Advances in Applied Mechanics Volume 26
Editorial Board T. BROOKE. B E N J A M I N
Y.
c. F U N C i
PAUL G E R M A I N RODNEY H I L L L. HOWARTH C.-S. Y I H (Editor, 1971-1982)
Contributors to Volume 26 D. Y. H S I E H KLAUS K I R C H C A S S N E R J. T. C. L I U
IC'HIROT A N A K A TAYLOR C. W A N G
ADVANCES IN
APPLIED MECHANICS Edited by Theodore Y. Wu
John W. Hutchinson D I V I S I O N OF A P P L I E D S C I E N C ' t S HARVARII UNIVERSITY CAMBRIDGE, MASSACHUSETTS
ENGINEERING SCIENCE DEPARTMENT C'ALIFORNIA INSTITUTE OF T E C H N O L O G Y P A S A D E N A , C A L I FORNl A
VOLUME 26
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Contents vii
ix
Equilibrium Shapes of Rotating Spheroids and Drop Shape Oscillations Tavlor G. W a n g I Introduction I I . Background 111. Experiments IV. Concluaion Acknowledgments References
On Dynamics of Bubbly Liquids
D. Y. Hsieh I . In t rod u ct io 11 I I . General Formulation of Dynatnical l.qu,itions 111. Dynamic Equations of Bubbly I iqiiid\ IV. Sound Waves in Bubbly Liquid V. Waves a n d lnstahility in Bubbly Liquid5 V I . Steady Flows V I I . Dynamic\ of a Liquid Containing Vapor Bubbles Appendix A Appendix L3 Appendix C' Appendix D References
64 66 75
88 96 1 ox 1 IS 124 115 118
130
132
Nonlinear Resonant Surface Waves and Homoclinic Bifurcation Klaus Kirchgassner 1. Introduction
135
I I . The Problem
142
v
vi
Contents
I I I . Transformations a n d Symmetry IV. The Method V. Reduction a n d Result5 V I . The Mathematics References
144 147 152 172 179
Contributions to the Understanding of Large-Scale Coherent Structures in Developing Free Turbulent Shear Flows
I. Introduction 11. Fundamental Equations a n d the Interpretation 111. Some Aspects of Quantitative Observations
184 188 21 1
IV. Variations o n the Amsden a n d Harlow Problem-The Temporal Mixing Layer 219 V. The Role of Linear Theory in Nonlinear Prohlenis 232 25 1 VI. Spatially Developing Free Shear Flows VII. Three-dimensional Nonlinear Effects in Large-Scale Coherent M o d e Interactions 284 298 VIII. Other Wave-Turbulence Interaction Problems 300 Appendix 302 Acknowledgment 302 References
Three-Dimensional Ship Boundary Layer and Wake
Ichiro Tanaka I . Introduction 11. Flow Around a S h i p Hull 111. Equations of the Three-Dimensional Turbulent Boundary Layer and Wake a n d Their Solutions IV. Scale Effects o f the Boundary Layer and Wake V. Concluding Remarks Acknowledgment References
31 1 313
AUrHOR
361 361
INIIFX
SLJBJFCT I N D E X
318 342 357 358 358
List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
TAYLORG. WANG( l ) , Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91 109 D. Y. HSIEH(64), Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 KLAUS KIRCHGASSNER (135), Math. Institut A, Universitat Stuttgart, Federal Republic of Germany J. T. C. Liu (184), Division of Engineering, Brown University, Providence, Rhode Island 02912 ICHIRO TANAKA (311), Department of Naval Architecture, Osaka University, Osaka, Japan
vii
This Page Intentionally Left Blank
Preface This volume contains five comprehensive articles covering several main active areas of applied mechanics. Taylor Wang’s paper on the rotation and bifurcation of a self-gravitating liquid d r o p relates to the long history dating back to the pioneering studies on theoretical models for simulating the revolution of planets by Newton, Dirichlet, Maclaurin, a n d Riemann. This article further presents a vivid report o n a n unprecedent Spacelab experiment carried out by Dr. Wang himself in the microgravitational environment of the Spacelab 3 during April 29-May 7, 1985, but only after surmounting single-handed a n equipment failure that occurred unexpectedly in flight. Klaus Kirchgassner’s contribution to the emerging field of resonant nonlinear waves a n d their bifurcation is timely. It raises some searching questions concerning forced generation of solitary waves and presents a new mathematical method with examples to project a general applicability. D. Y. Hsieh’s article provides a critical review on the state of the art regarding the theoretical models commonly used for analyzing two-phase flows and the mechanics of a liquid containing bubbles. Questionable points and prospects of theoretical improvement are examined to enhance further advances of “this complex and sometimes confusing subject.” J. T. C. Liu’s paper proceeds from Dryden’s article in Volume 1 of this series, where Dryden expressed the prescient conviction that in analyzing turbulent boundary layers it is important to separate the random processes from the non-random ones. This article then addresses the physical problem of large-scale coherent structures in real, developing turbulent free-shear flows with a n interpretation of the non-linear aspects of hydrodynamic stability theory and presents a discussion of the results on the basis of the conservation principles. Ichiro Tanaka’s work on three-dimensional turbulent boundary layers a n d wakes for ship-hull shaped bodies illustrates the challenging problems our profession is still to overcome. The experience acquired from practical applications is of value for further development of this complex subject. ix
X
Preface
We are grateful to these authors for their efforts in giving surveys of the present state of research work in various fields of applied mechanics. The authors have had full liberty to develop their personal views and to explore the promising directions of future advances. THEODORE Y. Wu
A D V A N C E S I N APPL ! L ! l M t ( IIANIC S , VOLUMI-
26
Equilibrium Shapes of Rotating Spheroids and Drop Shape Oscillations TAYLOR G. WANG Jet Propulsion Luhorutory CulIf’ornia Institute of’ Technology Pasadena, Caljfiwnia
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Equilibrium Shapes o f a Rotating D r o p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Stability C . Shape Oscillations _ . _ . . . . . _ . . . . . _ . . _ . . . . . . . . . . _ . . . . . t , r , . . . . . . _ , . . . . . D. Gravitational Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Drop Fission I l l . Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Figures o f a Rotating Drop in an lmmiscible System . . . . B. Drop Oscillations in an Immiscible System.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Oscillations o f a Rotating D r o p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , . , . . . , . , D. Compound Drop Oscillations. . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . , . , . , . . E. Drop Dynamics in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 3 4 6 9 10 10
10 20 30 38 43 57 58 58
I. Introduction The behavior of liquid drops has long been studied by many scientists. Two problems in particular, equilibrium shapes of rotating spheroids and drop shape oscillations, have been investigated for hundreds of years. The problem of equilibrium figures of rotating self-gravitating liquid masses originates from the Newton-Cassini controversy (over 300 years old) about the shape of the earth. In centuries since, the problem has interested many illustrious mathematical physicists. The basic mathematic 1
< op\rlpht < 19XY Academic I’re\s Inc All rights 01 r c p r r r d u ~ t i a n~n .iny lorin rererved I \ H N 0 I?002026 2
2
Taylor G. Wclrig
properties of instabilities and bifurcation solutions have been discovered, a n d the similarity of the dynamic behavior between surface-tension-dominated and self-gravitational-dominated systems of liquid masses was stressed by Chandrasekhar ( 1965). But until recently, comparison with experiment was not available. A reasonably complete theoretical formulation has been developed for small-amplitude d r o p oscillations under the influence of surface tension forces. Chandrasekhar gives a n extensive review of the theoretical work, and a large variety of experimental tests have been conducted to verify a n d support this theoretical work. The experimental procedures fall into three general categories: the liquid drop is suspended in a neutral, buoyant media; the drop is supported by a vertical gas flow; or the drop falls through a gas o r vacuum. The limitations of these methods limit the detailed quantitative comparison of theory and experiment and d o not extend to large-amplitude d r o p oscillations. This chapter reviews some of the recent work on equilibrium shapes of a rotating spheroid and d r o p shape oscillations.
11. Background
Experimental observation of the behavior of a rotating d r o p held together by surface tension goes beyond simply testing existing theory. The theory is in fact embedded in a larger theory which at o n e extreme embraces fluid masses held together by gravity-modeling the stars-and at the other extreme embraces uniformly electrically charged fluid masses-modeling atomic nuclei (Swiatecki, 1974). Consequently, any deviation in the observed behavior of ordinary liquid drops from predicted behavior questions the larger theory of the equilibrium figures of fluid masses. Conversely, if experiments on the equilibrium figures or ordinary drops agree with theoretical predictions, it strongly suggests a unified theory of the dynamics of fluid masses. The observed behavior of ordinary liquid drops would then help to frame this theory of dynamics, which in turn could be extended into the astronomical and nuclear realms. The rotation a n d oscillation experiments are precursors to experiments in which the oscillation of rotating drops are studied. The experiments are also precursors to ones in which the drops are electrically charged, electrically conducting, dielectric, non-Newtonian, o r superfluid, and ones where external fields are applied (electric, magnetic, electromagnetic, acoustic, or
Equ ilibriir m Shapes 0f’ Rota t irig Spheroids
3
thermal). I n addition, it is envisaged that multiple-drop experiments be performed in which the interactions of free drops can be observed. The following discussion traces the evolution of the current theory of the equilibrium shapes of a rotating fluid drop held together by its surface tension. Lord Rayleigh’s calculations ( 1914) of axially symmetric equilibrium shapes formed the basis of Appell’s work (1932), which extended and gave a more detailed and elegant description of these calculations, opening the discussion of the dynamics of shape change and the stability of these shapes. Chandrasekhar made a definitive study of the stability of the simply connected axisymmetric shapes, and in addition obtained the frequencies of their small-amplitude oscillations. Ross (1968) reviewed and extended some of the previous work on drops t o “bubbles”-fluid drops less dense than the surrounding medium. Cans ( 1974) examined small-amplitude oscillations about equilibrium shapes of compressible fluids. The equilibrium shapes of a drop containing a bubble were discussed by Bauer and Siekmann (1971), who also studied the shape of a rotating dielectric drop in an electric field (1974). Finally, Swiatecki, by inserting the theory of the equilibrium shapes of “surface-tension drops” into the more general theory, gave a fairly complete semiquantitative description of the stability of shapes for such drops as a function of their angular momentum, including a discussion of metastable shapes he called “saddle-point” shapes.
A. E Q U I L I B R I USHAPES M OF
A
ROTATINGDROP
The axisymmetric equilibrium shapes (see Figures 1 and 2) are conveniently described as a function of the dimensionless angular velocity (1, which is the angular velocity measured in units of the fundamental oscillation frequency of the resting drop. f Z = w / ( S c ~ / p a ~ ) ) ” where ’, w is the rotational angular velocity, (T is the surface tension, a is the equatorial radius, and p is the density. When R<0.7071, there are always two equilibrium shapes; the one of lower energy is simply connected, while the other is torus-like. For R = 0.7070, an additional “collapsed” shape appears in which zero thickness at the center yields a “figure-eight” cross section. For 0.7071 < R < 0.73, there are two torus-like shapes but still two simplyconnected shapes. When R = 0.73, there is only one torus-like shape? but still two simply-connected shapes. The torus-like shapes are lost once
+ Ross and Appell disagree on the number
of toroidal shapes of
< 0.7071
4
Taylor G. Wang
2 0.8
-
0
R
0.4
0.8
1.2
= 0.679
1.6
2.0
!-
n
0.8
=
0.707
I-
0 0.4
* I N THE UNITS OF
0.8
1.1
1.6
2.0
)';(: F I G . 1.
Axisyrnrnetric equilibrium shapes
R > 0.73, and when R = 0.7540 there remains only one simply connected shape; this is the greatest angular velocity that an axisymmetric equilibrium shape can have.
B. STABILITY The only detailed study of the stability of the equilibrium figures was made by Chandrasekhar, and only for simply connected shapes. He showed that for R=0.584 the drop can deform, without changing its energy, to another shape having rigid body rotation; thus the original shape is unstable. He presumed that the stable equilibrium shapes become nonaxisymmetric for R = 0.584. This presumption is based on an analogy with what is known
5
Equilibrium Shapes of Rotating Spheroids
fl
= 0.754
0.4
0.8
0.8
1.2
1
1.6
n
2.0
0.56
~
2 L 1 0 4 . 4
0
1.2
1,6
fl
0.8
0.4
0.8
~
2.0
0
0.4
0.707
0.8
1.2
1.6
fl
=
2.0
0.35
-
FIG. 2. Axisymmetric equilibrium shapes (to pinch-off).
to occur for a liquid mass held together by self-gravitation. There the equilibrium figures are true ellipsoids stable below a critical angular velocity, the triaxial ellipsoids being stable above. As shown in Figure 3, R = 0.584 the bifurcation point and the secular stability pass from the sequence of axisymmetric shapes to triaxial shapes. Berringer and Knox (1961) calculate that for the “surface-tension” drops the triaxial shapes are not ellipsoids, but the stability of the toroidal shapes has received no definitive treatment. However, the results of Wong (1973) on the toroidal shapes of charged liquid drops suggest that these equilibrium shapes may be “saddle-shapes”-shapes stable against deformations that preserve the axial symmetry but unstable against others such as varicose deformation against which fluid jets are unstable (the Rayleigh instability). A substantial extension of the theory to include the shapes and stabilities of nonaxisymmetric figures of equilibrium was undertaken by Brown (1979). Along the simply-connected sequence, the axisymmetric drop shape was stable to two-lobed perturbations for 2 <0.318 = C,, (Brown calculates
6
Taylor G. Wang
* O * * c VI .c
3
t.
.-e
B I FURCAT I ON
c
fl m
Id 1.22 1.13 1.0
0 AXISYMMETR IC
I
0.5
52,
0.584 0.679
ANGULAR VELOCITY IN UNITS OF
1.
au 1 / 2
(7) Pa
FIG. 3 .
Illustration of bifurcation points.
E l l = 0.313). At XI, the drop is neutrally stable to these perturbations and, above it, the axisymmetric shape becomes unstable. Similarly, Brown calculates bifurcation points to three- or four-lobed families from the simply connected sequence at Z I l 1= 0.5001 and XI, = 0.5668 (see Figure 4).
C. SHAPEOSCILLATIONS In 1879 Lord Rayleigh conducted one of the first investigations on the behavior of an oscillating liquid drop about its spherical equilibrium shape. He limited his study to the case where the oscillations were small-amplitude, axisymmetric and assumed that the internal motions were described by a potential flow field. He did not include viscous effects. The results of Rayleigh’s investigations can be expressed by a series expansion of Legendre polynomials: r = a,+C a,,P,(cos O),
(2.1)
where r is the radial coordinate, O is the polar angle measured from the pole of the drop and the coefficients a, are functions of time. In solving for the an’s that appear in Equation (2.1), it is necessary to limit the oscillations to small amplitudes, urrQ a,. It can be shown that if
Equilibrium Shapes of Rotating Spheroids
7
9.0
g az
9 IY
8.0
0 0
7.0 0 0
4 LOBES
u)
-0
u) v)
2
0
6.0
c> -O
U
n
n
0
_
- 0
0 0
w
N
5
3 LOBES
0
0 0
o n -
5.0
8 f
4.0
3.0
I
0
0.1
I
0.3
0.2
I
I
'
I
0.4
I
0.5
,5668
Cmox = 0.5685
z - (NORMALIZED ANGULAR VELOCITY)' FIG. 4. Calculated equilibrium shapes (from data given by Brown, supplemented by Chandrasekhar and Ross).
a,
= cos w,t
then w ', = n ( n - I)( n
u +2 ) 7 , Pa
where u is the surface tension, p is the density of the liquid, and a is the equilibrium spherical radius. It should be noted that cases where n = 0 and n = 1 correspond to rigid body motions. The fundamental mode of oscillation is given by n = 2 . The period for the fundamental mode is given by 72 = 5 7
JI". 2a
For a 2.5-cm-diameter water drop where p = 1 gm/cm' and u = 75 dyn/cm, l8 the period would be 7>=0.36 sec for the fundamental mode, ~ ~ = O . for n = 3 and ~ ~ = 0 . 1for 1 n=4. Foote ( 1971) performed extensive computer calculations of Rayleigh's equation and dropped the restriction of the calculations to small amplitudes.
Taylor G. Wang
8
The results of Foote's calculations are shown in Figure 5 for the cases of n = 2, 3, and 4. In all cases, the drop is started in motion at time t = 0 in the deformed shape with the internal flows at zero. The time is measured in units of 7r radians so at t = 1 the drop goes through one-half cycle. Even at large amplitudes the calculated drop shape appears to have the approximate shape observed in experiments. A detailed comparison with experimental data is impossible because the data quality is limited by experimental techniques now available. Rayleigh solutions are probably only true for amplitudes corresponding to f from 0.375 to 0.625. Viscous effects have not been included in the analysis so far. Lamb (1932) showed that for small viscosity the only effect on an oscillating spherical drop is the gradual damping of the oscillation amplitude. The normal-mode frequency is not affected by the viscosity. The decay of the amplitude, A, can be shown to be given by
A = A,, e-0~8'
(2.4)
where A. is the initial amplitude of the oscillation of the drop and T
=
T =
O.OO0
O.ooO
T = 0.500
T =
0.m
T
=
Pn is
1.ooO
0 T
=
1.ooO
0 T = 0.m
T
=
0.500
T = 1O . OO
FIG.5. Fundamental modes of oscillation. The axial ratio is 1.7 at the maximum distortion and the axis of symmetry is vertical.
Equilibrium Shapes of Rotating Spheroids
9
given by Pn
=
( n - 1)(2n + l ) u a’
where v is the kinematic viscosity of the liquid and a is the radius of the drop. For a drop of water 2.5 cm in diameter with v = 0.014 cm2/sec, oscillating in fundamental mode n = 2, P = 0.045. Thus a free-oscillating drop decays to 1% of its initial amplitude in 102 sec. Chandrasekhar showed that a periodically damped motion of the drop is possible for the fundamental mode if w,,RJ v is less than 3.69 cmz/sec. Therefore a drop 0.023 cm in radius (or smaller) would experience aperiodic motion. Prosperetti (1978) found that a drop initially critically damped should have an aperiodic decay for a short time and a damped oscillation motion later. He implies that the effective damping factor first increases and then decreases. These predictions have yet to be experimentally verified.
D. GRAVITATIONAL FORCES
In contrast to the case of small viscosity where w, is independent of viscosity, Happel and Brenner (1965) showed that for a system controlled by gravitational forces as v + 00 the normal mode oscillation can be given by 2
w=w,
2n+l a’ 2(n-1)(2n2+4n+3) Y
where a is the radius of the sphere and v is the kinematic viscosity. Foote (1971) noted (from his computer calculations) that the drop spends more time in the prolate configuration than in the oblate (57% vs. 43%). Montgomery (1968) observed a similar behavior in vertical wind tunnel measurements on drops. However, it is not clear how much effect the air streaming around the drop has on this unequal distribution of time in the prolate and oblate shapes. At small amplitudes the calculations show that the drop spends an equal amount of time in the two configurations. Another feature of the computer calculations is that the period of oscillation is not constant with large-amplitude oscillations but shows an increase in the period as the amplitude increases. For a large oscillation in which the ratio of the major to the minor axis is 1.7, the fundamental frequency increases approximately 9% for an ellipsoid-shaped drop and 5% for a Rayleigh-shaped drop.
Taylor G. Wang
10
Montgomery’s experimental work on small drops agrees qualitatively with these computer calculations. The computations show a smooth change in the increase of the period with increasing amplitude of the oscillation. The calculations do not take into account any turbulent flow within the drop which one might expect to find at large-amplitude oscillations. This turbulent flow might manifest itself as a break i n the curve of period vs. amplitude as one enters the turbulent flow region.
E. DROPFISSION As the drop oscillations grow to large amplitudes, a point is reached when fission of the drop is possible. A considerable amount of theoretical work on charged drops was stimulated by nuclear physicists’ attempts to model nuclear fission with the behavior of a charged drop. Diehl (1973) calculated the ternary fission for the liquid drop model and showed that there are two modes of fission, the prolate mode that has axial symmetry and the oblate mode. Alonso (1974) calculated the binary fission case and found that the neck connecting the two sections becomes very elongated and eventually develops a long thin neck that will not pinch off until it has extended to virtually no width. It is hypothesized that the pinch-off is actually initiated by a surface fluctuation in the neck. Thompson and Swiatecki observed the thin-necking in a neutral buoyant experiment on polarized drops. From their data it also appears that the drop attempts fission in the prolate ternary mode.
111. Experiments
In the following section some recent experiments on drop dynamics are discussed.
A. FIGURESOF
A
ROTATINGDROP I N
AN
IMMISCIBLE SYSTEM
A large (-15 cc) viscous liquid drop is formed around a disk and shaft, in a tank containing a much less viscous mixture having the same density as the drop. This supporting liquid and the drop are immiscible. If the shaft and disk were not present, the drop would float freely in the surrounding
Equilibrium Shapes qf Rotating Spheroids
11
medium and assume the shape of a sphere. With the drop attached and initially centered about the disk, the shaft and disk are set into rotation almost impulsively, reaching a final steady angular velocity within one-half to two revolutions. The drop deforms under rotation and develops into various shapes depending on shaft velocity. The process of spin-up, development, and decay (or fracture) to some final shape is common to all runs. In this system, gravity is diminished by introducing a supporting liquid which is viscous and which may be entrained by the motion of the drop, thereby allowing angular momentum to transfer from the drop. Nevertheless, comparison of this experiment's results to the theory of free rotating liquid drops is prompted by the fact that several novel families of drop shapes have been observed. 1. Immiscible System
The immiscible tank (see Figure 6) in which a drop is buoyantly supported and rotated consists of a Lucite cylinder contained in a cubical outer tank. Thus, cylindrical symmetry about the axis of rotation is obtained while lens-like distortion of the drop inside the cylindrical tank is minimized by the parallel-sided geometry of the outer tank and the water circulating between it and the inner tank. The circulating water is pumped into the system from a constanttemperature bath with a 15-liter capacity. By this means, the temperature of the system may be controlled to within 0.01"C or better. Such control is a critical factor in the performance of the experiment. FLYWHEEL
MIXTURE OF WATER AND
TEMPERATURE REGULATING
I
FIG.6.
I
Immiscible system apparatus.
Taylor G. Wang
12
The fluids used in this experiment are silicone oil (Dow Corning 200,100 centistoke [cst]) for the drop, and a 3-to-1 water-methanol mixture for the host. The physical properties of the mixture are highly dependent on temperature. Therefore, the equilibrium positions of the drop are extremely sensitive to the temperature gradient shown in Figure 7. The shapes of rotating spheroids and the fluid flows are recorded on a camera and digitized on a motion analyzer. The flow visualization inside the drop is accomplished by forming tracer particles out of the watermethanol mixture. 2. Shapes of Rotating Drops Five basic families of shapes are observed (Tagg, 1980). They are axisymmetric, two-lobed, three-lobed, four-lobed, and toroidal. Additionally, the off-axis single lobe is the final shape for all experimental runs except those in which the drop undergoes fracture. These shapes are shown in Figures 8 through 14, and in Figure 15.
4. 24.2 -
j:.a.. 24.0
8 POSITION OF THE DISC I
. ......... .. . I r,0,.0(01.80@
I
I
I
..5.
-
0s.
5
I
I
Equilibrium Shapes of Rotating Spheroids
FIG.8. Drop at rest. Note internal trace drops and external satellite drops.
FIG. 9. Axisymmetric oblate drop. Shaft angular velocity = 0.8 rps.
13
14
Taylor G. Wang
FIG. 10. Axisymmetric biconcave drop. Shaft angular velocity = 1.8 rps. Drop is still spinning up.
FIG. 11. Two-lobed shape. Shaft angular velocity = 1.8 rps.
Equilibrium Shapes of’Rotating Spheroids
FIG. 12. Three-lobed shape. Shaft angular velocity = 2.0 rps.
FIG. 13. Four-lobed shape. Shaft angular velocity = 3.8 rps.
15
16
Taylor G. Wang
F I G . 14. Torus. Shaft angular velocity= 4.8 rps.
FIG. 15. Break up of torus. Shaft is not rotating.
Equilibrium Shapes of Rotating Spheroids
17
Apart from the axisymmetric shapes at slow rotation rates, the three-lobed family is the easiest to obtain. This is due in part to the particular drop volumes and shaft dimension used in this experiment. The ease with which three-lobed shapes are generated is nevertheless remarkable; even in an early, very crude 1/4-scale version of the experiment, three-lobed shapes were readily obtained. Two-lobed shapes, which develop for slower shaft velocities ( < 2 rotations per second [rps]), may be harder to obtain because the decay processes which cause the drop to form into an asymmetric single lobe may set in before the drop can develop symmetric lobes. Four-lobed shapes, on the other hand, are obtained at generally higher shaft velocities (-4 rps) than the three-lobed shapes. When asymmetries develop in the drop at these angular velocities, fracture usually results. During the decay of higher nonaxisymmetric modes, one lobe generally rotates faster than the others, eventually catching up and joining with the lobe preceding it. Thus, three converge into two and two into one. This is not surprising, since the mass of the drop is never equally distributed among the lobes, and so one lobe is smaller and suffers less drag from the surrounding fluid. The presence of drag is immediately apparent from the pinwheel appearance of all of the lobed shapes, with the lobes curving backwards against the direction of rotation. A further effect, attributed to the motion of the outer fluid, appears in many runs in which two- and three-lobed shapes are produced. In the course of the drop’s development, the drop rises and becomes sessile on top of the disk (i.e., it only contacts the upper surface of the disk and shaft). Three-lobed drops decay to two-lobed drops which are sessile (Figure 16) and often persist for many seconds before decaying to single-lobe drops (also sessile). This drop rising occurs even when the level of exact density matching is below the disk by (for example) two centimeters. Furthermore, a different effect occurs above a well-defined shaft velocity midway in the range of velocities producing three-lobed shapes. The three-lobed drop still decays to a two-lobed drop, but one lobe forms above the disk and the other below, i.e., the drop is tilted (Figure 17). This appears to be a very stable geometry which can persist for minutes. Only a few instances of the toroidal shape are observed using this system. Nevertheless, striking examples have been photographed of the formation of a torus and its subsequent highly symmetric fracture into three or four large drops and a corresponding number of small satellite drops (Figures 14 and 18-19). The sequence of shapes (axisymmetric to two-lobed to
18
Taylor G. Wang
FIG. 16. rotating.
Break up of torus. Shaft is not
FIG. 17. Single lobe. Ultimate decay shape
Equilibrium Shapes of Rotating Spheroids
FIG. 18. Sessile two-lobed shape. The results ofdecay from a three-lobed shape; shaft is not rotating.
F I G . 19. Tilted two-lobed shape. Decay route for three-lobed shapes.
19
20
Taylor G. Wang
three-lobed to four-lobed to toroidal) seems to be linked to increasing spin-up velocity.
B. DROPOSCILLATIONS I N A N IMMISCIBLE SYSTEM In this section, we are interested in the measurement of the first few resonant frequencies and the damping constant in the small-amplitude region as both the drop size and the viscosity are varied. Viscosities between 1.3 and 130 cSt are studied, and the drop diameter ranges between 0.5 and 1.5 cm. The experimental method involves acoustic levitation and radiationpressure-force modulation (Marston and Apfel, 1979). In addition to quantitative information about the various resonance modes of the drop, a qualitative treatment of the characteristics of the internal fluid particle flow field has also been obtained. Comparison with available theoretical predictions is generally favorable when appropriate precautions are taken to satisfy the theoretical assumptions. It then appears, at least for the steady-state or long-time behavior, that the linear theory yields satisfactory answers to the problem at hand. 1. Experimental Approaches
A liquid drop can be trapped at a stable position in an acoustic standing wave existing in a resonant cavity filled with liquid. The static equilibrium shape of the drop can be controlled by varying the magnitude of the acoustic radiation force, which is proportional (to the second order) to the square of the magnitude of the first-order acoustic pressure. In this case, the cavity is of rectangular geometry, and the axis of symmetry is taken as its vertical axis. Depending upon the standing acoustic wave or the pressure intensity, a drop may be given the shape of a prolate spheroid, or an oblate spheroid, or a nearly perfect sphere. Shape oscillations are induced through low-frequency modulation of the acoustic radiation force. The excitation can correspond to either a periodic elongation or compression of the drop at the poles. The restoring force is, in the ideal case, provided only by the interfacial tension which tends to drive the drop back to the equilibrium shape. For low-amplitude vibrations both modes yield the same results for the resonant frequency, although this is not to be the case for large displacements (Trinh and Wang, 1980).
Equilibrium Shapes of Rotating Spheroids
21
Various liquids are used for both drop and host media. First, the combination of a phenetole drop suspended in a 1 :2 by volume mixture of methanol and distilled water is used. Next, various viscosity grades of Dow Corning silicone oil (5-200 cSt) are mixed with carbon tetrachloride (CCl,) to form drops which are suspended in distilled water. All drop liquids are colored with oil-soluble dyes. The drop oscillation amplitude is monitored using an optical detection technique: a slit either parallel or perpendicular to the axis of symmetry is uniformly illuminated, and the shadow of the drop is then centered across the slit, blocking some of the light. As the drop oscillates, the light intensity detected by a photo transistor varies periodically in phase. Up to a reasonably large amplitude, the response is approximately proportional to the crop's deformation. If the drop axis rotates, however, the detector no longer yields a true maximum amplitude variation. 2. Normal Modes
The normal mode of drop oscillations for various liquids has been measured as a function of size. Figure 20 is a logarithmic plot of the measured frequency (h) square as a function of the cube of the drop radius ( R ) for a phenetole drop. The result yields a power law f 2 - R-'.'' which is very close to Lamb's theory, f 2 R-'.50
-
FIG.20. Data for phenetole drops (density approx. 0.96 g/cm3) in distilled water-methanol mixture. The kinematic viscosity of phenetole is about 1.22 cSt, interfacial tension is about 16.5 dyn/cm.
22
Taylor G. Wang
Figure 21 displays similar logarithmic plots for a series of mixtures of silicone oil with CCI, of various viscosities. The drops were suspended in distilled water. A least-squares fit also yields coefficients very close to -1.5. All the above measurements are obtained with a 22 kHz standing wave used for levitation, and a modulated 66 kHz wave for the oscillation drive. For small amplitudes the results are within experimental uncertainty, independent of the nature of the acoustic field. I
40
I
I
I
I 36.8 cst.
I
03.2crt. 40
N
~
I
-
30 N ~
06.5cst.
0
-
0
0
20 -
0
0
-
I
A 120.4 crt.
0
0
20
468.6crt. -
30-
N N L LI N
16.5 cst.
0
40
I
I
30 -
I
I0
I
I
I
I 68.6 cost.
+
-
4
NL N L N
I
15
-
4
20 -
4
-
P I
I
I
30 N N LL N
N N L
I N 30
I
120.4 cost.
-
A
-
1 R3 cm3
0.3
0.4
0.1
-
A A A
10 -
0.2
1
-A
20-
15
0.1
I
40 -
16.5 cst.
-
A I 0.2
I
I
0.3 0.4 R3 crn3
FIG. 21. Experimental results for fundamental frequencies of drops of silicone-CCI, mixtures of various viscosities immersed in distilled water (density approx. 0.99 g/cm3, interfacial tension between 35 and 40 dyne/cm). f 2 is plotted as a function of R 3 . The kinematic viscosity ranges between 3.2 and 120.4 cSt. Drop radii range between 0.49 and 0.68 cm.
I
Equilibrium Shapes of Rotating Spheroids
23
The effects of more substantial distortions upon both the resonant frequency and the damping constant have also been investigated. This is done by adding a static levitating standing wave to the time-modulated field, and by increasing its intensity. Both static oblate and prolate distortions may be obtained by using the appropriate frequency. They show similar effects. Even for small-amplitude oscillations, the resonant frequencies for the fundamental and first-higher modes increase with distortion. Figure 22 displays the results for a 2.0 cm3 phenetole drop and a 1.5 cm3 silicone-CCI, 30
20 -
8
GI-
5
10
15
20
25
% W/H FIG.22. Variations of the fundamental resonant frequency with oblate deformation. Drops are statically deformed by the acoustic field. The resonant frequency is measured for smallamplitude oscillations: (a) silicone-CCI,, 1.5 cm3 drop; (b) phenetole, 1 cm3 drop.
24
Taylor G. Wang
mixture drop. Both drops are statically deformed into the oblate spheroid shape. The phenetole drop is driven into oscillation by a periodic compression at the poles, while the silicone-CCI, drop is driven by a periodic elongation at the poles. This distortion is characterized by the ratio of the long axis of the drop to the short axis ( W / H ) . The percentage of distortion is plotted c n the horizontal axis, and the percentage increase in the fundamental resonance frequency f2 on the vertical axis. An approximately linear variation of Af2/f 2 , with increasing distortion, is observed for both drops. The higher-order resonance modes are increasingly more damped than the fundamental; thus comparatively larger forces (i.e., acoustic pressures) are required to excite them. Under these conditions the measured frequencies are not those corresponding to a nearly spherical equilibrium shape as in the case of the fundamental mode. Figure 23 shows photographs of higher-order modes of drop oscillation. The oscillation amplitudes shown here are appreciable and cannot be viewed as small. However, the measured frequencies are in agreement (*lo%) with calculated values. 3 . Amplitude Dependence The experimental results are shown in Figure 24 for two characteristic drop sizes: 0.5 cm3 and 1.0 cm3 (the drop radii are equal to 0.49 cm and 0.62 cm respectively). The results are for drops of a silicone oil/CCl, mixture immersed in distilled water and for a prolate-biased initial oscillation drive. The percentage of relative change in the decay frequency is plotted as a function of the maximum deformation with the drop in the prolate shape. This latter characteristic is quantified by the ratio of the major over the
FIG. 23.
Photographs of drops oscillating in the I
= 2,
3, 4 axisymrnetric ( m = 0 ) modes.
25
Equilibrium Shapes of Rotating Spheroids I
m
I
o
I
I
I
I
I
I
0
O 0} 1 0 3 0.5 0 3
O 0
0
0 0
1.2
1.4
1.6
C
1.8
L/W
FIG.24. Relative change in the free-decay frequency as a function of the initial oscillation amplitude. The axial ratio I / W is measured at maximum deformation during the steady-state drive prior to the free decay phase. The drops are of a silicone oil-CCI, mixture, and are immersed in distilled water.
minor axis. The measured free decay frequency for very small amplitude oscillations is fo. The initial shape oscillations are provided by a modulated 66 kHz standing wave, and the equilibrium shape is slightly prolate. Similar results are obtained of the free decay frequency when the initial shape is slightly oblate.
4. Damping Constant Measurements The damping constant for the fundamental mode has been measured as both a function of drop size of the viscosities of the inner and outer liquids. For viscosities lower than 100 cSt and drop volumes larger than 0.5 cm3, data were obtained from photographs of oscilloscope traces taken during free decay. Figure 25 displays the photographs of a few decay curves for various viscosities.
26
Taylor G. Wang
la) PHCNETOLE -1.22 CST
(cJ
SILICONE-6.48CST
le)
SILICONE -36.78CS1
(hl
SILKONE -3.24 CST
(dl
SILICONE -16.51CST
if1 SILICONE -68.65CST
FIG. 25. Photographs of oscilloscope decay traces for drops of various viscosities. Phenetole: (a) 1.22 cSt. silicone; (b) 3.24 cSt; (c) 6.48 cSt; ( d ) 16.51 cSt; (e) 36.78 cSt; ( f ) 68.65 cSt.
Equilibrium Shapes of Rotating Spheroids
27
The experimental results on the damping constant for phenetole (1.22 cSt) and for a series of mixtures of silicone oil and CCl, are given as functions of the drop radius in Figure 26. Phenetole is characterized by the set of data with the lowest values. The increasing values of 7;' are shose for mixtures with kinematic viscosity equal to 3.25,6.5, 16.5,68.6, and 120.4 cSt, in this order. A linear least-squares fit of the phenetole data yields a power law 7;'
aR-'.~.
Although similar least-squares fits of the data were attempted for all other liquids, the correlation was not good.
5 . Internal Fluid Flow The motion of the fluid inside the drop can be made visible by the addition of quasi-neutrally buoyant tracer particles. The markers should be small enough to follow the motion of the fluid but large enough to scatter sufficient
I
.4 5
.53
.61
.69
R
I
(Cm)
FIG. 26. Damping constant for various viscosity grades as a function of drop radius. 0, 1.2; 0, 3.2 cSt; A, 6.5 cSt; 0, 16.5 cSt; M, 36.8 cSt; 36.8 cSt.
*,
28
Taylor G. Wang
L=2 I N T E R N A L FLOW PATTERNS
FIG.27. Streak patterns of suspended dye particles in the midplane of an oscillating drop in the I = 2 mode. The drop liquid is silicone oil (50 cSt), and the host liquid is a mixture of distilled water and methanol. The appearance of a steady drifting motion is visible as the oscillation amplitude is increased. The Reynolds number varies from about 2 to 10.
Equilibrium Shapes of Rotating Spheroids
29
light for detection. Here we disperse organic dye particles with size ranging between 5 and 50 micrometers (Fm) in the drop liquid. The particles close to the central midplane containing the symmetry axis are illuminated by a plane light sheet, and the 90" scattered light is photographed while the drop is driven into steady-state oscillations. The exposure usually lasts for a few oscillatory cycles. Figure 27 reproduces the resulting streak patterns of oscillating particles situated in the plane containing the symmetry axis. These photographs show the evolution of the fluid flow pattern as the oscillation amplitude is increased. All drops are undergoing steady-state oscillations in the fundamental mode. In Figure 27 the maximum oscillation amplitude is from 5 % to 20% of the equilibrium spherical radius. In photograph (a), the steady drifting motion of the fluid particles appears limited to the outermost parts of the drop. Photographs (b) and (c) reveal a spreading of this steady motion to the inner drop regions. Note, however, that even in photograph (c) the center of the drop remains motionlesss, and the particles along the two intersecting axes still undergo a strictly linear oscillatory motion. Together with the spreading circulatory motion, the velocity of this steady drift also increases as the amplitude grows. Photograph (d) is a seemingly erratic oscillatory motion. No distinct pattern can be distinguished, although the basic fourfold symmetry is still preserved. Similar phenomena may be observed within a drop oscillating in the I = 3 mode. Figure 28 reproduces photographs of the two different drops undergoing vibrations in the I = 3 mode. The upper row of photographs reveals a substantially more disordered pattern as the vibration amplitude is increased. The lower row shows a well-defined circulation together with the disappearance of the characteristic symmetry. 6 . Fission Associated with Oscillation
As the oscillation amplitude is greatly increased, the increasingly larger deformation may cause the drop to split. Radiation-pressure-induced drop fission was first observed with 2-mm-diameter drops (Marston and Apfel, 1980), and our results confirm that cavitation-free splitting of l-cm-diameter drops is also possible when the acoustic wavelength is comparable to the drop diameter. Figure 29 illustrates the stages of such dynamic fission for a l-cm3 silicone oil (50 cSt) drop in water. A third smaller satellite drop is often formed when the two main droplets split.
30
Taylor G. Wang L = 3 internal flow patterns
FIG. 28. Streak pattern within a 1.5 cm3 drop (3.2 cSt silicone oil-CCI, in water in the upper row; 50 cSt in water-methanol in the lower row) oscillating in the I = 3 mode.
C. OSCILLATIONS OF
A
ROTATINGDROP
The effects of rotation on the frequencies of the shape oscillations of a drop inside an immiscible system were investigated by Busse (1984). He assumed that the rotation-induced shape deformation would remain small and axisymmetric, that the amplitude of the oscillations would be small, and that no inertial waves would be excited in the host fluid. This approach yields analytical expressions for the effect of rotation on the resonance frequencies of both the axisymmetric and nonaxisymmetric modes of shape
31
Equilibrium Shapes of Rotating Spheroids
FIG. 29. Stages of dynamic drop fission by acoustic radiation pressure. The drop is 1.5 cm3 in volume.
oscillation. The results for axisymmetric oscillations can be written as AO, --
O(o)-
0 ,-
OY- r i a 2 ( A + B ) T
forwl>2a,
A = 2 [l(l+2)(21-I)
+
B=
14+213-412-51+6 - 12(. k 4 ) ~ ~ + ( 1 + 1 ) ~ ( 1 - 3 ) p ' 4[pi(l+l)+lp0]
P'
1.
(3.3) (3.4)
32
Taylor G, Wang
Figure 30 is a schematic representation of the experimental system. The heart of the apparatus is a transparent lucite cylindrical cell (i.e., r = 4.7 cm, h = 7.4 cm), fitted on a precision-machined turntable, and directly coupled to a piezoelectric transducer. The liquid-filled chamber is closed by a rigid top, firmly attached to the cylinder after all visible bubbles are removed. In order to avoid cavitation, the host liquid (distilled water) is thoroughly
THE OSCILLATIONS OF A ROTATING DROP
FIG. 30. Schematic of the experimental setup. 1. Cylindrical acoustic cell; 2. aluminum block attached to a piezoelectric transducer; 3. turntable; 4. slip rings for electrical contact; 5 . belt drive; 6 . light encoder; and 7. drive motor.
Equilibrium Shapes of Rotating Spheroids
33
outgassed and freed of suspended solid impurities. The angular speed is measured with the light encoder fixed to the rotating shaft. The drops are driven into oscillations through modulation of the acoustic forces deforming the shape of the fluid sphere. The voltage imposed across thee transducer is expressed (approximately) by V = V, sin(27rf,t)+ V, sin(27rft) cos(2n-fmt),
(3.5)
where V, drives the standing-wave mode used to position the drop (f,= 42 kHz), V, excites the mode used to induce drop-shape oscillations (f;.= 92 kHz), and f m is the low-modulation frequency. The shape oscillations of the liquid drop are closely observed using a video system. The resonant frequencies of the drop corresponding to the I = 2 and 1 = 3 modes are determined by slowly sweeping the low frequency signal (f m ) , and by locating the frequency of maximum response. 1. Resonant Frequencies Four different drop sizes ( r , = 0.36 cm, 0.41 cm, 0.46 cm, and 0.51 cm) were chosen for the experiment, the density difference between the drop and the host liquids varying from -8 to +6% of the host density. The drop liquid was a mixture of low-viscosity silicone oil ( v < l O c S ) and CC14, varying in concentrations to adjust the density. Oscillation amplitudes of up to 20% of the drop radius were used for the measurements and were determined at the rotation axis in the vertical direction where optical distortion was at a minimum. The typical uncertainty in the frequency measurements was kO.1 Hz, or about 2%. The angular speed remained within k0.05 rps of the set value. Generally, a sequence of measurements was taken by alternating readings in a nonrotating state with those at a determined rotation velocity in order to monitor any time-dependence of the interfacial tension. Figure 31 displays some of the experimental results for the fundamental mode for p' < p", p' -- p", and p' > pO,respectively. The experimental uncertainty was within the size of the symbols used in these figures. Figure 32 displays data for the 1 = 3 mode and for p ' -- p" and p' > p", respectively. The experimental results for the p' > p" case d o not extend to large values of the normalized angular speed due to wobbling induced by a less-thanperfect alignment of the cylinder, drop, and rotation axes. The solid lines shown in these figures represent the theory.
Taylor G. Wang
34 0.8
I
o
0.7
I
I
I
1
I
I
I
I
I 0.7
I 0.8
0.9
pi = 0.915 g / m ~ 0 p =0.995 g/ml pi = 1.007 g/ml
0.6
1
pi = 1.061 g/ml
0.5
Aw2 (0) 0.4 w2
0.3
0.2
0.1
0 0
0.1
I 0.2
I
I
I
I
0.3
0.4
0.5
0.6
I 1.0
FIG. 31. The relative shift in the resonant frequency as a function of the square of the normalized rotation rate for I = 2 and ( a ) p' < p". ( b ) p ' = p o , and ( c) p l > po. The solid lines were obtained from Equation (3.1).
The slopes of the solid lines in Figure 31 are about 0.85 and those in Figure 32 about 0.93. They reveal that the effect of rotation was felt more strongly in the three-lobed oscillations than in the two-lobed oscillations. They also show that in the range of density differences studied (i.e., for -8 to +6% of the host-liquid density), the resonant frequency shift was not significantly affected by the density of the drop. A close agreement of the data with the theoretical predictions verifies the linear dependence of the relative frequency shift on the square of the normalized angular speed. The oscillation amplitudes used in the experiments were not infinitesimal as was assumed in the theory. The resonant frequency is known to decrease with an increasing oscillation amplitude (Trinh and Wang, 1982). Such a decrease, due to finite amplitude excited in the experiments, is estimated to be about 3 to 5 % .
35
Equilibrium Shapes of Rotating Spheroids THE OSCILLATIONS OF A ROTATING DROP
0.8
0 Pi -
0.7
= 1.061 g/ml
0p i = 1.007 g / m ~ 0 p ~ 0 . 9 9 5g/ml
0.6
0.5
9 (0) 0.4 w2
0.3 0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
FIG. 32. The relative shift in the resonant frequency as a function of the square of the normalized rotation rate for I = 3 and ( a ) p' = po and (b) p' > p'. The solid lines were obtained from (3.1).
An intriguing experimental observation was made as the angular speed > '1.0. ) ~Evidence showed the of rotation was increased, i.e., for ( 0 / 0 1 ' ~ existence of more than one resonant frequency for the 1 = 2 mode for a given rotational velocity. Those results are displayed in Figure 33. Stroboscopic illumination revealed that they are all oscillations of the oblateprolate types. One must note that this rotational velocity limit corresponds to w , I2 0 , where the assumptions of the linear theory discussed above break down, and inertial waves could be excited in the chamber. The present
1.0
Taylor G. Wang
36
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
FIG. 33. Observed multiplicity in the resonant frequency at large rotation rates ( I = 2).
n
z
-
-I
56 0.1
-
f =11.2Hz 2
I
I
I
I
I
I
37
Equilibrium Shapes of Rotating Spheroids
experimental apparatus, however, does not readily lend itself to the observation of such wave phenomena. 2. Decay Constanf
The damping constant measurements were carried out for the I = 2 mode by recording on high speed cinefilm the decay process of an initially oscillating drop. Figure 34 reproduces experimental data obtained for a 0.20 ml drop rotating at 5 rps, initially driven into oscillation at a resonant frequency of 11.2 Hz. Figure 35 displays measured damping constants at various rotational velocities for the same 0.20 ml drop. A slight increasing trend may readily be observed. For larger drops ( V 2 0.40 ml), the observed decay process is no longer characterized by a simple exponential curve for rotation rates 3 rps. Typical results are reproduced in Figure 36 for the free decay of a 0.40 ml drop at 3 and 7 rps. The data obtained at the higher rotation rate reflect a sharp initial decrease in the amplitude during the first cycle, and a subsequently significantly slower decay process. A check on the behavior of 1
1
I
I
1
I
I
I
I
1
2
3
4
5
6
7
8
9
2.5
2.4
-
2.3 v)
LW 2.2
2.1
2.0 0
10
R,rev/s FIG. 35. The damping constant as a function of the rotation rate for a 0.20 rnl drop ( I
= 2).
Taylor G. Wang
38
NUMBER OF OSCILLATIONS FIG.36. The decay process of a 0.40 rnl drop at 3 and 7 rps ( l = 2).
the 0.20 ml drop shows a similar trend, but at 9 rps (f2 = 16 Hz). As discussed before, this complex decay process could be due to the non-negligible static distortion. One might again note that the onset of this discontinuity roughly coincides with reaching the limit w I 5 2a.
D. COMPOUND DROPOSCILLATIONS Saffren, Elleman, and Rhim (1981) studies the oscillation frequencies for a concentric three-fluid system as shown in Figure 37. Since 9 is a nonsecular solution of the Laplace equation, it can be expressed in the following forms in each region of the system:
q ( r , t ) = 1 [ A ( / ,rn; t ) r ' + B(Z, rn; t)r-"+"] YIm( 8, a)
(shell),
(3.6)
1, m
(core), and (3.7) (host).
(3.8)
By applying these solutions to the equations of motion with proper boundary conditions, the equations can be reduced to a eigenvalue equation which
Equilibrium Shapes of Rotating Spheroids
39
FIG. 37. A concentric three-fluid system used in the theory.
is essentially the same as a coupled harmonics oscillation. The normal mode frequencies are given by (3.9)
where (3.10)
J
+
+ Ap,)T ( ~ ’ + ’ )- A p i Ap07
([RrORq.,’)
(1-1)1(1+1)(1+2) (21+ 1 ) p
= (1 Ap,,)( 1
“2=
~
(2/+1)
,
(3.11) (3.12)
2
Taylor G. Wang
40
and
It is important to note that when the positive square root in Equation (3.9) corresponds to the normal mode with the higher frequency, the boundary oscillations are in phase. We call this high frequency (“+”) mode the “bubble” mode and the lower vrequency (“-”) mode the “sloshing” mode. The relative boundary displacement of the sloshing mode is out of phase. The relative boundary displacement of these two modes as they were observed in the neutral buoyancy tank is shown in Figure 38. The top figure shows the bubble mode in which the two boundaries oscillate in phase, and the bottom figure shows the sloshing mode in which the two boundaries move out of phase. In this compound drop, the core and the host were silicone oil, and the shell was water. The limiting cases of Equation (3.9) are shown below. (1) Simple drop (i.e., R, = 0) one gets
*
w, =
aOI(z+l)(l-l)(z+2)
which is Lamb’s result. (2) Thin Shell: (i.e., r = 1) then w -
*
w+ =
(3) Thick Shell (i.e.,
((To
+
T+CO) 2
w, =
(3.13)
[ I P O + ( I + 1)PlR;:
(Ti
)(I
-
=0
and
1 )( I + 2) I( I + 1)
(3.14)
then
4%-v,(I-l)l(l+l)(l+2) 3 VJkJ+ ( I + 1)Ptl
(3.15)
and w- =
4 ~ ( ~ , , ( 1 -1)1(1+ 1 ) ( I + 2 ) 3V,[bo+(l+l)~l
(3.16) ’
Today there are experimental data on compound drops obtained in an electrostatic chamber. The block diagram of this apparatus is shown in Figure 39. This method is similar in principle to more well-known transient techniques in nuclear magnetic resonance or optical spectroscopy. The basic idea of this technique is to simultaneously record capacitance variation as the drop evolves freely
Equilibrium Shapes of Rotating Spheroids
41
FIG.38. “Bubble” mode (top) and “sloshing” mode (bottom) oscillations of an oil-wateroil compound drop.
Taylor G. Wang
42
I
TRIGGER PULSE GENERATOR
t
SCOPE
- -
PULSE GENERATOR
-
CONTROL C IRCU IT
FIG.39. A block diagram of our apparatus which excites the drop electrostatically and detects the ensuing capacitance signal caused by the drop deformation.
toward an equilibrium state from an initial nonequilibrium range of frequency. Therefore, the Fourier transformation of this transient signal reveals characteristic frequencies simultaneously. Figure 40 shows a set of typical data obtained by this method. For the simple drop case, the signal is a damped monotonic oscillation which corresponds to a single peak in the frequency domain. Of course, the spectral line width is inversely proportional to the damping time constant of the time domain signal. The compound drop signal looks more complicated. However, Fourier transformation shows two well-resolved peaks corresponding to the bubble mode and the sloshing mode respectively. The
Equilibrium Shapes of Rotating Spheroids
FREE DECAY
43
FT SPECTRA
COMPOUND DROP
FIG.40.
Signals obtained by the capacitance bridge and their Fourier-transformed spectra.
preliminary data agree with the calculation reasonably well on a limited basis. Further experiments are needed to verify the validity of the calculation.
E. DROPDYNAMICS I N SPACE Space, with its extended zero-gravity environment, offers a unique opportunity to study the dynamics of a truly free drop without any host liquid to influence its outcome. However, when conducting experiments in spaceflight, one cannot escape residual gravitational effects. Therefore, an acoustic levitation technology and facility have been developed in order to position and manipulate drops without physical contacts. 1. Acoustic Technology
An acoustic method has been developed for controlling liquid samples without physical contact. This method utilizes the static pressure generated
Taylor G. Wang
44
by three orthogonal, acoustic standing waves excited within an enclosure. Furthermore, this method allows the sample to be rotated in the acoustic field. The laboratory model of the rectangular resonance chamber is a 44 inch x 4$ inch x 5 inch plexiglass rectangular box and three commercially made speaker drivers. These drivers are mounted at the center of the three orthogonal sides of the box. When the chamber is driven at one of its resonant modes by an acoustic compression driver, the ambient pressure is minimum at the pressure nodes of the wave and is maximum at the antinodes. Consequently, there is a tendency for liquids and particles introduced into such enclosures to be driven toward the nodes, where the materials collect and remain until excitation ceases. The average ambient pressure change is -
-
(AP) =---PL
Pu2pc2 2 '
(3.17)
a second-order effect in both the acoustic dynamic pressure and the particle velocity. (This effect vanishes within linear acoustics, as well as for an infinite traveling wave.) The bars represent time averages of each quantity. The main feature of the static pressure is that its node is a plane ( x = l x / 2 ) becoming a point if all three drivers are turned on. Because this is a three-dimensional system with independent control of each dimension, it has a great deal of versatility. It can be utilized to position a drop acoustically and to manipulate it; for example, drop oscillation and/or rotation can be induced. The physics associated with these capabilities is now briefly discussed. a. Acoustic Force in a Resonance Chamber The acoustic radiation force on a sphere in a standing wave field was first calculated by L. V. King (1934). King approached the problem of solving the linear wave equation with scattering corrections and found F to be F
5r P: 6 PC
=-7 ka'
sin 2 k x ,
(3.18)
where p , is the pressure amplitude of the fundamental frequency, k is the wave number, a is the sphere radius, x is its center position, p is the gas density, and c is the speed of sound. Placed in an acoustic resonance
Equilibrium Shapes of Rotating Spheroids
45
chamber of length L, and driven by a standing acoustic wave of length A =2L, the sphere experiences a maximum acoustic restoring force at x = L/4, 3L/4, and experiences no force at the center. This theory was tested experimentally first by Klein (1938)and later by Rudnick (1977)with reasonable agreement. Recently E. Leung et al. (1981)studied the acoustic force in a high-intensity standing wave situation where both scattering and harmonic generation due to nonlinear effects are equally important. At the resonance of the chamber, the output of the microphone was examined using a Hewlett Packard spectrum analyzer to determine the amplitude of the higher harmonics. A specially designed frequency mixer with phase adjustment was used to suppress the second harmonics. This was accomplished by adding an acoustic signal equal to the amplitude of the unwanted harmonic, and canceling it with an opposite phase. The experiments were conducted with and without suppression of the second harmonic. Since the amplitude of the third harmonic was small compared to that of the fundamental, suppression of harmonics higher than the second was not necessary. Figure 41 shows the acoustic force as a function of pressure levels without harmonic suppression at three different positions. The experimental data are observed to exceed the values predicted by King's theory near the walls, but are smaller than the values predicted by the theory near the center of the chamber, and progressively deviate from it as a function of sound intensity. As mentioned earlier, suppression of the second harmonic was achieved by adding an acoustic signal equal to the amplitude of the second harmonic and canceling it with an opposite phase. The data obtained from such measurements agree better with King's theory, as shown in Figure 42. b. Acoustic Rotation The chamber has been designed so that two sides have the same dimension. When the acoustic drivers for three directions are driven 90" out of phase, maximum torque is applied to the sample. This torque is in the (x, y ) plane and is proportional to the sine of the phase difference. This torque is a result of a viscous effect rather than of the Bernoulli effects as proposed by Lord Rayleigh (1882).Consider a thin disk with its axis in the z-axis and a radius ( r o ) very small compared to wavelength. When the disk is subjected to the influence of two orthogonal acoustic waves with 1/4wavelength phase lag, the disk is surrounded by a gas in which the particles move circularly. Thus it experiences a torque due to viscosity. The torque
Taylor G. W a n g
46
0
0.2
0.4
0.6
P2
rms
0.8
1.0
1.2
1.4
1.6
2 -1 O8 (dyn/cm2 )
FIG. 41. (a) Acoustic force vs. RMS acoustic fundamental pressure squared for 0.635 cm radius spheres positioned 1.2 cm from the end wall, away from the speaker; (b) acoustic force vs. RMS acoustic fundamental pressure squared for 0.635 cm radius sphere positioned 4.2 cm from the end wall, away from the speaker, where the effect of the second harmonic is minimum.
T can be computed: (3.19) where 1, is the viscous length defined as ( 2 v / ~ ) ’ ’ ~v , is the kinematic viscosity, A is the total surface of the disk, P, and P,. are pressure amplitudes of the orthogonal waves, and wt, is the phase angle between the orthogonal waves. In evaluating the integral over the radial dimension of the disk, zero disk thickness was assumed and higher-order terms were neglected in Equation (3.19). Note also that the contributions to the torque on both sides of the disk have been taken into account. As expected, the torque is proportional to the boundary layer thickness (I,,) and to the product of the pressure
Equilibrium Shapes of Rotating Spheroids
47
KING'S THEORY
-
2ND HARMONIC SUPPRESSION
0
-
hl 0
-
7
X c I
-
(3
-0.4 -
2
0 U
-0.8 -1.2 -1.6
-
-2.0
-
-2.4 -2.8 0
-3.2 -
I
I
1
2
I 3
I
I
I
-
I
I
I
I
I
I
I
I
I
4
5
6
7
8
9 1 0 1 1 1 2 1 3 1 4 1 5
DISTANCE IN cm FIG. 42. Acoustic force per weight of the sphere vs. sphere position for three levels of total sound pressure. Measurements with second harmonic suppression are compared with King's theory.
amplitudes of the two plane waves. A single plane wave cannot generate a torque, as is obvious from symmetry considerations. For the same reason, two plane waves at right angles cannot generate a torque if their pressure oscillations are either in phase or 180" out of phase. A general shift ( m i o ) other than these two values, however, gives rise to the generation of torque described by Equation (3.19). Figure 43 shows the acoustic torque as a function of the acoustic energy with r,= 1.27 cm and k = 0.247/cm. The points are the experimental values, and the dashed line represents the calculated values.
Taylor G. Wang
48 I
I
I
I
PHASE SHIFT,
1
1
I
1
1
= 90'
lo-*
I
I
I
1
I
1
10
I
1
1
1
ACOUSTIC ENERGY, P2 (lo6 cgs) FIG. 43. The acoustic torque as a function of the acoustic energy. The points represent the experimental values and the dashed line represents the calculated values of Equation (3.19). (The solid line is a guide for the eye.)
c. Acoustically Induced Oscillation To generate oscillations of a liquid drop, it is necessary to excite the drop surface at its normal-mode oscillation frequencies. The surface for the nth mode is described by r, = a + a,P,,(cos 0 )
(3.20)
as in the axisymmetric case. P, is the Legendre polynomial of order n. The modulation of the acoustic force is obtained electronically through a balanced modulator. In the simplest case, a sinusoidal signal (frequency f c ) is multiplied by a second sine wave of much lower frequency ( f m ) - ' .
Equilibrium Shapes of Rotating Spheroids
49
The voltage across the transducer terminals is then V,
=
V,. sin(2rht) cos(2rfmt).
(3.21)
The acoustic pressure is proportional to this voltage for linear operation of the transducer. The radiation pressure force is proportional to the time average of the acoustic pressure squared, and can be described as pr -- (P:couCtic) -c0s2(2rfmf).
Taking the time average of this force over
(3.22)
(fm)-'
( P r ) - ( ; ( l +cos(4.rrf,t))),
(3.23)
one sees that the resulting effect yields a steady-state force as well as a slowly varying force. In the zero gravity environment, this steady force is sufficient to trap the drop at a stable position, and the slowly varying force can introduce a shape oscillation of the drop. d. Acoustic Frequency Shift The acoustic properties of a resonance chamber are modified when an object is introduced into the chamber. Introducing an object modifies the boundary conditions of the problem and gives rise to volumetric and scattering effects due to the presence of the object. These effects cause changes in the resonant frequencies, perturbation of the normal modes, and drops in the quality factor ( 0 )of the cavity. These effects are expected to become more noticeable as the size of the object introduced into the chamber increases. The resonant frequency of the cavity can be calculated by solving the Helmholtz equation, V2@ + k 2 @= 0 (the spatial part of the linear wave equation). Multiplying y by the eigenfunction, integrating over volume and performing one partial integration, an equivalent form for the oscillation frequency ( w = C k ) is (3.24) It is seen that the numerator in Equation (3.24) is proportional to kinetic energy, and the denominator to potential energy. The sign of the resonant frequency change can be visualized graphically without calculation. When the object is placed near the walls, the kinetic energy is hardly changed, while the potential energy is reduced, resulting in an increase in oscillation
Taylor G. Wang
50
frequency. Similarly, when the object is introduced near the center of the chamber, the potential energy is hardly changed, while the kinetic energy is reduced, resulting in a lower oscillation frequency. This holds for the fundamental mode and can be trivially extended to higher modes as well. The acoustic frequency shift on the basis of volumetric and scattering effects has been calculated, and we obtain 6k - 4.rrR3’3 [(l-!(klR)2) kl Vchamber
cos2k,X+
(3.25) where 6k = k - k, is the shift in wave number due to the presence of the sphere at X. The first term in Equation (3.25) is the volumetric result and the last two account for the scattering contribution to the frequency shift. The last term in particular is responsible for the DC downward shift. Figure 44 shows the measured relative frequency shift of a half-inch radius sphere as a function of sphere position. The spatial dependence follows a sinusoidal behavior, and it is asymmetrical with respect to the unperturbed frequency. The solid lines are drawn on the basis of Equation (3.25). The agreement between measurement and theory is remarkable, since our theoretical model assumes the boundary walls are sufficiently removed to ignore secondary scattering effects. This summation does not hold true in all regions. 2. Spacelab Experiment From April 29 to May 7, 1985, a Drop Dynamic Module was flown on Spacelab 3 (see Figure 45). This module incorporated all the acoustic positioning and manipulation capabilities described earlier: rotating, oscillating, and shaping of a liquid drop without physical contacts. Failure and subsequent repair (in space) of the module took precious time; therefore only the experiment studying the shapes of rotating drops was completed and will be reported here. Drops of water or glycerin-water mixtures of various viscosities were deployed inside an acoustic position chamber while orbiting the earth inside the spacelab module carried in the payload bay of the Space Shuttle. The
1 .o
0.5
I
I
I
I
1
I
I
1
I
-
c
Y
m ,
-0.5 -
\
0
2
4
6
10
8
X / L (-101) FIG. 44. Position dependence of the relative resonant frequency shift for a sphere: ( a ) I = 1 mode; (b) I =
=2
mode.
Taylor G. Wung
52
FIG. 45.
Drop Dynamics Module flown on Spacelab-3 ( 1985).
Equilibrium Shapes of Rotating Spheroids
53
average level of residual acceleration was around times the standard earth gravitational field. The drops were held within a small region inside the chamber by three orthogonal acoustic standing waves generating steadystate forces on the order of 1 dyne, and were driven into rotation by acoustic torques. The drop volumes varied between 0.5 and 10cm3 with viscosities of 1, 10, 20, and 100cSt, densities between 1.0 and 1.22g/cm3, and surface tensions from 5 5 to 65 dyneslcm. An initially nonrotating drop was subjected to an acoustically generated torque which caused it to spin up with increasing velocity. The rotational velocity of the liquid was determined through the motion of immiscible tracer particles suspended within the primary drop. The time variation of the shape of the rotating drops was determined by recording the drop profile along three orthogonal views on 16 mm motion picture film. Data from a typical experimental sequence involving several spin-ups and spin-downs of a 1 cm3 100 cSt drop are shown in Figure 46, where the rotation speed is plotted along the y-axis, and time is measured along the x-axis. In the first phase of the spin-up the drop shape is still axisymmetric with respect to the rotation axis: the initially spherical drop deforms into a spheroid with increasing aspect ratio a / b, where a is the equatorial and b the polar radii. At this point the rotation velocity of the drop starts to decrease due to the larger deformation. If the acoustic torque continues to act on the drop, fission eventually takes place. On the other hand, if the torque is turned off, the two-lobed deformation will decrease, and a transition back to the axisymmetric configuration will take place at 430-490 seconds. The rotational velocity at which this reverse transition takes place is very close to f l ( I 1 ) . In the axisymmetric regime, the rotational velocity of the drop will slowly decay if only drag is acting on the liquid. A much faster decay-to-zero rotation rate can be induced when a reverse acoustic torque is applied at 480 seconds. Because of this reverse torque the drop will slow down quickly and start to rotate in the other direction, eventually repeating the same cycle. This sequence ends in the fission of the drop. In this particular instance, the reverse torque was of a higher magnitude than in the forward direction. This was accomplished by purposely raising the sound pressure level. Of particular interest here are both the value of fl(IT), the rotational velocity at bifurcation, and the deformation of the drop as a function of rotation speed, since these two characteristics have been precisely determined theoretically.
4
-
1
I
I
I
AX ISYMMETRIC
-
I
4
1
I
I
2-LOBED
I
1
I
4
I
I
I
I
1
2-LOBED
AXISYMMETRIC
-
VI
P
-
1
0
I-
0 0
0 0 0
0
c Q: I-
0
0
0 0
'F
0
t
3
FISSION
0
FORWARD TORQUE
0
-
0.
s
0
I 300
I
I
I
I
I
400
I
I
REVERSE TORQUE
lq? 500
I
I
I
600
TIME (sec) FIG.46. Experimental sequence studying d r o p rotation using a l-cm3 drop of a 100 cSt glycerin and water mixture. The drop went from an axisymmetric shape to a two-lobed shape under an applied acoustic torque (372 sec); from a two-lobed shape to axisymmetric shape when torque was removed (462 sec).
55
Equilibrium Shapes of Rotating Spheroids
Figure 47 shows experimental data for a 3 cm3, 100 cSt water-glycerin drop. This particular experiment involves the spin-up of the drop, its transition to the two-lobed shape, and finally its relaxation to the axisymmetric shape as the acoustic torque is turned off. The rotational velocity is normalized by the frequency of the nonrotating fundamental oblate-prolate resonant mode of shape oscillation. The relative deformation is the largest dimension in the equatorial plane of the drop divided by the diameter of the nonrotaing spherical drop. Both theoretical predictions and experimental results are shown in the plot. The most obvious difference is in the location of the bifurcation: O(I1) was experimentally determined to be 0.47 + 0.04 while it is calculated to be 0.56. In addition, the experimental rate of change of the relative deformation with rotation rate appears less than that predicted by calculations for the 2.5
1
I
1
1
I
I
1
I
I
I
I
I
I
1
I
EXPERIMENT (SPIN UP) 0 EXPERIMENT (SPIN DOWN) THEORY (BROWN 8, SCRIVEN, 1980)
2.0
0
v
w
3:
3
v
1.5
w
1 .o SHAPES
0.:
0
0.2
0.4
0.6
0.8
NORMALIZED ROTATION RATE
FIG. 47. Comparison of theoretical and experimental values for the relative deformation r(max)/r(O) as a function of the normalized rotation speed for a 3-cm3 drop (100 cSt waterglycerin mixture).
Taylor G. W a n g
56
two-lobed shape, while the agreement appears to be quite good for the axisymmetric shape. One might also note that, within experimental uncertainty, there is no clear evidence of the existence of different paths for spin-up and spin-down in the two-lobed regime (although the scatter of the experimental results increases when the data for spin-down are added). These results are representative of the data gathered using several drops of various volumes. When the deformation of the drop in the two-lobed regime is increased by imposing a larger torque, fission becomes inevitable beyond a certain value. Experimentally, fission generally produces two main drops of equal volume, and a satellite droplet at the center of mass. Figure 48 reproduces experimental data for a rotation sequence leading to the splitting of the drop. Fission occurred at a normalized rotational velocity of about 0.21
2.5
2.0
0
v
w
3
v
l3
w
\ 1 .o EXPERIMENT
- THEORY (BROWNa SCRIVEN) 1980
0.5 0
0.2
0.4
0.6
NORMALIZED ROTATION RATE FIG. 48. Theoretical and experimental relative deformation in the two-lobed shape up to fission as a function of rotation speed.
Equilibrium Shapes of Rotating Spheroids
57
(k0.04).There is no simple prediction of a critical velocity at fission, but one could compare the data to the velocity calculated for maximum drop deformation (0.26). Calculations also indicate the rotational kinetic and surface energies for a single two-lobed drop and for two separate drops matching the region where the normalized velocity is equal to 0.25.
IV. Conclusion Behavior of liquid drops of the equilibrium shapes of a rotating spheroid and drop shapes have been actively studied both in terrestrial and extraterrestrial laboratories. For equilibrium shapes of a rotating spheroid experiment the ground-based immiscible systems provide good qualitative information but limited quantitative comparison due to the presence of viscous stresses and drag exerted by the outer suspending liquid. Direct comparisons of experiment with the theory of a free rotating drop were possible only in the case of slowly rotating axisymmetric drops. The agreement between the two was better than expected; the qualitative shape of the equatorial area versus Z curves were similar, differing from theory by 30%. This is remarkable because the theory does not address the presence of an outer fluid. The generation and study of axisymmetric equilibrium shapes for higher rotational rates is difficult, because of the presence of the more stable n = 1 lobed nonaxisymmetric shape. This mechanism prohibits extracting the exact location of the bifurcation points between families of equilibrium shapes from the data. When generating n 2 2 lobed drops in a controlled manner, primarily two- and three-lobed shapes were obtained. The latter had not been observed before. The study of equilibrium configurations of these lobed shapes is made difficult by the presence of the outer fluid; as soon as the lobes occur, the interaction between the drop and the host liquid increases significantly and generates large secondary flows. The accelerated transfer of angular momentum from the drop in the lobed configurations gives rise to decay routes in which one lobe slows and is absorbed by the one trailing it; this continues until there is only one arm left. There were two exceptional types of decay in which either the whole drop would lift up (independently of the neutral buoyancy level) and become sessile on the disk, or the drop would tilt and form a slanted drop; in both cases, the shapes were stable and long-lived. The lobed-shape behavior was not easily compared to the free drop theory. The study of the angular velocities and momenta demonstrated that the development of the various lobed shapes takes similar paths,
58
Taylor G. Wang
but no evidence was found for the location of branch points between axisymmetric and triaxial behavior. The immiscible system provides a much better testing facility for shape oscillation experiments. Satisfactory agreements were obtained between the experimental data and the theoretical prediction. For small-amplitude oscillations, a drop suspended in a host liquid behaves very much like a simple damped linear oscillation; for example, with pure sinusoidal oscillation and 90" phase shift between the drive and response, the decay rate is linear with drop viscosity, and the relative shift in resonant frequency is linearly dependent on the square o f the relative rotational velocity. For large-amplitude oscillation, the experimental data suggests a soft nonlinearity in the fundamental resonant frequency, and a hint of an increase in the decay rate, but no definite quantitative conclusion can be drawn yet. The most exciting part of the experimental data has been obtained in the low-gravity environment of space. It has yielded a first experimental test in agreement with the various analytical and numerical predictions regarding the axisymmetric equilibrium shapes of rotating drops and evidence for the early onset of secular instability experienced by those shapes. This may mean that the drop shape is less stable with respect to fluctuations than the calculations have predicted. That no mode higher than the two-lobed shapes was observed also lends support to the diminished stability argument. These data also yield a good quantitative agreement between theory and experiment for the velocity at fission in the two-lobed region. However, due to its limited accessibility, only small portions o f the possible data have been obtained during the first flight. Acknowledgments This chapter represents the work o f a collection o f individuals too numerous to list here; however, the author wishes to acknowledge the contributions o f Dr. E. Trinh, Dr. D. Elleman, Mr. A. Croonquist, Mrs. E. Leung, Dr. W-K. Rhim, Dr. M . C. Lee, Dr. M . Saftren, and Dr. P. Annamalai. The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
References Alonso, C. T., Proceedings of the First International Colloquium on Drops and Bubbles, Pasadena, CA, August 29-31, 1974, Editors M . Plesset, M. Saffren, and D. Collins.
