 7, the mixing tube is provided with a conical diffuser, geometrically defined by
17
SUPERSONIC AIR EJECTORS
(a = 3.30°, CT = 2.92). In both cases the performance curves, corresponding to pi' = 5.5 p and consisting of a pure supersonic flow pattern followed by the saturated pattern, are identical, thus showing that the ejector operation is independent of the downstream conditions. For pi' = 4.5 p , with a downstream diffuser, the flow pattern is supersonic, then saturated; without diffuser it is mixed, then saturated. For pi' = 3.5 9, with a diffuser, the
FIG. 7. Comparison of theoretical and experimental performances for various primary pressures (Nozzle D. A,
=
5.45;
E
=
61.5;
IJ =
2.92, a
=
3.30").
mixed flow is followed by saturated flow; without diffuser, there is just a mixed flow pattern. Finally, for pi. = 2 . 5 p , the mixed flow pattern with primary separation can be observed in both cases. In general the theoretical calculations describe the experimental results well enough. The interpretation of the quite intricate practical performance curves becomes easy if one considers the theoretical discussion of the various flow patterns.
18
J . FABRI :\NU
a : siipersonic pattern, p,' = 7 p , b: supersonic pattern, pi' = 6.5 p , c : transition pattern, pi' = 6 p ,
I<. SIESTRUNCU
FIG. 8. rl: mixed flow, p;' = 6 p , e: mixed flow, pi' = 4p, f : mixed flow with separation,
pj'
= 3p.
SUPERSONIC AIR EJECTORS
19
It will be noted how strongly the performance curves of a given jet ejector differ when the primary stagnation pressure is higher or lower than the critical value 3,'. The determination of this criticai pressure of a set-up becomes therefore very important. This can be done much more easily in the particular case of zero induced flow (p = 0). 3. Operation without Induced Flow When the induced mass flow rate is zero, the secondary chamber pressure can be compared to the base pressure of a sudden enlargement of the primary flow a t section 1 [6]. For a given experimental set-up this base pressure depends only on pi', and the base pressure characteristic of the set-up will be the curve representing the variations of the non-dimensional base pressure pi"lp versus pj'/p. The various flow patterns already described, but now corresponding to zero induced flow conditions, are shown in Fig. 8*. Mixed flow patterns. When the supersonic flow, leaving the primary nozzle as a convergent (Fig. 8e) or a divergent (Fig. 8d) jet, breaks down to become subsonic in the enlarged section of the mixing tube, the flow pattern can be considered as a particular case of the mixed flow in a jet ejector. The aerodynamic theory of mixed flows gives only an overall description of the flow configuration, but the theoretical dependence of pi'' on pi', calculated by the one-dimensional assumption, gives correct predictions of the actual base pressures, if the following obvious simplifications are introduced in the equations of Section 11.2:
pi"
The description of mixed flows with primary separation (Fig. 8f) is derived from the equations of Section 11.3 by the additional assumption that the pressure Pjrf acts not only on the base of the sudden enlargement, but also in the part of the primary nozzle not filled by the jet. In such flows, the actual design conditions (pl', M > of the primary nozzle are not used in the performance calculations, and the base-pressure characteristics should be independent of the primary nozzle if the mixing tube is geometrically defined by the same value A, of the mixing-tube to throat section-area ratio. The results are represented in Fig. 9 where the same mixing tube is used with different primary nozzles having the same throat section. The theoretical base pressure characteristic is a single curve, and the experimental points follow this curve, a t least in the part where, for each primary nozzle, the conditions for the existence of mixed flow with separation are satisfied.
*
Geometric conditions similar to those of Fig. 2.
20
J. FABRI AND R. SIESTRUNCK
I n Fig. 9, the high-pressure limitation is represented for each nozzle by the straight line
-Pi” _
(3.2)
P
-2.5-(
pi’
m2 - M *’21
m2
9
)
Y / ( Y -1)
.
.8
.7 .6 .5
.4
.3
I
2
3
4
5
FIG.9. Base pressure vs. primary stagnation pressure for mixed flows with separation ( A , = 8.93; = 48; u = 1).
The low-pressure limitation of these flows is less well defined. Of course, the theoretical calculations lose their significance as soon as the separation line reaches the transonic part of the primary nozzle, where the breakdown of the flow is never simple. The transition from mixed flow (with or without separation) to subsonic flow seems therefore difficult to describe theoretically, and as the latter does not have much industrial application, we shall not study it here. Supersonic Flow Patterns. For mixed flows the base pressure decreases as the primary pressure increases, and the difference between the primary exit pressure and the base pressure increases as well. Eventually the primary exit pressure becomes high enough to allow an enlargement of the primary
21
SUPERSONIC AIR EJECTORS
jet up to the mixing tube walls (the flow configuration is very similar to Fig. 8c). The flow configuration may then become independent of the downstream part of the set-up, i.e. one may obtain the supersonic flow pattern. Thus, with the simplifications (3.1), the supersonic flow pattern may appear when
1 --
(3.3) \
I
m 2 - M,, 1
+ Mi2, 1-1
m2 - M,,' 2
(
m
.
4 ( Y - 1)
1
'
where M i represents the mean velocity of the primary flow of an isentropic one-dimensional expansion filling completely the section of the mixing tube. Actually this condition is not strict enough, as the mixed flow pattern still persists over some part of +he range indicated in (3.3). For these transition flows, the initial spreading of the primary jet in the mixing tube is already larger than that necessary to give the theoretical limit configuration of (3.3), but the jet breaks down and becomes subsonic before it reaches the wall (Fig. 8c). The dead air of the base region remains thus in contact with the downstream air, and the mixed flow equations give a correct description of these transition flows. In reality, the supersonic flow pattern appears for a pressure condition more drastic than (3.3) and depends principally on the section ratio 1. The corresponding base pressure characteristic is then given by an equation of the form
due to the similarity laws of Section 11.1, which become valid once more. The flow pattern is independent of the stagnation pressure as soon as the supersonic flow appears (Figs. 8a and b); it is also independent of the downstream geometry of the set-up (Fig. 10). As the overexpanded primary jet hits the mixing tube walls instead of making smooth contact, no elementary calculation can predict the theoretical value of the pressure ratio K,. The measurement of K, has sometimes been proposed [la] to help or to replace the study of base pressure measurements on truncated projectiles, since the flow patterns are very similar in both cases; also the wind-tunnel testing of sting-mounted high speed projectiles is difficult, whereas the base pressure measurement on sudden enlargements is easy. It has also been shown [13] that in this case Crocco's boundary layer-shock wave interaction theory [ l l ] gives a satisfactory description of the phenomenon.
22
J. FABRI A N D R. SIESTRUNCK
Critical Pressure &’. It is seen from the foregoing discussion that for 0 theory and experiment give two different limiting values (3.3) and (3.4)for the supersonic performance of jet ejectors. This explains the
p
N
FIG. 10. Base pressure characteristic of a sudden enlargement (Nozzle D,1,
E
=
5.45;
= 61.5; G = 1).
seemingly incorrect experimental verifications of the theoretical performance predictions for small induced flow coefficients (Figs. 6 and 7). The discrepancy disappears, however, as soon as the amount of induced flow increases, and even for quite smaIl values of ,U the experimental pressure measurements are close to the theoretical predictions. The critical pressure &’ defined above as characteristic of the beginning of the transition period is obtained (Fig. lo) a t the intersection of the mixedflow performance curve of the equivalent sudden enlargement and the straight line representing Eq. (3.3).
SUPERSONIC AIR EJECTORS
23
4. Study of the Main Parameters
The mixing tube length is an essential parameter of a jet-ejector set-up. The variations of the secondary stagnation pressure with the mixing tube
FIG.11. Mixed flow secondary pressure vs. mixing tube length for various induced flow coefficients (Nozzle D,1, = 5.45; u = 1; pi' = 3.5 p ) .
length for given values of the primary pressure and the secondary mass flow rate are shown in Fig. 11. For very small values of 6 the primary jet leaves the mixing tube and has practically no effect on the secondary flow. For very long mixing tubes both fluids form a single homogeneous flow a t the exit section, and the pressure drop along the mixing tube is considerable. Between these two cases, the secondary stagnation pressure has a minimum value for a mixing tube length 6,. This is the optimum exhaust pressure& for a given induced flow coefficient p at the operating conditions (pi'/+,
MI,,,
0,~).
24
J. FABRI AND R. SIESTRUNCK
The mixed-flow theory gives a correct represedation of the p v s & variation for E > tmonly.This critical value 5, actually depends on the primary stagnation pressure. For example, Figs. 2c and 2d show that for higher primary pressures the supersonic part of the primary jet extends further
FIG. 12. Influence of downstream diffuser on performance curves (Nozzle D , 1, = 5.45; 6 = 61.5; pi’ = 4.5 p ) .
downstream and the actual subsonic mixing length becomes shorter, increasing therefore the critical value tm. Similarly, for values of $ / / $ that are high enough to allow the supersonic flow pattern, the value of Em decreases and becomes independent of pi’/$. As a general rule, the theory represents always correctly the effect of mixing tube length for > 50. The downstream exhamt diffuser (parameter a) has an effect on the jet ejector performance only in the mixed flow patterns. However, the critical primary pressure for which the supersonic flow pattern appears depends on u. In Fig. 12 the performance curves of the same ejector with three
SUPERSONIC AIR EJECTORS
25
different exhaust diffusers are compared for the same primary conditions (see Fig. 10). It can be seen that a well-designed diffuser facilitates the appearance of the supersonic flow pattern arid improves therefore the theoretical and experimental performance of the ejector.
FIG. 13. a: influence of primary Mach numbers on performance curves (nozzles B , = 61.5; u = 1 ; pi’ = 4.5p). C, D , A, = 5.45; nozzle E , 1, = 5.43;
The mixing tube cross section, characterized by the ratio A, to the primary throat section, is a fundamental parameter of the ejector operation. The influence of A, is well known by the theory. Large values of A, give high induced mass flow rates but necessitate high primary pressures in order to maintain low secondary pressure conditions. The Primary Mach number has also a great effect on the performance of a given set-up. Large values of M ; , give low secondary pressures if high primary pressures are available. For moderate primary pressures, primary nozzles with weak expansions are more appropriate. Some examples of
26
J. FABRI AND R. SIESTRUNCK
performance curves as functions of the primary Mach number are shown in Figs. 13a and b. The effect of the geometrical shape of the primary nozzle is shown in Fig. 14 where the performances of nozzles E and E , are compared. These
Fig. 13 b : influence of primary Mach numbers on performance curves (nozzles B , C, D , 1, = 5.45; nozzle E , 1, = 5.43; 6 = 61.5; u = 1 ; p$' = 3.5 p ) .
two nozzles differ only in the shape of their divergent part and thus in their ability to give primary flow separation. The supersonic flow pattern remains therefore unchanged, and the high-pressure mixed-flow performance curves are also identical in both cases. For lower primary pressures, the conical nozzle gives mixed-flow performance with separated primary flow, while the aerodynamically better designed nozzle E , still gives pure mixed-flow performance. Actually the performance is better with separation than without. I t may happen for such low-pressure performance curves that the experimental verifications are not as good as usual: this is mainly due to the
SUPERSONIC AIR EJECTORS
27
fact that the flow patterns are less stable for partial or total separation of the primary flow. The empirical rule (2.19) defining the separation criterion does not hold universally either, as was assumed in the numerical calculations for the sake of simplification.
FIG. 14. Comparison of mixed flow performances with a n d without separation. (Nozzle E , I , = 8.93; nozzle E,, 2, = 9; 6 = 48; (I = 1; pi’ = 3 . 5 9 ) .
The elementary aerodynamic theory of jet ejectors cannot describe the effect of all parameters on the performance of the ejector. The effect of the location of the 9rimary-nozzle exit section along the mixing tube axis can only be determined experimentally. In the small-sized set-up described in section 111.1, the relative positions of the primary and secondary nozzles could be slightly changed. In particular it was easy to place the primary nozzle farther back, i.e. into the secondary chamber (Fig. 1). Although in this case the geometrical configuration of the set-up differs from the theoretical configuration assumed in the calculations, the measured perform-
28
J . FABRI AND R. SIESTRUNCK
ance curves show very little deviation from the theoretical predictions. Of course, this is true for large values of I only, that is, for primary nozzles having external dimensions which are small compared to the mixing tube
2
%i
I
0.527
'd
0
"
10
20
30
40
sox
FIG.15. Mixing-wall pressure distributions in supersonic flow pattern for various induced flow coefficients. (Nozzle D, I , = 5.45; 5 = 61.5; u = 1 ; p,' = 5.88p).
section. If the space between the outside of the primary nozzle and the inlet of the mixing tube is small, any displacement of the primary nozzle may change the geometrical configuration of the secondary duct, and in particular the geometry of the secondary throat section. Thus the performance curves can be profoundly changed, an effect which is most noticeable for the saturated supersonic flow patterns.
SUPERSONIC AIR EJECTORS
29
For some larger industrial set-ups designed for combustion research, it was possible to show that the energy level of both fluids (i.e. the stagnation temperatures Ti' and Ti") have the effect predicted by the similarity rules.
FIG.16. Mixing-wall pressure distributions in mixed flows for various induced flow coefficients. (Nozzle D, 1, = 5.45; = 61.5; 0 = 1 ; p,' = 3.54p).
5. Pressure Distribution along the Walls of the Mixing Tube
Basically, the one-dimensional theory of supersonic flow patterns in ejectors assumes that the induced secondary fluid accelerates as it flows along the primary jet. Thus the formation of a sonic section limits the amount of secondary flow which can be induced. Such an assumption can be justified by means of wall pressure measurements along the mixing tube.
30
J . FABRI AND R . SIESTRUNCK
Some longitudinal pressure distributions +"/pirr are shown in Fig. 15 as functions of the axial distance X from the primary nozzle exit*. These pressure distributions correspond to different induced mass flow rates a t the same primary stagnation pressure, which was chosen high enough to insure a supersonic flow pattern throughout the whole operating range of the ejector. For zero induced flow, the nearly periodic pressure fluctuations along the wall are due to the structure of the primary jet. A final pressure rise appears upon the subsonic breakdown of the jet. The pressure distribution remains periodic as long as the induced mass flow rate is small enough to retain boundary layer properties. For more larger induced flows the secondary pressure fluctuations disappear, and the secondary pressure attains the sonic value 9" = 0.527 pi' as predicted. For lower primary pressures the continuous increase of the wall pressure is characteristic of a mixed flow pattern (Fig. 16).
IV. OPTIMUMJET
EJECTOR
DESIGN
By means of appropriate diagrams [5], in which the similarity rules of Sections 11.1 and 11.2 were used to condense the performance curves into a small set of useful characteristic curves, the design of jet ejectors for given tasks becomes easy. As a great number of geometric configurations remains possible, it is still necessary to choose among them the one which gives the best results. The designer has generally at his disposal a given primary air supply with a fixed stagnation pressure and a maximum mass flow rate. He has then to choose the most satisfactory combination of primary nozzle and mixing h b e . The primary throat section is determined by the available air supply. The remaining free parameters are the primary nozzle speed M ; , and the ratio 1, of the mixing tube section and the primary throat area. However, this last parameter is very often determined by the available space, and can be considered as fixed. The above theoretical and experimental study of supersonic jet ejectors shows that, among the various flow patterns which can be observed during ejector operation, the supersonic flow pattern gives the best exhaust perfomance. But this very general condition only determines the minimum pressure necessary for a given set-up to operate under economical conditions; beyond that, it is desirable to determine the best ejector configuration for a given job.
* The reference length is one quarter of the mixing tube diameter. This same unit was used in the caIcuIation of the non-dimensionalmixing tube length E.
SUPERSONIC AIR EJECTORS
31
This optimum corresponds to the lowest secondary pressure for a fixed primary pressure and a given secondary mass flow rate; or to the highest secondary mass flow rate for a given secondary pressure, and of course, a given primary pressure.
FIG.17. Optimum performances of jet ejectors with supersonic flow patterns.
If one considers, for a given value of the parameter A*, the various performance curves corresponding to various primary nozzles, the set of such curves possesses an envelope which constitutes the optimum operation curve. The analytical definition of that envelope is derived from the analysis of Section 11.1 together with the contact condition
- _ac;, _ - - _ap_ _ _ a6 (44
aiwg, aa
aa
and can be written in the simple form (44
p,'
=
pl".
ap
aM;,
--o
32
J. FABRI AND R. SIESTRUNCK
I t will be recalled that this is also the limiting pressure condition of saturated supersonic flows, but the corresponding operation point is not on
FIG. 18. Pressure in the secondary exit section for supersonic flow patterns. (Nozzle D, a, = 5.45; E = 61.5; = 2.92; JI = 3.30’; pi’ = 5 . 3 3 ~ ) .
the envelope, since it is a singular point of the (p, &) parametric representation. The numerical solution of (4.2) shows that there exists on each performance curve a point of operation where plf = p i f and M i < 1. In the operating range between that point and the saturated flow limit, the ratio Pif/pl’ passes through a maximum value, larger than 1. A single p, &-diagram gives the set of optimum performances for different values of A, and M i l (Fig. 17). The theoretical and experimental variation
SUPERSONIC AIR EJECTORS
33
of the induced mass flow coefficient p and the pressure ratio 6 = pl”/p1’ are represented as functions of the secondary compression ratio 6 in Fig. 18 for a jet ejector having a secondary section of moderate width, which permits pressure measurements at the wall of the secondary duct, giving a valid indication of the mean value of pl”. The external wall of the primary nozzle is here parallel to the mixing-tube wall in order to avoid the formation of a secondary throat upstream of the primary exit section. There is good agreement between the theoretical predictions and the measured values of 6 in the vicinity of the optimum operation point (6 = 1). At this point the experimental primary and secondary pressures are nearly equal as predicted by the theory. This agreement no longer holds for the transition to saturated flows. However, the disagreement between theory and experiment can be attributed to a transverse velocity distribution in the secondary duct which becomes progressively less uniform. It is to be expected that for such flows the sonic line no longer coincides with the section 1 of the secondary duct; the wall pressure measurements then fail to give correct indications of the actual mean pressure in the fluid. On the other end of the 6 vs. &-curve where ,u is very small, it is interesting to note that the experimental values of 6, corresponding to an overexpansion of the primary jet, lie on the continuation of the theoretical 6, &-curve. Unfortunately, no physical explanation can be given for that. These comparisons of theoretical and experimental studies of supersonic jet ejectors show that the optimum design of a jet ejector for a given program presents no particular difficulties. Supersonic jet ejectors can now be designed with a relatively small amount of calculation with good accuracy in performance prediction. At the same time, the seemingly intricate performance curves of ejectors can be interpreted and simplified, if the various flow patterns which govern their operation be considered.
References 1. MELLANBY,A. L., Fluid jets and their practical applications, Trans. of the Institution of Chem. Eng. 6, 6G-84 (1928).
E. P., and LUSTWERK, F., An investigation of ejector 2. KEENAN,J . H., NEUMANN, design by analysis and experiment, Journ. A p p l . Mech. 17, 299-309 (1950). 3. JOHANNSEN, N. H., Ejector theory and experiments, Trans. of the Danish Academy of Techn. Sciences 1, (1951). Copenhague. 4. FABRI,J., LE G R I V ~ E., S , and SEESTRUNCK, R., Etude akrodynamique des trompes supersoniques, in Jahrbuch 1953 der Wissenschaftlichen Gesellschaft fur Luftfahrt, F. Vieweg u. Sohn, Braunschweig, pp. 101-110 (1954). 5. LE GRIVES,E., FABRI,J., and PAULON, J., Diagrammes pour le calcul des Bjecteurs supersoniques. O.N.E.R.A. N . T . No. 35, (1956). Paris.
34
J . FABRI AND R. SIESTRUNCK
6. FABRI,J.. and SIESTRUNCK, R., Etude des divers regimes d’Bcoulement dans l’elargissement brusque d’une veine supersonique, Revue GLnLrale des Sciences Appliqubs 2, 229-237 (1955), Bruxelles. 7. ROY,M., TuyBres, trompes, fusees et projectiles, Publ. Sc. et Techn. du Ministire de Z’Air No. 203 (1947). Paris. 8. FABRI, J., MBthode rapide de determination des caractBristiques d‘un Bcoulement gazeux B grande vitesse, O . N . E . R . A . , N . T . KO. 17, (1949). Paris. 9. VON KARMAN,TH., Mechanische Ahnlichkeit und Turbulenz, Nach. Ges. Wiss. Gottingen, Math.-Phys. Klasse 68 (1930). 10. SUMMERFIELD, M., FORSTER, C. R., and SWAN, W. C., Flow separation in overexpanded supersonic exhaust nozzles, Jet propulsion 2.7, 319-321 (1954). L., and LEES,L., A mixing theory of the interaction between dissipative 11. CROCCO, flows and nearly isentropic streams, J o u m . Aeron. Sc. 19, 647-676 (1952) 12. WICK,R. S., The effect of boundary layer on sonic flow through an abrupt crosssectional area change, Journ. Aeron. Sc. 20, 675-682 (1953). 13. KORST,H. M., and WICK,R. S., Comments on ref. 12. Journ. Aeron. Sc. 21, 568-569 (1954) and 22, 135-137 (1955).
Unsteady Airfoil Theory BY A . I . VAN DE VOOREN National Aeronautical Research Institute. Amsterdam
Page I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 I1. The Fundamental Equations . . . . . . . . . . . . . . . . . . . . 37 1. Conditions for Linearization . . . . . . . . . . . . . . . . . . . 37 2 . The Linearized Equations . . . . . . . . . . . . . . . . . . . . 39 3. Harmonic Oscillations . . . . . . . . . . . . . . . . . . . . . . 41 4. Edge Conditions and Radiation Condition . . . . . . . . . . . . . 42 42 5. Reciprocity Relations . . . . . . . . . . . . . . . . . . . . . . I11. The Oscillating Airfoil in Two-Dimensional Subsonic Flow . . . . . . . 44 1. Method of the Velocity Potential . . . . . . . . . . . . . . . . . 44 2 . Method of the Acceleration Potential . . . . . . . . . . . . . . . 47 3. Method of the Integral Equation . . . . . . . . . . . . . . . . . 48 4 . Semi-Empirical Methods . . . . . . . . . . . . . . . . . . . . . 50 I V . The Oscillating Airfoil in Three-Dimensional Subsonic Flow . . . . . . 52 1. Analytic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 52 2 . The Integral Equation . . . . . . . . . . . . . . . . . . . . . . 54 3. Approximations for Wings of High Aspect Ratio . . . . . . . . . . 55 4 . Wings of Very Low Aspect Ratio . . . . . . . . . . . . . . . . . 56 5. Wings of Low Aspect Ratio . . . . . . . . . . . . . . . . . . . 57 6. Swept Wings . . . . . . . . . . . . . . . . . . . . . . . . . . 58 V . The Oscillating Airfoil in Supersonic Flow (Supersonic Edges) . . . . . 59 1. Method of the Moving Sources . . . . . . . . . . . . . . . . . . 59 2 . Riemann’s Method . . . . . . . . . . . . . . . . . . . . . . . . 63 3. Operational Method . . . . . . . . . . . . . . . . . . . . . . . 63 4 . Calculation of Pressure ..................... 64 VI . The Oscillating Airfoil in Supersonic Flow (Subsonic Edges) . . . . . . 65 65 1. The Rectangular Wing Tip . . . . . . . . . . . . . . . . . . . . 68 2. Oblique Tips . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Delta Wing with Subsonic Leading Edges . . . . . . . . . . . 69 4 . Numerical Approach for Arbitrary Planform . . . . . . . . . . . . 70 71 5. The Integral Equation . . . . . . . . . . . . . . . . . . . . . . VII . Non-Linear Approximations . . . . . . . . . . . . . . . . . . . . . 72 1. Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . 72 2. Hypersonic Flow (Piston Theory) . . . . . . . . . . . . . . . . . 76 VIII . Indicial Functions . . . . . . . . . . . . . . . . . . . . . . . . . 77 1. Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . 78 2 . Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . 79 3. Calculation of Flutter Derivatives from Indicial Functions . . . . . 81 4. Calculation of Flutter Derivatives for Large Subsonic Mach Numbers 82 from Indicial Functions . . . . . . . . . . . . . . . . . . . . . 5. Indicial Functions for Finite Wing . . . . . . . . . . . . . . . . 84 I X . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
35
36
A.
I . VAN DE VOOREN NOTaTION
Coordinates fixed in the airfoil ( x chordwise, y spanwise, z downward) Coordinates fixed in the undisturbed fluid Time Lorentz coordinates, see 11, 2 Velocity potential Acceleration potential Reduced values of and see 11, 3 Pressure Pressure difference between upper and lower side of airfoil Air density Velocity of main stream (body) Downwash Frequency Reduced frequency = v l / U hM/I1 - M21, see (2.19) Mach number Speed of sound Semi chord Ratio of specific heats.
#.
A coordinate as subscript indicates partial differentiation with respect to t h a t coordinate. A subscript 0 denotes free stream values or, in case of harmonically varying quan. tities, the amplitude.
I. INTRODUCTION Unsteady airfoil theory as treated in the following review refers to that part of aerodynamics which considers the calculation of the pressure distribution over an airfoil moving with constant speed while performing an unsteady motion in the direction perpendicular to its plane. Since the publications in the middle 1930’s of Theodorsen [ l ] and Kiissner [2] on the oscillating airfoil in incompressible, two-dimensional flow, the subject has been extended widely by taking into account both compressibility and finite-span effects. Being in itself an interesting branch of applied mathematics, the technical importance of this field of aerodynamics lies in the fact that it permits the calculation of flutter speeds. These are defined as the speeds where the forces on the oscillating and deforming airfoil are such that the overall damping just vanishes and, hence, the oscillation tends to become unstable. In order to determine the pressure distribution over an airfoil performing an arbitrary motion normal to its filane, use is made of the indicial functions. These functions describe the lift or moment, or more generally the pressure distribution, as functions of the time, after a unit-step disturbance of either the angle of attack or the rate of pitch a t constant angle of attack has been applied. Although the forces for the oscillating airfoil can also be obtained by means of the indicial functions, the direct approach is often more convenient in this special case.
UNSTEADY AIRFOIL THEORY
37
The results obtained in the analysis of arbitrary normal motions are of importance for the calculation of gust loads. For this problem the variation of the pressure distribution with time, caused by the entrance of the airfoil into a gust front, should also be known. Nearly all available theoretical results are based upon certain simplifying assumptions, viz. (i) the neglect of viscosity (although the Kutta condition at the trailing edge reflects, in a certain sense, the influence of viscosity), (ii) the neglect of heat conduction, (in) the absence of shock waves of finite strength. A further simplification made in the greater part of all investigations in this field is that of linearization. The conditions under which linearization is justified are considered in Section 11. The assumptions mentioned above are the cause of discrepancies between theoretical and experimental results, but little progress has as yet been made in the elimination of these restrictions (with the exception of the second-order theory given in Section VI). However, if these assumptions are accepted, the theory has now entered into a fairly complete state, and it is the object of the present paper to give a review of what has been obtained. Lack of space prevents the inclusion of systems of airfoils which are of importance for compressor flutter as well as for wind-tunnel wall corrections, nor was it possible to consider slender bodies. The treatment is essentially restricted to single airfoils. Completeness of the already large list of references has not been intended; the references should be considered as illustrative rather than exhaustive. 11. THE FUNDAMENTAL EQUATIONS 1. Conditions for Linearization
The conditions for linearization of the equations governing the nonsteady motion of two-dimensional planar bodies have been established by Lin, Reissner and Tsien [3]. Later, Miles has extended these conditions to the cases of three-dimensional planar bodies [4] and to slender bodies [5]. Recently, Landahl, Mollo-Christensen and Ashley performed a similar investigation, including viscous flows in their considerations [6]. Let it be assumed that the flow can be described by a velocity potential y . This excludes viscosity, heat conduction, and shock waves of finite strength. The equation of continuity and Euler’s equations imply that p? satisfies the non-linear equation
38
A. I. VAN DE VOOREN
where the Cartesian coordinates x,y,z move with the flight velocity U of the body and c denotes the speed of sound. The speed of sound c as well as the pressure p are determined by Bernoulli's equation, which follows from Euler's equations and the isentropic equation of state as
here p denotes the density, y the specific-heat ratio, and the subscript 0 indicates free stream values. In the case of three-dimensional planar bodies (thin wings of finite span), the positive x-axis taken opposite to the direction of flight, the y-axis positive to starboard and the z-axis positive in downward direction, the potential y should satisfy the following boundary conditions : (i) at the wing surface, z = h(x,y,t), the condition of tangential flow (2.3)
pl*=k+pL+V,h.Y,
(ii) a t infinity the conditions v x =
u,
yy=o,
qx=O.
In order that the theory of small disturbances should be applicable, it is required that the departure of the velocity components and the pressure from the corresponding free stream quantities remains small. As is shown in [3] and [a], this is only possible under the following conditions : (2.4)
6 << 1,
kd << 1,
Md << 1,
kM6 << 1 ;
here 61 (I = semi-chord) denotes the amplitude of oscillation or the wing thickness (whichever is larger), k is a characteristic reduced frequency (for harmonic oscillations k = vL/U, v = angular frequency) and M is the Mach number. According to (2.3) violation of the condition S << 1 means that the vertical velocity at the airfoil would not be small. The same is true if kd is not small, since then the term hi in (2.3) d l become large. Near a subsonic leading edge the assumption of small disturbances is never satisfied. The validity of the small-disturbance theory in this region has been investigated by Lighthill [7]. If the condition Md << 1 is not satisfied, the flow may be called hypersonic. In this case the free-stream (static) pressure will be very small and the pressure disturbances near the airfoil will be large compared with that
39
UNSTEADY AIRFOIL THEORY
reference. In this sense, no small-disturbance theory exists for hypersonic flow. However, if the free-stream velocity pressure is taken as a reference, it may be said that a small-disturbance theory still exists, although it is no longer linear (van Dyke [8]). In a small-disturbance theory for planar bodies, the boundary condition (2.3) may be applied in first approximation a t z = 0 instead of being applied at the wing surface. This is permissible because of the small thickness (6 << 1) and the regular character of p near z = 0. The last condition requires that the boundary condition for slender bodies should always be applied a t the body surface. The conditions for linearization of (2.1) follow from a careful consideration of the order of magnitude of the various terms as it has been carried out in [3], [4] and [S]. The result is that linearization is permitted if any one of the three following conditions is satisfied
iM
(2.5)
-1
>> S2/q
k >> S2I3,
R << S-1/3.
The latter condition (R= aspect ratio) is not sufficient for linearization at large distance from the airfoil. For each of the conditions (2.5) there is next to pzzin (2.1) a second linear term that is more important than any of the nonlinear terms. If IM - 11 >> d2l3, this term is (1 - M2)p,,; if k >> d2/3it is the sum of the “unsteady terms” p t t / c 2 2 M p X r / c ; while for R << S1/3 it is the term qYr (slender-wing theory). For slender bodies a small-disturbance theory always leads to a linearized equation in first approximation, since then a condition similar to W << 8-’l3 is satisfied. The final conclusion is that a linear small-disturbance theory is always possible except for transonic, nearly two-dimensional, steady flows and for hypersonic flows.
+
2. The Linearized Equations Neglecting all quadratic and higher order terms in (2.1), we obtain the general linearized equation as (2.6)
(1 - M 2 ) p x x
M + pry + pzz - 2 qxt C
1 qtt
C
= 0,
where p may be thought of as designating the velocity potential of the disturbance only. The coordinate system is fixed in the body. Introducing the Galilei transformation
40
A. I . VAN D E VOOREN
with coordinates equation
XI,
y’, z’ fixed in the undisturbed fluid, we obtain the wave
Vx‘x‘
(2.8)
+
+
Vy‘y‘
1
FZ‘Z’
- 2 Vt’t‘ = 0. C
For subsonic flow a Lorentz transformation (Kussner [9]) X=
+
x’ I Ut‘ ,
Y = - yP’
I
’
Z = - zP’ 1
’
keeps the wave equation essentially invariant, viz. (2.10)
vxx
+
~
Y
+ vzz Y
Q ~ T T= 0,
and has the simultaneous advantage of giving the airfoil a fixed position in the Lorentz coordinates. In the supersonic case the transformation
X = x’
+ Ut‘ I
Y = - yP ,
I
Z = - zP’
I
’
I
’
(2.11)
T=
+U
ct’ - x’ C - ~.
I
’
P = v M 2 - 1,
is used, which brings (2.8) into the form (2.12)
vxx -~
Y
Y vzz - Q)TT = 0,
again with the wing fixed in the coordinate system. Linearization of the boundary condition (2.3) yields (2.13)
v z = ht
+ Uh.,,
which easily can be transformed into the other coordinate systems (see also (2.14)). In the linearized approximation the unsteady problem may be considered separately from the thickness and the lifting wing problems. It follows from (2.13) that for the unsteady problem a, will be antisymmetric in z.
41
UNSTEADY AIRFOIL THEORY
Linearization of (2.2) yields for the pressure
p
- p o = - po(vt
p-p
Po
I
O-
(2.14)
+ Vyz)
=
- Po pi’,
(vT+ M T ~ )if
01
M<1
and
M ( y ~ - M v x ) if M > 1 .
I
In the linearized theory an acceleration potential $ may be introduced by
$=-- P - P o ,
(2.15)
Po
which satisfies the same equation as q.x The known normal acceleration (obtained by differentiation of the normal velocity (2.13)) then serves as boundary condition. For a given acceleration potential the velocity potential follows by means of the relation (2.16)
with similar relations in the case of the Lorentz coordinates.
3. Harmonic Oscillations For harmonic oscillations of frequency v, the time dependence is given by (2.17)
p ( x , y J , t ) = yo(X,Y,4 exp id-
In Lorentz coordinates one may write (2.18)
p(x,y,z,t) = @ ( X , Y , Z )exp i x T ,
where (2.19)
X =
kM i f M < 1 1-M2
and
-
x=-
kM M2-1
if M > 1.
The reduced potential @ satisfies the equation (2.20)
@xx
+ @YY +
(2.21)
@xx
- @yy
@ZZ
- @ZZ
+x2@ =0
if M
< 1,
+
if M
> 1.
x2@
=0
42
A. I . VAN DE VOOREN
Solutions of these equations are sought which satisfy the boundary condition of prescribed GZ at the airfoil as well as the conditions to be mentioned under II,4. Both for M < l and M > l the relation between yo and @ is (2.22) Introducing a reduced acceleration potential Y which is connected with (2.16) into
4 in the same way as @ with p, see (2.18), we can transform X
(2.23)
@ ( X , Y , Z )= -exp U
(
-- i
. 1-M2
k
Fat.
--oo
for M 2 1. 4. Edge Conditions and Radiation Condition The physically important solutions of the boundary value problems formulated in II,2 and II,3 are, in general, not regular. In case of the velocity potential, the wake forms a discontinuity plane, the strength of which is determined by the Kutta condition. This condition prevents the pressure at the trailing edge from becoming infinitely large. For a subsonic trailing edge this.means that the pressure becomes equal to the free stream pressure & i.e. the acceleration potential vanishes (strong Kutta condition). For a supersonic trailing edge the acceleration potential may have a finite value (weak Kutta condition) in which case a shock will occur a t the trailing edge. In the wake the acceleration potential is zero since the pressure will be continuous there and should be antisymmetric in z. It follows from (2.14) that in the wake ytl vanishes which means that all vortices (discontinuities in y ) are carried off with the flow under retention of their strength. At a subsonic leading edge the pressure a t the airfoil surface becomes infinite as where s is the distance from the leading edge. At a supersonic leading edge the pressure remains finite, but may be discontinuous in the presence of a shock. Sommerfeld's radiation condition requires the solution to correspond to waves which are expanding from the wing toward infinity. This condition is meaningless in incompressible flow.
l/vL
5. Reciprocity Relations
Reciprocity theorems give relations between the aerodynamic properties of wings of the same planform in forward and reverse flows. They have been developed for steady flows by a number of authors including A. H. Flax,
43
UNSTEADY AIRFOIL THEORY
who extended them to unsteady flow by considering a harmonically oscillating wing [lo]. Heaslet and Spreiter [ll] treated the case of indicia1 functions, while Timman [12] gave a unified treatment for arbitrary unsteady flows. The reciprocity relations are based upon Green’s theorem
/I/
(2.24)
(udv - vdu) d t = /-/-(24?&-
vg)ag.
The function u is identified with a solution @ for the reduced velocity potential in forward subsonic flow satisfying (2.20). The function v is identified with a solution for the reduced acceleration potential in reverse flow, which is easily seen to satisfy also (2.20). It is required that for the reversed flow the Kutta condition is satisfied a t what was originally the leading edge. The volume of integration is the half space 2 > 0. All integrals in (2.24) appear to vanish except that over the surface Z = 0. Hence
/‘/-(@E-F@z)dxaY =o.
(2.25)
plane 2 = 0
It follows from (2.14), (2.15) and (2.18) that
The minus sign of the second term is due to the reversal of flow direction. Upon substituting (2.26) into (2.25) and performing a padial integration in X , it follows that
where the integration may be confined to the airfoil surface, since !P = = 0 outside the airfoil in Z = 0. In physical coordinates the relation may be written as (2.28) wing
no
wing
where and wo are the amplitudes of the pressure difference and the downwash for forward flow and and Go the same quantities for the reverse flow, which should have the same frequency and speed.
170
A.
44
I. VAN DE VOOREN
The reciprocity theorem (2.28) may be extended to arbitrary unsteady flows by remarking that 17,(x,y,z;v) may be considered as the Fourier transform of a function n ( x , y , z , t ) , viz.
Ilu(x,y,z;v) =
.i
e - i y ' 1 7 ( x ,Y&) dt.
--m
170,
Similarly w,,and G,, are the transforms of of the convolution theorem, (2.2s) becomes
17, w and E.
Hence, by means
where 17 and w now denote instantaneous values of pressure difference and downwash for arbitrary unsteady motions. By a slightly modified derivation the same relations can be shown to hold also for supersonic flow. The importance of the reciprocity theorems is that they allow the calculation of overall forces or moments, if a well-known flow is taken as reverse flow. For applications see [lo], [ l l ] and [12].
111. THE OSCILLATING AIRFOILI N TWO-DIMENSIONAL SUBSONIC FLOW This problem can be solved by any of the methods described in 111.1, 111.2 and 111.3. However, if x is large (eg. larger than 2) an alternative method given in VIII, 4 yields results more quickly. 1 . Method of the Velocity Potential
The complete solution for the reduced velocity potential Q can be separated into Q, Q2, where both @, and Q2 satisfy the two-dimensional form of the wave equation (2.20). Q1 is the regular solution corresponding to a flow without circulation and satisfying the boundary condition of prescribed normal velocity at the airfoil, i.e. Qlz = Qz = f ( X )if - 1 < X < 1. G2 has a vanishing normal derivative at the airfoil. This potential denotes a flow with circulation and, hence, is discontinuous in the wake. For the oscillating airfoil the wake is a vortex sheet of harmonically varying strength.
+
UNSTEADY AIRFOIL THEORY
45
The boundary value problem for @, can be solved by means of a Green's function G(X,Z;X1,Z,) of the second kind. Physically, this function denotes the potential a t the point X , 2 due to a source at X,, Z, (or reversely) in the presence of the boundary, where the normal derivative of the unknown potential is prescribed. I t will be clear from this definition that the normal derivative of G at the boundary vanishes. A source a t X,, Z, is defined as a solution of the pertaining equation, which becomes infinite as log R ( R = distance from X,,Z,). In a three-dimensional flow, G becomes infinite a t X,, Y,,Z, as R-l. The boundary for the present problem being the slit - 1 < X < 1 of the X-axis, it is convenient to introduce elliptic coordinates by
X
= cosh 6 cos q,
Z = sinh 5 sin q.
The solution for @, then can be written as*
(3.1)
@,(Ed
=
i
@l~(O,vl) G(E,q;0,qJ sirlq,dq,.
0
In incompressible flow the function G is a solution of the Laplace equation and can be obtained by conformal transformation (Theodorsen [l]) as
I n compressible flow where G should be a solution of the wave equation, no solution in closed form is known. However, it follows from work of Haskind [13], Reissner [14], Timman [15], Timman and van de Vooren [16] and Billington [17], who found the regular solution of the boundary value problem by separation of variables, that
(3-3)
where se,(q) and NeL2)(5)are solutions of the Mathieu and the modified Mathieu equation respectively (see Mc Lachlan [18]). The potential GiZhas been obtained by Theodorsen [l] by considering a vortex distribution in the wake. The corresponding field is easily obtained if the airfoil is mapped onto the unit circle. Here and further on, G has the slightly different significance of being the potential due t o the combined action of a source a t 0, ql and a sink of equal strength at 0, 2 n - q1 in presence of the double line segment - 1 < X < 1.
46
A. I. VAN DE VOOREN
An alternate procedure for determining G2 which is also possible in compressible flow is due to Haskind [13]. The reader is referred to [la] for a detailed description of this method. The Kutta condition is used to determine an as yet unknown factor which still could be added to G2. The resulting pressure distribution for the oscillating airfoil with flap in incompressible flow has been calculated by Theodorsen [l] and also by Kussner and Schwarz [19] in a different way. In order to illustrate the result it may be mentioned that the force K due to a harmonic translatory motion A e x p i v t is equal to
(3.4)
K
= (npl2V2- 2npUlivC) A ,
where K and -4 are positive in the same direction. The first term denotes the inertia of the air. This concept of virtual mass keeps its simple meaning only in incompressible flow. But for very high frequency (vl/c is of order 1) the fluid may not be considered as incompressible even if U = 0, since the time required for a disturbance to travel along the chord corresponds to an appreciable phase difference. Then the airfoil behaves as an acoustic radiator. It follows from (2.19) that for U = 0 the parameter vZ/c = kM determines whether the potential or the wave equation should be used. Therefore, physically, the first term in (3.4) is not valid for very high frequencies (see also Sec. 8.3). The second term contains Theodorsen’s C-function, given by
(3.5)
C(k) =
Hi2)(k) iHr’(k) Hi2’(k)’
+
where Ha) and Hi2) denote Hankel functions of the second kind. This function, the modulus of which lies between 0.5 and 1 for all real and positive values of k, gives the reduction of the circulatory part of the force compared with its quasi-steady value (C = 1). The function C appears also in the results for forces and moments due to harmonic motions with arbitrary downwash distribution along the chord.
It may be added that the theory is not only valid for an airfoil performing harmonic oscillations since t = - 00, but also for an airfoil performing oscillations of exponentially increasing amplitude. In this case v is complex but Im v < 0. However, for damped oscillations (Im v > 0) the theory does not hold since then the disturbance due to the initial phase of the motion is always large in comparison with that due to the oscillatory motion itself as has been shown by van de Vooren [20] and others.
47
UNSTEADY AIRFOIL THEORY
2. Method of the Acceleration Potential The complete solution for the reduced acceleration potential Y can be separated into Y1 Y,, where both Yl and Y, satisfy the two-dimensional wave equation. Yl is the regular solution corresponding to a flow with zero pressure a t leading and trailing edges and satisfying the boundary condition of prescribed normal acceleration. In general, Yl will correspond to a flow with circulation. Ylis obtained in exactly the same way as @j1, see (3.1). The way in which the singular solution Y2can be determined has first been given by Timman [16]. !Pzhas a vanishing normal derivative a t the airfoil. For incompressible flow, YZmust be identical with the acceleration potential of the steady flow past the flat plate a t incidence, since for this flow the normal velocity along the plate is constant and, hence, the normal acceleration is zero. In the case of compressible flow, however, Yzshould be a solution of the wave equation instead of Laplace’s equation. This is achieved by adding to the result of YZfor incompressible flow a series of correction terms. It has later been recognized by Kussner [21] and Timman 1221 that this singular solution is nothing else but
+
where G again denotes the Green’s function of the second kind. In order to explain this, it is first remarked that G is a solution of the wave equation and that its normal derivative a t the boundary vanishes. However, G has a logarithmic singularity a t q = ql, 6 = 0 which vanishes for q = q1 = n, as is seen from (3.2), valid for incompressible flow. Since the singularities of the wave equation and the Laplace equation are of the same type, this holds also for compressible flow. The proper singularity a t the leading edge is produced by differentiation with respect to ql, and since Gql is also a solution of the wave equation and has zero normal derivative a t the boundary, this function satisfies all requirements. The determination of the arbitrary factor a,, by which the singular solution may still be multiplied follows from the condition that the normal velocity a t the airfoil, corresponding to the acceleration potential Yl a,, Y,, should agree with the prescribed value. By the method of the acceleration potential numerical results for forces and moments acting on the oscillating airfoil in compressible flow have been obtained by Timman, van de Vooren and Greidanus [23] and [24]. A variation of this method is due to Hofsommer [25], who writes
+
48
A. I. VAN DE VOOREN
where @, and Y2have already been defined while Y3is a solution similar to Y2but with the singularity at the trailing edge. Although Q), is the flow without circulation, it will have pressure singularities at both edges. The coefficient a3 should be determined in such a way that the singularity a t the trailing edge disappears. Next, a2 follows from the condition that the normal alone yields the true normal velocity due to a2 Y2 a3 U3vanishes since velocity. Hofsommer then obtains the pressure distribution over the airfoil in incompressible flow as
+
n
(1
+ cosq‘) ~ ( 7 ‘dq’)
0
(3.6)
*
+ (ik - A?) sin q aq flog Jog
+
1 - cos (17 17‘) sin __ 1 - cos (7 - q’)
ql
w(r’) q,
0
where T = 2 C - 1, see ( 3 4 , while 17 and w are positive in the same direction. (3.6) has been generalized to compressible flow in [22] and [26]. 3. Method of the Integral Equation
In this method use is made of Green’s function of the first kind with the X-axis from - 00 to w as boundary. This function has similar mathematical properties as Green’s function of the second kind, but it vanishes at the boundary (instead of its normal derivative). Then, for compressible flow
+
a-
@(X,Z) = - - J@(X,,O+) [&Hi2’ {. V(X - X,)2 + ( Z 2
-
Z1)2)]
-m
z,
=0
ax,
if 2 > 0.
(3.7)
Since @ is antisymmetric in 2 and continuous ahead of the airfoil, the range of integration may be taken from - 1 to m. Upon differentiating the whole equation with respect to 2 and letting 2 approach 0, (3.7) becomes m
a2
(3.8) @z(X,O)= lim
z+o 2
@(X,,O+)z2
H!) { x v ( X - X,)2
+ Z2}dX,.
-1
If follows from the condition of zero pressure in the wake that
Q)(X,,O)= @(1,0)exp
49
UNSTEADY AIRFOIL THEORY
Since QZ is known for - 1 < X < 1, (3.8) is an integral equation for @ ( X l ,O+) in the interval -1 < X,< 1. Approximate solutions of this equation as well as results for forces and moments have been presented by W. P. Jones, [27] and [28]. For incompressible flow it can be shown that (3.7) becomes m
-m
if
z
> 0,
which by suitable reduction can be brought into the form m
(3.9)
pz(x,O) = 2 n
J’*
Y ( 4 = px(x1,0-) - p.(x1,0+).
dx,,
-1
Equation (3.9) expresses the downwash due to a vortex distribution at the airfoil and in the wake. From this equation exact solutions for the incompressible case have been obtained by Kussner [2], Schwarz [29] and other authors. Equation (3.7) holds also in terms of the acceleration potential Y instead of @, Then
(3.10)
Y ( X , Z )=
Y(Xl,O+)
a
Hi2’ { X V ( X - X J 2
+ z2}
-1
if Z
> 0.
By means of (2.23) the reduced velocity potential @ can be obtained. Differentiation with respect to Z yields then the famous Possio equation [30], viz.
(3.11)
a
*a -zH b 2 ’ { ~ V ( X - X l ) 2 + T 2 } d l d X l . This is an integral equation determining the pressure distribution. Various approximation methods have been based upon this equation. Collocation methods have been applied by Possio [30] and Frazer [31]. Schade [32]
50
A. I. VAN DE VOOREN
reduced the problem to the solution of a set of linear equations which are obtained by an expansion in Legendre functions. The method of Dietze [33] is probably the one most used. It is an iterative method using the known results for incompressible flow. Turner and Rabinowitz [34] have calculated further values using Dietze’s method. Fettis [35] has approximated the kernel in Possio’s equation and then succeeded in giving a solution in closed form. Convergence of the iterative methods becomes slower with increasing x. The analytic methods of II1,l and 2 are then also handicapped by the slow convergence of series such as (3.3). In this case an asymptotic solution of the wave equation (2.20) (asymptotic in X ) is possible as will be described in VIII,4. 4. Semi-Empirical Methods There exists a certain discrepancy when measured pressure distributions are compared with theoretical results. This is due to the idealizations which have been introduced in the theory, viz. the neglect of viscosity (save for the Kutta condition) and the linearization. In order to obtain pressure distributions which are closer to the experimental values a number of semiempirical methods have been proposed which make use of measured data for steady flow. Schwarz [36] and W. P. Jones [37] have introduced the concept of a “skeleton line” z(x), which is defined by aid of the measured pressure in steady flow by distribution n(5)
(3.12)
The pressure distribution as well as the skeleton line depend on the angle of incidence. For the oscillating airfoil, it is assumed that the skeleton line varies harmonically between the two skeleton lines corresponding to steady flow past the airfoil in the positions of maximum and minimum angle of incidence. After calculation of the normal velocity w of the flow for the oscillating skeleton line the pressure distribution over the oscillating airfoil follows by solving (3.12) for IT([). In this way, W. P. Jones [37] was also able to obtain results for the oscillating airfoil a t high mean angles of incidence. By expanding the pressure distribution along the airfoil in a Fourier series and taking into account only the first terms contributing t o lift and moment, the same procedure can also be applied approximately if only lift and moment are known.
UNSTEADY AIRFOIL THEORY
51
In general, formula (3.12) will lead to skeleton lines whose angle of incidence is smaller than the angle for which the pressure distribution has actually been measured. Another method for reducing the pressure distribution is to reduce the circulation, which requires the introduction of a trailing edge singularity. In this case the skeleton line is kept identical with the mean camber line of the airfoil. Rott [38] has proposed a method based on this idea. If, in steady flow, the Kutta condition is replaced by the condition that the total lift should agree with the measured value, the flow pattern is also completely determined. Leading-edge singularity S , and trailing-edge singularity S , then follow as functions of the angle of incidence a, Elimination of a leads to a relationship
St
= f(S1).
For unsteady flow, where both St and S, are functions of time t it is assumed that st(t t)= f{Sr(t)),
+
thus introducing a time lag t,due to the fact that it may take some time before the viscous effect of the change in the leading edge singularity is felt at the trailing edge. The time lag t is taken in the form t = Cl/U,
where c = 2 would mean that the influence of a variation in S, is carried along the chord with the speed U . Other assumptions may be made concerning c and a general form is
St
= M(K)
s,,
where M is a complex function of the reduced frequency k. A third theory has been presented by Woods [39], who besides viscosity effects takes also into account the potential-theoretical thickness effect. The latter effect multiplies the lift curve slope by 6. I t then follows that K should be replaced by an effective reduced frequency k6, while the various derivatives are also to be multiplied by powers of 6. Furthermore, Woods assumes that the wake vorticity moves downstream with the local velocity of the mean steady flow, which gives an important correction to the pressure for large values of k. The viscosity effects are introduced by assuming that the rear stagnation point performs a small chordwise motion in phase with the local angle of incidence a t the trailing edge and with such an amplitude that the lift-curve slope 2 7 4 1 - E ) is reproduced. The factor 1 - E is called the Joukowski efficiency of the profile. The measured value of the moment curve slope, ac,lau, is used for a correction to the velocity distribution.
A.
52
I. VAN DE VOOREN
Although these semi-empirical theories may in some cases give a n impressive improvement of the results, the agreement is still far from complete and the problem of the introduction of viscosity effects in unsteady flow is still open to further research. This is especially important for controlsurface oscillations.
AIRFOIL IN THREE-DIMENSIONAL SUBSONIC FLOW IV. THE OSCILLATING 1. Analytic Solutions
The method of separation of variables can be used as a first step in obtaining an analytic solution for the three-dimensional problem. Recently, Miles [40] has recalled the theorem that separation of (2.20) is only possible in eleven coordinate systems, whose coordinate surfaces are confocal nondegenerate or degenerate quadrics. Lifting surfaces of practical interest for which the boundary value problem can be described in one of these coordinate systems are, in subsonic flow, the two-dimensional wing and wings of circular or elliptic planforms. The circular lifting surface in incompressible flow has been considered by Schade [41] and recently by van Spiegel [42]. The orthogonal coordinate system for the circular wing is given by _____
(4.1) X = v 1 +q21/1 -p2cos 6,
<
<
Y
___ __= v 1 + q 2 V 1 -,u2sin
6,
Z =pq,
<
where 0 q, - 1 < p 1, 0 6 < 2 n so that the entire space is covered just once. The wing itself is given by q = 0, while the part of the x,y-plane outside the surface is given by p = 0. The problem is treated by van Spiegel by the method of the acceleration potential (Sec. 3.2). The regular solution is equal to m
n-1
where P r and Q: are Legendre's associated functions of the first and second kind. Since Y, should be an odd function of Z and p, the summation is restricted to m n = odd; this is indicated by the prime added to the summation over m in (4.2). The constants C z and S; are determined by the prescribed normal derivative a t the airfoil. By expanding Ylz = Yl,,/,u into a series of surface harmonics the constants are obtained. I t is found that
+
53
UNSTEADY AIRFOIL THEORY
where Green's function of the second kind is equal to
G(q,pu,6; O,Pl>8,) = 00
n=l
with Em=
1 if
m>O
and
E,,,=&
if
m=0.
Consider now
This is a solution of Laplace's equation having zero normal velocity a t the airfoil and a singularity at the point q = 0, p = 0, 6 = 6,,which lies either a t the leading or a t the trailing edge. The singularity is of the type ~(COS a)/r3, where r is the distance from a point on the airfoil to the singular point and il. the angle between the vector Y and the radius of the circle. The singular solution Y2is then given by
1
3 n/2
Y2(q,p,6)=
a(@,) Gpl(7,p,8;0,0,6,) d6,,
n/2
where the integration is along the leading edge only, since no singularities are admitted at the trailing edge. As in the two-dimensional case, the where Y is the distance from the leading singularity in Y2is of the type edge. The function ~ ( 6should ~ ) be determined from the condition that the normal velocity at all points of the leading edge due to the acceleration potential Yl Y2should agree with the prescribed function. In general, this will lead to an integral equation for a(6,) which can be solved by expanding a(&,) in a Fourier series. I t may be noted here that Kiissner 1261 has proposed a different singular solution which, however, fails to have the proper singularity a t the leading edge. Schade [all has treated the problem of the circular lifting surface in a different way, but only for the six downwash distributions up to the second degree in X and Y . The problem of the oscillating circular wing in compressible flow can be solved in an analogous way by means of spheroidal wave functions. Also the oscillating elliptic wing can be treated when Lam4 functions are used.
v$,
+
A.
54
I . VAN DE VOOREN
The main difficulty will be that for this purpose the knowledge of these functions has to be extended.
2. The Integral Equation All approximate treatments use the integral equation in two variables as a starting point. Using Green's function of the first kind for a plane, which is taken as boundary of the half space 2 > 0, one has in analogy to (3.10)
2-
Y ( X , Y , Z )= - 2 n / / - Y ( X l , Y I , O + ) (4.3)
wing
{&);t}
z,
=0
dX,dY,
if 2 > 0 , where Y =
V ( X - X,)2
+ ( Y - Y,)2 + (2- 2,)2.
A result which is the extension of Possio's equation to the three-dimensional case can be derived in the same way as (3.11) has been obtained from (3.10). This yields
(4.4)
-cc
which, by (2.22), can also be written in the form
11
(4.5) w ( X , Y ) = -
PU wing
D(X,,Y,) K ( X , - X , Y , - Y ;K,M) d X , dY,.
The kernel K in this equation has been expressed by Watkins, Runyan and Woolston [43] in terms of known functions and is being evaluated numerically at Harvard University. This will lead eventually to a completely numerical method. (4.3) holds also for the velocity potential @ or for QX, provided the region of integration is also extended over the wake. Such equations, too, have been used as starting points of numerical methods, e.g. by W. P. Jones [44] who uses the equation for @. The unknown @-distribution corresponds
UNSTEADY AIRFOIL THEORY
55
physically to a doublet distribution. As in two-dimensional flow, Jones uses a set of fundamental doublet distributions corresponding to fundamental downwash distributions. This correlation, simple as it is in the two-dimensional case, is much more intricate in three-dimensional flow. Although the downwash distribution can in principle be calculated to any desired accuracy by analytical methods, Jones recommends the use of Falkner’s method with corresponding tables. Applications of Jones’ procedure have been made by Miss Lehrian [45]. Work along similar lines, but restricted to the circulatory component of the flow, has been published by Dengler and Goland [46] as well as by Dengler [47]. 3. Approximations for Wings of High Aspect Ratio
Wings of aspect ratio larger than 3 or 4 are characterized by the fact that the inequality ly - yl(>> Ix - xll holds for most points ( x , y ) and (x1,y1) of the wing. The transformation (4.6)
6= X / l ,
=E Y~I?,
where E is a small parameter denoting ‘the ratio between root chord and span, introduces coordinates 6 and 7 of the same order of magnitude for most points of the wing. The expression for the downwash w may then be expanded in a series of increasing powers of E. There is always a term independent of E , and if the integral equation is solved by taking into account only this term, the results of “strip theory”* are obtained. In the case of straight wings in incompressible flow there is no term linear in E and the &2 log & &2 next terms are of order c210ge, E % , -- and - , see Eckhaus [48]. k k The terms of order loge and e2 are neglected in a theory for wings of high aspect ratio. The terms with a factor k in the denominator are the correction terms for oscillating wings of high aspect ratio. Hence it follows that these corrections increase in importance with decreasing frequency. If k becomes smaller than E , the estimate of the orders is no longer correct. Reissner [49] has presented a theory which is based more or less on the idea mentioned above. The method starts from (4.3)with QX instead of Y , which yields Biot and Savart’s equation after some transformations. Terms of order E~ log E and ~2 are again neglected, while in the terms of order & 2 log & &2 and - a certain approximation is introduced (viz. the replacement k k
* I n strip theory the flow in any chordwise strip is assunied to be two-dimensional, as i t would be if that strip were part of a two-dimensional wing.
56
A. I . VAN D E VOOREN
of the incomplete by the complete Cicala function). This approximation makes the method workable, the result being that the formulae for the aerodynamic forces in two-dimensional flow may be used provided a correction factor is added to Theodorsen’s C(k)-function. The correction factor, which depends upon the spanwise coordinate, is determined by an integral equation that resembles, but is less simple than, the integral equation for lifting-line theory in steady flow. Extensive numerical tables and results are given in [50]. Kussner [9] has independently come to the same results as Reissner by using (4.4) and replacing the upper boundary X by 0 in the integration over 6. Later, Reissner has extended his theory to compressible flow [51]. 4. Wings of V e r y Low Aspect Ratio
Wings of very low aspect ratio (less than 1/2) are characterized by the inequality 1% - xl/ >> Iy - y,l, valid for most points of wing and wake. The transformation (4.6) for high aspect ratio wings can be used again, but now E is large, and the expansion of the downwash must be made in increasing powers of 1/&. The first approximation to the integral equation is in the case of incompressible flow [48] blxl
(4.7)
which is the usual two-dimensional slender wing approximation. This equation can be inverted in the usual way for all spanwise sections lying ahead of the section of maximum span. The corresponding pressure distribution, which is of order 1 / ~ has , been calculated by Garrick [52]. One so obtains the complete solution for wings with a straight trailing edge, which is a t the same time the section of maximum span. Merbt and Landahl [53] have presented a theory which under certain conditions is also valid for compressible flow. They simplify (2.6) to
which is allowed under the conditions
The first condition annihilates the term (1 - M2)px, and the second condition makes the term with pxt in (2.6) small compared with the term ptt/c2. However, for incompressible or nearly steady flow, the term with p r t
UNSTEADY AIRFOIL THEORY
57
need not be small compared with tptt/c2, since this latter term can then itself be neglected. Therefore the results obtained from (4.8) are also valid if
the latter condition denoting that the difference between the Laplace and wave equations becomes unimportant, see eq. (2.20). The solution of (4.8) is obtained by means of elliptic coordnates and Mathieu functions. For incompressible flow, the results are equivalent to those of Garrick [52], but this method also permits one to find the pressure in sections aft of the maximum-span section provided these sections are simply-connected (unswept wing). The pressure there is given by the regular part of the solution of (4.8) written in terms of the acceleration potential. 5. Wings of Low Aspect Ratio
Solutions for wings of low aspect ratio (we mean here the range between 112 and 3) are the most difficult to obtain. Besides the general methods of Sec. IV,2 there exists a method due to Lawrence and Gerber [54] for calculating the force and rolling moment in a spanwise strip of a low-aspect ratio wing with straight trailing edge in incompressible flow. This method, which yields very reliable results a t least for rigid wings, starts from the same equation as Reissner’s high-aspect ratio theory, viz. from an equation expressing the downwash as a double integral over wing and wake, containing yx. This two-variable integral equation is reduced to a single-variable integral equation by multiplication with the weight function v b 2 ( x ) - y 2 and integration over the span b ( x ) . A further approximation is made regarding the distance, viz.
This makes it possible to perform the integration over y in closed form. The resulting integral equation in x is solved by collocation. One of the reasons for the success of this theory is that it yields the correct results in the limiting cases of both zero and infinite aspect ratio. The main drawback is that no pressure distribution is obtained, but only spanwise pressure integrals. It is interesting to consider the various approximations used for 7 in different theories. Reissner’s theory appears to be equivalent to replacing 7 by ly - yll on the wing. In the wake a similar approximation is of course impossible, and the wake integral is obtained by means of the complete Cicala function. In Garrick’s very-low-aspect ratio theory Y is replaced by Ix - xl/. The approximation used by Lawrence and Gerber, both for wing and wake, is given above. Laidlaw [55] has proposed another approximation, viz.
58
A. I . VAN DE VOOREN
V ( X
- X1I2
+ (Y- Y J 2 w 1 , ( m 1%
-
4+
IY - YlL
to be used for the wing integral only. The functions 1, and ,I2 are determined by the requirement that for any value of the aspect ratio the squared error {V(X
- X1)2
+ (Y-
Y1I2 - 1,(4
1%
- x11 -
1 2 w
IY - Y11I2
after integration over the 4 variables x,y,x,,y, should be a minimum. The function A, and A2 have been evaluated by Laidlaw for rectangular wings. The relation
1,(W) = In(A4-l) exists. If W = 0, one has 1, = 1 and A2 = 0 (Garrick) and if W = co, then I , = 0 and A, = 1 (2-dim.). The wake integral is evaluated exactly by Laidlaw using the incomplete Cicala function. The resulting equation for the vortex distribution is solved by a collocation method that uses Fourier series in both chordwise and spanwise directions.
A theory for swept wings of high aspect ratio can be obtained by the procedure described in Sec. IV,3, viz. by using the transformation (4.6) and expanding the expression for the downwash in powers of E . While there is always a term independent of E , there exists in the case of swept wings a term proportional to E sin A ( A = angle of sweep). This term is due to the fact that a linearly varying circulation a t the wing, which produces streamwise vortex lines in the wake whose strength is independent of their spanwise position, gives no contribution to the downwash in the case of straight wings, but does contribute in the case of swept wings. Van de Vooren and Eckhaus [56] have given a theory that takes this term into account. The strength of the streamwise vortices in the wake is put equal to the spanwise rate of change of the two-dimensional circulation, which is known. This introduces an error of order e2, but such terms are neglected. The same integral equation is obtained as in the two-dimensional case, but the downwash is corrected by the influence of the streamwise vortices in the wake. The result is again a strip theory, containing now also terms proportional to the local rates of change of the translational and rotational amplitudes and of the chord. In this way the strip theory for swept wings is brought to the same accuracy as that for straight wings. The evaluation is possible in closed form and [56] contains correction terms for lift and moment to be added for swept wings.
UNSTEADY AIRFOIL THEORY
59
By expanding the downwash as given by the three-dimensional Possio equation (4.4)in a series of increasing powers in E , Eckhaus [57] has succeeded in extending the above mentioned theory to the compressible subsonic case. For this case no numerical results are yet available. Swept wings can also be dealt with by the general theories valid for any planform, which were mentioned in Sec. IV, 2. In particular, the theories of Dengler and Goland [46] and of Dengler [47] have been presented with swept wings in mind.
V. THE OSCILLATING AIRFOILIN SUPERSONIC FLOW(SUPERSONIC EDGES)
If all edges are supersonic, there are no disturbances in regions outside the airfoil which affect the flow at the airfoil. Examples of such purely supersonic planforms are the two-dimensional wing and the delta wing with leading edges ahead of the Mach lines from the apex. In these cases the solution for the velocity potential is directly given by an integration over a part of the wing. The most general method (valid for arbitrary unsteady motions) of arriving at this solution is that of the moving sources (Sec. V,l). Two more mathematical methods, valid for harmonic oscillations and solving eq. (2.21), will shortly be considered in Secs. V,2 and 3. 1. Method of the Moving Sources
This method, originally due to Possio, has been brought into a more easily accessible form by Garrick and Rubinow [58] and [59]. Recently Heaslet and Lomax [60] used this method in a modified form by introducing the concept of the acoustic planform. The method is based upon the physical fact that a disturbance occurring a t a certain point is felt at another point only after a time rlc, r being the distance of the two points. Consider a source of variable strength Q(t) moving in the direction of the negative XI-axis with a speed U and situated a t the instant t a t the point A ( r , q ' , [ ' ) . All primed coordinates are fixed in the undisturbed medium. The potential a t P(x',y',z') a t the time t is then given by
where r1 and r2 denote the distances C,P and C2P (Fig. la), C, and C, being the centers of the spheres through P which are tangent to the Mach cone
60
A. I . VAN DE VOOREN
through A . Clearly AC, = M Y , and A C , = MY,, and it follows by simple geometry that
FIG.la. Plane through P and the x’-axis.
Let now A not be a single source but a point of the leading edge of a wing (= source distribution). The influence of points of the wing aft of A , but ahead of B ( B P = Mach line through P ) is given by a formula similar to (5.1) with Y denoting the distance from P to points between C, and C,. To B corresponds the point D ( D P 1B P ) . The action of points aft of B is not yet observed at P a t the time t. If the leading edge is perpendicular to the x‘-axis, all points from which P receives a signal a t the time t were situated within an ellipse with P as a focal point (Fig. l b ) when these signals were emitted. This area is called the acoustic planform. The total potential a t P becomes equal to
61
UNSTEADY AIRFOIL THEORY
where F is the acoustic planform, Y,, is the distance from P to any point of F , and q the strength of the source distribution per unit area. Since it can be shown that the normal velocity at a point of the distribution is exactly equal to half of the local source strength, the result can also be written as
’i
I
ACOU s T Ic PLAN FORM
FORWARD MACH COME O F P W I T H X‘ ,y’- PLANE
Fig. l b . Projection on the
%‘,
y’-plane. The acoustic planform for supersonic flow.
It may be added that for subsonic flow the acoustic planform is limited by one branch of a hyperbola, corresponding to the fact that only one “propagation”-sphere would pass through P. In the result of Garrick and Rubinow the integration is not over the acoustic planform but over the part S of the wing lying within the forward Mach cone of P (Fig. lb). The correspondence between the area element d t F dr]F of the acoustic planform and the element d5 dr] of s is given by dr]F = dr],
(5.4) d5F =
M(x’ - 5’) F V ( x / - 5 ’ ) 2 - ( M 2 - 1) { ( y ’ - r ] ’ ) 2 ( M 2 - 1)V ( x / - 5 ‘ ) 2 - ( M 2 - 1){ ( y ’ - r ] ’ ) 2
+
+
(2’
(2’
- 5’)21d5*
- 5’)2}
The preceding relation is found in the following way: if the point A (coordinate l‘) corresponds to C l , 2 (coordinate 5‘ MY^,^), a point with
+
62
A. I. VAN DE VOOREN
+
+
coordinate 6' df will correspond to points with coordinates f' dE + M ( r t 2 d ~ ~ where , ~ ) , d r , , can be expressed in d5 by differentiating (5.2) with respect to 5'. Since d f F = IdE 1, (5.4) follows. Upon substituting (5.2) and (5.4) into (5.3), introducing coordinates without primes (which are fixed in the wing), putting
+
+
and noticing that the transformation gives a double mapping of the acoustic planform onto the region S, the potential at P turns out to be
The values of vEused in (5.3) and (5.6)are exactly the same. Note also that
are fundamental solutions of the equation for q in x,y,z-coordinates, i.e. Eq. (2.6). For harmonic oscillations (5.6) becomes, by (2.17),
(5.7)
if
z>O.
Transforming to Lorentz-coordinates and introducing reduced potentials produces the final result @z(X',Y',O)cos x R d X ' d Y ' , R
(5.8) @ ( X , Y , Z )= -
~
S
where (5.9)
R
= V ( X - X')2 -
(Y - Y ' ) 2- Z 2
if Z > O ,
63
UNSTEADY AIRFOIL THEORY
In two-dimensional flow where QZ is independent of Y' the integration over Y' can be performed with the result n
(5.10)
@ ( X , Z )= -
I
@z(X',O)J o { x v ( X - X ' ) 2 -?z}dX'
if
Z
> 0;
the integration has to be carried out from the leading edge to the point X ' = x - 2. 2. Riemann's Method
This method, described in many textbooks on mathematical physics (eg. [61]), has first been applied to the two-dimensional oscillating wing in supersonic flow by Temple and Jahn [62]. It gives the solution of (2.21),viz. (5.11)
L(@)r @ x x- @zz
+
x2@ = 0
with prescribed values of QZ by making use of Green's theorem. A Riemann function Z ( X , Z ;X,,Z,) is introduced which may be considered as the hyperbolic counterpart of Green's function for elliptic boundary value problems. It has the properties of being a solution of (5.11) and assuming the constant value 1 if the points X,Z and X,,Z, are on the same characteristic. It appears that
z = J , { x V ( X - X,)2 - (2 - Z,)2}. I t can then be shown that (5.10) is the solution of the boundary value problem. Although the concept of the Riemann function cannot be extended to three-dimensional flow, W. P. Jones [63] has shown that a solution can still be obtained by application of Green's formula. This method leads to Eq. (5.8). 3. Operational Methods Equation (2.21) with the pertaining boundary condition can also be solved by means of Fourier transforms, This method which has been introduced by von KBrmBn, was followed by von Rorbely [64], Gunn [65] and Stewartson [66]. Miles [67] was the first to apply double transforms, once to the streamwise and once to the spanwise coordinate. The Fourier transformations reduce the partial differential equation to an ordinary differential equation in Z of which the solution satisfying the boundary condition can be written down immediately. The transformation back to the original X and Y coordinates appears to be possible, and the
64
A. I . VAN DE VOOREN
same results as derived in V,1 are again obtained. The method is treated extensively by Temple in his contribution on Unsteady Flow in Modern Developments in Fluid Dynamics [SS]. 4. Calculation of Pressure
According to (2.14) and (2.18) the pressure can be obtained from the formula
p -Po
C
=
1
(ix@ - M @ x ) e x p i x T .
For the two-dimensional case (5.10) should be substituted. Taking X a t the leading edge, the result is after some reductions
=0
X
$(X,O+) - p - - Po[ix@z(X’,O)- M@ZX~(X’,O)] O1c { /0- (5.12)
I
Jo [x(X - X ’ ) ] dX‘ - M @z(O,O) J o ( x X ) exp i x T . On the other hand, formula (5.10) also holds for Y instead of @, since Y satisfies the same differential equation as @. However, it should be kept in mind that the normal acceleration Yz is infinite a t the leading edge, since there the normal velocity jumps from zero to a finite value. It follows from (2.14), (2.15) and (2.18) that (5.13)
C
Y z = - - ( i x @z - Max).
The discontinuity in
1
QZ
as function of X a t X
=0
permits one to write
where S ( X ) denotes the Dirac &function. Substitution of (5.14) in (5.10), written for Y, leads again to (5.12), where the integration over X’ should be taken from O+ to X (i.e. not including the &function in Qzx/, since this is already taken into account by the term with Qz(0,O)). It follows from (5.12) that the pressure takes a finite value a t the leading edge as well as at the trailing edge, which is in agreement with II,4. The same considerations also hold for three-dimensional flow with a supersonic leading edge.
UNSTEADY AIRFOIL THEORY
65
VI. THEOSCILLATING AIRFOILIN SUPERSONIC FLOW (SUBSONIC EDGES) In the case of a subsonic edge the potential in points of a part of the wing is influenced by a region outside the wing, but in its plane. In such a region the upwash is not known a priori, but is determined by the condition that the potential (pressure) should vanish there. This leads to an integral equation for the upwash which, unlike to the steady case, has not yet been solved in closed form. Other methods not requiring the explicite determination of the upwash outside the wing have led to the solution in the case of a rectangular wing tip (V1,l). By aid of a suitable transformation the case of a raked-out or a raked-in wing tip as well as that of a delta wing with one supersonic and one subsonic leading edge can then be obtained (VI,2). A method using pseudo-orthogonal coordinates succeeds in solving the case of a delta with subsonic edges (VI,3). Besides these analytical methods numerical methods have been developed which are valid for any planform. These are the methods presented in VI, 4 and 5. 1. The Rectangular Wing Tip
This problem has been treated by Miles [69] using the Wiener-Hopf technique, by the same author [70] also by using a Laplace transformation, by Stewartson [66] with the aid of a different Laplace transformation method, by Goodman [71] using Gardner’s method, and by Rott [72] by means of a generalization of a method used by Lamb for the half-plane diffraction problem. In the following, in accordance with Goodman, Gardner’s method will be followed, and the result will be given in a simple form valid for an arbitrary downwash distribution a t the wing [73]. The idea of Gardner’s method, which has been described a.0. by Heaslet and Lomax [74], is to separate the four-variable boundary value problem (2.12) into two boundary value problems of three variables each, viz.
nxx
n,
- QTT -
nc ,
= 0,
- QYY - Qzz = 0.
It will be clear that any function Q(X,Y,.Z,T,() satisfying both (6.1) and (6.2) will also be a solution of (2.12) provided (, is kept constant. Accordingly, the function to be identified with the solution v ( X , Y , Z ,T ) of (2.12) will be taken as (6.3)
v ( X , Y , Z ,T ) = (Qdt =o.
In order to solve (6.1) and to comply a t the same time with the boundary condition of prescribed vz, a new variable (6.4)
a(X,Y,T,t)= (Qz)z=o
66
A. I . VAN DE VOOREN
is introduced. (6-5)
The equation for OXX
G
becomes
- aTT - GCE = 0.
By (6.3) and (6.4),the boundary condition which a should satisfy is that (6.6)
(VZ)Z=O
= (Qz&=o,z=IJ = (a&=o
has a prescribed value at the wing ( Y > 0, X > 0 ) , while a and uE vanish ahead of the wing, that is for X < 0. For Y > 0 the determination of a is identical with the solution of an Evvard problem. Its solution is obtained immediately from (5.8) by substituting x = 0 and replacing Y and 2 by T and 6 respectively. Hence
dX, dT, . v ( X - X,)2 - ( T - T J 2 - E2
t>o.
v(X- X,)2 - t2 and for X , The limits of integration for T , are T 0 and X - 5. The integration over T , for harmonic oscillations gives
This determines the boundary condition a = (Qz)z=o for all Y > 0 for equation (6.2). Thus we have again an Evvard problem with the flow in the direction of the negative &axis (Mach number X appears here as parameter (Fig. 2 ) . The wing occupies the area Y > 0, E < X , and here Qz = a is given by (6.8). For Y < 0 the condition Q = 0 is assumed since, by (6.3), this leads to g~ = 0. Hence the solution Q of this problem yields a function v satisfying all conditions. In analogy to (6.7), the solution is
vq;
The region of integration S is the triangle lying in the plane Z = 0 within the forward Mach cone of the point P ( Y , ( ) , Fig. 2. If Y E < X , this triangle contains a part of the upwash region a: the, side of the wing, viz. ADE. According to a well-known property of the steady flow problem, the
+
67
UNSTEADY AIRFOIL THEORY
contribution of d A D E in the integral of (6.9) is completely cancelled by that of AABC and hence, S is equal to the region PABC. Substituting (6.8) in (6.9), differentiating with respect to 5, putting E = 0, and performing the integration over 5, leads to the following result for v, as has been shown by van de Vooren in [73]. If Y > X , i.e. without tip effect, the usual formula reappears
where R = v(X- X , ) 2 - ( Y - Y , ) 2 , and the integration should be performed over the triangle cut by the forward Mach cone in the plane 2 = 0. I f 0 < Y < X , the integration in (6.10) should be performed over the region PABC (Fig. 3), while the triangle AOB gives a contribution to v(X,Y,O+,T) equal to
1 n12
x7c/ ~ v z ( X l , Y , , T )
(6.11)
J l ( x Rsin6,) d6,dX1dY,,
where
/
/
/
FIG.2. The plane 2 = 0 for the second problem (6.2).
FIG.3. Regions of integration for (6.10) and (6.11).
68
A. I . VAN DE VOOREN
This gives a confirmation of Stewart and Li's result [75] that the contribution of A O A B may be neglected when considering only zero and first order in frequency. The same solution but with a trivial change in the area of integration holds also for a supersonic leading edge that is not perpendicular to the flow direction. I t follows from (6.10) and (6.11) that pl -0 if P approaches the wing tip (Y .+ 0). Moreover, p l y becomes infinite as Y-'I2. The complete solution for a wing with supersonic leading and trailing edges and with streamwise tips can be built up from this solution, provided the Mach lines from the foremost points of the tips do not cut the tips at the other end. A solution for the rectangular wing with B V M 2 - 1 < 1 (Mach lines from leading-edge wing tips now intersect the opposite tips) has been given by Miles [76] using a Laplace transformation to eliminate the X-coordinate in (2.21). This leads to (6.12)
FYY
+ Fzz = (s2 + x 2 )F ,
where
dX. 0
Equation (6.12) with its boundary conditions is then solved by introducing elliptic coordinates instead of Y and 2. The function F is obtained in terms of Mathieu functions. The difficulty is to determine @ from F. This has been performed in [76] by an expansion in powers of A I v M 2 - 1, which suffices for the low-aspect-ratio wing (except for M near to 1, when the expansion is in powers of k Az). 2. Oblique
Tips
With the aid of a transformation originally given by Lagerstrom and applied to unsteady flow by Miles [77], it is possible to transform an oblique tip into a streamwise tip. Let the equation of the tip be Y = - m X, 0 if it is a where 0 m < 1 if the tip is a leading edge and - 1 < m trailing edge. New coordinates are defined by
<
<
69
UNSTEADY AIRFOIL THEORY
The transformation has the property of keeping the equation (2.12) as well as the hyperbolic distance R and the area element d X d Y invariant. Since the tip is a t Y’ = 0, the equation in the X’,Y’,Z,T-variables can be solved as described in VI,l, and the result is transformed back to the X , Y,Z, T-variables. This procedure yields the exact solution for a leading edge (0 m < 1) provided the function in the differential equation is the velocity potential. As mentioned in VI,1 one has at the tip q . = ~ 0 while q+?, and hence also vX, possess the required root singularity. This makes the pressure infinite at the subsonic leading edge tip. In the case of a trailing-edge tip the simplest method is to take the acceleration potential z,b as unknown function in (2.12). The solution a t the tip is then z,b = 0 and hence, the Kutta condition is automatically satisfied. However, the delta function in z,bz a t the leading edge should not be omitted when the pressure is calculated (see V,4). It may finally be added that the case of a delta wing with one supersonic and one subsonic leading edge and a supersonic trailing edge can also be solved in this way.
<
3. The Delta Wing with Subsonic Leading Edges
Miles [40] has shown that separation of the hyperbolic equation (2.21), like that of equation (2.20), is only possible in eleven coordinate systems. For supersonic flow an analytic solution by separation is possible for the delta wing with subsonic leading edges. Robinson [78] has described the general lines of the procedure. The coordinates to be used are the so-called hyperboloido-conal coordinates defined by
(6.13)
where
rJ,q form a system of pseudo-orthogonal coordinates, which means that the hyperbolic distance remains invariant in the transformation. With the specified intervals, the coordinates r, t and 17 can only represent points within the backward Mach cone of the apex ( X = Y = Z = 0). In particular, r = 0 denotes the Mach cone, E = k denotes the delta wing, & -P 00 approaches again the Mach cone, q = h denotes the part of the
70
A. I. VAN D E VOOREN
plane Y = 0 within the Mach cone, and 7 = k denotes the part of the plane 2 = 0 outside the wing but inside the Mach cone. The general solution of (2.21) in these coordinates is [78] m
where A," denote constants,
2nC1
Jn++
the Bessel function of order n
+ 4 and
F,"(t) and E r ( q ) are Lam6 functions in the definition of Hobson [79]. The constants A: are determined from the prescribed value of the normal velocity at the functions* and to expand the comparison of
airfoil. By using the orthogonality relations between Bessel also those between the Lam6 functions E,"(q), it is possible normal velocity in a double series of these functions. By this result with the finite values of lim \/t2- k 2 the I+k
constants are obtained. With a supersonic trailing edge, the Kutta condition does not affect the flow along the airfoil; thus no circulatory solution a2(as in I11 and IV,1) should be added. The solution found above has a discontinuity in the wake and, hence, introduces circulation into the flow. No numerical results obtained by this method are so far available.
4. Numerical Approach /or Arbitrary Plan form
A numerical method has been developed by Pines, Dugundji and Neuringer [81]. It is based upon formula (5.7) giving the velocity potential q~ as an integral over the normal velocity w = ye. The wing as well as the part of the wing plane where w differs from 0 (called the diaphragm) is divided into a grid of square boxes. w is assumed to be constant in each of these boxes. The pressure can now be written in the form
(6.15)
* It has been remarked by Germain and Bader [80] that the orthogonality relation between the Bessel functions as given in [78] is incorrect. They also indicate how to circumvent this difficulty.
71
UNSTEADY AIRFOIL THEORY
where r is given by (5.5), and the summation extends over all boxes, whose individual areas are denoted by Sj. The aerodynamic influence coefficients
are computed by introducing suitable approximations such as the expansion of the integrand in a Taylor series about the center of the box. With the coefficients A j assumed to be known, the relation (6.15) is of different type whether the point x,y is a t the wing or at the diaphragm. For points a t the diaphragm the pressure vanishes. The unknowns are the pressure a t the wing and the normal velocity at the diaphragm. The equations (6.15) referring to the wing points can be solved for w a t the diaphragm. When these values are substituted in the equations referring to the wing points, the pressure a t the wing is obtained. Unless a large number of boxes is accepted, the numerical accuracy will not be very high. Therefore, this method is specially adapted for use on electronic digital computers. According to [81], tables of the aerodynamic influence coefficients are being computed a t the Wright Air Development Center. This method has been further developed by Zartarian, Hsu and Voss [82]. They use rectangular boxes whose diagonals are Mach lines and, in another scheme, characteristic boxes, that is rhombuses formed by a set of Mach lines. This simplifies the calculation of the aerodynamic influence coefficients.
5. The Integral Equation
Watkins and Berman [83] have presented a method based upon the integral equation for the pressure distribution, which is completely analogous to the method of Watkins, Runyan and Woolston [43] for the subsonic case (IV,2). Similarly to (4.3) one has for the reduced acceleration potential
Y ( X , Y , Z )= - 7d
ss S
(6.16)
{ail
Y ( X , , Y , , O + ) -~ ,OS; R}z,= 0
dY1
if 2 > 0,
where
R
= v ( X - X , ) 2 - ( Y - Y,)2 - ( 2 - 2J2.
72
A. I. VAN DE VOOREN
The relation between downwash and pressure difference, which can be derived from this formula, is formally identical with (4.5), but the kernel is different. Application of this method awaits the numerical evaluation of the kernel. Earlier work of Watkins was based upon the analogue of Eq. (6.16) valid for the velocity potential. By differentiation with respect to Z it yields an integral equation for the distribution of velocity-potential doublets. This equation was solved by expansion in increasing powers of x . Numerical results are available both for the rectangular [84] and the delta wing [85], but, especially for Mach numbers slightly in excess of 1, they are limited to small values of the reduced frequency.
VII. NON-LINEAR APPROXIMATIONS
It was shown in 11,l that the transonic and the hypersonic regions of speed are critical for linearization. I n the transonic range no non-linear theory for an oscillating airfoil is yet available. In VII,2 the hypersonic case will be considered. However, also in the supersonic range, the errors made by neglecting thickness effects are larger than in the low subsonic range, a fact, which is essentially due to the non-linear character of the governing equation (2.1). Non-linear theories are here available for a twodimensional field of flow which is everywhere supersonic, which means that the Mach number should be at least equal to a value slightly above that where an attached bow wave appears. 1. Supersonic Flow
It has been shown by Busemann [86], that the pressure a t any point of an airfoil in a steady, two-dimensional supersonic stream is given by (7.1)
P - p0 = 4 po U i {?
+
Cl6 4-C2fi2 F c3 63fD 6,(0)3) O(@4).
where the subscript 0 denotes free stream values, 6 the local slope of the airfoil and 6(0) the value of 6 a t the leading edge. 6 is taken positive in the same direction as the angle of incidence. The upper sign should be taken for the upper surface and the lower sign for the lower surface. The terms with 6 are due to adiabatic compression or expansion, the term with 6(0) is due to a shock wave. If 6(0)is such that either a t the upper or a t the lower surface no shock exists, this term should be omitted for that side. The correct values of the coefficients Cl,C,,C3 and D have been given by W. P. Jones [87].
