Advances in Applied Mechanics, Volume 32

Advances in Applied Mechanics Volume 32

T T. BROOKEBENJAMIN DEPARTMENT OF MATHEMATICS OXFORD UNIVERSITY OXFORD, UNITEDKINGDOM Y. C. FUNG AMES DEPARTMENT OF CALIFORNIA, SANDIEGO UNIVERSITY LA JOLLA,CALIFORNIA PAULGERMAIN DES SCIENCES ACADEMIE PARIS,FRANCE RODNEYHILL DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS OF CAMBRIDGE UNIVERSIIY CAMBRIDGE, UNITEDKINGDOM PROFESSOR L. HOWARTH SCHOOL OF MATHEMATICS UNIVERSITY OF BRlSTOL BRISTOL, UNITEDKINGDOM C . 3 . YIH(Editor, 1971-1982)

Contributors to Volume 32 JEAN-LOUIS AURIAULT HSUEH-CHIA CHANG EVGENY A. DEMEKHIN NORDENE. HUANG R. LONG STEVEN CHIANG C. MEI CHIU-ON NG ZHENGSHEN J. M. Wu J. Z. Wu

ADVANCES IN

APPLIED MECHANICS Edited by John W I Hutchinson

Theodore YI Wu

DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS

DIVISION OF ENGINEERING AND APPLIED SCIENCE CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA

VOLUME 32

W ACADEMIC PRESS, INC. Boston San Diego New York London Sydney Tokyo Toronto

This book i s printed on acid-free paper.

@

Copyright C 1996 by ACADEMIC PRESS, INC All Rights Reserved. N o part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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International Standard Serial Number: 0065-2 165 International Standard Book Number: 0- 12-002032-7 PRINTED IN THE UNITED STATES OF AMERICA 96 97 9 8 9 9 00 0 l Q W 9 8 7 6 5

4

3 2 1

Contents vii

CONTRIBCJTORS

ix

PREFACE

Solitary Wave Formation and Dynamics on Falling Films Hsueh-Chia Chang and Eugeriy A. Demekhin 1. 11. 111. IV. V.

1 5

Introduction Model Equations Evolution toward Solitary Waves Solitary Waves Discussion and Future Work Acknowledgments References

13 27 53 55 56

The Mechanism for Frequency Downshift in Nonlinear Wave Evolution Norden E. Huang, Steven R. Long, and Zheng Shen Abstract I. Introduction 11. The Hilbert Transform: The Mcthodology 111. The Laboratory Experiment IV. The Field Experiment V. Discussions VI. Conclusions Acknowledgments References

60 60 65 75

98 111 114 115 115

Vorticity Dynamics on Boundaries J. Z. Wu and J. M. Wu I. 11. 111. IV. V. VI.

Introduction Splitting and Coupling of Fundamental Dynamic Processes General Theory of Vorticity Creation at Boundaries Vorticity Creation from a Solid Wall and Its Control Vorticity Creation from an Interface Total Force and Moment Acted on Closed Boundaries by Created Vorticity Fields V

120 127 148

168 198

224

’

Contents

vi

VII. Application to Vorticity Bascd Numerical Method? VIII. Concluding Remarks Acknowledgments References

247 264 267 267

Some Applications of the Homogenization Theory Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng 1. 11. 111. IV. V.

Introduction One-Dimensional Examples Seepage Flow in Rigid Porous Media Diffusion and Dispersion Other Applications Acknowledgments References

278 279 287 309 343 345 345

AUTHOR INDEX

349

SUBJECT INDEX

355

List of Contributors

Numbers in parentheses indicate the pages on which the authors' contributions begin.

JEAN-LOUISAURIAULT (277), Institut de Mecanique de Grenoble, 38041 Grenoble, France HSUEH-CHIA CHANG(11, Dcpartment of Chemical Engineering, University of Notre Dame, South Bend, Indiana 46556

EVGENY A. DEMEKHIN (1), Department of Applied Mathematics, Krasnodar Polytechnical Institute, Krasnodar, The Russian Republic NORDENE. HUANG(59), Ocean and Ice Branch, Laboratory for Hydrospheric Processes, NASA Goddard Space Flight Center, Greenbelt, Maryland 20771

STEVENR. LONG (59), Observational Science Branch, Laboratory for Hydrospheric Processes, NASA Goddard Space Flight Center/Wallops Flight Facility, Wallops Island, Virginia 23337 CHIANGC. MEI (277), Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 CHIU-ONNG (277), Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

ZHENG SHEN(59), Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland 21218

J. M. WU (119), The University of Tennessee Space Institute, Tullahoma, I) Tennessee 37388 J. Z. Wu (119), The University of Tennessee Space Institute, Tullahoma, Tennessee 37388

vii

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Preface

This volume of the Adilances in Applied Mechanics presents four treatises on subjects of fundamental importance and timely interest. Some of these subject areas appear to be in their budding stage in conception and formulation, yet are already receiving sound empirical and experimental supports and may soon be followed by dynamic development and applications. The chapter by Hsueh-Chia Chang and Evgeny A. Demekhin explores the phenomenon of solitary wave formation on a falling film, a highly dissipative system to which the inverse scattering transform does not apply. A new theory is introduced in light of recent experiments: that the chaotic dynamics of some spatio-temporal patterns can be driven by a set of coherent local structures, an interaction process which primarily involves only the nearest spectral neighbors, possibly with inelastic coalescence. It has promising applications to analogous systems. The chapter by Norden E. Huang, Steven R. Long, and Zheng Shen presents both a new discovery of great significance and an invention of a new tool, without which the discovery could not have been so definitively established. The discovery reveals that the frequency downshift in wind-wave and wave-wave interactions takes place in a narrow band wave train through wave fusion, with two neighboring waves coalescing into one. The new tool the authors developed is based on applying a nest of the Hilbert transforms through a sophisticated process that depends only on the intrinsic nature of the data itself being analyzed. The mechanism of wave fusion is shedding a completely new light on the frequency downshift phenomena in nonlinear wave evolution in general. Concerned with a central problem of vortex dynamics, J. Z. Wu and J. M. Wu address the topic of interaction between vortices and boundaries, whether a rigid or a deformable solid wall or an interface of two different fluids. This chapter elucidates the mechanism by which vorticity is generated near a boundary of a viscous compressible flow and subsequently transported away from it. This is a subject of increasing importance as fluid mechanics experiences applications with boundaries of all varieties. ix

X

Preface

The chapter by Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng reviews the theory of homogenization which has received active recent development for applications to poro-elastic media and flows involving multiple phases. For a great variety of heterogeneous bodies, the macroscopic constitutive properties and material coefficients can be calculated by using the mathematical technique of homogenization on multiple scales, which is especially effective for applications. It is hoped that the important new discoveries, improved understanding of interesting phenomena, and introduction of new methods brought forth by these authors in their fine scholarly work will serve as valuable sources of information, tools, and stimuli for making further advances in applied mechanics. Theodore Y. Wu and John W. Hutchinson

ADVANCES I N APPIJED MECHANICS. VOLUME 32

Solitary Wave Formation and Dynamics on Falling Films HSUEH-CHIA CHANG Depurimenl of Chemical Engineering U n i i w - s i ~of~ Notre Dame South Bend, Indiunu

and

EVGENY A. DEMEKHIN Depurinieril of Applied Muthemufic.c fiustiodur Polyrechnicul Institute Krusnodur, The Russian Republic

..................................... 11. M o d e l E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Evolution toward Solitary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . A. Linear Theory at Wave Inception . . . . . . . . . . . . . . . . . . . . . . .

13 13

B. Saturation, Subharmonic Secondary Instability, and Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

IV. SolitaryWaves.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Existence, Estimate, and Construction . . . . . . . . . . . . . . . . . . . . B. Symmetries and Coherent Structure Theory. . . . . . . . . . . . . . . . . C. Coalescence, Transition State, and Dynamics . . . . . . . . . . . . . . . .

27 27 32 42

............................ Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

1. Introduction..

V. Discussion and Future Work.

1

5

55 56

I. Introduction

This is an extension of the recent review by one of us (Chang, 1994) on falling-film wave dynamics. An extension in such a short time is justified in light of the recent experiments, mostly by Gollub’s group at Haverford 1 Copyright C) 1496 hy Acddcmic Press Inc All rightq ot reproduclion In any form re5ervzd ISBN (1 12 002012 7

2

Hsueh-Chia Chang and Ergeny A. Demekhin

College, on the formation and dynamics of solitary waves and our recent discovery that one of the most intriguing concepts in pattern formation can be applied, with appropriate modification, to analyze solitary wave dynamics on a falling film. This is the theory that the chaotic dynamics of some spatio-temporal patterns is driven by long-range interaction among a large number of very localized patterns called coherent structures (Coullet and Elphick, 1987; Elphick et al., 1989, 1990). A recent review on the general subject is offered by Balmforth (1995). As such, the “turbulent” dynamics of an extended system can be captured by a finite-dimensional dynamical system describing the interaction of these coherent structures. Typically, only nearest neighbor interaction needs to be considered, and hence the dynamics is akin to lattice gas dynamics. In the discussion of the previous review, it was speculated that such a theory is appropriate for solitary wave dynamics on a falling film. This has now been verified and we review the progress in this new direction thus far. In its most ambitious form, coherent structure theory is touted as a possible connection between real turbulence and deterministic chaos of dynamical systems theory (Moffat, 1989). Some experimental support to this claim has surfaced most recently. The delayed embedding analysis of a time series from the velocity measurement at a single probe shows that the dimension of the reconstructed attractor is rather small for boundary layer turbulence. This suggests that the dynamics near the probe is dominated by only a few localized structures. These structures are thought to be vortex filaments and some attempts to capture their shapes have been carried out by using the Karhunen-Loeve technique (Sirovich, 1987; Aubry et al., 1988). Kachanov’s group has indeed detected distinctly localized A-shaped filaments near the wall and solitary vortices further out in the transition region of a boundary layer (Kachanov, 1994). Localized structures are also observed during chaotic spatio-temporal dynamics in binary convection (Anderson and Behringer, 1990), liquid crystals (Joets and Ribotta, 19881, solidification (Coullet et al., 19891, nerve axons (Evans, 1975), and reaction-diffusion systems (Rubinstein et al., 1993). In some of these examples, they appear as spirals and shocks (kinks) as well as the more common pulses. However, the most dramatic and best studied localized structures are soliton interfacial waves. They include the historical KdV soliton observed by Russell in a Cambridge canal (Drazin, 1983; Wu, 1987) and solitary waves of the nonlinear Schrodinger equation for deep water waves or in Faraday instability (Wu et al., 1984). For such integrable systems,

Solitaly Wave Formation und Dynamics on Fulling Films

3

the dynamics on a real line can be deciphered analytically by the elegant inverse scattering transform (Kruskal, 197.5; Ablowitz and Segur, 1981). Qian et al. (1989) have recently demonstrated with the integrable Benjamin-Ono equation that the pole dynamics from even a four-pole expansion is indeed chaotic without any noise forcing. For highly dissipative thin films, the inverse scattering transform does not apply. Nevertheless, thin-film coherent structures do exist and they also dominate thin-film interfacial dynamics. In sheared horizontal films, for example, defects corresponding to spiral solutions of the complex Ginzburg-Landau equation (Sangalli, 199.5) appear to trigger the onset of transverse instability of two-dimensional traveling waves. However, the most dramatic solitary interfacial waves exist on a thin film falling down a plane under the force of gravity. The formation and dynamics of these solitary waves (also known as pulses or humps), which are evident in our numerical simulation shown in Figure 1, is the subject of this review. Unlike the KdV solitons, these highly dissipative solitary wavcs cannot be studied with the inverse scattering transform or a perturbed version of it. The description of its dynamics requires a very special inelastic version of the coherent structure theory that includes the possibility of coalescence. As seen in Figure 1, random forcing at the inlet triggers wave formation in an inception region within the first 50 units of film thickness downstream. Small time-periodic waves of a specific frequency are chosen from the broadband noise and this harmonic grows exponentially downstream. A blowup is shown in Figure 2 with periodic forcing at the inlet. The exponential growth is arrested around 50 film thickness units downstream, and the amplitude of the wavcs reaches a constant value. At around 100 0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400 26 24 22 20 18 16 14 12 10 08

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

26 24 22 20 18 16 14 12

10 08

FIG. 1. A s n a p h o t of the spatially cvolving waves due to random inlet forcing on a vertical falling film. The spatial unit is the Nu\\elt Rat film thickness h,, which IS about 1 mm. A ncar coalescence is seen at 380 units and the next pulse downstream is one that has just undergone coalescence.

4

Hsueh-Chia Chang and Evgeny A. Demekhin

FIG.2. A snapshot of the spatially evolving waves due to periodic forcing with a forcing period of 2 units. The waves tracing in time at locations x = 300, 530, 600, and 700 are also shown. The space units here are 0.2hN.

units in Figure 2, dramatic subharmonic and sideband (with zero-mode excitation) instabilities develop to drastically alter the shape of the waves. These secondary instabilities are far more rapid than the primary one, and the resulting wave amplitude is roughly twice the previous saturated value, as every crest has absorbed its neighbor on the average. The saturation

Solitary Wave Formation and Dynamics on Falling Films

5

and secondary instability also occurs at around 50 to 100 film thickness units in Figure 1. The exact locations of these transitions are sensitive to the nature of forcing; namely, its amplitude and frequency content. As seen in Figure 2, all the way to the onset of the secondary instability, the wave amplitudes are small (less than 30% of the flat film) and waves contain only a few discrete temporal harmonics as is evident from their sinusoidal shapes. This changes dramatically after the secondary instability. The subharmonic and sideband modes seem to trigger an explosion of many harmonics, and most surprisingly, these harmonics synchronize and form extremely large localized structures at about 150 units in Figure 1. Due to the narrow width (about 10 film thickness units) of these pulses, humps, or solitary waves, the harmonics of its wide frequency content cancel each other except in the immediate neighborhood of the pulse, where they enforce each other. More interestingly, these pulses retain roughly the same amplitude downstream but their time-averaged separation, which is on the order of 50 film thickness units, increases downstream as shown in Figure 3. The origin of this increase in separation is a net consumption of solitary waves by coalescence, as we shall expound in subsequent sections. Two pulses that have nearly coalesced are seen at 400 units in Figure 1, and a large pulse that was formed after the coalescence of two pulses is seen immediately downstream. Coalescence occurs more often near the inlet as is evident from the “world lines” in Figure 3 but a rare one at 1100 units is shown in Figure 1. We shall analyze the transition toward these solitary waves and the dynamics of the interacting solitary waves.

11. Model Equations

The analysis of falling film dynamics is typically carried out on a hierarchy of model equations for a film on an inclined plane at an angle 8 from the horizontal. Using h,, the Nusselt thickness for the flat film basic state as the characteristic length, the average velocity ( u ) = gh; sin 0/3 v as the characteristic velocity, the dimensionless Navier-Stokes equations of motion become dU

-

dt

+ u .vu =

-vp

+ -1v 2 u + -g3 R

R

(2.1)

Hsueh-Chia Chang and ELgeny A. Demekhin

6

6.5 -

-

7.0

5.5 6.0

4.5 5.0

-

4.0 3.5

-

2.5 2.0 1.5 3.0

-

1.00.5

0

l ' " l ' l ' l ' l ' l ' l ' l ~ l ' l ' l ' 20 40 60 80 100 120 140 160 180 200 220 240

0

200

400

600

800

1000

FIG.3. The average separation between pulses as a function of 7osition for the numerical experiment in Figure 1 . The world lines tracking the peaks are plol ed in the space (x)-time ( y ) coordinate. The space units are h,, = 1 mm.

where u = ( u , u , w ) is the velocity field, R = ( u ) h , / v is the Reynolds number, which measures the inertial force that drives the instability, and g = (1, -cot 0,O) is the gravitational acceleration. The coordinate system is chosen such that the upper normal of the inclined plane is y , x is along the downward tangent, and z is in the transverse direction. Let the

7

Solitaly Wave Formation and Dynamics on Falling Films free-surface position (x, t ) , where x nates, be defined by f(X,

t)=y

=

-

( x , y , z ) are the Cartesian coordi-

h ( x , 2, t )

=

0

(2.3)

and using zero as the pressure in the inert gas phase, one obtains the kinematic and stress conditions at the interface y = h: df

-

dt

+ u.Vf = 0

[ p - W K ( h ) ] n ,- t,,n,

=

0

(2.4)

i,j

=

(2.5)

1,2,3

where K is the interfacial curvature, t,, is the shear tensor l/R(du,/ dx, du,/dx,), and W = a/p(u>’h,,, is the Weber number. The unit vector n is normal to the free surface. There is also a no-slip condition at the wall

+

u=o

at

y=o

(2.6)

These equations represent the most complete model for falling film dynamics. Three independent parameters specify the problem-the Reynolds number R, the Weber number W , and the inclination angle 19. Because both R and W are dependent on the average velocity or the flow rate (u)h,,,, it is more convenient sometimes to replace W with the Kapitza number y = WR4/(3sin O ) + , so that when the flow is varied as a control parameter, y and 6, remain fixed. The Navier-Stokes equation is too complex to allow any analysis and a numerical study is required. We have developed the necessary numerical codes to tackle the most complex, three-dimensional, spatially evolving wave fields of this equation. (A preliminary description of the numerical approach is offered in Chang et al., 1993~).However, under conditions of high surface tension, more precisely, when the film parameter K , where K~ = WR/3 = a/pgh’, sin H is a large number, we (Chang et af., 1993a) have been able to show that the Navier-Stokes (NS) equation can be reduced to the boundary layer (BL) equation which is valid to O ( K ) : dU

-

dt

+ U -d U + U - d U + dx dy

]

7 +1 +3 ad Y2 u

-

dh 3xdx (2.7)

dW

-

dt

+ U -dw + U -dw + dX dZ

+73 3 dy -

dh x

(2.8) ~

8

Hsueh-Chia Chang and Evgeny A. Demekhin

where stretching in x , z , t , and u have been carried out, L‘ + I ~ / K and ( x , z , t ) + K ( X , z , t ) . The major simplification is that the long-wave expansion that is used to simplify the NS equation has explicitly related the pressure to the leading order interfacial curvature V 2 h , where V 2 h = d 2 / d x 2 + d 2 / d z 2 , and hence eliminates p from the equations. As such, the normal velocity u can be computed from the continuity equation (2.9) instead of the y-momentum equation, and the difficult interfacial conditions are simplified considerably. The NS equation has also been “parabolized” by the elimination of the ( d 2 / d x 2 ) uterms, which would require an elaborate shooting method in the extended downstream direction x if periodic boundary conditions were not used as in Figure 1. Equally important is the realization that when WR is large, the three parameters R, W , and 8 are reduced to two parameters:

As a result, the verticle film problem is parameterized by the normalized Reynolds number 6 only. This greatly simplifies the numerical analysis. Even when the full NS equation is used, we find the results for most fluids with R < 500 can be parameterized by 6 and y, only. We shall focus mainly on the case of a vertical film with x = 0. More drastic simplifications can be made for very thin films whose inertia as measured by 6 is small. The velocity field u then behaves quasi-statically in response to the film position h ( x , z , t ) , and it can be solved explicitly as a function of h . i t is instructive to carry out this low-inertia, long-wave expansion explicitly to decipher the physical origin of the instability. We shall focus on the two-dimensional problem without z dependence for a vertical film ( x = 0). For 6 small, the x-component of the velocity can be expanded in powers of 6,

such that one obtains the locally parabolic lubrication flow field that

Solitary Wave Formation and Dynamics on Falling Films

9

behaves quasi-statically to the dynamics of h ( x , t ) . Here, ug

=

3[1 + h,,,I(hy

-y2/2>

(2.12)

and the leading order flow rate is simply (2.13) It is clear from (2.12) and (2.13) that, without the capillary curvature term h,,,, a higher local film thickness yields a higher interfacial velocity u 0 ( y = h ) and a higher flow rate q,,. This is because a thicker interface feels the wall drag less and hence is driven at a higher velocity by gravity. By simple mass conservation arguments, one then realizes that if the local interface gradient is negative ( h , < O), the interface will rise whereas one with a positive local gradient will drop. The maxima and minima, in contrast, will translate in the x direction without amplitude modification. If the interface is a pure harmonic, the capillary term h,,, vanishes at the maxima and minima and the preceding observation still holds in the presence of capillary effect. These observations can be made more precise by invoking Liebnitz rule on the kinematic condition to yield

Upon linearizing about the flat film Nusselt solution, which is unity due to the scaling, h - 1 = q, one obtains

(2.15) The rise and fall of q in response to the local gradient due to the kinematic effect is seen in the first term. The usual normal mode expansion exp(ht + iax) then reveals the phase speed of a pure harmonic is exactly three times the average velocity due to the coupling between kinematics and the higher velocity of a thicker film. The factor of 3 in these kinematic waves arises because q scales as the cube power of h without capillary effect, as seen in (2.13). The second capillary term is seen to stabilize all waves but damps the shorter waves more than the longer waves due to their higher curvature. The stabilizing mechanism is purely capillary-the pressure below a crest is higher than the pressure underneath a trough of an interface. This capillary pressure gradient drives a flow from the crests to the troughs and hence has a stabilizing effect.

10

Hsueh-Chia Chang and Eqeny A. Demekhin

To obtain instability, one must then include the O(6) inertia terms and the next order equation is

n2u, dY2

=

56 - + U g -

duo dX

+ I’,)--

dY

1

(2.16)

where u , ( x , y , t ) is given by the continuity equation and u g can be obtained from it:

We note that the terms within the square bracket in (2.16) resemble a pressure gradient. For example, if we limit ourselves to long waves such that the capillary terms in (2.14) and (2.15) are small,

and the inertial pressure that results in (2.16) is -456~71. Unlike the capillary pressure, it is negative below a crest (7 > 0) and positive underneath a trough (71 < 0) and it is related to the zeroth derivative of 71 instead of the second derivative. It hence drives a flow from the trough to the crest and is destabilizing. The physical origin of this instability is clear from (2.16). Due to inertia, the velocity does not accelerate instantaneously in response to perturbations at the interface according to the quasi-static flow field of (2.12). Instead, the velocity at a particular location feels the effect of a interfacial perturbation in front of it due to this delay. At a crest, for example, u , “sees” the negative interfacial gradient ( h , < 0) in front, and hence part of the kinematic growth mechanism at the point with negative gradient is transferred to the crest. This delay is independent of curvature and hence related to the zeroth derivative of 77-long waves are destabilized more than the short ones. If all the inertial terms in (2.16) are included, one gets the inertial flow rate

11

Solita y Wave Formution und Dynamics on Falling Films and the extension of (2.14) yields

In the original coordinates of the NS equation and for the full threedimensional problem, this equation becomes the Benney's equation: dh

RWh'

a =

dt

0

(2.17) If one further assumes 7 = h - 1 is small, the weakly nonlinear Kuramoto-Sivashinsley (KS) equation results for the two-dimensional case:

H,

+ 4HHx + H,, + Hxxxx = 0

(2.18)

where t = 7(2SW/48R), 7 = 4(8R/25W)(SW/12)'/'H, x - 3t = (SW/12)'/2X, and ( x , t ) are the coordinates of the NS equation. We note that the factor 2/3 in q / ( x - 3 t ) = ;HJX corresponds to the fact that characteristic velocity used in the KS equation is the interfacial velocity of the flat film, which is two-thirds the average velocity ( u ) used in the earlier scalings. Because inertia destabilizes long waves and capillarity stabilizes short waves, the linear instability o f a flat film is a long-wave instability at relatively low R , which is consistent with the long-wave simplifications. The exp(iax + A t ) for the KS equation, usual normal mode expansion, H for example, yields a nondispersive parabolic growth rate that includes a band of unstable wave numbers that extends to 0. This unique growth rate curve is responsible for triggering the formation of solitary wave. However, as we shall show subsequently, the appropriate growth rate should correspond to a spatial growth as the falling film instability is a convective instability. One should apply the BL, Benney, and KS equations with care. There are essentially three independent parameters in the problem: K , x and 6 or R , y and 8. For the BL equation, we have carried out an expansion in K-' to O ( K - ' ) (see Chang et al., 1993a), which is a good approximation provided WR is large, namely, high surface tension fluids. For Benney's

-

12

Hsueh-Chia Chang and Eugeny A. Demekhin

equation, however, an expansion in 6 is also carried out. In fact, O(6) terms are retained while O ( K - * )terms are omitted. From (2.111, it is clear that this is consistent only if W O ( K )and R O(1)-we are restricted to thin films with high surface tension. In fact, the inclination angle y, should also be close to vertical for both cases. (The equation (2.17) is strictly for the vertical case.) The KS equation is even more restrictive due to the weakly nonlinear expansion. A careful evaluation of the region of validity of these equations can be found in the previous review (Chang, 1994). Recently, Salamon et al. (1994) have numerically compared all these approximate model equations to the NS equation. They found that the KS and Benney equations break down beyond R = 5 for W = 150, so that the solutions are not even qualitatively correct. The BL equation also deviates from the NS equation near some bifurcation points of the periodic wave solution branch due to the extreme parametric sensitivity at these points. However, away from these singular points, the BL solution agrees with the NS equation up to very large R. Although the KS equation is valid only for small-amplitude waves on large surface tension W fluids at low Reynolds number R , it offers several advantages. It is extremely simple, with only t h e dominant nonlinear term HH, that arises from kinematics-larger waves travel faster. More important, it contains no explicit dependence on the parameters. As a result, it implies certain self-similarity of the solution structure that is also approximately obeyed by the full NS equation at low R. We shall hence use it as a model equation to demonstrate some of the concepts of the coherent structure theory, although the latter will be shown to be strictly valid only for the other more complete equations. Another model equation we will use is an ad hoc but convenient simplification of (2.7) to (2.10) first introduced by Shkadov (1967) by assuming the self-similar profile of (2.12) is valid even at larger 6. The resulting integral boundary layer (IBL) equation o r averaged equation is, in the stretched coordinates of the BL equation, dq 6 d 1 ah dq - + - = 0 - + --(q2/h) - -(hh,,, + h - q/hz) = 0 dt 5 dx 56 dt dx (2.19)

-

-

Surprisingly, for 6 < 0.1 ( R < 10 for water), it yields excellent agreement with the BL and NS equations and is hence superior to the KS and Benney equations.

Solitary Waue Formation and Dynamics on Falling Films

13

111. Evolution toward Solitary Waves AT WAVE INCEPTION A. LINEARTHEORY

The linear stability of the flat film basic state h = 1 with a parabolic unidirectional velocity profile was first studied by Benjamin (1957) and Yih (1963). A comprehensive review of the earlier experimental and theoretical results on linear stability has been compiled by Lin (1983) and Lin and Wang (1985), and measurements following these reviews have been made by Liu et al. (1993). Two issues remain unresolved, however. Although it is clear from the linearized version of the Benney’s equation (2.17) and our earlier analysis (Chang et ul., 1993a) of the BL equation (2.7) to (2.10) that the Squire’s theorem holds for a liquid film on an inclined plane (namely, two-dimensional disturbances are more unstable than three-dimensional ones), this has never been shown for the NS equation for all conditions. The other issue concerns the more recent notion of convective instability. Joo and Davis (1992) and Liu et a/. (1992) have used Benney’s equation and the KS equation to show that a wave packet triggered by a delta function disturbance on the film will grow in space and not in time. This has the important implication that interfacial disturbances are all sustained by inlet noise, and they will propagate out of a system if forcing is stopped. It also brings up t h e distinct possibility that the turbulent interfacial dynamics one sees downstream are due to noise and hence are broadbanded and high dimensional (Deissler, 1989). Experimental evidence by Liu and Gollub (1994) has shown that the number of coherent structures and their initial separation are determined by the inlet noise but the downstream dynamics are dominated by the inelastic interaction of these coherent structures and not directly driven by noise. Hence, strongly nonlinear dynamics involving the solitary waves essentially obliterates the effect of noise and allows a deterministic description of the wave dynamics, although there is still a random component to the dynamics. In any case, both Joo and Davis and Liu et al. found that the film becomes absolutely unstable at sufficiently high R. This has never been observed, and because their equations breakdown beyond R = 5, as was discussed in the previous section, confirmation of the convective instability for all conditions by the linearized NS equation remained an open problem (Chang, 1994). Demonstration of Squire’s theorem and convective instability will be offered here.