Equilibrium Shapes of Rotating Spheroids
59
Andreas, J. M., Hauser, E. H., and Tucker, W. B. (1938). Boundary tension for pendant drops. J. Phy. Chem. 42, 1001. Annamalai, P., Trinh, E. H., and Wang, T. G. (1985). Experimental study of the oscillations of a rotating drop. J. Fluid Mech. 158. Appell, P. E. (1932). “Traite de Mecanique Rationelle”, Vol. 4, Chapter 9. Gauthier-Villars, Paris. Baldwin, C. M., Aldeida, R. M., and Mackenzie, J. D. (1981). J. Non-Crysi. Solids 43, 309. Bauer, H. F., and Siekmann, J. J. (1971). Appl. Math. and P h y ~ (ZAMP), . 22, 532. Berringer, R., and Knox, W. J . (1961). “Liquid-Drop Nuclear Model with High Angular Momentum,” Pliys. Rev. 121, 1195. Beyer, R. T. (1977). Nonlinear acoustics (Naval Sea Systems Command, Washington, D.C., 1974). J. Acousi. Soc. Am. 62, 20. Brown, R. (1979). The shape and stability of three-dimensional interfaces. Ph.D. Thesis, University of Minnesota, Minneapolis, MN. Brown, R. A,, and Scriven, L. E. (1980). Shape and stability of rotating liquid drops. Proc. R. Soc. Lond. A371, 331-357. Busse, F. H., and Wang, T. G. (1981 1. Torque generated by orthogonal acoustic waves-theory. J. Acousi. Soc. Am. 69, 1634-1639. Cammack, L. S. B. (1979). Measurement of the interfacial tension between silicone oil and water-methanol mixtures at 25°C. Preliminary Report, JPL Containerless Processing Group (an internal document). Carr, C., and Riddick, J . A. (1951). Physical properties of methanol-water system. Ind. Eng. Chem. 43, p. 692. Carruthers, J. R., and Grasso, M. (1972). The stabilities of floating liquid zones in simulated zero gravity. J. Cr.ysral Growth 13/14, 61. Carruthers, J . R., and Grasso, M. (1972). Studies of floating liquid zones in aero gravity. J. Appl. Ph.vs. 43, 436. Chandrasekhar, S. (1965). The stability of a rotating liquid drop. Proc. Roy, Soc. (London) A286, 1-26. Clayton, B. R., and Massey, B. S. (1967). Flow visualization in water: a review of techniques. J. Sci. Instrum. 44,2. Clifford, G., and Campbell, J. A. (1951). Densities in the methanol-water system at 25°C. J. Am. Chem. Soc. 73, 5449. Diehl, H., and Greiner, N. (1973). “Ternary Fission in the Liquid Drop Model,” Phys. Letter, 458, 35. Drexhage, M. G., Quinlan, K., Moynihan, C., and Saleh Boulos, M. (1981). In “Physics of Fiber Optics” (B. Bendow and S. S. Mitrd, eds.), p. 57. Columbus, OH. Dunstan, A. E., and Thole, F. B. (1909). The relation between viscosity and chemical constitution: Part IV. Viscosity and hydration in solution. J. Chem. SOC.95, p. 1556. Foote, B. (1971). A theoretical investigation of the dynamics of liquid drops. Ph.D. Thesis, University of Arizona Gans, R. S. (1974). “On the Poincare problem for compressible medium,” Journal of’ Fluid Mechanics, C.2, 657-675. Gibson, E. G. (1974). Skylab fluid mechanics demonstrations. Proceedings of International Colloquium on Drops and Bubbles. Pasadena, CA, p. 158. Goldberg, Z. A. (1971). Acoustic Radiation Pressure, in High-Intensity Ultrasonic Fields (L. D. Rozenberg, ed.). New York, NY. Gorkov, L. P. (1967). Solv. Phys. Dokl. 16, 773. Happel, J., and Brenner, H . (1965). Low. Reynolds Number H.vdrodynamics, Prentice-Hall, p. 62.
Taylor G. Wang Harkins, W. D., and Anderson, T. F. (1937). A simple accurate film balance of the vertical type for biological and chemical work, and a theoretical and experimental comparison with the horizontal type. J. A m . Chem. Soc. 59, p. 2189. Harkins, W. D., and Brown, F. E. (1918). The determination of surface tension (free surface energy), and The weight of falling drops: the surface tension of water and benzene by the capillary height method. J. A m . Chem. Soc. 41, p. 4923. Harkins, W. D., and Jordan, H. F. (1930). A method for the determination of surface and interfacial tension from the maximum pull on a ring. J. Am. Chem. Soc. 52, p. 1751. Hasegawa, T. (1977). J. Acoust. Soc. A m . 61, 1445. Huh, C., and Mason, S. G . (1975). A rigorous theory of ring tensiometry. Colloid Polymer 253, 566. Hyzer, W. G . (1965). “Photographic Instrumentation Science and Engineering,” US. Govt. Printing Office, Washington, D.C. Jacobi, N., Tagg, R., Dendall, J., Elleman, D. D., and Wang, T. G . (1979). Free oscillations of a large drop in space. 17th Aerospace Sciences Meeting, paper 79-225. King, L. V. (1934). On the acoustic radiation pressure on a sphere. Proc. R . Soc. London, Ser. A, 147, 212. Klein, E. (1938). J. Acoust. Soc. Am. 9, 312. Lamb, H. (1932). “Hydrodynamics.” Dover Publication, pp. 473-639. Cambridge University Press. Lauterborn, W. (1976). J. Acoust. Soc. A m . 59, 283. Lee, C. P., and Wang, T. G. (1984). The acoustic radiation force on a heated (or cooled) rigid sphere-Theory. J. Acoust. Soc. A m . 75, p. 8 8 . Leung, E., Jacobi, N., and Wang, T. G . (1981). Acoustic radiation force on a rigid sphere in a resonance pressure. J. Acoust. Soc. A m . 70, p. 1762. Leung, E., Lee. C. P., Jacobi, N., and Wang, T. G. (1982). Resonance frequency shift of an acoustic chamber containing a rigid sphere. J. Acousf. Soc. A m . 72, p. 615. Longsworth, L. G., and Maclnnes, D. A. (1939). lon conductances in water-methanol mixtures. J. Phys. Chem. 43, p. 239. Maidanik, G . , and Westervelt, P. J. (1957). J. Acoust. Soc. A m . 29, 936. Marston, P. L., and Apfel, R. (1979). Acoustically forced shape oscillation of hydrocarbon drops levitated in water. J. Coll. Inteqace Sci. 68, 280. Marston, P. L. (1980). Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stresses. J. Acoust. Soc. A m . 67, 15. Marston, P. L., and Apfel, R. (1980). Quadrupole resonance of drops driven by modulated acoustic radiation pressure-Experimental properties. J . Acoust. Soc. Am. 67, 27. Marston, P. L., Loporto Arione, S., and Pullen, G. (1981). Quadrupole projection of the radiation pressure on a compressible sphere. J. Acoust. Soc. Am. 69, 1499. Mikhail, S. Z., and Kimel, W. R. (1961). Densities and viscosities of methanol-water mixtures. J. Chem. Eng. D a t a 6 , p. 533. Miller, C., and Scriven, L. (1968). The oscillations of a fluid droplet immersed in another fluid. J. Nuid Mech. 32, 417. Montgomery, D. N. (1968). “Collisional phenomena of uncharged water drops in a vertical electric field.” (Ph.D. thesis), Univ. of Arizona (1968). Morgan, J. L. R., and Neidle, M. (1913). The weight of a falling drop and the laws of state, XVII. The drop weights, surface tensions and capillary constants of aqueous solutions of ethyl, methyl, and amyl alcohols and of acetic and formic acid. J . A m . Chem. Soc. 35, p. 1856. National Research Council (1930). “International Critical Tables,” Vol. 7. McGraw-Hill, New York, NY. (index of refraction @ 15”, 15.5”, 17”, 25°C).
Equilibrium Shapes of Rotating Spheroids
61
Neidig, H. A., Yingling, R. T., Lockwood, K. L., and Teates, T. G. (1965). Interaction in chemical systems 1. The methanol-water system. J. Chem. Educ. 42, p. 309 and Interaction in chemical systems 11. The KCI-methanol-water system. J. Chem. Educ. 42, p. 368. Nyborg, W. L. (1967). J. Acoust. Soc. Am. 42, 947. Plateau, J. A. F. (1863). Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the actions of gravity. The Annual Report o f t h e Board of Regents of the Smithsonian Institution (Cover Printing Office, Washington, DC), pp. 270-285. Princen, H. M. (1969). “The Equilibrium Shape of Interfaces, Drops, and Bubbles. Rigid and Deformable Particles and Interfaces.” In Surface and Golloid Science ( E . Matijevic, ed.), Vol. 3. Wiley (Interscience), New York, NY. Princen, H. M., Zia, Y., and Mason, S. G. (1967). Measurement of interfacial tension from the shape of a rotating drop. J. Colloid Interface Sci. 23, p. 99. Prosperetti, A. (1980a), J. M k c . 19, 149. Prosperetti, A. (1980b). Free oscillations of drops and bubbles; the initial value problem. J. Fluid Mech. 100, 333. Lord Rayleigh (1882). On the instrument capable of measuring the intensity of aerial vibration. Philos. Mag. 14, 186. Lord Rayleigh (1914). The equilibrium of revolving liquid under capillary force. Phil. Mag. 28, p. 161. Ross, D. K. (1968). The shape and energy of a revolving liquid mass held together by surface tension. Aust. J. Phys. 21, p. 823. Ross, D. K. (1968). The shape and energy of a revolving liquid mass held together by surface tension. Aust. J. Phys. 21, p. 837. Roe, R. J., Baccheta, V. L., and Wong, P. M. A. (1967). Refinement of pendant drop method for the measurement of surface tension of viscous liquid. J. Phys. Chem. 71, p. 4190. Rudnick, I. (1977). Measurements of acoustic radiation pressure o n a sphere in a standing wave field. J. Acoust. Soc. Am. 62, 20. Saffren, M., Elleman, D. D., Rhim, W. K . (1981). Normal modes of a compound drop. Proceedings of the 2nd International Colloquium on Drop and Bubbles. Pasadena, CA. Sperber, D. (1962). Equilibrium configuration and fission barrier for liquid drop nuclei with high angular momentum. Phys. Reu. 130, p. 468. Subnis, S. W., Bhagwat, W. V., and Kanugo, R. B. (1948). Application of mixture law to Rheochor-Part 11. J. Indian Chem. Soc. 25, p. 575. Swiatecki, W. J. (1974). The rotating, charged or gravitating liquid drop, and problems in nuclear physics and astronomy. In Proceedings of the Infernational Colloquium on Drops and Bubbles. Jet Propulsion Laboratory, Pasadena, CA, p. 52. Tagg, R., Cammack, L., Croonquist, A,, and Wang, T. G. (1979). Rotating liquid drops: Plateau’s experiment revisited. Report 900-954, Jet Propulsion Laboratory, Pasadena, CA. Takahashi, S., Shibata, S., Kanamori, T.. Mitachi, S., and Manabe, T. (1981). In “Physics of Fiber Optics” (B. Bendow and S. S. Mitra, eds.), p. 74. Columbus, OH. Thompson and Swiatecki, “Personal conversation between Wang, Thompson and Swiatecki.” Trinh, E. H., and Wang, T. (1980). A quantitative study of some non-linear aspects of drop shape oscillations. Paper presented at the 100th Meeting of the Acoustical Society of America. Trinh, E. H., Wang, T. G., and Robey, J . (1981). A non-uniformly heated resonant chamber for levitation studies. Acoust. Sac. Am. 70, p. 590. Trinh, E. H., Zwern, A,, and Wang, T. G . (1982). An experimental study of small amplitude drop oscillations in immiscible liquid systems. J. Fluid Mech. 115, 453.
62
Taylor G. Wang
Trinh, E. H., and Wang, T. G. (1982). Large amplitude free and driven drop shape oscillation: experimental observations. J. Fluid Mech. 122. U.S. National Bureau of Standards, (1924). Standard Densit-y and Volumetric Tables, Circ. 19 (density @15”C). Vonnegut, B. (1942). Rotating bubble method for the determination of surface and interfacial tensions. Reu. Sci. Inst. 13, p. 6 . Wang, T. G., Elleman, D. D., and Saffren, M. M. (1976). Dynamics of rotating and oscillating free drops: investigation and technical plan. JPL Document 701-238 (an internal document). Wang, T. G., Saffren, M. M., and Elleman, D. D. (1974). Drop dynamics in space. Proceedings of the International Colloquium on Drops and Bubbles, p. 266. Jet Propulsion Laboratory, Pasadena, CA. Wang, T. G. (1979). Acoustic levitation and manipulation for space applications. Proceedings of IEEE Ultrasonic Symposium, pp. 471-475. Wang, T. G., Saffren, M. M., and Elleman, D. D. (1974).Acoustic chamber for space processing applications, AIAA paper 7, 4. Wang, T. G., Saffren, M. M., and Elleman, D. D. (1977). Drop dynamics in space. I n “Materials Sciences in Space with Applications to Space Processing” (L. Steg, ed.). American Institute of Aeronautics and Astronautics, New York, NY. Wang, T. G., Trinh, E. H., Croonquist, A. P., and Elleman, D. D. (to be published). Shapes of rotating free drops: Spacelab experimental results. Phys. Rev. Lett. Wang, T. G., Kanber, H., and Rudnick, 1. (1977). First order torques and solid body spinning velocities in intense sound fields. Phys. Rev. Lett. 38, 128. Westervelt, P. J. (1950). J. Acoust. Sac. Am. 22, 319. Yosioka, K., and Kawasima, Y. (1955). Acustica 5, 167.
26
ADVANCES I N APPLIED MECHANICS, VOLUME
On Dynamics of Bubbly Liquids D. Y. HSIEH Division uf Applied Mathematics Brown University Providence, R. I .
I. Introduction
...........................................
11. General Formulation of Dynamical Equations A. Local Instantaneous Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Averaged Dynamical Equations . . . . . . . . . . . . . . . . . . . . C. Some Other Approaches.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
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I
111. Dynamic Equations of Bubbly Liquids ..... .. ...... . A. Single Species of Gas Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Pressure Terms and Rayleigh-Plesset Equation . . . . . . . . . . . . . . . . . . . . . . C. Mutual Interaction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The Self Force . . . . ......................... E. The Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Equation for the Fluid Mixture ................................. G. Dynamic Equation IV. Sound Waves in Bubbly Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Linearized Problem and the Characteristic Equation B. The “Dead Zone” and the Negative Damping.. . . . . . . . . . . . . . . . . . . . . . . . . . C. Comparison with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Waves and Instability in Bubbly Liquids. . ............... A. Finite Amplitude Waves in Locked Bub B. The Case with Small Mutual Friction . C. Instability of Slip Flow.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
64 66 66 69 73 75 76 78 80 84
85 86 87
88 88 93 95 96 96 103 105 108 V1. Steady Flows ............................................. A. One-Dimensional Steady Flow . . . .................... 109 B. One-Dimensional Flow with Disco . . . . . . . . . . . . . _ . . . . . . 113 VI1. Dynamics of A Liquid Containing Vapor Bubbles . . . . . . . . . . 115 A. The Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 115 B. The Source Te...................................................... 117 119 C. Waves in A Liquid Containing Locked Vapor Bubbles ..................................... 124 Appendix A . . . . Appendix B . . . . . . . . , . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . 125 63 Copyright 0 19x8 Academic Press, Inc. All right, of reproduction in any form reserved. ISRNo - I ? - o o ~ ~ - ?
D. Y. Hsieh
64
Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
I. Introduction The motion of a liquid containing many bubbles is a very complex phenomenon. This complexity may be appreciated first by looking at the problem of the motion of a single bubble in a fluid of infinite extent. Even for that idealized problem (Hsieh, 1965; Plesset and Prosperetti, 1977), we note that the bubble may move, oscillate, or collapse, we must include the processes of heat transfer, mass diffusion, viscous dissipation and compressibility, and the bubble may be gaseous or vaporous; so the subject matter is already very complex for a single spherical bubble. Not much work has been done on the motion of a single nonspherical bubble (Plesset and Mitchell, 1956; Hsieh, 1972), and even less on the motion of more than one bubble. Therefore, it is understandable that the subject of the dynamics of a moving liquid containing many bubbles is still in a very actively developing stage from the theoretical point of view. The theoretical problem is made even more difficult because of the scarcity of relevant experimental investigations, which may in turn have resulted from the lack of'guidance from theoretical understanding. It is natural that a statistical or continuum mechanical approach is usually adopted to deal with the complex phenomenon. At the basic level, before the special properties of bubbles have been taken into account, the approach and formulation are essentially the same as those for any two-phase flow problems in fluid mixtures. There have been many works on the general continuum theories of fluid mixtures. We shall just mention a few relatively recent studies. Most authors devoted their attention to the studies of solidfluid mixtures. Recent status of the general theoretical formulation are given by Bowen (1976) and Kenyon (1976a,b). For liquid-gas mixtures, the monographs by Wallis (1969) and Ishii (1975), and works by Delhaye (1975a, b, c, d ) and Boure (1978a, b, c) have offered extensive coverage on the basic formulation of the general two-phase flow problems as well as discussions of various applications. For liquids containing bubbles, an averaging scheme for deriving the basic continuum equations was proposed by Drew and Segal (Drew, 1971; Drew and Segal, 1971; Drew, 1976), while Drumheller and Bedford (1978, 1979, 1980) have approached the problem along the line of a variational method. One would also naturally try to deal with the problem of a liquid containing many bubbles from the approach
O n Dynamics of Bubbly Liquids
65
of multiple scattering, and one such recent endeavor was made by Varadan et al. (1985). Despite the variety of various formal approaches for the theoretical formulation of the basic theory, essentially the same set of governing equations is obtained. A crucial problem in these continuum formulations is to determine the “constitutive” relations. In some sense, this is the physics of the problem, which depends on our knowledge about how bubbles move in a liquid. The complexity of the motion of even a single bubble in a liquid, the nonlinear interaction among bubbles themselves and with the liquid, and our ignorance about the motion of more than one bubble underlie the difficulties and the unsatisfactory status of the problem of dynamics of bubbly liquids. The purpose of this paper is not to straighten out everything and present the subject matter in a neat and both mathematically and physically sound version. We may have to wait some time for that achievement. The purpose of this paper is rather to present the developments of the subject matter in as clear a way as possible so that one can understand what is going on. We shall start with a rational approach to the general problem and point out along the way how and where we make plausible or drastic approximations and assumptions. We shall also deal with some of the problems which have caught or still catch the attention of the practioners in the field. In the following, we present first the general formulations of dynamical equations for two-phase fluids in Section 11. There we follow essentially the averaging scheme developed by Drew (1971). Then in Section 111, we restrict the discussion to dynamic equations for bubbly liquids. Based on our knowledge about the motion of a single bubble in a liquid, we develop rationally the constitutive relations and the governing continuum equations for the bubbly liquids. Section I V is devoted to the topic of sound waves in bubbly liquids. Characteristic equations are derived, the subjects of “Dead Zone” and negative damping are discussed, and some comparisons with experimental results are made. In Section V, we discuss the finite amplitude waves in bubbly liquids as well as the instability problem which is associated with the “ill-posed” nature of the fundamental equations. A Burgers-KdV type equation is derived and the instability is demonstrated to be the Kelvin-Helmholtz type. Steady flow problems are discussed in Section VI. There we also show the mechanisms of the shock transition. Finally in Section VI, the dynamics of a liquid containing vapor bubbles are presented. The effects of evaporation-condensation are explicitly taken into account and problems of wave propagation are discussed.
D. Y. Hsieh
66
We emphasize again that the main purpose of this paper is to gain some understanding about this complex and sometimes confusing subject. We refer only to works relevant to the developments of the topics in the paper. They naturally reflect the author’s taste, biases and ignorance.
11. General Formulation of Dynamical Equations
The fluid system under consideration is a liquid containing bubbles. By bubbles we mean a small body of gases. Therefore the fluid system consists of fluids of two different phases. There are numerous works on the general formulations of dynamic equations of two-phase fluids, as we have mentioned in Section I. Generally speaking, all these works start with a local and instantaneous formulation of the fluid media, and then introduce some kind of averaging scheme to obtain a set of averaged governing equations. In the following, we shall present the local, instantaneous equation first, then introduce and discuss various kinds of “averaging” schemes, and finally present the “averaged” equations.
A. LOCALINSTANTANEOUSFORMULATION
Let $(x, t ) be the density of some property of the fluid, v(x, 1 ) the local velocity of the fluid, and let V be any region fixed in space and a V be the boundary surface of V. Then the general transfer equation can be written down as follows:
$1,
$(x, t ) d 3 x +
( $ v + f ) . n d’x=
s(x, t ) d3x,
(2.1)
where ($v+f) is the flux of $, which consists of the convective part $v and the nonconvective part f; n denotes the outward unit normal vector, and s is source density of $. If we denote by V, the material volume of the fluid, the transfer equation can also be put in the form (2.2) (2.1) and (2.2) are equivalent by the Leibniz rule. For homogeneous fluids, the integral form of the transfer equation can be transformed, by making
On Dynamics of Bubbly Liquids
67
use of the Gauss theorem, into the differential equation 9+V.($v)=-V.f+s. dt
(2.3)
Thus, if we identify CC, as p, the density of the fluid, then since f = 0 and s = O because of the principle of conservation of mass, we obtain
*a t+ v .
( p v ) =o,
(2.4)
which is the continuity equation. The momentum equation can be obtained by setting $ = pu,, f; = -u!,, and s = pb,, where (T,,or u is the stress tensor and b, or b represents the external body force. Thus we obtain
a
-(pv)+C. at
( p v v ) = V . at-pb.
(2.5)
Introduce the internal energy o f the fluid U, the heat flux q and the body heating Q. If we now identify $ with the total energy p ( U + ( u 2 / 2 ) ) ,then from the energy principle, we see that f = q - u . v and s = pb . v + pQ. Thus the energy equation is of the form
When the fluid is not homogeneous, as in the case of a fluid consisting o f two phases, the differential forms of the general transfer equation (2.3) are no longer valid, while the integral form of the general transfer equation (2.1) is still valid if we make some appropriate modifications. The main thing we need to accommodate now is the presence of interfaces between the constituent fluid phases. Let us denote the interfaces by A. These interfaces may intersect with a V Thus the intersection of A and aV, i.e., A n d V, is a curve L in space. Then the equation (2.1) should be modified to become
c
=
c
J
s(x, t ) d3x+ V
J
s,(x, 1 ) d 2 x ,
(2.7)
A
where $s is the surface density of the fluid variable, 5 represents some line flux of $I, and s, is a surface source density of $.
68
D. Y. Hsieh
For the mass equation, we again have $ = p, f = s = 0. The additional terms do not contribute, and we have GS=J; = s, = 0. For the momentum equation, we have $ = pv,, J; = --u,,, and s = pb, as before. No net momentum is created due to the presence of interfaces in the interior of V . Therefore $s = s, = 0. However, J; is present because of surface tension. Its direction will depend on the orientation of aV and L. We may denote J; = t. For the energy equation, we write $ = p ( U + ( v 2 / 2 ) ) , f = q - u . v and s = pb . V SQ as before. Now since there is interfacial energy associated with the interface, we have = u,which can be identified with the surface tension coefficient. When u is constant, t is proportional to u.When the gas phase consists of the vapor of the liquid, there will be a term s, which represents the heat released due to condensation, and which is proportional to the latent heat of evaporation-condensation. Now let us introduce the a-phase function pa by
+\
pa(X’
=
if x lies in phase a at time t, if x lies outside phase a at time t.
(2.8)
a = I when the phase is liquid, and a = g when the phase is gas. We may note that C,=,,, pa(x, t ) = 1, except when x is on L. Corresponding to the Equation (2.1) or (2.7), the general transfer equation for each separate phase can then be written down as: ~d[ ~ / 3 ~ $ d ~ x + [ ~ ~ P . ( g u + f ) * n d ~ x +
+ [ A p a $ ( ~ - ~ A ) - n d ’ x =VJ Pasd3x,
a=f,g,
(2.9)
where uA is the velocity of the surface, and in the surface integrals, the limiting values of the integrand as they approach the surface from the respective region are to be taken. From the point of view of each phase, d V and A are the same boundary surfaces, except that in general A refers to moving surfaces. Therefore the convective flux is P a $ ( v - v A ) on the surface A. The simplicity in form of (2.9), as compared with (2.7), is to be expected, since this is just the transfer equation for one fluid in the region it occupies. However, it differs from (2.1) in several aspects. First, A may not be a material surface as remarked above. Second, since in general the integrals are over non-connecting regions, we cannot in general apply the Gauss
On Dynamics of Bubbly Liquids
69
theorem to derive differential equations. Thirdly, only when V is contained entirely within A can differential equations like (2.3) be obtained for each phase; but then we need to specify interfacial conditions at the interface between phases A. It should also be remarked that we have ignored the cases in which part of d V may sometimes coincide with part of A. This can be accommodated so long as we do not take twice the integral over the same surface area.
B. AVERAGEDDYNAMICAL EQUATIONS Let us denote the average of any field quantityf(x, t ) by (f)(x, t ) . There are various kinds of average, e.g., ensemble average, time average, spatial average, and spatial-time average, in either Eulerian or Lagrangian formulation. Drew (1971) has even adopted a double spatial-time average defined in the following manner: (f)(x, t ) = C(1, t )
I,,,) [ da
I,(c)
d3k
da’ Sdx)
I,,,,,
d 3 k ’ f ( k ’ ,a’), (2.10)
where S,(x) is the sphere of radius 1 with center at x, I , ( t ) is the interval [ t - T, t + T],and C(1, T ) = [(4r/3)13. 2 ~ ] is - a~ normalization constant. The average is taken double to smooth out the irregularities in microscale, so that ( f )(x, t ) would be a function smooth enough for analytical purposes. For practical purposes, we may define ( ) in a somewhat vague sense, except that we should assume that the average quantities are smooth enough, and also that the operation of spatial integration commutes with the average operation, i.e., for any V : (2.11)
and
We shall also assume that
which can be justified again if ( f ) is smooth enough, i.e., the averaging scales 1 and T are small in comparison with the scale of variation of (x, t )
D. Y. Hsieh
70
in ( f ) ( x ,
t).
Similarly, for two functions f and g, we also have ((f)g)
= (f>(g>.
(2.14)
Now we can establish in some sense that:
"( dt
I
vfd'x) =
(2I
V
fd'x).
(2.15)
This can be explicitly demonstrated by the time average, because
Thus
In fact, for time averages, if g ( x , 1 ) is smooth in
t,
we have
Although in general f ( x , t ) may not be a smooth function, here j,f(x, T ) d 3 x is assumed to be smooth in T. Similarly, if g ( x , 7 )is smooth in x, by considering the spatial average we also have
The quantity j A f ( x , T ) d ' x , which is an integral over all the moving interface A inside V, is again a function of T. For a certain class of interfaces, this quantity should be proportional to the size of V. Therefore we shall write (2.16)
where f * ( x , T ) is an averaged quantity. Thus, taking the average of (2.9), and making use of (2.11), (2.12), (2.15) and (2.16), we obtain
On Dynamics of Bubbly Liquids
71
Since the averaged quantities are all smooth, and (2.17) is valid for any V fixed in space, therefore we can apply the Gauss Theorem and obtain the differential form of the averaged equations
In principle, if we know how to produce averaged quantities from local instantaneous variables, Equations (2.17) or (2.18) would contain all the necessary information provided that we know the physics of the constituent phases and their mutual interaction. In practice, the averaged quantities are often as obscure as new unknown variables. Still, the fact that they represent the average of certain definite variables can contribute to our physical understanding of the mechanisms involved and can be helpful to construct approximate models for the constitutive equation. Let us denote ( P a ) by Px, i.e., the average volume fraction of phase a, and let P j P a =(Pap),
(2.19)
Pa*PaV, = (PaPV).
(2.20)
Thus pa is the average density of phase a and v, is the average velocity of phase a. Now let us take $ = p ; since f = 0 and s = 0 for this case, Equation (2.18) becomes
where
When there is no mass transfer across the interface between phases, then v = v A , and we have S a ( p )= O . We are using the notation to denote the term arising from the interfacial transfer. Now let us identify $ as pv; then the corresponding identifications o f f and s are -u and pb respectively. Thus the Equation (2.18) takes the form A
A
72
D. Y. Hsieh
Let us denote
Then, noting that b represents external force field and is independent of the averaging process, and using (2.21), the Equation (2.23) can be written as
In the above equation, u, represents the average of the stress in phase a; 7 , is the stress term representing the diffusive momentum flux arising from the statistical average; P, is a mutual interactional force between the phases; and 6, is a mutual interactional force arising from the mass transfer across interface between phases. When there is no mass transfer across the interface between phases, then 6, = 0. Now let us take 14=p(LI+(v*/2)), f = q - a . v , and s = p b . v + p Q . Denote
where U, is the average internal energy per unit volume for phase a,q, is the average ath phase heat flux, and Qa represents the ath phase heat source. Then the Equation (2.18) takes the form
- [Pap
( u+);
(v - vA) n]
*.
(2.32)
73
O n Dynamics of Bubbly Liquids
By making use of (2.21), (2.23), (2.28) and relations such as (2.14), the equation can be put in the form: 1
- (PaP(V
-Val
+p:(a,
-7,):
( u ++))
{v-v
-
}2
- {v-v,})
Vv, - [ & ( q - a (v-v,)
p",] n]*
I*
.n
.
(2.33)
The Equations (2.21), (2.28) and (2.33) are the basic equations for the averaged fluid properties. Just like the fundamental transfer equation (2.3), or the fundamental equation for the homogeneous fluid, the number of unknown variables far exceeds the number of equations. Constitutive relations based on certain assumptions or models have to be supplied to make the system solvable. As we may appreciate from the form of the basic equations, the system of bubbly liquid is extremely complex. Therefore it is very difficult to derive constitutive relations by following the averaging process even with simplest models. Therefore, constitutive relations are usually assumed in some form analogous to those for homogeneous fluids with some adjustable parameters to be determined empirically. But we do not think that all our previous effort to derive these equations is just an empty exercise; although we cannot derive rigorously various averaged terms contained in those fundamental equations, an approximate estimate based on some simple model can usually be given. We also know what effect we have ignored when we neglect certain terms. Moreover, by carrying out the averaging procedure to derive these equations, we also have a clear idea what we mean by these newly introduced continuum variables. Thus when we write down the constitutive relations, we are on a relatively firm ground of rationality as to their physical meaning. C. SOMEOTHERAPPROACHES Various approaches are developed to arrive at the dynamical equations essentially equivalent to (2.21), (2.28) and (2.33). We may just mention a few.
74
D. Y. Hsieh
One class of approach is again based on the averaging of the “microscopic” local instantaneous equations. However, the averaging is applied to the differential form of the transfer equations for each phase, instead of the integral form (2.9) (Delhaye, 1977a, b, c, d; Ishii, 1975). The physical content of these approaches is essentially the same as the one presented above, which was first developed by Drew (1971) and Drew and Segal (1971). However, the approach we presented above seems to present a clearer picture and is simpler in details. Another class of approach treats the two phase fluids as a mixture of two superimposed continua. They are endowed with various properties based on physical considerations. Therefore it is intrinsically a “macroscopic” theory. Since our averaged dynamical equations are so incomplete that we have to appeal to other means to supply constitutive relations, it would seem that the averaged equations are really not any better than the purely “macroscopic” equations. However, there are some differences. First of all, although we may not be able to compute rigorously various average terms in the averaged equations, we do know what those terms stand for, and physical models can be constructed accordingly to make approximate calculations. For the mixture theorv, the connection between “macroscopic” and “microscopic” is more remote and artificial. Secondly, we know what we mean by the averaged variables. They are the average, i.e., time, space or ensemble average of a definite fluid property, which in principle can be measured in a definite manner. In mixture theory, the variables need to be defined to give unambiguous physical meaning. General theoretical formulations for mixtures have been given by Bowen (1976), Kenyon (1976), and Drumheller and Bedford (1978). Drumheller and Bedford have adopted a variational approach for a system to take care of dissipative terms, and they have applied their formulation specifically to the problem of bubbly liquids (1979, 1980). Another approach is to consider a bubbly liquid with small bubble volume fraction as essentially a liquid with only dilute dispersion of bubbles. Especially for wave propagation problems, the bubbles serve as scattering centers of a multiple scattering problem. When the inter-bubble distance is large compared with the bubble radius and the typical wave length is again large compared with the inter-bubble distances, a set of effective equations can be derived for this mixture (Caflisch et al., 1985). This approach, although of limited scope, is useful in that it provides a more rigorous demonstration of the validity of the physical models to be employed for the determination of some averaged terms in the dynamical equations. It
On Dynamics of Bubbly Liquids
75
may also indicate the range of applicability of certain assumptions involved for the construction of constitutive relations.
111. Dynamic Equations of Bubbly Liquids
The average dynamic equations (2.21), (2.28) and (2.33) are applicable to any two-phase media. They can be fluid-solid mixtures, liquid-gas mixtures or others. The shape of the microscopic interfaces between the phases can also be arbitrary. Now we want t o specialize our study to the problem of a liquid containing bubbles. As a first approximation, we take the bubbles to be all spherical. This is a valid assumption when the bubbles are small, since the surface tension would act to make them spherical. Let the average radius of the bubbles be R . Since R will change in response to the flow field, R ( x , t ) is a macroscopic field variable. Let us define another macroscopic variable n ( x , t ) as the average number density of the bubbles. Then it is clear that (3.1) where p,* is the average volume fraction of the gaseous phase. From now on we shall use subscript g to designate the gaseous phase and the subscript f to designate the liquid phase. Although the fluid inside the bubble consists in general of both the vapor of the liquid and other gases, we shall consider, for simplicity, two idealized cases, i.e., the gas bubbles with no vapor, and the vapor bubbles with vapor only inside the bubble. For the case of gas bubbles, the gas content of each bubble is practically constant, if one neglects the coalescence and splitting of bubbles, which we shall assume to be the case. Moreover the influence of heat transfer on the dynamics is usually secondary, and we can often bypass the energy equation and thus simplify the analysis a great deal. On the other hand, for the case of vapor bubbles, the thermal effect is of primary importance. Furthermore, there is mass transfer across the interface between two phases due to evaporation or condensation. The situation is thus much more complex. It is possible that there may be more than one species of bubbles, characterized by distinctly different radii, coexisting at the same “macroscopic” space-time. Then we can treat different species of bubbles as different phases and extend the two-phase analysis to multi-phase analysis.
76
D. Y. Hsieh
A. SINGLESPECIESO F GASBUBBLES We consider the case that there is only one species of gas bubbles, i.e., the total mass of gas in each bubble m is constant. Therefore the average density of gas pg is related to the average radius of bubbles R by 4rr - R 3 p , = m. 3
Since there is only one species of bubbles, we shall denote P,* by P for simplicity, and hence Pf* = (1 - P ) . For gas bubbles, there is no mass transfer across the interfaces. Thus i U ( p )= 0 for a = f ,g , and hence (2.21) becomes: (3.3) and
From (3.1)-(3.3), we can also obtain an -+ at
v . (nv,) = 0.
(3.5)
Equation (3.5) states that the number of bubbles is conserved, which is a natural consequence of the assumption that there is no coalescence or splitting of bubbles. Now let us consider the Equation (2.28). Since there is no mass transfer between phases, then 1
Gu=O,
Now consider the term V .
a=J;g.
(3.6)
We may recall that P%u
=(Pad.
(2.25)
For ordinary liquids and gases, we can write u = - P I + u’,
(3.7)
where p is pressure and u’ is the viscous stress term. We shall denote PXPa =(Pup),
(3.8)
PXUb = ( P a @ ‘ ) .
(3.9)
Then we have
PXU, = - p : p u l
+@Xu:.
(3.10)
On Dynamics of Bubbly Liquids
77
The meaning of pa is now fairly clear. pa is the average pressure in the phase a. For instance, if we take a representative bubble, the pressure of the gas inside the bubble will be p , . Now let us consider the term ka.From (2.26) and (3.7), we can also write f;, = -(P,pn)*
+kb,
(3.11)
where
From the definition of the interfacial average (2.16), it can be shown (Appendix A; see also Drew and Segal, 1971; Ishii, 1975) that if the phasic pressure is the same as the interfacial phasic pressure, or if PgVp, = (PaVp), then -(Papn)* = PaVPX.
(3.13)
The continuity of viscous stress across the interface, if we assume that the coefficient of surface tension is constant, requires that A
1 .
FI. = -Fh.
(3.14)
Presumably, we could make a similar analysis for u' as for p and obtain a relation similar to (3.13) for k:,. There is a physical reason to expect that (3.13) is valid, because p represents a volumetric force. On the other hand, kb represents a force arising from boundary interactions. Therefore, we shall leave k: as it is and be content with the relationship (3.14). With the above discussions, then the Equation (2.28) now takes the form:
and
A
+ V . [( 1 -P)(u;-T~)]-FL.
(3.16)
It is to be noted that the variables in these equations are averaged quantities. Thus it is meaningful to talk about terms like Vp,, even though p g represents pressure of the gas inside some bubbles, and the bubbles are not connected.
D. Y. Hsieh
78
B. THE PRESSURETERMSA N D UYLEIGH-PLESSET EQUATION The equations of state relating p , p and T of each phase are assumed to be known. The relation of the form p = p ( p , T) is valid of course only “microscopically.” It is not obvious that we also have “macroscopic” equations of state of the form, pa = p a ( p a , T o ) ,which holds for averaged variables of each separate phase. Partly due to our ignorance and inability, and partly because it seems plausible, we shall assume that such equations of state exist for each phase. Thus (3.17)
and Pi=Pr(Pr, Ti).
(3.18)
We may even assume, if nothing more reliable can be ascertained, that (3.17) and (3.18) are the same as the “microscopic” relations. Now what is the relationship between pp and pf? For a single spherical bubble with radius R in an incompressible fluid of infinite extent at rest at infinity, the equation governing the bubble motion is the well-known Rayleigh-Plesset equation (Hsieh, 1965)
where p is the density of the external incompressible fluid, p ’ is the pressure inside the bubble, pm is the pressure at infinity, 77 is the viscosity coefficient of the external fluid and (+ is the surface tension coefficient. Using this as a model, then we may identify p ’ as p g , pm as p r , p as pr, and 77 as Vr, the viscosity coefficient for the liquid. Since R is now a function of both position x and time t, we should replace d/dt by [a/dt +v, . V], the time derivation following the motion of the bubble. Therefore (3.19) becomes
where
(3.21)
On Dynamics of Bubbly Liquids
79
The adoption of this model can be justified when /3 is small and the liquid is nearly incompressible (Caflisch et al., 1985). It is plausible that the right-hand side will effectively depend on R only even for more general cases. That is why we use the notation P { R } to indicate the essence of this complex dependence. It may also be noted that a relationship between p g and p f is necessary, since this is in fact the equation to account for the new variable p or R introduced for this fluid mixture. It may be noted that we may also take into account the linear motion of the bubble as a whole in the liquid. Consider again a single spherical bubble with radius R moving with velocity v in an incompressible, inviscid fluid of infinite extent at rest at infinity. Then it can be shown that the pressure in the liquid on the bubble surface p ( R , t ) , where R denotes a point on the bubble surface, is given by p(R,t)=p,+p
[
d2R 3 dR ' 5 dR 1 R-+- --v-cosO+-R.d t 2 2( d t ) 2 dt 2
dv dt (3.22)
where 0 is the angle R makes with v, i.e., cos 0 = R * v / Rv. Thus if we average over the bubble surface, then (3.22) becomes d2R 3 dR p ( R , t ) = p , + p R-+dt2 2( d t ) 2 - : ] y
[
where p ( R, t ) =4
1;
(3.23)
p(R, t ) sin 0 d0
is the average pressure of p ( R, t ) . Therefore the Rayleigh-Plesset Equation (3.19) should be modified to take the form
R ) 2 = -: [ p t ( R ( t ) , t ) - p m ( t ) - 4-17 dR R -d+2-R 3 ( ddt2 2 dt R(dt)
--+2a R
p v 2 . (3.24) 4 1
Thus, (3.20) should also be modified to become (3.25) This additional term ( p f / 4 ) ( v , v f ) 2has also been suggested by Wijngaarden (1982).
80
D. Y. Hsieh
C. MUTUALINTERACTION FORCES Let us again consider the model of a single spherical bubble of radius R is in a liquid of infinite extent at rest at infinity. When the bubble is moving with ve1ocity.v in the liquid, it experiences various kinds of forces. There are at least the following types of forces. (1) Drug Force
For slow motions, the Stokes drag is given by
f d = -6i-r7)R~,
(3.26)
where according to the analysis, taking into account the internal fluid motion inside the bubble, 7 should be given by (3.27) But experimental findings indicate that 7 is in fact given approximately by (3.28)
7) = 7)f,
i.e., as if the bubble is a rigid sphere. The drag coefficient CD is usually defined as (3.29) Thus (3.26) can also be written as (3.30) For Stokes flow, we have (3.31) Let us denote the contribution to $; from the drag force by F:; then
F d, = n f
d
=-
3P
47rR'
where in f d, v is to be replaced by (vg - vr).
fd,
(3.32)
On Dynamics of Bubbly Liquids
81
Thus we obtain from (3.26) or (3.30) (3.33)
or (3.34) (3.33) and (3.34) are different expressions for the same thing. However, it appears that F," varies linearly with (vg-vf) according to (3.33), while it varies quadratically with (v,-vr) according to (3.34). The difference is due to the fact that C, varies inversely with Ivg-vfl according to the Stokes formula. For large Ivg-vflr C,, may become independent of Ivg-vfl. Then (3.34) would be a more appropriate form to adopt. For instance, for churn-turbulent flow, it is found (Hench and Johnston, 1972) that CI,= 109.8R(1 -/3)3,
(3.35)
where R is measured in meters. (2) Virtual Mass and Basset Force When a sphere of radius R moves slowly, i.e., Stokes flow, in an incompressible viscous fluid of infinite extent with arbitrary rectilinear velocity v( t ) , the force the sphere experience is (Landau and Lifshitz, 1959)
--]
3 ~ l v + l p R ~ ~ + 3 R & [ ' dv d7 -cc dr fi ' dt
(3.36)
where p is the density of the fluid and 7 is the viscosity coefficient of the fluid. In (3.36), the first term can be readily identified to be the drag force (3.26), while the second and third terms are known as the virtual mass force f " and Basset force f h respectively. The virtual mass force is independent of the viscosity of the fluid. The underlying physical mechanism is that when the sphere is moving in the fluid, besides its own momentum, it also generates momentum in the fluid because of the fluid motion it excites. That is why f" is proportional to the density of the fluid p. One way to derive f " is to consider the velocity potential @ for a sphere of radius R moving with v in a fluid with density p at rest at infinity. It is
82
D. Y. Hsieh
well known that @=
-'(!));.
.
2 r
(3.37)
where r is the position vector measured from the center of the sphere. With this @, the total kinetic energy of the fluid K , can be readily calculated and we find that (3.38) Equating
dKi -- -f" dt
. v,
(3.39)
we obtain the virtual mass force in (3.36). Now if we want to consider the case of a sphere of radius R moving with velocity v in a fluid which is moving with velocity u at infinity, the corresponding velocity potential Q, is (3.40) Then the total kinetic energy of the fluid is found to be UY,
(3.41)
where M is the total mass of the fluid. The interactive part of the kinetic energy due to the relative motion between the sphere and fluid, which we shall again denote by K i , is now given by (3.42) The symmetry of v and u in K , in (3.42) suggests that when we transit from f " to F:, which is the corresponding contribution to k:, we write as in (3.32),
F"=- 3p f " , 497R3
(3.43)
where now
Dg
f " = - -297 3 PfR3(L)1Vg--Vr
Dt Df
),
(3.44)
On Dynamics of Bubbly Liquids
83
and
D Dt
+:(
=
6 D f =+;(
vg . v),
Vf
. v),
(3.45)
are the material derivations with respect to time. The form (3.44) retains the symmetry with respect to vg and of, and also is consistent with the principle of objectivity (Drew, Cheng, and Lahey, 1979). Since K i is associated with the kinetic energy of the fluid motion outside the sphere, and the density of the external fluid is represented by [( 1 - p ) p f + pp,], then it seems natural that we should replace pf by [ ( l -p)pf+pp,] in (3.44). In other words, since (3.43) can be written as (3.46) with
c,,
=$,
(3.47)
now it is suggested that (Anderson, Astrop and Rothmann, 1976): C"M
="
(1
2
-p)+pS Pf
(3.48)
On the other hand, it could also be argued that because of the presence of other bubbles, the external liquid is not of infinite extent but is confined, say, within a sphere of radius R , , with ( R / R , ) ' = p . Then, based on this model, it is found that (Zuber, 1964)
c,,
=
!( 3) >1 -
2
1-p
2
(3.49)
It is difficult to ascertain on theoretical grounds what value CVMshould take, or whether (3.46) is really the correct expression for the virtual mass force, since we can only base our analysis on crude models. The important point is that something of this nature should be present, and ultimately the issue should be determined by experiment. But unfortunately the effect of virtual mass force on experiments carried out so far is so small that the question is still open. The Basset force f b is due to the transient effect of the boundary layer at the bubble surface because of the variation of the relative motion. To
84
D. Y. Hsieh
transit from f b to FL, the corresponding contribution to fi;, we will replace v by (v,-vf), p by p f , 77 by vr,and d l d r by d / d r . Thus, as in (3.32), we obtain (3.50) where f b = - 6rR2@ {‘
L[v,(x, dr
r)-vf(x, r
)
dr ]
~
.
(3.51)
We have used (d/dr)[v, - vf], instead of ( Dg/D r ) v g - ( Df/ D r ) v f , because the Bassett force is basically a type of viscous drag, and hence should belong to the category similar to F,” rather than the virtual mass force Fi.
D. THE SELF FORCE
The self-force term V [ P ( U ; - T ~ ) ]in (3.15), as we may recall from (2.24), (2.25) and (3.7), arises from the average of viscous stress and diffusive momentum transfer terms. It is difficult to derive the correct form of this term. Because of our ignorance, we thus resort to the assumption of analogy with the homogeneous fluids and treat the term like the one for a simple Newtonian fluid. Thus we assume that
and l,* are “average” viscosity coefficients for the gas phase. Similarly in (3.16), we assume that
in Cartesian coordinates, where 77:
Because of the uncertainty of theoretical basis and lack of experimental support for the appropriate form of this term, it has been variously suggested that given a stress term ug*like (3.52), instead of V . [pug*]in (3.15), we or V . u,*. Since the viscosity coefficients are usually should have pV . a,*, assumed to be functions of p, the differences between these various forms may be reconciled.
O n Dynamics of Bubbly Liquids
85
E. T H E ENERGYEQUATION We shall now drastically simplify the general energy equation (2.33), and appeal to physical arguments to obtain a manageable form for the bubbly liquid. Let us consider the case for the liquid phase first. We shall first take the reasonable assumption that Uf =
GTS,
(3.54)
where Cf is the specific heat at constant volume for the liquid phase. Then (2.33) can be written as
where we have lumped together all the other terms in a term associated with a generalized heat flux qf and a generalized heat source term. Again, because of our ignorance of the complex mechanisms involved, we shall assume
qf = - K fV Tf,
(3.56)
when Kf is an “average” thermal conductivity in the liquid phase. into two parts. Let us decompose
of
Qr=
QfS
+ Q,,
(3.57)
where Qfs is the heat due to external heat source and those generated by viscous and dispersive dissipations, while Q, is the heat exchange between the phases. We know very little about Qfc and it is probably not significant. When Tf= T g ,i.e., when the temperature of the liquid and gas phases are equal, it is reasonable to assume that Q, =O. When Ts- T g # 0, we shall assume that
Q, = h ( T g - Tf).
(3.58)
where h is a heat exchange coefficient. Let us consider a very crude model of a sphere of radius R with temperature Tg in a bath of temperature T f . Take R to be the characteristic diffusion length. Then the total heat transferred from the sphere to bath per unit time is roughly 4 r r R 2 K r (T,- Ts)/R. Since there are n = 3 P / 4 r R 3 bubble in a unit volume, we therefore obtain (3.59)
D. Y. Hsieh
86
It should be remarked that when we take R to be characteristic diffusion length, we have implicitly assumed that the process is relatively slow. collecting (3.56)-(3.59), (3.55) can be put in the form
(3.60)
Similarly for the gas phase, we have
For many dynamical problems with gas bubbles, both liquid and gas phases can be considered as barotropic, i.e., (3.17) and (3.18) are ofthe form (3.62)
Pr = ~ rP C( ) .
(3.63)
Then the dynamic equations are decoupled from the energy equations. Unless we are interested in the variation of temperature fields, there is no need to solve the energy equations (3.60) and (3.61).
F. EQUATIONFOR
THE
FLUIDMIXTURE
Let us summarize the previous discussions. Under reasonable assumptions, the equations governing the motion of a liquid containing many bubbles are (3.2), (3.3), (3.4), (3.15), (3.16), (3.17), (3.18), (3.25), (3.60) and (3.61). In (3.15) and (3.16), the term pL is given by
fi;=
F;+ F;+ F ; ,
(3.64)
where Fgd, FP and FB are given by (3.33), (3.46) and (3.50) respectively. It may be illuminating to recast these equations in terms of the fluid mixture as a whole. We shall define the density p, pressure p and velocity v of the fluid mixture as follows: P=PPg+(l-P)Prr
(3.65)
P=PPg+(l-P)Pf,
(3.66)
O n Dynamics of Bubbly Liquids
87
and PV = PPgVg + ( 1 - P )PfVf.
(3.67)
Then we obtain from (3.3) and (3.4)
*+v. at
(3.68)
(pv)=O.
Let us define the diffusive fluxes by j,=Pp,(v,-v)=-(1
(3.69)
-P)dvf-v)=-jf.
Then the Equations (3.3) and (3.4) can be recast in the following form: (3.70)
If we add Equations (3.15) and (3.16), then we obtain
where = P ( u L-7,)
+ ( 1- P )(a;- 4 - (vgj, + vfjf).
(3.73)
Equations (3.68) and (3.72) are useful in the sense that, except for the extra term (p,-pf)VP in (3.72), they are identical to the continuity and momentum equations for the homogeneous fluid. For certain flow problems it may be a valid approximation to take p g = p f . Then, if a constitutive equation relating 7~ to z7 could be found, the problem would be reduced to that of a homogeneous fluid.
G. DYNAMICAL EQUATIONS FOR MANYSPECIES OF GAS BUBBLES The formulation presented above can be readily generalized to deal with bubbly liquids of more than one species, say n species of gas bubbles. Take these equations for a single species of bubbles summarized in Section 1II.F. Formally, we need only to change the variables R, p,, m, P, v,, pp, u:, T,, @:,T,, F$, Fi, Fi, C, K,, Q g y , to R , , P,, m,, v,, P,. a:,7 , , T,, FP, F:,
@:,
88
D. Y. Hsieh
FP, C , , K , , Qlr respectively in (3.2), (3.3), (3.15), (3.17), (3.64), (3.25) and (3.61), and for i = 1,. . . ,n. In (3.4), (3.16) and (3.61), we shall take n
c
P =,=I
PI>
and
,=I
IV. Sound Waves in Bubbly Liquid
A. THE LINEARIZED PROBLEM AND
THE
CHARACTERISTIC EQUATION
By sound waves we mean the small amplitude waves propagating in the medium otherwise at rest. For those governing equations summarized in Section IILF, we shall find first the equilibrium state. Let us use the subscript 0 to denote the equilibrium quantities, e.g., Po, R,,, pgo, pfo, p g o , pfo, etc. It should be noted that vgo = vfo = 0. Then we set
P = P o ( l + P ' ) , etc.,
(4.1)
and substitute it into the governing equations. Now treat P', R ' , . . . as well as vg and vf as first-order small quantities, and neglect all the second- and higher-order terms; then we obtain a set of linearized equations. These equations are the governing equations for the sound waves. The dynamics of bubbly liquids are so complicated that even for the linearized problem, we would be content to obtain the dispersion relation or the characteristic equation for the sound wave problem. From the dispersion relation, we can deduce how the propagation velocity changes with frequency and how various mechanisms contribute to the damping of waves. When the damping is small, it is generally true that the damping effects due to various effects are additive. Thus we could isolate these various effects and deal with them individually. For liquids containing gas bubbles only, the thermal effects are not important dynamically. They would contribute to some damping, of course, but these contributions can be calculated separately. Thus to simplify the analysis so that we can gain a clearer understanding of the dynamics of the
On Dynamics of Bubb1.v Liquids
89
problem, we shall neglect these effects in our discussion. In other words, we shall take
instead of (3.17) and (3.18). Then the temperature fields are decoupled from the dynamical system. Hence there is no need to consider (3.60) and (3.61). Let us assume that the self forces are given by (3.52) and (3.53), and neglect the external body force b; then the linearized version of the equations (3.2), (3.3), (3.4), (3.15) and (3.16) become respectively
+
3 R' p;
a
-( p ' dt
= 0,
+ p t ) + v . vg = 0,
(4.4)
(4.5)
where
(4.9) (4.10)
(4.11) and 6; will be discussed presently. If we make use of (3.21), then the linearized version of (3.25) is
90
D. Y. Hsieh
where (4.13) is in general different from v I . For $,; we shall disregard the Basset force F i for the time being. For this linearized theory, we shall use (3.33) for the drag force Fi. Using (3.46) for the virtual mass term, we thus obtain the linearized k;:
,.
F; = - D " ( V r
a -Vf)
- ~ " * , P o P m - (vg - v o , at
(4.14)
where (4.15) Let us now investigate how sinusoidal waves will propagate in the bubbly liquid. We assume that all the perturbed quantities are proportional to the factor exp[i(k. x - w t ) ] . Let us now denote (4.16)
(4.18) (4.19) Then we obtain from (4.4) and (4.12) pi.= BR',
(4.20)
where (4.21) Using the above equations, after some straightforward calculations as shown in Appendix B, we obtain the characteristic Equation (B.8). since B is quadratic in w, Equation (B.8) is a polynomial equation of order five in w. The roots can be found for a given physical problem by numerical methods.
91
On Dynamics of Bubbly Liquids Now from (4.15) and (4.18), we may express w D as 9 a;
W,]
= - -.
(4.22)
2 Flq
It seems reasonable to assume a ; is of the same order as a , and a g . Furthermore since q < 1 for this formulation to be valid, a' and ag are ~ . we can neglect the terms in general small in comparison with E ~ wThus with af and a g in (B.8). Noticing also that < 1, E? < 1, and w i < w : / q , we thus obtain the following simplified characteristic Equation (B.11): 1 +-
+ iwD
[
( 1-
( +?) 1
wjw
fw2
(4.23) Now let us consider the following limiting cases: (i) wD B w, i.e., when the mutual friction is important. The approximate solutions are (4.24) and (4.25) Since we usually have q<<1, the mode associated with w ; is thus not a propagating mode and is not relevant to sound propagation. Damping coefficients can be estimated by making use of (4.23). Let us take w = wI - is,, and since w , Q w , , we obtain
(4.26)
92
D. Y. Hsieh
The frequency equation (4.24) is the most important one that is relevant to sound propagation in bubbly liquid, since for most cases the mutual friction is important. For q < 1, (4.24) yields the velocity of propagation (4.27)
which is usually an order of magnitude smaller than c,, the sound speed in the gaseous medium. It may be noted that c, does not depend on the parameter C V MThus . when w D > w , the effect of virtual mass is negligible on the sound speed in the bubbly liquid. However, the damping of the sound wave does depend on C V Mas , may be seen from (4.26). The frequency w \ as given by (4.25) is of the order of ( cf/ R,,),corresponding to the resonance frequency of a liquid sphere with radius R o , which is very high. Our formulation definitely needs to be modified at such high frequency. It may be of interest to note also that the corresponding 6 ; turns out to be negative. Negative damping implies instability. This instability further reinforces our view that it is not physically realistic to deal with such high frequency within the context of the present formulation. (ii) w D < w , i.e., when the mutual friction is insignificant. The approximate solutions are
(4.28)
and
L’
’
CVMJ
Note that for << 1, w ; = w ; . Thus this mode is again non-propagating. w 2 = O ( w , ) , while if we neglect It is interesting to note that for finite CV&,, CvM,i.e., letting CvM”0, then w : / w i = O ( F , ) ,i.e., w 2 is much larger than w , . We may remark here that as seen from (4.27), the dynamics of the sound wave associated with the mode w Iare that the inertia of the system is largely that of the liquid while the compressibility is mainly due to the gas bubbles.
On Dynumics of Bubbly Liquids
93
This is to be expected, since the liquid and gas move together when the mutual friction is important. When the mutual friction is insignificant, we expect that the liquid phase and gas phase would move independent of each other. Indeed if CvM= 0, we have w 2= c,k. But if CVMis finite, the virtual mass effect will couple the motions of liquid and gas together and again yields the sound speed of the same order of c,. Thus the investigation of sound speed at high frequency should reveal clearly the effect of virtual mass. A rough idea of the magnitude of w,, is useful at this stage. From (4.22) we have (4.30) Take water and air, and then T,-= lo-', pg= lop3;take Ro= lo-', p = 0.1, all in cgs units, and then we obtain w,, = 5 x lo3 sec-I. In comparison, the bubble resonance frequency w:=3~,c:/Ri, and thus w 0=2x lo4 sec-'.
B. THE "DEAD ZONE" A N D
THE
NEGATIVEDAMPING
Now, the characteristic Equation (4.23), when we consider it from the perspective of k, is actually only quadratic in k. Let us thus rearrange it accordingly and we can rewrite (4.23) as (see (B.16))
(4.31) where (4.32)
94
D. Y. Hsieh
It can be seen from (4.33) and (4.34) that for small w both m , and m2 are positive. As w increases to exceed a value of the order of w,, which we shall denote as W,,m , and m, become negative, and they stay negative for large w. Thus from (4.31), we can see that
Rlk2> 0,
for o < W,,or w > w,,
(4.35)
Rlk’ < 0,
for W,< w < w,.
(4.36)
and
< w < w,, k has a very large imaginary part, and hence the wave Thus for 6,. is not propagating. We shall call this frequency range the “Dead Zone.” 0, is associated with the resonance of a gas bubble in a liquid. The frequency w,, aside a factor P / ( l - P ) , is the resonance frequency of the liquid outside an enclosure of radius R o . w , is sometimes called the antiresonance frequency. We can visualize the whole picture as follows. For low frequencies, or equivalently when the wavelength is long compared with the bubble radius, the mode of propagation is essentially that due to the combined effect of the compressibility of gas and the inertia of liquid. We encounter the “Dead Zone” as w reaches the bubble resonance frequency 0,. For very high frequencies, the propagation mode is essentially that in a pure liquid, since the gas and liquid phase move independently. For this case, the value q = ( k R J 2 in fact is large. Thus (4.29) will yield a propagating mode for large q. As w decreases towards the anti-resonance frequency w,, the “Dead Zone” is again encountered. From (4.31), we may also notice that Imk2>0,
for w < w , ,
(4.37)
Imk’ < 0,
for w > w , .
(4.38)
Thus if other dissipation mechanisms, such as thermal damping, viscous damping, etc., are not included as in (4.31), there will be negative damping for w > w,. Is this negative damping real? We may trace the cause of this negative damping, by a more refined analysis, to the Rayleigh- Plesset equation. The linearized Rayleigh-Plesset equation in the plane wave form can be written as (4.39) where S ( w )= w2
On Dynamics of Bubbly Liquids
95
Thus for small w , p; and p;, and hence p ; and p ; , are of the same sign. However for very large w, pi! and PI. are of opposite signs, and lpfl* IpkI. That seems to be odd. It is more plausible that p ; = PI.for very large w. In any case, it is not likely that the Rayleigh-Plesset equation will still be valid for large w in the context of the continuum formulation of a bubbly liquid. One phenomenological way to deal with this anomaly is to assume s(w)zw2,
for w small,
S ( w ) + 0,
for
while w
large.
A function of the form (4.40)
could serve the purpose, and remedy the situation. Now, for an S ( w ) as given by (4.40), the term ( d 2 R / d t 2 )in the Rayleigh-Plesset equation would have to be replaced by a term with differential-integral operators on R. The complexity of the problem points to the inadequacy of this formulation to deal with high frequency phenomena in the bubbly liquids.
W I T H EXPERIMENTS C. COMPARISONS
Experimental investigations on sound propagations in a bubbly liquid are few and far apart. We still need to refer back to the work by Silberman (1957) for comparison. Since then, there have been experimental investigations by Barclay, Ledwidge and Cornfield (1969), and the dissertation by Hall (1971), among others. Based on the same formulation that we presented above, but including the Basset force and thermal dissipation in the liquid, Cheng (1983) in his dissertation made an exhaustive study of the problem. Here we have reproduced some figures from his dissertation to show the comparison between theory and the experimental results of Silberman. For this comparison, Po = 5.84 x and the bubble radius for the theoretical computation are taken to be Ro = 2.5 x lo-‘ cm and Ro = 1.92 x lo-’ cm respectively. It can be seen from Figures 1 and 3 that the existence of the “Dead Zone” is clearly revealed. On the other hand, the “negative damping” is apparently more than compensated for by the thermal damping, as can be seen from Figures 2 and 4. The general agreement between the theory and experiments
D. Y. Hsieh
96
PHASE VELOCITY
vs
FREQUENCY
2.5 x l d 3 m
Ro
p,= 5 . 8 4 x
0
1
W
m
a
L
-
10'.
I
I
I
I
I I l l
I
I
I
1
1
1
1
1
1
I
I
I
I l l
V. Waves and Instability in Bubbly Liquids A. FINITE AMPLITUDE WAVESI N LOCKEDBUBBLYLIQUIDS From the analysis of small-amplitude waves in bubbly liquids, it is clear that at least for low frequency cases, the dominant mode of propagation is
On Dynamics qf Bubbly Liquids
97
I "
M
W
+ la
I
1.98
0
x
IC
I
I oo
c
!
10'
I
I
I
IIII
I
1
, , , , , , I
I o2 lo3 FREQUENCY (Hz1
FIG. 2. Attenuation versus frequency, R,, 2.5 x lo-' rn, Po = 5 . 8 4 ~ Cheng (1983).)
1 lo4 (Courtesy of L.
that for which the compressibility is controlled by the gas and the inertia is controlled by the liquid. Therefore for a large class of problems we may make the approximations that pf is constant and E , < 1. Furthermore, as we see from the sound wave, the damping due to the slippage between phases is more important than the "viscous" damping in each phase. Thus we can also ignore the self-frictional forces. We shall again neglect the body force and the Basset force in this section.
98
D. Y. Hsieh lo4
PHASE V E L O C I T Y
vs
FREQUENCY Ro = I 9 2 X p =5 B ~ x I O - ~ 0 M E A N B U B B L E RADIUS ( R o ) , m
o
-
o
-E lo3 7
x IO-~ x 10-3 2 0 7 x 10-3 I 92
I I rc,,
I 98
o
u)
\
I I
a
2 50 x I O - ~
>-
n =o0
I
t 0
l
0
2 w
I
>
--cc,,=o
w
5
cn
a
E lo2:
10'
1
10'
I
I
1
I
I
I
I
1
I
I
I
I
I
I 1 1 1 1
lo3
lo2 FREQUENCY
I o4
(Hz)
FIG.3. Phase velocity versus frequency, R,, = 1.92 x lo-' m, L. Cheng (1983).)
Po = 5.84x
(Courtesy of
With these approximations, using ( 3 . 2 ) , (3.33), and (3.46), the equations ( 3 . 3 ) , (3.4), (3.15) and (3.16) can be written as: "(")+V.(".)=O, at R3
aP
-= at
R3
g
v . [ ( l -P)V,-],
(5.1)
(5.2)
(5.4)
99
On Dynamics of Bubbly Liquids 10
-z
10:
E m \
-0
0 I-
3 I0 Z W
Il-
a
10
IC
lo2
10‘
lo3
lo4
FREQUENCY (Hz) FIG. 4. Attenuation versus frequency, R,, = 1.92 x lo-’ rn, p,, = 5 . 8 4 ~I W 4 . (Courtesy of L. Cheng (1983).)
We shall take a simplified version of (3.25), just to retain the essential feature of the mechanisms involved, i.e.,
[=,-.,
R 9 + 3 3 ’ ] +$.
(5.5)
Let us also assume the equation of state for the gas is barotropic and y = 1 . For gas bubbles, there is evidence to support the isothermal
100
D. Y. Hsieh
assumption for a wide frequency range. Thus we have Pp = Pp"(
2y.
Equations (5.1)-(5.6) form a closed system which governs approximately the general motion of a liquid containing gas bubbles. Using (5.2), we can rewrite (5.1) as: aR_ --(v,*V)R+-V R 3P
at
* V ~ + [ ( V ~ - V ,V) ]. R - - VR 3P
*
[P(v~-v~)].
(5.7)
Using (5.5) and (5.6), (5.4) can be rewritten as
_ avf - -(Vr. V)Vf+ at
[
3pg0R' -2]VR+V{ ( 1 - P ) p r R 4 PfR2
R$+i($)2}.
(5.8)
Let us now consider the case in which the liquid and gas phases move together more or less. That will be realized if bubbles are small and the pressure gradient is small. Since (5.3) can be written as:
therefore to a first approximation, we have 2R2 (vr-v,)=-Vp,= 977
2P& 7 3 77R
(5.10)
If we substitute (5.10) into (5.7), we obtain: _dR - -(Vr . V) R
at
+ -VR
3P
. vf + 2p R'{$V. 377
[SVR]
-(?)I}.
(5.11)
The equations (5.2), (5.8) and (5.11) will be the basic governing equations for the ''locked'' bubbly liquids and will form the basis for our discussion of finite amplitude waves. The terms in the curly bracket in (5.11) are responsible for the dissipation due to mutual friction or mutual slippage, while the terms in the curly brackets in (5.8) will cause dispersion due to bubble oscillation.
On Dynamics of Bubbly Liquids
101
If we neglect the dissipation and dispersion terms, and denote c2 =
20
P& P,;B(1 - P ) R ’
(5.12)
3PfPR’
then the characteristic surfaces of the system $(x, t ) =constant can be determined. We found the equations for 4 of the following characteristic system:
a*
c,:
v* = 0,
-+vfat
(5.13)
and
c,:
(
v*)2 = C2(V*)2.
$+vr.
(5.14)
Across C , , p is continuous, but R and vf can be discontinuous, while R and vf are continuous across C, and p can be discontinuous. For one-dimensional waves, C, and C, reduce to:
c():ddtx = uf ’
(5.15)
-
c,:
dx dt
- -- U f i
c.
(5.16)
For the one-dimensional case, if we multiply (5.8) by * R / 3 p c and then add to (5.11), again neglecting the dissipation and dispersion terms in the curly brackets, we obtain: dR -+ dt
FdVf
aR
(qfc ) -f-1ax 3 p c a t
(Uff
c)
(5.17)
Thus we have dR- R _ -T-, dvf
3pc
on C , ,
(5.18)
which relates the change of R and vf on the wave characteristics. It is illuminating to consider the linear wave in this system of locked bubbly liquid. Let us take p = Po+ p , , R = R0+ R , , and treat p , , R , and
D. Y. Hsieh
102
vf as small quantities; then the linearized equations for (5.2), (5.8) and (5.1 1) become: aP1
-= (1 -Po)V
. Vf,
(5.19)
d2R, ~;vR+ , R,,V Ro at2 '
(5.20)
at
- 3Po
at
and (5.21) where c;
=
pgo -~2 u P o ( 1 - P 0 ) P f 3PoRoPr'
(5.22)
It is clear that (5.20) and (5.21) form an independent system for R, and P I passively from vf. By eliminating vf from (5.20) and (5.4), we obtain of, while (5.19) will yield
in which the terms responsible for dispersion and dissipation are clearly revealed. If we again consider sinusoidal plane waves with a factor ei(kx--wo, we obtain from (5.23): '
(5.24) Let us now consider the case of weakly nonlinear waves. The effects of dissipation and dispersion are also assumed to be small. It is equivalent to treat the radius of bubble R, to be a small parameter. Since the linear wave propagates with the speed co, thus for one-dimensional weakly nonlinear waves, we can take Uf = U f ( 5 , 71, (5.25) where [=x-cot,
T=Et,
(5.26)
and E is a small parameter. R and /3 have the same dependence. (5.25) means that there is a slow modulation of the basic wave form u f ( x - c o t ) .
On Dynamics of Bubbly Liquids
103
A standard multiple-scale expansion scheme can be applied to the onedimensional version of the system (5.2), (5.8) and (5.11) (see Appendix C). After finally setting E = 1, we obtain the following equation governing uf ( C. 16) :
(5.27) is a wave equation of the Burgers-KdV type. This type of equation has also been discussed by Wijngaarden (1972) from a somewhat different perspective. The derivation given in Appendix C is direct and systematic. The mechanisms involved here are dissipation due to mutual slippage and dispersion due to bubble oscillation. If the effect of surface tension is neglected, then the Equation (5.27) can be simplified to:
B. THE CASEW I T H SMALLMUTUALFRICTION When the mutual friction is small, we may neglect the left-hand side in equation (5.9). Let us denote Vd
= Vf
-
(5.29)
vs;
then (5.9) can be written approximately: aVd _ - (Vd . V ) V ,
- (Vf
.V)V,
at
- (Vd
. V)V,-
4
CVMPfR
VR,
(5.30)
while (5.7) is now of the form dR _- - ( v , . V ) R + - VR V ~ ~ ~ + ( V ~ . V ) R - R -V.((PV~). at
3P
(5.31)
3P
The Equations (5.2), (5.8), (5.30) and (5.31) will then be the governing equations for the case when the mutual friction is negligible. This system is dispersive but not dissipative. The term responsible for dispersion is the last term on the right-hand side of (5.8).
D. Y. Hsieh
104
Let us neglect the dispersion term, and consider the one-dimensional case. Then the characteristic equation from (5.2), (5.8), (5.30) and (5.31) is given by: dx + Vf dt
--
-(I -P)
0
0
_-3 P c 2 R
R
3p
R
R 3
--
-
3P
vd
-
dx -+ dt
=0,
vf-
(5.32)
Vd
where c2 is again given by (5.12). Let us denote (5.33)
Thus, if we neglect the surface tension term,
VM
The characteristic equation (5.32) can be expanded to give:
”3 +
c”+ ’ c )
($-vf)2[($-vg)2-(
( 1 -p,c2v: = 0.
(5.34)
Let us denote h ( A )= ( A
- vf)2[(A - vg)’ - (c’+
c”)].
Then for (uf( and lug( somewhat smaller than ( c ~ + c ’ * ) ’ / ~ the , curve h ( A ) can be schematically represented by Figure 5. Therefore, it is clear the Equation (5.34) has four real roots. For vd very small, the characteristics are given to the lowest order of ud by: dx dt
( 1 -p>c’ c2 - v:
+
(5.35)
c12
and (5.36)
If we do not make the approximation that p g / p f > 1 , then we should add to the right-hand side of (5.8) the term [ - ( P / ( l - P ) ) ( p , / p , ) A ] , and add to
On Dynamics of Bubbly Liquids
FIG. 5 .
105
Function h versus A.
the right-hand side of (5.30) the term [(A/ CvM)(p,/pf)], where A=
av at
(v, . V,)V,
Thus if CvM= 0(1), and (p,/p,.) is small, the effect of correction is indeed small. In particular, the characteristics are still all real. Prosperetti and Wijngaarden (1976), using a slightly different set of equations, argued that the critical condition for a nozzle flow can be obtained by setting d x / d t = 0, in (5.34). Their results are then compared with the experimental measurements by Muir and Eichhorn (1963). The discrepancies between theory and experiment are about 10%.
C . INSTABILITYO F SLIP FLOW If we neglect both the mutual friction and the virtual mass effect, we have the case of slip flow. For this case we should restore the gas inertial
106
D. Y. Hsieh
terms in (5.3) and have av,
-+
(v, V)V, '
=
1 --
vp,
(5.37)
PE
at
The Equation (5.4) is also changed to: av,. -+
(Vff V)v,=
1
--vp,..