73
UNSTEADY AIRFOIL THEORY
For slow oscillations, according to linear theory, the pressure difference is given by 2
M 0
as follows from (5.12). This result can also be obtained from (7.1) if in the linear term 6 is replaced by x
and all further terms are neglected. W. P. Jones makes the assumption that this same substitution can be performed for all terms (in fact, he makes this assumption for the vertical velocity w instead of 6 which in the higher terms makes some, but no essential, difference). However, the assumption has not been confirmed by later work, except in the hypersonic range, where Lighthill [88] found it to be correct. In a second report, W. P. Jones and Miss Skan [89] propose another method, which treats the flow about an oscillating airfoil a t incidence as only slightly different from the steady state flow at zero incidence. Only the linear terms of the difference are retained. The method is complicated and may be criticized on the ground that the leading edge pressure does not agree up to second order with eq. (7.1) when instead of 6 the dynamic local slope of the airfoil a t the leading edge is substituted. The most straightforward theory is due to van Dyke [go]. I t yields a solution correct to second order in thickness, but only to first order in the angle of attack. This is no serious limitation since second order terms in the angle of attack are equal on the upper and lower surfaces and hence, do not affect lift or moment. For the velocity potential the following expression is introduced p(x,z,t) = U1 { X
(7.3)
+ +(X,Z) + aoeix*di(X,Z)};
here (b(X,Z) denotes the mean steady flow and @(X,Z)the additional flow due to the oscillation in the angle of attack with amplitude u,,. Substituting (7.3) into the exact solution (2.1), using (2.2) for the local speed of sound, and neglecting all terms higher than second order leads t o (7.4)
4xx - +zz @XX
(7.5)
- @ZZ
+ x 2 di
+2 ix M
where N = ( y
= -M2
[ ( N- 1) 4;
- 2 M 2 [ ( N - 1) ~
1
[( 2 N - 1) +x@x
+ 1) M 2 / 2p2
+ &]x, +
X @ X
~Z@Z]X
+ N 4 ~ x +0 +z@z] + 2 N
lt2 $ X
@,
and M is the Mach number of the free stream.
A.
74
I. VAN DE VOOREN
The condition of tangential flow (2.3) must be applied at the surface of the airfoil given by z = I { E g(X)
(7.7)
+ a,, eivtk(X)},
where E is the thickness parameter, while k(X) = X - b for a rigid airfoil oscillating about the point X = b. By a Taylor series expansion the value of the normal velocity following from the tangency condition can be obtained a t z = 0. The result becomes (7.8)
P $ z = E ( ~+ ~ x ) ~ x - P ’ E & z ~ ,
(7.9) P O z = ((1
+).4
kx
+ i K Iz
-
pz+zz} e i x M X
+&{(OX
- i x M 0 ) gx - P2Ozzg).
By neglecting all second order terms, the problem reduces to the linear problem with solutions 4(l)and W). In order to solve the second order problem it is allowed to substitute $(l) and O(l) for 4 and 0 in all second order terms. The steady second order problem defined by (7.4) and (7.8) has been solved by van Dyke [91]. Complications arise in the neighborhood of shock waves, where the second order solution may produce discontinuities or regions of multi-valuedness in the potential. However, it follows from (7.1) that, to the second order, a shock wave is equivalent to the limit of a rapid continuous isentropic compression. This means that the airfoil contour may be smoothed near the leading edge so that the resulting shock wave occurs at so large a distance that it does not affect the pressure at the airfoil. The solution obtained for the smoothed shape yields by a suitable limiting procedure the proper result for the original airfoil except in the vicinity of the shock wave. At the airfoil surface, van Dyke’s solution becomes identical with Busemann’s. For the oscillating airfoil a similar procedure [go] is followed. Again the airfoil contour near the leading edge is smoothed, so that the tip of the extension (Fig. 4) is always in the direction of the free stream. This can only be achieved by a flexible tip corresponding to a complex function k(X) in (7.7). After having obtained the solution for the smoothed problem, h ( X ) is replaced by X - b, which again yields the proper result, except near the shock wave. The unsteady second order problem (7.5) with boundary condition (7.9) was solved by van Dyke by means of a Laplace transformation applied to the X-coordinate. Two cases are considered: an arbitrary profile for which the solution is given in the form of an expansion in increasing powers of the frequency (including the third) and a wedge for which a solution exact in frequency is given. In the first case the pressure coefficient a t the lower
UNSTEADY AIRFOIL THEORY
75
surface of the airfoil is, when only the first power of the frequency is retained, equal to
&$
2
= - (& gx
P
+ a ) + M 2 P2N - 2
when the airfoil is pitching about an axis a t a distance bl aft of the leading edge. PROF1 LE
I
FLEXIBLE TIP
Z FIG.4. Smoothing of an airfoil by means of a flexible tip.
In this formula, the first and second terms agree with Busemann’s result (7.1) for steady flow. The third term is known from linearized unsteady theory. The last tern1 represents the second order effect in unsteady flow. It is seen that (7.10) satisfies the condition that the leading-edge pressure is given by Busemann’s formula when the dynamic angle of attack Egx f a - ci bllU is taken into account. After calculation of the moment about the pitching axis from (7.10), those combinations of M and 6 can be determined for which unstable oscillations are possible. It appears that the non-linear thickness effect gives a small increase of the unstable region in the M,b-diagram.
76
A. I. VAN DE VOOREN
In general, the second-order effect is moderate in magnitude except for unduly thick wings or unless the Mach number is close to one. Non-linear effects of higher order are negligible as can be verified for the wedge, for which Carrier [92] (see also van Dyke [93]) has given a solution which is exact in thickness although still linearized in angle of attack. 2. Hypersonic Flow (Piston Theory)
The non-linear theory for the two-dimensional hypersonic case is greatly simplified by the so-called piston theory, first given for the oscillating airfoil by Lighthill [88]. In order to show the principle of piston theory, the linearized case for large values of M will first be dealt with. If M is sufficiently large, the linearized equation (2.6) may be replaced for twodimensional flow by
The general solution of this equation is = g(x - M Z) f ( x
(7.11)
- Ut),
where f and g are arbitrary functions. Hence the velocity potential together with its derivatives such as the normal velocity and the pressure are constant if x = Ut and x = M z are kept constant. This means that a perturbation is carried with the flow ( x = Ut), but at the same time is moving in z-direction with the speed of sound (z = x / M = ct). I t follows from (7.11) that
w
= qZ=
- M g'(x) f ( x
-
Ut)
if
z =0
and
p
-
p,
= -p0(p
+ U p )=
- po
U g ' ( x - M z ) f ( x - Ut).
By comparison, it is seen that a t the airfoil one has (7.12)
P
-Po
=fJ0cw.
This is identical with the pressure acting on a piston that is moving with a velocity w into a narrow (one-dimensional) cylinder. The piston analogy is not restricted to linearized flow but exists also in the general two-dimensional case, admitting shock waves with entropy changes. This has first been noticed by Hayes and has later been shown more extensively by Goldsworthy [94]. The relative error is of order M - a and corresponds to the neglect of the difference between M 2 and M 2 - 1.
UNSTEADY AIRFOIL THEORY
77
According to piston theory the disturbance of the airfoil is not perceptible ahead of the line x = Mz while in actual fact the Mach line x = z 1/M2 - 1 is the boundary of the disturbed region. The result of piston theory may be formulated by saying that, within an error of order M-2, any plane slab of fluid initially perpendicular to the undisturbed flow remains so and moves in its own plane under the laws of one-dimensional unsteady motion. In the exact (non-linear) theory the pressure due to a one-dimensional piston moving with instantaneous velocity w is given by y-
(7.13)
-P= ( l + T T )
1 zer
2Yh-1
,
Po
if no shock waves are present. This expression agrees with (7.1) if in (7.1) the shock-wave term is neglected and if the coefficients are calculated by retaining only the term of highest order in M . It has been noticed by Lighthill [88] that the first four terms of the binomial expansion of (7.13) constitute a good approximation also in the case that shock waves are present. The term in w3 is not exact, but improves the accuracy compared with retaining the quadratic terms only. By using the first four terms of the expansion of (7.13), Lighthill has calculated the aerodynamic stiffness and damping for a pitching airfoil. When all third order terms are neglected and in van Dyke’s result only the terms of highest order in M are retained, the two results become identical. Similarly to the way in which van Dyke solved the second order problem for low frequencies, Landahl [95] gave the solution for high values of M (in both cases x is small and a power-series expansion in x is possible). The result for @ [see (7.3)] is of the form
The terms with superscript (1) are the linear terms; they can be found by expanding (5.10) for small values of x . The terms with subscript 1 are the terms given by piston theory. It is seen that the second order term in piston theory, @i2),is larger than the error made by piston theory in the linear term, @), only if ~3 >> &-I. VIII. INDICIAL FUNCTIONS There exist certain analogies between two-dimensional unsteady flow problems and three-dimensional steady state problems, as has first been pointed out by Lomax, Heaslet, Fuller, and Sluder [60]. These analyses
78
A.
I. VAN DE VOOREN
are particularly useful for calculating the indicial functions, which give the aerodynamic action as a function of time after a sudden change in angle of attack (or in rate of pitch a t constant angle of attack). 1. Incomfiressible Flow
In coordinates x', y', z' fixed in the undisturbed fluid, the boundary value problem for two-dimensional, unsteady, incompressible flow is determined by the equation (8.1)
q7xfxl i-vz'a' = 0,
together with the boundary condition that qz< is prescribed at the airfoil and certain singularity conditions at the edges. In a x',t'-diagram (Fig. 5),
4
VORTEX L I NES
*
I
t', x
FIG.5 . Equivalence between two-dimensional indicial motion at M = 0 and threedimensional steady motion at M = 1.
the wing is given by the area A BCD for t' > 0. The position of the wing at time t' = t,' is given by the segment of the line t' = t,' lying between AB and CD. Aft of the trailing edge CD there is a system of starting vortices with axes parallel to the t'-axis. According to (2.14) the pressure is given by fi - Po = - Po%'*
79
UNSTEADY AIRFOIL THEORY
Consider now on the other hand a wing ABCD placed in a flow a t M = 1 in the direction of the positive x-axis. This problem is determined by the equation pry p z z = 0,
+
again with the boundary condition that yz is prescribed at the airfoil and with the same singularity conditions as above. The x,y,z-system is fixed in the wing. If in the two cases the same values of the normal derivatives are prescribed, i.e. if p/(x',t')
=p Z ( y , x )
for
x' = y
and
t'
= x,
the solutions for p are also identical. The pressure in the second problem is given by fi - Po = - po Ucpz (V = c). The starting vortices become identical to the tip vortices in the second problem. Hence the solution obtained by thin wing theory for the second problem gives also the solution of the first problem. According to thin wing theory, there exists an infinite loading along BC, which is given by
A$
=
- 2 po U 2 ad(x) V12 - y2,
where 6 ( x ) is the Dirac &function. Hence the loading along BC in the unsteady two-dimensional problem is
d p = - 2 p"
u s(t')V P - x2. Gc
The total force after a sudden change in angle of attack is denoted by
The indicia1 function k,(s) contains a term xS(s). At s = O+ it assumes the value x and then approaches asymptotically the value 2 n. The function has been tabulated in [96] and [97]. 2. Compressible Flow
The two-dimensional unsteady problem for compressible flow is formulated by the equation cpx'x'
+
1
cpz'x'
= --p cpt't'
with pertaining boundary conditions. This equation is of the same type as the equation for the three-dimensional steady, supersonic problem, viz. pyy
+
VZE
= (M2 -
1) p x x .
80
A. I . VAN DE VOOREN
vg
If the latter problem is solved for M = the solution will also satisfy (8.3) provided x,y,z are replaced by c t ’ , ~ ’ , ~ ’ .The planform corresponding to these problems is shown in Fig. 6 and is seen to be a swept-forward wing
--- x’, y
FIG. 6. Equivalence between two-dimensional indicia1 motion a t M # 0 and threedimensional steady motion at M =
tip. Leading and trailing edges are subsonic if the Mach number of the unsteady problem is smaller than 1 as in Fig. 6; they are supersonic if that Mach number is larger than 1. In the latter case the solution is simple since the potential in P is given by an integral of which the integrand is known everywhere. With subsonic edges one obtains an Evvard problem. It can be solved consecutively for all regions numbered in Fig. 6. The solution becomes increasingly more complicated for larger x (= ct’), except in the sonic case where the regions 1 and 3 cover the whole wing, allowing an exact solution. For the compressible case the k,-function contains no longer a deltafunction but assumes the value 4 alM. If a wing is entering a gust front (at x‘ = - I), the total force is given by a formula similar to (8.2), but with k,(s) replaced by k,(s). This function
81
UNSTEADY AIRFOIL THEORY
can also be calculated by considering the wing A B C D in Fig. 6, since the prescribed value of yZwill now be different from 0 only for x’ < - 1. Results are given by Lomax [98].
3. Calculation of Flutter Derivatives from Indicial Functions
+
vz
Indicial functions are obtained if is constant along lines x‘ Ut‘ = constant. If plz is varying along such lines by a factor exp ivt’, the flutter
-
m. However, the flutter derivatives can derivatives are obtained for t‘ also be calculated directly from the indicia1 functions. Let pind(x,t’)be the pressure a t the point x = x‘ Ut’ of the airfoil at time t‘ due to a vertical velocity ~ ( x acting ) since the moment t’ = 0. The pressure p(x,t’) due to a harmonic downwash ~ ( x exp ) ivt’ is then equal to
+
1‘
p(x,t’) = posc(x)e i V f -
dt,’
+ pid(x,t’)
if
t’
.--,
00,
0
where x has to be kept constant during the integration to t i . It then follows for Im Y 0 that
<
(8.4) After integration over x, one has for the total force
0
Substituting (3.4)for KO, and (8.2) for Kid, one obtains the result
s
W
2 z C(K) = i K
k,(s) e--iks ds,
0
where the term n6(s) in k,(s) is to be omitted, since this would reproduce the first term (aerodynamic inertia) of (3.4). The relation (8.6)has first been given by Gamck [99]. Equation (8.5) can be inverted with the result
82
A. I. VAN DE VOOREN
where the hook denotes integration along the real axis in the complex k-plane, indented at the singular point k = 0 in the lower halfplane (Im k < 0). Generalizing a result of Garrick [99], one can show that (8.7) may be replaced by any of the relations m
where K,, = Kos,(0) denotes the steady value of the force. The first Eq. (8.8) has been used by Mazelsky and Drischler [loo] to calculate indicial h n c tions. Similar relations are also valid for the moments. As a general result following from the theory of Laplace transforms, Eqs. (8.6) and (8.7) imply that the behavior of the indicial functions for small values of s is determined by the forces on the oscillating airfoil a t large k. On the other hand, the asymptotic behavior of the indicial functions is determined by the forces acting on a slowly oscillating airfoil. 60 the pressure at a point I t follows from (8.4) that in the limit of k of the airfoil is completely determined by the initial indicial pressure a t the same point. 'This pressure, in turn, is completely determined by the normal velocity at the same point and instant. For very high frequency the pressure is thus proportional to the local vertical velocity. The proportionality constant is such that (7.12) and, hence, the piston theory is again valid. Outside the airfoil, however, the field of flow deviates from that described by piston theory.
-
4. Calculation of Flutter Derivatives for Large Subsonic Mach Numbers from
Indicia1 Functions
A method related to that of VIII,Z, but using Lorentz coordinates and acceleration potential, has been presented by Burger [ l o l l ; it is based originally on an idea of Timman. The pertaining equation is +xx
+
+zz = + T T ,
and a solution is sought such that the prescribed normal acceleration a t the 1) is of the form airfoil (- 1 X
< <
U(n
+Zlz(X)
where U ( T ) is the unit step function.
UNSTEADY AIRFOIL THEORY
83
In the X,T-diagram (Fig. 7), this corresponds to an Evvard problem of a wing with infinite chord, placed in a flow of Mach number 1/5 The solution 5LiM(X,T) can be evaluated similarly to what was done in VIII,2. Accordingly, the solution for the oscillating airfoil with prescribed normal acceleration is given by
Y ( X )= m
=ix
l + i n d ( X , T )e - j X T d T . 0
(8.9)
The advantage of this form is that for large x results can be obtained by asymptotic expansion of the integral, requiring the knowledge of +&X,T) only for small values of T . For Mach numbers near to 1, 1z is large unless the frequency is very small. I t should be added that (8.9) gives only the regular part of !P(X), which vanishes at the leading edge and at the trailing edge. This part can be written in the form
I FIG. 7. Wing of infinite chord in supersonic flow ( M = 12).
1
r
where the Green’s function G(X,t) is found by the procedure described above as a series which is asymptotic in x . As in III,2 suitable differentiation of G ( X , t ) yields the singular part of the solution, while the normal velocity determines a yet unknown factor. By this method numerical values for the flutter derivatives a t high subsonic Mach numbers are being calculated by Eckhaus and Zandbergen [1021.
84
A. I. VAN DE VOOREN
5. Indicia1 Functions for Finite Wings
Calculations of indicial functions for finite wings are complicated. We restrict ourselves here to a short description of the available results. R. T. Jones [lo31 has calculated k,(s) and k,(s) for elliptic wings of W = 3 and 6 in incompressible flow. All other results refer to supersonic flow. Miles [lo41 calculated k,(s) and k,(s) for rectangular wings with tip Mach cones that do not intersect the opposite tip (W V M 2 - 1 < 1). Goodman [105], using Gardner’s method as in VI,1 and considering also rectangular wings for which Ai vM 2 -. 1 < 1, calculated K,(s) as well as the corresponding pressure distribution. This method has also been used by Lomax a.0. [lo61 to calculate generalized indicial functions for deforming rectangular wings in supersonic flow. The triangular wing with supersonic edges has been considered by Miles [107], Strang [108], and a.0. also by Lomax [60]. In the last reference the case of triangular wings with subsonic edges is also treated. For very slender wings the governing equation may be simplified to
which has the same solution as the problem considered in V111,2 provided x is replaced by ct. ACKNOWLEDGMENTS
This review has been written with the partial support of the Netherlands Aircraft Development Board (N.I.V.) and the National Aeronautical Research Institute (NLL). By the aid of the Advisory Group for Aeronautical Research and Development (AGARD), the author was able to discuss personally many subjects with leading scientists. The author wishes to thank Professor Timman, Mr. van Spiegel and Mr. Eckhaus for many valuable discussions during the preparation of this review and for having carefully read the manuscript.
References NACA ARC NLL KTH ARL
National Advisory Committee for Aeronautics, Washington. Aeronautical Research Council, London. = Nationaal Luchtvaartlaboratorium, Amsterdam. = Kungl. Tekniska Hogskolan, Stockholm. = Aeronautical Research Laboratories, Melbourne. = =
1. THEODORSEN, TH., General theory of aerodynamic instability and the mechanism of flutter, N A C A Rept. No. 496 (1935). 2. KUSSNER,H. G., Zusammenfassender Bericht iiber den instationaren Auftrieb von Fliigeln, Lllftfuhrtforschung 13, 410-424 (1936).
UNSTEADY AIRFOIL THEORY
85
3. LIN,C. C., REISSNER, E., and TSIEN,H. S., On two-dimensional non-steady motion of a slender wing in a compressible fluid, J . Math. and Phys. 27, 220-231 (1948). 4. MILES,J. W., Linearization of the equations of non-steady flow in a compressible fluid. J . Math. and Phys. 33, 135-143 (1954). 5. MILES,J. W., Slender body theory for supersonic unsteady flow, J . Aeronaut. Sci. 19, 280-281 (1952). 6. LANDAHL, M., MOLLO-CHRISTENSEN, E. L., and ASHLEY,H., Parametric studies of viscous and nonviscous unsteady flows, Fluid Dynamics Research Group Report No. 55-1, Mass. Inst. Techno]., Cambridge, Mass., 1955. 7. LIGHTHILL, M. J., A new approach to thin aerofoil theory; Aeronautical Quarterly 3, 193-210 (1951). 8. VAN DYKE,M. D., A study of hypersonic small-disturbance theory, N A C A R e p . No. 1194 (1954). 9. KUSSNER.H. G., Allgemeine Tragflachentheorie, Luftfahrtf. 1 7 , 370-378 (1940). 10. FLAX,A. H., Reverse flow and variational theorems for lifting surfaces in nonstationary compressible flow, Cal-42, Cornell Aero. Lab., Buffalo, N.Y.. 1952. 11. HEASLET,M. A., and SPREITER,J. R., Reciprocity relations in aerodynamics, N A C A Tech. Note No. 2700 (1952). 12. TIMMAN, R., Zum Reziprozitatssatz der Tragflachentheorie bei beliebiger instationarer Bewegung, Z. f . Flugwiss. l, 146-149 (1953). 13. HASKIND, M. D., Oscillations of a wing in a subsonic gas flow, Prikl. Mat. i Mekh. 11, 129-146 (1947); transl. as Translation A9-T-22. Air Mat. Command, Dayton, 0. (Brown Univ., Providence, R.I.). 14. REISSNER,E., On the application of Mathieu functions in the theory of subsonic compressible flow past oscillating airfoils, N A C A Tech. Note No. 2363 (1951). 15. TIMMAN, R., Beschouwingen over de luchtkrachten op trillende vliegtuigvleugels, Thesis, Technological University Delft (1946). R., and VAN DE VOOREN,A. I., Theory of the oscillating wing with 16. TIMMAN, aerodynamically balanced control surface in a two-dimensional subsonic compressible flow, N L L Report F. 54 (1949). 17. BILLINGTON, A. E., Harmonic oscillations of an aerofoil in subsonic flow, A R L Report A 65 (1949). 18. Mc LACHLAN, N. W., “Theory and application of Mathieu functions”, Oxford University Press, 1947. 19. K ~ S S N E R H., G., and SCHWARZ, L., Der schwingende Fliigel mit aerodynamisch ausgeglichenem Ruder, Luftfahrtf. 17, 337-354 (1940). 20. VAN DE VOOREN,A. I., Generalization of the Theodorsen function to stable oscillations. J . Aeronaut. Sci. 19, 209-21 1 (1952). 21. K ~ S S N E RH. , G., A review of the two-dimensional problem of unsteady lifting surface theory during the last thirty years, Inst. of F1. Dyn. and App. Math., Un. Maryland, Lecture Series No. 23 (1953). 22. TIMMAN, R., Linearized theory of the oscillating airfoil in compressible subsonic flow, J . Aeronaut. Sci. 21, 230-236 (1954). R.. VAN DE VOOREN,A. I,, and GREIDANUS,J. H., Aerodynamic 23. TIMMAN, coefficients of an oscillating airfoil in two-dimensional subsonic flow, J . Aeronaut. S G ~18, . 797-802 (1951). 24. TIMMAN, R., VAN DE VOOREN,A. I., and GREIDANUS, J. H., J . Aeronaut. Sci. 21, 499-500 (1964). 25. HOFSOMMER, D. J., Systematic representation of aerodynamic coefficients of an oscillating aerofoil in two-dimensional incompressible flow, N L L Report F. 61 (1950). 26. KOSSNER,H. G., A general method for solving problems of the unsteady lifting surface theory in the subsonic range, J . Aeronaut. Sci. 21, 17-27 (1954).
86
A. I. VAN D E VOOREN
27. JONES,W. P., Oscillating wings in compressible subsonic flow, A R C Report No. 14,336 (1951). 28. JONES,W. P., The oscillating aerofoil in subsonic flow, ARC Report No. 15,642 (1953). 29. SCHWARZ, L., Berechnung der Druckverteilung einer harmonisch slch rerformenden Tragflache in ebener Stromung, Luftfahrtf. 17, 379-386 (1940). 30. POSSIO,C., L’azione aerodinamica sul profilo oscillante in un fluido compressibile a velocita iposonora, L’aerotecnica 18, 441-458 (1938). R. A., Possio’s subsonic derivative theory and its application t o flexural31. FRAZER, torsional wing flutter, ARC R . & M . No. 2553 (1941). 32. SCHADE,The numerical solution of Possio’s integral equation for an oscillating aerofoil in a two-dimensional subsonic stream, A . R . C . Report 9506 (1946). 33. DIETZE, F., Die Luftkrafte des harmonisch schwingenden Fliigels im kompressiblen Medium bei Unterschallgeschwindigkeit, Deutsche Versuchsanstalt f . Luftfahrlforschung, Forschungsber. No. 1733 (1943). M. J.. and RABINOWITZ, S., Aerodynamic coefficients for an oscillating 34. TURNER, airfoil with hinged flap, with tables for a Mach number of 0.7, N A C A Tech. Note No. 2213 (1950). 35. FETTIS, H. E., An approximate method for the calculation of nonstationary air forces at subsonic speeds, Wright Air Development Center, Dayton, O., Tech. Rep. 52-56 (1952). 36. SCHWARZ, L., A semi-empirical method for determining unsteady pressure distributions, ARC Report 10,387 (1947). 37. JONES,W. P., Airfoil oscillations at high mean incidence, ARC R . 6 M . No. 2654 (1953). 38. ROTT,N., and GEORGE,M. B. T., An approach to the flutter problem in real fluids, Preprint No. 509. I.A.S., 1955; to be published in J . Aeronaut. Sci. 39. WOODS,L. C., The lift and moment acting on a thick aerofoil in unsteady motion, Phil. Trans. Roy. Soc. London, A , 247, 131-162 (1954). 40. MILES, J. W., On solving subsonic unsteady flow lifting surface problems by separating variables, J . Aeronaut. Sci. 21, 427-428 (1954). 41. SCHADE,TH., Theorie der schwingendea kreisformigen Tragflache auf potentialtheoretischer Grundlage, Luftfahrtf. 17, 387-400 (1940). 42. VAN SPIEGEL,E., Theory of the circular wing in steady incompressible flow, N L L Report F. 189 (1956). 43. WATKINS,CH. E., RUNYAN, H. L., and WOOLSTON, D. S., On the kernel function of the integral equation relating the lift and downwash distributions of oscillating finite wings in subsonic flow, N A C A Tech. Note No. 3131 (1954). 44. JONES,W. P., The calculation of aerodynamic derivative coefficients for wings of any plan form in non-uniform motion, ARC R . & M . No. 2470 (1952). 45. LEHRIAN,D. E.. Calculation of the damping for rolling oscillations of a swept wing, ARC Current Paper No. 51 (1951). 46. DENGLER,M. A.. and GOLAND,M., The subsonic calculation of circulatory spanwise loadings for oscillating airfoils by lifting-line techniques, J . Aeronaut. Sci. 19, 751-759 (1952). M. A., Development of charts for downwash coefficients of oscillating 47. DENGLER, wings of finite span and arbitrary plan form, J . Aeronaut. Sci. 21, 809-824 (1954). W., On the theory of oscillating airfoils of finite span in subsonic flow, 48. ECKHAUS, N L L Report F. 153 (1954). 49. REISSNER.E., Effect of finite span on the airload distributions for oscillating wings, Part I, N A C A Tech. Note No. 1194 (1947). 50. REISSNER,E.. and STEVENS,J. E., Effect of finite span on the airload distributions for oscillating wings, Part 11, N A C A Tech. Note 1195 (1947).
UNSTEADY AIRFOIL THEORY
87
51. REISSNER,E., On the theory of oscillating airfoils of finite span in subsonic compressible flow, N A C A Rept. No. 1002 (1950). 52. GARRICK,I. E., Some research on high-speed flutter. 3?d Anglo-American Aeronautical Conference, 419-446 (1951) published by the Royal Aeronautical Society, London. 53. MERBT, H.. and LANDAHL, M., Aerodynamic forces on oscillating low aspect ratio wings in compressible flow, K T H Aero Tech. Note No. 30 (1953). 54. LAWRENCE, H. R., and GERBER,E. H., The aerodynamic forces on low aspect ratio wings oscillating in an incompressible flow, J . Aeronaut. Sce. 19, 769-781 (1952). 55. LAIDLAW,W. R., Theoretical and experimental pressure distributions on oscillating low aspect ratio wings, Thesis, Mass. Inst. of Techn., Cambridge, Mass., 1954. 56. VAN DE VOOREN, A. I., and ECKHAUS, W., Strip theory for oscillating swept wings in incompressible flow, N L L Report F. 146 (1954). W., Strip theory for oscillating swept wings in compressible subsonic 57. ECKHAUS, flow, N L L Report F. 159 (1955). I. E., and RUBINOW, S. I., Flutter and oscillating air-force calculations 58. GARRICK, for an airfoil in a two-dimensional supersonic flow, N A C A Rept. No. 846 (1946). 59. GARRICK, I. E., and RUBINOW,S. I., Theoretical study of air forces on an oscillating or steady thin wing in a supersonic main stream. N A C A Rept. No. 872 (1947). H., HEASLET,M. A., FULLER,F. B., and SLUDER,L., Two- and three60. LOMAX, dimensional unsteady lift problems in high-speed flight, N A C A Rept. No. 1077 (1952). R., and HILBERT,D., “Methoden der mathematischen Physik’, Vol. 11, 61. COURANT, p. 311-317, Berlin, 1937. 62. TEMPLE,G., and JAHN,H. A,, Flutter at supersonic speeds. Derivative coefficients for a thin airfoil at zero incidence, A R C R . G. M . No. 2140 (1945). 63. JONES,W. P., Supersonic theory for oscillating wings of any plan form, A R C R . G. M . No. 2655 (1948). S., uber die Luftkrafte die auf einen harmonisch schwingenden 64. VON BORBELY, zweidimensionalen Fliigel bei uberschallgeschwindigkeit wirken. 2. f . angew. Math. u. Mech. 22, 190-205 (1942). 65. GUNN,J. C., Linearized supersonic airfoil theory, Phil. Trans. Roy. SOC.London. A , 240, 327-373 (1947). 66. STEWARTSON, K., On the linearized potential theory of unsteady supersonic motion, Quart. J. Mech. Appl. Math. 3, 182-199 (1950). 67. MILES, J. W., Transform and variational methods in supersonic aerodynamics, J . Aeronaut. Sci. 16, 252-253 (1949). 68. TEMPLE,G., Unsteady motion, in “Modern Developments in Fluid Dynamics, High Speed Flow”, Vol. I (L. Howarth, ed.), pp. 325-374, Oxford University Press, 1953. 69. MILES, J. W., The oscillating rectangular airfoil at supersonic speeds, Quart. A p p l . Math. 9, 47-66 (1951). 70. MILES, J. W., A general solution for the rectangular airfoil in supersonic flow, Quart. A p p l . Math. 11, 1-8 (1953). 71. GOODMAN, TH. R., The quarter infinite wing oscillating a t supersonic speeds, Quart. Appl. Math. 10, 189-192 (1952). 72. ROTT,N., On the unsteady motion of a thin rectangular airfoil in supersonic flow, J . Aeronaut. Scz. 18, 775-776 (1951). 73. VAN DE VOOREN.A. I., to be published.
88
A. I. VAN D E VOOREN
74. HEASLET,M., and LOMAX,H.,
75. 76. 77. 78. 79. 80.
81.
Supersonic and transonic small perturbation theory in High Speed Aerodynamics and Jet Propulsion, in “General Theory of High Speed Aerodynamics”, Vol. VI (W. R. Sears, ed.) Princeton University Press, 1954. LI, T. Y., and STEWART,H. J., On an integral equation in the supersonic oscillating wing theory, J . Aeronaut. Sci. 20, 724-726 (1953). MILES, J. W., On the low aspect ratio oscillating rectangular wing in supersonic flow, Aeronaut. Quart. 4, 231-244 (1953). MILES, J. W., A note on subsonic edges in unsteady supersonic flow, Quart. Appl. Math. 11, 363-367 (1953). ROBINSON, A., On some problems of unsteady supersonic aerofoil theory, Aeronaut. Coll. of Cranfield Report No. 16 (1948). HOBSON, E. W., ”The theory of spherical and ellipsoidal harmonics”, Cambridge University Press, 1931. GERMAIN, P., and BADER,R., Quelques remarques sur les mouvements vibratoires d’une aile en regime supersonique, La Recherche Aeronautique No. 11, Sept.-Oct. 1949, 3-14. PINES, S., DUGUNDJI,J., and NEURINGER, J., Aerodynamic flutter derivatives for a flexible wing with supersonic and subsonic edges, J . Aeronaut. Sci. 22,
693-700 (1955). G., Hsu, P., and Voss, H. M., Application of numerical integration 82. ZARTARIAN,
83.
84.
85.
86. 87. 88. 89. 90. 91. 92. 93. 94.
techniques t o the low aspect ratio flutter, Aeroel. and Struct. Res. Lab., Techn. Rept. 52-3, Mass. Inst. of Techn., Cambridge, Mass (1954). WATKINS,CH. E., and BERMAN,J. H., On the kernel function of the integral equation relating lift and downwash distributions of oscillating wings in supersonic flow, N A C A Tech. Note No. 3438 (1955). NELSON,H. C., RAINEY,R. A., and WATKINS,CH. E., Lift and moment coefficients expanded to the seventh power of frequency for oscillating rectangular wings in supersonic flow and applied t o a specific flutter problem, N A C A Tech. Note No. 3076 (1954). WATKINS, CH. E., and BERMAN, J. H., Velocity potential and air forces associated with a triangular wing in supersonic flow, with subsonic leading edges, and deforming harmonically according to a general quadratic equation, N A C A Tech. Note No. 3009 (1953). BUSEMANN,A., Aerodynamischer Auftrieb bei Uberschallgeschwindigkeit. Luftfahrtf. 12. 210-220 (1935). JONES,W. P., The influence of thicknesslchord ratio on supersonic derivatives for oscillating aerofoils, ARC Report 10,871 (1947). LIGHTHILL,M. J., Oscillating airfoils a t high Mach number. J . Aeronaut. Sci. 20, 402-406 (1953). JONES,W. P., and SKAN,S. W., Aerodynamic forces on biconvex aerofoils oscillating in a supersonic airstream, ARC R . G. M . No. 2749 (1953). VAN DYKE,M. D., Supersonic flow past oscillating airfoils including nonlinear thickness effects, N A C A Rept. No. 1183 (1954). VANDYKE,M. D., A study of second-order supersonic flow theory, N A C A Rept. No. 1081 (1952). CARRIER,G. F., The oscillating wedge in a supersonic stream, J . Aeronaut. Sci. 16, 150-152 (1949). VAN DYKE,M. D., On supersonic flow past an oscillating wedge, Quart. Appl. Math. 11, 360-363 (1933). GOLDSWORTHY, F. A,, Two-dimensional rotational flow at high Mach number past thin aerofoils. Quart. J. Mech. Appl. Math. 5 , 54-63 (1952).
UNSTEADY AIRFOIL THEORY
89
95. LANDAHL, M. T., Unsteady flow around thin wingsat highMach numbers, Aeroel. and Struct. Res. Lab., Techn. Rep. 55-3, Mass. Inst. of Techn., Cambridge, Mass., 1955. 96. KUSSNER,H. G., Das zweidimensionale Problem der beliebig bewegten Tragflache unter Berucksichtigung von Partialbewegungen der Fliissigkeit, Luftfahrtf. 17, 355-363 (1940). 97. ,JONES, W. P., Aerodynamic forces on wings in non-uniform motion, A R C R . 6 M . No. 2117 (1945). 98. LOMAX,H., Lift developed on unrestrained rectangular wings entering gusts a t subsonic and supersonic speeds, N A C A Tech. Note No. 2925 (1953). 99. GARRICK, I. E., On some reciprocal relations in the theory of non-stationary flow, N A C A Rept. No. 629 (1938). 100. MAZELSKY, B., and DRISCHLER, J. A., Numerical determination of indicial lift and moment functions for a two-dimensional sinking and pitching airfoil at Mach numbers 0.5 and 0.6, N A C A Tech. Note No. 2739 (1952). 101. BURGER,A. P., On the asymptotic solution of wave propagation and oscillation problems, N L L Report F. 157 (1955). 102. ECKHAUS, W., and ZANDBERGEN, P. J., t o be published. 103. JONES,R. T., The unsteady lift of a wing of finite aspect ratio, N A C A Rept. No. 681 (1940). 104. MILES,J. W., Transient loading of supersonic rectangular airfoils, J . Aeronaut. Sci. 17, 647-652 (1950). 105. GOODMAN, TH. R., Aerodynamics of a supersonic rectangular wing striking a sharp-edged gust, J . Aeronaut. Sci. 18, 519-526 (1951). 106. LOMAX,H., FULLER,F. B., and SLUDER,L., Generalized indicial forces on deforming rectangular wings in supersonic flight, N A C A Tech. Note No. 3286 (1954). 107. MILES, J. W., Transient loading of wide delta airfoils a t supersonic speeds. J . Aeronaut. Sci. 18, 543-554 (1951). 108. STRANG,W. J., Transient lift of three-dimensional purely supersonic wings, Proc. Roy. SOC.London, A, 202, 5 4 8 0 (1950).
This Page Intentionally Left Blank
The Theory of Distributions BY CHARLES SALTZER Case Institute of Technology, Cleveland, Ohio, and General Electric C o m p a n y , Electronics Laboratory, Syracuse, N e w York
Page I. 11. 111. IV. V. VI. VII. VIII. IX. X.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . The Theory of Distributions. . . . . . . . . . . . . . . . . . . . . The Singularity Functions and the Finite Part of an Integral . . . . A Heuristic Approach . . . . . . . . . . . . . . . . . . . . . . . Fourier Series and the Poisson Transformation . . . . . . . . . . . Ordinary Differential Equations . . . . . . . . . . . . . . . . . . Applications to Fourier Transforms . . . . . . . . . . . . . . . . Fourier Transforms of Distributions . . . . . . . . . . . . . . . . Generalized Harmonic Analysis and Stochastic Processes . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . .
91 92 $5 98
99 102 104 104 107
109 110
I. INTRODUCTION One of the central problems in analysis and its applications is that the processes used in the solution of problems frequently lead to nonsensical results, and a variety of techniques have been developed to deal with this situation. For example, in Fourier series and transform theory various summability procedures are needed, and in problems leading to integral representations devices such as the Cauchy principal part, the Hadamard finite part of an integral, and the Stieltjes integral must be used. Another related problem has been the very useful - and quite embarrassing mathematically - family of singularity functions, such as the Dirac delta function and its derivatives, which occur not only in analytical calculation but also as physical concepts, e.g., as inputs to servomechanisms. The theory of distributions created by Laurent Schwartz [I], [2], [3] which has been described by Temple [4] as “one of the great events in contemporary mathematics” attacks all of these problems a t once by generalizing the notion of a function, a derivative and an integral; and, in addition to providing a unified and rigorous account of the above concepts, also gives a simple and direct calculus for the application of the method. 91
92
CHARLES SALTZER
The term “distribution” was introduced as a generalization of a mass density which could be represented by a function. Such a distribution on the x-axis could be, for example, a unit mass a t the origin and zero density a t all other points of the line. The total mass of any segment including the origin is one; and, the total mass of any segment not containing the origin is zero. If this mass distribution is multiplied by a continuous density p ( x ) then the total mass of any segment containing the origin is p(O), and the total mass of any segment not containing the origin is zero. This example describes the &function. In the following section the theory of distributions will be developed by the very general method of Mikusinski [rj, 61 and the relation to the method of Schwartz will be indicated. The succeeding sections will deal with applications to the Dirac delta function and its derivatives, Fourier series, ordinary and partial differential equations, and generalized harmonic analysis. A detailed account of the applications can be found in [3, 7, 8, 10, la].
11. THE THEORYOF DISTRIBUTIONS The generalized functions will be defined as the closure of the space of ordinary functions in a way which is quite analogous to the Cantor definition of the real numbers as regular* sequences of rational numbers. I n Mikusinski’s method three sets F , @ , and C together with a composition subject to the following restrictions are considered. (i) The composition of an element f of F and an element 4 of @, denoted by ( f , I$), is an element of C and is defined for all f and I$ in F and Q5 respectively. (ii) @ is total; i.e., if ( f , 4) = (g, 4) for all 4 in @ then f = g. (iii) The set C is a convergence space ; for certain sequences cl, c,, . . . of C, the lim c, is defined in C, and if c, = co for n = 1, 2,. . then lim c, = co. We form the closure F of F in exactly the same way as we define the real numbers by regular sequences of rational numbers. A convergence is introduced in F by defining a sequence f l , f,,. . . of elements of F as weakly convergent to f in F if lim (fn,4)= (f,I$) for all 4 in @. We first redefine F as the set of all sequences which are “weakly” convergent to the elements of the original F. Two sequences are defined as equivalent if they converge “weakly” to the same limit. The elements of F consist of the equivalence classes of sequences. If f is in the original F , the sequence f , f , . . . converges “weakly” to f because of the convergence condition on C. Hence there is a
.
*
A sequence is regular if it satisfies the Cauchy convergence criterion.
THE THEORY OF DISTRIBUTIONS
93
one-to-one correspondence between the elements of the original F and the redefined F. We now define F a s the set of all “weakly” regular sequences* of elements of the original F regarding two sequences fl, f,, . . . and g, g,, .. as equivalent if the sequence ( f i - g,), (f, - gJ,. . . converges “weakly” to zero. The elements of F are denoted by A sequence of F which has no limit in the original F may be described as a “weak” point of accumulation of the redefined F. An element in F can be represented by any of the sequences which define it. I f j i s an element of F a n d if f t , f,, . . . is a “weakly” regular sequence of Fwhich defines we say that fis the weak limit of this sequence, and we write?= 1i”m f,.The composition for F and @ is defined by
.
7
7
(Lb) = lim (fn,4).
(2.1)
This new composition again satisfies conaitions (i) and (ii) above. Our procedure generalizes not only the space F but also the composition. We take 1. F as the set of all real functions f ( x ) defined for all real x and summable in every finite interval either in the sense of Riemann or Lebesgue, 2. @ as the set of all real functions $ ( x ) which vanish outside of some finite interval (n, b) and which are infinitely differentiable: i.e., have derivatives of all orders, 3. C as the real numbers, and 4. (f,4) = s f ( x ) + ( x ) d x where the limits of integration are to be understood as - 00 to 00 unless otherwise specified. An alternative choice for @ is the set of functions [13] called testing functions {+c,
where n is any positive integer, a &(x)
for
=0
d(x)}l’”
< c < x < d < b, and x< c
or
1
1
x>d,
(2.2) +c,dh)
= exp [.-.+-d]-
The concept of continuity for weak functions is given in terms of the null sequences in @ defined by Laurent Schwartz. A sequence @,,Q2,. . . of elements of @ is a null sequence provided 4:’ approaches zero uniformly as v goes to infinity for each non-negative integer n where 4:) denotes the nth derivative of $,,. A weak function 7 i s continuous means lim (f,+,,) = 0 for all null sequences in @. The space of continuous weak functions is precisely the space of distributions of Schwartz in the sense of the next paragraph.
*
i.e. sequences fn for which
(I,,,+)
is a regular sequence.
94
CHARLES SALTZER
The procedure of Schwartz is based on the concept of the linear functional. A linear functional T associates a number T(+) with functions and has the property
+
T(4,
+ b+Z)
=a
wl)
+ bT(+z).
A distribution on an open interval is defined as a linear functional which is a continuous linear functional on the set of functions @ for any closed subinterval of a given open interval (a,b). If the linear functional T can be represented by an integral (2.3)
the symbol f(+) or f is also used to denote the distribution T . Thus the relation of continuous weak functions to distributions can also be stated in the following way. Every distribution can be represented by the composition of a continuous weak function with elements of @ and conversely. The proof is given by Temple [4]. I t should be noted that a weak function cannot in general be identified with a point function but must be considered on an interval. The delta function which will be discussed in the next section is an example of such a weak function. The restriction to continuous weak functions simplifies limiting processes since we always have lim K,+)= (limfn,+).
(2.4)
One of the most important results of the theory concerns the weak derivatives of a weak function which is defined formally by
7
7
Because of the properties of is in @ and hence the composition is defined for all 4. If t i s a point function f ( x ) the above definition reduces to the usual definition since @,+I
+
and the limit of the right hand side as h 0 exists for all and is To show that? is in F, we now construct a sequence of elements of F which If f = lim f,, where /,(n = 1, 2 , . . .) are elements converges weakly to f’. of F then the functions -+
-
(/,+I).
95
THE THEORY OF DISTRIBUTIONS
are in F and
which by definition is (f’,$). Since Vmm4)
(-794’)
-
1(/mn,+)
- (- /*A’) -
[~n,+’)
- &+‘)I
it follows that a sequence (Ii,,,)k(n),+) can be chosen whose limit is - (f,+’) for all 4. Hence f‘ = limfi(n)k(n) and therefore is a weak function. In addition we have the result that the weak derivative of a continuous weak function is continuous since if . . is a null sequence lim(f’,h) = - lim (&’) = 0. It follows immediately that a continuous weak function has continuous weak derivatives of all orders, which are themselves continuous weak functions. If the summability used in the construction of weak functions has the property that for all f in F , lim (f,+,,) = 0 for every null sequence .. then the elements of F are continuous weak functions. This is the case for example, if the elements of F are square-integrable in the sense of Lebesgue. Hence the weak derivatives of the elements of F are also continuous weak functions, and it can be shown [13] that the class of continuous weak functions consists of the elements of F and their weak derivatives of all orders. Conversely any given functional which is continuous in the above sense can be represented by the composition of a fixed continuous weak function with elements of 4, and this remark is in effect a generalization of the representation theorem for linear functionals of F. Riesz. The theorem that every continuous weak function or distribution is locally a derivative was proved by J. R. Ravetz [16]. The relaxation of the restrictions on the interchange of limiting processes can be seen in the following theorem of Schwartz: If a sequence of weak functions converges weakly to a limit then the sequence of derivatives converges weakly to the weak derivative of the limit. The proof is nearly trivial. If is the kth element of the sequence and is the limit, then
- -
El, 4) = - &4’)
7
-
-
and lim which completes the proof.
7
-
-
(f2,4) = - lim (fnt 4’) = - (/, 4’) = (f’, 41,
111. THE SINGULARITY FUNCTIONS AND THE FINITE PARTOF Let U ( x ) be the Heaviside unit function U ( x ) = 0,
for
x<
U ( x ) = 1,
for
x
o
> 0.
AN
INTEGRAL
96
CHARLES SALTZER
Then U ( x ) is in F , and consequently in and thus has weak derivatives of all orders which, as we shall show, are the Dirac function and its derivatives. Let d(x) denote theweak derivativeof U ( x ) .Then jd(x), =--[U(x),+‘(x)] by the definition of the weak derivative, and
I).(+
W
0
+(%)I
The result for the weak derivatives of d(x), namely, [ 6 ’ ( x ) , = (-l)‘P(O), is immediate. The Cauchy principal value defines a continuous linear functional,
The associated weak function will be symbolically denoted by l / x . Since by a straight-forward calculation,
(3.4)
PV
s
+’@) log 1x1 a x
=
- PV
JF a x ,
the left hand side also defines a continuous linear functional. If we represent the associated weak function by log 1x1 we have
and by the definition of the derivative we may write
+
We may define the generalized function log x by log x = log 1x1 in U (- x ) , and by our previous results we obtain the Dirac relation (log x)’ = ( l / x ) - in d(x). It can be readily proved for these weak functions that x ( l / x )= 1. These considerations were extended to the Hadamard finite part of an integral by Temple [14]. Consider two functions f ( x ) and g ( x ) where g ( x ) and its first fi derivatives and f ( x ) are continuous on the closed interval (0, c). Also, let the first fi derivatives of f ( x ) be continuous but not integrable on the interval 0 < x c. We define l ( x ) by
<
(3.7)
Z(x) = g(x)fP-I(x)
- g’(x)fP-2(x)
+ . . . + (-
I)#-lgP-1 ( x ) f ( x ) ,
97
THE THEORY OF DISTRIBUTIONS
and denote the Hadamard finite part of an integral by H finite part of the following integral is defined by
s
.
The Hadamard
The motive for this definition is the fact that the identity obtained by integration by parts,
(3.9) E
gives a representation in which the integral has a limit as E 0. For simplicity, although this is not essential, we shall consider g ( x ) to be an element of @ associated with the interval (a, b) where a < 0 and b = c. Now f ( x ) is in F and is a distribution. We represent f ( x ) by -+
where U has been defined in (3.1). Following Temple we define m(x) in the same way in which we defined Z(x) with the exception that the derivatives of f ( x ) will be computed in the sense of distributions from the representation (3.10), and (3.9) becomes c
C
C
where all derivatives are now interpreted in the sense of distributions. Since differentiation and passage to the limit can be interchanged and since U ( x - E ) f ( x ) can be identified with a point function in a neighborhood of x = c that does not contain the origin, the derivatives of f can be evaluated a t these points. By formal differentiation in the weak sense
+
+
(3.12) f i x ) = lim [ U ( x - E ) ~ ( X ) rU’(x - & ) f ’ - l ( ~ ) . . . &+O
and hence
(3.13)
P(0)= 0,
+ U‘(x - e ) f ( x ) ] ,
98
CHARLES SALTZER
and f ( c ) coincides with the value of the usual derivative of f for r Thus m(0)= 0 and m(c) = Z(c) and (3.11) becomes
= 0, 1 , .
.. p .
c
But by (3.8) and (3.9),
H
s
g ( 4 f W a x = 44
0
+ (-
1)fi
s
g w
0
f ( 4a x
where the integrands on the right in both of the above equations are integrable point functions. Hence
1
H
s c
C
g ( x ) f Wa x =
0
0
g(4fW
where the integral on the right is to be interpreted in the sense of distributions.
APPROACH IV. A HEURISTIC One surprising result obtained by L. Schwartz is equivalent to the statement that any continuous weak function is a weak derivative of a function in the ordinary sense, and the latter class of functions can be restricted to absolutely continuous functions. The theory could then be developed formally as the set of continuous linear functionals consisting of continuous functions and their derivatives defined by integration by parts as in Eq. (2.6). In the light of this result weak functions can be introduced as in the following example. Let U ( x ) be the Heaviside unit function* and let & x ) be an element of @. We write formally the rule for integration by parts,
U(x)+’(x)a x
=-
U’(x)+(x)dx
and say that U’(x)is the weak derivative of U ( x ) ,the integral on the right being defined by the integral on the left. Thus, as in Section 3, m
*
U ( x ) is of course the derivative of ) ( x
+ Ix i ) .
THE THEORY OF DISTRIBUTIONS
99
This gives us the delta function again. The procedure is quite general and enables us to introduce weak functions heuristically by an operational calculus at an early stage.
V. FOURIER SERIESAND
THE
POISSON TRANSFORMATION
Consider the point function,
f(4=
f
2 cos 2nwx
-
(27242
dw.
1
If we regard this as a weak function, f i x ) , and differentiate twice in the weak sense we get /“(x) =
(5.2)
If we add the point function
i
2 cos ~
Z W Xdw.
2 cos 2 n w x dx to f ” ( x ) we get the new weak
function
-
g(x) =
(5.3)
i
~ C O S ~ ~dw. W X
0
This integral, which is not convergent in the usual sense is to be interpreted by its relation to elements of @ in the following way: m
If the point function gn is defined by
(5.5)
n
100
CHARLES SALTZER
then we seen that g(x) = Km g,,(x). with g,, is
Tn(+)=
(5.6)
J
But the distribution T,, associated
sin 2nnx nx
+(X)d%
which is a form of the classical Dirichlet integral, and we know that lim T*(+)= W). n+m
Hence by the definition of the delta function we have
(5.7)
We now consider the representation of a periodic distri-ution by its Fourier series. We first define the support of a function in di to be the closure of the set of points on which + ( x ) # 0. We then define a generalized functionxx) as zero in an open set if @, 4) = 0 for every in @ whose support is contained in the given open set. We finally define the support of a distribution and its associated weak function as the complement of the largest open set in whichxx) = 0. Although the product of two weak functions is not in general a weak function it is easy to show that the convolution of two weak functions, of which at least one has a bounded support, is again a weak function. The convolution of and 7 will be denoted by
+
+
7
In addition to this result we will also need the Poisson transformation. Consider the point function m
m
(5.9)
’
(1
=2ni - - x
1
+2nni
( n = O , & l , ...),
where the appropriate branch of the logarithmic function is to be selected. Since g ( x ) has the period 1 by definition and the branch points are at the
THE THEORY OF DISTRIBUTIONS
solutions of way :
e&jx
=1, we see that the branches must be chosen in the following
(1 1
g(x)=2ni - - - x (5.10)
101
*
=2 4 ;
-
=2 4 ;
-
.)+ .)
O<x
n],n < x <
72
+ 1,
- n < x < - n + 1.
-
Hence g(x)/2niis a function with period 1 whose value in the interval (0,1) is (1/2 - x ) , and the difference between the right and left hand limits at ( x = 0,& 1,. ..) is unity. If this function is regarded as a weak function then its weak derivative7 is m
-
f ( x ) = g’(x)j2ni = - 1
(5.11)
+ 2 d(x - j ) , -m
while from the series representation for g ( x ) m
co
If this is applied to a function
1
(5.14)
C# in Qi we obtain the Poisson identity
e2nikx
m
b ( x ) dx =
2+(i), -m
since the order of all the limit processes involved can be inverted for weak functions. Let be a weak function whose support is contained in the interval, (0,I), and let a h , the Fourier coefficients of7, be defined by a k = e-2nikx), (k = 0, f 1 , . ..). If we take the convolution of both sides of equation (5.13) with f i x ) we have,
3%)
m
(5.15)
m
2f * d ( x - j ) 2 f * =
-m
-m
e2nikx.
102
Since
CHARLES SALTZER
?* d(x - i ) =
s
f ( y ) 6 ( x - y - j ) d y , the left hand side becomes the
periodic function obtained by translating f i x ) which we will denote by & ( x ) , and from the right hand side we get (5.16)
Hence we have the result
(5.17)
i.e. the weak function, extended periodically, is represented by its Fourier Series. This also shows that Fourier series whose coefficients are polynomials in k, have meaning as weak functions. We also could have developed the theory directly by defining @ as the set of infinitely differentiable periodic functions of period one.
VI. ORDINARYDIFFERENTIAL EQUATIONS
The concept of a weak function as the solution of a differential equation presents an immediate generalization of the theory of differential equations. The power and simplicity of the use of distributions in the theory will be illustrated by a simple example, the application to linear equations with constant coefficients. We shall need a preliminary result. Let T ( x ) be an arbitrary weak function. Then, since 6 ( x - y) has a bounded support, namely the point y , the convolution of f a n d 6 is again a weak function, and indeed, it is readily shown by composition with elements of @, that f ( x ) * 6 ( x - y) = f ( y ) by setting 6 = x - y, 7 = y in the following integral:
THE THEORY OF DISTRIBUTIONS
103
Similarly
In general, if D is the derivative operator then
Consequently, if f a n d in this case,
2 are weak functions, by the easily proved associativity D'(T*
2) = D'd *
-
* 2)
* f) * 2
=
(D'S
=
(07)* 2 =7 * (D'X).
Thus if we wish to solve the differential equation
7.)
where P ( D ) is a polynomial in D , and is a given weak function, we consider the weak function f i x ) which is the solution of the differential equation
We show by substitution that the solution of the given equation is
since by the above relations and the linearity of P ( D ) ,
(6.8)
P(D)
(7.X) = [P(D)T]* -h = 6 * -h = -h.
104
CHARLES SALTZER
VII. APPLICATIONS TO FOURIER TRANSFORMS The Fourier integral of a weak function is defined as [ f ( x ) , e--8m'xY]. Since the kernel of the transformation is not in @, an extension of the theory is required which will be given in the next section. Here we show how some of the preceding results can be used in the treatment of Fourier transforms. The transform of the Dirac delta function is readily computed:
The Fourier transform of 1, which is also not defined in the classical theory, w of can be represented as the limit for n' --+
sin 2nnx Z X
-n
By the result in Section 4, we know that the limit of this sequence of functions is the weak function 6 ( x ) . If we differentiate the relation
a(%) = (7.3) n times we obtain the result (- 2x4.
(7-4)
s
s y e--2m'xYd
yne--2n+dy
= 6(4(%),
and by the definition of the nfhderivative of the Dirac delta function we have
These preliminary results enable us to extend the range of the theory of Fourier integrals so that, for example, in network theory, the problem of convergence of the Fourier transforms no longer arises, and the generalized Fourier transform allows us to dispense with the Laplace transform in favor of the physically more meaningful Fourier transform. VIII. FOURIER TRANSFORMS OF DISTRIBUTIONS As was indicated in Section 7, the class of functions chosen for @is inadequate for Fourier transform theory. We take as the space @ the class
105
THE THEORY OF DISTRIBUTIONS
of functions of rapid decrease, i.e. those functions which are infinitely differentiable and have the property
for all non-negative integers p and k. We introduce a topology by defining a null sequence (q$,) in @ as a sequence for which
uniformly in x for all non-negative integers p and k. If we take C as the real numbers and the composition as integration on the real axis we obtain the dual space of distributionsP called the space of temperate distributions or distributions of slow increase. Thus, by the new definition of @, a polynomial and, hence, a continuous function bounded by a polynomial are distributions. The classical Fourier transform (8.3)
maps @ on itself since by the classical theory the differentiability properties of 4 imply the required asymptotic properties of at infinity, and the asymptotic properties of 4 imply the required differentiability properties of 3+. If we denote the inverse transform by
w
s
3-14 = 4(y)eznixydy, then the inverse transform has the same properties and,
Also, if (A} is a null sequence in @ then 34,, -+ 0 and 3-Vn -+ 0. Finally we have the Parseval relation. If y51 = and i,hZ =
wl
We note that n
w2then
106
CHARLES SALTZEH
Hence the Parseval relation can be written
or also, from an alternative form of the Parseval relation,
To define the Fourier transform of a distribution we must define it in terms of a composition with elements of @ in such a way that it reduces to the classical definition when applied to functions. If is in @, and we define 42 by (8.10)
(42,
$1
=(418
3 4)
for all 4 in @, we see that the functional on the left is linear and is defined uniquely. But by the Parseval relation (A,34) = (34', 4), and hence $a = 341. A similar result holds for 3-'. Accordingly we define the Fourier and inverse Fourier transforms of a temperate distribution as the distributions 37 and 3-l7 for which
7
(8.11)
for all 4 in @. The linearity is obvious, but we have yet to show that the linear functional defined above is continuous; i.e., if {$,,} is a null sequence then (37,+), -+0 and (3-lT4") 0. This is a consequence of the fact that 4" - -0 implies 34,, - 0 and 3-'4,, - 0 uniformly, hence 34")-0, (f, 3-l+,,).+ 0, because of the properties of @ and the fact t h a t j i s a temperate distribution. Thus, if? is a temperate distribution, so are 37 and 3-lf since these are continuous linear functionals on @. We can also show that 33-1 = 3-'3 is the identity transformation. We have
-
(8.12)
(33-'7,4)= (3-'-/;34)= E3-'34)
=
K+),
and (8.13)
(3-'374 = (3E3-'4)= E33-'4) = (j;+) - -
and hence 33-l7= 3-'3f = f. These developments complete the considerations of Section 7 and provide a basis for studying generalized harmonic analysis and its application t o stochastic processes.
THE THEORY OF DISTRIBUTIONS
Ix. GENERALIZED HARMONIC ANALYSISAND
107
STOCHASTIC PROCESSES
The importance of generalized harmonic analysis in the stochastic processes associated with such subjects as turbulent flow, noise, Wiener filtering and prediction has created another area in which the simplification offered by the theory of distributions is useful to the engineer and physicist. If f ( t ) is a suitable time function, say a current or a velocity, the autocorrelation function is defined as T
(9.1)
where f* is the conjugate of f . The power-spectral density is defined by
F(A) =
s
p(h) e-2ziah dh = 3
P I
and the energy spectrum by
(9.3)
S(A) =
i
F
(A)dil.
-a
The mean-squared energy is p ( 0 ) since by the above definition T
(9.4)
In addition,
(9.5)
p(h) = 3-13p =
s
J(il) e2*iah dh,
and hence (9.6)
p(0) = 1 F @ )dA = S(00).
If p(h) can be represented as a Fourier transform, the classical theory suffices. But this class of functions is too restricted for practical applications. For example, if f ( t ) is a constant or a periodic function, the classical Fourier transform of the autocorrelation function does not exist since in the first case it is a constant, and in the second case it is a periodic function.
108
CHARLES SALTZER
The method of generalized harmonic analysis proceeds by considering the truncated function (9.7)
The autocorrelation function of the truncated function is
'S
P T ( k ) = 2T
fT(t
+ k)
fT*(t)
at,
the power-spectral density is (9.9)
FT(2) = 3 P T ,
and the energy spectrum becomes
(9.10)
The basic result, due to Wiener, is that for a large class of functions including most of the functions which occur in applications, the energy spectrum .",(A) of the truncated function approaches a limit S(A) as T bo such that -+
(9.11)
If we regard the autocorrelation functions and the spectral densities as distributions rather than as functions, the necessity of working with the truncated functions no longer exists. If for example (9.12)
f ( t ) = c,
then
(9.13) J
-cn
where U(A)is the Heaviside unit function.
THE THEORY OF DISTRIBUTIONS
109
If f ( t ) is the periodic function f(t) = c eZn*fi,
(9.14)
then T
and
s 1
F(1) = 3p(h) = =:
=
lc12 eZnitfi rznifi dh
!c12 e-Zni(l-/)h
lC126(1
dh
- f).
Hence (9.16)
S(1) = lC12U(A - f).
We can formulate simple theorems which assure the existence of S(A). For example, a necessary and sufficient condition for S(1) to be a pointfunction is that p(h) shall have a Fourier transform which is the derivative of a temperate distribution identifiable with a point-function. Engineers have already used operational rules [12] relating to &functions in generalized harmonic analysis in a purely formal way. The theory of distributions not only offers a justification for these rules but also offers guidance in their formulation and use.
X. CONCLUSION The simplicity and theoretical unification provided by the theory of weak functions and distributions has been indicated here in a brief sketch. Some of the applications which have been omitted are to integral equations, difference equations, partial differential equations [15] and the theory of functions of a complex variable. For example, the Green’s function for the two-dimensional Laplace equation may be defined as the solution of (10.1)
uxx
+
u y y
= - S ( x - x‘) S ( y - y’),
and the extension to functions of more variables is quite direct. Another example is the one-dimensional wave equation for a finite string. For the
110
CHARLES SALTZER
case in which the initial velocity is zero and the initial displacement is triangular, the formal solution by Fourier series leads to a function whose second derivatives do not exist in the desired region. However, if we consider the solution as a weak function, and differentiation as weak differentiation, the required derivatives exist and satisfy the wave equation. In general, one rule for deciding whether distributions should be considered in the treatment of a problem is the occurrence of a formal analytical procedure leading to meaningless results or the introduction of unusual forms of analytical processes. On the purely theoretical side there are, for example, applications to the theory of matrix functions [ll] and the theory of harmonic integrals [9]. We must also not overlook the new insight provided into physical problems since the distributions (or weak functions) are closer to physical concepts than point functions.
References 1. SCHWARTZ, L., “Thdorie des distributions”, Vol. I, Hermann, Paris, 1950. 2. SCHWARTZ, L., “ThBorie des distributions”, Vol. 2, Hermann, Paris. 1951. 3. SCHWARTZ, L., Gndralisation de la notion de fonction et de ddrivation, thdorie des distributions, Annales des Telecommunications 8, 135- 140 (1948). 4. TEMPLE,G., Theories and applications of generalized functions, J . London Math. SOC.28, 134-148 (1953). 5. MIKUSINSKI, J. G., Sur la mdthode de ghdralization de Laurent Schwartz et sur convergence faible, Fundamenta Mathematicae 36. 235- 239 (1948). 6. RYLL-NARDZEWSKI, C., Une remarque sur la convergence faible, Fundamenta Mathematicae 85, 240-241 (1948). B., “Techniques in Solving Partial Differential Equations”, Institute 7. FRIEDMAN, for Mathematics and Mechanics, New York University, 1950. 8. M ~ L E s E . G. B., Laurent Schwartz’s Theory of “Distributions” and some of its applications (Fourier Transformationl, Project Squid, Tech. Rept. 41, Johns Hopkins University, 1951. 9. DE RHAM,G., and KODAIRA, K., “Harmonic Integrals”, Revised Edition, Institute for Advanced Study, Stevens and Co., 1953. P., Notes on the theory of distributions, Graduate Divison of Applied 10. GERMAIN, Mathematics, Brown University, 1954. D. W., Functions of commutable linear transformations, Thesis, Case 11. ROBINSON, Institute of Technology, 1956. 12. JAMES, H. M., NICHOLS, N. B., and PHILLIPS, R. S., “Theory of Servomechanisms”, McGraw-Hill, 1947. I., “Introduction to the Theory of Distributions”, University of Toronto 13. HALPERIN, Press, 1952. 14. TEMPLE,G., Weak functions and the “finite part” of divergent integrals, i n “Studies in Mathematics and Mechanics (v. Mises Volume)” Academic Press, New York 1954. 15. O’KEEFE,J.. The initial value problem for the wave equation in distributions of Schwartz, The Quarterly Journal of Mechanics and Applied Mathematics 8, 442-434 (1955). 16. RAVETZ,J. R., Distributions defined as limits, I. Distributions as derivatives; continuity, Proc. of the Cambridge Phil. SOC.68, 76-92 (1957).
Stress Wave Propagation in Rods and Beams
BY H . N . ABRAMSON. H . J . PLASS.
AND
E . A . RIPPERGER
Southwest Research Institute. San Antonio. Texas. and The University of Texas. Austin. Texas Page I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 I1. Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2. Longitudinal Elastic Waves . . . . . . . . . . . . . . . . . . . . 114 Elementary Theory . . . . . . . . . . . . . . . . . . . . . . . 114 Pochhammer-Chree Theory . . . . . . . . . . . . . . . . . . . 115 Love’s Approximate Theory . . . . . . . . . . . . . . . . . . . 120 Other Approximate Theories . . . . . . . . . . . . . . . . . . 122 Experimental Results and Discussion . . . . . . . . . . . . . . 128 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 138 3 . Longitudinal Plastic Waves . . . . . . . . . . . . . . . . . . . . 139 Elementary Plastic Theory . . . . . . . . . . . . . . . . . . . 139 Strain-Rate Theory . . . . . . . . . . . . . . . . . . . . . . . 145 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 I11. Flexural Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2 . Flexural Elastic Waves . . . . . . . . . . . . . . . . . . . . . . 151 Elementary Theory . . . . . . . . . . . . . . . . . . . . . . . 151 Pochhammer-Chree Theory . . . . . . . . . . . . . . . . . . . 154 Timoshenko Theory . . . . . . . . . . . . . . . . . . . . . . 162 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 168 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . 175 3. Flexural Plastic Waves . . . . . . . . . . . . . . . . . . . . . . 175 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Elastic-Plastic Analyses . . . . . . . . . . . . . . . . . . . . . 177 Rigid-Plastic Analyses . . . . . . . . . . . . . . . . . . . . . 182 Comparisons Between Elastic-Plastic and Rig.d-Plastic Analyses . . 185 Strain-Rate Theory . . . . . . . . . . . . . . . . . . . . . . . 186 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . 188 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
I . INTRODUCTION The problem of the response of structures to dynamic loading is of great importance in a wide variety of technical applications . The loading of a 111
112
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
structure may be classified broadly from the viewpoint of time variations as (a) static, i.e. independent of time, (b) gradually applied, the magnitude of the load changing on a time scale measurable in minutes or seconds, (c) pulsating or alternating, the loading varying in some regular manner with time, generally a t a rate of a few hundred or few thousand times per second, or (d) dynamic, the loading being applied “suddenly,” i.e. within a very small fraction of a second. The dynamic loads themselves may be of widely differing kinds depending on the total duration of the pulse, the pulse shape, and the time of rise to a maximum value. Extremely short pulses, in the order of a few microseconds in duration, are usually developed by impulsive or impact loads. If a deformable body is subjected to a change in stress distribution, e.g. as a result of the application of a dynamic load, then the changes in stress distribution are propagated throughout the body as stress waves. During the transient phase the stress a t a point is determined by the arrival of stress waves, singly or in combination. The evaluation of the stress during this transient period is usually.very difficult because of the complexity of the wave forms, the interaction between the load, the structure and the material, and generally an inadequate knowledge of the dynamic loading, and also because of the great complexity of the governing equations. Many suggestions have been made for the classification of stress waves; the simplest classification seems to be a division into two types depending upon the stress level and a third type depending upon additional stresses. These three types are: (a) elastic waves in which the stresses are such that the material obeys Hooke’s law, (b) plastic waves in which the material undergoes permanent deformation as a result of being stressed beyond the elastic limit and (c) viscoelastic waves in which internal viscous stresses are not negligible and act in addition to elastic stresses. Such a simplified classification is not entirely satisfactory, however, as one may immediately think of viscoplastic waves and of various combinations of the three types listed. Nevertheless, the present state of knowledge is such that a more detailed classification would probably serve little purpose. Interest in stress wave propagation is not of recent origin. The problem was an object of intensive study by the classical elasticians. During the past few years as a result, in part, of the frequent occurrence of dynamic loading problems in technical applications, interest in this area has been constantly growing. No less than five excellent surveys and monographs [l--51 concerning stress waves have appeared within the last four years. On the other hand, the technical literature and knowledge relating to the response of structures to dynamic loading is now so vast that no single monograph can do justice to all, or even a large portion, of the various aspects of the problems of stress wave propagation. Thus we plan to limit ourselves in this paper to a somewhat more restricted view of the field but
STRESS WAVE PROPAGATION I N RODS AND BEAMS
113
a t the same time to present more detailed information, within this restricted area, than other authors have been able to do. Accordingly, the present survey is confined to a study of stress wave propagation in rods and beams. We hope to provide in this way a fairly complete and connected account of the present state of knowledge. Even so some additional limitations were necessary; those resulting from space restrictions are not insignificant in an area as active as this, and those resulting from the scope of our own researches are quite obviously of primary influence. The beam or rod problem is of intrinsic interest itself because it represents the simplest of all engineering structures. Although the structure itself is simple the propagation of stress waves is complicated by the presence of the free boundaries. Reflections from the boundaries seriously modify the propagation characteristics. The complexity of the problem calls for simplifying assumptions in many cases. Therefore many approximate theories exist, differing in a greater or lesser degree from one another. In other cases exact solutions are available, but only for infinite trains of sinusoidal waves ; and not even these solutions are available for beams of square or rectangular cross section. The present paper is restricted to discussions of the propagation of elastic and plastic waves, and viscous stresses are not treated in any detail. In view of the excellent survey of Davies [a], which gives a rather complete account of the experimental techniques in common use, this paper deals with experimental results rather than meth,ods.
11. LONGITUDINAL WAVES 1. Introduction
Of the two types of wave transmission in rods discussed in this paper, the longitudinal wave is the simpler to analyze. As a result, more has been written concerning both elastic and inelastic longitudinal wave theory. Also the problems of measurement of longitudinal disturbances are less difficult, hence more experimental data on longitudinal wave transmission have been published. These analytical and experimental studies have not been restricted to waves in elastic rods, although most of the investigations fall within this category. Considerable effort has also been put into analyzing and measuring the transmission of pulses whose magnitudes are large enough to produce plastic deformations in the rod. The discussions of longitudinal waves are divided into two sections; Section 11.2 is on elastic waves and Section 11.3 is on plastic waves. In each section various theories are discussed, from the simplest to the most nearly exact available. Also a survey of experimental work is given for each area
114
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
of investigation. Where comparisons can be made between theory and experiment, these are summarized. Not all published theoretical work could be included, nor all experimental results. In our opinion, however, most of the significant developments in the subject of longitudinal wave propagation appear in this review.
2. Longitudinal Elastic Waves
Elementary Theory. The simplest theory which describes approximately the phenomenon of the propagation of longitudinal waves in an elastic rod is well known. In this approximate theory it is assumed that plane cross
FIG. 1 . Element of rod transmitting a longitudinal disturbance.
sections remain plane, and that only axial stresses are present, being uniformly distributed over the cross section (Fig. 1). The equations comprising this theory are: aa, --
av,
ax - p a t
(equation of motion) (equation of continuity)
ax = E
(2-3)
E ~
(equation of material behavior).
In the above. a,
= axial
E,
= axial strain
stress
v, x
= axial
particle velocity
= axial coordinate
t =time E = modulus of elasticity p =mass density.
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
115
The above equations can be written as one second-order partial differential equation for one of the variables, say ux, by eliminating the other two. The resulting equation is the well known wave equation
in which co = 1/E/p, the “bar” velocity, or the “velocity of sound” in a uniform bar. The general solution for this equation is of the form (2.5)
0%= f ( x - cot)
+ F ( x + cot).
This solution represents the sum of two traveling waves. The first term represents a wave, whose shape is described by / ( x ) at t = 0, traveling in the positive x-direction, and the second term represents a similar wave, whose shape a t t = 0 is given by F ( x ) , traveling in the negative x-direction. It is seen that each traveling wave has its shape preserved as it proceeds along the bar a t the velocity co. A particular solution of (2.4) is that which describes the steady state longitudinal vibration of an infinitely long uniform rod. It is assumed that the axial stress crx varies sinusoidally with time and the axial coordinate as follows :
where A is the wave length of a sinusoidal wave, and c is the velocity with which some part of the wave, say a peak or a zero, is propagated. Upon substitution of Eq. (2.6) into the differential equation (2.4), it is found that the velocity c is equal to co regardless to the wave length A. Since any pulse can be described by means of a Fourier analysis in terms of its sinusoidal components, and each component sinusoidal part is propagated a t the same velocity co, no phase differences develop at remote stations along the bar, and hence no distortion of pulse shape results.
Pochhammer-Chree Theory. The elementary theory presented in the preceding is not exact, as it is based on some rather restrictive assumptions about the manner in which the bar deforms and is stressed. In principle it is possible to describe any problem concerned with the vibrations of elastic solids by means of the exact equations of motion governing such media. However, the complexity of such an analysis precludes the use of the theory in engineering practice and, therefore, recourse is made to some approximate, but much more convenient, theory. Even in those cases where the exact theory is unsuitable for engineering use, it may still serve a valuable purpose if sufficient information can be obtained from it to demonstrate the adequacy, or inadequacy, of a given approximate theory.
116
H . N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
In order to know under what conditions the conclusions of the previous subsection are valid, it is necessary to study the same problem more precisely. This has been done independently by Pochhammer [6] and Chree [7]. In their analyses, as given by Love [8],the bar has a solid circular cross section, is infinite in length, is made of homogeneous linear elastic material, and is free from tractions on the lateral surface. The equations of motion are
1 ad
(3,+2p)--Y
amr ax
2p-+
ae
a%e
am, ar
2p-=p-,
at2
2p amr a2ux ad 2p a (A + 2p) - --(roe) + _ _ = p~
r ar
ax
r
ae
~
at2
’
where A = - - i( a rur) r ar
aur ax
2wo=------,
am aux + -r1 ae + ax
I
au, a7
and where ur,He, u, are the displacements in the radial, tangential and axial directions, and 3, and p are the Lam6 constants, and the three rotation components satisfy the identity
The stresses are found in terms of the displacements by the following relations:
(2.10)
STRESS WAVE PROPAGATION I N RODS AND BEAMS
117
A solution for displacements is assumed in an axially symmetric form (i.e. independent of 0): ur = U(r)exp i ( f x (2.11)
ug
+pt),
= 0,
uZ= W(Y)exp i ( f x
+ pi).
In these relations, the frequency is p l 2 n and the wavelength is A = 2 4 f . When these assumed displacements are substituted into the partial differential equations (2.7), two ordinary differential equations are obtained for the functions U(r) and W(r). Solution of these equations yields
where A and C are arbitrary constants, Jo and the first kind, and It’ and K’ are given by
J1
are Bessel functions of
(2.13)
Substituting these solutions into the stress equations (2.10), results in two transcendental equations for zero surface stress:
here a is the radius of the cylinder. This is a set of two homogeneous equations in A and C. In order to have a nonvanishing solution, the determinant of the coefficients of A and C must vanish. The resulting equation is known as the frequency equation; it expresses a relation between p and A . The ratio p l j is the phase velocity c, and therefore the frequency equation supplies a relation between phase velocity and wave length. This equation has been studied in detail by Field [9], Bancroft [lo], Davies [ll] and others. Some curves of the phase
118
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
velocity-wave length relation are shown in Fig. 2. In this figure only two of the infinite number of branches for the Pochhammer-Chree theory are shown. From the figure it may be seen that a pulse which has a wide band of frequencies in its Fourier spectrum will undergo a change in shape
------LOVE
---
-a h
EXACT THEORY (FIRST TWO MODES, u = 0.29). THEORY.
RADIUS OF BAR WAVE LENGTH
FIG. 2. Phase velocity curves for longitudinal elastic waves in a solid circular cylinder.
as it propagates along the rod. The conclusion from the elementary theory discussed previously, that the form of the pulse is preserved, is therefore incorrect. It is nevertheless approximately correct for pulses of very long duration, that is, where the duration of the pulse is long compared with the time required for a wave front traveling a t velocity c,, to move a distance equal to the diameter of the bar. An analysis of this phenomenon of dispersion, may be made by employing Kelvin’s method of stationary phase. This has been discussed by Davies [ll]. In this method of analysis a pulse of infinite amplitude and zero duration is imagined to be applied to the infinite bar a t some point x a t t = 0. The Fourier spectrum of such a pulse, sometimes called an impulse function or Delta function, contains all frequencies with equal amplitudes. At the time of application of the pulse, t = 0, the entire set of sinusoidal components are phased such that they reinforce one another a t the point x to produce infinite amplitude, and interfere with one another a t all other locations to
STRESS WAVE PROPAGATION I N RODS AND BEAMS
119
produce zero amplitude. Because the Fourier components all propagate a t different phase velocities, it should be expected that a t some other station X' along the bar, wave groups with different wave lengths arrive a t different times. At certain times there could be a grouping of a considerable number
22 20 18
16 14
Tp -
a CURVE @
12
A
Ta' 10 8
6 4
2
8.0 1.2 1.4
1.6
1.8 - 2 . 0 2.2 2 . 4 2.6 2.8
t' -
3
To FIG. 3. Dominant period vs. arrival time for longitudinal elastic waves (From Davies [ll]).
of waves of nearly the same wave length or period which reinforce one another, producing a local disturbance large in comparison with the nearly completely canceled components surrounding it. This larger disturbance travels at the group velocity cg which is given by (2.15)
dc
c~Ec-A-. dA
The group velocity represents the velocity of propagation of a wave packet whose wave lengths do not differ greatly from that of the dominant wave of the group 112-141. To apply the method of stationary phase, a plot is made of the dominant period T p versus the time of arrival t' a t a certain station x'. Such a plot is shown in Fig. 3. Examination of this figure enables one to determine
120
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
what frequencies are to be expected in the dominant portion of a strain More is said about this method in a later record taken a t the station paragraph on experimental results for longitudinal elastic waves. The Pochhammer-Chree theory is applicable only to an infinitely long bar in which sinusoidal wave trains of infinite length are propagated in either direction. It cannot be used to develop a solution for a finite, or even semi-infinite, bar having an arbitrarily prescribed displacement or stress distribution on the end cross section, since the infinite set of modes of the theory do not form a complete set with respect to section coordinates. In addition, the complexity of the frequency equation introduces great difficulties in the analysis for the propagation of a pulse of a given form. As a result, approximate theories have been constructed which contain the essential features of the exact problem in a simplified form. These theories are discussed below. Methods different from the Pochhammer-Chree theory have been developed in which individual wave fronts are followed as they propagate at some angle with the boundary, are reflected and form new waves, etc. Harrison [15] has analyzed the problem of the propagation of waves in a slab (plane strain problem) and has shown that the same frequency equation as that of Lamb [16] is obtained. A similar approach has been used by Pondrom [17]for the slab and the circular cylinder. In addition to obtaining the frequency equation, Pondrom also has shown how the process of critical angle reflection, internally from the boundary, causes the initial pulse, an impulse function, to be separated into identifiable groups of waves of different periods arriving a t different times depending upon the number of such reflections. His results are in agreement with experimental results reported by Hughes, Pondrom and Mims [18],discussed later in this paper. XI.
Love's Approximate Theory. An approximate theory* given by Love [8] describes the stresses and motions within a bar in a more nearly correct manner than does the elementary theory discussed previously. In this theory it is assumed as in the elementary theory that plane cross sections remain plane, and that axial stresses are uniformly distributed over the cross sections, but terms are included which take account of the inertia associated with the lateral expansion or contraction connected with axial compression and extension, respectively. Love developed his equations by applying Hamilton's principle to an energy expression which includes a transverse inertia term. The assumption
* A correction for the frequency of steady state vibration which is the same as that obtained from the Love theory was given by Rayleigh. For that reason this equation is sometimes called the Rayleigh approximation.
121
STRESS WAVE PROPAGATION IN RODS AND BEAMS
is made that the displacement in the radial direction is proportional to the radial coordinate r and to the axial strain auJax; that is (2.16)
where u, is the axial displacement and v is Poisson's ratio. Thus the kinetic energy per unit axial length for the vibrating rod is (2.17)
where p is the density, A is the cross section area, and K is the polar radius of gyration. The strain energy V,, per unit length of the bar, is
(2) 2
V, = SEA
(2.18)
The potential energy due to loads is denoted by V L and the total energy is therefore given by I
1
(2.19)
According to Hamilton's Principle, the first variation of the total energy integrated between two fixed instants of time is zero. That is,
6/dt t.
/[$($)' +
(sr
(VK)~
0
(2.20)
"2"(:)"]
+-
~
dx+d
.li'
vLdt=O.
tl
On integrating the above by parts, the following differential equation is obtained for the interior of the bar: (2.21)
122
H . N. ABRAMSON, H. J . PLASS, AND E . A. RIPPERGER
The following (natural) boundary conditions assume that the end x is subject to a stress ox, and that the other end x = I , is free:
=0
(2.22)
(2.23) I t is evident upon examining (2.22) and (2.23) that the familiar proportionality between axial stress uz and axial strain au,/ax does not exist in this theory. I t seems that this fact was not noticed by Davies [ll], as he assumed that proportionality ox = E au,/ax did exist. A discussion of Davies’ solution for a finite bar and a correction of the error which he made is included in a report by Plass and Steyer [19]. Also included in this report is a solution for a semi-infinite rod struck a t x = 0 by a stress pulse having a time variation in the form of a half sine wave; some of these results are shown here in Fig. 4. The adequacy of Love’s theory can be estimated by a comparison with the exact theory by the use of dispersion curves. I t may be seen that the curve labeled “Love Theory” in Fig. 2 is in agreement with the lowest branch of the exact theory as long as the disturbances in the rod do not contain wave lengths less than about ten times the radius of the rod. It should also be noted that this theory predicts only one branch of the dispersion diagram. As an approximate theory, it has value only in a very limited region of application ; however, the fact that closer agreement between the dispersion curves is obtained here, as compared with the elementary theory, suggests that the inclusion of the inertia of lateral motion is indeed necessary, but is not sufficient to describe the physical phenomenon of wave propagation completely. Other Approximate Theories. The Love theory just discussed was based on the assumption that the radial movements of points in the bar are of importance in predicting the manner in which longitudinal stress waves are propagated. There is however a physical phenomenon of equal importance which clearly accompanies the lateral expansion or contraction of a bar in which there is a moving disturbance. In the neighborhood of the wave front there would be a rather large rate of change of lateral dimension, especially when short waves are present. With such changes in geometry are associated shearing strains of non-negligible magnitude. An approximate theory which considers these shearing strains and the accompanying shearing stresses, has been proposed by Mindlin and Herrmann [ 2 0 ] . As in Love’s theory, it is assumed that radial displacements are proportional to the radial coordinate, and that initially plane sections remain plane. While Love
STRESS WAVE PROPAGATION IN RODS AND BEAMS
123
postulated a direct proportionality between radial and axial strain, Mindlin and Herrmann do not prescribe such a relation in advance. They assume that (2.24)
Y
Ur ==
- u ( x , t), a
218 = 0,
Ha
= w ( x ,t ) .
Four “bar stress” components are defined as follows:
(2.25)
a
a
These definitions are suggested by the form of the strain energy expression. Under conditions of zero lateral surface stresses, the equations of motion become, when expressed in terms of the bar stresses, P,
_aQ_ ax
(2.26)
+ PO a
8% -- pa2 __ 4 at2 ’
-aP, _ - _-_ pa2 _ _ .3% ax
2
at2
The expressions connecting bar stresses with the displacement functions u and w are aw 2Pv= Z P ~ = Z a ( A + p ) u + a ~ i l - ~ ax
(2.27)
2 P , = 2aAp 4Q = a 2 p
+ a2(A+ 2 p ) -,aw ax
-.au ax
In order to allow more freedom in the corrections for shear and inertia, two unspecified factors, K and K~ are introduced into Eq. (2.27), SO that
(2.28)
2 p , = 2ailp
aw + a 2 ( L+ 2 p ) ax
9
124
H. N. ABRAMSON, H . J. PLASS, AND E. A. R I PPER G ER
Substitution of these equations into Eqs. (2.26) yields the following two equations of motion in terms of displacements :
(2.29)
Mindlin and Herrmann [20] obtain sinusoidal solutions of (2.29) in the form w = B exp i a ( x - ct).
u = A esp ix(x - c t ) ,
(2.30)
When Eqs. (2.30) are substituted into Eqs. (2.29), a pair of homogeneous algebraic equations in A and B is obtained. Setting the determinant of the coefficients of A and B equal to zero yields a relation between phase velocity and wave length. The curves labeled “Mindlin-Herrmann Theory” in Fig. 2 are obtained from this determinantal equation. In obtaining these curves the constant K~ was set equal to unity, and K was adjusted so to make the lower mode curve agree with the exact (Pochhammer-Chree) theory for very short wave lengths. MINDLIN-HERRMANN THEORY.
------ EXPERIMENT.
LOVE THEORY. +1.0
+0.4
,.--.
+0.2 I
YO-0.21
15
20
25
30
%’ a
FIG.4. Calculated and measured values of axial force a t x = 16 diameters. duration = 4.4 microseconds, bar diameter = 1/16 in.
Pulse
Other solutions of the Mindlin-Herrmann equations have been given by Herrmann [21], Miklowitz [22,23], and Plass and Steyer [19]. Herrmann [21] solved formally Eqs. (2.29) under conditions of free and forced vibration. Miklowitz [22, 231 obtained a solution by means of Laplace transform
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
125
techniques for the semi-infinite rod subjected to an axial force, P,, in form of a step function. The solution is expressed in terms of certain definite integrals arising from the inversion of the transform. These integrals are very complicated and difficult to evaluate, especially for large values of time, because of the oscillatory character of their integrands. Numerical solutions have been obtained using a high speed electronic digital computer [23]. Plass and Steyer [19] have rewritten the equations of the Mindlin-Herrmann theory as five first-order partial differential equations involving the five dependent variables P,, P,, Q, b , W , the dot indicating time differentiation. The families of characteristics are found, together with the differential equations corresponding to each characteristic in a manner similar to that used by Malvern [24] in a paper on longitudinal plastic waves. These differential equations are then rewritten as difference equations for values of the five variables at the points of intersection of the characteristic curves. A solution for the case of a semi-infinite rod subjected to a half-sine pulse has been obtained subsequently and is shown here in Fig. 4. The characteristics, and the differential equations which are associated with them, are useful not only for computational purposes, but also for purposes of understanding the manner in which wave fronts (or discontinuities) are propagated.* The basic first-order partial differential equations for longitudinal waves, taking account of radial inertia and radial shear, are [19]:
~
(2.31)
Ea 2
Ea2
z
v,= (1
-
ap, ap, v) __ - ~, at at (material properties and
avo - aP, ax
aP, 2-9
at
at
equations of continuity)
aQ - 2 P, _ - = - 2pa, aV a
ax
(2.32)
4
at (equations of motion)
_aP, -_ pa2 avo _ an.