Hsueh-Chia Chang and Evgeny A. Demekhin

14

The linearized NS equation or the Orr-Sommerfeld equation can be

y

=

1

~I,!I”’

-

3a(a2

+ p2)+h‘+ i a 2 R

-3i(a2

+ p2)cot 8

-i(a2

+ p2)RW=0

*” +

( a 2+

,8219

=

‘i

2 9’

3

4= c

3/2 (3.2) where +(y) is the complex stream function, U(y) = 3(y -y2/2) is the parabolic profile, a and p are the wave numbers in the x and z directions, and c is the complex wave speed. The Orr-Sommerfeld equation can be transformed by the following scaling aR

=

-

GR

cot % / a= cot 6 / G w/a2 =

(3.3)

w/2

where G 2 = a 2 + p * is the generalized wave number, to *Iv

- 2&2*1t

+ ~~9= i i i R [ ( U - c ) ( $ ‘ ~ y=o

-

~ 2 9-) ~

$

1(3.4)

*=*‘=O

I+bcI”f 629 = 3 $ t = c - 3/2 It is clear that if we set p to 0 in (3.1) and (3.2), the resulting equations coincide with (3.4) and (3.51, assuming R, W, and 0 are replaced by their wiggled counterparts. Because R is smaller than R and W and cot 6 are larger than W and cot 8 and because R is destabilizing but W and cot 6,

Solitary Wave Formation and Dynamics on Falling Films

15

the gravitational term, are stabilizing, it is clear that the flat film is more stable in three-dimensional disturbances than two-dimensional ones. Conversely, at the inception region in Figure 1, two-dimensional disturbances are preferentially excited over three-dimensional ones. This was verified by our numerical analysis of three-dimensional wave evolution (Chang et ul., 1994a). Three-dimensional disturbances will eventually appear, but in the first 1000 units of film thickness h , downstream, very little transverse variation is observed for small-amplitude forcing. In this review, therefore, we shall focus only on two-dimensional wave evolution. The preceding arguments relating to downstream evolution presupposes the waves grow in space and not in time. This convective instability can be ascertained by locating the saddle point a * , where dw =0 (3.6) da and w = ac is the wave frequency. If the imaginary wave frequency at a* is negative, the instability of a wave packet is convective (Huerre and Monkewitz, 1990). In Figure 4, we depict the computed saddle points from the Orr-Sommerfeld equation. As seen in Figure 5 , for R = 10, 19 = ~ / 2 , and y = 2850 for water, the first branch in Figure 4 is not a pinch point of the contour F (which was originally on the real axis) in the complex wave number space and is hence irrelevant due to causality (see Huerre and

-(a*)

1

I

I

I

I

I

I

0.8

0.6 0.4

9 '

I

I

I

-

0.2

2

I

0 /

-0.2

4 I

-0.4

50

I

I

I

1

I

I

I

100 150 200 250 300 350 400 450 500 R

FIG.4. The computed saddle point imaginary frequency as a function of R for 0 and y = 2850.

=

n/2

16

Hsueh-Chia Chang and Ergeny A. Demekhin

FIG. 5. The pinch points in the complex wave number space showing that branch 1 of Figure 4 is not a true pinch point at R = 10.

Monkewitz, 1990, for details of this analysis). The other branches are genuine pinched points, and we find them to be in the lower half of the complex plane for all R below 500 as shown in Figure 4. Hence, the convective instability persists even at high R. Any disturbance on a film is then driven by disturbances at the inlet, and Squire's theorem stipulates that the three-dimensional disturbances are damped in favor of twodimensional ones in this inception region. The spatial growth rate of the two-dimensional disturbances can be obtained from the solution of the Orr-Sommerfeld equation (3.1) and (3.2) by assuming a real frequency w and solving for the complex wave

Solitary Waue Formation and Dynamics on Falling Films

17

number a . Note that the usual Gaster transformation to convert temporal growth rate to spatial growth rate is invalid away from criticality ( R , = 0 for the vertical film) and explicit computation of the growth rate is necessary. We demonstrate this procedure with the IBL equation. Linearizing (2.19) about the basic state ( h ,q ) = ( 1 , l) and defining the vector y = ( q - 1, h - 1, h,, h,,)e"", one obtains

dY dx

-=Ay

(3.7)

where

A(w)

=

I

1 -56wi

iw 0

0 - 3 - t 126wi

-6 6

0

The spatial eigenvalues A,(@) of A(i = 1,2,3,4) then determines the spatial growth rate. However, not all the spatial eigenvalues are pertinent. If one uses periodic forcing at the inlet as shown in our numerical simulation of the IBL in Figure 2, it is evident that saturated, smallamplitude time-periodic waves exist just beyond inception. If spatial eigenvalues A , ( w ) at this frequency with positive real parts are in play, smallamplitude temporal harmonics can never appear, even if nonlinearities can arrest their spatial growth. The only tolerable spatial eigenvalues to ensure bounded waves are then ones with negative or slightly positive real parts. The latter dominate the spatial growth but can still be arrested by nonlinearities to form small-amplitude waves. [Consult Kirschgassner (1982), Iooss et al. (1988), and Roberts (1988) for a formulation of this spatial growth using center manifold theory.] There is only one branch A , ( w ) that is slightly positive for small w . In fact, A,(0) = 0 as shown in Figure 6. This is then the pertinent spatial mode that determines the spatial growth rate. It is also clear from Figure 6 that there is a band of unstable frequency between 0 and the neutral frequency w o , and the spatial growth rate is of the parabolic variety like the temporal growth rate of the KS equation. (Incidentally, the KS equation, which is appropriate for waves in a moving frame, does not permit the calculation of the spatial growth rate.) The maximum growing frequency w,, which is approximately w,)/ \/z at low 6, is the reason why a specific harmonic is chosen in Figure 1 out of all other harmonics generated by random forcing at the inlet.

18

Hsueh-Chia Chang and Evgeny A. Dernekhin

W

stable direction of bifurcation

v unstable

R-R,

R=Rc

FIG.6. Parabolic spatial growth rate A J w ) and neutral frequency curve

w,)

However, because all waves up to infinitely long waves are unstable, this harmonic quickly excites its subharmonic and sidebands and then triggers a whole band of harmonics that synchronize to form solitary waves. At low 6, wo and w, approach 9 m and 9 6 , respectively.

B. SATURATION, SUBHARMONIC SECONDARY INSTABILITY, AND SYNCHRONIZATION Although an entire band of unstable frequencies exist, linear filtering typically selects w, in the inception region even if the forcing is broadbanded. Hence, one can consider only the dynamics of this discrete fundamental mode and that of other discrete modes excited by the fundamental. The linearly selected fundamental, after it has grown to sufficient amplitude, induces a secondary instability and triggers an explosive growth of certain modes within and outside the unstable band by nonlinear excitation. If the forcing is periodic at the inlet, the fundamental mode is then at the forcing frequency but the subsequent excitation of the other modes still occurs if there is sufficient background noise. Let w , be the fundamental frequency that is within the unstable band, w1 E (0, w o ) . We shall also assume that the overtone 2 w l is stable. This is certainly true if w 1 is the maximum-growing mode w,, which is approximately w o / fi at low 6. For sufficiently small amplitudes, the quadratic nonlinearity is the dominant nonlinearity, and we are interested in which modes will be excited by the fundamental through quadratic interaction. Some consideration quickly reveals that a family of triads with frequencies

Solitary Wave Formation and Dynamics on Falling Films

+

19

+

{wl, w 2 , w 3 } that satisfy the resonant condition w , w2 w 3 = 0 will dominate the interaction. There are three subfamilies of triads: { w l ,w,/2,

w1/2} subharmonic

{ w , , w 1, 2 w , J

overtone

{ w l ,A w , , w l f A w l }

(3.8)

sideband

The sideband triads are parameterized by the sideband width A and the zero mode A w l . Strictly speaking, they include the subharmonic ( A = 1/21 and overtone ( A = 1) triads. They also include the second subharmonic triad { w 1 7w1/2, 3w1/2} for A = 1/2. We shall, however, distinguish the subharmonic and overtone interaction from the sideband interaction by limiting A to the interval (0,1/2). It is also clear that the first subharmonic triad interaction dominates the second because all three members are unstable in the former. The first triad to manifest itself is the overtone because it involves self-interaction of the fundarncntal. The others involve interaction of the fundamental with a possibly unstable mode. However, due to selective forcing or linear filtering or both, the other unstable modes are insignificant in amplitude, and the interaction is hence negligible. The overtone excitation is responsible for the downstream saturation of the fundamental seen in Figure 2. Energy transfer from the unstable fundamental to the stable overtone arrests the spatial growth of the former. Using the IBL equation as an example, we can expand the deviation variable y of (3.7) in terms of the spatial modes: 4

y

=

C a,(x)v,,

4

e'"l'

n= 1

+ C b,(x)w,, ei2wl' n=l

where vn is the eigenvector of a spatial mode n for A ( w ) and w, is t h e corresponding one for A(2 w ) . Substituting the expansion into the nonlinear version of (3.7) and taking inner product with the adjoint eigenvectors, one obtains the amplitude equations: a,

=

f(a, b)

b,

=

g(a, b)

Using the center manifold theory of Kirschgassner (1982) and Iooss et al. (1988), we include the small effects of the unstable and stable spatial modes on the nearly neutral mode a, of Figure 6 by carrying out the

20

Hsueh-Chia Chang and ELgeny A. Demekhin

proper adiabatic elimination to remove the unstable and stable modes. The detailed formulation can be found in Cheng and Chang (1992). The projected dynamics is described by the Stuart-Landau equation: (3.91 where the Landau constant R , receives contribution mostly from the overtone triad interaction between the nearly neutral fundamental with itself and with its overtone near the neutral branch A , ( w ) of Figure 6. The Landau constant is found to be positive for all 6 values, and hence a supercritical bifurcation exists. Physically, (3.9) implies that after an inception length of l / A i , the exponential growth in space of the wave is saturated at the amplitude IAi/Ri11’2, where the superscript r denotes the real part. This is clearly seen in Figure 2. Because the bifurcation is supercritical, the saturated wave is stable with respect to disturbances within the overtone triad. It is, however, unstable to both the sideband and the subharmonic interactions. Dramatic experimental evidence of these two dominant instabilities was reported by Liu and Gollub (1993). We can study the relative dominance of these two secondary instabilities by including the appropriate discrete modes in the expansion of y to obtain the amplitude equations

where a = ( a , , a , , a , ~ A and ) A is a diagonal matrix with the spatial growth rates A,(w,), A,(Aw,), and A,(w, - A w l ) as diagonal elements. The equations with the right sideband a , + & are similar to (3.10) with trivial changes in sign. Equation (3.10) also describes the subharmonic instability if A is set to 1 / 2 and if the last equation in (3.10) is omitted-it involves two-wave interaction instead of the three-wave interaction of the sideband instability with 0 < A < i. Combining our sideband and subharmonic stability theories (Cheng and Chang, 1990, 1992; Prokopiou et al., 19911, we are able to show that near-neutral fundamentals ( w l w o >are dominated by the sideband instability and lower frequency fundamentals are more unstable to the subharmonic instability (Cheng and Chang, 1995). Hence, a critical fundamental frequency separates the unstable

-

Solitary Wave Formation and L@narnics on Falling Films

21

band in Figure 6 into two regions, one whose saturated fundamental is dominated by the subharmonic instability and one by the sideband instability. This critical frequency has also been carefully measured by Liu and Gollub (1993) for an inclined plane. In Figure 7, the computed critical frequency from the preceding secondary stability theory is shown to be in favorable agreement with their experimental data. The conditions in Figure 2 correspond to a case below the critical frequency, and the dominant subharmonic instability is clearly evident in the figure, although the residual sideband instability introduces some gradual modulation of the wave amplitude. It is also evident from Figure 7 that the fastest growing mode w,, which approaches m,,/fi at low 6,also lies in the subharmonic dominant region. This is why, even for the randomly forced case of Figure 1, the selected wave w, in the inception region undergoes period doubling. The Fourier spectra measured at several spatial stations during the numerical experiment of Figure 2 are shown in Figure 8. The saturated fundamental with sharp peaks at the fundamental and the overtone are seen close to the inlet. The w1/2 and 3w,/2 subharmonics are clearly evident at the onset of the secondary instability, with the former more pronounced because of its linear instability. The A zero mode of the weaker sideband instability can also be seen. The convective nature of the secondary instability with spatial growths in the excited 1.o

0.8

who

0.6

t 0.4 1

2

3 RIR,

FIG. 7. Critical frequency measured by Liu and Gollub (1994) separating fundamental frequencies with dominant sideband secondary instability from those with dominant subharmonic instability. The theoretical curve from a weakly nonlinear analysis is also shown.

22

Hsueh-Chia Chang and ELgeny A. Demekhin

FIG.8. Fourier spectra of time tracings taken from the numerical experiment of Figure 2 at locations x = 300, 400, 600, and 700.

modes is also seen. To further demonstrate that the subharmonic instability is the preferred secondary mechanism even with broadband forcing, the evolution of a wave packet is studied numerically in Chang et al. (1993~) and shown in Figure 9. With this experiment, the evolution is more temporal than spatial and broadband forcing is replaced by a wave packet with a broad wave number content. It is clear from the leading edge of the wave packet in the third and fifth frames in Figure 9 that the subharmonic instability has eliminated every other peak, as in Figure 2 prior to the birth of a solitary pulse. According to the theory of Cheng and Chang (19921, the dominant subharmonic secondary instability can occur only if the subharmonic mode is linearly unstable, which is always the case for the falling film. Because this secondary instability triggers the formation of solitary waves, the linear instability of the subharmonic mode can well be a necessary condition for solitary wave formation in any system. Other conditions are probably also necessary to ensure the phase locking of the excited modes, but the

Solitary Wave Formation and Dynamics on Falling Films

23

1.25 I =

12

0.2

100

I

0.75

I

I

I

I

I

I

I

1

I

I

I

1.35

I 1.00

t

=

0.4

t

=

0.6

'

0.75 -

'

I

I

L

1.8 I

O

I

I

I

I

h

L

I .o 0.7

1 .o

I

I

I

I

I

I

I

,

I

I

I

I

I

unstable band of frequencies or wave numbers must be broad enough to allow a subharmonic instability. It is also reasonable to question whether it is valid to study this subharmonic instability by linearizing (3.10) about the saturated fundamental. If the saturated fundamental is unstable, why should the system approach it at all, as seen in Figure 2? The answer was

24

Hsueh-Chia Chang and Eugeny A. Demekhin

provided by Chang et al. (1994b). For almost all conditions, the unstable eigenvalue of the saturated fundamental is much smaller in the absolute value of its real part than the stable eigenvalues, one of them, -2A;, being the one associated with the overtone disturbance in (3.9). The unstable eigenvector also points in the direction of the subharmonic amplitude. Hence, if the initial amplitude of the subharmonic is small due to selective forcing or linear filtering, the system gets very close to the saturated fundamental as seen in Figure 2 and stays in its neighborhood for a duration of l / A i 1 2 before departing slowly due to subharmonic instability. The eigenvalue is the unstable one corresponding to the subharmonic instability and lA;,21 << I Ail. Near w,, our computed values (Chang et al., 1994b) indicate lA;,21 is typically one order of magnitude smaller than IAiI, which is consistent with the transition lengths of Figure 2. However, unlike the primary instability whose growth is decelerated by nonlinearity for the present supercritical case ( R , > 0 in (3.9)), nonlinearity tends to accelerate the slow linear growth of the secondary instability. This is evident from the numerical experiments in Figure 2 and Figure 9. Janssen (1986) has demonstrated that double-exponential growth in the subharmonic mode of a nondissipative system can be triggered by nonlinear subharmonic resonances; that is, if the subharmonic mode travels at the same speed as the fundamental. This condition is satisfied at the nondispersive limit of low 6 for the falling film although the doubleexponential growth is suppressed by dissipation here. Nevertheless, the absence of frequency mismatch still triggers a rapid growth of the subharmonic, as is evident in Figure 2. It physically corresponds to the fast drawing of fluid from one peak into its neighbor such that the number of peaks is reduced by half. The enlarged peak will distort its sinusoidal shape and, in the process, excite the stable overtones 2w,, 3 w , , and so forth and the second subharmonic 3w,/2, as is evident from the spectra in Figure 8. These numerous discrete modes begin to synchronize and form the distinct hump shape of the solitary wave. Its front steepens and individual pulses begin to be separated by a relatively flat film. The bow waves nucleate on these flat films and are swept up by the growing humps. Some adjustment of their amplitudes occur as fluid is drained into the main hump before they are permanently affixed to the front of the propagating solitary wave. Liu and Gollub (1994) have shown that these bow waves generate undergrowths around the peaks in the spectra. The space between the discrete peaks therefore begin to fill up rapidly by this broadband undergrowth. In the meantime, the growing and steepening hump excites more stable overtones. This spiked broadband spectrum resembles white noise, but the

Sojitary Wave Formation and Dynamics on Falling Films

25

patterns are not random at all. Except for the nucleating bow waves on the flat film, which exhibit more fluctuation, the fundamental, subharmonics, and overtones all phase lock (synchronize) to form the one hump localized structure. They are thus very phase coherent. If broadband forcing is used, the sideband instability and simple linear beating tend to localize the subharmonic instability such that one localized larger structure begins to absorb its front neighbor. The resulting structure is even bigger, and it begins to absorb the smaller peaks in front that have not undergone subharmonic coalescence. As the large structure absorbs the smaller peaks, it grows and steepens to form the one-hump solitary wave. Its distortion generates the overtones and the subsidiary bow waves trigger the broadband undergrowth as before. However, because the nucleation of the subharmonic coalescence is now localized and randomly spaced, the number and the spacing of the individual humps are random and highly dependent on the frequency content and amplitude of the forcing. There are other possible transitions to this hump structure. For low-frequency periodic forcing below o,,, the numerical experiments of Joo et al. (1991) indicate that the unstable overtone and even stable higher harmonics are first excited before the broadband growth begins. The subharmonics would probably have been excited if their domain of integration were sufficiently large to admit them. Until the rapid nucleation of solitary waves, which occurs when the excited hump begins to absorb the smaller ones, the overtone excitation that saturates the amplitude growth of the fundamental, the subharmonic excitation, and the resulting excited wave all behave quasi-stationarily. Even the solitary humps, which are created from a rapid process, behave as a stationary hump, as seen in Figure 9. One can therefore model these quasi-stationary stages during the evolution as stationary periodic waves. Construction of these periodic waves has been reviewed in the previous article (Chang, 1994). The essential result is that the saturated fundamental wave in Figure 2 corresponds to a y , family that travels slower than the linear phase speed of the fundamental. Hence, there is a nonlinear frequency and wave number correction due to the resulting compression, so that the saturated wave is slower and shorter than the fundamental. The excited hump after the subharmonic instability and the eventual solitary hump belong to the same y 2 wave family that travels much faster than the linear phase speed. These predictions have been verified by Liu and Gollub (1994). The unique synchronization of harmonics and subharmonics, first triggered by a global or local subharmonic instability to form solitary humps,

26

Hsueh-Chia Chang and Eiigeny A. Demekhin

is driven by the same kinematics we examined for wave inception. It is represented by the HH, term of the KS equation and physically implies that the particle velocity on the interface of the thicker film travels faster than a thinner one. In the KdV equation, this term introduces a focusing phenomenon that is balanced by dispersion. Here, it excites the overtone to stabilize the growing fundamental beyond wave inception. Physically, inertia first excites the fundamental and distorts the flat interface. The distorted interface, in turn, excites the overtone due to the nonlinear kinematic effect embodied by HH,. The overtone is stable because its higher curvature triggers a capillary driven flow that overwhelms the inertia-induced instability due to a delay in the acceleration of the fluid particles. In this manner, the energy of the unstable mode is transferred through nonlinear kinematic interaction to stable modes, where it is dissipated. Such a scenario is not strictly correct when many modes have been excited during the secondary instability. Nevertheless, the same kinematic mechanism still triggers front steepening and mode synchronization. It also implies that larger humps travel faster than the smaller waves. This, in turn, incites the absorption of the smaller waves by the larger humps that resulted from local subharmonic instabilities. The excitation and synchronization of a large band of modes to form phase-coherent localized structures with noiselike broadbanded spectra seem to occur in many turbulence transitions. Kachanov (1994) has observed synchronization of velocity Fourier modes in boundary layer transition to turbulence. The velocity modes there synchronize to form vortex filaments, which are then analogs of the solitary pulses on a falling film. Several comments are appropriate regarding this formation process of coherent structures. The resulting pulses are of the same amplitude as the flat film as seen in Figure 1. As such, the previous weakly nonlinear theory that allowed the scrutiny of secondary instability is no longer valid. In fact, many of the linearly stable slave modes that were filtered in the inception region and discarded by adiabatic elimination in the weakly nonlinear analysis are reexcited by the large solitary pulse as seen from the last spectrum in Figure 8. This “slave revolt” causes the formation of largeamplitude localized patterns. It seems to occur whenever the bandwidth of unstable frequency or wave number is large enough to allow subharmonic instability. This condition is always satisfied for the neutral curve of the falling film in Figure 6, but this is not the case if the long waves are stabilized, so that the subharmonic is stable and hence cannot induce secondary instability. Also because the slaves have revolted, master weakly nonlinear equations, like the coupled complex Ginzburg-Landau equa-

Solitary Wave Formution and Dynamics on Falling Films

27

tion, which covers small bandwidths near a finite number of master modes like those in (3.101, can never fully capture the solitary wave formation process. It would also be tempting to say that the broad spectrum in Figure 8 results from a Feigenbaum period doubling cascade. This is definitely not the case. The first period doubling is immediately followed by overtone excitation and broadband growths to form the “white noise” spectrum, as is evident in Figure 8. Successive period doubling can conceivably occur if the first subharmonic eliminates every other peak, the remaining peaks coalesce again, and so on. The coalesced peaks after each period doubling must be self-similar in shape to the ones before coalescence. This is clearly unlikely even with periodic forcing as the excited overtone, and the sideband modes distort the peaks after each coalescence. These two additional instabilities corrupt the idealized scenario of period doubling. With broadband forcing, the coalescence occurs at localized positions. The isolated coalesced peak then overtakes the smaller peaks in front in a pattern bifurcation that is distinctly not period doubling. Because the absorption involves draining of fluid from the smaller peaks into the localized structure, it is more appropriate to study this formation process through absorption as an intermediate blow-up phenomenon of the structure (Kalliadasis and Chang, 1994). In fact, it is difficult to quantify and probably unimportant to understand exactly how the Fourier modes are excited and synchronized during this absorption stage. They are excited and do synchronize, however, on a falling film over a very short distance, as is evident in Figure 1 and 9. I t is more meaningful to ask how many localized structures are generated for a given forcing. The forcing seems to determine whether a local peak will be sufficiently large to trigger a subharmonic instability, absorb its neighboring peaks, and initiate subsequent absorption to form a solitary wave. Liu and Gollub’s experiments indicate that the density of these nucleating sites for solitary waves and the distribution of their separation are functions of both the forcing amplitude and bandwidth.

IV. Solitary Waves A. EXISTENCE, ESTIMATE, AND

CONSTRUCTION

The solitary waves, with their large amplitudes and broad Fourier content, obviously cannot be described by weakly nonlinear theories with only a few Fourier modes. Their construction and dynamics can be

28

Hsueh-Chia Chang and E q e n y A. Demekhin

discerned only with strongly (global) nonlinear techniques in dynamical systems theory. In this respect, the falling film is arguably the best hydrodynamics example of how modern dynamical systems theory, especially global theories associated with homoclinic bifurcations, can be applied to understand its complex spatio-temporal dynamics. Although the chaotic dynamics of solitary pulses occur at relatively low Reynolds numbers due to the presence of destabilizing inertia force at scales different from the stabilizing capillary force, the dynamics share many of the characteristics of high Reynolds number shear flow turbulence. Subharmonic secondary instabilities, broadband excitation, synchronization, and coherent structure interactions seem to occur in all these open flow systems. With its low Reynolds number, the falling film instability is relatively easy to study numerically, and its solitary waves are simpler to construct. One need not resort to complex Ginzburg-Landau-type longwave equations whose premise of slow envelope modulation is often incompatible with the length scales of the localized coherent structures. For these reasons, falling-film interfacial ‘‘turbulence’’ is an excellent prototype for other open-flow hydrodynamic turbulence. Tsvelodub (19801, Pumir et al. (1983), and Lin and Suryadevara (1985) first demonstrated that the KS equation and the Benney’s equation admit localized solitary wave solutions that propagate at a constant speed without change in shape. Estimates of the speed and amplitude of small solitary waves of the KS equation were then obtained by Chang (1986) using normal form theory. Other stationary patterns have recently been uncovered by Sefik and Unal (1994) using the same theory. Nakaya (1989) demonstrated that the n-hump solitary waves of Benney’s equation first found by Pumir et al. (1983) can bc estimated by combining n one-hump solitary waves additively. Kawahara and Toh (1988) used a simple coherent structure theory to show that the one-hump waves of a generalized KS equation can interact dynamically, under certain conditions, to form the two-hump waves, which they called bounded states. Because a solitary wave is stationary in a moving frame shifting with its speed A, the partial differential equation (pde) (2.18) can be transformed to -AH*

+ 2(H*)’ + H: + H:,,

=

0

(4.1)

where use has been made of the fact that the solitary wave H * ( X ) must decay to zero at X + +=; that is, its true film thickness must approach the Nusselt thickness h,. The speed A is the deviation speed in the

29

Solitary Wave Formation and Dynamics on Falling Films moving coordinate of the original KS equation (2.18). By defining x ( H , H,, H,,), (4.1) can be written as a dynamical system:

x=

(g

0

A

-1

‘]x+ 0

0

( 0A 00

0 00]x-

[

=

(4.2)

2!];

where the overdot denotes derivative with respect to the spatial coordinate X . The fixed point at the origin corresponds to the Nusselt flat film and the solitary wave corresponds to a homoclinic orbit that connects the origin to itself. For A = 0, the linear Jacobian yields the spectrum (0, f i} and with a small A perturbation, the real eigenvalue becomes slightly positive at A and the complex ones have a small negative real part, -A/2. The homoclinic orbit then leaves the origin in the direction of the real unstable eigenvector and reenters the neighborhood of the origin on the twodimensional eigenspace spanned by the two complex stable eigenvectors. This immediately suggests that thc solitary wave has a smooth back slope but is preceded by small “bow” waves of wavelength 27-r in the X coordinate of the KS equation. This wavelength corresponds to the neutral wavelength of the flat film where inertia and capillary forces canal exactly. This qualitative description of the shape of the solitary wave is obviously consistent with our numerical simulations in Figure 1 and 9. The observation that the bow wave frequency is approximately the neutral frequency of the substrate film has been confirmed by Alekseenko et al. (1985). Liu and Gollub (1994) found it to be about twice the neutral frequency of h,. When there are numerous solitary waves, as in the periodically forced experiment of Liu and Gollub, the substrate thickness is thinner than h , near the inlet, as is evident in Figure 1. Consequently, the natural frequency of the substrate is higher than that of h,. This could explain Liu and Gollub’s observed inconsistency. Even though the bow wave characteristics can be obtained from a simple linear expansion near the substrate thickness, to decipher the speed A and amplitude of the solitary wave, one must construct the entire homoclinic orbit. Actually an infinite number of such solitary waves, with diffcrcnt values of the “nonlinear eigenvalue” A, can connect the unstable eigenvector to the stable eigenplane. Some of them can be better paramcterized by “unfolding” (3.1) with an additional dispersion term:

-AH*

+ ( 2 H * ) 2+ H; + 6 ’ H ; , + H;xx= 0

(4.3)

Hsueh-Chiu Chang and Eqqeny A. Demekhin

30

The computed homoclinic solution branch of (4.3) by Chang et al. (1993b) is shown in Figure 10. There are actually other branches isolated from this one, including the ones uncovered by Sefik and Unal (19941, but they are not pertinent because they have never been observed. Each point on this branch represents a homoclinic orbit/solitary wave. Further bifurcations to periodic waves from each homoclinic orbit can be deciphered using the PoincarC map technique of the Shilnikov theory (see Glendinning and Sparrow, 1984; Balmforth et al., 1993) and were explored numerically by Chang et al. (1993b). However, as is evident from Figure 1, despite the possible patches of equally spaced pulses, it is more accurate to describe the downstream pattern as interacting solitary pulses rather than periodic wave trains. This then requires a departure from the periodic stationary wave approach for the inception and secondary instabilities reviewed in the previous article (Chang, 1994). In fact, only two members of the infinite number of homoclinic orbits of the KS equation at 6 ' = 0 are pertinent: the one-hump solitary wave and the two-hump solitary wave on branches b and c of Figure 10. The one-hump solitary wave at A, = 1.216 with an amplitude H, = 0.70 is also the one observed with more elaborate model equations in Figure 1, and the two-hump solitary wave at A2 = 1.208 (the bounded state studied by Kawahara and Toh, 2

3

5

2

A.