(5.38)
PI.
dl
We may again employ (5.1) and (5.2). However, to consider the more general case, we shall treat the liquid as compressible and refer back to (3.3) and (3.4):
a - (PP,) at
+ V . (PP,V,)
= 0,
(5.39)
Let us now neglect the surface tension and effect of dispersion due to bubble oscillation; then (5.5) is simplified to Pg
(5.41)
= Pr = P.
Denote cf and c, as sound speeds for liquid and gas respectively; we then have dp=c~dp,=c~dp,.
(5.42)
Then (5.39) and (5.40) can be written as at
and
ld[;
~ + ( v , . v ~ P ] + p f {a-t ~ + V . [ ( l - p ) v ~ , } = o (5.44) .
Let us consider the one-dimensional case. Then the characteristic equation from (5.37), (5.38), (5.43) and (5.44)can be obtained to give:
(5.45)
On Dynamics of Bubbly Liquids
107
Stewart and Wendroff (1984) showed that the Equation (5.45) has complex roots for
The existence of complex characteristics implies that mathematically the Cauchy problems are ill-posed. I n other words, there is inherent instability in the flow system. It should be pointed out that the inclusion of dissipation mechanisms, e.g., viscosity of fluids, would change the system of equations to become parabolic, and thus mathematically well-posed. Furthermore, if the effect of virtual mass is included, as demonstrated in Section V.B, the characteristics are all real. That is also the case for locked bubbly liquid treated in Section V.A. Thus for real fluids, there seem to be no catastrophical instabilities. Still, this instability may be manifested in the form of unusually large disturbances. It has been speculated that there are instabilities like the KelvinHelmholtz instability hidden in this slip flow model (see Prosperetti and Wijngaarden, 1976). Indeed, it can be demonstrated (Hsieh, 1987) that the instability encountered here is exactly the same as the Kelvin-Helmholtz instability for two compressible fluids. It should be remarked that if we perform a linear stability analysis for the steady state with u f = V , and u,= V, for the unperturbed values, the criterion for instability is exactly the same as (5.46) except that uf and u, are changed to V, and V, respectively, and all other quantities are the unperturbed quantities. The Kelvin-Helmholtz stability for two compressible fluids of infinite depths was first treated by Landau (1944). If we modify Landau’s analysis to consider two fluid layers, designated by subscripts 1 and 2, of depths h, and h2 respectively, the dispersion relation can be shown to be (Hsieh, 1987):
where (5.48) and w and k are the frequency and the wave number of the perturbed waves respectively. For the problem with infinite depths, sinh k,h, = cosh k,h,, and (5.47) is reduced to the dispersion relation obtained by Landau (1944).
108
D. Y. Hsieh
On the other hand, for problems with shallow depths, i.e., when khi < 1, then since sinh kihi= kihi and cosh kihi = 1 , the dispersion relation (5.47) becomes, after using (5.48): p , h 2 ( f - V , ) 2 [ l - ; ( ~ -1 w
V2)*]+p2h,(f-
V2)’[l-k(f-
V,)’] = O .
(5.49) It is clear if we identify the subscripts ( 1 , 2 ) as ( g , f ) , h , as p, h2 as ( 1 - p ) , and w l k as d x l d t , then (5.49) is the same as (5.45). The underlying mechanism of Kelvin-Helmholtz instability is the Bernoulli effect, i.e., the higher the fluid flow speed, the lower the fluid pressure. For subsonic channel flows, the smaller the channel depth, the higher the fluid speed. Thus reduction in pressure tends to contract the channel further, and this is why the flow system is unstable by perturbation. On the other hand, for supersonic flows, the fluid speed increases with the channel depth. Therefore, an expansion in channel depth would encounter a reduction of pressure and hence revert back to the unperturbed configuration. That is the basic reason behind the criterion (5.46)-i.e., the instability is arrested when the speed becomes sufficiently supersonic. The manifestation of this type of instability would be the irregularities of the void fraction /3 along the flow direction. Even though dissipation and other mutual interactional effects would diminish the effect of the instability, it is still to be expected for slip flows that there is a tendency for the segregation of liquid and gaseous phases and the formation of periodic bubble concentrations along the flow channel.
VI. Steady Flows Let us again consider the liquid as incompressible, neglecting the selffrictional forces and the Basset force. Then for steady flow problems, the Equations (3.3), (3.4), (3.15) and (3.16) become
O n Dynamics of Bubbly Liquids
109
and ( 1 -P)Pf(Vr * V)vr= -[( 1 - P ) v p r +
P v ~ g l +[ ( I - P)pr+ Ppg1b
- PPr(Vg. V b , ,
(6.4)
where (3.21, (3.33) and (3.46) have been employed. Note that from ( 3 . 2 ) , we have
where the subscript 0 denotes the initial or equilibrium value. We shall take a simplified version of (3.25), i.e., Pg - Pf =
2a R
- 9
and assume the isothermal behavior for the bubble, i.e., P g = Pro( 3
2)
From (6.3) and (6.4), it may be seen that the uniform flow state is possible only when b = 0 and vf = vg. Let us now consider the case that pg/pf< 1. Then (6.3) and (6.4) can be simplified to become
and
where (6.6) and (6.7) have been used to eliminate pf and p g . Equations (6.1),(6.2), (6.8) and (6.9) are the governing equations to solve for P, R, vf, and vg.
A. O N E - D I M E N S I O N STEADY AL FLOW
For
one-dimensional problems, we have vf = ( uf(x),0, 0), ug = ( u , ( x ) , 0 , 0 ) , b = (b, 0 , 0), and all other quantities are functions of x only.
D. Y. Hsieh
110
Thus, Equations (6.1), (6.2), (6.8) and (6.9) become (6.10) (6.1 1 ) (6.12)
and (6.13)
From (6.10) and (6.1 l ) , we obtain (6.14)
With (6.14), the Equations (6.12) and (6.13) form an autonomous system of two first-order equations when b is a constant. A detailed calculation can be readily carried out. In order to grasp more clearly the essential features of the problem, let us consider a simpler case for which b = 0 and u = 0, i.e., when there is no externally applied force field, and the surface tension term is neglected. Then the Equation (6.13) can be immediately integrated to yield (6.15)
while (6.12) can be rewritten as (1 - P o ) U r o
duf -+dx
CVM
2
-
dx
( v f- u;)
977
= -- R 2 ( vr-v,).
(6.16)
Let us denote
(6.17)
(6.18) (6.19)
On Dynamics of Bubbly Liquids
111
where
Jiu,)
(1 - U , ) ( U M - 1 )
=
(6.20)
(uf-Um)(uM-~f)'
Figure 6 shows schematically the variation of J ( uf) with uf. Thus it is clear that we have u,,
(6.21)
5 U f SUM,
When uf = u M , we have r + co and ug+ co. When uf = u,, we have ug+ co, p = 0 , and r = [ ~ , M ~ ( U ~ - u ~ , ) l ~ " ~ . Let us denote j = t ~ ~ , , / z l ~ , , C = x / R , ; then the Equation (6.16) can be rewritten as (6.22) where Re = ;(.rluf,/ R,) is a Reynolds number The critical points of (6.22) are given by (6.23)
uf=jug.
Since J ( u f )is always positive for uf in the range given by (6.21), there is no critical point if j < O , i.e., if ugo and tIfo have opposite signs. Thus the flow will eventually become singular if j < 0. For j > 0, there are two critical points (see Appendix D) for
I
u,
u<
'0
'>
FIG. 6 . Function J versus u,
M'
Uf
D. Y. Hsieh
112
Let these two critical points be u, and u, ,with u , > u,. Using Equations (6.19) and (6.20), the Equation (6.22) can be written as (6.25)
where L( Uf) = u, and
+ C"MUf[
1 -j'J2(
. I[ ' + K ( u ~ ) =1--
1 uf-u,
Uf) K ( U f )
1,
(6.26)
(6.27)
UM-Uf
Figure 7 shows schematically the variation of K ( u f ) with uf. The zeros of L ( u f )are the singular points of the equation (6.25). Since J 2 K is monotonously increasing for ui> uo and ( 1 - j 2 J 2 K ) > 0 at u f = u o , there is one and only one zero of L ( u f )for uf> uo unless CVM=O. Let us call this zero u,. u, is the only singular point if J 2 K is a monotonously increasing function of uf. It can be shown (Appendix D) that a sufficient condition for J 2 K to be monotonously increasing is (UM -U,J<
10.
(6.28)
The requirement to insure that there is only one singular point in reality is much less restrictive than that given in (6.28).
'I '
/i
I
I I
I
--1---
I I I I uM
9 I
FIG.7.
Function K versus u,.
"f
On Dynamics of Bubbly Liquids
113
At the critical points, J ( u , ) is increasing at u f = u,, and decreasing at L(u,) > 0 (see Appendix D), u< is always a stable critical point. For the case CvM= 0, the situation is particularly simple. There is no singular point; since L( ur) = u,,, > 0, the critical point u , is stable and the critical point u , is unstable. The situation is analogous to the onedimensional flow of a single component fluid with dissipation. However, with CVM# 0, a singular point appears, and the stability of the critical points u, depends on the relative location of u , and u,. If u, < u,, then u , is a stable critical point. If u, > u , then u, is an unstable critical point. The situation may be summarized as follows: For u, < u, the initial flow such that 1 < u, will result in a flow u f + u as [ + 00, while the initial flow with 1 > u, will result in a flow Uy+ u as .$+a. For u, > u,, the initial flow 1 < u will result in a flow u f + u< as [+ 00, while the initial flow 1 > u., will lead to u, = u, at some [. At the point where u F = u , , a discontinuity in flow appears, or the flow chokes. u, = u,. Since
,
~
B. ONE-DIMENSIONAL FLOWWITH
DISCONTINUITY
When b = 0, one-dimensional uniform states can be connected by a discontinuity. The uniform flow states are possible only if v f = u g , as we have remarked before. With b = 0, and vl-= v g , the one-dimensional version of (6.1)-(6.4) can be rewritten as
d(p ) = o , dx R' "
(6.29) (6.30) (6.31)
and prur
dvl-+dx
dPf = 0. dx
(6.32)
Thus, with the aid of (6.5)-(6.7), we obtain (6.33)
114
D. Y. Hsieh [( 1 - p )u,-l = 0,
(6.34) (6.35)
(6.36) where [ ] denotes now the difference between the states separated by the discontinuity. Let the state to the left of discontinuity be denoted by the subscript 0 and the state to the right by the subscript 1. Since u f = vg, we thus obtain (6.37)
(6.39)
t [ ~ -f vz0] , = &!! Pf
[(2)3 -
I]
-L( 2a --L). 1 Rl Ro
(6.40)
From (6.39) and (6.40), we obtain a relation R , = R , ( R , ) . In fact, if the surface tension can be neglected, Le., if a = 0, we have (6.41) where r, is the root of the equation (6.42) Since p J p g o > 1 in general, we have r, < 1. Now we have six unknowns, i.e. P I , Po, u , , , ul0, R , and R,, for the four Equations (6.37)-(6.40). Therefore we cannot arbitrarily prescribe Po, ufo and Ro independently. This is not satisfactory from physical considerations. To overcome this difficulty, let us neglect the mutual frictional term to start with, thus relaxing the condition u g = ul for the uniform flow state. Then, if we neglect the body force b, and make the approximation p g / p lQ 1, the governing equations will be (6.1), (6.2), (6.8) and (6.9) with b = 0 and 77 = 0.
On Dynamics
nf Bubbly Liquids
115
Thus, instead of the Equations (6.33)-(6.36), the equations connecting two uniform flow states across a discontinuity will be (6.43)
(6.45)
(6.46) where [ ] denotes again the difference between states separated by the discontinuity. Consider the case that w = O . For weak discontinuity, it can be readily shown that
Mi=
1+ Cv,
= Mf,
(6.47)
where Mo is given by (6.17). In general, in one of the uniform states we have o f / ( p g O / p f< ) MT; while in the other, u f / ( p g o / p r> ) Mf. We shall call these uniform states subcritical and supercritical states respectively. The discussion in the last section shows that the subcritical state is in general stable, while the supercritical state is unstable. Therefore it is permissible only for the supercritical state to transit through the discontinuity, or the shock, to reach the subcritical state. Keeping in mind that in reality a uniform state is possible only for ug = u , , it is desirable to set ugo= ufo for the initial supercritical state. After the transition, the mutual friction will bring ( u s , - vg,) to zero eventually.
VII. Dynamics of a Liquid Containing Vapor Bubbles
A. THE D Y N A M I C ‘ A EQUATIONS L The governing equations for a liquid containing one species of gas bubbles, which are summarized in Section IILF, need to be modified when we are dealing with the dynamics of a liquid containing vapor bubbles. To illustrate the general approach and catch the essential feature of the
D. Y. Hsieh
116
modification, let us consider the case in which the bubbles are all pure vapor bubbles and there is only one species of vapor bubbles. By one species, we mean that macroscopically the bubbles are characterized by a single radius R ( x , t). But the total mass content of each bubble m is no longer a constant as in the case of a gas bubble. m ( x , t ) is now a variable also. However, even though m is not a constant, we shall again retain the conservation of the number of bubbles (3.5). Thus we take the view that the vapor bubbles grow out of definite nuclei and never disappear completely, albeit the nuclei may be infinitesimally small in terms of macroscopic scales. Since n = 3 P / 4 v R 3 , the conservation of number of bubbles can be expressed as
I"["
at
R'
I$[
+v.
=o .
(7.1)
Because of the evaporation and condensation processes going on all the time in a liquid containing vapor bubbles, the separate conservation of mass in liquid phase and the vapor phase is no longer valid. Instead of (3.3) and (3.4), we now have (7.2)
and
a
--[(I -P)prl+V at
. [(I - P ) P P r l =
-s,,
(7.3)
where S , is the rate of evaporation of the fluid mass per unit volume per unit time. Now, there is latent heat associated with the process of phase transformation. Let L be the latent heat of evaporation per unit mass. Thus there is a heat source term (-LS,,,) to be added in the energy equations. This heat source term usually will dominate over the heat generated by dissipation. Thus the terms Qgs and can be neglected in Equations (3.60) and (3.61). By adding together (3.60) and (3.61), we obtain
of>
=V
. [( 1 -p)KfV Tr+ PK,V T,] - LS,.
(7.4)
Now the pressure of the vapor is also related to its temperature by the process of phase transformation. We shall assume a local phase equilibrium
On Dynamics of Bubbly Liquids
117
relation, and thus have
Together with the unchanged equations (3.15), (3.16), (3.17), (3.18) and (3.25), we have a complete system of governing dynamical equations for a liquid containing vapor bubbles. The other equations are listed again in the following
p g- p f = P { R } -
( v,
-
vf)'.
(7.10)
B. THE SOURCETERM To construct a model for the determination of the source term, we shall make use of the knowledge of how a single spherical vapor bubble behaves in a superheated liquid (Plesset and Zwick, 1955; Hsieh, 1965). Although it is not realistic to expect precision from this model, the following construction has the appeal of intuitive understanding. For each bubble with temperature T,, the total heat supplied from the liquid per unit time is 47rR2Kf(T f - T,)/l, where 1 is a characteristic thermal diffusion length. This amount of heat will be spent for the evaporation, and thus the total amount of mass evaporated per unit time for each bubble is
(7.11)
D. Y. Hsieh
118
Hence the source term S , is (7.12)
Now the increased mass of vapor by evaporation will cause the growth of the vapor bubble. Thus we have (7.13)
where pg it taken to be constant, since it can be justified that in this thermally driven process the temperature T, will remain essentially constant as a function of the vapor pressure corresponding to the ambient pressure. Thus pg is also constant. We have also used t’ to emphasize that we are dealing with a “microscopic” time scale. Combining (7.13) and (7.11) we obtain
dR dt’
1-=-
K,( Tf-- T,). p,L
(7.14)
Now the thermal diffusion length is commonly defined as [Drt‘]’/2, where D f i s the thermal diffusivity of the liquid, i.e., D f = Kf/pfCf.If we substitute [Drt’]”2for 1 in (7.14), noting that ( T f - T,) is also constant with respect to the “microscopic” time t’, we obtain (7.15)
Thus (7.16)
We should note that (7.15) is actually only valid for the asymptotic phase of the growth of a vapor bubble in a superheated liquid. It is not valid for the case Tr< T,. However the expression 1 in (7.16) appears to have a more general validity; therefore we shall use it for both T,->T, and T,-< Tg.Since I is always positive, therefore we put an absolute value sign in ( T f - T,). Now for I Tr- T,l Q 1, it appears 1 will become infinite. A more reasonable
On Dynamics of Bubbly Liquids
119
upper bound for 1 seems to be R. Therefore we shall take (7.17)
l=R, and
(7.18) Substituting (7.17) and (7.18) into (7.12), we obtain
and
C . WAVESI N
A
LIQUID C O N T A I N I NLOCKED G VAPORBUBBLES
Let us consider the case of a liquid containing locked vapor bubbles. We shall follow the approach adopted in Section V.A, i.e., taking pf as constant, p g / p f < 1, and ignoring the self-frictional forces, the body force, and the Basset force. The summation of Equations (7.2) and (7.3),taking into account p g / p f < 1, leads t o
a’- = V . [ ( l at
P )bI.
(7.21)
Using (7.21), the Equation (7.1) can be rewritten as dR
-+
(vg . V ) R
R
= -V
3P
at
. [ ( 1 - P ) v ~ Pv,]. +
(7.22)
From (7.5) and (7.8), we may express both Tg and pg in terms of p g . Let us thus write these relations as
Tg = TAP,),
(7.23)
=PJPg).
(7.24)
Pg
D. Y. Hsieh
120
Equations (7.23) and (7.24) mean that the temperature T, is the boiling temperature corresponding to the equilibrium vapor pressure which is now pg, and p, is the corresponding vapor density. Denote the sound speed of vapor cg by 1
2
c g=
(7.25)
(dP”/ dP,).
Then the equation (7.2), using (7.21), can be rewritten as dp”+(v,.V)p,
1
+p,v * [ ( l - p ) v f + p v , l = s , .
(7.26)
Similar to the development in Section V.A, the equations (7.6) and (7.7) become (5.3) and (5.4), which we copy again here av V ) v f - l - ( v g . V)v,],
977 Vp,=-y(vf-v,)+prCvM 2R
at
(7.27)
and avf -+(vf.v)v,=-at
(7.28) Pf
Similarly, the Equation (7.10) becomes (5.5): Pg-Pf= Pf[ RT+Z(dr)*] d2R 3 dR
+$.
(7.29)
Since p g / p f < 1, the Equation (7.4) can also be simplified to become
(7.30) where (7.31) Now we make the assumption that the bubbles are essentially locked with the liquid phase, i.e., (v,--v,) is small. Then, we may replace (7.27) by the approximate equation vr-vg=-
2 R2 -Vp, 9 7
(7.32)
On Dynamics of Bubbly Liquids
121
With the aid of (7.32), Equations (7.22) and (7.26) can be rewritten as
dR -+
(Vf
*
V ) R =-
R
V
3P
3P
at
3
R R*(Vp,) . ( V R )-- V . (PR'Vp,) ,
*
(7.33)
"2+(Vf.1
9",14%'
V)p, +p,(V . v , . ) = -
c,
I
T ( V p , ) 2 + p p , V * [PR2Vp,] + S m . (7.34)
The equation (7.28), using (7.29), can be rewritten as av,. -+ at
(Vf
. V)vr=
-
1 (1 -P )pr.
If S, in (7.34) is given, then Equations (7.21), (7.33), (7.34) and (7.35) form a closed system. However S,,, as given by (7.19) and (7.20), is a function of (P, R, p , , T,.),and T, is to be determined from (7.30). Although this system of equations is quite complex, we may, following the insight gain in Section V.A, identify the terms associated with the factor ( 2 / 9 ~ )in Equations (7.33) and (7.34) to be those responsible for dissipation due to mutual slippage, and the last terms on the right-hand side of (7.35) to be those responsible for dispersion due to bubble oscillations. The term S , will also contribute to dissipation due to the evaporation and condensation effects. To illustrate these features, let us consider the propagation of small amplitude waves. The equilibrium state variables will be designated by the subscript 0. At equilibrium, we have
The liquid in the equilibrium state is just boiling if u = 0, and is superheated if u > 0 . Let us now take P = P o + p , , R = R o + R , , T f = T , + T f , ,p s = p g O + p g l and , treat P I , R , , Tfl, pgI and vf to be small quantities. Then the linearized equations for (7.21), (7.33), (7.34), (7.35) and (7.30) become (7.37)
122 (7.38)
(7.39)
(7.40)
(7.41)
(7.42)
(7.43)
(7.44) Now the term associated with S,,, in (7.43) has to be small in order that the wave can be propagating. Let us consider first the case that S,,,, = 0. Then prl = -(3pgocio/ R o ) R , ,and (7.44) becomes
where
(7.46) is practically the same as (5.22). The first term on the right-hand side of (7.45) will reveal the resonance effect, and the second term represents the damping due to the mutual slippage.
On Dynamics qf' Bubbly Liquids
123
To estimate the damping caused by the term Snllr let us consider the sinusoidal plane waves with a factor e''k'x-w'). Therefore (7.41) becomes
Thus we have (7.47) where
w
M(w,k )=
1-Po
(7.48)
l+iand D,-= K f / p , C f is the thermal diffusivity in the liquid. Using (7.43), we obtain (7.49)
On the other hand, (7.44) can be written now as
(7.50) Combining (7.49) and (7.50), using (7.46), we obtain the dispersion relation (7.51) where
D. Y. Hsieh
124
For small k, i.e., when D f k ’ / w < l , M ( w , k ) - 1. Thus there is positive damping due to the evaporation and condensation effect. When the damping is small, and if kR, > 1, then we have
The relative importance of the effect of evaporation and condensation versus the effect of mutual slippage, when M ( w , k ) is taken to be unity, is given by the ratio ( 2 7 7 ~KfT:,/2 ’ R:Lp,,,w’).
Appendix A From (2.16), we see that -
1
V
( P a p ) *d”x =
-(5, Pupn
d’x).
The boundary of the region occupied by phase a is A and those parts of aV where there is phase a ; thus after applying Gauss theorem, we obtain
On Dynamics of Bubbly Liquids
125
This quantity is zero if the average phasic pressure pa is the same as average of the interfacial phasic pressure.
Appendix B Using ( 4 . 4 ) and ( 4 . 2 0 ) ,( 4 . 5 ) and ( 4 . 6 ) become respectively: k . vg= w ( P ’ - 3 R ’ ) ,
(B.1) (B.2)
Thus from ( 4 . 1 4 ) ,we obtain k
= w [ -Do
[
+ iwCvMPopro]0 ” -(3+B)R1]. 1-P
1
(B.3)
Let Equation ( 4 . 7 ) and ( 4 . 8 ) make a dot product with k. Then we obtain: [ iw - ($vg+ V,) k 2 ] w( P ‘ - 3 R ‘ )+ i3k2ciR’
+-
w
(B.4)
PoPgo
[
[ i w - ( $ v f + iir)k2]w B R ’ -
-
w
(1 -Po)Pro
( f;)PI] --
- ik2cfBR’
[A-+ 1
[-Do+ iwCvMPopfo]
( 3 B ) R ’ = 0.
(B.5)
Using (4.16)-(4.18), ( B . 4 ) and (B.5) can be rewritten as {iw[l+
(1 -Po)&,
[
+ { - iw2 3 +&
(3
+B ) ]
El
+w[(l -Po)w,(3+B)+3ag]+3ik2~i
(B.6)
D. Y Hsieh
126
From ( B . 6 ) and (B.7), the characteristic relation, after some rearrangement, can be written as
{ -[
(:+A)] w3
[
- i ( 1 -Po+ & l p o ) w D +a,+ a,+
CVM
CVM
(B.8)
If we neglect simplified to:
Lyf
and a,, and also the term O ( E ~Equation ), ( B . 8 ) can be
On Dynamics of Bubbly Liquids
127
(B.13)
D. Y. Hsieh
128
With w , and w, defined by (4.32), we can rewrite (B.13) as
(B.14) CVM
Thus, with ml and m2 defined by (4.33) and (4.34), we obtain
+ iil
- p O ) w D ]( w ;
k 2= C VM
m , + iw,m,
3
-0 2 ) 9
(B.15)
or
Appendix C Consider the one-dimensional version of the Equations (5.2), (5.8) and (5.11), with c i defined by (5.22). Let us introduce the new independent variables: lf=x-c,t,
where
E
ic.1)
T'Et,
is a small parameter. Let us consider the following expansion: P=Po+EP(l)+E2p(2)+.
I-- - E v ( y l ) + & Z U ; Z ) + .
R
= Ro(l+ & I ) +
..
.. E2r"'+.
(C.2) (C.3)
. .),
(C.4)
where Ro and Po are constant. We shall also assume that
Ri= O ( E ) .
(C.5)
On Dynamics of Bubbly Liquids
129
Thus the expression c2 as given by (5.12) can be expressed as c 2 = c;7,+& a ” ’ + o ( E * ) ,
(C.6)
where a “’ =
Pgo
P o ( 1- po)2Pf P “ ’ - [Po(;’iolPf
3*-
(c.7)
+I).
3PoRoPf
Substituting (C.l)-(C.7)into (5.2), (5.8) and (5.11), we obtain the following equation to the orders o f
O( E ) :
and (C.10) 0(F2):
(C.12)
and
ar”’ -c
’ at
1 3Po
sup'
a6
ar”’
arCl)
a7
at
-
p(1)auiI) 2P,o -+ - R : 3p: a t 977
a2r(l’
at’ ’ (C.13)
where we have set F = 1 . If we are considering waves of finite extent, thus [+-a. Then we obtain (C.S)-(C.lO). v : ’ ) = -3/3,,cor“),
v i l )=
r ‘ ’ )= p‘” = 0 as
(C.14)
D. Y. Hsieh
130
(C.15) Now from (C.12) a n d (C.13), it is clear that in order that avj”/ag a n d ar‘”/a[ have non-trivial solutions, the right-hand side of (C.12) should be equal to the right-hand side o f (C.13) multiplied by the factor (3P,c,). Using (C.7), (C.14) a n d (C.15), we thus obtain, after some rearrangement: at$’) -+ dT
[ +(’ 1
4u
Pe”
2 4 Po(1 - Po)Pr 9P%)Pl (C.16)
Appendix D For j > 0, the critical points are given by
jJ(u f ) = 1. From (6.20), we see that
Thus J(u,) has a minimum at
Now
Thus ( D . l ) has roots if a n d only if
Denote
[
K ( u , ) = 1--
1 MI- u,
+I;
1
u&f - Uf
On Dynamics of Bubbly Liquids
131
then we obtain from (6.19) and (6.20)
Thus
K ( u,) is a monotonously increasing function for ui in the range [ u,,,, u , ~ ~ ] . We have also K ( u , , ) = 1 , and let us define u,, such that K ( u , ) = O . Then since J 2 is monotonously increasing for u, > ug and monotonously decreasing for u,< u,, we have J'K monotonously increasing for uf> u,) and u , < u,. Now d -(J2K)= dU,-
where we have made use of (D.2) and (D.5). For u , < u , < u , , w e h a v e 2 K ( K - I ) > - : . Now
Thus for u, < u,< u,,:
Therefore a sufficient condition for ( d / d u , ) ( J ' K )> 0 is (UL, -
u,,,)? < 10.
(D.7)
Now j J = 1 at u,= u, and u,-= uhl.Since K ( u c ) < 1 , it is clear that L( u- ) > 0. On the other hand, depending on the value of u, and C V ML, ( u ,) may be either positive or negative. Now since L(u, ) = 0, thus L(u,) S 0, for u f S u,. Thus L( u,) S 0, for u, S u,. From (6.26), we can see that if u,, > u , then L(u,-)> 0 for u f < u o , since j 2 J 2 K< 1 for u,< u". Let us explore this property for the particular case of j = 1, i.e., when the initial flow u,= 1 is also a critical point. Let u, = 1 ; then a straightforward calculation shows that K ( l )=
1 - umuM - um)(uM
-
l)
132
D. Y. Hsieh
Thus if (1 - u m u M )< 0, then K (u,) < 0, or u, > u,. Hence, for this particular case, or for j I1, another sufficient condition for existence of only one singular point in the flow is UmUM
> 1.
(D.8)
References Anderson, P. S., Astrop, P., and Rothmann, 0. (1976). Characteristics of a one-dimensional two-fluid model or two-phase flow. A study of added mass effects. Report N O R H A V-D-OJ 7, RISO, Denmark. Bedford, A,, and Drumheller, D. S. (1978). A variational theory of immiscible mixtures. Arch. Rat. Mech. Analys. 68, 37-51. Barclay, F. J., Ledwidge, T. J., and Cornfield, G . C. (1969). Some experiments on sonic velocity in two-phase one-component mixtures and some thoughts on the nature of two-phase critical flow. Symposium on Fluid Mechanics and Measurements in Two-Phase Systems, Proc. I Mech. 184, 3C. Boure, J. A. (1978a, b, c). Constitutive equations for two-phase flows, Critical two-phase flows, and Oscillatory two-phase flows. In “Two-Phase Flows and Heat Transfer with Application to Nuclear Reactor Design Problems” ( J . J . Ginoux, ed.), pp. 157-239. Hemisphere Pub. Corp., Washington. Bowen, R. M. (1976). Theory of mixtures. I n “Continuum Physics” Vol. 111 ( A . C. Eringen, ed.). Academic Press. Caflisch, R. E., Miksis, M. J . , Papanicolaou, G. C., and Ting, L. (1985). Effective equations for wave propagation in bubbly liquids. J. Fluid Mech. 153,259-273. Cheng, L. (1983). An analysis o f wave dispersion, sonic velocity and critical flow in two-phase mixtures. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York. Delhaye, J. M. (1976a, b, c, d). Local instantaneous equations, Instantaneous space-averaged equations, Local time-averaged equations, Space/time and time/space-averaged equations. In “Two-Phase Flows and Heat Transfer” Vol. 1. (S. Kakac and F. Mayinger, eds.), pp. 59-1 14. Hemisphere Pub. Corp., Washington. Drew, D. A. (1971). Averaged field equations for two-phase media. Sfudies in Appl. Math. L, 133-166. Drew, D. A,, and Segal, L. A. (1971). Averaged equations for two-phase flows. Studies in Appl. Math. L, 205-231. Drew, D. A. (1976). Two-phase flows: Constitutive equations for lift and Brownian motion and some basic flows. Arch. Rat. Mech. Analys. 62, 149-164. Drew, D., Cheng, L., and Lahey, Jr., R. T. (1979). The analysis of virtual mass effects in two-phase flow. I n f . J. Mulriphase Flow 5 , 233-242. Drumheller, D. S., and Bedford, A. (1979). A theory of bubbly liquids. J. Acousf. SOC.A m . 66, 197-208. Drumheller, D. S., and Bedford, A. (1980). A theory of liquids with vapor bubbles. J. Acoust. SOC.Am. 67, 186-200. Hall, P. (1971). The propagation of pressure waves and critical flow in two-phase mixtures. Ph.D. Thesis, Herriot-Watt University, Edinburgh, G.B. Hench, J . E., and Johnston, J . P. (1972). Two-dimensional diffuser performance with subsonic, two-phase, air-water flow. Trans. A S M E 94D, 105-121.
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Hsieh, D. Y. (1965). Some analytical aspects of bubble dynamics. Trans. A S M E 87D, 991-1005. Hsieh, D. Y. (1972). On the dynamics of nonspherical bubbles. Trans. A S M E 94D, 655-665. Hsieh, D. Y. (1987). Kelvin-Helmholtz stability and two-phase flow. To be published. Ishii, M. (1979. “Thermo-Fluid Dynamic Theory of Two-Phase Flow.” Eyrolles, Paris. Kenyon, D. E. (1976a). Thermostatics of solid-fluid mixtures. Arch. Rat. Mech. Analys. 62, 117-130. Kenyon, D. E. (1976b). The theory of an incompressible solid-fluid mixture. Arch. Rat. Mech. Analys. 62, 131-148. Landau, L. D. (1944). Stability of tangential discontinuities in compressible fluid. Camp. Rend. ( D o k l a d y ) Acad. Sci. URSS, 44, 139-141. Landau, L. D., and Lifshitz, E. M. (1959). “Fluid Mechanics.” Pergamon, Oxford. Muir, J. F., and Eichhorn, R. (1963). Compressible flow of an air-water mixture through a vertical two-dimensional, converging-diverging nozzle. Proc. 1963 Heat Transfer and Fluid Mechanics Institute, Stanford University, Stanford. Plesset, M. S., and Mitchell, T. P. (1954). On the stability of spherical shape of a vapor cavity in a liquid. Q. Appl. Math. 13, 419-430. Plesset, M. S., and Prosperetti, A. (1977). Bubble dynamics and cavitation. Ann. Reu. Fluid Mech. 9,145-185. Prosperetti, A,, and Wijngaarden, L. Van (1976). On the characteristics of the equations of motion for a bubbly flow and the related problem of critical flow. J. Eng. Math. 10,153- 162. Silberman, E. (1957). Sound velocity and attenuation in bubbly mixture measured in standing wave tubes. J. Acoust. Sac. Am. 29, 925-933. Spitzer, Jr., L. (1943). Acoustic properties of gas bubbles in a liquid. OSRD1705, NDRC 6.1-sr20-918. Stewart, H. B., and Wendroff, B. (1984). Two-phase flow: models and methods. J. Camp?. Phys. 56, 363-409. Varaden, V. K., Varadan, V. V., and Ma, Y. (1985). A propagator model for scattering of acoustic waves by bubbles in water. J. Acoust. Sac. Am. 78, 1879-1881. Wallis, G . B. (1969). “One-Dimensional Two-Phase Flow.” McCraw-Hill, New York. Wijngaarden, L. Van (1972). One-dimensional flow of liquids containing small gas bubbles. Ann. Rev. Fluid Mech. 4, 369-396. Wijngaarden, L. Van (1982). Bubble interactions in liquid/gas flows. Appl. Sci. Res. 38,33 1-339. Zuber, N. (1964). On the dispersed two-phase flow in the laminar flow regime. Che. Eng. Sci. 19, 897-917. Zwick, S. A., and Plesset, M. S. (1955). On the dynamics of small vapor bubbles in liquids. J. Math. Phvs. 33, 208-330.
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ADVANCES I N A P P L I E D MECHANICS, VOLUME
26
Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation KLAUS KIRCHGASSNER~ Marh. Instifur A Uniuersifat Stuttgarf Sfuftgarf,Federal Republic of Germany
................
I. Introduction
135
11. The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Transformations and Symmetry . . . . . . . IV. The Method ...................................................
142
V. Reduction and Results.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
A. Capillary-Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Capillary-Gravity Waves under Periodic Forcing ......................... C. Capillary-Gravity Waves under Local Forcing ........................... D. Forced Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152 159 163 168
VI. The Mathematics . . . . . . . . . . . . . . . . . . . . .
144 147
172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
I. Introduction The behavior of steady nonlinear water waves on the surface of an inviscid heavy fluid layer has received much attention during the past century, both from the mathematical and from the physical side. Despite a persistent scientific effort involving some of the best names of the time, a number of fundamental questions have not been completely answered. For instance, do solitary waves exist in the presence of surface tension, or, how does a fluid react to a localized pressure distribution moving over its surface with constant speed? Similarly, what are the possible motions under a moving pressure distribution which is spatially periodic? These problems are related to determining the flow of an inviscid fluid through a channel with a bump in its bottom, or its periodic dislocation. How far upstream, one may ask, t Research partially supported by the Deutsche Forschungsgemeinschaft under Ki 131/3-1. 135 Copyright @> 1Y8X Academic Pres Inc All rights of reproduction In any form rererved ISBN 0.12 OO2OZh 2
136
Klaus Kirchgassner
can the bump be felt, and how does this distance depend on the size of the obstacle? In this survey, recent results answering the time-independent aspects are described for motions of moderate amplitude. We do this from a mathematical point of view by exhibiting the totality of solutions of the underlying equations of motion, the Euler equations. Only two-dimensional flows which are steady, i.e., time-independent in a moving frame, are considered. The motion is then described by a quasilinear system of elliptic equations in an infinite cylindrical domain and, if the unbounded variable is interpreted as a dynamical variable, then the bounded solutions, obeying the appropriate boundary conditions, are orbits in an infinite-dimensional phase space. This space consists of functions living on the cross section of the cylinder. Solitary waves appear as homoclinic orbits and cnoidal waves as closed orbits. Restricting the flow to moderate amplitudes, the orbits live on a lowdimensional surface-the center-manifold-and are described by low-order ordinary differential equations. These are the analogues of the Landau equations and they contain all moderate amplitude solutions. The limiting case of a layer of infinite depth is excluded as well as instationary motions of any kind, e.g., the difficult stability question. Thus, the scope of this paper is rather narrow, but we treat the problems with rigor. Our main intention is to present a new mathematical method by analyzing a few examples yet unsolved in the literature, and to show how this method could be used for a wider range of problems. In its long history, the analysis of nonlinear surface waves has been promoted by scientists of various backgrounds, and a vast literature is available for the unforced case, i.e., when the pressure at the surface is constant. The reader who is interested in tracing back the main ideas and results is referred to the classical monographs of Stoker (1957) and Whitham (1974), and to the mathematical book of Zeidler (1971a, 1977). We also mention the article by Yuen and Lake (1984) as one of the more recent excellent surveys on the whole spectrum of this field. The influence of an external pressure distribution has already been analyzed in a linearized model in Stoker (1957) and, for an analogous situation-periodically deformed bottom of the fluid layer-in Zeidler (1971a) for some nonlinear cases (cf. also the literature given there). Among the many contributions which have appeared in the meantime, the brilliant discussion and justification of Stoker’s conjecture on the shape of surface waves of extreme form has to be mentioned. Here, Amick, Fraenkel and Toland (1981, 1982) seriously attack global aspects of a
Nonlinearly Resonant Surface Waves
137
qualitative theory, and Amick et al. (1984) complement these results by describing the extreme form of internal waves. As for the stability theorywhich we otherwise neglect here-we mention some important recent developments based on Hamiltonian formulations of the problem by Zakharov (1968) and Miles (1977), concerning the behavior of critical eigenvalues leaving the imaginary axis (cf. MacKay and Saffman, 1985). Attempts to describe the nonlinear interaction between external pressure waves and surface waves have been undertaken by Wu and Wu (1982), Akylas (1984), and by Grimshaw et al. (1985). As anticipated by the linear analysis, the resonant case, when the pressure speed coincides with the critical wave speed, becomes of particular importance and difficulty. All these authors try to calculate the far field behavior in space and time by treating model equations, e.g., forced KdV equations. Let us finally mention the mathematical results on cnoidal capillarygravity waves by Ter-Krikorov (1963), Beckert (1963), Zeidler (1971b), Beale (1979), and Jones and Toland (1986). Here we analyze this nonlinearly resonant interaction for the Euler equation in full generality with the sole assumption of moderate wave amplitudes (order of the mean depth of the fluid layer). We describe the space-like modulations for a periodic pressure wave as well as one having compact support, i.e., vanishing identically outside some bounded interval. The method with which we achieve this goal is based on ideas from the theory of dynamical systems as applied to elliptic equations. The idea to use “dynamical” arguments for solving nonlinear elliptic problems in a strip goes back in the linear case to Burak (1972) and was developed by Scheurle and the author (1981, 1982a, b). Later, Amick and the author (1987) showed how to incorporate free boundary problems, when proving that solitary surface waves exist in the presence of not-too-small surface tension, i.e., when the Bond number is greater than 1/3. The extension of the method to nonautonomous semilinear systems and finally to quasilinear systems is due to Mielke (1986a, 1 9 8 6 ~ )He . also proposed some of the transformations which we use throughout this paper. The method itself is a nonlinear separation of variables by which one can reduce the elliptic system to an ordinary differential system of minimal order, if bounded solutions of restricted amplitude are considered. There are recent global versions of this method for nonlinear parabolic systems known as “inertial manifolds” due to to Foias, Sell and TCmam (1986), but the necessary increase in the distance of consecutive eigenvalues of the linearized operator does not occur for the problems under consideration here.
138
Klaus Kirchgassner
The plan in this article is to present this reduction method for nonlinear surface waves in two dimensions. They are characterized by three parameters: A = g h / c 2 ,the inverse square of the Froude number, b = T/pgh', the Bond number, and by E , the dimensionless amplitude of the external pressure. T denotes the coefficient of surface tension, h the mean depth of the layer, p the density and g the gravity. For technical reasons we treat irrotational motions only, although the general case could be included without any principal difficulty, thus incorporating work of Beyer (1971) and Zeidler (1973). Capillary-gravity waves are described for h > 0. We have to distinguish between b < $and b > i. In the latter case, our reduction leads, in lowest order in p = A - 1, to the second order equation
where - a , ( x ) , x E R is the form of the free surface and & p 0 ( x )the external pressure distribution. For the complete equation see (5.4). Similarly one obtains for b = 0, the case of pure gravity waves, in lowest order of p = 1 -A,
a:
= 3 pa,
+ ;a; + 3 &PO.
For the complete equations see (5.36). The case b < 4 leads to a fourth-order system, which we have given explicitly at the end of Section V.A. However, its analysis, even for the unforced case E = 0, is essentially open. In both of the above equations, a , is of order O ( I/*./). The lowest-order approximation-i.e., all terms of order Ipl-of the velocity field v is given by u , = a,, v2 = 0. To determine approximations of higher order, one has to incorporate two almost identical transformations, first proposed by Benjamin (1966) and Mielke (1987c), and described here in Chapter 111. Up to this point, po has only to be bounded and smooth. The special assumptions of the local (compact support) and also of the global (periodic) nature of p o are made only for the discussion of (1.1) and (1.2). We could incorporate, for example, quasi-periodic forcing and apply recent results by Scheurle (1986). The above equations contain all bounded solutions of moderate amplitude of the original problem. For E = 0 they are reversible, i.e., invariant under x -+ -x. The breaking of reversibility by p o generates a nonstandard bifurcation. For a thorough discussion of the role of symmetry in this realm, see Zufiria (1986). In the local case, which has been analyzed for a related problem by Mielke (1986b), the limiting equation (cf. 5.26)), from which all solutions
Nonlinear!,, Resonant Surface Waues
139
can be constructed by perturbation, leads to the intersection of two homoclinic orbits being shifted along the a:,-axis in the two-dimensional phase space, The complete set of solutions can be depicted from Figure 7 in Section V.C, for p = A - 1 > 0, 6 > A similar analysis could be d o n e for p < 0 a n d for 6 = 0. Observe that the solutions shown in Figure 7 have a j u m p in the first derivative being proportional to the integral of p o . We also give a perturbation argument for higher-order approximations in the class of continuous a n d piecewise twice continuously differentiable functions with a possible j u m p at x = 0. However, for p f 0, these are smooth classical solutions of the full equations (5.26) (for previous work, cf. Keady, 1971). In the global case, when p o is periodic, the appearance of horseshoes has to be expected whenever a homoclinic orbit exists. Then solitary waves are broken into a spacelike chaos. Similar phenomena can occur for heteroclinic solutions in the form of bores. They have been analyzed for a two-phase-flow in a channel by Mielke (1987~).For the homoclinic case, see Turner (1981). The existence of a transverse homoclinic point, which is the cause for the chaotic behavior here, is implied by the Melnikov condition, a certain scalar condition in E a n d the free phase p of t h e homoclinic orbit. To show its validity here, we have to use the period of p o as a n extra parameter. The final conclusions are contained in Propositions 5.3 and 5.4. We also mention the existence of subharmonic bifurcations near the homoclinic orbit. The analysis of the nonlinear resonant reaction of cnoidal waves t o p o has been completely suppressed. It leads to a bifurcation of tori in a set of positive measure in the phase space. The flow on these tori is quasiperiodic a n d requires for its existence the application of K A M theory (cf. Moser, 1973; Scheurle, 1987). The appearance of chaotic phenomena in nonlinear wave motion has been predicted by several authors, e.g., Pumir et al. (1983) a n d Abarbanel (1983), who treat model equations for different physical systems. Previous work of the author on this subject can be found in Kirchgassner (1984,1985). In order to make this survey accessible to a wider readership, 1 have minimized the technicalities wherever the procedure is formally explained. Sometimes I have included sketches of proofs which seemed necessary for a deeper understanding of the subsequent material; there the mathematics is a bit more demanding. The real mathematical justifications are summarized in Section VI without compromise. Much of the material I have presented here in comprised form is d u e to A. Mielke, to whom I a m very indebted. My deep gratitude extends to J. K. Hale, to whom I owe my knowledge in the field of dynamical systems, a n d
:.
Klaus Kirchgassner
140
to C. Amick and R. Turner for introducing me to the field of nonlinear surface waves. I express my sincere thanks also to T. Y. Wu for his persistent encouragement and kind patience. Major parts of this manuscript were written when I was visiting the Department of Mathematics of the University of Utah, where I profited from the inspiring atmosphere and many helpful discussions with Frank Hoppensteadt, Jorge Ize and Klaus Schmitt. My thanks extend to Ms. A. Hackbarth and to W. Pluschke for the careful preparation of the manuscript.
SELECTED SYMBOLS 1. Parameters
g = acceleration of gravity
h =mean height of fluid layer c = speed of wave T = coefficient of surface tension p = mass density
2. Coordinates and Transformations
(5,7 )Cartesian
coordinates in moving frame
v = ( u , , u2)‘=
7 = z( 5) free surface,
(::)
velocity vector &pO(5) external pressure
a,, a, a, denote partial derivatives 9:stream function, i.e., a,Vr
= -u2,
a T 9 = u, ,
(x, y ) transformed coordinates: x = 5,
D
=Rx
TI,,=^ = o
y = 9((, 7)
( 0 , l ) transformed flow domain
W = ( W ,, W2)’transformed velocity field: v = g ( W ) (1, W 2 ) ’ -( 1 , O ) ’
Nonlinearly Resonant Surface Waves
141
where
3 . Spaces C k ( R ,E ) : space of k-times continuously differentiable functions from R into the (real) normed vector-space E
C h ( R )= C k ( R ,R)
Ck(R, E ) : subspace of C k ( R , E ) consisting of those functions which, together with their derivatives up to order k, are bounded
Ck,,(R, E ) : subspace of Ck(R, E ) consisting of functions which, together with their derivatives, are uniformly continuous on R k2(0,1) = M O , 1) x LAO, 11,
Norm llWll0
W'(0, 1) = H ' ( 0 , 1 ) x H ' ( 0 , l),
X
= R x L2(0, l),
llwllx
= (w,
Norm llWlll scalarproduct (w,ti)
w)''~,
defined in (3.5') Z = D(A),
llwIIa = IIwIIx
+ IIAwIIx
4. Operators
J=(;
71 =identity,
A ( A ) : linearization in w = 0;
A = A(Ao), D ( A ) = R x W'(0,l) n M,
A.
A = (A,
= (1,O)
where M
-A)
={
E )
resp. (A, b, E )
resp. (1, b, 0)
W,(O)= 0, Wl(1) = a } for b = 0,
and M = { W,(O)= 0 , W,( 1) = p }
for b>O
A
So eigenprojection commuting with A corresponding to eigenvalues with Re (T = 0; So = X, a r q ,
sl=n-so,
X,=S,X,
z,=s,z
(T
Klaus Kirchgassner
142
Nonlinear operators:
11. The Problem
Here we describe the basic equations for the interaction of traveling nonlinear surface waves with in-phase external pressure waves. An inviscid fluid layer of mean depth h is considered under gravity g. On its free upper surface, where capillary forces may also act, it supports nonlinear surface waves of permanent form, traveling from right to left with constant speed c. In a moving frame this phenomenon is stationary, and so is the external pressure &p0.We write the equilibrium equations in nondimensional form for irrotational flow in a 6, 7-system, where 6 is the unbounded coordinate div v = curl v = 0, u* = 0,
$IvI’+ p
+ Az = const.
v,a*z - u* = 0
0<7
7=0
I
17 = 4 5 )
Nonlinearly Resonant Surface Waves
143
where v = ( v , , v J ’ ; z describes the free boundary, p the (constant) density, p the pressure on the free surface and A = g h / c 2= (Froude number)-’. We T)/[ER, seek solutions which are bounded in the flow domain a={([, 0 < 7 < z ( [ ) } . Due to the moving frame we have a normalized flux Q = 1 through any simple cross section of the layer. Solitary waves satisfy the asymptotic condition
For cnoidal waves, (2.2) is replaced by the requirement of periodicity. For forced motions, the asymptotics are more complicated, but will follow naturally from our analysis. We restrict ourselves to moderate-amplitude solutions, i.e., some suitable norm is a priori bounded from above. This upper bound could be estimated explicitly by quantitatively showing the validity of subsequently described, almost identical transformations and estimates on noncritical eigenvalues of the operator A(A, b ) . We shall treat a number of what we think are instructive problems which are distinguished by the form of p , the pressure on the free surface. The Bernoulli constant C in (2.1) is set to A + $ and the following cases are considered. 1 . Forced gravity waves: p = E P ~ , , where p , ( [ ) is given. We distinguish between local forcing, i.e., E # 0 and p o has compact support (vanishes outside a bounded interval), and global forcing, i.e., E # 0, and p o is a given periodic function. The period is considered as an extra parameter.
2. Forced capillary-gravity waves:
where po is given as in 1. The dimensionless quantity b is the Bond-number
As the reader can see throughout the analysis, the assumption of irrotational flow could be removed at the expense of additional technical work. Also we could include more general types of forcing, e.g., quasiperiodic p o .
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Klaus Kirchgassner
111. Transformations and Symmetry
Here we formulate (2.1) in the canonical domain D = R x (0, 1) using a semi-Langrangean transformation proposed by Benjamin ( 1966). Then we apply an almost identical transformation of the velocity components to bring the boundary conditions into a suitable form for our method. This transformation is due to Mielke (1987~). Define the stream-function ? by
a,v=v,,
ag?=--uz,
then, since Q = 1, we have ?(& (x, y) E D via
x = 5,
~ ( 5 )=)1.
?I,=,,=o; We introduce new variables
y = w 5 , 771,
and set
Then (3.1) is invertible for small 121, I V1l, I V,l. We obtain for V = ( V , , V,)’
axV = M(V)a,,V, VI+$I V12+ p
+ AZ = 0
(1+ V,)a,Z- v , = o
(x, y ) E D, (3.1)
y=l,
where
M ( V )= Divide the first equation in (3.1) by ( 1 + V,)’ and integrate in y to obtain
v 1+ , v, --ax -
[l:vl]. ~
d.(Z+[&])=O,
y=l,
(3.2)
=I,!, Vl(y) dy.
where [ V , ]
The fact that, if external forces are not present, (3.1) is reflectionally
Nonlinearly Resonant Surface Waves
145
not distinguished, can be expressed by
symmetric in x, i.e., +aand -a
and
~ ( k vJ,J) iv)= -EM(V) a,v.
(3.3)
We call the vectorfield M ( V ) a, V reversible. 2 can be expressed by V , via (3.2). We simplify the first boundary condition at y = 1. This can be achieved by defining 2w,=(l+V,)’+v:-l,
w2=v*(1+v,)-’,
or, inversely, by
being valid for V , > -1. System (3.1) reads now a,W=K(W)a,.W, W,+p+hZ=0,
inD,
W2=0
fory=O
J,Z- W2=0,
fory=l
(3.4)
where
In addition, (3.4) implies via (3.2) (3.4’) is reversible with respect to I?.It is this formulation Observe, that K ( W ) a,>w which we are going to discuss. The idea behind our method is simple: we treat (3.4) as a dynamical system in the unbounded variable x, although the initial value problem is not solvable in general. However, with the “boundary condition” that W is bounded, (3.4) is well posed and we can apply concepts and ideas from the theory of dynamical systems. To obtain the final formulations we have to distinguish the two cases b = 0 and b > 0. The trace of W, resp. W2 at y = 1 is introduced as an extra variable.
Klaus Kirchgassner
146
Case b = 0, no surface tension: Define
),
-AJ aWz( , W1)
&Ah=(
w=(i),
J = ( *0
- 1o)
Then, if the first boundary condition for y = 1 is differentiated i n x, (3.4) can be written d , w = A ( h ) w + P ( t , x, w),
We seek solutions of (3.5) in X
6)= a;+
(w,
=R x
wE
D(A).
(3.5)
L2(0, 1) with the scalar product
i,: w,G,+ w,IF2)
dy.
(
(3.5‘)
Define D ( A )= R x W’(0, 1) n { W2(0)= 0, W , (1 ) = a } . Here W’(0, 1) = ( H ’ ( 0 , 1))2is the usual Sobolev-space; the traces of W, on y = 0 or 1 are defined. We call W E Cb(R, X ) n C”,R, D ( A ) ) a bounded solution of (3.5). Here, the suffix “b” indicates the boundedness of w, i.e., sup{llaxw(x)ll.
+IIw(x)IIA}<~,
It is easily seen that (3.5) is reversible for
E
+ IIAwllx
IIwIIA
= IIwIIx
= 0,
with respect to
X€R
(3.6) 0 0
-1
i.e.,
&AIR
= -R&A),
P(o, R W )
=
-RP(o, w).
Having determined solutions of (3.5), we find the free boundary via (3.4’).
Remark: Differentiating the first boundary condition at y = 1 introduces an artificial degeneracy into the problem. As we will see, an extra 0 eigenvalue of A ( A ) is generated. This is the price for obtaining a dynamical formulation for the boundary values of w. Case b > 0:
Define
147
Nonlinearly Resonant Surface Waves moreover, for X = (A, b, e ) ,
Then the basic equation follows from (3.4): d , w = i ( A , h)w+
k(X,X, w),
WE
D(A).
(3.7)
We work in X with scalar product as i n (3.5‘), but a replaced by
p.
D ( A )= R xH’(0, I ) n {W,(O) =0, WA1) = P I .
The underlying symmetry for
F
R=
=0
:I.
is determined by
[-:: 0 0
(3.7a)
-1
The notion of a bounded solution is understood as in the first case. It is readily seen that this notion implies the usual solvability of the original equations if the norms are sufficiently small. For later use we give F explicitly up to quadratic terms
1: -(-(3[
p=
wl - 3 W,d, wl+O ( (w(‘(d,w ( ) - w,a,w,+ w,a, wz+ O(l W121d,Wl) w,d,
In the following sections we will use the abbreviations
a
1
Wfl+[ Wil~+&P,+O(IWl’))
=
A( A,, , h,,),
F
=
F + A(A, b ) A, -
‘
(3.8)
(3.9)
where ( A ” , b,) is a “critical” point in the (A, 6)-parameter space, and similarly for A(h,) when b = 0 is considered.
IV. The Method
In a purely formal way, equations (3.5) and (3.7) describe the “evolution” of nonlinear waves in the spatial variable x. But rigorous conclusions can
148
Klaus Kirchgassner
be drawn from this concept: a solitary wave, e.g., is a curve in D ( A ) emanating and returning to the equilibrium w = O , i.e., it is a homoclinic orbit. Similarly, cnoidal waves correspond to closed orbits. Our aim is to show that bounded solutions of moderate amplitude are bound to a lowdimensional manifold in the phase space D ( A ) . Its dimension can be determined and coincides with the order of the reduced system of ordinary differential equations. It is this reduction which can be considered as a nonlinear separation of variables. It is reminiscent of the center-manifold approach in ordinary differential equations. In this realm the method is not standard; therefore we separate the formal aspects given here from their mathematical justification in Chapter VI. In view of the rich structure we treat the case b>O first. We have to investigate the spectrum of i ( A , b). It consists of eigenvalues only; we denote them by CT. A R = - R g implies that the spectrum is invariant under CT+ -u as well as under CT+ Cr (c.c.) in view of the reality of Thus, non-real and non-imaginary eigenvalues appear in quadruples. Simple eigenvalues can never leave the real and the imaginary axis. The eigenvalue problem itself reads
A.
-a,,w, = u w , , a,.w,= u w ~ , W*(O)= 0,
W,( 1) = p.
Observe that w = (p, W , , W2)’can be considered as a function of y only. (4.1) yields ( A - bu’) sin u = u cos CT. (4.2) It is a neat exercise to show the validity of the following picture. The curves C , , . . . , C, are the loci of multiple imaginary resp. real eigenvalues which are close to the imaginary axis. The analytic form of C , , C, near A = 1, b=fisgivenby4(A - l ) = 5 ( 3 6 - 1)’+0((3b-1)3).Therest ofthespectrum is bounded away from the imaginary axis. Bifurcation from the rest state w = 0 occurs when a point in the parameter-space traverses one of the curves C2,C 3 ,C,, not C1.Hence, understanding the full solution picture requires a complete analysis of possible solutions near the singular point A = 1, b = f . We will give the reduced equations for this case. There are numerical experiments of Hunter and Vanden-Broeck ( 1983) covering these parameter values, but without a definite conclusion whether solitary waves exist for
Nonlinearly Resonant Surface Waves
149
FIG. 1. Critical spectrum of A ( h , b ) . Simple eigenvalues are denoted by ”.”, multiple ones by “x”.
b < f , A < 1 or not. Existence of cnoidal waves has been shown by Beyer (1971), Zeidler (1973), and Beale (1979), even for nonpotential flow. Existence of solitary waves for b > A > 1 has been proved by Amick and the author (1987). Everything said so far holds only for F = 0. The case E # 0 is unsolved in major parts. The linear dispersion relation for cnoidal waves corresponds to the imaginary eigenvalues; e.g., in region I11 set u = iq, q E R; then K = q / h , o = q c / h yields the dispersion relation given in Whitham (1974, p. 446).
4,
How to reduce near C, ?
A
We choose ( A o , b o ) E C, for some j and set = A(Ao, bo). The steps which have to be performed are listed below. Their justification is given in Chapter VI. Determine the “critical” eigenvalues u of A with R e u = 0 and its generalized eigenfunctions ‘PJ. Their span is denoted by 2 0 . Moreover, calculate in X and the generalized eigenfunctions G k to the the adjoint A* of critical eigenvalues such that (cp,, + A ) = 8; holds. Define the projections A
Snw=C (w,
+‘h~t,
s, = n so, -
Klaus Kirchgassner
150
and w,=S,w,
X,=S,X,
Al,=A,,
j=O,l.
A,
Then the S, commute with i.e., S,a c AS,; X = X , + X I .We set 2,= XIn D ( A ) . The equations (3.5) and (3.7) assume the form 1
+ F,@,
W" + w 1 ),
(4.3a)
axw, = A , w , + F , ( X , . , w o + w , ) ,
(4.3b)
a h W"
= Aow,
. 1
A
where A = (A, b, E ) and F is as defined in (3.9), F, = S,F. Equation (4.3b) is bounded away from the can be inverted in 2, since the spectrum of imaginary axis. Therefore, it is rather obvious that w, should be functionally dependent on wo. However, one can prove more: if we restrict ourselves to solutions (wo, w , ) ( x ) which are, for all x E R, in some suitable neighborhood U of 0 in Z, x Z , , then w, is a pointwise function of w,,, i.e., there exists a smooth function h ( X , x, w,,) mapping A(] x R x Z,, into Z , such that
A,
w , ( x ) = h ( A , x, W,(X)),
x E R,
(4.4)
holds for all solutions with w ( x ) E U, x E R. A,, denotes a neighborhood of A0 .
We call h the reduction function. It satisfies h ( A 0 , . ,0) = d,,h(A(,, . , 0) = 0.
(4.5a)
Therefore h = o ( ( ~ A - ~ ~ l + I b - b ~ ~ l ) I I w ~ ) l l + I I w ~ ) l l ~ (Since + I ~ I ) . Z, is finite dimensional, we do not need to distinguish between different norms in Zo.) Define h,(X,w,)
= h ( A , .,WO)IF=o,
(4.5b)
hI@, * , w , ) = ( h - h , ) ( A , . , w o ) . Then h,, is independent of x. If we decompose R into its action in 2, and Z,, R = R,+ R , , we obtain h,(X, ROW") = R,ho(A, wo).
(4.52)
Moreover, if po is periodic in x with period d > 0, then the same holds for F, and F , and also for h :
h(A, x + d , wo) = h(A, x, w,).
(4.5d)
If p o has compact support and if A E A,,, wo E U,, then, for every y > 0, there exists a constant c ( y , A,, U,) such that Ilh(A,
X,
W O ) I I AC (~Y , Ao, Uo) e-'IXI.
(4.5e)
151
Nonlinearly Resonant Surface Waves
We have listed only those properties of h in (4.5) which are needed in the subsequent analysis. Observe that h can be computed from (4.3) by
d , , h ( ~ o w o + Fo)= A , h + F , ,
F, = F,(X, ,wo+ h ) , *
j =0,l
(4.6)
to any algebraic order. Although we do not need higher-order approximations, we give some examples at the end of Chapter VI nevertheless. With the aid of the reduction function one is able to reduce (4.3) to the system of ordinary differential equations dxWO
= A l W ( l + f O ( ~ ,. ,wo),
(4.7)
where f o ( k .,wo)=Fo(& .,wo+h(A,
*,Wo)).
We decompose f o into a reversible and a nonreversible part by defining foo=foIF=O,
All
=fo-foo,
f0l
= f o , ( A , 6, E , x, wo).
(4.8) fa0
= f o o ( A , 6, wo),
For the construction of the projection S,, one has to determine the adjoint A* of = bo). It is an elementary exercise to verify, with the aid of the scalarproduct (3.57,
a A(&,
i
1
-W,(l) A*w = -8, W,+ W,( 1 ) , d,Wl
(4.9)
{
D ( A * )= R x W'(0,l) n W,(O) = 0, W,( 1 ) = --/3 bl
l
.
The (generalized) eigenfunctions of the critical eigenvalues of A* can be chosen to be biorthogonal to the eigenfunctions of and S,, is easily constructed. Finally we list the corresponding facts for the case 6 = 0 . Here the eigenvalue relation A ( h ) w= (TW reads (w= ( a ,W)')
A,
-A W,( 1) = (T W ,( 1)
-a, a,
w, = u w, , w, = uw,,
W,(O)= 0,
(4.10)
W,(1) = a.
The eigenvalue relation is A sin u = u cos u.
(4.11 )
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Klaus Kirchgassner
(T = 0 is a simple eigenvalue for A # 1, and a triple eigenvalue for A = 1. For 0 < A < 1, all eigenvalues are real and simple, and for A > 1 there is a pair of imaginary simple eigenvalues whose values correspond to the dispersion relation for cnoidal waves; all other eigenvalues are real and simple. Therefore, A = 1 is the critical parameter value where bifurcation from the trivial solution w = O occurs. The reduced equation is of third order. The formal procedure is in full analogy to the one described for b>0. We define A = A ( 1 ) and F = . F + A ( A ) - A and obtain reduction via h = h , + h , as in (4.4). The final system corresponds to (4.7); b, however, is missing. To obtain the projection commuting with we use again the adjoint operator to here given by
A
A,
D(A*)= R x W'(0,l) n {a + W,( 1) = 0, W,(O)= 0},
(4.12)
w = ( a ,W)'.
V. Reduction and Results A. CAPILLARY-GRAVITY WAVES
With the method described in the last section we are able to determine the reduced equations of minimal order. We do this here for b > 0, E Z 0, i.e., for the case of forced capillary-gravity waves. In the discussion of the reduced equation we restrict ourselves to E = 0, postponing E # 0 to later sections. The available parameters (A, b ) are taken near the bifurcation curves C, and C3 of Figure 1. C, represents the simplest case, A = 1, b > f. We show the reduced phase space to be two-dimensional. A unique solitary wave exists for A > 1 as a wave of depression. In phase space it appears as a homoclinic orbit which is the envelope of a one-parameter family of periodic orbits (the cnoidal waves around the conjugate flow). For A < 1, 0 is a center, and a family of cnoidal waves bifurcates from 0. There is again a homoclinic envelope with its tail in the conjugate flow. But this has no physical meaning.
Nonlinearly Resonant Surface Waves
153
The situation near C3 is more complicated and essentially unsolved. The reduced phase space is four-dimensional and the existence of a homoclinic orbit is equivalent to the intersection of two curves-the stable and unstable manifold of O-in a four-dimensional space. Except for reversibility we have not found any inherent symmetry to guarantee such an intersection. Therefore we anticipate existence of solitary waves for (A, b ) on a curve in the parameter space. For completeness and for future research the explicit form of the reduced equations near A = 1, h = +is included. Reduction at C,: Set A = 1 + p, Ipl small, b > f . Since b is considered to be fixed, we suppress, if possible, its explicit notation. u=O is a double eigenvalue of = 1) with the generalized eigenfunctions
A A(
AQn=o,
Qn=(:).
AQI=Qo.
All other eigenvalues have nonzero real part. Observe, that R Q , = Q , , R Q , = - Q ~ ,R from (3.7a). We identify Z o , the linear span of Q , , ' p l ,with [w2 via W o = U o Q o + U , q 1 + + ( U n , u , ) ' . Then
The adjoint eigenfunctions are determined as
A*+'
= 0,
A*+" = + I ,
(Q,, + l k )
= 8:.
Observe that &R,= -RoAo, .f&)R,= -R,j;,,)holds. Therefore, f & must be in (3.8), (4.5), (4.8) we odd in a , , fAo even. Using the explicit form of obtain ( a : = a,a,) ab=al(l+3a,+rn,(y,a))+r",(~,~,a)
(5.la)
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Klaus Kirchgassner
where rk,= O(plal'+k+la12tk), r,, = O ( ~ p + ~ l a l k) ,= 0 , 1. The remainder terms rkn are even in a , . For small IpI, /&I, lal, a , can be determined as a function of a,, a; and the parameters, using (5.la), a1 =
where
Po0
a x 1 -3%+Pno(P*., (a,),)+Po,(E, P, (ao),)),
(5.2)
is even in ah and Poo=O(Plaoli+lanl:),
~ol=O(~~+~laol~).
Here, we use the notation ( a o ) ,= (a,, ah), Iaol, = 1 4+ lahl, (5.3) and similarly for (a,)2, when we include a,". Inserting (5.2) into (5.lb) finally yields
so, being even in ah, soo= 0(plaol:+la,12), sol= O ( E+~la,lJ. ~ It is this equation which we are going to discuss. It contains all information about the bounded, small-amplitude solutions of the original problem. Observe that (5.3) is autonomous for E = 0; explicit x-dependence is introduced by Po and s o , . We discuss (5.4) for E = 0. Since the phase space has dimension 2, all bounded solutions are either equilibria, connections of those, or closed orbits. There are only two rest points if Ipl and la& are small, namely
For p > 0, a, is a saddle, a,, a center, and for p < O vice versa. There is at most one saddle-saddle connection (homoclinic orbit). If it exists, it contains the center and all closed orbits in its interior (Poincark-Bendixson). Verifying this picture will yield the uniqueness of the constructed solutions. Scaling as follows
transforms (5.4), for
E
= 0,
into
A;'=signlpl. A,,-+A:+ R , ( p , (A,)*). R , is even in A&and satisfies R,,=O(p),with respect to IA&.
(5.6)
Nonlinearly Resonant Surface Waves
155
Proposition 5.1. Equation (5.6) has, for every suficiently small positive value of p, a unique nonzero even solution decaying to 0 at infinity.
The proof is elementary and could be left to the reader. But the argument is of some importance for the subsequent analysis; therefore we include it here. That there is at most one such solution follows from the fact that the stable and unstable manifold of (0,O) are one-dimensional. The assertion is true for p = 0 (inspect phase portrait). Call this solution Qo and set Q = Q,,+ z ; then z ” - z = -3Qoz+r(p,
*,
(z)~),
(5.7)
where r is a smooth function of its arguments and an even function of 5. Denote by C : the space of k-times continuously differentiable functions in R which, together with their derivatives up to order k, are bounded, and denote by I ( Z \ ( ~ the corresponding sup-norm. Then r obeys the estimate 11 rllnSc , p + ~~z~~~ for sufficiently small 1 1 ~ 1 1 ~Moreover . r maps even functions z into even functions. The left side of (5.7) has a bounded inverse from C t into C’, given by the kernelfunction
~ ( 5t ),= -4 For f
E
e-lt-rl ,
(5, t ) E R 2 .
CE we denote
( K f ) ( 5 )=
5‘
K ( 5 , t l f ( t ) dt. .x
Also, K preserves exponential decay up to exponent 1; i.e., continuous functions decaying like exp(-al[)) are mapped into C2-functions with the same decay, if 0 5 (Y < 1. Applying K to (5.7) yields Z=
Lz+Kr(p;,z),
L=-3KQo.
(5.8)
L : C:-+ C’, is a continuous, linear operator. Since Qo decays like exp(-151), L maps bounded sets in C: into sets of uniform exponential decay in C’,. An easy extension of Arzeli-Ascoli’s theorem yields the compactness of L
in CB. Therefore the spectrum of L consists of eigenvalues p. If we could exclude p = 1, (5.8) would be uniquely solvable for small 1p1 and 1 1 ~ 1 1 ~ However, . p = 1 is a simple eigenvalue with eigenfunction Q&.The simplicity follows by multiplying z “ = z - 3KQOz by Q&and integrating between -a and f. One obtains d ( z / Q h ) / d t = 0. We take Q o as an even function; thus Q&is odd.
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Klaus Kirchgassner
Now we eliminate 1 as an eigenvalue of L by a trick. Since r maps even functions into even functions and so does L, we see, by restricting (5.8) to Ci,e= {z E C’,/z(x) = z(-x)}, that 11- Lis continuouslyinvertible.Thus (5.8) is uniquely solvable in Cg,eif lpl and llzllr are sufficiently small. To obtain exponential decay for z at infinity, observe that the above arguments work for functions z, for which exp(l&l)z”’(()E C!, j = 0, 1,2; the assertion is thus proved. The above argument shows that it is the reversibility which makes (5.6) so stable. Its breaking by external forces changes the solution picture dramatically, as we will see. The proof has an additional consequence. Consider the perturbed version of (5.6) A,“= A(i-$A;+ R d p ,
(&I?)+ &f(O
(5.9)
where f~ CE. Under which condition is (5.8) solvable near Q, the unique even solution of Proposition 5.1? Q decays like exp(-Itl). If we set A. = Q + z we obtain z”-z = B(P, 0 ) z + &cL, 6, ( Z M + & f ( 5 ) , (5.10) where
and therefore
Invert (5.10), using K , to obtain
z - KB(Pu,Q ) z = K&P,
. , ( Z M + &Kf
(5.1 1)
It is easy to see that (5.11) is solvable if and only if ( q ,, K&(P, . , ( Z ) J + EKf)
(5.12)
= 0,
where q, is the adjoint eigenfunction to Q’ of the eigenvalue p relative to the scalar product
=1
of KB,
(5.13) In effect, K B is compact in C i and the simplicity of p = l is robust to changes in p. Normalize q , such that ( q , , Q’)= 1, and define z, = z - ( q , , z)Q‘. Then we can solve (5.11) for small lpl, I E ~ in the subspace
Non 1in ea rlji R eson a n t Su rface W aves
157
Ci,Lof Ci, with ( q l , z) = 0, if the right side is projected into C‘i,Las well, yielding z I = Z , ( E , p, zo), zo = ( q ,, z ) , llZ1112 = O( F + lzol’). Set Qf = K q , and observe that Qf = Q’+ O ( p ) , Q’= Q:,+ O ( p ) ;then we obtain from (5.12) the solvability condition for (5.1 1 )
(QT,
i o ( P , ‘ , zoQ’+z’,)+&f)=O.
Specialize to z,=O and replace QT by
Qh:
F(Q:I,.~)+O(&p++*)=0.
(5.14)
We could have included the case where z,), the projection of z on the span of Q’, is nonzero. But this does not yield new results. It is (5.14) which will lead to Melnikov’s condition in the next section. We summarize: Proposition 5.2. For the solvability of (5.9) in C i , it is suficient to solve the scalar equation (5.14) for small E and p.