2
at ’
~
~
* The physical interpretation of the characteristics is discussed a t some length by Malvern [24]; see also Plass [19, 251.
126
H . N . ABRAMSON, H . J . PLASS, AND E . A. RIPPERGER
where Vo is the axial particle velocity and Vl = aU/at, where the radial displacement is given by U rla. The characteristics for these five partial differential equations are found from the following differential equations: dx
(2.33)
=0
dx = c,dt, dx =
dx = $2,
- cldt,
dx =
-
c,dt,
where c1 = dilatational wave velocity =
c2 = shear wave velocity =
VTJ E ____
Corresponding to each of the above families of curves there is one ordinary differential equation, found by integrating Eqs. (2.31, 2.32) : dPx -
1-v ~
dP,
Eac +2 Vldt = 0, 2Cl
V
dPx -
Ea2cl dV 0 4(1 v)c,co
+
_ - v_ _ _COCl _ 2 (1 - V2)CZ
Vldt
= 0,
Ea2cl V dVo - - ‘OC1 V,dt = 0, d P x + 4(1 v)c2co 2 (1 - Y2)C2 (2.34) 2(1 - V)CZCO P d t = 0, dQ - -dV1Pa2 4% (1 +l@
+
+
dQ
+ --dVlPa2 4%
2(1 - v)c,c P,dt (1 V)ClU
+
= 0,
(along d x
= 0)
(along d x = cldt) (along d x
= - c,dt)
(along d x
= c2dt)
(along d x = - c,dt).
The characteristic equations (2.33)tell us that two actual wave-front propagation velocities are predicted by the Mindlin-Herrmann theory. One velocity is the dilatational velocity cl; the other is the shear velocity c2. Further analysis of Eqs. (2.34) leads to the conclusion that discontinuities of P , or V , travel at the dilatational velocity cl, while discontinuities in Q and V , travel at the shear velocity c2. An actual disturbance consists of many wave fronts, traveling at each of the two velocities, forward and backward. At a station of observation several diameters from the end, the effect of this wave mixing is the spreading out (dispersion) of the pulse applied a t the end.
STRESS WAVE PROPAGATION I N RODS AND BEAMS
127
Another approximate longitudinal wave theory has been given by Bishop [26], in which Love’s assumption concerning the relation between lateral displacements and axial strains is used. An additional term is included to take account of the shear stresses that result from the rather abrupt changes in diameter as the wave passes. The use of Love’s assumption is too restrictive, however, and the resulting equations are good only for the first branch of the dispersion family which corresponds to propagation of discontinuities in axial force. The second branch, corresponding to propagation of discontinuities in radial shear, is not well represented by this theory. Infinite velocities of propagation for short waves arise, which are not predicted by the exact theory. Physically speaking, Bishop’s theory does not allow a sudden change in radial shear without an accompanying sudden change in axial strain. The Mindlin-Herrmann equations, however, do have this necessary freedom. The form of the equations of the Mindlin-Herrmann theory can be derived directly from the exact equations of Pochhammer and Chree by the “method of internal constraints” due to Volterra [27]. Volterra assumes that the displacements occurring within the interior of the bar are restricted in certain prescribed ways (hence the name internal constraints). The strains are then computed, and substituted into the strain energy integral. By means of Hamilton’s Principle, the partial differential equations of Pochhammer and Chree are reduced to ordinary differential equations in terms of the variables describing the manner in which the displacements vary with the coordinates of the cross section. The Mindlin-Herrmann equations result when the displacement in the axial direction is assumed to be constant over the cross section, and when the radial displacement is assumed to be proportional to the radial coordinate. The parameters are the axial displacement and the rate of change of radial displacement with radial distance. The equations of Mindlin and Herrmann result, but the values of K and K~ are not arbitrary. The method of internal constraints has the advantage that higher orders of variation of displacements can be introduced in a series form. Volterra [28] has used higher orders of deformation variation to obtain a differential equation of sixth order. The dispersion curves (three of them) are shown in his report, and the first two agree quite well with the exact theory. Plass [29] uses a similar scheme to that of Volterra [27] to obtain exactly the same results. In the development of Plass [29], Hamilton’s Principle was not used but certain average stresses and strains were defined, the definitions for the averages being suggested by the assumed variation of displacements with the radial coordinate. The equations of motion, etc. were modified to equations in terms of these averages, and the wave equation was deduced. Only the Love and Mindlin-Herrmann theories have been used to compute transient or pulse behavior in a rod.
128
H. N . ABRAMSON, H . J.. PLASS, AND E. A. R I P P E R G E R
Experimental Results and Discussion. Experimental investigations of the propagation of longitudinal elastic waves in cylindrical bars have in general taken one or the other of two approaches. Either the steady-state vibrational characteristics of a bar are examined, or the pulse-propagation characteristics of the bar are studied. The two approaches are described in more detail below, and significant results obtained by both are discussed. The technique used for the steady state vibration studies consists essentially of setting a bar into resonant longitudinal vibration by various methods of excitation, such as a quartz crystal driven by a controlled frequency oscillator, or by magnetostriction of the. bar or a part of the bar. By using some indicator such as lycopodium powder the locations of the nodal points in the vibrating bar are determined and the wave length is measured. The phase velocity of propagation is the product of this wave length and the corresponding frequency. The principal contributors to this line of investigation have been Rohrich [30], Schoeneck [31], Shear and Focke [32], Stanford [33]. At the time the first three of these investigators were active (1930’s), Bancroft’s calculations [ 101 of phase velocities from the Pochhammer-Chree frequency equation were not available. Indeed, it appears that the results obtained by Shear and Focke [32] provided the stimulus for Bancroft’s calculations. As a consequence, the results of these early investigations were compared with calculated results based on both the Love theory (the Rayleigh correction) and the Giebe-Blechschmidt [34] theory. It was found that the experimentally determined velocities agreed with calculated velocities at wave lengths that were longer than the bar diameter, but as the wave lengths approached the bar diameter the agreement became progressively poorer. Shear and Focke [32] conducted steady state experiments using magnesium, silver, and nickel rods. The radii of these rods were approximately five millimeters and the lengths ranged from 15 to 25 centimeters. Excitation at frequencies ranging from 200 to 800 kc was supplied by a quartz crystal driver. The phase velocities obtained in this investigation are compared with the results predicted by the Pochhammer-Chree solution in Fig. 5 . It may be seen that, except for four scattered points, the measured values agree very well with the computed values corresponding to the lowest branch of the dispersion curve. The four points in exception do not fall on any of the higher branches of the dispersion curve, so presumably they must represent errors in observation, or, some mode of vibration not accounted for by the theory. Some difficulty was noted in the experimental work in that longitudinal, flexural, and torsional vibrations were all excited simultaneously to the extent that measurements could not be made unless the longitudinal mode sufficiently dominated the other modes. Under these conditions it would not be surprising to find some errors of observation.
129
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
The significance of these results goes beyond the remarkable agreement shown in Fig. 5. I t should be recalled that the Pochhammer-Chree solution assumes an infinitely long bar with two sets of waves distributed throughout the length of the bar, one set traveling to the right and one traveling to the left. The nature of these waves is such that they cannot be superimposed so as to cancel both normal and shear stresses at a given cross section. I. 0.
0.
0. 0. C 0.
LONG I TUDl NA L WAVE S.
CO
-
x DIAMETER I, 0 -
0.
OF ROD
= 4.615MM.
= 5.895MM. THEORETICAL CURVE (FOR u = 0.25).
0.
'I
'I
0.
0. I
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5 0
0.6
0.7
0.8
0.9
-
0
h
FIG. 5 . First branch phase velocity curve for longitudinal elastic waves in a solid circular cylinder, with experimental results (From Kolsky [el).
Therefore, this solution cannot satisfy completely the boundary conditions at the ends of a bar of finite length. The solution is obviously incomplete. Nevertheless, experiments conducted on finite bars have given phase velocities which agree with the velocities predicted by this theoretical solution. Further experimental evidence supporting the view that, although the Pochhammer-Chree solution is incomplete, the deficiency affects results only near the ends of the bar has been provided by Bancroft [lo]. In these experiments, phase velocities were determined using bars 4 to 3 wave lengths long. The results are shown below in Table 1.
130
H. N . ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
TABLE1.
O B S E R V E D VELOCITIES I N STEEL B A R S O F
DIFFERENT LENGTHS
Bar length cm
wave length cm
resonant frequency
velocity m/sec
15.235 5.083 15.235 3.812 15.235 2.540
10.157 10.166 7.518 7.624 5.078 5.081
-50473 50426 67184 67162 100390 100364
5126 5126 5118 5120 5098 5099
The wave lengths shown in the second column of this table were not actually measured, but were computed from the bar lengths and the number of nodes along the length of the bar. The velocities were computed using the assumed wave lengths and the measured frequencies. It should be expected that the end effects would be more significant for the shorter bars, but the variation in velocity with bar length is well within the experimental error. Additional experimental evidence of the accuracy of phase velocities predicted by the Pochhammer-Chree solution was obtained by Stanford [33]. His experiments were conducted with aluminum bars having diameters between 1* and 12 inches. Using exciting frequencies between 5 and 50 kc he found good agreement between the measured and computed phase velocities. He also found some velocities which appear to belong to the second and third branches of the dispersion curve family. The results of experiments using the steady state experimental technique have demonstrated the adequacy of the Pochhammer-Chree solution for predicting phase velocities of propagation as a function of wave length. Nevertheless, it could not be concluded from these results that pulse propagation characteristics in an elastic bar might be predicted by using the Pochhammer-Chree solution, as the equations are too formidable to permit any hope of an analytical solution. for a given set of initial conditions. Interest in the effects of dispersion on a pulse propagating along a bar began with the development of the Hopkinson pressure bar as a device for measuring large forces of short durations. This device, which was conceived by B. Hopkinson [35], and described by him in 1914, consists of a cylindrical bar with a short piece of the same diameter and material wrung on to the end of the bar. This small end piece flies off when the bar is subjected to an impulsive force at the opposite end. By catching the piece in a ballistic pendulum the momentum trapped can be determined. From this measured momentum some information concerning the magnitude of the applied force can be deduced. By the use of this bar Hopkinson was
S T R E S S W A V E PROPAGATION I N RODS A N D BEAMS
131
able to make satisfactory measurements of the maximum pressures applied and the time during which the applied force exceeded a certain level.* The development of this technique by Hopkinson represented a considerable advance in the study of stress waves by experimental methods since earlier workers had been able to measure only the durations of impacts. His method does not, however, give the complete force-time record for an impact, hence its usefulness in the investigation of stress pulse propagation is limited. Furthermore, the pressure bar, in its original form, does not give reliable results if the applied force is not large, and it obviously can yield only qualitative information concerning the distortion of a pulse as it progresses along a bar. Hopkinson attempted to determine this distortion in a pulse propagated along a one inch bar by varying the length of the end piece and the length of the bar. He concluded from a series of measurements that there was no systematic difference between the results obtained with a 15 inch and a 45 inch bar. The duration of the impact which he studied was about 50 microseconds. Hopkinson’s work was continued after his death by Landon and Quinney [36], but the basic limitations of the measurement technique prevented them from adding anything of particular significance to the understanding of pulse propagation in bars. They did, however, measure in a qualitative way the distortion caused by the dispersive process by observing the momentum trapped in a one inch end piece as the length of the pressure bar was varied. The loss of momentum which was observed as the length of the bar was increased was interpreted as a flattening of the pulse as it propagated through the bar. About thirty years later Davies [ll] modernized the experiment by replacing the end piece with a capacitance-type gage which measured the displacement at the end. The signal from this gage was recorded as a function of time by using a cathode ray oscilloscope and a camera. Thus, a detailed picture of the form of a pulse after propagation through a bar became available for the first time. In addition to the gage for the end displacements, Davies also used gages for measuring radial displacement that could be placed anywhere along the length of the bar. With these radial gages he presumably could have made measurements of the pulse as it passed two different points along the bar. Since he did not do this, he had to find the extent of the dispersion which had occurred by computing the nature of the applied pulse at the end of the bar. By plotting calculated phase velocitiest as a function of wave length in nondimensional form, as shown in Fig. 2, and using the well known rela-
* The discrepancy between measured and calculated values for the maximum force of impact of a rifle bullet was reported to be only 24 percent. t These phase velocities were computed by Davies before the publication of Bancroft’s [lo] results.
132
H . N . ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER
tionship for group velocity (2.15), Davies computed the group velocities which are shown here in Fig. 6. By regarding the initial distribution of stress as the superposition of an infinite number of trains of sinusoidal stress waves of equal amplitude, all in phase at the origin but out of phase a t every other point in the bar so as to interfere destructively and give zero stress, the propagation of a stress can be examined theoretically through Kelvin's method of stationary phase as discussed in a previous section.
FIG.6. Group velocity curves for longitudinal elastic waves in a solid circular cylinder (From Davies [ll]).
The curves shown in Fig. 3 can be used to predict, in general terms, the nature and extent of the dispersion to be expected under a given set of circumstances. For example, from curve (1) in Fig. 3 it may be seen that t' has all the values between 4 To and 2.64 To/2. This shows that a pulse which is initially infinitely short becomes distorted into a pulse having a duration of 1.64 T0/2. Also it may be noted that for t' less than 1.73 To/2 only one dominant group of waves arrives a t a given point a t a given instant. The
S T R E S S W A V E PROPAGATION I N RODS AND B E A M S
133
period of this group decreases as t’ increases. After t’ exceeds 1.73 To/2, two dominant groups arrive simultaneously at the given point. Thus the tail immediately following the main pulse consists of a series of high frequency oscillations which are approximately sinusoidal in form. The frequency decreases rapidly at first and then more slowly. After t’ = 1.73 To/2, the oscillations suddenly become more complex because of the arrival of the second dominant group, simultaneously with the first dominant group. To extend these results to a disturbance of finite duration, it is necessary to superimpose a number of disturbances of the simple type considered.
FIG.7. Comparison of pressure variation a t one station on a bar transmitting a longitudinal pressure pulse with dominant period of oscillation (From Davies [ l l ] ) .
Davies’ 1111 analysis of an oscillogram of a pulse generated by the impact of a bullet is shown in Fig. 7 . Values of T , for the first mode are shown by the small sine curves in this figure. Since the duration of the impact of the bullet is about 35 microseconds, and the duration of a disturbance which originated as an infinitely thin pulse at the pressure end of the bar 115 centimeters away would be 378 microseconds, the total duration, allowing for the finite time of impact of the bullet, should be about 413 microseconds. Fig. 7 extends only to 340 microseconds; therefore, it does not include the whole of the disturbance and does not include that part of the tail at which
134
H . N . ABRAMSON, H . J. PLASS, A N D E. A. RIPPERGER
the two dominant waves amve simultaneously. From these considerations it appears that if the vibrations excited by the bullet impact belong to the first mode then the whole of the tail in Fig. 7 should be composed of simple harmonic vibrations gradually decreasing in period from 15.7 microseconds a t t’ = 255 microseconds, to 104 microseconds at t’ = 350 microseconds. If, on the other hand, the disturbance included vibrations from the higher modes, vibrations with shorter periods will be superposed on the single vibration contributed by the first mode. In that case the curve should show sudden changes of slope which are characteristic of curves compounded of two or more vibrations varying in period and phase. There is no indication oY this in the tail of Fig. 7. Thus Davies concludes that if any vibrations of the second mode are present their amplitudes are negligibly small in comparison with those from the first mode. This is one of the important conclusions of the investigation. Davies carefully analyzed the rising portion of some measured pulses produced by applied forces, which rose practically instantaneously from zero to a finite value, and found small oscillations superimposed on the rising portion of the pulse, and a zero slope a t the beginning. Increasing the time of rise of the applied force reduced the amplitude of the oscillations in the rising portion of the curve. From these experiments it was concluded that the pressure bar could not measure impacts whose magnitudes changed in time of the order of one microsecond. These conclusions and results are consistent with the observation that it is the very short wave lengths in a pulse that are associated with a rapid rise, and the short wave lengths are the ones most affected by the dispersive process. By assuming a periodic trapezoidally-shaped input pulse at the end of the bar, Davies was able to compute the shape of the stress pulse a t a given point in the bar by inserting, in each of the Fourier components, the appropriate phase velocity and then recombiriing the components. This was done using the phase velocities obtained from a solution of the Love equation and also for the phase velocities from the Pochhammer-Chree solution. Comparison of these calculated pulse shapes with some experimentally measured pulse shapes led to the conclusion that the Pochhammer-Chree solution predicts more accurately the distortion that will occur than does the Love equation. Davies [37] was also concerned with the distribution in a pulse of stress over the cross section of the bar. To determine experimentally the extent of non-uniformity in the distribution, he conducted a very ingenious experiment in which small steel balls were lined up on the upper end surface of a 29 inch diameter bar. When a bullet was fired against the lower end of the bar the balls flew off and their trajectories were photographed. The analysis of these trajectories showed that the stress in the pulse is uniformly distributed across the cross section after the pulse has traveled four to five diameters from the point of impact.
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
135
A further study of propagation of pulses in the pressure bar was reported by Ripperger, in 1953 1381. In this study, pulses were generated by the impact of small steel balls on the end of the bar. By allowing the balls to strike the end of the bar a t very low velocities it was possible to avoid causing plastic deformation a t the point of impact and to generate the extremely short pulses desired. The small strains associated with these pulses were measured by means of piezoelectric gages cemented to the surface of the bar. The sensitivity of these gages is high enough to make the measurement of strains as small as 1 x feasible. Also, since these gages are equally sensitive to the longitudinal and lateral strain components, they give a better approximation of the average strain over the cross section of the bar, when very short pulses are being measured.* Measurements a t various points along the bar allowed the pulse to be studied as it progressed along the bar. Thus, distortion by dispersion could be directly observed. A systematic study of the relationship between pulse durations and bar diameters revealed that the shortest Hertz-type pulse which will be transmitted through a given cylindrical bar without appreciable distortion by dispersion has a duration equal to the time required for a wave traveling a t the bar velocity to traverse a distance of 7 to 8 times the bar diameter. Measurements of the velocity of propagation of pulses of various durations in bars of various diameters indicated that the peak of the pulse invariably propagated at the velocity No part of the disturbance in the pulses could be detected traveling a t the dilatational velocity or a t any velocity greater than the bar velocity. Miklowitz and Nisewanger [40] measured propagational velocities and announced results which are somewhat contradictory to Ripperger’s [38] conclusions. In their experiments a step pressure was applied to the end of a one inch diameter aluminum rod. Radial displacements and axial strains were measured a t points along the rod. These measurements showed that the initial part of the disturbance traveled a t a velocity greater than the bar velocity. At x = 20 inches, the most distant station from the point of force application, the velocity of the leading part of the disturbance was about 1.1 1/E/p. For the Hertz-type pulses studied by Ripperger, the slopes of
1/%.
* Petersson [39] compared the applied impulse to the impulse computed from measured strains, using wire resistance strain gages on the surface of the bar, and concluded that a surface gage sensitive only to longitudinal strain would not give accurate results for pulses shorter than 5 microseconds. He then pointed out that, in view of the theoretical distribution of strain given by Davies [ll], gages arranged t o measure both lateral and longitudinal strains would, when their outputs were added together, be sufficiently self-compensating t o extend appreciably the range over which accurate measurements could be made.
136
H. N . ABRAMSON, H . J . PLASS, A N D E . A. RIPPERGER
the leading edges of the pulses were very flat after they had traveled a few diameters away from the impact point. This made it impossible to determine the velocity of the leading edge with any degree of accuracy. Also the waves generated by the impact were three-dimensional in nature a t the beginning whereas in the shock-tube method used by Miklowitz and Nisewanger [40j a two-dimensional system of waves was generated. Miklowitz and Nisewanger also concluded, from a study of the activity recorded a t each gage station for a period of time after the initial arrival, that the higher modes in the Pochhammer-Chree solution were present. I n 1949 Hughes, Pondrom, and Mims [lS], while studying methods for measuring the elastic constants of solids, developed a technique, using a quartz crystal as a driver, for generating pulses in elastic bars, with times of rise of the order of 0.2 to 0.6 microseconds. These pulses consisted of a group of plane waves of dilatation traveling nearly parallel to the walls of the bar. At the receiving end of the bar the original pulse appeared as a series of distinct wave groups. The arrival times of these wave groups indicated that the first group to arrive had traveled straight through the bar a t the dilatational velocity. The second group to arrive had been delayed by a time just equal to the time that it would require for a shear wave to travel a distance of one bar diameter. The third group to arrive had been delayed by just twice that time, the fourth group three times as much, and so forth. From these results, and the well-known fact that the reflection of a plane dilatational wave a t a free boundary produces both a dilatational wave and a shear wave, it was concluded that the second wave group represented a group that had traveled along the bar as a dilatational wave, then after being reflected from the boundary had traveled across the bar almost along the normal to the bar axis, and again after reflection traveled parallel to the bar axis for the remainder of the distance. The third group had traveled across the bar twice as a shear wave and then the remainder of the distance as a dilatational wave, and so on, The nature of these wave groups arriving a t the receiving end of the bar in one of the sets of experiments performed by Hughes, Pondrom and Mims [l8] are shown here in Fig. 8. These results were the first to indicate the mechanism through which the boundaries of a cylindrical bar affect the propagation of longitudinal waves through that bar. The Pochhammer-Chree solution as evaluated by Davies [ l l ] indicates that the maximum group velocity is the bar velocity. Thus the fastest part of the pulse would travel a t the bar velocity. However, it seems physically reasonable to expect that some wave propagation a t the dilatational velocity would occur. Studies of wave propagation in slabs by Holden [41] and Harrison [15] indicate that the phase velocity curves may have horizontal slopes at the dilatational velocity in the higher branches; if this were true also in the case of the circular cylinder, then the maximum group velocity would be the dilatational velocity. The percentage of the total energy,
STRESS WAVE PROPAGATION I N RODS AND BEAMS
137
however, propagated a t the dilatational velocity in pulses of the type studied and discussed by the various experimenters is extremely small.
/ -=
10 MICROSECOND MARKERS
m V m 20.32 25.40
-
O 0 (L
P 15.24
A
LL
I
II--
l
10.16 m
I.
0 5.05.08
z
W
II
\
10 MICROSECOND MARKERS
FIG. 8. Oscillograph records of pressure variation in a bar transmitting a sharp pressure pulse (From Hughes, Pondrom, Mims [18]).
Kolsky [42a], in 1954, reported the results of some experimental work which was intended to extend and verify the work of Hughes, Pondrom, and Mims[18]. In his studies, Kolsky found that the motion of the detecting end of the bar appeared to be continuous if the length of the bar was 15 centimeters and the radius 1.25 centimeters. The discrete type of arrivals which Hughes, Pondrom, and Mims had observed were not detectable but the leading edge of the pulse traveled at the dilatational velocity. However, for a bar length of 10.4 centimeters and a radius of 7.6 centimeters the discrete arrivals could be identified a t the detecting end, and the times of these arrivals agreed remarkably well with computed values. It should be noted that in Kolsky's work the pulses were generated by exploding a small charge at the' center of the end face of the bar, and as a consequence the initial wave system set up was not quite the same as that generated by the quartz crystal used in the experiments of Hughes, Pondrom, and Mims. Kolsky also concluded that for cylinders in which the ratio of length to radius is large, most of the energy in a pulse will travel with a velocity
138
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
less than the bar velocity. Experimental evidence in support of this statement was not presented. In that connection it will be recalled that Ripperger’s [38] measurements indicated that most of the energy would be propagated a t the bar velocity. In our previous discussion of the arrival times of the dominant waves making up the various parts of the pulse it was pointed out that, considering only the first mode of propagation in the Pochhammer-Chree solution, the last dominant wave to arrive a t a given point should arrive a t a time 2.6 t where t is the time of arrival of the head of the pulse. I n Kolsky’s measurements the pulse did not end at 2.6 t, but rather low frequency oscillations continued to arrive up to at least 5 t , and possibly later. These later arrivals can be accounted for by the Pochhammer-Chree solution only by assuming that the higher modes of vibration have been excited. It will be recalled that Davies had concluded that these higher modes of vibration were not excited; on the other hand, Miklowitz and Nisewanger [40] came to the opposite conclusion. Additional experimental results obtained by ultrasonic pulse techniques have recently been reported by Tu, Brennan and Sauer [42b]. This investigation was concerned with the dispersion of longitudinal and shear waves in several elastic rods of different materials. The authors found good agreement with the predictions of the Pochhammer-Chree theory for a/A < 0.8, in the lowest mode of transmission. For large values of u / A it was found that the group velocity approached the dilatational velocity. The results for large a / A seem to be in agreement with the upper branch of the MindlinHerrmann theory, as evaluated by Miklowitz [22]. Direct comparisons between experiment and approximate theories (Love and Mindlin-Herrmann) are shown in Fig. 4 which is taken, in part, from [19]. These appear to be the only direct comparisons made on the basis of pulse propagation, that is, by comparing experiment and theory for amplitude and shape. The general shape is predicted by both theories - but the size of the amplitudes is not predicted very accurately. Concluding Remarks. It may be said that the implications of the exact solution of the longitudinal pulse propagation problem are not too well known. This does not mean that understanding of the basic mechanism of propagation is lacking. Experimental studies have helped to indicate the nature of the mechanism while the highly developed approximate theories, combined with the nearly exact Pochhammer-Chree solution, have made it possible to predict analytically the essential propagational characteristics. Since analysis by the exact theory is extremely difficult mathematically, progress in this direction will be slow, at least until wider use is made of high-speed computing machinery. Evidence of such progress is contained in a very recent paper by Skalak [43]. In this paper the exact solution was found for the step-type impact resulting from the collision of two semi-
STRESS WAVE PROPAGATION IN RODS AND BEAMS
139
infinite rods of equal size and composition. Numerical results were found approximately (i.e., by employing the Rayleigh approximation) and are correct only for large values of time after the initial impact. In the future we should expect to see more effort placed on obtaining exact solutions. Progress on the experimental side is also being made. For example, photoelastic techniques have been applied to stress propagation problems for some time [44-471; the extent to which this technique has been developed is indicated in a recent paper by Durelli and Riley [48]. Furthermore, experimental techniques, such as those of [38], have been sufficiently standardized to permit the investigation of such problems as the effects of discontinuities in cross section [49], [50], curved or bent rods and rods with discontinuities in elastic properties [51]. The high degree of refinement of the ultrasonic technique is illustrated in [42b]. 3. Longitudinal Plastic Waves
Elementary Plastic Theory. If the end of a bar is subjected to an impact of sufficient magnitude to produce stresses in the range where Hooke’s Law
Ex
-
FIG.9. Broken-line stress-strain curve (Donnell [52]).
no longer applies, plastic strains are produced. A simple hypothesis for the propagation of plastic waves which was given by Donnell [52] is as follows: Suppose the material has a stress strain curve such as that shown in Fig. 9. If the impact produces a stress (rl, the compression zone will move along _the bar at a velocity vE,/p, as discussed previously. Now, if the impact
140
H. N. ABRAMSON, H. J . PLASS, A N D E. A. R I P P E R G E R
produces a total stress czthere will be two steps in the front of the compression __ zone. One__of these will travel at the velocity 1/E,/p and the other at the velocity VE,/p. The wave front will appear as in Fig. 10. The gap between the two parts of the front will continually increase with time. If the stress strain relationship is a smooth curve, rather than a broken straight line, it can be considered to consist of a series of small straight line segments.
Q2
X-
FIG. 10. Stress distribution in the rod (Donne11 [52]).
Thus each stress increment will propagate at a different velocity and the front of the compression zone will have a curvature that continually changes as the front propagates. If, upon unloading, the stress strain curve follows a straight line, the rear of the compression zone will be vertical and will propagate at a velocity depending on the slope of the unloading curve. A formal solution for the problem of plastic wave propagation based on Donnell's hypothesis was obtained independently by G. I. Taylor in England [53] and Th. v. KBrm6n in America [54j at about the same time. The problem was also studied by White and Griffis [55], [56]. I n these solutions it is assumed that the relation between longitudinal stress and strain, plastic as well as eiastic, is the same for dynamic and for static loading, and that the unloading wave is entirely elastic. Also, lateral inertia effects are neglected. The equation of motion is of the same form as (2.4), with E replaced by do,lde,. Thus,
(2.35)
141
STRESS WAVE PROPAGATION I N RODS AND BEAMS
This equation is to be solved for a bar extending from - 00 to zero. The end conditions are taken as (2.35a)
u,= v,t
at
x
=0
and
ux= 0
at
x=-
00.
If it is assumed that (2.36) where
Ex =
=
/(El
x/t, the displacement u, has the form
(2.37) By differentiating one obtains
(2.38)
Substituting into (2.35) gives (2.39) from which
(2.40)
f‘(8= 0 is obtained. If it is assumed that the a, vs. E , curve is concave downward with a maximum slope E , (2.40) can be interpreted as follows: (a) for x > c, t , E , = 0, i.e., the disturbance has not had time to reach this region. (b) for c’ t < x < ,c, t , then do,/de, = p ( x / t ) 2 and the propagation follows Donnell’s analysis. c‘ is a velocity which depends on the velocity of impact. (c) for x < c‘ t, E , = is constant. This comes from the second of Eqs. (2.40).
142
H . N . ABRAMSON, H. J . PLASS, A N D E. A. RIPPERGER
To determine c', we write 0
(2.41) -m
Making use of (2.37) and changing variables, we find E.
e.
(2.42)
da,/de, is a given function of strain; hence E, is defined by this equation as a function of V,. If da,/d&, approaches zero at large values of E,, as it does in most materials, and the material breaks a t some value of F,, the integral in (2.42) has a limiting value. This is the critical velocity of impact. An impact a t any greater velocity would cause an instantaneous fracture. One explanation which has been advanced [55] for this phenomenon is that, since da,/d&, is zero, the velocity of propagation is zero and only a part of the wave is able to travel down the bar. Most of the energy of impact has to be absorbed, therefore, in the neighborhood of the impact end. Experiments have been made to verify (a) the existence of a plastic of the wave front of a given amplitude, (b) the relationship between plastic wave front and the velocity of impact and (c) the distribution of plastic strain between the plastic and elastic fronts [54], [57]. In this experimental work copper wires 100 inches long and .071 inch in diameter were loaded in tension by the impact of a weight guided between two rails and accelerated by rubber bands. A static stress strain curve obtained, using one of the copper wires in the test series, was used to obtain do,/&, as a function of E, for use in the calculations. The critical impact velocity was calculated to be 150 ft/sec. Deformation was obtained by measuring between marks inscribed on the wire before the test, and again after the wire had been subjected to impact. The rate of change with x of the difference between these two values is essentially the plastic strain. Elastic recovery was neglected because it is very small relative to the inelastic or permanent strain. The velocity of propagation corresponding to a given value of E, was computed from the da,/dEx vs. &,-data. For annealed copper the velocity of elastic waves is about 12,500 ft/sec. In the plastic range the velocity decreases rapidly and is only about 1800 ft/sec for a strain of 0.5 percent, and a t a strain of 16 percent the velocity vanishes abruptly. Also, the uniform strain E, a t the impact end of the bar can be computed from the plot of dux/&, vs. 8,. Experiments conducted to verify this computed relationship are in fairly good agreement.
143
STRESS WAVE PROPAGATION I N RODS AND BEAMS
At an impact velocity of 171 ft/sec a specimen broke within an inch of the end but the rupture was not brittle and a plastic strain of 2 or 3 percent was observed over a distance of 20 inches from the point of impact. This indicates that a plastic wave was propagated down the wire before failure occurred. The difference between the calculated and observed critical velocities therefore throws considerable doubt on this particular part of the theory. 148 -IMPACT VELOCITY
k
"0
- FT.
PER SEC.
20 40 60 DISTANCE FROM IMPACTED END, INCHES
80
FIG. 11. Permanent strain distribution and impact velocity vs. permanent uniform strain in a long tensile specimen whose end is suddenly subjected to a constant velocity (From Duwez and Clark [57]).
Measurements of strain distribution along the wire show a general agreement with theoretical predictions. There is a region of constant strain the magnitude of which depends on the impact velocity near the impact end (see Fig. 11). Thus the plastic wave front of constant amplitude predicted by the theory is evident.
144
H. N. ABRAMSON,
K. J. PLASS, A N D E. A. R I P P E R G E R
The strain ahead of the plastic wave front diminishes gradually but the experimental values and those calculated from (2.40) are not in good agreement (see Fig. 12). The shape of this part of the curve appears to be influenced considerably by the manner in which . the pulse-producing striking mass is stopped in order to limit the pulse duration. I t is also
D
DISTANCE FROM IMPACTED END, INCHES FIG. 12. Theoretical and experimental permanent strain distribution in rod impacted in tension with a sudden constant velocity (From Duwez and Clark [57]).
conceivable that the discrepancy is due in part to a strain-rate effect not considered in the theory. I t was noted that in impact tests on iron wire specimens no plastic strain occurred at the impact end until the stress wave reached a value three times the static yield stress. Further evidence of a strain-rate effect was obtained by Sternglass and Stuart [58]. In this investigation, a longitudinal impact pulse of short duration and of sufficient magnitude to cause permanent deformation was applied to a bar, and direct observations were made of propagational velocity and changes in the shape of the pulse as it traveled along the specimen. To make these measurements, flat strips of material (4 x $in.) were mounted in a testing machine and prestressed in tension into the plastic region. A longitudinal impact was then applied at the end of the specimen by a falling weight. Measurements of the propagational velocity showed that the wave front moves with the bar velocity c,. This result was obtained regardless of whether the loading was continued or stopped during the impact.* Whereas pulses of the duration
*
This result was also confirmed by Bell [59].
STRESS W A V E PROPAGATION I N RODS AND BEAMS
145
used in the study, i.e., 400 to 800 microseconds, travel without appreciable change in shape in the elastic region, in these experiments they were distorted by dispersion. It was also observed that the distortion depended on the original pulse duration ; this suggests that the dispersion is frequencydependent. The velocity of propagation of every part of the pulse was always much greater than the velocity based on the tangent modulus at the point of the stress-strain curve to which the material was prestressed. Also it was found the measured strain agreed with that computed elastically except for the longest pulses; consequently, Eq. (2.42)does not predict the maximum strain. The results of these experiments indicate that the strain-rate effect cannot be ignored and that the stress-strain curve obtained from staticloading measurements should not be used in calculations for plastic wave propagation.
Strain-Rate Theory. Malvern [SO], [24] modified the theory of plastic wave propagation to introduce the effect of rate of strain on the stress-strain relation. A number of investigators [61-641 have suggested that the relation between nominal tensile stress ux, plastic strain E ~ ' ' (permanent strain), and plastic strain rate kz'' should be (2.43)
csx
Ex").
=Ip(Ex'l,
Deutler [63] found experimentally that (2.44)
Ex" B
ux = u1 + A In _ _ ,
where csl is the stress for ix"= B. Prandtl [62] had derived this law theoretically from a physical theory of plastic flow, and Ludwik [61] had proposed it on empirical grounds almost twenty years earlier. By putting Eq. (2.44) in the form (2.45)
ox =
+ a In (1 + b i n " ) ,
where u, = I(&,) is the static stress-strain relation, the static stress may be compared with dynamic stress. Thus,
=I(.
ax - / ( e x )
(2.46)
E,"
b
a
- I).
This equation suggests that the plastic strain-rate effect is a function of ux - f(E,), which is the excess of instantaneous stress over the stress at the
same strain in a static test. A more general law obtained from (2.43) is (2.47)
E , ix'! = g(o,,
Ex).
146
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
Since EoEx’ = 6 x ,
(2.48)
for the elastic strain-rate, we can write (2.49)
Eo E x
=
5,
+ OX, ex)
for the flow law when plastic deformation occurs. I t is assumed that plastic flow occurs only when O, > / ( e x ) , otherwise (2.48) applies. For his calculations Malvern [SO, 241 assumed that g(ux, E x ) = k [a. - / ( e x ) ] .
(2.50)
Thus, (2.51)
EoEx = u
x
+ k[u. -
/(Ex)],
and has the interpretation given above. Starting with the three first order equations (equation of motion)
a&,
(2.52)
at
EO-
aEx at
- av, ax =
~
aoX at
(equation of continuity)
+ k [ ~ -x
(material property)
/(ex)]
and using (2.53)
10
f ( E ) = 20,000 - EX
to approximate / ( e x ) for hardened aluminium specimens, Malvern calculated the propagational characteristics of a pulse in a semi-infinite bar.* A comparison between strain-rate and elementary plastic theories on the basis of calculated results is shown in Fig. 13. These calculations show that (a) the wave fronts are all propagated with the velocity of elastic waves co, regardless of what the strain-rate function g(o,, e x ) may be (b) the stresstime variation is in better agreement with experiment than the predictions of the elementary plastic theory and (c) the permanent strain dstribution is in worse agreement than the elementary plastic theory. The strain-rate theory does not predict the zone of permanent strain near the impact end. Sokolovsky [65] solved this problem in closed form for an idealized perfectly plastic material.
STRESS WAVE PROPAGATION I N RODS AND BEAMS
147
Malvern’s comment regarding this point is “. . . a more realistic speed law than the linear law, . . . might give more satisfactory results; but it is not likely that any law of the type considered would give a constant strain region. ”
t
X ( I NGHES)
FIG. 13. Strain distribution at t = 102.4 microseconds. Solid line is from strain-rate effect solution, while broken line is from solution neglecting strain-rate effect (From Malvern [go]).
Plass [66] has extended Malvern’s work by comparing the linear strainrate law with an exponential law and has investigated the effects of different shapes of the static stress-strain curve. The effect of strain-rate on critical velocity was investigated and found to be mainly a delay of the failure, whereas the elementary plastic theory predicts instantaneous failure. This result is in agreement with the experimental results reported by Duwez and Clark [57], previously discussed. Plass [66] concluded that there is no essential difference in the effects produced by the two strain-rate laws (i.e. linear and exponential). For the example which he presented, the excess in stress over the static value lasted longer in the material with the exponential strain-rate law, but this was due to the choice of the constant k in the linear strain-rate law. A decrease in the value of this constant causes the reduction in the excess stress (relaxation) to take place at a slower rate. More experimental work on the problem of material properties appears to be required. Riparbelli [67, 681 has reported some experimental results which indicate that : (a) Annealed copper exhibits a strain rate proportional to the difference in actual stress and the stress under static conditions at the same strain. This supports Malvern’s simplifying assumption.
148
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
(b) The constant k in the strain rate expression varies from 3.5 x to 7.0 x 10-3 1b-lsec-lin2. (c) For impact durations of 0.3 x 10-3 sec, no region of constant plastic strain develops near the impact and of the specimen. These results are shown in Fig. 14. (For longer durations this region would no doubt appear.)
I " TRAVEL 140 fps
0 5.00
GUIDE PIPE
/
4.00
t
3.00 2.00
0
0
FIG. 14. Permanent strain in a copper wire under longitudinal impact Riparbelli [67]).
(From
(d) When the stress vs. strain-data are plotted, the points fall in a region bounded on one side by an extension of the linear portion of the static stress-strain curve, and on the other side by the static stress-strain curve beyond the yield point. (e) The initial stress in the wire, at the higher impact velocities, is about twice the ultimate static strength of the wire. The Sternglass-Stuart experiment [58] was, in effect, repeated by Alter and Curtis [69] using a unique method for superimposing one dynamic strain on another. For these measurements they used a lead bar 4" in diameter and 13" long which was held between a steel anvil bar and a steel transducer bar, each of which was 10 feet long. The stress was generated by the longitudinal impact of another steel bar 10 feet long on the transducer bar. This impact produced a pressure on the specimen which lasted for more than 1 millisecond before the unloading wave from the free end of the anvil bar began to relieve the pressure. To produce a stepped pulse which would be the equivalent of superimposing one strain on another,
STRESS WAVE PROPAGATION I N RODS AND BEAMS
149
an impact bar made up of two uniform sections of different diameters was used. The first section was 3' long and had a diameter of the same as the transducer bar and the specimen. The second section, 7 feet long, had a diameter of 1 inch. This abrupt change in cross section causes a part of the pressure pulse traveling along the striking bar, as a result of the impact, &'I,
I I
2
w
I
1
II
I
I
IL
w
0
zs!?
0
I
!
D
z
c
5-,p-I I
5a
p-I
I
I
I
I I
I
I
I I
--.t
TIME
3.5
(500 MICRO-SEC. INTERVALS)
FIG. 15. Strain vs. time-curves in a lead bar impacted by a double step pressure pulse (From Alter and Curtis [69]).
to be reflected back to the specimen, producing an increase in pressure 360 microseconds after the first step in pressure arrives. Typical results of these measurements, and those in which a single pulse was used, are shown in Fig. 15. For the single impact type loading, which was produced by a striking bar 1 inch in diameter, it was noted that the first arrival at each gage station had traveled at the elastic velocity, c,,. This elastic front is so small, however, that it is difficult to detect. Following this elastic front the strain gradually increases to a value of about 0.5 percent. This strain is not relieved to any appreciable extent when the anvil bar is removed; hence, it is definitely a plastic strain. As the pulse travels along the bar the slope of the plastic part becomes flatter indicating that the higher strains are propagating at a slower velocity. I t is estimated that the velocity for the average strain in the pulse is only 15 percent of the elastic velocity. Both the elementary plastic theory, and the strain-rate theory predict this general type of behavior. The results of the double impact type loading show an elastic front for both steps in the pulse. This result must be regarded as a confirmation of
150
H . N. ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER
the Sternglass-Stuart results. As in the single impact pulses the elastic fronts are very small and difficult to detect. The plateau between the two steps in the pulse rapidly disappears and is replaced by a sloping line that extends from practically no strain to the maximum strain. The elementary plastic theory indicates that this plateau will be maintained while the strainrate theory predicts that the two steps will coalesce. A comparison of the travel times of the average strains in the first and second steps of the pulse, with the same strain levels as in the single pulse, shows that they definitely are not the same. The elementary plastic theory requires these travel times to be the same. According to the strain-rate theory the strains near the mean of the first step should travel slower than the same strain in the single pulse, and the mean strain of the second step should travel faster than the same strain in the single pulse. This is precisely the relationship observed. The predictions of the strain rate theory are borne out in a qualitative sense only by these measurements, but there is a definite indication of the inadequacy of the elementary plastic theory for predicting correctly the basic features of plastic wave propagation.
Summary. Experimental results indicate that there is a strain-rate effect which should be considered in the propagation of plastic waves. Including this strain-rate effect in the analysis leads to results which are checked, in part, by experimental results. On the other hand, the strainrate theory does not predict a region of constant strain near the impact end as the simple theory does, and this region has definitely been observed by a number of experimenters. I t should be noted that neither of these theories considers lateral inertia. It has been observed in studies of elastic waves that lateral inertia has an important influence on propagation when the wave lengths involved are of the same order of magnitude as the bar diameter or smaller. Such wave lengths are to be expected in the wave fronts that have been discussed. Thus it appears that if the plastic wave theory is to be improved some method will have to be found for introducing the lateral inertia effects. The study of plastic wave propagation is difficult at present due to the paucity of data concerning material properties that may be pertinent to the problem. I t has been assumed in the elementary plastic theory that the dynamic stress-strain curve is identical with the static curve, whereas in the strain-rate theory it is assumed that the rate of relaxation depends on the difference between dynamic stress and static stress at the same strain. Some of the available experimental evidence supports this view, but on the other hand there are also indications that a single dynamic stress-strain relation cannot be used to describe the behavior of a material at all impact stresses [70a, 70bl. In particular, higher order effects cannot be represented by these simple relations.
STRESS WAVE PROPAGATION IN RODS AND BEAMS
151
Further progress in the study of plastic wave propagation, it appears, will depend on a more precise knowledge than is now available of the relationship between stress and strain under dynamic conditions of loading.
111. FLEXURAL WAVES 1. Introduction
An off-center longitudinal impact applied to the end of a beam and the transverse impact of a lateral force on a beam must be similar in nature, for it is clear that in either case a system of stress must be induced which has both symmetrical and anti-symmetrical components about the beam mid-plane. Thus we are led to consider the propagation of stress waves that are classed either as longitudinal or flexural. In speaking of the propagation of flexural waves we mean that type of vibration of a beam or rod in which portions of the beam are in flexure and elements of the neutral axis move laterally during the motion. The problem of the flexural vibrations of beams is somewhat more difficult than that of the longitudinal vibrations discussed earlier because the elastic deformations involved are more complex, depending upon a fourth order rather than a second order differential equation as in longitudinal vibrations. Further, it will be seen that even the elementary theory of flexural wave propagation leads to dispersion. The present discussion will follow approximately the sequence used for the longitudinal waves. A brief discussion of the elementary theory preceeds an analysis based on the exact theory of Pochhammer and Chree and this is followed by a discussion of the approximate theory of Timoshenko; a discussion of pertinent experimental results concludes the section. The next section discusses the various approximate theories of plastic wave propagation, with special emphasis on a higher-order theory which considers strain-rate effects, and again concludes with the pertinent experimental results. 2. Flexural Elastic Waves Elementary Theory. Application of the elementary theory of beam bending to the determination of transient stresses in beams has generally been made by means of the mode-superposition method employing the natural vibration modes of the beam [71,72] ; however, it is well known that this technique has many shortcomings, particularly for sharp impacts [73, 741. An interesting survey of the impact response of simple structures, based essentially on the modal method, was published by Frankland [75].
152
H. N . ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
An alternative treatment, based on the elementary beam theory, can be developed by consideration of the energy relations involved when a beam is struck by a falling mass [76, 711. This approach is valid for cases in which the mass of the beam is negligible in comparison with the mass of the falling body, the beam is thin, and the displacement at the point of impact is proportional to the force. An extension of the analysis may be made which accounts approximately for the mass of the beam. A third method of analysis employing the elementary beam theory was also developed by Timoshenko [77]. Here the local deformation at the point of impact is accounted for by the analysis of Hertz [78], and the deflection is determined by integration of Lagrange’s equation of motion. The solution is expressed by employing the Duhamel integral relation, in the form of a Volterra integral equation. Eringen [79] has studied the impact of beams and plates by numerically evaluating the integral equation by several different methods. Some effort has been made to develop a method of analysis based not on normal mode procedures, but on a wave type analysis wherein the beam is matched with an image system and solutions are obtained by operational methods, the motion being analyzed by wave forms. This study [80], however, still employs elementary beam theory as its basis. I t is of interest to consider wave type solutions based on the elementary theory of beam bending. Considering the vibration of a uniform beam, the governing equation, due principally to Bernoulli and Euler, is of the form
where E I is the flexural rigidity, p the density, A the cross-sectional area, u the transverse displacement, and x the coordinate in the direction of the spanwise axis. This elementary theory is based primarily upon the assumtions that the cross-sectional dimensions are small compared with the length of the beam and that transverse sections of the beam, originally plane, remain plane and normal to the longitudinal fibers of the beam after bending. A wave type solution of (3.1) may be taken in the form (3.2)
24
= Ccos(f5t - f X ) ,
where C is the amplitude and
A being the wave length and c the wave (or phase) velocity. Introducing (3.2) into (3.1) one obtains
(3.4)
c = 2nR c0/A,
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
15s
where
R2= I / A .
(3.5)
I t may be noted that co is the same wave velocity for longitudinal waves of infinitely long wave length that has been referred to previously as the “bar velocity.’’ The phase velocity c is inversely proportional to the wave length so that one must conclude that for waves of infinitely short wave length the velocity is infinite. Therefore, a disturbance containing extremely short wave lengths, prominent in a very sharp impact, must be propagated throughout the beam almost instantaneously - a conclusion a t variance with physical reason.
I .o
0
FIG. 16. Phase velocity curves for flexural elastic waves.
The foregoing relation for the phase velocity is readily described by plotting Eq. (3.4)as c/co versus RIA, as shown in Fig. 16; this type of “dispersion diagram” was used previously in the discussion of longitudinal waves. Lamb [16] was apparently the first to point out that the elementary beam theory is inadequate for impact type loads since it leads to the physically impossible conclusion that disturbances are propagated instantaneously
154
H. N . ABRAMSON, H. J. PLASS, A N D E. A. RIPPERGER
throughout the beam. Therefore, it is evident that the elementary beam theory requires modification in order to be applicable to impact type problems, and these modifications must be such as to modify or remove the assumptions of the elementary theory stated earlier. These refinements to the elementary theory have been given by Rayleigh [81] and Timoshenko [82, 831; the correction offered by Rayleigh is to account for rotation of beam cross sections and that offered by Timoshenko is to account for the transverse deformation due to the shear force. The correction for rotatory inertia of the beam cross sections results in a differential equation of the form a42t
CO
R2-
ax4
a42t
- R2ax2
at2
a2U
+--0. at2
If Eq. (3.2) is again taken as a solution, the dispersion relation is found to be (3.7) and the corresponding dispersion curve is shown also in Fig. 16. The Rayleigh theory appears to be more satisfactory than the strictly elementary theory for very short wave lengths, as a finite phase velocity equal to the bar velocity is predicted; for very long wave lengths the two theories agree as expected. The Timoshenko theory will be discussed in detail in a later section. Pochhammer-Chree Theory. I t would seem that further discussion of the problem of impact on beams should be based on the equations of the theory of elasticity; a general discussion concerning this approach was given in the earlier section on longitudinal waves and also by Goland, Wickersham, and Dengler 1841. The essentials of that discussion are repeated here as an aid to understanding the phenomena involved, and as an introduction to the application of the exact theory (Pochhammer-Chree theory) which will follow. A t the instant that a load is applied a t a point on the surface of a beam the disturbance is propagated in all directions within the beam from that point by means of elastic waves of stress. These waves depart from the original point of disturbance and move along the beam, being reflected from the boundaries of the material in a complex manner. While the load is acting, the stress pattern in the immediate vicinity of the load is highly three-dimensional in character; away from this localized region the “beam action” begins to assert itself, and the stress distribution becomes approximately two-dimensional, with strong variations of stress occurring only
STRESS WAVE PROPAGATION I N RODS AND BEAMS
155
along the beam length and in the direction of the beam depth. Once the load is removed, the beam-like action extends over the (previously) loaded region as well. A first simplification would neglect the three-dimensional character of the stresses immediately near the load. As a consequence the stresses cannot be estimated properly in the immediate vicinity of the applied load as long as the load is acting. However, the situation near the load is such that the Hertz impact theory mentioned earlier may be used to approximate the stresses in that region. If the stress distribution within the beam is treated in two-dimensional approximation, then the considerations become those of a problem of plane strain or plane stress, depending on the dimensions. A problem in plane strain was .treated by Lamb [l6]; he showed that the natural vibration modes of an elastic plate may be separated into two classes: those which have cross-sectional stress distributions symmetrical about the beam midheight plane, and those with anti-symmetrical distribution about this plane. The discussion [84] continues with a consideration of the characteristics of these modal solutions, pointing out that for an infinitely long beam the modal forms display a continuous natural frequency variation with change of wave length along the beam. Further, it is pointed out that in the low frequency region i.e., for long waves, the stress distribution at each beam cross section is very nearly linear and therefore the results reduce to the modes predicted by the elementary beam theory. The problem now at hand is to study the propagation of flexural waves from the viewpoint of the equations of the theory of elasticity. The wellknown equations of motion of an isotropic elastic solid in cylindrical polar coordinates were given previously in Eqs. (2.7). The customary analysis would now require that solutions of these equations be sought which would satisfy the boundary conditions for the particular problem under study. Such a procedure would not prove fruitful here, however, for even the simplest case of a beam of finite length would be found to be much too difficult.* Further, to solve the equations of motion for an arbitrary cross-sectional shape appears to be beyond our ability. Therefore, for the sake of simplicity we study beams of infinite length with some selected cross-sectional shape. The case of an infinitely long beam of uniform circular cross section was investigated by Pochhammer [6] and independently by Chree [7]. The elements of the theory are summarized by Love [8], but some of the basic equations are reproduced here for convenience.
* In fact, solutions of Eqs. (2.7) which satisfy all boundary conditions have not been found. The solutions to be described below satisfy boundary conditions at the lateral surface exactly, but only approximately at the end of a beam; therefore, these solutions may be held strictly applicable only to beams of infinite length.
156
H. N. ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER
If the axis of the circular cylinder is oriented in the x-direction, the three radial stress components take the form given by (2.10). Solutions of (2.7) may be taken in the form
+ pt), ue = V sin 8 exp i ( f x + f i t ) , utC= W cos 8 exp i(fx + p t ) , uy= U cos 8 exp i ( f x
(3.8)
where U , V , W are functions of Y only according to
(3.9)
W
= ifAJ,(h’r) - ~ B K ’ ~ J ~ ( K ’ Y ) .
Here A , B , C, are arbitrary constants, kind, and
J1
is the Bessel function of the first
(3.10)
which are identical with Eqs. (2.13). The equations for the displacements may now be expressed in terms of the arbitrary constants A , B , C and then introduced into the expressions (2.10)for the radial stresses. The boundary conditions then specify that the radial stresses are zero at the surface Y = a so that there are left three homogeneous linear equations in A , B , C. The determinant of the coefficients must vanish, and expansion of the determinant yields an equation dependent upon the elastic constants L and p , the density p, the radius of the rod a, the frequency of the waves 912n,and the wave length 2nlf. By expressing the equation in non-dimensional form, however, an equation in two variables c/co and a/A results, with Poisson’s ratio as a parameter. Consequently, it is possible to represent the results in the same type of dispersion diagram as was shown earlier for the elementary theory (see also, the previous discussion of longitudinal waves). The frequency equation for the case of flexural vibrations was first given by Bancroft [ l o ] in the form of a determinant but was not evaluated.
157
STRESS WAVE PROPAGATION I N RODS AND BEAMS
Hudson [85] was able to perform some numerical computations with this determinant but unfortunately overlooked the existence of all modes or branches of the dispersion curve above the fundamental.* By certain simplifying steps, McSkimin [88] evaluated the determinantal equation, 2.8
---
2.4
EXACT THEORY. ELEMENTARY THEORY. -
2.0
f
1.6
C
CO
1.2
0.8
0.4
I I
0.2
0.4
0.6
1
1
0.8
I .o
I .2
AA FIG. 17. Phase velocity curves for flexural elastic waves in a solid circular cylinder (From Abramson [89]).
including some of the higher modes, for the high frequency range. Abramson [89] noted the convenient non-dimensional form of the determinant and calculated the three lowest modes for a value of'Poisson's ratio v = 0.29 over the significant frequency range; these are shown in Fig. 17. Curve (1) of Fig. 17 corresponds to the result given by Hudson [85] while curves (2) and (3) represent higher branches; there are, of course, an infinite number of branches lying beyond these. Previous discussions of flexural waves in beams have often been based on the results for plates
* This error
has been carried over into more recent work [2, 4, 11, 86, 871.
158
H . N. ABRAMSON, H . J. PLASS, A N D E. A . RIPPERGER
obtained from the general elastic equations [go, 911; as expected, the present results confirm the qualitative correctness of those discussions because of the similarity of the present curves to those obtained for the plate. Thus far, only phase velocities have been mentioned; that is, the velocity of propagation of surfaces of constant phase, such as defined by the equation 2n(x - ct) = constant,
(3.11)
as in (3.8). In a dispersive medium, however, energy is transmitted not a t the phase velocity, but at the group velocity as defined in (2.15). It may be noted that elastic substances are not inherently dispersive; however, a dispersion effect is exhibited by waves traveling through rods and beams as already discussed. This dispersion effect in beams arises solely as a result of wave reflections from the boundaries, and therefore this effect in the case of an elastic medium is simply an interface phenomenon and not a physical property of the material. I .6
I .2
t
- 0.8 C9 CO
0.4
C
a
--t
h
FIG. 18. Group velocity curves for flexural elastic waves in a solid circular cylinder (From Abramson [89]).
Fig. 18 shows group velocity curves for the two lowest phase velocity curves of Fig. 17. The phase velocities in the higher branches exceed the dilatational velocity (at long wave lengths), but the group velocities do not; note that the greatest possible velocity of energy transmission is the dilatational velocity, for both flexural and longitudinal waves.
STRESS WAVE PROPAGATION IN RODS AND BEAMS
159
It may be noted from Fig. 18 that for the first branch the group velocity reaches a maximum where a/A is between .25 and .30. This implies that when a pulse is propagated along a beam in accordance with the first branch, Fourier components of wave length about .25 or .30 times the radius will be found a t the head of the pulse. This general subject will be discussed in greater detail in the paragraphs which follow.
FIG.19. Periods of dominant groups for flexural elastic waves in a solid circular cylinder (From Abramson [SS]).
The procedure for studying the propagation of a pulse in a solid circular cylinder based on Kelvin’s method of stationary phase, described by Davies [ll] and discussed earlier in this paper is also applicable here. As before, let the period of the dominant group of waves be denoted by T p and let T , and 3 To be the times required for a longitudinal wave of infinite
160
H . N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
wave length to traverse the radius a and the distance x , respectively. The various quantities may be written as two non-dimensional ratios (3.12)
(3.13)
-t‘_-- co Ta
cg
The plot of T,/T, vs. t ‘/ ( + To), as in the case of longitudinal waves (Fig. 3), is shown in Fig. 19 [89], where the number designations of the curves correspond to those of Fig. 17. The curve (1) also represents the result given by Davies [ l l ] which he described as follows: The faster components arrive at the cross section x at time t’= 1.56 To/2; the period of these components is 5.29 Ta and their wave length is equal to 2.7 a. Between this value of t’ and t‘ = 1.73 To/2, two groups of different wave lengths and periods arrive simultaneously at each value of t ’ ; a t t’ = 1.73 To/2, the Rayleigh surface waves and waves of period 17.3 T, arrive simultaneously. When t‘ exceeds 1.73 To/2, only one group will arrive at each instant. The situation is considerably different, however, for the second branch. Here the faster components arrive at the cross section x at time t’ = 0.9 T0/2; the period of these components is 5.8 T, and their wave length is 10 a. Between this value of t’ and t‘ = 1.5 To/2, two groups of waves appear, while for 1.5 To/2 < t‘ < 1.62 To/2 four groups of different wave lengths and periods arrive simultaneously at each value of t’. For values of t’ greater than 1.62 To/2, but less than 3.33 To/2, three groups arrive simultaneously at each value of t‘, while for t’ > 3.33 To/2 only one wave group appears. On the basis of the lowest dispersion curve alone, Davies concluded that no flexural displacements will occur at the cross section x until t’ 1.56 To/2, i.e., t’ 3 1.56 x/co. However, on the basis of the second dispersion curve, flexural displacements may be expected to occur at considerably earlier times, i.e. t‘ = 0.9 To/2; and since there are an infinite number of higher branches, and the maximum group velocity is the dilatational velocity, it is clear that the first arrival will occur at t’ = 0.887 T,,/2, followed by an infinite number of wave arrivals. At this place it may be worth-while to comment on the physical nature of these vibrations. An important question concerns the existence of phase velocities greater than the dilatational velocity, although no disturbance can be propagated with a velocity greater than that. A disturbance propagated into an elastic medium, in the general case, originates as one composed of both dilatational and distortional waves.
>
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
161
These waves reflect from the boundaries in a very complex manner [92], thus making it difficult to follow the mode interactions that take place. In any event, the situation is closely analogous to that obtained in electrical wave guides and may be discussed in similar terms. The essential point is, of course, that energy is propagated at the group velocity rather than the phase velocity, and group velocities greater than the dilatational velocity will not occur. The phase velocities spoken of earlier represent the velocities of propagation in the axial direction of loci of constant phase in the mode pattern.* One of the principal assumptions of all approximate theories of beam bending is that of plane sections remaining plane. The exact theory may provide the means to judge the validity of that assumption. With the relationships between phase velocity and wave length established in Fig. 17, it is a fairly straightforward procedure to study the cross-sectional distortion and, for that matter, the distribution of normal and shear stresses, as well. For a given combination of values of a/A and c/co, the ratios A / C and B/C may be solved for from the three homogeneous algebraic equations comprising the boundary conditions. The three functions U , V , W may then be evaluated from (3.9) in terms of a single amplitudinal constant, e.g. C, and the radius Y. The displacements u", uo,u, can then be obtained from (3.8), and the corresponding stresses from (2.10). I t may be noted from (3.8) that 26, is essentially dependent upon the function W; further, from (3.10), W is a complex quantity dependent upon the constants A and B. Therefore, the results are presented in Fig. 20 in the form of curves with the modulus of W as abscissa and the ratio Y/a as ordinate. The curves shown [89] have been normalized by means of the largest value of W . The three curves at the top of Fig. 20 show the distortion variation, for three values of a/A, corresponding to the first branch of the phase velocity relation (curve 1 of Fig. 17). For the smallest value of a / A the distortion is nearly linear, confirming again the validity of elementary beam theory for long waves; however, for 0.4 < a/A < 0.8, a nodal plane appears as the curve departs from linearity.? The lower three curves of Fig. 20 show the distortion variation for corresponding values of a/A for the second branch (curve (2) of Fig. 17). Three features are of interest here: (a) the maximum displacement occurs at about Y = 0.8 a for a/A = 0.2, but occurs at the surface Y = a for a / A = 0.8; (b) the slopes of the curves are quite different from those based on the first branch; (c) if a nodal plane occurs at all, it will be for a/A > 0.8.
* Some additional comments of this general nature may be found in [91], [93], and [94].
t
A similar result, in the case of longitudinal waves, was noted by Davies [ I l l .
162
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
These few curves give some idea of cross-sectional distortion for the two lowest dispersion branches as a function of ./A and also for the change with c/co as a/A is held fixed.
(0)
BASED ON CURVE
I
0
OF FIG. 17.
t
I.
-=0.2 A
(
iJ
a A
'=0.4 A
BASED ON CURVE
0
0.8
:
OF FIG. 17.
FIG. 20. Cross-sectional distortion curves associated with the first two branches of Fig. 17 (From Abramson [SS]).
Timoshenko Theory. The Pochhammer-Chree theory just described is too complicated mathematically to be useful as an engineering theory for predicting stresses and displacements in rods and beams subjected to bending impacts. On the other hand, the elementary theory is too limited to describe adequately the transient behavior resulting from sharp bending impacts. A comparison of the dispersion curves in Fig. 17 makes this evident because the (single) curve of the elementary theory agrees with the exact theory
STRESS WAVE PROPAGATION I N RODS AND BEAMS
163
curve for the lowest mode only for very long waves. No higher branches are predicted by the elementary theory. The gap between the over-simplified and the too highly complex theories is very well bridged by an intermediate theory due to Timoshenko 182, 831. His theory adds terms which correct for shear strains and for rotatory inertia of beam elements. Whereas the elementary theory contains the assumption that plane sections remain plane and perpendicular to the deformed middle curve of the beam, the Timoshenko theory is based on a less restrictive assumption : the sections remain plane but can be inclined a t different angles to the deformed middle curve. Also, whereas the elementary theory assumes the conventional equilibrium relation between bending moments and shear forces, the Timoshenko theory includes the inertia term arising from rotation of beam elements in the plane of vibration of the beam.
I
d
x
-!I
d
FIG.21. Element of a beam in bending.
A discussion of the essential details of the Timoshenko theory now follows. The forces, moments, and directions of motion of a beam element are shown in Fig. 21. From the figure it can be shown that the equations describing the behavior of the element are as follows:
164
H. N. ABRAMSON, H . J. PLASS, AND E. A. RIPPERGER
aK am -_- _ at
= EIK
M
(bending)
at
(bending)
(3..16)
Q
= As7
(shear)
i
1
of continuity)
(equations of material behavior),
where
M Q K
= moment = shear force = axial rate of
change of section ahgle
y = shear strain = awlax - # o = angular velocity of section ZI
E p p
I A
A,
=
-
= -
a#/ax
a#jat
= transverse velocity = awlat = Modulus of Elasticity = Shear Modulus = density = section moment of inertia =
section area
=
z ) dA s s y ( true variation of y over the area. area parameter defined by
= yA,,
where y(z) is the
When all but one variable, say M , is eliminated from the six equations above, a single fourth-order partial differential equation results. This equation is
where (3.18)
This is the form usually designated as the Timoshenko equation. The quantities Elp and p A , / p A have the dimension of a velocity squared. The quantity \ / E T has appeared before, as the velocity of propagation of longitudinal waves according to the elementary theory. The other quantity, ,uA,/pA, is a modified form of the square of the shear velocity, which is ,u/p. The modification results from the assumption of plane cross sections.
STRESS WAVE PROPAGATION IN RODS AND BEAMS
165
The steady state solution to the Timoshenko equation is found by substituting into it the following form of the solution: (3.19)
M = Moexpi[ar(x- ct)].
The resulting equation is
or (3.21)
The soiution of this equation for c/co vs. a/A, where a is the radius of the bar, and M = 2n/A, yields two separate branches. They are plotted on Fig. 17 and labeled “Timoshenko Equation”. One of the branches agrees quite well with the solution given by Hudson [85] using the PochhammerChree theory discussed previously. Better agreement can be had if the constant A , in the Timoshenko equation is adjusted to produce a velocity CQ = V p m i n agreement with the asymptotic value of the lowest mode of the exact theory. Such adjustment is not contrary to the basic physical assumptions in the Timoshenko theory, as the distribution of shear strains over the cross section is not an ingredient of Eqs. (3.14) - (3.16), and this distribution is precisely what affects the value of A,. The second branch, however, agrees only in form with the corresponding branch of the exact theory. The asymptotic value of the phase velocity in the Timoshenko theory is the bar velocity; however, the asymptotic value in the exact theory is the velocity of Rayleigh surface waves along the cylinder (approximately the same as the shear velocity, Other attempts to develop similar theories have been made by Mindlin [95] (actually for plates), Volterra [27], with his method of internal constraints, and Plass [29], using a method similar to that of Mindlin. In all of these theoretical approaches the exact equations are used as a starting point, assumptions about displacements are introduced, and the equations subject to these restraints are found. In the work of Volterra and Plass it is found that the velocities appearing in the final equation are the dilatational and the shear velocities instead of those mentioned above. The reason for this difference is that in the displacement assumptions used by Volterra and Plass, no account was taken of the Poisson’s ratio effect, that is, of the motion of particles perpendicular to the plane of bending. An interpretation ____ of the velocities co = V E / p and cQ = VpAJpA, appearing in the Timoshenko
v&).
166
H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER
equation, was given by Fliigge [96]. He showed that co is the velocity with which discontinuities of moment or section angular velocity are propagated, and that cQ is the velocity of propagation of discontinuities of shear force, or transverse velocity. Fliigge also pointed out that the disturbance occurring a t a particular station bears no resemblance, timewise, to the applied disturbance because of the complicated mixing of waves of both bending and shear types. The Timoshenko equation has received a great deal of attention in recent years. Solutions for different kinds of impacts have been found by Uflyand [97], Dengler and Goland [98], Miklowitz [99], Boley and Chao [loo], Zajac [loll, and Plass and Steyer [19]. The solutions of Uflyand and Dengler and Goland are for transverse velocity inputs in the form of impulse functions, and each contain certain errors, later corrected. Miklowitz [99] has solved the impact problem of a step function moment by Laplace transform techniques. Boley and Chao [loo] have presented a rather complete set of solutions for step-function type impacts of transverse velocity, moment, shear, and angular velocity. They again use the Laplace transform methods. An important part of their paper is the table showing the interrelation between solutions corresponding to various boundary conditions. Zajac [loll studied in detail the problem of response to a step moment impact. Asymptotic solutions for large time or great distance from the impact end are developed. Also a series solution is developed for the behavior of the beam just behind a discontinuous moment wave front. The work of Plass and Steyer [19] is aimed directly at obtaining solutions for a problem easily duplicated experimentally. The problem is that of a moment impact whose time variation is in the form of a single half sine wave. The problem is solved, for a limited range, by evaluation of integrals arising from a Laplace transform solution. A large number of solutions, for half sine impacts of moment, shear, angular velocity, and transverse velocity, are obtained by means of the method of characteristics. Some of these solutions are compared with experimental results of Ripperger [102], to be described in more detail in the next section. Some solutions for a bar of finite length were obtained by Leonard and Budiansky [103]. In their work they compare modal solutions (i.e., superposition of vibration modes) with corresponding solutions obtained by the method of characteristics. Other investigations relating to this problem are presented in [104-1141. The solution of the Timoshenko equation by the method of characteristics is very similar to the solution of the Mindlin-Herrmann equations discussed in Section 11. The characteristics are solutions of the differential equations (3.22)
167
STRESS WAVE PROPAGATION I N RODS AND BEAMS
These equations have as solutions the families of straight lines, in the x - t plane, (3.23)
where C, and C, are constants. The equations (3.14) - (3.16) can be rewritten as differential equations along the characteristics as follows : along
d x - codt = 0
along
dx
dQ - pAcQdw = p A c Q o d t ,
along
d x - cQdt = 0
+ pAcQdw = p A c i o d t ,
along
dx
dM - PIC,&
dM
= COQdt,
+ p I c o d o = - coQdt,
+ codt = 0
(3.24) 2
dQ
+ CQdt = 0.
EXPERIMENT
Iz W
I 0 I
TIMOSHENKO THEORY
22
+0.4+0.2 -
=z
--
5
-0.4
oz
&-
0
-v.'c
A 2
v 4
-
6
10
8
I
1
12
14
16
CO +
(0)
D
x = 2 DIAMETERS, D.0.516 INCH. PULSE DURATION = 6 MICROSECONDS (HALF SINE WAVE FORM).
TIMOSHENKO THEORY
Iz W
EXPERIMENT
0 I
t0.2 I
5
5
-0.2 -0.4
CO'
a
D
El%-
(b) x = 2 DIAMETERS, D = 0.516 INCH. PULSE DURATION= 24 MICROSECONDS (HALF SINE WAVE FORM).
E
FIG.22. Moment vs. time-graphs for two different impact durations (From Plass and Steyer [19] and Ripperger [102]).
168
H. N . ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER
It is seen from the above equations that waves involving M and w are associated with the propagation velocity c,, while waves involving Q and ZJ are associated with cQ. The two g;ioups are not isolated from one another, however, as can be seen when the right-hand sides of the above equations are examined. The changes in the M - w waves depend upon Q, and the changes in the Q - v waves depend upon w. Physically, this is to be expected, as moment and shear are closely connected to each other through the equations of motion. The solutions of the Timoshenko equation for a semi-infinite beam subject to Cnd-moment impacts in half sine wave form, given in the report of Plass and Steyer [19], have a small initial wave train, the front of which travels with the velocity co. The main part of the moment wave occurs, for a particular station, at a later time than that required for a wave traveling at velocity c, to reach that station (see Fig. 22). I t is not possible to assign a unique velocity to this greater part of the traveling wave, as its shape continually canges. This main part of the wave can be thought of as a group of smaller waves, some faster, some slower than the speed of the peak. As the wave travels along the rod, the membership of the group changes from a population corresponding to one velocity to that of another velocity. Experimental Results. Experimental investigations bf the propagational characteristics of short bending wave pulses have not been nearly so extensive as the investigations of the propagation of longitudinal waves. Most of the early investigations were concerned with the impact of a mass on the beam and the measurement of the force acting between the beam and the mass, or the deflection of the beam [115 - 1171. The results of such investigations as these do not elucidate the problem of pulse propagation and therefore will not be discussed here. Shear and Focke [32], in the investigation previously mentioned, carried out a few measurements of phase velocity vs. wave length which fall on the lowest branch of the family of dispersion curves shown in Fig. 23. These points are so closely grouped that they do not verify the accuracy of the theoretical dispersion curves, but they do suggest that there is some confirmation of theory by experiment, a t least for the lowest branch of the curve. The first account of an experimental study of bending wave pulse propagation appears to have been given by Dohrenwend, Drucker, and Moore, in 1944 [118]. The stated purpose of this paper was “to demonstrate that usable experimental techniques exist for the determination of transverse impact transients.” A series of oscillograph records show the manner in which a bending wave pulse changes its form as it propagates along a beam away from the impact point. The pulse was about 300 microseconds long and hence would be considered in the range of long pulses. Nevertheless, there is a very appreciable change in form of this pulse as it travels along the beam. The anti-symmetrical nature of these pulses was shown by
169
STRESS WAVE PROPAGATION I N RODS AND BEAMS
measurements made on opposite sides of the beam. Propagated moments were computed from measured strains and were compared with computed values based on the elementary beam theory and an assumed initial distribution of lateral velocity in the beam. The agreement is very good but the significance of the agreement is questionable since the initial velocity distribution was selected arbitrarily to give the best fit to the experimental data. 1.c 0.1 -
0.E
FLEXURAL WAVES.
-
X
-DIAMETER OF ROD = 4.615MM. " " = 5.895MM. THEORETICAL CURVE (FOR u = 0.25). ,I
0 -
0.i -
0.E C 0.e CO
0.4
0.; 0.2 0.I
C
I
I
I
I
I
1
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 -
1.0
h
FIG. 23. First branch of phase velocity curve for flexural elastic waves in a solid circular cylinder, with experimental results (From Kolsky [2]).
Even more questionable was the assumption of an initial velocity over a finite length of the beam. This assumption violates the basic concepts of wave propagation in that the effects of the impact are required to be transmitted instantaneously to remote parts of the beam. The results of this investigation are of a qualitative nature, but significant in that they show that these bending strains can be satisfactorily measured. Quantitative data concerning the propagation of bending wave pulses were presented by Dengler, Goland and Wickersham [84] in 1952. Bending wave pulses were generated in their experiments by the transverse impact
170
H. N. ABRAMSON, H. J. PLASS, A N D E. A. R I P P E R G E R
of steel balls on steel beams. The durations of the impacts ranged from 11.5 to 37 microseconds. Peak forces developed by the impact ranged from 8 to 64 pounds. The maximum strains developed were of the order of 0.5 microinches per inch. In order to measure these small strains the authors found it necessary to use piezoelectric barium titanate strain gages similar to those previously discussed. Measured values of bending strain were compared to computed values, the computation based on the Timoshenko equation. The agreement is in general very good as may be seen in Fig. 24. .I
f r ao EZ
1
0
-
‘i.-.I
..2
. EXPERIMENT
..3
-COMPUTATION
- .4 -3 0
TIME
- MICROSECONDS
FIG.24. Measured and calculated strain vs. time-record a t 4 diameters away from station impacted transversely. Solid line is measured curve, broken line is Calculated curve from Timoshenko equation (From Dengler, Goland, and Wickersham [84]).
The Timoshenko theory indicates that a sharp pulse would have its front propagated at the bar velocity and the experimental results presented appear to indicate that this is true. However, the more exact theory (PochhammerChree) predicts a first arrival traveling at the dilatational velocity, followed by waves traveling at all velocities between the dilatational and about 30 percent of the bar velocity. The first arrivals in the experimental work seem to have been propagated at a velocity equal to or slightly greater than the bar velocity, but the amplitudes of the arrivals are too small to allow accurate velocity measurements. It must be pointed out here that the gages were not arranged to eliminate the symmetrical strain components, generated along with anti-symmetrical components by the impact. Hence, the form of the pulses observed may have been altered by the presence of symmetrical components. The authors have shown, however, by means of a set measured outer fiber strains on the top and bottom surfaces of the beam, that the strains are essentially anti-symmetrical. These curves are almost mirror images of each other except at the later times (see Fig. 25). At the later times a high frequency component appears which the Timoshenko theory does not predict and which is not completely anti-symmetrical. This component either represents transmission in one or more of the higher branches which are not provided by the Timoshenko equation, or it arises
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
171
from the symmetrical strains previously mentioned. In any event, these results indicate that the Timoshenko equation predicts transmission characteristics adequately enough for engineering purposes.
1
1
l
1
1
1
1 "1
1
1
1
1
Measurements of short bending wave pulses were reported by Ripperger [102].' These pulses were generated by the eccentric impact of small steel balls on the end of the beam. The strains were measured by means of wire resistance gages (i"gage length) connected to cancel out the symmetrical strain components. Pulse durations ranged from 3.5 to 30 microseconds, and measurements were made at 4, 6, 10, 16, 22, and 28 diameters from the impact end on a 3" diameter bar. These measurements, like those of previous investigators, showed that the change of pulse form during the propagation along the beam is essentially independent of the pulse duration. This continual change, which is shown in Fig. 26, makes it impossible to determine a definite velocity of propagation. The time of arrival of the initial disturbance was too indefinite to be measured; however, if the first positive peak in the pulse is used as a reference point, propagation appears to be a t about 10,000 ft/sec. Comparisons of Ripperger's data with computed results based on the Timoshenko theory were made by Plass and Steyer [19]. The agreement was good for all but the shortest pulses. The elementary theory did not predict the propagational characteristics satisfactorily for any of the pulses (see Fig. 22). In this investigation [lo21 a series of measurements was also made of pulses propagating in beams that were pinned at the impact end. I t was expected that this end condition would alter the pulse forms that appear as a consequence of the effect of the support on the shear in the beam.
172
H. N . ABRAMSON, 13. J . PLASS, A N D E . A. RII’PEKGER
The recorded pulses for the beam with the pinned end appear, however, to be practically identical with those for the free end beam. This may be an (a) Applied Mornen t
Transmitted Moments
(b)x= 2d
(C)X =
6d
( d ) x= 10d
(C)X
=
16d
(f)x = 22d
( g )= ~ 28d
t
FIG. 26. Oscillograph records of bending strains in a simply supported cylindrical steel bar (From Ripperger [102]).
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
173
indication that the end of the beam was not restrained sufficiently to prevent lateral movement, or the shear loading introduced by the support is negligible under the conditions of the experiment.
-
time
10-6 secs
FIG.27. Oscillograph records of pulse transmission of a bending wave in a cylindrical steel bar (From Ripperger and Abramson [119]).
These experiments were later extended by Ripperger and Abramson [119] to pulses having a duration of about 2 microseconds applied to a 4 inch diameter rod. These pulses were generated by the eccentric impact of a 1 le inch steel ball with an impact velocity of 460 ft/sec. Parts of the disturbance were detected traveling at (a) the dilatational velocity, (b) the bar velocity and (c) the shear velocity. These results are shown in Fig. 27. The dilatational disturbance disappears by the time the pulse has traveled 8 diameters. The conclusions drawn from these measurements, when compared with the Pochhammer-Chree theory [89], were that the Timoshenko
'0-20 -
--TIME AFTER PULSE ARRIVAL
M
---
I
x.0
x.+
I 1.
I
It
I
I X.
l
3
l
I x.4f
I
1
I X I
l 6
t
.
I
b (zfz
f*
I
22' CLAMPED DRILL ROD $.3O'SIMPCY SUPPORTED MILD COLD ROLLED STEEL I
x=ri
x = DISTANCE ALONG B E A M ,
I x=g
.
"
I
,
x.12
INCHES
FIG. 28. Progress of major portion of bending pulse in a circular cyIindrical rod (From Cunningham and Goldsmith [IZO]).
i
STRESS WAVE PROPAGATION IN RODS AND BEAMS
175
theory predicts arrival times quite accurately with the exception of the first arrival traveling at the dilatational velocity. This is not serious since the amplitude of this portion of the disturbance is very small. A detailed investigation of the manner in which the form of a bending wave pulse changes as it propagates along the bar was reported by Cunningham and Goldsmith 11201. In this study an effective gage spacing of inch was obtained by making a series of measurements in which the
A
point of impact was shifted inch farther from the gage point for each impact. The results of these measurements are shown in Fig. 28. Note how the pulse, which is initially positive, gradually changes, until a t a point 2 inches from the impact end it is almost completely negative. These results confirm the observations of other investigators qualitatively but add more details. From measurements of the time of arrival of the maximum strain at successive gage stations Goldsmith and Cunningham [121] concluded that the average velocity of propagation of the predominant group is 13,500 ft/sec. Measurements of the earliest arrivals in the pulse indicated that this part of the pulse had been traveling a t the bar velocity. Since the technique used did not eliminate the symmetrical strain component, these early arrivals could well have been influenced by them. Vigness [122] studied the bending waves which were generated in a beam by suddenly putting one end of the beam in motion. The results which he obtained largely confirm what other investigators found.
Concluding Remark. Recent experimental and theoretical investigations [84,119] haveservedtoestablish the adequacy of the Timoshenkobending mechanism for describing the response of beams to sharp impacts. Other investigations, e.g. [98 - 1011, have provided solutions of the Timoshenko equation for a wide variety of impact loads and boundary conditions. In view of these two advances it would appear, as in the case of longitudinal vibrations, that the theory has attained sufficient strength so that attention may be directed in the near future to studies of more complicated beams, such as those containing cross-sectional or material discontinuities 1491. 3. Flexural Plastic Waves
Introduction. Interest in the problem of plastic deformation due to impulsive loading is of comparatively recent origin. Investigations of the longitudinal plastic deformation in a rod under end loading were undertaken in the early 1940’s by Taylor, KBrman, and others as discussed earlier. The first attempt to deal with transverse plastic deformations seems to be that of Duwez, Clark and Bohnenblust [123], reported in 1950. In the ensuing six years this area of investigation has been very active; more than a score of papers have appeared in the unclassified literature.
176
H . N. ABRAMSON, H. J. PLASS, AND E. A. R I P P E R G E R
Efforts in the transverse deformation problem have been directed, with but one exception, towards analyses based on simple beam theory with the moment-curvature relationship being considered generally as elastic-plastic
k ( a ) MOMENT - CURVATURE RELATION:
ELASTIC- PLASTIC.
k (b) MOMENT-CURVATURE RELATION:
RIGID -PLASTIC.
lV'E~~E~~~~l M
N
PERF ECT LY PLAST IC
k ( c ) PERFECTLY PLASTIC AND LINEAR STRAIN HARDENING CHARACTERISTICS. FIG.29. Moment vs. curvature-curves.
(Fig. 29a) or rigid-plastic (Fig. 29b), and the plasticity being described generally as either perfectly plastic, as in Fig. 29a and b, or as linear strain hardening (Fig. 29c). The single exception is a paper that considers shear deformation and rotatory inertia effects and also the effect of strain-rate; it will be discussed in detail later.
STRESS WAVE PROPAGATION I N RODS AND BEAMS
177
I t is indeed unfortunate that the majority of the papers on this subject are simply solutions of special problems or, at best, of special classes of problems; this of course is evidence for the difficulty of the general problem. Because of the difficulty involved in obtaining solutions on the basis of elastic-plastic analyses, it is not yet possible to compare directly the relative usefulness and accuracy of the elastic-plastic and rigid-plastic theories. Further, the importance of the higher order bending effects and strain-rate is not known, and experimental evidence is meager. Practically all of the work to date has emphasized the permanent deformation resulting from the impulsive loading rather than the propagation of the deformation along the beam. The concept of plastic hinge formation has been employed, and while some discussion has been made of the case of a moving hinge, the question of wave propagation alone has not yet been studied to any great extent. Nevertheless, the analyses available at present should be discussed briefly here, not so much for their direct relation to the wave propagation problem, but rather to complete the general picture of the impulsive loading problem, in order to provide an introduction to the transverse plastic-wave propagation analysis based on a more exact theory which follows later. Elastic-Plastic Analyses. As stated previously, the first investigation concerned with the plastic deformation of beams under transverse dynamic loading seems to be that of Duwez, Clark and Bohnenblust [123]. The problem considered is that of an infinite beam subjected to a dynamic transverse constant-velocity loading. The early solution of Boussinesq [124] for elastic beams is extended to the plastic problem by replacing the elastic bending moment-curvature relation by a more general expression. An elastic-plastic analysis is developed which considers both perfect plasticity and linear strain hardening. The analysis is valid only for the time during which the impulse is acting. We repeat here the essentials of the theory, which is due to Bohnenblust, as it is basic to analyses presented in other papers. The equations of motion, neglecting rotatory inertia, are (3.25)
Q=
aM ax
_ - I
(3.26)
The curvature relation is (3.27)
M
= Elk.
178
H. N . ABRAMSON, H. J. PLASS, AND E. .4. KIPPERGER
Using Eq. (3.27), Boussinesq found for the elastic beam that ujt is proportional to x2/t; this is true also in the case of the plastic beam. Therefore, a new function r j is defined by (3.28)
A solution of the form u =tfh)
(3.29)
is assumed, so that
(3.31) Also, (3.32)
Now, a quantity S is introduced by (3.33)
Eq. (3.26) thus becomes (3.34)
S'
+ l/17(2aak - f ' )
= 0,
and by differentiating and using (3.31) and (3.33), (3.35)
S"
dk + E I -S dM
= 0,
is obtained, which is the fundamental differential equation of the theory. From the previous relations, the following expressions may be obtained : m
(3.36)
(3.37)
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
179
(3.38)
(3.39)
This relation makes it possible to show that the constant impact velocity is m
(3.40)
The corresponding impact force P and energy W are given by (3.41)
(3.42)
EI P=-S(0) a3
w = 2EI a3 --
1
-,
V t
VJ(0)
v.
Since all of these quantities depend upon S ( q ) it is clear that a solution of (3.35) is a solution of the problem. Further, it is shown in the paper [123] that these relations satisfy all boundary conditions of the problem so that the assumed form of the solution u = t f(q) is valid. The essential point is the solution of Eq. (3.35), which is complicated because of the nonlinear function d k / d M . A procedure is given in [I231 for a numerical solution of this equation. The authors' conclusion that strain is not propagated along the beam at a constant velocity is consistent with our earlier observation that no definite velocity can be associated with the propagation of an elastic bending wave pulse. I t should be noted, however, that the analysis presented neglects the effects of shear and rotatory inertia. Neglecting these factors in the elastic case led to a physically unrealistic result, and the final equation did not predict propagational characteristics satisfactorily under any circumstances. Under these conditions it should not be expected that the simplified theory for filastic propagation would lead to a satisfactory representation. In an experimental investigation designed to test the theoretical results, cold rolled steel and copper beams 10 ft. long with rectangular cross sections were used. These beams were pinned at the ends and struck at the midpoint with a heavy hammer moving at a measured velocity. The deflection curve at the end of the impact was photographed for comparison with calculated deflection curves.