1

8'

FIG.10. Solitary wave solution branch of the generalized KS equation. The original KS equation has a one-hump solitary wave at A A = 1.208.

=

1.216 and a two-hump bounded state at

Solitary Wave Formation and Dynamics on Falling Films

31

1988) will be important in our study of coalescence between two one-hump solitary waves. The amplitudes of the two humps are approximately identical and equal to the H , of the one-hump solitary wave. The other members correspond to multihump solitary waves, which have never been observed. The exact shape of the solitary wave must be deciphered by a numerical analysis or estimated by nonlinear normal form transformations, which simplifies (4.2) (Chang, 19861, but the most important information on the amplitude-speed correlation can be obtained with little effort. Equation (4.2) has a fixed point at x = (A/2,0,0). It is a “conjugate” fixed point (Chang, 1986, 1989) whose amplitude depends on the speed A. A shock solution would correspond to a heteroclinic orbit that connects the fixed point at the origin to the conjugate fixed point. Hence, the existence of this conjugate fixed point is necessary for the existence of steady shocks. This conjugate fixed point arises from simple mass balance from the leading order kinematic term H H , of the KS equation. For a shock (hydraulic jump) to remain stationary in a moving frame with speed A, the flow rates in the moving frame within the two flat portions must be identical. Although a solitary wave does not strictly have another flat film region distinct from h,, the Nusselt thickness, its long-wave characteristics imply that, neglecting the higher order surface tension and inertia effects that introduce curvature, the region near the maximum of the hump is nearly flat and hence the solitary waves should have a deviation amplitude close to A/2. From our numerical study of the homoclinic orbits, we found that the humps correspond to loops around the conjugate fixed point before reattaching to the origin. Consequently, the speed-amplitude correlation can be conveniently estimated by the conjugate fixed point as H = A/2 or, in the original coordinates, 3(h

-

1)

=

c

-

3

(4.4)

For example, the one-hump solitary wave has a speed of A, = 1.216 and an amplitude of H , = 0.70, which is close to A,/2. For significantly large 6, the KS equation breaks down and one has to use the BL equation and the NS equation. Nevertheless, the one-hump solitary waves of the KS equation can be followed into the large 6 region numerically. This has been done for the BL equation by Chang et al. (1993a), and Salamon et al. (1994) have recently constructed the one-hump solitary waves of the NS equation. For 6 less than about 5 ( R < 300 for water), there is very little quantitative difference in these solitary waves

32

Hsueh-Chia Chang and ELgeny A. Demekhin

and they are in good agreement with the photographs and experimental tracings of Kapitza and Kapitza (1949, 1965) and Nakoryakov et al. (1985). However, if Benney’s equation is used, the branch of the one-hump solitary wave exhibits a turning point at about R = 5 (Pumir et al., 1983; Nakaya, 1989) such that solitary waves do not exist beyond this value of R. This is erroneous because both the BL and NS equations yield solitary branches that do not exhibit a turning point and experimental data suggest that solitary waves continue to exist for R > 500. It should be noted that the IBL equation also yields solitary waves for all 6. The one-hump solitary wave solution branch of the IBL equation is shown in Figure 11. The self-similarity of their shapes, a legacy of the fact that the KS equation is parameter-free and yet has a unique homoclinic orbit, is evident in the figure. The flat film height at infinity for each solitary wave has been normalized to unity. Surprisingly, the correlation (4.4) applies for this family of solitary waves even though the KS equation would be valid for only very small 6 values. It can be shown that the conjugate fixed point still exists for the NS equation and other equations. The validity of (4.4) for solitary waves at relatively large 6 suggests that the hump of the solitary pulse still comes very close to this fixed point in the phase space. In the recent measurement of Liu and Gollub (19941, the speed-amplitude correlation has a slope close to (4.4) but there is shift of the intercept for different R. (With their scaling, the slope is 2 instead of 3.) The scatter is again because they used the speed and Nusselt thickness of the flat film near the inlet and not the flat film connected to the solitary wave, as is assumed in the theory. If there is only one solitary wave, these quantities are identical, but for a train of solitary waves, they become distinct as is evident in Figure 1, where the “substrate” on which the solitary waves appear has a thickness different from the flat film at the inlet. In Figure 12, we use the data of Alekseenko et al. (19851, where the solitary pulses are isolated and hence the substrate thickness has been properly accounted for, to demonstrate the validity of (4.4) even for large-amplitude solitary waves at large 6.

B. SYMMETRIES AND COHERENT STRUCTURE THEORY For any given condition-namely, if R,W, and 13 or 6 and y, are specified-there is a unique one-hump solitary wave that approaches unit film thickness, scaled with respect to h , , at positive and negative infinity.

I

r

0.064

0.06

0.056

0.052 0.040 0.044

0.04 0.036 0.032

0.020 0.024

1

0.02 0

15

30

45

60

X FIG.11. One-hump solitary solution branch of the IBL equation. The substrate thickness has been normalized to unity.

Hsueh-Chia Chang and ELgeny A. Demekhin

34

/ 0.0

I

1.o

FIG.12. Comparison of the correlation 3(h

-

1)

=

c

-

:

h-’

3 to the data of Alekseenko et al.

(1985).

For vertical films at x = 0, the one-parameter family of single-hump solitary waves of the IBL equation is shown in Figure 11. For the KS equation, there is a unique one-hump solitary wave at A , = 1.216, as shown in Figure 10. However, the KS equation is valid for any small S that has been absorbed into the scaling. The unique one-hump solitary wave of the KS equation should then generate an entire one-parameter family of these solitary waves. This is true due to a Galilean symmetry of the KS equation first pointed out by Elphick et ul. (1991). Taking the derivative of (4.11, which defines the solitary wave, one obtains

-AH$

+ 4H*H; + H$x+ H:xxx

=

0

(4.5)

with H * ( X + +-_..) = 0. It is clear that the transformation

H*+H*+@

A+A+4@

(4.6)

Solitary Wacie Formation and Dynamics on Falling Films

35

leaves (4.5) invariant and hence a one parameter family of solitary waves is generated by simply shifting the substrate thickness by @ and the speed by 4@. The origin of this symmetry is a readjustment of the characteristic length h , in the original scaling. If we use a particular S as a reference and set the dimensionless film thickness and dimensionless interfacial velocity at that condition to unity, then at a different 6 with a correction @ to the film thickness, the interfacial velocity that scales as square of thickness is increased by factor of 1 + 2 @ at the leading order resolution of the KS equation. The solitary wave speed, which is scaled by the interfacial velocity in the KS equation, is roughly twice the interfacial velocity. As a result, for the new 6, the solitary wave speed increases by an amount of 4@ units of the reference 6. We note that the relationship H,, = A,/2, or (4.4) with different dimensionless variables, for t h e onehump solitary wave remains the same for the thicker substrate provided the new thickness and interfacial or average velocity are used in the scaling, respectively. The Galilean symmetry implies a degeneracy of the solitary wave solution. There is also the translational symmetry that also generates a family of solutions. Because the KS equation is invariant to a shift in X , an arbitrary translation of the solitary wave solution H * ( X - X,) is also a solution. These two symmetries are related to the stability of a solitary wave. Consider the translational symmetry first. If one perturbs a solitary wave solution by translating it slightly, the degeneracy implies that this motion corresponds to an eigenfunction with a zero eigenvalue. More precisely, for small X,,,

H*(X

-

-

X,)) H * ( X )-X"H$(X)

(4.7)

and we expect i,h, = -H; t o be a null eigenfunction. This is true because the linearized operator for the stability of the KS solitary wave is

and differentiating the defining equation (4.5) for the solitary wave shows that 9 H ; = 0. Similarly, if one solves the solitary wave solution of the KS equation with a substrate thickness of @, H * ( X ; @), we expect J j 2 = H: ( X ; 0) to also be a null eigenfunction. This derivative is unity but the constant eigenfunction can be conveniently set to 1/4. It turns out to be a generalized eigenfunction because Y(1/4) = - H;. The solitary wave

36

Hsueh-Chia Chang and Eugeny A . Demekhin

solution of the KS equation then has two zero eigenvalues but the null space has only a geometric multiplicity of 1. There are other discrete eigenvalues but, unlike $*, they all decay to 0 at both infinities:

A simple integration of (4.9) then shows that /?%$,,dX = 0 for n 2 3. In fact, the only eigenfunction with an unbounded integral is the generalized eigenfunction q ! ~ ~Hence, . if the disturbances to the solitary wave decays to 0 at the infinities, J!,~I will never be excited. However, if there are permanent baseline disturbances that are not integrable, qh2 should be included. We shall show, however, that this is incompatible with the KS equation. Also, if { are negative (stable), then the dynamics of a particular solitary wave in response to the disturbances is dominated by the two neutral modes $, and $z. This premise is, unfortunately, untrue for the KS equation but we shall continue to use this simple equation to demonstrate how coherent structure theory can be used to derive the equations of motion for these two modes. Let us define the adjoint operator to 9 as

~,}r=~

9'= -Al-

d

dX

d

+ 4H*-dX

-

d2 ~

dX2

-

d4 ~

dX4

(4.10)

It is defined with respect to the inner product ( f , g ) = / " f ( X ) g ( X ) d X such that (9f, g ) = ( f ,9's) with appropriate boundary conditions for f and g at the infinities. Elphick et al. (1991) first constructed a null adjoint eigenfunction 2'iqJl =

0

with cpl(m) = - 4 - M) # 0 such that (cp,, = an,. It is quite obvious from (4.10) that (p2 = 1 is also a null eigenfunction and ( q 2 ,$,,) = 0 for n # 2 by virtue of the fact that $,, has a zero mean if n # 2. Note, however, ( q 2 $z) , = -t. The mathematical origin of this infinite inner product stems from the proper functional space that can be spanned by the eigenfunctions $,, of the operator 3.For the eigenfunctions to form a complete basis set of this functional space, the adjoint operator Yt should have a null space that is of the same dimension as the kernel of 9. This is clearly not true as both q1 and ( p 2 correspond to simple zeros; that is, Pat has a two-dimensional

Solitary Wave Formation cind Dynamics on Fulling Films

37

null space and 9has a one-dimensional one. The only consistent formulation is then to omit both 4k2 and 'p2 so that only disturbances that decay to zero at the infinities are tolerable. This restriction also defines the proper functional space for the disturbances. This was actually obvious from the fact that integration of the equation Pt'pl= 0 yields (p,, GI) = -A,['p,(m) - p l ( - ~ ) ]= -2A,'pl(-x) # 0. In fact, ( p l , $,) is unity through normalization of 'p,. The fact that ('p,, $,I f 0 implies that LYt does not have a generalized zero eigenvalue and one should remove the generalized zero eigenvalue of Y . Physically, this inability to include disturbances that do not decay to zero at the infinities originate from the assumptions made in deriving the KS equation. These disturbances correspond to step changes in the average interfacial height or substrate thickness that generate effective point sources or sinks of liquid in the moving frame with the speed of t h e solitary wave. As a result, the total liquid mass around a solitary pulse will change in time. Because the liquid in the substrate immediately below a pulse is negligible compared to the mass carried in the pulse, the net change in the liquid mass must come from the pulse that is undergoing a quasi-steady evolution through the solitary wave family of different substrate thickness shown in Figure 11. Unfortunately, with the weakly nonlinear approximation and subsequent scalings, the KS equation has only one solitary wave and the family of solitary waves it generates through transformation (4.6) all have the same area above the substrate. As a result, the liquid lost or gained cannot be accounted for by adjusting the substrate thickness, and $z must then be removed from consideration. However, the BL, NS, and IBL model equations generate solitary waves that contain different amounts of mass when the substrate thickness is varied, so disturbances corresponding to changes in substrate thickness that are spanned by $2 can indeed be excited. As a result, (p,, $1) = 0 and L?'+ also has a generalized 0 whose generalized eigenfunction p2 has a finite and nonzero inner product with $,. In fact, we shall define the generalized adjoint eigenfunction as 'pl and the null eigenfunction as 'p2 for consistency in notation. Nevertheless, we shall continue to use the KS equation as a simple example to demonstrate the dynamics of the two dominant modes of a solitary wave, even though one of them is never excited. We are now in the position to project any disturbance onto the center manifold spanned by the translation mode which is also a position mode, and the substrate mode $*, which is a speed mode, due to the substrate/speed correlation o f (4.6). Consider the perturbation f (X ) to

38

Hsueh-Chia Chang and ELgeny A. Demekhin

the speed and position of a solitary wave

Upon substituting this expression into the KS equation in the moving frame with the unperturbed solitary wave speed A , ,

H,

=

A, H , - 4HH, - H,,

- Hxxxx

(4.12)

and taking inner product with respect to p,, we obtain the equations of motion for the solitary wave in response to the perturbation (4.13) where the diagonal capacitance matrix A is simply A,, = (cp,, $,) with = 00. We note that X , and C are the perturbation A , , = 1 and position and speed to the original solitary wave. They are coupled through the Jordan form due to the obvious relationship between position and speed, which also accounts for the generalized zero eigenvalues. The projection of the perturbation f onto these two modes are

g, = (Pf4ffXx,(P,) g,

=

(2f4f f x , %)

Consider first perturbations that do not alter the baseline; namely, = 0. Because H * ( X ) is a homoclinic orbit in the phase space of ( H , H,, Hxx), the Melnikov function g , corresponds to how a perturbation f ( X ) to the homoclinic orbit changes the trajectory in the phase pace if we suppress the time derivative in (4.13) or replace it by a constant. The constant is then a correction to the original solitary wave speed A,. Consider, for example, the perturbation imposed by the presence of another solitary wave at a distance L in front, f ( X > = H * ( X - L). Because 9f - 4ffx 4( H "f), vanishes for this perturbation, the Melnikov function g, becomes -(4[H*f],, cp,) = ( 4 H * f ,%) and the position of the first solitary wave is then described by

g,

+

Solitary Waue Formation and Dynamics on Falling Films

39

Because the solitary waves decay exponentially to 0, the integral can be evaluated from 0 to positive infinity. If the two solitary waves are sufficiently far apart, only the two tails contribute to the integral in (4.14)

H * ( X ) + A cos(wX

+ & ) e - m lx

H*(x)

x+

+

X

+ ~0

--so

The one-hump solitary wave is located at A = A, = 1.216, so the quantities m,,m,, and w can be computed from the characteristic polynomial of the Jacobian in (4.21, a 3 + cr - A, = 0, which yields m, = 0.38, m2 = 0.76, and w = 1.20. Considering the width of the KS one-hump pulse is about 5 in the X coordinate, these values of m , and m2 indicate that the interaction between KS solitary waves is very weak and significant only when the pulses are one or two widths apart; that is, they are short-range forces. This remains true at higher 6 values when the IBL, BL, or NS equations are used. For two pulses that are sufficiently close to feel this short-range interaction, their binary interaction can then be shown to be described by the dynamical systems d

-Xo dr

=

Ff(X, -Xo)

d -Xl dr

=

F,(XI

-

Xo>

(4.15)

when F, originates from the Melnikov function F ( L ) of (4.14) such that

F’(L)

=

41; H * ( x ) H * ( x

-

~ 1d4)I - a dX

- - a emlL

where a , p, and E are positive constants. It is clear that the back pulse is always repelled by the front because dX,)/dr is negative, but the front one can be attracted or repelled by the back one depending on the separation L. A “bounded” state with equilibrium separation L* is also possible with a perturbation speed A’ defined by Ff(L*)- F,(L*) = A‘. A countable infinite number of such bounded state exists at a given A’, as is evident from the Shilnikov functional form of Fs and Fb (see Glendinning and Sparrow, 1984, for details). Every other one can be shown to be stable from simple arguments. The shortest bounded state is the one at A, = 1.208 in Figure 10. It is unstable, as is expected because we observe from the numerical experiments that two extremely close pulses will coalesce instead of repelling each other and form a bounded state. Our first order

40

Hsueh-Chia Chang and Eugeny A. Demekhin

theory here was unable to determine A’ specifically, but a second order one would allow an estimate of A’ = 1.208 - 1.216 = -0.008, obtained numerically. This particular bounded state will play an important role in our theory. The other two-hump bounded states with larger L* are not shown in Figure 10. As mentioned earlier, however, the Shilnikov functional form of the Melnikov function suggests that their unfolded branch through the generalized KS equation (3.3) will have the same distinctive wiggles shown in Figure 10. These simple arguments can also be extended to a train of pulses whose dynamics is governed by

A periodic train with equally spaced pulses is then defined by A’

=

F f ( L * )- F,(L*)

which clearly has the same structure for Shilnikov bifurcation. The wiggles of the periodic wave branches of the KS equation and the unfolded KS equation (4.3) have been shown numerically by Chang et al. (1993b), but their origin from a single solitary wave can be understood from the preceding simple analysis. Its mathematical origin stems from the real and complex eigenvalues of the fixed point, which give rise to the Melnikov functions of (4.15). For a more detailed description of this bifurcation analysis, see Glendinning and Sparrow (1984). For its application to the generalized KS equations, consult the recent review by Balmforth (1993, which also includes a review on the use of coherent structure theory to generate irregularly spaced pulses that all travel at the same speed (Balmforth et af., 1993)-irregular multihurnp bounded states. Because A,, = (&, (p2) = m, the substrate/speed mode & of a KS solitary wave can never be excited. When the IBL, BL, and NS equations are used, A,, does not blow up if a proper weight is used in the inner product (Chang, Demekhin, and Kalaidin, 19951, and we can include interfacial disturbances that do not decay to 0 at the infinities, which affect the speed and amplitude of the solitary wave. For the KS equation, however, as Balmforth et af. (1993) have realized, the solitary waves are inertialess-they can never accelerate! Acceleration or growth/decay in response to step disturbances in the film height is suppressed. We shall show that this missing speed mode is essential for the dynamics seen in Figure 1.

Solitary Wave Formation and Dynamics on Falling Films

41

There is yet another problem with the simple KS equation. We have assumed that the two neutral modes, which do not alter the shape of the solitary wave but their position and speed-amplitude, are the dominant modes of the solitary pulse, although the speed mode has been shown to be unexcitable. The other eigenvalues of the operator L , which correspond to shape-distorting eigenfunctions, have been assumed to be stable. We realize, of course, that there will be a continuous subset of the spectrum whose eigenfunctions approach finite-amplitude oscillations at X = Jr to, and we expect part of this continuous spectrum to be unstable because the flat films far from the pulse are unstable to periodic disturbances of certain wavelength. Nevertheless, we expect these “radiation” modes to account for the fluctuations seen by Liu and Gollub (1994) near the bow waves. They are swept up by the faster moving humps and are hence hopefully unimportant to the dynamics of the pulses. We have, however, also assumed that all localized eigenfunctions that decay to zero in both directions yield stable eigenvalues p,,. These localized disturbances correspond to point (discrete) eigenvalues, and their stability must be satisfied to ensure the integrity of the humps. This is, unfortunately, not true for the KS equation. In Figure 13, we plot the computed two leading localized eigenvalues of the one-hump solitary wave, other than the two neutral ones, of the IBL equation as a function of 6. As is evident in the low 6 limit of the KS equation, this mode becomes unstable. The radiation modes proved to be destabilizing also at the KS limit (Chang, Demekhin, and Kopelevich, 1995). This explains why integration of the KS equation (Frisch et al., 1986; Jayaprakash et al., 1993) does not show dynamics

-0.1 -0.2 -0.3

-0.4 -0.5 -0.6 I

1

0.02

0.03

I 0.04

I

I

I

0.05

0.06

0.07

FIG. 13. Variation of the point spectra of the solitary waves of the IBL equation in Figure 11. Coherent structure theory is valid only for 6 > 0.02.

Hsueh-Chia Chang and Eugeny A . Demekhin

42

dominated by the solitary pulses of Figure 10. Instead, irregular cellular structures unlike the solitary waves are observed. Although Toh (1987) has still tried to use the solitary waves to derive the KS spectrum and others have attempted to derive other statistical description of KS chaos based on this theory, application of coherent structure theory to the KS equation is approximate at best. The multihump solitary waves, the periodic trains, and the aperiodic steady trains are all unstable because each individual hump will disintegrate. We note that the extended KS equation of (4.3) does indeed have stable pulses at a sufficiently large 6 ' and coherent structure theory is quite appropriate for it (Kawahara and Toh, 1988; Balmforth, 1999, although the substrate mode is still unexcitable.

c. COALESCENCE, TRANSITION STATE,

AND

DYNAMICS

From Figure 13, we note that the solitary humps become stable to the full problem when 6 is sufficiently large. Hence, coherent structure theory becomes appropriate for the large solitary waves of Figure 11, which is not surprising in view of our numerical experiment in Figure 1. The machinery for reducing the dynamics to dynamical systems like (4.13) becomes more elaborate due to the increased complexity of the model equations. We shall demonstrate the coherent structure theory for high-inertia films with the IBL equations for convenience. The same approach can also be applied to the BL or NS equations, albeit with considerable more algebraic and numerical effort. Let the solitary wave solution of the IBL (2.191, first constructed by Tsvelodub (1980) be given by ( q * ,h * ) ( x ;6 ) where the dependence on 6 is also denoted. The values of ( q * ,h*) approach unity as x approaches +=. Then the linear operator dictating the stability of the solitary wave is (4.16)

where

* + 56 h*-du3 + h:xx + 1 + 2q*/h*3

9= c-

-1;"2=

-

-1

1 6 d 2q* .56h*2 - 5*[(TI

--

Solitary Wave Formation and Dynamics on Falling Films

43

The null eigenfunction and generalized null eigenfunction are simply $, = (q:, h: ) and Q2 = (q;, hg 1, where ch: = q: by the kinematic condition. If we freeze the 6 value at a reference value and vary the substrate thickness from unity, the solitary wave family of Figure 11 would then be parameterized by the substrate thickness X , which is analogous to @ of (4.6). This can be done because the steady version of (2.19) in the moving frame with speed c is invariant to the transformation

h(x; 6 , x)

+

q(x; 6 , X )

+ /&(xx-1/3;

c(6, X )

+

X h ( x X - 1 / 3 ., 6*11/3,1)

6x"/3,1)

(4.17)

/yZc(6/y1'/3,1)

where x is the substrate thickness at infinity. This symmetry has the same origin and replaces (4.6) of the KS equation. In Figure 11, where we have set y, to 1, the speed c of each member is then a function of 6. Alternatively, the symmetry of (4.17) implies that the same family can be generated by holding 6 constant and vary X-it is a one-parameter family. The speed then would be a function of the substrate thickness c( x). One can also parameterize the family by c and solve for y, or 6. The derivative d c / d X is approximately 3 by (4.4). The generalized eigenfunction
0

= (P2

@(PI

(4.18)

where ($1,

(P,) = ( $ 2 ,

($,, $5)

= ($2,

=

1

(PI) =

0

(Pz)

where the inner product is now defined for vector functions with a proper weighting function (Chang, Demekhin, and Kalaidin, 1995). Because of this vectorial form due to the inclusion of fluid inertia and the use of is now bounded and normalized to unity. We weighting functions, (p2, also note that the null eigenfunction and the null adjoint are actually orthogonal to each other, which is again distinct from the KS equation.