We return to the discussion of (5.4) for F = 0 resp. its scaled version (5.6). According to Proposition 5.1 we know the solutions for p > 0. For p < 0, set a,= a:, + bo, where a:, = 2 p / 3 + O ( p 2 )as given in (5.5), and proceed for b, as for a, before. Thus, one obtains a complete description of all possible solutions, as indicated in Figure 2. Explicit formulae can be given by tracing back the transformations in 111. The free surface z ( x ) = 1 + Z ( x ) is then given by Z ( x ) = -lplA,(
(””’3 b)- 1
I/*
x)
+ O(p2).
(5.15)
For the solitary wave we have
Similarly, approximations for the cnoidal waves can be determined for < 0. Solve
p
A,”= - A o - $ A ; , w=1+6wI+
A,,( [ )= B ( w ( ) ,
...,
B=6B,+6*B2+
where 6 is an amplitude parameter and B can be considered even. One obtains in a straightforward way 62
A , ( ( ) = 6 cos((i -~6z)t)-:s2+-cos((i 4
(5.17) - ~ 6 ~ ) 2 5 )0(tj3), +
which yields the form of the free surface via (5.15) for each fixed p < O .
Klaus Kirchgassner
158 ob
t
t aI
n.
A
Finally we derive the reduced equations near A = 1, b =;. Define = A ( l , f ) , A* as in (4.9) and proceed as before. Again, a = O is the only eigenvalue of with R e a = 0. It has multiplicity 4.The generalized eigenfunctions are
a
with ai’pi= RI, = Ro
!I.
+, = 0. We identify Zo with R4 and obtain for Ao=[
0
0
1
0),
0 0 0 1 0 0 0 0 The adjoint eigenfunctions are
-A
0 0
0 1 0 0 Ro=[i
0 0
-1
=
Lo,
Nonlinearly Resonant Sugace Waves
159
B. CAPILLARY-GRAVITY WAVESU N D E R PERIODICFORCING External periodic forces interacting with nonlinear waves may lead to chaotic phenomena. This is the theme of this section. From the point of view of dynamical systems, chaos appears here as a consequence of a transverse homoclinic point bifurcating from a homoclinic orbit. Thus, the interactions of periodic forces with solitary waves are expected to yield chaotic behavior. For certain second-order equations, these phenomena have been analyzed (cf. Holmes and Marsden, 1982; Chow et al., 1980; Guckenheimer and Holmes, 1983). However, the situation is not as easy as model equations may suggest. The “dirt” generated by the real equations and hidden in so” and sol in (5.4) is small in E , p, la&, but it is only algebraically small. It turns out that the condition for a transverse homoclinic point to exist is not robust to perturbations of this sort. To overcome this difficulty, we introduce the period d of the external pressure as an additional parameter.
160
Klaus Kirchgassner
where P o ( 0 = PO(X), Observe that Po and R , have the period d(31p1/(3b- 1))”2= d, (cf. Theorem 6.1). Consider the case p > 0; then A,= A&= 0 is a saddle point of (5.18) for r] = 0. Such a critical point is robust under small bounded perturbations, and S > 0, there exists a i.e., given p > 0, then for all sufficiently small unique solution A; of (5.18) satisfying l(A:)I2< 6. This solution is periodic with period d,. It is given by A:
= -r]KPo+
O( v2).
(5.19)
The proof is a simple application of the implicit function theorem. We seek further solutions of (5.18) near Q, the unique even homoclinic solution of (5.18) for 77 = 0 , which was constructed in the last section. Taking advantage of the translational invariance of (5.18) for r] = 0 , we introduce a free phase P and define
P )+z(5+ P). Ao(5) = Q(t+ Now we can follow the analysis ofthe last section ((5.10) to (5.14)), replacing ~f by -r]P,+ R , . Thus we obtain the existence of a solution close to Q for r ] # 0, if
( Q &P, o ( ’ - P ) ) + O ( p +
r]) =
k ( P , p ) =0.
(5.20)
Let us set
--s
and observe that Q&and Po are explicitly known. If ko has a simple zero for some P = Po, p = 0, i.e., (5.21) we can solve (5.20) for (p, p ) near (Po,0). (5.21) is known as the “Melnikov condition.”
Nonlinearly Resonant Surface Waves
161
Before we solve the Melnikov condition, let us discuss its consequences. Since )Ao- QI2(.$)is small and since Q decays to 0 at infinity, is small for large 151. Thus A. must lie in the intersection of the stable and unstable manifold of A,*. In effect, this intersection is transverse, i.e., with linear independent tangent spaces (cf. Chow et al., 1980). The following figure shows the well-known intersection properties of these manifolds for the PoincarC map T, which takes a point y € R 2 into A,(d,,y), where A o = ( A o ,A:)) solves (5.18) with initial condition y at t = O . The points of intersections Fk, k E Z,satisfy TI;, = Fk+land thus form an orbit of the diffeomorphism T The set M := { Ph/ k E Z}u { P z } is T-invariant and compact. Here P z = ( A * ( O ) ,A*’(O)),P, = ( A ( & ) , A’(&,)). In each point there exists a natural local coordinate system given by the tangent vectors to the invariant manifolds at 4. The action of T is strictly contracting in the stable and strictly expanding in the unstable direction. Thus M forms what is called a hyperbolic set. The dynamics of such sets are well known (cf. Newhouse, 1980, for details). We extract from the wealth of possible consequences just one significant result: Attach to M a T-invariant neighborhood U (M ) and assume that there is a 6-pseudoorbit { Qk/k E h} in U ( M ) , i.e., there exists a positive 6 such that
I
Qkt l -
T ( Qk
)I < 8,
k Z;
then, for each sufficiently small positive 6 there exists an r > 0 and an orbit { Ph/ k c Z} c U (M ) such that IPk-QhI
holds for all k E Z .
This is known as the shadowing lemma. Its proof shows that r = O(6). In view of Figure 3 we can generate 8-pseudoorbits (Qk)in M by jumping from the stable to the unstable manifold in a &neighborhood of P $ . Since
FIG. 3
162
Klaus Kirchgassner
-
A, = Q + O( 7 )and Q exp( -161) one needs, starting at some for 6 = 0, an interval of order ln(1/6) for entering a 6-neighborhood of P Z . Being on the stable manifold, the orbit ( P k ) shadowing ( Q k )must follow M . Thus we obtain Proposition 5.3. Assume that (5.21) holds. Take 6, O < 6 < 1. To each natural number N a n d to each sequence ( a ,,. . . , a N - , ) ofpositive real numbers satisfying a k + ,- a k s In 1 / 6 , k = 1, there exists a solution of (5.18) with at least N extrema. The position of N extrema can be chosen such that the distance between two consecutive ones is at least a k .
Therefore we have found solutions of the original problem (5.4) with arbitrarily many extrema. Their amplitude is of order O( [ A - 11). For large 1x1 they follow the d-periodic solution a t , which is of order I E ~ . An example is shown in Figure4. It is to be expected that these solutions are unstable and form transient states in an otherwise chaotic motion. But this is pure speculation. As is well known, such a homoclinic bifurcation is accompanied by subharmonic bifurcations of the periodic orbits inside the homoclinic orbit (cf. Chow et al., 1980). These arguments carry over immediately to Equation ( 5 . 1 8 ) ;we leave the details to the interested reader. The case p < 0 can be treated by translating A . into the conjugate flow, which is a saddle then. The analysis is similar to the one given above. Now we solve k , ( P ) = 0. Take P o ( [ )= A cos w6,
FIG. 4
w
2T =-.
4
Nonlinearly Resonant Surface Waves
163
Then, since Qo is even, k,(O) = 0 . To calculate kh(0) we determine the residues of
Integration over a large rectangle with base [ - R , R ] and height up to i2k7r, k E N, and taking the limit R + CO, k + a,yields kb(0)=w2Re
-
-
laX
e-xw 1 - e-'""'
f ( 5 ) d5=2.rrw3
Since w d,' p - l I 2 ,kb vanishes of exponential order for p = 0 . Therefore we use the period d of po as an additional parameter. Set d = d o p - l / ' ; then the Melnikov condition is satisfied for p < p", 1771 < 77" implying Proposition 5.3 for all sufficiently large periods d. Proposition 5.4. Assumep,(x) = b c o s ( ( 2 . r r l d ) x ) Then . thereexistpositive numbers po, vo such that, if d > ( p o ) - ' / ' ,O < A - 1 < p o , I E ~< ~ ~ k ( Pp) has a simple zero for p = 0 and thus, Proposition 5.3 holds.
The above analysis shows that the behavior of liquid layers may become chaotic under periodic external pressure waves. The same is true for more general pressure distributions such as quasiperiodic ones (cf. Scheurle, 1986). We have also seen how delicate the dependence on parameters may be, which should introduce some scepticism towards the results obtained using model equations. Of course we have discussed very special cases only. We have not included, for example, a serious analysis of small amplitude effects for A < 1, i.e., the effect of an external pressure on cnoidal waves.
c . CAPILLARY-GRAVITY WAVES U N D E R
LOCAL FORCING
The external pressure is assumed to vanish outside some bounded interval. Moreover, we suppose (5.22)
~
,
Klaus Kirchgassner
164
It is shown that the lowest order approximation of every solution of (5.4) has a jump in the first derivative which is of order O ( E ) .We describe the complete solution set. The importance of the limiting equation (5.26) was discovered by Mielke (1986b), when he studied the steady flow through a channel with an obstacle. His analysis and conclusions carry over to our problem, since the reduced equations are the same. The set of solutions can be found by the intersection of two shifted homoclinic orbits and their interior, as will be seen below. As was pointed out in Section IV, h , inherits the exponential decay from Fo,. Therefore, if I(a)l,s y, IA - 115 y, F i ) I y, y sufficiently small, and if A > 0 is arbitrarily chosen, then there exists a c( A, y ) such that
1&1<
I I ~ , ( A ,E , x,ao)II 5 c(A, 7) e-""lI&l,
&().
(5.23)
This again implies a similar inequality for the remainder term so, in (5.4) IsOI(&, p, x, (aoL)l< c l ( ~Y) , e - A " ' ( l w l + l ~ Ila(,lz).
(5.24)
We define rl = "I/--"*(b -4)-I'*(Po),
as a parameter replacing
E.
Pil(5) = p o ( x )
Scaling as in (5.6), (5.4) leads to (5.25)
where the remainder terms satisfy ( A ' = A( 6 -4)) IRob,
(A")2)15
r"lP1 IAolz,
l ~ , ( pT ,, t , ( A , , ) ~ ) ~rS, ~ p ~ e"- A2' l~~ l~l ~~'
Observe that Po/(Po) converges to 0 for every 5 2 0 , and its mean is (Po/(Po))= 1. Therefore it is natural to consider the following limiting equation A,"- sign( p )A,,+ :A: - T&,
= 0,
(5.26)
where So is the Dirac functional concentrated at 0. It is this equation which governs the solution behavior of our problem. Mielke (1986b) has shown that (5.26) yields the complete unfolding of the original equations in the (A, &)-parameterspace. This requires a discussion of penetration properties of stable and unstable manifolds. Here we settle for a less ambitious task and show that every solution of (5.26) with A,(-m)=O can be extended by perturbations of the order
Nonlinearl-v Resonant Surface Waves
165
O(lpl”2) to a solution of (5.25) and thus to the full equations. We restrict the analysis to p > 0 and leave the analogous calculations for p < 0 to the interested reader. We work in the space Y of continuous functions which are twice con01 and [0, a), bounded, together with their tinuously differentiable in (-a, derivatives, and decay to 0 at -alike exp([/2).
where [W+
= [0, a), [W- = (-m,
01, and
-A&(-O) = 77 which Solving (5.26) means finding A o € Y with A&(+O) satisfies (5.26) with 77 = 0 for all 5 # 0. Using the kernel K in Section V.B, and K S , = -exp( -(51)/2 we can write (5.25) as
Observe the validity of the following estimates:
(5.28)
for bounded 7, p, [ A & . R, and R ,map Y into the space of functions being continuous and bounded in (-00,Ol and [O,CO) with decay exp([/2) at 5 = -03, but may have a jump at 0. As is easily seen, K maps these functions back into Y . The composed map is Lipschitz continuous, but not differentiable, in view of the possible jump in the derivatives of A,. Thus we have
166
Klaus Kirchgassner
Now take any solution Bo€ Y of (5.26). Set A,= B o + Z and obtain 2
+3KBoZ = -+KZ2+ 7
1
) + K ( Ro+ R l ) (Bo+
-KP,+ ie-Ic-'
(,Po)
2).
In view of (5.28), (5.29), the right side defines, for small p and llZll y, a contraction in Y. It remains to be shown that 1 + 3 K B o has a bounded inverse in Y. In fact, observe that B ; = d r B o lies in the nullspace. We can argue, as in Section V.A, that every function in the nullspace of 1+ 3 K B o is proportional to Bt, for 5 < 0. If we have in addition B , ( + a ) = 0, then the argument works for E > O also, and thus -3KB,, has a one-dimensional eigenspace to the eigenvalue 1. As will be shown when discussing (5.26) in detail, the only other case to be considered is Bo periodic for [> 0. Observe that -3KBo is still compact in C : by a direct application of Ascoli-Arzela's theorem. If Z + 3 K B o Z = 0 , Z E Y, multiply by B &and integrate from 0 to [ and obtain
( B & Z ' -B ; Z ) ( ( ) - ( BhZ' - B;IZ)(+O)= C for 5 > 0. Set 2 = BA W and conclude W ' = C / B,!;. If C # 0 , Z would have a singularity of the order (5- to) In([- 5,))whenever tois a simple zero of Bt, contradicting Z E Ci([O,a)). But all Bo being periodic for [ > 0 have infinitely many 5 where this holds. Thus, (5.27) has, for every solution Bee Y of (5.26) and for every sufficiently small p > 0, a unique solution in Y close to B o , and similarly for p
FIG.5. Solutions of limiting equation ( 5 . 2 6 ) for
p>O, O <
7 <4/3&
167
Nonlinearly Resonant Surface Waves
FIG. 6.
Solutions of limiting equation (5.26) for p > 0, -4/3&<
7 <0
Proposition 5.5. Given any of’thesolutions of (5.26), shown in Figures 5 to 7, then there exist positive constants po and y such that, f o r every p with lpl< p o , there exists a unique solution A. of (5.27) satisfying IIAo- Boll < ylpl”’. Moreover, if Bo has a limit for [++a(homoclinic or heteroclinic solutions), then so has Ao. Both are even functions.
It is easy to classify all possible cases. For 7 = 2/3&, p > 0, there exists a heteroclinic solution, being equal to fi for [> 0. Comparing the above results with the Propositions 5.1 and 5.2, we see that the orbits I always correspond to the response of A,,=O to the perturbation measured in the parameter 7,whereas I1 reveals the response of the homoclinic orbit (Proposition 5.2). Of particular interest are the solutions 111 in Figure 5 which are even, homoclinic and “bifurcating from infinity.” The distance of their
I FIG. 7.
Solutions of limiting equation (5.26) for p
Klaus Kirchgassner
168
maxima to [ = O is given by
where ((s) is the smallest positive 0 of small values of 9 > 0 a n d
€'-t3= '7'.
Therefore
5-
7 for
T ( v ) = l l n s1+0(1). Hence we obtain the following "paradox": The smaller the amplitude of the external pressure, the further upstream the influence of the local pressure distribution can be felt.
D. FORCEDGRAVITYWAVES The analysis of forced pure gravity waves ( h = 0) proceeds along the previously described route, except for two new features: the effect of a n artificial eigenvalue 0 a n d the need to consider a n integral of the system (3.5). Both problems have the same source, namely the fact that we use the Bernoulli equation on the free boundary in differentiated form. Since the methods by which we overcome these problems are of general interest, we present them here. The basic equations were given in (3.5). First we have to determine the spectrum of A(A)
~;,").'O~=UW,
A ( A ) w = ( -A Wz(1 )
w h e r e & = W l ( l ) , W,(O)=O.Weobtainasin(4.11): a c o s a = A sin c r ; a ~ @ , resulting in the following spectral picture.
0
<
1
x = 1
Fic;. 8
x
:,
1
Nonlinearly Resonant Surface Waves
169
The symmetry of this picture is caused by the commutation property (3.6) of A(A). Nonreal eigenvalues exist only for A > 1 as a pair of simple Via the identifications w = eigenvalues f iq. We have q / A + 1 for A + +a. qc/h, k = q / h , one obtains from (4.11) the dispersion relations (13.25), p. 438 in Whitham (1974). The only multiple eigenvalue occurs for A = 1; it has multiplicity 3. Therefore we unfold the solution set near A = 1 . Set
F=F+X(A)-A,
A=&i),
A = I - ~ :
then we have
F ( E, A,
X, W) =
1
PWAX, 1) - & & P O ( X ) Wy3, W , - 3 Wla,W , ,
(
-
w,a,wl+ w,a,,w,
up to terms of order O(lWl:+plWl:). We define as in (4.7), (4.8): A,= foo+fol. The generalized eigenfunctions of A to u = O are (A'p,= ( P ~ - ~ , 'p-1
=O)
Their span 2, is identified with R'. The operators restricted to Zo read
"=(; ; ;), 0
The adjoint A* and are ( A * + k= Jlk+'; IJJ'
with ( q j ,+ k )
= 6;.
1
0
.;i;-!
a and R
(cf. (3.6))
A was given in (4.12). Its generalized eigenfunctions =0 ) ,
Define
Klaus Kirchgassner
170
Acccording to (4.3), (4.4) we have w = wo
+ ha( A, a) + h,( E , A, x, a)
(5.32)
for all sufficiently small llwllA. Moreover, ho(A, Roa) = R,ho(A,a),
R , = RIZ,
ha = O(lA - I I la1+ la12),
In view of the reversibility, f we obtain
and
9
h , = O(E).
f i 0are odd in a , , and fA0 is even. Thus
ah= a , ( l + ~ p + $ u o - ~ u z + r , ) o ( pa))+&pb+ , r o , ( p ,E, x,a),
(5.33a)
a : = a 2 + & a : - a O a 2 -a:+ r I 0 ( p a , ) + r I I ( p E, , x, a),
(5.33b)
a ~ = a , ( 3 p - ~ a 2 + 9 a o + r z o ( p , a ) ) + 3 ~ p ~ , + r ~, , ,( xp , a )
(5.33c)
where A
and
= 1 - p,
ph = d,po, and where lk0=O(Ipl(al+la12),
k=0,2,
r l 0 = 0(Ip1la1~+la1~),
are even in a , ,
r k l = O(Elal+
w),
k
= 0, 1,
(5.33)'
2.
Due to the fact that we have used one of the basic equations in a differentiated form, the above equations are not independent. This can be seen by the validity of (5.34) which follows from (3.2) and (3.5a). First we observe that
Moreover we have from (5.30) and (5.32) W, = ao-;y2a2+ h i , W, = - Y U ,
+ h',
where h = ( h a , h i , h2)'. An elementary calculation leads to 4 ( 2 + p ) ~ 2 +~
~~+~u:+;u:-~u,,u,+ h'(~ 1) u~ [+h l ]
+ &pa+ O( laI3+ p (a12+ ~p+ E la1) = C.
(5.35)
Nonlinear1.v Resonant Surface Waves
171
Now (5.33a, b, c) imply (5.35) for some C. On the other hand, near a =0, E = p = C = 0, we can express a2 as a unique function of a,, a , , p, F and C. Thus we can replace ( 5 . 3 3 ~by ) (5.35) and reduce (5.33) to a second-order equation. Another consideration concerns the term h ’ ( l ) - [ h ’ ] in (5.35). It seems that one should calculate h up to order e+plal+la12. However, since h maps into Z , , and 2, is the orthogonal complement of the @, we obtain, using (+*, h ) = 0 , h’(1) - [ A ’ ] = 0. Concerning the free constant C, we This includes observe that C = O for E = 0 , if a tends to 0 for ,$+-a. homoclinic solutions for E = 0 as well as all solutions for F f 0 if p o has compact support. In the global case, p ( , being periodic, C may be of order F . But this constant can be incorporated into p o . Therefore we set C = 0. Using (5.35) for C = 0 and h’ - [ h ’ ] = 0 to eliminate a, in (5.33), we obtain (5.36)
to
is of the same order as r,,, and r,, is as in (5.33)’. The similarity where between (5.36) and (5.1) is evident. In particular, the justification of the scaling as well as the truncation can be accomplished as in Section V.A, B, C. Therefore we restrict our analysis to the main steps and the results. Scaling as follows a d x ) = IpIAdl),
a , ( x )= 1pI”’AI(5),
5 = (pII”x,
leads to the limiting equation 3.5 A , “ = 3 sign(p)A(,+;A;+yP,,
(5.37)
P
where Po(5) = p o ( x ) . For E = 0, we can draw the same conclusions as in proposition 5.1 ( p > 0) and (5.17) ( p (0). However, here p > O means A < 1 and we have A,
(5.38)
Thus, for p > 0 we have a unique even solitary wave of elevation. But (5.38) is valid also for the one-parameter family of cnoidal waves bifurcating from 0 for p < 0.
172
Klaus Kirchgassner
. the propositions 5.3 If F # 0 and po is periodic, we set 7 = 3 & / p 2Then and 5.4 hold. In the local case, when po has compact support and ( p o ) # 0, we define 7 = 3 ~ ( p ~ ) / l pand l ~ proceed '~ as in Section V.C. Then proposition 5.5 holds, when the signs of A, in Figures 5 to 7 are interchanged. The formula (5.38) for the free surface is valid.
VI. The Mathematics
In this final chapter we describe the mathematical basis for the reduction method, which we have formally applied in the previous sections. Moreover we show how higher order approximations of the reduction function h can be computed. The reduction method described in Section IV and used in Section V is reminiscent of the center manifold approach for ordinary differential equations, which was first proved by Pliss (1964) and Kelley (1967) for autonomous (x-independent) equations. The extension to nonautonomous equations in the extended phase space can be found in Aulbach (1982). Generalization to partial differential equations are well known for semiflows, i.e., the parabolic case (cf. Henry, 1981), and for hyperbolic equations (cf. Carr, 1981). For elliptic systems, as they are considered here, this method was first formulated by the author (1982) in a special situation. Fischer (1984) proved its validity for general semilinear autonomous systems. The first application to free boundary value problems was given by Amick and the author (1987), the version for semilinear, nonautonomous elliptic systems by Mielke (1986a), and for quasilinear systems by Mielke (1987a). We shall use Mielke's formulation and prove two properties HI, H2 of the basic equations (3.5) and (3.7). One can use them as axioms for the validity of the reduction method. In the following lemma we shall treat-pars pro toto-the system (3.5) in the real Hilbert space X = R x L2(0,1) x L 2 ( 0 ,l ) , with the norm 11. Moreover Z = D ( A ) =R x H ' ( 0 , 1) x H ' ( O , 1) n { W,(0)= 0, W , (1) = a } and 11. / I A = 11 + IlA. The norm of a scalar function W in L, resp. HI is denoted b y 1 WI, resp. I WI,, and 1 * I is the Euclidean distance in R". We set X = (A, F ) , Xo = ( 1 , O ) and = i ( 1 ) . The operator is linear, closed and densely defined in X and has a compact inverse. Thus, its spectrum consists of eigenvalues with finite multiplicities. These properties are more or less trivial.
tix.
]Ix
[Ix.
A
A
Nonlinearly Resonant Surface Waves Lemma 6.1. Consider the natural complex$cation Then there exist positive numbers qo, yo such that
II(A
-
zI)-lyk+k
of
173
a in k
Yo
=X
+ iX. (6.1)
5-
IZI
holdsfor all z = iq, 1912 qo, q E R. ProoJ
We have to solve -
W,( 1 ) -ZW,(l) -a,w,-zw,= a,w,-zw,=
= a,
(6.2a)
VI,
(6.2b)
v,.
(6.2~)
From (6.2b) and ( 6 . 2 ~ we ) obtain
I VII;+ I VI, i= I
J,
WII i+ IJ,. WzI ?I+ lZ12(IWII
+ 1 W21;)
+2q I m ( r n W 2 ( 1 ) ) . Multiplying (6.2a) by W,( 1) yields
2q Im( W,(l) W,(l)) = -21 W2(1)12-2 Re(GW,(l)). We estimate [ W,] via ( 6 . 2 ~ )
and
(6.3)
174
Klaus Kirchgassner
where
c,=1+-(2+9,) lz12
(1+-d,)
,
Therefore, if qo> 0 is chosen sufficiently large, and e l , E? sufficiently small, one obtains for all IzI 2 qo, Rez = 0, v = (a, V , , V,)',
l~,wll:,+la,w2l:,~c:llvll'x. Moreover we obtain from (6.2a)
I wI( 1)I 5
1 IZI
(la I + I w2(111).
If we estimate I W2(l)l by (6.4), the assertion follows for any IzI 2 qo. The estimate (6.1) can be extended to a cone IRe z I 5 61Im zI for O < 6 < Y O 1 , 141= IIm ZI 3 4 0 .
II(A- z U ) - ~ I I ~ + =~ ]]((a - ( z
-
i q ) ( A- iqU)-')-'(A- iqU)-']\z+x
A.
Therefore, the line z = iq contains at most finitely many eigenvalues of Denote by Sothe real eigenprojection: S,,A c ASoand S , = U - SO,?()X = X o , S , X = X , , A , = A l x , , j = O , l . Then we have X = X , , O X , , a = A , O A , . A
H1:
I
The space XO has jinite dimension. I f u E Z &-the spectrum of Ao-then Re (T = 0. There exists a p > 0 such that (T E 1 implies )Re (TI 2 /I. To each positive p' < p there exists a y , ( / I ' ) such that the inequality
A,
holds for all z E C with (RezI s p'
Nonlinearly Resonant Surface Waves
175
Now, we turn to the nonlinearity in (3.5) which we write F(X, ., W ) = ( i ( A ) - i ( l ) ) w +
where h = ( A , E ) , ho=(l,O),and
P=(
-&axPo ( K ( W ) -m,w
Observe, that W E D ( A ) implies W E H ' ( 0 , 1) x H ' ( 0 , 1 ) . Since H ' ( 0 , 1 ) is embedded in Co[O,11, g ( W ) and thus K ( W ) defined in 111 are Ck-mappings from ( H ' ( 0 ,1))2 into (Co[O,11)' for each EN. Therefore, if p 0 € Ci,,,(R), the space of k-times differentiable functions in R with bounded and uniformly continuous derivatives, then F E Ci,,(A x R x D ( A ) ,X ) where A is some neighborhood of A,,= (1,O). Moreover, there exists a bounded function y 2 ( r ) for r > 0 such that, if J J w J < J Ar, we have IIF(h, . , w ) I l x s
-11
Y2(r)(lEI+lA
IIWIIA+IlW112A).
(6.6)
We can decompose the system (3.5) as follows: W", W I ) ,
(6.7a)
= ~ i , w , + f 1 ( X ,. r W O , W I ) ,
(6.7b)
a x w o = Aow,,+f,(& axw1
where wJ E X, n D ( A ) ,f; = SJF,j
H2:
*
1
= 0, 1.
There exist neighborhoods U I , c X o , U ~ DC( A ) n X l , A o f A O ~ R 2 , and some k E N such that f = ( f o , f , )E Ck,z'(Ax R x U ~ UX; , X 0 x XI) holds. Furthermore f(A,,, . ,0) = 0, d,f(A,, , . ,0) = 0.
Since the projections are continuous in X and in D ( A ) , we conclude from the previous considerations that H2 is fulfilled for the nonlinearity F from (3.5).The same can be shown for (3.7). The value Xo is given by A = 1, F = 0 in the first, and by A = 1 , b > 0 fixed, E = 0 in the second case. Let us remark that (6.5) cannot be improved in the power of IzI as is well known for elliptic systems. Using results of Burak (1972) one could show, by verifying that the Agmon condition for the boundary values holds, that projections S:, S; exist corresponding to the positive and negative part of Z and that A: := AISFgenerate holomophic semigroups for x s 0 resp. x 2 0. Thus, we could invert (6.7b) for f l E CB(R, X ) .
A,
176
Klaus Kirchgassner
-A,
However, the invertibility of a, can be shown without the projections S:. The inequality (6.5) leads to a logarithmic singularity of the inverse which can be handled. This approach has the advantage that the reduction method can be extended to the case where F maps into some closed subspace of X . For the implications of this generalization see Fischer (1984) and Mielke (1987b). Both systems, (3.5) as well as (3.7), have a quasilinear structure, i.e., the highest derivative d,W has coefficients depending on w. The inversion of d,-A, in (6.7b) leads to a loss of regularity due to the singularity of exp( -All[() at [ = 0. For semilinear systems, when F maps D ( A Y ) y, < 1, into X , this loss can be compensated by the gain in regularity between D ( A Y )and D ( A ) .Thus the extension from finite to infinite dimensions is relatively straightforward in the semilinear case (Fischer, 1984; Mielke, 1986a). For quasilinear systems, Mielke (1987a) has shown that this difficulty can be overcome by a result of “maximal regularity” for the linear equations
which correspond to (6.7). One constructs a space Y over X , x XI such that (6.8) is uniquely solvable for (v,,, go, g , ) E X o x Y with a solution satisfying (d,wo, d,w,) E Y. Mielke (1987b) constructed Sobolev-spaces with exponential weight leading to maximal regularity for (6.8). Thus he obtained the following: Theorem 6.1. Let the assumptions H 1 and H 2 be valid. Then there are neighborhoods of zero Uoc U ; c XI,, U2c U ; c D ( A ) n X I , a neighborhood A o c A of ha and a function
~ = ~ ( A , x , ~ ~ ) ~ C ~ U,) ( A ~ X R X U ~ , with the following properties: (i) The set
MA ={(x,wo, h ( X , x , w o ) ~ R x X o x ( D ( A ) n X , ) l ( x , w o Uo) )~~x is a local integral manifold for (6.7) for A E Ao. ( i i ) Every solution of (6.7) with A € A o and ( w o , w ~ ) ( x ) EU o x U 2 ,xER, belongs to M A . (iii) We have h ( A o , x, 0) = dw,,h(Ao,x, 0) = 0 for all x E R. (iv) Zf fo and f l in (6.7) are periodic in x, then so is h with the same period.
Nonlinearly Resonant Surface Waves
177
( v ) I f there are linear isometries R , :X , + X n , i = 0 , 1, and a constant K E {-I, l} such that
J;(&
KX,
ROWO, Riwi) = KR,J;(A,X , wo, w i ) ,
A,R,= KR,,&,
i =o, 1,
then h ( A , K X , Rowo)= R ,h ( A , x, wo) holds. For the proof see Mielke (1987a). The formulae (4.4) to (4.7) follow by setting
h,(A, b, wo) = h(A", w,,),
where l o = (A, b, E = O ) ,
h , ( A , b, &, x, wo) = h ( A , x, wo) - h(A", wo). The effect of reversibility is described by R, = R I , , K = -1. In Section IV we have claimed exponential decay of h in x of any order, if p o has compact support. This follows from Corollary 6.2. Assume there exists a function f ( A , w ) E Ci,T'(Ax U ~ X U i , X ) and some 0, d > O such that
~ l f (X ,~W,) -
f ( ~ w)llX , 5 D e-d'x',
x
E
R,
f o r all w = w,+ w , , W,,E U:,, w, E US. Then U,,, U, and A,, in Theorem 6.1 can be chosen such that a function L(A, wo)E C : ( A o x U,,, U 2 )exists satisfying Ilh(A, x, w d - i ( A , wO)IIx,5 A d ) e-"lx',
X E
R,
f o r some y ( d ) and all w,,E U,,, A E A,, . The proof follows from Theorem 3.3 i n Mielke (1986a) and the proof of Theorem 6.1. If, as in the case considered in Section V.C, f is independent of x for all sufficiently large 1x1, the choice of d is arbitrary.
Computational Aspects We close with some remarks concerning the computation of the reduction function h. Although h is not unique, in general, it has a unique Taylor expansion about wo = 0 and A = A,,, i.e., the coefficients of the Taylor jet of order k are uniquely determined by the properties o f h ; different h differ only in terms which are exponentially small in wo and /L (cf. Sijbrand, 1981).
178
Klaus Kirchgassner
The computation of this Taylor jet is conceptually simple but actually quite tedious. In the cases we have been considering here, where 0 is the only critical eigenvalue having geometric multiplicity 1, the calculations lead to a sequence of recursively solvable linear equations. As an example we treat the case of gravity waves ( b = 0). We wish to determine the terms of order O( E + plal+ of h. Remember that we identified W,,E Z , with a € R3, A = (A, E ) , A = 1 - p.
h ( A , w 0 ) = ~ h 0 ( x ) + p h i ( a ) + h 2 ( a ) +... , where h2(a) = ht2’(a,a), and h‘*’(a, b) is a symmetric bilinear form over R3 with values in D ( A )n X I . According to Theorem 6.1, wI(x)= h(A, x, W~,(X)) holds for all small bounded solutions. Inserting into (6.7b) and collecting the terms of O( 8 ) of F yields
Using the explicit forms of F = ( F 0 , Fl, F2)’
‘p,
+’ in (5.30) resp. (5.31) we obtain for
and
- 4 6 - A[ F1I + 3b’ Fi I 10
2Y ~
~
0
+
~
-
~
+
~
~
2
~
~
~
l
l
(6.10) + 3 ~
F~ - ~ Y [ Y F , I This implies FO1 - (1 5,
2 i +n
’ 0)’-
32 ~ ’ - ,
(6.11)
and thus (hy, h:) = = 0 implies @(x, 1) = [@](x) and vice versa. Thus we have to solve From (6.9) and (6.11) we conclude that d,h:=d,hy
(a,@, d y @ ) = V@ for some scalar function @. Moreover (+’,h’) V2@=(-3+3 10
d,@(x, 0) = 0,
2Y
2
)axPo
(6.12)
@(x,1 ) = [@](x).
The solution of (6.12) is uniquely determined up to a constant for bounded @, and thus we obtain h0 = (a,@(. , l ) , a,@, a,@)‘.
To calculate h’(a) we observe that hl defines a linear functional from X , into D ( A )n X I . Since X , is spanned by cp,, ‘pl, ‘p2 it suffices to determine
~
2
~
i
1
179
Nonlinearly Resonant Surface Waves
hf
= h'(9,).
Define for A,
=(1
- p,
0)
Then one obtains
Moreover, we conclude from (5.33): a:)= a , , a : = a2 and a; = 0 up to terms of order O(&+pIal+lal2). Therefore d , h l = A l h l + F : leads to
Alh:)=O, h;=O,
A,h:+FIl=O,
h:=(O,O,-&y+;y')',
h l2 -- ( - L
A,h:=h:; 27-L20Y
175, 1400
'+!8Y
9
O)'.
Similarly one could calculate the quadratic approximations h2(a),for which it suffices to determine h?, = h"'(cp,, c p , ) . Since Acp, = q l - l ,Q-, = 0, this can be achieved recursively. We leave the lengthy calculations to the reader as an exercise. The results are
References Abarbanel, H. D. I. (1983). Universality and strange attractors in internal-wave dynamics. J. Fluid Mech. 135, 407-434. Akylas, T. R. (1984). On the excitation of long nonlinear water waves by a moving pressure distribution. Y. FIuid Mech. 141, 455-466. Amick, C. J., and Toland, J. F. (1981). On solitary water-waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9-95.
Amick, C . J., Fraenkel, L. E., and Toland, J. F. (1982). On the Stokes conjecture for the wave of extreme form. Acra M a r h . 148, 193-214. Amick, C . J., and Toland, J. F. (1984). The limiting form of internal waves. Proc. Roy. Soc. London, A 394, 329-344.
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Amick, C. J., and Kirchgassner, K. (1987). Solitary water-waves in the presence of surface tension. Manuscript. Aulbach, B. (1982). A reduction principle for nonautonomous differential equations. Arch. Math. 39, 217-232. Beale, J. T. (1979). The existence of cnoidal water waves with surface tension. J. Dig Eqn. 31, 230-263. Beckert, H. (1963). Existenzbeweis fur permanente Kapillarwellen einer schweren Flussigkeit. Arch. Rat. Mech. Anal. 13, 15-45. Benjamin, T. B. (1966). Intenal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241-270. Benjamin, T. B. (1967). Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559-592. Beyer, K. (1971). Existenzbeweise fur nicht wirbelfreie Schwerewellen endlicher und unendlicher Tiefe. Dissertation B, Leipzig. Burak, T. (1972). On semigroups generated by restrictions of elliptic operators to invariant subspaces. Israel J. Math. 12, 79-93. Carr, J. (1981). “Applications of center manifold theory.” Appl. Math. Sci., Vol. 3 5 . Springer, New York/Berlin. Chow, S. N., Hale, J. K., and Mallet-Paret, J. (1980). An example of bifurcation to homoclinic orbits. J. Dig Eqn. 37, 351-373. Fischer, G. (1984). Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen. Math. Nachr. 115, 137-157. Foias C., Nicolaenko, B., Sell, G. R., and TCmam, R. (1986). Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. To appear in J. Maths. Pures et Applique‘es. Grimshaw, R. H. J., and Smyth, N. (1985). Resonant flow of a stratified fluid over topography. Res. Rep. No. 14, Dep. Math., Univ. Melbourne. Guckenheimer, J., and Holmes, Ph. (1983). “Nonlinear oscillations, dynamical systems, and bifurcations of vector fields.” Appl. Math. Sci., Vol. 42, Springer, New York/ Berlin. Henry, D. (1981 ). “Geometric theory of semilinear parabolic equations.” Lecture Notes in Mathematics, Vol. 840. Springer, New York/Berlin. Holmes, P. J., and Marsden, J. E. (1982). Horeshoes in perturbations of Hamiltonians with two degrees of freedom. Comrn. Math. Phys. 82, 523-544. Hunter, J. K., and Vanden-Broeck, J. M. (1983). Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 134, 205-219. Jones, M., and Toland, J. (1986). Symmetry and bifurcation of capillary-gravity waves. Arch. Rat. Mech. Anal. 96, 29-54. Keady, G. (1971). Upstream influence in a two-fluid system. J. Fluid Mech. 49, 373-384. Kelley, A. (1967). The stable, center stable, center, center unstable and unstable manifolds. J. Dig Eqn. 3, 546-570. Kirchgassner, K. (1982a). Wave solutions of reversible systems and applications. J. Dig Eqn. 45, 113-127. Kirchgassner, K. (1982b). Homoclinic bifurcation of perturbed reversible systems. In “Lecture Notes in Mathematics” (W. Knobloch and K. Schmitt, eds.), Vol. 1017, pp. 328-363. Springer, New York/Berlin. Kirchgassner, K. (1984). Solitary waves under external forcing. In “Lecture Notes in Physics” (P. G. Ciarlet and M. Roseau, eds.), Vol. 195, pp. 211-234. Springer, New York/Berlin. Kirchgassner, K. (1985). Nonlinear wave motion and homoclinic bifurcation. In “Theoretical and Applied Mechanics” (F. I . Niordson and N. Olhoff, eds.), pp. 219-231. Elsevier Science Publ. B. V., IUTAM.
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ADVANCES IN APPLIED MECHANICS, VOLUME
26
Contributions to the Understanding of Large-Scale Coherent Structures in Developing Free Turbulent Shear Flows J. T. C. L I U t The Diiinon of' Engineering, Laboratory .for Fluid Mechanics, Turbulence and Computation Brown University Providence. Rhode Island
I. Introduction . . . . . . . . . . . ... .............. 11. Fundamental Equations and Their Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Description and Averaging Procedures ............. B. Equations of M o t i o n . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Kinetic Energy Balance . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Vorticity Considerations . . . . . . . . . . . . . . . . . . . . . . . . . , . ........... E. The Pressure F i e l d . . . . , . , . , , . , . , . . . . . . . . . . . . . . . . . . . , . , . . . , . . . , , . . . . . F. The Reynolds and Modulated Stresses. . . . . . . . . . . ...... 111. Some Aspects of Quantitative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
184 188 188 193 195 199 206 207
IV. Variations on the Amsden and Harlow Problem-The Temporal Mixing ,ayer A. Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The "Turbulent" Amsden-Harlow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Diagnostics of Numerical Results via Reynolds Averaging . . . . , . . . . . . . . . D. Evolution of Length Scales . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . E. Some Structural Details . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . , , . , . , . . . . . . . . V. The Role of Linear Theory in Nonlinear Problems . . . . . . . . . , . . , . , . , . . , . . . . .
219 219 220 224 229 230 232
Introductory Comments ...................................... Normalization of the Wa litude .................... Global Energy Evolution Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subsidiary Problems. The Role o f the Linearized Theory.. . . . . . . . . . . . . , . . Nonlinear WdVe-EnVelOpe Dynamics. . . . . . . . . . . . . . . . . ... The Mechanics of Energy Exchange Between Coherent Mode and FineGrained Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Wave Envelope and Turbulence Energy Trajectories. A Simple Illustration VI. Spatially Developing Two-Dimensional Coherent Structures. . . . . , . . . . . , . , . , ,
232 235 236 237 243
A. B. C. D. E. F.
211
244 248 25 1
1- On sabbatical leave 1987-88 at the Department of Mathematics, Imperial College, London SW7 2BZ, U.K. 183 Copyright 0 198X Academic P r e u . Inc. All rights of reproduction i n any form rcwrved.
ISBN 0- IZ-OO?OZh-?
J. T. C. Liu
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A. General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Single Coherent Mode in Free Turbulent Shear Flows.. . . . . . . . . . . . . . C. Coherent Mode Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Multiple Subharmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Three-Dimensional Nonlinear Effects in Large-Scale Coherent Mode Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Discussion ................. B. Parallel Flows.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Spatially Developing Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Energy Exchange Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Nonlinear Amplitude Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Relation to Temporal Mixing Layer Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII. Other Wave-Turbulence Interaction Problems . . . . . . . . . . . . Appendix . . . . . . . . ............................................ Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25 1 253 258 280 284 284 286 287 289 293 296 298 300 302 302
I. Introduction
In his article on recent advances in the mechanics of boundary layer flow, published in Volume 1 of this series, Dryden (1948) recalls that at the Fifth International Congress for Applied Mechanics, von K6rmAn (1938) pointed out the difficulties in reconciling a scalar mixing length with turbulence measurements made in a channel by Wattendorf and by Reichardt. In the discussions that followed, which were not precisely recorded in the 1938 Proceedings, Dryden (1948) pointed out that both Tollmien and Prandtl suggested that the measured fluctuations include both random and nonrandom elements and that the correctness of these ideas is borne out by later turbulence measurements in the boundary layer, which were conducted at the National Bureau of Standards. It is important to note that Dryden (1948) emphasized that ". . . it is necessary to separate the random processes from the non-random processes," but concluded that ". .. as yet there is no known procedure either experimental or theoretical for separating them." In the early fifties, Liepmann (1952) surveyed aspects of the turbulence problem and pointed out the importance of the presence of a secondary, large-scale structure superimposed upon turbulent shear flows, citing as examples measurements of Corrsin (1943) and Townsend (1947) in free turbulent flows, Pai (1939, 1943) and MacPhail (1941, 1946) in the flow between rotating cylinders and Roshko (1952; see also 1954, 1961) in the far turbulent wake behind a cylinder. Liepmann (1952) concluded that although the details of the large-scale structure may be in doubt, such
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structures cannot be ignored in many of the technological problems in aerodynamic sound, in combustion and in mixing controlled problems in general. More quantitative discussions of the large-scale structure in free turbulent flows were initiated by Townsend (1956, Section 6.5) in the first edition of his monograph on the structure of turbulent shear flows. He considered the total flow to consist of a mean motion and fluctuations consisting of a large-scale disturbance and the balance of the motion to be fine-scaled fluctuations. The scales are taken to be nonoverlapping so that the spatial, volume integral of the products of the disparate-sized fluctuations vanishes. The resulting global energy balance of the large-scale structure (Townsend, 1956) gave the essence of the physical interpretation that the large-scale structure gains energy from the mean flow and exchanges energy with the fine-grained turbulence by the rate of working of the large-scale motion against the excess Reynolds stress owing to its presence. Townsend (1956) hypothesized certain kinematical details of the large-scale motion but ruled out motions of the hydrodynamical instability type. The splitting of fluctuations into large-scale structures and fine-grained turbulence was further underscored by Liepmann (1962) in his discussion of free turbulent flows. He advanced the idea that the large-scale motion could be attributable to the hydrodynamic instability of the prevailing mean flow. It was still not clear then how the large-scale motions could be sorted out, either experimentally or theoretically, from the total fluctuations. Liepmann ( 1962) emphasized, however, that the large-scale structures in turbulent shear flows ought to be studied in a well-controlled manner, similar to the studies of the Tollmien-Schlichting waves leading to transition in a laminar flow (Schubauer and Skramstad, 1948). The well-controlled experiments suggested by Liepmann (1962) in terms of perturbing or enhancing the periodicity in a turbulent shear flow when the usual Reynolds (1895) average is accompanied by a form of conditional averaging (Kovasznay et al., 1970) now widely known as the phase average geared to the periodicity, allow fluctuations measured at a point to be split into coherent and random parts. In principle, this procedure takes the jittering out of the phases of otherwise coherent fluctuations (e.g., Thomas and Brown, 1977), and is similar to the Schubauer and Skramstad (1948) experiments that place the Tollmien-Schlichting wave at a desired location. The pioneering experiments leading to the recognition of coherent oscillations in turbulent shear flows were associated with Bradshaw (1966), Bradshaw et al. (1964), Davis et al. (1963) and Mollo-Christensen (1967).
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Experiments on well-controlled coherent oscillations in turbulent free flows began with Crow and Champagne (1971) and Binder and Favre-Marinet (1973) for the round jet, Hussain and Reynolds (1970a) for turbulent channel flow and Kendall (1970) for a wavy wall perturbation beneath a turbulent boundary. The primary advantage of the phase-averaging procedure (Binder and Favre-Marinet, 1973; Hussain and Reynolds, 1970a), from a theoretical point of view, is that it allows the systematic derivation of the coupled fundamental equations for the mean flow, the large-scale coherent fluctuations with a dominant periodicity, and the fine-grained turbulence. The presentation of these equations for a homogeneous, incompressible fluid may be found in Hussain and Reynolds (1970b), Elswick (1971), Reynolds and Hussain (1972) and Favre-Marinet (1975). The description of the perturbed turbulent shear flow problem is entirely similar to the limited-time (or space) averaging procedure for educing naturally occurring coherent features in turbulent shear flows (Blackwelder and Kaplan, 1972, 1976) and the fundamental equations from this point of view are given at the 1970 von KQrman Lecture by Mollo-Christensen (1971), who discussed many facets of interactions between disparate scales of motion in the turbulent boundary layer problem. Lumley (1967) developed a more formal definition of the large-scale motions and obtained their dynamical equations, using “conventional” (as compared to “conditional”) averaging methods. As in Townsend (1956), Lumley (1967) suggested that the effect of the motion of smaller scales in the dynamical equations for the large-scale motion be represented by a constitutive relation. I n the lowest-order approximation, the large-scale motion satisfies the Orr-Sommerfeld equation for small disturbances. Lumley (1967) further suggested that the mean velocity profile could be neutrally stable, corresponding to the minimum Reynolds number maintained by an eddy viscosity. This is reminiscent of the marginal stability ideas for wall-bounded turbulent shear flows put forth by Malkus (1956), in which the turbulent velocity fluctuations are represented by a collection of neutral wave solutions of the Orr-Sommerfeld equation. This idea was extended by Landahl ( 1967) to the superposition of wave solutions satisfying a non-homogeneous Orr-Sommerfeld equation; the nonlinearities are assumed weak and prescribed. However, free-wave disturbances corresponding to the standard turbulent eddy viscosity in wall-bounded turbulent shear flows are strongly decaying. Thus the presence of these waves is attributed to a continuous driving mechanism arising from variations of the turbulent Reynolds stresses. In general, this class of theoretical problems
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is linear, and some are associated with the eddy-viscosity representations of the effect of the background turbulence. Further discussions of the role of wavelike representations in turbulent shear flows are given by Moffatt (1967, 1969), Lighthill (1969), Phillips (1967, 1969), and Kovasznay (1970). The experiments of Hussain and Reynolds (1970a) on imposed monochromatic disturbances in turbulent channel flow indicate that such disturbances propagate like Tollmien-Schlichting waves but that they decay strongly downstream, as would be expected from theoretical considerations (Reynolds and Tiederman, 1967). As we now appreciate, the coherent large-scale motions in wall-bounded turbulent shear flows are much more involved than free turbulent shear flows (see, for instance, the review by Cantwell, 1981). However, some of the theoretical ideas that evolved in the above discussions are more relevant to the free shearflow problem, which is the main subject of this article. For free turbulent shear flows it is not necessary to conjecture that the local fine-grained turbulence rearranges itself to give bursts of white noise in order to maintain the hydrodynamically “unstable” waves as for wallbounded shear flows, nor does there appear to be experimental evidence indicating such a mechanism. It is easily seen that the existence of large-scale coherent motions in free turbulent shear flows would be a manifestation of hydrodynamic instability associated with local inflectional mean velocity profiles. This would account for the observed pronounced large-scale-and what now appear to be wavelike-structures in this class of flows (Corrsin, 1943; Townsend, 1947; Roshko, 1954, 1961; Grant, 1958; Bradshaw et al., 1964; Mollo-Christensen, 1967; Brown and Roshko, 1974; Papailiou and Lykoudis, 1974). The present impetus regarding the existence and importance of large-scale coherent structures in free turbulent shear flows has been brought about essentially by optical observations of such flows (e.g., Brown and Roshko, 1971, 1972, 1974) in which such structures have been almost obscured by previous correlation measurements. Prior to the more recent recognition of the role of coherent structures in turbulent free shear flows, it was widely thought that such Rows were independent of initial and environmental conditions (Townsend, 1956; Laufer, 1975). The experiments of Crow and Champagne (1971) and Binder and Favre-Marinet (1973) pointed out the distinct possibilities of controlling the downstream development of the jet-flow oscillations via the upstream forcing of the large-scale coherent structure. These findings have enormous implications with regard to technological applications such as jet noise suppression (Bishop et al., 1971; Liu, 1974a; Mankbadi and Liu, 1981, 1984), mixing and instabilities
J. T. C. Liu in combustion chambers and chemical lasers (Carrier et al., 1975; Marble and Broadwell, 1977; Broadwell and Breidenthal, 1982), to mention a few. Thus the study of large-scale coherent structures in free-turbulent shear flows is of technological interest not only because such structures directly and indirectly affect the local mixing but also because they render the downstream flow controllable. The present article is intended to address the physical problem of largescale coherent structures in real, developing free turbulent shear flows from the point of view of a broader-minded interpretation of the nonlinear aspects of hydrodynamic stability. Indeed, this interpretation has to be the case in light of the presence of fine-grained turbulence in the problem; even in its absence, there exists the distinct lack of a small parameter. We shall present the discussion on the basis of conservation principles and thus on the dynamics of the problem. The discussion is directed towards extracting the most physical information with the least necessary computations, and thus must necessarily involve approximations. As such, the discussions presented here are seen to supplement other works using methods such as numerical simulation or straightforward inviscid linearized stability theory and other kinematical interpretations.
11. Fundamental Equations and Their Interpretation
A. GENERAL DESCRIPTION A N D AVERAGING PROCEDURES Both visual observations and unconditioned quantitative measurements of turbulent flows sample the total flow quantities. Flows that occur naturally or in the laboratory do so without regard to the artificial separation into mean and fluctuating quantities. On the other hand, for purposes of understanding and particularly for possible flow control, the Reynolds (1895) type splitting of the flow into mean Row and fluctuations is helpful, particularly in problems of hydrodynamic stability (Lin, 1955). Flow instabilities are efficient extractors of energy from the mean motion under certain conditions, and thus it is not overly simplistic to say that instabilities can be controlled via appropriate alterations of the mean motion. Gaining insights into the problem would be most difficult if it were viewed on an overall basis without regard to such Reynolds splitting. With the present widespread recognition of the important role of large-scale coherent struc-
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tures in turbulent shear flows, the usual Reynolds splitting has become inadequate in that it blends the coherent structures and the “real” finegrained turbulence. While the latter is most likely to be “universal,” the former definitely is not, particularly if it is argued (Liu, 1981) that the large-scale coherent structures in turbulent shear flows are a manifestation of hydrodynamical instabilities. Such instabilities are attributable to different specific mechanisms such as inertial instabilities associated with inflexional mean flows, centrifugal instability in the Taylor and Gortler vortex problems, viscous instabilities in wall bounded shear flows and so on. Thus it is not at all surprising that in Reynolds stress modeling for turbulent shear flows that include all fluctuations as “turbulence,” the closure constants are by no means universal but are dependent upon the problem concerned. Of course, one would generally not entertain ideas of using such closure methods for nonlinear hydrodynamical stability problems. This should also be the case for the coherent structure problem in turbulent shear flows. The suggestion of Liepmann (1962) that perhaps the properties of largescale structures could best be studied by well-controlled forcing, similar to the experimental study of Tollmien-Schlichting waves, leads us to the natural synthesis of numerous theoretical ideas. With the fixing of the phase of the large-scale motions, appropriate conservation and transport equations could be derived for the large-scale coherent motions, the modulated finegrained turbulent stresses and the mean motion problem. The relevant description of the development of the large-scale motion is inherently nonlinear, from which a broader interpretation of ideas from nonlinear hydrodynamic stability theory (Stuart, 1958, 1960, 1962a, b, 1971a) will naturally follow. This interpretation would be coupled with the fine-grained turbulence problem through the modulated- and Reynolds-mean stresses from which the large-scale coherent motions have already been separated out. In this case, Reynolds-stress closure ideas (see for instance Lumley, 1978) could be judiciously applied to the fine-grained turbulence. Lumley somewhat anticipated this earlier (1967, 1970). The formalism leading to the derivation of the conservation and transport equations for the monochromatic perturbation problem, originally intended for the study of imposed Tollmien-Schlichting waves in a turbulent channel flow (Hussain and Reynolds, 1970a; Reynolds and Hussain, 1972) is more relevant as the starting point for the study of large-scale coherent motions in free-turbulent shearflows (Elswick, 1971). In the subsequent exploration of the consequences of the basic equations, we shall make use of the richness of ideas from
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nonlinear hydrodynamic stability, particularly in the interpretation of observations. The study of a monochromatic large-scale disturbance in a turbulent shear flow is of considerable difficulty in itself, since any such study relevant to observations must necessarily take into account interaction of the disturbance with the fine-grained turbulence as well as the mean motion (Liu and Merkine, 1976; Alper and Liu, 1978; Gatski and Liu, 1980; Mankbadi and Liu, 1981; Liu, 1981). We shall, however, present the derivation of the more general fundamental equations with multiple large-scale-mode interactions in mind. To this end, the idea (Stuart, 1962a) of splitting the coherent modes into odd modes and even modes is used. Originally Stuart (1962a) used this framework to illustrate the energy transfer mechanism between the fundamental disturbance and its harmonic. For the subharmonic problem, one can in turn reinterpret that the previous first harmonic mode is now the fundamental component and that the previous fundamental mode is now the present subharmonic component. In mixing regions and jets, it is now well known that spatially occurring subharmonics take place (see, for instance, Freymuth, 1966; Miksad, 1972, 1973; Winant and Browand, 1974; Ho and Huang, 1982; Hussain, 1983). Accordingly, we shall consider that any flow quantity q can be split into
where Q denotes the mean flow quantity obtained by Reynolds averaging, 4 the odd modes, $ the even modes and q’ the fine-grained turbulence. In the usual Reynolds framework, (G+ 4 + q’) would be considered as turbulence. The form of the Reynolds averaging procedure would be attached to the type of periodicity associated with (4 + i ) .In the hydrodynamic stability sense the spatial problem, as is usually found in laboratory wind tunnels or water channels, is where the mean flow develops and spreads spatially and the amplitudes of coherent modes (or wave envelopes) grow and decay in the streamwise direction; the periodicities are in time. Consequently, the time average, denoted by an overbar, over at least the longest period T (of frequency p ) would be the appropriate Reynolds average
In this case we denote the special conditional average, which here is the
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phase average geared to the frequency P, by ( )
where x, is the spatial coordinate, t is the time. A “layman’s” interpretation of this can best be visualized by considering that hot wire signals, taken at a given spatial location, are recorded as a continuous function of time on tape. The average is performed by adding the signals at N number of the interval T (or p - ’ ) and then dividing by N. This procedure is somewhat related to the limited-time-averaging procedure used in turbulent boundary layers where the phase is not fixed by forcing (see, for instance, Blackwelder and Kaplan, 1972). The average (2.3) will pick up all the coherent mode contributions from frequencies rnp, where m is an integer. The phase average of linearly occurring fine-grained turbulence signals is zero, ( 4 ’ )= 0, while (Q) = Q. Thus
(4)=Q+4+4.
(2.4)
The sum of odd and even modes is obtained from ( 4 ) - ij
=
4 + 4.
(2.5)
We denote further a similar phase average tied in with frequency 2 p by ( ( i ) ) = O , the even modes are obtained from
(( )) so that, with
((i+ 4))= 4.
(2.6)
The 2P-phase average picks up all the m(2P) contributions, with rn being an integer. The odd modes are then explicitly obtained by subtracting (2.6) from (2.5). For linearly occurring flow quantities, (2.6) is equivalent to the procedure in directly performing the 2P-phase average upon the total signal ((4))- Q = 4.However, for nonlinear quantities this latter procedure would give rise to the introduction of the Reynolds average of partially modulated fine-grained turbulence stresses which are to be necessarily augmented by their corresponding transport equations, thereby unnecessarily complicating the issue further. To anticipate the more straightforward procedure indicated by (2.6), the corresponding modulated turbulent stress, ?,, and ?,, are obtained from the products of fine-grained turbulence velocity fluctuations through -
(u:.;)
-
u ; u ; = ?,,
+ i?,,,
and, applying the 2P-phase average to both sides of (2.7), we obtain
(2.7)
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In this case, only the appropriate Reynolds stresses - (uluj)= u;u;
would occur in the nonlinear equations. (In the undesirable procedure, ( ( u : ~ ; ) )which , is not equivalent to u:u:, would be introduced.) The temporal problem is illustrated by the tilting tube experiment, where a lighter liquid is placed on top of a heavier one (e.g., Thorpe, 1971); a slight tilt sets up a mean shear layer, homogeneous in the “horizontal” direction, which then spreads vertically as a function of time. In this case, the coherent modes are spatially periodic and the amplitudes or wave envelopes develop in time. The appropriate Reynolds average would be the horizontal average over the longest spatial wavelength A ~
q=
1
s,:
q dx.
The appropriate ( )-phase average in this case is
(d=
l
cN q b + nA, Y , z, t ) .
n =0
The subharmonic in this case would have wavelength 2A. The (( ))-phase average in obtaining the even modes is entirely similar to the spatial problem. The temporal problem is similar to the prevailing numerical simulation techniques in that the Reynolds average is taken with respect to the spatial direction and the Reynolds mean flow grows or decays in time. In the laboratory situation, the Reynolds averaging procedure is with respect to time. The contrasting situations have been referred to as the “temporal” and “spatial” problems, respectively, in the hydrodynamic stability literature. The transformations between the two cases are given significant discussions (Gaster, 1962,1965,1968) for linearized problems. For nonlinear problems the transformation between the two situations is achievable only on a “mimicking” basis. There is no suitable convection velocity to achieve a one-to-one physical correspondence between the temporal and spatial problems, particularly for the large scales. However, the temporal problem remains useful because of its simplicity. In cases where “three-dimensional” coherent modes are important, such as the spanwise periodicities in the plane shear layer (Huang, 1985; Corcos and Lin, 1984; Jimenez, 1983) or the helical modes in the round jet (Mankbadi and Liu, 1981, 1984), the averages already discussed would
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have to be supplemented by those pertaining to the spatial periodicities of the “three dimensionality” problem. For instance, as part of the Reynolds average these would introduce spanwise averaging pertaining to the spanwise periodicities in an otherwise basic two-dimensional flow, or circumferential averaging pertaining to helical coherent modes in an otherwise round jet.
B. EQUATIONS O F MOTION We begin with the continuity and Navier-Stokes equations for an incompressible homogeneous fluid
(2.10) where v is the kinematic viscosity; the density has been absorbed into the pressure p . If we substitute the splitting of flow quantities (2.1) into (2.9) and (2.10), the Reynolds average of the total flow produces the mean flow problem (2.11)
Du, --
- aP _-+-,v a’u, a (@, + il,il, + u:u;,, ~
-
ax,
Dt
ax;
ax,
~
(2.12)
+
where D/ Dt = a / a t U, alax,. If we deal with the spatial problem, the mean flow is steady, and D / Dt = U, alax,. For the temporal problem, D/ Dr = a / a t according to the discussions of Section 1I.A. In the subsequent section we will retain such usage and interpretation of DlDt. After the ( )-phase averaging of the total flow and subtracting out the mean flow, the overall large-scale motion is given by (2.13)
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194
D a( Jj+ p^ ) (ii, + ii,) + (ii, + ii,) a-=u, -~ Dt ax, ax,
-
a %
- -[ ( ii,
+
a*( 6, + ii,) ax;
+ ii!) ( tiJ+ CJ)
-
( ii, + ii, ) ( ii,
+ ti,)] (2.14)
where the modulated fine-grained turbulent stresses are already defined in (2.7). In obtaining (2.14), the property that the coherent motions and the turbulent fluctuations are uncorrelated is used. Equations (2.13) and (2.14) for the overall large-scale motion (G,+ G , ) appear in the same form as that for a monochromatic disturbance (e.g., Hussain and Reynolds, 1970b). Following the procedure indicated by (2.6) and (2.8), we perform the (( ))-phase average on (2.13) and (2.14) to obtain the conservation equations for the even modes:
a ii, -- 0, ax, D
-
ii
Dt '
au, --+ ap^ + u* -= ax,
ax,
a%,
a
l . I
(2.15)
a
v 2 - (U,U,- u p , ) -- (ii& ax, ax, ax/
__
- Ij,IjJ) --,
"rJ
ax,
(2.16) We note that the products of odd modes, such as GIGJ, contribute to the even modes and thus ((fi,G,)) reproduces itself, The nonlinear effects of even-mode self-interaction, C$,, produce even modes as well. If we subtract (2.15) and (2.16) from (2.13) and (2.14), respectively, the conservation equations for the odd modes are obtained:
a ii, _ - 0, ax,
(2.17)
It is noted here that nonlinear effects formed by the products of even modes with odd modes, .^,GI and u"i.^,, give rise to odd-mode contributions. The system (2.15) through (2.18) forms the starting point for studying nonlinear interactions between coherent modes themselves and between coherent modes and fine-grained turbulence. The second term on the left of (2.16) and (2.18) is the advection of mean flow momentum by the coherent motion
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and forms the basic mechanism of shear flow hydrodynamic instabilities (Lin, 1955). The mechanism of viscous diffusion of momentum is augmented by the modulated stresses of the fine-grained turbulence. The transport equation for these stresses will be obtained in the sections to follow. The nonlinear effects, which are appropriately split into even- and odd-mode contributions in (2.16) and (2.18), respectively, contribute to coherent-mode amplitude-limiting mechanisms, as ideas from nonlinear hydrodynamic stability would indicate (Stuart, 1958, 1960, 1971). The momentum equation for the mean motion (2.12) indicates that finite-coherent-mode disturbances, as would the fine-grained turbulence, affect the mean motion through their respective Reynolds stresses. We also note that the effect of the fine-grained turbulence on the mean motion and on the coherent motion occurs in the form of stresses, through the Reynolds average and the phase average, respectively. The detailed, instantaneous fine-grained turbulence motions are thus not directly involved. However, for purposes of obtaining the Reynolds stresses and modulated stresses, the conservation equations for the instantaneous turbulent fluctions are stated here; they are obtained from the continuity and Navier-Stokes equations for the total flow quantity by subtraction of the contributions from the mean flow and coherent modes, (2.19)
D
a u,
Dt ’
ax,
a d u; a (G,+i,)= --+ ap’ v a2u:z +u^ )-+
-u ’ + u ’ - + ( G I
1
ax,
ax,
ax,
a --
ax,
(u:u; - ( u : u ; ) ) .
(2.20)
ax, C. KINETIC. ENERGYBALANCE The physical mechanisms underlying the coupling between different scales of motion indicated by the momentum equations can be better illustrated by energy considerations. Although the fluctuation kinetic energy equation can be obtained from its Reynolds stress equation by equating indices, we prefer to deal with the Reynolds stresses and the modulated stresses separately in the subsequent section. Here, we shall obtain the kinetic energy equations directly for the various scales of motion by multiplying the relevant ith-component momentum equation by the corresponding ith-component velocity and summing.
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The mean flow energy equation is obtained by multiplying (2.12) by U,,
exchange
transport
(2.21) A comment about the viscous terms in (2.21) is warranted. These are common to similar terms in the energy equations for the other components of the flow. The form in (2.21) is written for convenience, the first viscous term being interpretable as the viscous diffusion of kinetic energy. The second viscous term, though the negative of positive-definite quantity, is not the actual viscous dissipation rate, The less convenient but physically meaningful form of the viscous effects is as follows. The rate of viscous dissipation is of the form
(2.22) and is combined with the “viscous diffusion” term in the form
where, through the use of continuity,
a2u1q au,acr, ax, ax,
ax, ax,
(see, for instance, Townsend, 1976). The form appearing in (2.21) will be used throughout, with the physical interpretation through (2.22) and (2.23) kept in mind. The first group of terms on the right of (2.21) include the pressure work and the transport of mean flow energy by the Reynolds stresses of the even- and odd-coherent modes and the fine-grained turbulence. The second group of terms is the energy exchange mechanism between the mean flow and the fluctuations consisting of the coherent modes and the turbulence. If
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then there is a net energy transfer from the mean flow to the overall fluctuations; the opposite is true if the sign is negative. Of course, this interpretation holds for the individual components of the fluctuations as well. The energy equation for the odd modes is obtained from (2.18) by multiplying by Gi and then performing the Reynolds average,
transport
exchange
(2.24) The contributions within the first group of terms on the right represent, respectively, the pressure work, the transport of odd-mode energy by the even modes, and the transport of odd-mode energy by the modulated fine-grained turbulence. The second group of terms includes the mechanisms of energy exchange between the odd modes and, respectively, the mean flow, the fine-grained turbulence and the even modes. If
then energy is transferred from the mean flow to the odd modes and this term has the opposite sign as that occurring in the mean flow energy equation (2.21). If
then energy is transferred from the odd modes to the fine-grained turbulence via the work done by the modulated stresses against the odd-mode rates of strain ati,/ax,. If
then energy is transferred from the odd modes to the even modes. The viscous terms are similar to those occurring in the mean flow equations and have similar interpretations.
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The energy equation for the even modes is similarly obtained from (2.16), ~D -u,/2 2 = --d -[fz, + ti,lif/2+ lilt$, + i?,i,,]
-
Dt
ax,
transport
exchange
(2.25)
Again, the first group of terms on the right of (2.25) represents pressure work, transport of even-mode energy by itself, by the odd modes and by the modulated fine-grained turbulence. The second group represents energy exchanges between the even modes and, respectively, the mean flow, the fine-grained turbulence and the odd modes. The first and third of these have opposite signs to similar terms in (2.21) and (2.24), respectively. The viscous terms need no further comment. The kinetic energy equation of the fine-grained turbulence is obtained from (2.20) by multiplying by u : , first ( )-phase averaging and then Reynolds averaging, Du:*/2 Dt
-
”-
= --
ax,
p’uj+ u;u:’/2+ -
11,c 7 + ( -Y21
[ -;; ( - $) + -u:u;-+
6,
?,,
transport
~
+
,;
-rp,-
(-G)]
exchange
(2.26) The first group of terms includes the usual pressure work and self-transport and the transport of fine-grained turbulence energy by the coherent fluctuations. The first term in the second group of terms, commonly known as the turbulence production mechanism, has the sign opposite to that of the similar mechanism in the mean flow energy equation (2.21). The second and third energy exchange terms are the mechanisms involving the odd and even modes, respectively; they have opposite signs to their counterparts in (2.24) and (2.25), respectively. The combined viscous effects include, again, “diffusion” and rate of viscous dissipation previously interpreted.
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We note here that the advective mechanism in the momentum equations provides, in the kinetic energy equations, mechanisms of transport and of energy exchanges among the various scales of motion. From the structure of the latter mechanism occurring in the same form but of opposite sign in a “binary” interaction, we have emphasized energy exchanges rather than “production.” The latter perhaps too often implies the regulation of the direction of energy transfer in terms of a (positive) eddy-viscosity effect. For instance, from hydrodynamic stability it is well known that energy could return from fluctuating motions to the mean flow (a “damped” disturbance in the inviscid sense). In the next section we shall explore the consequences of vorticity considerations. One would expect that vorticitymagnitude exchanges among the different scales of motion would arise from advective effects, but that no such exchanges would result from the vorticitystretching and tilting effects in three-dimensional motions.
D. VORTICITYCONSIDERATIONS There is an extensive discussion of the role of mean and fluctuating vorticity, within the context of the Reynolds splitting procedure in turbulent flows, in Tennekes and Lumley (1972). Some aspects of the role of coherent-mode vorticity in turbulent shear flows and the resulting interactions between different scales is given attention in Mollo-Christensen (1971). The vorticity equation, which is obtained by taking the curl of the momentum equation (2.10), is in a way simpler in form for the description of fluid motion in that it is devoid of the presence of the pressure. Let us define the overall vorticity in the “shorthand” notation,
where & , k m is the alternating tensor. It has the property that etkm= 0 if any two of ikm are the same; if all ikm are different and in cyclic order, then &,km = 1 ; but E~~~ = -1 if the cyclic order is disrupted by the interchange of any two numbers. The overall vorticity equation, obtained by taking the curl of (2.10), is -aw, + u - = ”am -+at
’ax,
au,
’ax,
a2W,
ax;.
(2.27)
In addition to the continuity condition duj/axj = 0, we shall also make use
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of the condition awllax, = 0 in the splitting procedure to follow. The nonlinear advective term on the left of (2.27) will anticipate the transport of vorticity and vorticity exchanges among the different scales of motion, similar to the interpretations of the kinetic energy balances. However, the vorticity stretching ( i = j ) and tilting ( i # j ) mechanism on the right of (2.27) will anticipate net intensification of vorticity; although the mechanism of vorticity exchanges is present even for plane (coherent) motions, the net intensification mechanism is necessarily a three-dimensional phenomenon. Similar to the overall velocity splitting, we let
+ G, + 6,+ w : ,
w , = 0,
where R,, G,, 6 ,and w : are the mean vorticity, odd- and even-mode vorticity and turbulent vorticity, respectively. The procedure for obtaining the individual vorticity equations is similar to that for the momentum equations. At this stage it is helpful to introduce the symmetrical, rate-of-strain tensor s,,=-
rut -+-
2 ax, ax, specifically for use in the vorticity stretching/tilting mechanism. Thus
a4
w, -= wp,,,
ax1
to which the antisymmetrical, rotational part of au,/ax, makes no contribution. The occurrence of s,, in the present context then readily identifies the stretching/tilting mechanism, whereas the occurrence of u, identifies the advective role of the fluid velocity. In what follows, the stretching/tilting mechanism will be referred as “stretching” for simplicity. The splitting of sv into appropriate flow components readily follows. The mean flow vorticity equation is then
D --R Dt
a - - -
= -’ 3x1
(G,G, + GI&, + U ; W : ) + [ n , s , , transport
~~-
+ (G,iv + +,+ w ; s : , ) ] + v-+.a2R stretching
ax,
(2.28)
The first group of terms on the right of (2.28) is the transport of vorticity by the fluctuating motions; the second group includes the net intensification of mean vorticity by the rates of strain of the mean flow and that of the fluctuations. Equation (2.28) differs from the vorticity equation in a laminar viscous flow, which would have the same form as (2.27), through the fluctuation contributions to vorticity transport and stretching in the mean.