180
H. N. ABRAMSON, H. J . PLASS, AND E. A. R I P P E R G E R
The relationship between bending moment and curvature required for the solution of (3.35)was obtained by calculation based on experimentally determined stress-strain curves. The calculated curve was then approximated by two straight lines for convenience in solving the equation. The calculated and approximate curves for the two materials are shown in Fig. 30. A comparison of measured and calculated deflections is shown in Fig. 31. For steel the calculated curve based on the elastic theory gives the best
M
?-k COLD-ROLLED
M
STEEL
k ANNEALED COPPER
FIG.30. Actual and approximate moment-curvature curves.
i
EXPERIMENT.
----THEORY. --THEORY (ELASTICI.
I
COLD-ROLLED STEEL
1
ANNEALED COPPER
FIG.31. Theoretical and experimental deflection curves for a long bar subjected t o a transverse impact sufficiently large to produce plastic strains (From Duwez, Clark and Bohnenblust [123]).
STRESS WAVE PROPAGATION I N RODS AND BEAMS
181
fit but for copper the plastic theory gives the best fit. The authors interpret this to mean that the influence of plasticity is much greater for copper than for steel, and point out that in the copper beams plastic strains are found some distance from the point of impact, whereas in the steel the plastic deformation was localized near the impact. The calculations show that the distance from the point of impact to the point where u = 0 varies as a relation which is independent of the shape of the moment-curvature curve. Experimental results indicate that the variation of xo with V t i s linear but the slope of the line representing this variation is much greater than had been calculated. By taking a different approximation for the moment-curvature relationship a better agreement could have been obtained as the authors point out, but this would have made the fit for the deflections much worse. It seems doubtful that much significance can be attached to any agreement, or lack of agreement between the computed deflection curves and the experimental results in view of (a) the neglect of shear and rotatory inertia in setting up the basic equations, (b) the strong possibility that the measured deflection curves were affected by reflections, not accounted for in the theory, from the pinned end, and (c) the approximate nature of the momentcurvature relationship used in the computations. A somewhat different type of elastic-plastic analysis has been developed by Bleich and Salvadori [125]. The analysis is applied to free-free beams moving under dynamic forces and possessing perfect plasticity (Fig. 29a). The motion is expressed in terms of the normal modes of the beam, when the beam is subjected to a given symmetric initial velocity distribution. The final deformation is determined on the basis of the formation of a single plastic hinge at the impact point. By use of the proper continuity relations between displacements and velocities the solutions are established, and the final deformation, in terms of the plastic angle a t the impact point, may be determined for a particular problem. Conroy [126], by a method similar to that of [123], has treated the problem of a semi-infinite beam, subject to a constant bending moment and a transverse force of magnitude inversely proportional to vtapplied a t the free end. Under this type of loading the free end of the beam moves a t a constant velocity, which accounts for the similarity of solutions. A solution for the problem of an elastic-plastic beam (Fig. 29a) subjected to a velocity impulse at the middle cross section, when the acceleration time is small but finite, has been given in [127]. The permanent angle of rotation at the struck cross section is computed on the basis of a single plastic hinge. The permanent deflection predicted by this theory exceeds that observed in recent experiments [128] by an appreciable amount. In [129] a simply supported uniform beam of ductile material, subjected t o impulsive loading such that the initial velocity is a half-sine wave, is
vc
182
H . N . ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER
considered. The elastic-plastic analysis is based on that of Bleich and Salvadori [125] and retains the assumption of a single plastic hinge. I t is pointed out that the principal difficulty of the elastic-plastic problem is that of satisfying the moment-plasticity condition throughout the beam. The authors of [129] suggest that one way of obtaining an adequate approximate solution to elastic-plastic problems might be to insert, at some stage of the deformation, a central plastic zone whose length is chosen arbitrarily. Thomson [130] has employed a normal mode superposition method to several elastic-plastic beam problems, the solutions being obtained on an analog computer. The plastic action is again simulated by plastic hinges.
1'
I
I
FIG. 32. Schematic description of rigid plastic bending.
Rigid-Plastic Analyses. Following a suggestion by Prager, an essentially different type of analysis was developed by Lee and Symonds [131]. If the final strains in a beam are large compared with the elastic strains, it may be sufficient to consider the behavior characterized by a rigid-plastic beam (Fig. 29b). The basic assumptions of such an analysis are: (a) negligible elastic strains, (b) segments of the beam undergoing rigid-body motion are joined by plastic hinges where the entire relative motion is assumed to take place. This analysis, in contrast to the elastic-plastic analysis of Bohnenblust [123], includes also the motion after impact. The motion is described in three phases. In phase I, M < M,, so that there is only rigd-body translation. In phase 11, M = M , at the central impact point so that the two halves of the beam rotate with respect to each other as though the beam were hinged in the middle. In phase 111, the bending moment at cross sections a certain distance from the mid-point
STRESS WAVE PROPAGATION I N RODS A N D BEAMS
183
attains the value M , with sense opposite to M , at the midpoint. Thus there are three plastic hinges to be considered; one remains at the struck point while the other two move outward. A pictorial representation of the three phases of the motion is shown in Fig. 32. The analysis consists essentially of writing the dynamical equations for each stage of the motion, as described above, writing the kinematic conditions at the hinge, and finally writing the equations of motion of beam segments on either side of the hinge. Continuity conditions at a traveling hinge are also discussed. The theory is applied to a free-free beam of finite length under the action of a triangular impulse. It may be noted that the solution is to be obtained by successive approximations, although the analysis presented does not go beyond the first such approximation. A major objective of the paper is an attempt to formulate a criterion for the use of the rigid-plastic analysis for finite beams. The criterion given is (3.43)
where t is the fundamental period of elastic vibration of the free-free beam, 8, is the final central angle of deformation, T is the duration of the impact, and m is the beam mass per unit length. For a given beam and a given value of maximum force P,, this relation sets a lower limit for T , above which the present analysis can be expected to give satisfactory results. Pian [132] employed the same basic analysis to determine the plastic strains in a simply supported beam subjected to a concentrated impulsive load at the center of the beam. For a rectangular impulse, the solution is obtained in closed form. The problem treated by Bohnenblust [123] has also been studied by a rigid-plastic analysis [133]. This leads to a discussion of the two outward traveling hinges in the case of an infinitely long beam. Both perfectly plastic and linear strain-hardening materials are studied, although no analytical results are obtained for the latter. Symonds [134] also considered the rigid-plastic analysis [131] when a force function of square wave type is applied. An empirical formula is offered by which the plastic deformation can be estimated from knowledge of the impulse shape and the peak value. This same analysis was further extended [ 1351 to provide numerical results for concentrated force pulses of rectangular, half-sine, and triangular shape. The calculations show that the central angle of permanent deformation for all three cases can be obtained from the empirical relation
8, = C J 2 P?, (3.44) where J is the impulse and C depends on the dimensions and proportions of the beam and a numerical factor (slightly different for each pulse shape).
184
H. N. ABRAMSON, H. J. PLASS, AND E . A. RIPPERGER
This result is valid as long as M does not reach M , at any section other than at the center or the two traveling hinges. A true impact problem was studied by Symonds and Leth [136] who treated a finite beam whose mid-section acquires a given velocity in zero time. The basic method of Lee and Symonds [131] was again employed. The case of a distributed load studied by a rigid-plastic analysis was presented by Seiler and Symonds [137], the pulse being rectangular in shape. The same problem was treated by Salvadori and Di Maggio [138], for a smoother but more complicated load distribution function ; the normalmode method [125] was employed. Several cases of simply supported and built-in beams, loaded by uniform pressure or by a concentrated central force that is a specified function of time, were solved by Symonds [139]. A problem involving an impact velocity variable in time, with consideration of the motion following unloading, was presented by Hopkins [140]. An interesting study of the case of linear strain hardening by the use of the rigid-plastic theory was made by Conroy [141]. Considering finite beams, the author noted that if the rate of change of curvature caused by the loading is of the same sign (or zero) along the entire beam, the differential equation has the same form as in the elastic case. With this idea in mind, three problems are discussed : (a) initial-motion problem using superposition, (b) initial-stress problem for a simply supported beam using superposition, and (c) a free-boundary problem for a uniform beam. In the first two problems the beam was initially plastic and remained entirely plastic for all time so that no rigid-plastic boundaries were present. The third problem typifies a case in which the beam is initially rigid; the application of a load produces a moving rigid-plastic boundary. The determination of the free boundary in the third problem was made by the inverse method. Cotter and Symonds [142] treated an earlier problem [125] by considering various zones of plastic distortion along the beam. I t was found that these precede the ultimate rigid-body motion a t constant velocity. A rigidplastic analysis, considering only a single plastic hinge, was also made for this problem by the original authors [125]. The method of Bleich and Salvadori was also applied to a simply supported rigid-plastic beam by Symonds [143] ; the specific problem was one treated previously [lag]. The solution of Symonds and Leth [136] was extended in [144] to include finite acceleration time. Mentel [145] applied the analysis of Lee and Symonds [131] to simply supported and built-in beams with an attached mass and subjected to a uniform pressure pulse of rectangular shape. Rigid-plastic analyses presented in connection with elastic-plastic analyses are contained in several papers [125 - 1271.
STRESS WAVE PROPAGATION IN RODS AND BEAMS
185
The foregoing represents a rather formidable array of special solutions ; they have all been summarized in Table 2 for the convenience of the reader. TABLE2. type of analysis Ref.
SUMMARY O F SOLUTIONS
plasticity
beam
loading elasticplastic
rigid- perfect strain plastic hardening
123 infinite 125 finite 126 semi-infinite
x
X
x
127 finite
X
X
X
129 130 131 132 133 134 135 136 137 138
finite finite finite finite infinite finite finite finite finite finite
X
X
X
139 140 141 142 143 144 145
finite finite infinite finite finite finite finite plus added mass
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
x
X
X
X
X
X
X
X
X
X
X X
constant velocity velocity impact constant moment plus force velocity impulse with finite acceleration time half-sine wave velocity concentrated force rectangular pulse square pulse constant velocity square pulse distributed load variously shaped pulses constant velocity distributed load, rectangular pulse variously shaped pulses various time-dependent velocity velocity distribution pressure pulse half-sine wave velocity finite acceleration time
Comfiarisons Between Elastic-Plastic and Rigid-Plastic Analyses. It is to be emphasized at the outset that both the elastic-plastic and the rigidplastic theories have their usefulness. For those cases in which the final strains are not large compared with the elastic strains, the elastic-plastic theory must be employed. Several attempts to establish criteria for the applicability of the rigid-plastic theory have already been mentioned; here we will mention only those few cases in which a direct comparison of the two theories was attempted. A direct comparison was made in [125] but no definitive statement could be made regarding the comparison. Again, no conclusive statement was formulated in the comparison of Conroy [126].
186
H. N. ABRAMSON, H. J . PLASS, AND E. A. R I P P E R G E R
Alverson [127] compared his elastic-plastic solution with the rigidplastic solutions of [136] and [144], but again the comparison was not conclusive. A direct comparison was also attempted by Seiler, Cotter and Symonds [129], but it was found again that the solutions have regions of validity which do not overlap so that comparison is difficult. Strain-Rate Theory. The theories of plastic bending deformation under impulsive loads are all inadequate, when the effects of suddenly applied transverse or bending loads are to be investigated, because of their dependence on elementary beam theory. I t was pointed out previously that the elementary theory predicts an infinite propagational velocity while physical observations reveal finite propagational velocities. A theory which considers these significant influences of rotatory inertia and finite shear rigidity on plastic wave propagation has been given by Plass [25]. In addition to the above mentioned effects, the theory also includes the dependence of the material properties on the rate of strain. Although this last inclusion might at first appear to complicate the theory, it really has the opposite effect. Calculations of stresses and velocities are made easier because of the fact that straight-line characteristics independent of the state of strain are predicted, with consequent simplification in the numerical work. The assumptions about the strain-rate effect are generalizations of the linearized form used by Malvern 1241 for longitudinal waves. It is shown by Plass that the equations connecting moment and shear with the strains and velocities are as follows (see Fig. 21) :
EJ
aK aM at = __ + k ( M - M ) , at material behavior)
(3.45) /@s
ay
-=
at
aQ %
+ k(Q
-
Q);
}
(3.46)
at
(equation of motion)
ax
(equation of continuity),
(3.47)
at
ax
187
STRESS WAVE PROPAGATION I N RODS AND BEAMS
where
E,I ,uoA, k _M,Q
= static flexural rigidity = static shear rigidity = strain-rate constant = static
values of moment and shear for a state of strain corresponding to the one existing dynamically; the yield condition of von Mises is assumed,
and the other symbols are as previously defined. Equations (3.45)apply to loading; for unloading the terms containiiig k are set equal to zero. The above equations can be solved by means of the method of characteristics in very much the same way as the Timoshenko equation was solved. The characteristics are integrals of the following differential equations: d x = 0, dx =
(3.48)
[twice) codt,
dx = f CQdt.
I t is easy to see that the characteristics are all families of straight lines in the x - t plane. Corresponding to each family of characteristics is an ordinary differential equation as follows: EoIdK - dM
= k(M - M ) d t ,
along
dx=O
p o A & - dQ
= k(Q - Q ) d t ,
along
dx=O
along
d x = codt
coQ - k ( M - @))it,
along
d x = - codt
dQ - pAcQdv = [ p A c t w - k(Q - Q)]dt,
along
d x = cQdt
along
d x = - cQdt
dM - plcodw = [coQ - k ( M - M ) ] d t , (3.49)
dM
dQ
+ plcodw = [-
+ pAcQdv = [ p A c i w - k(Q -a]&.
Again, the above equations are used for loading; for unloading, the terms which are multiplied by k are set equal to zero. Details of the boundary conditions and a method of solution for the problem of a step moment and step angular-velocity impact on the end (assumed supported by a pin to a rigid base) of a semi-infinite beam are found in the paper by Plass [25]. It is of interest to point out here that the zones of plastic flow, i.e., permanent deformation, occur only near the impact end and in a thin zone near the bending wave front. The remainder of the beam behaves elastically.
188
H . N. ABRAMSON, H . J. PLASS, AND E. A. RIPPERGER
In the case of the applied step moment, essentially no plastic strains occur, as unloading occurs almost immediately after the passage of the wave front. However, in the case of the step angular velocity, a zone of plastic strain forms near the impact end and lengthens with time, but a t a slower rate than the velocity of propagation c,, of the bending wave front. The theory just described would need to be applied only when very sudden loads are involved. When the time of rise of the load or moment, from zero to maximum value, is of the order of the time required for a bending wave (velocity c,,) to travel from the impact end to the far end and back again, an elementary theory similar to that of Duwez, Clark, and Bohnenblust [123] should be adequate to describe the behavior of the beam. Concluding Remark. The inadequacy of experimental data makes it very difficult to judge the present state of development of plastic bending wave theory. Even the theoretical analyses of the final permanent deformation in a beam resulting from an impulsive load, discussed a t some length here, have not been subject to critical appraisal in the light of experimental results. Progress in the development of an adequate theory of wave propagation has been made [25], but this has come about almost exclusively from knowledge of plastic longitudinal and elastic flexural wave propagational characteristics. Further progress must rest on the availability of reliable and adequate experimental data.
ACKNOWLEDGMENT
Portions of the work reported in this paper were supported by the Bureau of Ordnance, Department of the Navy, under a contract with the University of Texas, Defense Research Laboratory. The authors’ institutions generously provided funds and services to assist in the preparation of this paper.
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103. LEONARD, R. W., and BUDIANSKY, B., On traveling waves in beams, Natl. Advisory Comm. Aeronaut. Tech. Note 2874 (1953). 104. ANDERSON, R. A,, Flexural vibrations in uniform beams according to the Timoshenko theory, J . A p p . Mechs. 20, 504-510 (1953). 105. ANDERSON, R. A,, Wave groups in the flexural motion of beams predicted by the Timoshenko theory, J . A p p . Mechs. 21, 388-394 (1954). 106. DOLPH,C. L., On the Timoshenko theory of transverse beam vibration, Quart. A p p . Math. 12, 175-187 (1954). 107. HOWE,C. E., and HOWE,R. M., Application of the differential analyzer to the oscillation of beams, including shear and rotatory inertia, J . A p p . Mechs. 22, 13-19 (1955). 108. IMACHI,I., On the lateral mass impact applied to a long uniform bar with two flexural freedoms - bending and shearing, Mem. Fac. Engvg., Nagoya Uniu. 2 (1950). J., Flexural wave solutions of coupled equations representing the 109. MIKLOWITZ, more exact theory of bending, J . A p p . Mechs. 20, 511-514 (1953). 110. MINDLIN, R. D., and DERESIEWICZ, H., Timoshenko’s shear coefficient for flexural vibrations of beams, Proc. 2nd National Cong. A p p . Mechs., pp. 175-178 (1955). hl. K., Effect of rotatory inertia and shear on maximum strain in 111. NEWMAN, cantilever impact excitations, J . Aeronaut. Sci. 22, 313-320 (1955). R. W., and COLLAR,A. R., The Effects of shear flexibility and 112. TRAILL-NASH, rotatory inertia on the bending vibrations of beams, Quart. Jour. Mechs. and A p p . Math. 6, 186-222 (1953). 113. JONES,R. P. N., Transient flexural stresses in an infinite beam, Quart. Jour. hlechs. and App. Math. 8, 373-384 (1955). 114. BARR,A. D. S., Some notes on the resonance of Timoshenko beams and the effects of lateral inertia on flexural vibration, 9th Int. Conf. on A p p . Mech., Brussels (1956). 115. MASON,H. L., J . A p p . Mechs. 3, 55 (1936). 116. ARNOLD,R. N., Impact stresses in a freely supported beam, Proc. Inst. Mech. Engrs. 137, 217 (1937). 117. LEE, E. H., J . Appl. Mechs. 7, 129 (1940). C. O., DRUCKER,D. C., and MOORE, P., Transverse impact 118. DOHRENWEND, transients, Proc. S E S A 6, 1-11 (1943). 119. RIPPERGER, E. A., and ABRXMSON, H. N., A Study of the Propagation of Flexural Waves in Elastic Beams, Univ. of Texas Defense Research Lab. Rept. DRL-378, CM-864 (1956). Also J. A p p . Mechs. 24, 431-434 (1957). 120. CUNNINGHAM, D. M., and GOLDSMITH, W., An experimental investigation of beam stresses produced by oblique impact of a steel sphere, J . A p p . Mechs. 23, 606-611 (1956). 121. GOLDSMITH, W., and CUNNINGHAM, D. M., Kinematic phenomena observed during the oblique impact of a sphere on a beam, J . A p p . Mechs. 23, 612-616 (1956). 122. VIGMESS,I., Transverse waves in beams, Proc. S E S A 8, 69-82 (1951). 123. DUWEZ,P. E., CLARK,D. S. and BOHNENBLUST, H. F., The behavior of long beams under impact loading, J . A p p . Mechs. 17, 27-34 (1950). 124. BOUSSINESQ, J., “Applications des Potentiels”. Paris, 1855. 125. BLEICH,H. H., and SALVADORI, M. G., Impulsive motion of elastic-plastic beams, ASCE Sep. 287 (1953). 126. CONROY, M. F., Plastic deformation of semi-infinite beams subject to transverse impact loading a t the free end, J . A p p . Mechs. 23, 239-243 (1956). 127. ALVERSON, R. C., Impact with finite acceleration time of elastic and elasticplastic beams, J. A p p . Mechs. 23, 411-415 (1956).
194
H. N. ABRAMSON, H. J. PLASS, A N D E. A. RIPPERGER
128. LMENTEL, T. J., GREEN.D. J., and SYMONDS, P. S., Plastic deformation of beams in impact - theory and preliminary tests, Brown Univ. Tech. Rept. ONR N7-onr-35801 (1956). 129. SEILER,J. A., COTTER,B. A., and SYMONDS, P. S., Impulsive loading of elasticplastic beams, J . A p p . Mechs. 23, 515-521 (1956). W. T., Impulsive response of beams in the elastic and plastic regions, 130. THOMSON, J . A p p . Mechs. 21, 27-287 (1954). 131. LEE, E. H., and SYMONDS, P. S., Large plastic deformations of beams under transverse impact, J. A p p . Mechs. 19. 308-314 (1952). 132. PIAN,T. H. H., A note on large plastic deformations of beams under transverse impact, Proc. 8th International Congress of A p p . Mech. (1952). M. F., Plastic-rigid analysis of long beams under transverse impact 133. CONROY, loading, J . A p p . Mechs. 19, 465-470 (1952). P. S., The influence of load characteristics on plastic deformations of 134. SYMONDS, beams under concentrated dynamic loading, Brown Univ. Tech. Rept. NO. 9, ONR N7-onr-35810 (1952). 135. SYMONDS, P. S., Dynamic load characteristics in plastic bending of beams, J. A p p . Mechs. 20, 475-481 (1953). 136. SYMONDS, P. S., and LETH, C. F. A., Impact of finite beams of ductile metal, J . Mechs. Phys. Solids 2, 92-102 (1954). 137. SEILER,J. A., and SYMONDS, P. S., Plastic deformation of beams under distributed dynamic loads, J . A p p . Phys. 25, 556-563 (1954). 138. SALVADORI, M. G., and DIMAGGIO,F., On the development of plastic hinges in rigid-plastic beams, Quart. A p p . Math. 11, 223-230 (1953). P. S., Large plastic deformations of beams under blast type loading, 139. SYMONDS, Proc. 2nd US. National Congress of A p p . Mechs., pp. 505-515 (1954). 140. HOPKINS, H. G., On the behavior of infinitely long rigid-plastic beams under transverse concentrated load, J . Mechs. Phys. Solids 4, 38-52 (1955). 141. CONROY.M. F., Plastic-rigid analysis of a special class of problems involving beams subject t o dynamic transverse loading, J. APPZ. Mechs. 22, 48-52 (1955). 142. COTTER, B. A,, and SYMONDS, P. S., Plastic deformations of a beam under transverse loading, ASCE Sep. 675, 1955. P. S., Simple solutions of impulsive loading and impact problems of 143. SYMONDS, plastic beams and plates, Brown Univ. Rept. No. 3, N 1895-1756 A (1955). 144. GREEN,D. J., The effect of acceleration time on plastic deformation of beams under transverse impact loading, Brown Univ. Rept. A 11-112, ONR N7onr-35801 (1956). 145. MENTEL,T. J., Plastic deformations due t o dynamic loading of a beam with an attached mass, Canad. Jour Tech. 33, 237-255 (1955).
Problems in Hydromagnetics BY EDWARD A . FRIEMAN
AND
RUSSELL M . KULSRUD
Princeton University. Princeton. New Jersey Page I . Introduction . . . . . . . . . . . . . . . . . . . . I1. Fundamental Equations . . . . . . . . . . . . . . . 1. The Equations and Their Validity . . . . . . . . 2 . The Boundary Conditions . . . . . . . . . . . . I11. General Processes . . . . . . . . . . . . . . . . . . 1 . Motion of Magnetic Lines of Force . . . . . . . . 2 . Validity of Approximation of Infinite Conductivity . 3 . Conservation of Energy . . . . . . . . . . . . . 4 . Three Limits . . . . . . . . . . . . . . . . . . 5 . Hydromagnetic Equilibria . . . . . . . . . . . . IV. Stability of Hydromagnetic Equilibria . . . . . . . . 1. Normal Mode and Energy Methods . . . . . . . . 2 . The 6W Formalism . . . . . . . . . . . . . . . 3 . Applications of the SW Formalism . . . . . . . . 4 . The Pinch Effect . . . . . . . . . . . . . . . . 5 . Minimization Technique for Expansion Problems . 6. An Axisymmetric Problem; Coordinate System . . . 7. An Axisymmetric Problem; Stability . . . . . . . 8. Physical Interpretation . . . . . . . . . . . . . . V. Hydromagnetic Waves . . . . . . . . . . . . . . . . 1. Introductory Remarks . . . . . . . . . . . . . . 2 . Three Modes . . . . . . . . . . . . . . . . . . 3 . Equation of Propagation . . . . . . . . . . . . . 4 . Energy and Wave Generation . . . . . . . . . . 5 . Group Velocity . . . . . . . . . . . . . . . . . 6. Reflection and Refraction ; Boundary Conditions . . 7. Angles of Reflection and Refraction . . . . . . . . 8. Transmission and Reflection Coefficients . . . . . . 9 . Damping . . . . . . . . . . . . . . . . . . . . 10. More General Waves . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
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216
. 217 . 220 . 221 . 222
. 223 . 225 . 227 . 229 . 231
I . INTRODUCTION The subject of hydromagnetics is concerned with the interaction between magnetic fields and the motions of a highly conducting fluid . This field of physics is governed by classical equations of motion. and its phenomena 195
196
EDWARD
A.
FRIEMAN AND RUSSELL M. KULSRUD
could have been investigated by the 19th century physicists following Maxwell since all the necessary concepts were known at that time. However, it is only in the past 20 years that hydromagnetics has been actively considered. A possible explanation for the late development of this field is that one impetus for its study has come from modern astrophysics where the importance of magnetic fields in astrophysical phenomena has only recently been appreciated. Hydromagnetics has also become important in applications to such varied phenomena as controlled thermonuclear reactors, the theory of the aurora, terrestrial magnetism, and flow of liquid metals. There are five excellent review articles in this field; Elsasser [l, 21, Lundquist [3], Chandrasekhar [a], and a book by Spitzer [5]. In consideration of these review articles, no attempt at completeness has been made in this paper. We have chosen rather to treat certain branches of the subject with some detail indicating the approach to the solution of problems. Section IV on the Stability of Hydromagnetic Equilibria and Section V on Hydromagnetic Waves contains much previously unpublished material and material which has not previously been reviewed.
11. FUNDAMENTAL EQUATIONS 1. The Equations and Their V a l i d i t y We consider a highly conducting fluid in which magnetic fields may be present. The equations governing such a fluid can be taken to be
C7
dv dt
x
B
p -= -
4n
=-j, C
x B op + jC
dp =--pV.V, dt E + - v- -x, B - j C
CT
197
PROBLEMS IN HYDROMAGNETICS
We use Gaussian units; E is the electric field, B the magnetic field, j the current density, p the mass density, v the fluid velocity, the pressure, c the velocity light, (T the electrical conductivity, and y the ratio of specific heats. We use the convention that d/dt is the total time derivative following the motion (“material derivative”). The first three equations are the usual Maxwell equations with displacement currents neglected. This approximation is valid if the fluid velocity is small compared to c. The term in the momentum balance equation (2.4) arising from the action of a field E on the charge density is also relativistically small and is neglected. Therefore, the Maxwell equation E = 4 n ~where E is the charge density, serves only to determine E . We have further neglected the viscosity terms in (2.4) and have assumed that the pressure # is always a scalar. Equation (2.5) is the usual equation of continuity. Equation (2.6) is Ohm’s Law for a moving fluid. The term v x Blc arises fiom the fact that the current is produced by the electric field seen by the moving matter. Equation (2.7) states that the change of state is isentropic and assumes a perfect (compressible) gas in which heat conduction is negligible. For an incompressible fluid no such equation is needed to complete the system. For a highly ionized gas these equations apply in many situations. See, for instance, the book by Spitzer [ 5 ] .
+
v*
2. The Boundary Conditions
The system is described by fifteen scalar equations in fifteen unknowns. In a vacuum region adjacent to a hydromagnetic fluid (2.1) - (2.3) apply with j = 0. At the interface the two sets of equations are connected by the following set of boundary conditions. Let n be a unit normal on the interface pointing into the vacuum and let {X}= X,, - X,,,. Then
(2.9) (2.10) (2.11) (2.12)
I-“,“
{ n . { B } =o,
n x B --K,
{#
+} ;
= 0,
n . { v }=o.
The above boundary conditions are obtained by integrating the respective equations (2.1) - (2.5) across the interface. Relations (2.6) and (2.7) yield
198
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
no boundary conditions. K is the surface current density. We have assumed no surface distribution of mass. Therefore, if B has a component normal to the surface, K must be chosen to be zero since otherwise infinite accelerations would arise. Thus, in this case, by (2.9)and (2.10)B must be continuous and no refraction of the lines of force is possible. If n * B is zero, on the other hand, a jump in the tangential component of B is allowed. I n this case K need not be zero, and (2.9) merely serves to determine it. Equation (2.12) is not significant at a fluid-vacuum interface since v is undefined in the vacuum. It should be noted that the above set of boundary conditions apply equally well to any interface in the fluid. 111. GENERALPROCESSES 1. Motion of Magnetic Lines of Force
Almost by definition, the subject of hydromagnetics, is concerned with these equations in the limit of large or infinite electrical conductivity. This is because the interaction between the fields and the fluid mass motion only becomes strong in this limit. I n the opposite limit, of small conductivity, the motion of the system is treated by ordinary hydrodynamics, while the behavior of the fields is treated by electromagnetic theory. If the conductivity can be taken to be infinite, the matter can be regarded as strictly tied to the lines of force of the magnetic field. Thus the motion of the fluid completely determines the behavior of the field. To see this we combine (2.1) and (2.6) to obtain
Using (2.2) we obtain
-1-a-B_-_ 1 c
at
0
c
1
4n
B x B x B- -V C
x
(V
x B),
and by (2.3) we finally get (3.3)
aB =
at
vx
(V
x B) +-
C2
4na
V2B.
Assume that (T is so large that the second term on the right hand side of (3.3) may be neglected. Let us consider two fluid particles separated by a distance A f on the same line of force. Thus (3.4)
df x B = 0 .
199
PROBLEMS IN HYDROMAGNETICS
We wish to show that a t any later time the line of force which passes through one particle will pass through the other. That is, that (3.5)
d -AZ at
xB
= 0. \
+d
Note from Figure 1 that
\
From (3.3)
(3.7)
& at
U
(v+Avldt,
aB at
- B = -+v.VB
FIG. 1. Diagram illustrating the change of distance A1 on a line of force.
=B.VV-BV-W. Thus
(&Al)
x B + A Z x dB dt
= ( A Z * V V )x B + A l x [ ( B * V ) U - B V * W ] . From (3.4) we see that (3.8) vanishes, which proves that adjacent particles stick to the same line of force and, therefore, that any two particles originally on the same line of force will remain on that line. This result enables us to identify lines of force at different instants, giving the lines of force a type of physical reality which they do not in geqeral possess, since there is usually no meaningful way to label them from instant to instant. The elementary definition of lines of force demands that the number of lines crossing a unit area perpendicular to them, i.e. the flux, is equal to the magnitude of the field. A sensible labeling of the lines should preserve this definition in the course of time. Such a labeling is called flux-preserving. The labeling given by the particles on the lines is of this type. This subject has been elaborated by Newcomb [6], and he gives the conditions that a general electromagnetic field must satisfy to have a flux-preserving labeling. To show that our labeling is flux preserving, consider a small area A S which follows the motion of the fluid. If A@ is the flux through this area we must show that (3.9)
d -A@ at
d
= -B
at
*
AS
= 0.
200
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
To calculate d A S / d t consider the volume dt conservation 1 d 1 dP O=--pddt=-----dt+-Az. P dt P dt
(3.10)
a
-AT=
(3.11)
= A S . AZ.
Then from mass
d
at
V.YAz=dZ.(VY).dS+dZ.-~S, d
at
at
where we have used (3.6) for dAZ/dt. Thus from (3.3) and (3.11) (3.12)
a
-B * A S = [B * VY - BV
at
*
V]
+
* A S B * d SV
*
Y
- B * ( VV)* d S = 0,
where we have used the fact that B and AZ are parallel. 2. Validity of Approximation of Infinite Conductivity
We now ask when the conductivity is large enough for the above considerations to apply. Or alternatively, when can the third term in (3.3) be neglected ? Define T to be a characteristic time and L to be a characteristic length for the system. We can then write (3.3) in order of magnitude as (3.13)
where (3.14)
The criterion for large conductivity is that T << TDecay or Llv << Toecay. is the time in which a static magnetic field decays on account of the TDecay production of joulean heat in the hydromagnetic fluid. The first criterion may be written in the alternative fashion (3.15)
T TDecay
-=
6 -<<1, L
where 6 is the usual “skin depth” corresponding to a time T . As an example of the application of these criteria we consider liquid sodium. The conductivity a t 700 OF is 4.5 x l0ls esu. Thus (3.14) yields (3.16)
TD,y = 6.3 X
L2sec.
201
PROBLEMS IN HYDROMAGNETICS
Choosing L to be 1 meter, we find TDecay = 6.3 seconds; thus for times short compared to 6 seconds liquid sodium behaves as though it had infinite conductivity. This criterion can also be conveniently described in terms of a magnetic Reynold's number R, VL RM=--,
(3.17)
VkI
where v M = 4ng/c2 is the magnetic viscosity and RM >> 1 corresponds to large conductivity. For more discussion of this point of view, see Elsasser [I, 21.
3. Conservation of Energy
In the infinite conductivity limit, the system of equations (2.1) - (2.7) and their associated boundary conditions is conservative if we assume the fluid and vacuum regions to be surrounded by an infinitely conducting rigid wall. Since the wall is rigid, no mechanical work can be done on the wall, and since the wall is perfectly conducting, no joulean heating can occur. Hence, no energy can be transferred from the system to the wall. We shall demonstrate this property explicitly for the case that B * n = 0 a t the wall and the system consists of only a hydromagnetic fluid. The total energy & of the system is (3.18)
and
From (2.5), (2.4), (3.3) with infinite conductivity and (2.7)
s=
-vp+7-
\(--v.(p 1v ) v 2 + Y .
dt
Y
,
2
(3.20)
B 4n
1
+---*FX
pv' vv
j x B ( v x B ) + - ~ -[- v . Y-1
I
\
v$l- y $ l v . v ]I at.
Collecting terms and integrating by parts we have (3.21)
/(-
g = at
1
Y
- F . ( p v v 2 ) --F*(pv)+-V* 2 Y-1
1
4n
[(v x B) x B]
202
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
By Gauss’ theorem (3.22)
dd&t = ~ d S ( - $ p v . n u 2 - - p vY~ n + - n - [ (1v x Y-1
B) x B ]
4n
and we see that this vanishes upon using the conditions (3.23)
4. Three Limits The energy integral, equation (3.18), shows that the energy appears in three forms: el = 4 pv2,
The kinetic energy per unit volume,
B2 e2 = -- , 8n
The magnetic field energy per unit volume,
e3 =
~
*
Y-1
, The compressional or “internal” energy per unit volume.
Three interesting types of situations arise as each one of these kinds of energy is negligible compared to the others. If e2 is small, the reaction of the field on the matter is small, and the velocity field is determined hydrodynamically. The field can then be found from equation (3.3). This process can be viewed as an expansion in powers of e2. The next order in v can be found from the lowest order B etc. Secondly, if el is small, the situation in lowest order is a static equilibrium. The higher order terms in the motion can be treated by the normal-mode method and are closely bound up with the stability of the equilibrium state. This is treated in more detail in Section IV. Third, if e3 is negligible, hydromagnetic wave propagation can arise. This is treated in Section V. 5. Hydromagnetic Equilibria
Before proceeding to the study of hydromagnetic phenomena changing in time, let us consider static situations. These situations fall under the second case in which the kinetic energy el is negligible. We restrict ourselves to situations in which the conductivity is infinite and the fluid velocity is everywhere zero. This defines a static equilibrium
203
PROBLEMS I N HYDROMAGNETICS
state. I t is to be noted that if u were not infinite the only static equilibrium is the one with uniform field. Otherwise, the lines of force would diffuse through the matter until a uniform field is reached. In static equilibrium, (2.1) - (2.7) reduce to (3.24)
j x B pp=---.
(3.25)
4nj VxB=----,
(3.26)
V*B=O,
(3.27)
E = 0.
C
C
Note that the density p disappears from the equations so that in equilibrium any density distribution may be assumed. There exist some interesting consequences of these equations. From (3.25) we see that V - j = 0 so that there exist current “lines” in analogy with the magnetic lines of force. From (3.24) it follows that B . =0 and j v p = 0 so that the surfaces of constant pressure contain the j and B lines. Equations (3.24) and (3.25) can be combined to yield
v$
-
(3.28)
1
VP = 4,z (V X B ) x B,
or (3.29)
The term V B 2 corresponds to what is usually termed the magnetic pressure and represents the resistance of the lines to compression. The B VB term corresponds to a magnetic tension and represents the resistance of the lines to bending. Let us consider an infinitely long cylinder with the magnetic field in the axial direction and with a non-vanishing pressure. Since B does not vary along the cylinder, (3.26) is trivially satisfied. Further, B * VB vanishes and (3.29) is satisfied by choosing
-
(3.30)
B2
P+,n=
const.
204
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
By (3.25),j is in the azimuthal direction. If we choose 9 to decrease radially and vanish a t some finite radius, B must increase radially and the hydromagnetic fluid can thus be confined by the magnetic field. Other examples of equilibrium states will be given in Section IV.
IV. STABILITY OF HYDROMAGNETIC EQUILIBRIA 1. Normal Mode and Energy Methods
A method of obtaining controlled release of thermonuclear energy involves the containment of a hot, fully ionized gas, deuterium or tritium, by a magnetic field. The equations governing the equilibrium state in such a gas have been discussed in Section 111. We turn now to the question of the stability of these equilibrium states. Instability is used here to mean that any perturbation applied to the system results in an increasing departure of the system from the equilibrium state. The determination of the stability of a system has, historically, been accomplished in two major ways. The first and more usual method is the normal-mode analysis. In this case the linearized perturbation equations are solved for the normal modes of oscillation and instability about a given equilibrium state. The energy method, on the other hand, examines the sign of the change in the potential energy of the system under all displacements. If the potential energy decreases under some displacement, then kinetic energy is available for motion away from the equilibrium state. The advantage of this second method is that much more difficult qnd more general problems may be solved. However, we do give up some knowledge because the energy principle tells us neither the exact structure of the normal modes nor accurate eigenfrequencies. Lundquist [7] was the first to apply the energy method to hydromagnetics. We shall review the extension and generalization of his work which was done by Bernstein, Frieman, Kruskal and Kulsrud [S]. We may illustrate the method by a trivial example from dynamics. Consider the stability of a ball on an arbitrary surface. Let W ( x ) be the potential energy a t x . Then
where xo represents the equilibrium point. Define
sw = W ( x )- W ( x o ) , 6= x - xo;
205
PROBLEMS I N HYDROMAGNETICS
then (4.1) becomes
aw
+2
4 2 a2w - 7 ( X o ) .
6 W = t - (ax xo)
(4.3)
ax
The equilibrium state is defined by the vanishing of the force - aW!ax. Thus 6W finally becomes (4.4)
Note that GW is a quadratic function of the displacement E and depends on properties of the equilibrium state. If 6W is negative then the ball is unstable at the position xo and will move away from it. 2. The GW-Formalism
In a similar way, we now construct 6W for a hydromagnetic fluid in which the fluid velocity is zero in the equilibrium state and which is surrounded by a rigid perfectly conducting wall. In analogy to classical mechanics we assume 6W to be
fjw=
(4.5)
z
- 1
/
5 * SF(E,)dz,
where 5 is the vector displacement from the equilibrium state, F is the force acting a t the undisplaced position
and the integral is carried out over the undisplaced fluid volume. The change in a quantity X a t a fixed point in space is denoted by 6 X . The first order changes in p , j and B caused by the displacement E, can be obtained from (2.1) - (2.7). Thus, by definition, (4.7)
and we see that v
=
aE,/at to lowest order in
E,. From
aB -=VX(VXB)
at
so that to first order in (4.9)
,E SB = V x (5 x B ) G Q .
(3.3)
206
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
Similarly we obtain
sp= (4.10)
-ypP.g-g*Pp, C
Sj= - V 4n
xQ.
We, therefore, can write
@ xC B
(4.11)
+ j-]ar. x6B C
J
Integrating by parts, we
As in the demonstration of energy conservation in Section 111, the surface terms vanish in consequence of our assumptions that the boundary is rigid and perfectly conducting. We now quote the extension of the principle to the case where there is a vacuum region in which there exists a first order vector potential A surrounding the hydromagnetic fluid. The vacuum region is in turn enclosed by a rigid, perfectly conducting wall. In this case 6W is given by
(4.13)
6W
= SWf
+ 6WS + 6W",
where
(4.14) and
(4.15)
's
6Wv = 87C
x A)'.
The boundary condition which must be satisfied a t the fluid-vacuum interface is
(4.16)
n x A=
- (n.g)B,
while that at the rigid wall surrounding the vacuum is
(4.17)
nxA=0.
The utility of this form of 6W lies in the fact that one does not require g to satisfy the condition that the sum of the material and magnetic pressures
PROBLEMS IN HYDROMAGNETICS
207
be continuous which is demanded in the normal mode analysis. This requirement follows here as a natural variational condition. We term (4.13) and its associated boundary conditions the extended energy principle.
3. A$$lications of the 6 W-Formalism We now turn to the application of the energy principle. There are two ways in which the principle is commonly used. The first consists of using trial functions for ( which are obtained by physical reasoning or examination of the structure of 6W. The second, which must be used for more complicated problems, consists in minimizing 6 W over all possible displacements. Since 6W is a quadratic form in (, it is necessary to normalize ( in a positive definite way before minimizing. If the normalization is chosen to be (4.18)
ts
p52dz = 1,
then the frequency or instability rate can always be obtained after the minimization is completed. We represent the displacement ((r,t)= g(r) exp (wt) and find the value of w from Rayleigh’s Principle [9] which states that o is the stationary value for (4.19)
If for analytical simplicity any other normalization condition is used in the minimization, an approximate value of w is obtained by substituting the minimizing 5 in (4.19). We illustrate each of the two methods by the examples below. 4. The Pinch Effect
Figure 2 depicts an idealized version of a pinch effect equilibrium. The hydromagnetic fluid pressure is constant within the radius R and zero outside and is supported by the field B, produced by the surface current it.. I n this case 6W simplifies to
(4.20)
’s
+8n
VaC
(I7 x
208
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
v
The obvious way to make 6W in (4.20) negative is to choose both x A and V - zero. The choices listed below accomplish this and, in addition, satisfy the boundary conditions at R and S
FIG. 2. The pinch effect equilibrium.
aA -
ae
=o A , = t v BO
at R,
A, =0
at S.
We estimate the rate of growth of the instability by assuming the density to be constant and 6,of the form (4.21)
We immediately find (4.22)
r
& = &(R) sin k 3.
PROBLEMS I N HYDROMAGNETICS
209
A physical explanation of the instability we have demonstrated is as follows. After the deformation, the system appears as in Figure 3. Since the deformation leaves the vacuum field unchanged, Be llr, and the maxima are
-
&FIG.3. Sketch of the distortion of the plasma surface produced by the displacement
5.
now in regions of weaker field strength, the minima in regions of stronger field strength. Since fi is not changed by the deformation, the unbalanced forces are in the direction to increase the growth of the deformation. The demonstration of the instability of the pinch is originally due to Kruskal and Schwarzschild [lo]. 5. Minimization Technique for Exflansion Problems
Before turning to a problem where we must minimize 6W, we first digress to cover a point in the minimization technique. It should be noted that in many cases which one meets in practice, difficulty is encountered in solving the equilibrium equations. This difficulty arises from the non-linearity of the equations. For this reason, it is often necessary to solve the equilibrium equations by perturbation methods applied to some known or easily determined solution. If the problem is treated by perturbation theory, e.g. an expansion in a dimensionless quantity 3, << 1, then we expect 6W to appear in the form (4.23)
6w = 6wo + a6wa + avwaa -+ . . . .
We then minimize each term in 6W successively until we reach the first non-vanishing term. Since 3, is small, we expect that the succeeding terms are negligible. It should be noted that in the expansion procedure it is also necessary to expand the E’s. That this is necessary can be made clear by a simple example of a ball on a two-dimensional surface. Let us assume that 6W is given by (4.24)
where I is the expansion parameter.
210
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
First assume that 5, and EY do not depend on 1. The obvious choice would then be to minimize 6W by choosing 5, = 0 so as to make the largest term as small as possible. We are then left with 6W = A2 ty2.We would then say that the system is stable. However, let us now proceed to expand 5, and tYin 1. We assume
5,
=
(4.25)
EO,
+ 155 +
5,=5;+15:+
* * *,
a
*
.
,
where tXo, tXa,. . ., are arbitrary, and write
6W
(4.26)
= 6WO
+ 16W + A26WAA+ . . .,
where
6W0 =
(4.27)
[53?
Choose 62 = 0 to minimize 6Wo. We then find 6Wa SE 0. We must proceed to find the lowest non-vanishing term. 6WAais given by (4.28)
We can minimize 6WaAwith respect to 5; by the choice (4.29)
which leads to (4.30)
and thus to instability. This example shows that the 5 which minimizes 6W can depend on A. Since 1 enters the description of the equilibrium, we describe a variety of equilibrium states as 1 varies continuously. We are minimizing 6W for all these equilibria simultaneously. We cannot expect that a 5 which does not change as 1 changes will be able to make 6W negative. Therefore, we must let 5 depend on 1 and thus be expanded as a power series in 1. This consideration is due to Kruskal [ll]. 6. An Axis ymmetric Problem; Coordinate System
We turn now to the treatment of a general axisymmetric system in which BB= 0. In this problem we shall consider the ratio of the material pressure to the magnetic pressure to be small and use it as our expansion parameter.
PROBLEMS I N HYDROMAGNETICS
211
The symmetry of the problem allows the introduction of a stream function i,h which satisfies
B*V+=O.
(4.31)
In fact, the choices
(4.32)
satisfy (4.31) and, in addition, satisfy V. B = 0. The magnetic flux can be shown to be 2 ~ 4 .We now construct a right-handed orthogonal coordinate system #, 8, x which is the natural system for axisymmetric problems. B is, of course, in the 0’ direction. The unit vectors for this system are (4.33)
and the Jacobian of the transformation from Cartesian coordinates is (4.34)
The expressions for gradient, divergence, and curl are given by
Vf = e 4 y B - af +
a+
ee af 1 - - + e --, r ae JB
af ax
(4.35)
1
a
We now impose the condition that the equilibrium equations be satisfied in this coordinate system. Thus from (3.25) (4.3 $1
. = leee . =-_ egcr a _-
J
4n J
a+
BJ2*
212
E D W A R D A. F R I E M A N A N D RUSSELL M. KULSRUD
From (3.24) we obtain
(4.37)
aP = 0, a0
We see that p is a function of $ alone. Define p‘ of (4.37) then determines the Jacobian
= dp/d$.
The first equation
(4.38)
x.
x
where g is an arbitrary function of We note that g(x) = 1 makes go over to the magnetic scalar potential as p + O . Note further that (4.37) implies the interesting relation (4.39)
P I = - .
10
CY
7. An Axisymmetric Problem; Stability
Turning now to the stability we assume our system is periodic in x and is surrounded by a perfectly conducting, rigid wall which is tangent to a constant $ surface. We denote the ratio of the fluid pressure to the magnetic pressure by ,!I. To lowest order in ,!I, 6W is given by
213
PROBLEMS IN HYDROMAGNETICS
The minimization of 6Wo is trivial since all the terms are positive. We find
a -
ax
(YBO~;) = 0,
(4.41)
a a*
-YBO~O
*
+
a i;
-- =
a8
0.
Y
Since 6Wo vanishes we must examine the sign of 6WB. We have
x
Note that the last term in 6WB integrates to zero in by virtue of the periodicity assumption and the first of equations (4.41). Use of the third of equations (4.41)allows us to write 6Wp as
We now minimize 6W* with respect to @ B holding Euler equation is
62 fixed.
The resulting
(4.44) We immediately obtain
(4.45) where FP(+,O) is arbitrary but must be chosen so as to satisfy the periodicity requirements. Since E,O/Bomust be periodic, we rearrange (4.45)in the form
(4.46)
214
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
and integrate over a period in
x. We obtain
(4.47) We can interpret the two integrals on the right hand side of (4.47)in terms of characteristic properties of the equilibrium state. The volume interior to a surface of constant 4, having a length equal to a period in is
x,
(4.48) Thus the differential volume between two such surfaces separated by di,h is
(4.49) Therefore
(4.50) [ VO]” 2n
We thus obtain FP iil the form
(4.51) We return to dWb which now becomes
The general axisymmetric case with
p finite is given in Reference
[8].
8. Physical Interpretation
If we assume that fi = ApY in the equilibrium state, then the quantity in square brackets in (4.52)can be written as MO”/MO’ where dM = p dV is the amount of mass in the flux shell whose volume is dV. Note that if Mo“V0“ is anywhere negative then l2 can be chosen to be finite there and zero everywhere else since no derivatives of appear which would link the values of tz on one surface of constant t,h with the values on another.
PROBLEMS IN HYDROMAGNETICS
2 16
The physical interpretation of this instability is as follows. Since Qo vanishes both 6B and 6j vanish. Thus the source of the energy driving the instability must be the compressional energy in the gas. It follows from the fact that 6B = 0 that a line of force in the equilibrium is moved into the position of another line of force by the perturbation. On closer examination we see that the flow pattern of the 5's shows a tendency to interchange two flux shells. Let us compute the change in compressional energy which arises from such an interchange of two adjacent flux shells (subscripts 1 and 2 ) . Denote by i and f the initial and final quantities characteristic of the flux shells. Then the change in compressional energy under an interchange is (4.53)
( y - 1) 6W
= pzrdvzj
+ plidvlj - PzidVzi - PlBVli,
where dV is the volume in the flux shell ahd p is the pressure. Since dV is proportional to d,h we keep the magnetic energy unchanged by carrying out an interchange such that (4.54)
The last form on the right of (4.54) follows from the fact that the two layers are adjacent. From the adiabatic law (4.55)
PZrdV'b = PZidVL
we obtain (4.56)
By an exactly similar calculation we get (4.57)
We substitute these relations in (4.53) to obtain
Upon using the relations (4.59)
216
ED W ARD A. FRIEMAN AND RUSSELL
M.
KULSRUD
we get (4.60)
M“ 6W = ypv” -( 4 4 3 , M’
which is proportional to the energy we obtained in (4.52). Note, however, that these are finite interchanges as opposed to the infinitesimal interchanges arising from the g-deformation in (4.52). This agreement lends some weight t o our description of the instability as an interchange of two flux shells.
V. HYDROMAGNETIC WAVES 1. Introductory Remarks
An electromagnetic wave cannot propagate very far into a conducting medium but is soon damped. In the infinitely conducting medium we are considering, it is damped immediately. A sound wave in an infinitely conducting medium is unchanged in the absence of a magnetic field but is strongly affected in the presence of a magnetic field, which provides an interaction between the electromagnetic and velocity fields. The result of this interaction is a new set of waves, the hydromagnetic waves. The simplest example of such a wave occurs in the case of a uniform magnetic field imposed on a uniform, infinitely conducting medium, the wave propagating in the direction of the field with arbitrary amplitude. This wave was first discovered by AlfvCn [12]. The velocity and the perturbed magnetic field are perpendicular to the main magnetic field. Since the lines of force follow the matter they are bent by the sidewise motion. The tendency to become straight provides the restoring force to make the wave propagate / ~ , the AlfvCn velocity. The wave is with the velocity B ( 4 7 ~ p ) - ~called analogous to a wave propagating in a stretched string in which the tension is B2/8n and the mass per unit length is p.
2. Three Modes If we confine ourselves to a uniform medium with a uniform field and small disturbances, we can treat the general case of a wave propagating in arbitrary directions with arbitrary polarizations. These more general waves were first treated by Van de Hulst [13]. It is found that in any direction there are three modes of propagation. One mode, in which the mass velocity is perpendicular to the plane containing the field and the direction of propagation, is just the simple Alfvbn wave mentioned before, traveling with a modified velocity. The remaining two modes, in which the velocity lies in
PROBLEMS IN HYDROMAGNETICS
21 7
this plane, are the result of coupling between the AlfvCn mode and a mode corresponding to a sound wave. If the propagation direction is parallel to the direction of the field, one finds two pure AlfvCn modes and an unaffected sound wave. If the propagation direction is perpendicular to the field, one mode is a sound mode with its velocity of propagation enhanced by the additional magnetic pressure and the other two modes disappear.
3. Equation of ProPagation Let us assume infinite conductivity and zero viscosity so that there are no dissipative mechanisms in our system. Differentiating (2.4) with respect to time and neglecting products of perturbed quantities as small, we can write
4n
where B is our uniform field, a is the velocity of sound, b is the perturbed field, u the disturbance velocity, and we have used (2.2) to expressj in terms of b. From (2.1) and (2.6) ab--
at
vx
(u
x B),
so that (5.3)
a2u
p,-a2pV(P.u)--[Vx
at
1 4n
V x ( u x B ) ]x B = 0 ,
an equation in u alone which yields all the waves. Consider a wave with propagation vector k and let u = vk exp (ik* r ) .
Then
-
a2(k* n)(k uk)n where B (5.5)
=
n IBl, and
+ a2(k n)%, = 0, *
218
EDWARD ‘4. FRIEMAN AND RUSSELL M. KULSRUD
Let the y axis be in the B direction, choose the z axis so that k lies in the y,z-plane, and let k make the angle 8 with B. Then, in components, (5.4) becomes (5.6)
(5.7) (5.8)
+ a 2k2cos2Ov, = 0, + k2a2cos28vy + k2a2sin 8 cos 8v, = 0, i, + k2a2sin 8 cos 8vy + k2(a2+ a2sin2O)v, = 0. i;,
..v
Here we have suppressed the k subscript, and dots denote time derivatives. Equation (5.6) represents the ordinary AlfvCn mode with velocity a cos 8, while (5.7) and (5.8) are coupled. When analyzed into normal modes they yield the modified AlfvCn and the modified sound waves. Introduce normal coordinates A + and A - by (5.9)
v y = A + cos p - A - sin p,
(5.10)
v, = A + s i n p
+ A - cosp.
Equations (5.7) and (5.8) become
+ c:k2A+ = 0, A - + Cyk2A- = 0
(5.11)
A+
(5.12)
where 2
tan p
(5.13)
=
c+ - a2 cos2 8 ’ a2 sin 8 cos 8
and c+ and c- are the two positive roots of (5.14)
c4
- (a2+ a2)c2+ a2a2cos28 = 0.
The quantities c+ and c- are the velocities of the fast and slow modes respectively,
As a -+O we have (5.16)
c+ = a
(5.17)
c- = acos8
a2 + -sin28 + . . ., 2a a3 2a2
- -cos
8sin28
+ . . .,
PROBLEMS IN HYDROMAGNETICS
219
so that c- becomes the velocity of a modified AlfvCn mode, and c+ tends to the velocity of sound. A + then represents a sound wave with mass velocity in the direction k, while A - represents an AlfvCn mode with the difference that the velocity is perpendicular to k rather than B, since the two modes 8 must be perpendicular to each other. This follows from (5.13) since p as a + 0. As a increases the modes change their character, and for a much larger than a, the angle p is nearly 90". The velocities of propagation are then -+
(5.18)
c+ = a
(5.19)
c-=acosO-
a2 + --sin28 + ... 2a a3 --cosOsin2O+ 2a2
...,
and now the fast mode is an AlfvCn mode with its velocity perpendicular to B , while the slow mode has its velocity along B and travels with a modified sound velocity. A plot of these velocities for varying a is given in Fig. 4 for 8 = 45".
FIG.4. c + and c- for 0
=
45'.
There are two other limits in which the character of the waves becomes clear. If 8 = go", that is, with velocity of propagation perpendicular to B , c + 2 = a2 a2, c- = 0 and p = 90". The fast mode is an enhanced sound wave with its velocity in the k direction. The magnetic field provides an increased effective pressure to drive the wave. The slow mode disappears.
+
220
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
The mode with velocity in the x-direction also disappears. This is physically reasonable since for velocities in these directions the lines of force can move freely over each other and no wave can propagate. Finally, if k is parallel to B, 8 = 0, c+ is the maximum of a and 01, c- is the minimum, and /3 = 90" or 0" according as u or a is larger. Let u be larger than a. Then c+ is an AlfvCn wave with velocity perpendicular to B , and c- is a sound wave with velocity along B and is thus unaffected by B.
4. Energy and Wave Generation
Each of these modes carries magnetic energy as well as kinetic energy and compressional energy. The electrical energy is relativistically small. The sum of the magnetic energy and the compressional energy is
while the kinetic energy is (5.21)
for a given distribution in k of the waves, where c, = u cos 8 . It is seen from (5.6), (5.11), and (5.12) that the compressional and magnetic energies are just balanced by an equal kinetic energy. This set of modes is complete. If we are given at any initial time a distribution of B , p , and v compatible with (2.1), (2.3), (2.5), and (2.6), we can determine v and b by (2.2) and (2.4). By Fourier analysis one finds initial values for vk and v k . Further, by the normal mode analysis of (5.9) and (5.10) one can find initialconditions for A , , A -, vkand their time derivatives. Consequently, one knows the amplitude of each mode of every plane wave, and a t any later time, by recombining them, the total fields and velocities can be found. For example, assume a t some time that v = 0 and b is in the x-direction and is a strongly peaked function about the origin with amplitude b,. Then by (2.2) and (2.4)
.
(5.22)
V =
( V x b )x B 4nP
or (5.23)
B
ab,
v x = ~, 437P aY
',=O,
d,=O
PROBLEMS IN HYDROMAGNETICS
221
To obtain propagation with k in the y - z plane write exp (ikycos 8
+ ik,sin 8)
(5.24)
where A t is the volume in which b(0)is concentrated and we assume Kdt << 1. The amplitude of the wave is thus (5.25)
In a similar manner the generation of waves by an external force may be computed. The force is Fourier-analyzed and broken up into normal modes so that (5.6), (5.11), and (5.12) become (5.26)
zkx
2
-k k 2 C x V k x
=pkx,
(5.27) (5.28)
A_,
+k
2 2
C-A-k
=$k-,
where # k x , p k + , and p k - are the normal mode components of the external force. Each wave behaves as an oscillator driven by an external force. The energy generated into each mode can be easily calculated. Such a calculation has been carried out by Kulsrud [14].
5. Grot@ Velocity
Although each of the plane waves seems to act independently, the energy of these waves is actually transported in the direction of the group velocity rather than in the k direction. Each of the modes has its own group velocity which can be computed as the gradient of its angular frequency o,with respect to k. It is to be noted that w depends on the direction as well as the magnitude of k. For the AlfvCn mode we have (5.29)
rkW
= rh(C~k= ) r k ( a
*
k ) = a = B/(4np)’”.
That is, the energy is transmitted along the field with AlfvCn velocity. The plane waves associated with this mode are “phase waves.” Since their velocity is in the x-direction, any motion of the lines in a plane z = const.,
222
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
is completely independent of their motion in another plane z = const. By correlating the motions in these planes properly, a phase wave moving in any propagation direction k in the y,z-plane may be constructed. It is clear why these phase waves move with the velocity a cos 8. Although it is true for the other two modes that the group velocity and phase velocities differ in direction and magnitude, no such simple interpretation exists for them, since the motions of different lines are no longer independent. One finds from (5.14) with W* = kc, that (5.30)
% *=
Denote by (5.31)
x*
+ az),o: 7 a 2 ( a- k)2]k 7 a2k2(a k)a [(a2+ a2)k4- 4a2(a k)2k2]1/2
[(a2
W~
*
*
the angle which vg* makes with the y axis. Then ta '
If 8 = 0 both direction.
x+
and
x-
equal zero, and the group velocity is in the B
6. Reflection and Refraction of Waves; Boundary Conditions
The reflection and refraction of incompressible hydromagnetic waves a t an interface of discontinuity has been treated by several authors. Ferraro [15] has treated the case of finite amplitude waves traveling in the B direction incident on an interface making an arbitrary angle with B . Roberts [16] has treated the case of waves traveling a t an arbitrary angle to B. The case of compressible waves can be treated in a similar manner. One first asks what interfaces are possible. On an interface the jump conditions (2.8) - (2.12) must be satisfied with v = 0. As pointed out in Section 11, if B * n # 0, there can be no surface current since a surface current would lead to infinite tangential accelerations. Thus, in this case B must be continuous, and by (2.11) p must be continuous; hence only a jump in p is possible. However, if B is tangentiai to the boundary, a surface current is allowable since it leads to no tangential forces. Therefore, the only jump $ B 2 ) = 0, and arbitrary jumps in 9, B , and p are permitcondition is {fI ted consistent with this condition. We shall restrict ourselves to this case and compute the reflection and refraction for incident waves of each of the three modes. In general, a pure mode incident on an interface is expected to be coupled with other modes upon reflection or refraction a t the interface. Therefore we might expect that the reflected and refracted waves will consist of more
+
PROBLEMS IN HYDROMAGNETICS
223
than one mode. When k lies in the plane of B and n, however, the reflected wave consists of a single mode of the same type as the incident mode. The refracted mode need not exist in the case of total reflection, or may exist as a single mode. This single mode may be of the incident type or not. If the interface is moved by the wave, we must express the boundary conditions on the moving interface. Denote by subscript 1 quantities in the region of incidence and reflection, by subscript 2 quantities in the region of refraction, and let n point from region 1 to region 2, The expanded boundary conditions become (5.32)
-{B}
- (Vv) n + n - (6) = 0, *
(5.33)
{v * n} = 0,
(5.34)
{p + y}
=0,
where in (5.32) the rate of change of the normal n, due to a motion of the surface (5.35)
i a = - (VV)
+
-
* n n[n. ( V V ) n],
has been employed. Let the interface be z = 0 and let there be jumps in p , B , and p satisfying (2.11). Then for an incident AlfvCn mode with 6 and v in the x-direction the reflected and refracted modes must also be of this type. For such modes, (5.32) - (5.34) give no conditions, so that any reflection and refraction is possible. This is compatible with the previous remark that this mode is a phase wave for which motions on adjacent lines are independent. Hence, these motions may be arranged to give any reflection and refraction. 7. Angles of Reflection and Refraction
Let us now consider the incidence of a fast mode whose propagation vector k lies in the plane of n and B and makes an angle 8 with B . Its mass velocity lies in the plane of incidence. For this mode, (5.32) and (5.33) y;eld identical conditions as would be expected from the tact that the lines must follow the matter. If any motion moves the boundary, the lines must move so as to remain tangent to the boundary. It would seem that the two independent conditions (5.32) and (5.34) are not sufficient to determine the amplitudes of the two possible reflected and refracted modes. However, since these conditions must be satisfied over the entire interface we must have for the reflected mode (5.36)
k cos 8 = k' cos O',
224
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
where k‘ and 0‘ are values for the reflected mode. But for the slow mode k‘ = o/c-(0’) while k = w/c+(0) so that (5.37)
Further, from (5.14) c + c~ - ~= a2 cos 20 so the square of (5.37) becomes (5.38)
which is impossible by (5.15); hence the slow mode cannot be reflected. Also for a reflected fast mode the reflected angle is (-0). For the two refracted modes we must have (5.39)
,,
I’
k + cos 8+
=’k
I’
cos 8-
=k
cos 0,
where double primes indicate refraction. From the same argument applied to medium 2, it is seen that a t most one mode can satisfy (5.39), so we have a single reflected mode, and at most a single refracted mode. This result is partly due to Bernstein [17]. The angle of refraction for a fast mode is determined by (5.40)
But if u2> a2, c2-2 lies between 0 and a22so that the fast mode is refracted if (5.41)
Similarly, a slow mode is refracted if 2
(5.42)
If (5.43)
or (5.44)
-< a:
1
cl-(e) < a$x; a2
1 -t3;. 42
225
PROBLEMS IN HYDROMAGNETICS
there is no refracted mode and we have total reflection. By replacing c1-2 by c1+2 we get similar relations for an incident slow mode. These relations are made clear by Fig. 5 where k cos 8 is plotted against 8 for a > a in both regions. The lower branch refers to the fast mode and the upper branch t o
FIG. 5. k cos 0 for
Q
= 2a.
the slow mode. Suppose a fast mode is incident with angle 8. We find the value of k cos 8 on the lower branch and look for this value of k cos 8 on a. corresponding set of curves for region 2. If no such value exists, the wave is totally reflected. If the value is found on the lower branch, a fast mode is refracted and its angle can be found. If the value is on the upper branch, a slow mode is refracted. If in either region a > a, their values are merely interchanged in determining these curves for that region. 8. Transmission and Reflection Coefficients
We now proceed to determine the amplitudes of the reflected and refracted modes resulting from an incident fast mode. It is to be noted from (5.13) that p’ = - p. Equation (5.34) than can be written for an incident fast mode (5.45)
p1 sin p cos 8RZ+( A+
+ A ’+)
= p2 sin P,’
cos O”k”2;’A
y,
226
EDWARD A. FRIEMAN AND RUSSELL M . KULSRUD
or, since k cos 0 (5.46)
=
k" cos O",
plsinpZ+(A+ +A:)
=p,sinP"ZyA:,
where (5.47)
Z, = at (cot p
+ tan e) +
with a corresponding expression for 2;. (5.48)
42
tan 8,
Equation (5.32) becomes
s i n p ( A + - A > ) =sinP"Ay.
Hence, for the reflected wave (5.49)
and for the refracted wave (5.50)
By use of (5.13),Z, simplifies to
(5.51)
z+=
- u2 cos2 8 - c:
sin 0 cos 8
Near a critical angle for total reflection c'+' + u2, 0" + 0, and 2; 00, while Z, remains finite. Hence A; -+ 0 and A: A+. For normal incidence ,8 --*go", 8 - 4 2 , both 2 ' s become infinite and (5.49) and (5.50) become indeterminate expressions. Their values are obtained by passing to the limit. Let 8 = 4 2 - p where p is small. Then -+
---+
(5.52)
while (5.40) gives (5.53)
PROBLEMS IN HYDROMAGNETICS
227
so that
(5.54)
(5.56)
The expressions for reflection and refraction when a slow mode is involved in either medium are obtained by replacing Z, by Z- in that medium. Z- is obtained by replacing c- by c+ in 2,. p is replaced by (90"- p). 9. DamPing
So far we have neglected any terms retarding the wave motion. If viscous terms are present in (2.4) they will slow down the motion while still allowing the lines of force to follow the matter, and will damp the wave eventually either in time or space. If we consider the motion to be sinusoidal in space, it must damp out in time. The presence of a non-infinite conductivity will likewise produce damping. The lines need no longer follow the matter exactly and can move relative to the matter getting out of phase with the motion and retarding it. We will assume in this subsection that both viscosity and resistivity are finite but so small that the wave is not appreciably damped over many wave lengths or during many oscillations. We compute the damping effects to first order neglecting products of viscosity and resistivity terms. It will be found that the velocity has an imaginary part but that its real part is unchanged. If 6 is the skin depth, I the wave length of the wave, and the viscosity is zero, it will be found that the wave damps appreciably in A2/b2 wave lengths or oscillations. The equation of motion (2.4) with viscosity included is (5.57)
where p is the viscosity. In Ohm's Law (2.6) the term ]/a must now be included. With these modifications (5.6), (5.7), and (5.8) become (5.58)
i
i =~iwVx ~ cos2 ~ e,
228
(5.59) (5.60)
EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD
+ Z2k2sin 6 cos Ov,, W2v, = Z2k2sin 8 cos Ov, + ( G2 + Z 2 sin2 6) k2vz, W2v, = Z2k2cos2Ov,
where (5.61)
(5.62)
(5.63)
Thus (5.6), (5.7), and (5.8) take on the same form in these new variables. (5.15) becomes (5.64)
-c 2 = 7
W2
k
1 =-{Z2+ 2
&'&((a2+
% 2 ) 2 - 4 Z 2 & 2 ~)' ~I 2 I~. 2 8
The quantity p is also modified but we do not include this change since it merely rotates the plane of polarization. Let (5.65)
w -
k
=
5 + ir.
Expanding (5.64) we find is just the velocity for zero damping. That is, the velocity is unchanged to first order. The expression for q is
(5.66)
where c+ and c- are the velocitiesof the fast and slow modes for zero damping. If a and u are roughly equal and p = 0 (5.67)
2
c&q+ M c,k2d2,
where 6 = ~ / ( 4 n o w )is~the ' ~ skin depth. If one regards w as real, the amplitude of the wave is proportional to (5.68)
PROBLEMS IN HYDROMAGNETICS
229
and the wave damps in a distance (5.69)
If k is real, w is complex and the damping time is (5.70)
If c is infinite and p finite, the damping length is (5.71)
the damping time is (5.72)
2u2p 1 Ti=-------. "P
10. More General Waves
There are many other different types of waves that may exist in situations in which the main field is no longer uniform but varies in space, or in which the boundary conditions are different. Because of the increased complexity, it is difficult to treat such waves in as much detail as in the plane-wave case. In investigating these more complicated waves, the dW-formalism of Section IV leads to the equations for them in a simple and direct manner. One case of such waves is the situation in which uniform field, pressure, and density are present in an infinitely long cylindrical tube with rigid infinitely conducting boundaries. The magnetic field is parallel to the axis of the tube. The boundary conditions are: v * n and b * n must vanish on the boundary. If the propagation direction k is parallel to the axis of the tube, there are three simple modes. For the mode in which the mass velocity is along k, an undisturbed sound wave propagates with velocity a. For the mode in which the velocity is radial, no wave can propagate if its wave length is much greater than the radius of the tube. Thus, a cut-off frequency exists for this mode in analogy with microwaves in a wave guide. A third mode has its velocity in the 8 direction and propagates with velocity u. This is a torsional wave. Since each cylindrical shell oscillates independently of its neighbors any radial distribution of velocity is possible. No cut-off exists for this mode. For propagation in other than axial directions the waves become more complicated.
230
EDWAHD A. FRIEMAN AND RUSSELL M. KULSRUD
To illustrate the application of the GW-formalism in determining the equations of motion let us consider the radial pulsations of a cylinder of radius R and length L in which the field is parallel to the axis and the pressure and field are functions of the radius. Introducing cylindrical coordinates we find t o= 6, = 0 and from (4.9)
Qr = Qe = 0.
(5.73)
Since everything depends only on the radial coordinate, 6W can be written R
sw=?C,srdr{(g
+YP)[($)"$]+(f
+.p);,}.
1
at2
0
(5.74)
We have dropped the subscript r on 6,. Since t vanishes a t term may be integrated by parts to give
Y =
R, the last
R
t 2 (W- YW') &t'2+ 2 7
(5.75) 0
where primes denote differentiation with respect to r and (5.76)
W =-+ y p . B2
4n
The motion is determined by the condition that (5.77)
must be stationary, and the motion is proportional to exp [iwt]. Carrying out the variation we find (5.78)
W'
If i;, is constant the equation simplifies and (5.79)
1
PROBLEMS IN HYDROMAGNETICS
w is determined
231
by the condition 5 ( R ) = 0, to be
(5.80)
where yi is the i-th zero of
J1.
References 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
ELSASSER,W. M., Hydromagnetism I, a review, A m . J. Phys. 23, 590 (1955). ELSASSER,W. M., Hydromagnetism 11, a review, A m . J. Phys. 24, 85 (1956). LUNDQUIST, S., Studies in magneto-hydrodynamics, Arkiv For Fysik 6 , 297 (1952). CHANDRASEKHAR, S., Problems of stability in hydrodynamics and hydrodynamic astrophysics, Mon. Not. Roy. SOC. 113, No. 6, 667 (1953). SPITZER,L., JR., “Physics of Fully Ionized Gases”, Interscience Publishers, New York, 1956. NEWCOMB, W. A., Motion of magnetic lines of force, Princeton University Observatory Technical Report No. 1, 1955. LUNDQUIST, S., On the stability of magneto-hydrostatic fields, Phys. Rev. 88, 307 (1951). BERNSTEIN,I. B., FRIEMAN, E. A., KRUSKAL,M. D., KULSRUD,R. M., t o be published (1957). LORDRAYLEIGH, “The Theory of Sound”. KRUSKAL,M. D., and SCHWARZSCHILD, M., Some instabilities of a completely ionized plasma, Proc. Roy. SOC.A, 223, 348 (1954). KRUSKAL, M. D., (Private Communication). A L F V ~ NH., , “Cosmical Electrodynamics”, Clarendon Press, Oxford, 1950. VANDE HULST,H. C., Chapter VI, in Problems of cosmical aerodynamics, Central Air Documents Office, Dayton, Ohio, 1951. KULSRUD, R. M., Effect of magnetic fields on the generation of noise by isotropic turbulence, Astrophys. J. 121, 461 (1955). FERRARO, V. C. A., On the reflection and refraction of Alfven waves, Astrophys. J. 119, 393 (1954). ROBERTS,P. H., On the reflection and refraction of hydromagnetic waves, Astrophys. J. 121, 720 (1955). BERNSTEIN,1. B., (Private Communication).
This Page Intentionally Left Blank
Mechanics of Granular Matter *
BY H . DERESIEWICZ
.
Columbia University. New York N . Y . Page 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 I1. Geometry of a Granular Mass . . . . . . . . . . . . . . . . . . . . 236 A . External Shape . . . . . . . . . . . . . . . . . . . . . . . . . 236 237 B . Internal Configuration . . . . . . . . . . . . . . . . . . . . . . 1. Arrangements of Equal Spheres . . . . . . . . . . . . . . . . . 237 2. Densest Packing of Equal Spheres . . . . . . . . . . . . . . . 239 3. Effect of Interstitial Spheres on Density . . . . . . . . . . . . 245 4 . Packing of Non-Spherical Bodies . . . . . . . . . . . . . . . . 249 111. Some Recent Results of Contact Theory . . . . . . . . . . . . . . . 2.51 A . Normal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 251 253 B . Tangential Forces . . . . . . . . . . . . . . . . . . . . . . . . 1.Increasing. . . . . . . . . . . . . . . . . . . . . . . . . . . 253 256 2. Decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Oscillating . . . . . . . . . . . . . . . . . . . . . . . . . . 258 C. Oblique 'Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 263 D . Twisting Couples . . . . . . . . . . . . . . . . . . . . . . . . . 265 I V . Mechanical Response of Granular Assemblages . . . . . . . . . . . . . 267 A. Continuous Model . . . . . . . . . . . . . . . . . . . . . . . . 267 R . Discrete Model . . . . . . . . . . . . . . . . . . . . . . . . . . 270 1. Rigid Granules . . . . . . . . . . . . . . . . . . . . . . . . 270 2a . Elastic Granules. Normal Contact Forces . . . . . . . . . . . . 271 2b. Elastic Granules. Oblique Contact Forces . . . . . . . . . . . . 285 V . Suggestions for Further Research . . . . . . . . . . . . . . . . . . . 300 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
I. INTRODUCTION Perhaps the most striking characteristic of a granular material is its dual nature which prohibits rigid classification as either solid or fluid . For. although the individual particles of the medium are solid. the tendency of the entire mass to flow readily is a property shared with fluids . That is. in contrast to the essentially fixed geometric form of a solid. both granular
* This study was supported by the Office of Naval Research under contract Nonr-266(09) with Columbia University . 233
234
H . DERESIEWICZ
and fluid media assume a form dependent on imposed external boundaries: each is capable of indefinite shear. On the other hand, when a liquid is subjected to no active (as opposed to reactive) forces other than the force of gravity, its traction-free surface is a horizontal plane. Under similar circumstances, however, a granular medium may take on a variety of shapes, the particular one assumed depending on the geometry of the supporting surface. Before proceeding further it will be well to set down a few basic definitions. By “granular medium” we will understand an aggregate of discrete, solid granules in contact. A given volume of material will thus consist, in part, of contiguous granules, the remaining portion being taken up by voids. The granules may be free to displace with respect to their neighbors and the voids may be filled with gas, as in a sample of dry sand, or the voids may be filled with liquid, as in a sample of wet sand. Or it may happen that some or all of the voids are filled with a binding material which holds the granules in more or less fixed positions, as, for example, in sandstone or concrete. Following the geologists, we call such material “porous granular.”* Finally, we shall denote a material “porous” if it consists of a continuous solid matrix which surrounds either isolated or interconnected voids filled with gas, liquid, or a solid binder. Although the study of the mechanical behavior of porous materials is of great interest and a good deal of progress has recently been made in this field, we will not concern ourselves with it here. What are the problems peculiar to granular media and what are some of the applications and uses of such materials ? In attempting to answer these questions we uncover an enormous variety of technological fields in which granular matter plays a major role. The oldest and perhaps best known of these is soil mechanics, with its problems of foundations, grades, retaining walls, etc. Such problems have been treated on a more or less scientific level for a t least two and a half centuries, in the course of which they have attracted the attention of a large number of architects, engineers and physicists; among the most prominent of these we may cite Coulomb, Navier, Poncelet, Rankine, Boussinesq, St.-Venant, Flamant, and Winkler. A detailed account of the early work in this field will be found in a review article by Kotter [l]; for later work, reference may be made to Terzaghi [ZIT. Similarly, problems of computing static
* It is to be noted that no stipulation was made concerning the size of the individual granules but, for our purposes, we may assume that we are dealing with essentially convex particles whose mean diameter exceeds lW4 cm. Thus, we exclude colloidal particles from our definition, although some questions touching upon granular media are pertinent even to problems of molecular configuration.
t Added at press-tima: For an exhaustive treatise on theoretical soil mechanics, see the book by Caquot and Kerisel [84].
MECHANICS OF GRANULAR MATTER
235
pressure exerted by a granular mass arise in the design and construction of silos (Janssen [3]). The dynamical response of granular materials has been much investigated in connection with the propagation of seismic waves and with petroleum prospecting (a bibliography on these subjects may be found in a paper by Brandt [4]). The problem of the vibration of a granular assemblage arises in the analysisof the performanceof awidely used type of microphone (Sell [5]) of which an aggregate of carbon granules is the essential constituent. The internal configuration of a granular mass is of interest wherever problems of ordering and packing appear. Thus, it is found that the most favorable results are obtained with densest packing of an aggregate in such widely separated fields as the manufacture of ceramics, refractories, and grinding wheels (Westman and Hugill [6]), of cement and mortar products (Anderegg [7]), in powder metallurgy (Wise [8]), and, in connection with the strength of asphalt, in highway engineering (Horsfield [9]). The study of the packing of granules is intimately connected with problems in crystallography (Schonflies [lo]) and cytology (Marvin [ll]),and belongs to the branch of pure mathematics called extremal problems of geometry (Fejes T6th [12]). Problems of hydrodynamics and aerodynamics arise, among others, in the analysis of the movement of ground water, petroleum, and natural gas through sand and rock (Muskat [13]), in studies of drainage, seepage, and filtration (Polubarinova-Kochina and Falkovich [14]), in attempts to predict distribution of hydrothermal mineral deposits (Fraser [15]), and in questions pertaining to combustion in a solid fuel bed (Bennett and Brown [16]). Finally, granular matter has been employed in an attempt to explain formation of islands (Terada and Miyabe [17]). The purpose of the present article is to discuss some recent progress made in the investigation of the static and dynamic response of granular media to external forces, that is, of small deformations and vibrations of such materials and of the propagation of elastic waves through them. To facilitate their task, the authors of most modern theories assume such granular matter depicted by means of models in which the individual granules are represented by spheres or other convex shapes in contact in, for the most part, regular arrays. Accordingly, Section 11. of this review is devoted to a summary of the geometry of packing. The contiguous granules in many of the models are considered elastic, so that the theory of contact of elastic bodies must be invoked to furnish a description of some of the phenomena. However, the classical theory of contact due to normal forces does not, in general, suffice for the purpose. The extension of the theory to account for contact due to oblique forces and due to twisting couples is quite recent; it is discussed in Section 111. The account of studies of static and dynamic behavior of granular media constitutes the content of Section IV. I t was found desirable and convenient
236
H . DERESIEWICZ
in this portion of the article to group the various investigations not by subject matter, but rather by the types of models used in approaching the subject matter. A brief outline of some of the unsolved problems in the fields treated in Sections 11.-IV., given in Section V., and a list of references conclude the paper.
11. GEOMETRYOF
A
GRANULAR MASS
A . External Shafie Mention has already been made of the diversity of forms which a granular aggregate may assume when it is subjected solely to gravitational forces. An extensive experimental investigation of this subject of “equilibrium figures” was carried out by Auerbach [18] who came to the following conclusions: a) The traction-free surface of a dry granular medium bounded by a horizontal base and a vertical wall, both of indefinite extent, is a plane whose slope depends on the size, shape, density and surface roughness of the constituent granules.* If the pressure in the interior of the medium were transmitted by all the granules through the height of the mass, we should expect the free surface to be concave rather than plane, that is, its trace in a plane normal to the fixed bounding planes would be a curve whose slope increases with elevation above the base. Accordingly, we must conclude that the pressure is transmitted only by a limited portion of the granules.? b) The equilibrium figure over a circular base may be approximated by the lower sheet of a hyperboloid of two sheets; the figure over a base with a circular hole is a crater generated by rotating a portion of the upper half of one branch of a horizontally oriented hyperbola about the vertical axis through the center of the hole. c) The figures over bases having the shape of regular polygons are pyramids with rounded edges and vertices. The rounding of the vertex is greatest over a circular base, smallest over a triangular base, and varies directly as the size of the granules. Further, the steeper an edge, the greater the amount by which it is rounded.$
* Somewhat related experiments were performed by Terada and Miyabe [19] with a different end in mind. They started with a rectangular box of sand with a free horizontal surface and, by slowly receding one wall, were able to simulate step-faulting. This phenomenon was found to be strongly sensitive to the coefficients of friction of the granular mass. With the same setup as above, an investigation was also made of the effect, on the medium, of the wall pushing in on the sand and, further, of reciprocating motion of the wall [20]. t We shall have occasion to return to this question in Section IV. B, 1. t I t may be noted that such equilibrium figures, obtained with infinitesimal granules, represent the stress surface for the case of complete yielding of a twisted bar of the same cross-section as the base of the heap (Nadai [21]).
MECHANICS OF GRANULAR MATTER
237
B. Internal Configuration 1. Arrangements of Equal Spheres. Two basic questions arise in connection with the arrangement of granules in an aggregate: (a) What is the actual packing of granules in a well-agitated mass ? (b) How can densest packing be attained? The first of these questions can, clearly, be answered solely on the basis of experimental evidence; the second is susceptible to analytical formulation.
(a) Simple Cubic
(c)TetMgonal Sphenoidal
(b) Cubical -Tetrahedral
(dl Pyramidal
(el Tetrahedral
FIG.1 . Modes of regular packing of equal spheres.
Let us first consider the type of model in which the granules are represented by equal spheres in contact with each other in such a way that they form a regular pattern in space.* The simplest of all such arrays is formed by the sim9le cubic packing (Fig. l a ) in which the centers of spheres of radius R form a space lattice generated by a cube of side 2 R . Thus, every sphere is in contact with four spheres with centers in one plane and with one sphere in each adjacent parallel layers, i.e., a total of six spheres. The number of spheres in contact with any given sphereis called the “coordination
* The first to represent symmetric structures (crystals) by regular packings of like spheres was Barlow [22]; for a bibliography of the early work on this subject, see the article by Schonflies [lo]. A d d e d at press-time: It appers that Barlow had been anticipated by Wollaston [85], who employed not only spheres but oblate and prolate spheroids in his models, and, to some extent, by Hooke [86].
238
H. DERESIEWICZ
number” of the packing. The density, D ,of the packing is defined as the ratio of the volume of space occupied by solid matter to the total volume. Since each primary cube of volume V = (ZR)3 contains one octant of each of eight spheres, the density of simple cubic packing is D = n / 6 m 0.5236. In cubical-tetrahedral, or single stagger, packing (Fig. l b ) each sphere is in contact with six others in the same layer and, as in the simple cubic packing, the highest points of spheres in one layer are in contact with the lowest points of spheres in the layer above. Hence, each sphere is in contact with eight others. The volume of a unit prism containing segments equal in total volume to one full sphere is V = ( 2 R ) 2 ( R the density of the packing is D = n / 3 v F m 0.6046. Tetragonal-sphenoidal, or double stagger, packing (Fig. lc) is similar to single stagger packing in that each sphere is in contact with six spheres of the same layer, but now each sphere in a given layer rests in the depression between two adjacent spheres in an adjoining layer. In this case, then, the coordination number is ten; the volume of a unit prism is V = (2R)(RV3)2 and the density of the packing is D = 2x19 M 0.6981. In pyramidal packing (Fig. Id) each sphere is in contact with four neighbors in the same layer (as in the simple cubic arrangement) and each of the spheres in a given layer nestle in the hollows formed by four contiguous spheres in the next layer. Accordingly, each sphere is in contact with four others in each of three adjacent layers, or a total of twelve spheres. The unit volume is V = (2R)2(Rv5jand the density of the packing is D = ~ / 3 V $ m0.7405. Finally, in tetrahedral packing (Fig. le) each sphere is in contact with six others in one layer (as in the cubical-tetrahedral array) and is lodged in the hollows formed by three adjacent spheres in each of the layers immediately above and below. Each sphere is, therefore, in contact with twelve others. The unit volume is V = (2R)(RVg(ZRv2/3), so that the density is D = z/3vFm 0.7405. We may note that in each of the packings described above, the density is independent of the size of the spheres, provided that the packing is extended over a very large region so that the effect of boundaries is negligible.* The results are summarized in Table 1, in which the column marked “porosity” indicates the percentage of voids in a unit vo1ume.t
vg;
* The problem of packing of spheres in a finite region is considerably more difficult: packing in a circular cylinder has been discussed by Supnik [23] ; inside a hollow sphere, by Hadwiger [24]. t A detailed study of these packings may be found in a paper by Graton and Fraser [25].
239
MECHANICS OF GRANULAR MATTER
2. Densest Packing of Equal Spheres. It isseen that, among the arrangements listed in Table 1, the greatest density is attained in the pyramidal and tetrahedral structures. That these represent the closest of all regular packings* of like spheres may be shown by the following construction. TABLE1
Coordination number
Spacing of layers
Simple cubic
6
2R
Cubical-tetrahedral
8
2R
Type
Of
packing
Volume of unit prism
4v3Rs 6R3
Density
Porosity
(%I
(431j3) 0.6046 (2749) 0.6981
39,54
Tetragonal-sphenoidal
10
Pyramidal
12
RvB
4v2R3
(ni3Vz)
25.95
Tetrahedral
12
2R1/2/3
4v2R3
(n’3v2)
25.95
0.7405 0.7405
30.19
Arrange the spheres, their centers a distance 2R apart, in rows in a horizontal plane and order these rows in such a way that the lines connecting the centers in each row are parallel and as closely spaced as possible. In this position the spheres in a given row will fit in the notches between two contiguous spheres in an adjacent row, so that the generating lattice of centers in the plane is a rhombus whose short diagonal divides it into two equilateral triangles. This layer .represents the densest packing in the plane, each sphere being in contact with six others.? Now take a similar layer and place it on top of the first so as to cause the planes of the centers in each of the two layers to be the smallest possible distance apart; that is, the spheres in the second layer will just fit into the
* Proof that the largest number of spheres in contact with a given sphere is twelve, regardless of the regularity of the packing, has been given only recently by Boerdijk [26], Schiitte and van der Waerden [27], and Leech [28]. In addition, Boerdijk demonstrates configurations which, locally, are denser than the close-packed systems. A discussion of loosest packing of like spheres (in which the mobility of the spheres is completely restricted) is given by Hilbert and Cohn-Vossen [29]. The coordination number of such a packing is four, but the density is a matter of conjecture ( D 0.123). Added at press-time: A further study of arrays of spheres having large porosity is due to Heesch and Laves [87]. t Hence, the density of closest packing of the horizontal great circles (packing in 0.9069. two dimensions) is D = 4 2
<
240
H. DERESIEWICZ
depressions of the first layer.* Thus, each sphere of the second layer will be in contact with three spheres of the first layer. A third layer of spheres may now be placed on top of the second layer so that, once again, the distance between the respective planes of centers is minimal. This operation may, however, be carried out in two different ways. The spheres of the third layer may be inserted in the hollows of the second layer located directly over the spheres of the first layer; that is, the first and third layers will be symmetric with respect to the second one. This is the ordering which results in our previously discussed tetrahedral packing (also commonly referred to as close-packed hexagonal).