44

Hsueh-Chia Chang and Evgeny A. Dernekhin

This leads to a generalized adjoint eigenfunction and the different indexing of the adjoint eigenfunctions here. The h and q components of the vector eigenfunction cp,(x) are shown in Figure 14. Like (p, of the KS solitary wave, they approach constants of opposite sign at the infinities. Note also the intrinsic mass balance condition ch, = q, yields the conjugate condition h = -cq observed in the two components of cp2(x). The adjoint cp,(x) allows the projection of disturbances that do not decay to 0 at the infinities. In particular, it measures the effect of a distant step change in the substrate thickness, which introduces an impulse change in the flow rate, as we shall demonstrate, on a solitary wave. It is impossible for us to ascertain the location of this impulse but the fact that cpz approaches a constant value rapidly outside the pulse implies that the exact location is unimportant. As long as the jump is not right at the pulse, its effects on the amplitude and speed of the pulse are position indepen-

1

104

5-

+I

-0.1

1

I'\I'

1

-I

I

-0.3 -0.4

-5

-20

0

20

40

-20

0

20

40

I 0

I 5

10

15

20

25

30

35

40

X

6 = 0.07

14. The null adjoint eigenfunction cp2 the generalized eigenfunction d h / & . FIG.

=

( h , q ) , the null eigenfunction dh/din, and

Solitary Wave Formation and Dynamics on Falling Films

4s

dent. In this manner, excitation of the substrate mode involves long-range interaction between the solitary pulse and a step change in the height. We shall first focus on the interaction between two solitary waves. Like in the KS equation, the repulsion felt by a solitary wave from another one in front of it on the same substrate is weak. The computed values of m , and m2 in the Melnikov functions of (4.15) never exceeded 0.3 and decreases with increasing 6 for the IBL equation. Hence, long-range dynamics is dominated by the speed/substrate mode. This is consistent with Liu and Gollub’s observation (1994) that the solitary humps are almost independent, except when they are very close to each other. However, the excited speed/substrate mode also triggers the position mode due to the coupling through the Jordan form in (4.13). This position dynamics is initiated not by the short-range interaction of solitary waves but by excitation of the substrate thickness. A substrate correction occurs when mass is added to a solitary wave after coalescence, for example. If too much mass is added, the liquid mass can break up into several pulses (Alekseenko et al., 1985). The mass built up after coalescence between two pulses seems to be too small and only a single, larger pulse results. The liquid mass after coalescence quickly approaches the shape of a solitary wave because the slave modes that distort its shape quickly decay away as indicated by their eigenvalues in Figure 13. The remaining two masters modes, the substrate and translational modes, alter the baseline or shift the hump with corresponding quasi-steady change to its shape. Because mass has been added to the solitary wave, the resulting “excited” pulse is not a solitary wave solution of the original substrate but of another one with a higher value. It then wants to decay by lowering its baseline to that of the surrounding substrate. This is done by expelling fluid from the back of the hump. The expulsion occurs at the back because the excited hump travels at a much higher speed than the original one. As a result, an evolving shock of the type depicted in Figure 15 results. It resembles a solitary hump perched on a hydraulic jump. However, it is not a stationary shock and its amplitude or substrate thickness decreases steadily downstream. The nonsteadiness is due to the fact that the difference in the mass flows at the two different substrate thicknesses is not compatible with the solitary wave speed, and a slow drainage of fluid from the hump must occur. This fluid is then expelled in the form of a small shock that propagates away from the hump as seen in Figures 16 and 17. Because the shape-distorting modes are all stable, the solitary hump perched on the jump behaves quasi-

Hsueh-Chia Chang and Eugeny A. Demekhin

46

FIG.15. An excited pulse with different substrates.

statically during the drainage process such that it simply evolves through the solitary wave solution of Figure 11. It is important to realize that the drainage is mainly from the solitary hump, and the back substrate thickness simply readjusts its thickness quasi-statically by expelling fluid in the form of the back-propagating shock. The exact location of the hydraulic jump is not important as long as it is not right on the hump. Except for the translation and decaying of the hump, the rest of the interface remains quasi-steady due to the stability of the shape-distorting slave modes. Consequently, the instantaneous drainage from the hump is always equal to the difference in the mass flows of the two substrates in the moving frame, regardless of where the hydraulic jump is located. This is reflected in the shape of the adjoint (p2(x), which governs the projection onto the substrate/speed mode. As is evident from Figure 14, cp2(x) decays to constant values immediately outside the pulse. It therefore measures the drainage rate from the pulse due to a step change in the substrate thickness.

1.6 1.4

1.2 1.o

0

20

40

60

80

100

FIG. 16. The decay of an excited pulse 1 toward a stationary saturated pulse 2 in a frame moving with the speed of the latter. Coalescence occurs resulting in a new excited pulse 3. Note the linear decay in the amplitude of pulse 1 and the higher back substrate of pulses 1 and 3.

Solitary Wave Formation and Dynamics on Falling Films

47

FIG. 17. Two numerical experiments showing coalescence and no coalescence when an excited pulse approaches a stationary saturated pulse. Note the small shocks that propagate backward in both cases.

That the solitary hump behaves quasi-steadily as it goes through the family of solitary waves in Figure 11 is evident in our numerical experiment of Figure 18. The speed and amplitude reference points are based on incompatible values and hence (4.4) is not satisfied but d c / d h is clearly close to 3 and c and h are linearly related as the substrate thins. To quantify this decay of an excited pulse, we realize that the hydraulic jump

48

7.0 0

Hsueh-Chia Chang and Evgeny A. Demekhin

I

I

I

I

10

20

30

40

50

I

60

0

I

I

I

I

1

10

20

30

40

50

I

.

60

FIG.18. Tracking the evolution of the speed c, amplitude h,,, and position x of an excited pulse. Curves 1 are the theoretical predictions and curves 2 are numerical results. The amplitude and speed decay exponentially in time but linearly in space with d c / d h approximately 3.

in Figure 15 causes a flow rate change of approximately (c" - 3)( y, - 1) for shocks, where co is the speed relative to the laboratory frame of an unperturbed solitary wave on the unit substrate in front and y, is the substrate thickness in the back. This introduces a source term - 6 ( x * ) ( c , - 3)(x - 1) to the kinematic equation of (2.19) in the moving frame: ah at

-

=

dh dx

c-

dq ax

- - - NX")(C,, -

3)( ,y

-

1)

(4.19)

The hydraulic jump location x * is unimportant due to the shape of the h component of ( p 2 ( x ) shown in Figure 14. We expand in terms of the two neutral modes of the solitary wave as usual:

The substrate mode can be replaced by the speed mode (qf ,hT ) and the deviation speed coefficient c^, which measures the perturbation from the nominal solitary wave speed, is approximately 3(x - 1) due to (4.4). Substituting (4.20) into the flow equation in (2.19) and the kinematic equation in

Solitury Wave Formation and Dynamics on Fulling Films (4.19), taking the inner product with d

-

-(:)

dt

=

0 (0

(p,

and

o)(:)

as before, one obtains

(p2

1 -

49

(y";)

(4.21)

where y p2(m)(cU- 3) measures the drainage rate out of the solitary hump and (p2(x)is the value of the h component of the null adjoint eigenfunction at positive infinity. The decay dynamics of the excited pulse is now clear from (4.21). The perturbation speed and the amplitude/substrate decays exponentially in time

t,

,y

-

I,

it

- exp(-yt)

(4.22)

and the pulse position varies as (4.23) where 3 0 ) denotes the initial speed. Such exponential long-range behavior in the time of the excited pulse in Figure 16 is clearly evident in Figure 18. In Figure 19, the decay of the initial excited pulse of Figure 16 and the decay of the coalesced pulse are both shown to obey (4.22) with the same exponent y . The faster transient immediately after coalescence corresponds to the decay of the shapc-distorting slave modes whose eigenvalues are shown in Figure 13. The excited hump that is formed after this transient with a higher back substrate is also clearly seen in Figure 16. The time behavior of (4.22) and (4.23) predict a linear decay in space

c;

- h(0)

-

yx,/C^(O)

(4.24)

which is also seen in Figures 16 and 18. The value of y is typically about 0.01, representing the long decay seen in the figures. If one includes the short-range forces that enter when the excited pulse is within a few pulsewidths of the unperturbed solitary wave in front, the dynamics becomes (4.25) where 1 = x , -x,) is the separation between the two pulses and f ( 1 ) = a exp( -m21) - ( p sin ul - c: cos ul)exp( - m , l ) is the short-range force analogous to Fh - F f of (4.15). In the resolution of (4.29, the bounded

50

Hsueh-Chia Chang and Eugeny A. Demekhin

FIG. 19. The amplitudes of the excited pulse and the coalesced pulse before and after the coalescence at t = 12.5 of Figure 16.

pair or two-hump solitary wave travels at the same speed as the one-hump solitary wave, and its equilibrium separation is defined by the fixed point (1,;) = (I*,0). The quantity I* satisfies f ( l * ) = 0. A simple computation shows that f ’ ( l * ) is negative, corresponding to instability with respect to 1 disturbances, and the two-hump fixed point of (4.25) is a saddle point as shown in Figure 20. The stable manifold demarcates the phase space into two regions: one where the excited pulse is not sufficiently large and is repelled after a close encounter, and one where the separation I decreases below I*. The coherent structure theory breaks down when 1 is below I* because the two pulses now overlap considerably. Coalescence occurs in this region. That the two-hump solitary wave is a “transition state” is clearly evident in Figures 16 and 17. It is also seen in Liu and Gollub’s experimental tracings (Liu and Gollub, 1994). In Figure 21, we have numerically excited a solitary wave exactly on the stable manifold. It is seen that the two-hump solitary wave is approached and remains coherent forever without the coalescence or repulsion of Figure 17.

Solitaly Wave Formation and b n a m i c s on Falling Films

51

FIG,20. The phase space for binary interaction. The two-hump bounded states serves as a transition state. The stable manifold can be estimated by (4.271, which is in good agreement with numerical experiments represented by the open circles for K O ) = 25.

The stable manifold can bc estimated analytically by exploiting the different length scales of the long-range and short-range forces. Away from the fixed point, 1 and c^ decay as the long-range dynamics of (4.22) and (4.23) such that

where c^(0)and 1(O) are the initial speed of the excited pulse and the initial separation. This is a good description of the stable manifold except within a small neighborhood near the two-hump fixed point at (l*,O) where the short-range forces become important. However, because K O ) >> 1* and 8 0 ) x=-0, it is a good approximation to assume (4.26) goes through the fixed point to obtain an estimate of the stable manifold and hence a

52

Hsueh-Chia Chang and Eugeny A. Demekhin

FIG. 21. Approach of an excited pulse toward a saturated one to form a two-hump bounded state. The initial condition corresponds exactly to the stable manifold of Figure 20.

Solitary Wave Formation and Dynamics on Falling Films

53

condition for coalescence

t(0)= y ( l ( 0 )- I * )

(4.27)

We have carried out a series of numerical experiments with an excited pulse at exactly l(0) = 25 units behind an unperturbed solitary wave. The initial amplitude of the excited wave can be correlated to 2(0) and the results of Figure 20 show that their critical magnitudes for coalescence are in good agreement with (4.27). The parameters y and I* are computed from the coherent structure theory for each S value. The two-hump solitary wave of Figure 21 corresponds to an initial excited pulse just on the border for coalescence. A simple criterion to predict how coherent structures interact with each other inelastically has hence been derived. It essentially means that the amplitude of the excited pulse will be unaffected by the front pulse and decay linearly in space with a slope of y , as is evident in Figure 16. If its amplitude is approximately that of the front unperturbed solitary wave at separation 1*, the two pulses will form a bounded two-hump solitary wave as in Figure 21. If it is bigger, coalescence results; and if it is smaller, repulsion ensues as seen in Figure 17.

V. Discussion and Future Work The binary interaction theory of the previous section can be used to construct a simple model for the dynamics seen in Figures 1 and 3. The coalescence of two identical pulses form one with about twice the original amplitude. Hence, the saturated wavelength in Figure 3 must correspond to the critical separation l,, at which a coalesced pulse with twice the amplitude of the steady pulse will decay to the steady pulse amplitude when the separation is I*; that is,

(5.1) A

where h,, is the amplitude of the unperturbed solitary wave measured from the substrate. In most cases, I* <( I,, and the saturated wavelength is satisfactorily estimated by (5.2) that is, about the range of the long decay. It is typically two orders of magnitude higher than the Nusselt film thickness.

Hsueh-Chia Chang and Eugeny A. Demekhin

54

The linear increase in x of the time-average separation can also be described by a simple model. The increase is a result of the separation 1 less than I,. Assuming that it is just below 1,, so that the coalesced pulse is approximately 2 k m after each coalescence. This coalesced pulse will decay but has just enough speed to capture the next one and form another coalesced wave with again an approximate amplitude of 2 k m . The process then repeats with the coalescence of the next pulse. With each elimination of a pulse, the average wavelength increases. Let L be the average separation between two excited pulses that are pursuing their equally spaced smaller neighbors in front whose speeds are unaffected by the coalescence, and let m be the initial number of small pulses between them. Then the average wavelength after the nth capture is 1, = L / ( m n ) I(1 + n / m ) for n << m , and 1 = L / m is the initial average separation. Now each capture corresponds to roughly the same elapsed time A t = l/Ac, where Ac is the average differential speed of the excited pulse and the regular pulses between captures. Hence, the elapsed time after n captures is t, = nAt = nl/Ac and the distance traveled by the excited pulse is x = c,t, = nlc,/Ac, where c, is the average speed of the excited pulse. Because the excited pulse is about twice as high as the regular pulse and hence twice as fact and its speed decays linearly in space according to our theory, c, 3/2c, where c is the speed of the regular pulse and Ac = c, - c = c/2. As a result,

-

-

1,

- 1[1 + x / 3 L ]

(5.3)

Hence, the slope of the increase in the average separation is directly correlated to the density of the excited pulses. This density, of course, is sensitive to the amplitude and bandwidth of the forcing and is determined during the subharmonic secondary instability stage. Correlation between the forcing statistics and L seems to be obtainable only through numerical experiments and it is a worthy open problem to pursue. The primitive models for I , and how 1 increases with x can also be improved with a more detailed statistical model. After a certain distance downstream, the three-dimensional disturbances that have been suppressed by the Squire selection mechanism begin to distort the crests of the pulses in the transverse direction. They cause the crests to pinch off in yet another highly nonstationary transition, involving three-dimensional wave patterns (Chang et al., 1994a). However, beyond this transition, three-dimensional localized structures, called scallop waues, in the form of crescent-shaped waves appear and begin to interact each

Solitary Wave Formation and Dynamics on Falling Films

FIG. 22. A three-dimensional coherent structure-the inclined plane channel.

scallop wave-observed

55

in our

other. A particular scallop wave captured on video in our laboratory is shown in Figure 22. The downstream cross-section of such waves resembles the two-dimensional solitary waves upstream but the crests bow into the crescent or scallop shapes. The scallop waves’ amplitude goes to 0 in the positive and negative downstream and transversed directions. The transverse length scale is much longer than the pulsewidth but still much shorter than the average separation between the structures where the film is extremely flat and waveless. We have observed these structures to also coalesce and, like their two-dimensional counterparts, the coalesced waves lack sufficient mass to break into two structures. Instead, they expel fluid in the back and swim toward the smaller ones in front. There is, hence, again an interaction dynamics dominated by coalescence that reduces the number of coherent structures. It is our goal to study this dynamics among scallop waves in the near future.

Acknowledgments

The work reported here was supported by NSF-PYI, TAPPI and the Department of Energy. We are indebted to our students M. Cheng, E. Kalaidin, S. Kalliadasis, D. I. Kopelevich, and M. Sangalli for the results

56

Hsueh-Chia Chang and Ergeny A. Demekhin

reported here. HCC would also like to thank H. K. Moffat and his colleagues at DAMTP, Cambridge, for their hospitality during the preparation of this review. References Ablowitz, M. J., and Segur, H. (1981). Solitons and the inclerse scattering transform. SIAM, Philadelphia. Alekseenko, S. V., Nakoryakov, V. E., and Pokusaev, B. G. (1985). Wave formation on a vertical falling liquid film. AIChE J . 31, 1446-1460. Anderson, K. E., and Behringer, R. P. (1990). Long timescales in traveling-wave convection patterns. Phys. Lett. A 145A. 323-338. Aubry, N., Holmes, P., Lumley, J . L., and Stone, E. (1988). The dynamics of coherent structures in the wall region of a turbulent boundary layer. J . Fluid Mech. 192, 115-173. Balmforth, N. J. (1995). Solitary waves and homoclinic orbits. Annu. Re[). Fluid Mech. 27, -373. Balmforth, N. J., Ierley, G. R., and Spiegel, E. A. (1993). Chaotic pulse trains. SLAM J . Appl. Math. 54, 1291-1334. Benjamin, T. B. (1957). Wave formation in laminar flow down an inclined plane. J . Fluid Mech. 2, 554-574. Chang, H.-C. (1986). Traveling waves on fluid interfaces: Normal form analysis of the Kuramoto-Sivashinsky equation. Phys. Fluids 29, 3142-3147. Chang, H.-C. (1989). Onset of nonlinear waves on falling films. Phys. FluidsA 1, 1314-1327. Chang, H.-C. (1994). Wave evolution on a falling film. Annu. Rec. Fluid Mech. 26, 1035136. Chang, H.-C., Demekhin, E. A., and Kopelevich, D. 1. (19931). Nonlinear evolution of waves on a vertically falling film. J . Fluid Mech. 250, 433-480. Chang, H.-C., Demekhin, E. A,, and Kopelevich, D. 1. (1993b). Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physicu D ( Ansterdum) 63,299-320. Chang, H.-C., Demekhin, E. A,, and Kopelevich, D. I . (1993~).Construction of stationary waves on a falling film. Comput. Mech. 11, 313-332. Chang, H.-C., Cheng, M., Demekhin, E. A,, and Kopelevich, D. 1. (1994a). Secondary and tertiary excitation of three-dimensional patterns on a falling film. J . Fluid Mech. 270, 2.5 1-275. Chang, H.-C., Cheng, M., Demekhin, E. A,, and Kalaidin, E. (1994b). Quasi-stationary wave evolution on a falling film. In: Nonlirieurinstability of nonpurullelflows (S. P. Lin, ed.). pp. 407-424, Springer-Verlag, New York. Chang, H.-C., Demekhin, E. A., and Kdlaidin, E. N. (1999. Interaction dynamics of solitary waves on a falling film. J . Fluid Mech. 294, 123-154. Chang, H.-C., Demekhin, E. A,, and Kopelevich, D. I. (1995). Stability of a solitary pulse against wavepacket disturbances in an active medium. Piiys. Re(,. Left. 75, 1747-1751. Cheng, M., and Chang, H.-C. (1990). A generalized sideband stability theory via center manifold projection. Phys. Fluids A 2, 1364- 1379. Cheng, M., and Chang, H.-C. (1992). Subharmonic instabilities of finite-amplitude monochromatic waves. Phys. Fluids A 4, 505-523. Cheng, M., and Chang, H.-C. (1995). Competition between sideband and subharmonic secondary instability on a falling film. P h y . Fluids 7, 34-54. Coullet, P., and Elphick, C. (1987). Topological defect dynamics and Melnikov’s theory. Phys. Lett. A 121A, 233-236.

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Liu, J., and Gollub, J. P. (1993). Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. 70, 2289-2292. Liu, J., Paul, J. D., Banilower, E., and Gollub, J. P. (1992). In: Proceedings of the First Experimental Chaos Conference (S. Vohra, M. Span, M. Shlesinger, L. M. Pecova, and W. Dittos, eds.). World Scientific, Singapore, pp. 225-239. Liu, J., and Gollub, J. P. (1994). Solitary wave dynamics of film flows. Phys. Fluids 6, 1702-1712. Liu, J., Paul, J. D., and Gollub, J. P. (1993). Measurement of the primary instab flows. J . Fluid Mech. 220, 69- 101. Moffat, H. K. (1989). Fixed points of turbulent dynamical systems and suppression of nonlinearity. In: Whither turbulence? Turbulence at the crossroads (J. L. Lumley, ed.). Springer-Verlag, Berlin, pp. 250-257. Nakaya, C. (1989). Waves on a viscous liquid film down a vertical wall. Phys. Fluids A 1, 1143-1154. Nakoryakov, V. E., Pokusaev, B. G., and Radev, K. B. (1985). Influence of waves on convective gas diffusion in falling liquid films. In: Hydrodynamics and heat and nzass transfer of free-surface flows. Siberian USSR Acad. Sci., Novosibirsk, pp. 5-32. Prokopiou, T., Cheng, M., and Chang, H.-C. (1991). Long waves on inclined films at high Reynolds number. J . Fluid Mech. 222, 665-691. Pumir, A., Manneville, P., and Pomeau, Y. (1983). On solitary waves running down an inclined plane. J . Fluid Mech. 135, 27-50. Qian, S., Lee, Y. C., and Chen, H. H. (1989). A study of nonlinear dynamical models of plasma turbulence. Phys. Fluids R 1. 87-98. Roberts, A. J. (1988). The application of center-manifold theory to the evolutions of systems which vary slowly in space. J . Ausf. Math. Soc. Ser. B 29, 480-500. Rubinstein, J., Sternberg, P., and Keller, J. B. (1993). Front interaction and nonhomogeneous equilibria for tristable reaction-diffusion equationa. SIAM J . Appl. Math. 53, 1669-1685. Salamon, T. R., Armstrong, R. C., and Brown, R. A. (1994). Traveling waves on inclined films: Numerical analysis by the finite element method. Phys. Fluids 6, 2202-2220. Sangalli, M. (1995). A study of weakly nonlinear waves in stratified fluid-fluid flows and distributed reactors. Ph.D. Thesis, University of Notre Dame, Notre Dame, IN. Sefik, B., and Unal, G. (1994). Travelling waves exhibiting spatio-temporal chaos on the surface of a vertically falling film. Int. J . Eng. Sci. 32, 721-742. Shkadov, W. Ya. (1967). Wave conditions in the flow of thin layer of a viscous liquid under the action of gravity. IzL:. Akad. Nuuk SSSR, Mekh. Zhidk. Gaza 1, 43-50. Sirovich, L. (1987). Turbulence and the dynamics of coherent structures: I, I1 and 111. &. J . Appl. Math. 45,561-590. Toh, S.(1987). Statistical model with localized structures describing the spatio-temporal chaos of Kuramoto-Sivashinsky equation. J . Phys. Soc. Jpn. 56, 949-962. Tsvelodub, 0.Yu. (1980). Steady traveling waves on a vertical film of fluid. IzL'.Akud. Nuuk SSSR, Mekh. Zhidk. Gaza 4, 142-146. Wu, J., Keolian, R., and Rudnick, I. (1984). Observation of a nonpropagating hydrodynamic soliton. Phys. Reu. Lett. 52, 1411-1424. Wu, T. Y.-T. (1987). Generation of upstream advancing solitons by moving disturbances. J . Fluid Mech. 184, 75-99. Yih, C.-S. (1963). Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321-334.

ADVANCES IN APPLIED MECHANICS, VOLUME 32

The Mechanism for Frequency Downshift in Nonlinear Wave Evolution NORDEN E. HUANG Oceun und Ice Branch Laboratory for Hydrospheric Processes NASA Gorldard Space Flight Center Greenbelt, Maryland

STEVEN R. LONG Obsenuiionul Science Branch Laborutory for Hydrospheric Processes NASA GSFC/ Wallops Flight Facility Wallops Island, Virginia

and ZHENG SHEN Department 01 Eurth and Planetary Sciences The Johns Hopkins University Bultimose, Maryland

........................................ I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. The Hilbert Transform: The Methodology . . . . . . . . . . . . . . . . . . . . 111. The Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Field Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract..

60

60

65 75

98 111 114 115 115

59 Copyright 0 1996 hy Academic Press, Inc All rights of reproduction in any form reserved. ISBN o-12-oozo32-7

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Norden E. Huang et a].

Abstract

It has long been recognized that the frequency downshift in the wave field evolution is a consequence of nonlinear wave-wave interactions and that the frequency downshift is also necessary for the wind wave field to grow. Yet the detailed mechanism for the frequency change is still unknown: Is the process continuous and gradual? Or, is the frequency of a wave train varying gradually and continuously? Recently, Huang et al. (1995) found that the frequency downshift for a narrow band wave train is through wave fusion, an event described as two waves merging to form one wave, or n waves merging to form n - 1 waves. The process was seen tobe local, abrupt, and discrete. Such an event cannot be studied by the traditional Fourier analysis. Using a Hilbert transform to produce the phase-amplitude diagram and the Hilbert Spectrum, we found that in addition to the narrow band waves, the wave fusion event could occur in finite band widths and wind wave fields as well; and it is indeed the mechanism responsible for the frequency downshift in nonlinear wave evolution in general. Specifically, the frequency downshift is an accumulation of wave fusion events, which is also the same phenomena of the “lost crest” observed by Lake and Yuen (1978), and the “crest pairing” observed by Ramamonjiarisoa and Mollo-Christensen (1979). We have made quantitative measure of this fusion through Hilbert analysis techniques. Other than the fusion process, the local frequency can have small variations due to the amplitude modulations. Because of the abrupt and discrete localized variations of wave frequency, a new paradigm is needed to describe the nonlinear wave evolution processes.

I. Introduction The starting point of modern water wave theory can be rightly placed at the discovery of the weakly nonlinear wave-wave interaction mechanism. This seminal idea was introduced by Phillips (1960), who discovered that, in addition to the nonlinear effects of bounded harmonic distortions, there were even weaker nonlinear interactions among different free wave components. Although this new nonlinear effect is an order of magnitude weaker than the self-interactions, its effects can be observed through an accumulation over the span of hundreds of wave periods in time, or hundreds of wavelengths in space. The most conspicuous result of this is

Frequency Downs/$

in Nonlinear WaL>eErlolution

61

the frequency downshift. Thus, the new nonlinear effect really governs the instability of the Stokes waves, the modulation of the wave envelope, and in fact, all of the wave field evolution processes. An alternative way to study the wave evolution is through the envelope function. Discovered independently by Benny and Newell (1967) and Zakharov (1968), the governing equation for the envelope was found to be the nonlinear Schrodinger equation,

in which A is the complex valued amplitude of the waves, w g and k,, are the frequency and wave number of the carrier waves, and x and t are the spatial and temporal variables, and i is This elegant approach has been extended later by various investigators including Chu and Mei (1970), Hashimoto and Ono (1972), Davey and Stewartson (19741, Yuen and Lake (19751, and Ablowitz and Segur (1979). Later, Phillips (1981) showed that the nonlinear Schrodinger equation could be derived as a consequence of the weakly nonlinear wave-wave interactions. Recently, Zakharov and Kuznetsov (1986) and Tracy et al. (1988) have shown that the shallow water waves are also governed by the same nonlinear Schrodinger equation. This connection between the water wave equation and the nonlinear Schrodinger equation has greatly inspired the wave community, chiefly because the inverse scattering transform method can give a closed form solution in the form of solitons. Furthermore, t h e close relationship between the wave envelope equation and the nonlinear Schrodinger equation has also expanded the horizons of wave study: The wave evolution problem is no longer limited to the water waves, for it has close analogies in plasma physics and nonlinear optics. As a result, the problem of wave evolution has received the close attention of the fluid mechanics researchers, physicists, and mathematicians recently, and it has held the center stage in modern wave studies. Appealing as the nonlinear SchrGdinger equation is, there is a subtle limitation to its applications: The nonlinear Schrodinger equation is derived based on an expansion around a constant frequency carrier wave. Although the carrier frequency can be modulated slightly, it cannot change to a different value totally. Yet the most conspicuous feature of water wave evolution is the frequency downshift. As the study of water waves became increasingly sophisticated, this limitation became more obvious and simply could not be ignored. To illustrate this point, let us examine

m.

