Advances in Applied Mechanics, Volume 26

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 ...

Author: John W. Hutchinson | Theodore Y. Wu

38 downloads 788 Views 14MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

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

A C A D E M I C PRESS, INC. Harcourt Brace Jovanovich, Publishers

Boston San Diego New York Berkeley London Sydney Tokyo Toronto

C O P Y R I G H0 T 1988

H V ACAI>I-MI( PRESS. I N ( ALI. R I G H T S R t S t R V E I > N O [’.ART O F T t I I S P I J I I L I C A T I O N M A Y B E K f S P K O I > l ~ (L I I O K T R A N S M I T T E D I N A N Y F O R M O R B Y 4 N Y hlLANS. I I I-(‘TKONIC O K M C C ‘ H A N I C ’ A L , I N C L C I I I I N C i I’HOTOC~OI’Y, RFC O R I > I N < i . O R A N Y I N ~ O K M A T I O N STORAGE. ANII K t i - i K i r : v . ~ i s \ s r m h i . M’I I . H O I ~ I I’kRMISSION I N W R I T I N G F R O M 7 H t I’IJBLISIII R

A C A D E M I C PRESS, INC. 1250 Siuth Avenue. San Diego. (’A Y 2 1 0 1

United Kingdom Edition published by A C A D E M I C PRESS I N C . ( L O N D O N ) LTD 24-28 Oval Road. London N W I 7 D X

ISBN 0-12-002026-2 I’KINTE13 I N T H I

88899091

U N I T € II \ T A T 1 \ O F A M E R I C A

9 8 7 6 5 4 3 2 1

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, there are two critical points (see Appendix D) for

I

u,

u

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
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


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


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)


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


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


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)


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


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.


133

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.

,


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 ) ’


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

0w 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} 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


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


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 )

152

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.


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)

154

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)


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.

156

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.”


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 ( 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
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'


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.


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


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.

180

Klaus Kirchgassner

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.

Nodinearl-y Resonant Surface Waves

181

Kirchgassner, K., and Scheurle, J. (1981). Bifurcation of non-periodic solutions of some semilinear equations in unbounded domains. I n “Applications of nonlinear analysis in the physical sciences” (Amann, Bazley, Kirchgassner, eds.), pp. 41-59. Pitman. MacKay, R. S., and Saffman, P. G. (1985). Stability of water waves. Manuscript. Miles, J. W. (1977). On Hamilton’s principle for surface waves. J. Fluid Mech. 83, 153-158. Mielke, A. (1986a). A reduction principle for nonautonomous systems in infinite-dimensional spaces. J. Difl Eqn. 65, 68-88. Mielke, A. (1986h). Steady flows of inviscid fluids under localized perturbations. J. Difl Eqn. 65, 89-116. Mielke, A. (1987a). Reduction of quasilinear elliptic equations in cylindrical domains with applications. To appear in Math. Merh. Appl. Sci. Mielke, A. ( 1987h). Uher maximale L‘’-Regularitat fur Differentialgleichungen in Banachund Hilbert-Raumen. T o appear in Math. Annalen. Mielke, A. ( 1 9 8 7 ~ ) .Homokline und heterokline Losungen hei Zwei-Phasen-Stromungen. Manuscript. Moser, J. (1973). “Stable and random motions in dynamical systems.” Princeton Univ. Press, Princeton, N.J. Newhouse, S. E. (1980). Lecture on dynamical systems. In “Dynamical Systems” (J. Guckenheimer, J. Moser, S. E. Newhouse, eds.), pp. 1-1 14. Birkhauser, Boston/Basel. Pliss, V. A. (1964). A reduction principle in the theory of stability of motion. Izv. Akad. Nauk SSSR Ser. Mat. 28, 1297-1324. Pumir, A., Manneville, P., and Pomeau, Y. ( 1983). On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27-50. Scheurle, J. (1986). Chaotic solutions of systems with almost periodic forcing. Manuscript. Scheurle, J. (1987). Bifurcation of quasi-periodic solutions from equilibrium points of reversible dynamical systems. Arch. Rat. Mech. Anal. 97, 103-139. Sijhrand, J. (1981). Studies in non-linear stability and bifurcation theory. Proefschrift, Univ. Utrecht. Ter-Krikorov, A. M. (1963). ThBorie exacte des ondes longues stationnaires dans un liquide h6tBrogtne. J. de Micanique 2, 351-376. Stoker, J. J. (1957). “Water Waves.” Interscience Puhl., New York. Turner, R. E. L. (1981). Internal waves in fluids with rapidly varying density. Ann. Scuola Norm. Sup. Pisa, Ser. IV, Vol. 8, 513-573. Whitham, G. B. (1974). “Linear and nonlinear waves.” J. Wiley, New York. Wu, D.-M., and Wu, T. Y. (1982). I n “Proc. 14the Symp. o n Naval Hydrodyn.”, Ann Arbor. Yuen, H. C., and Lake, B. M. (1984). Nonlinear dynamics of deep-water gravity waves. Advances in Nonlinear Mechanics 22, 68-229. Zakharov, V. E. (1968). Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190-194. Zeidler, E. (1971a). “Beitrage zur Theorie und Praxis freier Randwertaufgahen.” AkademieVerlag, Berlin. Zeidler, E. (1971h). Existenzbeweis fur cnoidal waves unter Berucksichtigung der Oberflachenspannung. Arch. Rat. Mech. Anal. 41, 81-107. Zeidler, E. (1973). Existenzheweis fur Kapillar-Schwerewellen mit allgemeinen Wirhelverteilungen. Arch. Rat. Mech. Anal. 50, 34-72. Zeidler, E. (1977). Bifurcation theory and permanent waves. I n “Applications of Bifurcation Theory” (P.. Rahinowitz, ed.), pp. 203-223. Academic Press. Zufiria, J . A. (1986). Nonsymmetric gravity waves on water of finite depth. Manuscript.


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

184

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

Understanding Large-Scale Coherent Structures

185

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).

186

J. T. C. Liu

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


187

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-


189

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

190

J. 1 C. Liu

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


191

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)

J. T. C. Liu

192

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


193

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)

J. T. C. Liu

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


195

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.

196

J. T. C. Liu

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


197

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.

J. T. C. Liu

198

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.


199

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

J. T. C. Liu

200

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.


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 ~

~

J. T. C. Liu

202

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


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)

J. T. C. Liu

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 &, - 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


"production" from mean

21 1

work done by mean 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


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).

J. T. C. Liu

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).


215

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”).


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). ~

218

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


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

220

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


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


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,,

224

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


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 ( 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,,

J. T. C. Liu

250

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.


r

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

J. T. C. Liu

252

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


253

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

254

J. T. C. Liu

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),


255

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).

J. T. C. Liu

I

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).


257

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

258

J. T. C. Liu

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


259

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

J. T. C. Liu

260

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/


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

J. T. C. Liu

262

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


263

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

264

J. T. C. L i u

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


265

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 ~

266

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.


‘-1

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.

268

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


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


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

Understanding Large-Scale Coherent Structures 10 lo

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


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.


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


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.


277

x lo-*

"1


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


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


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


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


285

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,


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


289

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

J. T. C. Liu

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.


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.

J. T. C. Liu

292

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.


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

J. T. C. Liu

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


295

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.

296

J. T. C. Liu

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


297

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.


299

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


301

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, 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.


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


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.


359

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.


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

Advances in Applied Mechanics, Volume 26

Advances in Applied Mechanics, Volume 26

Advances in Applied Microbiology, Volume 26