FIG.2. Construction of densest packings of equal spheres. of sphere centers in third layer resulting in the face-centered cubic array. x: Location of sphere centers in third layer resulting in the close-packed hexagonal array. 0 : Location
Alternatively, the spheres of the third layer may be placed in those hollows of the second which are located over the unfilled hollows of the first; hence, the translation which brings the first to the position of the second layer causes the second to move into the position of the third layer (Fig. 2). This is the pyramidal packing viewed from a different vantage point. The array is commonly referred to as face-centered cubic since it consists of a simple cubic lattice distended to accomodate a sphere with its center a t the intersection of each pair of face diagonals of the cube.t The two close-packed arrays are shown in Fig. 3. In continuing the packing process by superposing additional layers, we may change from one arrangement to the other in any number of ways. Thus, if we denote by A the first layer of (closely-packed) spheres, by B the second layer and by C the third layer which arises as a result of the
*
In this process only the alternate depressions can be filled. If we were t o surround each sphere of a close packing by a convex body in a fashion such that all space were filled homogeneously, we would find the unit cell corresponding to a face-centered cubic array t o be a rhombic dodecahedron and the one corresponding t o a close-packed hexagonal system a trapezo-rhombic dodecahedron, the latter being a solid bounded by six rhombuses and six isosceles trapezoids (Fejes T6th [12]).
t
MECHANICS OF GRANULAR MATTER
241
identical translation of B as was performed on A t o obtain B, we have the scheme of densest packings given in Table 2.
FIG. 3. The two close-packed arrays. Left, face-centered cubic; right, close-packed hexagonal.
TABLE2*
Order
sequence of lattice planes
densest packing
1
A
nonet
2
AB
c. p. hex.
3
A BC
f. c. cubic
4
A BCB
5
ABCBC a
A BCBCB
h
ABCACB
6
I
II
packings of higher ordert
etc.
* After Menzer [30]. t This is the cubical-tetrahedral
packing of Fig. lb. A packing of “higher order” is defined (by Menzer [30j) as one in which the sequence of lattice planes has a periodicity greater than three. An interpretation of it as a systematic multiple twinning of the close-packed array is given by Graton and Fraser [25]. It is pointed out by Melmore [31] that, of all the higher-order packings, only four are homogeneous in space.
242
H . DERESIEWICZ
Experimental studies by Westman and Hugill [6], who packed lead shot in a cylindrical container, appear to confirm the theoretical result that the closeness of packing of large volumes by equal spherical particles is independent of particle size. The data indicate that such particles may be packed with a porosity of about 37 percent. With sized, rounded, and washed sand, the porosity was found to vary between 37.7 and 46.5 percent, the packing improving with decrease in particle size. Similar experiments by White and Walton [32] on silicon carbide, aluminum oxide, and mullite (all rounded) yielded porosities of 44.7, 40.7, and 43.5 percent, respectively.
FIG.4. Distribution of the coordination number for several porosities, experimentally (Smith, Foote, and Busang [34]).
obtained
The effect of moisture on sand is, at first sight, perhaps unexpected. Thus, Litehiser [33] found, upon mixing sand with water before packing, that the density of the aggregate decreased as the moisture increased, the lowest value being reached at 3-5 percent of water (by weight). A size effect appears here, since the smaller the particles the greater the maximum drop in density and the greater the amount of water required to reach this drop. The phenomenon is known as “bulking” and is due to the formation of a capillary film between the granules which binds them and thus counteracts their tendency to settle when the mass is agitated. In Litehiser’s experiments a further increase in moisture reversed the trend until, at about 18-20 percent of water (by weight), somewhat beyond the point of saturation, the density had risen to the value corresponding to rodded dry sand (about 9 percent above the density of loose dry sand). Again, the higher numerical value of the moisture pertains to the smaller particles. The reversal of the trend is due to progressive weakening of the capillary bond;
243
MECHANICS O F GRANULAR MATTER
the increase in density, a t saturation, above the density of the initial dry state may be accounted for by the lubricating action of the water after the capillary film has been destroyed. In an experiment designed to determine the coordination number in a well-packed aggregate, Smith, Foote, and Busang [34] stacked lead shot in a large cylindrical vessel and filled the voids with acetic acid. The formation of lead acetate caused a white patch to appear on each contact so that the total number of contacts in a given volume (away from the boundary of the container*) was easily counted. Their findings are given in Table 3, in which N denotes the total number of spheres in a sample, P the porosity of the sample, and w the average number of contacts per sphere. A statistical presentation of the distribution of the number of contacts per sphere (Fig. 4) reveals that, at low porosities, this distribution resembles a Gaussian one, but a t higher porosities there is a decided shift toward the higher coordination numbers. TABLE3
Sample N
A B C
D E
1562 1494 887 906 905
P
.359 .372 .426 .440 .447
Distribution of contacts per sphere n
9.14 9.51 8.06 7.34 6.92
4
5
6
7
8
9
10
11
12
1 0 0 3 6
13 14 14 54 78
77 86 69 173 243
245 192 182 309 328
322 233 316 233 200
310 193 212 118 48
208 161 87 14 2
194 226 7 2 0
192 389 0 0 0
If we now assume with the authors that, statistically, an aggregate of well-agitated like spheres may be regarded as an arrangement of separate clusters of close-packed and simple cubic arrays, each present in a proportion to yield the observed porosity, we may compute the average number of contacts per sphere corresponding to a given por0sity.t Thus, if x represents the fraction of close-packed spheres, then
* Graton and Fraser [25] present evidence that the effect of a container, especially one with curved walls, is propagated a considerable distance into the aggregate.
t Although this assumption represents an over-simplification of the actual situation, it has the great merit of leading rapidly to numerical estimates. It should be noted that, in natural assemblages, one may find zones of disorderly arrangements in which, on account of “bridging” of groups of particles, the porosity exceeds that of the simple cubic array.
244
H. DERESIEWICZ
-
where, from Table 1.’ Pc.fi.= 1 -n/31/2 and if V is the unit volume of a given array,
Pcub.
= 1 - n/6.
Further,
Again, from Table l., nc,p,== 12, = 6, vc,p, = 4@R3, V c u b . = 8 R 3 ; hence, (2.1) and (2.2) enable us to find the desired relation between the average number of contacts per sphere and the porosity: n--------[6(2v%-1)-i_p] 1
v2-
nV2
1
26.4858 - 10.7262/(1 - P ) .
We may compare values of n obtained from (2.3) with the observed values given in Table 3. This is done in Table 4. TABLE4
Sample
Porosity
Observed n
Computed n
A
.359 ,372 .426 ,440 ,447
9.14 9.51 8.06 7.34 6.92
9.75 9.39 7.79 7.32
B C D E
i.X
In an effort to obtain the number of contacts in an aggregate devoid of interstices, Marvin [ l l ] packed lead shot in a cylinder and applied axial compression. With the spheres packed randomly, a moderate initial compression (1000 lb over a circular area 25 mm. in diameter) resulted in a n average coordination number of 8.41; a compression sufficient to eliminate all voids (35,000 lb) yielded an average number of contacts per sphere slightly over 14 (specifically, 14.16).* With the spheres initially
* Hopes that the deformed shape would turn out to be Lord Kelvin’s orthic tetrakaidecahedron [35], [36] were not realized. This figure represents the most economical surface-volume configuration in the form of a polyhedron with plane faces permitting homogeneous division of all space. It has eight hexagonal and six square faces, all of the edges of the faces being equal. To mineralogists this body is known as the cubo-octahedron 1371. However, in the fourteen-sided figure obtained in the experiments, pentagonal faces were the most common ones.
MECHANICS OF GRANULAR MATTER
245
arranged in a close packed array (face-centered cubic), the initial and final number of contacts were each 12 and each deformed sphere assumed the shape of a rhombic dodecahedron. In carrying out similar experiments on lead shot of two sizes (ratio of diameters 2:1, grading 4 : l by volume), Matzke [38] observed an increase (from 14) in the number of contacts per sphere for the larger size, a decrease (from 14) for the smaller size.
FIG.5 . Spherical section through centers of spheres arranged in configurations of densest packing round a kernel sphere, illustrating the distribution of the two kinds of voids : left, face-centered cubic; right, close-packed hexagonal (Hudson [39]).
3. Effect of Interstitial Spheres on Density. We come now to consider an interesting computation by Horsfield 191 who investigated the effect on the density of inserting into a close-packed array successively smaller spheres just large enough to fit into the interstices. The spheres (of radius yI) comprising the initial (close) packing will be referred to as “primary” spheres. There exist two different kinds of voids in a close-packed array, the larger one having a “square,” the smaller, a “triangular,” shape* (see Fig. 5).t The largest sphere which will fit into a square pocket is called a “secondary” sphere, the largest sphere which can be inserted into a triangular pocket
* Graton and Fraser [ 2 5 ] give a lengthy discussion of the geometry of voids in the several packings with a view to shedding light on permeability t o fluid flow. They term the two kinds of interstices “concave cube” and “concave tetrahedron,” respectively. It is worthy of note that all the voids are interconnected in such a manner that the circuit is passable by any sphere not exceeding ( 2 / 1 F - 1) % 0.1537 v1 in radius (Hudson [39]). t After Hudson [39]; the number of the two kinds of interstices is the same in each of the two close-packed systems, but their distribution is different.
246
H. DERESIEWICZ
is called a “tertiary” sphere. Finally, the two next largest spheres which can be introduced after the secondary and tertiary spheres are in place are called “quaternary” and “quinary,” respectively. TABLE5 Type of sphere
primary secondary
tertiary quaternary quinary
filler
r;/rl, i = 1, 2, 3, 4, 5* 1 0.4142 0.2247 0.1766 0.1163 0.0000 Relative number of spheres 1 1 2 8 8 Volume of sphere/r? 4.1888 0.2977 0.0475 0.0231 0.0066 Cumulative volume of spheres/?? 4.1888 4.4865 4.5815 4.7663 4.8191 5.4394 Porosity of aggregate (yo) 25.95 20.69 19.01 15.74 14.81 3.84 True volume of spheres in aggregate (Yo) 77.01 5.47 1.75 3.40 0.97 11.40 Cumulative increment in 0.07103 0.09375 0.13790 0.15076 0.29879 density
-
* The exact values of the radii of the interstitial spheres are: rz/rl = v 2 - 1; r,/rl=1/3/2-l;r4/~1=(1+3cra)/6(1+cc),a=1+~-~;rs/rl= l-l/r2-r4/rl. Horsfield’s results are given in Table 5,* in which the radii of the successive interstitial spheres are denoted by ri, i = 2, 3, 4 , 5 . It is seen that, by using spheres of five sizes, the porosity may be reduced to 14.81%. If, in addition, the remaining voids were close-packed with a filler of spheres so small that the boundary effects could be neglected, the remaining porosity would have only 25.95% of its value prior to the insertion of the filler, i.e., 3.84% (see last column in Table 5). These theoretical values are predicated, of course, on the precise placing of each sphere in its appropriate void. Since this is manifestly impossible in any practical experiment, the values obtained above can a t best serve only as an indication of favorable proportioning of an aggregate. This is borne out by experiment. For example, White and Walton [32] obtained a porosity of 32.2% for a theoretically graded mixture of aluminum oxide, although when they mixed primary lead shot with graded silicon carbide fines, they were able to reduce porosity to 15,1%; in a rammed silicon carbide and clay mix, the porosity was further reduced to 10.7%. Of interest are the data cited by Muskat [13] which indicate that, in packed assemblages found in nature, groups of coarse particles tend to pack to greater density than similar groups of fine particles. A different approach to the problem of filling the voids in the primary close-packed structure by spheres of smaller size was taken by Hudson [40]. * Similar computations were carried out by White and Walton tetragonal-sphenoidal system.
[32] for the
247
MECHANICS OF GRANULAR MATTER
He investigated the increase in the density of the mixture due to the insertion in the two types of interstices of n spheres of equal radii, Y(< Y J , arranged in patterns of cubic symmetry. Some of his results are shown in Table 6. Of the cases considered, the greatest total density increment occurs when each square interstice contains 21 spheres, each of radius Y = 0.1782 Y ~ and each triangular interstice contains 4 spheres of the next smaller size, i.e., Y = 0.1547 rl. The total density increment in this case is 0.14854; the system corresponds to a mixture of 87.1% coarse, 12.9% fine*.
,
TABLE6 (a) DENSITY ~ N C R E M E N T S GOVERNED BY DIMENSIONS OF SQUARE INTERSTICE
Square interstice
Triangular interstice
density increment n
rlrl
1 2 4 6 8 9 14 16 17 21 26 27
0.4142 0.2753 0.2583 0.1716 0.2288 0.2166 0.1716 0.1693 0.1652 0.1782 0.1547 0.1381
n (r/r,)
0.07106 0.04170 0.06896 0.03028 0.09590 0.09150 0.07074 0.07 768 0.07660 0.11892 0.09626 0.07 108
density increment n
0 0 0 4
Total density increment
2n(r/ri)3
0.04038
0
-
1 4 4 4 1 4 5
0.02034 0.04042 0.03882 0.03605 0.01132 0.02962 0.02632
0.07106 0.04170 0.06896 0.07066 0.09590 0.11 184 0.111 16 0.11647 0.11265 0.13025 0.12588 0.09740
(b) DENSITYINCREMENTS GOVERNED BY DIMENSIONS OF TRIANGULAR INTERSTICE
Triangular interstice
Square interstice
density increment
n
rlr1
1 4 5
0.22475 0.1716 0.1421
2n(r~)3
0.02271 0.04042 0.02868
density increment n
8 21 26
Total density
increment
n(r/4s
0.09083 0.10611 0.07457
0.11354 0.14653 0.10325
* A d d e d at press-time: To be noted is the work of Manegold, Hofmann, and Solf [88], who discussed the size of the largest interstitial sphere possible in each of the several regular packings of equal spheres, and of HrnbiSek [89], who was interested in the void structure of packings.
248
H. DERESIEWICZ
In experiments by Westman and Hugill [6] referred to previously it was found that, for a mixture of two sizes, the density of the aggregate increased as the ratio of the diameters of the two sizes increased, and the best mix was obtained with a grading of 70% coarse, 30% fine. There were three grades of round, washed sand - coarse, medium and fine - with average diameters in the ratios 50.5:8: 1. Individually, the coarse aggregate packed to a porosity P = 37.7y0, the medium, to P = 38.2y0, the fine, to P = 42.5%. In the two-size mixture 70% coarse-30% medium (ratio of diameters = 6.3) the resultant porosity was 26.2y0, in the mixture 70% coarse-30yo fine (ratio of diameters = 50.5), the porosity was depressed to 18.5%. In a mixture of all three \ \ /' \ components the highest densities of \ packing were attained with mixtures of 70% coarse-lOyo medium-20% fine, '\ \ with P = 15.5y0,and 70% coarse-20yo \ medium-lO~ofine, with P = 16.8%. I Once again, it should not surprise that the theoretical densities are not I matched experimentally, since the random tamping precludes any possibil\ / ity of precise placement of each sphere, / '\\ and any deviation from the exact theoretical arrangement destroys the FIG.6. Construction of dense random packing. M ~ the packing of spheres (Wise [S]). effect of the boundary may not always be negligible [25]. The random character underlying any practical method of packing of a granular aggregate is taken into account in a study by Wise [8]. He considers an array of spheres of unequal size in a random packing such as may be obtained by the following procedure. A large sphere A is chosen and other, smaller, spheres are laid on its surface. The first two of these are placed in contact with each other and each subsequent sphere is caused to touch, in addition to sphere A , at least two others which are in contact with each other. An aggregate such as this is shown in Fig. 6. By connecting the centers of all spheres neighboring a given sphere, a set of tetrahedra with vertices located a t the sphere centers is obtained which is associated with the given sphere. The array will be considered densely packed if in every tetrahedron all the spheres are in mutual contact. This means that gaps, such as G in Fig. 6, are excluded from consideration.* The properties \
.He---
,'(
.---/'
* It should be noted that this procedure will not create dense packing when all the spheres are of equal size.
~
~
MECHANICS O F GRANULAR MATTER
249
of such a packing are expressed in terms of a probability distribution function for the four radii in each tetrahedron. For the case in which the logarithms of the radii of the aggregate obey a normal distribution of standard deviation a = 0.4 (Fig. 7a, b), the mean radius is 1.08 and the mean density of the packing is estimated at 0.8. Further, the mean number of spheres in contact with a given sphere has a distribution as shown in Fig. 7c.
FIG.7. Dense random packing of spheres having radii obeying a log-normal distribution of standard deviation u = 0.4 (Wise [S]) : (a) distribution of spheres of radius Y ; (b) proportion of spheres having radii smaller than Y ; (c) mean coordination number of a sphere of radius Y .
A computation of the size of interstitial spheres (one in each tetrahedron) indicates another normal distribution of standard deviation cr = 0.178 and a mean radius 0.2742. The mean volume per sphere of the “coarse” aggregate is 8.50, of the “fine” filler it is 0.0973. With the filler occupying 5.84y0 of the volume of the coarse aggregate, the mean density of the final mixture is approximately 0.85. 4. Packing of Non-Spherical Bodies. Many granular media, such as round sand, some powdered metals, and even the carbon packing in a carbon microphone may be assumed to be represented adequately by a model composed of spherical elements. Others, however, such as coal in a fuel bed or a granular catalyst in a packed tower, deviate radically from an array of particles of spherical shape; for these, the models previously discussed are, at best, inadequate. Experimental work on such irregular particles indicates that they pack in a manner different from that of spherical particles (Brown and
250
H. DERESIEWICZ
Hawksley [41]). Thus, the porosity is, in general, greater, possibly as a result of “bridging” of groups of particles, and the frequency distribution of the number of contacts is different. I t was found by Bennett and Brown [16] in studies on lumps of starch that the mean number of contacts per lump* in a packing was 6.5 and that the packing of closely graded lumps was much more uniform than that of lumps covering a wide size range. Furnas [42], [43], in his work on flow of gases through arrays of irregular particles, found it convenient to introduce, as a measure of the true size of particles, the “equivalent spherical diameter” ;this he defined as the diameter of a sphere of volume equal to the average volume of a particle. In addition, he employed a shape factor, given by the ratio of the equivalent spherical diameter to the average screen size as measured by square-mesh screens. He suggested that if the material is poured into place slowly and from a small height above the bed then the aggregate tends to form with a definite percentage of voids characteristic of the material. Furnas further assumed that if extremely small particles were introduced into a bed of large pieces, the small pieces would fill the voids without changing the total volume of the system. Hence, if P denotes the porosity of a given sized? aggregate, the volume of matter in a unit volume will be V , = 1 - P. If a small filler is introduced, which also packs to a porosity P , its volume per unit volume will be V 2 = P(l - P). Hence, the proportion of the larger size, measured with respect to total particle volume, is 1/(1 P). Experimentally, Furnas found that these relations hold as long as the ratio of diameters is greater than five. The reasoning can be extended to a mixture of n sized components, each packing to a porosity P and having elements much smaller than those of the preceding component. Then, the ratios of the volumes per unit volume of the successive components are given by
+
v, : v2 : v,: . . . v,=
1 : P : P2:.
.
We conclude this section by calling attention to some work on densest packing of non-spherical bodies. Thus, Lord Kelvin [44], [45] discussed the homogeneous close-packing of convex bodies of equal size, shape, and orientation in space. Having assumed that, as in the case of spheres, in closest packing four bodies are always in mutual contact, he found several possible variations of a single packing, each having a coordination number twelve, but only one, in general, forming a configuration of greatest density. A more exhaustive analysis of the problem of regular densest packing of convex bodies was made by Minkowski [46] who found, in addition to the packing given by Kelvin, two others, of which one has a coordination number fourteen.
* The contacts were determined from discoloration caused by passing iodine vapor through the aggregate. t A material is considered “sized” if it passes one screen and is retained on another which differs from the first by a factor of
MECHANICS O F GRANULAR MATTER
251
The other packing, exemplified by a set of certain polyhedra bounded by twelve planes, is distinguished by the fact that the number of bodies in mutual contact is not four.*
111. SOMERECENTRESULTSOF CONTACTTHEORY
A . Normal Forces The various attempts at an analytical description of the mechanical behavior of granular materials may be classified according to whether the medium is represented by means of a model in which the material is assumed to be continuous or by one which consists of discrete elements. Many of the theories in the latter category assume the constituent granules of the model to be in direct, elastic contact with one another. A consideration of the forces and deformations a t each contact surface serves as the point of departure in developing such theories. t Accordingly, the present section contains a summary of the pertinent aspects of the problem, and some of the results of its solution, of two elastic bodies subjected to a variety of static and dynamic loading programs. We start with the simplest case, that of two like spheres, each of radius R , compressed statically by a force, N , which is directed along their line of centers, i.e., normal to their initial common tangent plane. The theory, due to Hertz [as], predicts a plane, circular contact of radius a = (6NR)lI3; here 6 = 3(1 - v2)/4E, and v, E denote, respectively, Poisson’s ratio and Young’s modulus of the material. The normal pressure on the contact area is given by (see Fig. 8) (T=-
3N (a2 - p 2 ) Y 2 , 2na3
where p represents the radial distance from the center of the contact circle. Finally, the relative approach of the spheres is (3.3)
a = 2(6N/R1/2)213;
* In the two packings characterized by having four bodies in mutual contact, the centers of the constituent elements in the arrays having twelve contacts per element are located at the vertices of cubo-octahedrons; those in arrays having fourteen contacts per element, at the vertices of rhombic dodecahedrons (Minkowski [as]). t These are discussed in detail in Section IV. B; for a brief account, see the review article by Mindlin [47].
252
H. DERESIEWICZ
this leads to the normal compliance of the contact (3.4)
where p is the shear modulus of the material.* The inherent non-linearity of the relations (3.1) and (3.3) offers a first indication of difficulties to be encountered in the application of contact theory to the study of granular media.
FIG. 8. Distributions of normal (a)and tangential (t)components of traction on the contact surface of two like, spheres subjected to a normal force followed by a monotonic tangential force (Mindlin [ 4 9 ] ) .
* The Hertz solution for a pair of non-spherical bodies, compressed by a force N normal to their common initial tangent plane, predicts [as] a plane contact, bounded by an ellipse of semi-axes a and b, and a normal pressure on the contact of magnitude u = ( 3 N / 2 n a b ) [ l- (%/a)* - (y/b)z]*/*.The semi-axes of the elliptic contact area and the normal approach can, in general, be evaluated only numerically. The normal compliance of the contact has been given by Mindlin [49] in terms of the complete of modulus k , = (1 - b2/aZ)lla,b < a, elliptic integral of the first kind,
a,,
(3.5)
C
= a(@,
+ 6,)K l / 3 n a ,
where the subscripts on 6 refer to the value of this quantity appropriate t o each of the two bodies.
MECHANICS OF GRANULAR MATTER
253
It is useful to recall the assumptions on which the Hertz theory is based: (1) The dimensions of the surface of contact are assumed small compared
with the principal radii of curvature of either body a t the initial contact point. It is, therefore, adequate, for a first approximation, to consider the points on each body which come into contact as a result of compression, as well as neighboring points, to lie, before compression, on a surface of the second degree, the coordinates of each of the two surfaces being referred to an origin located at the point of initial contact.* (2) The warping of the surface of contact due to unequal geometric and elastic properties is neglected, so that the contact surface is taken to be plane. As a consequence of assumptions (1) and ( 2 ) the problem is considered as a problem of the semiinfinite solid bounded by a plane. (3) The tangential components of the traction across the contact are neglected.
R. Tangential Forces 1. Increasing Tangential Force. Suppose now that the system of like spheres, initially compressed by a constant normal force, N , is subjected to an additional force, T , which acts in the plane of contact and whose magnitude rises monotonically from zero to a given value. Because of symmetry the distribution of the normal pressure remains unchanged. If it is assumed that there is no slip? on the contact, then, on account of symmetry, the displacement of the contact surface in its plane is constant (i.e., that of a rigid body). The solution of the appropriate boundary-value problem, due to Cattaneo [51] and Mindlin [49], yields the tangential component of traction, t,on the contact surface and the displacement, 6, of points in one sphere remote from the contact with respect to similarly situated points in the other sphere. The tangential traction is parallel to the displacement (and to the applied force T ) , axially symmetric in magnitude, and increases without limit on the bounding curve of the contact area (Fig. 8). The relative displacement is proportional to the applied force,
this relation being illustrated in Fig. 9.
* A singularity at the point of initial contact is thereby excluded. An extension of the Hertz theory to the compression of two bodies of revolution each of whose surfaces in the neighborhood of the initial contact point are assumed to be, before deformation, of the fourth degree has been considered by Cattaneo [50]. t By “slip” we mean a relative displacement of contiguous points on a portion of the contact surface. We distinguish between “slip” and the term “sliding,” which we reserve to denote relative displacement over the entire contact.
254
H. DERESIEWICZ
As a singularity in traction is ruled out on physical grounds we may expect slip to occur, no matter how small the applied tangential force. It is reasonable to suppose the slip to be initiated at the edge of the contact since it is there the singularity in traction takes place in the absence of slip.
0
0.I
az
a3
0.4
,ua6/N
FIG.9. Tangential force-displacement relation; comparison of equations (3.6) and (3.8) with results nf static tests (Johnson [52]).
Further, since this traction is symmetric in the absence of slip, slip is assumed to progress radially inward, covering an annular area. On this annulus it is assumed, as a first approximation, that the tangential component of traction is in the direction of the applied force and is related to the normal (Hertz) component of traction, already present, in accordance with Coulomb’s law of sliding friction, i.e., z = fa,where f is a constant coefficient of static friction and 5 is given by (3.2). On the remainder of the contact area, called the “adhered portion,” considerations of symmetry indicate that the tangential component of the displacement is constant. Thus, the boundary conditions may be specified for this problem in elasticity: the normal component of traction (zero) and tangential component of displacement (constant) are given on the adhered portion, and all the components of traction are given on the remainder of the boundary (normal component zero, tangential component proportional to Hertz pressure on slip annulus ; traction zero elsewhere). The solution of this problem, obtained by Cattaneo [51] and by Mindlin [as], predicts the relation between the radius
MECHANICS OF GRANULAR MATTER
255
of the adhered portion (or, what is the same thing, the inner radius of the annulus of slip) and the applied tangential force,* (3-7)
c
=~
(-l T/fN)V3,
the distribution of the tangential component of traction on the contact (Fig. 8), and the relative tangential displacement of points in the two spheres a large distance from the contact,
The relationship between the displacement and applied force is illustrated in Fig. 9 , along with experimental results obtained by Johnson [52] which confirm the validity of (3.8). The tangential compliance of the contact is given by
(3.9) and represents the reciprocal of the slope of the curve in Fig. 9 . It should be noted that, as the applied force T approaches f N , the inner radius of the slip annulus tends to zero in accordance with (3.7) and the tangential compliance tends to infinity in accordance with (3.9); hence, when T > f N , the displacement 6 becomes indeterminate. This is a mathematical expression of the physical phenomenon of rigid body sliding, defined earlier. The initial value of the tangential compliance (i.e., its value a t T = 0 ) , obtained from (3.9), is (3.10)
s = (2 - Y)/4pu
and should be compared with the value of the normal compliance (3.4). That these two are of the same order of magnitude is apparent from the fact that the ratio of the initial tangential to the normal compliance is % = (2 - ~ ) / 2 (1 Y ) ; that is, % ranges from unity, for v = 0, to 3 / 2 , for v = 112. It is seen, further, that the compliance obtained on the basis of the assumption of no slip [from (3.6)] is identical with the initial compliance obtained from the solution which takes slip into account (3.10). This is of
* A small lateral component of relative tangential displacement is found to accompany the major slip in the direction of the applied force. The lateral tangential traction which accompanies the lateral slip is- neglected.
256
H. DERESIEWICZ
significance in applications of the theory to cases in which the additional force is small (see Section IV. B, 2b).* 2. Decreasing Tangential Force. Mindlin, Mason, Osmer, and Deresiewicz [54] consider the effect on the system of two spheres of a reduction of the
tangential force from a peak value T*, 0 < T * < fN. This time, if slip were prevented, the tangential traction would tend to negative infinity on the edge of the contact area. Therefore slip, once again, may be presumed to occur, but its direction will be opposite to that of the initial slip. In a manner analogous to that discussed above, an annulus of counter-slip will
* The problem of contact of two non-spherical bodies has been discussed by Cattaneo [51] and Mindlin [as]. The latter also computed the ratio of the initial tangential to the normal compliance for bodies of like elastic properties : 4 2 - v) I4(l-v)
vN 2n2(2-v)k
]
a
’
(3.11)
a = b,
I
vN1
2 7 4 2 - Y)kl
4(1 - V )
]
’
a > b,
where N = 4n [ ( 2 - k a ) H / k - 2 E / k ] , and N, is obtained from N by inserting the subscripts on k , K, and E . H and E denote, respectively, complete elliptic integrals of the first and second kind of modulus k = (1 - aa/be)ll2,K, and Elare analogous integrals of modulus k, = (1 - b2/aa)1/a. The relations of (3.11) axe shown in Fig. 10.
2.0 I.6
\o 0.8
0.4 0
103
lo*
10’
I a2/b2
10’
18
lo3
FIG. 10. Ratio of the initial tangential to the normal compliance of a pair of contiguous, non-spherical bodies of like elastic properties (Mindlin [as]). The distribution of tangential traction on the contact due to the application of a monotonic tangential force has also been computed for the case of two bodies of revolution whose undeformed surfaces in the neighborhood of initial contact are taken to be of the fourth degree (Cattaneo [ 6 3 ] ) .
257
MECHANICS O F GRANULAR MATTER
be formed and will spread radially inward as the tangential force is gradually decreased. Its inner radius is t3.12) The corresponding relative displacement of distant points of the two spheres during unloading is (3.13)
6,
T* -
=
1)2;'
-
- l-m
and is depicted by curve PRS in Fig. 11. Here is manifested a further complication in addition to the previously noted nonlinearity, namely, the inelastic character of the unloading process. When the tangential force is completely removed ( T = O), the annulus of counter-slip does not vanish [as may be seen from (3.12)]; thus a permanent set appears* (point R in Fig. 11) and can be removed only by applying a tangential force in the reverse direction. For T = - T*, that is, when the tangential force is fully reversed, a comparison of (3.7) and (3.12) shows that b = c, which indicates that the counter-slip has penetrated to the depth of the original slip. The traction a t this loading is identical with that at T = T* except for reversal of sign. Thus, the situation corresponding to point s Of Fig. 11 is identical with that at -point P except for reversal of sign. The compliance during unloading is '(3.14)
( "p"
s=-(24- vP
1-
T
-6
s FIG.11. Theoretical hysteresis loop due to oscillating tangential force at constant normal force (Mindlin and Deresiewicz [55]).
T*2fN -
We may note that the initial compliance on unloading [T = T* in (3.14)] is the same as the initial compliance on loading (3.10).
*
The corresponding traction is self-equilibrating but not identically zero.
258
H. DERESIEWICZ
3. Oscillating Tangential Force. A subsequent increase of T from -T* to T* will give rise to identical events as occurred in the course of reduction of T from T* to - T* except for reversal of sign. The appropriate displacement during this loading process will be dL = - &(- T ) and is illustrated
c
FIG.12. Example of an annulus of wear (Mindlin, Mason, Osmer, and Deresiewicz [54]).
by curve S U P in Fig. 11; the corresponding tangential compliancg is given by (3.14) with the sign of T reversed. Thus, the load-displacement curve forms the closed loop P R S U P (Fig. 11) and this path will be traversed during subsequent oscillation of T between the limits T* as long as N is maintained constant. The area enclosed in the loop represents the frictional energy dissipated in each cycle of loading: (3.15)
F
=
{ (
9(2 - v) (fN)2 1lopa
[ (
5T* I - - T*)5’3 _ _ _ 1+ fN 6fN
1---
T*)z/3]}
fN
.
MECHANICS OF GRANULAR MATTER
259
For small amplitudes of loading, that is, when T*/fN << 1, (3.15) reduces to (3.16)
F=
(2 - Y ) T * ~ 36pafN ’
indicating that the energy loss per cycle vanes as the cube of the amplitude of the tangential force.
FIG.13. Dimensions of annuli of wear; comparison of equation (3.7) with experiment (Mindlin, Mason, Osmer, and Deresiewicz [54]).
Many of the conclusions of the preceding theory were subjected t o experiment. Tests, reported by Mindlin, Mason, Osmer, and Deresiewicz [54] , were made with polished glass lenses compressed by a constant normal force and subjected to a transverse force whose direction reversed at the rate of 60 cps. Since relative displacement at the contact is confined to an annular region, a pattern of wear on such a region is to be expected. One example of the observed annuli is shown in Fig. 12; a comparison of experimentally and theoretically determined dimensions, the latter computed from (3.7), is made in Fig.l3* and indicates close agreement. Measurements
* In connection with Fig. 13 i t may be noted that the coefficient of friction appears to remain sensibly constant, as was assumed in the theory.
260
H . DERESIEWICZ
of energy dissipation a t large amplitudes of tangential force agreed with values computed from (3.15); however, a t very small amplitudes, the energy loss was found to vary as the square of the amplitude rather than as the cube as predicted by (3.16). In the experiments referred to earlier in this section, Johnson [52] employed a system consisting of steel balls in contact with a steel plane.*
FIG. 14. Hysteresis loops obtained from static tests (Johnson [52])
The static tests included loading and over-loading (Fig. 9), unloading and cyclic loading (Fig. 14) ; once again the agreement of theory with experiment was strikingly good. Johnson’s dynamic tests, carried out a t 46.5 cps. with a variety of sphere diameters and normal loads, resulted in the tangential force-displacement relation plotted in Fig. 15. A t first sight it may seem strange to find such excellent agreement between the prediction of statical theory and the results of dynamic tests. This will not be surprising, however, when it is recognized that the dynamic tests of Johnson, as well as the earlier ones of Mindlin, Mason, Osmer, and Deresiewicz, imposed reversals of stress whose period was long when compared with the time of travel of the stress
*
For this case the Hertz theory predicts a contact radius [48]
a
=
[3(1 - v 2 ) NR/2E]11S.
MECHANICS O F GRANULAR MATTER
261
waves through a distance of two diameters of either of the spheres; that is, sufficient time was available for stress equilibrium to be reached. On the other hand, data obtained from damping tests (Fig. 16) indicate that, a t low force amplitudes, the energy loss is proportional to the square of the amplitude, although, a t large amplitudes (near the levels of gross sliding)
FIG. 15. Tangential force-displacement relation; comparison of equation (3.8) with results of dynamic tests (Johnson [52]).
the predictions of the theory are approached. Furthermore, at intermediate amplitudes, the energy loss appears strongly dependent on the diameter of the spheres and, hence, on the radius of the contact area, in that it decreases as the contact radius increases. An explanation of the difference between theoretical and experimental values of the energy dissipation may be sought in the fact that real contact between two bodies occurs only over the surface asperities. It is suggested by Johnson [52] that, for small tangential forces, the tangential displacement necessary to relieve the singularity in traction takes the form of an elastic deformation of the asperities; the damage to the surface is small and the energy loss is largely due to elastic hysteresis. An increase in applied tangential force causes the asperities a t the edge of the contact surface to deform plastically through relatively large strains, a process which leads to marked increase in energy dissipation and to severe damage to the surfaces.
262
H. DERESIEWICZ
It is thought that in intimate metal-to-metal contact the surfaces form cold-welded junctures which, when subjected to oscillating stress, work
I ENERGY LOSS FROM
STATIC HYSTERESIS (N.20.5 LE., 0=0.375")
0
0.1
0.2
T*
fw
0.3
0.4
0.5
0.6
FIG.16. Relation between energy dissipated per cycle and amplitude of tangential force; comparison of equation (3.15) with experimental data (Johnson [52]).
harden and ultimately develop fatigue failure. Finally, near values of the tangential force sufficiently large to cause sliding, the agreement between the measured energy loss and that computed on the assumption of slip between the two surfaces may be due to the possibility that the magnitude of the plastic strains undergone by the surface asperities is controlled by the stress distribution in the elastic region of the spheres. We may offer an alternative, and perhaps simpler, explanation by supposing that, a t small amplitudes of the tangential force, energy is dissipated as a result of plastic deformation of a small portion of the contact surface, whereas, a t large amplitudes, the Coulomb-sliding effect predominates.
MECHANICS OF GRANULAR MATTER
263
C. Oblique Forces In general, when a granular assemblage is acted upon by varying external forces or when it is in a state of internal vibration the individual contact surfaces are subjected to forces whose normal as well as tangential components change simultaneously. I t has been shown by Mindlin and Deresiewicz 1551 that, under such circumstances, and as a result of the inelastic character of the tangential force-displacement relation at constant normal force, the relation between the instantaneous tangential forces and displacements depends not only on the initial state of loading, but also on the entire past history of normal and tangential forces. Moreover, a variety of phenomena are involved which depend upon: whether either the normal or the tangential force is held constant while the other varies; whether they both vary and whether the sense of variation is such that one increases while the other decreases, both increase, or both decrease; whether the relative rate of change of the two forces is greater or less than the coefficient of friction; whether the immediately preceding history of loading was in the same or in the opposite sense as the current loading. Suppose, for example, that a pair of like spheres in Hertz contact due to a normal force Nois acted upon by additional normal and tangential forces which are increased at an arbitrary relative rate. In this instance (3.9) is no longer valid and must be replaced by
where N denotes the total normal force and a represents the instantaneous, radius of the contact circle. Another example of a problem involving a varying oblique force is given by the following situation: To a pair of spheres in Hertz contact due to N o apply a tangential force, T*, then reduce it to T ; now increase the normal force, at the same time decreasing the tangential force at an arbitrary relative rate. This time the tangential compliance of the contact is
264
H. DERESIEWICZ
Since, in (3.18), d N / d T is inherently negative, its absolute value must be employed in the expression.* We will touch upon one other loading schedule which is of interest in connection with the theory of deformation of granular media. Here, the contact surfaces of two spheres, initially compressed by a normal T force No (which gives rise to a contact radius ao), are acted upon by additional normal and tangential forces whose resultant, R, oscillates in magnitude between values fR* but maintains a constant direction in space. That is, its tangential scalar component, T , oscillates between f T * while its normal component varies in such a way that dN/d T is constant. For the case where dNldT l / f , a hysteresis loop, illustrated in Fig. 17, is obtained, in which the force-displacement S relation, after traversing curve OPRYS, stabilizes along path
<
FIG. 17. Theoretical hysteresis loop due to oscillating force having constant direction (Mindlin and Deresiewicz [55]).
suvws-
The tangentia1 compliance during initial loading (path OP) is given by
(3.19) and during initial unloading (paths PRY and Y S ) by
(3.20)
In these expressions, L = T / f N o , L* = T*/fNo; also, 8 = f/P, where = dT/dN f, so that 0 8 1. The constant 1,in terms of 8, becomes
p
>
< <
* These and a variety of other cases are discussed in detail in the paper by Mindlin and Deresiewicz [ M I . Added at press-time: The problem of contact of nonspherical elastic bodies under oscillating tangential forces, and under varying obliqe forces, has recently been discussed by Deresiewicz [go].
265
MECHANICS OF GRANULAR M A T T E R
+ <
il = (0 - l ) / ( 0 1 ) 0. The tangential compliance during loading in the stabilized cycle (path S U V ) is
The corresponding compliance during unloading (path V W S ) may be obtained from (3.21) by reversing the signs of 0 and L , and is seen to be identical with the first of (3.20). In all cases t,he initial value of the compliance is given by (3.10). The associated energy loss per cycle is found to be
and, for small values of L*, reduces to (3.23)
< < <
I t should be noted that the condition d N / d T l/f imposes a limitation on the value of T* which may be expressed as 0 L* 1/(1 0). In the case for which 1 0 bo, i.e., d N / d T 2 l / f , it is found that, again, the path in the load-displacement plane stabilizes after three-quarters of a cycle, but that no loop is formed; that is, no frictional energy loss is involved. The tangential compliance during loading and unloading in the stabilized cycle is given by (3.10). Finally, since, for 0 = 1 , (3.22) yields F = 0, it appears that d T / d N must exceed the coefficient of friction if energy is to be dissipated.
< <
+
D. Twisting Couples Certain types of loading are responsible for the presence, on the individual contact surfaces in a granular medium, of twisting moments about the line of centers of curvature of the contiguous portions of a pair of granules. The procedure in tackling this type of contact problem, and its results, are very similar to those encountered in the study of tangential contact. Consider a pair of like spheres compressed.by a constant normal force N . Now apply a twisting couple, about the axis of N , which rises monotonically to a given value M . It was found by Mindlin [49] that, in the absence of slip, the contact surface rotates as a rigid body about the line of centers and
266
H. DERESIEWICZ
the circumferential shearing stress tends to infinity at the edge of the contact.* The torsional compliance of the contact is given by (3.24)t
C, = 3/8pa3.
Assumptions and considerations similar to those made in the course of the solution of the tangential contact problem (part B, article 1 of this section) lead to the conclusion that slip in a circumferential direction takes place over an annular area, and the remaining portion of the contact undergoes rigid body rotation about the line of centers. Solution of the appropriate boundary-value problem, due to Lubkin [ 5 7 ] ,yields the distribution of the circumferential shear stress on the contact, as well as expressions, in terms of quadratures, relating the twisting moment and the angle of twist to the (unknown) inner radius of the slip annulus.$ From the latter, numerical values of the torque-twist dependence yield a curve which is qualitatively the same as the tangential force-displacement curve of Fig. 9. Explicit expressions, valid for small values of MlfNa, have been computed by Deresiewicz [59] which yield the dimensions of the slip annulus and the angle of twist resulting from a given applied moment. Thus, the inner radius of the slip annulus is (3.26)
2fNa
and the torsional compliance of the entire contact is given by (3.27)
-4 2
3 8pa3
Once again, as in the case of tangential contact, the initial compliance obtained on the assumption of slip [ M = 0 in (3.27)]is identical with the value of the compliance in the absence of slip (3.24).
* A mathematically equivalent problem was first solved by E. Reissner and Sagoci [56]. t The torsional compliance for general contact, in the absence of slip, is (Mindlin [as]) (3.25)
c,=
(”)
8pbS
8[BD - YCE] n[E - 4v(l - k2)C]
where, D = (K - E)/ka, B = [E - (1 - k * ) R ] / k , and C K, E, and k have been defined in the footnote on p. 256
’ =
[(Z - kZ)R- 2E]/k4;
2 The corresponding solution for a pair of non-spherical bodies in torsional contact is given, approximately, by Cattaneo [58].
267
MECHANICS OF GRANULAR MATTER
With the analysis still limited by the approximation MlfNa << 1, the effects of decreasing and oscillating applied torque may be studied. The phenomena are again analogous to those occurring in the case of tangential contact (B, 2 and 3 of this section). Oscillation of the applied moment between the limits fM* is shown by Deresiewicz to give rise to a hysteresis loop which is similar in appearance to the one shown in Fig. 11. The torsional compliance during the unloading portion of the cycle is
[(
C,&=- 3
(3.28)
Spa3
2 1 - - 3 M* -M)-1'22 2fNa
1]
while during the loading portion the compliance is given by (3.28) with the sign of M reversed. The energy dissipated per cycle is (3.29)
[ (
)"I F: [
FA-- 2(fN)218 - 1 - 1 -3M* Pa 19 2fNa
-
-~
If
(
1 - - 3M*)1'2]} 2fNa '
which, in view of the limitation M*/fNa << 1, reduces to (3.30)
F
= 3M*3/16pfNa4.
IV. MECHANICALRESPONSEOF GRANULARASSEMBLAGES
A. Continuous Model Most of the early work on granular media concerned itself with problems of soil mechanics, in particular with evaluating earth pressure on bounding surfaces of the medium. The theories, a detailed account of which is given in the survey articles by Kotter [l] and H. 'Reissner [60], were invariably based on a representation of the material as a continuum subjected to additional constraint conditions and relations. Of these constraint conditions the most widely used, due to Coulomb, may be stated as follows: The angle formed by the traction acting on the surface of a volume element of the medium and the inward-drawn normal may not exceed a certain value pl. In the interior of the medium this limiting value is equal to the angle of internal friction of the material; a t the boundaries, to the angle of friction between the wall and the granular medium. It is implied that n m e of the principal stresses a t any point in a cohesionless assemblage is tensile. Coulomb's relation may be expressed analytically in terms of the largest and smallest principal stresses or in terms of the stress components in the plane ( x y ) of these two principal stresses:
268
H . DERESIEWICZ
As an example of the use of Coulomb’s inequality, extended by means of a variational formulation, we may cite Kotter’s solution of the problem of determining the vertical pressure exerted by a medium on the circular base of a cylindrical container [61]. A related problem in the theory of earth pressure is concerned with the shape of the surface along which sliding of a portion of the medium occurs as a result of a small displacement of the containing boundary, and with the magnitude of the traction across this surface. In dealing with the case of plane strain in a quarter-infinite medium bounded by a vertical wall, Kotter [62] derived a relation between the magnitude of the traction, q, at the surface of sliding and the angle of inclination, a, of the tangent to this surface to the horizontal: (4.2)
dq/ds - 29 tang, dalds =.p sin ( a - g,) ;
here, e, is the angle of friction, p the density of the material, and s the coordinate measured along the curve of sliding.* An additional condition has been given by JBky [63] in the form of a non-linear differential equation of the second order the solution of which, unfortunately, presents formidable mathematical difficulty. I t has been shown by H. Reissnert [64] that the sliding surfaces of a cohesionless, heavy medium are characteristics of the differential equation obtained by inserting in the equality of (4.1) the expressions for the components of stress in terms of Airy’s stress function. Approximations to the magnitude of the traction have been obtained from Kotter’s equation (4.2)by assuming the curve of sliding to be represented variously by portions of a straight line, circular arc, parabola, logarithmic spiral, etc. In considering a different problem, Nutting 1651 starts with the empirical result that the product of the depth, z, and the void ratiot, R,, is constanttt; i.e., zR, = C. Hence, the ratio of the (variable) density of the material, p, to the (constant) density of the grains, pg, is given by
* The curve of sliding represents the intersection of the surface of sliding with the vertical plane normal t o the wall. In relation t o this problem the reader is referred to the footnote on page 236.
t
Reissner credits J. Massau with prior discovery.
The void ratio of an aggregate is defined as the ratio of the volume of voids to the volume occupied by solid matter.
$2 This relation was obtained from well borings taken in compacted granular material of fairly uniform grain size.
MECHANICS OF GRANULAR MATTER
269
Now, let v (= l / p ) and vg (= l / p g ) represent, respectively, the specific volume of the material and of the grains. Since the pressure a t any given I
depth is given by fl
=spdz,
the relation between pressure and specific
0
volume is (4.4)
[p - C
(1 -
31
(v - V d =
c.
We note that, for large specific volumes, (4.4) reduces to flv = const., a relation reminiscent of Boyle's law for ideal gases. On the other hand, for very large pressures, v + v g ; i.e., the granular material approaches a solid whose density is equal to that of the grains. Further, the compressibility p = - a v/vdfl may be obtained from (4.4):
it tends to zero as the granular material approaches homogeneity. A recent analysis by ChambrC [66], which falls within the category of representation of an aggregate by a continuous model, deals with the determination of the speed of a compressional wave traveling through a mixture. If po and p denote, respectively, the original and instantaneous densities of the medium, then p o / p = 1 E , where E = a x / & - 1 is the strain defined in terms of the displacement x of a particle from its original undisturbed position a. At this point the author assumes that the stress, u, is a function only of the strain, E , and of a parameter, 7,characteristic of the composition of the medium. He further assumes that the composition of the medium remains constant during the motion. The speed of sound in such a medium, expressed in terms of the pressure, fl (= - u ) , is given by
+
Consider now a mixture of two components, of partial volumes, masses, and densities Vl, m,, p1 and V,, m2,p,, respectively. Let the characteristic parameter be
and let us assume that the density of the mixture is given by the ideal mixture law
270
H . DERESIEWICZ
Then, on performing in (4.8) the differentiation indicated in (4.6), we find the speed of sound in the mixture in terms of the parameter 7 : (4.9)
where c1 = (dfi/dpl)1/2and c2 = (dfi/dpz)1/2represent the speed of sound in each of the two constituents. If we introduce the ratio of the volume of the first component to the total volume of the mixture, i.e., (4.10)
we may reduce (4.9) to the form (4.11)
c/c2 =
[(Oa
+ 1) (8b + 1)]-112,
where a = p1/p2 - 1 and b = pzc;/plc; - 1.
B. Discrete Model 1. Rigid Granules. A distinction between solids and fluids on one hand and granular assemblages on the other has been made on the basis of a property of the latter which Reynolds [67] called “dilatancy”. Considering a granular mass to be made up of smooth, rigid particles, Reynolds suggested that such a medium undergoes “. . .a definite change of bulk consequent on a definite change of shape or distortional strain, any disturbance whatever causing a change of volume and generally dilatation.” He arrived a t this conclusion by assuming that the position of each internal particle becomes fixed when the positions of the surrounding particles are fixed. It follows that a granule in the interior cannot change its position by passing between its contiguous neighbors without disturbing them, so that a given particle will always remain in the same neighborhood. Consequently, any distortion of the boundary will affect the density of the medium, and this in a way which will depend on the arrangement (packing) of the granules.* It appears, then, that when the constituent granules of a medium are arranged in a position of maximum density, any subsequent deformation of the medium must be such as to result in a decrease in the density. A classic application of the hypothesis of dilatancy is furnished by Reynolds in explaining the following well-known phenomenon: If one steps on wet sand a t the sea-shore, the sand around one’s foot whitens, indicating that the sand has momentarily dried; subsequent removal of the foot causes a puddle of water to appear on the surface of the sand. It is assumed
* The
effect of friction is neglected in Reynolds’ discussion.
MECHANICS OF GRANULAR MATTER
27 1
that the motion of the tides cadses the particles of sand to be deposited in a condition of densest packing. In this state the sand is saturated with water, capillary forces causing the surfaces of the water and the sand to be a t equal heights. Now, according to the principle of dilatancy, the pressure applied by the foot creates an enlargement of the void spaces between the granules. More water is required to fill the distended interstices, which may be accomplished by drawing off water from the surrounding sand. But this process takes time to effect, so that, temporarily, the surface of the water is depressed below that of the sand, leaving the sand dry and white.* When a sufficient amount of water has flowed in, the sand, once again, becomes wet. Finally, removal of the foot restores, a t least partially, the original dense packing, which results in the excess water being left above the sand. Jenkin [68] applied the hypothesis of dilatancy to discuss several problems of soil mechanics. Confining himself to a two-dimensional model composed of rigid circular disks of equal diameters, he computed the forces exerted by the medium on the horizontal base and vertical wall of a rectangular container filled with such disks in a state of densest packing. This was done for the two cases where friction was and was not neglected. Further, using the same model, he investigated the effect on the boundary traction of “bridging” or “arching” of a portion of the granules. Experimental work on sand and glass balls confirmed the following computed results: a) The angle between the force acting on the wall and the normal to the wall is equal to the angle of friction. b) The position of the center of pressure is indeterminate and may be considerably higher than one-third of the height of the wall above the base. That is, the pressure is not transmitted uniformly and does not vary linearly with height above the base.? c) The results are strongly dependent on the pattern of packing and on the shape of the grains.
2a. Elastic Granules, Normal Contact Forces. A model consisting of rigid particles is manifestly unsuited for problems of vibration and wave propagation. On the other hand, the concept of an array of elastic particles leads easily to an evaluation, a t least in a first approximation, of the phase velocity of a wave traversing such a medium. An early attempt along these lines was made by Hara [69] who, having in mind the behavior of a carbon
* To eliminate possible doubt that the sand immediately below the area of compression is momentarily deprived of the interstitial water, Galt [45] employed a transparent glass plate in the place of an opaque foot and found that the anticipated phenomenon did indeed take place. t The reader will recall a similar conclusion by Auerbach [18] mentioned in Section 11. A.
272
H . DERESIEWICZ
microphone,* considered the propagation of compressional waves in simple cubic and face-centered cubic arrangements of like spheres. Confining himself to disturbances.of long wave-length, t in each case propagating in the direction of one of the edges of the unit lattice (e.g., the (100) direction), he replaced the spherical granules in his models by mass-spring systems in series, in which the stiffness of each spring was computed from Hertz’s theory of normal contact.* In this manner, the velocity of compressional waves was found to be (4.12)
c1 a ( N / R Z ) V ,
where N is the normal contact force and R the radius of each sphere. Consequently, in the case of a vertical columnar array of spheres, compressed by its own weight, the velocity of propagation is proportional to the sixth root of the height of the column. An expression for this velocity is given by Iida [70] and is based on the value of N being the mean value of the external force (proportional to the height h of the column and the square of the radius R ) :
where Q is a constant which increases with the density of the packing; we note that the velocity is independent of the size of the spheres. Experimental work by Iida [70], [71] bears out the “sixth root” relation of (4.13) but indicates a slight rise of velocity with grain size. The velocity is also found to decrease with increasing porosity of the packing and with increasing moisture present in the medium. Similar results were obtained by Iida for torsional waves through the array, but, although the author derived an expression, similar to (4.13), for the corresponding velocity, he was not in possession of a formula for the torsional stiffness of a contact of two spheres which would enable him to perform numerical computations. The preceding discussion may be generalized and refined to account for the exact distribution of spheres in a systematic packing. Takahashi and Sat6 [72], having in mind seismological applications, consider an array of like spheres, each of radius R and mass m, arranged such that their centers form a space lattice. Let R, denote the position vector of the center of any given sphere, So, with respect to a fixed origin, and Rki similar vectors
* The operation of the carbon microphone is based on the fact that the electrical resistance of a carbon-to-carbon contact is a continuous and monotonic function of the contact pressure. t Long
*
in comparison with the diameter of the spheres.
The tangential stiffness of the contact was neglected in this computation. The normal stiffness is given by reciprocal of C in (3.4).
MECHANICS OF GRANULAR MATTER
273
corresponding to the n neighboring spheres, S,,, j = 1, 2,. . ., 7212. If nj represents a unit vector directed along the line of centers from So to S,,then (4.14)
R*i - Ro = f bnj,
where b = 2R - a is the distance between centers of contiguous spheres in the deformed state. Further, if rj denotes the displacement of the center of the j-th neighboring sphere, and if at each contact only normal forces are considered, then the contact force across the j-th contact of sphere So is
-
dNj = - k [nj (r, - ro)]nj.
(4.15)
Provided that the ANj are the only external forces acting on sphere So, the equation of motion of this sphere becomes:
*i
(4.16)
42
=
k
2
[(rj
+ r-,
-
2r0) * n,]nj.
j=1
Now set
r+j = A exp {27ci(a - R+i - wt)},
(4.17)
where w is the frequency and a the wave-number vector. Insertion of (4.14) in (4.17) yields
rf j
(4.18)
= ro exp { f 2niba *
nj},
which, when in turn inserted in (4.16), leads to the equation
(4.19)
nlz
..
[l - cos (2nba * nj)] (ro * nj)nj.
mro = - 2k j=l
If now attention is confined to wave-lengths which are large in comparison with the radius of a sphere (i.e., la1 << R ) , (4.19) reduces to
(4.20)
..
mro
nl2
+ 47c2b2k 2 (a nj) *
j= 1
(ro* nj)nj = 0.
274
H. DERESIEWICZ DERESIEWICZ H.
Finally, by virtue of the expression for ro given by (4.17) with = 0, (4.20) may be transformed into a set of algebraic equations in the components of the vector amplitude A : nl2
(4.21)
(v/vo)z~
2 (s
nj)2 (A
- n,)ni = 0;
j=1
in this equation v = w/laI is the phase velocity, vo = b ( k / n ~ ) lis/ ~the velocity of long waves in a one-dimensional lattice, and s = a/lai is a unit vector parallel to a. Hence, the secular equation may be written '11
(4.22)
- (v/vo)2
'12 '22
'12
'13
'13
- (v/vo)2 '23
= 0,
'23 '33
- (v/vo)2
and the nji are the components of nj. It is also possible to derive expressions for equivalent elastic constants of the assemblage. Thus, on setting r, = - ro/4n2w2in (4.20) and integrating once with respect to time, it is found that 42
2 (S
(4.24)
2e/m = (vo/v)2
* nj)2
(io *
j= 1
. .
in which E = ( m / 2 )(ro-ro) is the energy per sphere. The material may now be treated as a continuous medium and the components of strain defined by means of the relations* (4.25)
Eii =
- SiZii/V,
&ik =
- (SiZik
+ skici)/2v,
i, k = 1, 2, 3,
where the si and hi are the components of s and r,, respectively. (4.24) may be written
Hence,
42
(4.26)
2e/mv02 =
2 (elln:l +
2 Ez2ni2
+ . . . + 2E12
njlnjz)2.
j= 1
* See Love [73], Article 208; the difference between Love's expressions and those appearing in (4.25) is due to the fact that the latter represent components of the strain tensor.
MECHANICS OF GRANULAR MATTER
275
Now let p* be the apparent density and let the components of strain E ~ ~. ,.,. 2q2, be denoted by xl, xz,. . ., x6. Then, analogous to a standard procedure in the theory of elasticity [73],
ell,
(4.27)
where cicj is to be replaced by the elastic constant cij in the expanded form of (4.27). Comparison of (4.26) and (4.27) yields the equivalent elastic constants 4 2
(4.28)
Cik = p*V2
2
i, k = 1, 2,. . ., 6 ,
titk,
i=l
where
2
2
tl = nil, t 2= ni2,
. . .,
t6 = nil q2.
Among other efforts to arrive at a satisfactory theoretical description of wave propagation through the earth’s crust we may mention a study by Gassmann [74] in which the crust is idealized as a hexagonal close-packed array of like spheres. The system may be described by the coordinates of the centers of the spheres in the k-th horizontal layer:
+ i + (( ~ V 3 / 4[4i ) + (-
+ 11,
x = (W)[4i
(4.29)
Y=
z =2RkV2/3,
1)k
+ 11,
i , j = l , 2 , 3 ,...,
where the positive z direction is taken vertically downward. The spheres in a given layer are assumed to be subjected only to the weight of the spheres lying above this layer and, as in previously discussed computations, the tangential stiffness of the contacts is neglected. For small variations of the stress from its initial value, the increments in the components of stress and strain are linearly related, the matrix of the elastic constants being appropriate to a system with transverse isotropy; in Love’s notation [73, article 1101, (4.30)
A=T,
F=L=47,
C = 167,
N=0,
where 7, obtained from the Hertz compliance (3.4), is
(4.31)
-c=--[1 6
]
6E2Nk 113 (1 - v ~ ) ~ R ~
276
H. DERESIEWICZ
E being Young’s modulus of the spheres. If no load is assumed above the first layer, then the force on the contact between spheres of the k-th and (K 1)-th layers is
+
Nk =nR2(p^- P)gz/3,
(4.32)
where and are, respectively, the densities of the spheres and of the fluid filling the voids, and g is the magnitude of the acceleration of gravity. Substitution of (4.32) in (4.31) results in (4.33)
With = 0, jI4.301 and [4.33] furnish the elastic constants of an assemblage whose void spaces are filled with air (“dry system”). The elastic constants appropriate to an assemblage whose voids are filled with a liquid (“wet system”) may be obtained from the elastic constants of the dry system by means of Gassmann’s relations [75]*
(4.34)
A
=7
C
=
+
167
ci bI2/D,
F
= 47
+ ab, b3/D,
+ abS2/D,
L
= 47,
N
where
= 0,
+ Ti2+ Ci3)/3k, h
bi = ~i - (Ti1
(4.35)
El
= E2 = E3 = 1,
Ep
= E, = E, = 0,
h
k and k“ are the moduli of compression of the solid and liquid, respectively, and 9 is the porosity of the assemblage. The Cij appearing in the first of (4.35) represent the elastic constants of the dry system so that, in the present case, h
b1 = b2 = 1 - 2 7 / k ,
(4.36)
h
b3 = 1 b,
-
8C/k,
= b, =
b, = 0.
With the values of the elastic constants determined by (4.30) (dry system) or (4.34) (wet system), the velocities of propagation of plane waves
* The derivation of these relations is achieved by consideration of porous media and is omitted here.
MECHANICS OF GRANULAR MATTER
277
through the assemblage may be -VELOCITY V, IN M / S E C found in the usual manner from , the equations of motion of an I elastic solid. In general three OZ distinct wave velocities exist, corN responding to three different I waves, whose magnitudes are IQ w functions of the direction of wave propagation. Gassmann [74] carried out computations for a packing of spheres whose elastic constants were those of granite. Fig. 18 illustrates the variation of the largest of the three velocities with depth for the dry system and for the sys00 tem when filled with water; in each instance the normal to the wave front is parallel to the zaxis, i.e., vertically downward. It is seen that, in the dry system, the velocity varies as the sixth root of the pressure, so that the velocity tends to zero as the pressure approaches zero. On the other hand, the velocity in the wet system does not, of course, vanish when the pressure approaches zero; moreover, a t any given value of the too pressure, it is greater than the corresponding velocity in the FIG. 18. Variation of velocity of plane dry system*. elastic waves with depth in a closeConsiderable experimental packed hexagonal array of like spheres, direction of propagation being normal work h& been done with a view t o the surface of the packing (Gassmann to ascertaining the relation beP'41). tween velocity, pressure and water content, particularly in reference to materials of interest to geophysicists. I n experiments on rocks of very low porosity (< lye), subjected to hydrostatic pressures
-
1
Added at press-time: Experimental data obtained by Matsukawa and Hunter [91] on the dependence of velocity of compressional waves on stress in dry sand (porosity P = 390i0) reveal a relation qualitatively similar to that predicted by Gassmann (see Fig. 18), but sensitive to the size of granules.
278
H. DERESIEWICZ
ranging from 50-6000 bars,* Hughes and Jones [76] found that, a t low pressures, the dilatational velocity increased due to the presence of fluid (oil) but, at high pressures, the velocity decreased.? Data on limestone (porosity 4%) and sandstone (porosity 5%), gathered by Hughes and Cross [77], showed the variation of velocity with pressure to be small and linear for saturated samples of both materials as well as for dry samples of the limestone. Further, for the limestone, a t low pressures, the velocity in the dry sample was about 5% greater then in the saturated sample, while at high pressures (5000 bars) the velocity in the saturated system exceeded by about 10% that in the dry system. The situation was found to be reversed for the sandstone; there, a t low pressures, the velocity in the dry rock was much smaller than in the saturated rock, while at high pressures it exceeded the velocity in the saturated rock. In addition, in marked distinction to the behavior of dry and saturated limestone and of saturated sandstone, the velocity in the dry sandstone rose sharply with pressure. In still another set of experiments, Hughes and Kelly [78] measured the dilatational velocity as a function of pressure in sandstone in which the porosity varied from 8-20y0. They found that, a t low pressures (50 bars), the velocity increased rapidly a t low values of water content ( O - l O ~ o of saturation), remained constant a t values of water content ranging from 10-90% of saturation, and then decreased, as the water content approached lOOyoof saturation, to a value higher than that in the dry specimen. As the pressure was increased, the rate of the rise in velocity a t low water content decreased, vanishing completely a t a pressure of 500 bars. That is, a t this pressure, the velocity was found to be practically constant with water content until the value of the latter reached about 90% of saturation, a t which point the velocity decreased rapidly as total saturation was approached. The general rise in velocity a t low water content gives an indication that the water might be regarded as a cementing material which provides better acoustic coupling between the rock particles. At high water content the water becomes mobile and the bond between fluid and solid is, to a large extent, destroyed. All of the investigators whose work we have discussed so far limited their analyses to models consisting of regular arrays of like spheres. Such restriction is seen to be both advisable and prudent in initial attempts to explore this extraordinarily complex field. Nevertheless, as a result of this restriction, the efforts can be expected to furnish only an indication of the qualitative aspects of the solution.
* It will be recalled that 1 bar mosphere.
=
1W dynes/cm2 = 14.504 lb./in2 = 0.98692 at-
t A rise in temperature was found to depress the velocity at any given value of the pressure.
MECHANICS OF GRANULAR MATTER
279
A large step in the right direction was taken by Brandt [4]who abandoned the idea of regular arrays and, availing himself of some empirical results, examined the behavior of an irregular stacking of particles. To follow his argument, consider a model composed of spherical particles of several sizes but of the same material. Spheres of largest radius are packed at random into an assemblage of porosity P . Spheres of the next smaller size are assumed small enough to be packed at random, and again with a porosity P, in the interstices of the primary array without being influenced appreciably by the effects of boundaries (cf. Section 11. B, 4 ) . This process may be continued, the interstices created by the secondary spheres being filled at random to the same porosity P by still smaller spheres, until a total of m sets of spheres is contained in the model. The aggregate is enclosed in a volume large enough so that the boundary effects of the enclosure on the primary packing can be neglected.* Hence, if there are k, primary spheres of radius R,, then the volume of the enclosure is expressed by (4.37)
V = 4nk1R:/3(1- P ) .
Further, if the primary interstices contain k, secondary spheres of radii R,, then k,/k, = PR:/Ri. Similarly, for each successively smaller set of k j spheres having radii Ri we find (4.38)
k;/k, = P i - l R:IRB
= (PU3);-l,
i = 1 , 2 , . . .,m,
the assumption having been made that the ratio of the radii of spheres in successive sets is constant, i.e., RclIRi = U. Now, let the volume of the enclosure be decreased, thus causing a deformation of the individual spheres. Once again, it is assumed that the relative displacement of the spheres is purely normal and is given by the Hertz theory of contact. If the average reduction of the radii of primary spheres is A,, then to quantities? of the order A,, the bulk volume of the aggregate is reduced to (4.39)
V = 4nk1R:/3(1- P ) - 4 n k 1 R ~ A 1 / (l P),
where A , is given by a/2 of (3.3). Further, the change in volume of the particles due to deformation is a small quantity of order higher than A,, so that the entire reduction in bulk volume, expressed by the second term on the right-hand side of (4.39),represents a decrease in the void volume of the primary array. As a result of the decrease of the latter there is an equal reduction in the bulk volume of the secondary assemblage. Similar considerations hold for each subsequent set of spheres. Hence, the average deformation
*
See, however, the first footnote on p. 243. of the Hertz theory that A / R
t This is consistent with the assumption
<< 1.
280
H. DERESIEWICZ
of spheres in the i-th set is related to the average deformation of the primary spheres by means of
i = 1 , 2 , . . ., m.
Ai/A, = (PU)- (i-l),
(4.40)
In order to make the theoretical model conform to experimental techniques, consider the entire assemblage of spheres enclosed within flexible walls which transmit an external pressure Po to the aggregate. A tube is inserted through one of the walls, permitting a liquid under pressure p , to be added to the array, as shown in Fig. 19. In this manner the model is capable of simulating experiments in which the material is either exposed to liquid under pressure or is enclosed by a jacket and pressurized. An expression may be written relating the energy required to compress the assemblage to the elastic energy stored in the solid particles and to the energy in the compressed liquid. The energy required to decrease 'the bulk volume of the assemblage is 4
(4.41)
E.r. =
1-P 0
PL
FIG. 19. Model of a granular medium filled with liquid under pressure ( p ~ and ) subjected to an isotropic compression (p0) ( ~ r a n d t~41).
The energy of deformation of the spheres is
2 ni ki
1
i=l
0
m
(4.42)
Es =
'i
N , ax,
where tzi denotes the average coordination number of a sphere in the i-th set and Ni represents the average force on a contact of such a sphere. However, since each set of spheres is assumed packed to the same porosity P, the respective coordination numbers are taken to be equal, i.e., ni = n. Moreover, if account is taken of (3.3), (4.38), and (4.40),then (4.42)may be written
/-
4
(4.43)
E, = nk, C ,
N,dx,
0
m
where ,C
=
2 i=1
P-3($-1)/z
is a porosity factor.
281
MECHANICS OF GRANULAR MATTER
Finally, since the bulk modulus of the interstitial liquid is defined by the relation B = - Vd$/dV, the change in the pressure of the liquid due to a change in its volume is given by
P-$L=
- Blog
void volume of aggregate after deformation void volume of aggregate beforedeformation
(4.44)
Hence, the change in the energy of the liquid due to compression is
I-[ Am
EL=
(
$ ~ - B l o g 1 - - P3% R m )]4nkrnRK 1 - P dx
0
(4.45) -% ? L .? !!!
1-P
f[fiL
- B log (1 -
&)] dx.
0
+
The law of conservation of energy requires that ET = E , E L and, since this relation must hold for an arbitrarily small value of A,, it gives rise to an equation connecting the integrands in (4.41), (4.43), and (4.45). To be consistent with the assumptions underlying the Hertz theory, small terms of orders higher than A , are discarded; the equation of conservation of energy thus yields -~ -
1-P (4.46)
-
P"(1- P)
(4.46) is a cubic equation for the determination of the contact force N,. Brandt takes the value of the porosity of a random packing of like spheres to be P = 0.392, in accordance with experimental data obtained by Westman and Hugill [6] (cf. Section 11. B, 2 ) . The corresponding average coordination number of a sphere, determined by the method of Smith, Foote and Busang [34], is found from (2.3) to be rt = 8.84. Further, the term CVm 1/3 p?, where yrn= Pmis the porosity of the aggregate of m sets of spheres,
turns out to be essentially constant. Its value is taken to be (4.47)
C;' p1I2m 213,
and the subscript m discarded.
282
H. DERESIEWICZ
An approximate solution of (4.46),computed on the assumption that the term in (4.46) involving CVmis small,* is given by (4.48)
N,
= 2.34 R;(P0 - PL)/C,L,
where (4.49)
C q L = C,[l
B3T2(l- v2) + 30.75E(P0 PL)
1
and C, may be computed from (4.47) for any given value of the porosity of the aggregate. With the value of the average contact force given by (4.48) and the corresponding average value of the reduction in radius obtained from (3.3), the bulk volume of the aggregate is found from (4.39): (4.50)
vbdk =
1.75 (1 - v2)($0 - PL) 3(1 - P )
The speed of a dilatational wave in an unbounded, isotropic medium is (4.51)
where 7 is Poisson's ratio and 7 the density of the aggregate, respectively; the latter is given in terms of the density of the particles, p, and of the liquid, p , by the expression
Hence, the speed of a pressure wave in the granular medium may be found from (4.50) and (4.51), account being taken of (4.49) and (4.47); for low pressures it is given by (4.53)
where k
= 1.75 (1 - v2)/E
and
(4.54) In support of this assumption Dr. Brandt has kindly communicated t o me some numerical results obtained for sandstone saturated with kerosene. If the radii of the largest particles are taken to be R, = .01 in. and the porosity of the aggregate pl = .lo, then, for a pressure difference po - p~ = 10apsi, the deviation of the approximate solution (4.48) from the exact value (N,= .00175 lb), determined from (4.46), is w 1%; for a pressure difference Po - p~ = lodpsi, the deviation from the exact value (N,= .0457 lb) is w 10%.
MECHANICS O F GRANULAR MATTER
283
The preceding theory, developed for packings of spherical granules and resulting in (4.50) and (4.53), is extended by Brandt to predict the behavior of assemblages of non-spherical particles. The shapes of these particles are limited to those for which the Hertz theory is valid. The speed of a pressure wave in such an aggregate will depend on the relation between the average radius of curvature and the volume of the granules, the average coordination number of each granule, and on the void volume of the aggregate. All of these effects are related to the precise shapes of the granules, giving rise to a number of shape-dependent factors which, like the elastic constants of the aggregate, may be determined experimentally for each particular granular medium. Thus, on the assumption that the porosity of an array of sized, non-spherical granules is essentially that given above for spherical particles, (4.53), with the value of k to be obtained empirically, may be employed for non-spherical assemblages. The expression for the speed of a dilatational wave in a dry aggregate reduces to the particularly simple form : + I
(4.55)
In a medium such as consolidated sandstone, in which only a portion of the voids is accessible to the interstitial liquid, (4.53) must be modified by replacing the term (Po - 9,) by ifo - cf,), where c is another empirical parameter whose value is less than unity. Like some of the work discussed previously in this section, the present theory predicts that the speed of a pressure wave in a dry medium is proportional to the one-sixth power of the external pressure. This result agrees reasonably well with experimental data obtained at low pressures by Nasu [79], who worked with gravel mixed with clay, and by Hughes and Kelly [78], who worked with several sandstones. A t higher pressures, the velocity in the sandstones increases less rapidly than the sixth root of the pressure. I t is not unlikely that the point of transition between the behavior at low and at high pressures indicates the upper limit of the range in which Hertz's theory of contact is applicable. Beyond this point the contact surfaces between the granules become larger than is consistent with the basic assumptions of classical contact theory. It is interesting to note that the ratio of the contact radius to the radius of curvature of a sphere, which may be employed as a criterion of the validity of the Hertz theory,* turns out to be independent of the porosity of the sample: (4.56)
a / R = (d/R)'/'
= 1.5 P1/'[1.75
(1 - y2)po/E]'f3.
This fact is borne out by data on the sandstones mentioned above.
* We
will recall that this ratio is assumed to be small (Section 111. A).
284
H. DERESIEWICZ
The variation of porosity with pressure may be found from the relation Porosity a t pressure fi,, Porosity at zero pressure
-(
void volume q~ bulk volume
= 1
Numerical computations indicate a 3% decrease in porosity due to a pressure of 5000 psi, which represents an average of the range 1yo--5% of four reservoir rocks found experimentally by Fatt [SO] *.
mI
lO3X I6
EXTERNAL PRESSURE,FSI
10
12x18
FIG. 20. Speed of sound through sandstone as a function of pressure (Brandt 141). The solid curves represent theoretical predictions, the dashed curves experimental results, for sandstone saturated with oil; the heavy solid and dashed curves refer, respectively, to theoretical and experimental results pertaining to dry sandstone.
The effect of external pressure on the speed of sound in sandstone is shown in Fig. 20. The heavy lines, dotted and solid, represent, respectively, the theoretical and experimental variation in a dry medium. The light lines, of which the dotted and solid refer, respectively, to theoretical and experimental results, depict the speed-pressure relation in sandstone saturated with oil at various pressures. It is of interest that, a t a given external
* Added at pvess-time: ’rests by Fatt [92] on like spheres (of neoprene, voids filled with kerosene: of steel, voids filled with water), randomly packed in a cylindrical container and ‘compressed axially. indicate a linear relation between bulk volume and the 2/3 power of the pressure, a result which is in good agreement with the predictions of e a n d t [see Equation (4.50)].
MECHANICS O F GRANULAR MATTER
285
pressure, the speed of sound is diminished by an increase in the pressure of the interstitial fluid. 2b. Elastic Granules, Oblique Contact Forces. In spite of the elaborate and ingenious nature of several of the theoretical endeavors we have discussed, and in spite of the success of some of them in predicting qualitative behavior of the mechanical response of granular materials, they invariably have been found wanting in the attempt to achieve quantitative agreement with experiment. Thus, for example, the theories predict wave velocities proportional to the sixth root of the applied isotropic pressure and to the cube root of the elastic modulus, a result essentially corroborated experimentally, but the magnitude of these velocities is generally too low. Such differences indicate a basic deficiency inherent in the several models, and their cause must be sought in the assumptions made and the factors omitted in each model. In a real granular medium under an arbitrary external loading or in a state of vibration, the contact surfaces of the individual granules are subjected to local forces which, in general, are directed obliquely to these contact surfaces. Also present may be twisting couples at each contact. However, as has been pointed out in several places in this section, the various authors, by virtue of having employed the Hertz theory of contact exclusively, had overlooked the effects of all but the normal components of the contact forces. A major reason for the omission has been the unavailability, until recently, of solutions of the equations of elasticity for oblique contact forces and for twisting moments. It was, in fact, the intention to take account of forces and couples hitherto neglected which led Mindlin and his collaborators to undertake their studies of the contact problem described in Section 111. The corrections stemming from these additional factors will be of greater than academic importance, as is %vident from the fact that, even in the simplest case of contact due to oblique forces, the tangential stiffness of a contact is of the same order of magnitude as its normal stiffness [see (3.11)]. The price paid for the extension and improvement of the theory is the greater mathematical complication, which appears because the presence of tangential forces and twisting moments a t the contacts gives rise to load-displacement relations which are not only non-linear but also inelastic. The mechanical response of the medium must, therefore, be expected to depend on the entire past history of loading. Hence, the stress-strain relations a t any point in the medium have to be expressed in terms of the corresponding increments in these quantities; that is, differential rather than total stressstrain relations must be written. A first approximation to the structure of a real medium in terms of regular packings of spheres was seen to consist of clusters of simple cubic and face-centered cubic arrays (cf. Section 11. B, 2). Accordingly, the initial efforts within the framework of the extended theory may profitably be
286
H . DERESIEWICZ
directed a t an analysis of models having the configuration of either of these lattices. We shall begin by following Duffy and Mindlin [81] in considering a medium composed of a face-centered cubic array of like spherical granules. An elementary cube of this medium is acted upon by incremental forces whose components dP, are shown in Fig. 21. The incremental stress components are defined by the relations doij = dPij/8R2 where, R being the radius of each sphere, the denominator represents the area of a face of the block.
FIG.21.
Elementary cube of a face-centered cubic array of like spheres subjected to incremental forces dPij.
The application of incremental stresses will cause incremental deformation of the volume element which can be computed if the incremental forces a t each contact are known. I t is the evaluation of the contact forces which creates the first difficulty. Since each sphere in the model is in contact with twelve other spheres, there are thirty six unknown rectangular components of contact forces; the assumption of a homogeneous state of incremental stress reduces this number to eighteen, of which six are normal and twelve are tangential components. The conditions which express the equilibrium of each sphere and each portion of a sphere give rise to nine independent equations for the determination of the eighteen force components ; that is, the problem is statically indeterminate. The additional nine equations are obtained by introducing conditions which require the displacements of the centers of the spheres to be single-valued (equations of compatibility of
MECHANICS OF GRANULAR MATTER
287
relative displacements of spheres). The latter equations, when expressed in terms of the unknown components of the incremental contact forces, involve the normal and tangential compliances at each contact which, in turn, depend on the contact forces and on the previous loading history. A simple loading program which is amenable to exact solution consists of an arbitrary incremental stress applied subsequent to an initial isotropic compressive stress a,. In this case the initial contact forces are purely normal and all are equal. Further, if the incremental stresses remain small compared with the initial stress, the compliances are sensibly the same for all contacts and are satisfactorily approximated by their initial values; i.e., the normal compliance is given by (3.4),the tangential compliance by (3.10). In each instance the contact radius is uo due to the presence of an initial normal contact force N o = 1/FR2uo. Hence, the compliances are constant and the incremental contact forces are easily determined. A knowledge of the forces leads to expressions for relative incremental displacements of the centers of the spheres in terms of the applied incremental forces. These, in turn, yield the incremental stress-strain relations, which, when referred to rectangular coordinate axes parallel to the edges of the elementary block (Fig. 21), read
(4.57)
where (4.58)
2(4 - 3 Y) c11 =
Thus, consequent upon the prescribed system of loading, the differential stress-strain relations have a form corresponding to those for a crystal with cubic symmetry. The incremental stress-strain relations (4.57) are useful in correlating the theory with an experimental study of the problem of wave propagation, providing the variations in stress which accompany the disturbance are small in comparison with the initial stress in the medium. For purposes of such an experiment granular bars were constructed of like steel balls, arrayed in a face-centered cubic lattice inside a thin rubber sheet container, and held in place by an external isotropic pressure attained by a partial evacuation
288
H. DERESIEWICZ
of the container (Fig. 22). The balls in the various bars were arranged in such a manner that the longitudinal axis of the bars was parallel either to the (100) or to the (1 lo} direction of the lattice,* this in order to avoid coupling
FIG. 22. Bar made of 1/8" steel balls arranged in a face-centered cubic packing whose {loo} direction coincides with the axis of the bar (Duffy and Mindlin [Sl]).
of the compressional with the flexural waves. For a bar with a cross section whose linear dimensions are small compared to the wave-length of the disturbance, the elementary theory of longitudinal vibrations yields a satisfactory value of the velocity; in the present case the use of (4.57) results in 2c:2
2
p"00 = c11 -
c11+ c12
(4.59)
2
pull0 = -
4c44(cii
2CllC4,
+
[
- 2 (8 - 7 ~ ) 3p2a0 8 - 5~ 2 (1 - Y)'
-c~2)
+
(51 - 5 2 ) (51
1'"
,
2ci2)
+ 2ClJ
2 (4 - 3 ~ (8) - 7 ~ ) (4 - 3 ~ ) ' + (2 - Y) (8 - 7 ~ )
where p is the density of the medium. A comparison may be made with corresponding velocities computed on the basis of the Hertz theory of contact,
* That is, respectively
parallel to an edge and a face diagonal of the elementary cube.
289
MECHANICS O F GRANULAR MATTER
i.e., obtained by neglecting the effect of the tangential components of the contact forces. The expressions in this case are (4.60)
3p&0/2 = p&o=
[3p2~,b/2(1- Y)2]1/3.
Both theories predict proportionality of the velocity to the sixth root of the external pressure and to the cube root of the elastic modulus. However, the values obtained from (4.59) are greater than the corresponding values INITIAL NORMAL CONTACT FORCE (POUNDSXIOe) .6 .7 .8 .9 I 2 3 4 5 6 INITIAL PRESSURE ON BAR (PSI)
7
8
FIG.23. Resonant frequency and wave velocity of the first compressional mode of the bar of Fig. 22 as a function of the initial pressure (Duffy and Mindlin [ S l] ) : (a) prediction of theory which accounts for effect of tangential contact forces; (b) prediction of theory which omits effect of tangential contact forces.
given by (4.60).An indication of the difference may be had from numerical values appropriate to steel balls. Thus, from (4.59), vloo = 13300;" and vllo = 1350 ail6, while, from (4.60),vloo = 800 o;" and vll0 = 980 a;", the units in each case being ft./sec. for O, in psi. A comparison of the theoretical and experimental results, given in Fig. 23, indicates close agreement of experiment with the predictions obtained by use of (4.59). The two sets of data pertain to bars containing balls of high and low tolerance the diameters of which are in the range 0.125 -j= 10-5in. and 0.125 f 5 x lO-5in., respectively. I t is seen that the larger the dimensional tolerance and the lower the initial pressure the greater the deviation of the experimental from the theoretical curves. This phenomenon is due to the fact that the spheres in an actual configuration, being unequal in size, are subjected to unequal initial contact forces some of which may even be zero. The stiffness of the assemblage is thereby decreased, causing the frequency of vibration to be depressed. The influence of the magnitude of the tolerance on the packing of the lattice becomes evident when the tolerance is compared with the diminution in diameter of each ball due to
290
H. DERESIEWICZ
initial pressure. Thus, when uo = 2 psi, a = 1.96 x in.; when a = 7.39 x in., while the tolerances are f in. and & 5 x 10-5in. Data obtained from observation of the decay of free vibrations of the bar indicate that the energy dissipated per stress cycle is proportional to the square of the (small) amplitude of the displacement. This result conforms to results of previous measurements of energy loss, although it is a t variance with the prediction of the contact theory. The discrepancy has already been discussed in Section 111. B, 3. The imposition of an initial isotropic pressure causes the initial compliances to be the same at all contacts. This fact obviates the need for distinguishing between different contacts in the array and leads directly to the establishment of the differential stress-strain relations (4.57). As a consequence, however, integration of (4.57) is limited to increments in which the stress remains at all times isotropic, since any other finite stress increments will create a variation of forces and, hence, of compliances, from contact to contact. Accordingly, if finite stress-strain relations are sought for a loading other than isotropic, it is necessary, in the derivation of the incremental stress-strain relations, to keep track of contacts with different loading histories. Duffy and Mindlin have also studied the effects of loading which gives rise to a state of stress, initial as well as subsequent incremental, consisting of an isotropic pressure uo and a uniaxial pressure ua parallel to one edge of the elementary cube, say in the x,-direction (Fig. 21). Provided the loading is achieved by variations of uo and ua alone, the differential stress-strain relations in this case assume the form appropriate to those of a tetragonal crystal with six independent elastic moduli. Once again, the moduli are functions of the contact compliances and, hence, of the stress history. The loading creates two types of contacts, differing from one another by the loading history which occurs a t each of them. The incremental forces a t these contacts, expressed in terms of the increments of the applied stresses, are
uo = 14.7 psi,
where (4.62)
and where the subscript 1 refers to contacts whose normal lies in the x1x2 plane and the subscript 2 refers to contacts whose normal lies in either of
29 1
MECHANICS OF GRANULAR MATTER
the other two coordinate planes. For a given variation of applied loading, (4.61) must be integrated in order that the contact compliances and, by virtue of these, the elastic moduli in the differential stress-strain relations, be known. But the equations (4.61) themselves contain the unknown compliances; that is, (4.61) comprise a system of simultaneous, non-linear, integro-differential equations. An inverse procedure is employed by Thurston and Deresiewicz [82] to a m v e a t a solution of (4.61). Instead of starting with specified external loading, these authors seek a variation /(ao, aa) = 0 which makes (4.61) tractable. At the outset, in order that available theory may furnish the form of the contact compliances, they assume, a t each contact, d T / d N = P, a constant. Further, they consider only the case in which the increments in the normal force, d N , are positive, and, to facilitate the integration, choose 3/ < f , f being the coefficient of static friction at a contact. As a result, the tangential compliances are given by (3.10); the normal compliances are expressed by (3.4). Finally, to avoid dealing with an integro-differential system, they assume that the initial state of stress is purely isotropic, the magnitude of the pressure being o,, and that this state was reached homothetically.* The value of the complibnces in (4.61), a t an arbitrary level of subsequent isotropic plus uniaxial loading, is
+ Nz)--’13, = K ” ( N o + N*)-’l3,
C, = (1 - Y ) / ~ , U C Z = ~ K’(No (4.63)
S, = (2 - Y)/~,uu,,
i = 1,2,
where N o = 1/FR2a, is the initial, purely normal, contact force and N i the normal component of the additional force on contact i ; further, K‘ = (1 - Y ) / ~ , u ( ~ R and ) ~K“ / ~ = (2 - Y ) / ~ , u ( ~ Rwhere ) ~ / ~ ,6 was defined in conjunction with (3.1). With the compliances given by (4.63), and subject to the proviso that d T , / d N , = P, equations (4.61),when integrated, yield: (a) the relation between the components of the finite additional contact forces, (1 N1/No)2/3 = PK (1 - PK) (1 N2/N0)213, (4.64) T , = 0, T2= PN2,
+
+
+
where K = K“/K’ = (2 - v)/2(1 - Y ) ; (b) the relation between the additional uniaxial pressure aa and the force components N , and N , , (4.65)
* Two states of stress are said to be “homothetic” if the corresponding stress quadrics are similar and similarly oriented. The loading giving rise to homothetic states of stress is sometimes referred to as “radial,” since, in stress space, the loading path is a straight line through the origin.