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some typical examples of wave evolution. The first one is a laboratory experiment conducted by Lake et al. (19771, in which they generated a simple monochromatic wave train mechanically and studied its evolution down the channel. Their observations showed that the weakly nonlinear effects caused the wave envelope to modulate, but eventually to return to its original sinusoidal form (Yuen and Lake, 1982). Those experimental results were treated as the prima facie evidence of the recurrence predicted by the nonlinear Schrodinger equation. On a closer examination, however, one can easily see the frequency downshift from either the location of the peak frequency of the wave spectra or the number of waves in a given time span in the data. Interestingly, the frequency change in the process was precisely the amount predicted by the weakly nonlinear wave-wave interaction theory. Thus the frequency downshift is firmly linked to nonlinear wave interaction theory, yet the equation governing it remains a problem: Long before the wave returns to its initial sinusoidal state, the governing equation, the nonlinear Schrodinger equation, would cease to operate because of the frequency downshift. The second example is a group of spectra measured in the field by surface contour radar (Walsh er al., 19891, as shown in Figure 1. The gradual frequency downshift of the spectral peak is again clearly shown. This frequency downshift is necessary for the waves to grow: The downshift enables the waves to become longer; therefore, they could have the potential to grow higher. Otherwise, the waves would break and drain the accumulated wave energy into turbulence. Only by downshifting to a lower frequency and a longer wavelength can the waves avoid being too steep to be stable. As discussed above, the frequency downshift is not only a consequence of weak nonlinear wave-wave interaction, but also a necessary mechanism for wave growth under the action of the wind. The importance of frequency downshift is fully realized by the theoreticians as well as modelers. In fact, one of the key reasons for the present success of wave models can be attributed to having codified the nonlinear wave-wave interactions as formulated by Hasselmann (1962, 1963a, 1963b) or Zakharov (19681, that are directly responsible for the frequency downshift. Unfortunately, we still do not know the detailed mechanism of the frequency downshift process. In a narrow band wave field, the frequency downshift can be examined through the well-known Benjamin-Feir instability (1967). Subsequent experiments by Lake and Yuen (19781, Yuen and Lake (19821, and Huang et al. (1986) all indicated that the lower side lobe of the sidebands would

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Fiti. 1. Frequency spectra measured using the surface contour radar by Walsh el a/. (1989) with increasing fetch from Long Island. As the wave field grows, the peak frequency downshifts continuously to lower valucs.

grow and eventually overtake the main one and become the main energy containing peak. Although this movement of the peak location to a lower value has been regarded as frequency downshift, past investigations have never addressed the details of the processes: How is the wave energy transferred to the low frequency sideband? Specifically, when does the downshift process start? Does it start suddenly only after the lower sideband becomes the main energy-containing peak? Or, does it occur earlier and gradually and continuously? Furthermore, in the downshift process, do all the waves grow a little longer uniformly as the lower sideband grows? Or, do some waves grow more than others but only give the appearance of sideband growth as a mathematical consequence from Fourier analysis? Although most investigators have taken it for granted that the process of the frequency downshift is global, gradual, and continuous, some clues to the contrary have been reported by Lake and Yuen (1978) and Ramamonjiarisoa and Mollo-Christensen (1979). Lake and Yuen first reported “lost” peaks in the nonlinear modulations of mechanically generated waves. They proposed two possibilities, both associated with the steepness of the waves.

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The first scenario is for the frequency change to occur when the waves in the unmodulated state are sufficiently steep ( a k > 0.3). These waves will become highly modulated. When the modulation pushes some of the wave to attain unmanageable steepness values, local breaking would occur. Thus, downshifting is associated with wave breaking. The second scenario is for the frequency change to occur when the amplitude in the modulated state is reduced to 0, producing a wave crest which is simply “lost” to the wave train. Even though the two possibilities cover the two extreme states of the wave motions, their data could not show which of the two is the real one. Nor did they show the details of the frequency downshifting processes. They did, however, identify statistically that the frequency downshifted to the lower sideband of the Benjamin-Feir instability and also showed the corresponding increase in phase velocity. Later, Ramamonjiarisoa and Mollo-Christensen (1979) observed “crest pairing” as a sporadic event in the wind wave evolution process. The phenomenon was described as one crest simply overtaking the previous crest and forcing one of them to disappear. Through this “crest pairing,” a wave of twice the dominant wave period would be produced momentarily and the overall frequency would downshift. They identified t h e crest pairing as an event closely related to the sudden increase in group velocity and a sudden decrease in energy density, all occurring locally. They further claimed that the crest pairing phenomenon could occur only near breaking; therefore, the phenomenon could not be associated with linear or nonlinear wave theory. This is a very interesting observation, for if crest pairing indeed is responsible for frequency downshifting, the process would not be global, gradual, and continuous. Apparently, they have fully realized the implications of their observations, for they suggested succinctly that it would be unlikely to be able to use perturbation methods to model this sudden shift of frequency; an ad hoc transition rule would be required as an alternative. Although both observations were qualitative, these researchers indicated that two waves could fuse to become one, as if one crest were lost in the process. To study this process quantitatively, the traditional Fourier analysis would not be helpful, because in Fourier analysis the wave number and frequency have always been assumed to span the whole time domain. Consequently, the frequency distribution is uniform globally. Here, a new data analysis method is needed to examine the local frequency change as it occurs wave by wave. Fortunately, the Hilbert transform fulfills this requirement perfectly.

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11. The Hilbert Transform: The Methodology

The Hilbert transform has been widely used in theoretical mechanics (Muskhelishvili, 19531, geophysics (Tanner et al., 1979; Aki and Richards, 19801, and signal processing (Oppenheim and Schafer, 1975; Bendat and Piersol, 1986; Cohen, 1999. In theoretical mechanics, it is applied to elasticity and aerodynamics to solve boundary value problems, especially the ones with discontinuous coefficients and boundary conditions. In geophysics, it is used to detect phase shifts caused by supercritical reflection and transmission through geophysical discontinuities such as the plate boundary and ocean bottom. In data processing, it is used to calculate the envelope function and, more specifically, the envelope of the correlation function to evaluate the timc delays in energy propagation. In most applications, it is applied to data of narrow band width, or simple oscillatory functions (see, for example, Cohen, 1995). As a result, many users assume that the Hilbert transform can only be applied to narrow band signals. The Hilbert transform in fact can be applied to a very general class of functions belonging to L". With it one can examine the local properties of the time series. A problem arises only in the definition of the instantaneous frequency, to be discussed later. In the following paragraphs, we will give a brief description of the transform according to that given in Titchmarsh (1986). For any function f ( t >of LP class, we can have

where a(w>

b(w)

=

=

l - . -/ f(r)cos wrd7 I = -/ ,f(T)sin w r d r rr

-z

7T

-%

(3)

For this function f ( t ) , there is an allied integral, defined as g(t)

=

/I[b(w)cos 0

wt -

a(w)sin w t l d w (4)

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Equation (4) can be obtained easily from (2) by replacing a ( w ) and b ( w ) by b( w ) and -a( w ) . It can also be shown that the allied integral for g ( t ) is - f ( t ) . Thus the relationship between f ( t ) and g ( t ) is skew reciprocal. By definition, the Fourier transformations of , f ( t ) and its inverse are

(5)

Formally, we also have, from eqs. ( 3 ) and (5),

therefore,

Hence, the Fourier transform of g ( t ) will be G(w)

=

(8)

-iF(w)sgn w

where F( w ) and G( w ) are the Fourier transform of by definition,

f(t)

and g ( t ) . Again,

which is the Fourier transformation of l / t ; therefore, from eqs. (8) and (9), we have g ( t ) as the convolution of f ( t ) and ( - l/n-t):

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Or, we can write the alternative forms for eq. (10) as

(11)

where P indicates the Cauchy principal value. Equation (11) is the Cauchy-type integral. It is treated as a singular integral and applied to problems in theoretical mechanics by Muskhelishvili (1953). Sometimes, it is also used as the definition for the Hilbert transform, even though the reciprocity expressions given in eqs. (2) and (4) were first noticed by Hilbert, and the two functions so connected are now called the Hilbert transform. In the form of eq. ( 1 I), one can see that the Hilbert transform really emphasizes the properties of the function at the location t ; thus, it can be used to examine the local properties of a function. In practical applications, eq. (7) actually offers an easy way to write the algorithm for numerical computation of the Hilbert transform. Essentially, the practical implementation of the Hilbert transform can be achieved by using the following steps: Perform a Fourier transform of the data, and set all the Fourier coefficients with negative frequency to be zero. Multiply the result by two, and perform an inverse Fourier transform. The result is the complex valued Hilbert transform. More details of the numerical computation can be found in Bendat and Piersol (1986). As shown in Titchmarsh (1986), the requirement on the regularity of the function is quite general: It is valid for any function of L” class. Because f ( t ) and g ( t ) are related as a conjugate pair, they can be treated as the real and the imaginary parts of an analytic function Z ( t ) as

z(r) = f ( t )

+ ig(t>

=

A ( t ) exp[ix(t)l

(12)

Thus, the local amplitude of the analytic function is A(t)=

[f’W+ g2(t)1”2

(13)

and its phase is

According to Cohen (19951, the local frequency, w , can be defined as w=-

dX dl

(15)

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From eq. (12), it is clear that the instantaneous frequency is the frequency of a cosine wave that best fits the data locally. Analytically, this statement can be expressed as follows: Let the Fourier transform of z ( t ) be Z ( w ) , then

Therefore, by the stationary phase method, -( d

x - wt) = 0

df

(17)

gives the most contributions to Z( w ) locally. Hence eq. (15) follows. This is a much better justification for the definition of the instantaneous frequency than the one given in Cohen (1995), which invoked an averaging integral. Even with this justification, both the amplitude and the phase functions are single valued functions of time; thus, this definition can be applied only to a simple oscillatory function. As discussed in Huang et al. (19951, the simple oscillatory function restriction alone is not sufficient. An additional requirement of zero local mean is also necessary for a meaningful definition of the instantaneous frequency. This can be illustrated by a simple example, as follows. Let us consider the following single sine waves: f,(t)

=

sin

f2(f)

=

sin w,,r

w,t

+ 0.8

(18)

The phase functions are shown in Figure 2(a): The phase function for f,(t) is a straight line, while the phase function for f,(t) is a wavy line. The two lines are entwined together, indicating that the mean frequency of the two functions is identical. As for the instantaneous frequency, the value for fl(t) is a constant as expected, but that for f 2 ( t )has highly variable values. This problem of nonzero local mean becomes crucial when we encounter the data with smaller waves riding on larger waves. To overcome this difficulty, the signal has to be decomposed into simple oscillatory functions

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FIG. 2. Phase functions and instantaneous frequencies of single component sine waves with zero mean and with nonzero mean. (a) The phase function of the sine wave with zero mean is indicatcd by the straight (dashed) line, while the phase function of the sine waves with nonzero mean is a wavy (solid) line. (h) The instantaneous frequencies of the sine wave with zero mean are shown by a constant (dashed line), while the ones with a nonzero mean vary greatly (solid line). The mean frequency, however, is the same.

with zero local mean. For this, we will use what we call the Characteristic Scale Decomposition Method (CSDM) as a preprocessing step to extract exactly such simple oscillatory function elements from the total signal, as in Huang et al. (1995). The details of this method will be discussed in a separate paper. The essence of it is briefly summarized as follows: The procedure is to first find all the local extrema in the signal being analyzed. Then connect all the local maxima and minima separately by the cubic spline method to form the upper and lower envelopes of the signal. From these envelopes, one can find the local mean, rn,(t), say. The first simple oscillatory function extracted, C , ( t ) ,is then given by f ( t ) - rn,(t)= C J t )

(19)

Since rn,(t) still might contain information on the wave motion, it could still have local extrema. Thus, we repeat the above procedures by applying

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it to rn,(t). We will then get rn,(t) and C,(t> as the second tier mean and extracted component, respectively. Repeat this process again and again, and stop after n steps. Then, we will have

rn,,-,(t> - rn,(t>

=

C',(t>

when either rn,(t> or C,(t> is less than a preassigned criterion for stopping the procedure. Now, if we sum up eqs. (19) and (201, we have f(t>

=

C C , ( t ) + rn,,(t>

(21)

This completes the decomposition. By virtue of this decomposition, each subcomponent must be a simple oscillatory function, and it also must have a zero mean. These elements, however, are not necessarily narrow band in the Fourier sense. It is a simple oscillatory function signal in the sense that there is only one extrema between two zero-crossings. Most importantly, the elements so extracted are locally orthogonal, since by definition they represent the signals and their local means. Now, we should have no problem applying the Hilbert transform to all the components. After all the components have been through the Hilbert transform, we will have the local amplitudes and instantaneous frequencies. Then, the local amplitudes and frequencies can be combined into a joint distribution that can be contoured to produce a time-frequency distribution, here designated as the Hilbert spectrum. As shown by Huang et al. (199.51, the Hilbert spectrum has a better resolution in local frequency than the product of wavelet analysis. It is crucial at the beginning to point out that the local frequency, even when it exists, has a different meaning than the ones obtained through Fourier analysis. In the narrow band case, the frequency defined by the Hilbert transform would coincide with that from the Fourier analysis. In more general cases, any signal in Fourier analysis is expressed as a summation of sine and cosine components, each spanning the whole time interval:

Compared to eq. (12), one immediately sees the difference. From Fourier analysis, one would always get more than one wave component at a given

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time even for the simple oscillatory function obtained here. Therefore, there would usually be more than one frequency at a time, with each frequency component spanning the whole time interval. From the Hilbert transform, one would get only one wave component at a time. Therefore, there would usually be more than one frequency at a time, with each frequency component spanning the whole time interval. From the Hilbert transform, one would get only one frequency at a time even if the signal is not narrow band in the usual sense. In other words, the Hilbert transform gives the best local fit with a single sine or cosine wave to the signal; consequently, the amplitude and frequency make sense only locally. On the other hand, Fourier analysis gives the best fit of signal with a sum of sine and cosine waves, each spanning the whole data range. Because of this, the signal at a particular time will have exactly the same frequency contents as at any other time in the Fourier approach. Thus, the frequency content in the Fourier representation can make sense only if the signal is stationary. In fact, the homogeneity is a fundamental assumption of Fourier analysis. From the above discussion, one can see that the Hilbert transform departs drastically from the Fourier transform. For a transient signal with time varying frequency, Fourier analysis should not be used at all, as the condition violates the assumption of the Fourier transform. In that case, the frequency will contain spurious components introduced by the requirement that each component has to span the whole time domain. Under this condition, only the Hilbert transform should be used. Fourier analysis should be reserved for the global properties of stationary processes. To capture the sharp local frequency, there is a price to pay: The frequency so obtained could be very noisy, for the phase function could be a fractal function, as shown by Huang (1992), Huang et al. (1993). To circumvent this difficulty, one can smooth the data by filters before applying the Hilbert transform, as Shum and Melville (1984) have done. A n alternate is to avoid point-by-point differentiation by using a finite time step. Under this circumstance, the selection of the time step serves as a filter. Even with the finite time step, the results obtained should still be better than the bandpass filter used by Melville (1983). A more simple Hilbert transform has been used by Huang et al. (1992, 1993, 1999, Long et al. (1995) to study the nonstationary time series. The characteristic scale decomposition method (CSD) described here can increase the range of application of the Hilbert spectrum. Although the data we have here seem to be stationary, Huang et al. (1992, 1995) used the

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Hilbert transform to examine the variations of the local phase and frequency and found that the wave data do indeed involve time varying frequency. In addition to examining the frequency itself, Huang et al. (1995) introduced the phase-amplitude diagram to quantify the phase changes in the downshift process. The phase-amplitude diagram is based on the variations of the phase with respect to a fixed reference state and the local amplitude. As an indicator for the wave evolution, it helps us determine quantitative variations of the phase changes and their relationship with the amplitude modulations. Here we will use all the available tools to analyze wave records as nonstationary data. In the past, the Hilbert transform has been used by many investigators in wave studies. Tayfun and Lo (1989, 1990) used it to study the phase distribution and found, to the first order, the phase function with modulus 27r to be nearly uniformly distributed. Shum and Melville (1984) used it to study the joint amplitude and period distribution of ocean waves, in which the local period was defined as the inverse of the time derivative of the phase function. Most relevantly, however, are the two studies by Melville (1983) and Chereskin and Mollo-Christensen (1985). Melville (1983) used the Hilbert transform to study local frequency modulation and breakdown of uniform wave trains. However, he assumed that the Hilbert transform works only for narrowband data. Furthermore, he also assumed that the phenomena of interest in a nonlinear wave train are all located in the neighborhood of the fundamental frequency and its higher harmonics. As a result, he applied a bandpass filter to the data in the frequency bands defined by

where w,, is the peak frequency of the energy spectrum. The cases studied were all mechanically generated monochromatic wave trains of various amplitudes with a driving frequency at 2 Hz. Thus, the fundamental band was defined as 1 to 3 Hz. He concentrated on the variations of the instantaneous frequency and identified large phase jumps and “phase reversals”. It turns out that both the large phase jumps and the phase reversals are all artifacts of the Hilbert transform improperly applied, as discussed in Huang et al. (1995). This led him to associate the increasingly asymmetrical modulations in the harmonics and eventually the wave breaking as the mechanism for the frequency downshift.

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Other than Melville (1983), Chereskin and Mollo-Christensen (1985) again used the Hilbert transform to examine modulation and evolution of nonlinear gravity wave groups. Following Melville’s (1983) practice, they also bandpassed their data. The most important finding from their study was again the identification of the local modulational instability as the mechanism of “crest pairing.” They also observed the phase reversals in the wave group evolution and found local frequencies outside the passed band as did Melville (1983). Both of these observations should serve as indications for problems associated with the observation of the instantaneous frequency. Negative frequency, of course, does not make sense; the frequency values outside the filter band should not exist either, for they should have also been filtered out. The practice of preprocessing the data with Fourier band-pass filtering clearly introduces error. In Chereskin and Mollo-Christensen (19851, the largest phase jump identified was 2.3 radians, which was associated with crest pairing. If the crest pairing is caused by a phase jump of 2.3 radians, only a partial crest can be lost in the process. Such partial crest loss has also been inferred by Melville (1983) through the instantaneous frequency calculations, which led him to question crest pairing as an event of period doubling and conclude that “the most important question to be answered concerns their [crest pairings] possible role in the shift to lower frequency following wave breaking” remained unresolved. Melville (1983) relied on the instantaneous frequency, which amplified the local singularities of the phase function. Chereskin and Mollo-Christensen (1985) used the envelope solitons which did not show the frequency downshift as clearly as from a uniform reference state. They did make a more definite statement to link the phase jump with the frequency downshift. They identified these phase reversals as the cause of the frequency downshift, which we believe now to be only partially true. Most importantly, they suggested that the downshift process was irreversible. Huang et ul. (1995) studied the phase changes starting from a uniform reference state and quantified the crest pairing as a local loss of 271. in phase value, or a loss of exactly one wave. In this paper, we will use two methods to study the wave evolution processes. The first one is the phase-amplitude diagram method applied to the unprocessed data. This is a much better way to quantify the phase variations than the time derivative, which may be misleading. The actual values of phase change are related to the total number of waves in the raw wave elevation data. Effectively, the phase diagram method performs a

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quantitative zero-crossing analysis. We will show that the wave frequency downshift is achieved through a local process: Two waves combine and become one or three waves combine and become two. In general, n waves combine to become ( n - 1) waves. Other than the waves involved actively in the local process, all the other waves are left unaffected. Each time, only one wave will disappear locally within the time span of only one wave period. Following Huang et al. (1999, it is again designated here as a fusion process. In the initial two-to-one stage, it is identical to the lost crest, crest pairing, and the phase reversals reported before, but the subsequent variations are more general. We will show the process quantitatively through the phase function. We believe the fusion process is what makes the wave evolution described here drastically different from the accepted view of a global, gradual, and continuously varying process. It is thus on the contrary, that we describe the process as local, abrupt, and discrete. The frequency change either happens to only one wave at a time or is confined within one wave group. The cumulative effect of one of the waves disappearing suddenly within one wave period eventually produces the frequency downshift. We believe this is true not only for the laboratory waves but also for the random wave field in the open ocean. The second method is the characteristic scale decomposition (CSD) method and the Hilbert spectral analysis. We will use the Hilbert spectrum to study the frequency modulations at locations other than where the active fusion is taking place. In construction of the Hilbert spectrum, we have to preprocess the data by the CSD method. The emphasis on the Hilbert spectrum is that it allows us to show that the wave frequency can have some modulation other than the jump locations. These modulations might produce recurrence modulation patterns locally. Yet, throughout that modulation phase, the overall mean wave frequency should stay a constant. It is still unclear whether there is a threshold frequency or amplitude change to trigger the fusion process. The trend is that the fusion will occur when the amplitude modulation deepens to full amplitude scale, while the process is not at the point of zero amplitude. In the next sections, we will use both laboratory and field data to demonstrate the detailed frequency downshift process through wave fusion. The narrow band laboratory data illustrate the process clearly. The wide band field data, even random in nature, still retain enough of the fusion characteristics. We will discuss the laboratory data first.

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111. The Laboratory Experiment

The laboratory experiment was conducted in the NASA Air-Sea Interaction Research Facility located at NASA GSFC/Wallops Flight Facility, Wallops Island, Virginia. The facility is the same as that used in Huang and Long (19801, which consists of a wind-wave tank with a test section 91.5 cm in width, 122 cm in height, and 1830 cm in length, with an operating water depth at 75 cm. A more detailed description of the facility can be found in Long (1992). For the present experiments, only mechanically generated waves are used. The wave maker is driven by a signal generator at one end of the tank, and a beach to absorb the waves is located at the opposite end. For this study, the signal generator was set at 2.5 and 3.0 Hz. Both wave conditions have the same initial wave amplitude at 0.8 cm, which gives initial wave steepness, Ak,, at 0.20 and 0.29 respectively. Data on the surface elevation at eight stations along the length of the tank were collected as functions of time simultaneously for 1 min after the wave maker has run for more than 10 min to reach a steady state. Let us first discuss the data set used by Huang et al. (1995) with wave maker frequency of 2.5 Hz and the initial steepness at 0.20 first. To give an overall view, the raw data are shown in Figure 3, in which the vertical axis indicates the distance from the wave maker in meters with the signal in arbitrary units. As shown in the figure, the shortest fetch is around 3 m, while the longest fetch is around 15 m. The group velocity is 31.2 cm/sec for the wave at 2.5 Hz. For the typical station spacing of 1.5 m, it will take 5 sec for the wave group to propagate from one station to the next; therefore, to identify and follow a specific event, the time axis would have to be shifted accordingly. The raw data show that the waves at the shortest fetch are quite uniform, but they already have slight amplitude modulations. As the waves evolve, the amplitude modulations grow with the fetch. To examine the frequency variations, the spectra at stations 1, 4, 5, and 8 are shown in Figure 4. The spectral analysis shows only the sideband growth, and eventually, the lower lobe of the sideband overtakes the main peak to become the most energetic peak at station 8. Our tank is not long enough for us to examine the evolution any further. Will the sideband growth reverse and return to the original state? The details of the frequency modulation will be discussed later.

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FIG.3. Wave elevation data measured in the laboratory with wave maker set at 2.5 Hz and an initial steepness at 0.2. The vertical axis indicates the distance from the wave maker. Notice the increasing modulation of the wave envelope as the waves evolve.

With these data, the process of the frequency downshift will be examined with the Hilbert spectrum as in Huang et al. (1995). After applying the Hilbert transform to all the data, the unwrapped phase is plotted in Figure 5. As shown in eq. (151, the overall slope of the phase function should be the mean frequency. The decrease in the overall slope of the phase function indicates the decrease of the mean frequency. At this scale, the trend seems to suggest a global, gradual, and continuous change of the phase function or frequency. The slope of the first four stations, in fact, stays almost identical to that of the first station. The first time the phase function shows any deviation from this initially closely clustered group is station 5, but even this change in phase seems only to be able to give a small and gradual change in frequency. To examine the variations of phase in more detail, we will take two different approaches. The first approach is to examine the phase change relative to the mean at each station. Figure 6 is the detrended unwrapped phase functions for stations 2, 4, 5 , and 8. The phase changes for station 2 are limited to kO.4 rad around the mean. These fluctuations are the consequence of amplitude variations and the superharmonic distortions of the wave profiles. Even with these fluctuations, to the first order approxi-

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FIG. 4. Selected spectra from data at different stations shown in Figure 3: (a) station 1, (b) station 4, (c) station 5, and (d) station 8. Notice the energy density growth of the lower sideband. By station 8, it has overtaken the main peak.

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FIG.4.-continued.

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FIG. 5. The unwrapped phase functions for all eight stations from the data shown in Figure 3. At this scale, the variations of the phase functions seem to be gradual and continuous.

mation, the frequency is constant; therefore, the nonlinear Schrodinger equation could be used to model it with acceptable accuracy. Conceivably, a given pattern could repeat itself and give the impression of recurrence, as observed by Lake et al. (1977). As the amplitude modulations deepen, the frequency variations will increase, too. Somewhere between stations 4 and 5 , the phase function jumped by 27r as shown in Figure 6(b). Once the phase jump occurs, the nonlinear Schrodinger will cease to operate there. Our data also show that, after the jump, the phase would never jump back. Furthermore, the number of jumps also increase as the waves evolve, as shown in Figure 6(c) for station 8. The spectra of the phase deviations from the mean are shown in Figure 7. A prominent time scale at the frequency of the wave train, 2 Hz in our present case, can be easily identified before any jump has occurred. The 2 Hz peak, however, is not the most energetic, for a broad ramp at lower frequency can be seen. Longer data than what we have now are needed to resolve the peak. It will be investigated in the future. Once the jumps occur, the spectra assume a smooth power law form without any particular time scale. They are very similar to the wide band cases studied by Huang et al. (1993), an indication that the occurrence of the jumps is not predictable.

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FIG.6. The detrended unwrapped phase functions for the selected stations: (a) station 2; (b) stations 4 and 5; and (c) station 8. The range o f the phase function changes is most evident in (b), in which the first phase jumps occur at station 5. The phase jump is 2i7.

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The second approach is the one used by Huang et al. (1995), who have examined the phase changes by taking the difference between the phase functions of all stations with respect to that of the first station, which we call the initial and the reference state. The difference is shown in Figure 8. From this figure, we can see that the changes of the phase functions consist of a series of steps, with the sharp jumps confined in very short time spans. Thus, in seemingly stationary data, we find local transient phenomena. The characteristics of these type of jumps is the reason why the Hilbert transform is chosen for this study. The traditional ways of examining the Hilbert transform products are to plot either the amplitude envelope or the complex amplitude, as shown in Figure 9. In Figure 9(a), the complex amplitude from data at station 1 shows a regular winding pattern around the origin with each turn introducing an increase of 27r in phase. This pattern indicates a nearly sinusoidal oscillation. In Figure 9(b), however, the winding of the data from station 7 becomes irregular and forms local loops, which could bypass the origin. These are the “phase reversals” reported by Melville (1983). H e also noticed qualitatively that at the phase reversals, the data gave a large

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FIG.7. The spectra for the detrended unwrapped phase functions at stations 2, 4, 5, and 8. The spectra before the jumps first occurred show prominent time scales. Once the phase jumps start, the spectra show only a gentle slope with n o time scale.

negative frequency and wave number. He further asserted that these phase anomalies occurred always in the neighborhood of local minima in the wave amplitude and only in the breaking region. As discussed in the previous section on the methodology, we now know that the phase reversal is only an indicator of a riding wave, or multi-extrema, between two consecutive zero crossings. It could be the initial stage of a crest pairing, but it is neither a necessary nor sufficient condition for a complete crest pairing as discussed in Huang et al. (1995). To have a quantitative assessment of the phase anomalies, we decided to use the phase-amplitude diagram, a method new to water waves as introduced by Huang et al. (1995). The phase-amplitude diagram is a plot of the amplitude envelope as a function of the phase normalized by 2 ~ . Because the phase function increases monotonically, the only manageable way to plot it is to select the deviation of the phase from a reference state in order to amplify the local anomalies. Thus, the phase-amplitude diagram will utilize the two main products of the Hilbert transform, the

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FIG. 8. The phase function deviations from a reference state chosen as station 1. Notice the localized jumps of the phase functions at this scale. The amount of each phase jump is 2T.

amplitude and the phase. In applying this method, the selection of the reference is critical to the final presentation. We will return to this point later. Let us first try to quantify the phase anomalies with respect to that of the first station in phase-amplitude diagrams, as in Figure 10 for stations 2, 4, 5, 6, 7, and 8. The phase-amplitude diagram for station 2 is shown in Figure 10(a), in which we detect no large variation in either phase or amplitude. But as we move to station 4, even though the phase variation is still small, the amplitude variation becomes very large. By the time the wave reaches station 5, however, the phase starts to jump, which reflects the sudden shift of the phase values shown in Figure 8. Two interesting characteristics should be especially noted. First, the jumps are discrete, as reported by Huang et al. (1995). These phase changes are very similar to the phase dislocations found in shear flow by Browand and Ho (1987) and Huang (1992). The number of phase jumps increases as the wave propagates down the tank as shown in Figure 10(d) for station 6, and Figures 10(e) and (f) for stations 7 and 8. As the number of jumps increases, the phase function seems to become a continuous and almost smoothly sloped line. But, even for those cases, the amount of phase change in each jump is

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FIG.9. The complex amplitude plots. (a) For station 1, this complex amplitude diagram represents a mostly regular sinusoidal oscillation. (b) For station 7, this complex amplitude diagram represents a highly modulated oscillation. A local loop is also formed, as seen near the center, which is associated with the phase jump.