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201
The vorticity equations for the odd- and even-modes are, respectively,
D a --;,=--[l?,R,+l?,~,+li,&,+&,,] Dt ax, transport
(2.29) stretching
D Dt
-4,
a =--[6,111+(ii,&,-C,&,)+(li,&,-li,&,)+rii,,] ~
ax,
~
transport
Similar to the introduction of the modulated fine-grained turbulence stresses and f,,, we have defined and used the modulated fine-grained turbulenceproduced transport and stretching effects, respectively
y;
The vorticity transport effects, reflected by the first group of terms on the right of (2.29) and (2.30), are due to interactions with the mean flow, mode interactions, and the fine-grained turbulence. The second group of terms in (2.29) and (2.30) is due to vorticity stretching and tilting. In the odd-mode +&,);, are due to the stretching vorticity equation (2.29), the effects of (a, of the mean and also the even-mode vorticity by the odd-mode rates of strain, while &,(S,, +$,) is the stretching of odd-mode vorticity by the rates of strain of the mean flow and of the odd modes; ?, is the contribution from modulated stretching effects due to the fine-grained turbulence. Similar interpretations hold for a$,, and &,S,, found in (2.30). However, the nonlinear effects of odd-mode vorticity stretching by the odd-mode rates of strain (&$,, - &,&) give rise to even-mode contributions, similar to the nonlinear effects present in the even-mode momentum equation (2.16). The vorticity stretching due to self-straining effects of the even-mode - L,.fy) give rise to even contributions. Similar odd-mode and even-mode selfinteractions give rise to the nonlinear transport effects in (2.30). These two nonlinear self-interaction effects are peculiar to the even-mode vorticity ~
~
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only, whereas similar stretching and transport effects for the odd-mode vorticity come from even-odd mode interactions only. In (2.30), i?, is again the even part of the modulated fine-grained turbulence vorticity stretching effects. Finally, the diffusion of vorticity by viscosity is the last term in (2.29) and in (2.30). In the description of the evolution of the vorticity of the mean flow and enters into the of the odd and even modes, the fine-grained turbulence problem through Reynolds averaged quantities u ; w : and w : s : , in (2.28), and through the modulated quantities h,,,Et and hi,,, c!, in (2.29) and (2.30), respectively. The transport equations for such quantities could be readily obtained, if desired, through the instantaneous equation for w : in conjuncbe stated here, tion with that of u : given by (2.20). The equation for w : willwhich will subsequently be used to obtain the magnitude w:’/2. The finegrained turbulence vorticity equation is obtained in a similar way as that for u : :
D
a
-w : = -- [ u ; ( n , Dt ax,
+ (3, +G!)+(G, + li,)w:+ u ; w : - ( u ; w : ) ] transport
stretching
I
(2.31)
The transport effects are immediately obvious, being due to turbulent transport of the total coherent vorticity present, the transport of turbulent vorticity by the coherent fluctuations (transport by the mean flow is already accounted for in the left side of (2.31)), and effects of self-transport. The turbulent vorticity stretching is contributed by the presence of total coherent vorticity in the rate-of-strain field of the turbulence, the presence of turbulent vorticity in the total coherent rate-of-strain field, and the self-stretching effects indicated by w:s’, - ( w J s ’ , ) .The viscous diffusion of turbulent vorticity is obvious. While the physical understanding of the interactions among the various scales of motion was provided by the energy considerations in Section II.C, a similar understanding of interactions between the mean and the various scales of fluctuatinuorticities would be provided by the “magnitude” of vorticities nf, &f, Gf and w : 2 , known as the enstrophy (e.g., Pedlosky, 1979). The derivation of transport equations for such quantities is similar to that of the energy equations. The mean flow problem is obtained from
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203
(2.28) by multiplying by R,, with some rearrangements, -
D
a
Dt
ax,
-0 f / 2 = - -
-
~ ~
-
-I
aR [a, (t?,&, + GJLt + u ;w 3 3 + (t?,&> + GILT+ u ; w ;) transport
-
ax,
exchange
-~~
a* +[n,(&,f,,++/.;,, +W:S:,)+n,R,s,,]+,Rf/2(shared)
(self)
v
ax,
stretching
(2.32) The transport of RZf/2 by the fluctuations, indicated by the first group of terms on the right, is entirely analogous to that for the mean kinetic energy. The exchange of vorticity with the fluctuations is indicated by the second group of terms on the right, and these are analogous to the similar exchange mechanisms for the kinetic energy. As we have emphasized already, the mechanisms of transport and exchange of the square of vorticity are affected by the advection mechanism in the momentum equation. The third group of terms on the right of (2.32) is the intensification of R5/2 due to the effect of stretching of fluctuation vorticity by the rates of strain of the fluctuations and the stretching of mean vorticity by the mean rates of strain. The viscosity effects, indicated by the sum of the fourth and fifth terms on the right, include the viscous diffusion of i l f / 2 and its rate of viscous dissipation. If the mean flow is two-dimensional, then the self-stretching mechanism R,R,S,, vanishes. If the coherent modes are also two-dimensional, the intensification of R f / 2 due to stretching of the coherent-mode vorticity R , ( & j ~+l ,GI$,) by the coherent-mode rates of strain would also vanish, leaving only the stretching mechanism due to the turbulent fluctuations R,ois:/ (e.g., where i = 3 and the motion is in the 1-2 plane). The equations for the square of the odd- and even-mode vorticities are respectively
transport - - _ _ - -
,&,&+&,?,)+
exchange
(;,wJ~~,+o,L,~,j)+(3,&,s,J]
(shared)
(self)
(other)
stretch1 ng
+ v-dX,’d2 &f/2- v
(2)l
(2.33)
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204
transport
exchange
~- - - +[(Q,&,i,, +&,C,) +(&,&,& +&,&,s,,) &,&J,,]
+
(shared)
(self)
(other)
stretching
(2.34)
In the above two equations, (2.33) and (2.34), the first group of terms on the right represent the transport of the mean square coherent-vorticity fluctuations by the coherent-mode fluctuations and by the modulated turbulent fluctuations. The latter is associated with the modulated turbulent vorticity transport f i J , and &,<. These transport effects are similar to those for the transport of coherent-mode kinetic energies. The second group of terms on the right side includes the exchange of coherent-mode mean square vorticities with the square of the mean flow vorticity associated with dR,/dx,. The signs of these effects in (2.33) and (2.34) are opposite to those in (2.32). Similar exchanges exist between odd- and even-mode mean square vorticities as indicated by the opposite signs of C,&, d&,/dx, in (2.33) and (2.34). The exchange mechanisms with the fine-grained turbulence, as will be anticipated in the transport equation for ~ : ~ to / 2follow, are given by 6iJ,d&,/ax, and h,, d&,/dx,. The form of these exchange mechanisms have in common the product between the vorticity flux of one component of flow and the vorticity gradient of another. These are analogous to the kinetic energy exchange mechanisms due to the product of a stress and a velocity on the right - side of gradient or rate of strain. The third group of terms (2.33) and (2.34) is the effect of intensification of Gf/2 and &f/2, respectively, due to vorticity stretching. The effect due to interaction between the mean vorticity and fluctuating rates of strain of the coherent mode, flJ4,gLand Q,&,.?,,, gives rise to an overall intensification rate of Gf/2 and &f/2, respectively, thatis the same as that for Q f / 2 in (2.32), which is fl,(&,fv+ &$,,). Both GI;,and &,Z, are due to the modulated turbulent vorticity and rates-of-strain fluctuations. As will be apparent subEquently, 2. the sum of these rates of intensification is the same as that for ~ : ~ / The stretching effects due to the mean flow rates of strain, &t&JS,,and &,&,SV, are not “shared.” Finally, the middle group of terms in the stretching group is due to coherent-mode rates of strain fluctuations themselves. Except for ~
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205
the other three self-stretching effects are due to odd-even mode interactions. The sum of the last two terms in (2.33) and (2.34) is again due to the viscous diffusion and dissipation. Finally, the evolution equation for wi2/2 is
transport
exchange
(shared)
(self)
(other)
stretching
(2.35) We have defined ( w : w : ) in terms of the sum of its Reynolds mean and modulated parts -
-
1
(w:w:,= w:w:+5,,+LIJ.
Both terms describing the transport of 3 1 2 on the right side of (2.35) are analogous to that for the turbulent kinetic energy (2.26); they are due to the turbulent fluctuations and the coherent-mode fluctuations. The firstterm in the vorticity energy exchange mechanism reflects an exchange of ~ : ~ / 2 with n f / 2 , with the same term having opposite signs in (2.35) and (2.32); third terms in this group are the vorticity the second and- exchange - mechanisms between ~ : ~ and / 2 the odd and even modes, Gf/2 and ;f/2, respectively. Again, these terms have opposite signs in (2.33) and (2.34). The intensification of ~ : ~ due / 2 to vorticity stretching is again grouped into three effects. The first is the total rate of intensification, which is shared by - of flow and is due to fluctuations of the turbulent rates other components f / 2 ) and to the of strain n , w : s ; (which is in common with that for R-~ (which modulated fluctuations of the turbulent rates of strain (GI;+ is in common with the same stretching mechanism for the overall coherentmode vorticity intensities). The second effect in the stretching mechanism group is due to self-stretching. The last effect in this group consists of the stretching mechanism of rates of strain of the mean flow and of the coherent modes. The last two terms are the familiar viscous effects. If the coherent
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206
fluctuations are predominantly two-dimensional in a two-dimensional mean flow, the only coherent-mode vorticity intensification from stretching effects is due to the modulated-stretching effects of the turbulence, and &,cZ. Such two-dimensional coherent motions, however, fully participate in the transport of vorticity and, particularly, in the exchanges of vorticity with other scales of motions, as is evident in (2.33) and (2.34). ~
E. THE PRESSURE FIELD Mollo-Christensen ( 1973) emphasized that the pressure fluctuations associated with one scale of velocity fluctuations may in fact have scales larger than the scale of such velocities. The pressure depends on the entire flow field, since it is given by an equation of Poisson’s type (Townsend, 1956) in terms of the double spatial derivatives of the “stress tensor,” u,u, (a special case of Lighthill’s stress tensor T,,for the sound pressure generated by fluid motions (Lighthill, 1952, 1962)). The question that naturally arises is what the role of the pressure field is in the light of the splitting procedure for flow quantities that we already used. We begin with the momentum and continuity equations. Taking the divergence of (2.10) and using (2.9), the equation satisfied by the pressure is (2.36) Following similar splitting and averaging procedures in deriving the momentum equations, we obtain the components of the pressure corresponding to that of the mean flow, coherent and turbulent fluctuations. The mean flow pressure field is given by
a2P
a’
ax;-
ax, ax,
_ _ - -
(U,U,+ ‘,‘,+ti,’,+
(2.37)
u:u;,,
that of the odd-coherent modes by (2.38) and that of the even modes by
a’$
a*
axf - ax, ax,
[ U,’,
+ U,’! + (I&,’
-
~
- GIG,)
+ ( G,’,
-
G,’,)
+ ;,,I.
(2.39)
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207
The instantaneous turbulent pressure fluctuations are given by
d2p’ -ax:-
--
a2
dx,dx,
[( u,+ii, + ; , , u ; + ( U /+ ii,+
~
f , ) u : + ( u : u ; -u : u ; ) ] . (2.40)
The above individual “Poisson’s equation” could also have been obtained from their respective original momentum equations by taking the divergence and then using the continuity relation. The individual Poisson’s equation has solution of the form (2.41) where p represents any of the pressures above in (2.37)-(2.40) and [T,,] is the corresponding “stress tensor” on the right side of the appropriate Poisson’s equation; the pressure takes the field coordinates at x , whereas [T,,]takes the same coordinates at x‘, and d R ( x ‘ ) is a volume element occupied by the “sources.” This illustrates that the pressure, although it could be split consistently into mean, coherent and turbulent contributions, is a field quantity that depends on the appropriate overall flow field. In the present context one is tempted to argue that even if U,, ii, and f, flow fields were absent, j7 and p* will be different from zero because of, respectively, the modulated fine-grained turbulence stresses y; and ;,. However, the modulated stresses are set up by the flow fields of the coherent modes, ii, and f,. The contributions of large-scale structures to the far pressure field, or aerodynamic sound (Lighthill, 1952, 1962), was recently addressed by Mankbadi and Liu (1984), supplementing earlier works on contributions from eddies of relatively low correlation radius.
F. THEREYNOLDS A N D MODULATED STRESSES The importance of Reynolds stresses is illustrated in Section 1I.B through the transport of mean flow momentum by the sum of all the Reynolds stresses of the fluctuations, much in the same way that the modulated stresses transport the coherent-mode momentum. In the energy considerations of Section ILC, the Reynolds stresses of all the fluctuations do work against the rates of strain of the mean flow, thereby effecting energy exchanges between the mean flow and the fluctuations. In a similar manner, the
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208
modulated stresses d o work against the rates of strain of the coherent modes, resulting in the energy exchange between coherent motions and the turbulence. The interactions between the coherent modes and the mean flow and between the coherent modes themselves involve coherent-mode stresses, and these are taken into account in principle by the explicit consideration of the coherent-mode motions. In this section we shall obtain and interpret the transport equations of the Reynolds stresses and the modulated stresses of the fine-grained turbulence. We begin with the momentum equation for the instantaneous turbulent fluctuations u : given by (2.20) and multiply by u i ; then we add a similar equation through exchanging indices i and j , first ( )-phase averaging and then Reynolds averaging, and obtain
transport
“production” from mean
“production” from coherent modes
action of pressure gradients
viscous effects
The kinetic energy equation (2.26), which is a contraction of (2.42), yields similar interpretations for (2.42). Thus the development of Reynolds stresses is dictated by the balance on the right side of (2.42) between transport, “production” from the mean flow and from the coherent motions, the action of pressure gradients and viscous effects. The transport equation of the total modulated stresses ( Fv + ?,,) is obtained from that of ( u i u ; ) by subtracting out the Reynolds mean It has the same form as that obtained by Hussain and Reynolds (1970b) for their monochromatic modulated stresses,
G.
D Dt
-(;.
+ F..) =
’
”
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209
“production” from mean flow
work done by mean atresses against coherent rates of strain
work done by modulated stresses against coherent rates of strain ~~
action of pressure gradients
(2.43) viscous effects
The physical interpretation of (2.43) is similar to that of (2.42). The right side of (2.43) indicates that the transport of the modulated stresses is due to the turbulent fluctuations in terms of the triple correlations and the coherent velocity fluctuations; these are the nonlinear contributions. The linear contribution to transport is due to the advection of the mean stresses by the coherent velocity fluctuations. The “production” of the modulated stresses is due to the work done by the modulated stresses against the mean flow rates of strain and by the mean stresses against the coherent rates of strain; these two mechanisms are ‘‘linear’’ effects. The third “production” mechanism is the nonlinear effect of work done by the modulated stresses against the coherent rates of strain. The remainder in the balance includes the action of the pressure gradients and viscous effects. Upon (( ))-phase averaging, the transport equation for + ?rr)would yield that for ?,. Upon subtraction of the latter from the former, the transport equation for F1, would be obtained. Before stating the individual transport
(c,
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210
equations for iyand ?,/, we shall define certain symbols for ease of presentaand we define tion. Following similar splitting of ( u i u ; )- U J U ; into the simplifying symbolic representations for the tr[ple correlations ~
( u l , u : u : > - u ; u : u := &,,+4aI,
(2.44)
for the action of the pressure gradients (2.45)
(u:
g)
a.,
- u: dP’ = p‘,/
+6,,
(2.46)
and for the viscous “dissipation” (2.47)
The transport equations for the odd-mode i,,and the even-mode ?, are, respectively,
transport
“production” from mean
work done by mean stresses against coherent rates of strain
work done by modulated stresses against coherent rates of strain
(2.48) action of pressure gradients
and
viscous etIects
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"production" from mean
21 1
work done by mean stresses against coherent rates of strain
work done by modulated stresses against coherent rates of strain
work done by modulated stresses against coherent rates of strain
a'
-(p^i;+p^ji)+v--T?i,-2v~,,. ax;, actions of pressure gradients
(2.49)
viscous eltects
Their interpretations are similar to that for (2.43). We note again that the products between even modes and between odd modes give rise to even modes, whereas the product between even and odd modes gives rise to odd modes. This accounts for the nonlinear transport effects as well as the nonlinear production effects in (2.48) and (2.49). The self-interaction of odd modes produce effects upon the even modes and the mixed products of odd/even modes produce effects upon the odd modes. These modeinteraction mechanisms are already noted in the energy considerations.
111. Some Aspects of Quantitative Observations
In order to set the stage for using certain of the conservation principles of Section I 1 to describe the large-scale structures in sections following
212
J. T. C. Liu
Section IV, we shall now discuss some of the features of quantitative results from experiments that would be susceptible to interpretation, either qualitative or quantitative, from a dynamical point of view. This dynamical interpretation would certainly supplement, if not be preferable to, the purely kinematic interpretations and artistic descriptions of the observations. The present section is not intended to be a complete survey of experimental results (see also the more recent surveys of observations by Roshko, 1976; Browand, 1980; Cantwell, 1981; Hussain, 1983; and Wygnanski and Petersen, 1987). We shall place emphasis on the development of the large-scale coherent structures in free turbulent flows as they evolve through interactions with the mean flow, among themselves and with fine-grained turbulence. The coherent-mode amplitudes would evolve in the streamwise direction for the spatial problem, as in the mixing region established in a wind tunnel or water channel. In this case, the coherent-mode periodicities are in terms of frequencies and the mean flow spreads along in the streamwise direction. This situation would correspond to that of most technological applications. Mimicking this situation is the temporal problem, such as the tilting-tube experiment or numerical simulations, where the periodicities are in the streamwise direction, and the mean flow spreading rate is time-dependent, as is the evolution of the coherent-mode amplitudes. The nonlinear temporal problem yields theoretical and computational conveniences, but we have already emphasized that there is no one-to-one transformation to the spatial problem. The emphasis on development and evolution is mainly because of the strong dependence on initial conditions on the part of the coherent modes in turbulent shear flows, recognized theoretically some time ago (Liu, 1971b, 1974a) and for which widespread experimental evidence now exists. In the spatial problem, the coherent-mode amplitudes have spectrally-dependent fixed streamwise distributions. The amplitudes (or wave envelopes) grow and decay, with the lower-frequency components peaking further downstream and higher-frequncy modes peaking closer to the initiation of the free turbulent flow for a given initial energy level (e.g., Liu, 1974a). Under the spatially fixed amplitude or wave envelope, the propagating coherent modes enter from its region of initiation and exit downstream, if at all. The nature of such modes and the spatial distribution of their envelope depend on a number of factors in addition to their own spectral content and initial amplitudes; these factors include the fine-grained turbulence level, the initial mean flow distribution and the length scale in forming the initial Strouhal
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213
number. As such, the description of the local structure of coherent motions would be meaningful only if it is placed in an overall context in order to fix the identity of their otherwise apparent nonuniversalities. In order to illustrate the coherent-mode amplitude development, we show in Figure 1 the results from Favre-Marinet and Binder (1979), who forced a turbulent jet at rather large initial coherent-mode amplitudes. The open circles indicate the root mean square of measured streamwise velocity of the coherent mode, obtained via phase averaging, at the Strouhal number St =f d / U, of 0.18, wheref is the forcing frequency, d the jet diameter and U, the mean velocity at the nozzle exit centerline. The signals were measured on the jet axis. The evolution in terms of x / d , where x is the streamwise distance from the nozzle exit, shows that the signal, which is indicative of the coherent mode amplitude behavior, amplifies and then decays. The turbulence signal, again on the jet axis, is characterized by the root mean square of the turbulent streamwise velocity, and is shown as blackened circles for the case without forcing and as open triangles for the case with forcing. There is an indication that if the turbulence is enhanced, then the jet spreading rate and centerline mean flow decay are also enhanced. On the basis of the theoretical discussions in Section 11, the questions that
0.4
0.3
0.2
0.I
0 FIG. 1.
10
20
Coherent mode and turbulence measurements o n the jet centerline. 0 :unforced,
0, A forced at Strouhal number Sf =0.18 (Favre-Marinet and Binder, 1979).
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214
naturally arise are 1) what is the role of the coherent mode in the enhancement of the turbulence and mean flow spreading rate, and 2) what are the mechanisms leading to the amplification and decay of the coherent mode? To illustrate the coherent mode energy production (and destruction) mechanism through its interaction with the mean flow, we show in Figure 2 the measurements by Fiedler et al. (1981) of this mechanism along the line of most intense mean velocity gradient in a controlled, one-sided turbulent mixing layer. Here w is the vertical velocity, z is the vertical coordinate and U , is the free stream velocity. However, the coherent signal was obtained by filtering at the controlled frequency rather than by phase averaging. One can argue that if the monochromatic coherent signal is as energetic as the overall broadband turbulence, then the energy content of the turbulence at the coherent signal frequency conceivably could be relatively weak. Filtering would then produce a similar result to that derived from phase averaging. Fiedler et al. (1981) compared the measured coherent structure production mechanism, as shown in Figure 2, with that of the total fluctuation production mechanism along the line of maximum mean shear. While the random fluctuation production remained positive, that of the coherent structure increased, reflecting the energy extraction process, and then decreased below
0.10
0.05
0
I
I FIG.
I
I\
20
40
loss
2. Measured streamwise development of fluctuation production mechanism along the
line of most intense mean velocity gradient in a turbulent mixing layer. x: coherent mode production mechanism; 0 : overall production mechanism ( Fiedler, Dziomba, Mensing, and Rosgen, 1981).
Understanding Large-Scale Coherent Structures
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the axis, indicating the negative production or return of kinetic energy to the mean shear flow. These results typify similar negative production mechanisms observed by Hussain and Zaman (1980), Oster and Wygnanski (1982), and Weisbrot (1984) (see also Hussain, 1983). Such observations are not entirely surprising from the perspective of ideas from hydrodynamic stability for developing shear flows. The development of this energy exchange mechanism between the mean flow and coherent structure is very similar to that in a laminar free shear flow (KO, Kubota and Lees, 1970; Liu, 1971b), except that the rate of this development is significantly modified in the more rapidly spreading turbulent shear flow. Not only the “negative production” mechanism itself, but also the observed evolution of the coherent mode and mean flow energy exchange rate as seen in Figure 2, is entirely expected from theoretical considerations (e.g., Liu, 1971b; Gatski and Liu, 1980; Mankbadi and Liu, 1981). This negative production mechanism is only partially responsible for the decay of the coherent mode. Fiedler et al. (1981) also showed that the shear layer spreading rate is altered by the enhanced coherent mode. However, we shall illustrate a similar observed effect of coherent-mode development through the use of results from Ho and Huang (1982). Although the shear layer in Ho and Huang (1982) is one undergoing transition, it is used here to illustrate the role of fluctuations on the mean flow spreading rate. (A collection of spreading rates from various laboratories, though not exhaustive, appears in Ho and Huerre, 1984, Figure 24.) Ho and Huang (1982) measured mean shear flow thickness developing as a function of the streamwise distance; their results are shown in Figure 3. The conditions correspond to their “Mode 11,” in which the subharmonic component (2.15 Hz) is forced at a streamwise velocity (route mean square) of about O.lOo/~of the averaged upper and lower free streams and at an R parameter (ratio of the upper and lower stream velocity difference to the sum) value of 0.31. The steplike structure of the mean flow thickness is fairly obvious. The thickness of disturbed turbulent shear layers also exhibits such steplike behavior (Fiedler et al., 1981; Weisbrot, 1984; Fiedler and Mensing, 1985; see also Wygnanski and Petersen, 1987). The corresponding coherent-mode energy measured by Ho and Huang (1982) is shown in Figure 4, where E(f) is the kinetic energy due to the streamwise velocity fluctuation associated with each of the frequencies, integrated across the shear layer. As we shall see later, such a quantity, including all the contributions to the coherent-mode kinetic energy integral, is related to the amplitude or wave envelope of each mode. Figure 4 indicates that the peak of the fundamental component (4.30 Hz)
J. T. C. Liu
216
0.25
0 FIG. 3. I I”).
Streamwise development of mixing layer thickness ( H o and Huang, 1982, “Mode
10-1 2.15 Hz 10-2
I 0-3
4 . 3 0 Hz
E (f) I 0-4
6.45 H z
I 0-5
I
I
10
20
I
I
x(cm)
30
FIG. 4. Streamwise development of coherent mode energy (u-component only), corresponding to the shear layer thickness development in Figure 3 (Ho and Huang, 1982, “Mode 11”).
Understanding Large-Scale Coherent Structures
217
energy is associated with the first plateau of the shear layer thickness in Figure 3; the peak of the subharmonic energy is associated with the second plateau in the shear layer thickness further downstream. The linear growth far downstream is attributable to turbulence dominating the rate of spread. As will be shown more formally in the next section, it is not difficult to demonstrate that the shear layer spreading rate d 6 / d x can be obtained from the mean flow kinetic energy equation, integrated across the shear layer (see, for instance, Liu a n d Merkine, 1976; Alper a n d Liu, 1978), with a change in sign and retaining only the dominant energy exchange mechanisms:
In a purely laminar viscous flow, the shear layer will spread as long as kinetic energy is removed from the mean flow via viscous dissipation. In a transitional shear flow, this viscous spreading rate would be augmented by the emergence of finite-amplitude coherent disturbances. In a turbulent shear flow, a highly enhanced coherent mode would similarly augment the turbulent spreading rates. If we denote - the fundamental disturbance-mode Reynolds stress contribution by -126,the magnitude of the energy exchange mechanism -69 d U / d Z very nearly follows the development of the wave envelope as appearing in Figure 4. Its value along the line of most intense mean shear, illustrated by the measurement of Fiedler et al. (1981), very nearly represents the entire sectional integral of this quantity. Thus the first peak of d 6 / d x is associated with the vigorous transfer of energy from the mean flow to the fundamental. The shear layer thickness itself, which is a running streamwise integral of the energy exchange mechanism, reaches a plateau after the streamwise peak of the fundamental component. The second, distinct plateau follows from similar reasoning for the subharmonicmode energy transfer mechanism -I,% d UaZ. Apparently, after the coherent modes have subsided relative to the turbulence, the linear growth is attributable to -u‘w’ d UdZ. The development of the negative production mechanism on the part of the coherent mode discussed earlier, which corresponds to “damped disturbances” in the hydrodynamic stability sense for dynamically unstable flows, would make a negative contribution to d 6 / d x , thus contributing t o a decrease in 6. This decrease in 6 would be obviously observable if the negative production rate were the dominant energy exchange mechanism within a streamwise region (see Weisbrot, 1984; Fiedler a n d Mensing, 1985). ~
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J. T. C. Liu
Although not decoupled from the direct interactions between coherent modes and fine-grained turbulence, the production of the fine-grained turbulence by the mean motion appears both experimentally (e.g., Fiedler et al, 1981) and theoretically (e.g., Liu and Merkine, 1976; Alper and Liu, 1978; Mankbadi and Liu, 1981) to be devoid of the large-scale amplification and negative production as was found for the coherent modes. Consequently, the turbulence energy, excluding the coherent-mode contributions, appears to be developing monotonically, if at all, even in the nonequilibrium region of coherent mode/turbulence/mean flow interactions. The contribution of -u’w’ d U / d Z to the shear layer spreading rate eventually becomes very nearly constant along the streamwise direction, so that the linear spread of the shear layer is due to this mechanism. For the transition problem (e.g., Ho and Huang, 1982) or the forced turbulent shear layer (e.g., Weisbrot, 1984; Fiedler and Mensing, 1985), the initial steep steplike development of the shear layer is thus conclusively reasoned from the above discussion to be due to vigorous energy transfer to the coherent modes. The arrest of this steep development is due to the decay of the coherent disturbance in the downstream region where production becomes small or negative. The existence of the plateau region between steep increases of 6(x) indicates that the production mechanism of the first mode has subsided prior to the rise in production of the subsequent mode (or finegrained turbulence). The downstream persistent linear growth of the shear layer, again from our present discussions, indicates that the coherent mode activities have subsided and that fine-grained turbulence is now responsible for the shear layer spreading rate. This spreading rate is not necessarily universal in that it has a n upstream dependence on whatever nonlinear coherent mode interactions have taken place (e.g., Alper and Liu, 1978; Mankbadi and Liu, 1981).This lack of universality in the measured turbulent shear layer spreading rate, summarized, for instance, by Brown and Roshko (1974) and by Ho and Huerre (1984), is thus not surprising but expected. The basic two-dimensional free shear flow appears to support dominantly two-dimensional coherent modes, with its vorticity axis perpendicular to the mean motion. In the following sections, our theoretical discussions will interpret the role of such observed dominant modes as well as the threedimensional coherent modes in terms of observed spanwise standing waves (e.g., Konrad, 1977; Bernal, 1981; Breidenthal, 1982; Jimenez, 1983; Browand and Troutt, 1980, 1984). An issue to be addressed with the three-dimensional modes is that the spanwise wavelengths appear to increase
Understanding Large-Scale Coherent Structures
219
downstream, in a somewhat similar fashion to the formation of longer, streamwise wavelength of frequency subharmonics.
IV. Variations on the Amsden and Harlow Problem-The Temporal Mixing Layer
A. INTRODUCTORY COMMENTS
Amsden and Harlow (1964) considered the “temporal” mixing layer formed by parallel opposite streams. The disturbance is two-dimensional and horizontally periodic. The growh in amplitude and the spreading of the Reynolds mean motion is in time. However, Amsden and Harlow (1964) considered the entire flow velocity as a single dependent variable, encompassing the Reynolds mean and the disturbance, and solved the unsteady Navier-Stokes equations with horizontal periodic boundary conditions. The study of mean flow and disturbance interactions can always be obtained from the numerical result by performing the Reynolds average, which is the horizontal average in this case. The utility of the idea of using the total flow quantity as the dependent-dynamical variable is particularly apparent for the simple temporal mixing layer problem, and has been fully exploited by Patnaik et al. (1976) in the case of stratified flow. The two-dimensional problem (Amsden and Harlow, 1964) for a homogeneous fluid, including the consideration of passive scalar advection and diffusion, was given greater detailed consideration by Corcos and Sherman ( 1984). The secondary instabilities in the form of spanwise periodicities, solved on the basis of linearizing about the two-dimensional motion, was considered by Corcos and Lin (1984) and Lin and Corcos (1984). These are still relatively low Reynolds number problems, and the participation of fine-grained turbulence was not intended. The dominant two-dimensional coherent mode problems in turbulent shear layers have been studied by Knight (1979) and Gatski and Liu (1977, 1980) using different closure models for the fine-grained turbulence; the coherent-mode agglomeration problem was studied by Murray ( 1980) and Knight and Murray (1981) with an eddy-viscosity model. The basic aim of the present section, through Reynolds-averaged diagnostics obtained from results recovered from the numerical solutions, is to motivate the subsequent
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J. T C. Liu
approximate considerations directed towards spatially developing turbulent free shear flows. This will naturally lead to the discussion of the role of linear theory in Section V, which in turn will form a bridge to the discussion of spatially developing free shear flows in Section VI.
B. THE“TURBULENT” AMSDEN-HARLOW PROBLEM The problem of the presence of a coherent structure in a turbulent mixing layer, as considered by Gatski and Liu (1980) in the “spirit” of Amsden and Harlow (1964), shall be given some attention because of the physical information that can be extracted from the results. In the present context, the coherent flow variables to be solved would be %,= U , + ( u ’ , + i , ) ,
?7J = P + (p’+p^).
Their governing equations are obtainable from the continuity and NavierStokes equations, (2.9) and (2.10), by substituting
+
u, = 021, u : ,
p
=P+p‘,
(4.2)
and taking only the phase average ( ); the Reynolds average is not performed at the outset. The resulting equation for 021, would be coupled to the phase-averaged, total stresses
9,, =(u:u;)=
u:u;+(;,,+F,).
(4.3)
which involves no explicit Reynolds stresses u : u j ,
The system OU,, 9,Pi?, is identical in form to those obtained by Reynolds (1985) for U , , P and (u’, + 12, + u : ) (u’, + G, + u ; ) with the phase average replaced by the Reynolds average, as was pointed out earlier (Gatski and Liu, 1980; Liu, 1981). The OU,, 9,%,, system thus has the large-scale coherent structure taken out and considered explicitly, as suggested by Dryden (1948). The stresses Pi?, = (u:u:) involve only the “real turbulence”; thus second-order closure models, when suitably found, would most likely be more universal than the prevailing closure models for the Reynolds stresses (u’, + G, + u : ) ( u’, + 6, + 4;) that include the contributions from the large-scale coherent structures. The latter are now well recognized as being non-universal because of the non-universality of hydrodynamic instability mechanisms (Liu, 1981). As was shown
Understanding Large-Scale Coherent Structures
22 1
in Section 11, conservation equations can always be obtained for U,, U, and z2,, and transport equations for u : u : , F, and ?,. Within the %!, B, 3,, framework, however, the study of mean flow, coherent mode and finegrained turbulence interactions can always be obtained by performing the Reynolds average after the results are found. As we have emphasized already, this procedure is really only practicable for the simplest problem, that is, the time-dependent, mixing layer between two opposite parallel streams. This problem was considered by Gatski and Liu (1980). The problem consists of the interaction of a monochromatic component of the large-scale coherent structure (so that U, + 6, reduces only to u”,, for example) with the fine-grained turbulence in a temporal mixing layer of horizontally homogeneous and oppositely directed streams. The coherent mode is horizontally periodic and develops in time. The physical significance of this class of problems is that it strongly resembles, but does not exactly correspond to, the spatially developing free shear layer in observations. The coherence enters into the horizontal periodic boundary conditions, and the numerical problem is thus well-defined, in contrast to the numerical problem for the spatially developing mixing layer, which is not as well-defined because of the unknown but necessary downstream boundary conditions. In this problem the vorticity axis of the large-scale structure lies in the spanwise, y-direction, with the velocities %, W i n the streamwise and vertical directions, x, y, respectively. The spanwise velocity ‘V is taken to be zero. (Relaxation of the monochromatic two-dimensional coherent structure is certainly possible in order to accommodate subharmonics, the coherent streamwise vortical structures and the resulting generation of ‘V and spanwise variations of 021 and %”.) Here, all spanwise gradients of the phaseaveraged quantities are also zero. Because of the two-dimensional coherent motions, it is possible to define the stream function ~
The vorticity is then related to the stream function via 0 = - V 2 q , where
v’ = d2/dX’ + d2/dz‘
(4.5)
is the Laplacian in the x, z plane. The nonlinear, total-coherent vorticity equation then gives
V’?,
+ ‘ P z V 2 q l -‘PXV”Pz
= ( U ’ W ’ ) , ~- ( u ‘ w ‘ ) ~+, ( ( W ’ ~ ) - ( U ’ ~ ) ) ~ = (4.6) ,
J. T. C. Liu
222
where subscripts indicate the appropriate partial differentiation. If we were to study the transition problem, (4.6) will then be augmented by the viscous diffusion mechanism V 4 W / R e on the right side. Here all velocities and coordinates are made dimensionless by the free stream velocity and the initial shear layer thickness (the pressure is made dimensionless by the free stream dynamic pressure). Viscosity effects have been neglected in the large-scale structure vorticity equation (4.6). Thus the phase-averaged stresses on the right of (4.6) take the place of viscous diffusion in the turbulent shear layer problem. The two-dimensional vorticity equation (4.6) for 0 = - V 2 V is merely the y-component of the total coherent vorticity (52, + &, 4,) given by Equations (2.28)-(2.30) in the absence of the vorticity stretching/tilting mechanism and with the viscosity effects neglected. The net phase-averaged vorticity transport contributions from the turbulence on the right sides of (2.28)(2.30) would give
+
a
- -(U;,
:).
axf Its two-dimensional form, through the use of the continuity condition, reduces to the form on the right side of (4.6), in terms of the phase-averaged stresses ( u : u i ) . Their transport equations are identical in form to the Reynolds system for u : u ; (Gatski and Liu, 1980; Liu, 1981), ~
production
redistribution
transport
dissipation
where the Reynolds number Re is based on the free stream velocity and initial shear layer thickness, and 8,, is the usual Kronecker delta. In the present problem, (4.7) is equivalent to the sum of (2.42) and (2.43) for ( u : u ; )= u : u ; + ,(; + ?,,). In Catski and Liu’s (1980) framework, the dominant large-scale coherent structure is sorted out distinctly from the fine-grained turbulence through phase averaging at the outset. This procedure is in
Understanding Large-Scale Coherent Structures
223
contrast to the prevalent numerical simulation methods, where the entire flow is decomposed into succeeding, neighboring Fourier modes corresponding to the horizontal periodic boundary condition. For lower Reynolds numbers the simulation is “exact,” whereas for high Reynolds numbers an eddy-viscosity subgrid closure is invoked (Reynolds, 1976; Riley et al., 1981 ) . In order to discover coherent structures, additional limited spatial averaging is needed, as was done for the turbulent boundary layer problem by Kim (1983, 1984) and Moin (1984). In the simpler, explicit calculation of the dominant coherent structure in the mixing region (Gatski and Liu, 1980), the phase-averaged, fine-grained turbulent stresses appear in the coherent structure vorticity equation, as would the Reynolds-averaged stresses in the Reynolds (1985) system. In this case, some form of the Reynolds stress closure arguments (e.g., Lumley, 1978) could conceivably be adapted to the closure problem for (4.7). The eddy-viscosity models were purposely avoided, primarily because the consequences of such a model imply the a priori regulation of the direction of energy transfer to the smaller scales. Gatski and Liu (1980) used the formalism of Launder et al. (1975) to obtain the energy transfer mechanism between the coherent mode and turbulence, i,,(du”,/dx, + d u ” , / d x , ) , on the basis that the coherent mode dynamics are obtained from conservation equations, coupled to the turbulent stresses via their transport equations. The functional forms of the Reynolds stress closure should apply, though the detailed closure constants might not. However, the behavior of the fine-grained turbulence, with the non-universal coherent structure subtracted out, would be much more universal than the treatment of all oscillations, including the coherent structures, as “turbulence.” In Gatski and Liu (1980), the transport equations for phase-averaged stresses include those for the single shear stress ( u ‘ w ’ ) , three normal stresses ( u ” ) , ( u ” ) and (w”) and a modeled transport equation for the rate of viscous dissipation ( E ) . The fine-grained turbulence is three-dimensional, but the spanwise derivatives, d( u : u : ) / d y , vanish. The vertical boundary conditions require that all flow quantities vanish far away from the shear layer. Horizontal periodic boundary conditions are applied to all phase-averaged quantities, with the periodicity dictated by the wavelength of the initial coherent mode chosen. The initial conditions are arrived at through an initialization process described in Gatski and Liu (1980). In absence of the coherent structure, the Reynolds problem, consisting of the hyperbolic tangent type mean shear flow and the Reynoldsaveraged stresses and dissipation rate, is solved from “ t = -a’’ to t = t,,
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J. T. C. Liu
when self-preservation is very nearly achieved. This solution ensures selfconsistency among the Reynolds-mean flow quantities when the initial conditions are to be imposed. The coherent disturbance imposed initially is obtained from the Rayleigh (“inviscid”) equation corresponding to the initialized mean velocity profile at an initial wave number corresponding to the most amplified mode for this profile ( a ~ 0 . 2 7 5 ) The . initial kinetic energy content of the turbulence used in the computations was E,(O) = 1.2 x lo-’, obtained from the initialization process; that of the coherent mode was ~ 5 , ( 0 ) = 1 0 - ~where , E, and El are defined by (4.8) and (4.9), respectively. The disturbance is considered to be suddenly imposed, with the corresponding phase-averaged stresses and dissipation rate (which require finite time to respond) set equal to zero. The interaction between the Reynolds mean motion U, the coherent structure 6, and the Reynoldsaveraged fine-grained turbulence 1.4: u i can be studied after the numerical results are obtained as already emphasized. In the following, the physical features of the problem are discussed in terms of the computational results of Gatski and Liu (1980).
C. DIAGNOSTICS O F N U M E R I C ARESULTS L V I A REYNOLDS AVERAGING The two-dimensional large-scale structure energy content is (4.8) where the overbar is the Reynolds average and is here the horizontal average over one wave length. Similarly, the fine-grained turbulence energy content is
_ _ _ (4.9) The mean flow kinetic energy defect is defined as
where the dimensionless outerstream velocities are U,, = *l in the present notation. The development of El, E, and Em with time provides the insight into the nonequilibrium interactions among the three “components” of the
Understanding Large-Scale Coherent Structures
225
energy. To this end, the diagnostics of the exact energy integral equations and the energy exchange mechanisms are obtained from the computational results via Reynolds averaging. The energy integral equations, which follow for a single mode from (2.21), (2.24)+ (2.25) and (2.26), are:
-
dE,,, - -I, dt
--
-
I;,
-
dE/ - I/>- I / , , dt
_-
(4.11) (4.12) (4.13)
We note that (4.11)-(4.13) were the starting point for an approximate consideration of the problem discussed in Liu and Merkine (1976). The energy exchanges between the mean flow and the fluctuations are given by the integrals (4.14) (4.15)
the energy exchange between the large-scale coherent structure and finegrained turbulence is given by the integral
The integrands in (4.14)-(4.16) have in common the product of stresses with the appropriate rates of strain. The fine-grained turbulence dissipation integral is, with the integrand identified with the last term on the right of (2.26), = '
5'
Pdz.
(4.17)
-1
Consistent with (4.6), viscosity effects on the large-scales are not included. The sum of (4.11)-(4.13) gives
d (E,,,+ E / + E , ) = -F; dt
-
(4.18)
the overall kinetic energy decays according to the rate of viscous dissipation of the fine-grained turbulence.
J. T. C. Liu
226
In spite of the local regions where energy is transferred from the finegrained turbulence to the large-scale coherent structures, as indicated by structural results (see Figures 9 and 11-13 in Gatski and Liu, 1980), the integral I,, > 0 indicates that the global energy transfer is from the large to the fine scales of fluctuations. The time development of this integral is shown in Figure 5, which indicates that I,, peaks in the vicinity when the global energy transfer from the mean motion to the coherent mode changes sign. This latter mechanism, which is the integral of the energy exchange mechanism between the mean flow and the large-scale coherent structure, is also shown in Figure 5; f,, first increases, with energy feeding from the mean flow into the coherent mode, and then decreases to below the axis as time increases, indicating an energy transfer back to the mean flow. The evolution of such features is a familiar one in hydrodynamic stability problems of developing shear flows whether fine-grained turbulence is present or not. In laminar flows, the development of a positive and then a negative disturbance production mechanism was first uncovered by KO, Kubota and Lees (1970) in their approximate consideration of spatial, finite-disturbances in the laminar wake problem. Similar features were also recovered in the extensions of the Amsden and Harlow (1964) computational problem by Patnaik et a/. (1976). It was also anticipated and shown that 0.04
r
0.02
0
- 0.02 FIG. 5 . Evolution o f energy exchange mechanisms between the large-scale structure a n d the mean Row (<,) a n d the fine-grained turbulence ( I , , ) .
Understanding Large-Scale Coherent Structures
227
the development of the positive and then negative coherent structure production mechanism would also exist in free turbulent shear flows (Liu, 1971; Mankbadi a n d Liu, 1981). This is essentially a n “inviscid” or “dynamical” instability phenomenon in the hydrodynamic stability sense and can be anticipated when the kinematics of the growth rates from linear hydrodynamic stability theory are applied to the developing free shear flow through scaling by the local shear flow thickness. In the temporal problem for a fixed wave number disturbance, the local wave number increases as the shear layer grows in time, causing the growth rates eventually to become negative. Similar interpretations also hold for the spatial problem, where the local rescaled frequency increases with the downstream distance and the disturbance is eventually advected into the damped region. This linear reasoning anticipates the nonlinear results in Figure 5. The experiments of Fiedler et al. (1981) reproduced in Figure 2, taken along the line of most intense mean shear, very nearly approximates the integral I,, (see also Weisbrot, 1984). The theoretical results from the dynamics of the problem (see also Liu, 1971; Mankbadi and Liu, 1981) a n d the experimentally observed evolution of this energy exchange mechanism are strikingly similar. There is no mistaking that this “damped disturbance” phenomenon is one derived from ideas in hydrodynamic stability theory. The kinematical interpretation in terms of possible eddy orientations (Browand, 1980) are summarized in Hussain (1983). The time evolution of the coherent-mode energy d E , / d t is thus the difference f,,-Z,, according to (4.12), and is also shown in Figure 5. It is clear from this numerical example that the fine-grained turbulence produces a global “turbulent dissipation” and augments the “damped disturbance” mechanism in causing the demise of coherent energy with time shown in Figure 6. The earlier vigorous amplification is due to extraction of energy from the Reynolds mean motion. The peak in the coherent-mode energy El/ E,, corresponds to the vicinity when f,, changes sign and I,, is maximum. The fine-grained turbulence production rate 1; starts out slightly larger than the dissipation rate F , as shown in Figure 7. This accounts for the initial small rate of growth of the turbulent kinetic energy E l / El,,shown in Figure 8. The production of turbulence from the mean flow is made more efficient by the presence of the coherent mode in the initial development stage, although the direct energy transfer to the turbulence from the coherent mode is relatively small. But the net difference between production and destruction gives rise to a nonequilibrium development of the turbulence energy that evolves from a n initial self-similar behavior to a new, higher
-
J. T. C. Liu
228
200
-
100
-
0 W
-
\
W
0
F I G .6
0.15
2
4
t
Evolution of large-scale coherent structure energy.
r
0.10
0.05
0
-0.05
-0.10 FIG. 7. Evolution of fine-grained turbulence energy production ( I ; , ) , viscous dissipation (4’) and energy transfer from the large-scale coherent structure ( I , , ) .
20 0
c
W \
J I0
FIG. 8. Evolution of fine-grained tur. bulence energy.
0
2
4
t
Understanding Large-Scale Coherent Structures
229
level of self-similar behavior as shown in Figure 8. In terms of time development, the “burst” of fine-grained turbulence has taken place at the expense of the coherent mode. The physical pictures derived here strongly suggest that similar physical mechanisms, except for details, hold for the turbulent jet experiments of Favre-Marinet and Binder (1979) depicted in Figure 1, where the observed coherent mode grows and decays while the turbulence is enhanced. As far as the coherent mode is concerned, the production and “dissipation” are generally not in balance during the time evolution in free shear flow problems. Thus marginal stability ideas would not be as useful here as they would be for confined flow problems (e.g., Barcilon et a/., 1979).
D. EVOLUTION OF LENGTHSCALES The following definition of the shear layer thickness (which is not unique) is used by Gatski and Liu (1980): S(t)=
[
z2 a U l a z d;]‘’’ U l d z dz
It is normalized by its initial value and is shown in Figure 9. Initially, the growth is self-similar in that S t. Its subsequent modification is due to the
-
12
8
4
0
2
4
FIG.9. Evolution of length scales: shear layer thickness ( 6 ) , fine-grained turbulence scale ( L e ) , large-scale coherent structure closed streamline height ( H ) .
230
J. T. C. Liu
nonlinear, nonequilibrium interactions that the Reynolds mean flow directly engages with the coherent mode and the fine-grained turbulence. Eventually, after the coherent mode subsides, the spreading is self-similar again ( 6 t ) . From the structural results, the height of the closed streamline H,normalized by its initial value H , , is also shown in Figure 9; it reaches a maximum at about t = 2 and subsequently decreases and is similar to the development of the coherent-mode energy with time. There is another “detail” of observations that can be qualitatively understood from Gatski and Liu (1980), namely that in the optical observations of Brown and Roshko (1974), the graininess of the fine-grained turbulence appears t o enlarge as the shear layer spreads downstream. The size L, of the fine-grained turbulence from Gatski and Liu (1980) can be estimated by using a local equilibrium argument such that the eddy energy transfer rate down to the size L, just balances the viscous dissipation rate. This leads to L, E;’*/F. Shown in Figure 9 is L, normalizing by its initial value LEOas it evolves in time. Although L, rapidly decreases initially, as the coherent-mode energy El passes its maximum at about t = 2 (Figure 6 ) , the scale of the fine-grained turbulence begins to increase with time at a rate similar to that for the shear layer thickness 6. In fact, L,/6 remains very nearly constant after t 2 1.50. The coarsening of the graininess of the fine-grained turbulence derived here accompanies the spreading of the shear layer (Gatski and Liu, 1980). This appears to be entirely consistent with observations of Brown and Roshko ( 1974) that, as the observed “strength” of the coherent mode weakens, the spreading of the shear layer is maintained via the coarsening of the graininess of the fine-grained turbulence.
-
DETAILS E. SOMESTRUCTURAL
We shall refer to Gatski and Liu (1980) for the details of the time evolution of structural results in terms of the phase-averaged stream function and vorticity contours. We will illustrate here the instantaneous T and fl contours in Figures 10 and 11, respectively, for t = 1.50 when the coherent-mode energy is at its maximum and d E , / d r = 0. There are strong vorticity nonuniformities within the “cat’s eye” as would be expected for the t = 0 ( 1 ) nonequilibrium stages of development. A similar nonlinear critical-layer theory (Benney and Bergeron, 1969) for the present class of problems would require the vorticity within the cat’s eye region to be uniform. This might be achieved as t + co for the idealized single event of the monochromatic
23 1
Understanding Large-Scale Coherent Structures 2.076 x 10-1,
z
36 9 . 4 . 6 2 2 x 10-I
t
= 1.5
1 . 8 7 5x
10-1,
,1.487x
10-1
\
3.409~10-~
0
+ 4.619 x
- 0.5836 X
80
lo-'
2 . 2 8 4 x 10-11
'I 012 x
1.1
10-1
FIG.10. Large-scale coherent structure streamlines at I = 1.50.
problem as the fine-grained turbulence smooths out the inner coherent vorticity distribution. The coherent structure at the t + 00 neutral stage would have been so significantly weakened that its participation in the shear layer dynamics would be of questionable interest. t
= 1.5
z = O 5836
0
-0.5836 x
1
I
=o
X'I
FIG. 11. Large-scale coherent structure vorticity at
I =
1.50.
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J. T. C. Liu
Other interesting structural details of the phase-averaged quantities are those pertaining to the energy conversion mechanisms; the consequences of their Reynolds average have already been discussed. At the phaseaveraged level, the conversion of overall coherent mode energy to the horizontal fine-grained turbulence energy (u”)/2 is achieved primarily through the work done by the modulated turbulent shear stress against the coherent rate of strain, -(u’w’)dU/az. The contours of both of these quantities are shown in Figure 12a, b for the instant t = 1.50. The conversion mechanism due to the normal stress -(u”)dU/dx is significantly weaker in this case and is not shown (see Gatski and Liu, 1980). The rate of energy transfer in Figure 12a shows that there are local regions where turbulence energy is converted back to the coherent-mode. The contributions to the vertical part of the turbulence energy (w”)/2 come from the dominant normal stress conversion mechanism -( w”)d W / d x . These patterns are similar to those for (uf2)/2, and we refer again to Gatski and Liu (1980) for details. These are the direct energy transfer mechanisms between the fine-grained turbulence and the two-dimensional overall coherent mode. The three-dimensional turbulence also includes the spanwise contribution to its energy ( ~ ’ ~ ) /This 2 . is produced and maintained via the isotropizing mechanism of the pressure-velocity strain correlation ( p ’ d u ’ / d y ) .The contours of these quantities are shown in Figure 13 for t = 1.50. The quantity ( p’dv’/dy), however, was not explicitly calculated but was approximated via closure arguments, including the effects attributable to local rapid distortion due to the large-scale coherent structure (Gatski and Liu, 1980; Launder et a/., 1975). Although this mechanism converts energy to (u”)/2 on an overall basis, there are nevertheless local regions in which this energy conversion mechanism reverses sign.
V. The Role of Linear Theory in Nonlinear Problems
A. INTRODUCTORYCOMMENTS
The role of linearized theory in finite-amplitude, weakly nonlinear hydrodynamic stability problems is well known (Stuart, 1958, 1960, 1962a, b, 1967, 1971a, 1972); in particular, the parallel flow problem there serves as a valuable guide to the class of problems of interest here. In order to gain the necessary perspective on how the linear hydrodynamic stability problems
Understanding Large-Scale Coherent Structures t = I
233
5
z = 0 6079
0
- 0 6079
=o
x
a=I
z = O 60I79
0
179 -0.60 .
I
x =I
X I 0
(b) FIG. 12. Horizontal contribution of the phase-averaged turbulent kinetic energy and its dominant production mechanism at I = 130: ( a ) ( u ' * ) / 2 ;(b ) -(u'w')dU/dz.
fit into and can be utilized in nonlinear problems involving developing flows, we purposely preceded this section by the discussion of a simple nonlinear problem in Section IV. The temporal mixing layer discussed in that section contains a vast richness of physical processes which could still be explored further. However, numerical extensions of the problem, while
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t’l
210.6079
5
0
-0.6079
=o
x
x =I
~10.6079
0,
926
I
1
0
7
I
-0 6 0 7 9 1
x=o
x - l
(b) FIG. 13. Spanwise contribution of the phase-averaged turbulent kinetic energy and its “production” mechanism at t = 1.50: ( a ) (uf2)/2; ( b ) ( p ’ a u ’ / d y ) .
readily possible, are nevertheless tedious. It would be most worthwhile to explore certain ideas and concepts derivable from the problem of Gatski and Liu (1980) in order to make progress, via simplification, towards the ambitious possibilities of describing the class of problems that involve real, spatially developing flows found in the laboratory and in practical devices involving mixing-controlled situations. For the purpose of introducing ideas,
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we shall mingle in our discussions ideas derived from the temporal problem with observable quantities in the laboratory without further qualifications. The energy content of the coherent mode E l , introduced in Section IV, equation (4.8), is a quantity measurable in the laboratory (Ho and Huang, 1982) for different modes. For a given initial energy level and shear layer thickness, this quantity depends on the mode content in its nonequilibrium evolution. Thus in the laboratory, the energy content of high frequency modes peaks further upstream than those of lower frequency modes (or, the longer wavelength disturbances peak later in time). As such, El is essentially related to an amplitude of the disturbance. It is the “slowly varying” wave envelope bounding the “fast” oscillations of the wave motion. There is strong observational evidence that for nonlinear problems, although the wave envelope has to be obtained from a nonlinear theory with the physics of the problem participating fully, the wave characteristics are obtainable from the kinematics of a locally linearized theory (Michalke, 1971). However, considerable confusion concerning the role of the linear theory would still result from the nonuniqueness of the “amplitude” associated with the linear solution, the lack of distinguishability of the “wave envelope” from the “wave function” and the relative sensitivity of the wave envelope to the real physical mechanisms in the shear flow evolution. Thus, an extended discussion along these lines might help unravel some of the possible confusion that might result from reading the current literature (Wygnanski and Petersen, 1987).
B. NORMALIZATION OF T H E WAVEAMPLITUDE
In order to bring in the role of the linear theory, it is essential that we provide the distinguishable roles of the wave envelope or amplitude, the hydrodynamical instability wave functions and the physically sensible manner in which such functions are to be normalized. If we introduce a coherent-mode energy density as E l / 6 = IAI’, where IAl2would be a function of the mode number and the mean motion evolutionary variable (wave number and time for the temporal problem, frequency and the streamwise distance for the spatial problem), then according to our definition, (4.8) gives
We are using the two-dimensional coherent mode for simplified illustration;
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these ideas are easily extendable to include three-dimensional coherent modes. I f we further assume that the velocities fi, G can be represented by the linear hydrodynamic stability theory, then the linear eigenfunctions are represented by the disturbance stream function 4 in terms of local variables 6 = x/6, 5 = z / 6 , with the local wave number a referred to 6. Then, in terms of the temporal mixing layer notation, for instance,
(5.2) where 4’ is the 6-derivative of 4, the local [-derivative of 4 is given by -ia4, and C.C. denotes the complex conjugate. In this case, the physical wavenumber is fixed but the local wavenumber changes as the shear layer thickness, 6, grows. If we substitute (5.2) into (5.1), then (5.3) Thus, in order that we consistently attribute IAl’ = E , / S as the energy density, the local eigenfunctions must be normalized locally by the condition
(5.4) This addresses an appropriate normalization for the wave functions of the local linear theory for which the wave envelope would be given consistent physical meaning. Such would not be the case with the “equal area” normalization (e.g., Wygnanski and Petersen, 1987).
C. GLOBALENERGYEVOLUTION EQUATIONS The above discussions follow those ideas put forth by KO, Kubota and Lees (1970) in their generalization of the shape assumption ideas of Stuart (1958) to real, developing free laminar shear flows with strongly amplified disturbances. While the cross-stream shape of the coherent mode would be given by the (properly normalized) linear theory, the overall evolution via must be solved by the nonlinear theory with the the wave envelope, [A( proper physics involved. In KO, Kubota and Lees (1970), the evolution of (AI2follows naturally from the disturbance energy integral equation. This is solved jointly with 6, which follows from the mean flow kinetic energy equation following a similar shape assumption for the mean velocity profile.
?) I 2,
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The problem is relatively much simpler in the absence of the participation of fine-grained turbulence in the dynamics. From the diagnosis of the numerical results of Gatski and Liu (1980) for the turbulent shear layer, we see that if we were to obtain the evolution of ( A ( ’ = E l / & then the “exact” envelope equations (4.1 1)-(4.13) would tell us that
(5.7) From the diagnosed numerical results (Gatski and Liu, 1980) discussed in Section IV, any approximate calculation for the wave envelope IA12 must necessarily involve the participation of the turbulent kinetic energy content E , and the mean flow kinetic energy defect Em. Thus, strong nonlinear interactions occur among the “envelopes” IAI’, El and Em independently of whatever version of the linear hydrodynamical stability equations might have been used to generate the eigenfunctions 4 in a possible assumption such as (5.2). We emphasize here that for nonlinear problems, the wave envelope must necessarily be obtained with the simultaneous nonlinear interactions between the mean motion, fine-grained turbulence and largescale coherent structure properly (though approximately) taken into account.
D. SUBSIDIARY PROBLEMS. THE ROLEO F
THE
LINEARIZED THEORY
To further interpret (5.5)-(5.7) in terms of approximate considerations, and in order to make practical the modeling of the “envelope” evolution problem, we further postulate, again using the temporal mixing layer as illustration (Liu and Merkine, 1976), that the mean flow behaves like
J. T. C. Liu
23 8
where F ( 5 ) could conveniently be tanh 5 or other function of 5. From observations, the similarity behavior is almost established with the establishment of the mixing region profile. We introduce a similar energy density for the fine-grained turbulence as E = E , / 6 , where E, was defined by (4.9). We postulate (Liu and Merkine, 1976) that, similar to the shape assumption for the coherent mode, the Reynolds stresses of the fine-grained turbulence can be represented by ~
u ; u ;= E ( f ) R , , ( S ) ,
(5.9)
such that the energy density E ( t ) , like IA(t)l’, bears the burden of the history of the nonequilibrium interactions, while the local shape functions R , ( l ) behaves according to observations c,, exp(-L’). The constants c,, would reflect the proper ratio between the turbulent kinetic energy and Reynolds shear stress, as well as the necessary normalization to show that E = E , / 6 is indeed the turbulence energy density. The rate of energy transfer between the large-scale coherent structure and fine-grained turbulence is provided by the integral I,, in (5.6), which is defined by (4.16). As the numerical results of Gatski and Liu (1980) illustrate, I,,contributes significantly towards the energy balances determining the evolution of the wave envelope lAl’ or 61Al’ in competition with the “inviscid” mechanism of energy exchanges between the coherent mode and the mean flow, &. Thus the participation of modulated fine-grained which occur in the integrand of I,,, must be taken turbulence stresses, iz,, into account. From the general considerations discussed in Section I1 and illustrated by (2.14), the problems of u“, and ;, are coupled through the action of the modulated stresses on the momentum problem of the large-scale coherent structure. Concurrently, !,; is given by its own transport equations, illustrated by (2.43).Thus, following the manner in which the coherent mode velocities were represented by the shape assumption as in (5.2), with the cross-stream shape given by the linear theory, ;, would necessarily take the form (Liu and Merkine, 1976) i , = A ( t ) E ( t ) r , , ( (a)exp(ia()+c.c., ,
(5.10)
with r,,(<, a ) given by the local linear theory, jointly with $([, a ) . Prior to discussing the nonlinear “envelope” problem, we shall briefly discuss the subsidiary, appropriate linear problem for $ and rv that has to be solved. The nonlinear problem that we have discussed thus far places
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the auxiliary linear problem in the proper perspective. The linear problem for a monochromatic large-scale disturbance follows directly from the linearized form of (2.13), (2.14) and (2.43). It was considered by Elswick (1971), Reynolds and Hussain (1972), and Legner and Finson (1980) in various forms. Liu and Merkine ( 1976), Alper and Liu (1978), and Mankbadi and Liu (1981) considered the linear theory as an implement in nonlinear problems involving coherent mode-turbulence interactions. The local linear theory is obtained through the substitution of (5.2), (5.9) and (5.10) into the linearized vorticity and transport equations for ,,; as already discussed. In terms of local variables, we obtain ia [ ( U - c)( 4 ” - a ’4) - 4 U ” ]= E [ - a2rr= - ryz
Advection by mean flow
Transport (vertical advection of mean stresses by wave)
“Production” from mean
+ ia ( rzz
-
r x y ) ’ ] , (5.1 1 )
Work done by mean stresses against wave rates of strain
(5.12)
where c is the wave speed, Re is a local Reynolds number, primes denote differentiation with respect to the local vertical variable l, and the local &differentiation is replaced by ia. The subscripts x, y , z are associated with the streamwise, spanwise and vertical coordinates, respectively. The effect of viscous diffusion, which could be included, has been omitted from ( 5 . 1 1 ) and (5.12). The form of the linear problem given by ( 5 . 1 1 ) and (5.12) holds for either the temporal problem ( c complex, a real) or the spatial problem ( a c =frequency, real) or the “wave packet” problem (Gaster, 1981). The linearized vorticity equation in terms of the stream function, (5.1 l ) , immediately bears resemblance to the nonlinear vorticity equation (4.6). If we subtract the Reynolds average of (4.6) from (4.6) itself and linearize, the resulting linear equation then forms the basis for (5.11). The right side of (5.1 l ) , in terms of differentiation with respect to local variables, has the same interpretation as the right side of (4.6). The linearized version of the transport equations for the modulated stresses (5.12) is written in a form in which the right side resembles that of
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(2.43) (see also (2.44) and (2.45)). Comparing the forms of (2.43) and (5.12), the linearization circumvents the triple correlations as well as the transport of ;u by the fluctuations ilk in the transport mechanisms, so that for local parallel flow the sole surviving transport effect is the advection of the mean stresses by the coherent vertical velocity. In the mechanism of “production” from the mean, the only effect comes from the shear rate of strain of the mean flow, U’. The third group of terms on the right of (5.12) is the work done by the mean stresses against the coherent (wave) rates of strain. Absent in (5.12) is the work done by the modulated stresses against the coherent rates of strain in (2.43), which is a nonlinear effect. No empiricisms were present in these first three groups of effects. The action of the pressure gradients is represented by ( ptr+ p , , ) , defined (prior to the wave amplitude/wave function assumption) by (2.45) and (2.46). Similarly, the viscous dissipation rate in (5.12) is related to the definition in (2.47). These two mechanisms, if included, would require closure arguments. Even without the effects of the action of pressure gradients and viscous dissipation, it is obvious from only the first three mechanisms on the right of (5.12) (a form of “rapid distortion” theory, free from empiricisms (Hunt, 1973)) that r,, would not necessarily be in phase with rates of strain of the coherent mode. Thus any eddy-viscosity assumption in relating r,, to the rates of strain of the coherent mode might, according to (5.12), render such eddy viscosities complex, with magnitudes changing sign depending on the location across the shear layer and the local coherent-mode number. The implications of the relative phases between the coherent mode velocity gradients and the modulated stresses in energy transfer will be discussed subsequently. In order to make practical usage ofthe system (5.1 1) and (5.12), statements about the viscous dissipaton rate and pressure-gradient action must be made. Since the linear theory here is thought of as a valuable implement in the approximate consideration of the nonlinear “wave envelope” problem, the simplest form of such closure statements would suffice. However, for good reasons already discussed, we have precluded an overall eddy viscosity treatment of ru as was done by Reynolds (1972) and Reynolds and Hussain (1972) for the linear problem. Elswick (1971) considered the wave-modulated stresses as a perturbation upon the mean stress. In so doing, Elswick (1971) also considered the closure of the linear problem as a perturbation of the closure statements upon the mean motion problem. However, Elswick (1971) neglected viscous effects altogether in (5.12), including viscous dissipation. He also neglected the “transport” effect that partially constitutes the sources or sinks for the wave-modulated stresses. +t,
Understanding Large-Scale Coherent Structures
24 1
The perturbed form for the common assumption (e.g., Lumley, 1970, 1978) about (b,,+b,/)appears for (5.12) in the form
where C, denotes the sum of the three normal stresses, and T-’ = sU’ is the time scale for return to isotropy where the constant of proportionality is of the order of unity ( s -- 1.445). Elswick (1971) also partially perturbed this time scale. In the linearized form of the transport equations for r,/ presented by Reynolds and Hussain (1972), prior to their eddy-viscosity assumption, the pressure-gradient action and the viscous dissipation were neglected; however, the viscous diffusion, ( R e - ’ ) a’r,,/axf,,was retained for the wall-bounded shear flow problem. The perturbed form of the viscous dissipation rate in (5.12) would be of the form
where the constant is of the order d -- 0.1 (see, for instance Liu and Merkine, 1976; Alper and Liu, 1978). For turbulent free shear flows, the presence of a mean inflectional profile strongly suggests the consideration of the coherent oscillations 6, in terms of “dynamical” or “inertial” instabilities (Liepmann, 1962; Liu, 1971b, 1974a)-that is, arguments to this effect (Liu and Merkine, 1976) lead to the “inviscid” or Rayleigh equation in place of (5.11). T o this end, Elswick (1971) discussed an expansion procedure in inverse powers of an appropriately defined turbulent Reynolds number, which comes from “proper” scaling. The scale in our case here is set from the normalizations. I n (5.11) all quantities were made dimensionless by the velocities associated with the free stream and the initial shear layer thickness. An examination of the “sources” or “sinks” for ?,/ reveals that the modulated stresses, scale according to IAI’E. For instance, this scaling is naturally suggested by the transport mechanism in terms of the vertical advection of the mean stresses ( - E ) by the coherent motion ( - A ) . The presence of the local value of the mean turbulence energy density E = E , / 6 , defined by (4.9), o n the right
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side of (5.11) suggests a similar scaling discussed by Elswick (1971). The energy density E is estimated essentially by the ratio of the sum of the mean normal stresses to a mean velocity squared, and therefore has the interpretation of a n inverse (local) turbulent Reynolds number, E R;’. From the numerical example of Gatski a n d Liu (1980), RT 30 when E is maximum a n d RT 100 a t the “initialized” initial condition. However, these numbers are not necessarily representative of the actual turbulent Reynolds numbers. Nevertheless, if we expand
-
-
4 = 4(”’+R f ’ d “ ) + . . .
-
(5.13) (5.14)
then 4“’ satisfies the Rayleigh equation and immediately becomes uncoupled from r,,. The first approximation for the shape of the modulated stresses r:) satisfies (5.12) but with the coherent wave streamfunction there replaced by 4‘”’(Liu a n d Merkine, 1976). In this approximation, the outer boundary conditions for 4“” follow those of the Rayleigh equation, a n d one seeks the outgoing wave solution. Generally, because of the presence of the turbulent-nonturbulent interface associated with the outer “boundary,” Reynolds (1972) formulated the necessary interfacial conditions for the more general problem. However, because the mean velocity is essentially continuous across the turbulentnonturbulent flow interface according to measurements, this continuity is to the order of RT‘. The interface is actually “transparent” as far as the 4‘”’eigenvalue problem is concerned, a n d the coherent-mode velocities and pressure are continuous to the order of R f l . The “instability” properties are primarily attributed t o dynamical instabilities, associated with the inflectional mean velocity profile, that occur well within the turbulent fluid, a n d thus the outer boundary conditions are indeed those for the Rayleigh problem of decaying outgoing waves. The interface, if of interest, would be the subject of study at the higher-order, 4 “ ) a n d r:,’),level of description. This result is expected from a physical viewpoint as well, since the interface region is much less important energetically because (1) sharp gradients in the mean velocity are absent in that region and (2) the fluctuations are much less energetic there than in the vicinity of the mean velocity inflection point in the interior of the shear layer. The role of the linear theory, (5.11) a n d (5.12), or its approximate, “dynamical instability” form, is now clear. The local eigenfunction 4’ generates the local shape of the large-scale coherent velocity distributions
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across the shear layer. Michalke (1971) was the first to find that the local linear theory was able to generate the coherent velocity fluctuations that compare favorably with observations. However, in using the linear theory as a “curve fit,” the local mean flow velocity and shear layer growth rate are considered as given (from measurements, for example). As such, the theory does not, and could not, address the wave envelope or amplitude evolution problem. Recent improvements on the linear theory that account for slight flow divergence (Crighton and Gaster, 1976) have been applied, in the same spirit as that of Michalke (1971), to the turbulent mixing layer problem (Weisbrot, 1984; Gaster, Kit and Wygnanski, 1985; Wygnanski and Petersen, 1987). Similar good fits were found between the eigenfunctions and experiments with those generated by the Rayleigh equation. However, in the normalization of such eigenfunctions, the local “area” under the root mean square of the streamwise fluctuation velocity was set equal to that from measurements. This precludes the possibility of giving the wave envelope the physical interpretation discussed earlier. The wave amplitude problem follows a higher-order correction due to slight flow divergence, but excludes the essential physics of the turbulent shear flow problem that we have discussed. In regions where the coherent mode has grown to significant amplitudes so as to change the mean flow spreading rate, such a “weak disturbance” procedure would not suffice. We shall soon see the role of the linear theory in the strongly nonlinear problem that is typical of free shear flows.
E. NONLINEAR WAVE-ENVELOPE DYNAMICS We follow Liu and Merkine (1976) as a simple example. The nonlinear problem concerns the “wave envelope” development. From the coupled system (5.5)-(5.7), with the substitution of (5.2) and (5.8)-(5.10), we obtain (5.15)
(5.16) d 6 E = IAl’EI,, ,( a ) EZ L, dt
-
+
-
4’.