292
H . DERESIEWICZ
where (4.66)
(c) the relation ,&ween the additional isotropic pressure a, and L e additional uniaxial compressive stress a,, expressed parametrically in terms of 5, 2X
= .(YE3
- 1)/(8- [ 2 ) 3 1 2 - 3p,
(4.67) 2y
=K [ ( 1
-P)E3
+ l]/(6-
E2)3/2
- (2 - B),
14
12
10
a bx
&
E
4
I
C
FIG. 24. Relation between additional isotropic compressive stress (ao)and additional uniaxial compressive stress (a,) for which the incremental stress-strain law of a facecentered cubic array of like spheres is integrable in closed form (Thurston and Deresiewicz [82]).
MECHANICS O F GRANULAR MATTER
293
-
+
where y = 1 3P, 6 = (1 - PK)-1, K = [PK/(l - ,8K)]3/2,x = o a /, and y = oo/om. Thus, (4.67), which prescribes the interdependence of a. and a,, represents the sought-after finite loading to be applied subsequent to an initial isotropic pressure . ,0 It is seen from Fig. 24, in which this relation is illustrated, that the variation of oo with a, in the range shown is almost linear, despite the complex appearance of (4.67). The loading being specified by (4.67), the contact forces may be computed from (4.64), (4.65), and (4.66); these, when inserted in (4.63), lead in turn to the compliances expressed in terms of the applied stresses. Further, as has been noted previously, the compliances enter into the expressions for the elastic moduli of the differential stress-strain relations. The remaining step in the solution is the integration of these stress-strain relations, of which a pertinent example is given by
+
= c13d~1,
+
~13d~Zz
C33dE33 9
(4.68)
with ‘13 =
1 21/2R (&
-
i) ’
FIG.25. Relation between components of additional stress and strain in the direction of the additional uniaxial stress for a face-centered cubic array subjected to the loading illustrated in Fig. 24 (Thurston and Deresiewicz [SZ]).
(4.69)
Considerations of the geometry of an elementary cube, together with the first of the relations (4.64), reveal that (4.70)
dell
= dEZ2 = d ~ 3 3(1
- PK)/( 1 + PK).
294
H . DERESIEWICZ
The knowledge of relations (4.70) completes the information needed for the integration of (4.68). The easiest way to express the component of the integrated additional strain is, once again, in terms of the parameter 6,e.g., (4.71)
+
where K' = [ ( l PK)/(l - PK)] [3(1 - ~)/41/&]'/~. Thus, the integrated stress-strain relation is given by (4.67)and equations such as (4.71). Reference may be made to Fig. 25 in which the dimensionless additional stress a33 is plotted against the corresponding strain e%, with the initial isotropic pressure ,a entering as a parameter. An attempt a t experimental verification of this relation is in progress.
FIG.26. Elementary cube of a simple cubic array of like spheres subjected to incremental forces dPij.
In distinction to the face-centered cubic array, analysis of which involves inherent complexities, the simple cubic lattice of like spheres is shown by Deresiewicz 1831 to afford an example of a model which can be handled without difficulty. Consider the assemblage shown in Fig. 26, which is assumed to be under an initial isotropic compression, arrived at homothetically, and subjected subsequently to an arbitrary incremental loading dP,. Since each sphere is in contact with six other spheres, the
295
MECHANICS OF GRANULAR MATTER
present arrangement gives rise to eighteen rectangular components of contact forces per sphere. However, the assumption of a homogeneous state of stress reduces the number of independent force components to nine, of which three are normal and six are tangential. Conditions of equilibrium in this case lead directly to the relations dNii = d P j j , i = 1, 2, 3; i.e., the components of the incremental contact forces are equal to the corresponding components of the incremental applied forces. It should be noted that the structure is statically determinate and the conditions of geometric compatibility are satisfied identically. As in the case of the face-centered cubic array discussed above, the incremental stress components are defined as the ratios of the corresponding incremental forces to the area of a face of the cube, or doii = dP,J4R2. By means of the geometry of the assemblage the components of incremental strain may be expressed in terms of the contact compliances and the components of incremental contact forces, and, by virtue of the latter, in terms of the components of incremental stress: dcjj
(4.72) dEij
=R
= 2RCj dgji,
+
[(SiiTij/tjk)dtjk (SjlJji/tik)dTik],
i# #k
=
1, 2, 3,
where Zjk, for example, denotes the resultant shear stress on a plane whose normal is parallel to the i direction, i.e., t j k = (cJ;~ &)1‘2. The subscript on the symbol for each of the compliances, Ci and S j , indicates the normal component of the contact force, in addition to which, in the Si, the subscript serves to identify the resultant tangential force on the contact; e.g., the tangential force which corresponds to Si is N;k)1/2. Once again we wish to avail ourselves of the expressions for the tangential compliances which are a t our disposal. Consequently, we shall discuss the following loading program: subsequent to the application of the initial isotropic pressure 4R200,which gives rise to normal contact forces No and results in an initial radius of contact a,,, the additional loading is to be applied in such a manner that, on each contact surface, the normal component of the force increases, the relative rate of normal and (resultant) tangential loading is constant and greatern that he coefficient of friction, and the force retains its direction in space. Under these circumstances, the tangential compliance per contact, Si, is given by (3.19) in which a, 8 and L have each acquired the subscript i, and Oj = f/&, Pi = d [(N,”i N;k)1/2]/dNjj, L j = (N:;. N~k)”2/fN,,. The specified loading conditions restrict the stresses to homothetic quadrics; accordingly, we must postulate that the stress components obey the relations
+
+
+
+
(4.72)
&;,/doll = ;Zij (a constant),
i, j = 1,2,3.
296
H. DERESIEWICZ
The compliances may now be written (4.74)
(4.75)
+
where, e.g., Substitution of (4.74) in the first set of = (a;i &)”’. (4.72), and of expressions of the type (4.75), together with (4.73), in the second of (4.72), yields, upon integration between the desired levels (from zero at the initial state to the final values), the total strain-stress relations
Having attained by means of the loading process described above a given level of additional stress, a ,: we may derive the stress-strain relations which are valid for a similarly conducted process of unloading. That is to say, the stress a ,; which was reached homothetically, is now reduced to a lower level of homothetic stress aii 0. The procedure is similar to that carried out for the loading process except that the tangential compliance is now given by the first of (3.20) with the subscript i in the appropriate places and with LT = ( N ; 2 N:2)1/2/fNo. The final form of the strainstress relations, valid during unloading, may be written
+
+ terms
obtained by interchanging i and j , with i # j .
297
MECHANICS O F GRANULAR MATTER
We note that the expression foi-.sii which holds during unloading, (4.77), is identical with the corresponding expression valid during loading, (4.76). This is due to the fact that no dissipation takes place in the structure in the course of pure dilatational deformation. Further, as the loading is reduced to its initial level, i.e., aii = 0, the normal strains, eii, vanish. On the other hand, the shearing strains, .sij,are not removed completely when oij = 0 gwing to the frictional dissipation mechanism inherent in the tangential deformations. Their values in this case are given by
(4.78)
-
+ terms obtained by interchanging i and j . We shall speak of failure of the simple cubic stwcture when sliding occurs between any pair of contiguous spheres. To obtain an analytical criterion of failure during loading, we note that in the expression for eij in (4.76) each of the quantities raised to the 213 power must be non-negative. This requirement, which stems from consideration of the tangential loaddisplacement relation for two spheres in contact, assures that the radius of the adhered area is not negative (cf. 3.7). If, in addition, we take account of (4.73), we find that sliding will occur whenever the inequality (4.79)
Zjk
> /Co/(l
- fAii/&k)
is satisfied. An example is given by Deresiewicz [83] which serves to illustrate the foregoing theory. Let the simple cubic array, under initial isotropic stress, o,, be compressed in a direction parallel to one of its face diagonals. The loading may be shown to be homothetic, the A, of (4.73) having the values A,, = A,, = A,, = 1, A,, = A,, = &, =,O. It is convenient to introduce a set of rectangular coordinate axes xl, x,', x,', obtained from the original axes xl, x,, x3, (Fig. 26) by a rotation through an angle of 45" about the x,-axis; that is, the xl' and x,' axes coincide with the diagonals of the face of the elementary cube which lies in the xlx, plane. If the x,'-axis is taken parallel to the direction of the applied compressive stress, a;,, then the non-zero strain components, referred to the primed coordinate system, are given by (8pu0/3R00)(&, &) = [2(1 - Y) f/ ( Z - Y)] (1 (4.80) - 2(1 - Y)
+~ ~ ~ / 2 u ~ ) ~ / ~
/ ( 2 - v) [l - (l/f- l)a;1/2a012/3,
298
H. DERESIEWICZ
valid during loading, and
FIG. 27. Stress-strain relation for a simple cubic array of like spheres compressed in the { 110) direction: loading portion of cycle (Deresiewicz [SS]).
valid during unloading. The relation between ail and &, (4.80),is illustrated in Fig. 27 for several values of the coefficient of friction, and demonstrates clearly the non-linear character of the stress-strain law. Each of the curves terminates with a horizontal slope a t 0~~/20,, = f/(l- f ) , a t which point failure (sliding) occurs in the lattice. The relation between a;, and corresponding to a cycle of loading and unloading [(4.80) and (4.81)]is depicted in Fig. 28 for a single value of the coefficient of friction but for various values
&il
299
MECHANICS OF GRANULAR MATTER
<
of the loading amplitude a : /2a0 f/(l- f). The inception of a hysteresis ; loop may be noted and is due to the frictional energy loss in the course of tangential displacements. The problem just discussed may be generalized by considering an arbitrary angle of rotation, 8, about the x,-axis. In this case, the Aij of (4.73)
9
FIG. 28. Stress-strain relation for a simple cubic array of like spheres compressed in the { 1 lo} direction : loading-unloading cycle (Deresiewicz [83]).
are A,, = 1, A,, occur if
= tan
8, A,,
all
-> 2a0
= tan2 8,
A,,
flsin 28 1 -ftane ’
=
Am
=
A,,
= 0,
and failure will
o < e <+,
(4.82)
These equations are plotted in Fig. 29. The angle at which the resistance to failure is smallest depends, on the value of the coefficient of friction, and is given by the roots of
300
H. DERESIEWICZ
(4.83)
tan20
=
& l/f;
the corresponding least failure-stress is given by
(4.84)
(c1;/2c,Jrnin.
=
w + V1+ f2)
3
8 FIG.29. Compressive stress causing failure in a simple cubic array of like spheres as a function of the angle between the direction of loading (in the x l x l plane) and the { 100) direction (Deresiewicz [83]).
V. SUGGESTIONS FOR FURTHER RESEARCH In conclusion it may be appropriate and useful to mention some of the problems in the fields we have been discussing which remain unsolved a t the time of writing.
I. A major problem in the theory of packing of convex bodies may be stated as follows: (1) How can n spheres, of equal radii, be packed into a finite cylindrical or cubical container in densest fashion ? A generalization of this question may be obtained by inquiring: (2) How can n spheres of radii yi, i = 1,2,. . ., k , be packed into a finite container of arbitrary shape in densest fashion?
MECHANICS OF GRANULAR MATTER
301
11. We recall that the Hertz theory of contact of elastic bodies is based on the assumptions that the shear stresses on the contact are negligible and that the dimensions of the contact surface are small enough in comparison with the principal radii of curvature at the initial point of contact so that the surfaces of the bodies may be regarded as planes of indefinite extent. Elimination of one or the other of these assumptions leads to the following problems : (3) Considering the compression of two elastic solids by forces normal t o their initial tangent plane as a “problem of the plane,” what is the effect on the Hertz solution *of the presence of frictional forces at the contact ? In this case the Hertz condition of the vanishing of the shear stresses might be replaced by a relation between the normal and tangential components of traction which expresses Coulomb’s law of dry friction. (4) What is the exact solution (within the scope of the linear theory of elasticity) of the Hertz problem, considering the radii of curvature of the surfaces to be finite ? In particular, problems (3) and (4) should be examined for spherical bodies.
The theory of contact of elastic bodies under tangential forces leads in the case of an oscillating tangential force (the normal force remaining canstant) to values of the frictional energy dissipation which, for small amplitudes of the oscillating force, do not agree with experimental results. This disagreement leads naturally to the questions : (5a) Since the law of sliding friction does not adequately describe the situation a t small amplitudes of the oscillating tangential force, how should the boundary conditions for the problem of contact under constant normal and increasing tangential forces be modified? (5b) What is the solution of the resulting boundary-value problem ? The theory of contact due to oblique forces has been worked out in detail only for the case of a force having constant direction in space. It may, however, be of interest to have solutions available for the cases of
(6) contact due to a force whose inclination to the contact surface remains the same, but whose line of action varies in space, and this in such a way that it generates a cone whose vertex lies a t the point of application of the force; (7) contact due to a force whose magnitude and direction vary in accordance with an arbitrary prescribed law. In connection with the theory of oblique contact it would be useful to (8) perform experimental work with a view toward checking such theoretical results as are available. Analogous problems may be stated in regard to the response of a pair of elastic solids to applied torsional efforts.
302
H. DERESIEWICZ
111. As pertains to assemblages of contiguous elastic bodies, evidently the problems consist of taking account of local tangential forces in the course of the establishment of the stress-strain relations for such assemblages. Thus, a “simple” problem would consist of
(9) the derivation of total (as opposed to incremental) stress-strain relations which describe the response of indefinitely extended regular arrays of like spherical particles, in particular the configurations of densest packing, to an arbitrary applied loading. A step in another direction would be accomplished by (loa) the working out of the relations between increments gf stress and increments of strain due to specified loading of an indefinitely extended medium composed of isotropically arranged zones containing dense (facecentered cubic) and loose (simple cubic) packings of like spheres; (lob) the subsequent integration of such relations to yield a total stress-strain law for the medium; (1Oc) the setting up of criteria for the failure of such an assemblage. The next problem in this series is (11)the establishment of stress-strain relations and criteria of failure for a material composed of like spherical granules in a random configuration,
account being taken of all the forces a t the local contacts. An extension of this task may be made by considering a set of spheres inserted in the interstices of the primary spheres, followed by other sets of spheres each inserted, in turn, in the interstices of the previous set. At the end lies the problem which requires (12) the determination of the stress-strain relations and criteria of failure appropriate to a medium composed of a randomly packed array of granules of arbitrary size and shape, the latter quantities being specified perhaps by empirical distribution functions. That none of the questions listed above possess easily accessible answers is evident and almost gratuitous to state. It is probably equally evident that each of them is of considerable importance, either from a purely theoretical or from a practical point of view. It is hoped that the description of some of the problems which have been attacked, with varying degrees of success, as well as the listing of others, as yet untouched, will serve to stimulate more interest in this relatively obscure field of mechanics.
ACKNOWLEDGMENT
It is a pleasure to express my thanks to Professor R. D. Mindlin for his helpful suggestions and to Professor H. D. Baker who photographed the models shown in Fig. 3.
MECHANICS OF GRANULAR MATTER
303
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306
H. DERESIEWICZ
77. HUGHES,D. S., and CROSS,H. J.. Elastic wave velocities in rocks a t high pressures and temperatures, Geophysics 16, 577-593 (1951). 78. HUGHES,D. S., and KELLY,J . L., Variation of elastic wave velocity with saturation in sandstone, Geophysics 17, 739-752 (1952). 79. NASU. N., Studies on the propagation of an artificial earthquake wave through superficial soil or sand layers, and the elasticity of soil and sand (in Japanese), Bull. Earthquake Res. Znst. (Tokyo Univ.) 18, 289-304 (1940). 80. FATT,I., The effect of overburden pressure on relative permeability, J . Petroleum Technology, Tech. Note 194, 15-16 (1953). 81. DUFFY,J., and MINDLIN, R. D., Stress-strain relations and vibrations of a granular medium, J. Appl. Mech., Paper No. 57-APM-39. 82. THURSTON, C. W., and DERESIEWICZ, H., Analysis of a compression test of a facecentered cubic array of elastic spheres; in preparation. 83. DERESIEWICZ, H.. Stress-strain relations of a simple-cubic array of elastic spheres, J . Appl. Mech., Paper No. 57-A-90. 84. CAQUOT,A, and KERISEL,J., “Trait6 de MBcanique des Sols”, third edition. Gauthier-Villars. Paris, 1956. 85. WOLLASTON, W. H., The Bakerian Lecture. O n the elementary Particles of certain Crystals, Phil. Trans. Roy. Soc. London 103. 51-63 (1813). 86. HOOKE,R., “Micrographia”, pp. 85-86. Martyn and Allestry, London, 1665. 87. HEESCH,H., and LAVES,F.. o b e r diinne Kugelpackungen, Z. Krist. 86, 443-453 (1933). 88. MANEGOLD, E., HOFMANN, R., and SOLF, K., o b e r Kapillarsysteme XII. I. Die mathematische Behandlung idealer Kugelpackungen und das Hohlraumvolumen realer Geriiststrukturen, Kolloid-2. 66, 142-159 (1931). 89. HRUBfgEK, J., Filtrationsgeometrie der I(ugelsysteme, Kolloid Beihefte 63, 385-452 (1941). 90. DERESIEWICZ, H., Oblique Contact of Nonspherical Elastic Bodies, t o appear in J. Appl. Mech. 91. MATSUKAWA, E., and HUNTKR, A. N., The Variation of Sound Velocity with Stress in Sand, Proc. Phys. Soc. (London), Ser. R , 69. 847-848 (1956). 92. FATT.I., Compressibility of a Sphere Pack - Comparison of Theory and Experiment, J . Appl. Mech. 24, 148-149 (1957).
Condensation in Supersonic and Hypersonic Wind Tunnels BY P . P. WEGENER
AND
L . M . MACK
Jet Propulsion Laboratory. California Institute of Technology. Pasadena. California
Page I . Equilibrium Condensation Limits . . . . . . . . . . . . . . . . . . . 1 . Equation of State. Phase Rule. and Coexistence Line . . . . . . . 2. Equilibrium Limits of Isentropic Expansions . . . . . . . . . . . I1. Condensation of Water Vapor in Supersonic Nozzles . . . . . . . . . 1 . Definitions of Humidity and Expansion of Moist Air . . . . . . . 2. Historical Note. Description of Phenomena. and Empirical Correlatioh 3 . Summary of Observed Effects . . . . . . . . . . . . . . . . I11 . Condensation in Steam and Hypersonic Nozzles . . . . . . . . . 1. Steam Nozzles . . . . . . . . . . . . . . . . . . . . . . . 2 . Expansion of Nitrogen with and without Vapor Impurities . 3. Expansion of Air . . . . . . . . . . . . . . . . . . . . . I V . Diabatic Flows and Thermodynamics of Condensation . . . . . . 1. Derivation of One-dimensional Condensation Equations . . . . 2 . Analysis of the Condensation Shock . . . . . . . . . . . . . 3 . Condensation as Weak Detonation . . . . . . . . . . . . . . 4 . Saturated Equilibrium Expansion . . . . . . . . . . . . . . 5. Shock Waves with Vaporization . . . . . . . . . . . . . . . V Kinetics of Condensation . . . . . . . . . . . . . . . . . . . . 1 . Condensation Nuclei and Classification of Condensation Processes 2 . Spontaneous Nucleation . . . . . . . . . . . . . . . . . . 3. Droplet Behavior . . . . . . . . . . . . . . . . . . . . . 4 . Comparison with Experiment . . . . . . . . . . . . . . . . VI . Experimental Methods . . . . . . . . . . . . . . . . . . . . . 1. Effect of Condensation on Measurements in Wind Tunnels . . . . 2 . Notes on Methods of Air Drying. Heating. and Special Techniques for Condensation Studies . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
I.
EQUILIBRIUM
307 307 312 320 320 326 342 343 344 347 353 365 366 373 383 387 397 403 404 412 421 428 433 433 440 442
CONDENSATIONLIMITS
1. Equation of State. Phase Rule. and Coexistence Line
In most problems of compressible fluid dynamics we are accustomed to dealing with a thermodynamically pure substance in the gaseous phase . 307
308
PETER P. WEGENER AND LESLIE M . MACK
We may then completely describe the thermodynamic state of the gas at any point in the flow by specifying any two of the properties, temperature, pressure, and density. In addition to the conservation equations of mass, momentum, and energy, we therefore require an equation of state of the form
Any two of the properties in (1.1) are independent variables as long as a change of thermodynamic state does not lead to a chemical reaction or phase change in the flow field. Under these circumstances a mixture of gases such as air may also be treated as a thermodynamically pure substance. Chemical reactions or condensation are observed if either very high or low temperatures occur in the flow field. Many interesting flow processes take place in a moderate pressure and temperature range where it has been found empirically, for many gases of technical interest, that the equation of state (1.1) takes the form of the perfect gas law, (1.2)
R p=p-T. P
( R = universal gas constant, p = molecular weight). We refer to a gas which obeys (1.2) as thermally perfect. Equation (1.2) can be derived from the kinetic theory of gases, if i t is postulated that the molecules are point masses and, on the average, so far apart that the intermolecular forces can be neglected. These assumptions become unrealistic for increasing values of gas density. It is important, therefore, to find the error introduced by the use of the perfect gas law and to determine at what point a more realistic equation of state, such as the semiempirical Van der Waals, Berthelot, or other equations [e.g. 1, 21, must be employed. In particular, this check must be carried out where the thermodynamic state approaches the region of liquefaction. We can write (1.2) as
where Z is the comfiressibility factor, and experimental values of the thermodynamic properties of a substance may be used to determine 2. Experimental values of the compressibility factor for air [3] are given in Table 1 ; it is apparent that appreciable densities or very low temperatures are needed to cause significant departures from the perfect gas law. Such departures do not ordinarily occur in actual flight, but they are quite possible in the supply sections of high-density supersonic and hypersonic wind tunnels,
309
CONDENSATION IN WIND TUNNELS
or in the low-temperature free stream of hypersonic tunnels. also occur* in the driver sections of high-pressure shock tubes. TABLE1. COMPRESSIBILITY FACTOR, Z,
FOR
They may
AIR
T"K
0.01 atmos
0.1 atmos
1 atmos
10 atmos
100 atmos
50 100 150 200 250 300 400 500 600 700 800 900 1000
0.99871 0.99981 0.99994 0.99998 0.99999 1.00000 1 .ooooo 1.00000 1 .ooooo 1 .ooooo 1.00000 1 .ooooo 1 .ooooo
0.99813 0.99941 0.99977 0.99991 0.99997 1.00002 1.00003 1.00004 1.00004 1.00004 1.00004 1.00003
0.98090 0.99407 0.99767 0.99906 0.99970 1.00019 1.00034 1.00038 1.00038 1.00037 1.00035 1.00033
0.9378 0.97666 0.99082 0.99717 1.00205 1.00348 1.00385 1.00385 1.0037 1 1.00351 1.0033 1
0.8105 0.9417 0.9933 1.0299 1.0393 1.0408 1.0397 1.0379 1.0356 1.0333
In addition to using the perfect gas law, it is also usually permissible in fluid dynamics to assume that the fluid is calorically peufect, i.e., for the gaseous phase of the substance or mixture of substances,
(1.4)
CP _ --
- y = constant
cv
throughout the flow field. The value of y depends on the nature of the molecules in question, their degree of internal excitation, and, in the case of a gas mixture such as air, on its composition. For air in the low temperature and low pressure range where condensation effects occur, we know from spectroscopic investigations that while the rotational degrees of freedom of the diatomic molecules of nitrogen and oxygen are fully excited, practically none of the internal energy is stored in the vibrational mode. This again is no longer necessarily true in the supply sections of high-density hypersonic wind tunnels. At constant temperature there is a pressure effect on y , which is appreciable at room temperature [3]. On the other hand, if the temperature of the air supply is raised, the combined effect of the
* It is found that the right hand side of (1.3) deviates from unity a t very high temperatures when dissociation, ionization, or chemical transformation of the gas or gas mixture takes place. This change of 2, however, is due t o an increase of the number of particles in a given initial volume and shall not concern us here.
310
PETER P. WEGENER AND LESLIE M. MACK
excited vibrational mode and the pressure is slight, and it is again permissible to employ a constant y-value [a]. For more accurate work, however, these effects must be taken into account [e.g., 2, 51. - As long as (1.2) and (1.4) are applicable it can be shown that internal energy and enthalpy of the gas or gas mixture are linear functions of temperature. Condensation phenomena may occur in regions of a flow field where the temperature and pressure are low enough so that a phase change of one or more of the participating substances (= components) may be expected. A part of the gas will then become a liquid or solid. This may happen when a gas undergoes an expansion to a lower temperature as in a nozzle or around an airfoil in flight. Conversely, a partially liquefied or solidified vapor may be transformed into a pure vapor by an isentropic compression, a shock wave, or a shear flow. Wherever two or more phases of one or more components are present, the availability of independent thermodynamic properties in the flow field is reduced.* Gibbs’ well-known phase rule [e.g., 61 states that
N
=n
+ 2 -z;
(1.5)
here N is the number of available independent variables, and the system contains n components and z phases. A phase diagram for the single substance, water, is shownin Fig. 1.The phase rule assumes particular importance in flow processes with gas and vapor mixtures as initial participants. Of main interest to us ,oo 2oo 3oo 4oo are the liquid-gaseous or -100 0 T CC) solid-gaseous equilibrium lines,- the dividing lines FIG. 1. Phase diagram of water. in a phase diagram that may be crossed, thermodynamically speaking, by a flow process. By (1.5) there exists for the coexistence of vapor and liquid (or solid) in thermodynamic equilibrium a single function
* For example, in a single component substance such as water for a given arbitrary temperature below the critical point the liquid and vapor can be in equilibrium a t only one pressure.
31 1
CONDENSATION IN WIND TUNNELS
relating p,, the vapor pressure determined in the presence of a plane liquid surface, to the temperature. (The subscript 00 denotes the infinite radius of curvature of that plane.) The function (1.6) is given by the ClausiusClapeyron equation [e.g., 6, 71
where L ( T ) is the heat of vaporization a t temperature T and the indices 1 and 2 distinguish the specific volumes, v = l/p, in the gaseous and liquid phases. Ordinarily, we may assume that the liquid volume is negligible with respect to the gaseous one so that vl - v2 m v1 and, further, that the heat of vaporization is constant in an appreciable temperature range. With these restrictions (1.7), together with (1.2), yields
or
Experimental determinations of vapor pressures may be approximately represented by (1.9); a tabulation of values for the coefficients A and B for several gases is given in Table 2. These values, with the exception of air, have been taken from standard tables [3, 8, 91. TABLE2. COEFFICIENTS FOR COEXISTENCE LINESIN (1.9) Symbol
A [“K]
B [atoms]
B [mm Hg]
Range [“K]
State
Nitrogen
N,
Oxygen
0,
314.22 359.093 386.0 437.5 1367.3 403.91 2681.0 336.34
4.0679 4.77813 4.370 5.250 7.027 4.653e 7.588 4.114
6.9487 7.65894 7.251 8.131 9.908 7.5344 10.469 6.995
63.2- 125 35 - 63.2 54.4-90 45- 54.4 138-216 65-83.8 175-273
Liquid Solid Liquid Solid Solid Solid Solid
Substance
Carbon dioxide Argon Water Air
CO, A H,O
See text
Experimental vapor pressure data are generally more numerous a t the higher temperatures, where a representation more accurate than (1.9) is warranted. However, at the low temperatures occurring in flow through nozzles or in the restricted temperature range of the vapor-solid equilibriums given in Table 2, the simple vapor-pressure function (1.9) ordinarily conforms with the measuring accuracy of thermodynamic parameters in supersonic flows. Tabulated and critically faired values of vapor pressure collected
312
PETER P. WEGENER AND LESLIE M. MACK
from many observers are given in convenient form in the NBS tables [3] if higher accuracy is desired. The special case of air will be discussed later in more detail. 2. Equilibrium Limits of Isentvopic Expansions
We will first describe the equilibrium limits of isentropic expansions of pure vapors that obey the perfect gas law and have a constant ratio of specific heats. Pressure and temperature are then related by (1.10)
x=($) P
Y/(Y - 1) *
In order to compare the slopes of the isentropes with the slope of the vapor pressure curve, or coexistence line, we obtain from (1.10) (1.11)
P Y Ty-1’
where the subscript s denotes that the entropy is constant during the process. We may rewrite the Clausius-Clapeyron equation, (1.7), using (1..2), as (1.12)
where we have again neglected the specific volume of the liquid and taken L = constant. The ratio of the slopes is therefore (1.13)
$/(x),
L
=
7’
where (1.4)and R / p = cp - c, have been used. For all vapors with LIT > cp, the slope ratio (1.13) is larger than one: the vapor pressure curve is steeper than the isentropes. In other words, the saturation vapor pressure decreases with decreasing temperature more rapidly than the pressure of the vapor in an isentropic expansion, and the saturation ratio p = p / p , increases with decreasing temperature until saturation is reached. This leads to an intersection of the isentropes with the coexistence line and, consequently, the expansion process passes into the coexistence region. The liquefaction of part of the vapor is to be expected from thermodynamic equilibrium considerations. Head [lo] gives the slope ratio (1.13) for several vapors as a function of temperature. For a slope ratio of 2, which might be considered as a typical value in the range where a pure vapor may become saturated in steam or nitrogen nozzles operated at customary reservoir conditions, we find the corresponding temperatures to be 482 OK for H20,
313
CONDENSATION IN WIND TUNNELS
222 OK for CO,, 113 OK for O,, and 87 OK for N,. Since the heat of vaporization increases with decreasing temperature and cp = constant, we see from (1.13) that the slope ratio becomes larger as the temperature decreases for the vapors quoted. The inverse of the slope ratio is important in the theory of the saturated isentropic expansion which is developed in Section IV,4. A plot of this quantity for air a t various To and Po is given in Fig. 54. 20c
p0 = 5 atm To * 300°K IOC
8C 6C
4c
y = I. 4 lsentrope
-
-20 I E E
n
10 8 6
4
M=6 2
I
I 25
I 75
I
50
I
100
I2
T (OK) FIG.2. Typical supersonic wind tunnel isentrope and coexistence line for air.
An isentropic expansion in a typical supersonic wind tunnel which operates with supply conditions of Po = 5 atmospheres and To = 300 O K is shown in Fig. 2. Also given is the coexistence line for air, Eq. (1.9), with
314
PETER P. WEGENER AND LESLIE M. MACK
coefficients from Table 2, with air assumed to be a single substance or pure vapor. The isentrope intersects the vapor pressure curve a t a low pressure and temperature and is extended into the coexistence region, the fact that condensation may take place being neglected. Application of the perfect gas law (compare Table 1) is legitimate even close to, or at, the coexistence line since the vapor density is low in this region. Thus isentropes and vapor pressure curves in expansions in nozzles ordinarily intersect a t large specific volumes [4]in marked contrast to the thermodynamic states occurring in industrial liquefaction processes which are carried out near the critical region of the vapor. Static pressure and temperature, as a function of Mach number and supply conditions, are given by the well-known formulas
(1.14) and
(1.15)
6
5
I4
s
2
I
I
2
5
7
10
po( a h )
-
20
50
70
100
FIG.3. Mach number of equilibrium condensation for expansion of carbon dioxide.
The Mach numbers given by either of these formulas are marked on the isentrope in Fig. 2. Let us define the Mach number of equilibrium condensation, M,, of a given isentropic expansion of a given vapor as that Mach
315
CONDENSATION IN WIND TUNNELS
number where the static pressure and temperature of the flow are equal to the saturation vapor pressure and temperature. Then, from (1.14) and (1.9), (1.16)
log10P = ~0g10P0-
'
Y-
log,, (1
+
yG
Mz)
= -
A
+ B.
Substituting (1.15) into (1.16) results in
M:)
+ '2 log,, (1 +G ' M: ) . -1
(1.17) 12
I I 10
I::
5"
6
5 4
3
2
5
7
10
20
SO
70
100
po (atrnl-
FIG. 4. Mach number of equilibrium condensation for expansion of nitrogen.
Figures 3, 4, and 5 show (1.17) graphically for expansions of carbon dioxide, nitrogen, and air, based on the coefficients of Table 2. I n the case of carbon dioxide, the calculation was made for y = 1.33. Inspection of Fig. 2 shows that M , is increased if To is increased a t a given Po, since this results in a shift of the isentrope to the right and, therefore, away from the coexistence region. Figures 3, 4, and 5 serve to determine the nozzle supply conditions which will prevent the gas in the test section from liquefying. A somewhat different picture is presented when a vapor mixture undergoes an isentropic expansion. Head [lo] has derived the analogue of (1.11) for the vapor components. As an example, we will discuss the case of moist air. A t temperatures below about 1500 OK (the temperature where chemical transformations become apparent) a unit volume of dry air is composed of the following volumetric fractions of gases [3]: 0.7809 nitrogen, 0.2095 oxygen, 0.0093 argon, 0.0003 carbon dioxide. The average
316
PETER P.
WEGENER A N D LESLIE M. MACK
molecular weight of this perfect gas mixture is then 28.966. Since the diatomic molecules, N, and 0,, comprise the major part of the mixture, the ratio of specific heats may be taken as y = 1.400, for which value most compressible flow tables [e.g., 11 have been computed. In addition, air may contain varying amounts of water vapor. The vapor pressure curves of these components are shown in Fig. 6 for the pressure range of interest.
Our previous wind tunnel isentrope now has to be replaced by separate curves, each of which gives the variation of a component partial pressure with temperature. Assuming that no condensation occurs, saturation would be attained in succession by H,O, CO,, A, 0,, and N,. Fortunately, carbon dioxide and argon are present only in traces, and both have modest heats of vaporization. Our considerations in the next two sections will therefore be restricted to water vapor in air, steam, and the major air components. Figures 2 and 5, and the application of the vapor pressure relation (1.9) directly to the gas mixture “dry air,” deserve further comments because of their great importance for the design of hypersonic wind tunnels. From the wind-tunnel point of view, it is of primary importance to determine the test section pressure and temperature, or Mach number, at which the first non-gaseous phase might be formed. It is only of secondary interest if this non-gaseous phase is solid or liquid, oxygen or nitrogen. Experimental data on the equilibrium condensation line of air are available from three sources [ l l , 12, 131. The data of Furukawa and McCoskey !13] extend to a lower temperature range than the other sources, i.e., 60 OK. A comparison of their results, within the stated accuracy, with the simple relation (1.9) using the constants given in Table 2, shows that the assumption of a temperature-independent heat of vaporization is sufficiently accurate for engineering applications near the Mach number of equilibrium condensation.
317
CONDENSATION IN WIND TUNNELS
For more accurate investigations, the direct measurements, which are reproduced in Table 3, may be used.
Air
A
b
25
20 c Cll
I E '5 E Y
n 10
5
0
0
50
200
150
I00
T
250
300
(OK)
FIG.6. Vapor pressure curves of air and its components. TABLE3. EXPERIMENTAL RESULTS ON
60.00 64.97 69.01 69.94 75.00 80.00 83.00 85.01
'JFurukawa and McCoskey [13].
CONDENSATION
22.5 63 133 150 335 628 89 1 1105
LINE OF AIR"
318
PETER P. WEGENER AND LESLIE M . MACK
However, designers of hypersonic tunnels are faced with the problem that static temperatures in the test section can fall below 60 OK (see Fig. 2). Extrapolation of (1.9) to 50 OK or below, which was done to obtain the curves shown on Fig. 5, leads to considerable uncertainty. On the other hand, it is possible to compute the condensation line of air from the ideal mixture theory [14] by assuming air to be a mixture of, say, 79% N, and 21% O,, and using the experimental results for the condensation lines of these two constituents individually. Such data are available at the low temperatures of interest [8, 9, 151, and Wagner [16] has carried out such calculations down to 40 OK. Furukawa and McCoskey /13] compare their results with the same theory in the range of their experiments. Figure 6 shows that at a given pressure pure oxygen has a higher saturation temperature than “air.” At first it appears surprising to see that pure oxygen should condense at a higher pressure than the oxygen mixed with nitrogen in air. The explanation is to be found in the fact that the vapor pressure of a certain vapor mixed with other vapors is directly proportional to the mole fraction of that vapor in the mixture, provided the vapors obey Raoult’s law 1141, which states that at a given temperature (1.18)
PU’
= xpu,
Here p,’ is the equilibrium vapor pressure of the vapor in the mixture, and 9, is the saturation pressure of the pure vapor whose mole fraction n is present in the mixture. Raoult’s law is applicable if the concentration of the selected vapor in the mixture is sufficiently low, a point that is in some question for oxygen in air. Furukawa’s calculations give a value of the saturation pressure at a given temperature below that measured, and this same result will probably hold when experiments become available for comparison with Wagner’s calculations. Saturation pressures computed from (1.9), with the coefficients of Table 2, lie above Wagner’s curve. I t is interesting to note that Wagner predicts solid oxygen would be the first condensate encountered if the isentropic expansion intersects the coexistence line of air below T = 52 OK, which is below the triple point of oxygen. Finally, recent experimental work by Armstrong et al. on air [17] to determine activity coefficients [14] demonstrates that air may not be a sufficiently ideal mixture to permit application of (1.18) a t the low temperatures where solid nitrogen and oxygen are in equilibrium with the vapor mixture. It is therefore useful at present to consider (1.9), which is compared with the experiment and ideal mixture theory in Fig. 7, as a tentative standard of comparison with wind-tunnel air condensation experiments until further research has been completed on the condensation line of air to 50 OK or lower. All previous remarks on pure vapors or vapor mixtures were based on vapor pressure curves obtained experimentally in the presence of a plane liquid phase at thermodynamic equilibrium. Neither one of these conditions
319
CONDENSATION I N W I N D TUNNELS
is necessarily fulfilled in the environment of a supersonic tunnel, a steam turbine nozzle, or in any other rapid expansion process. The isentropic flow
4
I-
Furukawa and McCoskey Wagner's Calculation
(1.9) 2-
0.I
40
I 50
I
I
I
I 70
60
I
I 80
I 90
T("W FIG. 7. Estimated coexistence line of air at low temperatures.
in the center of a supersonic tunnel nozzle is surrounded on all four walls by a boundary layer with a large temperature increase at constant pressure
320
PETER P. WEGENER AND LESLIE M. MACK
in the direction of the nozzle wall. In fact, the wall temperature is of the order of the supply temperature. Inspection of (1.9) shows that since the gas layer near the wall is a t a much lower pressure than the supply pressure, but at nearly the same temperature, it will be even further removed from the coexistence line than the tunnel supply. Also, supersonic tunnel expansions to seconds) whose time scale is are rapid expansions (order of governed by the nozzle geometry and size, and it is a priori doubtful if thermodynamic condensation equilibrium is established. These questions will be investigated in some detail later. However, the equilibrium conditions discussed here form the basis of all further considerations. A study of Fig. 5 indicates the technological problems with which the designer of a hypersonic wind tunnel is confronted. In order to retain the relatively high test-section Reynolds numbers needed to achieve a test environment in the continuum regime of fluid dynamics a t high Mach numbers, high supply pressures are required. This fact, in turn, lowers M , for a given supply temperature and leads to high supply temperatures if thermodynamic states everywhere in the flow are to be removed from the danger of liquefaction. (We assume a t this point that no model testing is possible with a partially liquefied vapor as test medium, and that liquefaction occurs instantly if Mc is attained.) I t is then apparent from Fig. 5 that the operation of a hypersonic wind tunnel a t M = 10 and p , = 100 atmospheres requires a supply air temperature of about 1050 OK. Operation of wind tunnels under such circumstances represents a formidable technical problem in the design of heating and cooling systems [e.g., 18, 191. This practical problem has led some [20, 21, 221 to the choice of helium as a working fluid, in spite of the fact that it is a monatomic gas with y = 1.67, since M , of helium is very high for expansions starting at room temperature and high pressures. Turning now to a phenomenological description of observed condensation phenomena in nozzles, we shall see that condensation in nozzles is actually delayed with respect to our previously developed equilibrium notions. Therefore, since we find that we deal with a time-dependent phenomenon, methods other than those of equilibrium thermodynamics must be applied to the solution of the condensation problem in high speed flows. 11. CONDENSATION OF WATERVAPORIN SUPERSONIC NOZZLES 1. Definitions of Humidity and Expansion of Moist A i r
Relative humidity of moist air is defined as the saturation ratio expressed in per cent:
32 1
CONDENSATION IN WIND TUNNELS
where p , is the partial pressure of water vapor in air. Since # m = p,(T) from (1.8), and T and p,, are functions of M , the relative humidity is a function of local Mach number for a given moisture content, pressure, and temperature in the nozzle supply section. If (b > lOOyo in the flow field, and no actual condensation is observed, we call the flow supersaturated. Conditions of > 100% (or a, > 1) may also be expressed in terms of the supercooling
A T = T,- T ,
(2.2)
?! p
Td c
B E c”
pv k
T
Saturation Line
Log Entropy FIG. 8. Schematic temperature-entropy diagram and nomenclature.
where T , is the saturation temperature of a given isentropic expansion of moist air, T is the local static flow temperature on the isentrope, and T < T,. The temperature at which condensation actually starts will be designated by T k . Then $ b k , v k , and ATk may be used interchangeably to characterize the condition of supersaturation at which the collapse of the thermodynamically metastable supersaturated state (see Section V) takes place. Some of these definitions are illustrated on the schematic temperatureentropy diagram in Fig. 8. The partial pressure ratio is also an expression
322
PETER P. WEGENER A N D LESLIE M. MACK
of the absolute moisture content of the air, as it is proportional to the mass ratio of water vapor (subscript u ) to dry air (subscript a). This ratio is called the mixing ratio, x, and is given by
We have taken p v = 18.02, pa = 28.966, and p , << p . The mixing ratio, in contrast to the relative humidity, remains constant throughout any change of state provided there is no condensation and no mixing with air masses of different moisture content. A quantity which is closely related to the mixing ratio is the specific humidity o. It is defined as the ratio of the mass of water vapor to the mass of moist air and is given by
The simplest practical method of determining the humidity of windtunnel supply air is to measure the dewpoint in addition to supply temperature and pressure. In the supply we have p = po, and by cooling air samples slowly and isobarically, the relative humidity increases until @ = 1 0 0 ~ o This . happens at the dewpoint, Td (Fig. 8), and since at saturation p , = p,, (2.3) becomes
With our known vapor pressure data pm = $,(Td) we can calculate the mixing ratio as a function of dewpoint for various tunnel supply pressures; such results are shown on Table 4. Values of 1000 x are given for convenience, expressing the mixing ratio in g water per kg dry air. The left-hand column, where 9 = 1 atm., is of special importance as it is usual to measure dew points at atmospheric pressure. Next, some aspects of the expansion of moist air will de discussed. If the expansion of a vapor takes place in a well-designed nozzle, and we except the narrow region of the boundary layer on the nozzle walls, the entropy will be constant until condensation occurs. However, if a vapor mixture is expanded isentropically, it can be shown that the entropy of the individual constituents is not constant [7, 101. Let us take the isentropic expansion of moist air as an example and consider the air itself as a pure vapor for the moment. With s denoting the specific entropy, the entropy change of the mixture is given by (2.6)
+ mvdsv= 0.
mds = madsa
323
CONDENSATION I N W I ND T U N N E L S
TABLE 4. MIXINGRATIOx
IN
g H,O/kg AI R .is A FUNCTION OF DEWPOINT TEMPERATURE, Td, A N D P R E S S U R E
p (atmospheres) Td ["CI - 30
- 25 - 20 - 15 - 10 -2
0 5 10 15 20 25 30 35
1 0.234 0.390 0.636 1.017 1.600 2.476 3.770 5.401 7.629 10.645 14.689 20.071 27.183 36..546
3
0.0467 0.0779 0.127 0.203 0.319 0.493 0.750 1.07 1.51 2.10 2.86 3.91 5.25 6.98
10
0.0635 0.101 0.160 0.247 0.375 0.536 0.755 1.05 1.44 1.95 2.62 3.47
20
30
40
50
0.0318 0.0508 0.0798 0.123 0.187 0.268 0.377 0.524 0.718 0.974 1.30 1.73
0.0212 0.0339 0.0532 0.0822 0.125 0.179 0.251 0.349 0.479 0.649 0.869 1.15
0.0159 0.0254 0.0399 0.0617 0.0937 0.134 0.188 0.262 0.359 0.486 0.652 0.864
0.0127 0.0203 0.0319 0.0493 0.0750 0.107 0.151 0.209 0.287 0.389 0.521 0.691
Introducing the mixing ratio (2.3) into (2.6) we obtain (2.7)
asa
+ xds, = 0.
If we assume the vapors to be thermally and calorically perfect we may write the entropy change as
where cp is the heat capacity per unit mass. Substituting (2.8) into (2.7) we obtain
for the isentropic expansion of the vapor mixture. Again, the mixing ratio x remains constant if no condensation takes place; therefore by (2.4) (2.10)
By introducing (2.10) into (2.9) we obtain for the vapor mixture
(2.11)
324
PETER P. WEGENER AND LESLIE M. MACK
We can now find the entropy changes of the air and water vapor from (2.8) and (2.11). For air we obtain
(2.12)
where a
(2.13)
= pulpa.
Similarly, the change in the entropy of the water vapor is
a--
as, = - -
kJP
From (2.7), (2.12), and (2.13) we find (2.14)
ds, -
dsa
1
x
-
1
B,
a B,’
where B, and B, represent the quantities in brackets of (2.12) and (2.13) respectively. This analysis shows that water vapor and air would undergo individual isentropic expansions only if a = cp /cp,, orp,cfi, = pacp,. That is, only when the molar specific heats are equal do (2.12) and (2.13) yield ds, = ds, = 0 for any x > 0. This is quite evidently not the case for the water vapor-air mixture; in fact, a = 0.622 [cf. (2.3)], while cpa/cpv= 0.539 at 300 OK [3]. As a result, we find for the typical mixing ratio x = 0.02 that in (2.13) B, = 0.149. This mixing ratio corresponds approximately to saturated atmospheric air at T = 25 “C. Therefore, the entropy of the water vapor increases somewhat, while that of the air decreases. The numerical value of the bracket B, in (2.12) is only - 0.0048 because of the small value of x, and the entropy decrease is small. However, we have shown that it is incorrect to assume that in an expansion of moist air the entropy of the water vapor at the onset of condensation is the same as the entropy in the supply section. Dry air was previously treated as a pure vapor even though oxygen and nitrogen, its two major components, constitute a vapor mixture. Fortunately, both nitrogen and oxygen are very similar diatomic molecules, with the result that their molar specific heats differ by less than 1% [3]. Therefore, when air expands isentropically, both the nitrogen and oxygen undergo individual isentropic expansions. The presence of water vapor in the expansion of moist air does not affect the pressure and temperature as functions of Mach number within ordinary
.CONDENSATION
IN WIND TUNNELS
325
measuring accuracy, as shown in detail by Head [lo]. This fact may also be demonstrated by considering the specific heats of a gas mixture which are given by 1 cp = m
(2.15)
2 micpi
t
and (2.16)
c,
1 m
=-
2 micOi,
where i denotes all participating substances. Hence, the ratio of specific heats for moist air is given by (2.17)
Even for saturated air at ordinary temperatures and pressures, x << 1, and consequently y = cPa/cua. For example, at p , = 1 atm. and To = 300 OK, y = 1.3999 for dry air in the ideal gas state [3]. With a relative humidity Qo = 35% at this temperature, x = 0.01, and y = 1.3991, which certainly permits most flow calculations to be made with the conventional y = 1.400 tables [l]. The same reasoning applies to the determination of the gas constant of the mixture, which can be taken as that of air alone. We may now write for the isentropic expansion of moist air up to the point of actual condensation (2.18)
+
+
where fi = p,, pa and p , = p , @ao. We recall that the slope of (2.18) is less steep than that of the saturation line (1.8),and we expect Q to increase rapidly with decreasing pressure and temperature. Applying (1.9) to the supply condition and to some other point in the expansion, we obtain (2.19)
As long as no phase change occurs, the partial pressure ratio remains unchanged. From (2.1) we have (2.20)
Y - P Yo Po
Pa, Pa
326
PETER P. WEGENER A N D LESLIE M. MACK
Combining (2.19) and (2.20),we find (2.21)
Then, with (1.15) we have finally (2.22)
This shows that the relative humidity in expansions of moist air increases exponentially with the square of the Mach number [23] apart from the relatively small variation of p/p0. Since the mixing ratio itself remains unchanged, it is convenient for further work to express the saturation ratio, p, directly in terms of the flow pressure, temperature, and mixing ratio. From (2.3) we have (2.23)
log,,
p,, = log,, P X - log,,
(0.622
+ x).
With the definition of the saturation ratio and (1.9), we can write (2.24)
lOg,op
= log10 P X - log,, (0.622
+ + AT X)
--
- B.
For most cases of interest x << 0.622, which is equivalent to saying that Then we can simplify (2.24), insert the constants for water vapor from Table 2 , and finally obtain
p , << p. (2.25)
where $ is given in mm mercury and T in degrees Kelvin. The substitution of typical values of Mach number and flow conditions for supersonic wind tunnel nozzles operated with atmospheric humidity in the supply into (2.22) or (2.25) shows that saturation will practically always be attained ahead of the sonic throat. In fact, in an atmospheric supply tunnel which is operated at ordinary humidities, the moist air will be more than ten times supersaturated at the nozzle throat. We will soon see that condensation is nearly always observed at supersonic Mach numbers and therefore takes place at high supersaturation. 2 . Historical Note ; Description of Phenomena ; Empirical Correlation
Early investigators of air flow through supersonic nozzles were puzzled b y the observation of two oblique shock-like disturbances which occurred
CONDENSATION IN WI ND TUNNELS
327
just downstream of the nozzle throat. Prandtl [24, p. 1671 showed a beautiful schlieren picture of such a disturbance at the Volta Congress in 1935. Wieselsberger remarked in the ensuing discussion [l.c., p. 5581 that the position of these apparent shock waves was probably dependent upon the initial humidity of the air. The first systematic investigation of this relationship (Aachen 1934 to 1936) was published by Hermann [25] who demonstrated that these disturbances were indeed caused by condensation of the water vapor in the air, and, consequently, because of their optical appearance,
FIG.9. Schlieren photographs of water vapor condensation from Eber and Gruenewald [26] ; 40 X 40 cm supersonic tunnel, atmospheric supply. (a) Do = 36%. x = 6.2 g/kg; (b) a, = 67%, % = Sg/kg.
they became known as “condensation shocks“. (In the subsequent discussion we retain the accepted expression “condensation shock” for historical reasons, although we shall see later that no shock process is actually involved.) Shocks close to the throat and nearly normal to the flow direction were seen at high humidities, while they appeared further downstream and more x-shaped for lower moisture contents. Also, these condensation phenomena were accompanied by pressure non-uniformities. Two typical schlieren pictures [26] which were taken a t different initial humidities are shown in Fig. 9.
328
PETER P. WEGENER A N D LESLIE M. MACK
Schlieren pictures of this same general type were taken by Hermann and since then, for example, by Lukasiewicz and Royle [27], Burgess and Seashore [28], and Wegener [29]. All of these photographs from nozzles of different size and geometry roughly follow the pattern outlined above. The important discovery of the Aachen group that the visible shocks near the nozzle throat are due to humidity effects led to the introduction of airdrying equipment into supersonic wind tunnel circuits. This was necessary because it proved to be impossible to conduct careful experiments in an environment where Mach number, static pressure, etc. showed daily variations because of the random changes of atmospheric humidity. Subsequent operational experience with the 40 x 40 cm Peenemuende supersonic wind tunnels [26] showed that drying the supply air to n 0.5 g waterlkg air was needed to render the condensation shocks practically invisible in the schlieren system, and to allow most tests to be conducted free of humidity effects (see Section VI).
<
Several years went by before the connection with the long known similar phenomena of condensation in the supersaturated state in Wilson cloud chambers [30] and steam I0 nozzles [e.g., 311 was recognized. Heybey [25, 321 08 solved the purely fluid dynamical aspects of the condensation process on * 06 the basis of one-dimensional flow, with the condensation considered to be X 04 a discontinuity in which M =2 29 an unspecified amount M = I 70 of heat is released (see 0.2 M = 40 Section IV, 2). Oswatitsch [33, 34, 35, 361 was the I I I first to present a unified 0 0 20 40 60 80 100 kinetic and thermodynama0(%) ic picture of moist air and steam condensation FIG. 10. Dimensionless distance of water vapor (see Sections Iv and v). condensation shock from nozzle throat as a funcHis qualitatively correct tion of initial relative humidity from Hermann [25]. picture has not yet been superseded, in spite of much further work by a theory that would permit accurate quantitative predictions of the onset of condensation without the empirical adjustment of coefficients. This situation is much like that presented in the prediction of chemical reaction rates.
'
,
I
329
CONDENSATION I N W I N D TU N N ELS
Following these publications, we find a large number of papers on the subject which may be broadly classified into two types of work. One group gives results that were obtained incidentally in connection with the aerodynamic calibration of nozzles, checks of drying equipment, and force and pressure measurements on models, all directed towards the elimination of condensation effects. The other group made the condensation process itself the subject of experimental and theoretical study. Some results of this group will be reviewed by giving typical samples without attempting to quote all investigations. 0.8
0 @ @
0.7
0.6
\
Isentropic Curve From Area Ratio Dry Nitrogen Calibration po= 2790 mm
Hg, @,, = 74.4%
@
2040
56.7
@
1260
32.2
0.5
op 0.4
\
a
0.3
0.2
0.I
-
0
0
2
I
3
~(inches)
FIG.11. Static pressure on nozzle centerline with a n d without water vapor condensation from Head [lo].
330
PETER P. WEGENER AND LESLIE M. MACK
Hermann [25] found that in a given nozzle the dimensionless condensation shock position on the centerline, X , varied with the initial relative humidity as shown for three nozzles in Fig. 10. The three faired curves were obtained from an evaluation of the shock position in about 150 schlieren photographs. The static pressure disturbances which are associated with these condensation shocks can be readily measured. Typical results obtained by Head [lo] are shown in Fig. 11 in comparison with an isentropic expansion. Similar to the shock in the pictures, the pressure disturbance shifts position downstream with decreasing relative humidity in the air supply. Farther downstream, after nearly all water vapor has condensed, the static pressures are nearly the same in all cases. This is related to the fact that since the mixing ratio was held constant in Head's experiments, the total heat released by condensation during the entire process was about equal in all three cases. In fact, the pressure increase through the disturbance, or condensation shock strength, has been found by Hermann [25] to be largely governed by the mixing ratio. In larger tunnels, with smaller Mach number gradients in the flow direction, the static pressure can actually increase in the condensation region and then decrease again after nearly all the condensation has taken place [eg. 10, 29, 371. 0
0 CD 1.8 -0 0
-
Eber and Gruenewald
A3x3cm? 0
1.6
40 x 40 cm
Oswatitsch
14 x 14 cm
Lukasiewicz
A Small
*O 0
Head
Q fl.4
00
-
OA 1.2
.A
0
-
B A 0 0 00
a0
A
O '
1.0 -
0.8
A
I
I
I
I
I
I
I
A A
I
I
FIG. 12. Mach number at the onset of water vapor condensation as a function of initial relative humidity.
The supersonic Mach number a t which condensation is observed can be found approximately from either the condensation shock location on a photograph or the position of the static pressure disturbance. In both cases
331
CONDENSATION IN WIND TUNNELS
the nozzle must have been previously calibrated with dry air. This Mach number, M,, is shown in Fig. 12 as evaluated from results of several experimenters who worked with nozzles of varying size with, however, a roughly atmospheric supply in common. Due to uncertainty in the exact 00
0
0
75 0
70
'
0 0
000
co 03 65
0 0 0
5
k s
a
0
0
60
0 0 0 0 0 0 0
55
0
do 0 0
50 0
(AT/Ax),,- 3 OK/cm 40 x 40 cm Tunnel
45
40
0
20
60
40
00
100
Q (%I FIG. 13. Supercooling as a function of initial relative humidity for water vapor condensation in a M = 1.86 nozzle with atmospheric supply from Eber and Gruenewald 1261.
flow parameters, and for reasons to be discussed later, considerable scatter is apparent. Nevertheless, the initial relative humidity appears to be an important parameter in the correlation.
332
PETER P. WEGENER A N D LESLIE M. MACK
In a manner similar to the determination of the Mach number for the onset of condensation, the supercooling (2.2), or saturation ratio (2.1), (2.25), at the shock position may be estimated. The supercooling, as evaluated from Eber's photographs [26], increases with decreasing relative humidity in a given nozzle as shown in Fig. 13. Furthermore, for a given initial relative humidity, supercooling increases with an increase of the average temperature gradient of the expansion from the point in the nozzle at which saturation occurs to the point of condensation. The average temperature gradient is defined by (2.26) 80
60
-
5 40 kX
a
o 2.5 x 2.5 cm v 3x3cm? 20
-
0
A
4Ox40cm I4 x 14cm
Wegener Oswatitsch Eber and Gruenewold Lu k asi ew icz
humidities 65 < m0< 90% OL
0
I
20
I 40
I
60
80
(AT/AX),["K/cm) FIG. 14. Supercooling as a function of nozzle temperature gradient for water vapor condensation with high relative humidity.
For a set of nozzles of given length and exit height which cover a Mach number range, the temperature gradient increases with increasing Mach number. Therefore, the temperature-gradient effect on condensation may already be inferred from Hermann's original data (Fig. 10). These data were obtained in nozzles of the same length, and it was found that the dimensionless distance of the shock from the throat, X , was largest for the
CONDENSATION IN WIND TUNNELS
333
high Mach number nozzle. Some of the scatter of the data in Fig. 12 is a result of the size difference in the nozzles and the resulting temperature gradient effect. Head [lo] collected evidence from several sources before 1949, including his own data from three supersonic tunnels of different size, and related what he terms the threshold supersaturation ratio, where condensation becomes noticeable, with the static temperature at condensation. This threshold supersaturation may be computed from (2.25). Head found that the temperature gradient (in a form different from that given in 2.26) correlated the experimental data, including some results from steam nozzles (see Section 111, 1) and cloud chamber experiments [38]. The effect of temperature gradient on supercooling, estimated from pressure measurements and shadowgraph observations of the condensation shock in a small nozzle with plane diverging walls [39, 401 is shown in Fig. 14. Experiments by Oswatitsch [34], Eber and Gruenewald [26], and Lukasiewicz and Royle [27] are also indicated. All of the results were obtained in a narrow range of high humidities. Again, considerable scatter among different tunnels is noticeable, a fact that has plagued all investigators of condensation phenomena. During the discussion on the kinetics of condensation (see Section V) it will become apparent that the collapse of the supersaturated state observed in our previous findings is highly sensitive to small environmental changes. In general, temperature gradients of about one to ten degrees Kelvin per centimeter are found in the larger supersonic wind tunnels which operate at 1 < M < 5, and amounts of supercooling of about 50 OK are common at the higher humidities [27]. However, for the larger temperature gradients which exist in small experimental nozzles [lo, 39,401, amounts of supercoolings up to 80 OK (or more at lower relative humidities) are observed. This rather extreme supersaturation is considerably above the values measured in conventional observations of condensation phenomena made in cloud chambers or in the atmosphere. Among the further features of condensation effects in nozzles operated with moist air, we find that at higher humidities the entire flow appears to be foggy downstream from the condensation shock location. Near the nozzle walls a clear zone is observed, since vaporization in the boundary layers (see Section I) causes disappearance of the mist. At lower humidities the fog droplets are less dense and initially smaller, particularly in short nozzles. However, the droplets can be made visible by observing the light scattered when a collimated beam of light is passed through the nozzle (see Section 171). Light scattering investigations [lo, 291 show the fog to appear rather abruptly downstream from the location of the condensation shock as seen in schlieren pictures. The condensation shock appears bent in nozzles designed to produce uniform flow, since the flow parameters in planes perpendicular to the centerline are highly nonuniform in the throat region. Condensation would be expected to occur first at the nozzle wall
334
PETER P. WEGENER A N D LESLIE M. MACK
where the temperature and pressure are lowest, and then to propagate towards the nozzle center [36]. The intersection of these disturbances and their propagation downstream gave rise to the early nomenclature, x-shock [25], because of the resultant x-shaped appearance in a photograph. An investigation of the thermodynamic states near the wall just outside the boundary layer and on the centerline of a highly curved nozzle [29], in the light of empirical experience concerning the location of condensation in onedimensional flow [40], showed that the whole first branch of the x-shaped disturbance can actually mark the beginning of a high rate of condensation in agreement with the light scattering findings. The oblique condensation disturbance and its attendant pressure disturbance are reflected from the walls of the nozzle, and some schlieren pictures [27, 281 indicate successive reflections from one wall to the other. Lukasiewicz and Royle [27] found the pressure to be nonuniform on the entire nozzle centerline. This system of condensation shock and successive oblique shocks is commonly observed to oscillate longitudinally. The effects of condensation near the throat of the nozzle on flow properties in the test section will be studied in Section VI. The fact that a sudden condensation occurs in a narrow zone, but not as a shock wave, can be inferred from interferometric observations of the flow. Two such interferograms [29] are shown in Fig. 15, where the interference patterns were taken with a Mach-Zehnder interferometer using monochromatic light. The flow conditions were identical for the two pictures, but the spacing of the interference fringes was different. I n both cases the condensation zone is visible as an x-shaped region of bent fringes. However, every fringe is continuous through the entire condensation zone in marked contrast to interferograms of shock waves in wind tunnels [e.g. 41j. In the latter case, the fringes are discontinuously displaced across the shock, whose thickness is of the order of the mean free path of the molecules. No shock wave in the ordinary sense can be present in the initial condensation zone which extends a distance in the flow direction of the order of lo4 times the mean free path of the molecules. The fact that condensation zones are shaped differently in different n o d e geometries, and that most experimental data refer to centerline conditions either by pressure measurement or photographic observation, is another factor explaining the large scatter when the data of several investigators are compared. Continuous operation of nozzles with a thermally insulated glass sidewall whose temperature, T,, is in the vicinity of the freezing point, leads to heavy icing of the tunnel interior [28]. The supply temperature where this effect must be expected may be computed from the temperature recovery factor, which is defined as (2.27)
CONDENSATION I N W I N D T U N N E L S
335
FIG. 15. Interferograms of water vapor condensation taken with different fringe spacing. U. S. Naval Ordnance Laboratory 18 x 18 cm supersonic tunnel, atmospheric upp ply, Do = 47%. x = 4g/kg.
336
PETER P. WEGENER AND LESLIE M. MACK
where Tkis the flow temperature at M,. Therefore, the supply temperature at which the wall ices is (2.27 a)
1
+ Tk.
To = - (273 - Tk) 7
i.05
6 Tot I .oo
9L POI
3.95
2
3
xkm) FIG. 16. Measured and computed flow parameters through a water vapor condensation zone.
For all except the smallest nozzlcs, the boundary layer is turbulent, and we have Y = 0.9 [42]. With Mk = 1.2 for medium humidities (Fig. 12), we expect icing for To 7 280 O K . Also, condensation in the free stream leads to a complicated interaction with the boundary layer on the tunnel walls which forms condensate deposits of peculiar shape, as shown bv Guienne 1431.
CONDENSATION IN WIND TUNNELS
337
All of the previously described experimental results were based on directly measurable parameters. It will be shown in Section I V that we may determine all flow parameters through a condensation zone in a diverging nozzle from the measurement of the initial conditions, the static pressure along the line of flow, and the knowledge of the area increase in the flow direction r(4.10, 4.27, 4.29, 4.30)]. The area may be inferred from a nozzle calibration made with dry air, if we assume that the boundary-layer displacement thickness is essentially unaffected by the condensation process. The primary uncertainty in this procedure is the fact that condensation produces a strong adverse pressure gradient which interacts with the turbulent boundary layer on the wall and may affect the pressure measured by orifices at the wall. The results of such a calculation, and the experimental data on which it is based 1291, are shown in Fig. 16. Some simple thermodynamic facts about the condensation process of *ater vapor in air can then be readily obtained. All parameters in Fig. 16 have been made dimensionless by the corresponding parameter in the nozzle supply (subscript 01). We see that up to the nozzle position marked “start”, the dry and moist air pressure distributions, and therefore all other parameters, coincide. Once condensation begins, all flow parameters are drastically affected in a narrow zone in the flow direction, after which the expansion is resumed a t different levels of static pressure, temperature, etc. The entropy increase of the dry air in the condensation zone may be computed from the stagnation temperature and pressure by means of the integrated form of (2.8), (2.28)
We have neglected the presence of water vapor, except, of course, that the whole process is based on the heat released by water vapor condensation. I n an adiabatic shock wave, To would be constant, hence the first term on the right hand side of (2.28) would vanish. The change of stagnation enthalpy per unit mass across the condensation zone is a measure of the heat released by condensation and, for a perfect gas, is given by (2.29)
9 = ho, - h0l = c,(To, - TOIL
where cp, the specific heat of moist air, is very nearly cp,. If we now assume that all of the water vapor present condenses in the condensation zone, the maximum heat added a t the end of the zone is (2.30)
m V
g e = ma
+ m, L = o L z z x L ,
338
PETER P. WEGENER A N D LESLIE M. MACK
where L is a constant. We can compare (2.30) with (2.29) by somewhat arbitrarily defining the end of the condensation zone as the point where the expansion is resumed. We then find, for the data of Fig. 16 1291, that
(2.31) which indicates that nearly all condensatiop is confined to a narrow zone. Indeed, very little water vapor is needed to keep the flow saturated during the further expansion at these low temperatures, and we observe from Fig. 16 that the stagnation pressure and temperature again become nearly const ant . It is by now apparent that the collapse of the supersaturated state of the water vapor is a time dependent, kinetic phenomenon of an irreversible nature. In fact, the moist air is subjected to a cooling rate of the order lo5 to lo7 "Klsec. Cooling rates of this order exist in expansions in rocket nozzles and lead to "frozen flow" where most chemical recombination reactions are impossible [a]. Consequently, a substantial departure from the equilibrium condensation process must be expected in nozzles, as droplet formation is a time dependent phenomenon. Without discussing the kinetics of the problem a t this point, we may determine a few facts concerning the condensation delay. I t is interesting to determine the approximate time interval in which a volume of moist air travels from the point of saturation to that of observed condensation in cases of large supercooling. This time interval is shown in Fig. 17 and is based on experiments in a small adjustable nozzle [40].The approximate transit time, t,is computed from
(2.32) where the subscripts on the position, x , and on the velocity, w , refer again to saturation and condensation. The relative humidity of the supply in these experiments was about constant, and the temperature gradients and amounts of supercooling are indicated in the upper and lower parts of the figure. For the extreme gradients, the transit time was less than 50 microseconds with a corresponding large supercooling. From Eber and Gruenewald's experiments [26] at the same humidity, but with dTldx = 3 "K/cm, we can estimate t = 900 microseconds, which is typical for a large nozzle. The curves in Fig. 17 at the largest temperature gradients are nearly vertical. This, in connection with the time dependence of the phase transition, has led to the conjecture that for extremely short duration of the expansion condensation is impossible. In fact, Head [lo] was unable to observe condensation a t low humidities in short, high Mach number nozzles; and Cwilong [45], studying rapid moist air expansions in a cloud chamber to temperatures below 150 "K, was also unable to note a phase transition.
339
CONDENSATION I N W I N D TUNNELS
On the other hand, Wegener and Lundquist [46] used a shock tube in connection with sensitive electronic light scattering detection and showed
50
I00
I 50
2 00
T(micr0seconds) FIG. 17. Supercooling and nozzle temperature gradient as a function of time available for water vapor condensation from Wegener [39].
that water vapor in air undergoing a rapid, unsteady, isentropic expansion certainly condensed a t temperatures as low as about 130 "K. Since these
340
PETER P. WEGENER AND LESLIE M. MACK
expansions were of shorter duration than those encountered in ordinary nozzles, there appears to be no lower threshold for condensation of water vapor in the observed temperature range.
32.5
0
Nozzle M.1.22 1.56 1.86 2.50
A
3.31
Symbol
2.0-
0
1.5
1.0
-
0
I
I
I
I
I
I
190
200
210
220
230
240
I
I
180
170
251
T,(OK) FIG. 18. Correlation of Eber and Gruenewald’s experiments on water vapor condensation in a 40 x 40cm tunnel.
Finally, it would be desirable to have an approximate empirical expression to determine the location of condensation in a given nozzle as a function of the supply conditions. This correlation would not be a criterion to obtain condensation-free flows (see Section 171). Specifically, for a medium-sized series of nozzles of equal length operated a t essentially atmospheric supply temperature and pressure, it is possible to express the saturation ratio, V k , at which condensation occurs as a function of several parameters. Such a limited and approximate empirical correlation is based on about 40 photographs from Eber and Gruenewald’s experiments [26] in a 40 x 40cm tunnel, and is given by
(2.34)
+ 4.23 loglorpk = - 0.0415 Tk + 11.24
(2.35)
10glopk = 2.3’5X
(2.33)
lOg,oVk
=
- 315 x
+ 1.38
for for for
< x < .01, 175 < Tk < 245 0.1 < X < 1.3. .001
O K .
341
CONDENSATION IN WIND TUNNELS
The mixing ratio, x , was chosen for (2.33) rather than the relative humidity because it is less dependent on supply pressure. Again, X is the dimensionless distance of the condensation shock from the throat. Comparing (2.34), which is shown by the solid line in Fig. 18, with the previously mentioned similar correlation of Head [lo] leads to reasonable agreement.
40 to 70 12 to I 6
7 3 0
Eber and Gruenewald Lukasiewicz Eber and Gruenewald, Lukasiewicz, Oswotitsch
A+
Wegener
Q(%) FIG. 19. Approximate correlation of supercooling as a function of initial relative humidity and nozzle temperature gradient for water vapor condensation.
Again the largest and smallest tunnels [lo] are not correlated. eliminate the saturation ratio from these equations to obtain (2.36)
T R= 7590 x
+ 169,
and (2.87)
X = - 134 x
+ 1.21.
We may
342
PETER P. WEGENER AND LESLIE M. MACK
Therefore, within the limitations given above we are able to estimate the temperature at condensation and the shock location directly from the mixing ratio in the nozzle supply. In more general terms, our previous considerations of the experimental evidence have shown us that there may be an empirical function (2.38)
which relates the supercooling, or supersaturation, to the initial relative humidity and temperature gradient. In Fig. 19 various data have been collected in order to express (2.38) empirically. However, the scatter, particularly for large values of the temperature gradient where only one group of experiments [39, 401 is available, precludes a mathematical formulation. On the other hand, a study of the individual points shows the general validity of (2.38). A theoretical derivation of this type of equation based on first principles is necessarily a task for kinetic theory rather than thermodynamics. Once condensation appears, it is possible to employ thermodynamics and fluid mechanics to find the flow parameters at the end of the condensation zone, and to predict the further course of the expansion to the end of the nozzle. Although much theoretical work has been expended on the first phase of the problem, we shall see later that only a qualitative picture of the collapse of the supersaturated state can be given, although more accurate predictions of the onset of condensation are possible by adjusting empirical coefficients according to experiment. The second phase of the process can, however, be adequately treated by thermodynamic and fluid-dynamic methods. 3. Summary of Observed Effects
Summarizing the variety of theoretical and experimental results which we have reviewed, we can list the following qualitative statements. In an isentropic expansion of a vapor mixture, the entropy of the individual components is not necessarily constant. In an expansion of moist air, the presence of the water vapor may be neglected until condensation occurs. The relative humidity of moist air during an expansion in a supersonic nozzle increases very rapidly with increasing Mach number, and saturation, with respect to equilibrium conditions, is reached for all normal humidities ahead of the nozzle throat. The shocklike disturbances and pressure nonuniformities which are seen near the throat of a supersonic nozzle operated with moist air are due to the condensation of the water vapor and the attendant heat release in the flow. Actual condensation is nearly always observed at supersonic speeds which demonstrates that, before condensing,
CONDENSATION I N W I N D T U N N ELS
343
the water vapor is in a highly supersaturated state. In a given nozzle the location of the condensation disturbance is governed by the initial relative humidity, and the disturbance moves downstream with decreasing relative humidity. The strength of the disturbance is controlled by the mixing ratio. The initial condensation process, although confined to a narrow zone, is not connected with a shock wave in the ordinary sense. For a given initial relative humidity, the nozzle geometry and size (or temperature gradient) control the supercooling of the water vapor, with large temperature gradients producing large supercoolings. Evaluation of condensation data shows an entropy and stagnation enthalpy increase of the flow through the condensation zone. Fog formation and wall icing may occur as side effects of condensation. Moist air expanding in a nozzle undergoes an extreme cooling rate, and condensation under these circumstances is a highly time-dependent phenomenon. Empirical correlation of the available experimental results is difficult due to the scatter inherent in any observations of the rapid collapse of the supersaturated state. Finally, it is apparent that the prediction of condensation effects in nozzles operated with moist air is a subject for kinetic theory. However, once the condensation environment is known, thermodynamic and fluid dynamic considerations will be applicable.
111. CONDENSATIONIN STEAM AND HYPERSONIC NOZZLES Owing to the early introduction of de Lava1 nozzles (1883) into steam turbine design, condensation investigations in converging-diverging steam nozzles have been made for a long time. In general, the condensation phenomena in steam are not markedly different from those in moist air described previously. However, the small size of most steam nozzles makes a quantitative description even more difficult, partly because friction losses in the boundary layers produce an appreciable deviation of the isentropic core flow from that predicted by the area ratio. Summaries of the work with steam are given in several books [e.g., 7, 311, and we shall therefore review only a few basic observations. The study of condensation phenomena in aeronautics is much more recent. The first work on water vapor condensation dates from about 1934, while the first experiments on the condensation of air were not undertaken until about 1943. The motivation for this latter work was the advent of hypersonic wind tunnels which operate above Mach numbers of five, and therefore attain low test-section temperatures. Many observations of condensation in air or nitrogen are in sharp contrast to those described in connection with moist air expansions and lead us to suspect that somewhat different processes are involved.
344
PETER
P.
WEGENER AND LESLIE M . MACK
We have seen previously that the major manifestation of condensation effects in supersonic flow is the change of the flow parameters as a result of the heat of vaporization which is released when some vapor undergoes a phase transition to the liquid or solid state. We expect, therefore, that the magnitude of the condensation effects is partially governed by the amount of the heat of vaporization of the condensing vapor in relation to the specific heat of the remaining vapor mixture. From equations (2.29) and (2.30), we can write the change in the stagnation enthalpy of the flow, which results from the condensation of a small amount of vapor mL,as
Ah,
= h,
- h,, =
Lr-? L%,L mu -k mL Inu mL
where the subscript, v , denotes the remaining vapor, and L must be chosen according to the temperature range where condensation takes place. By the perfect gas law, (3.1) becomes
We now assume that a change of 1% in the stagnation enthalpy is measurable by, say, a static pressure change. Using typical values of Tk,Tol, etc., we find from (3.2) that the condensation of only 1 g of water vapor per kg dry air is sufficient to cause a change of 1% in the stagnation enthalpy. The corresponding amounts of condensate for other mixtures are 4 g H,O/kg steam, 11 g N,/kg air, and finally, 13 g O,/kg air. This simple consideration leads us to expect less drastic changes in the flow parameters in cases other than moist air. Indeed, the heats of vaporization of the major components of air are less than 1/10 that of water vapor a t the respective condensation temperatures [S, 91. In the following discussions it will be found that the onset of condensation is governed by the availability of condensation nuclei in the flow, in contrast to the moist air condensation phenomena described previously. It will therefore be necessary to understand some of the kinetic concepts developed in Section V, 1. 1. Steam Nozzles
Hirn and Cazin in 1866 [31, 471 expanded superheated steam contained in a pipe by opening a valve, and the cloud formation due to condensation was observed through a glass window in the pipe. These investigators found that the steam condensed immediately upon reaching equilibrium saturation. Stodola [31], in 1913, published his first results obtained with a steam jet which issued from a well-rounded orifice, and introduced the method of light scattering observations as a means of determining condensation, i.e.,
CONDENSATION I N W I N D T U N N E L S
345
of making the fog visible to the eye. In these classical experiments, Stodola found for his orifice and later for transparent, slender, converging-diverging nozzles operated with saturated or superheated steam that appreciable supersaturation occurred before condensation set in. He also observed that in cases for which the pressure ratio across a nozzle was supercritical and the flow in the diverging nozzle section was supersonic, condensation occurred past the throat in the supersonic flow. Stodola assumed that his laboratory steam was equally provided with condensation nuclei as a result of foreign impurities (see Section V, 1) as the steam in Him and Cazin's experiments, and when he obtained such contrasting results he repeated the Hirn-Cazin experiment. When he did this he also found condensation a t the saturation line. Stodola then suggested that the answer to this discrepancy lay in the different time scales of the experiments. In contrast to Him and Cazin's expansions (and his own repetition of them) which lasted for about 1/20 of seconds to a second, the steam in the nozzle case required only 4 x traverse the expansion zone. This time was too short to permit equilibrium conditions with respect to the foreign nuclei to be established. He also found that for increasing nozzle 'O length, everything else remaining 60 O O the same, the supersaturation decreased. Therefore, Stodola was 50 the first to recognize that rapid expansions in supersonic flow, leadY o, 40ing to large supersaturations, are WX processes very different from those encountered where relatively slow a 30 changes of the thermodynamic state 20 0 Binnie ond Woods (such as those in the atmosphere, expansion through valves, etc.) are 0 Yellot observed. Foreign nuclei such as 10 dust, etc. are unimportant in the case of high-speed condensation. -0 20 40 60 80 100 In the two volumes of Stodola's eo (%) classical work [31], these and other experiments are described, along FIG. 20. Supercooling in steam nozzles as with a discussion of much theoreta function of initial relativu humidity. ical information on the kinetics of droplets.
I
-
Many later investigations of supersaturation in steam nozzles were principally based on the observation of light scattering. From these experiments it was established that the order of the supercooling is about 45 < d Tk< 60 "C. as shown in Fig. 20, which is taken from the experiments
346
PETER P. WEGENER AND LESLIE M. MACK
of Yellott [48,491 and Binnie and Woods [50]. I t is interesting to note that in contrast to the condensation of water vapor in air (Fig. 13),the supercooling is roughly constant with increasing humidity. On the other hand, Stodola’s experiments in nozzles of varying length demonstrated that supercooling increases with increasing temperature gradient as in the moist air experiments.
FIG. 21. Comparison of moist air and steam experiments on the onset of condensation.
Although the saturation and condensa,tion temperatures were much higher in the steam experiments than in those with m d s t air, the saturation ratio can be correlated as shown in Fig. 21 as a function of condensation temperature for both a small moist air nozzle [lo] and a steam nozzle [50] as suggested by Head. For steam, the saturation ratio at condensation, P)k, changes but little with the condensation temperature. In fact, for the data shown we see that for steam 5 < P)k< 7 in a range of initial relative humidities of 90% > @ ,, > 20%. In contrast to this, 20 < fpk < 100 in the moist air nozzle. The reason for the difference between these two cases will become apparent in our later discussion of the kinetics of the process. Steam condensation again leads to non-uniformities of the pressure in a narrow zone in the flow direction [49, 501. From a practical viewpoint, it
CONDENSATION IN WIND TUNNELS
347
is found that the expansions in steam nozzles follow isentropes into the coexistence region and that, generally, for industrial steam and the typical geometries of steam-turbine nozzles condensation appears at a well defined limit. The practical limit of condensation in steam nozzles employed in the design of steam turbines is called the Wilson line [7]. The Wilson line is the locus of the supersaturated states at which a dense fog of small liquid drops forms in a steam nozzle. This line is located in an enthalpy-entropy (or Mollier) diagram below and approximately parallel to the coexistence line [3]. I t is shifted downward from the latter curve by an enthalpy difference of about 33 kcal/kg. I t is more difficult to follow the steam condensation process in detail thermodynamically than in the moist air case because even in the superheated state the thermodynamic state of the steam may be near the critical region where an equation of state different from the perfect gas law must be used [31]. For other than design purposes, owing to some sensitivity of condensation effects to purification of the steam and other factors, the Wilson line should be replaced by the “Wilson zone” according to Rettaliata [51]. The Wilson zone includes the supersaturated states in the range of about 3 < p k < 8. 2. Expansion
of
Nitrogen with and without Vapor Impurities
During the period of 1948 to 1950 it became increasingly apparent that dry air, when expanded in high Mach number nozzles, exhibited little or no supersaturation before its major components began to condense. In view of the mass of experience with moist air, where large amounts of supercooling were found, and in the light of kinetic theory calculations (Section V) on the condensation of pure nitrogen and oxygen, this was a t first a rather startling discovery. As a possible cause of the near absence of supersaturation in air, vapor impurities which served as condensation nuclei were soon suspected, and in order to obtain further information on this point, experiments with pure nitrogen were carried out. Four papers on this subject were published in succession in 1952. The nozzles used by these investigators ranged in size from about 2 to 5 cm exit height. Faro, Small, and Hill [52] used a conical nozzle, while Hansen and Nothwang [53] selected a contoured one. Arthur [54] and Willmarth and Nagamatsu [55] worked with plane diverging walls. The Mach numbers at which condensation was observed extended as high as nine. The size of the equipment was chosen so that commercially pure bottled nitrogen could be used. Traces of vapor impurities of this nitrogen can vary from batch to batch and also varied among the different investigators. Both chemical and mass-spectrograph analyses were employed [54, 551 to determine the exact composition of the working fluid. The average composition of the nitrogen used, e.g., by Arthur [54] over a period of two years varied in the following ranges:
348
PETER P. WEGENER AND LESLIE M. MACK
99.6 to 99.9% N,, 0.1 to 0.2% O,, 0.05 to 0.1% A, 0.002 to 0.0080/0 CO,, 0.0006 to 0.00270 H,O, traces of He, H,, and hydrocarbons.
I
Stort of Condensation
From Area Experimental Isentrope I
I
5
10
I!
Distance From Throat, xkml FIG.22. Static pressure distribution in a pure nitrogen nozzle from Arthur [54] Po = 8.83 stm, To = 294 OK.
Figure 22 shows a typical pressure distribution from Arthur’s nozzle. An isentrope calculated from the area ratio shows the large boundary layer displacement effect of such a small high-Mach number nozzle. Although condensation begins at the location marked in the figure, it would not be possible to discern this fact easily from static pressure measurements alone, as no obvious pressure non-uniformity is noticeable. Also shown as a dashed line, however, is an experimental, condensation-free isentrope which indeed deviates from the pressure measured in the presence of condensation. Such an experimental isentrope can be found either by increasing the supply temperature to avoid condensation $21 or by measuring the
349
CONDENSATION IN WIND TUNNELS
pitot pressure [54, 551 which is insensitive to condensation effects as will be shown in Section IV. The onset of condensation of the nitrogen can also be found by detecting nitrogen “fog” with light-scattering apparatus [55]. The relatively small isentropic core flow area (similar to that in small steam nozzles) makes experimental determinations of the isentrope mandatory. I oc
-7 t-
Triple point
50
0
M =5
10 c
CI,
I
5.0 v
P
0 Fora,Small and Hill 6 Hansen and Nothwang 0 Willmarth and Nagamatsu
1.0
0.5
0.2 4
40
60
80
loo
FIG.23. Thermodynamic state at onset of pure nitrogen condensation in nozzles and typical isentrope.
In fact, Faro, Small and Hill were able to change the Mach number from 8.5 to above 10 by changing the supply pressure from 10 to 200 atmospheres in the same fixed nozzle. A detailed discussion of Reynolds-number effects on Condensation phenomena in small nozzles is given by Arthur. The pressure and temperature a t which condensation first occurred in pure nitrogen is shown in Fig. 23 as evaluated from the results of all four investigators. A sample isentrope is given to indicate the fact that appreciable supersaturation was observed. The vapor pressure curve is from (1.9) and Table 2,
350
PETER P. WEGENER AND LESLIE M. MACK
and is in good agreement with Aoyama and Kanda’s [15] experimental points. All isentropes intersect the equilibrium vapor pressure curve below the triple point, which suggests that the condensate might be solid nitrogen. More will be said about this point later. Considering the wide thermodynamic range of these experiments, the dubious nature of the determination of the onset of condensation, and the variable purity of the nitrogen, the results show remarkably little scatter. (Hansen and Nothwang’s one experiment was made in a contoured nozzle, possibly with a different temperature-gradient history of the flow.) The average supercooling from Arthur’s experiments is ATk = 16.4 OK, and is practically constant for his restricted range of initial conditions. Faro, Small and Hill, a t higher Mach numbers and in a larger pressure and temperature range, find on the average AT, = 22 OK.
FIG.24. Mach number at onset of pure nitrogen condensation as a function of Mach number of equilibrium condensation.
When we determine the Mach number of condensation and the Mach number of equilibrium condensation (1.17) on a common basis, we arrive a t Fig. 24, which shows a condensation delay of about 1 to 2 in Mach number. According to our remarks in Section I, this is a practically important supersaturation and results in an appreciable reduction in the heating of the nitrogen supply which is required to achieve a given Mach number. Unfortunately, however,
351
CONDENSATION IN WIND TUNNELS
it is impractical to operate a nitrogen tunnel on a large scale. Systematic experiments on the effect of the temperature gradient on AT, are not available. Willmarth and Nagamatsu changed the area ratio for one experiment by a factor of two, and found that the duration of the expansion up to the point of condensation varied from 2 to 5 x seconds, while the supercooling remained essentially unchanged.