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Normalized Phase Radian/2*pi

FIG. 10. Selected phase-amplitude diagrams for data shown in Figure 3: (a) station I , (b) station 4, (c) station 5, (d) station 6 , (e) station 7, and (f) station 8. For station I , both amplitude and phase variation are small. For station 4, the amplitude variation becomes large although the phase function stays constant. For station 5, the phase starts to jump at the discrete 27r steps. The discrete jumps increase for stations 7 and 8.

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FIG.10.-continued.

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Fici. 10.-continued.

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still 27~.This 2 r jump is precisely the quantity needed in the phase change for a “lost crest” or “crest pairing.” It represents a loss of one whole wave rather than part of a wave. The local period doubling is consistent with the observation of Ramamonjiarisoa and Mollo-Christensen (1979). In the process, each wave acts exactly like a “quantum” or a “particle.” Surprisingly, the wave motion, although a macroscale phenomenon in a continuum media, still retains quantum or particle properties. The second characteristic to be especially noted is that the jumps from one phase value to the next always occurs at the relatively low amplitude locations. This agrees with one of the observations of Melville (1983). On the other hand, if the phase jumps indeed occur at the points with low amplitudqs, they cannot also be associated with or be a consequence of wave breaking, for the local breaking always occurs when the local steepness is too large. Low amplitude just cannot provide a steepness large enough for breaking. The phase jumps shown here are cumulative, counting relative to the base of the reference state. On careful examination, one can see that there are two types of phase jumps: (1) The simple one involving two waves fusing into one; and (2) the complex one involving n waves fusing into ( n - 1) waves. Now, let us examine these two types of jumps from the raw wave elevation data in detail at the phase jump points. The first case involves a simple jump. A section of the time series wave elevation data from station 5 is expanded and displayed in Figure 11 in a two-way comparison. When we superimposed the data from station 5 with its phase difference from station 1, we found the time domain associated with the jump in phase could be easily confined within a single wave. Then we superimposed the raw elevation data from station 5 on those from station 1, as shown in Figure ll(b). One can immediately see that all the wave peaks line up except at the location of the jump, where two waves fuse into one. Thus, in the time span of two waves at station 1 we find that later it contains only one wave locally at station 5. The period is doubled; the local frequency is, therefore, half the reference state. This local change in frequency also indicates group velocity change as reported by Melville (1983). With these results, we have provided a quantitative measure to the qualitative observations by Lake and Yuen (1978), Ramamonjiarisoa and Mollo-Christensen (1979); we have also removed the uncertainty expressed by Melville (1983) on whether the period indeed doubled at the “crest pairing.” Here we see that the crest pairing is clearly demonstrated as a combination of two waves into one, where the phase loss is precisely 2rr.

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a

I

4,

-3

'

4000

89

4200

4400

4600

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5000

I

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5400

5600

5200

5400

5600

Time in Seconds*100

b 4

-3 4000

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4400

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FIG.11. Detailed comparisons for selected section of data at station 5. (a) Raw data are superimposed on the corresponding phase function for the same time period. The time span covering the phase jump is confined within one wave period near the 49th second. (b) The same raw data are superimposed on the data from station 1, serving as a time scale here. The wave near the 49th second shows a clear two-to-one simple fusion event.

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After the initial two-to-one fusion, the process of evolution will continue in a different fashion to become the complex jump, as illustrated next. In Figure 12, the data from station 6 is superimposed on those of station 1. Since the group velocity is 31.2 cm/sec, the jump at station 5 shown in Figure ll(a) will take about 5 sec to propagate to station 6 shown here. The changes in frequency all show up toward the end of the data span selected here. Although there is a loss of only one wave in each phase jump event, there are two types of fusion at these three-phase jump locations. There is one simple jump shown between time span of 50 to 52 sec; and two complex jumps are shown between time spans 52 to 54 sec and beyond 54 sec, as can be identified from Figure 12(b). The simple jump is still two to one; there is a new 25-r phase loss. The complex jumps are between 52 to 54 sec and 54 sec and beyond. The jump at 52 to 54 is at the downstream location of the event shown in Figure 12(b). No new loss of phase values occurs, but the waves enter a new stage of adjustment: The two neighboring waves are trying to equalize their frequencies. The combination process is more than simple crest pairing; it is a process of n waves being fused into ( n - 1) waves. The local frequency will adjust in such a way that the fused waves in the local group will have the same frequency. We designate this type of jump as complex. The sum of the simple and complex jumps gives the final picture of wave evolution. Therefore, we decided to use a new term, waue fusion, to describe the phenomenon rather than the simple term of crest pairing. Based on these discussions, one can see that the jump at 54 sec and beyond is also a complex event: The waves are entering a readjustment stage. Somewhere upstream, there must be a simple event our widely spaced stations failed to record. Unfortunately, we did not attempt closely spaced measurements here. Melville (1983) did measure the wave elevation at a spacing of 8 cm and found the phase jumps to occur locally in space. So it is certainly possible to have such a jump in between stations 5 and 6. Having seen jumps localized in time, coupled with the jumps localized in space, we conclude that the phase and, therefore, the frequency changes are local. Both of these characteristics are drastic departures from the traditional view of the wave motions. Other than the limited time duration where fusion is underway, the mean frequency remains unchanged from the initial state. The fusion region expands discretely and locally, and eventually it covers the whole time axis. Each fusion can either be a crest pairing or a trough pairing. In the case studied here, we can see the local frequency from the trough changes as well. The subtle difference can be

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FIG. 12. Detailed comparisons for sclccted section of data at station 6. (a) Raw data are superimposed on the corresponding phase function for the same time period. Notice the time spans covering the phase jumps are confined either within one wave period or over one wave group near the end of the data. (b) The same raw data are superimposed on the data from station 1, serving as a time scale here. The wave near the 51st second shows a clear two-to-one simple fusion event. The wavcs beyond the 52nd second show complex three-to-two jump events.

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discerned in the complex amplitude diagram similar to that shown in Figure 9. The precise definition will be given later in the discussion section. To show that the phase jump is the cause of frequency downshift, we have to show that the frequency of the waves indeed have changed. The frequency of the waves can be measured in three independent methods. The first method uses the moments of the frequency spectra, in which we define the mean frequency as

where + ( w ) is the frequency spectrum, and w, is the cutoff frequency. Because the wave elevation data were measured by a contact probe, we have adopted a cutoff frequency at 13 Hz to minimize the potential complication of the surface tension. With this cutoff frequency, we have included the main peak and all the wave components up to its third harmonics. Although the mean frequency can be defined by higher moments, we decided to use only the lowest moment possible, for our interest here is in the main peak area. The second method is by using the phase function derived from the Hilbert transform. We can either obtain the total number of waves in the time series by dividing the final value of the unwrapped phase function by 27r or obtain the total number of waves lost by counting the total number of the 2.rr phase jumps from the phase-amplitude diagrams as given in Figure 10. Either way, the results give us the final number of waves in the record. In a way, the phase function method is equivalent to the zerocrossing method used widely in wave statistical studies (see, for example, Huang et al., 1990). The third method is just identifying the peak of the spectrum. This is the crudest way to identify the frequency downshift, but it is also the most widely adopted. Unfortunately, it is not very sensitive in defining the frequency variation. For example, in our present data the only spectrum showing a peak frequency change is the last station. Yet the wave is continuously evolving. Therefore, there is no place for this method in the detailed study of the frequency downshift processes, other than the crude applications in the field. We will not discuss this method any further.

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Having processed the frequency data with both the moment method and the phase method, we normalized the mean frequency at each station with respect to the value measured at station 1. The results are presented in Figure 13. Encumbered by the contribution from the higher frequency components, the mean frequency from the moment method shows some scattering at short fetches, and the amounts of the downshifts are also smaller than those obtained from the phase method. Frequency variations from both methods show downshifts, and the trends from both methods are similar: Downshift, starting around k , X = 200 or after 30 wavelengths, coincides with the first phase jump or the first fusion event. If we had a longer tank, the downshift would continue and reach the state realized by Lgke et al. (19771, when the lower sideband becomes the sole prominent energy peak. At that stage, the wave train could be uniform again with some amplitude modulations, and the normalized frequency would then be approximately w

_ - 1-Ak,

(25)

0 0

In our case, this value would be 0.8. The important point to be made here, however, is that the frequency downshift is the consequence of a discrete wave fusion process. Other than the discrete phase jumps, the phase values stay in a very narrow range as shown by the width of the line clusters in Figure 10. Although the phase jumps are the prominent features of the frequency changes, there are also the narrow-ranged phase fluctuations, which will be discussed here. In order to quantify the fluctuation, we proposed the Hilbert spectrum, which is defined as the joint energy-frequency-time distribution as discussed in Section 11. From eq. (211, any time series can be expressed as a group of space curves in the amplitude-frequency-time space. When these space curves are projected onto the frequency-time plane with the third axis representing the amplitude as a proxy for the local energy density, we have a time varying spectrum. Because the Hilbert transform gives a single-valued function for frequency and amplitude, the raw Hilbert spectrum has values only along the lines representing the components from the CSD method’s process. A Laplacian filter is applied to produce a smoothly contoured spectrum similar to the wavelet results.

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Normalized Distance (k,,X) FIG. 13. The frequency variations as measured by the moments of the Fourier spectrum ( + ) and by the phase jumps (0). Both measures indicate a general trend of downshift long before the peak location changes.

As our data is collected with a contact probe, we cannot detect any wave that is capillary-force dominated. Therefore, a frequency cutoff at 13 Hz is reasonable. With the digitizing rate at 100 Hz, using every fifth point to determine frequency will smooth the result and will still give us a frequency resolution of 20 Hz, that is nearly twice the frequency we can possibly detect. The final amplitude contours in linear scale are produced by a 5 X 5 Laplacian filter less the corner points to give the Hilbert spectra shown in Figure 14 (see color plates) for stations 2, 5, and 6, respectively. The corresponding wavelet spectra with the Morlet wavelet are given in Figure 15 (see color plates). Comparing the products from the two methods, one can see that the Hilbert spectra give a much sharper result and local information that also makes direct physical sense. Both results show the appearance of the higher harmonics produced by the growth of individual wave amplitude through modulations. The amplitude modulations also cause the mean wave frequency to vary. As a consequence, the mean frequency band is

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widened as the waves evolve. The widening is less prominent in the Hilbert spectra than in the wavelet analysis results. For the wavelet analysis to show local change at a fusion event, we have to examine the high frequency range, for only there is the wavelet information local. Yet since the energy is moving toward a lower frequency, the indication of the event at higher frequency is counter intuitive. Thus the wavelet results not only make little physical sense, but also render the final product difficult to interpret. From a phyiscal point of view, the most important difference should be in the low frequency range. The wavelet results actually show a decrease in the low frequency energy density at the locations of the phase jumps. The Hilbert spectra, on the other hand, clearly show the localized frequency variations at the precise locations of the phase jumps. While the mean value of the frequency may have remained relatively unchanged over a narrow range, there are fluctuations. Another way to look at the Hilbert spectrum is to examine the marginal function by integrating the spectrum with respect to time. The result is a frequency-amplitude distribution as shown in Figures 16(a-c). Here the marginal spectra are similar to the ordinary Fourier spectra. By virtue of the decomposition method used, the resulting spectra reveal very rich information on the low frequency range, which suggests the existence of the subharmonics (Longuet-Higgins, 1978). These low frequency components increase with the modulation as the waves evolve. The full dynamical implication needs to be explored in the future. Now, let us examine another laboratory case with an initial frequency of 3 Hz and a higher initial wave steepness of Ak,, = 0.29. Data of the surface elevation for all eight stations along the length of the tank are shown in Figure 17 in the same format as that of Figure 3. Four selected spectra from stations 1, 6, 7, and 8 are shown in Figure 18. Because of the steeper initial waves, the modulation is stronger. By the time the waves reached station 6, the spectrum is almost wide band; yet the peak is still at the initial value of 3 Hz. The spectrum at station 7 show a drastic change of the peak location: It is downshifted to nearly 2 Hz. To examine this set of data with Hilbert transform method, the unwrapped phase functions of all eight stations referenced to that of station 1 are summarized in Figure 19. For the first six stations the reference can still be traced to the frequency of the first station. But, because of the downshift of the spectral peak at stations 7 and 8 and increases in the spectral band, the phase difference to the first station shows a general slope very different from that of the original one.

Frequency (Hz) FIG.16. The marginal spectra of amplitude-frequency from the Hilbert spectra: (a) station 2, (b) station 5 , and (c) station 6. The marginal spectra show the gradual growth of the low frequency components as the wave train evolves. The energy of the low frequency component suggests the subharmonic instability.

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Frequency (Hz) FIG.1h.-consinued.

Four phase-amplitude diagrams for station 3 , 6, 7, and 8 referencing station 1 are shown in Figures 2Na-d). As the frequency shifted, the phase-amplitude diagrams for stations 7 and 8, shown in Figure 15, no longer show any clear 2.rr jumps. Here we encounter the difficulty of the reference state, as mentioned before. Clearly, the mean frequency has shifted, as shown in the spectra as well as the phase variations in Figure 19. If a new reference state is selected, the phase-amplitude diagram would be different. Only in the one with the proper reference can one see the regular jumps. After some trials, we selected a new reference state at 2.11 Hz, which is the peak frequency of the spectrum. The phase difference from this new reference is shown in Figure 21, and the corresponding phase-amplitude diagram is shown in Figure 22. Here the jumps return to the 2.rr intervals. This critical dependence of reference state also clarifies the frequency downshift process further. The downshift is accomplished through fusion in two stages. The initial stage is again the two-to-one fusion. When this longer wave interacts with the third wave and forms the three-to-two wave fusion, the frequency of the fused wave would become almost equal. As

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T i m i n Seconds/100

FIG. 17. Wave elevation data measured in the laboratory with wave maker set at 3.0 Hz and an initial steepness at 0.29. The vertical axis indicates the distance from the wave maker. Notice the development of even faster modulation of the wave envelope as the waves evolve than in the data shown in Figure 3.

this process continues, to the state of n fusing into ( n - 1) waves, the internal readjustment of frequency will eventually make a majority of the waves become longer and the Fourier spectrum would shift downward. At this time, the reference state would have to be changed. The selection of the reference for a wide band spectrum data, as in the case here, should be done with great care. So far, the laboratory cases show us that the waves with an initially narrow band spectrum can evolve into a finite band width one. Yet the process of wave fusion is still clearly illustrated. Next we will examine the random wind waves in the field to see how they evolve. Do they also evolve through this fusion process? This will be the subject of the next section.

IV. The Field Experiment The field data were collected during the surface wave dynamics experiment (SWADE) off the Virginia Coast in October 1990. The data used here are the identical set used in Huang et ul. (1992); they were collected

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FIG. 18. Selected spectra from data shown in Figure 17: (a) station 1; (b) station 6; (c) station 7; and (d) station 8. Notice the energy density growth of the lower sideband. By

station 6, the spectrum becomes almost finite in band width, but the main peak is still located at 3 Hz. By stations 7 and 8, the lower sideband has overtaken the main peak.

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Frequency

FIG. 18.--conrtnued.

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101

Time SectindsllO

FIG.19. The phase function deviation from a reference state chosen as station I , from the data shown in Figure 17. The localized jumps f o r the first six stations are similar to that in Figure 8. But the phase functions for the last two stations show a general slope and jumps indicating both the downshift of the pcak frequency and the effects of finite band width.

with a 3-m buoy that recordcd the heave, pitch, and roll motions of the buoy. Wave elevation is obtained by double integration of the vertical acceleration. This type of data has always been treated as a stationary process. We will examine the data in detail to show that even in this seemingly stationary data, abrupt local changes are common. Now, let us first look at the raw data set given in Figure 23, which is digitized at 1 Hz, for a total of 90 min. Traditionally, this type of data is treated as a stationary random variable, and the Fourier analysis is applied routinely to obtain the spectrum. But Huang et al. (1992) have shown that these data are actually nonstationary, with time varying frequency. The variations in frequency can be seen from the unwrapped phase function as given in Figure 24. Presented in this format, the phase function looked rather smooth. The mean slope of the data is 0.7 rad/sec, which is equivalent to a mean frequency of 0.1 Hz or a mean period of 10 sec. After detrending the phase, the residual, or the deviation from the mean, is given in Figure 25. This function is very jagged; it has been shown to have the properties of a fractal function (Huang et al., 1992; Berry and Lewis, 1980).

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Normalized Phase radiani2*pi

Normalized Phase radiani2*pi

FIG.20. Selected phase-amplitude diagrams for data shown in Figure 17: (a) station 3; (b) station 6; (c) station 7; and (d) station 8. For station 3, the phase starts to jump at the discrete 27r steps. This regular pattern persists to station 6. By station 7 and 8, the combined effects of frequency downshift and the finite bandwidth make the reference state at the station unuseable. The discrete jumps are no longer at the discrete 27r steps, when thus referenced.

Frequency Downshift in Nonlinear War:e Euolution

Normalized Phase radian/2*pi

Normalized Phase radiadZ*pi

FIG.20.-continued.

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Time second\* 100

FIG.21. The phase function deviation from a reference state chosen as at 2.11 Hz. Notice the localized jumps of the phase function again show the discrete 2n steps.

Nomlalized Frequency (cycles)

FIG.22. The phase-amplitude diagram for station 8 referenced to 2.1 1 Hz. The discrete 2n steps show up again. This figure illustrates the sensitivity of the selection of the reference state for the construction of the phase-amplitude diagrams.

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Time (sec)

23. Raw wind wave elevation data measured with a 3-m buoy from SWADE (Huang 1992). This data set represents typical wind wave data traditionally treated as a stationary random process. FIG.

el nl.,

Examining the residual detrended phase function along the time axis, one can also see that the positive and negative slope sections persisted for quite long time spans. Here any section with a positive slope indicates a higher local frequency than the mean; on the other hand, any section with a negative slope indicates a lower local frequency than the mean. The long runs of slope of one sign or the other mean that the local frequencies of the waves are not at all random but are quite coherent. Each run of the slope at a nearly straight line corresponds to a group of waves of similar frequency. Indeed, the nearly constant frequency for a number of waves is a necessary condition for the formation of wave groups. Furthermore, all the positive slope sections seem to have the same slope value of the phase function versus time; the negative slope sections show this as well. Let us examine the time varying frequency property of the data in even more detail. Figure 26 shows a section of the phase function expanded together with the raw surface elevation data. At subsections A and B, the wave frequencies are quite different. Section A resides on the upward slope of the residual phase function; therefore, the local frequencies are visibly higher. Section B resides on the downward slope of the residual

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Time in Seconds

FIG.24. The unwrapped phase function (solid line, -) for the data shown in Figure 23. A reference state with frequency of 0.0917 Hz (dashed line, - - - - - - ), and the difference between the phase function and the reference state (dotted line, ... ... ).

Time

in

Second,

FIG.25. The detrended unwrapped phase function, showing persistent positive and negative slope runs of the phase function.

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Time

in

107

Seconds

FIG.26. Selected data section togethcr with the detrended phase function. The differcncc of the apparent frequency in section A (corresponding to a positive slope run in thc phase) and section B (corresponding to a negative slope run in the phase) is obvious.

phase function; therefore, the local frequencies are visibly lower. To show this phenomenon more clearly, we computed the Hilbert spectrum after the CSD method was applied. The result for the same section as in Figure 2(b) is shown in Figure 27. The contours in this figure indeed indicate regions of local frequency coherence and an overall frequency fluctuation. Clearly, the whole wave data should be looked at as if they are frequency modulated as well as amplitude modulated. Therefore, the stationary assumption is not true at all: The wave data are random variables with time varying frequency. The variations of the frequency are consecutive in time but with no time scale, a case for which the Fourier analysis should not be applied. As a result, the interpretation of the Fourier spectrum should be examined critically. With a detailed examination of the residual phase, it is clear that the phase can maintain an overall mean constant slope for a considerable length in time before switching to another value. But there are only two prominent values. The lower of the two frequencies, determined by averaging the lower slope values, is 0.0917 Hz. From this reference, the deviation of the phase for a selected section of the results is shown in Figure 28. Again, in plotting the phase-amplitude diagram in Figure 29, we recovered

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Time (sec) FIG 27. The Hilbert spectrum for the field data given in Figure 23. Notice the changes in frequency from one time section to another.

a phase jump result similar to that from the finite bandwidth laboratory case. The result here is more random, yet the jumps of 255- are still clearly visible. Because the ocean wave fields usually consist of different wave systems from different storms, there may be more than one wave train coexisting at any given time and location. This may be the reason why the jumps appear somewhat random in the phase-amplitude diagram, yet the basic properties of localized abrupt phase changes in discrete amounts are still clear. Here again, let us demonstrate the sensitivity of the selection of the reference state. From Figure 24, it is tempting to adopt the mean frequency as the reference. If this reference is selected, the same data would give a phase-amplitude diagram, as shown in Figure 30. Here the lines are completely chaotic; one would not be able to see any order at all. Having seen some of the phase-amplitude diagrams, we can come back to the question of selection of the reference state, as the phase-amplitude diagrams used here are very sensitive to this selection. In most cases

Frequency Downshiji in Nonlinear Waiie Ervlution

10’)

Time in Seconds

FIG.28. The pha5e deviation reference to 0.0917 Hz. The discrete jump steps are clear now.

Phase Angle

wrt

Low Frequency Components in Radian5

FIG.29. Phase-amplitude diagram for the ficld data. Notice the 27r jumps indicated by the short vertical lines.

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Detrended Phase Angle

in

Radians

FIG.30. Phase-amplitude diagram for the field data reference to the mean frequency. No 27r jumps can be discerned. This again illustrates the importance of the reference state

sclection.

studied so far, the initial state has been selected as the reference. This worked well if the deviation from the initial state is small and if the spectrum is narrow band. Once the spectrum exhibits a finite band width, the choice of the reference will be much more difficult. A slightly different choice of the reference would totally mask the results, as shown in the case of the last example. It is, however, certain that the mean frequency of a finite band width cannot be the reference state, as demonstrated by both laboratory and field cases with finite band width. Another manifestation of the time varying frequency property of the ocean wave data is that data from the field can maintain correlation for a long time, for after 2 n v jumps, the waves will be in phase again. This is exactly the situation observed by Shen and Mei (1993). Using wavelet analysis, they found the correlation of any wave components to be persistent for as long as their data lasted. Based on the persistent coherence, they concluded that the wave spectra are multifractal functions. Although they speculated that the high correlation over large time separation was the result of large-scale motions, our present analysis indicates that the phase lock is actually due to the dynamics of the waves. The results of

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Shen and Mei (1993) not only lend further support to the present results, but also indicate that the phase jumps prevail in all wave data.

V. Discussions As frequency downshift is not only a natural phenomenon but also a critical process for ocean wave field evolution, we must understand it fully. Contrary to the traditional view, the present results are surprising in many aspects. The downshift of frequency is shown to be an accumulation of individual discrete jumps in phase. Although these jumps are not necessarily the frequency anomalies observed by Melville (1983), they are local events. Concentrating on frequency variations, Melville (1983) failed to realize the jumps are all at the same value of 27r. These events occur at a very early stage of the wave evolution. The first occurrence of jump for a narrow band case appears at k , , X = 200, where the spectrum shows nothing but moderate side band developments. Once the initial phase jump starts, the subsequent development is similar to the propagation of defects observed in shear instability (Browand and Ho, 1987; Huang, 1992) and in other electrohydrodynamic convection and liquid crystal stability phenomena (see, for example, Gaponov-Grekhov and Rabinovich, 1992). From our data, we have shown that the phase jump is an indication of local wave fusion and that it is the key mechanism for frequency downshift. But an additional step should be involved in the fusion of n waves into ( n - 1) waves: a readjustment process to make the fused waves to be of nearly uniform frequency. Based on our observations, all fusions occur at the points of local amplitude minima. In fact, for two waves to fuse, the process will have to pass through a stage of singularity in the local frequency defined by (15). But not all the singularities in the local frequency will result in a wave fusion event. Based on the definitions given for amplitude and phase in Eqs. (13) and (141, it can be shown that the equation relating them is simply

_ -

~

(26)

Here we would have two possible singularities. The first possibility is at f = 0, the zero-crossing points. But when f = 0, tan x would be infinite.