(5.17)
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v
We will discuss the form of the dissipation integral subsequently. The initial conditions are S(0) = 1, IA(O)l’= IAI: and E ( 0 )= E,,. The mean flow kinetic energy defect integral (4.10) became simply (-6). The energy exchange mechanism between the mean flow a n d the coherent mode given where the by the integral defined in (4.14) has now become fp = IA12frr(a), integral f,,( a ) involves integration over the eigenfunctions of the linear theory a n d the mean velocity gradient (Liu a n d Merkine, 1976) a n d thus depends on the local wavenumber (in spatial problems, it would be the local frequency). See the Appendix for definitions of such integrals. The turbulence energy production integral, defined by (4.15), now becomes I ; = E I i y , where I : , involves the integral over the shape distribution of the Reynolds shear stress R,, and the mean velocity gradient and is a constant (Liu a n d Merkine, 1976). The fine-grained turbulence, viscous dissipation integral was defined in (4.17). If we follow the standard local equilibrium argument for large Reynolds numbers (Townsend, 1956), then = E 1’2Z;, where I ; is a constant. For simplicity, Liu a n d Merkine (1976) argued about the Reynolds-average shape function (5.9) on the basis of a locally homogeneous-shear problem (Champagne, Harris and Corrsin, 1970) so that E I ; a n d Zip = 1;. In this case, the nonlinear interaction problem is somewhat simplified in that the only mechanism causing the change of 6 E would be its interaction with the coherent structure through IAI’EI,,,( a ) . In the context of the numerical work of Gatski a n d Liu (1980), only at the later stages of development would II, -- 1;. In the present discussion of the approximate considerations of the wave-envelope evolution, the simplified version of Liu a n d Merkine (1976), I ; = I ; , will be continued for the purpose of illustrating ideas, leaving a fuller account of 1; # II, to subsequent discussions of application to real, spatially developing flows. The integrals I,,, Iiy,and I ; a s well as I,,, introduced subsequently are defined in the Appendix and discussed in detail in Liu a n d Merkine (1976).
v=
I
F. THE MECHANISMS OF E N E R G YE X C H A N G E BETWEENCOHERENT D MODE A N D F I N E - G R A I N ETURBULENCE The energy exchange between the large-scale coherent structure and fine-grained turbulence is given by the integral I,, defined in (4.16), which now becomes I,, = lA12EIx,,(a),where the integral I,, (Liu and Merkine, 1976) involves the shape functions of the modulated stresses a n d those of the rate of strain of the coherent mode. The importance of the relative
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phases between the modulated stresses and the coherent mode rates of strain comes from the energy exchange mechanism discussed in Sections 111 and IV,
which comprises the integrand of I,,. In the present context of using linearized theory to study the nonlinear “wave envelope” development, the integrand of I,,., consists of (Liu and Merkine, 1976) (5.18a)
(5.18b)
(5.18~)
(5.18d) The form above implies that we have represented complex shape functions of the modulated stresses, r,, and the coherent-mode eigenfunctions from the linear theory in the vector form in terms of magnitude and direction. Here 0, with the appropriate subscript, is the phase angle. In this representation the energy transfer then consists of the scalar products between modulated stresses and the appropriate coherent mode rates of strain. It is clear that the relative phases determine the directions of energy transfer. To illustrate this, the vector representation of the modulated stresses and coherent rates of strain is presented in Figure 14 for a wave number a = 0.4446, which corresponds to the most amplified mode for the hyperbolic tangent mean velocity profile. The qualitative behavior is similar for other values of a. The appropriate scalar products of the vectors in Figure 14, given by (5.18a)-(5.18d), are shown in Figure 15. In Figure 14 the curves represent the locus of vectors at different vertical positions across the shear layer. For instance, shown in Figure 14a are the vectors 2rx, and a4‘. At 5 = 0, Orxx = 0 and 0, = n-12, thus giving a negative FxXaC/ax, indicating a local transfer of energy from the coherent mode to the fine-grained turbulence. At 5 0.23, Or,, and O<,, are out of phase by n- so that FATd t i / d x -+ 0. 2-
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,
. \
90-
(c) FIG. 14. Locus of vectors representing the shape distribution of modulated turbulent stresses and coherent-mode rates of strain across the mixing layer. u = 0.4446. ( a ) Streamwise normal stress and rate of strain; (b) Shear stress and shear rates of strain; ( c ) Vertical normal stress and rate of strain.
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0.6
0.4
t
0.2
kinetic energy transfer from lorge to smoll
Il
kinetic energy tronsfer from smoll to lorge scale
1
o
-0.2
FIG. 15. Relative contributions to the coherent mode and fine-grained turbulence energy exchange mechanisms. a = 0.4446.
For 5 > 0.23, 04,lags behind Or,x so that locally energy is transferred from the fine-grained turbulence to the coherent mode with a maximum at about 5- 1. Shown in Figure 14b are the vectors 2r,,, 4’’ and a’@. Because U”(0)= 0, then @ ” = a’# at 5 = 0 according to the Rayleigh equation. Thus F,.. dG/dx and Fxzdii/az are equal there. While the latter of these remains positive, the former becomes negative after 5 ~ 0 . 1 5 The . vectors 2rz, and a4’ are shown in Figure 14c. Their scalar product, FzzdG/az, shown in Figure 15, is very nearly equal and opposite in sign to Frz dG/dx. Figure 15 shows that the mechanism of modulated horizontal normal stress-normal rate of strain dominates the energy transfer near the center of the shear layer ( 5 = 0), while the mechanism of modulated shear stress-shear rate of strain dominates the energy transfer away from the center of the shear layer. The net result of these four contributions is shown by the dot-dash line in Figure 15, which is positive over most of the shear layer, indicating that for this case energy transfer is from the coherent mode to the fine-grained turbulence. The dot-dash line would fall slightly below the axis in the outer
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regions of the shear layer, but the magnitude is not distinguishable within the width of the curve itself. We emphasize that from this consideration, there is significant energy transfer, within local vertical regions of the shear layer, from the turbulence to the coherent motion contributed by the is then twice the area under the individual mechanisms. The integral lM,, dot-dash curve, the 4‘ distribution being symmetrical about 5 = 0. In principle, I,,,,(a) depends on the local wave number, which, in turn, is scaled by the local developing shear layer thickness 8. It is clear that, in general, the wave-envelope development (5.16) is coupled to the spreading of the mean flow and the development of the fine-grained turbulence energy as indicated by (5.15) and (5.17), respectively. These approximate forms of the nonlinear interaction could be said to have been motivated by and to bear strong resemblance to the diagnostics of the numerical problem (Gatski and Liu, 1980) given by (4.11)-(4.13). The mean flow energy defect evolution (4.1 1) now reduces to the statement (5.15) that as long as energy is transferred to the fluctuating motions, d S / d t > 0. When f r , ( c y ) becomes negative, as in the “damped disturbance” regime discussed in Section IV ( fp < 0), the contribution to d 6 / d t would be to arrest the growth of the shear layer or even decrease its growth (Weisbrot, 1984; Fiedler and Mensing, 1985) depending on the relative magnitude between the coherent mode and turbulence contributions. The steplike behavior of 6 would come from peaking of IA12fr,,as has been anticipated in Section 111. These facts would account for the observed steplike shear layer thickness development observed by Ho and Huang (1982), Fiedler and Mensing (1985) and Wygnanski and Petersen (1987). The observed momentary depression (Weisbrot, 1984; Fiedler and Mensing, 1985) in the shear layer thickness is attributed to the dominance of the “damped disturbance” mechanism relative to others (such as turbulence and viscosity) affecting the spreading rate. Some of these aspects will be quantitively addressed in the sections to follow.
G. WAVEENVELOPE AND TURBULENC ENERGY E TRAJECTORIES. A S I M P L EILLUSTRATION Another feature of the wave-envelope problem exhibited by the observations depicted in Figure 1 could be qualitatively deduced from the much simplified framework here. If we assume that the right side of (5.15) is in some sense “small” so that the shear layer growth rate is correspondingly
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small (d6/dt + 0), the change in 6 is then ignored entirely. Thus 6 remains * at the initial value 6 = 1 and the interaction integrals I,, I , , are fixed by the initial wave number a. In this case, (5.15-5.17) reduce to
d - / A ) ’ = lA12f,5- IA I 2 E I W , dt
1
dE dt
-= IAI’EI,.,
(5.19) (5.20)
We haved retained the approximation (Liu and Merkine, 1976) that locally EI:, = F. The simple essentials here state that the energy transfer from the coherent structure to the fine-grained turbulence is the only mechanism causing E to change from its original value. The evolution of the coherent structure amplitude is determined by the local balances between energy exctraction from the mean flow and energy transfer to the fine-grained turbulence. In the IA12-E plane the system (5.19),(5.20) admits the solution (5.21) where Lo= EoZ,,,/f,, and M,)= lAli/ E,. The dimensionless time t is obtained from (5.22) where x = I n E / E,. In this special example the equilibrium values (denoted by the subscript e ) for IA12,E are such that IAI: = 0 is deduced from setting the right sides of (5.19) and (5.20) to zero, and the values for E, came directly from (5.21) L, exp( - L , ) = Lo exp[ - Lo(1+ M o ) ] , where L, = E,I,,,,/ fry.We expect that L, > Lo because we found I , , > 0 and the fine-grained turbulence energy would be increased, E, > E o , due to the presence of the coherent structure. For a fixed ratio of initial amplitudes M o , as L, = EoZ,,.,/fr5increases, the turbulence equilibrium amplitude ratio E , / E , , decreases. This can be interpreted as follows. If we fix the wave number, then I,,,/fr, is fixed, so that as Eo is increased more energy is transferred to the turbulence from the coherent motion, thereby limiting the coherent mode amplitude. This in turn decreases the efficiency of the coherent mode as an intermediary in taking energy from the mean flow and transferring it to the turbulence. On the other hand, if Eo is fixed and Zn,/f,,
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is increased, then the energy transfer from the coherent mode to the turbulence becomes more efficient than that from the mean motion to the coherent mode. This again gives a lower E,. If Lo and Eo are fixed a n d M o is increased through increasing IAI& E, is increased because the coherent mode is made more efficient in drawing energy from the mean flow a n d transferring it to the turbulence. In this special consideration, the equilibrium amplitude of the coherent mode IAl;-+O as long as E,,>0, and is independent of initial conditions. From the physical considerations discussed, E, is not independent of initial conditions. From (5.21) it is seen that Lo a n d Mo fix the trajectory in the lA12//lA1i,E / E o plane. The wave envelope o r amplitude lA12/lA\ireaches a maximum when E / Eo= ] / L ofor L,,< 1, whereas l A I * / / A ldecays ~ at the outset for L,,> 1. The latter situation occurs because energy transfer to the turbulence overwhelms that extracted from the mean flow. The trajectories in the lA12/lAl~-E / E,) plane are shown in Figure 16 for M,= 1 a n d various values of Lo< 1. The time development begins at (1, I), and follows the trajectory. Not shown are the decaying lA12/lA(itrajectories starting at ( 1 , 1) for the strong initial turbulence ( L o > 1) situation. The interesting physical picture that emerges is shown in Figure 17 for initial conditions where the coherent mode amplifies, its amplitude first grows “exponentially” d u e to extraction of energy from the mean motion a n d subsequently decays d u e t o energy transfer to the fine-grained turbulence. The fine-grained turbulence energy relaxes from an original equili-
10
I
E/E, FIG. 16. Coherent mode and fine-grained turbulence energy trajectories for the parallel flow model. M,, = 1.
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W
FIG. 17. Evolution of coherent mode and fine-grained turbulence energy for a given wavenumber ( a = 0.4446),parallel Row model.
25 1
t
brium level to a final, higher level due to energy supplied by the coherent mode. This recovers some of the physical mechanisms derived more laboriously from the numerical work of Gatski and Liu (1980), and could, in part, explain the observations depicting large-scale coherent structures interacting with turbulence, as reported, for instance, by Favre-Marinet and Binder (1979) and shown in Figure 1 . Other, semi-analytical models of this equilibration picture are given in Liu and Merkine (1976) for the temporal mixing layer. We have already appreciated the shortcomings of the temporal mixing layer relative to the real, laboratory situations of the spatially developing free turbulent shear flows. The expected lack of a legitimate one-to-one transformation (rather than mimicking) coincides with the similar situation in hydrodynamic stability theory (Caster, 1962, 1965, 1968). However, the physical similarities between the relatively simple approximate considerations of “wave envelopes” and the numerical computational results strongly encourage the further development of the former, directed at a simpler description of the realistic spatially-developing free shear flows.
VI. Spatially Developing Free Shear Flows
A. CENE.RAL COMMENTS Some aspects of the quantitative observations of turbulent free shear flows discussed in Section 111 pertain to laboratory, spatially developing flows. Although certain qualitative explanations of physical features are
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possible from the considerations of Sections IV and V, we shall directly address the spatial problem in this section. N o attempt will be made here for a complete survey of the literature, but aspects of the literature will again be drawn upon t o put forth a consistent “point of view” for the problem of large-scale coherent structures in free turbulent shear flows. Because many of the symptoms of such structures in turbulent flows share those of hydrodynamically unstable disturbances is an otherwise laminar flow, many of the physical features of the former can be inferred from the latter. In the context of Sections IV and V, such inferences must necessarily b e made with considerable care rather than with unaffected simplicity. For instance, one must differentiate carefully between (1) the dynamical instability mechanism for the “fast oscillations” that could generate local coherentmode velocity profiles from linear wave functions and (2) the slowly varying wave-envelope or amplitude distribution that necessarily requires the participation of the real physics of the problem, including turbulence, nonlinearities a n d mean flow development. In the case of finite amplitude disturbances, J. T. Stuart (1958) advanced the idea that the kinematics a n d shape of the disturbances in shear flow instability could be approximated by the linear theory, but that the amplitude o r wave envelope is to be obtained by the nonlinear theory. Its observational basis a n d application to the turbulent free shear layer problem has been discussed in Section V in connection with the work of Liu a n d Merkine (1976). The generalization of Stuart (1958) to the finite disturbance problem in a spatially developing free (wake) laminar shear flow was given by KO, Kubota a n d Lees (1970). Some of their results are worth emphasizing, since they anticipated many of the obvious aspects of the coherent structure problem in turbulent shear layers. Although only a single (fundamental) physical frequency was considered, these authors have shown how the nonlinear disturbance a n d the coupled mean flow would respond to several parameters. A simplified version of the wave-envelope problem of KO et al. (1970) (in the absence of fine-grained turbulence), in the context of the mixing region problem, appears in the form - d6
I-= dx
IAl’fr,(S)+LRe K,,/S,
- d 1 ?( 6) - ( 6 IAI’) = lAI’frT( 6 ) - - f+ ( S)lAl’/ 6, dx Re
where x is the dimensionless streamwise distance, f, f( 6 ) are the mean flow
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and fluctuation advection integrals, and id and f d ( 6 ) are the mean flow and fluctuation viscous dissipation integrals. Integrals involving instability modes are dependent on the shear layer thickness 6 ( x ) through the dependence of local instability properties on the local frequency parameter /3, whereas mean flow integrals and are constant for the similar mean flow shape distribution. Since f ( S ) > 0 and is slowly varying, it is replaced by a mean value indicated in (6.2). We refer to the Appendix for further details regarding the integrals. Here, the Reynolds number is Re = l%,,/ v, where U is the average over the upper and lower free stream velocities. I n the incipient instability region IA12+ 0, so the second term on the right of (6.1) initially dominates and provides the basic viscous shear layer spreading S - A . The deviation from this parabolic spreading would indicate the onset of finite disturbance levels as the first term on the right of (6.1) competes with the second. This is indeed the case found theoretically by KO et al. (1970) and experimentally by Sat0 and Kuriki (1961) for the wake problem. Thus a dominating peak in the energy extraction from the mean flow would bring about a steplike development of 6(x). The observed steplike growth of transitional shear layers (e.g., Ho and Huang, 1982), and forced turbulent shear layers (Fiedler et al., 1981; see also Wygnanski and Petersen, 1987) is attributed to this mechanism. However, in the turbulent shear layer problem, the basic spreading of the shear layer is due to the fine-grained turbulence with the mechanism depicted by E l Lx and discussed in Section IV, which tends to give a linear growth in the absence of other “nonequilibrium” energy loss from the mean flow. KO et al. (1970) found that for a fixed Reynolds number and initial wake thickness, the peak in the fluctuation energy density, IAl’, moves closer to the start of the wake as the initial fluctuation level is increased. For the same initial fluctuation energy level, the growth, peak and decay process is hastened in the streamwise direction as the Reynolds number is increased. Accompanying these properties of IAI2 would be the moving upstream of the steplike growth of the shear layer.
c,,
B. THE SINGLECOHERENT MODE I N FREE TURBULENT S H E A R FLOWS
The observed growth and decay of a single dominant coherent mode in turbulent free flows, the coherent mode “negative” production mechanism, and the eventual increase in the fine-grained turbulence level, illustrated in
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Figures 1 and 2, were explainable by the single-mode considerations of Section V. Several more detailed features of experimental observations could be explained within the considerations of this section. Following the forced plane turbulent mixing layer experiments of Oster and Wygnanski (1982), Weisbrot (1984) continued with quantitative measurements of the coherentmode energy exchange with the mean motion, in addition to the mean flow spreading rate, at high amplitudes of forcing. However, subsequent subharmonic formation was not detected further downstream. Although higher harmonics of the forcing frequency were present, these decayed rapidly with distance downstream. A significant rise in the level of the background broadband turbulence occurred with increasing downstream distance. The coherent mode at the forcing frequency appeared to be functioning as a monochromatic disturbance in the turbulent mixing layer. As anticipated in the discussions in Section V, even if the comparison of measured disturbance velocity distributions across the shear layer with those obtained from a local inviscid linear stability theory appeared good, the same “theory” is not capable of describing the amplitude or wave-envelope evolution in the streamwise direction. The nonlinear wave-envelope problem for a single coherent mode in a spatially developing turbulent shear layer, in the spirit of Section V, is in the form (Alper and Liu, 1978) (6.3)
dS E I ’ -= EZis + IA~’ EI,, (6) - E’l’Z;, dx
(6.5)
where & I ‘ are I+, and the mean flow energy advection and turbulence energy advection and dissipation integrals, respectively, and are constants for a nominally similar mean velocity and Reynolds stress profile; the local shear layer thickness-dependent, coherent-mode integrals were previously defined. We again refer to the Appendix for details of the integrals. Mean motion and coherent mode viscous dissipation have not been included for the turbulent shear layer problem. The observed behavior (Weisbrot, 1984) of the spreading rate of the “highly excited” turbulent mixing layer can be diagnosed directly by (6.3),
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which is obtained from kinetic energy considerations. The sum (IAl’f,, + E I ; 5 ) is the integral of the total energy exchange mechanism between the mean flow and the coherent plus turbulent fluctuations, across the shear layer. It has been evaluated from measurements by Weisbrot (1984) as a function of the streamwise distance. In his notation,
where U , + U,, Up, + U ,, z + y, w -+ v. We have assumed, for simplicity, that the mean flow develops similarly so that I = constant = 2R2(3/2- In 2) for a hyperbolic tangent profile, where for U - = > U,, R= (U-,- Urn)/(Up,+ U r ) . Thus, the shear layer thickness obtained from (6.3) becomes (6.7)
If nonsimilarities of the mean velocity profile were to be included, then r ( x ) would appear in (6.3) within the differential d ( J 6 ) l d x . In the experiments, the mean velocity profiles were indeed not entirely similar. In order to make use of the idea developed from energy considerations that the mean flow will spread as long as energy is taken away and will contract if energy is supplied to it by “damped” disturbances, we integrate the “raw” experimental data (Weisbrot, 1984, Figure 5.3.1) to obtain the features of shear layer growth (and contraction) via
The multiplication of the velocity ratio factor makes (6.8) consistent with the way in which i was originally made dimensionless. The subscript exp denotes the experimental data mentioned. Here both 0 and x are considered dimensional. We show the integral (6.8) in Figure 18. It amazingly resembles that of the measured shear layer momentum thickness given in Figure 5.1.1 of Weisbrot ( 1984)t. We have deliberately avoided “matching constants” leading to direct comparisons. Weisbrot (1984) also obtained the “phase locked” contribution to the shear stress “production” mechanism. From these considerations, it is thus shown conclusively that the excited coherent t The modified data was provided by I. Wygnanski (private communication, December 1987).
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0
1
I000
2000
x(rnrn)
FIG. 18. Illustrating that observed growth and contraction of observed shear thickness is attributed to wave disturbance energy extraction from and supply to the mean flow. exp: Weisbrot, 1984; “theory”: present explanation.
fluctuation causes the shear layer to spread rapidly and even in the “damped” region dominates the overall energy extraction/supply rate to the mean motion and causes the shear layer to contract. The eventual linear spreading rate is due to the broad-band turbulence. The features of the evolution of the coherent-mode energy “production” mechanism are similar to that of Fiedler et al. (1981), as shown in Figure 2, and were anticipated by the calculations of Gatski and Liu (1980), as shown in Figure 5. In the formulation (6.3)-(6.5), only the dominant energy exchange mechanism between the mean flow and the fluctuations was retained. Because the mean flow is rapidly expanding and changing in the streamwise direction in the experiments, the remaining energy exchange mechanisms for a twodimensional mean flow (in the present notation), - - d U ( u 2 - w2) -+
dX
,aw u w -,
dX
would need to be assessed in the diagnosis of the observed spreading rate in Figure 18. The dominant energy exchange mechanism included in (6.3)(6.5),as well as that measured by Weisbrot (1984), was sufficient to uncover the basic effect but was not intended for an “accurate prediction.” Of the mechanisms responsible for the coherent mode wave-envelope evolution depicted in (6.4), only \AI2f,., is relatively easily measured. The measurement of the wave-turbulence energy transfer mechanism, depicted by lA12EZw,in (6.4) or Z,,in (4.12) and (4.16), is difficult (see, for instance, Hussain, 1983).
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It would involve taking spatial derivatives of phase-averaged quantities and the subtraction between large numbers. Nevertheless, it is a n important mechanism in the turbulent shear flow problem. In this situation we must rely on the insights developed from theoretical considerations, such as in Sections IV a n d V, to help us towards the understanding of the coherent mode wave-envelope evolution problem (Alper and Liu, 1978). The shear layer growth, which is explained here from dynamical considerations, is the result of the overall energy drain o r resupply to the mean kinetic energy. The spectrum of Weisbrot’s (1984) observation indicates that several higher frequency harmonics undergo growth and decay processes earlier in the streamwise distance than the component at the forced frequency. A “phase-locked’’ subharmonic was not observed over the length of the streamwise distance measured. We shall delay to the following section a discussion of the theoretical aspect of multiple-coherent mode interactions. The growth a n d decay of higher-frequency coherent modes occurring in regions closer to the start of the mixing layer a n d lower frequency components further downstream from such observations have been borne out by theoretical considerations (e.g., Liu, 1974a; Merkine and Liu, 1975; Alper a n d Liu, 1978; Mankbadi a n d Liu, 1981, 1984) on the basis of single, independent modes interacting with fine-grained turbulence. The effect of initial conditions on single, independent coherent-mode development in terms of the initial Strouhal frequency, coherent-mode amplitude a n d turbulence level were discussed by Alper a n d Liu (1978). For the same initial energy levels, the higher-frequency coherent components which have shorter streamwise lifetimes attain higher wave-envelope peaks than lower frequency components. However, the higher frequency modes may not necessarily enhance the fine-grained turbulence energy as vigorously as the lower frequency modes because the mode-turbulence energy transfer depends not only on the magnitude of lA12 but also on the lifetime of the coherent mode. For the same frequency, increasing the initial mode amplitude moves the peak of IAl’ upstream. Control of large-scale coherent structures can also be achieved through the use of fine-grained turbulence (Alper a n d Liu, 1978). For the same coherent mode frequency but different initial turbulence energy levels, the higher turbulence level case suppresses the coherent-mode downstream development. Consequently, the fine-grained turbulence would achieve a relative lower enhancement downstream. The very large initial coherent-mode amplitude forcing would effect a subsequent decay of the coherent mode, a n d this limiting amplitude effect has also been found experimentally by Fiedler
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and Mensing (1985). Although the calculations were performed for coherent modes in a round turbulent jet, Mankbadi and Liu (1981) theoretically found that such an initial-amplitude threshold effect does indeed exist. We shall refer to Mankbadi and Liu (1981) for the elucidation of initial condition effects and the possible control of the free turbulent shear flow. They provided comparison with experiments on detailed flow properties, including both coherent structure and fine-grained turbulence as well as kinematic properties.
C . COHERENT-MODE INTERACTIONS To begin the discussion of mode interactions, it would be helpful to recall the streakline patterns obtained calculationally by Williams and Hama (1980) from the superposition of kinematically obtained wavy disturbances of the fundamental mode and its subharmonic upon a hyperbolic tangent mean velocity profile. Streaklines are also obtained from the local eigenfunctions of inviscid linear theory by Weisbrot (1984) (see also Wygnanski and Petersen, 1987; Wygnanski and Weisbrot, 1987), resolving in some sense the usefulness of the local linear theory in mimicking flow visualization (the quantitative wave-envelope problem is not resolvable from this consideration). We shall discuss Williams and Hama ( 1980) for illustrative purposes. They obtained streakline patterns from the superposition of subharmonic to fundamental with certain constant-amplitude ratios. These patterns bear a striking resemblance to the visual observation of dye streak behavior in a mixing layer (e.g., Freymuth, 1966; Winant and Browand, 1974: Ho and Huang, 1982). However, the streakline calculations of Williams and Hama (1980) come from a linear superposition of two constant amplitude wave disturbances; the pairing and roll-up are the consequence of wave interference. The simulated wave amplitudes of the fundamental and subharmonic are both constant, and the abrupt switching of modal structure, as the visual appearance of streaklines would suggest, is entirely absent. We are thus cautioned by this illustration that dye streak behavior is not necessarily indicative of unique physical circumstances; we must also refer to simultaneous quantitative measurements. Quantitative measurements suggesting mode-mode interactions between the fundamental disturbance wave and its subharmonic in a shear layer are reported by Ho and Huang (1982). Their shear layer is essentially one undergoing transition, and the presence of such distinct modes is brought
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about by forcing at the subharmonic frequency. The significance of Ho and Huang’s (1982) work lies in the identification of the visually observed location of “pairing,” indicated by the accumulation of dye streaks, with the occurrence of the measured cross-sectional energy maximum of the subharmonic (actually, they measured the kinetic energy associated with the streamwise velocity fluctuation, integrated across the shear layer). There was no abrupt switching from the fundamental frequency and wavelength to those of the subharmonic. Reproduced in Figure 4, corresponding to Mode I1 of Ho and Huang (1982), is the evolution of the measured sectional energy associated with the streamwise velocity fluctuation. The 2.15 Hz curve corresponds to the forced, subharmonic component; the 4.30 Hz curve is the fundamental. Although the peak amplitudes of the two modes are distinct, the fading in of the subharmonic occurs in regions of active fundamental development and, in turn, the fading out of the fundamental takes place in regions where the subharmonic is active. The measurements suggest a natural occurrence of the switch-on and switch-off processes, in contrast to the suggestive, abrupt switch in the modal content from visual observations of dye streaks alone (Freymuth, 1966; Winant and Browand, 1974). The theoretical formulation of mode-mode interactions in a spatially developing shear layer was undertaken for a laminar viscous shear flow by Nikitopoulos (1982), and by Liu and Nikitopoulos (1982). The same problem was considered by Kaptanoglu (1984) and Liu and Kaptanoglu (1984); Mankbadi (1985) considered the round jet problem with axisymmetric modes with the involvement of the fine-grained turbulence. The measurable sectional energy content of each mode is essentially 61A12, related to the square of the amplitude of the coherent structure. The cross-sectional energy content ( H o and Huang, 1982) thus reconciles measurements with the theoretical ideas about wave-envelope evolution. For each frequency and the same initial conditions, the amplitude is a fixed streamwise envelope under which the propagating wavy disturbance enters from its initiation upstream and exits downstream. The aim here is to understand the direction of energy transfer between the modes, its effect on establishing the spatial distribution of wave envelopes and the consequent rate of spread of the shear flow. Following the general discussions of Section 11, we first consider that an ensemble of disturbances exists in a shear flow and split the modes into “odd” (denoted by 4) and “even” (denoted by $). Then the rate of energy transfer from the even to the odd modes is given by (Stuart, 1962a; see also
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Section 1I.C)
where the average is taken over the largest periodicity of the disturbances. The mechanism is the work done (by the stresses of the odd modes) against the appropriate rate of strain (of the even modes). It is clear that the phase relation between the stresses and the rate of strain determines the direction of energy transfer and that the amplitudes determine the strength of this transfer (the Kelly mechanism, Kelly, 1967; Liu, 1981). For a spatially developing shear layer, Liu and Nikitopoulos (1982) considered the interaction between the subharmonic mode (a single “odd” mode) and its fundamental (a single “even” mode). If the energy content of the fundamental mode across the shear layer is denoted by E2 = 8IA2I2 and that of the subharmonic by E , = SIA,12,then the overall energy transfer mechanism between the modes is proportional to IA,121A,(.In contrast, the respective fluctuation energy production rate from the mean flow is proportional to [ A , ] ’and to IA2I2.The rate of viscous dissipation scales like (A12/S. The dimensionless energy density IAl’ is much less than unity, according to observations. In this case, the estimate shows that the individual energy production from the mean motion would seem to dominate over that of the mode-mode energy transfer except in regions where the former changes sign at a later stage of development. In the early stages of development, the mode interactions are dominated by implicit nonlinear interactions via the mean motion rather than by the more explicit direct energy transfer mechanism. At the later stages, mode interactions are most certainly important in affecting the details of the amplitude distribution in the streamwise direction. In the experiments of Ho and Huang (1982), modes other than the fundamental and the subharmonic are present, including initially weak fine-grained turbulence disturbances, and these are not included in this initial analysis. We refer to Nikitopoulos and Liu (1987a) for a more complete discussion of the “laminar” problem. We shall consider the turbulent problem in the following (Kaptanoglu, 1984). To begin, we make use of the kinetic energy equations (2.21), and (2.24)-(2.26) in Section II.C, with the spatial interpretation of the advective Dt. We address the wave-envelope problem and specialize derivative the odd modes to a single-plane subharmonic mode and the even modes to the plane fundamental. To this end, we again integrate the appropriate kinetic energy equations across the plane shear layer. But we need to discuss the new phenomenon of mode interactions first.
o/
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26 1
On the right side of the integrated subharmonic energy equation, the rate of energy exchange with the fundamental appear as the integral
and on the right side of the integrated fundamental energy equation would appear the similar integral -
1
All quantities are made dimensionless in the manner previously discussed. We recall that x is the streamwise coordinate measured from the start of the mixing layer, z is the vertical coordinate measured from the center of the mixing layer, u, w are the x, z fluctuation velocities, and U is the mean denoting the upper and lower streams, respectively. velocity with +.OO Following earlier work (see, for instance, Liu, 1981, and Section V), the disturbances are assumed to take the separable form of the product of an unknown amplitude A , ( x ) with a vertical distribution function given by the local linear stability theory (which has found experimental justification; e.g., Michalke, 1971; Weisbrot, 1984), as was done for the single mode in (5.2) (6.11)
The modulated stresses Fv, ?,,, following Section V (5.10), appear as
;v
= A , ( x ) E ( x ) r , ,e-Ipt+ c.c.,
;,
= A 2 ( x ) E ( x ) r , , ?e-2'Pr-'s
+ c.c. .
(6.13) (6.14)
We again recall that 4, r,,,, denote the eigenfunction and modulated stresses of the local linear theory and are functions of the rescaled vertical variable 5 = z / 6 ( x ) , where 6 ( x ) is a length scale of the mean flow (to be identified as the half-vorticity thickness); ( )' denotes differentiation with respect to 5 ; p = 2 ~ r f F ( x )U/ is the dimensionless local frequency; a n d f i s the physical frequency. We also recall that U = ( U p + U-,)/2; the local wavenumbers
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a are also scaled by 6(x); 0 is the relative phase between the fundamental component ( 2 p ) and its subharmonic ( p ) ; and C.C. denotes the complex conjugate. We are again reminded that the velocities and lengths are conand 6” (so that 6(0) = l ) , and time sidered to be made dimensionless by The turbulent energy density E ( x )and the Reynolds stresses --u;uJ by 6,/ 0. are already discussed in Section V.D. The mean velocity profile is taken to be the hyperbolic tangent profile U = 1 - R tanh 5. The mode sectionalenergy content is defined as in (5.1):
u
E,,(x)= I A , ( X ) l ’ ~ ( X ) .
(6.15)
This is similar to E ( f ) as measured by Ho and Huang (1982), except that their sectional energy refers to the contribution by u alone. The normalization of the local eigenfunctions according to (5.4) is implied, which allow us to relate the energy content to the amplitude or wave envelope. Alternatively, the square of the amplitude is an “energy density.” With the shape assumptions included, the integrated energy equations then yield four first-order nonlinear differential equations describing the streamwise evolution of 6, IAl12,IA,l2 (or in the alternative form E l , E,) and E : mean pow: - d6 I-= dx
1 Ia/S Re
I,,,AS+ I , , , A f +I:,E +-
(6.16)
subharmonic:
fundamen tall
turbulence: d6E I, -= I:,E dx
+ ( A:lw,,,+ ATI,., I )E - I ,
I
E3’*.
(6.19)
The relevant integrals in (6.16)-(6.19) are again defined in the Appendix. The “slowly varying” advection integrals I , ( S ) and I,( 6) are approximated by their “mean” values. The overall mode interaction integral (6.9), upon the shape assumption, has become f,, = A:A,Z,,. Not previously introduced are the mode-energy exchange integrals 6 ) and the viscous dissipation
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integrals Z+,,(8). The Reynolds number is again Re = l%,/v. The subscripts 1 and 2 denote the subharmonic and fundamental, respectively. Following arguments of inertial or dynamical instability reasoning (Section V), it is sufficient to use the Rayleigh equation to obtain the characteristics of such integrals (see, for instance, Liu and Merkine, 1976), and thus they are not functions of the Reynolds number. Equations (6.16)-(6.19) are subject to the initial conditions E , ( O )= El,,,E,(O) = E ( 0 ) = E, and 8 ( 0 )= 1, with p ( 0 ) = Po chosen to correspond to the physical frequency of the subharmonic (or any other mode), the specified U and the initial physical length scale of the mean flow &,. This length scale has been identified with the initial half-maximum slope thickness. There are many other less dominant disturbance modes present in the experiments of Ho and Huang (1982), including weak fine-grained turbulence, to which the shear layer is sensitive. The relative phase between the fundamental and subharmonic is left arbitrary in the experiments. Thus, the details of the real shear layer are not expected to be described by the idealized two-mode problem in the absence of weak fine-grained turbulence and other (not necessarily weak) modes. The problem solved by Nikitopoulos (1982) and Liu and Nikitopoulos (1982) for E, = 0 brings out the dominant physical mechanisms in the growth and decay and the effect of the relative phases of the overlapping fundamental and subharmonic disturbances in the absence of other complications. Some of these earlier qualitative results were discussed by Ho and Huerre (1984). Subsequent calculations and quantitative comparisons with experiments (Nikitopoulos and Liu, 1987a) are discussed there. We can understand the resulting growth of the shear layer thickness from the measurements of Ho and Huang (1982). The first plateau is due to the peak in the fundamental, the second due to the peaking of the subharmonic and the subsequent linear growth due to the turbulence according to (6.16). We will discuss this in more detail subsequently. Because the interaction between the mean flow and the amplified disturbances is strong, the rapid spreading rate is a part of the nonlinear interaction process and thus ought not be presumed as a known input for the nonlinear amplitude problem. This significant interaction feature, which is lacking in the “small divergence theory” (Gaster, Kit and Wygnanski, 1985; Wygnanski and Petersen, 1987; Weisbrot, 1984), is essential for the wave-envelope problem for strongly amplified coherent modes in developing free shear flows. The plateaus are clearly attributed to the net energy loss from the mean flow directly to the disturbances according to (6.16). The interaction between the coherent modes has only an indirect
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effect. In the absence of any fluctuations, of course, the shear flow spreads because of viscosity alone, as is evident from (6.16). In Liu (1981), the Kelly mechanism was discussed in a much broader context than the parallel flow theory from which it was obtained, as is illustrated here. In order to show consistency with the pioneering work of Kelly (1967) for parallel flows, Nikitopoulos and Liu (1987a) discussed the properties of the mode interaction integral Z2, in detail. We shall summarize here that Zz, < O for small /3 and 8, covering the range of /3 when the fundamental is most amplified and when 8 = 0” (Kelly, 1967), indicating that the fundamental energy is transferred to the subharmonic. As /3 increases, this energy transfer mechanism changes sign for the same 8, a feature attributable to the developing, spatial problem. For large 0 and small /3, energy is transferred from the subharmonic to the fundamental and again, this transfer mechanism changes sign as /3 increases. In the context of strongly amplified disturbances in a developing mean shear pow, however, the original Kelly mechanism for parallel j k ~ w sis largely academic, since the integral I>, changes sign as the flow evolves. However, in the broader sense, the Kelly mechanism is interpreted as having demonstrated the importance of both the relative phase and amplitudes in the subharmonic-fundamental mode interactions. Nikitopoulos and Liu (1984; 1987b) have also studied the three-mode interaction problem which will appear elsewhere. We have already emphasized that the spreading rate of the mean flow is proportional to the rate at which energy is removed from the mean flow. For a purely laminar viscous flow, only viscous dissipation contributes to the spreading rate I , / ( J R e 6 ) as indicated by (6.16); thus 6 -& as expec- rate of energy transfer ted. For a laminar flow undergoing transition, the to orginally small disturbances, reflected by the -6; Reynolds stress conversion mechanism (including, for simplicity in notation, an “ensemble” of coherent modes), now competes with the viscous dissipation. When the disturbances have become sufficiently finite, a marked deviation from the purely viscous spreading rate would be noticed (see, for instance, Sat0 and Kuriki, 1961; KO, Kubota and Lees, 1970). In the presence of both a fundamental disturbance and its subharmonic, such as the case discussed here (Ho and Huang, 1982), where the peaks in the finite amplitudes are distinctively separated in space, the growth of the shear layer undergoes successive plateaus; the vigorous shear layer growth regions are associated with active energy extraction from the mean flow for the disturbance amplification, and the plateau regions are associated with decaying disturbance amplitudes. In Ho and Huang’s (1982) experiments, the shear layer
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continues to spread after the plateau regions. A transition to fine-grained turbulence has most likely taken place in that the existing fine-grained turbulence, having been sufficiently strained by the coherent structures, is now contributing to the mean flow spreading rate via their Reynolds stress fine-grained turbulence -u‘w‘. -For large-scale coherent structures in a turbulent shear flow, both -tG and -u’w’, depending on their relative strengths, contribute to the growth of the mean shear flow. In the downstream coherent-structure mode has rearranged its velocity region where a particular distribution such that -66 is opposite the sign of aU/az, then energy is returned to the mean motion from this particular mode and thus contributes to the decrease of the spreading rate. We have already seen this process, using Weisbrot’s (1984) observation as an example. We now return to the problem of Kaptanoglu (1984) and Liu and Kaptanoglu (1984). They studied the dominant two-dimensional coherent-mode interactions in a two-dimensional turbulent mixing layer by extension of the corresponding problem in a laminar, viscous layer (Nikitopoulos, 1982; Liu and Nikitopoulos, 1982; Nikitopoulos and Liu, 1987a). The individual mode-turbulence interactions are entirely similar to the single coherentmode problem discussed in Section V and Section V1.B. Of particular interest is the application of these ideas to the transition problem (e.g., Ho and Huang, 1982), in which the initial fine-grained turbulence is sufficiently weak to allow coherent mode-interactions to develop initially unhindered by the fine-grained turbulence. Depending on the initial level of the turbulence and the relative strengths of the initial coherent-mode energy levels and the initial mode content, the fine-grained turbulence would eventually be amplified to a fully participating role in the dynamics of the shear layer through energy transfer from the mean flow and the coherent modes. We emphasize here that Kaptanoglu’s model still retains the dominance of the simple two-dimensional coherent modes without considering the spanwise standing waves found to exist observationally as streamwise “streaks.” Consequently, the comparison with observations (e.g., Huang, 1985) is not likely to be meaningful, since the three-dimensional wave disturbances are starting to play a significant role in the dynamics of the shear layer. We shall address this problem in Section VII. Nevertheless, we shall be content here to illustrate the transition problem via the simple two-dimensional coherent mode-interaction model in the presence of fine-grained turbulence. Kaptanoglu (1984) and Liu and Kaptanoglu (1984) first consider an “experiment” in which the “fundamental” mode is initiated at a relatively higher energy level A:,= 17 x 10 at the initial frequency 2p0, whereas its ~
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J. T. C. Liu
“subharmonic” at the initial frequency PI, is initiated at a lower level A:o = 3 x other parameters are set at R = 0.31, Re = 62, 6 = O”, and Eo= The initial Strouhal frequency was chosen to be Po= 0.149 so that 2Po = 0.298. The latter is slightly less than the Strouhal frequency of 0.4426 for the maximum initial amplification rate according to the linear theory. We shall continue to refer to the initial 2P,-mode as the fundamental and the initial &-mode as the subharmonic even if 2po # 0.4426 and Po # 0.2213. The numerical values of the above parameters are fixed and each variation from fixed values will be explicitly stated. The results from the above fixed set of parameters are shown in Figure 19. The energy densities in Figure 19a are denoted by “2” for A f (the initial 2P,,-mode), “1” for A: (the initial &-mode) and “0” for E. The shear layer thickness (normalized by the initial shear layer thickness) is shown in Figure 19b. For this set of parameters, the maximum magnitudes of At and Af reaches approximately the same level; in terms of maximum “amplification,” (A:/A&),,, = 206 and (A:/A?J = 1200. The respective coherent-mode amplitudes grow by extraction of energy from the mean flow; and decay by return of energy to the mean flow (“negative production”), viscous dissipation and energy transfers to the fine-grained turbulence. The relative phase was 6 = O”, so that initially energy is transferred from the 2PI,-mode to the @(,-mode,and this reverses sign with increasing streamwise distance. The mode interaction effect, which is proportional to amplitude cubed, is relatively effective in the vicinity where the mean Row production of wave-disturbance, proportional to amplitude squared, is nearly zero and about to reverse in sign. The production of fine-grained turbulence is slightly larger than its viscous dissipation; the turbulence growth is augmented by the energy transfer from the coherent modes, giving rise to the mild but noticeable maximum in the turbulence energy density in Figure 19a. The noticeable two bumps in the shear layer thickness in Figure 19b are due to the peaking of the energy transfer to the two coherent modes. The eventual linear growth is due to the fine-grained turbulence. In the far downstream region, the balance between the fine-grained turbulence production, dissipation and the effect of shear layer spreading gives an equilibrium fine-grained-turbulenceenergy density E, = 0.18 R 2 and an equilibrium spreading rate d 6 / d x -- 0.025 R due to the fine-grained turbulence. The effect of mean flow dissipation, not being important, was neglected. These estimations derive from the appropriate equation for d 6 l d x and d 6 E l d x with the coherent modes having equilibrated to zero in this case. We see that the equilibrium behavior of E and d 6 / d x in Figure 19 very nearly’follows from the estimates given.
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267
Energy Densities
x lop2
16‘ojShear Layer Thickness
o
. 0
o
,
,
200
, 400
+
,
,
600
l 000
,
I 1000
,
,
, 1200
I
l
~
1400
X
(b) FIG. 19. Evolution of (a) coherent mode and fine-grained turbulence energy densities and ( b ) shear layer thickness for a “standard experiment.” 2: A f , 1: A:, 0: E.
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J. T. C. Liu
We consider next the effect of initial turbulence levels, Eo, on the subsequent shear layer development. When the turbulence energy level is exceedingly weak ( E o= lo-”), we see in Figure 20 that the coherent modes a n d the initial shear layer development are essentially unaffected by the turbulence. The subsequent linear spreading rate far downstream is caused by the rising turbulence energy level. As the initial turbulence level is increased to Eo= lo-* in Figure 21, the linear spreading rate and steep rise in turbulence energy level moves upstream, with the coherent modes still somewhat unaffected. These results are to be compared to the “standard experiment” for Eo= lo-‘ in Figure 19, where the coherent modes are already modified by the fine-grained turbulence. As the initial turbulence in Figure 22, the maximum-A: level is energy is increased to E,,= lowered a n d turbulence development is moved upstream; the maximum- A: level a n d location is slightly modified. As the initial turbulence level is increased to Eo= lo-’ in Figure 23, corresponding to r.m.s. velocity ratios of about 7% of the averaged mean velocity, the coherent modes’ energy levels are significantly reduced. The steplike growth of the shear layer thickness is very nearly obliterated by the strong turbulence levels. The qualitative effects are consistent with observations of Browand a n d Latigo (1979). I n the experiments, however, it is difficult to preserve the same Po while changing the initial turbulence levels. In general, as the turbulence level is increased, the coherent-mode peaks tend to move upstream. With all other parameters fixed as in the “standard experiment” of Figure 19, the Reynolds number is increased to Re = 500 in Figure 24. Results for Re>500 shows only very modest differences. In this case, the viscous dissipation of the coherent modes and of the mean flow is not important. This results in a significant development of the 2Po-mode and, consequently, because 8 = O”, there is significant energy transfer from the @,,-moderesulting in the suppression of the latter. The “nonequilibrium” peak in the turbulence energy level (Figure 24a) is d u e to energy transferred from the coherent modes. The pronounced first step in the shear layer thickness (Figure 24b) is due to the pronounced peak in the 2P,,-mode. The second step, merging immediately into the linear growth region, is attributed to the combined peaks of the P,-mode a n d turbulence. As the Reynolds number is lowered to Re = 100 in Figure 25, the At level is lowered due to viscous dissipation, and the suppression of the Af level from mode-interaction is thus lessened; the turbulence level development is milder as shown in Figure 25a. In this case, the pronounced steplike growth (Figure 24b) has become milder (Figure 25b). These findings are to be compared, again, to the “standard
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269
Energy Densities
-2
Shear Layer Thickness lS'03
0.0
( , , , , , , , , , , , , 1, , , I , , , , , , , , , , , 0
260
SO0
760
1000
1260
,
1600
, /
, l""l""l
1760
2000
2260
X
(b) FIG.20. Shear layer development at a weak initial turbulence level E,,= lo-". ( a ) Energy densities; (b) Shear layer thickness.
J. T. C. Liu
270
Energy Densities
x
o
200
400
800
800
iaw
1000
1400
1800
ieoo
Shear Laver Layer Thickness
/ / 0.0
1
'
200
1 400
'
1 800
'
1
'
1 1000
000
X
(b)
'
1
'
iaoo
1400
FIG.21. Shear layer development at a weak initial turbulence level EI,= densities; (b) Shear layer thickness.
1800
( a ) Energy
27 1
Understanding Large-Scale Coherent Structures
Energy Densities
\
/
Y/-
U
--------
I
'"Of
I
1000
0
I 1200
Shear Layer Thickness
FIG. 22. Shear layer development at a moderate initial turbulence level E,= Energy densities; (b) Sheai layer thickness.
(a)
J. T. C. Liu
272 1.0-
I
x10-* 1.6-
1 .o-
0.6-
1
0.0 0
200
400
800
1000
iaoo
X
(a)
FIG.23. Shear layer development at a strong initial turbulence level E,, = lo-*. ( a ) Energy densities; ( b ) Shear layer thickness.
273
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Energy Densities
'
/-\
x
\. 2
8-
4-
4 4 , 0
a-
-
1
0
1 I
I
0
100
400
800
800
1200
lo00
FIG.24. High Reynolds number effect in the shear layer development, Re densities; ( b ) Shear layer thickness.
=
500. ( a ) Energy
J. T. C. Liu
274
,?Energy Densities
0
0
l”o~
0
200
400
600
800
1000
1200
1400
800
1000
1200
1400
Shear Layer Thickness
200 X
400
600
(b) FIG. 2.5. “Moderate” Reynolds number effect in the shear layer development, Re = 100 ( a ) Energy densities; (b) Shear layer thickness.
Understanding Large-Scale Coherent Structures
275
experiment” of Re = 62 shown in Figure 19. As the Reynolds number is lowered to Re = 40, the 2P,-mode is significantly suppressed at the outset due to viscous dissipation, and the &-mode, in the presence of weak intermode energy drain, is allowed to develop as shown in Figure 26a. The pronounced step in the shear layer thickness is due to the peak in A:. We note that as the Reynolds number is increased, the location of the peak of the 2&-mode moves upstream, whereas that of the P,,-mode remains more or less unchanged. The “standard experiment” (Figure 19) was initiated at the initial dimensionless frequencies Po = 0.149 and 2p0 = 0.298; both modes are on the lower frequency side of the most amplified frequency of 0.4426. Shown in Figure 27 is the case when Po = 0.25 and 2p0 = 0.50, the latter falling to the higher frequency side of 0.4426. Consequently, the 2p0 travels only a little downstream before it is advected into the “negative production” region, and it is thus unable to develop to any significant extent, as shown in Figure 27a; the second mild peak is due to the energy transfer from the PO-mode. In this case, the @,-mode develops almost independently of the 2Po-mode and gives rise to the single pronounced steplike shear layer thickness in Figure 27b prior to the linear growth region. As the initial frequencies are lowered to Po = 0.2 and 2p0 = 0.4, the A; is able to develop further before being advected into the “damped” region shown in Figure 28a, but the steplike structure in the shear layer thickness is still due to the strong levels of A: (Figure 28b). In the “low frequency” initiation at po=0.05 and 2p0 = 0.10, the 2P,,-mode is able to develop significantly and consequently suppresses the p,,-mode via mode interaction (Figure 29a). The pronounced step in the shear layer thickness (Figure 29b) and the peak in the turbulence level (Figure 29a) are attributed to the 2P,-mode. The initially lower frequency modes are stretched out in their streamwise evolution compared to the higher frequency modes, as was expected (Liu, 1974a; Mankbadi and Liu, 1981, 1984) from single-mode considerations. Although not shown, imposing very large initial amplitudes upon one of the modes causes the maximum of that mode to be precisely the initial amplitude, whereas the maximum amplification is achieved by imposing very small initial amplitudes. The amplification of the remaining other mode is only moderately affected. Such resulting properties of mode-forcing upon single, independent modes had already been obtained by Mankbadi and Liu (1981) in connection with the round turbulent jet problem. The recent experiments of Fiedler and Mensing (1985) also indicate interesting properties of possible control. Similar mode interactions in a round turbulent jet
J. T. C. Liu
276
Energy Densities
0
0
200
lb.0-
400
800
1200
1000
1400
Shear Layer Thickness
1a.b-
10.0-
?.b-
6.0-
a.s,
0.0
0
I 100
I
) 400
I
I 800
(
I 800
I 1000
I
I 1200
I 1400
FIG. 26. “Low” Reynolds number effect in the shear layer development, Re = 40. Energy densities; (b) Shear layer thickness.
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277
x lo-*
"1
Shear Layer Thickness
FIG.27. "High" initial frequency effect on shear layer development, Po = 0.25. ( a ) Energy densities; ( b ) Shear layer thickness.
278
J. T. C. Liu Energy Densities
00
12-
1
I
400
200
800
1100
lo00
Shear Layer Thickness
10.
8.
8
4
2
0
I
200 X
,
I
I
400
800
I
800
I 1300
1
1200
(b) FIG. 28. “Moderate” initial frequency effect on shear layer development, Po = 0.20. ( a) Energy densities; (b) Shear layer thickness.
279
Understanding Large-Scale Coherent Structures 10-
x 0-
8-
4-
a-
,,,,~"1,"'1""1""1,"'1",'~"'',
0
o
MX)
1000
1600
aooo
a m
SOOQ
s600
4000
S h e a r Layer Thickness
FIG.29. "Low" initial frequency effect on shear layer development, Po = 0.05. ( a ) Energy densities; (b) Shear layer thickness.
280
J. T. C. Liu
between two-frequency, axially symmetric ( n = 0) modes were recently considered by Mankbadi (1985). The interactions between axially symmetric and helical modes ( n # 0) in a round jet are very much similar to mode interactions involving two-dimensional and spanwise-periodic threedimensional modes in an otherwise two-dimensional shear layer. The issues with regard to such three-dimensional effects are addressed in the next section.
D. MULTIPLE SUBHARMONICS A kinematical model of developing shear layers, in the absence of finegrained turbulence, was suggested by Ho (1981). A fundamental and its succeeding subharmonics, each peaking at further downstream locations, are responsible for the streamwise development of the shear layer. It is an experimental fact that ( 1 ) the fundamental, being of higher frequency, peaks earlier in the streamwise direction than the first subharmonic; their measured energy levels, or wave envelopes, do not switch abruptly but, rather, perform a fade-in and fade-out overlap in the streamwise direction; and (2) the observed lower frequency modes peak further and further downstream. From these observations, for purposes of constructing a dynamical model of multiple subharmonic evolution in a developing shear layer, one can advance the idea that only binary-frequency interactions need to be taken into account in the streamwise development of wave envelopes. For the multiple subharmonic evolution model, this amounts to saying that only the interaction between immediate spatially neighboring wave envelopes needs to be accounted for. Consider, then, the flow disturbance beginning with the fundamental; it acts as an “even” mode to the first subharmonic which in turn is the “odd” mode in the first binary interaction. The first subharmonic, which has twice the frequency of the second subharmonic, then enters into another even and odd binary-mode interaction, and so on. In this case, the binary interaction integral, I ? , ,once tabulated (Nikitopoulos and Liu, 1987a), can be used for such successive interactions. In constructing such a multiple-subharmonic model, we introduce the following easily recognizable notation: Let the subscriptf denote the fundamental and the subscript sn ( n = 1 , 2 , . . .) denote the subharmonics. Thus their respective Reynolds-stress production integral becomes, respectively, Zrr, and I,,,,. The binary interaction mechanism, which was denoted by 12,A;A2in (6.17) and (6.18), uses the subscript 2 to denote the even and 1
Understanding Large-Scale Coherent Structures
28 1
to denote the odd mode. That the rates of strain are provided by the even mode while the stresses are provided by the odd mode is discussed in Section 1I.C on energy balance. This accounts for the powers of the amplitude occurring as A: and A , . If we make the “high frequency” cutoff at the fundamental (although higher harmonics can certainly be taken into account through binaryfrequency interactions), then the fundamental mode will have only one interaction term, that with the first subharmonic. In this case, the waveenvelope equation for the fundamental is similar to (6.18), but with the notation changed according to the discussion above,
f, dx
= I,fA:.
+ I,,,. ,Af ,A, - I,,,,,-EAF -
1
Z, A,;./6.
(6.20)
The first subharmonic wave-envelope equation, similar to (6.17) but with an additional interaction term to connect with the second subharmonic ( A J , then appears as - d6A:,
-~
r , , r -
1 Re
- ~ t s I A f l -~,ls:Af>A,l ~, -In,,IEAS,--l~~JIA;?IIS.
r s y l ~ ? l
(6.21) The nth subharmonic wave-envelope equation then becomes - d6Af,
1,n-
dx
-
I,,,, A:,, - I -
5 ( ,I
In,xnEASn -
~
I 1 \I!
1
A f,, A\
n
~
1)
+ I Fnc(
,1+
I
,AS,r, + 1 ,A,,
l+)nA?ttI6.
(6.22)
If the practical streamwise region of interest precludes consideration of say, the ( n + 1)-subharmonic, then the n - ( n + 1) interaction term would be absent in (6.22). The modifications to the fine-grained turbulence and mean flow equations are straightforward. Thus the second term on the right of (6.19) is now replaced by
and terms one and two on the right of (6.16) are replaced by
The initial conditions are similar to the two-mode interaction problem earlier.
J. T. C. Liu
282
This dynamical, multiple subharmonic model is explored by Liu and Kaptanoglu (1987) as an initial value problem in studying the spatialevolutionary properties of coherent structure wave envelopes in a developing mixing layer as well as possibilities for free shear layer control. A “standard” case is computed as the basis for comparison with results from variations of controlling parameters. The standard case is fixed as follows: Po = 0.4985 now denotes the dimensionless frequency of the fundamental mode, which is almost most amplified mode at R = 0.69 (Po for all subharmonics are halved). The Reynolds number is Re = 968. Three subharmonics are taken as “representative.” The initial conditions are expressed as E f , = 2.1 x EFIO = 0.925 x = E,,()= Eo = The relative phase angles are chosen to be O,,= 180”, so that energy transfers from high to lower frequencies for R = 0.69. The initial conditions are applied at ~ ~ / 6 ~ Subsequently = 7 . both 6 and x are normalized by 6,). The flow conditions here would correspond approximately to U , = 363 cm/s, U - , = 1980 cm/s, f = 750 Hz, So= 0.124 cm in air (Huang, 1985). The standard case is shown in Figures 30 and 31 as the solid line. In Figure 30 we show the effect of forcing the fundamental. With all other conditions fixed, it is known theoretically that for the single-mode 12.00
11.00 10.00
9.00
1
1
7.00
6’ool 5.00
2.00i
3.00
1.00
0.00 0 .
1
0
/
1
I
1
50.0
I
I
I
I
100.0
l
X
l
/
’
150.0
‘ 1 -
200.0
250.0
FIG. 30. The effect of forcing the fundamental component at various initial amplitudes
Understanding Large-Scale Coherent Structures
283
problem (KOet al., 1970) if the initial disturbance amplitude were increased, the development of the coherent mode and the enhanced spreading of the mean flow would be moved upstream. This effect illustrated through 6 in Figure 30, comparing the initial amplitudes Ef0- lo-' and with the standard case of -lop4. For E,()the relative amplification is larger for the fundamental energy density than for the &,- lo-' case. The exceedingly larger Efo case effects a choking of its own energy supply through the weakening of the mean flow. Not only the relative amplification is weakened; the streamwise lifetime is shortened in the high initial amplitude forcing. Though not shown here, more severe forcing causes the immediate decay downstream, thus rendering the forcing amplitude to be the maximum attachable amplitude (Mankbadi and Liu, 1981). This also effects the weakening of the subharmonics due to mode interactions. Although the subsequent shear layer growth rate is the same, because of the weakening of the coherent modes, the larger initial fundamental amplitude causes the shear layer to spread not as widely as the weaker initial fundamental amplitude cases. All the activities are confined to the x 5 100 region. To illustrate the effect of forcing lower frequency components, the forcing of the second subharmonic is shown in Figure 31. Because of the imposed 12.007
\
5.00
0.0
50.0
100.0
X/b,
150.0
200.0
FIG. 31. The effect of forcing the second subharmonic at various initial amplitudes
250.0
J. T. C. Liu relative phase angles, a strong lower frequency component takes energy away from the higher frequency components. Thus the higher frequency first harmonic and the fundamental are both weakened by the second subharmonic. (This is also the case with higher frequency components in the forcing of the third subharmonic.) The subsequent resurgence of a higher frequency component (not shown here) is due to the resurgency of production from the mean flow relative to other subsiding energy sinks such as the binary frequency transfer mechanism to lower frequencies for 8, = 180". We note that, in Figure 31, the subsequent spreading rate is somewhat dramatically enhanced. However, the fine-grained turbulence level always seems to settle to an equilibrium level within the region of interest. Though not shown here, the spreading rates associated with the third subharmonic forcing is most dramatic (Liu and Kaptanoglu, 1987). For a forcing level of E,?,,-there is a doubling of the shear layer thickness around x 200 standard case. The doubling in thickness is compared to the EF3"achieved much earlier, at about x 100, when E,,,,- lo-'. In the latter case, the forcing velocity ratio would be about 30%, a severe case approaching that of Favre-Marinet and Binder's forcing of a round jet. For further exploration of the problem, we refer to Liu and Kaptanoglu (1987).
-
VII. Three-Dimensional Nonlinear Effects in Large-Scale Coherent-Mode Interactions
A. GENERAL DISCUSSION
In the previous sections we have discussed the mechanisms of interaction between plane, large-scale coherent modes and the three-dimensional finegrained turbulence. Although the two-dimensional coherent structures are still the dominant coherent modes in two-dimensional shear flows, there is increasing observational evidence that three-dimensional coherent modes, in the form of spanwise periodicities or standing waves, persist (Miksad, 1972; Bernal et al., 1980; Bernal, 1981; Bernal and Roshko, 1986; Breidenthal, 1981; Browand and Troutt, 1980, 1984; Roshko, 1981; Konrad, 1977; Jimenez, 1983; Jimenez et al., 1985; Alvarez and Martinez-Val, 1984; Huang, 1985; Lasheras et al., 1986). The experiments dealt primarily with transitional shear layers, and coherent three-dimensionality is clearly most likely
Understanding Large-Scale Coherent Structures
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to provide additional sites for the straining and amplification of preexisting fine-grained turbulence, however initially weak (Huang, 1985). This would augment the direct production of fine-grained turbulence from the mean flow and from the two-dimensional coherent motions. The three-dimensional coherent motions persist well into the region where fine-grained turbulence has become active (Bernal, 1981; Roshko, 1981). On the basis of the discussions in the previous sections, it is entirely conceivable that such spanwise periodicities, again appearing as a manifestation of hydrodynamic instability, would also develop in an initially turbulent shear layer, depending on the balances between mechanisms of energy supply and “dissipation.” From this discussion, we are led to distinguish carefully the two very distinct three-dimensional motions. One is the fine-grained turbulence, and the other is the large-scale coherent motion in the form of spanwise standing waves in a two-dimensional mean shear flow or helical modes in the round jet (e.g., Mankbadi and Liu, 1981, 1984). It is an experimental fact that the spanwise wavelength of the three-dimensional coherent modes increases further downstream (Barnel, 1981; Jimenez, 1983; Huang, 1985), as if evolving through the emergence of a spanwise subharmonic formation, much in the same spirit as the subharmonic formation in terms of frequency and streamwise wavelength for two-dimensional coherent modes (Freymuth, 1966; Winant and Browand, 1974). Quantitative observations (e.g., Jimenez, 1983; Huang, 1985) indicate that the combined spanwise, three-dimensional modes develop downstream in a nonequilibrium fashion resembling, though not in detail, that of the two-dimensional modes. The wave-envelope dimensional disturbances are imposed by upstream perturbations such as the inherent waviness of the trailing edge of the plate separating the two streams or the screens placed upstream of the trailing edge. Consequently, the upstream initial conditions on the spanwise modes are uncontrolled. Unlike the situation with the wavenumber or frequency selection mechanism for the two-dimensional coherent modes, the spanwise wavenumber selection mechanism is still unsettled in spite of recent works on the temporal mixing layer from the point of view of computational-hydrodynamicstability (Pierrehumbert and Widnall, 1982; Corcos and Lin, 1984) and numerical simulation (Riley and Metcalfe, 1980; Cain et al., 1981; Couet and Leonard, 1981; Metcalfe el al., 1987). Corcos and Lin (1984) suggest that perhaps the nonlinear interactions between spanwise modes and the role of initial conditions might uncover the mechanism of the spanwise wave number selection. To this end, we shall return to a brief discussion of the classical nonlinear analyses of three-dimensional disturbances in shear flows. This
J. T C. Liu would form the basis for a discussion of real, spatially developing shear flows.
B. PARALLELFLOWS Three-dimensional disturbance effects in temporal, parallel shear flows have been studied by Benney and Lin (1960) and Benney (1961). This body of work is a second-order theory rather than one of finite amplitude in that the amplitudes are taken as exponentials. The authors considered the temporal problem consisting of two interacting fundamentals, a twodimensional wave disturbance of the form exp( i a ( x - cIt ) ) and a threedimensional disturbance of the form exp(ia(x - c z t ) )cos yy, where y is the spanwise wave number, c1 and c2 are the complex phase velocities and a is the streamwise wave number associated with the fundamental twodimensional disturbance. For simplicity, Benney and Lin ( 1960) assumed that cI = c2 for a given Reynolds number, which leads to harmonics that are stationary rather than periodic in time. Other second-order effects include the formation of harmonics of the two fundamentals and the distortion of the mean flow. The combination of nonlinear effects on amplitude and three-dimensional wave disturbance effects were studied by Stuart (1962b) and presented at the 1960 Second International Congress in Aeronautical Science in Zurich. Stuart (1962b) found that there are at least eight physically distinct “modes.” This can best be characterized by attaching the subscripts m and n to the relevant flow quantities, say, the velocity umni(where i is retained to indicate the components of the velocity). The first subscript m indicates the streamwise wave number for the temporal problem, whereas n would indicate the spanwise wave number. For instance, rn = 1 denotes the fundamental streamwise wavenumber a, m = 2 its first harmonic 2 a ; n = 1 denotes the cos y y mode and n = 2 denotes the cos 2 yy mode. The three streamwise nonperiodic modes consist of the 00, 01 and 02 modes. The first refers to the modification of the temporal mean motion, which is here the combined streamwise- and spanwise-averaged flow. The 01 and 02 modes are the streamwise-independent but spanwise-periodic harmonics generated by the three-dimensional wave disturbance. The 10 and 20 modes are the two-dimensional fundamental and harmonic components, respectively. The 11 mode is the three-dimensional fundamental and the 22 and 21 modes are the associated harmonics. Following earlier work on finite-amplitude effects for two-dimensional disturbances (Stuart,
Understanding Large-Scale Coherent Structures
287
1960), Stuart (1962b) obtained amplitude equations for the two complex two-amplitude functions A(t) and B(t) for the temporal two- and threedimensional disturbances, respectively, in a parallel flow:
dA - = A ( a , + a ~ ” l A I ’ + a ~ 2 ’ I B 1 2 + .. *)+a‘,’’ZB’+. dt dB
-=B(b,+bI”IA12+bl”lBI’+. dt
*
* *
,
* ) + b \ 3 ’ 6 A 2 +*** .
(6.19) (6.20)
In (6.19) and (6.20), the constants a, = -icuc, and b,= -icuc, come from the linear theory, and the remaining constants from the nonlinear theory. Stuart (1962b) showed how these constants could be evaluated. He argued that for finite values of the spanwise wave number y, the constants a?’ and b‘,” may be chosen to be zero. In this case, the “wave envelope” equations then appear in the form (6.21) (6.22) where the subscripts i and r denote imaginary and real parts, respectively. The amplitude equations from weakly nonlinear theory are stated here for later reference for purposes of showing the contrast with the wave-envelope equations of three-dimensional disturbances in spatially developing shear flows for strongly amplified disturbances.
C. SPATIALLY DEVELOPING SHEARFLOWS We have seen in Section V1.C how ideas from weakly nonlinear theory could be used as a valuable guide for mode interactions in a developing shear flow. There the single odd- and even-mode were given their individual amplitudes, as would be motivated by observations (e.g., Ho and Huang, 1982), rather than in terms of an expansion, which is in terms of ascending powers of a single amplitude function of the weak, nonlinear theory. The nonlinear effects, being of amplitude to the fourth power, reflect such an expansion procedure. This will be contrasted to the anticipated third power in amplitude for the present class of problems. In order to study the interaction between an initial fundamental component and its subharmonic
J. T. C. Liu in the spatial problem, the mode interaction is in terms of frequency and calls for the reinterpretation of the single even-mode as the fundamental component and the single-odd mode as its subharmonic at half the fundamental frequency. The same interpretation is used to denote threedimensional wave disturbance interactions. The even and odd modes here refer to the frequency only and the basic equations developed in Section I1 apply. The Reynolds mean motion by definition, is obtained via averaging over all periodicities. In this case, the average is taken with respect to time and over the spanwise distance for two-dimensional shear flows. For round jets, the latter average is replaced by the circumferential average. The conditional average used to separate the coherent modes and the fine-grained turbulence is still the phase-averaging procedure geared to the coherent frequencies or periods for the spatial problem. In order to study subharmonic/fundamental interactions (in the frequency sense) and the downstream evolution of at least two spanwise periodic scales for the spatially developing shear Bow, it is not difficult to confirm that the minimum number of frequency-periodic modes required is five. Using similar notation to that of Stuart (1962b), we denote the coherent dynamical quantities as qmn (with u,,,,, as the velocity, i as the component indicator); m refers to the frequency and n to the spanwise-periodicity. The even-frequency mode is denoted by m = 2 (reinterpreted as the fundamental mode in frequency) and the odd-frequency mode by m = 1 (the reinterpreted subharmonic mode in frequency). The two-dimensional modes are denoted by n = 0. It is not essential to take the spanwise periodicity indication n # 0 literally as long as we identify modes with n = 1 to have spanwise wavelengths twice that of the modes with n = 2. For instance, n = 2 and 1 could be taken to indicate cos 2yy and cos -yy, respectively, or cos yy and cos(y/2)y, respectively. In both cases, the spanwise wavelength ( A , l ) is such that A , = 2Az. In observations (e.g., Jimenez, 1983; Huang, 1985), A , eventually prevails over A z downstream. The five minimum frequency periodic ( m# 0) modes consistent with Stuart (1962b) would be three modes belonging to the fundamental frequency (even, m = 2) 20, 21, 22 and two modes belonging to the subharmonic frequency (odd, rn = 1) 10, 11. These modes still belong to the family of binary-frequency interactions (Liu and Nikitopoulos, 1982; Nikitopoulos, 1982). Inclusion of other rn # 0 modes would necessitate tertiary-frequency interactions, but these could still be formulated from the basic equations of Section 11, as was done for triplefrequency mode interactions for two-dimensional wave disturbances (Nikitopoulos and Liu, 1984, 1987). The remaining frequency-independent
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modes (00,Ol and 02) are modifications to the time-averaged mean flow; the 01 and 02 are modifications prior to spanwise averaging. Before we continue with the three-dimensional wave disturbance problem, we shall insert a brief comment about accounting only for binary-frequency interactions which shows that it could be more general than would be anticipated. The basis for our implicit hypothesis that only binary-frequency mode interactions suffice for the spatially developing shear flow lies in the earlier theoretical confirmation (Liu, 1974a) of observations that progressively lower frequency modes develop and peak further downstream relative to higher frequency modes. For mode-interactions of the sub- and superharmonic type to take place, modes of only integral multiples of the frequency participate. As demonstrated by Ho and Huang (.1982),the peaks of the fundamental and subharmonic do not overlap. The first subharmonic, peaking further downstream than the fundamental, would eventually serve as the fundamental to the second subharmonic, but in a region where the original fundamental has significantly weakened. In this case interactions between neighboring frequency modes would dominate. Situations where binary-frequency interactions would not suffice are elucidated by Nikitopoulos and Liu (1984, 1987). We already saw the utility of the binary-frequency interaction model in developing multiple subharmonics in Section V1.D.
D. ENERGYEXCHANGE MECHANISMS The spanwise periodicities are considered to be standing waves. To help understand the physical mechanisms of mode interactions within the limited framework described, we obtain and state the energy equations for the five coherent modes. The energy equations of the even-frequency modes are obtained from (2.25). These modes are specialized to be the twodimensional, fundamental-frequency mode; its energy equation is D -
Dt
T
u20,/2=
d - - ~ P 2 0 ~ 2 0 , + ~ ~ , 0 ~ ~ l , , + ~ l I ~ ~ , l , ~ ~ 2 " ~ + ~ 2 0 ~ ~ 2 0 ~ 1
ax,
The averaging, as already discussed, is with respect to both time and
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290
spanwise distance. The symbol (A), denoting even modes in Section 11, is identified here with the first subscript rn = 2 denoting the fundamental frequency, whereas (-), denoting the odd modes, is identified here with rn = 1 as the frequency-subharmonic. In the second group of terms on the right side of (6.23), there are direct energy exchanges between the 20-mode with the mean flow and the fine-grained turbulence, as well as direct energy exchanges with the two-dimensional and three-dimensional ( n = 1) subharmonic modes, 10 and 11, respectively. The fundamental frequency, n = 1 three-dimensional 21-mode energy equation is
,51
-u 2 , , / 2 =
Dt
a -
--
ax1
-
v
[p21u211+ ( U I O , U , , ,
+ U I I , ~ I 0 , ) U 2 1 r+ U Z l , ~ Z l , , I
(%)*.
(6.24)
Direct energy exchanges of the 21-mode energy with the mean flow and fine-grained turbulence are obvious in the second group of terms on the right of (6.24). The last item in this group reflects, as will be confirmed subsequently, the energy exchange between the 21-mode with the 10-mode through interference of the 11-mode -ul~,,ull,auZl,/C3xJ, and with the 11; net rate mode through interference of the 10-mode - u , , , u I o , d u 2 , , / ~ x ,the of these energy exchanges is the same. The fundamental-frequency, n = 2 three-dimensional 22-mode energy equation is
D Dt
y '22!12
a = --
ax,
[P22Ur2,
+ '1
1tu1
1IU22t
+ u?2!r22!Jl
(6.25) Again, in addition to energy exchanges with the mean flow and fine-grained turbulence, the last term in the second group of energy exchange mechanisms on the right of (6.25) reflects a direct energy exchange between the 22-mode and the 11-mode. We note that there are no direct energy exchanges between the three fundamentals 20. 21 and 22.
Understanding Large-Scale Coherent Structures
29 1
The energy equation for the two-dimensional subharmonic 10-mode is D-
/2=--
-uz
Dt
lo'
a ax,
bloUlor+u20u:,,/2+
(
+ v- a2 u:,,,/2. 2 )%.v! ! ! ~
-
ulo,rloyl
(6.26)
ax1
The rate of energy exchanges with other components of flow are again given by the second group of terms o n the right of (6.26). The 10-mode exchanges energy with the frequency-subharmonic, three-dimensional 11-mode via the interference of the fundamental 21-mode. It exchanges energy with the fundamental two-dimensional 20-mode directly but with the fundamental three-dimensional 21-mode via interference by the 11-mode. The frequencysubharmonic, n = 1 three-dimensional 11-mode energy equation is
(6.27) The energy exchange with other modes is given in the second group of terms on the right side of (6.27). The 11-mode exchanges energy with the two-dimensional 10-mode through the interference of the 21-mode and with the 21-mode through the interference of the 10-mode. As already noted, the 11-mode exchanges energy directly with the 20- and 22-mode. Again, the 11-mode exchanges energy directly with the mean flow and fine-grained turbulence as depicted, respectively, by the first two terms in this same group.