I
D
FIG. 25.
The effect of water vapor addition on supersaturation Arthur [54]. Po = 8.34 atm, To -. 290 “K.
in nitrogen from
In the reduction of all these experimental results the “onset of condensation’’ was taken as that point where a deviation of the pressure from the isentropic expansion could first be detected. This is a definition of interest to the aerodynamicist, since it means that at this point the heat release due to condensation has affected his measurements. Actually, condensation may well be present ahead of this point, but the amount condensed will be small. Light scattering observations are not accurate enough in establishing the actual location of the first phase transition in a nozzle, and in all our observations we are forced to base our deductions of supercooling, etc. on the occurrence of a measurable pressure change in the fluid.
352
PETER P. WEGENER AND LESLIE M. MACK
In summary, we find that pure nitrogen exhibits a supercooling from 16 to 22 "K when expanded in small nozzles. These values are below those
observed in steam or moist air nozzles. Condensation is not marked by strong non-uniformities in the flow field, and it is more difficult to detect than in the case of moist air.
2 50
200
150
-
zN
-
4
loo
50
C
3
0
-
FIG. 26. The efiect of carbon dioxide addition on supersaturation in nitrogen. Po = 8.33 atm, To 294 OK.
We now turn to the effect of traces of vapor impurities on the condensation delay in pure nitrogen. Those vapors are of greatest interest which are naturally present in air and whose vapor pressure curves are crossed a t higher temperatures than those of the major constituents (Fig. 6). We assume that these vapors condense before nitrogen reaches its Mach number of equilibrium condensation and that the small droplets or solid aggregates, as the case may be, serve as centers of condensation for the nitrogen. If this is true, then the injection of small amounts of vapor impurities into the supply should decrease the supersaturation which is observed with the pure vapor. Figure 25, which is taken from Arthur, shows this effect clearly for the addition of water vapor. Even at the small mixing ratio of about
353
CONDENSATION IN WIND TUNNELS
x = 0.5 g/kg, all supersaturation of the nitrogen is eliminated. (This is still sufficiently dry so that for most experiments no water vapor condensation effects would be noticeable.) The corresponding relative humidity in the supply is also shown. It is rather high, owing to the fact that the supply pressure is about 8 atmospheres. In Fig. 26 we see the similar effect of carbon dioxide addition as measured by Willmarth and Nagamatsu, and Arthur. The scatter in these measurements is not surprising, since it is difficult to meter accurately such small amounts of added gas. In fact, Arthur believes for this and other reasons such as the accuracy of the determination of the onset of condensation, that the saturation ratio a t condensation, vk,may not be known better than to about &40yo. However, Fig. 26 clearly shows a n effect similar to that described for water-vapor addition. It is interesting to note that the nitrogen saturation ratio a t condensation is still about q k = 40 (& 16) at the point where 0.03% of CO, was added to the supply. This addition corresponds to the normal CO, content in air. We therefore find that the addition of small amounts of H,O and CO, is very effective in reducing the condensation delay in otherwise pure nitrogen. This is not true if argon is added to nitrogen, as was found by Arthur. In the range of argon addition from 0.31 to 4.3% by volume, the supercooling was only reduced about 2 OK. We shall see in Section V that the mere availability of droplets or crystals of some substance does not necessarily mean that they are effective condensation nuclei for all vapors.
3. Expansion of Air
The first experimental attempt to generate a high Mach number air stream was made by S. Erdmann in 1943/44 a t the supersonic tunnel laboratory a t Peenemuende (unpublished, described in [4]). In a 30 x 40 cm uniform-flow nozzle designed for M == 8.83, and with Po 60 atm. and To 280 OK, he measured a centerline Mach number of 8.5 in the test section with a pitot tube. No apparent “condensation shocks” were seen in schlieren photographs, although a “fog” was visible to the eye. We now know that air condensation was present in the flow under the circumstances, and we have become familiar with the insensitivity of the pitot Mach number to condensation effects. This Mach number is evaluated from
-
-
where Po’ is the pressure measured by a pitot tube. Implied in (3.3) is a flow process which comprises (a) an isentropic expansion at constant stagnation enthalpy from rest to Mach number M , (b) a normal adiabatic shock, and (c) an isentropic compression to rest behind this shock. We shall see in the next section that the pitot pressure is nearly unaltered a t a given
354
PETER P. WEGENER AND LESLIE M. MACK
location in a nozzle when condensation is present, even though the flow process in that case is different from the one just described. Therefore, the actual flow Mach number with condensation present is not obtained from (3.3). Historically speaking, it was initially presumed that air condensation would produce strong flow non-uniformities similar to those experienced with moist air and steam. Also kinetic calculations (see Section V) led to the expectation of considerable supersaturation. Many of the first experimental investigations were directed towards a demonstration of the absence or presence of air condensation. However, experimental evidence which was evaluated on the basis of the above expectations was apt to be misleading in the sense that condensation could be easily overlooked if it were a gradual process. A new factor in wind tunnel operation, the variable supply temperature, had to be introduced [56], and experiments were directed towards determining the sensitivity of conventional measurements of supersonic flow parameters to air condensation. The first published demonstration of the fact that air condensation occurs a t or near the equilibrium saturation Mach number M,, and does not exhibit strong flow non-uniformities came from the group a t the NACA Langley Laboratory directed by Becker [57]. Similar results were simultaneously obtained a t other laboratories. We shall describe here a few typical results taken from many investigations but evaluated in a common frame of reference. In addition to pitot tube measurements, static pressures can be measured by the use of probes or sidewall orifices. It is then customary to determine the Mach number from (1.14). This formula was derived on the assumption that the flow isisentropic withconstant stagnation enthalpy between reservoir and point of measurement. From our previous considerations, we can expect M , determined from (1.14), to be erroneous if air condensation and heat release are present. Flows with condensation (or evaporation) cannot have constant stagnation enthalpy. However, the flow of a gradually condensing vapor may still be isentropic (see Section IV, 4) under certain circumstances. Due to the heat release alone, however, we expect the static pressure to be higher a t a given effective area ratio in the nozzle than the static pressure at the same location without condensation. The static pressure of the flow undergoing an isentropic expansion with constant stagnation enthalpy can be expressed in terms of potential flow area ratio from (1.14) and the well known relation
where A* is the nozzle throat area. Deviations of the measured static pressure from that predicted from the geometry should be indicative of condensation effects.
355
CONDENSATION I N W I N D TUNNELS
-
This is shopn in Fig. 27, which is taken from the experiments by 7. McLellan and Williams [58] in a nozzle of about 28 cm exit height at M
0.OOOII 0
'
'
10
'
'
20
'
' I 30 40 x (cm)
'
I
50
'
I 60
N 7 nozzle at Po = 29 atm from McLellan and Williams [58].
FIG.27. Static pressure distribution in a M
The static pressures for the low supply temperature do not even approach the potential-flow design value of the two-dimensional uniform flow nozzle downstream from the throat. Mc as computed from (1.17) is indicated, but no strong pressure non-uniformities are noticeable. With To increased to a value where M , is larger than the design Mach number, we see that the expansion follows the potential flow values more closely. The difference between the two measured expansions is that, with the low supply temperature, an increasing portion of the air is condensed as the expansion proceeds. The sensitivity of static pressure to air condensation is also scown in Fig. 28, taken from Wegener, Reed, Stollenwerk and Lundquist [59] where, at a fixed location near the exit of a 12 x 12 cm plane-walled diverging nozzle, the static pressure was measured over a range of supply temperatures.
356
PETER P. WEGENER AND LESLIE M. MACK
A definite discontinuity of the slope in the pressures indicates that for To> 140 "C the static pressure is approximately constant, and the flow at this location is free of air condensation for temperatures above 140 "C.
-
FIG. 28. Static-supply pressure ratio as a function of supply temperature at a fixed 7.6 nozzle from Wegener et al. [59]. p , = 7.1 atm. nozzle location in a M
(The slight decrease of the pressure ratio a t the higher temperatures is due to thermal distortion of the narrow, uncooled nozzle throat.) The Mach number a t To 140 "C is about M = 7.1, and if condensation actually occurred a t M = M,, the supply temperature would have to be raised higher than 140 "C to eliminate it, which shows that some supersaturation was present. In contrast to the static pressure, the pitot pressure a t the same fixed location remains relatively unchanged as shown in Fig. 29. In hypersonic nozzles it is not possible, as seen in the previous figures, to compare the geometrical area ratio directly with the measured pressure. The boundarylayer displacement thickness is large at these high Mach numbers and relatively low Reynolds numbers. Therefore, the effective area ratio is quite different from that given by the nozzle geometry. If we accept, for the moment, the insensitivity of the pitot pressure to condensation effects, a pitot pressure survey can be used to obtain the condensation-free expansion. This assumption also implies that the displacement thickness of the usually turbulent boundary layers is not seriously affected by condensation in the free stream in the range of supply temperature under discussion, as shown by Wegener [go].
357
CONDENSATION I N W I N D T U N N E L S
A third method for the determination of the Mach number is to eliminate (1.14) and (3.3),which results in
Po from
0.016
-
9 0.014-
8~
0 0 0
8
o0
0 0
o ~ o O o o O 0
0 0
oo
4
8 0
PO
0.012
B o o
-
0
I
1
I
0
50
I00
I
I50 To(OC)
I
I
I
200
2 50
300
-
FIG.20. Pitot-supply pressure ratio as a function of supply temperature a t a fixed nozzle location in a M 7.6 nozzle from Wegener e t al. [59]. p , = 7.1. atm.
the well known Rayleigh pitot formula. The advantage of (3.5) is that the past history of the flow to the point of measurement is immaterial in the determination of M . However, in flows with condensation, this M must be erroneous since adiabatic flow is assumed across the shock wave. When the Mach numbers from (1.14), (3.3), and (3.5) a t a given point in a nozzle are the same, the flow is free of condensation. If condensation is present the three Mach numbers must be different. The Mach number derived from (3.5) is the most affected since the static pressure, which is significantly changed by condensation, is a larger fraction of the pitot pressure than of the supply pressure. Therefore, in the divergence of the Mach numbers determined from the three different pressure ratios, we have an accurate indication of the onset of condensation. This is shown in Fig. 30, which is taken from Hansen and Nothwang's experiments [53] in a 25 x 3Gcm tunnel. The Mach numbers are shown as functions of the Mach number determined from the nozzle geometry (3.4), and there is quite definitely a point in the expansion where the three indicated Mach numbers diverge. If we again assume that the pitot Mach number (3.3) is representative of the free stream expansion both with and without condensation, we see that it is close to the potential flow curve, as expected in a relatively large tunnel. The point
358
PETER P. WEGENER AND LESLIE M. MACK
where the three Mach numbers diverge is very shortly after M , (1.17) has been reached. At this point air condensation begins, and any further conventional Mach number determinations are physically meaningless. In the next section we shall discuss the question of the actual Mach number of a condensing flow.
FIG. 30. Conventional Mach number determination in the presence of air condensation from Hansen and Nothwang [53]. Po = 6 atm, To = 288 OK.
Since the pitot-static pressure ratio is the one that is most affected by condensation, Buhler [61] suggested that this ratio be plotted against the insensitive pitot-supply pressure ratio. In this manner only actual measured values are used, and the onset of condensation is shown very clearly. A typical example is given in Fig. 31, which is taken from the experiments performed by Grey in a 13 x 13 cm tunnel [62]. In this nozzle the expansion was free of condensation up to a Mach number of 5.4, which is larger than M , = 4.7. At M = 5.4, the experimental points deviate from the condensation-free isentrope, and after a collapse-like region not unlike the moist air case, the expansion proceeds a t a higher pressure level. I n contrast to the experiments of Figs. 27 and 30, those of Figs. 28, 29, and 31 show some degree of supersaturation. The only obvious difference between these sets of results is the tunnel size, with the larger nozzles exhibiting
359
CONDENSATION IN WIND TUNNELS
little or no supercooling. In cases where supersaturation is observed, it is still permissible to determine all flow properties up to the onset of condensation from pressure measurements and conventional flow tables. This fact permits us to compare later experimental observations on supercooling without resorting to a discussion of expansions with condensation present.
0.04
0.0:
PIP:,
0.02
-
M=6.0
I
0.03
I 0.04
1 0.05
I
I
0.06 0.07
I
I
0.a 0.09 0.10
P h FIG.31. F’itot and static pressure measurements through supercooling and condensation zone from Grey [62]. Po = 5 atm, To = 300 OK.
We shall see in Section I V that the angle of a shock produced by a flow deflection is changed when condensation is present. This has been shown for wedges and cones by several investigators [59, 62, 63, 641, and in Fig. 32 we present results obtained by Stever and Rathbun [64]. In these experiments a wedge was held fixed in a nozzle of 5.7 x 7 cm exit section. At the lower supply temperatures air condensation was present, and for increasing temperature the shock angle decreased until it reached a constant value. For To> 485 O K no condensation effects were present. Finally, many observations of light scattering have been made to detect the presence of air condensation [53, 57, 58, 59, 63, 64, 651. Stever and Rathbun used this technique extensively to determine the onset of condensation as a function of varying supply conditions. The method depends on a small ray of collimated light, and is usually less accurate than pressure but it offers the practical advantage of rapidity measurements in finding Mk,
360
PETER P. WEGENER AND LESLIE M. MACK
and simplicity of operation. Quantitative investigations on the nature of the condensation process were made by Durbin [65] and McLellan and Williams, and we shall discuss these results later.
1
cn
..
il6-
; ; - ISIS 'c)
P 14: o 14P 0
a? 0 o
{ 1312-
-G No condensation
I
I
1
1
I
I
I
FIG. 32. Measurement of shock angle on a 11' included angle wedge as a function of supply temperature in a M = 6.7 nozzle from Stever and Rathbun [64]. 9, = 68.6 atm.
The preceding observations have shown that air condensation is likely to occur when the Mach number of equilibrium condensation is exceeded. The air condensation takes place gradually, just as observed in nitrogen, and is a process different from that of moist air and steam condensation. We shall next attempt to estimate the thermodynamic state at the onset of condensation, again choosing a state where the stagnation enthalpy has changed sufficiently to show deviations of the three Mach numbers determined from (1.14), (3.3), and (3.5). We have tried to apply similar criteria of divergence of these Mach numbers as indicative of condensation for six different series of experiments. One series [64] based on detection of condensation by light scattering will also be shown. After M Rhas been determined in this fashion, supercooling, pressure, and temperature a t M kcan be found from isentropic flow tables. Figure 33 shows that in general M k> M,, and the difference increases with increasing Mach number. A substantial Mach number increase without condensation beyond M , occurs only a t the highest Mach numbers. The experiments shown in this and the next two figures were made in a large range of environmental conditions. The nozzle heights varied from 2.5 to 36 cm, and the distance from the throat to the point a t which M , occurred was very dependent on the nozzle design and size. I n two cases plane diverging walls [54, 601, which resulted in a continuous expansion, were used. Some observations were made a t a fixed location in the flow and the supply temperature was varied, while in other cases the supply conditions remained constant and the observation point was moved along the centerline or wall. Supply pressures ranged from 3 to
36 1
CONDENSATION IN WIND TUNNELS
70 atmospheres, and supply temperatures from 285 to 760 O K . The highest pressure of about 70 atm. was used by Stever and Rathbun, and at their lowest temperatures the application of the perfect gas law and y = constant is not strictly permissible (Table 1). However, the uncertainty of determining the onset of condensation accurately by light scattering in a small nozzle
0 0
A
v 0 0
A A
-0
5
Arthur Grey Hansen and Nothwang Kubota Mckllan and Williams Stever and Rothlwn
10
MC
FIG.33. Mach number at the onset of air condensation as a function of Mach number of equilibrium condensation from experiments in a wide range of environments.
prompted us to evaluate their data like the others. The moisture content of the air supplies was low compared with that in a conventional supersonic tunnel, but it varied greatly. In some cases the air had been stored in highpressure vessels at, say, 200 atmospheres which resulted in very low dewpoints; in other cases, continuously operated power-plants with driers in the circuit were employed. Even the carbon dioxide content differed, since in compressing air the normal fraction is reduced by the increased solubility of carbon dioxide in water at high pressures [58]. The highest purity with respect to traces of vapors was probably obtained by Arthur [54],
362
PETER P. WEGENER AND LESLIE M. MACK
who produced “synthetic” air by mixing 0, and N, from bottles. Various types of electric heating equipment may perhaps have produced foreign nuclei by oxidation of the heating elements; powdered desiccants may well have been present in some plants: and in some experiments oil vapor from the flow machinery was present. The uniformity of the supply-temperature distribution ahead of the nozzle throats may have been poor in some experiments. Finally, considering that it is difficult to apply uniform judgment to the raw data in the estimate of the onset of condensation, we are not surprised to see the scatter exhibited in Fig. 34, where the pressure and
‘F
/
A
10
0
0 0
n
0 I .o
0.31: 0
0.51
0
0
0.2
0.11
I 35
I 40
/
/ I 45 T(”K)
V Arthur
IGrey 0 HonsenondNothwang 0 Kuboto A McLellan and Williams A Steverand Rothbun 0 Wegener et ol I
I
50
55
a
FIG.34. Thermodynamic state at the onset of air condensation from experiments in a wide range of environments.
temperature at M R are indicated. The corresponding isentropes are not shown (they are much steeper than the coexistence line). When we determine the supercooling for the same group of experiments, we arrive a t Fig. 35. The fact that M , was determined graphically from Fig. 5 contributes to the scatter. Also, with the inaccuracv of the experimental Mb.the cluantitv
363
CONDENSATION I N W I N D TUNNELS
ATk as determined from (2.2) and (1.15) is only known within a few degrees. For 5 < M k< 8, the range that includes the largest number of experiments, no trend is discernible, and for all observations 0 < AT, < 24 "C. The largest supercooling is below that observed for moist air (Fig. 19) and steam (Fig. 20) and is of the order of that found for pure nitrogen. 0
V Arthur
0
-
0 A A
0
Grey Hansen and Nothwang Kubota McLellan and Williams Stever and Rathbun Wegener et at
A
0 O.
0
0
0
0.
0 0 1
1
I
I
1
-
1
5 -
I
I
I
I
I
10
Mk FIG. 35. Supercooling as a function of the Mach number at the onset of air condensation from experiments in a wide range of environments.
It is difficult, if not impossible, to correlate the experimental evidence on air condensation in other than qualitative terms. The supercooling is likely to be a function of several parameters, just as in our previous studies. We expect the temperature and pressure at condensation, the nozzle temperature gradient, and the availability of surfaces of condensation to be dominant factors. I t is just these three conditions that varied over such a wide range in the experiments shown in Figs. 33, 34, and 35. I n general the larger tunnels showed less supercooling. The temperature-gradient effect appears to be the only variable that can be studied separately,
364
PETER P. WEGENER AND LESLIE XI. MACK
thanks to the experiments of Kubota [66]. In Fig. 36 we show the supercooling observed by him in two nozzles, one short and one long, with the same exit height of about 13 cm, but with throat radii of about 0.3 and 8 cm, respectively. Fortunately, both nozzles were tested in the same
0 4
5
7
6
8
9
Mc FIG. 36. Effect of temperature gradient (or nozzle length) on supercooling of air for a 13
x 13 cm nozzle a t fixed initial conditions from Kubota [66].
powerplant under presumably identical conditions of traces of vapor impurities. We see that the short nozzle with the small throat radius exhibits a much higher supercooling. This is comparable in effect, though not in magnitude, to our experience with moist air subjected to large temperature gradients. In general, however, as a result of the experiments with nitrogen and traces of vapors, we expect the impurity level to be the dominant factor governing any possible supersaturation in the case of air. In fact, if the impurity level is high enough to produce condensation right a t the equilibrium Mach number, we might use the supersonic tunnel as an instrument to study the condensation line of air. In qualitative terms, the experimental evidence on air condensation suggests that for the larger hypersonic wind tunnels it will be necessary to preheat the supply air to a temperature such that the highest Mach number expected in the flow field about some model in the test section is less than M,.
CONDENSATION I N W I N D TUNNELS
365
Iv. DIABATICFLOWS A N D THERMODYNAMICS OF CONDENSATION In this section we investigate to what extent one-dimensional fluid mechanics and thermodynamics can describe the condensation phenomena without recourse to kinetic considerations. When condensation and vaporization occur in a flow, we have an example of what is called diabatic flow. Many authors [e.g., 67, 68, 69, 701 have analyzed one-dimensional diabatic flows; Oswatitsch [34] has derived a set of equations specifically applicable to the condensation of steam and of water vapor in air. We derive here a different form of these equations which can be applied either to the condensation of a vapor trace, or a large amount of pure vapor. However, even at the outset we make the simplifying assumptions that the liquid (or solid) condensed mass is uniformly distributed throughout the gaseous components and has the same speed and temperature as the stream. The equations which result from the application of the conservation laws of mass, momentum, and energy, and the equation of state are not sufficient. The actual computation of a condensing flow requires an additional equation for the condensed mass, or else an experimental measurement of one of the flow variables. Even when the equations are supplemented in this manner to permit a particular flow to be calculated numerically, they do not allow many general conclusions to be drawn. For the purpose of a general discussion we consider what may be thought of as two limiting situations. The first of these is shown in a $ - v diagram, Fig. 37 (for simplicity we have drawn the diagram for a pure vapor). The flow continues along a dry isentrope into the coexistence region, until a t E the supersaturated state collapses instanteneously to state B. This picture leads to the analysis of the condensation shock, which can be carried out in some detail, but does not lead to a complete solution of the condensation problem as the location of the shock and amount of heat released must be specified separately. The other limiting situation is the complete absence of supersaturation. Condensation starts a t the coexistence line (point C, in Fig. 37), and the flow continues along the saturated isentrope, C, - F,. The analysis of this saturated isentropic, or equilibrium, expansion is particularly simple and valuable, as fluid mechanics and thermodynamics are sufficient to describe the flow completely. Most actual condensation processes lie somewhere in between these two idealized situations. For instance, the supersaturated state may start to break down a t D. The whole collapse region then extends to the point where the saturated isentrope C, - F , is reached. Here the vapor is again saturated, and if thermodynamic equilibrium is maintained, the subsequent flow is another saturated isentropic expansion a t a higher entropy level than C, - Fl. When the vapor behind the condensation shock, point B, is saturated, that flow may also continue along an isentrope C, - F, a t a still higher entropy level than C, - F,.
366
PETER P. WEGENER AND LESLIE M. MACK
When a shock wave occurs in a condensed flow, the temperature rise behind it vaporizes some or all of the condensate. As a result the conditions behind a shock differ from their value in dry flow. The effect of this vaporization on the pressure measured by a pitot tube is of particular interest. Also, the angle of an oblique shock will change, which affords an experimental tool for the investigation of condensed flow. These matters are briefly discussed in the last part of this section.
I
v = I/P FIG. 37. Schematic
p -u
diagram of condensation processes.
1. Derivation of One-dimensional Condensation Equations We derive here a set of one-dimensional, steady-flow equations which can be applied to either the case of an inert carrier gas with a condensable component, or to a pure vapor (or a mixture of vapors) which itself condenses. We assume that the gaseous components are calorically and thermally perfect and allow the molecular weight and specific heat of the mixture t o change as condensation takes place. There are five equations in all: the equations of continuity, momentum, energy, and state, and the condensationrate equation. The latter, however, can only be formulated here; its complete derivation must await Section V. Three .terms enter the continuity equation: the mass of inert carrier gas, ma, which passes a given location in the nozzle in unit time; the mass
CONDENSATION IN WIND TUNNELS
367
per unit time of the condensible vapor, mu; and the mass per unit time of the condensed phase, mL. The condensed phase can be either liquid or solid, but for convenience we shall usually refer to it as if it were liquid. The sum of the three mass rates must be constant, which gives (4.1)
ma
+ m, + mL = m = constant
Since the mass of inert carrier gas remains constant, and the condensate grows at the expense of the vapor, we have dma = 0, and dm, = - amL. For water vapor in air, all three terms are present, but for steam and air condensation there is no inert carrier, and ma = 0. The density of the liquid phase is much larger than that of the vapor phase, and we can, to a very good approximation, neglect the volume occupied by the liquid. Further, we can define a density pL’, where the condensed mass is referred to the same volume as the carrier gas and vapor. Equation (4.1) implies for the density of the mixture at any cross section (4.2)
+ pv + PL’.
p = pa
Using the density p we can write the continuity equation in terms of the velocity, w , and area, A, pwA = m,
(4.3) or, in differential form, dp
- + - + -dw= o .
(4.4)
P
W
dA A
The equation of state for a mixture of two perfect gases is (4.5) When we introduce the molecular weight of the mixture, p, we can write (4.5) as
For ,u we have (4.7)
1ma p ma+m,
-
-1+ ,ua
mu
1 ma+mu pv
where we have replaced the ratios of the densities by ratios of the masses.
368
PETER P. WEGENER AND LESLIE M. MACK
Let us consider a fixed mass, m, of mixture. The sum of the vapor and liquid mass is constant and equal to the vapor mass in the nozzle supply condition, muo. Then, from (4.1) we have
or, (4.9)
In (4.9) we have introduced the initial specific humidity, w,,, from (2.4), and the mass fraction of condensed phase, g = mLlm. With reference to Fig. 37 we recall from thermodynamics that g at, say, point B equals the ratio BC’IAC’. By (4.2) we can write p a pv in terms of p and g to obtain
+
(4.10)
or, with (4.9), (4.11)
P=P
(7+ u) RT.
1 -0 0
Pa
The differential form of this equation of state is (4.12)
With the assumption that the condensate moves with the stream velocity, the momentum equation is simply (4.13)
- pwdw = ap.
The energy equation can be obtained from the first law of thermodynamics, which, applied to a unit mass of the mixture, is (4.14)
0 = du
+ pav.
The left hand side is zero since there are no external heat sources. In terms of the specific enthalpy, and with (4.13), we have (4.15)
0 = d (h
+) ;
.
CONDENSATION IN WIND TUNNELS
369
When we apply (4.15) to the mass m of the mixture and write out the enthalpy in terms of the enthalpies of the separate components, we have (4.16)
1
+ maha+ m,h, + mLhL = 0.
I n general, the liquid and vapor are a t different temperatures. When the two phases are in thermodynamic equilibrium the enthalpy difference of the phases is simply equal to the heat of vaporization [e.g., 71. However, when the temperatures are different there is an additional internal energy term for the liquid phase. Therefore, we have for the enthalpy of the liquid, (4.17)
h L = h,
-L
-
cL(T,- TL),
where cL is the specific heat of the liquid. Since the last term is generally only a few percent of the other terms, we shall consider the liquid and vapor to be a t the same temperature in order to simplify the analysis. Substituting (4.17) in (4.16), we then obtain (4.18)
or (4.19)
where
the specific heat of the vapor mixture in the supply. Actually, the liquid phase is present in the form of droplets, and a mixture of vapor and droplets is a dispersive medium for the propagation of sound. Therefore, the speed of sound is a function of frequency. Oswatitsch [71] has made a detailed analysis of sound propagation in fog, and Buhler, Jackson, and Nagamatsu [72] have considered the same problem for a pure vapor fog. In both cases the droplets act as inert particles in the limit of a very high frequency sound wave, and the sound speed is given by the perfect gas formula in terms of the vapor density. It is this sound speed which we shall introduce here. We have, therefore, (4.21)
370
PETER P . WEGENER AND LESLIE M. MACK
and we define the Mach number in terms of this speed of sound. Then (4.19) becomes y _ H _
(4.22)
CP,
M 2 -w
P
d(gL)
dT
dw +
T
-
~
CPoT
= 0.
The specific heat of the vapor mixture is, from (2.15), (4.23)
(ma
+ m&p
+ mucpu,
= macpa
or, in terms of the initial specific humidity and the mass fraction of condensate, (4.24)
With RIP
= cp - cu,
(4.23) becomes
CP dw dT -(y-1)M2-+----=00. w T cpO
(4.25)
d(Lg) cpoT
The four equations, (4.4), (4.12), (4.13),and (4.25) involve six unknowns, T , w ,g, and A . Considering the area as given and g related to the other variables in a manner yet to be determined, we can solve for the pressure, density, velocity, and temperature differentials in terms of dA and dg. The results for d p and dp are
9, p,
9-
YM2 [yM' - ( 1 - g) - CP ( 1 - g) (Y - 1 ) M 2 CP0 - gL'
(4.26)
x dP
I-?+[ 1
1
L (CP,
Pv 1 - g
-
1 - ( 1 - g) CP,
- .@'
(4.27)
x
{-
Cp(1
[yM2 -
-
1
g) (y - 1 ) M 2 d A ]Af[(cptF;:))T
where L' = dL/dT. The velocity differential can be found from the continuity or momentum equations, and the temperature differential from the energy equation or equation of state.
CONDENSATION IN WIND TUNNELS
371
The stagnation temperature can be defined in the usual manner and is given by (4.28)
~p,To= cp,T
+ 21 w2. -
When we introduce (4.28) into the energy equation (4.19) we can integrate to obtain (4.29)
where To, is the stagnation temperature before condensation starts. The stagnation pressure can also be defined as usual from the isentropic relation (1.10). From an experimental viewpoint, the static pressure distribution of the flow is easily measured when it is assumed that the pressure at the nozzle wall is equal to the free-stream pressure. Then if the area distribution is known in some manner, as described in the previous section, all of the other flow quantities can be obtained from the preceding equations once the pressure has been measured. In fact, when we solve for dg, we find
dg =-
1
Therefore, with a measured static pressure distribution, known area and heat of vaporization, the entire flow can be computed from (4.30), (4.27), (4.4), and (4.12) by a step-by-step numerical calculation. This has been carried out for the experimental results given by Wegener [29] for moist air, and is shown in Fig. 16. The heat of vaporization was assumed constant in the small temperature interval under discussion, and equal to 688 cal/g, which is a value suitable for the vapor-ice transition. Further, since the amount of water vapor in this experiment was small, the molecular weight and specific heat were assumed constant and equal to their values for air alone. The missing equation for g can be obtained in two ways. The simpler one is to assume that thermodynamic equilibrium exists a t all times, with the result that the Clausius-Clapeyron equation (1.7) can be used. This leads to the analysis of the saturated isentropic expansion which is presented in
372
PETER P. WEGENER AND LESLIE M. MACK
Section IV, 4 for a pure vapor. However, in many instances some supersaturation is present, and the assumption of thermodynamic equilibrium is not correct. We must then turn to the kinetics of droplet formation and obtain what is called the condensation-rate equation, which relates g to the other flow variables. First we formulate the condensation-rate equation where condensation occurs as a result of self-nucleation. The number of droplets of critical size which are formed per second in volume Ad5 at position 6 in the nozzle is
where J is the nucleation rate, that is the number of droplets of critical size produced per unit time and volume. Once a droplet is formed which is able to grow, its further growth is determined by a droplet-growth law, G ( t , x ) . The growth law expresses the mass increase of a critical-size droplet as it passes downstream with the flow. We consider each droplet to grow separately, and not to coalesce with other droplets. The quantity G(E,x) is the mass a t position x of the droplet which was formed at 6. Hence, the mass fraction of condensed phase a t position x must be
-m
This is the condensation-rate equation 1341; it sums up the contributions to g from all critical-size droplets formed upstream of x. When self-nucleation is unimportant and condensation takes place on foreign nuclei, the number of droplets does not change through the condensation region, when it is again assumed that the droplets do not coalesce. The condensation-rate equation simplifies to
where N is the number of foreign nuclei per cm3, and G ( x ) is the mass of vapor condensed on a nucleus a t position x. In order to have such a simple equation, N must refer to some average-size nucleus. Thus, with G and J expressed in terms of the other flow variables, and the area known, we are able to calculate the whole flow when the condensation occurs as a result of self-nucleation. This is true for steam and water vapor in air as will be seen in Section V. For the condensation of air, where the foreign nuclei are important, N must be determined in addition. When these nuclei are themselves the result of condensation of vapor impurities this means, for a complete calculation from first principles, that the computation of another condensation process must be carried out previously.
CONDENSATION I N W I N D TUNNELS
373
We can also write a continuity equation for the number of droplets per cm3, N , present in the flow. The difference between the number of droplets flowing into a volume A d x in unit time and the number flowing out of it, must be equal to the number created in the volume per unit time, provided that the droplets grow only a t the expense of the existing vapor and do not combine into larger droplets as a result of collisions. Thus, neglecting the volume occupied by the droplets, we have (4.34)
~ ( N w A=) J A d x ,
where J is the nucleation rate. When N refers to foreign nuclei, J and (4.34) gives N = constant, as it should.
=
0,
2. Analysis of the Condensation Shock
We have seen in Section I1 that the supersaturation that occurs during the expansion of steam or moist air collapses in a short distance in the flow direction (e.g., Fig. 11). The disturbance marking this collapse is customarily called a condensation shock. Because of the short distance involved, the area change in the nozzle is usually small. Also, the mass of condensate is often small enough so that the mass of the gaseous part of the flow can be considered unchanged. With these simplifications, the only effect of condensation on the flow is the release of the heat of vaporization, and we can simplify the analysis to the consideration of a discontinuity front with heat release. This type of analysis, which is called condensation-shock theory, was first presented by Hermann [25], Heybey [32], and Oswatitsch [35], and later by many authors, including Lukasiewicz and Royle [27], Charyk [73], and Samaras [74]. The governing equations, which are the equation of state, and the three conservation equations of mass, momentum, and energy, can be written in integrated form as (4.35) (4.36) (4.37)
PlWl=
Pl + P1W:
P2W2
= P2
+
2 P2W2
(4.38)
where q = Lg, the heat added, or subtracted, per unit mass. Subscripts 1 and 2 refer to conditions in front of and behind the shock respectively.
374
PETER P. WEGENER AND LESLIE M. MACK
There are several possible solutions to these equations, corresponding to the various cases of supersonic or subsonic initial and final flow, and heat release and heat extraction. Two graphical representations are very useful in organizing the various solutions: the Rayleigh lines and the Hugoniot lines. A Rayleigh line connects states that satisfy the continuity and momentum equations, and it is specified when the mass flow and the initial thermodynamic state (pl,pl) are given. A Hugoniot line is obtained by eliminating the velocity from (4.36)-(4.38) ; thus it represents the possible thermodynamic states which can be reached from the given initial state.
v/v, FIG.38. Rayleigh lines and Hugoniot curves in
p - v diagram.
The intersection point of the Rayleigh and Hugoniot lines in a particular case satisfies all of the flow equations and initial conditions and gives the actual final state.
375
CONDENSATION IN WIND TUNNELS
From (4.36) and (4.37) we have for the Rayleigh line
1 P - P = - p2l 2q =- y k M : ,
(4.39)
v - v,
V1
where v = l/p, the specific volume. The subscript 2 has been dropped since the final state is as yet unspecified. We see that the Rayleigh line is a straight line in a P - v diagram, and for all M,, except zero and infinity, has a negative slope (Fig. 38). Therefore, a pressure increase is always accompanied by a density increase, and a pressure decrease by a density decrease. In an enthalpy-entropy diagram, the Rayleigh line always has the general appearance shown in Fig. 39, drawn for a supersonic initial flow, M,. The point of maximum entropy correspnds to M = 1, the lower branch to supersonic Mach numbers, and the upper branch to subsonic Mach numbers. When M , > 1, heating leads to a lower supersonic Mach number, and cooling to a higher Mach number. Correspondingly, when M,< 1, heating leads to a higher subsonic Mach number, i.e., heating always drives M towards unity.
-1.6
-1.2
-0.8
-0.4
0s-sO.4
I
0.8
1.2
1.6
2.0
R
FIG.39. Rayleigh lines and Hugoniot curves in h - s diagram.
We see immediately from Fig. 39 that there is a maximum amount of heat that can be released, and this amount of heat results in the maximum possible entropy increase and sonic flow downstream from the shock. Since all of our experience has been that condensation shocks are observed only a t supersonic speeds, with supersonic or sonic flow downstream, it is the lower branch of the Rayleigh line which represents the condensation-shock
376
PETER P. WEGENER AND LESLIE M. MACK
solutions. A normal adiabatic shock causes transition from the lower to the upper branch (point S in Fig. 39). Subsequent heating or cooling would then move the final state along the upper branch. An example of this type of flow is provided by a normal shock in a stream in which condensation has already occurred. The temperature rise through the shock vaporizes some of the condensate and cools the flow, with the result that the Mach number behind the shock is lower than with an adiabatic shock. When we eliminate the velocity from (4.36), (4.37),and (4.38),we obtain for the Hugoniot curve
using the equation of state to eliminate the temperature, we have
y f l (pv Y--l
(4.41)
(PV,
- P1V) - 2
q = 0.
In the - v diagram the Hugoniot curves are a family of hyperbolas with q as the parameter and common asymptotes given by #J
(4.42)
p=
2Ylp l , +l
v =Y - l v , .
Y+l
A family of Hugoniot curves, with y = 1.40, is plotted in Fig. 38 for q 2 0, together with some typical Rayleigh lines. The ratio of the slope of the Rayleigh line to the slope of the q = 0 Hugoniot curve a t p,, vl is easily found to be (4.43)
Therefore, the Ml
=1
Rayleigh line is tangent to the Hugoniot curve a t
PlJV l .
The meaning of the various intersection points in Fig. 38 can perhaps be seen more clearly by examining Fig. 39, where the Hugoniot curves are drawn in the h -s diagram for the initial state, M,. The intersection point with the lower branch of the Rayleigh line, 3, is the condensation shock solution, and M2 > 1 ; the intersection point with the upper branch, 3', is the solution for a normal shock followed by heat release, and M 2 < 1. The Hugoniot curve which is just tangent to the Rayleigh line a t C - J corresponds to the maximum heat release, and M 2= 1. This state can be attained either by a condensation shock, or by a normal shock with heat release. We can now locate these intersection points in Fig. 38. We start at PI, v1 and follow the Rayleigh line upward. By reference to the correspond-
377
CONDENSATION I N W I N D T U N N E L S
ing path in Fig. 39, we see that points 1, 2, and 3 represent the condensation shock solutions. C - J is the maximum entropy point, where M , = 1. The points l', 2', and 3' represent the solutions for a normal adiabatic shock followed by heating. Point S represents the solution for the adiabatic shock. Similar results are true when the initial flow is subsonic, except that in this case transitions to supersonic speeds must be excluded because they would require an entropy decrease. Therefore, the only permissible exothermic subsonic solutions are those which correspond to the condensation shock in supersonic flow, i.e., subsonic initial flows, which, upon heating,
0
2
I
3
MI
FIG.40. M ,
-
M , diagram for constant area flow with heat addition or subtraction [67].
become subsonic flows a t a higher Mach number. In Fig. 38 the permissible solutions are represented by points A and D ; the excluded solutions by A' and B. If the Hugoniot curves for q < 0 are drawn, still more solutions show up. One such curve is drawn in Fig. 39, and the upper branch intersection point, V, represents a normal shock with vaporization. The analytical results which correspond to the intersection points can be obtained from (4.35)-(4.38), and have been given by many authors.
378
P E T E R P. W E G E N E R A N D LESLIE M. MACK
With all of the flow variables expressed in terms of the initial Mach number, the results [27] are:
FIG.
(4.46)
(4.47)
_ -- --, P1P1 M;
P 2 $2
379
CONDENSATION IN WIND TUNNELS
where (4.48)
(4.49)
7
I
2
3
4
5
6
7
MI FIG. 42. Pressure ratio across condensation shock as a function of initial Mach number.
When we choose the lower sign in (4.44) and (4.45), we find that M , = MI for Q = 0. This is the condensation shock solution, since in the absence of condensation we must have continuous flow. When we choose the upper sign. we find that for Q = 0, the equations reduce to those for a normal adiabatic shock. It is evident that Qmax is the maximum amount of heat that can be added to the flow, since imaginary results are obtained for Q > Qmax. This corresponds to the fact that no intersection points exist in
380
PETER P. WEGENER A N D LESLIE M. MACK
the diagrams when Q > Qmax. The release of Qmaxresults in the state given by the maximum entropy point in the h - s diagram, where the Rayleigh and Hugoniot lines are tangent and M , = 1. An instructive way to represent the solutions of these last equations is afforded by (4.47) when lines of constant Q are drawn in an M,- M, diagram. Such a diagram is given in Fig. 40 [67]. The shaded area is the region of physically impossible solutions (entropy decrease). All condensa-
I
2
3
4
5
6
7
MI FIG. 43. Temperature ratio across condensation shock as a function of initial Mach number.
tion-shock solutions lie between the line Q = 0 (no-shock line) and the line M, = 1 ; the solutions for the normal shock with vaporization are between the normal-shock line and the M,-axis; and the solutions for the normal shock followed by heat release are on the other side of the shock line up to the line M , = 1. In Fig. 41 Qmax is plotted as a function of M I , from (4.49), for y = 1.40; we see that there is an asymptote as M , m,
-
CONDENSATION IN WIND TUNNELS
381
and only a finite amount of heat can be added to a steady, supersonic flow regardless of how large the Mach number becomes. In Figs. 42, 43, 44, and 45, we show pressure, temperature, Mach number, and stagnation pressure ratios as functions of M , for a few values of Q/Qmaxfor y = 1.40. In Fig. 44 we have for Q/Qmax= 1, M , = 1 for all values of M , . The condensation-shock analysis leaves two questions unanswered : the location of the condensation and the amount of heat released. In Section V we shall see that although kinetic theory offers qualitative information
FIG.44. Mach number behind condensation shock as a function of initial Mach number.
about the location of the condensation, it does not allow a quantitative prediction of condensation of water vapor in air. In the absence of any definite information, the correlation presented in (2.35) may be used as a guide. To obtain Q, the most reasonable assumption is that the vapor behind the shock is in equilibrium saturation. Lukasiewicz and Royle [27] found that this was a satisfactory assumption for some of their experiments, and that it resulted in a value of Q very close to Qmx. However, when Keenan [7]
382
PETER P. WEGENER AND LESLIE M. MACK
analyzed Yellott’s experimental results with steam on this basis, he found that the theory gave too large a pressure rise. Only by the assumption of a definite, small droplet size and, thus, a certain amount of supersaturation retained at the end of the collapse zone, could he fit the experimental results. Also, it is found that, when M , is near one and Qmaxis small, only a portion of the water vapor in air can be condensed in the condensation shock. The remainder, which can easily be over half of the original content, must condense downstream from the shock, where the flow will be a saturated expansion of the type considered in meteorology [75]. We have indicated in Fig. 46 the results of applying the condensationshock theory to the experiment of Wegener for which the detailed calculation
FIG.45. Stagnation pressure ratio across condensation shock as a function of initial Mach number.
of Fig. 16 was carried out. We have assumed a dry isentropic expansion up to the shock, then applied the shock equations, and finally assumed another isentropic expansion from the new pressure and temperature level. Two assumed positions are shown for the shock. The first is at x = 1.58 cm, where M I = 1.28. At this location the release of Qmax also satisfies the condition of saturation behind the shock. However, the total amount of heat released is only q = 2.69 cal/g compared to the value of q = 3.06 cal/g
383
CONDENSATION IN WIND TUNNELS
given by the step-by-step calculation (Fig. 16). We see that the pressures behind the shock fall slightly below the experimental points. The second shock was placed at the experimental peak-pressure point, where, from the isentrope, M , = 1.36. Enough heat was released to saturate the flow, which meant Q/Qmax = 0.724. For this condition, q = 2.96 cal/g, which is close to the result of the detailed calculation, and we see that the pressures a short distance behind the shock are very close to the experimental points. In summary we can say that the condensation shock theory can be used to fit experimental data, but that it is not too helpful in predicting details of the flow. However, it can be quite useful in estimating the order of magnitude of condensation effects. 0.6
-
0 .5
-
Condensation shock at MI = I. 28 q = qmox= 2.68 caI/g
P Pol
0.4
Condensation shock at MI = 1.36 q = 0.724 qmm= 2.96 . d / g
-
0.3 -
@0=53'/0 Calculated q = 3 . 0 6 calh I I
0
0.2
I
I
2
I
3
3. Condensation as a Weak Detonation The preceding analysis of the condensation shock can be placed in a more general framework when we realize that the same type of analysis is used in the theory of flow processes in which chemical reactions occur across sharply defined fronts [76}. The flow in front of the shock corresponds to the unburnt gas, the shock itself to the detonation (considered stationary),
384
PETER P. WEGENER AND LESLIE M. MACK
or slow combustion front, and the downstream flow of vapor and droplets to the burnt gas. As heat is released by the chemical reactionstheseflows must be among the solutions given by the various intersection points in Figs. 38 and 39. The solutions with subsonic initial velocity are the slow combustion processes, sometimes called deflagrations. When the final velocity is subsonic [point A in Fig. 38) they are called weak deflagrations; when it is supersonic (point A ’ ) , strong deflagrations. The strong deflagration can be ruled out on physical grounds, but none of these processes is of interest to us because they occur in subsonic flow. The reaction fronts with supersonic initial velocity are called strong detonations when the final velocity is subsonic (points l‘, 2‘, 3‘) and weak detonations (points 1, 2, 3) when it is supersonic. The detonation front that releases the maximum amount of heat, Qmax, and has a sonic final velocity (point C - J)isof special importance. It is called the Chapman-Jouguet detonation and can be shown [77] to be the only detonation that is expected to occur in practice. All observed chemical detonations appear to be Chapman- Jouguet detonations. From Fig. 38 we clearly see that the condensation shock appears to be an example of a weak detonation, since both processes axe transitions from a supersonic velocity to a (lower) supersonic velocity. It was formerly thought that weak detonations were impossible, being excluded on the basis of the shock-ignition model for the reaction. In this model, a normal shock is supposed to precede the detonation and raise the temperature sufficiently so that the reaction can take place. That is, the sequence of states within the reaction zone is assumed to be as follows: first, a shock which changes the state from PI, v1 to the state given by point S on the q = 0 Hugoniot curve; then, as the reaction proceeds, the state moves back along the Rayleigh line to the Hugoniot curve for the final amount of heat released, qf. This processes is a strong detonation, and with this model the weak detonation can never occur because it would be necessary to traverse states for which no Hugoniot curves exist (q > 4,). However, Friedrichs’ [78] analysis of the detonation process did include a weak detonation solution, which was linked to a requirement for a “very fast” reaction rate. Later, Burgers [79] speculated that the condensation shock might be an example of a weak detonation, and more recently, Reed [80] and Heybey and Reed [Sl] established that it is indeed a weak detonation. Oswatitsch [82], and Hall [83], also had similar thoughts. The analysis of Heybey and Reed will be followed here. In Fig. 38 we now consider the various Hugoniot curves to represent intermediate stages within the condensation zone of finite width. The q for each curve is the total amount of heat released up to that stage of the process. The reaction proceeds along the Rayleigh line from fil,vl to the final state, say 3, and is therefore a weak detonation. The strong detonation point, 3’. cannot be reached from 3 since this would require passage through states in
385
CONDENSATION IN WIND TUNNELS
which more heat than qf is released. Instead, as previously described, 3' can only be reached by a normal shock transition to S, followed by successive steps along the Rayleigh line to 3', where the reaction must stop. As qf is made continuously smaller the weak detonation solution approaches the initial state; the strong detonation approaches S. The Chapman-Jouguet detonation can be approached from either side and is a stable state, i.e., the reaction cannot proceed past it.
I
fi
1.30
0.901
I
I
I
I
"I
FIG. 47. Experimental demonstration of water vapor condensation as weak detonation.
The fact that the condensation of water vapor in air is a weak detonation has experimental backing from the previously discussed experiment of Wegener (Fig. 16). When we convert the actual flow process as measured in a diverging nozzle into a fictitious one in a constant-area channel by a series of isentropic compressions, we obtain the sequence of points shown in Fig. 47 [29]. We find that the points start from p,,v, and then move up the Rayleigh line towards the Hugoniot curve for qmax. This establishes the condensation process experimentally as a weak detonation.
386
PETER P. WEGENER A N D LESLIE M. MACK
Instead of considering a family of Hugoniot curves, each for a fixed q as in Fig. 38, we can introduce the idea of an equilibrium Hugoniot curve. That is, we add the integrated Clausius-Clapeyron equation, (1.9), to our system of equations, (4.35), (4.36),(4.37), and (4.38), and require that it be satisfied behind the condensation region. Then we have only a single Hugoniot for the given initial state, and each point on it represents a state of thermodynamic equilibrium. Such a Hugoniot is shown schematically in Fig. 48 for a pure vapor, although the situation is similar for water vapor in air. Outside of the coexistence region, the Hugoniot curve is the usual one with q = 0. At the coexistence line there is a discontinuity in slope since within the coexistence region the amount of condensate g must be appropriate to an equilibrium state. The amount of heat that must be released for the transition from pl,v, to each point on the curve is q = Lg.
Coexistence line P
Isentrope, no condensation
V’
I/P
FIG. 48. Equilibrium Hugoniot curve in p - v diagram and condensation process in pure vapor.
In Fig. 48 we have also drawn a family of Hugoniot curves to represent the intermediate non-equilibrium states. We assume that each curve can be characterized by a parameter E , where 0 < E < 1, and represents a fixed departure from equilibrium. The flow follows the dry vapor isentrope (shown dotted) to Z,, where the supersaturated state collapses. It then
CONDENSATION I N WIND TUNNELS
387
proceeds along the Rayleigh line through the non-equilibrium states 1, 2, and 3, until the equilibrium Hugoniot is reached a t the weak detonation point 2,. The strong detonation point 2,’ cannot be reached, nor can Z, be reached from Z2’ in a continuous manner, because the necessary intermediate states do not exist. Heybey and Reed have also pointed out that an adiabatic shock at Z,, and a weak detonation a t Z, followed by a shock a t Z,, both lead to the same final state 2,’.
4. Saturated Equilibrium Expansion
In a saturated equilibrium expansion condensation occurs as a gradual process that starts a t the condensation line and maintains thermodynamic equilibrium throughout. We can expect this type of condensation process to occur when supersaturation is practically absent, as in some experiments on nitrogen and air condensation (e.g., Figs. 27 and 30). We shall see later that some of the experimental results obtained with nitrogen and air do actually agree with the theory of the saturated equilibrium expansion within a few percent. Since the flow during such an expansion is always in thermodynamic equilibrium, the analysis is greatly simplified. We do not have to obtain a condensation-rate equation from kinetic theory, but can use instead the Clausius-Clapeyron equation to complete the system of equations. Thetheory along these lines has been developed by Reed [59, 631, and in particular detail by Buhler [61]. The equations for the flow of a pure vapor in equilibrium with its condensed phase axe the conservation equations of mass, momentum, and energy, the equation of state, and the Clausius-Clapeyron equation. From our previous results, they are
(4-4)
(4.50)
(4.51)
(4.52)
(4.53)
dp
dw -+-+-=o ,
P
dp
w
-
_
P
dw (y - 1 ) W -
w
dA *
y M 2 dw 1--g w
_
_
dT +T
-
p
L c~T
-dg
-
= 0,
388
PETER P. WEGENER AND LESLIE M. MACK
The momentum equation is obtained from (4.13) by the introduction of the Mach number defined in terms of the speed of sound at infinite frequency, (4.21). The energy equation is a simplified form of (4.25) for constant heat of vaporization and with c - c,, = constant since we have a pure vapor. Po -
FIG. 49. Ratio of zero-frequency speed of sound t o infinite frequency speed of sound ih pure vapor as a function of temperature.
The equation of state is likewise (4.12) with p = p,,. The last equation is an alternative form of the Clausius-Clapeyron equation (1.7). These equations can be solved in terms of the area, which we can regard as known. The results are
389
CONDENSATION IN WIND TUNNELS
(4.55)
dA A '
dP ---
-
P
T/ Tc FIG.50. Mass fraction of condensate in pure vapor as a function of temperature from saturated equilibrium expansion theory.
(4.66)
390
P E T E R P. W E G E N E R AND L E S L I E M . MACK
I .o-
0.0 0.6 -
0.4
0.2
-
P PC
0.1 0.00 0.06
0.04
0.02
-
-
-
0.010.6
0.7
0.8
0.9
I .o
T/ Tc FIG.51. Pressure ratio in pure vapor as a function of temperature from saturated equilibrium expansion theory.
The entropy of the mixture must be constant since thermodynamic equilibrium exists throughout the expansion. Thus, (4.54) and (4.55) give dpldp at constant entropy,
391
CONDENSATION I N W I N D TUNNELS
This, however, is the square of the zero-frequency speed of sound, i.e., the speed of sound when the frequency is sufficiently low so that thermodynamic
0.6
0.7
0.0
0.9
1.0
T/Tc FIG.52. Density ratio in pure vapor as a function of temperature from saturated equilibrium expansion theory.
equilibrium can exist a t all times. Both Reed [84] and Buhler [61] have obtained (4.58) ; the former by means of purely thermodynamic arguments.
392
PETER P. WEGENER AND LESLIE M. MACK
The square of the ratio of this speed of sound, sound speed, Vy(R/,u)T,is given by (4.59)
i2
the infinite frequency
1-g
-
___ R
i, to
~
0 0.68
0.76
0.84
0.92
I .oo
T/T, FIG. 53. Velocity ratio in pure vapor as a function of temperature from saturated equilibrium expansion theory.
We have plotted (4.59) for y = 1.40 in Fig. 49 as a function of TITc,where Tc is the temperature of equilibrium condensation (Fig. 8). The ratio has a discontinuity at the coexistence line, where TIT, = 1. Equations (4.56) and (4.57) yield a simple differential equation for g as a function of T whose integral is (4.60)
CONDENSATION I N W I N D TUNNELS
393
as was first found by Buhler [ S l ] . This means the saturated expansion theory can be given completely in terms of the integrated relations: (4.61)
FIG.54.
(4.62)
(4.64) Figures 50, 51, 52, and 53 give g, p/p,, pc/p, and W as functions of TITc for y = 1.40 and several values of the parameter cpTc/L. This parameter,
394
PETER P. WEGENER AND LESLIE M. MACK
which occurred previously in (1.13), is plotted for air as a function of To for several Po in Fig. 54. We see that for large values of M,, say 5 or 6, the velocity is nearly constant during the expansion, with the result that the area ratio is closely approximated by the inverse of the density ratio. This 0.01c 0.OOE
0
Arthur, Nitrogen with 0.26 O/O C02
p, = 8.33 atm
O.OOE
T =292 O K 0.004
0.00:
Saturated Expansion Theory
0.002
P Po
0.00I 0.0008 0.0006
Isentrope for no condensation from measured impact pressure 0.0004
0.0003
I
I
1
I
I
x(in1 FIG.55. Comparison of saturated equilibrium expansion theory with Arthur’s experiments [54] with nitrogen.
means that Fig. 52 can be used to find TIT, for a given position in a nozzle; we are then able to obtain rapidly all the other flow variables. We can now compare the saturated expansion theory with a few experiments. In Fig. 55 are shown the experimental points of Arthur [54] for nitrogen with 0.26% by volume of CO,. In this experiment, nitrogen condensation began very close to M,. The measured pitot pressures were assumed to be those that would exist for no condensation, as discussed in Section IV, 5. This defines the effective area ratio as well as the constant stagna-
395
CONDENSATION IN WIND TUNNELS
tion enthalpyisentrope. Theintersection of thisisentrope with the vapor pressure curve of solid nitrogen establishes 9, and T,. The value of the parameter cpTclL is 0.2576. With this parameter known, all of the flow quantities follow from the preceding equations or plots. The pressure ratio is shown in Fig. 55 in comparison with the experimental measurements. The agreement is within about 7% and demonstrates that the highly idealized saturated isentropic expansion theory bears a close relation to the actual expansion in this particular experiment. 0.030
0.026
Condensation shock plus
0.022
-
P -
pd 0.018
-
0.014
-
0.010
-
I
I
o Kubota,Air po = 14.6atm To = 394 OK I
I
FIG. 56. Comparison of saturated equilibrium expansion theory with Kubota's experiments [66] with air.
In Fig. 56 we compare the saturated expansion theory with experimental results obtained by Kubota [66] in air. We have used the Buhler method of plotting (e.g., Fig. 31). This example is of interest because some supersaturation was present. We see that the experimental points follow the zero condensation isentrope for some distance past M,, and then move toward the saturated expansion curve and finally above it. However, the collapse of the supersaturated state is not an isentropic process, and there is no reason why the experimental points should ever follow the saturated isentropic curve. We have attempted to include an entropy increase in the theory by
396
PETER P. WEGENER AND LESLIE M. MACK
st. supposing a process in which the expansion follows the zero condensation isentrope to a point where a condensation shock occurs. The shock returns
4.0
4.5
5.0
5.5
6.0
65
M(pitot) FIG. 57. Calculation of flow parameters from saturated equilibrium expansion theory for conditions of Hansen and Nothwang’s experiment [53] with air.
the flow to saturation conditions, and the subsequent flow i s again a saturated isentropic expansion. We see from Fig. 56 that such a curve lies above the simple isentropic curve, and in the direction to which the experimental points appear to tend. Finally, we have in Fig. 57 a comparison of the theory with the experiments of Hansen and Nothwang [53], previously shown in Fig. 30. These experiments were camed out with air in a 25 x 36cm tunnel, which is larger than the ones used by Arthur and Kubota. We see that almost no
CONDENSATION I N W I N D TUNNELS
397
supersaturation was present ; the experimental points lie above the theoretical curve with a somewhat different slope. Actually we cannot expect experimental results obtained with air to follow exactly the theory as calculated. What we have taken as the vapor-pressure curve of air is really the condensation line, i.e., the locus of states where condensation first occurs (see Section I). As condensation continues, the proportions of the components in the air will change, and the actual vapor-pressure curve will depart from the condensation line. Also in Fig. 57, we have shown all of the other flow quantities calculated by the saturated expansion theory for the conditions of the Hansen-Nothwang experiment. We show for comparison the same quantities as calculated from the measured pitot-stagnation pressure ratio, assuming no condensation. An interesting point is that the density is almost identical in the two cases. The velocity changes only a few percent during the expansion, as mentioned previously, and is little different from that in the expansion without condensation. 5. Shock Waves with Vaporization
When a normal or oblique shock wave occurs in a flow in which condensation has taken place, the temperature rise behind the shock will cause some, or all, of the condensate to vaporize. For this discussion we assume that vaporization will take place instantaneously, and, therefore, the vapor and condensate are in thermodynamic equilibrium behind the shock wave. The vaporization will change the flow variables from their usual values in condensation free flow. In addition, the angle of an oblique shock wave for a fixed flow deflection will be changed. We have shown in Fig. 39 the intersection point, V , representing a normal shock wave with vaporization. The Mach number behind the shock is smaller, the pressure is higher, and the temperature lower, than for the corresponding adiabatic shock. This solution is obtained from the equations (4.35), (4.36), (4.37), and (4.38), when q < 0, in accordance with the physical fact that heat must be supplied to vaporize the condensate. The equations are suitable for moist air and steam because the condensed mass is usually small. However, for nitrogen and air condensation the condensed mass fraction may be as large as 10-20%, and we must account for the change in the vapor mass across the shock. Such a generalized analysis has been performed by Buhler [61] and Ross [GI. We first discuss the simple situation where the mass change can be neglected. The effect of condensation on the measured pitot pressure, a problem first considered by Taylor [86], is of considerable interest. If we assume that complete vaporization occurs, Eqs. (4.44), (4.45), (4.46), (4.47), (4.48), and (4.49) provide, when we choose the plus signs, a solution of this problem for M , and amount of condensate being fixed. However, at a fixed
398
PETER P. WEGENER AND LESLIE M. MACK
position in a wind tunnel downstream from a condensation region, the Mach number and other flow variables differ from their values in condensation free flow. To find the combined effect of the condensation and subsequent vaporization at the pitot tube on the measured pitot pressure, we must first consider the effect of a condensation shock on the downstream flow. This is done in Section‘VI, and we defer the discussion of the pitot pressure until then. A second point of interest is the effect of vaporization on the angle of inclination of an oblique shock for a fixed flow deflection. For a definite amount of heat absorbed, Head [lo] calculated the apparent Mach number obtained from the measured shock angle and the usual oblique shock theory, as a function of true Mach number for several different flow deflection angles. He found that the shock with vaporization is inclined at a smaller angle to the flow than otherwise, and that the apparent Mach number for sizable flow deflections can be much larger than the true Mach number. This same result can also be obtained from the work of Samaras [74] and the more general analysis of Ross [85], where a sample shock polar diagram is given. We turn now to normal and oblique shock waves in partially condensed nitrogen and air and shall follow closely the analysis of Buhler. The equations are just those used for the condensation shock (4.36), (4.37), and (4.38) except that the equation of state must be in the form (4.10),and q = - Lg. We consider only those cases where complete vaporization takes place. At low Mach numbers it is possible that the temperature rise is not sufficient to satisfy this condition. It is convenient in this analysis to introduce a new Mach number which is proportional to the ratio of dynamic to static pressure, (4.65)
According to the saturated expansion theory, this Mach number happens to be very close to the Rayleigh Mach number found from (3.5) for the measured pitot-static pressure ratio [61]. From the continuity, momentum, and state equations, (4.36), (4.37), and (4.10),we can obtain (4.66)
where subscripts 3 and 4 refer to conditions in front of and behind the shock respectively. The state and continuity equations give (4.67)
CONDENSATION I N W I N D TUNNELS
399
Then, from (4.66) and (4.67) we find (4.68)
T/ T, FIG.58. Shock theory parameter as a function of temperature.
When we solve (4.67) for ua/u3,use (4.68), and substitute in the energy equation (4.38),we obtain
where (4.70)
400
PETER P. WEGENER AND LESLIE M. MACK
For complete vaporization behind the shock, g, = 0 and M , = M,. Therefore, when the conditions in front of the shock are known, (4.69) can be solved
FIG. 59. Effect of vaporization on m e pressure ratio across a normalshockinair [61].
for M,, and the pressure ratio across the shock follows from (4.68). In Fig. 58, we have plotted E in the usual manner as a function of TIT, for several values of c,T,/L. In Fig. 59 [61] we show the ratio of the pressure ratio across a shock with vaporization to the pressure ratio across an adiabatic shock as a function of M3. The dotted lines represent the limits above which the assumption of complete evaporatiQn is invalid. We note that a t high Mach numbers, evaporation has only a slight effect on the pressure ratio. Information on the assumption of vaporization immediately behind a normal shock is supplied by an experiment of Grey [62] with air. He
CONDENSATION I N WIND TUNNELS
401
established a normal shock in front of a cylindrical duct of large diameter in which the flow was choked. A pitot survey in the region behind the shock showed that the pitot pressure was constant. If vaporization did not occur until some distance behind the shock, a change in pitot pressure should have been observed. Light scattering experiments of Hansen and Nothwang [53] also indicated that vaporization occurs right a t the shock. We are now in a position to find the pitot pressure in a saturated expansion, and compare it with the pitot pressure at the same location in a nozzle with no condensation. We may write this ratio as (4.71)
We specify the effective area ratio and find the corresponding flow conditions from the saturated expansion theory. Then the first factor on the right is obtained from isentropic flow tables once M , has been determined from (4.69). The pressure ratio across the shock is found from (4.68). The third factor is known from the saturated expansion theory. The fourth is determined by the nozzle supply conditions and the vapor pressure curve. Finally, the last factor is obtained from isentropic flow tables for the given area ratio. We have carried out this calculation for the conditions of the Hansen and Nothwang experiment presented in Fig. 57. The result is that a t an area ratio of A / A * = 70, the pitot pressure for the saturated expansion is 1.1% higher than for the expansion without condensation. At smaller area ratios, this number is much smaller. Calculations for other conditions indicate that even a t area ratios which correspond to M = 10, the saturated expansion produces a pitot pressure only about 3% higher than in the absence of condensation. Therefore, experiments performed with heated and unheated supply conditions as well as equilibrium saturated expansion theory agree in the result that for the same effective area ratio the measured pitot pressure is almost unaffected by condensation. This validates our previous procedure of determining experimental isentropic and constant stagnation enthalpy expansions from pitot-pressure measurements in condensed flow. The theory developed for the normal shock wave can easily be extended to oblique shocks by the usual device of considering the flow normal and tangential to the oblique shock. The normal flow is governed by the normal shock relations ; the tangential flow requirement is that the tangential velocity component be equal on both sides of the shock. An interesting question concerns the strength of the oblique shock necessary for complete evaporation. The results of a calculation by Grey for a fixed supply temperature and two supply pressures are shown in Fig. 60. We see that for the larger pseudo Mach numbers % a large flow deflection is needed to produce an oblique shock of sufficient strength to cause complete vaporization.
402
PETER P. WEGENER AND LESLIE M . MACK
In the limit of very small flow deflection, 6, the oblique shock reduces to a Mach wave. From the equality of the tangential velocity components, and the momentum equation for the normal components, we obtain for an infinitesimal oblique wave [61],
9= Y M tan ~ eas,
(4.72)
P
4.0
4.5
5.0
-
5.5
6.0
M
FIG. 60. Supply conditions required for complete vaporization behind oplique shock waves [62].
where 8 is the angle of the wave. This equation permits an experimental determination of the effective speed of propagation of infinitesimai waves. Since is very nearly the Rayleigh Mach number, a measurement of dpld6 for dp -+0 gives tan 0, and, therefore, the effective Mach number. If the
CONDENSATION IN WIND TUNNELS
403
proper Mach number is the one defined in terms of the low frequency speed of sound i, (4.72) can be written (4.73)
This relation can be used to predict the pressure coefficient on slender bodies in condensed flow.
V. KINETICSOF CONDENSATION
We have seen in Section I V that purely thermodynamic considerations are not adequate for a complete description of condensation except in those flow processes where thermodynamic equilibrium exists at all times. This equilibrium does not prevail in the supersaturated state and subsequent collapse region (C,DF2 in Fig. 37), which makes prediction of the onset and course of the condensation process by thermodynamic methods impossible. The description of this part of the condensation problem, as well as the prediction of whether or not supersaturation is present, is of necessity a problem outside the scope of thermodynamics. In this section we shall first show that condensation phenomena in high speed flows, like other condensation processes, are linked to the availability of surfaces of condensation. These surfaces, or condensation nuclei, come principally from three different sources: a) inert impurities in the nozzle supply such as dust, etc.; b) small drops (or crystals) formed in the flow by condensation of vapor impurities of higher vapor pressure than that of the main constituents; and c) small drops (or crystals) formed in the condensing vapor itself by statistical cluster formation due to fluctuations. We must first investigate which of these three possibilities is applicable in each typical process, and determine the available number and rate of production of the nuclei. The prediction of the availability of nuclei will provide a determination of the point in the expansion where condensation should occur (point D in Fig. 37). Once condensation nuclei are present in sufficient numbers, it is necessary to understand their further growth in order to predict the thermodynamic state in the collapse region. We shall find that the prediction of nucleation rates and droplet growth still defies exact calculation except in the simplest cases. The predictions are difficult not only because the detailed condensation mechanism is still the subject of some speculation, but also since many of the thermodynamic properties required in the calculations are in doubt or unknown. However, the kinetics of condensation during expansions in nozzles is qualitatively well described,
404
PETER P. WEGENER AND LESLIE M. MACK
and approximate answers for the functions J ( l and ) G ( E , x ) in the condensation rate equation (4.32) can be given. Because of its limited application in flow problems, it is unnecessary to discuss the details of the nucleation theory, which has been summarized by Volmer [87], Frenkel [6], and recently Becker [88]. 1. Condensation Nuclei and Classification of Condensation Processes
Surface phenomena play a decisive role in processes of phase transition. Vapor in a nozzle cannot condense upon reaching saturation with respect to a plane liquid phase if either this phase is absent or no “cold” container walls are available to serve as surfaces of condensation. Owing to the presence of “hot” boundary layers surrounding the flow, the nozzle wall is ruled out as a surface of condensation. Thomson [89], von Helmholtz [go], and Gibbs [91] found that the pressure of a vapor in equilibrium with a small spherical drop of the liquid a t a given temperature increases with decreasing droplet radius Y. Therefore, a vapor may be supersaturated with respect to a drop of large or even infinite radius, and not saturated with respect to small droplets. For the mechanical equilibrium of a spherical droplet with the surrounding vapor [e.g., 61, we have p,-p,=2-.
(T
Y
The pressure in the drop, p,, is larger than the surrounding vapor pressure, $, by the amount 2017, where (I is the surface tension of the liquid, with dimensions forcellength, or work/length2. The surface tension is generally a linearly decreasing function of temperature, but, unfortunately, it is not known experimentally at low temperatures for the highly supercooled liquids of interest to us. To extrapolate the available experimental results to lower temperatures, we may resort to the semi-empirical expression of Eotvos and Ramsay-Shields [e.g., 6, 141,
where vL is the specific volume of the liquid and Tcrit.the critical temperature of the vapor. The surface tension vanishes when T = Tcrit,;for many liquids the ‘constant is 2.12. Although many factors such as impurities, solvents, etc. affect surface tension strongly [92], it is little changed by the nature of the surrounding vapor. A t T = 20 “C, the surface tension of water in water vapor is 70.60 erg/cm2,while in air it is 72.75 erg/cm2. For 0, and N, a t 70 OK, the values are 18.43 and 10.72 erg/cm2, respectively, according to the newest work by Keilly and Furukawa [93]. Extrapolations to lower
CONDENSATION IN WIND TUNNELS
405
temperatures based on (5.2) or obtained by other methods are available for H,O [e.g., 27, 281, and for 0, and N, [eg., 61, 641. Application of the stability criterion that the thermodynamic potential of the system of the spherical drop of liquid and the surrounding vapor must be zero leads, with (5.1), to the familiar Thomson-Gibbs equation [eg., 61 for the vapor pressure of the droplet, (5.3)
where p L is the liquid density [8,9, 931. From (5.3) we obtain for the radius of a droplet in thermodynamic equilibrium with the surrounding vapor (5.4)
y*
2up 1 ~ L R lTn v
=____,
where y is the saturation ratio, (2.1). In the limit of pv = fim,or y = 1, = 0 0 , (the droplet is infinitely large). At constant temperature, decreasing droplet size leads to increasing equilibrium vapor pressures, as shown for water vapor in Fig. 61. The solid curve was computed from (5.4) (1 Angstrtim = 10-8 cm). We have applied (5.4) down to very small droplet radii without concern about its region of validity. The range of very small droplets, with radii of the order of the molecular radius, is of concern to us in extreme cases of supersaturation, as shown in Fig. 62. In this diagram the moist-air condensation experiments [39, 401, previously shown in Fig. 17, are plotted as saturation ratio a t condensation against the temperature a t condensation. Also shown (solid lines) are values of stable droplet radii computed from (5.4), which indicate that a t condensation the droplets were of the order of the size of the molecule. The molecular radius of H,O is about 2.3 8,as determined from viscosity measurements [eg., 941. However, we cannot expect (5.4) to hold for Y < lO-'cm, say since the Thomson-Gibbs formula was derived on a continuum basis and with a constant value of surface tension. Actually, for very small droplets, surface tension is a decreasing function of the drop radius. Much effort has been expended to account for surface tension correctly on thermodynamic grounds or to determine the equilibrium droplet conditions statistically in order to find expressions corresponding to (5.4) for small clusters of molecules. Tolman [95] extended earlier ideas of Gibbs [91, p. 2321, and arrived by thermodynamic methods a t a first approximation for the surface tension of a drop of radius r p , , which is smaller than the surface tension of the plane surface urn: Y*
(5.5)
406
PETER P. WEGENEK AND LESLIE M . MACK
where 6 is a constant for a given liquid. This constant is of order 1 A, which is approximately equal to the intermolecular distance in the liquid. Kirkwood and Ruff [96] showed qualitatively by statistical methods that the trend of (5.5)is correct, as was earlier indicated for the free surface energy of crystals by Kossel [97]. For greater accuracy higher approximations must be found when only a few molecules are in one drop; Gilmore [98] discusses such results. Also shown in Figs. 61 and 62 are dashed lines for the equilibrium I00
I
I I I
---Ir Pa, 0
0
- P” --
p,
with surfoce tension correction
------------I 50
I I00
r(A)
FIG. 61. Vapor pressure of a water droplet in water vapor at 30 “C as a function of drop radius.
vapor pressure of a water drop and its radius, respectively, with surface tension based on (5.5); and large deviations from the result computed with (5.4) for small radii are seen. Stever and Rathbun [64] arrived at qualitatively similar answers by considering spherical molecules packed into spherical drops. All of these treatments are approximations which extend continuum notions to small droplets. Reed and Herzfeld [99, 1001 made calculations of equilibrium droplet conditions by building up nitrogen drops from individual molecules and determined the molecular cluster arrangement of lowest energy under the assumption of a Lennard- Jones potential. It is interesting to note that for nitrogen in the range of 2 to 8 molecules per “drop”, the results fair reasonably well into those obtained from macroscopic
407
CONDENSATION I N WIND TUNNELS
considerations without a surface tension correction. Finally, Kuhrt [loll recently demonstrated that very small clusters, even when surface tension corrections are applied or treatments like those of Reed are used, cannot be considered as drops at rest in an infinitely extended vapor space. Small clusters are similar to very large molecules, thus they are subject to Brownian motion. When the translational and rotational motion of the extremely small drop is included, the Thomson-Gibbs formula (5.3) changes to
0
210
2 20
230
240
2 50
T(OK) FIG.62. Droplet size in the range of water vapor condensation with large supersaturation from Wegener [39]. Droplet radii from the Thomson-Gibbs formula (solid line) and with a Tolman surface tension correction (dashed line).
where n denotes the number of molecules in the drop. A substantial correction appears if n is small. It has long been known from Wilson’s classical experiments in cloud chambers [30] that fog formation in vapors occurs at varying degrees of supersaturation, depending on a variety of conditions in the cloud chamber. Also von Helmholtz, as early as 1886 [go], observed that a jet of saturated steam issuing into the atmosphere remained clear for some distance before
408
PETER P. WEGENER A N D LESLIE M. MACK
clouding. Whenever condensation is actually observed, it takes place on condensation nuclei whose size approximately corresponds to that given by (5.4)for a saturated equilibrium condition. For condensation phenomena in high speed flows we must, therefore, investigate the nature of the condensation nuclei that initiate immediate or delayed condensation. The slightly delayed condensation in cloud chambers takes place on particles such as dust, ions, etc. The same process is found in the atmosphere, where cloud formation is possible only in the presence of condensation nuclei; in fact, clouds may be produced artificially by introducing just such nuclei (e.g., silver iodide or solid carbon dioxide) into air layers mixed with slightly supersaturated water vapor. The number of foreign particles ordinarily present in the atmosphere is variously found to range from, say, lo2 to lo6 particles per cubic centimeter [31], depending on many extraneous factors. Wind-tunnel air is first drawn from the atmosphere, and may in addition carry small particles produced by the dessicants of the drier systems. I n hypersonic wind tunnels, oxidation of heating elements may contribute small pieces of foreign matter. Oswatitsch [33] first demonstrated that these customary condensation muclei must be discounted as being responsible for the collapse of the supersaturated state in nozzles operated with moist air because of their insufficient number. A conservative estimate would be that N = 105/cm3particles are present in a nozzle supplied with laboratory air. In a typical supersonic nozzle an element of the flow may travel 10 cm between saturation and condensation in about 3 x 10-4sec. We assume that during this flight time enough water vapor condenses on every particle cm. At the end of this to result in the unlikely large drop size of Y = flight path, there will be lo5 spherical droplets of Y = cm per cubic centimeter with a total amount of water equal to about 4 x 10-'g/cm3. The heat of vaporization released in the flow by this phase transition is not sufficient to change the stagnation enthalpy of the flow measurably, and no flow parameters could be affected, in contrast to the actual observations. Therefore many more nuclei are needed for condensation than are provided by impurities in the supply. This is even more true for nitrogen or air expansions owing to the lower heat of vaporization. Indirect evidence for the fact that foreign particles do not play the customary role of condensation nuclei in moist air expansions rnay be seen by inspection of water vapor condensation data obtained in a great number of experimental facilities. Correlation of the results in Fig. 19 was possible without reference to the foreign nuclei present. A discussion of the size, distribution, and number of particles measured by the light scattering technique in a 'hypersonic air nozzle by McLellan and Williams [58] provides evidence that foreign matter nuclei do not enter the air condensation process. The situation is somewhat different for steam [31]. In expansions with small temperature gradients of industrial steam containing many foreign nuclei foreign nucleation may occasionally
CONDENSATION I N WIND TUNNELS
409
affect the onset of condensation. This is also apparent in cloud chamber experiments, whose time scale is much “slower” than that in nozzles [30]. The situation is different when we consider the number of nuclei produced by the condensation of vapor impurities which are present in hypersonic wind tunnel nozzles. Figure 6 shows that possibly H,O, CO,, and A will condense before the Mach number of equilibrium condensation for air is reached in a nozzle. The mere fact that liquid droplets or crystals of some substance are present when another vapor enters the coexistence region is not an obvious condition for condensation. A particular combination of the geometrical structure and size of the condensation nucleus and the molecular force field of its surface layer with respect to the condensing substance is needed to initiate the adherence of the condensate [6]. These conditions are difficult to predict in detail, and we must resort to our previous empirical experience concerning the effectiveness of carbon dioxide and water vapor droplets as condensation nuclei for nitrogen as shown in Figs. 25 and 26. We shall apply this experience directly to air since nitrogen is its major component. In particular, we would like to explain those observations where air condensation began at or near M,. At an early stage several investigators [59, 53, 61, 581 suspected that previous condensation of impurities initiates air condensation. Simple considerations indeed show that an extremely high number of nuclei can be produced. As an example, we can estimate the number of water droplets present a t the point of air condensation. By the definition of the mixing ratio, (2.3), we may write for the vapor density pv (5.7)
If we assume all water vapor to have condensed into N spherical drops per cm3, we obtain 4 pv = % n r 3 p ~ N .
The air density a t M , is (5.9)
and, with (5.7), (5.8), and (5.9), we find for the estimated number of water vapor droplets at M,, (5.10)
410
PETER P. WEGENER AND LESLIE M. MACK
For a given nozzle, x is known, and M , can be found for given supply conditions from Fig. 5. Figure 63 shows the results obtained from (5.10), with
x ~ 2 . 1x
2.6 x
10-3
10-4
,2.4 x 1 0 - 5
FIG.63. Estimate of number of water droplets present in a nozzle as a function of radius and mixing ratio a t M , (air) for Po = 7 atmos, To = 288 "K.
10 8
6
4
P" 43
2
1
FIG.64. Critical droplet radius as function of saturation ratio for air at 64 OK from the Thomson-Gibbs formula.
Po = 7 atm,
To = 15 "C, and a typical range of low mixing ratios. Application of (4.44) shows that condensation of such a small amount of water
CONDENSATION IN WIND TUNNELS
411
vapor cannot affect the pressure of the flow measurably in the water-vapor condensation range, 1.2 < M , < 2. The further growth of water droplets up to M,-4.2 cannot possibly be calculated accurately. For instance, Hansen and Nothwang [53] estimate the droplets at M , to have a radius of about 30 Angstrom, and Buhler [Sl] and Arthur [54] give such calculations in more detail. However, light scattering observations of McLellan and Williams [58] indicate droplet sizes of the order of 200-400 Angstroms a t M,. In their experiment carbon dioxide, whose normal concentration x = 0.46 g CO,/kg air, was present and would provide a number of droplets equal to the number shown in Fig. 63 for water vapor when x = 2.6 x In any case, Fig. 63 shows that from lo7 to lo1, centers of condensation are present a t M,, a number that vastly exceeds the number of nuclei due to foreign matter. Figure 64 gives the critical droplet radius of air at M,, obtained from (5.4),as a function of saturation ratio [63]. Since we know empirically that water vapor droplets are effective nuclei a t least for nitrogen, we can now match McLellan and Williams’ observed radii of 2 to 4 x cm with those of Fig. 64, and see that such drops already correspond to practically plane condensation surfaces. Therefore, we are not surprised to find very little supercooling of air in the larger nozzles with ordinary supply conditions. We are now able to classify the previously described experimental results for all types of condensation from the nucleation point of view:
A. Process Absence, or unimportance, of all foreign impurities and traces of vapors of lower vapor pressure. Collapse of supersaturated state due to self-nucleation in the condensing vapor. Degree of supersaturation controlled by initial conditions and time scale (nozzle geometry).
Examfdes: Water vapor in air (nozzle Mach number M Pure steam. Pure nitrogen.
< M , for air).
References: Moist air: [lo, 25, 26, 27, 28, 29, 33, 34, 37, 39, 40, 431. c Steam: [7, 31, 47, 48, 49, 50, 511. Nitrogen: [52, 53, 54, 551. B. Process: Large numbers of condensation nuclei present in droplet or crystal form because of previous condensation of vapor impurities such as H,O and CO,. Little or no supersaturation observed.
412
PETER P. WEGENER AND LESLIE &I MACK .
Examples: Air in the larger hypersonic nozzles with ordinary drying methods of supply air. References: [e.g., 53, 57, 581. C . Process Varying numbers of condensation nuclei present in flow, depending on vapor purity, self-nucleation, and time scale (nozzle geometry). Supersaturation controlled by processes A and B may range between both limits. Examples. Air in the smaller hypersonic nozzles. Nitrogen with varying degrees of impurities. References: [54, 55, 59, 60, 62, 63, 64, 661. 2. Spontaneous Nucleation
In the absence of foreign condensation nuclei of all kinds centers of condensation must be created in the supersaturated vapor itself. A new phase may appear first in microscopic dimensions without any change of the thermodynamic parameters of the macroscopic system. The phase change is then due to a local parameter change only. On the molecular scale, the number of molecules in any small volume of a gas will continually fluctuate as molecules enter or leave the volume. In fact, the very basis for a state of thermodynamic equilibrium is the transport of molecules and energy from one place in the system to another. This continual exchange is needed for the adjustment to equilibrium after each change of the macroscopic parameters. From this molecular interplay the steady state of a large volume of gas at uniform density results, with the entropy having a maximum value. This maximum value of entropy, however, may never be assigned to a very small subvolume in which we observe entropy fluctuations as a result of the changing number of molecules. Molecular collision processes in the presence of entropy fluctuations may then lead to accidental cluster formation in which two or more molecules may form an embryonic drop for some period of time. We call such a cluster a nucleus, or droplet of critical size, if it has reached a stable size in its given macroscopic environment. This stable size is given by the Thomson-Gibbs formula (5.4), or its modifications. When the Boltzmann relationship between entropy and the probability of a certain state is solved for the probability, we find it to be proportional
CONDENSATION IN WIND TUNNELS
413
to the Boltzmann factor, exp (- A S / k ) , where A S is the entropy decrease and k is the Boltzmann constant. Volmer [87] reasoned that this relationship, well-known in kinetic theory, is also applicable to the problem of formation of critical-size droplets in a supersaturated vapor. We can write the entropy decrease, A S , as
W A S = -,
(5.11)
T
where W is the work required to produce a droplet of critical size reversibly and isothermally. Our interest in the formation of a new phase in the vapor concerns the rate of production of stable nuclei per unit volume, J , which we may now write (5.12)
J
=K
exp
(- $)
=Kexp
(- % : ).
The Boltzmann constant is given by k = R/NA, where N A equals 6.02 x loz3 molecules per mole. The exponential term describes the probability of the formation of one stable drop, and K is a constant of proportionality which we expect to depend on the thermodynamic state of the vapor. Next we derive, on thermodynamic grounds, the work needed to produce a spherical liquid drop from the vapor phase, as first shown by Gibbs [91]. In the process of producing a droplet, work must be done against the surface tension to increase the drop surface area from zero to 4nr2. This work is
W , = 4nr20,
(5.13)
where (T is taken to be independent of the radius. Furthermore, during droplet formation the volume is increased from zero to 4nr3/3. From (5.1) we remember that the pressure inside the drop, p,, is higher than that of the surrounding vapor p,, by the amount 2alr. The work expended in the growth is, therefore, equal to the product of the final volume and the pressure differential between the inside and outside of the drop, (5.14)
8 w,= (p, - p") 45 ?zr3 =,nrz(T.