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The singularity is removed. Indeed the phase function suffers no difficulty at any zero-crossing point. The second possibility is at A = 0, which requires both the real and imaginary parts to be 0. For a narrow band signal, when the variation of A is small, this condition can be viewed as requiring both

f = 0 and

df -

rlt

=o

(27)

In other words, the singularity requires not just the signal to be 0, but also the local extremum to be 0. These conditions can be realized when either the local maximum or the local minimum is 0. The “crest pairing” is equivalent to the case of local minimum being 0; the “trough pairing” is equivalent to the case of local maximum being 0. At these precise times when a crest or a trough is lost in fusion, the local loop as seen near the center in Figure 9(b) would touch the origin. The wave profile, however, still shows two waves distinctly. If this local loop resides totally in the first quadrant of Figure 9(b), this would be a crest pairing event; if the local loop resides totally in the third quadrant, this would be a trough pairing occurrence. But the probability of recording the precise moment is not very good in practice. After that moment, the waves could move ever closer and fuse together. The final configuration will not have any singularity. Both cases can be easily identified in our data, as shown in Figure 12(b). For this reason, we prefer the term wave fusion. Although all the fusion events will pass through a singularity, not all the singularities will end up in a completed fusion. Two waves may get closer but then be separated. Therefore, a singularity in the phase function derivative cannot be treated as the crest pairing event, for the crests at this stage are still identifiable. As for the phase reversal, it could well be a consequence of riding waves. The wave fusion phenomenon observed here is different from the linear beating of two dispersive or nondispersive wave trains. Under the linear beating, at each nodal point, f is precisely 0. The Hilbert transform of the signal will show a phase jump confined to a single point, which is the mathematical consequence of the Hilbert transform. The jumps could be either n- or 2n- depending on the configuration of the wave profile. In our case, however, the jump is not confined to a point but spread over a whole wave period. Over the jump region, the amplitude is not identically 0, although it has the tendency to occur near the local low amplitude. The phase jump is more physical than mathematical, for after the jump, a wave

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of different period will merge. Unfortunately, we have not found the underlying dynamics involved in the process. It is tempting to invoke t h e action conservation law to explain the association of the jump near the low-amplitude region. The annihilation of a wave in fusion reveals particle properties. But a straightforward application of the conservation law is problematic here: The disappearance of a wave iriolates the kinematic conservation of the waves! For lack of spatial data, we cannot even determine the basic conservation laws across the jumps precisely. The process is much too complicated for us to draw any inference concerning the conservational laws. We do, however, believe that the overall conservation of wave action should hold. Future studies are needed. Knowing what is involved in a downshift, we can ask about the possibility of upshift. From the energy point of view, to achieve an upshift, the local energy density must incrcase based on the law of conservation of action. For lack of a local energy source, we do not expect a spontaneous upshift process. Based on our observations and those of others, we have not seen any upshift at all, without which the general type of recurrence would be impossible and the wave evolution should be irreversible. It should be pointed out, however, even without general type of recurrence, a special type of recurrence without frequency downshift is still possible, as shown by Lake et al. (1977). In the special recurrence, the wave pattern will repeat after the modulations undergo some variations without frequency downshift. Such phenomena can be realized only when the steepness of the wave is low, for then the nonlinear effects are extremely weak. Next, let us address the breaking events associated with downshift. There seems to be a perfistcnt belief that breaking is necessary for frequency downshift (Melville, 1983). Based on our observations we have found no evidence to support such a contention. A recent study by Hara and Mei (1991) has excluded breaking and obtained definite downshifting. Fully aware of this issue, we have kept the initial wave steepness values low to avoid breaking in our laboratory experiments. We have observed no obvious breaking. Furthermore, wave breaking is usually associated with local high values of amplitude in a group (Donelan et al., 1972). Based on our observations, all the phase jumps were seen to occur after the waves reach nearly full modulation or occur near the local low-amplitude region, which is not conducive to breaking at all. We believe that the frequency downshift is not a consequence of breaking; rather, it is the result of modulation. From our data, the location of the event can be traced from the instant when the modulation started to grow. Though the dynamics of

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the intriguing phase-amplitude coupling itself deserves more study; nevertheless, the fact that the phase jump is not associated with breaking is quite clear. The phase jumps show that the wave evolution process is not global, gradual, and continuous, but local, abrupt, and discrete. None of these properties is to be expected according to the accepted concept of wave motion. This contrast to the traditional beliefs leads us back to the nonlinear Schrodinger equation. Ever since the theoretical connection between the water wave governing equation and the nonlinear Schrodinger equation were established, experimenters have been looking for observational confirmations. Up to now, the only success for this observational confirmation came mostly from the shallow water waves by Osborne (1993a, b), Osborne and Petti (19931, and Osborne and Segr6 (1993). Even though the agreements are impressive, the theory requires that t h e carrier wave number and the frequency both remain constant. The situation in deep water is certainly different. Having seen the present results, we have to question whether the assumption is inconsistent with the observed frequency down shift. No doubt, the wave modulation prior to the first phase jump can be modeled by the weak nonlinear mechanism governing by the cubic Schrodinger equation. But, at the jumps, none of the available mathematical models can offer any help, for the current theoretical paradigm of water waves is based on slow and gradual variations due to a weak nonlinear mechanism. Thus we contend that new models are now needed to describe the wave evolution processes.

VI. Conclusions Using the Hilbert transform, we can examine the evolution of the weakly nonlinear wave trains in detail. We found two frequency variation modes. The first is a low-magnitude modulation which does not contribute to the frequency downshift and conceivably could reverse itself. But even this low-level modulation still violates the condition of the Schrodinger equation as given in Infeld and Rowlands (1990). The second mode is the discrete frequency jump. Although the frequency has always been assumed to be stationary globally, the Hilbert transform reveals that the frequency variations are abrupt and local. Such variations cannot be analyzed adequately with Fourier analysis. Furthermore, the local variations are also discrete, having the typical characteristics of particle or quantum varia-

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tions and the propagation of defects detected in many other hydrodynamic and electrohydrodynamic phenomena. This raises an interesting question: Do all wave motions exhibit quantum characteristics? We do not presently know. But we do notice that the frequency changes in the water waves are not symmetric: Of all the cases examined, not a single case of frequency upshift has been identified, which means that there is only a natural tendency for downshifting but no symmetric disposition for an upshift of frequency necessary for recurrent cycles. The frequency downshift is a cumulative effect of the fusion of two or more waves and a readjustment of local frequency. These processes are irreversible. This asymmetry is due to the preferred direction of downward energy flow. Our results also raise concerns about the past assumptions regarding slowly varying wave trains. It will be a new challenge to represent the abrupt and discrete changes analytically. These issues need to be studied in more detail in the future.

Acknowledgments We would like to express our thanks to Professors 0. M. Phillips of the Johns Hopkins University, T. Y. Wu of the California Institute of Technology, V. Zakharov and A. Newell of the University of Arizona, E. MolloChristensen, M. Shlesinger of ONR, and B. Lake and M. Caponi of TRW for their valuable comments and encouragement. We would also like to thank Dr. M. Donelan for the wave data from the SWADE experiment. This research is supported in part through the RTOP program from NASA (NEH and SRL) and in part by grants from both the Physical Oceanography and Coastal Programs (NEH, Z S ) of ONR.

References Ablowitz, M. J., and Segur, H. (1979). On the evolution of packets of water waves. J . Fluid Mech. 92, 691 -715. Aki, K., and Richards, P. G. (1980). Quuniiiatuv .seismologytheory und methods. Freeman, San Francisco. Bendat, J. S., and Piersol, A. G. (19x6). Random datu: Anuljsis and measurenzent procedures, 2nd ed. Wiley (Interscience), New York. Benjamin, T. B., and Feir, J. E. (1967). The disintegration of wave trains on deep water. 1. Theory. J . Fluid Mech. 21, 417-430. Benny, D. J., and Newell, A. C. (lY67). The propagation of nonlinear wave envelopes. J . Muih. Phy.7. 46, 133-139.

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Berry, M. V., and Lewis, 2. V. (1980). On the Weierstrass-Mandelbrot fractal function. Proc. R. Soc. London, Ser. A 370, 459-484. Browand, F. K., and Ho, C. M. (1987). Forced unbounded shear flows. Nucl. f'hy,s. B , Proc. Suppl. 2, 139-158. Chereskin, T. K., and Mollo-Christensen, E. (1985). Modulational dcvelopnient of nonlinear gravity-wave groups. J . Fluid Mech. 151, 365-377. Chu, V. C., and Mei, C. C. (1970). On the slowly varying Stokes waves. .I. Fluid Mech. 41, 873-887. Cohen, L. (1995). Time-Frequency Ana/ysis, Prentice-Hall, Englewood Cliffs, New Jersey. Davey, A,, and Stewartson, K. (1974). On three dimensional packets of surface waves. Proc. R. SOC.London, Ser. A 338, 101-110. Donclan, M. A., hnguet-Higgins, M. S., and Turner, J . S . (1972). Whitecaps. Nature (London)239, 449-451. Gaponov-Grekhov, A. V., and Rabinovich, M. I . ( 1992). Nonlineurities in action: Qscillutions, chaos, order, fractals. Springer-Verlag, Berlin. Hara, T., and Mei, C. C . (1991). Frequency downshift in narrowbandcd surface waves under the influence of wind. J . Fluid Mech. 230, 429-477. Hashimoto, H., and Ono, H. (1972). Nonlinear modulation of gravity wavcs. J . Pllys. Soc. Jpn. 33,805-811. Hasselmann, K. (1962). On the nonlinear energy transfer in a gravity wave spectrum. Part I. J . Fluid Meek. 12, 481-500. Hasselmann, K. (1963a). J . Fluid Mech. 15, 273-281. Hasselmann, K. (1963b3. J . Fluid Mech. 15, 385-398. Huang, N. E. (1992). Laboratory investigations of ocean surface roughness generation. In: Nonlineur dynamics of ocean wuclcs (A. Brandt, S. E. Rambcrg, and M. F. Shlcsinger, eds.). World Scientific, Singapore, pp. 128- 149. Huang, N. E., and Long, S. R. (1980). An experimental study of surface elevation probability distribution and statistics of wind generated waves. J . Fluid Mech. 101, 179-200. Huang, N. E., Long, S. R., and Bliven, L. F. (1986). An experimental study of thc statistical properties of wind-generated gravity waves (K. Hasselmann and 0. M. Phillips, eds.). In: Wacz dynamics and radio probing of the ocean s ~ ~ f a c ePlenum, . New York, pp. 129-144. Huang, N. E., Tung, C. C., and Long, S. R. (1990). The probability structure of the ocean surface. In: The sea: Ocean engineering science (B. Le Mehaute and D. M. Haincs. eds.), Vol. 9. Wiley, New York, pp. 335-366. Huang, N. E., Long, S. R., Tung, C. C., Donclan, M. A,, Yuan, Y., and Lai, R. J. (l9Y2). The local properties of ocean waves by the phase-time method. Geophys. Kes. L e t / . 19, 685-688. Huang, N. E., Long, S. R., and Tung, C. C. (1093). The local properties of transicnt stochastic data by the phase-time method (A. H.-D. Cheng and C. Y . Yang, eds.). In: Co~npu/utional stochastic mechanics, Vol. 1. Elsevier, Essex, England, pp. 253-279. Huang, N. E., Long, S. R., Lin, R. Q., and Shen, Z. (1995). Wave fusion as a mechanism for nonlinear evolution o f water waves. J . Fluid Meclz. (in press). Infeld, E., and Rowlands, G. (1990). Nonlinrur wai'es, solitons and chaos. Cambridge Univ. Press, New York. Lake, B. M., and Yuen, H. C. (1978). A new model for nonlinear gravity waves. Part 1. Physical model and experimental evidence. J . Fluid Mech. 88 Lake, B. M., Yuen, H. C., Rundgaldier, H., and Ferguson, W. E. (1977). Nonlinear deep-water waves: Theory and experiment. Part 2. Evolution of a continuous wave train. J . /+id Mech. 83, 49-74.

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Long, S. R. (1992). Wallops flight facility air-sea interaction research facility. NASA Ref. Pub/. RP-1227. Long, S. R., Huang, N. E., Tung, C:. C., Wu, M. L., Lin, R. Q., Mollo-Christcnsen. E., and Yuan, Y. (1995). lEEE Geoscience and Remote Sensing Soc. Newslerlrr 3, 6-11. Longuet-Higgens, M. S. (1978). Proc. R. Soc. 1,ond. A 360, 489-505. Melville, W. K. (1983). Wave modulation and breakdown. J . Fluid Mech. 128, 489-506. Muskhelishvili, N. I. (1953). Singular in/cgu/ equations. Noordhoff, Holland. Oppenheim, A. V., and Schafer, R . W. (1975). Drgiral signal processing. Prentice-Hall, Englewood Cliffs, NJ. Osborne, A. R. (1993a). Numerical inverse scattering transform for the periodic, defocusing nonlinear Schrodinger equation. l Y 1 . y ~ . Let/. A 176A, 75-84. Osborne, A. R. (l993b). Behavior of solitons in random-function solutions of the periodic Korteweg-de Vries equation. I'hys. Rev. Lelt. 71, 31 15-31 18. Osborne, A. R., and Petti, M. ( I 993). Numerical inverse-scattering-transform analysis of laboratory-generated surface wave trains. Phys. Rei'. E 47, 1035-1037. Osborne. A. R., and SegrC, E. (1993). The numerical inverse scattering transform for the periodic Kortcweg-de Vries equation. N y s . Let/. A 173A, 131-142. Phillips, 0. M. (1960). On the dynamics of unsteady gravity waves of finite amplitude. Part I. . I . Fluid Mech. 9, 193-217. Phillips, 0. M. (1981). Wave interactions-the evolution of an idea. J . Fluid Mech. 106, 215-227. Ramamonjiarisoa, A,, and Mollo-Christensen, E. (1979). Modulation characteristics of sea surface waves. J . Geophys. Rex 84, 7769-7775. Shen, Z., and Mei, L. (1993). Equilibrium spectra of water waves forced by intermittent wind turbulence. J . Phys. Oceanogr. 23, 2019-2026. Shum, K. T., and Melville, W. K. (1984). Estimates of the joint statistics of amplitudes and periods of ocean waves using an integral transform technique. J . Geophys. Res. 89, 6467-6476. Tanner, M. T., Koehler, F., and Sheriff, R. E. (1979). Geophysics 44, 1041-1063. Tayfun, A,, and Lo, J. M. (1989). Envclopc, phase and narrow-band models of sea wavcs. J . Waterway Port, Coasts/ Ocean Div., Am. Soc. Ciii. Eng. 115, 594-613. Tayfun, A,, and Lo, J. M. (1990). Nonlinear effects on wave envelope and phase sea wavcs. J . Waterway, PUIT,Coastul Ocecin Di!,., Atn. Soc. Citi. Eng. 116, 79-100. Titchmarsh, E. C. (1986). In/roduc/iori 10 /he /heoty of Fourier integra/.y. Clarendon Prcss, Oxford, England. Tracy, E. R., Larson, J. W., Osborne, A. R., and Bergdmasco, L. (1988). O n the nonlinear Schrodinger limit of the Kortewcg-dc Vries equation. Physica D 32, 83-106. Walsh, E. J., Hancock, D. W., Hines, D. E., Swift, R. N., and Scott, J. F. (19X9). An observation of the directional wavc spectrum evolution from shoreline to fully dcveloped. J . Phys. OceanoLqr.19. 670-690. Yuen, H. C., and Lake, B. M. (1975). Nonlinear deep water waves: Theory and experiment. Phys. Huids 18, 956-960. Yuen, H. C., and Lake, B. M. (1982). Nonlinear dynamics of deepwater gravity waves. Ad(,. Appl. Mech. 22, 67-229. Zakharov, V. E. (1968). Stability of periodic waves of finite amplitude on the surface o f a deep fluid. J . Appl. Tech. Phys. 9. Xh-94. Zakharov, V. E., and Kuznetsov, E. A. (19%). Multi-scale expansions in the theory of systems integrable by the inverse scattering transform. Physica D ( Amslrrdam) 18, 455-463.

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FIG.14. The Hilbert spectra for stations 2. 5. and 6. These results are very \imilar to those produced by wavelet analysis: (a) station 2: ( b ) rialion 5: and ( c ) station 6 . The range of frequency variations is relatively low in (a). By the time the wiives reach station 5. the range of the frequency variationh becomes much greater. There are also two low values at near half of the niean occurring at the exact locationc of the phase jumps. The Iocatioiis Ibr low values increase a t station 6.

FIG. IS. The wavelet analysis results for [he same stiitioiis a s i n Figure 14: (a) $talioil 2: (I?) station 5 : and ( c ) station 6 . The wavelet result with the modulus expressed in logarithmic scale as i n the

Hilbert spectruiii. The wavelet results fail 10 show the frequency change\ of the fusion events. In gen eral. they are less sharp and ;11so le\\ I(ic:ili~etlthan in the Hilberi \pectrum.

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ADVANCES IN APPLIED MECHANICS. VOLUME 32

Vorticity Dynamics on Boundaries J . Z . WU and J . M . WU

.

The UniL'errity of Tennesscv Space Institute. Tullahoma Tennessee

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Development of Boundary Vorticity Dynamics . . . . . . . . . . . . . . . B. Plan of Prescntation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I1. Splitting and Coupling of Fundarncntal Dynamic Processes . . . . . . . . . A . Dynamic Processes and Boundary Conditions . . . . . . . . . . . . . . . .

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121 125

128

B. The Splitting and Coupling of Dynamic Processes . . . . . . . . . . . . . C . Splitting and Coupling inside thc Fluid: The Helical-Wave Decomposition Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Splitting and Coupling on Boundaries: A Model Problem . . . . . . . .

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I11. General Theory of Vorticity Creation at Boundaries . . . . . . . . . . . . . A . Boundary Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Boundary Fluxes of Vorticity and Enstrophy . . . . . . . . . . . . . . . . C . Creation of Boundary Vortex Sheets . . . . . . . . . . . . . . . . . . . . .

148 148

IV . Vorticity Creation from a Solid Wall and Its Control . . . . . . . . . . . A . The Effect of the Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . B . The Effect of Wall Acceleration . . . . . . . . . . . . . . . . . . . . . . . . C . Three-Dimensional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Vorticity-Creation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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151 161

..

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V . Vorticity Creation from an Interface . . . . . . . . . . . . . . . . . . . . . . . . 198 A . Dimensionless Parameters on a Viscous Interface . . . . . . . . . . . . . 199 B. Flat Interface and Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . 203 207 C . Free-Surface Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . D . Complex Vortex-Interface Interaction and Surfactant Effect . . . . . 219 VI . Total Force and Moment Acted on Closed Boundaries by Created Vorticity Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Vorticity Moment and Kutta-Joukowski Formula . . . . . . . . . . B. Total Force and Convective Vorticity Flux on a Wake Plane . . . . . . C. Force and Moment in Terms of Boundary Vorticity Flux . . . . . . . . . VII . Application to Vorticity Based Numerical Methods . . . . . . . . . . . . . . A . A n Anatomy of Vorticity Based Methods . . . . . . . . . . . . . . . . . . B. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

224 226 231 239 247

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Copyright 0 1996 hy Acadcmic Press Inc . All rights of reproduction in any form reserved. ISBN 0-12-002032-7

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VIII. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgrncnts

I. Introduction One of the central problems in vorticity and vortex dynamics is the interaction of vorticity field and boundaries. The boundary can be a rigid wall, a flexible solid wall, or an interface of two different fluids. For example, an aerodynamicist may be concerned with the formation of the three-dimensional boundary layer on a wing surface, its interaction with shock waves, transition to turbulence, separation and control, as well as the effect of these on the wing lift and drag. A biomechanicist may be concerned with how a fish generates a Khrmhn vortex street with signs opposite to those behind a cylinder to gain thrust most efficiently. And a naval hydrodynamicist may be concerned with ship-wake vortices and how they interact with a wavy water surface. These problems, although apparently belonging to different fields, fall into the same category of vortex-boundary interaction at a fundamental level. In any case, boundaries are the most basic source of vorticity (in particular, the unique source for incompressible flow of a homogeneous fluid in a conservative external body-force field), and the whole life of a vortex usually begins at a boundary. Due to the continuous creation of vorticity from boundaries, a bounded vortical flow is much more complicated than an unbounded flow, in particular in the regions near the boundary. The complexity of vorticity dynamics caused by a boundary is further seen in turbulent boundary layer, where extremely abundant coherent vortical structures occur (c.g., Robinson, 1991), and in various vorticity-based numerical methods, where a difficult issue is how to formulate proper boundary conditions for the vorticity (Majda, 1987; Hald, 1991; Gresho, 1991, 1992). In fact, in most practical cases, unbounded flows are merely idealized approximations, and ultimately dealing with a boundary is inevitable. It is perhaps too ambitious to give a comprehensive review on vorticity-boundary interactions in a single chapter. However, the problem can be decomposed at least into two subjects. The first is ~ o v t e xdynamics inside a flow field, which is dominated by the highly nonlinear advection as

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well as diffusion, such as vortex stretching, instability, and breakdown; interacting vortex systems; and so forth; as well as their complicated consequences, including turbulent coherent structures. On that subject many excellent reviews are available. The second subject is the orti ti city dynamics on boundaries (or “boundary vorticity dynamics,” for short), which concentrates on the vorticity creation from a boundary and the reaction of the created vorticity to the boundary. It is this subject that makes a bounded vortical flow differ from an unbounded one and hence is an indispensable fundamental constituent and necessary prerequest in dealing with any vorticity-boundary interactions. This important subject has never been systematically reviewed before. In this chapter, we first present a unified general theory of the vorticity dynamics on various boundaries for viscous compressible flows and then review its applications to specific problems, which at this writing are mainly confined to incompressible flows. We shall consider two types of boundaries: a solid boundary, either rigid or flexible; and an interface of two immiscible fluids. Both are regarded as sharp material surfaces. Some other boundaries, such as a porous wall, regrettably are omitted, though they are also important in practice. In what follows we briefly review the development of boundary vorticity dynamics, then introduce the contents of the chapter.

A. DEVELOPMENT OF BOUNDARY VORTICITY DYNAMICS The study of boundary vorticity dynamics was first concentrated on the solid-wall type of boundary pioneered by Lighthill (1963). He called the normal diffusion flux on a solid wall (denoted by dB), d o u = u n - O o = udn

on

dB

(1.1)

as the iwficity source strength (per unit of time and per unit of area). Here o = V X u is the vorticity, v the kinematic viscosity, and n is the unit normal vector pointing out of the fluid domain. With this flux (referred to as the bounduly iwticity flux hereafter ), Lighthill expounded the physics of two-dimensional vorticity creation from a stationary wall, in particular its direct dependence on the tangential gradient of pressure, which resulted

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from applying the tangent component of the Navier-Stokes equation to dB: dw 1 dp (T= v-on d B (1.2) dn p dx Lighthill went on to present the whole boundary layer theory in terms of vorticity.’ Then, Batchelor (1967) devoted the main body of his classic book to incompressible rotational flows, and he gave a detailed but descriptive explanation of the vorticity creation from a solid wall, mainly in terms the formation of boundary vorticity rather than its flux. The importance of the vorticity creation and these pioneer works had not been fully recognized until the late 1970s. Lighthill (19791, in a survey paper on the waves and hydrodynamic loading, restressed the idea he raised 16 years before, which now stands at the center of his book (Lighthill, 1986). Then, Lighthill’s analysis was adopted by Panton’s (1984) textbook, which emphasizes both boundary vorticity and its flux. Morton (1984) criticized the common lack of fundamental understanding of vorticity generation mechanism-in the paper he still cited only Lighthill (1963) and Batchelor (1967). Morton tried to compromise the views of Lighthill and Batchelor and noted that for a moving wall its tangential acceleration should be added to (1.2) as another constituent of vorticity source. Then, in a review of unsteady, driven separated flows, Reynolds and Carr (1985) qualitatively explained the physics of many different forced, unsteady, separated flows in a unified way, based on the boundary vorticity flux, and added one more term due to the boundary transpiration to (1.2). They stressed the importance of a good understanding of vorticity production and transport in designing effective mechanisms for separation control. On the other hand, owing to the need for developing numerical vortex methods, eq. (1.2) and its normal counterpart,’

1 dp

do

-- - -uP dn dX I

.

on

dB

(1.3)

Lighthill (1963) found that “although momentum considerations suffice to explain the local behavior in a boundary layer, vorticity considerations are needed to place the boundary layer correctly in the flow as a whole.” He showed that the vorticity considerations “illuminate the detailed development of the boundary layer just as clearly as do momentum considerations.” Therefore, Lighthill has in fact placed the whole boundary layer theory into vorticity dynamics. ’Note that (1.2) and (1.3) are a pair of Cauchy-Riernann equations. Thus, on a hvo-dimensional stationary boundary, p and vu constitute the real and imaginary parts of a n analytical function. For a Stokes flow with advection being completely ignored, that complex function is also analytic inside the flow field.

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have been repeatedly rederived, widely cited, slightly extended as state earlier, or partly utilized by Pearson (1969, Roache (1972), Leonard (1980, 198.9, J. C. Wu et al. (1984), Leconinte and Piquet (1984), among others. Even though this development had been confined basically to twodimensional flow over a flat platc, progress appeared at the fundamental level as J. Z. Wu (1986) extended (1.2) to three-dimensional flows over an arbitrarily curved stationary wall, and found extra contributors to u, such as the skin friction (or equivalently, boundary vorticity) and surface curvature, which exist in three dimensions only. Lighthill (1963) has mentioned that the effect of the third dimension and wall curvature is of smaller order, but exceptions exists near a sharp edge or a spiral point of the skin-friction lines. More important, Wu’s extension enabled him to study the u distribution on a closed surface, which leads to a novel total force formula exclusively in terms of the vectorial moment of the boundary vorticity fluxes due to these effects as well as pressure gradient (J. Z. Wu, 1987). Therefore, the creation of vorticity, that is, the action of a solid body to the vorticity field, and the reaction of created vorticity to the body reached an intrinsic unity on the basis of boundary vorticity flux. These results were soon extended to moving wall and compressible flows by J. Z. Wu et al. (1987, 1988a). A himilar extension, based o n a novel approach, applicable to any continuous media, was made by Hornung (1989, 19901, independently and almost simultaneously. But a detailed presentation of the whole general theory, along with an in-depth physical discussion, appeared in monograph and journals only recently (J. Z. Wu et al., 1993a; J. Z. Wu and Wu, 1993). During this period, the general theory was applied to various specific problems as well, of which some will be reviewed in this chapter. Owing to these efforts, the theoretical foundation of vorticity dynamics on solid boundaries has now been well established. Parallel to the preceding development, the vorticity dynamics on an immiscible interface has its own history. Although it has long been known that, on a free surface, there is a viscous boundary layer (e.g., Lamb, 1932; Wehausen and Laitone, 1960; Moore, 1959, 1963; Lundgren 1989), works from the point of view of vorticity dynamics were relatively rare. The physical interpretation of vorticity formation on an interface started from Longuet-Higgins (1953; see also Longuet-Higgins, 1992) as cited by Batchelor (1967). Longuet-Higgins showed how the vorticity appears on a free surface as a direct consequence of the continuity of tangent stress across the surface and expressed this boundary vorticity in terms of the surface

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motion and geometry. For example, in steady two-dimensional flows, the vorticity on a free surface S simply reads

where U is the tangent velocity of S and K twice of thc mean curvature. But once again, this type of boundary vorticity dynamics had not received sufficient attention until the 1980s. Due to the great interest in the interaction between vortices and a free surface that causes the surface to significantly deform (see, e.g., the review of Sarpkaya, 1992a, b), rapidly growing works in publications have appeared. Like the solid-wall case, one of the key mechanisms involved in the interaction is the creation of new vorticity or the loss of existing vorticity (a negative creation) from the surface. This vorticity creation is inherently a viscous process and in most cases highly three-dimensional. How to understand the process and identify its role in observed complicated surface-deformation patterns during the interaction became utmost important. This practical need motivated corresponding theoretical studies at fundamental level. In a study of local flow properties on a viscous interface, Lugt (1987, 1989a) made a distinction between the roles of the vorticity on the surface and its diffusion flux across the surface. He pointed out that the surface vorticity does not provide information on the rate of production of vorticity or on the diffusion of vorticity into the interior of the fluid; this information should be furnished by the boundary vorticity flux. On a two-dimensional free surface, if the flow is steady, Lugt showed that the flux is (in our notation, s is the arc length along the surface)

(1.5)

being the total head. Equation (1.2) was thereby extended to an interface for the first time, and theories for two different boundaries started to merge. We shall see in Section 111 that a combination of (1.4) and (1.5) is indeed the simplest prototype of the entire vorticity dynamics on interface. Later, Rood (1991) stressed the importance of u without giving details. H e then extended Lugt’s approach to unsteady flow and attempted to explain a series of observed phenomena (Rood, 1994a, b). In fact, extending the vorticity dynamics on a solid wall to include an interface is

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12s

straightforward; a general three-dimensional theory was presented by J. Z. Wu (1995). Hence, now the development of a theoretical foundation of vorticity dynamics on an interface is almost as mature as its solid-wall counterpart.

B.