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We have, in Figure 32, depicted the mn-mode energy transfer mechanisms. The direction of the arrow in the figure is associated only with the manner in which the sign of the energy exchange term occur3 in the individual energy equation, not the actual direction of the individual energy exchange mechanism. As we have learned from our previous considerations, the direction of energy transfer lies in the relative phase relations between the fluctuations that make u p this mechanism. The energy exchange rate between the coherent modes and the finegrained turbulence is summarized in Table 1. The n = O two-dimensional mode energy exchanges between the coherent modes and with fine-grained turbulence have been the subject of discussions in Sections IV and V and in the present section. It is not difficult to see that the n = 1 , 2 threedimensional modes provide additional modulated turbulent stresses and coherent rates of strain for such exchange mechanisms. The energy exchange
two-dimensional n = O
three-dimensional n = I
three-dimensional I1 =
2
FIG.32. Two- and three-dimensional coherent mn-mode energy transfer mechanisms.
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293
TABLE 1 ENERGYE X C H A N G ~WITH S FINEGRAINED TURBULENCE n=O
n=2
n=l
m=l m=2
mechanisms with the mean flow are summarized in Table 2. We have already shown how energy extraction by two-dimensional modes from the mean flow causes its thickness to grow. The additional mechanisms due to the three-dimensional modes would augment this spreading rate if wave disturbances continued to take energy from the mean flow.
E.
NONLINEAR
AMPLITUDE EVOLUTIONEQUATIONS
From the special form of (6.23)-(6.27) for which the mean flow is two-dimensional, we can obtain the spatial evolution equations for the five mode amplitudes, and, in addition, those of the fine-grained turbulence energy and the mean flow thickness similar to the two-dimensional coherent mode problem. The notation used for the advection, interaction and dissipation integrals is similar to those previously defined except for the subscripts mn, where m = 1 , 2 and n = 0 , 1 , 2 (but there is no 12-mode within the present framework). The wave-envelope equations for the five modes can
TABLE 2 ENERGYEX< H A N G E S n =O
WITH
MEAN FLOW
n=l
n=2
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294
be written in terms of the square of the amplitudes, A,,
energy exchange with mean flow
energy exchange with turbulence
viscous dissipation
(6.28) energy exchange with other modes
The mode-mode energy exchange mechanisms, 9:, , are summarized in Table 3, where I : : , I::, etc. are the interaction integrals corresponding to : , occur in equal magnitude but of mechanisms indicated in Figure 32. 9 opposite sign in the binary interaction between mode ( m n )and mode ( k l ) , the actual direction of energy transfer depending on the relative phase of the participating coherent modes. The mean flow kinetic energy equation gives (6.29) energy exchange with overall coherent modes
energy exchange with turbulence
viscous dissipation
The fine-grained turbulence kinetic energy equation gives (6.30) energy exchange with mean flow
energy exchange with overall coherent modes
viscous dissipation
TABLE 3 MODE-MODE E N E R G Y E X C H A N G t M E C H A N I S M S9:Ln n=O
n=l
n =2
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Equations (6.28)-(6.30) would be subjected to the initial conditions AL,(0) = 6(0) = 1, E ( 0 ) = Eo, supplemented by choosing the initial frequency of the wave disturbance Po (and 2Po), the relative spanwise wave number y / P o and the relative phases between the coherent modes. We comment here that in the case of the round jet the physical mechanisms (except for details with regard to curvature effects in the downstream region) and formulation appear in the same form as in (6.28)-(6.30), with n = O identified with the axially symmetric modes and n # 0 with helical modes. Although the numerical aspects of this problem are under active pursuit (Lee and Liu, 1985, 1987), a number of relevant and meaningful interpretations can be directly inferred from the formulation and preliminary results. It is now well known that higher frequency wave disturbances grow, peak and decay in a region closer to the start of the shear flow than lower frequency disturbances. In this situation, the entire rn = 2 higher fundamental frequency group of 20, 21 and 22 modes accomplish such growth and decay activities earlier on in the streamwise direction than the rn = 1 group of 10 and 11 modes for not disparately different initial mode-energy levels. Within the rn = 2 group, it is expected that the n = 0 dimensional 20-mode would persist longer in the streamwise distance than the 21-mode; the latter in turn would prevail over the 22-mode. In this case, although the cos 2yy and cos yy modes would initially develop at about the same level, the shorter wavelength (25-/2 y ) , three-dimensional spanwise mode disappears first, giving way to the longer wavelength (27r/ y) spanwise mode associated with the higher, fundamental frequency group. Eventually in the streamwise development, the rn = 2 frequency group of modes give way to the rn = 1 subharmonic frequency group of 10 and 1 1 modes. The development of the 1l-mode, of wavelength 25-/ y , then persists further downstream (until it succumbs to subsequent subharmonics or turbulence). Thus, the present multiple-mode interaction model gives the important observational feature (Bernal, 1981; Jimenez, 1983; Huang, 1985) that the number of streamwise, longitudinal streaks lessens with the downstream distance. This important feature is inferred from the formulation of the problem and preliminary numerical results (Lee and Liu, 1985, 1987) confirm it. Characteristically, the problem with coherent modes in developing shear flows is one of nonequilibrium interactions and is sensitive to initial conditions. Perhaps, when the full numerical results become available, a study based on the variation of initial conditions and mode numbers might provide us with an understanding of the spanwise-mode selection mechanisms in developing shear flows.
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In the recent measurements of Huang ( 1985), frequency-fundamental and subharmonic-mode energies were obtained, but without differentiating between two- and three-dimensional modes in the present context. Thus, the sectional energy measured, in terms of the present interpretation, reflects the sum within each frequency-group of modes: ( E2,+ E21+ E2J for the fundamental and ( Elo+ El ,) for the subharmonic. Further decomposition along the lines discussed here would be helpful in understanding the important modal-interaction mechanisms that we have elucidated.
F. RELATIONTO TEMPORAL M I X I N GLAYERSTUDIES There are several temporal mixing layer studies that would be of interest from the present point of view. We delay discussions of these until the present nonlinear interaction problem has been fully stated. In this case, we will be able to place these temporal problems in proper perspective with respect to the spatial problem that we have discussed. To this end, the mode number in the temporal problem refers to the streamwise wave number and is taken to be analogous to the frequency in the spatial problem. This “common” mode number will be denoted by m in the mn-notation. The spanwise mode number is identical in both cases and is denoted by n. Pierrehumbert and Widnall (1982) studied the linear three-dimensional stability of a class of finite-amplitude, steady two-dimensional solutions to the Liouville equation obtained by Stuart (1967, 1971b). The class of solutions is obtained by variations of a so-called vorticity concentration parameter, E , which when set equal to zero gives us the hyperbolic tangent profile, which could be considered as the mean flow. For small but finite E, an expansion in powers of E reveals that the mean flow is perturbed by a steady, spatially periodic fundamental component at the E order; at the E~ order there is a first harmonic component and a correction to the mean flow, and so on. When E + 1, the flow due to a row of point vortices is recovered. The E + 0 range is relevant to our discussion. Because the flow is steady, the problem is neutral in that no energy exchange exists among the disturbance components and the mean flow. In our notation, in addition to the mean flow, this basic flow also consists of neutrally noninteracting 20 and 10 components (where we now revert to interpreting 20 as the first harmonic and 10 as the fundamental). The translative mode corresponds to a three-dimensional perturbation at the same m. In this case, the modes consist of the basic 20- and 10-mode plus the 1l-mode. In the linear problem
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only direct energy transfers are possible. From Figure 32 we see that there is no direct connection between the 11-mode perturbation with the basic 10-mode in absence of the 21-mode, but that there is a direct connection between the 11-mode with the basic 20-mode. Thus the amplification of the 11-mode comes from the basic mean flow and the 20-mode, while the 10-mode remains dormant in this process. As the parameter F is further lowered, the present first harmonic, the 20-mode, being of order E ’ , becomes unimportant, so that the only energy supply to the 11-mode would be the mean flow. This loss of an additional source of energy supply for E + 0 may well be the reason why the 11-mode amplification rate is lowered with decreasing values of E in the Pierrehumbert and Widnall (1982) translativemode problem (see also Ho and Huerre, 1984). This translative mode is not equivalent to the second-order interactions described by Benney and Lin (1960) and Benney (1961) in that they included the 21-mode, which interacts with and causes interaction between the 10- and 11-mode. To interpret the linearized helical-mode instability of Pierrehumbert and Widnall (1982), we now reinterpret the 20-mode as the two-dimensional fundamental and 11-mode as the subharmonic, three-dimensional perturbation. From Figure 32, there is a direct interaction between the 20- and the 11-mode, in addition to the direct participation of the mean flow. Corcos and Lin (1984) studied three-dimensional linear perturbations upon a time-evolving two-dimensional flow consisting equivalently of the mutually interacting mean shear flow and the two-dimensional mo-modes. In the equivalent translative mode interactions, they included the 20- and 21-mode, or alternatively, the 10- and 11-mode (cases 1 through 4); in these cases there are no direct mode interactions, but the three-dimensional mode derives its energy from the mean flow. In the translative-mode interaction with the presence of a subharmonic, the 20-, 10- and 21-mode (cases 7-10) are included; again, there are no direct three- and two-dimensional mode interactions in absence of the 1 1-mode. In the helical-mode interaction, modes 20,lO and 11 were involved (cases 5,6), and there is direct interaction between the 20- and 11-modes but none between 10- and 11-modes in absence of the 21-mode. The rate of energy supply to the three-dimensional disturbance given by Corcos and Lin (1984) is the overall rate, and thus does not elucidate these important individual mechanisms. The resonant triad of Craik (1971, 1980), originally discussed in terms of boundary layers, is essentially a two-mode interaction in the context of spanwise standing-wave disturbances, involving the 20- and 1I-mode for which there is a direct interaction (Figure 32). For a discussion of the related
298
J. T. C. Liu
unpublished work by Raetz on resonant interactions between threedimensional disturbances, we refer to Stuart (1962a). See also Craik (1985). Metcalfe el al. (1987) presented numerical simulations of the temporal mixing layer via the spectral method in the range Re 10’. I n principle, all modes included “participate” to a certain extent in the dynamics. However, they assigned prominent initial values to the amplitude of certain modes and thus singled these out as the participating coherent structures. It is thus possible to discuss such simulations in terms of Figure 32. Again, in terms of the present notation ( m = 1: subharmonic, m = 2 : fundamental), they considered the following interactions: 20, 10; 20, 21; 10, 21; and 20, 10, 21. If all other modes are less prominent and practically not participating, we see from Figure 32 that the only direct inter-mode energy transfer is between the 20- and 10-modes similar to that discussed by Nikitopoulos and Liu (1987) and Mankbadi (1985). The three-dimensional 21-mode interacts with the 20- and 10-modes implicitly via the mean flow. Metcalfe el al. (1987) also gave results from an initial random noise field simulation. The streamwise vorticity at a given time instant and a given spanwise cross section strongly resembled the flow visualization picture of Bernal (1981) at a given streamwise location. The evolutionary aspect of coherent structure properties, however, was not emphasized.
-
VIII. Other Wave-Turbulence Interaction Problems It seems appropriate to conclude this article by briefly pointing out a few examples to confirm that “. . .the more research in mechanics? expands, the more interconnections of seemingly far distant fields become apparent.” This was an observation and a spirit infused into this series by the founding editors, von K5rm5n and von Mises, in their preface to the first volume. In the structural aspects of the turbulent boundary layer, there is no dearth of problems involving the interactions between various scales of large-scale motions and fine-grained turbulence (Willmarth, 1975).Although the situation is considerably more complicated and involved relative to the free shear flows, many of the interaction ideas share the same fundamental basis. The prospects of control naturally lead to the attempt to understand various perturbations upon turbulent boundary layers. One such perturba1- In the present case, research in the large-scale organized aspects in free turbulent shear flows.
Understanding Large-Scale Coherent Structures
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tion occurs through the interaction of sound with wall turbulent shear layers (Howe, 1986), and some progress toward understanding it is beginning to take place (Quinn and Liu, 1985). Interaction between wave motions and turbulence has recently taken on an important role in the meteorological context in mesospheric dynamics (Holton and Matsuno, 1984; Fritts, 1984), and in the oceanographic context in the mixing mechanism in the interior ocean and the microstructure problem. In fact Munk (1981) underscores the connection between internal waves and small-scale processes as “where the key is.” Recent laboratory experiments (Stillinger et al., 1983) conducted in stratified fluid point to the necessity of separating waves and turbulence in order to understand their internal interaction processes. As an illustration of the turbulencemodified internal wave problem, similar conditional averaging procedures can be used to obtain the equation for linear internal waves (Quinn and Liu, 1986):
where G is the vertical wave velocity, V L the horizontal Laplacian, z is the vertical coordinate, x and y the horizontal coordinates, N 2 the Brunt frequency taken as constant, g the acceleration of gravity, T, the temperature of the undisturbed (hydrostatic) fluid taken as constant as far as the wave motion is concerned, and 6, the wave-modulated turbulence heat flux vector; rv has the same meaning as in the previous discussions. Equation (7.1) would be augmented by the transport equations for t,, G, and for the wave-modulated square of the turbulence temperature fluctuation 6. These would be a rational replacement of the standard eddy-viscosity assumptions where, particularly in geophysical problems, the magnitude and sign of such viscosities are difficult to estimate. Wave-turbulence interaction problems in the lower atmosphere in the vicinity of the atmospheric boundary layer have received attention (Einaudi and Finnegan, 1981; Finnegan and Einaudi, 1981; Fua et al., 1982). The onset of turbulence in KelvinHelmholtz billows is addressed by Sykes and Lewellen (1982) and by Klaassen and Peltier (1989, similar to the temporal homogeneous fluid problem of Gatski and Liu (1980).
J. T. C. Liu
300 Appendix
The integrals for the spatially developing plane turbulent mixing layer are explicitly defined here for completeness. These integrals are similar in form to certain of those that occur in the temporal problem except that there the integrals involving the eigenfunctions depend on the local wave number. The dominant coherent mode here is also taken as twodimensional, and the spatial eigenfunctions are evaluated ''locally'' and depend on the local frequency parameter. The mean velocity profile and be of the form (5.8) and (5.9), respectively. Reynolds stresses are taken toSpecifically, we have taken uiuj e-' and U = 1 - R tank 5, where R = (UrnU r n ) /Urn+ ( Urn),5 = z / 6 ( x ) . Generalizations to other profiles are certainly possible. The local shear layer thickness 6 ( x ) is measured in terms of the initial shear layer thickness (&), and is half of the maximum slope or mean vorticity thickness
-
where R = - a U / d z (see Brown and Roshko, 1974); 6 ( x ) is also twice the momentum thickness (Winant and Browand, 1974) for the hyperbolic tangent profile. The appropriate initial Reynolds number is Re = Sou/v, where is the average velocity (U-,+ U,,)/2. All velocities are normalized by 0 and lengths by 6". The integrals involving the local eigenfunctions reflect the normalization defined by (5.4).
u
( 1 ) Kinetic energy advection integrals
Mean p o w :
-f(
I=
+
( 1 - R tanh
lorn
(1 - R tanh
= ( 3 - 21n2)R2.
Coherent mode:
5)[(1- R tanh g)'-(l+ R)']
5)[(1- R tanh 5)2- ( 1 - R)2] d5
dl
I
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In the binary mode interactions, I? is associated with eigenfunctions with subscript 2, I , with subscript 1. In general Z2 and I, do not change sign, are very nearly “constant” and will be replaced by their respective meanvalue over the range of 6 of interest. Fine-grained turbulence: zl=zr
I‘
(1 +tanh 5 ) e p i ’ d5 = 1.
- T
( 2 ) Fluctuation “production” integrals
Coherent mode: fr,(8)=2R
5
il(
9m(a+T)sech2
-v
+.
where 6denotes the complex conjugate of In the “damped” disturbance region 9m(a+&’)changes sign and I,, < 0. In binary mode interactions, Zr,’ will be associated with a*, & . . .a nd with a , , 4 , , . . . . Fine-grained turbulence:
(3) Viscous dissipation integrals Mean pow: TI
4R’ sech4 5 d5 = -. 3
Coherent mode:
Fine-grained turbulence:
(4) Coherent mode-turbulence energy exchange integral
1,,(6)=-2
[
a3
~ ~ [ r x x ( - “ Y ~ ’ ) + r , , ( & ’ ’ + ( Y 2 & ) + r Z , ( id5. ti~’)]
J. T. C. Liu
3 02
(5) Binary-coherent mode energy exchange integral rw
The integrands of I,, and 12, are grouped to reflect “similar” stress-rates of strain products.
Acknowledgments This work is partially supported by the Defense Advanced Research Projects AgencyApplied and Computational Mathematics Program through its University Research Initiative; the National Science Foundation, Fluid Dynamics and Hydraulics Program, Grant MSM8320307; and the National Aeronautics and Space Administration, Langley Research Center Grant NAG1-379 and Lewis Research Center Grant NAG3-673. The partial support of NATO Research Grant 343/85, the National Science Foundation, U.S.-China Cooperative Research Program, Grant 1NT85-14196 and the United Kingdom Science and Engineering Research Council through its Visiting Fellowship Programme is also acknowledged. The numerous beneficial conversations with J. T. Stuart, F.R.S., his hospitality and that of the Department of Mathematics, Imperial College are gratefully appreciated.
References Alper, A,, and Liu, J. T. C. (1978). On the interactions between large-scale structure and fine-grained turbulence in a free shear flow. Part 11. The development of spatial interactions in the mean. Proc. Royal SOC.Lond. A 359, 497-523. Alvarez, C., and Martinez-Val, R. (1984). Visual measurement of streamwise vorticity in the mixing layer. Phys. Fluids 27, 2367-2368. Amsden, A. A,, and Harlow, F. H. (1964). Slip instability. Phys. Nuids 7, 327-334. Barcilon, A,, Brindley, J., Lessen, M., and Mobbs, F. R. (1979). Marginal instability i n Taylor-Couette flows at a very high Taylor number. J. Fluid Mech. 94, 453-463. Benney, D. J . (1961). A non-linear theory for oscillations in a parallel flow. J. Fhid Mech. 10, 209-236. Benney, D. J., and Lin, C. C. (1960). On the secondary motion induced by oscillations in a shear flow. Phys. Nuids 3, 656-657. Benney, D. J., and Bergeron, R. F., Jr. (1969). A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48, 181-204. Bernal, L. P., Breidenthal, R. E., Brown, G . L., Konrdd, J. H., and Roshko, A. (1980). On the development of three-dimensional small scales in turbulent mixing layers. In “Turbulent Shear Flows,” Vol. 11, pp. 305-3 13. Springer-Verlag, Berlin. Bernal, L. P. (1981). The coherent structure of turbulent mixing layers: 11. Secondary streamwise vortex structure. Ph.D. Thesis, California Institute of Technology. Bernal, L. P., and Roshko, A. (1986). Streamwise vortex structure in plane mixing layer. J. Fluid Mech. 170, 499-525.
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ADVANCES I N A P P L I E D MECHANICS, V O L U M E
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Three-Dimensional Ship Boundary Layer and Wake ICHIRO TANAKA Department qf Naval Architecture Osaka University Osaka, Japan
.................. I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. 11. Flow Around a Ship Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Equations of the Three-Dimensional Turbulent Boundary ayer and Wake and .................. Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Equations of the Turbulent Boundary Layer and Wake . . . . . . . . . . . . . . . . . . .................. B. Solution by the Integral Method .................... C. Some Examples of Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Equations of the Higher-Order Boundary Layer and Wake . . . . . . . . . . . . . . . . 1V. Scale Effects of the Boundary Layer and Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Some Examples of Scale Etfects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Methods of Estimation of Scale Effects on Wake Distribution . . . . . . . . . . . . . C. Scale Effects of the Boundary Layer and Wake with Longitudinal Vortices V. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References ...................................................
358 358
313 318 3 18 321 324 330 342 342 345 350 351
I. Introduction This article describes several features of the 3D (three-dimensional) ship turbulent boundary layer and wake, partly in theory and partly in experiment. The article is not intended to be a general review of the researches conducted up to the present. Instead, it intends to shed light on the main features of flow characteristics, to outline them briefly, and to describe attempts for predicting flow field and problems still unsolved. In some part of this article the terminology in naval architecture is used, but it is hoped that the meaning will be obvious. As to the characteristics of the ship boundary layer and wake, there are three predominant features, two of them being physical and one being a parameter of flow. They are: 311 C opyrighl
(01Y8X
Academic Pres, Inc
All right\ of reproduction in any form reserved ISBN 0 12-002026-?
3 12
Ichiro Tanaka
1. The boundary layer and wake are fully 3D over most of the ship hull. Ship form is a typical 3D form. It is not similar to the wing of the aeroplane. It is not similar to the body of revolution. Therefore the boundary layer over the ship surface exhibits a fully 3D nature; e.g., large crossflow occurs in some parts and, in more exaggerated cases, separation of the boundary layer of 3D nature occurs. 2. Near the stern there are some places where the boundary layer develops very rapidly and also very unevenly, sometimes including strong longitudinal vortices generated by 3D separation. For this area the ordinary boundary layer concept breaks down, so we have to develop some higher-order or alternative method to analyze the characteristics of flow. 3. As the flow parameter defining the viscous flow, the Reynolds number is very large, being on the order of lo9, the largest value in engineering, while the majority of experimental data to support the theoretical considerations are obtained at Reynolds numbers of lo6-- 10’. Therefore the difference between models and full ships should bring various problems in the correlation of data between models and ships. This is called the scale effect, the effect of size, correlation law, etc. For every characteristic of flow field, the scale effect should be checked. In this article, with this point in mind, the discussion of the methods of calculation of the turbulent boundary layer and wake will be the first central theme. Among various methods of calculation, the most convenient will be the integral method, because it has a long history and many data are based on this method. Furthermore it is rather simple both in concept and practical procedure. Coincidence of the theoretical results with experiments is also reasonable for most cases. Considering these factors, the integral method is mainly explained as a tool of the calculation of 3D turbulent boundary layer and wake here. The discussion on scale effects of velocity distribution in the boundary layer and wake will be the second central theme. If these effects are correctly and clearly indicated by experiments or by theories, it will be a very important knowledge for understanding the flow structure and for practical application in engineering. Finally, throughout this article, a turbulent boundary layer is assumed. This is obvious for a full-scale ship, considering the Reynolds number. However, with models the boundary layer starts from the laminar layer at least at the stem (front edge). Somewhere downstream at the bow of the model, the boundary layer will develop into the fully turbulent one, as is well known. Therefore, strictly speaking, it is always necessary for the
Three-Dimensional Ship Boundary Layer and Wake
313
development of the boundary layer to include the laminar-turbulent transition. However, this theme is not discussed in this article. For the present purpose, there is no important defect in omitting the consideration of transition. Similarly, the occurrence of wave-making at free surface is also neglected. It is obvious that the formation of a wave around the ship hull influences the characteristics of the viscous flow field and vice versa. But the essential characteristics of the viscous field manifest themselves in no-wave-making conditions. Therefore we will discuss the present theme with this assumption.
11. Flow Around a Ship Hull
The flow field around a ship hull consists of a typical 3D boundary layer and wake. Surface streamlines at the bow show strong divergence, while at the stern they flow in a convergent manner as shown in Figure 1 . Pressure gradients at the surface streamlines exist both in the direction of streamlines and in the direction orthogonal to them. Hence the boundary layer velocity distribution differs from 2D (two-dimensional) cases and exhibits crossflow (or secondary flow) distribution. This is one of the features of 3D boundary layers. A usual distribution of the crossflow is, as in Figure 2, the same in sign in the direction throughout the boundary layer thickness. However, at the streamlines passing fore and aft of the bilge parts (the edges of the ship bottom), a so-called 3D separation occurs due to large pressure gradients along the streamlines, as well as in the crosswise direction. This type of
FIG. 1.
Illustration of potential surface streamlines and bilge vortices.
Ichiro Tanaka
3 14
ord i ria r y (I i s t r i bu t ion
crossflow velocity Iwundary l a y e
coiiipoiie 11 t
thickness S-type ( r e v e r s e d ) crossflow distribut ion
FIG. 2.
Illustration of crossflow distributions.
separation develops into a vortex havings its axis along the streamlines, which is called a bilge or a longitudinal vortex. The actual flow field is much more complicated, but here the description is made simple for the sake of clarity. In Figure 1 two typical vortices are shown; one is a bow bilge vortex and the other is a stern bilge vortex. In such a flow field, the crossflow in the boundary layer shows S-type or reversed crossflow distribution, as in Figure 2. We can clearly observe the existence of bilge vortices, especially the stern bilge vortices, by flow visualization or measurements for full-form ships like tankers, but in fine-form ships like container ships of high speed, bilge vortices are usually weak, sometimes unnoticeable. Pairs of longitudinal vortices, if they exist, stream down from the ship stern into the wake, forming trailing vortices behind the ship. Because of the extensive research already conducted on the subject, we now know fairly clearly the structure of the boundary layer and wake represented by velocity distributions. An example is shown here. Velocity (average in time) measurement was made for a 2 m double model of an ore-carrier at Osaka University Wind Tunnel, as sketched in Figure 3 (Okajima et al., 1985). Wind speed was 20 m/s, so the Reynolds number was 2.4 x lo6. Measurement was made with a 5-hole Pitot tube by traversing around the stern. By differentiating the measured velocity distribution, the researchers also obtained the vorticity distribution. The results are shown in Figure 4, in which the equivelocity contours of the streamwise velocity component, the distribution of the transverse velocity component, and the equivorticity contours of streamwise and transverse vortex components are shown. Velocity is non-dimensionalized by uniform flow. The values of
Three-Dimensional Ship Boundary Layer and Wake
315
/ FIG.3
Apparatus for experiment (Okajima et al., 1985).
vorticity components are expressed in l/sec. S.S. means square station. The total length of the model is divided into tenths, and S.S.5 coincides with the midships (center of the model length). S.S.l corresponds to the place 10% of the model length from the stern end. As is seen in the figure, the
S.S.
2
LO
10
I (a) FIG.4.
(a) Measured velocity and vorticity distributions (Okajima et al., 1985).
60 80
./
./8
-20
-30 -40
-M
40
I
(b) FIG.4.
(b) Measured velocity and vorticity distributions (Okajima e t a / . , 1985)
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Ichiro Tanaka
development of the boundary layer becomes large at the stern. The equivelocity contours are also distorted considerably, so we can notice a quite uneven development of the boundary layer around the ship hull. Near the stern end we also observe the formation of a pair of vorticies appearing near the propeller shaft center, as schematically drawn in Figure 1. Furthermore, it is easily understood that if we draw a normal to the hull surface and obtain the crossflow velocity component along this line, we get S-type distribution of the crossflow at the places where a longitudinal vortex appears. The example shown above is the result of an experiment conducted in the wind tunnel. Therefore, in this experiment, there is no free surface and no wave-making, so the obtained results do not necessarily correspond to the actual flow field around ships. However, as was said before, the essential nature of the flow field is unchanged.
111. Equations of the Three-Dimensional Turbulent
Boundary Layer and Wake and Their Solutions A. EQUATIONS OF
THE
TURBULENT BOUNDARY LAYERA N D WAKE
Based on the results of calculations and measurements up to the present, the boundary layer and wake around a ship hull may be conveniently designated as in Figure 5. The majority of the boundary layer from the bow shows a typical thin nature in 3D space, because the boundary layer is very thin as compared to the representative length scale of the flow, as well as local length. Therefore, generally speaking, the ordinary first-order boundary layer equations are applicable to analyze that part of the flow. Near the end of the stern, the boundary layer becomes very thick. Therefore the region near the end of the stern is sometimes called the thick boundary layer. The word wake is usually used for designating the flow field at the downstream of the body, as shown in the upper part of the figure, but sometimes used more widely for designating the flow field including the boundary layer near the stern end, as shown in the lower part of the figure. One reason for this usage may be the complexity of flow geometry usually encountered in blunt sterns. Let us take an orthogonal coordinate system as shown in Figure 6. In these coordinates, as is well known, 3D turbulent boundary layer equations
Three-Dimensional Ship Boundary Layer and Wake
319
w a k e at propellor 1 oca t i o n thick boundary layer
FIG. 5. Schematic illustration of boundary layer and wake.
I
' a m 1in e s
FIG. 6 . Curvilinear orthogonal coordinates and velocity components.
are written down as follows, under the assumption that the boundary layer thickness 6 is small as compared with the representative length of the flow. 5 direction:
u au
u au
au
--t--+w-+(K,,u-K,,v)u hl a6 h 2 d T al (3.1)*
* A s to KIz and K , , , see p. 331.
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Ichiro Tunaku
7 direction:
(3.2) f direction:
The continuity equation is given by
Here, as the normal coordinate to the surface, the actual normal distance f is taken. Primes denote components due to turbulence, and upper bars denote time average. Quantities without bars are the mean values in time. h , and h, are metric coefficients of _the_6 and 17axes, respectively. In the boundary layer theory, uf2, v”, and w ‘ are ~ usually omitted and does not appear, so only two Reynolds stresses, u)wI and VIW),appear as unknown quantities. p is also a known quantity from (3.3). Therefore we have three equations to obtain five unknown quantities: u, v, w, and VIW).Two additional equations describing the relationship between the unknowns are necessary to solve the problem. Usually relationships between shear stresses and velocity components are assumed as
m,
-
u ’ w ’ = f ( u , u, w,. . .),
~
v ‘ w ’ = g ( u , u, w, . . .).
(3.5)
The point is how these equations express correctly the real situations of the turbulent boundary layer and also how we can solve these systems of equations correctly. Presently there are two directions for obtaining the solution of these nonlinear, simultaneous, partial, differential equations, as is well known. One is the integral method originated by Th. von Karman, and the other is the differential method-direct numerical calculation of differential equations by computer. In naval architecture, the integral method was prevailing about ten years ago. Numerical calculation was still in its infancy at that time because enormous computer capacity was necessary for the calculation. However, today the situation is changing. Because of the very rapid development of computer performance, we can perform numerical calculation far more easily, quickly, and accurately than
Three-Dimensional Ship Boundary Layer and Wake
321
before. So the differential method is gradually gaining popularity. It is also true that the differential method is more general and more powerful than the integral one. However, it is also to be noted that many things still need to be checked, improved, and solved in the future with reference to numerical calculation techniques or the mathematics behind them. Recently, the LES (Large Eddy Simulation) technique has also been tried as a new approach to the turbulent flow analysis, but the application of this to the flow around a ship hull is in the future, if possible at all. More importantly, the present article is not intended to be a review of the detailed discussion on the various methods of calculation of the boundary layer and wake. Therefore in the following, the solution of the boundary layer and wake will be pursued mainly by the integral method. B. SOLUTION RY
THE
INTEGRAL METHOD
As is well known, the integral method of the boundary layer theory is to apply the momentum theorem to the whole bulk of flow elements inside the boundary layer, i.e.,
(momentum change of the flow element) = (pressure components) +(frictional stress components), if we omit the component due to body force. Usually a friction law describing the relationship between the shear stress at the wall with other known and unknown flow quantities is added as a required condition. Furthermore, another restriction, usually called an auxiliary equation, is used to describe the change of shape of the velocity distribution. Integral methods are conveniently divided into groups according to several factors such as (i) coordinates, (ii) the velocity distribution model in the streamwise direction, (iii) whether the crossflow is small or not, (iv) the velocity distribution model in crossflow, and (v) the auxiliary equation. With reference to (i), the streamline coordinates are most common because by adopting these we obtain very simple governing equations. Taking the 6 axis (Figure 6) along the surface potential streamline,
Ichiro Tanaka
322
where rs and r., are the frictional stress components along 5 and 77, respectively. Parameters S1, S 2 , e l l ,612, 021,1 9 are ~ ~defined as follows:
m
u v
Otherwise, if we adopt a different coordinate system, governing equations are more complicated. Item (ii) relates to how to analytically express u, the distribution of streamwise velocity in the boundary layer. The most common model to be used is the power law distribution, but, as the method of solution develops over time, more accurate distributions to express the scale effect as well as the effect of pressure gradient are used. Item (iii) relates to the assumption < u. If this is adopted, the governing equations are simpler:
(3.9)
a hl a t
-(
u2e2,) + 2 ~ , ~~0~~ , -K , ~ U ~ + (B 6,)~=I.. ,
(3.10)
P
Item (iv), selection of the crossflow model, is also an important consideration in the integral method. Each quantity 6 , , etc. in the relations above is expressed by fundamental unknown quantities 6, r6, r, if the velocity distribution inside the boundary layer is assumed. The solution differs according to the expression of crossflow v. Especially at the place where long upstream history accumulates, e.g., at the stern, flow develops under the complicated effect of pressure gradients. In this case, an S-type crossflow model may be more suitable for the expression of actual velocity distribution. If 3D separation exists, the tendency will be more exaggerated.
Three-Dimensional Ship Boundary Layer and Wake
323
The most common model for crossflow is the Mager model, which is written U -
U
=f
($)tan p,
(3.11)
where f =(1 - {/a)’. p is the angle between the viscous surface streamline (also called the frictional stress line or limiting streamline) and the potential surface streamline. 6 is the boundary layer thickness. This expression is the most fundamental one representing the essential behavior of crossflow, but obviously it cannot allow the S-type distribution. Okuno (1976) proposed the distribution: (3.12) He made a calculation using this expression and obtained a better result in comparison with the experimental results. Eichelbrenner-Peube’s model (1966) is u =tan p .f(u),
(3.13)
which makes it possible to obtain an S-type distribution. In other examples, f i n the above is expressed in a cubic. As a different model, Himeno-Tanaka (1973, 1975) used a vector model based on Coles’ wall-wake law. This is shown in Figure 7, and expressed as (3.14) where U = a resultant vector composed of u and u, U, = a frictional stress vector having the direction of stress line and the magnitude, u * = ~ : ) / p ,U, = a vector to be determined in such a way that U is to coincide with potential flow at { = 6, f = the function to express the wall law, and w = t h e function to express the wake function. The calculation using this expression gives a fairly good result, but S-type distribution is not necessarily possible at any time; therefore this will be a restriction. Regarding item (v), there are many proposals about the relationships to be used for calculation, such as the moment of momentum equation, the entrainment equation, the turbulent energy equations, etc. However, it is difficult to point out their respective advantages and disadvantages, because usually these depend on various factors included in each equation, e.g., on
J(T$+
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Ichiro Tanaka
wake function
b
FIG. 7.
Coles’ vector model.
the empirical values to be used, degree of approximation, degree of fitness to the configuration of flow field, etc. To calculate the velocity distribution in wakes under ordinary boundary layer assumptions, we use the same equations as in the boundary layer problems, with different boundary conditions at the inner boundary, i.e., centerline in the wake. The velocity is not zero there, while the stress is zero. The assumptions for velocity distributions as well as auxiliary equations are changed following the changes in boundary conditions. To obtain a good result from the calculation, the initial condition at the start of the calculation of wake should naturally be as correct as possible. Other things are generally similar to the method of solution of the boundary layer, so it is omitted here. However, it is to be noted that the calculation based on the integral method is not suitable for the calculation of wake, when it involves longitudinal vortices inside it, which is the usual phenomenon in the wake of a full-form ship.
OF CALCULATION C. EXAMPLES
The boundary layer developments of many ship forms have been calculated and compared with experiments up to the present. Here some of the typical, somewhat old but pioneering works are described. In the boundary layer calculation, the potential flow field is to be given as input. For the flow around a ship hull, it is obvious that wave-making is a clearly observable phenomenon, and accurately speaking, the effect of the wave has to be accounted for in the calculation of the potential flow.
Three-Dimensional Ship Boundary Layer and Wake
325
However, large-scale computation including the wave-making has not been attempted by many researchers until now. Instead, the wave-making phenomena as input potential flow is usually neglected. The reason for this is that the essential structure of the boundary layer and wake around ships can be considered by the flow condition without the wave, especially at low speed. For the calculation of the potential flow field, the celebrated Hess-Smith method is widely used. Figures 8 to 11 show the results of calculations as well as experiment by Okuno (1976). The experiment was done at the circulating water channel using a model of 1.8 m length. Velocity distribution was measured by a 3-hole tube. Because the size of the model is somewhat small, the accuracy of some data may not be enough to discuss detailed points, but the general view of the flow field around ship hull is concisely demonstrated by this experiment. As is shown, S-type crossflow distribution appears at the stern. Figure 10 shows measured results of streamwise and crossflow velocities as well as the vorticity component with its axis along streamwise direction. In the figure, results are also shown for two kinds of calculations, one with Mager model and the other with S-type velocity distribution. At some places the difference of the calculated results is noted. After obtaining u and v as a function of l,we then obtain the distribution p of the viscous surface streamline as a limit l = 0. This is shown in Figure 11. The upper diagram shows the figure at the stern profile and the lower at the bottom of the bow. The arrows show the measured distribution of shearing stress at the surface, obtained with a Preston tube. Solid lines show the surface viscous streamlines obtained from the stress distribution, and dotted lines show the potential surface streamlines. Calculated curves show two kinds of envelopes. These are considered to show a 3D separation line and an attachment line, respectively. This result coincides with the experimental result of shearing stress distribution pattern, at least qualitatively, although the locations of the separation and attachment lines do not necessarily coincide with the experiment. In order to obtain satisfactory results on these matters, we have to include the effect of separated downstream flow on the upstream boundary layer. This problem has not been attempted by many. From the lower diagram we can also notice the generation of a bow bilge vortex at the bottom of the bow. The above examples of calculation are the ones obtained by numerical integration of the boundary layer equations. To make the solution easier to foresee, Tanaka-Himeno (1975) studied the possibility of obtaining a
o
measured
--- p o t e n t i a l
stream l i n e ( H e s s
FIG.8. Streamwise velocity profiles measured in boundary layer (Okuno, 1976).
&
Smith)
0
4 I I I
:* Im I
I
/
tF
rJY
N
1
u/
I
n
In0 0
11 -
1
328
Ichiro Tanaka
u/u 1.0
1.0
-0.1
v/u
-0.4
u s 0.2
/
f
stream line B
-0.1
1.0 stream
v/u
1 1%oj 4 4-
u/u
l.'O
-0'.1 'v/u
-0.4
0.2
iJ S
u/u
- calculated
1.0
"/U
0.2
-0.2
0.2
using model A
_ _ - - calculated using Mager model o
measured FIG. 10. Velocity and vorticity distributions at S.S. 1/2 (Okuno, 1976).
solution in an analytically expressed form. Assuming that 3D nature is small, they expanded flow quantities in perturbational form as follows.
u = uo+u,+.. . (3.15)
---- p o t e n t i a l
stream l i n e (liess
Smith)
&
-v i s c o u s l i m i t i n g s t r e a m l i n e + measured w a l l s h e a r stress, c
------
TW
= f
~
1/2Pt
---. 7l/2
Sq.St.
8
81/4
8112
83/4
9
9l/2
forc body ( b o t t o m vi ew)
FIG. 11.
Viscous limiting streamlines and distribution of wall shear stress (Series 60, Cb = 0.7) (Okuno, 1976).
F.P
330
Ichiro Tanaka
Then, the solution of equations (3.9), (3.10) is easily written in closed, explicit form by given velocity. Therefore the result is very well suited for the understanding and foresight of flow characteristics. The coincidence of the calculated result is, in general, good except for the region near the stern end. One example of this calculation result is shown in Figures 12 and 13. Figure 12 is the body plan of the ship model used for the calculation. Figure 13 shows the result of the calculation performed for streamline No. 3. In Figure 12, the result of calculation by a different small crossflow method by Larsson (1974) is also included, together with the experimental results. The flow field near the stern end, especially the flow field immediately after it, is usually called the near wake. There is no body surface here, so the flow moves under the sudden disappearance of no slip condition at the body surface. Therefore it is quite obvious that the correct prediction of the flow field at near wake will be a difficult task in 3D boundary layer and wake theory. From an engineering point of view, it is important to predict wake field properties such as the distribution of magnitude of inflow velocity and its direction, because the propeller is installed there. In order to design the propeller geometry and to predict its working performance, it is essential to know the inflow field to the propeller. Tanaka et al. (1973) made an attempt to obtain the distribution of inflow velocity to the propeller at the propeller location by an engineering method. In this attempt, for the sake of convenience in the calculation, an imaginary flat plate was placed between the stern end and the propeller, and the 3D boundary layer calculation was performed to find the velocity distribution at the propeller location. The distance between the stern end and the propeller is of the order of 1/100 of the ship length, so, if the result is judged with proper care, the calculated result will give us data on the inflow velocity distribution to the propeller. Figure 14 is a result of this calculation as compared with the experimental values.
D. EQUATIONS OF HIGHER-ORDER BOUNDARY LAYERA N D WAKE In the preceding sections, typical ordinary boundary layer approaches are discussed. The basic notion in developing the boundary layer theory was that the boundary layer is very thin as compared with the representative length of the body. Based on this understanding, the ordinary 3D boundary layer theory considers the effect of 2D mobility of the flow element over the body surface. This means the flow element streams down tangentially
W
x
0 rl d
rl
1 I C
In
z
LT,
W
P
I/
C
rrr u E
D 7-
4 I/
0 'j
E
m Q N
P
0 II
E
0
II
N
Three-Dimensional Ship Boundary Layer and Wake
r
--
33 1
Ichiro Tunaku
332
, u : measured by Larsson _ _ _ - : calculated by Larsson according t o small cross flow approximat ion : calculated by Tanaka Himeno according tc small cross flow approximation
0 ,A
5
4
-?, skin friction coefficient c f
3 2 4
1
0
P 0
1-4
8
6
(mm)
0
4 .momentum thickness
2
0
-16
8
,./ -- _ = ZX/Lpp, --a-
w
0.5
1 .o
midship
1.5
I
2 .o A.P.
FIG. 13. Boundary layer solutions along No. 3 streamline (Tanaka and Himeno, 1975).
FIG. 14. Wake contours in propeller disk plane at Sq. St. 0.094 (Series 60, Cb=0.7) (Tanaka era/., 1973).
L1.00
Three-Dimensional Ship Boundary Layer and Wake
333
to the surface, but can flow in a convergent (or divergent) manner as well as with curvature. The local curvatures of the body surface themselves are not included in the formulation. Let us call this category the first-level boundary layer theory. Now let us consider local curvatures of the body surface. If these values are large (local radii of curvatures are small), the boundary layer development will be greatly affected. First, consider the case where the surface curvature transverse to the streamwise direction becomes large in the downstream direction. In this case, the pressure gradient changes according to the change of configuration of the body along streamlines, but this is a common phenomenon occurring even in level 1. A new feature of this flow is rather the fact that the convergence of the surface streamline makes the fluid element a fan-shaped configuration as the thickness of the boundary layer increases. In other words, more flux flows at the outside of the boundary layer as compared with the case of level 1, in which the elementary shape is a 2D type. We can also call this a thick boundary layer problem as compared with the local radius of curvature. Therefore the basic governing equations become different from the ordinary ones. Next consider another case where the surface curvature in the main stream direction becomes large in the downstream. In this case, if the thickness of the boundary layer is also large as compared with the said radius of curvature, the elementary shape of the fluid element is again different from the case of level 1 . So the governing equations will again be different from the ordinary ones. Simultaneously, due to the effect of centrifugal forcewhich differs considerably from one fluid element to the other-on the fluid element inside the boundary layer, the pressure gradient inside the boundary layer changes appreciably. Therefore this effect is also to be included in the equations. Thus, in these two cases the characteristics of the boundary layer are different from the ordinary case. It is important to note that these additional effects are caused by the thick boundary layer as compared with the local radius of curvature. In this sense, as is mentioned above, the theory that considers these effects is called thick boundary layer theory. However, in this article, from a somewhat different viewpoint, let us call this category level 2. The idea is that, without using the word “thick,” which is the quantity initially unknown, we differentiate the categories of methods of calculation only by the geometrical quantities to be considered in the method.
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Ichiro Tanaka
One further comment should be added to the case of level 2. In this case the boundary layer thickness develops quickly. This means that the boundary layer exerts a large displacement effect upon the outside potential flow. Obviously this is not a one-way process; a strong interaction occurs between the outside inviscid flow and the inside viscous flow. Therefore, generally speaking we have to consider this viscous-inviscid interaction when we consider the effect of the body surface curvatures. Recently many methods of calculation at level 2 have been energetically developed by many researchers. Among these efforts, researches on the boundary layer around bodies of revolution are central (e.g., Patel, 1974; Nakayama et al., 1976) due to its fundamental importance. For the 3 D boundary layer, Larsson (1975), Hatano et al. (1977, 1978), Himeno et al. (1979), Nagamatsu (1980, 1985), Larsson et al. (1980), Ikehata et al. (1982), Hinatsu ( 1984), and many others have published papers. These researches all concern the higher-order boundary layer equations, but, as is easily imagined, there are some differences between them in detailed terms, expressions, approximations, etc. Therefore, as a representative example, Toda et aL’s report (1985) will be mentioned here. As shown in Figure 15, orthogonal curvilinear coordinates are taken over the ship hull surface. The 6 axis is along the potential surface streamline. 5 are u, v, w, respectively. The The velocity components along the 6, ~ , axes metrices along them are to be labelled h , ,h Z ,h3 respectively. It is also assumed that the 5 axis is approximated by a straight line normal to the surface, so h, = 1 . Let R be the local radius of curvature of the ship surface. Now, if we consider the terms of the first order of 6/ R in the boundary layer equations, we obtain the following equations.
u au v au au --+--+w--+(K,,u-K,,v)v+K,,uw h , a t h2 a77 al
(3.17) P
K , 3 u 2 + K 2 3 ~ 2a= - P( - ) . 86 P
(3.18)
Three-Dimensional Ship Boundary Layer and Wake
335
FIG. 15. Coordinate system and curvatures.
The continuity equation is 1 du
-
-+-
hl a6
1 av aw - + - + K 2 , u + K 1 2 v + ( K l ~ + Kz3)w=0. h2d77 a l
(3.19)
Here the following definitions are used
K
K
1 ah,
713
1 12 -
h,h, a( ’ = pe ($-K13U),
ah,
K
h,h, a17’
1 ah,
---,
I3-hl a5
,
r23= p,
1 ah, K23=--, h2 86
pe:effective
viscosity.
The values of the curvatures at the ship surface are written as follows.
Kl
= KZI(5 = 01,
K3= K23(5=0),
K2
=&2(l
= 01,
K4=KI3(l=0).
(3.20)
336
Ichiro Tanaka
Metrices at the surface are written as HI = h , ( 5 = 0 ) ,
H 2 = h2(6 = 0).
(3.21)
K , and K 2 are the two basic quantities representing the divergence-convergence rate and geodesic curvature of the potential surface streamlines respectively. They appeared already in the boundary layer equations in level 1 in the preceding section. (In this chapter KI2and K 2 ,are used as functions of 5, while in Section 111 they are used as independent of 5. Therefore K , here = K 2 , in Section 111 and K , here = K I 2in Section 111.) K, and K4 are newly added terms showing the two curvatures of the ship surface as represented by the curvatures of 7 and 6 in two normal planes including 7 and 5 or 6 and 5, respectively. Since the basic boundary layer equations are obtained, the momentum integral equations are written down in a straightforward way, following the ordinary procedure of the integral method, as follows.
a HI 86
au us:-+Hla5
( u2e;,)+
a H2a7
(
au u’e;,)+ us;H2
a7
+ K , u2(e:, ei2)+ K~ u2(e;, + e;,) -;u2cf, -
- K , [ o s ~ ( l + K 3 5 d) 5 - K 4 U 2 0 i 3 ,
-K,
los
(1+ K45) d 5 - K 3 U 2 8 ; 3 ,
(3.22)
(3.23)
Three-Dimensional Ship Boundary Layer and Wake
337
7230
Gi = ( 1/2)pU2’
0i2=
los-$ +
(1 K45) d[,
t
(3.24)
Here the quantities with prime denote the inclusion of the terms ( 1 + K 3 1 ) , ( 1 K4{) in the integrand. This inclusion is one of the features of this method and makes the momentum integral equations slightly different from the expressions of other papers. However, we do not discuss this further, because the discussion is too involved to be suitable here. We only point out that, if we put K 3 = K 4 = 0, (3.22) reduces to the ordinary boundary layer equations already stated above. If we reduce them to the axisymmetric case, they coincide with the expression by Pate1 et a!. (except for a slight difference, which is not essential in this context). Toda et al. applied this method for the calculation of the boundary layer development in several ship forms. Here, as an example to show the influence of the consideration of the surface curvatures, the result of calculation of pressure variation inside the boundary layer is shown in Figures 16 and 17 (Toda et al., 1985). Both figures show the comparison between the measured values of surface pressure and the calculated values. The ship model is called the Wigley model, which is a fine, mathematical model and has been used as one of the suitable models for the research of resistance, especially of wave-making. Figure 16 shows the distributions of surface pressure in the depthwise direction for several square stations. The marks are measured values. Three curves obtained by calculation are: (1) solid lines to show surface pressure by boundary layer calculation; (2) dotted lines to show
+
Ichiro Tanaka
338 Z/d
-.05
0
0-.05
0
.05 0
.05
.10 CP
o.2-[ 0.4. , O
0.6.
i
x/L
!
.7
"I 0.81
01
1
.o-
\
cpw=
----cpO=
-. - .- . Cpe=
o
Pw-P1/ 2 p UrnP Po - P1/2pu-" Pe-P1/2pu-2
Rn=3.695x106
Measurements Kajit ani e t a(.
FIG. 16. Pressure distribution (Wigley model) (Toda er al., 1985).
potential surface pressure; and ( 3 ) dot-dash lines to show potential pressure at the edge of the boundary layer. Figure 17 is the same data along some waterlines. As explained earlier, in order to investigate the variation of pressure inside the boundary layer, we have to check firstly the effect of the boundary layer on the potential flow outside the boundary layer and secondly the effect of centrifugal force, which is caused by the curvature of the body surface, inside the boundary layer flow element. In this particular example, the first effect was found to be small, so the majority of change in the pressure distribution seems to come from the second cause. Therefore we have to understand the results as showing the effect of the curvature of the body surface. As shown in the figures, the degree of coincidence is improved by considering the higher-order terms. Now let us consider the prediction of wake distribution in some of the higher-order type approaches. In the preceding section the method by Tanaka et al. (1973) was described. This method was the first attempt to
Three-Dimensional Ship Boundary Layer and Wake Pw-Fm
-
.10
1/2pUmf
cpO=
1/2pu-2
Cpe=
1/2pu.=2'
Po-P-
I I
Measurements K O j l t G n l e ?i a[. zz/d=0.12 / d = 0.12 Fn=0.1043 0 . 5 0 . 6 0.7O 0.8
t
lo
Rn=3.695x106
Pe-P-
---
-
O
cpw=
339
L
I
I
z / d = 0.5 2
0.5 0 . 6 0.7 0.8
-.lo
1
FIG. 17. Pressure distribution (Wigley model) (Toda et a l , 1985).
predict the propeller inflow distribution at that time, but it may be too much like an engineering approach. To obtain a more reasonable result, it will be necessary to follow a calculation procedure that takes into account, the proper characteristics of the wake, rather than to follow the simple boundary layer approach. Mori et al. (1978) developed this kind of approach; i.e., they attempted to solve vorticity transport equations for the flow field near the stern, introducing an idea somewhat expanded from Stewartson's tripledeck structure theory in laminar boundary layer. Although the calculation for a given ship form becomes complicated, the basic idea is orthodox, and so this will be one of the promising approaches. An example of the results is shown in Figure 18. As is shown in this figure and also in Figure 14, the boundary layer and near wake behavior near the stern end is pretty complicated. The thickness of the boundary layer is large and uneven in the transverse plane to the uniform upstream velocity. A vortex-like distribution is usually observed, especially with a full-form ship. By considering this, Okajima et al. (1985)
W
P
0
FIG. 18. Wake distribution
at 1/80
L aft from A.P. (Mori and Doi, 1978)
Three-Dimensional Ship Boundary Layer and Wake
341
developed a different method involving longitudinal vortex to predict the flow field. The main points of this method are as follows. By the analysis of measured quantities, the total head loss is found to be not large along streamlines which pass through the viscous region at the stern; therefore, the inviscid flow model with vorticity can be a good approximation to the flow near the stern. Based on this assumption, Okajima et al. (1985) developed a procedure to calculate flow field characteristics starting from the initial data on velocity distribution given at some upstream square station. Some of the calculated results are shown in Figure 19, in which we see that the coincidence between the measured and calculated values is quite good in spite of the complicated nature of the near wake. Strictly speaking, several points in this method need to be improved upon in the future. For example, in this particular example, pressure distribution is given as a known quantity. This point, as well as other approximations, will need to be studied further.
---__Calculated
I
FIG. 19. Comparison between measured and calculated velocity distributions at S.S. A, $, (Okajima et a/., 1985).
i,
Ichiro Tanaka
3 42
IV. Scale Effects of the Boundary Layer and Wake A. SOME EXAMPLES OF SCALE EFFECTS
The purpose of developing the methods of calculation of 3D boundary layer and wake is naturally to predict viscous flow characteristics around a ship hull. However, it is quite obvious that if this calculation is only performed for models it is far from satisfactory. The final purpose, of course, is to predict the characteristics of full-scale ships with good accuracy. Therefore these methods should be checked by the data in the model scale as well as in the ship scale. This task is obviously difficult, especially in the latter case, because we have to perform a measurement of velocity distribution in actual ships. Therefore, if these data are not available, it is preferable to obtain data for geosims (geometrically similar models). Actually, there are some data obtained by experiments for velocity distributions, etc. in full-scale ships or geosims. In the following we show two examples of this type of comparison. The first example is a comparison of boundary layer velocity distribution of nearly 2D boundary layer (Rep. SR107, 1973). The second one describes the velocity distribution in the boundary layer and wake for geosims, ranging from 3.5 m to 12 m, of a tanker (Tanaka et aL, 1985). Figure 20 shows the comparison of velocity distributions in the main stream direction for a tanker. Measurement was made for an actual ship of 285m length and a model of 6.5m at the location 40% of the length
1 .o
u_ U measure
0.5
calculates-
model
0
0 005
0.015
FIG. 20. Comparison of velocity distributions between ship and model (Himeno and Tanaka, 1975).
Three-Dimensional Ship Boundary Layer and Wake
343
from the bow and approximately at the keel line. Velocity measurements were made by a Pitot tube for the model and by an electromagnetic-type velocimeter for the ship. In the figure the measured values are compared with the values calculated by the 3D turbulent boundary layer theory of integral method both for the model and the ship. In this case, because the velocity distribution is of rather 2D, flat-plate nature, the comparison shows that the usefulness of distribution expressed by the log-law type used in this calculation is proved to be correct for practical purposes for 2D, flat-plate-type velocity distribution. Measurement of velocity distribution in the boundary layer near the stern and wake was carried out for four geosims of a tanker model. The lengths of the four models were 3.5 m, 4.714 m, 7 m and 12 m, and their parent ship was a test ship of Nippon Kokan Co., Ltd., called Daioh. The measurement was made for velocity distributions at three different places, i.e., propeller plane, 10% of the length ahead of the stern end and 10% astern of it, with spherical- or NPL- (National Physical Laboratory) type velocimeter for each model. The measurement was made with no propeller fitted; therefore the result at the propeller plane is called the nominal wake distribution in the terminology of naval architecture. The body plan of the models is shown in Figure 21. As representative examples of the measured results, two additional figures are included. Figure 22 shows the measured results of wake (or, more customarily, the velocity defect from ship speed) in the boundary layer at 10% of the length ahead of the stern end. Figure 23 the same quantity at the propeller plane. As can be seen in these figures, the widths of both the boundary layer and wake become narrower as the dimension of the model increases. In Figure 23,
FIG. 21. Body plan of DAIOH.
u
P P
e FIG. 22. Comparison of measured wake distributions at S.S.1 (Tanaka el a/., 1984).
Three-Dimensional Ship Boundary Layer and Wake
12
--
345
I
7 4.7 m
e
FIG. 23. Comparison of measured wake distributions at propeller section (Tanaka et a!., 1984).
the decrease of the peak value of the wake is also observable as the dimension increases. These two features are the indications of scale effects in the streamwise velocity distribution.
B. METHODSO F ESTIMATION O F SCALEEFFECTSON WAKE DISTRIBUTION As shown in the previous section, the scale effect of the wake distribution seems to appear not only in the width of domain but also in the flow velocity itself, especially along the direction of the main stream. To predict such scale effects, many methods based on the boundary layer theories have been proposed as guidelines. However, most are complicated in procedure or solvable only by numerical calculation. Therefore it is difficult to obtain clear information from these methods about the qualitative nature of the flow characteristics, such as the scale effects or the effect of change of ship form on flow characteristics. On the other hand, if we use the integral methods we can obtain more direct information about the scale effects of flow characteristics in a more straightforward way. In this section three
Ichiro Tanaka
346
methods, namely those of Sasajima-Tanaka, Tanaka, Himeno, are described, and comparisons with the experimental results of geosims mentioned above (Tanaka et al., 1984) are presented. The experimental data of four models are used for this purpose. First, a brief description about the procedures of the three methods is given below. Sasajima- Tanaka’s Method Sasajima and Tanaka proposed the following formulas for the wake distribution of actual ships based upon the concept of the 2-dimensional boundary layer theory: Thickness of the boundary layer: 6
-a C,, L
where C, is the frictional resistance coefficient of the ship or approximately of the corresponding flat plate. Velocity along the streamline:
where U is the streamwise velocity at the edge of the boundary layer. In this method, the scale effect is considered only in the breadth of the wake and not in the flow velocity within the wake. Tanaka’s Method Tanaka proposed the following formulas for the scale effect of wake distribution of ships, based upon the theories of the boundary layer around the 2-dimensional body, the wake far behind the body and the wake far behind the body of revolution: Thickness of the boundary layer:
L
Velocity defect:
Three-Dimensional Ship Boundary Layer and Wake
347
In applying this method to the present example, the following formulas are used under the assumption that at the propeller disc plane the strong influence of the upstream boundary layer still remains: 6
-a L
cg2
(4.5)
In these approximation formulas, the scale effect is considered not only on the thickness of the boundary layer, but also on the velocity component in the streamline direction. Himeno's Method Himeno proposed approximation formulas for the scale effect as shown below, based on the 3D turbulent boundary layer theory: Thickess of the boundary layer: 6 -a L
Cy
(4.7)
Velocity defect:
Crossflow velocity: u a u . CF. Normal velocity to the wall: W
-a C,. U
(4.10)
In the formulas, the scale effect is taken into consideration for the breadth of the wake and for the velocity components in the directions of the main stream, the crossflow, and the normal. The essential idea which forms the basis of this proposal is described in Tanaka-Himeno's report (1975). The relationship above is a concise expression for practical prediction of scale effects.
348
Ichiro Tanaka
In order to examine the accuracy of each method, the estimated values of the wake distribution of the 12 m model are calculated from the experimental values of the 3.5 m model, the 4.7 m model and the 7 m model by each method respectively and compared with the experimental values. (1) Examination at S.S. 1
First, the comparison with the result of calculation of the boundary layer by Okuno’s method is presented. Figure 21 shows the streamlines along which the calculation is performed. Figure 24 shows the comparison of the velocity components along the directions of the coordinates between the experimental values and those obtained by the calculation. The calculated values and the experimental ones for the crossflow component do not always conform with each other, but the difference between the 4.7 m model and the 12 m model, with regard to the velocity component along the direction of the streamline, is of the same order as the experimental value. Also, the results of estimation of the values for the 12 m model from the data of the 4.7 m model by three different methods show nearly the same values. These facts indicate that for the usual boundary layer, contracting the thickness proportionally to the value of CF by Sasajima-Tanaka’s method is almost equivalent to the combined operation of contracting the thickness proportionally to Cy2 or C v and increasing the flow velocity, as proposed by the two other approximation methods. However, for the distribution which has a peak inside the wake contour, as in the propeller disc plane, differences appear between the former method and the latter two methods. (2) Comparison at Propeller Disc Plane
On the three horizontal planes-the plane passing through the center of the shaft, and the planes 100 mm above and below it (in 12 m model scale)-the measured values are plotted and the comparison is made among the three methods by estimating the values of the large model (12 m) from those of the smaller models. The results of estimation of V, (longitudinal velocity component) are shown in Figure 2 5 . From these two figures, the following conclusions may be drawn: With respect to V,, little difference is observed between the results obtained by the three methods in the domain where the velocity distribution has a gradient. However, around the peak value of the wake distribution, the larger model generally gives a higher flow velocity. This would indicate
Three-Dimensional Ship Boundary Layer and Wake
Cal.
- - A
Meas.
----
------
12m (NKK) 4.7m(NKK)
349
Estimated 12m Model by o : Sasajima-Tanaka's Method* A : Tanaka's Method* 0 : Himeno's Method *For v , Namimatsu-like assumption is applied
1 .o U -
"M
A
a
0.1
0.5
Streamline No. 6
V -
VM 0
0
1 .o U -
0
A
VM
0
0.5
Streamline No. 10
V -
s.s.l
Vf4 0.1 0
0.05 0
FIG. 24. Comparison of velocity distributions between measured and estimated distributions of 12 m model from 4.7 m model at S.S.1 (Tanaka e l al., 1984).
Ichiro Tanaka
350 1.0
Z=100 rnrn
!LL "M
0.5
0
0
0
FIG. 25. Comparison of velocity distributions between measured and estimated distributions of 12 m model from 4.7 m model at propeller plane (Tanaka et a/., 1984).
that the scale effect appears in the breadth of the wake as well as the velocity defect. As to the scale effect of the velocity components V, and V, in the transverse plane, we cannot notice the effects clearly because the magnitude of the velocity components and hence the magnitude of their scale effects are both so small that they are unnoticeable.
C. SCALEEFFECTSOF BOUNDARYLAYER A N D WAKE WITH LONGITUDINAL VORTICES
This section describes the scale effects on the wake distribution of ships with bilge vortices. In the previous section we described the scale effects on wake distribution without the occurrence of bilge vortices. However, as explained earlier, we have to anticipate the generation of bilge vortices at the stern of a full-form ship. Therefore it seems to be necessary to further investigate if such boundary-layer-like approaches are also applicable to the flow with vortices of longitudinal axis inside the boundary layer. The boundary layer and wake of 3D bodies are expressed as the distributions of transverse and longitudinal vortices to the flow over the surface.
Three-Dimensional Ship Boundary Layer and Wake
351
In the case without 3D separation, both vortices adhere to the surface of the body and do not separate from it. Hence, in this case, the vortices of the longitudinal component are simply the appearance of a characteristic of the 3D boundary layer and wake. They are not concentrated at a particular point. However, on the bodies with 3D separation, there are concentrations of longitudinal vortices due to separation from the surface, like bilge vortices of ships. For this case, the simplest model of flow field is that the boundary layer and wake consist of two parts, one the transverse vortices representing the usual boundary layer and wake without separation and the other the longitudinal separating vortices representing bilge vortices. Both vortices orthogonally intersect, so the characteristics of each vortex are probably approximated to be independent of each other; i.e., the existence of longitudinal vortices does not affect the principal nature of the boundary layer and wake and, vice versa, the boundary layer and wake does not change the nature of longitudinal vortices as a first approximation. With this idea, Tanaka (1983) developed a method to predict the wake field with bilge vortices of a full-scale ship from the wake of models. In the following, only part of the method will be explained and the result of the prediction will be compared with the result of the full-scale experiment of a ship. ( 1 ) Scale Efiects of Longitudinal Vortex
If we consider a longitudinal vortex placed in a uniform flow, a possible method to correlate the laminar solution with the turbulent one is to replace kinematic viscosity u with eddy viscosity Yr. In principle uT surely changes its value according to the scale, configuration and characteristics of the flow. But, as is well known, the change is gentle and this method of correlation has proved to be effective in many boundary layer and wake problems. Therefore we try to apply this method to the present problem. Clauser found a useful expression for YT in the 2D boundary layer, i.e., US * -- 56 = cY(constant). ur
(4.11)
The process to obtain the Re dependency of turbulent solution is to use uT instead of Y in the laminar solution obtained by Batchelor (1964), and then to use the relation S * / L a C,. In this subsection, different notations from preceding subsections are used, as shown in Figure 26. A longitudinal vortex is placed in a uniform flow. The x axis is at the center of the vortex whose origin coincides with
Ichiro Tanaka
352
r
FIG 26. Coordinate axes and velocity components for longitudinal vortex (Tanaka, 1983).
the origin of the vortex and the r axis expresses the distance from the center. All things are axially symmetric. u, v, w are the velocity components along x, r, and circumferential directions respectively. U is the x-wise velocity at r -+ co. Batchelor’s solution is written as follows in downstream. x-wise velocity: u=U
-T(32rrr2vx
l o g y ) eF” +Higher Order Terms,
(4.12)
where (4.13)
and r is the circulation of the vortex at r + co. Circulation: 2 m w = r(1 - e - “ ) .
(4.14)
Longitudinal vorticity component having x- wise axis: (4.15) All these solutions are derived under the boundary layer approximation: UQU,
VQU,
a a -
(4.16)
Three-Dimensional Ship Boundary Layer and Wake
353
These solutions show us many interesting and important characteristics of the longitudinal vortex, such as x-wise or r-wise velocity distributions. However, in the following, we concentrate our attention on the effect of Re on r-wise distribution in velocity and vorticity. Namely, we compare various quantities between two Res corresponding to model and ship conditions at the same value of x/L, where L is the length of model or ship. Now, putting (4.1 1) into (4.13), we obtain the expression of 7 for turbulent flow as follows.
Ur2 a r'
rl=4v,x-4xs*-
-
const x
(rlL)* x / L * s*/L'
(4.17)
, in model scale and the other in ship scale. Suffixes We take two ~ sone m and s are used to signify the respective cases. Let us consider the two points where T, = q5and call them the corresponding points for each other. For these points, the relationship between (r/L), and ( r / L ) s is given as follows, remembering that we compare the points with the same x/ L:
To derive the last term from the middle in (4.18), the relation S * / L a C , was used. The corresponding points play a crucial role in the correlation of velocity and vorticity distributions between model and ship, because the r-wise distributions of them are all uniquely decided by 7.As shown in (4.18), if we contract the model abscissa ( r / L ) , in the ratio of -, we obtain the ship abscissa (r/ L), to achieve the similarity in this direction. In other words, the radius of the vortex in ship scale is smaller than the model in the ratio of Next, let us check the vorticity distribution. From (4.15) we obtain the following:
m.
W L _ - const X
U
r
~
1 e-". UL S*/L. x / L
In our present problem, it is proved that we can put therefore
(4.19)
r/ UL = const., and (4.20)
Ichiro Tanaka
354
at the same value of x / L . Therefore, at the corresponding points between model and ship
the ordinate of w L / U in ship scale is CFm/CFstimes the value of wL/ U in model scale, i.e., the ship’s vorticity in non-dimensional form is larger than the model’s value. Figure 27 is the illustration of the correlation of vorticity between model and ship. For the wake ( U - u ) / U, we obtain
~
u-u U
= const x
L)
2 1 1 (L) --x log (5 UL x / L S*/L L 6*/L
+ Higher Order Terms.
e-7 (4.22)
In (4.22) Re dependency through 6* appears in two places, at the outside of log and the inside of it. Re dependency from the latter makes the discussion slightly obscure, but we can advance the discussion by several approximations and considerations. However, it is not our aim to discuss these points here again (see original paper). For the present purpose, only the utilization of Batchelor’s solution is stressed, and the rest will be described briefly. After some consideration we obtain the next equation: (4.23)
at the same value of x / L, which is again the same as (4.20). Therefore, like OLl u,
=
u-u c,, (Y>, xc,,
(4.24)
at the corresponding points of model and ship. Thus the similarity law of the wake distribution is the same as the vorticity in non-dimensional form. Figure 27 also serves as the illustration of wake correlation between model and ship.
Three-Dimensional Ship Boundary Layer and Wake
355
WL u-u u * u
d
‘Corresponding points FIG.27. Correlation method for vorticity and wake of longitudinal vortex between model and ship (Tanaka, 1983).
( 2 ) Scale Efects of the Boundary Layer and Wake with Longitudinal Vortices
We superimpose the characteristics of the boundary layer and wake obtained in the previous section and the ones of the longitudinal vortex explained above, attempting to obtain the correlation law between the model and ship scale. This idea means that the longitudinal vortex streams down on the base flow which corresponds to the ordinary boundary layer velocity distribution in the present problem. As was explained in the previous section, there are several methods to predict the scale effects of the velocity distribution of the boundary layer and wake. Slight differences are also noticed between the methods. Therefore, here, only the basic idea for obtaining the flow field including longitudinal vortices is mentioned and details are omitted for the sake of brevity. Now let us apply this method to the result of experiments conducted by using three geosim models and a full-scale ship by Panel SR107, Ship Research Association of Japan (1973). The result of comparison between the measured values and the theoretically predicted values is shown in Figure 28. The word previous in the figure means the method in the previous
Ichiro Tanaka
356
calculated potential velocity ,
(
1 .o
(
------.----4.,_
predicted from model experiment i n previous method measured i n
praulLLru
uy
present meth,
-..2 .,65m W.L.
------_
Ship
Model
L
(302m) Em)
1
7 m BL
6
i
( lZm)with propeller
(
5
I
I
,
4
3
2
ux: velocity in the direction o f ship's
I
1
1
1
p.
center line
FIG. 28. Comparison of distributions between the measured full-scale experiment and the predicted distributions from model experiments (Niizuru Maru, SR 107) (Tanaka, 1983).
report by Tanaka (1979), which does not consider the effect of longitudinal vortices on the prediction of streamwise velocity distribution. In the new method considering the vortices effect, the feature is to consider a dent (or waving) in velocity distribution as the longitudinal vortex wake. In the figure there are many curves and marks, so we need several explanations to understand them. But we omit the detailed explanation about them. We say only that the extrapolation was made by using the data of 8 m model, because there were no noticeable final differences in the predicted values between the three models. Predicted streamwise velocities are shown in
Three-Dimensional Ship Boundary Layer and Wake
351
three thick dotted lines in the figure. The top line is the portion due to the ordinary wake. The lower two lines show the values, corrected for the longitudinal vortex wake, that correspond to two slightly different methods of prediction for the location of vortex, for which discussion is omitted here. Generally speaking, the idea of superimposing the wake of the ordinary boundary layer and of the longitudinal vortex is promising for understanding the wakes of full-form ships.
V. Concluding Remarks 1. The features of 3D ship turbulent boundary layer and wake have been briefly described. The features are characterized by, firstly, the existence of a fully 3D boundary layer over most of the ship hull surface except for the stern end of full-form ships, and secondly, at the said stern, the generation of pairs of trailing vortices due to 3D separation in the thick near wake. 2 . Applicability, usefulness and limitations of the integral methods of calculation of the 3D boundary layer and wake have been explained. The integral methods are described as being useful for the prediction of the development of the ordinary boundary layer and wake without appreciable 3D separation vortices. 3. To predict the characteristics of the flow field near the stern end of full ship forms, simple integral methods are not applicable, and some ingenious techniques, or some other methods based on direct numerical calculation, or some different methods which stress the behavior of the longitudinal vortices have to be developed. 4. The importance of scale effects of the characteristics of the 3D turbulent boundary layer and wake is explained with experimental measurements. The applicability and usefulness of the calculation based on the integral methods to the discussion of scale effects are demonstrated by attempts to predict the scale effects of velocity distribution in the boundary layer and wake. 5. In the future, one of the central problems in the area of 3D ship boundary layers and wakes will be the research on the flow structure of longitudinal vortices imbedded in or shot out of the boundary layer and wake, as well as the correct prediction of the flow field including the scale effects. It is also to be noted that this problem inevitably leads to the simultaneous investigation of a strong viscous-inviscid interaction of the flow field at the stern.