The net work required for the formation of the droplet is then (5.15)
W
=
4 W1 - W - -nr%. ,-3
The work needed to produce one stable droplet of critical size is, by (5.4), (5.16)
W=
16na3,u 3R2T2pLzln2y '
414
PETER P. WEGENER A N D LESLIE M. MACK
We observe that (5.15) may also be written as 1 W=-Ao. 3
(5.17)
Volmer [87] has shown that this result is also true for the formation of a stable solid cluster of molecules, or crystal of quite general shape. The surface tension must then be replaced by the free surface energy of the crystal in contact with a vapor. Some assumptions must also be made on the shape of the cluster [38]. Becker and Doring [lo21 point out that the droplet, or crystal, growth in the preceding analysis must be regarded as expressing the mean growth of the surface. None of the above considerations are applicable to single molecules, or even very small clusters, because of the thermodynamic nature of the reasoning. If there are only a few molecules in one cluster, the surface growth proceeds in a large discrete step when a single molecule is added or removed. Only a difference equation, such as that applied by Reed [99], can describe this initial cluster formation, while all other analyses treat a cluster as a uniformly extended mass of some arbitrary shape. Mathematical investigations of phase stability [e.g., 61 cannot be discussed here beyond a simplified physical model. If a drop of exactly the size prescribed by Thomson’s formula, (5.4), accidentally absorbs just one more molecule, the surrounding vapor will be supersaturated with respect to the drop, which will then grow. Conversely, if the drop loses one molecule by accidental evaporation, the surrounding vapor will be superheated with respect to the drop, which will then evaporate completely. Droplets of critical size, or larger, can grow, which leads to a collapse of the supersaturated state. I t is this metastable condition of the supersaturated state that leads to the collapse of the equilibrium with respect to the small nuclei formed according to (5.12) and explains the sudden condensation observed in those high speed flows where no particles of foreign matter or previously condensed vapor impurities are initially present. The rate of formation of clusters of critical size from (5.12) and (5.17) is (5.18)
where A* is the .surface of a cluster in unstable equilibrium with a given environment. For the special case of spherical liquid droplets of critical size, whose work of formation is expressed by (5.16), we obtain as nucleation rate (5.19)
J=Kexp
[
16n N A
-__ 3-
0
1
Rd(FY($)21nY(p[ipmll
CONDENSATION IN WIND TUNNELS
415
or
J
(5.20)
=K
[
($(:)&]
exp - 17.49 -
According to (5.20) there is a completely defined nucleation rate for a given supersaturation. The very strong dependence of J on the supersaturation q.~may be illustrated, as done by Oswatitsch [33], for a moist-air expansion in a supersonic nozzle with an atmospheric supply. In terms of the supercooling rather than the supersaturation, we compute (for some assumed value of K ) that for A T = 30 "C less than one nucleus is formed per cm3 for one centimeter of travel in the flow direction. For A T = 40 "C, this number is lo6, and for AT = 50 "C, it is loll. In other words, the moist air would have to travel about 10 meters at slightly supersonic speeds to contain lo3 nuclei/cm3 if A T = 30 "C, while for a supercooling of AT = 40 "C, the same number is formed after millimeter travel. For AT = 50 "C, 108 nuclei would be produced for every millimeter travel. In Section I1 we have seen that this is indeed the range of supercooling observed. Obviously, the number of 106 foreign condensation particles per cm3 is insignificant with respect to the nuclei formed statistically in the vapor itself. Any uncertainty in the knowledge of K in (5.20) simply shifts the location of rapidly increasing nucleation rate to a slightly different flow temperature. This shift will be small with respect to the extension of the collapse zone of the supersaturated state in the flow direction. The same uncertainty appears in the prediction of delayed condensation in cloud chambers [%I. Becker [88]proposes the following simple physical picture for a rough estimate of the determination of K . Let us consider nucleation as a game in which each collision of two molecules can lead to the formation of one nucleus. The Boltzmann factor may then be regarded as an expression for the chance to win, i.e., the probability that a given collision actually results in a cluster formation. K must then simply be the number of gas-kinetic collisions per unit volume and time (this result has also been stated by Volmer [87]). To gain insight into the magnitude of the various terms, we estimate on this basis the minimum saturation ratio q ~ ,below which no condensation in pure water vapor could take place in the absence of foreign nuclei. In Figs. 20 and 21 we saw that steam condensed in Binnie and Woods' nozzle 50 "C and a temperature slightly above room temperature. At a t AT, normal pressure and temperature, the collision frequency of the molecules is p 1O1O sec-1 and NL = 2.69 x 101g/cm3is the number of molecules per cm3. From Becker's argument we find K = pNL loas sec-l ~ r n - ~ .Inserting the proper numerical values for water vapor into (5.20), we obtain N
N
(5.21)
-
416
PETER P. WEGENER AND LESLIE M . MACK
-
We may now assume J(lng,) 2 , say, as the beginning of nucleation, since we saw before that a “reasonably” arbitrary choice of J would result only - ~ in (5.21), in a small error in this estimate. To obtain J = 1 ~ m sec-l we must have ln2g,= 1.7, or loglog, -0.6. Indeed this value of the 300 OK supersaturation, g, 4,is in the range of the observed values a t T in Fig. 21. Historically, the first successful attempt to solve (5.18) for the case of liquid droplets formed in the vapor was made by Volmer and Weber [103]. In this treatment the simplifying assumption was made that the macroscopic state of the vapor is maintained in the system by adding a number of molecules equal to the number absorbed by every drop that has reached t h e critical size given by (5.4). Furthermore, it was assumed that every molecule striking a small cluster is absorbed when its center of gravity has penetrated a spherical shell about the drop in a stand-off distance of the order of the molecular diameter. Finally, all evaporation was prohibited. For the first assumption to be valid, the droplet growth must be slow with respect to the time scale of molecular collisions. Ordinarily, this assumption of a quasisteady state is valid in flow processes, since the time scale for molecular collisions is of the order 10-lo seconds. The thermodynamic treatment requires first the knowledge of the Boltzmann distribution of molecular aggregates of all sizes. Throughout the entire system of droplets and vapor, there is then steady transport of molecules from the vapor phase into clusters, which are removed after they reach the critical size and the number of molecules in the clusters is replaced by vapor. Volmer’s formula was improved by Farkas [lo41 on the basis of kinetic, rather than thermodynamic, calculations, and his results are equal to Volmer’s for the first instant of production of nuclei as long as the boundary conditions have not changed. This work was again improved by Volmer [87]. Volmer’s final expression, in which he has left out a number of terms that approximately cancel each other, is
-
(5.22)
-
K
ZDv
(-)
=2
r*
3W nkT
[cm-3 sec-11.
2 is the number of molecules per unit volume, kept constant in the system,
v, is the volume of one liquid molecule, and D is the number of molecules impinging on unit area per unit time. In Volmer’s treatment the condensation coefficient, a, which gives the ratio of molecules absorbed in the liquid drop on impact to those not absorbed has been set equal to one, although physically it is possible that 0 a 1. W has previously been obtained in (5.16)Volmer shows further that
< <
(5.23)
Dv, Y*
-
1 t*
417
CONDENSATION I N WIND TUNNELS
expresses the reciprocal time of quasi-steady formation of one critical droplet. According to kinetic theory [e.g., 941, for a Maxwellian velocity distribution i n a gas,
D=
(5.24)
P (2nm,kT)ll2 '
where m, is the mass of one molecule. In (5.24) we may set p = p , for the equilibrium vapor pressure of the condensable vapor with respect to stable droplets. A single molecule is regarded in this thermodynamic treatment as a liquid sphere of volume v, = 4nrn3/3, where Y, is the molecular radius. Thus the time of formation of one drop in the quasi-steady state follows from ( 5 . 4 ) ,(5.23),and (5.24) as (5.25)
To get a physical feeling for the magnitudes involved, we turn to our example of steam condensation at normal pressure and temperature, with pk-4. From (5.4) we find the radius of a stable critical droplet to be Y* 8 8, or y * / r , 3.5, where Y,, 2.3 8 for H,O. This critical droplet contains about 400 molecules, and for our estimate we may omit the surface seconds. tension correction (see Fig. 61). From (5.25) we find z* This time is indeed short with respect to the time scale of most expansions in the supersaturated region. With
-
-
-
-
(5.26)
where A ,
= 4nyn2is
the surface area of the molecule, we may rewrite (5.22)as
(5.27)
When we insert numerical values from our steam example in (5.27) we find that (5.28)
which reduces (5.27) to (5.29)
K = ZDA,.
418
PETER P. WEGENER AND LESLIE M. MACK
Expression (5.29)gives again the number of gas-kinetic collisions of the vapor molecules per unit volume and time; hence the first guess a t the factor K was surprisingly accurate. We may rewrite J by inserting (5.22)in (5.20). Expressing all factors in terms of directly measurable quantities and universal constants [94] we obtain Volmer's second nucleation rate equation [87]
(5.30) or In J
= 59.82
+ 2 In
2
+ In
(5.31) where p , must be inserted in millimeters mercury. In (5.31)we have an expression for J that can be used in the condensation rate equation (4.32). Equation (5.31) has been successfully applied to the prediction of condensation in cloud chamber experiments by Volmer and Flood [lo51 who studied water vapor and several organic vapors. Other successful comparisons with cloud chamber results have also been made [30,38, 1021. In every instance, the predicted critical supersaturation was slightly below the observed one. However, the initial condensation is not necessarily visible, and some fog must have accumulated for observation. Many subsequent efforts have led to improvements of the kinetic foundation of (5.31). Becker and Doring [lo21 recognized the importance of an undetermined constant in Farkas' [lo41 treatment and they calculated K by an ingenious electric network analogy, including accidental vaporization during the formation of a nucleus. In particular, they determined (5.22)for crystal formation, a problem on which Volmer [87] had also worked extensively. Other theoretical work for drops and crystals was done by Frenkel [6], Stranski and Kaischew [log],Zeldovich [107], Sander and Damkohler [38], and others. The most recent refined nucleation rate equation by Kuhrt [108], where small clusters were treated as giant molecules, has not yet been applied to calculations of condensation in high speed flows. The bulk of these efforts was directed towards more rigorous expressions for K , yet many of the final results give numerically nearly equal answers for J . Sander and Damkohler [38] have shown that Volmer's second equation, (5.31),is practically equal to Becker and Doring's [lo21expression, and Probstein [log] has reduced Frenkel's modifications [6] also to Becker and Doring's expression. Since we have seen that the nucleation rate is remarkably
419
CONDENSATION IN WIND TUNNELS
insensitive to fine points in the derivation of the K factor, we shall be content with Volmer’s second equation for our purposes, when transition to a liquid is suspected. In Fig. 65 the nucleation rates for water vapor, nitrogen, and oxygen in inert nitrogen are shown. The results for H,O shown [4] were computed from (5.31) and are in numerical agreement with those calculated by the Becker-Doring method [lo]. The 0, nucleation rates in inert N, [a] and N, nucleation rates [110, 11 13 are also based on Volmer’s work. Figure 66, taken from [4], shows H,O nucleation rates based on (5.31) in more detail. 6
5-
,+ “20
J = 10”
J=lOO
50
I00
150
200
250
FIG.65. ru’ucleation rates from Volmer’s equation, (5.31), for different vapors.
Further calculations may be found in the aeronautical literature where the Becker-Doring method, or slight modifications, has been used for H,O [33, 34, 731, “air” [53], and 0, and N, [64]. The Volmer method has been used for H,O [28], and 0, in air [l6, 112, 1131. Rather than investigate K in more detail, we should direct any criticism of nucleation-rate expressions to the important exponential term, and to those inherent assumptions which may be poorly fulfilled in rapid nozzle expansions. For very rapid expansions, we might first question the assumption of quasi-steady state for. which J was determined. A study of the diffusion process that leads to the build-up of the steady-state nucleation rate was carried out by Kantrowitz [114], who applied his result to the extremely rapid expansion shown in Fig. 17. Using certain assumptions on the value of the condensation coefficient he obtained qualitative agreement with the trend of these data. Probstein [lo91 extended Kantrowitz’s treatment and also included a first approximation to the unsteady solution for the work term (5.17). His numerical calculations indicate that for H,O and N,, with a reasonable choice of the condensation coefficient, the steady-state nuclea-
420
PETER P. WEGENER AND LESLIE M . MACK
tion rate is attained after at most a few microseconds. This is a slightly longer duration than estimated by us with (5.25) based on the quasi-steady build-up of one droplet. Gilmore [98] obtained solutions to the unsteady problem, and it appears from this and Wakeshima’s work [115] that the steady-state assumption of Volmer et al. is perfectly adequate for the treatment of condensation in nozzle expansions. Our major problems concern the accurate determination of the terms in the expression for the work needed to form a stable droplet. In Section I11 we saw that condensation occurs in many cases a t temperatures below the triple point, and it is by no means evident whether supercooled liquid drops or crystals will first be formed. In any case, the choice between surface tension and free surface energy of crystals must be made. Even if we stipulate liquid drop formation, we see that the nucleation rate is tremendously affected by the value of 0 that occurs in the third power in (5.19). I t is also important to know the liquid density accurately. There are no direct experiments available on these two properties for the vapors and temperature ranges of interest to us, and we must, rely on extrapolations such as (5.2). When we make use of a semiempirical expression for the behavior of surface tension with temperature, we may collect the poorly known properties in the work term of (5.19) as
d+)
2/3
(5.32)
for the purpose of extrapolating to lower temperatures [28]. For water vapor, d(T) ia a nearly linear function for 265 < T < 315 OK, which may be extrapolated to T < 265 OK. In addition, we must face the problem of the dependence of surface tension on droplet radius discussed previously. In our steam example, 8 A . In there were about 400 molecules in the critical droplet when r* Fig. 62 the critical radii are much smaller for water vapor condensation in a very rapid expansion of moist air. For Y* = 3, 4, and 5 A, we have only about n = 2 , 5 , and 10 molecules, respectively, per critical drop. We therefore expect our continuous growth ideas to fail a t some 12 as shown previously. In this case (5.4) should be replaced by (5.6). We also doubt that expressions like (5.5) give surface tension correctly for extremely small drops, which forces us to adopt methods such as those used by Reed [99]. A 10% error in surface tension in (5.31) results in an error of 1000 in the nucleation rate of water vapor at 0 “C, and Head [lo] has shown that neglecting the effect of drop curvature on surface tension results in the prediction of nucleation rates that are too low a t a given supersaturation. Figure 66 also gives the nucleation rates for water vapor which was calculated by Head with the Becker-Doring method, but including Tolman’s surface
-
42 1
CONDENSATION I N W I N D TUNNELS
tension correction (5.5); the difference in the results with and without the surface tension correction is substantial. Calculations of nucleation rates, including a surface-tension correction, may be carried out by first finding r* from (5.4) for a given T and pl. Then am is given for the temperature in question by (5.2),or a table, and a, is found from (5.5)with 7 = Y * . Finally, a, is inserted in (5.31) for the given pl and T . 5
I 20=10gJ
4
-
I70
2 50
200
2 70
T(OK) FIG.66. Nucleation rates from Volmer’s equation, (5.31). for water vapor (solid lines), and from Becker and Doring’s equation including a Tolman surface tension correction (dashed lines).
3. Droplet Behavior
Once droplets of critical size are formed, they are carried with the flow and continue to grow. If these droplets lagged appreciably with respect t o the flow velocity, W , their drag would result in an entropy increase. Such a lag has not been included in our system of equations, (4.4),(4.13),and (4.25). We now derive an expression for the limiting acceleration of the flow for which the assumption that the droplet velocity, w,, is approximately equal to ze, is still justified [33]. Since the Reynolds number based on droplet radius and relative droplet velocity is less than one, we have for the drag of one drop (5.33)
4 D,= 6 n q ( ~ w,)7 = -Z 3
73pL
dW,
-, at
422
PETER P. WEGENER AND LESLIE M. MACK
where we have used Stokes' law [ll6], and the fact that the drag must equal the mass of one droplet times its acceleration. The quantity q is the gas viscosity. With dw,/dt = w,dwJdx, we obtain from (5.33), (5.34)
dw, dx
-
9 2 pL
If we assume that dw,ldx and
E G
1 wy2
(w,
')-
w,/w are both constant, then
(5.35) When we require the drop and flow velocity to differ by not more than 1%. or E = 0.99, and insert numerical values in (5.35) suitable for water drops carried in air a t approximately room temperature, we have r2 dwldx
-
cm2sec-l.
From this we estimate: r (cm)
d w / d x (sec-I-)
10-2
lW1 10s 107 10"
10-4
10-6 10-8
In a small nozzle with large velocity gradients, such as Head's (Fig. ll), dwldx is of the order of 103 sec-l in the region where condensation is observed. The droplets formed in this region have a radius much smaller than 7 = cm, which is the maximum permissible droplet radius for less than 1% velocity difference. Since steam, nitrogen, and air condensation ordinarily occur a t nozzle locations where dwldx < 103, we conclude that we may safely assume that the droplets move a t all times with the velocity of the gas. During the expansion and condensation processes, the droplets move into regions of different temperature. The questions arise whether the surface temperature of the drop attains equilibrium with respect to the surrounding vapor in a time that is short with respect to the time scale of the flow process, and whether temperature equilibrium in the drop itself is quickly established. Stodola [31] found that for the time scale of steam expansions temperature equalization within the drop and between the drop and the surrounding steam is indeed extremely rapid, which permits the application of equilibrium formulas such as (5.3) and (5.30). Oswatitsch 1331 has discussed the time of temperature adjustment of a small water drop in the surrounding air. He
CONDENSATION IN W I N D TUNNELS
423
called the time in which an initial temperature difference between the drop and a remote small air volume is reduced to half its original value, t(l/2). Dimensional analysis gives for this time p L CL
(5.36)
t(4) = const. __ k y2,
-
where k is the thermal conductivity of the air. With the constant equal to 0.231 by Oswatitsch's calculation, we find that t(1/2) lo-* and lo-* seconds for droplet radii of and cm, respectively. Buhler [61] demonstrated that for liquid nitrogen drops of Y = cm the temperature difference between the drops and surrounding nitrogen vapor is reduced to less than 10% of the vapor temperature in 10" seconds. He obtained for this time the approximate formula, t(lOyo) = 0 . 4 ~ where , Y is in cm. For a water drop cm (containing < 105 molecules), the maximum possible temof Y < perature difference in the drop is less than 0.02 "C [98]. We conclude, therefore, that in all situations of high-speed flow condensation temperature adjustment is practically instantaneous. We now relate the initial drop radius, yo, to the molecular mean free path of the condensing vapor in the thermodynamic state where the droplet begins to grow. The mean free path, 1,can be computed from the viscosity [e.g., 941 (5.37)
q
=
tp4
where (5.38)
is the average speed of the molecules. By (5.37), (5.38), and the perfect gas law, we can express 1 in terms of pressure and temperature as (5.39)
For water vapor in air, p in (5.39) refers to the partial pressure p,, and all collisions with air molecules are simply neglected. When we estimate 1 from (5.39) a t the point where condensation begins, we find for steam (Fig. 21) that 1 cm; for nitrogen (Fig. 22), I cm; and 5x cm. In all for water vapor in air (Fig. 11, Curve 4), 1 (H,O) three processes, a t the inception of condensation, we estimate from (5.4) that yo = Y* lo-' cm, and lo2< l/ro< 103. Therefore, the initial droplet growth must be computed on the basis of free molecular flow, for which 1 >> Y. Farther downstream, 1 Y,and the droplet growth must be computed from macroscopic considerations.
-
-
-
-
424
PETER P. WEGENER A N D LESLIE M. MACK
The following remarks are first restricted to pure vapors, and the nature of the initial droplet of radius r0 remains unspecified. It may have been produced by self-nucleation, previous condensation of a trace of vapor, or it could be a nucleus of foreign matter. Although we shall first follow the growth of a single droplet, we must bear in mind that the expressions to be derived will be valid only in the average taken over an actually large number of droplets. Finally, the laws of droplet growth are strictly valid only if the given thermodynamic environment is unchanged during the period of growth. The application of these laws to the rapidly changing states in an expanding flow results in a quasi-steady analysis. For free molecular flow we can determine the mass added to a droplet from the rate of impingement given by (5.24) [e.g., 33, 1111. With (5.38) and the equation of state, we find for the mass impinging on unit area in unit time (5.40)
Dmn= f p v c ,
where m, is again the mass of one molecule. We can write (5.40) in terms of the speed of sound as (5.41)
The mass increase of a spherical drop with volume v (5.42)
=
4nr3/3 is given by
dG -
dv dv dr dr - P L - = p~ - - = p~ 4nr2 - .
dt
dr dt
dt
We can express the same quantity by (5.43)
dG - = ctDmn4nr2, dt
where ct, the condensation coefficient expresses again the fraction of the impinging molecules that actually stick to the drop surface. From (5.41), (5.42), and (5.43), we obtain for the increase of the drop radius per second,
(5.44)
With the quasi-steady assumption, dt Mach number, we have finally (5.45)
= dx/w,
and the definition of the
425
CONDENSATION IN WIND TUNNELS
suitable for calculations in a nozzle (the quantities on the right hand side are now functions of x ) . In obtaining (5.45),we have used the condensation coefficient and thus avoided discussion of the detailed molecular processes a t the drop surface. Oswatitsch [33] and, particularly, Buhler [61] have given a refined treatment of droplet growth. Buhler’s quasi-steady solution, which includes an expression for u, is (5.46)
5
.
-
I
II
I
5
6
,
I
I
7
8
9
M FIG.67. Droplet growth rates in air for r < 1, as a function of Mach number from (5.46) as given by Buhler [61].
where T,, the surface temperature of the drop, may be chosen as the equilibrium saturation temperature of the vapor with respect to the drop. Droplet growth in air, computed from (5.46) by Buhler, is shown in Fig. 67. Buhler also derived an approximate solution for the non-steady case, which results in appreciable differences in the initial phases of growth. Once the droplets are of the order of the mean free path, further growth is governed by macroscopic processes. The drop absorbs the heat of vaporization when condensation takes place on its surface, and its growth will be limited unless this heat can be carried away by conduction, since radiation
426
PETER P. WEGENER A N D LESLIE M. MACK
is negligible for the small temperature differences involved. balance may then be written as
The energy
(5.47)
where k is the heat conductivity of the vapor, and 7 is a radial coordinate. At the drop surface 7 = r , and if we assume a linear temperature gradient a t the drop surface, we may set Ts - T r
(5.48)
which is in accordance with the accurate solution [ l l l ] . With (5.42) and (5.48), we can then write (5.47) as (5.49)
or also (5.50)
The important difference between the molecular (5.44) and the macroscopic (5.49) growth laws is that in the latter case the drop radius grows with the square root of the time in a constant environment. For a complete calculation of the condensation process, we must decide which growth law to use in (4.32) by comparing the order of magnitude of r with 1 a t every position in the nozzle. However, the x-coordinate a t which the changeover is made may be somewhat in error without seriously affecting the final result because of the small size of the droplets a t this point. When we apply the droplet-growth laws, on the other hand, to condensation of air, which takes place on nuclei produced by previous condensation of vapor impurities, it will be much more important to know the number of nuclei available a t M , than precise knowledge of the changeover point [61]. When we estimate the droplet growth for moist air, rather than a pure vapor, we use pv and y for water vapor in (5.45). I n the macroscopic growth law (5.50),the heat conductivity of air is used for k. More discussion of both laws and other variants for moist air condensation are given by Oswatitsch [33] and Wakeshima [115]. There is considerable uncertainty in choosing a suitable value for the condensation coefficient a in (5.45). I n the formation of clouds, it is found that a 0.03 [e.g., 1171, but there is reason to suspect that the value is 0.3 or higher in situations where extreme supersaturation is observed [log, 114, 1151. On the other hand, a may be adjusted to fit the data for a given
-
CONDENSATION IN WIND TUNNELS
427
experiment. In particular, this must be done when the molecular growth law must be applied without knowing whether we are dealing with liquid drops or crystals. There are no known direct physical measurements on droplet growth in the molecular regime. Drop growth in the macroscopic region has been indirectly measured in cloud chambers [30]. The velocity of the drops after the expansion is measured; and, assuming that they fall with their terminal velocity, Stokes' law (5.33)is used to compute the radius.
FIG. 68. Droplet growth of water droplets in air in a rapid expansion from (5.45) and (5.50).
Hazen [118] found good agreement of (5.49) with experiments on water and alcohol droplet growth in permanent gases, and Barrett and Germain [119], in particular, studied water drops in air. They found that from 0.2 to 0.6 seconds after the expansion in their cloud chamber, r 2 varied linearly with time. They predicted, from an expression similar to (5.49), a growth value of 4.4 x 10" cm2/sec, and found experimentally that 7.2 x 10" < dr2/dt < 7.8 x 10" cm2/sec. During the initial 0.2 seconds r < 2, and the drops grew more slowly; however, this may have been because they had not yet reached their terminal velocity. Frossling [117] also found good agreement cm. As an of (5.49) with the growth in clouds of large drops of r 2 example for an extremely rapid expansion we show in Fig. 68 the results
428
PETER P. WEGENER AND LESLIE M. MACK
of a calculation with both laws for water drops in a small moist air nozzle (Fig. 17). The expansion started from atmospheric supply conditions, and the mixing ratio was 12g/kg. The starting point of the growth estimate was arbitrarily chosen at J = 1 in (5.31), and ro = r* 7 A. We took a = 1, and used L for ice. The location of observed condensation is indicated,
-
and up to that point all flow parameters could be computed from an isentropic expansion. The droplet radius from (5.45) does not reach ilbefore condensation sets in. We shall see later that because of this and other difficulties no quantitative prediction of the onset of water vapor condensation in moist air expansions is yet possible.
4. Comparison with Experiment
Self-nucleation: In the aeronautical literature attempts have been made to predict the onset of condensation in high speed flows by determining a so-called “critical nucleation rate”. From the definition of the saturation ratio and (1.9), we may write (5.51)
where p, is the partial pressure of the condensing vapor, or p, = p in pure vapor expansions. Using (5.31)and (5.51),we can plot curves of J = constant in a p,T-diagram and compare thern with experimental results of observed condensation, such as those in Figs. 23 and 34. Criteria like Jcritical = lo3 were then sought after to define the thermodynamic state of the onset of condensation quite generally. Such criteria have no physical meaning whatsoever. The point in a nozzle where a vapor has condensed to an extent that measurable heating effects occur, can only be determined by the set of equations, (4.4), (4.25), (4.26), (4.27), and (4.32); and, therefore, only a complete discussion of the history of the flow is physically meaningful. We have further seen in our discussions of various condensation processes that use of (5.31) or its variants in the condensation rate equation (4.32) is justified only in expansions of pure vapors where no condensation nuclei from any other source are present a t saturation. However, we may use (5.31) and (5.51) to help us determine the validity of application of selfnucleation rate calculations. To demonstrate this, we show in Fig. 69 nucleation rates for 0, in N, [a] in comparison with air condensation experiments from Fig. 34. The pressure has been converted to oxygen partial pressure by Po, = 0.2 p. The nucleation theory is obviously not applicable, since condensation was observed for 0 < J < 1 ; we again conclude that self-nucleation plays no role in the air condensation process.
429
CONDENSATION I N W I N D T U N N E L S
Stever and Rathbun [64] computed nucleation rates, including a surface tension correction. When we apply these results or those of Charyk and Lees [110] without such a correction to condensation experiments in pure nitrogen (e.g., Fig. 23), we find that the experimental location of condensation coincides with high nucleation rates. Therefore, we are justified in believing that condensation in pure nitrogen expansions occurs as a result of selfnucleation, and the application of (5.31) in (4.32) is warranted. Unfortunately, such calculations have not been performed. However, we have seen, for these two contrasting examples, that the application of nucleationrate expressions to observed condensation states assists us to classify the physical nature of the processes, even though they do not produce quantitative predictions of the location of condensation.
30
40
50
60
70
T(OK) FIG.69. Comparison of nucleation rates of 0, in K, from Volmer’s equation with experimental results on air condensation.
Steam condensation: The only successful complete kinetic and thermodynamic prediction of a condensation process due to self-nucleation in supersonic flow was carried out for steam by Oswatitsch [34]. He employed a set of equations similar to those in Section IV, Becker and Doring’s nucleation rate expression, and suitable droplet-growth laws. In Fig. 70, we show his results as computed for Yellott’s experiments [48]. The arrows indicate the extent of the condensation zone, xk to xe, found from pressure distribution measurements. Oswatitsch calculated the rate of change of the number of nuclei, i.e. essentially the nucleation rate, as a function of distance from (4.34) and the nucleation rate equation. In the first part of the condensation zone, the production rate increases rapidly and then falls to zero because, with condensation, the supersaturation‘ and, therefore, the J-rate drops. It is this first steep increase in the nucleation rate which is decisive, since these
430
PETER P. WEGENER AND LESLIE M. MACK
are nuclei that grow to appreciable size and contain the major part of the condensed liquid. Additional large numbers of nuclei formed further downstream contribute little to the condensed mass fraction, owing to their small size [lo]. Also shown as a function of distance is the actual number of stable drops present. This number likewise increases enormously in the initial phase of the process and remains constant after the production rate has dropped to zero. Finally, the total mass fraction condensed, g, computed from the amount of liquid in all droplets, increases rapidly throughout the condensation zone. After x, has been reached, the existing drops are unchanged in number and they grow in thermodynamic
I 0.4
/;
.I
01
0.6
0.8
1 1.0
P
rk
I
I
I
I
1.2
1.4
1.6
1.8
2.0
xkm) FIG.70. Kinetic calculations of Oswatitsch applied to Yellott’s experiments on condensation in a steam nozzle. (-) Production of critical droplets per unit length. (- - -) Number of stable droplets present per unit volume. (- .-) Mass fraction of liquid phase.
equilibrium as the expansion proceeds. This stepwise solution of the kinetic and thermodynamic equations also yields the complete pressure distribution. The result calculated by Oswatitsch [34] for Binnie and Woods’ [50] experiment is shown in Fig. 71. The agreement between theory and experiment is excellent. We have previously linked these experiments with those by Head in moist air in Fig. 21 by a curve for log J = 26 (calculated from Becker and Doring’s result with the Tolman surface tension correction by Head). The success of the kinetic theory in the steam case is due to three factors. First, we see in Fig. 21 that Tk is relatively high, and we expect only water droplets (rather than ice crystals) to be formed. This permits us to use
431
CONDENSATION I N WIND TUNNELS
immediately (5.31) or a similar expression. Furthermore, we saw previously that for the steam experiments the critical size drops, according to (5.4), have diameters of the order of 10 hi and contain about lo2 molecules. Therefore, surface-tension corrections, with all their inherent uncertainties, are unimportant. Finally, again because of the relatively large initial droplet size and the high density and short mean free path of the steam, the major part of the drop growth follows the well-established macroscopic dropletgrowth law, (5.50). The heat conductivity of the steam like all other properties needed is well known in this temperature range. 1.01
I
I
I
0.8
0.7 0.6
/
P 0.5 pol 0.4 0.3
eThroat
I
* I
'I
1.
I .
I *
I
*
' B
-
I
I
,
0 .I - 6 - 4
*
-2
0
Isentropic Expansion
o
pol :1.44atm,To:42OoK 2
4
6
8
101
x (cm) FIG.71. Predicted pressure distribution in a steam nozzle with condensation. Theory: Oswatitsch [34]. Experiment: Binnie and Woods [ 5 0 ] .
Condensation of water vapor in air: Comparisons of experiments on condensation in moist air with the kinetic and thermodynamic theory of self-nucleation have so far been quantitatively unsuccessful. Oswatitsch [34] has computed pressure distributions, similar to those in Fig. 71, for his experiments in a small nozzle. He was able to correlate the point of deviation from the isentropic expansion for three different relative humidities when he adjusted the surface tension in the nucleation rate expression to fit the experimental results. The further trend of the collapse, however, could not be calculated. He had to choose values of the surface tension, or free surface energy, in the range expected for ice crystals. From Fig. 18 we see that for higher degrees of supersaturations condensation occurs a t Tk< 210 OK,
432
PETER P. WEGENER AND LESLIE M . MACK
a temperature below which Sander and Damkohler [38] believe they observed the first formation of ice crystals in their cloud chamber experiments. Wegener [29] applied (2.31) to a condensation zone and found that agreement with the experiment could only be obtained if the latent heat for ice was chosen, which gives another hint that ice crystals are probably first formed in the nozzle. Under these circumstances, the application of nucleation rate and crystal growth expressions is indeed extremely difficult.
Theory 0.50
0.40 I
I
0
5
I
10 15
,
0.34 I
,
20 25 30 Microsec.
FIG. 73. Pressure rise through the condensation zone in moist air. Theory: Wakeshima [115]. Experiment: Head [lo].
Wakeshima [ 1151 applied kinetic and thermodynamic calculations to one of Head’s experiments (Curve 3, Fig. 11) and computed the pressure increase with respect to the isentropic pressure distribution. He took the surface tension as independent of the droplet radius and u equal to one; it appears from Fig. 72 that the errors due to these simplifications roughly cancel each other. Wakeshima also applied unsteady solutions for the nucleation rate to this experiment, and he found that about 2 x seconds elapse before steady-state nucleation occurs. This time is short with respect to the duration of the condensation process, and we see again that the quasi-steady nucleation equation is sufficient. Oswatitsch [33] obtained a semi-empirical equation, including nucleationrate and droplet-growth considerations, which determines, for a given initial state, when condensation will first be noticeable. His final equation, in our notation, is (5.52)
(- )-.i.
1039pn5exp
WNA = const.
(g) 4
(CGS units).
The individual terms of (5.52) may be plotted as a function of temperature, and in this manner Tkcan be found. From his own experiments, Oswatitsch chose the constant equal to 2.2. The interesting feature of (5.52) is the strong dependence of the condensation temperature (or supercooling) on the temperature gradient of the flow. We previously noted this dependence in empirical terms, and now find it verified by kinetic considerations.
CONDENSATION I N W I N D TUNNELS
433
Air and nitrogen Condensation: Buhler [61] has given a simplified theory of the collapse of the supersaturated state that is applicable for air and nitrogen expansions, such as the one shown in Fig. 31. This theory requires a knowledge of the number and average size of the nuclei present a t the point in the flow where the observed expansion deviates from the condensation-free isentrope. Once this starting condition has been determined by optical measurements [58] or by a self-nucleation calculation for the vapor impurities from (5.30) or by an estimate of the nuclei present from the measured mixing ratio and (2.3) and (5.10), the further growth of these nuclei must be calculated. Buhler was able to derive an equation for this growth, essentially based on the droplet-growth law (5.46); hence he could compute g ( x ) and the pressure. The calculated pressure was found to be in general agreement with the experimental results [54, 551.
VI. .EXPERIMENTAL METHODS
1. Effect of Condensation on Measurements i n Wind Tunnels
The effect of water vapor condensation on the flow in a wind-tunnel test section can be estimated from the condensation shock analysis of Section IV, 2. For this one-dimensional analysis to be applicable, the condensation region must be nearly normal to the flow direction. In addition, we assume that almost all of the water vapor is condensed out a t the condensation shock in order to have an adiabatic flow between the shock and the test section. Such a treatment was first given by Heybey [32]. Because of the entropy change in the condensation shock, the throat area downstream from the shock that is required for sonic flow is changed. We can determine this new area from the continuity equation
(A*p*a*)l = ( A * ~ * u * ) ~ the equation of state, and the speed of sound relation (4.21), and obtain
Since po2< pol, and Q > 0, A,* is larger than the actual nozzle throat area A,*, and the Mach number at a fixed position in the nozzle behind the condensation shock is, consequently, reduced. The sonic-throat area ratio, (6.2), is a convenient parameter to represent the strength of the condensation shock, and we have plotted it against M , for several Q/Qmaxin Fig. 73.
434
PETER P. WEGENER AND LESLIE M. MACK
We can obtain an expression for M3, the Mach number in the test section when there is condensation upstream in the nozzle, n terms of A2*/A1* from the isentropic area relation (3.4). The result is
1.0
I .4
I .8
2.2
2.6
MI FIG.73. Sonic throat area ratio across condensation shock as a function of Mach number.
where M is the Mach number when there is no condensation. From (6.3) we have plotted, in Fig. 74, the ratio M 3 / M as a function of M for several values of A2*/A,*. We observe from the figure that in a given nozzle with a fixed condensation shock, the Mach number change due to condensation becomes less with increasing distance downstream from the shock. We can obtain the static pressure with condensation, p3, from (1.14) and (6.2). The result is
CONDENSATION IN WIND TUNNELS
435
where p is the pressure with no condensation. We have plotted the left hand side of (6.4) in Fig. 75 as a function of M for several values of A,*/A,*. The same effect as noted for M3 is evident. With increasing distance along the nozzle, the effect of the condensation shock on the pressure becomes smaller.
M
FIG. 74. Effect of water vapor condensation on test-section Mach number.
We now have a method to determine the flow in the test section for a given humidity of the supply air. From the supply relative humidity and nozzle temperature gradient, the supercooling can be estimated from Fig. 19, or the shock position obtained directly from (2.37) in terms of the mixing ratio. This establishes the Mach number M,, which we may take equal to M k , at which the condensation shock occurs. Some of the available experimental information on the shock location has been used by Wyker [120] to construct a chart for the prediction of M,, knowing Po, To,w,, and dTldx. For q, we can take either Lw, (approximately L x ) , or qmX, whichever is smaller. However, if L x is appreciably larger then qmax,or if T2is sufficiently high to require the presence of a large amount of vapor for saturation, the analysis cannot be valid since a significant amount of vapor will condense downstream from the shock. With q known, and Qmax found from Fig. 41,
436
PETER P. WEGENER A N D LESLIE M. MACK
we obtain p,,/fi,, from Fig. 45, and A2*/A,* from Fig. 73. Then (6.3), or Fig. 74, gives the Mach number in the test section, and (6.4), or Fig. 75, the static pressure. Condensation effects are completely avoided in a wind-tunnel test section when M , is greater than the test section Mach number. The previously mentioned chart of Wyker and a nomograph constructed by Smolderen [lSl] can be used to find the allowable humidity of the supply air to meet this requirement. However, for M > 2 , practically impossible degrees of dryness are necessary, and a more important quantity is the maximum humidity which results in a negligible effect on the measurements in the test section.
FIG.75. Effect of water vapor condensation on test-section static pressure.
This humidity is expected to be quite low, with the result that the condensation shock occurs at a high enough M , for qmx to be much larger than Lx. Also the temperature T , is low (for unheated supply air), and the assumption that all of the water vapor condenses a t the shock is well satisfied. With Q << Qmax, we can simplify the condensation shock solutions by expanding in powers of Q/Qmax and retaining only the first term [lo]. Then, with the assumption of isentropic flow downstream from the shock, simple expressions are obtained for the static pressure and Mach number in the test section. These formulas have been obtained by Monaghan [122] and Smolderen [121], and are
(6.5)
M - M M , 1 (1 M =BQ
+ y M ? ) (1 + '%M2 M2- 1
I,
CONDENSATION IN WI ND TUNNELS
437
and
We may now set some arbitrary value of the pressure disturbance as the maximum that we wish to allow, and use (6.6) to find the corresponding humidity as a function of Mach number. In Fig. 76, we have plotted the dew point (at 1 atm) required to keep A$/$ below 1% as a function of Mach number for the typical case of a wind tunnel with Po = 5 atm and To= 310 OK. +I0
Limit of no
I
2
3
4
M FIG. 76. Permissible dew point temperatures in typical supersonic wind tunnel.
We used a constant supercooling of 65 "C and an extrapolated vapor-pressure curve for undercooled water to locate the condensation shock, and took the heat of vaporization for ice, extrapolated from the values given in [27], to obtain Q. The curve on the left gives the dew point required for no condensation at all. We see that above about M = 2 we cannot completely eliminate condensation with ordinary drying equipment, but it is possible to keep the pressure disturbance to a low value. Again, we note the trend that as M increases, A$/$ for a given humidity becomes smaller, a fact that has been well verified experimentally [e.g., 1221.
438
PETER P. WEGENER AND LESLIE M . MACK
With the flow behind the condensation shock known, we are able to find the effect of water vapor condensation on the pitot pressure. We may write the ratio of the pitot pressure with condensation, Po,', to the pitot pressure with no condensation, Po', as
where subscript 4 refers to conditions behind the pitot shock. To obtain the quantities on the right side of (6.7) we assume, as in the calculation of pitot pressure in the saturated expansion of a pure vapor, that complete vaporization takes place right behind the shock. We first compute M 3 for the given condensation shock at M I . The pressure ratio across the pitot shock is given by (4.44) when we choose the plus sign and q < 0. The value of q, with the assumption of isentropic flow behind the condensation shock and complete vaporization at the pitot shock, is equal to the heat released at the condensation shock. The Mach number behind the pitot shock, M4, is obtained from (4.47). The staticlstagnation pressure ratios p4/po4 and $,/Po, for the Mach numbers M4 and M3 are obtained from isentropic flow tables, along with the pitotlstagnation pressure ratio p i / $ , , for M , the Mach number with no condensation. The stagnation pressure, po3, is the same as that behind the condensation shock, po2, and po,/po, is found from Fig. 45. Therefore, we can determine Po3'/po' as a function of M,, q, and M . The results of several calculations show that the pitot pressure with condensation is always greater than in the absence of condensation, except immediately behind the condensation shock. The ratio (6.7) increases sharply with M and then levels off. For an extreme case, where M , = 1.85 and q = qma,,,,which is only possible with a heated air supply and high mixing ratio, f i o i / p i = 1.10 at M = 5. For a typical case of atmospheric supply with high humidity, the ratio i s only 1.03 at M = 5. Finally, for a low humidity such as required for A$/$ = 1% in Fig. 67, the pitot pressure is only a fraction of a percent higher than with completely dry air. The insensitivity of the pitot pressure to condensation has been noted experimentally [ e g , 27, 371. The effect of humidity on model testing is not amenable to a generalized treatment. If condensation is present in the free stream, we expect the stagnation enthalpy to vary in the flow field about a model because of local shock waves, expansions, wakes, etc. These enthalpy variations may affect the various aerodynamic force and moment components differently. Cawthon [37] found that, for a cone-cylinder model with a cruciform wing configuration, the slope of the normal-force coefficient was most sensitive to humidity effects. However, he found that a higher humidity was allowable for a certain percentage error in the slope of the normal-force coefficient than for the same percentage error in the free-stream static pressure. This
CONDENSATION I N W I N D T U N N E L S
439
observation indicates that if the pressure errors are minimized by sufficient drying, errors in the force and moment data will be even smaller. Dayman [123] investigated humidity effects on the force and moment coefficients of a typical cruciform-finned guided missile model with an ogive nose. He found that errors due to water vapor condensation in his 46 x 51 cm nozzle first appeared in the moment coefficients. Furthermore, condensation effects appeared at lower humidities for larger angles of attack. He determined the “critical dew point” of the tunnel supply air as that dew point temperature, measured at 1 atmosphere, where the pitching moment curve (at an angle of attack of 20 degrees) deviated by about 1% from that measured with dry air. This critical dew point is shown as a function of Mach number in Fig. 76. As in Cawthon’s experiments, this 1% pitching moment error criterion permits operation a t higher dew points than required for an equal pressure error. The computed limiting curve for no condensation is in very good agreement with Dayman’s experimental results, which shows that the supercooling in his experiments is approximately the one used in the calculation. Finally, Dayman found that for blunt-nosed missile models the error in the force and moment coefficients at a given dew point were much below those for the pointed model. It is likely that in such a situation most of the water vaporizes in the strong bow shock. Laufer and Marte [124] found that the recovery temperature at the surface of an insulated cone increased when the humidity was increased at constant supply pressure and temperature. This increase amounted to about 10 “C in the laminar region a t M 4 for an increase of the mixing ratio from 0.54 to 3.7 g/kg. Vaporization of the droplets undoubtedly occurs in the boundary layer, and at first it appears surprising that the recovery temperature is higher in the presence of heat removal. However, the free stream temperature is increased by condensation and both increases are of the same order. Laufer and Marte further found the Reynolds number of transition on the cone to increase slightly with an increase of humidity. In summary, it appears that a criterion of maximum permissible humidity based on the maximum permissible error in static pressure will permit most other types of testing to be conducted with negligible errors. In turn, the pressure-error criterion should be based on the attainable flow quality of a given nozzle with dry air, rather than on some absolute standard. In this manner the amount of air drying, with all of its difficulties, can be minimized. N
The flow in the test section of a hypersonic wind tunnel operated with dry air can be kept free of air condensation by choosing the supply conditions according to Fig. 5 . If partially condensed flow is of interest, the methods of Section IV permit the calculation of all the flow parameters. However, the effect of air condensation on the aerodynamic coefficients of models is at present unknown. A discussion of aerodynamic testing in partially condensed air I ~ V V Vis given by Grey [62]. In addition, the effect of air condensation
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on laminar skin friction and heat transfer has been determined theoretically by Eimer [125]. 2. Notes on Methods of A i r Drying and Heating, and Special Techniques for
Condensation Studies Finally, we shall make a few remarks on the engineering and instrumentation aspects of condensation in high-speed flow. The spurious effects of water vapor condensation on the flow parameters and aerodynamic coefficients of models are avoided by drying the air fed to the nozzle supply. To achieve the necessary dryness is a formidable problem for large tunnels because of their high mass flow and the magnitude of the mixing ratio of atmospheric air, which may range from 5 to 25 g/kg depending on climate and season. All air drying is accomplished by one of three basic methods: compression, cooling, and absorption. Since the relative humidity (2.1) of a given air sample a t fixed temperature increases with increasing pressure, d, becomes 100% at a sufficiently high pressure, and the water in the storage container may be drained. Also, the relative humidity increases if a given air sample is cooled at constant pressure, and, again, the condensed water can be drained. Drying by refrigeration, however, is usually technically impractical for larger tunnels, but drying by compression is ordinarily sufficient for hypersonic tunnel operation from a storage reservoir. The customary wind tunnel drier makes use of the absorption of water vapor by a desiccant or drying agent, such as silica gel (essentially SO,) [e.g., 1261 or alumina (essentially A1,0,) [e.g., 1271. The absorption process in these materials is a physical rather than chemical one. Owing to their high porosity and large surface area, water is retained by capillary forces. After the material is saturated it may be reactivated by heating. The various methods of air drying are discussed in detail by Smolderen [ l e l ] . Halvorsen [128] describes the design procedure for an absorption drying system, and Gruenewald [129] has investigated the absorption and pressure-drop characteristics of desiccants for wind tunnel operation. Determination of the actual composition of the gas mixtures in windtunnel supplies is possible by various means. Quantitative chemical analysis and mass spectrography have been used [54,551, and different types of hygrometers to determine the humidity Leg., 121, 1301 are employed. Most commonly, the supply-section dew point is measured by cooling an air sample at constant and known pressure. When condensation is visually observed on a mirror, the mirror temperature is measured and the mixing ratio may be determined from (2.4)or Table 4. Once x is known, p , may be found from (2.3), and the relative humidity can be computed from (2.1) and vapor pressure data.
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To determine the purity of wind-tunnel supplies with respect to foreign particles which may serve as condensation nuclei, a nuclei counter of the Aitken type may be used. A sample of supply air is expanded in a cloud chamber and the resulting number of water drops falling on a graduated surface may be counted. Such a counting chamber may be built to record continuously the relative number of condensation nuclei present j1311. Light scattering and absorption measurements have been used by many investigators of condensation. Such observations make it possible to determine if small drops or crystals, not visible to the eye, are present in the flow. If a beam of monochromatic light of intensity I , is passed through a flow of density p containing droplets whose diameter is small with respect to the wave length A, light scattering is observed. Because of the presence of the small particles, the incident light beam undergoes changes of angle and intensity. If we assume all drops to be of equal volume v , and assume a constant number per unit volume N, Rayleigh's classical formula [132] gives
where I B is the intensity of the scattered light observed at the angle p with respect to the incident beam, and d is the distance of the observer from the medium under investigation. Since droplets at the inception of condensation may have radii of the order of 10, to lo2 A, while A is of the order of lo3 to lo4A, light scattering permits the determination of the presence of fine fog. If we observe a t right angle to the incident light and assume all drops to be spherical, (6.8) becomes
With A = 4800A, d = 10 cm, and p = g/cm3 for a typical moist air condensation experiment, we obtain approximately I&I0 = NrS [lo]. With the further assumption that (6.9) is applicable up to the drop radius r = lo2A, for which the linear dimension of a drop 2r 0.04 A, we find, for 0.1 < I9,,./I0< 1, that 10l2< N < 1013. Quantitative application of (6.9) to determine N or r directly for a known A is impossible until an independent measurement, or calculation, of either the radius or number of drops is made. Head [lo] has proposed to determine the product Nr3 from pressure distribution measurements, since Nr3 must be proportional to the heat released, which we can calculate from the pressure disturbance. The ratio of (6.9) to this value of Nr3 yields r3. Other methods of using (6.8) include an added measurement of light absorption. Durbin [65] has applied ingenious techniques of this sort to determine N and r for air condensation
-
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in a hypersonic nozzle; McLellan and Williams’ data [58] obtained from this method were discussed in Section V. Stodola [31] made early estimates of droplet size based on light scattering and color observations in steam nozzles. Finally, the application of these techniques to condensation experiments in high-speed flow has been well summarized by E. M. Winkler [132].
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CONDENSATION IN W I N D TUNNELS
447
115. WAKESHIMA, H., Spontaneous condensation in a supersonic flow of humid air, J . P h y s . SOC.J a p a n 10, 141 (1955). 116. STOKES,G. G., On the effect of the internal friction of fluids on the motion of pendulums, T r a n s . Cambridge Phil. SOC. 9 (1850). 117. FROSLING,N., Uber die Verdunstung fallender Tropfen, Gerland’s Beitr. Geophys. 53, 170 (1938). 118. HAZEN,W. E., Some operating characteristics of the Wilson cloud chamber, Rev. Scz. Instr. 13, 247 (1942). 119. BARRETT, E. O., and GERMAIN, L. S., Growth of drops formed in a Wilson cloud chamber, Rev. Sci. Instr. 18, 84 (1947). 120. WYKER,H., Charts for the estimation of the permissible humidity in supersonic wind tunnels, in “MBmoires sur la MBcanique des Fluides offerts & M Dimitri P. Riabouchinsky”, Ministere de l’Air, Paris, 1954. 121. SMOLDEREN, J. J ., Condensation effects and air drying systems for supersonic wind tunnels, AGARDograph No. 17, North Atlantic Treaty Orgdnization, Paris, 1956. 122. MONAGHAN, R. J., Tests of humidity effects on flow in a wind tunnel a t Mach numbers between 2.48 and 4, Aeronaut. Research Council, Current Paper No. 247 (1956). 123. DAYMAN, B., JR., Effects of tunnel dew point on force data in the 20” supersonic wind tunnel of the Jet Propulsion Laboratory, Calif. Inst. of Technol., Pasadena, California. Private communication, 1954. 124. LAUFER,J., and MARTE,J. E., Results and a critical discussion of transitionReynolds-number measurements on insulated cones and flat plates in supersonic wind tunnels, Rept. No. 20-96, Jet Propulsion Lab., Calif. Inst. of Technol., Pasadena, California, 1955. 125. EIMER,F., Direct measurement of laminar skin friction at hypersonic speeds, Ph. D. Thesis, Calif. Inst. of Technol., Pasadena, California, 1953. 126. Anon., Dehydration of air and gas with Davison silica gel, Davison Chemical Co., Tech. Bulletin No. 201, Baltimore, Md. 127. Anon., Activated alumina, its properties and uses, Aluminum Co. of America, Chemicals Div., Pittsburgh, Pa., 1949. 128. HALVORSEN, G. S., Supersonic wind tunnel condensation shock design problems, Thesis, Univ. of Florida, Gainesville, Fla., 1952. 129. GRUENEWALD, K. H., Investigation of several desiccants with regard t o their use in the drier system of the 40 x 40 cm supersonic wind tunnels of the Naval Ordnance Laboratory, U. S. Naval Ord. Lab. Mem. No. 10518, Silver Spring, Md., 1950. 130. DEAN,R. C., JR., et al., Aerodynamic measurements, Gas Turbine Lab., Mass. Inst. of Technol., Cambridge, Mass., 1953. 131. VONNEGUT, B., Continuous recording condensation nuclei meter, Occasional Rept. No. 19, RL-300, General Electric Res. Lab., 1950. 132. WINKLER, E. M., Condensation study by absorption or scattering of light. Part I, G in “Physical Measurements in Gas Dynamics and Combustion”, p. 289. (R. W. Ladenburg, ed.) Princeton University Press, Princeton, New Jersey, 1954.
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Author Index Numbers in parantheses are reference numbers and are included to assist in locating references where the author’s name is not mentioned in the text. Numbers in italics refer to the page on which the reference is listed. Numbers in hold-face type indicate the pages of the article in this volume written by the author mentioned.
A Abramson, H. N., 111, 139(49), 157, 158, 159, 160(89), 161(89), 162, 173(89), 175(49), 790, 797, 792 AlfvCn, H., 216, 231 Alter, B. E. K., 148, 149, 797 Alverson, R. C., 181(127), 184(127), 186, 193 Anderegg, F. O., 235, 303 Anderson, R. A,, 166(104. 105), 793 Aoyama, S., 318(15), 350, 442 Armstrong, G. T., 318(17), 442 Arnold, R. N., 168(116), 193 Arthur, P. D., 347(54), 348, 349(54), 351, 360(54), 361, 394, 411(54), 412(54), 433(54), 440(54), 444 Ashley, H., 37, 39(6), 85 Auerbach, F., 236, 271, 303
B Bader, R., 70, 88 Bancroft, D., 117, 128, 129, 131, 156, 789 Barlow, W., 237, 303 Barr, A. D. S., 166(114), 793 Barrett, E. O., 427, 447 Becker, J. V., 354, 359(57), 412(57), 444 Becker, R., 404, 414, 415, 418(102), 445, 446 Beckett, C. W., 308(3), 309(3), 311(3), 312(3), 315(3), 324(3), 325(3), 347(3), 442 Bell, J. F., 144, 797 Bennett, J. G., 235, 250, 303 Berman, J . H., 71, 72(85), 88 Bernstein, I. B., 204, 214(8), 224, 237 Billington, A. E., 45, 85 Binnie, A. M., 346(50), 411(50), 430. 431, 444 Biot, M. A,, 151(72), 791 Bishop, R. E. D., 127, 789 Bisplinghoff, R. L., 151(72). 797
Blechschmidt, E., 128, 790 Bleich, H.H., 181, 182, 184(125), 185(125), 193 Boerdijk, A. H., 239, 304 Bornstein, R., 311(9), 318(9), 344(9), 405(9), 442 Bogdonoff, S. M., 320(20), 419(111). 424(111), 426(111), 443 Bohnenblust, H. F., 175, 177, 179(123), 180, 181(123), 183, 188, 793 Boley, B. A., 166, 175(100), 792 Boussinesq, J., 177, 793 Brandt, H., 235, 279, 280, 284, 303 Brennan, J . N., 138, 139(42b), 190 Broberg, K. B., 112(5), 188 Brown, R. I,., 235, 249, 250, 303, 304 Budiansky, B.. 166, 793 Buff, F. P., 406, 446 Buhler, R. D., 358, 369, 387, 391, 393, 397, 398(61), 400(61), 402(61), 405(61), 409(61), 411, 423, 425, 426(61), 433, 444, 445 Bullen, K. E., 161(92), 792 Burger, A. P., 82, 89 Burgers, J. M., 384, 445 Burgess, W. C., Jr., 328, 334(28), 405(28), 411(28), 419(28), 420(28), 443 Busang, P. F., 242, 243, 281, 304 Busemann. A., 72, 88
C Carrier, G. F., 76, 88 Cattaneo, C., 253, 254, 256, 266, 304, 305 Cawthon, J. A., 330(37),411(37), 438(37), 443 Cazin, A , , 344(47), 411(47), 444 Chambr6, P. L., 269, 305 Chambr6, R., 365(70), 445 Chandrasekhar, S., 196, 231 Chao, C. C., 166, 175(100), 792 Charyk, J. F., 373, 419(73, 110). 429, 445. 446
4.49
450
AUTHOR INDEX
Chree, C., 116, 155, 189 Clark, D. S., 142(57), 143, 144, 147, 150(70a), 175, 177, 179(123), 180, 181(123), 183(123), 188, 797, 793 Cohen, H., 365(67), 377(67), 380(67), 445 Cohn-Vossen, S., 239, 304 Collar, A. R., 166(112), 193 Conroy, M. F., 181, 183(133), 184(126), 185, 793, 194 Conway, H. D., 157(86), 792 Cooper, J. L. B., 158(91), 161(91), 792 Cotter, B.A., 181(129), 182(129), 184(129), 186, 794 Coulson, C. A,, 119(12), 789 Courant, R., 63(61), 87, 383(76), 445 Cox, H., 152(76), 197 Cremer, L., 158(90), 792 Crocco, L., 12(11), 21(11), 34 Cross, H. J., 278, 306 Crown, J. C., 310(5), 442 Cunningham, D. M., 174, 175, 793 Curtis, C. W., 148, 149, 797 Cwilong, B. M., 338, 444
D Damkohler, G., 333(38), 414(38), 415(38), 418(38), 432, 443 Das Gupta, N. N., 328(30), 407(30), 409(30), 418(30), 427(30), 443 Davies, R. M., 112(3, 4). 113(4), 117, 118, 119, 122, 131, 132, 133, 134, 135, 136, 157(4, 11). 159, 160, 161, 788, 789, 190 Dayman, B., Jr.. 439, 447 Dean, R. C., Jr., 440(130), 447 Dengler, M. A., 55, 59, 86, 154, 155(84), 166, 169, 170, 171, 175(84, 98), 792 Deresiewicz, H., 166(110), 793, 233, 256, 257, 258, 259, 263, 264, 266, 291, 292, 293, 294, 297, 298, 299, 300, 305, 306 de Rham, G., 110(9), 110 Deutler, H., 145(63), 197 Dietze, F., 50, 86 Di Maggio, F., 184, 794 Dodge, B. F., 316(12), 442 Doring, W., 414, 418(102), 446 Dohrenwend. C. 0.. 168, 793 Dolph, C. L., 166(106), 793 Donnell, L. H., 139, 140, 790 Dorsey, N. E., 404(92), 446
Drischler, J. A., 82, 89 Drucker, D. C., 168, 793 Duffy, J., 286, 288, 289, 306 Dugundji, J., 70, 71(81), 88 Dunbar, A. K., 316(12), 442 Durbin, E. J., 360, 441, 445 Durelli, -4. J., 139, 790 Duwez, P. E., 140(54), 142(54, 57), 143, 144, 147, 175, 177, 179(123), 180, 181(123), 183(123), 188, 790, 797, 793
E Eber, G., 327(26), 328(26), 331, 332, 333, 338, 340, 411(26), 443 Eckhaus, W., 55, 56(48), 58, 59, 83, 86, 87, 89 Eggers, A. J., Jr., 308(2), 310(2), 442 Eimer, F., 440, 447 Elsasser, W. M., 196, 201, 237 Eringen, A. C., 152, 192
F Fabri, J., 1, 1(4), 10(8), 19(6), 30(5), 33, 34 Falkovich, S. B., 235, 303 Farkas, L., 416, 418, 446 Faro, I. D., 347, 348(52), 411(52), 444 Fatt, I., 284, 306 Feder, J. C., 139(47), 790 Fejes T6th. L., 235, 240, 303 Ferraro, V. C. A,, 222, 231 Fettis, H. E., 50, 86 Field, G. S., 117, 789 Fischer, H. C., 139(50), 790 Flax, A. H., 43(10). 44(10), 85 Flood, H., 418, 446 Fliigge, W. 166, 792 Flynn, P. D., 139(46), 190 Focke, A. F., 128, 168, 789 Foote, P. D., 242, 243, 281, 304 Forster, C. R., ll(10). 34 Frankland, J. M., 151, 797 Fraser, H. J., 235, 238, 241, 243, 245, 248(25), 303, 304 Frazer, R. A., 49, 86 Frenkel, J., 310(6), 311(6), 404(6), 405(6), 409(6), 414(6). 418, 442 Friedman, B., 92(7), 110 Friedrichs, K. O., 383(76), 384, 445 Frieman, E. A., 196, 204, 214(8), 237 Frocht, M. M., 139(46), 790
AUTHOR INDEX
FrBssling, N., 426(117), 427, 447 Fuller, F. B.,59(60), 77, 84(60, 106), 87, 89 Furnas, C . C., 250, 304 Furukawa, G. T., 316(13), 317, 318, 404, 405(93), 442, 446
Garrick, I. E., 56, 57, 59, 82, 87, 89 Gassmann, F., 275, 276, 277, 305 George, M. B. T., 51(38), 86 Gerber, E. H., 57, 87 Germain, L. S., 427, 447 Germain, P., 70, 88, 92(10), 110 Ghosh, S. K., 328(30), 407(30), 409(30), 418(30), 427(30), 443 Gibbons, R. A., 139(47), 790 Gibbs, J. W., 404, 405, 413, 446 Gilbert, J. T., 139(47), 790 Gilmore, F., 406, 420, 423(98), 446 Glasstone, S., 318(14), 404(14), 442 Goland, M., 55, 59, 86, 154, 155(84), 166, 169, 170, 171, 175(84, 98), 792 Goldsmith, W., 174, 175, 193 Goldstein, J. M., 318(17), 442 Goldsworthy, F . A., 76, 81, 88 Goodier, J. N., 251(48), 252(48), 260(48), 304 Goodman, Th. R., 65, 84, 87, 89 Graton, L. C., 238, 241, 243, 245, 248(25), 304 Green, D. J.. 181(128), 184(144), 186(144), 194 Greidanus, J. H., 47, 85 Grey, J., 358, 359(62), 400, 402(62), 412(62), 439, 444 Griffis, L., 140, 142(55), 790, 191 Gruenewald, K. H., 327(26), 328(26), 331, 333, 338, 340, 411(26), 419(112), 440, 443, 446, 447 Guienne, P., 336, 411(43), 444 Gunn, J. C., 63, 8 7
H Hadwiger, H., 238, 303 Hall, N; A., 384, 445 Halperin, I., 95(13), 770 Halvorsen, G. S., 440, 447 Hammitt, A. G., 320(20), 443 Hansen, C. F.. 347, 357, 358, 359(53), 396, 401, 409(53). 411(53), 412(53), 419(53), 444
45 1
Hara, G., 271, 305 Harrison, M., 120, 136, 189 Haskind, M. D., 45, 46, 85 Haurwitz, B., 382i7.5). 445 Havelock, T. H., 119(14), 789 Hawksley, P. G. W., 250, 304 Hawthorne, W. R., 365(67, 69), 377(67), 380(67), 445 Hazen, W. E., 427, 447 Head, R. M., 312, 315, 322(10), 325, 329, 330(10), 333(10), 338, 341(10), 346(10), 398, 411(10), 419(10), 420, 430(10), 432, 436(10), 441(10), 442 Heaslet, M. A ,, 43, 44(11), 59, 65, 77, 84(60, 85, 87, 88 Hermann, R., 327, 328(25), 330, 334(25), 373, 411(25), 443 Herrmann, G., 122, 124, 189 Hertz, H., 152, 792 Herzfeld, K. F., 406, 446 Heybey, W. H., 328, 373, 384, 433, 443, 445 Hicks, B. L., 365(68), 445 Hilbert, D., 63(61), 87, 239, 304 Hill, F . K., 320(22), 347, 348(52), 411(62), 443, 444 Hilsenrath, J., 308(3), 309(3), 311(3), 312(3), 315(3), 324(3), 325(3), 347(3), 442 Hirn, G. A,, 344(47), 411(47), 444 Hobson, E. W., 70, 88 Hofsommer, D. J., 47, 85 Holden, A. N., 136, 190 Holland, C. K., 346(49), 411(49), 444 Hopkins, H. G.. 184, 794 Hopkinson, B., 130, 190 Horsfield, H . T., 235, 245, 303 Howe, C. E., 166(107), 193 Howe, R. M., 166(107), 193 Hsu, P., 71, 88 Hudson, D. R.. 245, 247, 304 Hudson, T. E., 157, 165, 192 Hughes, D. S., 120, 136, 137, 189, 278, 283, 305, 306 Hugill, H . R., 235, 242, 248, 281, 303
Iida, K., 272, 305 Imachi, I., 166(108), 793 Inglis, J. K. H., 316(11), 442
452
AUTHOR INDEX
J Jackson, P. H., 369, 445 Jahn, H. A , , 63, 87 J i k y , J., 268, 305 James, H. M., 109(12), 110 Janssen, H. A., 234, 303 Jenkin, C. F., 271, 305 Johannsen, N. H., l(3). 33 Johnson, J. E., 150(70a), 191 Johnson, K. L., 254, 255, 260, 261, 262, 305 Jones, H. J., 278, 305 Jones, R. P. N., 152(80), 166(113), 192, 193 Jones, R. T., 84, 89 Jones, W. P., 49, 50, 54, 63, 72, 73, 79(97), 86, 87, 88, 89
H Kaischew, R., 418, 446 Kanda, E., 318(15), 350, 442 Kantrowitz, A , , 419, 426(114), 446 Keenan, J. H., l(2). 14(2), 33, 311(7), 322(7), 343(7), 347(7), 369(7), 381, 411(7), 442 Kelly, J. L., 278, 283, 306 Kennard, E. H., 405(94), 417(94), 418(94), 423(94), 446 Kirkwood, J. G., 406, 446 Kodaira, K., llO(9). 710 Kijtter, F., 234, 267, 268(61), 303, 305 Kolsky, H., 112(2), 129, 137, 150(70b), 157(2), 169, 188, 190, 191 Karst, H. M., 21(13), 34 Kossel, W., 406, 446 Kruskal, M. D., 204, 209, 210, 214(8), 231 Kruszewski, E. T., 151(74), 191 Kubota, T., 364, 395, 412(66), 445 Kiissner, H. G . , 36, 40, 46, 47, 48(26), 49, 53, 56, 79(96), 84, 85, 89 Kuhrt, F., 407, 418, 446 Kulsrud, R. M., 195, 204, 214(8), 221, 231
L Laidlaw, W. R., 57, 87 Lamb, H., 119(13), 120, 153, 155, 189 Landahl, M. T., 37, 39(6), 56, 77, 85, 87, 89 Landolt, H. H., 311(9), 318(9), 344(9), 405(9), 442 Landon, J. W., 131, 190 Laufer, J., 439, 447 Lawrence, H. R., 57, 87
Lee, E. H., 168(117), 182, 183(131), 184, 193, 194 Leech, J., 239, 304 Lees, L., 12(11), 21(11), 34, 419(110, l l l ) , 424(111), 426(111), 429, 446 le GrivBs, E., l(4). 30(5), 33 Lehrian, D. E.. 55, 86 Leonard, R. W., 166, 193 Leth, C. F. A., 184, 186(136), 194 Li, T. Y., 68, 88 Liepmann, H. W., 326(23), 334(41), 443 Lighthill. M. J., 38, 73, 76, 77, 85, 88 Lin, C . C., 37, 38(3), 39(3), 85, 365(70), 445 Litehiser, R. R., 242, 304 Lobb, R. K., 320(19), 442 Lomax, H., 59, 65, 77, 81, SC(60, 106), 87, 88, 98 Lord Kelvin (Thomson, W.), 244(35), ._ 250(44), 271(45), 304, 404, 4d.5 Lord Rayleigh (Rayleigh, J . W. S), 154, 192, 207, 231 Love, A. E. H., 116, 120, 155, 189, 274, 275(73), 305 Lubkin, J . L., 266, 305 Ludwik, P., 145(61), 197 Lukasiewicz, J., 328, 333(27), 334(27), 373, 378(27), 381, 405(27), 411(27), 437(27), 438(27), 443 Lundquist, G., 339, 355, 356(59), 357(59), 359(59, 63), 387(59, 63), 409(59), 411(63), 412(59, 63), 444 Lundquist, S., 196, 204, 231 Lustwerk, F., 1(2), 14(2), 33
M McCoskey, R. E., 316(13), 317, 318, 442 Mack, L. M., 807, 336(42), 444 McLachlan, N. W., 45, 85 Mc Lellan, C. H., 320(18), 355, 359(58), 361(58), 408, 409(58), 411, 412(58), 433(58), 442, 442, 444 McSkimin, H. J., 157, 192 Malvern, L. E., 125, 145(24, 60), 146, 147, 186, 189, 191 Marte, J. E., 439, 447 Marvin, J. W., 235, 244, 303 Mason, H. L., 168(115), 193 Mason, W. P., 256, 258, 259, 305 Matzke, E. B., 245, 304 Mazelsky, B., 82, 89 MBlkse, G. B., 92(8), 110
453
AUTHOR INDEX
Mellanby, A. L., l ( 1 ) . 33 Melmore, S., 241, 304 Mentel, T. J., 181(128), 184, 794 Menzer, G., 241, 304 Merbt, H., 56, 87 Miklowitz. J., 124, 125(23), 135, 136, 138, 166(109), 175(99), 789, 790, 792, 793 Mikusinski, J . G.. 92, 770 Miles, J. W., 37, 38(4). 39(4), 52, 63, 65, 68, 69, 84, 85, 86, 87, 88, 89 Mims, R. L.. 120, 136, 137, 789 Mindlin, R. D., 122, 124, 165, 166(110), 789, 792, 793, 251, 252, 253, 254, 256, 257, 268, 259, 263, 264, 265, 266, 286, 288, 289, 304, 305, 306 Minkowski, H., 250, 251, 304 Miyabe, N., 235, 236(20), 303 Mollo-Christensen, E. L.. 37, 39(6), 85 Monaghan, R. J., 436, 437(122), 447 Moore, P., 168, 793 Moynihan, J. R., 157(86), 792 Munson, A. G., 320(21), 443 Muskat, M., 235, 246, 303
P Paulon, J., 30(5), 33 Pearson. J., 112(1), 788 Penner, S. S., 338(44), 444 Peterson, S., 135, 790 Phillips, R. S., 109(12), 770 Pian, T. H. H., 183, 794 Pines, S., 70, 71(81), 88 Plass, H. J., Jr., 111, 122, 124, 125, 127, 138, 147, 165, 166, 167, 168, 171, 186, 187, 188, 789, 797 Pochhammer, L., 116, 155, 788 Polubarinova-Kochina, P. Ya., 235, 303 Pondrom, W. L.. Jr.. 120, 136, 137, 789 Possio, C., 49, 86 Prandtl, L., 145(62), 797 Prescott, J., 161(93), 792 Probstein, R. T., 418, 419, 426(109), 446 Puckett, A. E., 326(23), 419(113), 443, 446
Q Quinney, H., 131, 790
N Nbdai, A.. 237, 303 Nagamatsu, H. T., 347(55), 369, 411(55), 412(55), 433(55). 440(55), 444, 445 Nasu, N., 283, 306 Nelson, H. C., 72(84), 88 Neumann, E. P., 1(2), 14(2), 33 Neuringer, J., 70, 71(81), 88 Newcomb. W. A., 199, 237 Newman, M. K., 166(111), 793 Nichols, N. B., 109(12), 770 Nisewanger, C. R., 135, 136, 138, 790 Nisida, M., 139(44), 790 Nothwang, G. J., 347, 357, 358, 359(53), 396, 401, 409(53), 411(53), 412(53), 419(53), 444 Nutting, P. G., 268, 305
0 Offenbacher, E. L., 139(47), 790 O’Keefe, J., 109(15), 770 Osmer, T. F., 256, 258, 259, 305 Oswatitsch, K., 328, 333, 334(36), 365, 369, 372, 373, 384, 408, 411(33, 34), 416,419(33,34),421(33),422, a24(33). 425, 426, 429, 430, 431, 432, 443, 445
R Rabinowitz, S., 50, 86 Rainey, R. A., 72(84), 88 Ramberg, W., 151(73), 797 Rathbun, K. C., 359(64), 360(64), 405(64), 406, 412(64), 419(64), 429, 444 Ravetz, J. R., 95, 770 Rayleigh, J. W. S., see Lord Rayleigh Reed, S. J.. Jr., 355, 356(59), 357(59), 359(59, 63), 384, 387(59, 63), 391, 406, 409(59), 411(63), 412(59, 63). 414, 420, 444, 445, 446 Reilly, M. L., 404, 405(93), 446 Reissner, E., 37, 38(3), 39(3), 45, 46, 55, 56(50, 51), 85, 86, 87, 266, 305 Reissner, H., 267, 268, 305 Rettaliata, J. T . , 347, 411(51), 444 Reynolds, 0..270, 305 Riley, W. F., 139, 790 Rinehart. J. S:, 112(1), 788 Riparbelli, C., 147, 148, 797 Ripperger, E. A., 111, 135, 138, 139(38, 49, 51). 161(94), 166, 167. 171, 172, 173, 175(49, 119), 790, 792, 793 Roberts, D, E., 318(17), 442 Roberts, P. H . , 222, 237 Robinson, A., 69, 70(78), 88
454
AUTHOR INDEX
Robinson, D. W., 110(11), 170 Rogers, S., 157(86), 792 Rohrich, K., 128, 789 Roshko, A., 334(41), 443 Ross, F. W., 397, 398, 445 Rott, N., 51, 65, 86, 87 Roy, M., 34 Royle, J. K., 328, 333(27), 334(27), 373, 378(27), 381, 405(27), 411(27), 437(27), 438(27), 443 Rubinow, S. I., 59, 87 Runyan, H. L., 54, 71, 86 Ryll-Nardzewski, C., 92(5, 6). 170
S Sagoci, H. F., 266, 305 Saltzer, C., 91 Salvadori, M. G., 181, 182, 184(125), 185(125), 793, 794 Samaras, J. G., 373, 398, 445 Sander, A,, 333(38), 414(38), 415(38), 418(38), 432, 443 %to, Y . , 272, 305 Sauer, J. A., 138, 139(42b), 190 Schade, Th., 49, 52, 53, 86 Sehamberg, R., 419(113), 446 Schoeneck, H., 128, 189 Schonflies, A., 235, 237, 303 Schiitte, K., 239, 304 Schwartz, L., 91, 92(3), 770 Schwarz, L., 46, 49, 50, 85, 86 Schwarzschild, M., 209, 231 Seashore, F. L., 328, 334(28), 405(28), 411(28), 419(28), 420(28), 443 Seiler, J. A,, 181(129), 182(129), 184(129), 186, 194 Sell, H., 235, 303 Senior, D. A., 139(45), 190 Shapiro, A. H., 365(69), 445 Shear, S. K., 128, 16s’ 789 Sibulkin, M., 320(19), 442 Siestrunck, R., 1, l(4). 19(6), 33, 34 Skalak, R., 138, 190 Skan, S. W., 73, 88 Sluder, L., 59(60). 77, 84(60, 106), 87, 89 Small, T. R., 347, 348(52), 411(52), 444 Smelt, R.. 333(40). 334(40), 338(40), 342(40), 405(40), 411(40), 443 Smith, W. 0.. 242, 243, 281, 304 Smolderen, J. J., 436, 440(121), 447 Sokolovsky, V. V., 146, 191
Spitzer, L., Jr., 196, 197, 231 Spreiter, J. R., 43, 44(11), 85 Staab, H., 320(19), 442 Stanford, E. G., 128, 130, 189 Sternglass, E. J., 144, 148, 791 Stevens, J. E., 56(50), 86 Stever, H. G., 359(64), 360(64), 405(64), 406, 412(64), 419(64), 429, 444 Stewart, H. J., 68, 88 Stewartson, K., 63, 65, 87 Steyer, C. C., 122, 124, 125, 138, 166, 167, 168, 171, 789 Stodola, A., 328(31), 343(31), 344(31), 345, 347(31), 408(31), 411(31), 422, 442, 443 Stokes, G. G., 422, 447 Stollenwerk, E., 355, 356(59), 357(59), 359(59, 63). 387(59, 63), 409(59), 411(63), 412(59, 63). 444 Strang, W. J., 84, 89 Stranski, I., 418, 446 Stuart, D. A ,, 144, 148, 191 Summerfield, M., 11(10), 34 Supnik, F., 238, 303 Swan, W. C., 11(10), 34 Symonds, P. S., 181(128, 129), 182(129, 131).183(131,135),184(129),186(136), 194
T Takahashi, T., 272, 305 Taylor, G. I., 140, 790, 397, 445 Temple, G., 63, 64(68), 87, 91, 92(14), 94(4), 96(14), 110 Terada, T., 235, 236(20), 303 Terzaghi, K., 234, 303 Theodorsen, Th., 36, 45, 46, 84 Thomson, W., see Lord Kelvin Thomson, W. T., 182, 794 Thurston, C. W., 291, 292, 293, 306 Timman, R., 43, 44(12), 45, 47, 48(22), 85 Timoshenko, S. P., 151(71), 152(71), 154, 163, 191, 792, 251(48), 252(48), 260(48), 304 Tolman, R. C., 405, 446 Traill-Nash, R. W., 166(112), 193 Tsien, H. S., 37, 38(3), 39(3). 85 Tu, L. Y . , 138, 139(42b), 790 Turner, M. J., 50, 86 Tutton, A. E. H., 244(37), 304 Tuzi, Z., 139(44), 190
455
AUTHOR INDEX
U Uflyand, Ya. S., 166, 192
V Van de Hulst, H. C . , 216, 237 van der Waerden, B. L., 239, 304 van de Vooren, A. I., 35, 45, 46, 47, 58, 65(73), 67, 85, 87 van Dyke, M. D., 39, 73, 74(90, 91), 76, 85, 88 van Spiegel, E., 52, 86 Vigness, I., 175, 193 Volmer, M., 404, 413, 414, 415, 416, 418, 445, 446 Volterra, E. G., 127, 157(87), 165, 789, 192 von Borbely, S., 63, 87 von Giebe, E., 128, 190 von Helmholtz, H., 404, 407, 446 von KBrmBn, Th., 10(9), 34, 140, 142(54), 190 Vonnegut, B., 441(131), 447 von Neumann, J., 384(77), 445 Voss, H. M., 71, 88
359(59, 387(59, 411(29, 419(4),
63). 360(60), 371, 385(29), 63), 405(39, 40). 407, 409(59), 39, 40, 63). 412(59, 60, 63), 428(4), 432, 442, 443, 444 Wells, A. A., 139(45), 790 Westman, A. E. R., 235, 242, 248. 281, 303 White, H. E., 242, 246, 304 White, M. P., 140, 142(55), 790, 797 Wick, R. S., 21(12,13), 34 Wickersham, P. D., 154, 155(84), 169, 170, 171, 175(84), 192 Williams, T. W., 355, 359(58), 361(58), 408, 409(58), 411, 412(58), 433(58), 442, 444 Willmarth, W. W., 347(55), 349(55), 411(55), 412(55), 433(55), 440(55), 444 Winkler, E. M., 320(19), 441f132), 442, 442, 447 Wise, M. E., 235, 248, 249, 303 Wood, D. S., 150(70a), 197 Woods, L. C., 51, 86 Woods, M. W., 346(50), 411(50), 430, 431, 444 Woolston, D. S., 54, 71, 86 Wyker, H., 435, 447
W Wagner, C . , 318, 419(16), 442 Wakeshima, H., 420, 426(115), 432, 447 Walton, S. F., 242, 246, 304 Watkins, Ch. E., 54, 71, 72(84, 85), 86, 88 Weber, A., 416, 446 Wegener, P., 307, 310(4), 314(4), 320(19), 328, 330(29), 333(29, 39, 40), 334(29, 40), 337(29), 338(29,40), 339, 342(39, 40). 353(4), 354(56), 355, 356, 357,
Y Yellot, J. I., 346(49), 411(48. 49). 429, 444
z Zajac, E. E., 166, 175(101), 192 Zandbergen, P. J., 83, 89 Zartarian, G., 71, 88 Zeldovich, Y. B., 418, 446
Subject Index A
Condensation equations, one-dimensional, 366 ff. Condensation nuclei, formation of, 404 ff. Condensation shock described, 327 Condensation shock, analysis of, 373 pressure ratio across, 379 stagnation pressure ratio across, 382 temperature ratio across, 380 Mach number behind, 381 Conservation of energy (hydromagnetics), 201 Contact theory, recent results of (granular matter), 251 ff. Continuous model for granular matter, 267 ff. Coordination number, 238 Critical droplet, size of, 417
Acceleration potential, method of, 47 Air drying and heating, methods of, 440 ff. Air, expansion of (condensation), 353 ff. Alfv6n mode (hydromagnetics), 218 f. Annulus of wear, 258 f. Approximate theories (waves in rods), 122 ff. Arbitrary planform, numerical approach for (unsteady airfoil theory), 70 Arrangements of equal spheres, 237 ff. Aspect ratio, low (unsteady airfoil theory), 56 ff. high (unst. airf. th.), 55 f. Autocorrelation functions as distributions, 108 Axisymmetric problem in hydromagnetics, 210 ff. stability of, 212 ff.
D Damping of waves (hydromagnetics), 227 Delta wing, 69 f. Densest packing of equa.1 spheres, 239 Dew point temperatures permissible in wind tunnel, 437 Diabatic flow (condensation), 365 ff. Dirac’s delta function (th. of distrib.), 06 Discrete models of granular matter, 270 ff. Dispersion (waves in rods), 118, 158 Distributions, theory of, 92 ff. and weak functions, equivalence of, 93f. Double stagger packing, 238 Droplet radius, critical, 410 Droplet, behavior of, 421 ff. Droplet formation, kinetics of, 372 Droplet growth, macroscopic, 426 f. molecular, 424 ff. Droplet growth-rates in air, 425
B Bar stress components defined, 123 Boundary conditions (hydromagnetics), 197 f. Bulking (granular matter), 242
C Calorically perfect gas defined, 309 Chapman- Jouguet detonation defined,384 Coexistence lines, coefficients for, 311 Compressibility factor, 308 f. Condensation, of air and nitrogen, 433 effect on measurements in wind tunnels, 433 ff. kinetics of, 403 as a weak detonation, 383 ff. of water vapor in air, 431 of water vapor in supersonic nozzles, 320 ff. Condensation effects observed, 342 f., 328 ff. Condensation in nitrogen, onset of, 349, 348 f.
E Elastic granule model with oblique contact forces, 285 ff. with normal contact forces, 271 ff. Elastic-plastic analysis (waves in rods), 177 ff. Energy generation (hydromagnetics),220f.
456
SUBJEC
Equilibrium expansion, saturated, 387 ff. Equilibrium limits forcondensation, 307 ff. for isentropic expansions, 312 ff. Equivalent spherical diameter (granular matter), 250
F Failure in simple cubic array (granular matter), 300 Finite part of an integral, 95 ff. Flexible tip (unsteady airf. th.), 74 f. Flexural elastic waves in rods, elementary theory, 151 ff. Flexural plastic waves in rods, 175 ff. Fourier series in distr. th., 99 ff. Fourier transforms in distr. th., 104 ff.
G Generalized harmonic analysis in distr. th., 107 Geometry of granular mass, 236 ff. Granular assemblages, mechanical response, 267 ff. Granular bars, experiments with, 287 ff. Granular medium defined, 234 Group velocity (waves in rods), 119, 158 Group velocity (hydromagnetics), 221 f.
H Hydromagnetic equations. 196 ff. validity of, 197 Hydromagnetic equilibria, 202 ff. stability of, 204 ff. Hydromagnetic instability, physical interpretation of, 214 ff. Hydromagnetic waves, 216 f f . modes of, 216 f. equations of propagation, 217 ff. Hypersonic Flow (unsteady airf. th.), 76 f. Hysteresis loops (granular matter), 260, 264 Humidity effect on model testing, 438
I Indicia1 functions, 77 ff. for finite wings, 84 and flutter derivatives, 81 ff. Infinite conductivity, approximation of, 198 ff.
457
INDEX
Integral equation method (unsteady airf. th.), 48 ff., 54 f., 71 f. Internal constraints, method of (waves in rods), 127 Interstitial spheres, effect of, on density, 245 ff. Isotropic pressure and uniaxial pressure superimposed (granular matter), 290 ff.
J Jet ejectors, constant pressure mixing process, 14 constant section mixing process, 14 general performance curves, 16 ff. main parameters, influence of, 23 ff. mixed flow patterns in, 8 ff. with primary separation, 11 f. similarity laws of, 10 Jet ejectors, operation without induced flow, 19 ff. optimum design, 30 ff. pressure distribution along mixing tube, 29 supersonic flow patterns, 4 ff. saturated, 7 similarity laws of, 7 Jet ejectors, theoretical performance curves, 12 f.
H Kutta condition, strong and weak, 42
L Light scattering and condensation, 359 ff. Longitudinal waves in rods, 113 ff. elementary theory of, 114 ff. dispersion of, 118 f. Longitudinal plastic waves in rods, 139 ff. Lorentz transformation (unsteady airf. th.), 40 Love’s theory (waves in rods), 120 ff.
M Mach number, determination of (condensation), 354 ff. conventional determination of, 358 Magnetic lines of force, motion of, 198 ff.
458
SUBJECT INDEX
Mindlin-Herrmann theory (waves in rods), 124 f. Minimization technique (hydromagnetics), 209 f. Mixing ratio H,O - air as a function of dew point, 323 Moving sources, method of, 59 ff.
Refraction of waves (hydromagnetics), 222 ff. Riemann’s method (unsteady airf. th.), 63 Rigid granules model, 270 f. Rigid-plastic analysis (waves in rods), 182
N
Shock waves with vaporization, 397 ff. Simple cubic packing, 237 Single stagger packing, 238 Singularity function, 95 ff. Skeleton line defined (unsteady airf. th.), 50 Spectral densities as distributions, 108 Spontaneous nucleation, 412 ff. Steam condensation, 429 ff. Steam nozzles, condensation in, 344 ff. Strain-rate theory (waves in rods), 186 ff. Stress-strain relation for a simple cubic array (granular matter), 298 f. Stochastic processes in distr. th., 107 Supercooling in steam nozzles, 345 f. Supersonic edges (unsteady airf. th.), 59 ff., 65 ff. Supply conditions for complete vaporization, 402 Support of a function defined, 100 Swept wings (unsteady airfoil theory), 58
Nitrogen, expansion of, 347 ff. Non-Linear approximations in supersonic flow (unsteady airf. th.), 72 ff. Normal forces (granular matter), 251 f. Nucleation rates for vapors, 419
0 Oblique forces (granular matter), 263 ff. Operational methods (unsteady airf. th.), 63 Oscillating airfoil in subsonic twodimensional flow, 44 ff. in subsonic three-dimensional flow, 52 in supersonic flow, 59 ff. Oscillating tangential Force (granular matter), 258 ff.
P Packing of non-spherical bodies, 249 ff. Phase rule, 310 Pinch effect (hydromagnetics), 207 ff. Piston theory, 76 f. Plastic waves in rods, elementary theory, 139 ff. strain-rate theory, 145 ff. Pochhammer-Chree theory for flexural waves, 155 f. for longitudinal waves, 115 ff. Poisson transformation in distr. th., 99 ff. Porous defined, 234 Pressure, calculation of (unsteady airf. th.), 64 Pyramidal packing, 238
R Rayleigh’s correction (waves in rods), 154 Reciprocity relations (unsteady airf. th.), 42 ff. Reflection of waves (hydromagnetics), 222 ff.
S
T Tangential forces (granular matter), 253 ff. Tetrahedral packing, 238 Thermally perfect gas defined, 308 Timoshenko theory (waves in rods), 162 ff. Transmission and reflection coefficients (hydromagnetics), 225 ff. Twisting couples (granular matter), 265
U Unsteady airfoil theory, conditions for linearization, 37 ff. edge conditions, 42 equations of, 39 ff. semi-empirical methods in, 50 ff. radiation condition, 42
V Vapor pressure curves of air and its components, 317 of water droplet, 406
459
SUCJECT INDEX
Velocity potential, method of (unsteady airf. th.), 44 ff.
W Wave generation (hydromagnetics), 220 f. Waves in assemblages of non-spherical particles, 283 in granular matter, 272 ff. in irregular arrays, 279 ff. in regular arrays, 272 ff. Water vapor affecting test-section Mach number, 435
affecting tes -section static pressure, 436
Weakly convergent defined, 92 Weak derivative defined, 94 Weak function as solution of differential equations, 103 Weak point of accumulation defined, 93 Wing tip, rectangular, 65 ff. oblique, 68 f.
X X-shock, 334
459
SUCJECT INDEX
Velocity method (unsteady This Pagepotential, Intentionally Leftof Blank airf. th.), 44 ff.
W Wave generation (hydromagnetics), 220 f. Waves in assemblages of non-spherical particles, 283 in granular matter, 272 ff. in irregular arrays, 279 ff. in regular arrays, 272 ff. Water vapor affecting test-section Mach number, 435
affecting tes -section static pressure, 436
Weakly convergent defined, 92 Weak derivative defined, 94 Weak function as solution of differential equations, 103 Weak point of accumulation defined, 93 Wing tip, rectangular, 65 ff. oblique, 68 f.
X X-shock, 334