Ichiro Tanaka Acknowledgments The author wishes to express his sincere gratitude to Professor T. Y. Wu of the California Institute of Technology, who has kindly advised and encouraged him to write this article. He also wishes to express his gratitude to Drs. T. Suzuki, K. Matsumura and Y. Toda for their comments and discussions about the manuscript, to staff and students for preparing the figures, to Mrs. J. Azuma for typing the manuscript, and to the authors and publishers who kindly permitted the reproduction of figures from their publications. He wishes finally to apologize to Professor Wu and to the publisher for his extreme delay in writing.
References Batchelor, 0 . K. (1964). Axial flow in trailing line vortices. Jour. Fluid Mech. 20, 645-658. Eichelbrenner, E. A., and Peube, J. L. (1966). The role of S-shaped cross-flow profiles in three-dimensional boundary layer theory. Final Report, Laboratoire de Mechanique des Fluides, Poitiers. Hatano, S., Mori, K., and Hotta, T. (1978). Experiments of ship boundary layer flows and considerations on assumptions in boundary layer calculation. Trans. West-Japan Soc. Naval Arch. 56, 73-92 (in Japanese). Himeno, Y., and Okuno, T. (1979). Pressure distribution in ship boundary layer and its displacement effect. Jour. of the Kansai SOC.of Naval Architects, Japan 174, 57-68 (in Japanese). Himeno, Y., and Tanaka, 1. (1975). An exact integral for solving three-dimensional turbulent boundary layer equation around ship hull. Jour. of the Kansai Soc. of Naval Architects, Japan 159, 65-73 (in Japanese). Hinatsu, M. (1984). Thick turbulent boundary layer calculation and its application to evaluation of effective wake. Report ofS.R.T., Vol. 21, No. 1. Ikehata, H., Nagase, Y., and Maruo, H. (1982). An improved method of turbulent boundary layer theory to solve viscous flow around ship stern. Jour. of the SOC. of Naval Architects of Japan 152, 44-54 (in Japanese). Larsson, L. (1974). Boundary layer of ships, part I-IV. SSPA Allman Rep. No. 44-47. Larsson, L., and Chang, M . 3 . (1980). Numerical viscous and wave resistance calculations including interaction. Proc. of 23th Symposium on Naval Hydrodynamics, 707-728. Mori, K., and Doi, Y. (1978). Approximate prediction of flow field around ship stern by asymptotic expansion method. Jour. of the SOC.of Naval Architects, Japan 144, 11-20. Mori, K., Ohkuma, K., and Okuno, T. (1981). On practical method to predict ship wake distribution by vorticity shedding approximation. Trans. West-Japan SOC.Naval Architects 62, 1-12. Nagamatsu, T. (1979). A method of predicting ship wake from model wake. Jour. of the SOC. of Naval Architects of Japan 146, 43-52. Nagamatsu, T. (1980). Calculation of viscous pressure resistance of ship based on a higher order boundary layer theory. Jour. of the Soc. Naval Architects, Japan 147 (in Japanese). Nagamatsu, T. (1985). Calculation of ship viscous resistance by integral method and its application. Proc. of Second International Symposium on Ship Viscous Resistance, March 18-20, Goteborg, Sweden. Nakayama, A., Patel, V. C., and Landweber, L. (1976). Flow interaction near the tail of a body of revolution: Part 1 and Part 2. Jour. of Fluids Engineering, Trans. ASME, Vol. 98, 538-547.
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Okajima, R., Toda, Y., and Suzuki, T. (1985). On a stern flow field with bilge vorticies. Jour. of the Kansai Soc. of Naval Architects, Japan 197, 87-96 (in Japanese). Okuno, T. (1976). Distribution of wall shear stress and cross flow in three dimensional turbulent boundary layer on ship hull. Jour. of the Soc. of Naval Architects of Japan 139, 1-12 (in Japanese). Patel, V. C. (1974). A simple integral method for the calculation of thick axisymmetric turbulent boundary layers. The Aeronautical Quarterly 25, 47-58. Rep. SR107 (1973). Investigation into the speed measurement and improvement of accuracy in powering of full ships. Rep 73 (in Japanese). Sasajima, H., and Tanaka, 1. (1966a).On the estimation of wake of ships. Proc. 11th International Towing Tank Conference, Tokyo, 140-143. Sasajima, H., Tanaka, I., and Suzuki, T. (1966b). Wake distribution of full ships. Jour. of the Soc. of Naval Architects of Japan 120, 1-9 (in Japanese). Tanaka, I., Himeno, Y., and Matsumoto, No. (1973). Calculation of viscous flow field around ship hull with special reference to stern wake distribution. Jour. of the Kansai Soc. of Naval Architects, Japan 150, 19-26 (in Japanese). Tanaka, I., and Himeno, Y. (1975). First order approximation to three-dimensional turbulent boundary layer and its application to model-ship correlation. Jour. of the Soc. of Naval Architects of Japan 138, 65-75 (in Japanese). English translation in “Selected Papers from the Journal of the Society of Naval Architects of Japan” (1976), Vol. 14, pp. 1-12. Tanaka, 1. (1979). Scale effects on wake distribution and viscous pressure resistance of ships. Jour. of the Soc. of Naval Architects of Japan 146, 53-60. Tanaka, I . (1983). Scale effects on wake distribution of ships with bilge vortices. Jour. ofthe Soc. of Naval Architects of Japan 154, 78-85. Tanaka, I., Suzuki, T., Himeno, Y., Takahei, T., Tsuda, T., Sakao, M., Yamazaki, Y., Kasahara, M., and Takagi, M. (1984). Investigation of scale effects on wake distribution using geosim models. Jour. of the Kansai Soc. of Naval Architects, Japan 192, 103-120. Toda, Y., Tanaka, I., and Otsuka, Y. (1985). An integral method for calculating threedimensional boundary layer with higher order effect. Proc. uf Osaka International Colloquium on Ship Viscous Now, October 23-25, Osaka, Japan.
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Author Index Numbers in italics refer to the pages o n which the complete references are listed.
A Akylas, T. R., 137, 179 Aldeida, R. M., 59 Alonso, C., 10 Alper, A., 189, 215, 217, 238, 240, 253, 256, 299 Alvarez, C., 282. 299 Amick, C. J., 136, 137, 140, 149, 172, 179 Amsden, A. A,, 218, 226, 299 Anderson, P. S.,83, 132 Anderson, T. F., 59 Andreas, J. M . , -58 Annamalai, P., 59 Apfel, R., 20, 29, 60 Appel, P. E., 3, 59 Arhahanel, H . D. I., 139, 180 Astrop, P., 83, 132
Binder, G., 185, 187, 212, 228, 250, 299, 300 Bishop, K. A,, 187, 299 Blackwelder, R. F., 186, 190, 299, 302 Boure, J . A,, 64, 132 Bowen, R. M., 64, 74, 132 Bradshaw, P., 185, 187, 299 Breindenthal, R. E., 187, 218, 282, 299 Brindley, J., 299 Broadwell, J. E., 187, 299 Browand, F. K., 190, 211, 218, 226, 257, 272, 282, 297, 299, 305 Brown, G. L., 185, 187, 217, 228, 229, 297, 299, 305 Brown, R., 5, 6, 59 Burak, T., 137, 180 Busse, F. H . , 30, 59
B Baccheta, V. L., 61 Baldwin, C. M., 59 Barcilon, A., 228, 299 Barclay, F. J., 95, 132 Barratt, M. J., 300 Batchelor, G. K., 348, 354 Bauer, H., 3 Beale, J. T., 137, 149, 180 Beckert, H., 137, 180 Bedford, A,, 64, 74, 132 Benjamin, T. B., 138, 144, 180 Benney, D. J., 230, 283, 293, 299 Bergeron, R. F., Jr., 230, 299 Bernal, L. P., 218, 282, 291, 299 Berringer, R., 5 Beyer, K., 138, 149, 180 Beyer, R. T., 59 Bhagwat, W. V., 61
C
Caflisch, R. E., 74, 79, 132 Cain, A. B., 283, 300 Cammack, L. S. B., 59, 61 Campbell, J. A., 59 Cantwell, B. J., 187, 211, 300 Carr, C., 59 Carr, J., 172, 180 Carrier, G. F., 187, 300 Carruthers, J. R., 59 Champagne, F. H., 185, 187, 243, 300 Chandrasekhar, S., 2, 3, 4, 7, 9, 59 Chang, M.-S., 354 Cheng, L., 83, 95, 98, 99, 132 Chimonas, E. F., 300 Chow, S. N., 159, 161, 162, 180 Chuh, T., 12 Clayton, B. R., 59 Clifford, G., 59 361
Author Index
362
Corcos, G. M., 192, 218, 283, 293, 300, 302, 304 Cornfield, G. C., 95, 132 Corrsin, S., 184, 187, 243, 300 Couet, B., 283, 300 Craik, A. D. D., 295, 300 Crighton, D. G., 242, 300 Croonquist, A,, 61, 62 Crow, S. C., 185, 187, 300
D
Davis, P. 0. A. L., 185, 300 Delhaye, J. M., 64, 74, 132 Dendall, J., 60 Diehl, H., 10 Doi, Y., 336, 354 Drew, D. A., 64, 65, 69, 74, 17, 83, 132 Drexhage, M. G., 59 Drumheller, D. S., 64, 74, 132 Dryden, H. L., 184, 219, 300 Dunstan, A. E., 59 Dziomba, B., 213, 300
E
Eichelbrenner, E. A,, 319, 354 Eichhorn, R., 105, I33 Einaudi, F., 296, 300 Elleman, D. D., 60, 61, 62 Elswick, R. C., Jr., 186, 189, 238, 239, 240, 300
F
Favre-Marinet, M., 185, 186, 187, 212, 228, 250, 299, 300 Fendell, F. E., 300 Ferriss, D. H., 299 Ferzigen, J. H., 300 Ffowcs Williams, J. E., 299 Fiedler, H. E., 213, 214, 215, 216, 217, 226, 241, 252, 254, 256, 278, 300 Finnegan, J. J., 296, 300 Finson, M. L., 238, 302 Fischer, G., 172, 176, 180 Fisher, M. J., 300 Foias, C . , 137, 180 Foote, B., 9, 59 Fraenkel, L. E., 136, 179
Freymuth, P., 190, 257, 282, 300 Fritts, D. C., 295 Fua, D., 296, 300 G
Cans, R., 3 Gaster, M., 192, 238, 242, 250, 264, 300, 301 Gatski, M., 218, 301 Gatski, T. B., 189, 214, 219, 220, 221, 222, 223, 225, 228, 229, 231, 233, 236, 237, 241, 243, 247, 250, 255, 296 Gibson, E. G., 59 Goldberg, Z. A., 59 Gorkov, L. P., 59 Grant, H. L., 187, 301 Grasso, M., 5 Y Grimshaw, R. H. J., 137, 180 Guckenheimer, J., 159, 280
H Hackbarth, A,, 140 Hale, J. K., 139, 180 Hall, P., 95, 132 Hama, F. R., 257, 305 Happel, J., 9 Harkins, W. D., 59 Harlow, F. H., 218, 219, 226, 299 Harris, V. G., 243, 300 Hasegawa, T., 59 Hatano, S., 330, 354 Hauser. E. H., 58 Hench, J. E., 81, 132 Henry, D., 172, 180 Himeno, Y., 319, 321, 327, 328, 330, 338, 342, 343, 345, 354, 35.5 Hinatsu, M., 330, 354 Ho, C'. M., 190, 214, 215, 217, 234, 241, 252, 257, 258, 259, 261, 262, 264, 266, 267, 284, 286, 293, 301 Holmes, P. J., 1.59, 180 Holton, J . R., 295 Hoppensteadt, F., 140 Hotta, T., 354 Howe, M. S., 295, 301 Hsieh, D. H., 64, 78, 107, 117, 133 Huang, L. S., 190, 192, 214, 215, 217, 234, 247, 252, 257, 258, 259, 261, 262, 264, 266, 267, 282, 284, 285, 286, 291, 292, 301
Author Index Huerre, P., 214, 217, 262, 293, 301 Huh, C., 59 Hunt, J. C. R., 239, 301 Hunter, J. K., 148, 180 Hussain, A. M. F. K., 185, 186, 189, 190, 192, 211, 214, 238, 239, 240, 255, 301, 304
Hyzer, W. G., 60
I Ikehata, H., 330, 354 Ishii, M., 64, 14, 77, 133 Ize, J., 140 J Jacobi, N., 60 Jimenez, J., 192, 218, 282, 285, 291, 301 Johnson, R. F., 299 Johnston, J . P., 81, 132 Jones, M., 137, 180 Jordan, H. F., 59
K Kanamori, T., 61 Kanber, H., 62 Kanugo, R. B., 61 Kaplan, R. E., 186, 191, 299 Kappel, F., 181 Kaptanoglu, H. T., 267, 301, 303 Kasahara, M., 355 Keady, G., 139, 180 Kelley, A., 172 Kelly, R. E., 259, 264, 266, 301 Kendall, J. M., 185, 301 Kenyon, D. E., 64, 74, 133 Kibens, V., 302 Kim, J., 222, 301 Kimel, W. R., 60 King, L. V., 44, 45, 47. 60 Kirchgassner, K., 139, 179, 1x0 Kit, E., 242, 264, 301 Klassen, G . P., 296, 301 Klein, E., 45, 60 Knight, D. D., 218, 219, 301, 302 Knobloch, H. W., 172, 181 Knox, W., 5 KO, D. R. S., 214, 225, 235, 251, 252, 302 Konrad, J. H., 218, 282, 299, 302
363
Kovasznay, L. S. G., 185, 186, 302 Kubota, T., 214, 225, 235, 251, 266, 302 Kuriki, K., 252, 266, 305 1
Lahey, R. T., Jr., 83, 132 Lake, B. M., 136, 181 Lamb, H., 40, 60 Lamdahl, M. T., 186, 302 Landau, L. D., 81, 107, 133 Landweber, L., 354 Larsson, L., 326, 328, 330, 354 Latigo, B. O., 272, 299 Laufer, J., 187, 302 Launder, B. E., 222, 231, 302 Lauterborn, W., 60 Ledwidge, T. J., 95, 132 Lee, C. P., 60 Lee, S. S., 291, 291, 292, 302 Lees, L., 214, 225, 235, 251, 266, 302 Legner, H. H., 238, 302 Leonard, A,, 283, 300 Lessen, M., 299 Leung, E., 45, 60 Lewellen, W. S., 296, 305 Liepmann, H. W., 184, 189, 240, 302 Lighthill, M. J., 186, 205, 207, 302 Lin, C. C., 188, 194, 293, 299, 300, 302 Lin, S. J., 192, 218, 283, 294, 302 Lipschitz, E. M., 81, 133, 165 Liu, J. T. C., 187, 188, 189, 192, 207, 211, 212, 214, 215, 217, 219, 220, 221, 222, 223, 224, 225, 226, 228, 229, 231, 233, 236, 237, 238, 240, 241, 243, 244, 247, 248, 250, 251, 252, 255, 256, 257, 258, 259, 260, 262, 264, 266, 267, 278, 282, 286, 291, 292, 295, 296, 299, 301, 302, 303, 304
Lockwood, K. L., 60 Longsworth, L. G., 60 Loporto Arione, S., 60 Lumley, J. L., 186, 187, 199, 222, 239, 303, 30s
Lykoudis, P. S., 187, 304
M Ma, Y., 133 MacKay, R. S., 137, I81 Mackenzie, J. D., 59 Maclnnes, D. A,, 60
Author Index
364
MacPhail, D. C., 184, 303 Maidanik, G., 60 Malkus, W . V. R., 186, 303 Mallet-Paret, J., 180 Manabe, T., 61 Mankbadi, R., 187, 189, 192, 207, 214, 217, 226, 238, 256, 257, 278, 282, 303 Manneville, P., 181 Marble, F. E., 187, 300, 303 Marsden, J . E., 159, 180 Marston, P. L., 20, 29, 60 Martinez-Val, R., 282, 299 Maruo, H., 354 Mason, S . G., 59, 60 Massey, B. S., 59 Matsumoto, O., 355 Matsuno, T., 295 Mensing, P., 213, 215, 217, 247, 256, 278, 300 Merkine, L., 189, 215, 217, 224, 236, 237, 238, 240, 243, 244, 248, 250, 251, 256, 262, 302, 303 Metcalfe, R. W., 283, 303, 304 Michalke, A,, 234, 241, 242, 260, 303 Mielke, A,, 137, 138, 139, 144, 164, 172, 176, 177, 181 Mikhail, S . Z., 60 Miksad, R. W., 190, 282, 303 Miksis, M. J., 132 Miles, J . W., 137, 181 Miller, C., 60 Mitachi, S., 61 Mitchell, T . P., 64, 133 Mobbs, F. R., 299 Moffatt, H. K., 186, 303 Moin, P., 222, 303 Mollo-Christensen, E., 185, 186, 187, 199, 205, 303 Montgomery, D., 9, 10 Morgan, J . L. R., 60 Mori, K., 335, 336, 354 Moser, J., 139, 181 Moynihan, C., 59 Muir, J . F., 105, 133 Munk, 295 Murray, B. T., 219, 302, 304 N
Nagamatsu, T., 330, 354 Nagase, Y., 354
Nakayama, A., 330, 354 Neidig, H. A,, 60 Neidle, M., 60 Newhouse, S. E., 181 Nicolaenko, B., 180 Nikitopoulos, D. E., 258, 259, 262, 264, 266, 267, 286, 303, 304 Nyborg, W . L., 60 0
Ohkuma, K . , 354 Okajima, R., 310, 312, 313, 335, 337, 354 Okuno, T., 321, 322, 323, 324, 325, 354, 355 Oster, D., 214, 252, 304 Otsuka, Y., 355
P Pai, S. I., 184, 304 Papailiou, D. D., 187, 304 Papanicolaou, G. C., 132 Patel, V. C., 330, 333, 354, 355 Patnaik, P. C., 218, 226, 304 Pedlosky, J., 202, 304 Peltier, W. R., 296, 301 Petersen, R. A., 211, 215, 234, 235, 242, 247, 252, 257, 264, 305 Peube, J . L., 319, 354 Phillips, 0. M., 186, 304 Pierrehumbert, R. T., 283, 292, 293, 304 Plateau, J. A. F., 60 Plesset, M. S., 64, 78, 79, 94, 95, 117, 133 Pliss, V. A., 172, 181 Pluschke, W., 140 Pomeau, Y., 181 Princen, H . M., 60 Prosperetti, A., 9, 60, 61, 64, 105, 107, 133 Pullen, G., 60 Pumir, A,, 139, 181
Q Quinlan, K., 59 Quinn, M . C., 295, 304 R
Rayleigh, L., 3, 5, 6, 7, 8, 9, 45, 61 Reece, G. J . , 302 Reynolds, O., 188, 304 Reynolds, W. C., 185, 186, 189, 194, 222, 238, 239, 240, 241, 300, 301, 304
Author Index Rhim, W. K., 61 Riddick, J. A,, 59 Riley, J. J., 222, 283, 304 Robey, J., 61 Rodi, W., 302 Roe, R. J., 61 Rosgen, T., 213, 300 Roshko, A., 184, 187, 211, 217, 228, 229, 282, 297, 299, 304 Ross, D. K., 3, 7, 61 Rothmann, O., 83, 132 Rudnick, I., 45, 61, 62 C Y
Saffrnan, P. G., 137, 181 Saffren, M., 61, 62 Sakao, M., 355 Saleh Boulos, M., 59 Sasajima, H., 342, 344, 345, 355, 3.75 Sato, H., 252, 305 Scheurle, J., 137, 138, 139, 163, 180, 181 Schmitt, K., 140 Schubauer, G. B., 185, 30.5 Scriven, L. E., 59, 60 Segal, L. A,, 64, 74, 77, 132 Sell, G. R., 137, 180 Sherman, F. S., 218, 300, 304 Shibata, S., 61 Siekmann, J., 3 Sijbrand, J., 181 Silberrnan, E., 95, 133 Skrdrnsted, H. K., 185, 305 Smith, W., 299 Srnyth, N., 180 Sperber, D., 61 Spitzer, L., Jr., 96, 133 Stewart, H. B., 107, 133 Stillinger, H. K. N., 295, 305 Stoker, J. J., 136, 181 Stuart, J. T., 189, 190, 194, 231, 235, 251, 258, 283, 284, 285, 292, 295, 30-5 Subnis, S. W., 61 Suzuki, T., 354, 355 Swiatecki, W. J., 2, 3, 10, 61 Sykes, R. I., 296, 305 T
Tagg, R., 12, 60, 61 Takagi, M., 355
365
Takahashi, S., 61 Takahei, T., 355 Tanaka, I., 319, 321,326, 327, 328, 334, 338, 340, 341, 342, 343, 344, 345, 346, 347, 348, 351, 352, 354, 355 Teates, T. G . , 60 TCmam, R., 137, 180 Tennekes, H., 199, 305 Ter-Krikorov, A. M., 137, 181 Thole, F. B., 59 Thomas, A. S. W., 185, 305 Thompson, 10 Thorpe, S. A,, 191, 305 Tiederman, W. G., 186, 304 Ting, L., 132 Toda, Y., 330, 333, 354, 355 Toland, J. F., 136, 137, 179, 180 Townsend, A. A., 184, 186, 187, 196, 305 Trinh, E. H., 20, 34, 59, 61, 62 Troutt, T. R., 218, 282, 299 Tsuda, T., 355 Tucker, W. B., 58 Turner, R. E. L., 139, 140, 181 V
Van Atta, C. W., 305 Vanden-Broeck, J. M., 148, 180 Varadan, V. K., 65, 133 Varadan, V. V., 133 Von KBrman, Th., 184, 305, 316 Vonnegut, B., 61
w Wallis, G. B., 64, 133 Wang, T. G., 20, 34, 59, 60, 61, 62 Weisbrot, I., 214, 215, 217, 226, 242, 247, 252, 253, 254, 255, 256, 257, 260, 264, 266, 305 Weissman, M. A,, 304 Wendroff, B., 107, 133 Westervelt, P. J., 60, 62 Whitham, G. B., 136, 149, 181 Widnall, S. G., 283, 292, 293, 304 Wijngaarden, L., 79, 105, 107, 133 Williams, D. R., 257, 305 Willmarth, W. W., 295, 305 Winant, C. D., 190, 257, 282, 297, 30-7 Wong, P. M. A,, 5, 61 Wu, D. M., 137, 181
Author lndex
366
Wu, T. Y., 137, 140, 181 Wygnanski, I., 211, 214, 215, 234, 235, 242, 247, 252, 257, 264, 301, 304, 305
Y Yamazaki, Y., 355 Yingling, R. T., 60 Yosioka, K., 62 Yuen, H . C., 136, 181
Z
Zakharov, V. E., 137, 181 Zarnan, K. B. M . Q., 214, 301 Zeidler, E., 136, 137, 138, 149, 181 Zeman, O., 300 Zia, Y., 60 Zuber, N., 83, 133 Zufiria, J. A., 138, 181 Zwern, A,, 61 Zwick, S. A., 117, 133
Subject Index
A (-)-phase averaging, 193, 198, 208, 220 (( ))-phase average, 194, 209
Abrupt switching, 259 of modal structure, 258, 259 Acoustic force, 33, 44-48 Acoustic frequency shift, 49 Acoustic radiation force, 44 Acoustic rotation, 45 Action of pressure gradients, 210, 240 Active fundamental development, 259 Advection integrals, 262, 293 Advection mechanism, 199. 203 Advection of mean flow momentum, 194 Advection of the mean stresses, 240 Advective derivative, 260 Advective effects, 199 Aerodynamic sound, 185, 207 Alternating tensor, 199 Amplification, 214, 297 of preexisting fine-grained turbulence. 285 Amplified disturbances, 263 Amplitude dependence, 24 Amplitude distribution, 260 Amplitude equations, 287 Amplitude evolution problem, 243. 293-196 Amplitudes, 192, 212, 215, 219, 235, 250. 252, 254, 259, 260, 262, 264, 286, 287, 298 of coherent modes, 190 Amplitude/wave function, 240 Antiresonance frequency, 94 Approximate considerations, 220 Artistic descriptions of the observations, 212 Atmospheric boundary layer, 299 Axially symmetric modes, 295 Auxiliary linear problem, 239 Averaged dynamical equations, 66, 69, 7 1
Average density, 7 1, 76 Average dynamic equations, 75 Average number density of bubbles, 75 Average pressure, 77, 79 Average radius of bubbles, 75, 76 Average thermal conductivity, 85 Average velocity, 71 Average volume fraction of gaseous phase, 75 Axially symmetric ( n = 0 ) modes, 280
B Background broadband turbulence, 254 Baratropic liquid phase, 86, 99 Baratropic gas phase, 86, 99 Basset force, 81, 83, 84, 90, 95, 97, 108, 119 Bifurcation, 2, 53, 55, 138, 139, 148, 152, 162 Bifurcation: (2(11), 55 Bifurcation points, 5, 6, 57 Binary-frequency interactions, 280, 281, 288, 289 Binary frequency transfer mechanism, 284 Binary interaction, 1Y9, 280, 294 Binary interaction integral, 280 Binder forcing, 284 Bond number, 137, 138, 143 Boundary conditions, 223 horizontal periodic, 219, 221, 223 Boundary layers, 184, 297 of ships, see Ship boundary layer and wake Broad-band turbulence, 256 Brunt frequency, 299 Bubble oscillation, 100, 103, 106, 121 Burgers-KdV type equation, 65, 103 Burst. 229 367
368
Subject index C
Capillary-gravity waves, 137, 138, 152-168 Cat’s eye, 230 Cauchy problems, 107 Center-manifold approach, 136, 172 Centerline mean flow decay, 213 Centrifugal instability, 189 Chaos, 159 Chaotic behavior, 130, 159, 163 Chaotic motion, 162 Characteristic equations, 65, 88, 90, 91, 93, 104, 106 Chokes, 113 Circumferential average, 193, 288 Closure arguments, 232, 240 Closure methods, 189 Closure models, 219 for Reynold stresses, 220 second-order, 220 Closure of the linear problem, 240 Cnoidal waves, 135, 143, 149, 152, 157, 171 nonlinear resonant reaction of. 139 Coherent contributions, 207 Coherent energy, 227 Coherent fluctuations, 185, 202, 206 two-dimensional, 206 Coherent frequencies, 288 Coherent mode, 207, 208, 212-215, 217, 218, 221, 223, 224, 226, 227, 230, 238, 240, 243-251, 253, 263, 265, 266, 268, 283, 288, 292, 295 energy level of, 268 horizontally periodic, 221 in two-dimensional shear flows, 284 relative phase of, 294 spatially periodic, 192 three-dimensional, 192, 218. 236, 284, 285 two-dimensional modes, 203, 218, 219, 235, 265, 285, 293, 300 Coherent-mode agglomeration, 219 Coherent-mode amplitude-limiting mechanisms, 195 Coherent-mode amplitudes, 212, 213, 249. 257, 266 development of, 213 Coherent-mode contributions, 191 Coherent-mode development. 2 15 Coherent-mode eigenfunctions, 245 Coherent-mode energy, 215, 227, 230, 232
Coherent-mode energy density, 235 Coherent-mode energy exchange. 254 Coherent-mode energy production, 214, 256 Coherent-mode fluctuations, 204, 205 Coherent-mode integrals, 254 Coherent-mode interactions. 258-280 two-dimensional, 265 Coherent-mode kinetic energies, transport of, 204 Coherent-mode kinetic energy integral, 215 Coherent-mode mean square vorticities, 204 Coherent-mode negative production mechanism, 253 Coherent-mode peaks, 268 Coherent-mode periodicities, 2 12 Coherent mode rates of strain, 203, 204, 245, 292 Coherent-mode stresses, 208 Coherent mode-turbulence energy exchange integral, 301 Coherent-mode velocities 242 Coherent-mode vorticity, 199 stretching of, 203 Coherent-mode vorticity intensification, 206 Coherent motions. 194, 208, 248 three-dimensional, 285 two-dimensional, 206, 221, 285 Coherent oscillations, 185 well-controlled ( i n turbulent free flows), I86
Coherent rates of strain, 209, 232, 240, 245 Coherent signal frequency, 214 Coherent streamwise structures, 22 I Coherent structure amplitude, 249 Coherent structure problem, 252 Coherent structure production mechanism, 214 Coherent structure properties, evolutionary aspects of, 298 Coherent structure vorticity equation, 223 Coherent structure wave envelopes, 282 Coherent structures, 187, 214, 215, 220. 222-224. 244, 249, 259, 265, 298 two-dimensional. 284 Coherent three-dimensionality, 284 Coherent velocity distributions, across the shear layer, 242 Coherent velocity fluctuations, 209, 243 Coherent vorticity distribution, 23 1 Coherent wave stream function, 242
Subject index Combustion, 185 Complex amplitude functions, 287 Complex characteristics, 107 Complex phase velocity, 286 Compound drop oscillations, 38 Compressibility, 64, 92, 94, 97 Compressible fluids, 107 Computational conveniences, 2 12 Computational-hydrodynamic stability, 285 Computational results, 225, 251 Condensation, 65, 68, 75, 116, 121, 124 Conditional average, 190, 288 Conditional averaging, 185, 299 Confined flow problems, 229 Conservation of mass, 76, 116 of number of bubbles, 116 Constant amplitude wave disturbances, 258 Constitutive equation, 71, 87 Constitutive relations, 65, 73-75 Continuity equations, 64, 65, 67, 87, 193, 195 Control, 275, 298 Controlled frequency, 214 Conventional averaging methods, 186 Correction to the mean flow, 296 Critical points, Ill-113, 130, 131 Cross-sectional energy, 259 Cross-stream shape, 238 of the coherent mode, 236
D Damped disturbances, 199, 217. 227, 24X. 255 Damped disturbances mechanism, 248 Damped region, 221, 256, 275 Damping, 88, 92, 122-124 coefficients, 91 constant, 20, 23, 25, 27, 37 Damping time constant, 42 Dead Zone, 65, 93-95 Decay, 214 Decay constant, 37 Decay of the coherent mode, 215 Decaying disturbance amplitudes, 264 Decaying outgoing waves, 242 Density, 66, 67, 78, 81, 83, 86 Ikveloping Hows. 233
3 69
Developing mean shear flow, 264 Developing mixing layer, 282 Developing shcar flow, 263, 287 Diffusion, 198 Diffusion of vorticity, 202 Ditfusive momentum flux, 72, 84 Dimensionless frequency, 282 Direct energy exchanges, 290 Direct energy transfer mechanisms, 232, 260, 297 Direct interaction, 297 Directions of energy transfer, 223, 245, 260, 294 Directions of the individual energy exchange mechanism, 292 Discontinuity, 113-1 15 Dispersion, 100-104, 106 Dispersion relation, 88, 107, 108, 123 Dissipation, 85, 94, 95, 100-103, 107, 108, 113, 116, 229, 254, 285 viscous, 196, 205, 210, 217, 225, 240, 241, 254, 260, 264, 266, 268, 275 Dissipation rate, 224 viscous, 196, 198, 203, 223, 230, 240, 241 Dissipational integrals, 244, 293 Distinct three-dimensional motions, 285 Disturbance, 219 two-dimensional, 286 wavy, 258 weak, 243 Disturbance amplification, 264 Disturbance components, 296 Disturbance energy integral equation, 236 Disturbance modes, 263 Disturbance stream function, 236 Disturbed turbulent shear layers, 215 Dominant energy exchange mechanisms, 217 Doromant mode, 297 Downstream boundary conditions, 21 1 Downstream evolution, 288 Downstream flow control, 188 Downstream region, 266 Drag force, 80, 8 1, 90 Drop fission, 10, 29 Drop oscillations, 2, 20, 21, 24 Drop shape oscillations, 1-58 Drop shapes, 1-58 Dye streak behavior, visual observations of, 258
Subject index
3 70
Dynamical equations, averaged, 69-73 general formulation of, 65-75, 115-117 of bubbly liquids, 75-88 Dynamical instabilities, 227, 241, 242, 263 Dynamical instability mechanism, 252 Dynamical model of multiple suhharmonic evolution, 280 Dynamical point of view, 212 Dynamical, multiple subharmonic model, 282 Dynamically unstable flows, 21 7
E Eddies of low correlation radius, 207 Eddy energy transfer rate, 230 Eddy viscosity treatment, 240 Eddy viscosity, 186, 187, 199, 223, 240. 299 model, 219 Eddy-viscosity assumption, 241 Eddy-viscosity subgrid closure, 223 Eigenfunctions, 237, 243, 244, 261, 300 Eigenvalue problem, 242 Energy balances. 238 Energy considerations, 195, 255 Energy content, 262 of the coherent mode, 235 Energy conversion mechanisms. 232 Energy density, 236, 238, 242, 260, 262 Energy equations, 67, 68, 75, 85, 86, 290. 292 for the even modes, 198 for the odd modes, 197 Energy exchange mechanisms, 196- 199, 214, 217, 226. 227, 244, 245, 255, 256, 289-293 Energy exchanges, 197-199, 207, 208, 225, 238, 244-248, 290-292 Energy extraction from the mean flow, 249, 264 Energy extraction process, 214 Energy extraction/supply rate, 256 Energy integral equations, 225 Energy levels, 280 Energy production, 199, 260 Energy production rate, 260 Energy supply, 285 Energy supply, source of, 297
Energy transfer, 190, 197, 199, 218. 223, 227. 240, 245. 247-250, 260, 264, 265, 266, 275. 292 rate of, 232 wave-turbulence, 256 Energy transfer between modes, 259 Energy transfer to the fine-grained turbulences, 249 Enhanced coherent mode, 217 Enhancement of the turbulence, 214 Ensemble of disturbances, 259 Enstrophy, 202 Envelope equations, 237 Envelope evolution, 237 Envelopes, 237 Environmental conditions, 187 Equation of Poisson’s type, 206 Equilibrium amplitude of the coherent mode. 250 Equilibrium figures of fluid masqes, 2 stability of, 4, 5 Equilibrium find-grained-turbulence energy density, 266 Equilibrium level, 250, 284 Equilibrium shapes or rotating spheroids, 1-58 Equilibrium spreading rate, 266 Equilibrium values, 249 Equilibrium vapor pressure, 120 Euler equations, 136, 137 Evaporation. 65, 75, 116-118, 121, 124 Even binary-mode interaction, 280 Even modes. 190, 194, 197, 202, 205, 206, 2 I I , 260, 280 Even-odd mode interactions, 202 Even-coherent modes, 196 Even-frequency mode, 288, 289 Even-mode r,,,, 210 Even-mode contributions, 195 Even-mode mean square vorticities, 204 Even-mode self-interaction, nonlinear effects of, 194 Even-mode vorticity, 200 €\en-mode vorticity equations, 201 Exchange mechanisms, 204 Eschangeb o f vorticity, 203, 2 0 6 Excited coherent fluctuation, 256 Experimental results, 2 12 External pressure waves, 137 External pressure waves, 142
Subject index F Far pressure field, 207 Fast oscillations, 235, 252 Favre-Marinet forcing, 284 Field quantity, 207 average of, 69 Filtering, 214 Fine-grained turbulence, 185, 187-191. 195197, 201, 202, 204, 208, 218, 219, ??I227, 230, 231, 237, 238, 244-250, 252, 253, 257, 260, 263, 265, 266, 268, 280, 281, 285, 288, 290, 291-293 graininess, 230 production of, 218 three-dimensional, 223, 284 Fine-grained turbulence energy, 248-250. 257 horizontal, 232 Fine-grained turbulence level, 253, 284 Fine-grained turbulence production, 266. 284 Fine-grained turbulence production rate, 227 Fine-grained turbulence vorticity equation, 202 Fine-grained turbulent stresses, 189 phase-averaged, 223 Fine-scaled fluctuations, 185 Finite-amplitude coherent disturbances, 217 Finite-amplitude disturbances, 252 Finite-amplitude effects, 286 Finite amplitudes, 264, 286 Finite-amplitude waves, in (locked) bubbly liquids, 65, 96-103 Finite disturbance levels, 253 Finite disturbance problem, 252 First binary interaction, 280 First harmonic, 284, 296, 297 First subharmonic, 280, 281, 289 First subharmonic wave-envelope equation, 28 I Fission, 10, 29, 53, 58 Flow control, 188 Flow instabilities, 188 Flow visualization, 298 Flow with discontinuity, I13 Fluctuating rates of strain, 204 Fluctuating vorticity, 199 Fluctuation advection integrals, 253
37 1
Fluctuation energy density, 253 Fluctuation kinetic energy equation, I95 Fluctuation production integrals, 301 Fluctuations, 225, 253, 292 Fluctuation velocities, 261 Fluid masses, equilibrium figures of, 2 Fluid mixture, equation for, 86, 87 Forces capillary-gravity waves, 143, 152 Forced gravity waves, 143, 152, 168-172 Forced turbulent shear layer, 218 Forcing local, 143 quasiperiodic, 138, 143 well-controlled, 189 Forcing amplitude, 283 Forcing frequency, 254 Forcing level, 284 Free laminar shear flows, development of, 236 Free shear flow, 229 development of, 227 two-dimensional, 218 Free turbulent flows, 185, 187, 212 Free turbulent shear flows, 188, 189, 227, 25 2-2 5 8 development of, 183-301 control of, 258 Frequency, 191, 212, 227, 235, 257 Frequency-fundamental energies, 296 Frequency-independent modes, 288 Frequency-periodic modes, 288 Frequency selection mechanism, 285 Frequency-subharmonic energy equation. 29 1 Frequent subharmonics, 219, 290, 291 Froude number, 138, 143 Fundamental component, 190, 215, 217, 262 Fundamental disturbance, 190, 264 Fundamental disturbance-mode Reynolds stress, 217 Fundamental disturbance wave, 258 Fundamental energy density, 283 Fundamental energy equation, 261 Fundamental frequency, 259, 288, 290 Fundamental frequency group, 295 Fundamental frequence mode, 289 Fundamental mode, 24, 25, 50. 190, 258. 260, 265, 281, 282 in frequency, 288 of oscillation, 7, 8 oscillating in, 9
3 72
Subject index
Fundamental physical frequency, 252 Fundamentals, 258-261, 263, 264, 280, 281, 284, 286-289, 296, 298 forcing of, 282 two-dimensional, 286, 297 Fundamental streamwise wave number, 286 Fundamental three-dimensional mode, 291 Fundamental two-dimensional disturbance, 286 Fundamental two-dimensional mode, 291
G Gas bubbles, 75-77 dynamical equations for, 87, 88 species of, 75, 76, 1 IS Generalized heat flux, 85 Generalized heat source, 85 General transfer equation, 66-68 Geophysical problems, 299 Global energy evolution, 236. 237-243 Global energy transfer, 226 Global forcing, 143 Gravity waves, 138, 178
H Half-vorticity thickness, 261 Harmonics, 190, 257, 286 Heat exchange, 85 Heat exchange coefficient, 85 Heat flux, 72 Helical coherent modes, 193 Helical modes, 192, 280, 285, 295 Helical modes interaction, 297 Helmholtz billows, 299 Heteroclinic solutions, 139, 167 High amplitudes of forcing, 254 Higher frequency coherent modes, 257 Higher frequency components, 284 Higher frequency first harmonic, 284 Higher frequency modes, 257, 275 Higher frequency side, 275 Higher frequency wave disturbances, 295 Higher harmonics, 254, 281 Higher-order equations o f turbulent boundary layer, integral method of, 330-341 High frequency cutoff, 281 High frequency modes, 235
Homoclinic bifurcation. 135-179 Homoclinic orbits, 136. 139. 152-154, 159, 162, 164, 167 Homoclinic solutions, 160, 167, 171 Horizontal average, 192, 219, 224 Horizontally periodic. (disturbance). 219 Horizontally periodic coherent mode, 221 Hydrodynamic stability, 188, 190, 192. 195, 199. 215. 217, 226, 227. 232 weakly nonlinear, 232 theory, 227. 251 Hydrodynamic(a1) instability, 185, 187, 189. 2x5 non-universality of, 220 Hydrodynaniical instability wave l'unctions. normalization of, 235 Hydrodynamically unstable disturbances, 252 Hyperbolic tangent mean velocity profile, 745
I Ill-posed Cauchy problems, 107 Immiscible system, 11, 12, 20, 30, 57, 58 Incipient instability region, 253 Incompressible homogeneous fluid, 193 Inertia. 92, 97 Inertia o f liquid, 94 Inertial instabilities, 189, 241, 263 Inflectional mean velocity, 187 Inflectional mean velocity profile, 242 Inflexional mean flows, 189 Inherent waviness, 285 Initial Strouhal number, 213 Initial amplitudes, 212, 249, 275, 283 very large, 275 Initial coherent mode, 223 Initial coherent-mode amplitude forcing, 257 Initial coherent-mode energy levels, 265 Initial conditions, 187, 212, 223, 224, 244, 250, 259, 263, 281, 282, 295 eHect of. 257, 258 role of, 285 Initial dimensionless frequencies, 275 Initial disturbance amplitude, 283 Initial energy level, 212, 235, 257 Initial fine-grained turbulence, 265 Initial fluctuation level, 253
Subject index Initial frequency, 265, 266, 275, 295 Initial kinetic energy content, 224 Initial level, 265 Initial mean flow distribution, 212 Initial m o d e content, 265 Initial mode-energy levels, 295 Initial random noise field, 298 Initial shear layer thicknes\,, 222, 241. 7 0 0 Initial turbulence, 250 Initial turbulence energy levels, 257. 2hX Initial turbulence levels, 268 Initial value, 229, 230, 249, 298 Initial value problem, 282 Initial wake thickness, 253 Initial wave number, 224, 249 Initial-amplitude threshold, 258 Initialization process, 223. 224 Initialized initial condition, 242 Initialized mean velocity profile, 224 Initially lower frequency modes, 275 Initially turbulent shear layer, 285 Instability, 65, 187 in bubbly liquids, 96-108 of slip flow, 105-108 viscous, 18Y Instantaneous turbulent pressure fluctuations, 207 Integral method, of three-dimensional turbulent boundary layer a n d wake. 32 1-341 of higher-order equations of turbulent boundary layer, 330-341 Integrated energy equations, 262 Integrated subharmonic energy equation, 261 Intensification, 203 Intensification mechanism, 200 Intensification of 0 7 / 2 , 203 Intensification of vorticity, 200 Intensification rate of (see symbols in text), 204 Interaction between wave motions a n d turbulence, 299 Interaction integrals, 249, 262, 293, 294 Interaction model, 295 Interactions between different scales. 199 Interaction of s o u n d with wall turbulent shear layers, 299 Interfaces, 67-12, 75-71 Interfacial conditions, 242
373
Interfacial transfer, 71 Interference, 290, 291 Interior ocian, 299 Inter-mode energy transfer, 298 Internal energy. 67, 72 Internal fluid flow, 27 Internal interaction processes, 299 Internal waves. 299 lnviscid instability, 227 lnviscid linear theory, 258 Isothermal behavior, 99. 109 lsotropizing mechanism, 232
J Jet-llow oscillations, 187 Jet noise suppression, 187 Jets, 190, 213 Jet spreading rate, 213
K Kelly mechanism, 260, 264 Kelvin-Helmholtz instability, 65. 107, 108 Kinematical model, 280 Kinematic interpretations, 212 Kinematics of a locally linearized theory, 23 5 Kinetic advection integrals, 300 Kinetic energy, 82, 83, 217 Kinetic energy balances, 195-200 Kinetic energy considerations, 255 Kinetic energy equation of the fine-grained turbulence, 198 Kinetic energy equations, 195, 208, 260 Kinetic energy exchange mechanisms, 204
L Lack of universality, 218 Laminar flows, 226, 252 Laminar free shear flow, 215 Laminar problem, 260 Laminar viscous flow, 200, 217, 264 Laminar viscous shear flow, 259 Laminar wake problem, 226 Laplacian, 221, 299 Large initial coherent-mode amplitudes, 213 Large-scale coherent-mode interactions, three-dimensional nonlinear effects in, 284-298
374
Subject index
Large-scale coherent modes, 284 Large-scale coherent motion, 285 Large-scale coherent structures, 183-301 control of, 257 non-universal. 220 Large-scale dihturbance, 185 Large-scale mode interactions, multiple, 190 Large-scale motions, 185-187, 189, 194 Large-scale organized aspects, 298 Large-scale structures, 184, 185, 207. 221 two-dimensional, 224 Large-scale structure vorticity equation, 222 Latent heat, 68, I16 Leibniz rule, 66 Length scale of the mean flow, 261 Lighthill's stress tensor, 206 Limited spatial averaging, 223 Limited-time-averaging procedure, I91 Limited-time ( o r space) averaging procedure, 186 Limiting amplitude. 257 Linear effects, 209 Linear eigenfunctions, 236 Linear growth, 263, 266 Linear growth far downstream, 2 I 7 Linear growth region, 268, 275 Linear hydrodynamic(a1) stability, 237 theory, 227, 236 Linear internal waves, 299 Linear problem, 238-240, 296 Linear spreading rate, 256, 164 Linear stability theory, 254 Linear theory, 220, 232-252, 266, 287 Linear three-dimensional stability, 296 Linear wave functions, 252 Linear waves, 101, 102 Linearized helical-model instability, 297 Linearized theory, 232, 245 role of, 237 Linearized vorticity equation, 239 Liouville equation. 296 Liquid drops, 2, 11 oscillations of, 48 Local wave number, 227, 236, 244, 261 Local coherent-mode number, 240 Local coherent-mode velocity proliles. 252 Local eigenfunctions, 236, 242, 254, 300 Local equilibrium argument, 230, 244 Local flow shape distribution, 253 Local forcing. 143. 163
Local frequency. 244 parameter. 253, 300 Local instability properties, 253 Local instantaneous formulation, 66-69, 71, 72 Local linear stability theory, 236, 238, 239, 243, 258, 261 Local parallel flow, 240 Local phase equilibrium. 116 Local rapid distortion, 232 Local shape functions, 238 Local shear flow thickness, 227 Local variables, 236, 239 Locally homogeneous-shear problem. 244 Locked bubbly liquids, 100, 101. 107 Locked vapor bubbles, I19 Longer wavelength, 295 Longer wavelength disturbances, 235 Longitudinal streaks, 295 Lower atmosphere, 299 Lower frequency components, 257, 283, 284 Lower frequency modes, 235 Lower frequency side, 275 Lower frequency wave disturbances, 295
M Macroscopic equations, 74, 7 5 . 78 Marginal stability, 229 Ma\> content, I I6 Mass transfer, 71, 72. 75, 76 Ma xi m um amplification, 27 5 Maximum initial amplification rate, 266 Maximum slope, 300 Mean flow, 190, 206, 208, 215, 217. 219, 221, 224, 226, 237, 244, 248-250. 252, 253,255,256,260,263-266, 268,281, 284-286, 289-291, 293, 296, 297 interactions with, 201 spreading of. 248, 283 two-dimensional, 203, 206, 256, 293 Mean flow advection integrals, 252. 253 Mean tlow development, 252 Mean flow energy, flow of, 196 Mean Row energy advection, 254 Mean flow energy equation, 196-198 Mean (low kinetic energy, 237 Mean flow kinetic energy defect integral, 244 Mean flow kinetic energy equation, 217, 236
Subject index Mean flow spreading rate, 212, 243. 254 Mean flow spreading rate, role of fluctuations on, 215 Mean flow thickness, structure of, 21 5 Mean flow vorticity equation, 200 Mean inflectional profile, 241 Mean kinetic energy, 257 Mean motion (problem), 189, 190. 218, 226. 237, 250, 254, 256, 260, 265 Mean motion evolutionary variable, 235 Mean rates flow rates of strain. 209 Mean rates of strain, 203 Mean shear (layer), 192 Mean shear flow, 223, 265 return of kinetic energy to, 215 two-dimensional, 285 Mean square coherent-vorticity fluctu;itions, 204 Mean stresses, 209, 240 Mean velocity, 244, 254, 261. 268 Mean velocity gradient, 244 Mean velocity profiles, 236, 255. 262. 300 Mean vorticity, 200, 204 Mean vorticity thickness, 300 Melnikov condition, 139, 157, 160-162 Mena flow kinetic energy, 224 Mesopheric dynamics, 299 Metereological context, 299 Microstructure problem, 299 Mimicking flow visualization, 258 Mixing, 187 Mixing controlled problems, 185 Mixing layer, 214, 221, 258, 261 Mixing length, 184 Mixing mechanism, 299 Mixing regions, 190, 212, 223, 238, 2.52 Modal-interaction mechanism, 296 Mode energy transfer, 260 Mode interaction integral, 264 Mode interactions, 201, 258-260, 2 6 8 , 283. 288, 289 subharmonic type, 289 superharmonic type, 289 two-dimensional. 297 Mode number, 23.5, 295, 296 Mode-forcing, 275 Mode-interaction mechanism, 21 1 Mode-mode energy exchange mechani\m, 294 Mode-turbulence energy transfer, 257
375
Mode-turbulence interactions, 265 Modulated line-grained turbulence stresses, 194, 201, 207, 238 Modulated fine-grained turbulence \orticity stretching eHects, 202 Modulated fine-grained turbulenceproduced transport. 201 Modulated horizontal normal stress-normal rate of strain, 247 Modulated quantities, 202 Modulated stresses, 189, 195, 197, 207-21 I . 238-240, 245, 261 production of, 209 shape of, 242 Modulated stretching etIects, 201 Modulated turbulent shear stress, 232 Modulated turbulent stresses, 191, 292 Modulated turbulent vorticity, 204 Modulated turbulent vorticity transport, 204 Momentum equations. 67, 68. 87 Momentum thickness, 300 Monochromatic coherent signal, 214 Monochromatic component, 221 Monochromatic disturbance, 194 Monochromatic large-scale disturbance, 190, 239 Monochromatic modulated stresses. 20X Monochromatic problem, 230 Monochromatic two-dimensional coherent structure, 221 Most amplified frequency. 275 Most amplified mode, 224. 245, 282 Multiple coherent-mode interactions, 257 Multiple subharmonic (evolution), dynamical model of, 280-284 Mutual friction, 91-93, 100, 103, 105. I I S Mutual interaction, 71, 108 Mutual interaction forces, 72. 80-84 Mutual slippage, 100, 103. 121, 122, 124
N Navier-Stokes equations, 193, 194, 220 unsteady, 219 Negative coherent structure production mechanism, 227 Negative damping, 65, 92-95 Negative disturbance production mechanism 226 Negative production, 215, 217, 21X. 266
376
Subject index
Negative production rate, 217 Negative production region, 275 Neighboring frequency modes, 289 Net intensification of mean \orticity, 200 Neutral problem, 296 Neutral stage, 231 Newtonian fluid, 84 n - ( n + 1 ) interaction, 281 Nonequilibrium, 253, 268, 285 Nonequilibrium development, 227 Nonequilibrium evolution, 235 Nonequilibrium interactions, 224, 230 history of, 238 Nonequilibrium region, 218 Nonequilibrium stages of development, 230 Nonlinear amplitude problem, 263, 293 Nonlinear coherent mode interactions, 21 8 Nonlinear contributions. 209 Nonlinear critical-layer theory, 230 Nonlinear disturbance, 252 Nonlinear effects, 209, 240, 287 Nonlinear envelope problem, 238 Nonlinear hydrodynamic stability, 195 Nonlinear hydrodynamic stability theory, 189 Nonlinear hydrodynamic stability problems, I89 Nonlinear interaction problem, 244 Nonlinear interaction process, 263 Nonlinear interactions, 237, 248, 260, 295, 296 between coherent modes, I94 between spanwise modes, 285 Nonlinear problems, 232-25 I Nonlinear production effects, 21 I Nonlinear resonant reaction, o f cnoidal waves, 139 Nonlinear surface waves, 138, 140, 142 Nonlinear theory, 235. 236, 252, 287 weak, 287 Nonlinear transport etiects, 201, 21 1 Nonlinear water waves, 135 Nonlinear wave envelope development 245 Nonlinear wave-envelope dynamics, 243, 244 Nonlinear wave-envelope problem, 240. 2.54 Nonlinearities. 252 Nonlinearly resonant interaction, 137 Nonlinearly resonant surface waves, 135179 Nonturbulent Row interface. 241
Non-universal structure, 223 Nonuniversalities, 213 Normal-mode frequency, 8, 39, 49 Normal-mode oscillation, 9, 21, 48 Normal stresses, 223, 232, 241 Normalization, 243, 300 o f wave amplitude, 235, 236. 238, 241 Normalization condition, 236 Normalization for the wave functions, 236 Normalization of the local eigenfunctions, 26-1 Nozzle, 213 ( n + 1 i-subharmonic, 281 nth subharmonic wave-envelope equation, 28 1 Numerical problem, 248 Numerical simulations, 192. 212, 223, 285, 298
0 Oceanographic context, 299 O d d binary-mode interaction, 280 Odd-coherent modes, 196, 206 Odd-even mode interactions, 205 Odd-frequency mode, 288 Odd-mode P , ) . 210 Odd-mode contributions, 19.5 O d d - m o d e mean square vorticities, 204 O d d - m o d e rates of strain, 197, 201 O d d modes, 190, 194, 197. 202, 205. 211, 259, 260. 280, 288, 290 O d d - m o d e vorticity, 200, 202 O d d - m o d e torticity equations, 201 O d d - m o d e Lorticity stretching, 201 One-dimensional steady flow, 109 Orr-Sommerfeld equation, 186 Oscillations d r o p shape, I o f a liquid drop. 6. 28, 37, 48 of a rotating drop, 30 Outer boundary condition, 242 Outgoing wave solution, 242 Overall Ilom field, 207
P Pairing, 258. 259 Para b o Ii c spreading, 2 5 3
Subject index Parallel flows, 232, 264, 286, 287 Peak amplitudes, 259 Peak in the turbulence level, 275 Periodic external pressure waves, I63 Periodic forces, 159 Periodic forcing, 159-163 Periodic orbits, 162 Periodicities, 212, 288 Periodicity o f the disturbances, 260 Periods, 288 Perturbations u p o n turbulent boundary layers, 298 Perturbed turbulent shear flow, 186 Phase angles, 245, 282, 284 Phase average, 191, 195, 220 Phase-averaged quantities, 221, 223, 232, 257
Phase-averaged stresses, 222-224 Phase-averaged vorticity transport, 222 Phahe averaging, 186, 213, 214, 222, 2 X X Phase-locked contribution, 255 Phase-locked subharmonic, 257 Phase o f the large-scale motions, 180 Phase relation between the stresses and the rate o f strain, 260 Phase transformation, 116 Physical frequency (fundamental 1, 252. 761 Plane fundamental, 260 Plane motions, 200 Plane shear layer, 192, 260 Plane subharmonic mode, 260 Plateau regions, 264, 265 Poisson's equation, 207 Practical streamwise region o f intere\t. 2 x 1 Pressure, 199, 206, 207. 242 Pressure field, 206, 207 Pressure fluctuations, 206 Pressure gradient, 209, 240, 241 Pressure terms. 78, 79 Pressure-velocity strain correlation, 2 3 2 Pressure work, 196-198 Production mechanism, 209, 218, 227, 229, 240
Propagating coherent modes, 212 Propagating wavy disturbance. 259 Propagation velocity, 88
Q Qualitative results from experiments. 2 1 2 Quantitatit e measurements, 258
377
Quantitative observations, 21 1-219, 285 Quasiperiodic forcing, 138, 143
R Random fluctuation production, 214 Random fluctuations, 185 Rapid distortion, 240 Rate o f energy exchange, 261 Rate o f energy transfer, 232. 238, 259 Rate o f evaporation, 116 Rate of spread, 217 of the shear flow, 259 Rate-of-strain field of the turbulence, 202 Rate-of-strain tensor, 200 Rate of viscous dissipation, .see Viscous dissipation rate Rates o f intensification, 204 Rates of strain. 204, 225, 240, 260 in phase with, 240 o f the fluctuations, 203 of the mean flow, 201 of the o d d modes, 201 of the fluctuations, 203 Rates-of-strain fluctuations, 204 Rayleigh equation, 224, 241-243, 247, 263 Rayleigh-Plesset Equation, 78, 79, 94, 95 Kayleigh problem, 242 Real physical mechanisms, 235 Reduction method, 138 Regions where the Fundamental is active, 259
Relative amplication, 283 Relative phase relations, 292 Relative phases. 240, 244, 245, 263, 264. 266, 295 of coherent modes, 294 Relative spanwise wave number, 295 Rescaled vertical variable, 261 Resonance elfect, 122 R e w n a n c e frequency, 20, 23, 30, 33-37. 49, 58, 93, 0 4
Resonant case, 137 Resonant frequency shift, 51. 5X Kesonant interactions, 298 Resonant triad, 297 Rebersihility, 138, 156 Reiersihle equations, 138 Rebersihle vectorfield, 145 Reynold mean motion, 224. 227 Reynolds average. 190-193, 195, 219-221, 224, 232. 239
378
Subject index
Reynolds-average shape function, 244 Reynolds-averaged diagnostics, 219 Reynolds-averaged quantities, 202 Reynolds-averaged stresses, 223 Reynolds averaging, 190, 192, 198, 208. 224-229
Reynolds fine-grained turbulence, 224 Reynolds mean, 205, 219 Reynolds mean flow, 192, 230 Reynolds mean flow quantities, 224 Reynolds mean motion, 288 Reynolds mean problem, 223 Reynold5 mean stresses, 189 Reynolds n u m b e r problems, relatively l o w , 219
Reynolds numbers, 222, 223, 239, 244, 253. 263, 268, 275, 300
Reynolds Reynolds Reynolds Reynolds Reynolds Reynolds
shear stress, 238, 244 splitting, 188, 199 stress closure, 189, 223 stress conversion mechanism, 264 stress equation, 195 stresses, 185, 186, 192, 195, 196,
207-21 1, 220, 238, 262, 300
Reynolds stress modeling, 189 Reynolds stress profile, 254 Reynolds stress production integral, 280 Role of initial conditions, 285 Roll-up, 258 Rotating drops, 2, 10 oscillations of. 30 shapes of, 12-20 Rotating spheroids, equilibrium shapes of, 1-58 Round jet, 186, 192, 193, 280, 284, 285, 288 Round turbulent jet, 258, 275, 295 Round turbulent jet problem, 275 Row of point vortices. 296
S Scale effects, of ship boundary layer and wake, 342-357 Scale of the fine-grained turbulence, 230 Scaling, 242 Scaling of modulated stresse\. 241 Second-order etfects, 286 Second-order theory, 286 Second subharmonic, 280, 281, 283, 284, 289
Secondary instabilities, 219 Sectional energy, 259, 262, 296 Self force, 84. 89. 119 Self-interaction, 21 1 Self-preservation, 224 Self-stretching etfects, 202, 205 Self-atretching mechanism, 203 Self-transport, of fine-grained turbulence energy, 198 Shape assumption, 236, 238, 262 S h a p e oscillations, 6-9, 20, 25, 30, 33, 55, 58 of liquid drops, 33 Shapes o f rotating drops, 12. 50, 53 Shear flow, 264 development of, 215, 226, 295 classical analyses of three-dimensional disturbances in, 2x5 temporal, parallel, 286 Shear flow evolution, 235 Shear Row instability, 252 Shear layer, 217. 218, 223, 230, 240, 245, 247, 248, 254, 256, 258, 260, 263-265, 268. 283
control, 282 growth, 248. 257, 264 growth rate, 248, 283 linear growth of, 218 spreading. 215, 266 spreading rate, 217, 218, 253 steplike de\elopment of, 218 t \I 0 -d i m e n s i o n a I, 2 80 viscous, 253 Shear layer thickness. 217, 229. 235, 236, 248. 253. 254, 263. 266. 275, 284
lirst plateau of, 217 second plateau in. 217 steplike growth of, 253, 268 steplike structure in, 275 Shear rate of strain, 240, 247 Shear \tress, 223, 247 production mechanism, 255 S h i p boundary layer a n d wake, measured results, 313-359 method o f calculation. 318-359 scale etfects, 342-359 Simulated wave amplitudes, 258 Single coherent mode, 253-258 Single coherent-mode problem, 265 Single-mode considerations. 254, 275
Subject index
379
Single-mode problem, 282 Singular points, 112, 113, 132 Slight flow divergence, 243 Slip flow, 105, 107, 108 Slippage, 97 Slowly varying wave envelope, 235, 252 Small disturbances, 264 Small divergence theory, 263 Small initial amplitudes, 275 Small mutual friction, 103-15 Small-scale processes, 299 Solitary waves, 135-137, 139, 143, 149, 152,
Spatially developing shear flows, 286-289 Spatially developing shear layer, 259, 260 Spatially developing turbulent free shear flows, 220, 251 Spatially developing turbulent shear layer,
153, 157, 159, 171 Sound pressure, 206 Sound speed, 92, 93, 120 Sound waves, 88, 92 in bubbly liquids, 65, 88-96 Source term, 117-119 Spacelike chaos. 139 Spanwise-averaged flow, 286 Spanwise averaging, 193, 289
Species of bubbles, 75, 76, I15 Specific heat, 85 Spectral content, 212 Spectral method, 298 Splitting procedure, 200, 206 Spreading rate. 218, 248, 256. 263-265, 268,
288, 289
Spanwise standing-wave disturbances, 297 Spanwise standing waves, 218, 282 Spanwise subharmonic formation, 285 Spanwise three-dimensional (combinetl) modes, 285 Spanwise wavelengths, 218, 285, 288 Spanwise wave number, 286, 287 selection mechanism, 285 Spatial distribution of wave envelopes, 259
284
Spatially developing free shear layer. 221 Spatially developing mixing layer. 221
296
Spatial periodicities, 192, 193 Spatial problems, 190, 192, 193, 212. 227, 235, 239, 244, 264, 288, 296
284, 293
Spanwise contribution to energy. 232 Spanwise mode number, 296 Spanwise-mode selection mechanism, 295 Spanwise modes, 285, 295 Spanwise-periodic harmonics, 286 Spanwise-periodic scales, 288 Spanwise-periodic three-dimensional modes, 280 Spanwise periodicities, 192, 219. 284, 2x5.
Spatial evolution equations, 293 Spatial-evolutionary properties, 282 Spatially developing flows, 234, 251 Spatially developing free laminar shear flow, 252 Spatially developing free shear flows, 25 I
254
Spatially occurring subharmonics, 190 Spatially periodic fundamental component,
~
o f the highly excited turbulent mixing layer, 254 viscous, 217, 264 Standing waves, 284, 289 Steady flows, 65, 108-115 Steady two-dimensional solutions, 296 Steplike behavior, thickness of, 215 of 6, 248 of 6 ( x ) , 253 Steplike shear layer thickness, 253, 275 Stokes drag. 80 Straining, 285 Stratified flow, 219 Stratified fluid, 299 Streakline calculations, 258 Streakline patterns, 258 Streaklines, visual appearance of, 258 Stream function, 221, 230, 239 Streamwise-averaged flow, 286 Streamwise development, 295 of the shear layer, 280 Streamwise distance, 235, 266 Streamwise envelope, 259 Streamwise nonperiodic modes, 286 Streamwise velocity fluctuation, 259 Streamwise wavelength, 219. 285 Streamwise wave number, 286, 296 Stress, 72 Stress gradient, 204 Stress tensor. 67, 206, 207
380
Subjecl index
Stretching, 200 see also Vorticity stretching, Vorticity tilting effects of, 201, 203, 206 Stretching mechanism, 203, 205 Stretching o f the mean, 201 Stretching o f the mean vorticity, 203 Stretching o f the o d d - m o d e vorticity, 201 Strongly amplified coherent modes, 263 Strongly amplified disturbances, 236, 264, 287 Strong turbulence levels, 268 Strouhal frequency, 257, 266 Strouhal number, 213 Subharmonic component, 190, 215 Subharmonic energy, 217 Subharmonic formation, 254, 285 Subharmonic frequency, 259, 288 Subharmonic frequency group, 295 Subharmonic-fundamental m o d e interactions, 264, 288 Subharmonic-mode energies, 295 Subharmonic-mode infrequency, 288 Subharmonic-mode transfer mechanism, 21 7 Subharmonic problem, 190 Subharmonics, 192, 221, 258-260, 262-264. 266, 280, 282. 283, 287-289, 295-298 three-dimensional perturbation, 297 Successive interactions, 280 Surface density, 67 Surface tension, 3, 7, 68, 75, 103, 104, 106, 110, 114, 137 coefficient of, 77, 78 Switch-off processes, 259 Switch-on processes, 259
T Taylor a n d Gortler vortex problem, 189 Temperature, 85, 86, 89, 117, 1 1 8, I20 Temporal homogeneous fluid problem, 299 Temporal mean motion, 286 Temporal mixing layer, 219-233, 236, 237, 251, 285, 296-298 Temporal problem, 192, 193, 212, 227, 235, 239, 286, 296 nonlinear, 212 Tertiary-frequency interactions, 288 Thermal diffusion length, 117, 118 Thermal diffusivity, 118, 123
Third subharmonic, 284 Third subharmonic forcing. 284 Three-dimensional disturbances, 286, 287, 297
classical nonlinear analyses of. 285 Three-dimensional ettects, 280 Three-dimensional linear perturbation, 297 Three-dimensional modes, 2 18, 292, 293, 296, 297 Three-dimensional motions, 199 Three-dimensional ( n = I ) subharmonic modes, 290 Three-dimensional nonlinear effects, 284298 Three-dimensional perturbation, 296 Three-dimen\ional phenomenon, 200 Three-dimensional spanwise mode, 295 Three-dimensional turbulent boundary layer a n d wake. integral method of, 321-341 Three-dimensional wave disturbance interactions , 2 88 Three-dimensional wave disturbances, 265, 2 6 , 289 Tilting effects, 199 Tilting tube experiment, 192, 212 Time development, 226 Time evolution, 229 Time-evolving two-dimensional flow, 297 Time scale for return to isotropy. 241 Tollmien-Schlichting wave. 185, 187, 189 Total coherent rate-of-strain field, 202 Total coherent vorticity, 202 Total fluctuation production mechanism, 214 Total rate of intensification, 205 Total stresses, 220 Traditional shear layers, 253 Trailing edge. 285 Transfer mechanism, 264 Transfer of energy from the mean flow to the fundamental, 217 Transition, 185, 215, 218, 258, 264, 265 problem, 265 Transitional shear flow, 217 Transitional shear layers, 284 Translative mode, 296, 297 Translative m o d e interaction, 297 Transport, 199 Transport effects, 202 Transport equations, 222, 239, 299
38 1
Subject index Transport of even-mode energy, I98 Transport of fine-grained turbulence energy, I98 Transport of odd-mode energy. 197 Transport of vorticity, 200, 206 Transverse homoclinic point, 139, 159 Triple correlations, 209, 210. 240 Triple-frequency mode interactions, 7x8 Turbulence. 217, 2 2 3 , 248-252, 263, 29.5 Turbulance energy, 218, 248-251, 2hX advection, 254 density, 238, 241, 266 level, 26X production integral, 244 vertical part of, 2 3 2 Turbulence equilibrium amplitude ratio. 249 Turbulence level, 257, 268 Turbulence-modilied internal wave problem, 299 Turbulence production mechanism, I98 Turbulent boundary, 186 Turbulent boundary layers, 186, 191. 2 2 2 , 298 Turbulent channel flow, 186, 189 Turbulent contributions, 207 Turbulent dissipation, 227 Turbulent energy density, 262 Turbulent Ilows, 252 Turbulent fluctuations, 194, 203, 205. 206. 209 Turbulent free shear flows, 187, 241, 251 Turbulent free shear layer, 252 Turbulent jet, 213 Turbulent jet experiments, 229 Turbulent kinetic energy, 205, 227, 237, 238 Turbulent mixing layer, 220, 243, 254, 300 two-dimensional, 265 Turbulent problem, 260 Turbulent rates of strain, 205 modulated Huctuations of, 205 Turbulent Reynolds number, 241 Turbulent shear flow problem, 243, 257 Turbulent shear flows, 184-190, 199, 212, 215, 217 Turbulent shear layers, 219, 222, 237, 7 5 2 . 253, 254 Turbulent spreading rates. 217 Turbulent vorticity, 200, 202 Turbulent vorticity stretching, 202 stretching, 202
7-urhulent wake, 184 2-P-phase average, 191 Two-dimensional disturbances, 286 Two-dimensional flow, 193 Two-dimensional fundamental, 286, 297 Two-dimensional mean flow, 203, 206. 256 Two-dimensional modes, 285, 288, 293 Two-dimensional mo-modes, 297 Two-dimensional ( n = I ) subharmonic modes, 290 Two-dimensional problem for a homogeneous fluid, 219 Two-dimensional shear flows, 284, 288 coherent modes in, 284 Two-mode interaction prohlem, 281, 297 Two-phase flow, 64 Two-phase fluids, 65-67. 74
U Unstable critical point, 113 Upstream dependence, 218 Upstream forcing, 187 Upstream initial conditions, 285 Upstream perturbations, 285
V Vapor bubbles, 65, 75, 115-124 Vapor pressure, 116, 188 Velocity fluctuations, 206 Velocity of propagation, 92 Velocity transport etIects, 201 Virtual mass, 81, YO, 92, 93, 105, 107 Virtual force, 82-84 Viscosity coefficient, 78, 81, 84 Viscosity effects, 222 Viscous damping, 94, 97 Viscous ditfusion, 195, 196, 202, 203, 205, 222, 239, 241 Viscous dissipation, 196, 205. 210, 217, 2 2 S , 240, 241, 254, 260, 264, 266, 268, 275 Viscous dissipation integrals, 244, 262, 301 Viscous dissipation rate, 196, 198, 203, 223, 230, 240, 241 Viscous effects, 209, 240 Viscous instabilities, 189 Viscous shear layer, 253 Viscous spreading rate, 217, 264
382
Subject index
Vortex, longitudinal vortex in ship wake, 311-359 Vorticity. 199-206, 217. 221. 230, 29X concentration parameter, 296 magnitude of, 202 Vorticity considerations, 199
Vorticity energy exchange mechanism, 205 Vorticity equations, 199, 200. 221 Vorticity exchange mechanisms, 200, 205 Vorticity exchanges, among the ditferent scales of motion. 200 Vorticity flux, 204 Vorticity gradient, 204 Vorticity nonuniformities, 230 Vorticity problem, two-dimensional, 222 Vorticity shorthand notation, 199 Vorticity stretching, 199, 200, 201, 204, 205. 222
b e e ulso Stretching) d u e to self-straining etfects, 201 Vorticity tilting, 200, 201, 222 ( s e e U / W J Stretching) Vorticity transport, 200
W Wake, of ships, we Ships boundary layer a n d wake Wake problem, 253 Wall-bounded shear flow problem, 241 Wall-bounded turbulent shear flows, 187 Water channels, 190, 212 Water waves, nonlinear. 135 Wave amplitude, 243 Wave characteristics, 235
Wave disturbances, 293, 295 two-dimensional, 286, 288 Wave-envelope development, 248 Wave-envelope equations, 281, 287, 293 Wave-envelope evolution, 238. 244. 254, 256, 227, 259
Wave-envelope peaks, 257 Wave-envelope problem, 248, 252, 2 5 8 , 260, 263
Wave enbelopes, 190, 192, 212, 215, 217, 235-237.243.248-252.
262, 280
Wa\e-envelope three-dimensional disturbances, 285 Wave function, 235 Wave interference. 258 Wavelength, 192, 223, 224, 259, 295 Wavelike representations. 187 Wavelike structures, 187 WaLe-modulated square of the turbulence temperature fluctuation, 299 Wave-modulated stresses, 240 Wabe-modulated turbulence heat tlux \ector, 299 Wave number, 227, 235. 245, 249, 300 WaLe-number selection mechanism, 285 Wave packet problem, 239 Waves, 186 in bubbly liquids, 96-108 weakly nonlinear, 102 Waves and turbulence, separation of, 299 Wave-turbulence energy transfer mechanism. 256 Wave-turbulence interaction. 298-300 Wavy disturbances. 258 Wavy wall. 186 Weakly nonlinear theory, 287 Wind tunnels, 190, 212