PLAN OF PRESENTATION

This chapter consists of eight parts. In Section 11, an overall observation is made on three fundamental fluid dynamic processes driven by surface forces: the longitudinal compressing-expanding process, the transverse shearing process, and the surface-deformation process. The shearing process represented by vorticity is our main concern, but is coupled with the other two. In particular, the coupling on a boundary, as revealed by the Cauchy-Riemann equations (1.2) and (1.3) on a solid wall, or as (1.4) and (1.5) on a free surface, crucially affects the vorticity formation on the boundary. The coupling of the three processes as well as their splitting is the foundation of the entire body of boundary vorticity dynamics. We use a novel triple decomposition of stress tensor for Newtonian fluid to analyze the stress constituents, followed by a unified treatment of boundary conditions. These allow one to study the splitting and coupling of dynamic processes in both momentum balance inside the fluid and surface stress balance on a boundary. The former is further explored in terms of helical-wave decomposition, and the latter, exemplified by a unidirectional flow with both rigid and interfacial boundaries. Section 111 gives the general theory of vorticity dynamics on any material boundaries of viscous compressible flows. We first present the general formulas for the jump of normal stress and tangent vorticity across a boundary surface. Then, we present Hornung’s (1989, 1990) definition of vorticity flux and the derivation of its general formula. For Newtonian fluid with constant dynamic viscosity, definition (1.1) is recovered. Formulas for boundary vorticity and its flux are decomposed into tangent and normal directions to show the specific physical implication of each term. The difference between a rigid or flexible wall and an interface is addressed. As a complement to the boundary vorticity flux, the boundary enstrophy flux is introduced, and the central role of viscosity in the formation of both boundary vorticity and its flux is stressed. As the Reynolds number approaches infinity (the Euler limit of viscous flow), the vorticity creation

126

J. 2. Wu and J. M . Wu

from boundaries manifests itself as the boundary-vortex sheet creation, whose three-dimensional dynamic equation and velocity in a general circumstance are derived. In Sections IV and V we enter various specific physical mechanisms responsible to vorticity creation from a solid wall and an interface, respectively. Some easily misleading conceptual issues are clarified. In Section IV, many model problems and applications are reviewed, either briefly or in detail. Several guiding principles for controlling vorticity creation from solid walls, of great interest in applications, are reviewed and exemplified. In Section V, we start with a general dimensionless formulation of the interfacial vorticity and its creation rate, followed by simplified approaches, along with a few worked out examples. These include the flat interface or free surface and the free-surface boundary layer at large Reynolds numbers. The section ends with a brief observation of vorticity creation in complex vortex-interface interactions and the surfactant effect on the creation. We turn to the reaction of created vorticity field to boundaries in Section VI. It is shown that, due to the coupling of shearing and compressing processes, the total force and moment acted on a closed boundary can always be expressed as vorticity-based formulas, even if the flow is dominated by a compressing process, as long as the Mach number is not in hypersonic regime. This is done systematically as an observer moves from the very remote far field to near field, until to the closed-boundary surface. Correspondingly, the resulting formulas arc increasingly accurate and general: from the classic Kutta-Joukowski formula applicable to the Euler limit of steady viscous flows, to the accurate near-wake plane analysis for arbitrary steady viscous flows, to the total force and moment exclusively in terms of boundary vorticity fluxes. The unique characters and great potential in application of these formulas, compared with conventional force and moment formulas, are addressed. Examples are given to illustrate the implication of new formulas in aerodynamic diagnostics and optimization. One of the main applications of boundary vorticity dynamics is to provide proper boundary conditions for vorticity-based numerical methods. The relevant theoretical analysis and numerical examples are given in Section VII. The success of computations further confirms the power of the theory and the importance of correct physical understanding. We conclude the chapter with Section VIII. According to the preceding definition of the boundary vorticity dynamics, in the main body of this chapter we shall not go deeply into the interior

Vorticity Djn~inzicson Boundaries

127

of the flow field, and hence in most cases the advection effect is avoided. This confinement makes a unified general theory possible. However, this confinement also makes the theory alone insufficient to give a complete answer to any vorticity-boundary interaction problem where nonlinear advection is involved. Eventually, one has to rely on experiments or computations for every specific problem. The chapter includes some fully resolved vorticity-boundary interaction problems merely to illustrate the general theory. More examples of this type of interactions can be found in the review of Doligalski et 01. (1994) for rigid boundary and that of Sarpkaya (1996) for interfacial boundary. The material selection of this chapter inevitably reflects the authors’ personal experience. Some topics seldom mentioned in the chapter are by no means less important. On the contrary, they may precisely be the place where the boundary vorticity dynamics has great potential to apply. For example, the application to flow over a flexible wall is illustrated only once, but almost the entire field of biofluiddynamics, such as animal flight and swimming and blood flow (e.g., T. Y. Wu, 1971; Fung, 1971; Lighthill, 1975; T. Y. Wu et al., 1975; Childress, I981), falls into this category. Clearly, the vorticity creation and reaction on such a flexible boundary, either active or compliant, is of vital importance and hence the theory has a room to grow including its extension to non-Newtonian fluids. We believe that a combination of the boundary vorticity dynamics with existing relevant theories, or a re-examination of these theories from the viewpoint of vorticity dynamics, will open a very fruitful new avenue.

11. Splitting and Coupling of Fundamental Dynamic Processes

A vorticity field d x , t ) characterizes the most abundant dynamic process in a flow field-the transverse shearing process. This process coexists and is coupled with other fundamental dynamic processes as well as the thermodynamic process, both inside the flow field and on its boundaries. In particular, the specific behavior of vorticity dynamics on a boundary depends on the type of boundary, which to a large extent determines with what process the shearing is coupled. To provide a general background for the theory of boundary vorticity dynamics, therefore, in this section we make a systematic examination of the coupling and splitting of these processes.

128

J. Z. Wu and J. M. Wu A. DYNAMIC PROCESSES A N D BOUNDARY CONDITIONS

We first clarify how many fundamental dynamic processes coexist in a Newtonian fluid and what kinematic and dynamic conditions are imposed on different boundaries. 1. The Triple Decomposition of Stress Tensor For a Newtonian fluid, the viscous stress tensor is proportional to the strain-rate tensor D = D', where the superscript means transpose. Thus, to identify all possible dynamic processes driven by surface forces explici t l ~one , ~ needs to first decompose D into its corresponding constituents, in particular on boundaries. This, for a rigid stationary wall dB, was first studied by Caswell (1967), who obtained an elegant formula that can be easily extended to an arbitrarily moving wall with angular velocity W ( t ) (J. Z. Wu and Wu, 1993): 2D

=

26nn

+ n(w, X n> + ( w , X n)n

on

dB

(2.1)

where 6 = V . u is the dilatation and w, = o - 2W, the relutiL!e Liorticity. However, for an arbitrary surface, the following novel but much simpler decomposition of D is most appropriate. Let R = - R T be the antisymmetric spin tensor such that V u = D + R . Then, because VuT = D - R , there is D = 61

+R

-

B

(2.2)

where I is the unit tensor and

B = 61 - Vu'

with V . B

=

0

(2.3a, b)

is called the su~ace-strain-ratetensor (Dishington, 1969, (where the definition of B differs from (2.3a) by a transpose) since for any surface element dS = n dS there is (for component form see, e.g., Truesdell, 1954; Batchelor, 1967) D -dS Dt

=

n . B dS

(2.3~)

'There are also dynamic processes driven by body forces and thermodynamic processes. The conservative forces like gravity can be absorbed into pressure force as we shall do later, and nonconservative forces like those in magnctohydroclynamics are beyond our present concern. The thermodynamic process introduces additional complications that will be touched upon but not our main concern here.

Vorticity Dynumics on Boundaries

129

Now, let p and A be the first and second dynamic viscosities. The implication of (2.2) in the dynamics of a Newtonian fluid is immediately evident by substituting it into the Cauchy-Poisson constitutive equation: T

=

+ A6)I + 2 p D

(-p

which yields an intrinsic triple decomposition of the stress tensor T (J. Z . Wu and Wu, 1992, 1993; J. Z. Wu, 1999:

+ 2pcL.n - 2 p B

T = -111

(2.4)

where

n

-p

-

(A

+ 2p)6

(2.5)

is the isotropic part of T, which characterizes the compressing-expanding process (compressing process, for short) and will be referred to as the compression itanable? Then, on any surface S with normal n, either inside the fluid or on a boundary, the surface stress also has an intrinsic triple decomposition (2.6a)

t - n . T = -nn+.r+tt,

where T = p.cr, X

n,

t,

= -2pn.B

1 D

=

-2p--dS dS Dt

(2.6b,c)

Therefore, the surface stress of a Newtonian fluid consists of three parts: the normal stress -Hn, which i s dominated by the pressure p, plus a viscous compressible correction; t h e shear stress T , which is proportional to the tangent vorticity; and the surface-deformation stress t , , which represents the viscous resistance of a surface element of unit area to its strain rate. Each stress drives a fundamental dynamic process. Thus, through the stress balance, the shearing process may couple with other two dynamic processes as well as the thermodynamic process. Note that usually t , is neither parallel nor perpendicular to d S , see Figure 1. Among the three stresses the surface-deformation stress t, is less familiar and deserves a further analysis. From its physical meaning, we expect that t , should depend exclusively on quantities defined on the surface S. Indeed, due to the vector identity,’ (b

X

V)

X

c

=

(VC). b

-

b ( V . C)

=

b . { ( V C ) ~- ( V . c)I}

(2.7)

‘The compression variablc does not have a unique definition and specific name (this explains why the hydroacoustic variable is not uniquely defined either), hut the shearing i~uriahleis always the vorticity. 5 W e thank Professor T. Y. Wu’s suggestion on using this identity, which simplified relevant mathematic manipulation.

130

J. Z. Wu and J. M. Wid

FIG.I . The triple decomposition of the surface stress.

it immediately follows that

where the right-hand side contains only tangent derivatives. Hereafter, we specifically denote the surface velocity by a capital U, which is feasible only if no normal derivative of u is involved. In studying the boundary vorticity dynamics, we often need to decompose a vector, including the gradient operator, into normal and tangent components on a surface S. We use suffix rr to denote the tangent components of a vector; thus, for instance, n x V = n X V,. Then, let K = -V,n

=

-(Vnn)

7

and

K

= -V;n

be the (symmetric) cunlature tensor of surface S, of which only four tangent components are nonzero, and twice the mean curcature, respectively (for a neat and convenient technique of calculating various tangent derivatives of an arbitrary tensor defined on a surface, see an appendix of J. Z. Wu, 1995). Correspondingly, we write U = U, + n u , . Then, we may further split the normal and tangent components of (2.8a) explicitly (J. Z. Wu, 1995):

Note that from the generalized Stokes theorem

13 1

VorticityL?ynamics on Boundaries

for any tensor 9and any admissible tensor product 0 , eq. (2.8a) implies that the integral of n . B over a closed S (of which dS vanishes) must be zero, as is directly seen from (2.3b) and the Gauss theorem.

2. Primary and Deriiwl Boundury Conditions The kinematic and dynamic conditions on different boundaries can be stated in a unified way. We denote a material boundary of a viscous fluid by ~ 8which , can be either a rigid or flexible wall, or an immiscible interface of two different fluids. When there is a need to distinguish between a solid wall and an interface, we set : != dB for the former and .8 = S for the latter. Let us agree that the unit normal vector n points out of t h e flow domain on dB and from fluid 1 to fluid 2 on S. Let [ Y ] denote the jump of any quantity 9right across .%', so that on a solid boundary [u] = u - b, where b is the solid velocity, whereas on an interface between fluid 1 and fluid 2, say, [u] = u , - u 2 . Then, the boundary conditions should ensure the continuity of velocity and that of stress with allowance of surface tension (Wehausen and Laitone, 1960; Batchelor, 1967). Therefore, the velocity continuity implies [u] = 0 on :2,or, in decomposed form, the no-through and no-slip conditions: n . [ul

=

0,

n x [u] = 0 on

A?

(2.10a, b)

Similarly, let T be the surface tension that vanishes on dB, then the stress continuity across ~2? implies n . [t] + TK = 0,

n x [t] = 0 on

9

(2.1 l a , b)

where K is the mean curvature of 9 defined before. We call (2.10) and (2.11) the primary boundary conditions. Note that, with surface tension T = 0, they apply equally well to any material surface inside a viscous flow. Now, because the vorticity transport equation is one order higher than the momentum equation, in addition to these primary conditions we need some deriued conditions for vorticity dynamics, which are corollaries of (2.10). First, eq. (2.10b) directly implies the well-known continuity of normal uorticity : n.[co]=Vo,x[u,l=O

on

9

(2.12)

1. Z. Wu and J. M. Wu

132

Thus, on a nonrotating M there must be n . ~ r = ) 0; and a viscous vortex tube with ~ r ) n. f 0 cannot terminate at LB but will penetrate 94. The latter occurs if a solid wall is rotating (so the vorticity goes into the solid body as twice its angular velocity), or if a viscous tornadolike vortex intersects an interface. The second derived condition concerns the surface acceleration. If, at an initial time r = 0, a fluid particle sticks to a point of a solid wall or a particle of fluid 1 sticks to a particle of fluid 2 at an interface, by (2.10) the stickiness will continue as time goes on. Therefore, there must be the continuity of acceleration:

[a] = 0

forall

and x €9. a =

t

Du ~

Dt

Inversely, if this condition holds and if in addition

[u,,] = 0 at

t

=

0 and all

x

E&‘

then (2.10) is guaranteed. Therefore, the adherence condition (2.1 Oa, b) can be equivalently stated as n . [a] = 0,

n . [u,,] = 0

n x [a3 = 0,

n x [u,)]= 0

1

on

9

(2.13a, b)

respectively (J. Z. Wu et al., 1990). We shall see the key role of this acceleration condition in boundary vorticity dynamics. Obviously, (2.12) and (2.13) will be redundant if one confines oneself to primitive variables. In a recent article on interface dynamics, Yeh (1995) attempted to derive the continuation of velocity u, stress t (confined to the case without surface tension T ) , and acceleration a across %‘ from dynamic equations. While some of Yeh’s argument is incomplete, it may be easily improved to an extent, as briefly reviewed below. At this stage, we do not consider Yeh’s approach as a superior alternative to the conventional one; what makes it interesting is that it may further confirm the consistency between each boundary condition and a corresponding dynamic aspect of the fluid motion. In particular, the no-slip condition (2.10b) has been a hypothesis based on physical observation, but it now appears to be consistent with the energy balance.

Vorticity L?ynamics on Boundaries

133

Suppose that on both sides of a material boundary 9’a dynamic equation of the following general form holds:

dF -

Dt

+ FV.U+ V .S + V

X

A

+G

=

0

where F, S, A, and G can be any tensor provided that all terms are of the same rank and are smooth functions of (x,t ) on each side. Consider a domain 9 =8, + g 2 across c(8,where 9,and 9i2are subdomains on sides 1 and 2 of S‘, respectively. Take the integral of the above equation over 23.Then, using the weak-solution technique to handle the possible discontinuity across 9, Yeh ( 1995) arrived at a general jump condition: n . [Sl

+n X

[A]

=

0 on

9

Now, we first specify the above dynamic equation as the Cauchy motion equation (see (2.14) below), i.c., taking S = T and A = 0. Then the jump condition immediately gives (2.11) with T = 0. Second, by taking the divergence and curl of the motion equation per unit mass, a similar argument may lead to n . [a] = 0 and n X [a] = 0, respectively. Finally, we specify F as the kinetic energy, of which the equation has a single divergence term V * ( u . t), leading to [ ~ . t =] [ u ] . t

=

o

on

LB

due to the continuity o f t . This result implies that the mechanical energy cannot be stored on a surface clement without volume. Writing t = nt,i + t,, we see that (2.10a) follows as long as t,, # 0, and (2.1%) follows as long as the tangent stress t, is nonzero and not perpendicular to the tangent velocity u, on at least one side. Although a rigorous proof of this last condition is not yet available and exception at some special isolated boundary points cannot be excluded, physically it should be generically true on a material boundary. Note that for inviscid flow with t, = 0 there is no restriction to [u,], again consistent with the common result.

B. THE SPLITTING AND

C O U P L I N G OF

DYNAMIC PROCESSES

The shearing process or vorticity field is not always coupled with all other dynamic and thermodynamic processes. Quite often, one or more processes are unimportant or decoupled from the shearing. Then the

134

J. Z. Wu and J. M. Wu

relevant physics and analysis can be simpler. We now examine when and in what sense this situation happens. 1. Splitting and Coupling in Momentum Balance The most important use of the triple decomposition (2.4) is its combination with the Cauchy motion equation (where f is any external body force per unit mass) pa=pf+V.T

(2.14)

yielding a corresponding triple decomposition of the Navier-Stokes equation (J. Z. Wu and Wu, 1993) p(a

-

f)

=

- ~ nv x ( p a ) -

-

2 v p . ~

(2.154

where the right-hand side represents three surface forces that balance the body forces (inertial and external). The third term on the right has a simple physical interpretation: because p is a function of temperature, eq. (2.3~)indicates that this term is the viscous resistance (per unit volume) of isothermal sugaces to their deformation caused by dynamic-thermodynamic interaction. This point can be made clearer by writing

where dS, is an isothermal-surface element with normal n,, and S, = d(log p)/d(log T ) , usually of 0(1), is the dimensionless sensiticity (here T denotes temperature only in this single context). This effect is rather weak in comparison with other surface forces, except when heat transfer is extremely strong. So, from now on we shall always assume p = constant for simplicity. Then the surface-deformation process is absent from the momentum balance, and (2.15a) implies a natural Stokes-Helmholtz decomposition of the body force (Truesdell, 1954): p(a - f) =

-VIT

-

V x (pa>

(2.1%)

with II and pw being the scalar and vector potentials. Only the compressing and shearing processes are involved, which contain only three independent components because puw is solenoidal. Therefore, as first noticed by

Vorticity Dynamics on Boundaries

135

J. Z. Wu and Wu (1992), so f a r (is the momentum balance with constant Liscosity is concerned, the six-component stress tensor T can always be replaced by a three-component tensor

T

=

-n1+ 2 p a

(2.16)

which consists of only the isotropic and antisymmetric parts of T and will be referred to as the reduced stress tensor. This replacement is feasible in most applications and implies a big simplification.6 In fact, in (2.14), T can be replaced by any T’ as long as V . (T - T’) = 0, and 5. is the simplest among these infinitely many T’. We digress to observe that T is nothing but a tensor expression of the Stokes-Helmholtz potentials I1 and p w . Mathematically, for any tensor S (symmetric or not), one can always find a three-component tensor S such that S and S have the same divergence, which amounts to finding the Stokes-Helmholtz decomposition of that divergence. In the preceding case, T automatically emerged owing to the intrinsic decomposition (2.4) or the physically natural Stokes-Helmholtz decomposition (2.15). In other situations, the reduced tensor will be not so simple, although it still exists. For example, for an incompressible homogeneous turbulence, in the wave-number space the conventional six-component turbulent stress tensor T I j ( k ) can be replaced by a threc-component one (J. Z. Wu et al., 199Sb):

We return to (2.1Sb) and ask whether and when the remaining two surface forces can be further decoupled, at least approximately. Mathematically, the transverse and longitudinal vector fields V X w and V n in (2.13 are said to be orthogonal in a functional space, if (2.18) over the flow domain 53; in this case they can be solved independently (assuming a is not further split-the coupling in nonlinear advection is of ‘Some saving in computational fluid dynamics has been gained by implicitly using the reduced stress tensor: In the stress computation o f three-dimensional viscous incompressible flows by a finite-volume method, the number of grid points in a cubic element can be reduced from 27 t o 7 (Eraslan el al., 1983). The CPU time of overall computation can thereby be reduced by 40%.

136

J. 2. Wu andJ. M. Wu

a different nature, see Section 1 1 . 0 . Then, by the Gauss theorem, we have

where = d 9 is the closed boundary of 53.Hence, eq. (2.18) holds if one of the following conditions is satisfied:

n=O,

n ~ w = 0 , (nxV).w=0,

n x V n = O , on

.B

(2.19) Therefore, in an unbounded flow the ( w , I I ) decoupling is always possible. For a bounded flow, then, the (w, II) coupling inside the flow amounts to that on the boundary, clearly indicating the crucial importance of boundary vorticity dynamics. The third and fourth conditions of (2.19) are special cases of (1.3) and (1.2) and represent a homogeneous Neumann condition for pressure and vorticity equations, respectively. An obvious example where the fourth condition exactly holds is the Stokes's first and second problems, to be examined in Section II.D, including their generalization to rotating circular cylinder. A further example of practical interest is the incompressible Blasius boundary layer, in which d p / d x = 0 for sufficiently large x, and all the vorticity inside the whole layer must be created near the leading edge, where the full Navier-Stokes equation has to be applied. More generally, any attached boundary-layer approximation with known external main flow is a decoupled approximation. On the other hand, the coupling becomes very strong at a small Reynolds number, where, as noted in a note 2 to Section I, p and pu are simply the real and imaginary parts of an analytical function. This observation suggests that the strength of the (a, p ) coupling depends on the Reynolds number; which will be confirmed more unambiguously in Section VII. For an incompressible flow with a rigid boundary, the dynamics in the interior of the flow is further reduced to the shearing process alone (an incompressible potential flow belongs to kinematics), whose coupling with the compressing process (the pressure force) occurs merely at a boundary dB as indicated by (1.2) and (1.3). In other words, to compute an incompressible flow based on vorticity only, the tangent pressure force on dB needs to be solved (in numerical computation even this boundary ( w , p ) coupling can be bypassed, see Section VI1.A).

Vorticity Dytianiics on Boundaries

137

It should be stressed that we are not proposing a new “constitutive equation” (2.16) to replace (2.4). Obviously, the surface-deformation process may appear once we go beyond the momentum balance. One example is the angular momentum balance, but there the effect of t still amounts to shearing. In fact, in a fluid element of volume V bounded by dV, it can be shown that

where x is the position vector. This effect of t , again reduces to vorticity and vanishes if the boundary is not rotating. Another example is the dissipation rate a.From an identity (Truesdell, 1954) 1

D,,D,, = (I2 + --w? - (B,,u,),, 2 it follows that

and hence

Thus, all three dynamic processes contribute to the dissipation. However, substituting (2.21) into the energy equation immediately leads to the cancellation of t h e part due to t , . 2. Splitting and Coupling in Stress Balance Contrary to (2.20) and (2.20, where t , adds merely a small part to the total effect, this stress may play an role in the stress t = n . T on boundaries, because no divergence of T is taken. However, an exception still occurs on a rigid wall dB with angular velocity W(t); in that case, the boundary velocity can be written as u = U,,(t> W ( t >X x, and from (2.8) it follows that t , = - 2 p W x n, which can be absorbed into the shear stress. This yields the familiar formula, which also directly follows from (2.1):

+

t = -nn+pW,Xn

on

rlB

(2.22)

138

J. Z. Wu and J. M. Wid

where the relative vorticity or is nothing but [m], satisfying n . m, = 0 due to (2.12). Therefore, along with (2.15b), we see that except for strongly heat-conducting fiows, in the entire rorticity dynamics with rigid boundary, only the coupling of shearing and compressing processes is important (J. Z. Wu and Wu, 1993). Note that on a rigid wall the stress balance (2.11b) occurs between solid and fluid; should the solid stress be known, then so would be the boundary vorticity. But the real situation is, of course, precisely the opposite. In contrast, on a flexible boundary, either solid or fluid, the role of surface-deformation stress t , becomes very active. A n extreme case opposite to rigid wall is free surface, where by (2.11b) there must be n x t = 0, and hence right on a free surface the tangent r!orticityis balanced solely by the surface deformation, of which (1.4) is a simple example. We leave a full exploration of this issue to Sections I11 and V. So far we have met two types of couplings of shearing process and other processes o n a boundary: one is due to momentum balance, which leads to a coupling like (1.2); and the other is duc to stress balance, which leads to that like (1.4). Further decouplings may happen in both types, if (2.19) holds for the former and if the free surface is flat for the latter (where, by (1.41, the surface vorticity vanishes). It should be stressed that these two types represent different physics. The former is of one order higher than the latter ( V . T versus n . T ) but, as will be shown later, the former determines the latter. INSIDE THE FLUID: C. SPLITTINGAND COUPLING THE HELICAL-WAVE DECOMPOSITION APPROACH

We now further examine the splitting and coupling inside a flow field. Consider a slightly compressible, isentropic but viscous flow (the incompressible flow model is oversimplified for compressing process). In this case we assume the variation of the kinematic viscosity v = p / p is negligible and so are the external body force f and the viscous dilatation term ( A + 2 p ) 6 in II. Let h = / d p / p be the enthalpy. Then (2.1%) reduces to a = -Vh - vV x o (2.23) If the flow is unbounded or the material boundary of the flow domain is at rest, it is more convenient to write (2.23) as (2.24)

VorticifyLlynamics on Boundaries where L

5

139

o X u is the Lamb vector and

(2.25)

is the total enthalpy that now takes the place of compression variable. It is well known that to split the shearing and compressing processes one simply takes the curl and divergence of (2.23) or (2.24). This gives the vorticity equation and the “compression equation.” It will then be clear that the shearing and compressing processes are governed by the Reynolds number and Mach number, respectively; and as the range of these parameters changes, a hierarchy of approximations of this pair of equations can be constructed (J. Z. Wu and Wu, 1989a). Although some new results of these well-known curl and divergence operations will be presented later in Section VILA, here we introduce a less familiar but physically very appealing approach to the splitting. Let the Stokes-Helmholtz decomposition be also applied to the velocity and the Lamb vector and their respective vector potentials be made divergenceless:

Then (2.24) yields

)

+ G X [ Z + J + v w

(2.26)

For the present purpose, it is beneficial to work in the wave-number space, because then the spatial derivatives are simplified to multiplications by a wave vector k. Denoting the Fourier transform of any vectorial function f(x) by F{f(x)) = f(k), there is

Note that this pair of operations clearly indicates that dilatation waves are always longitudinal and vorticity waves are always transverse. Therefore, just by taking the inner and vector products of the Fourier transform of (2.26) with k/k, we immediately obtain the longitudinal and transverse

J. Z. Wu and J. M. Wu

140

"Bernoulli integrals" in the wave-number space: d

-4(k,t) at

d

-+(k,t) dt

+ X(k,t) + H ( k , t ) = 0

+ J(k,t) + u o ( k , t )

=

along k

(2.28a)

0 normal to k

(2.28b)

which govern the compressing and shearing processes, respectively. The decomposition (2.28), however, has not yet reached the finest fundamental building blocks of fluid dynamic interactions. Moses (1971) proved that the vector potential in a general Stokes-Helmholtz decomposition can be further intrinsically split into two, representing the righthanded and left-handed helical states. This is possible because the curl operator and the vector potential are both axial or pseudo vectors, which have handedness or polurity as an intrinsic property or dimension.' Thus, in (2.24) and similarly in (2.28), the vorticity and velocity can also be further split. Mathematically, this is achieved by the so-called helical-wave decomposition (HWD for short), first studied by Moses (1971) and Lesieur (1972; see also Lesieur, 1990). The decomposition uses the eigenvectors of the curl operator as the basis. In the wave number space the HWD basis becomes eigenvectors of the operator i k X , denoted by QA(k):

The eigenvalues for each k are -t k and 0. To see the structure of this basis explicitly, let e,(k) be an arbitrary real unit vector perpendicular to k, and e,(k) = k X e , / k , such that ( e , , e , , k/k) form an intrinsic Cartesian basis. Then a simple representation of Q*(k) is 1 QA(k) = -[e,(k)

fi

Q"(k) =

k k

for

+ iAe2(k)] A

=

0

for

A

=

I

(2.30)

'The relation between handedness and polarity is that a circularly polarized wave has only a single handedness, and a linearly polarized wavc has no handedness: that is, the right-handed and left-handed components are the same. A superposition of sufficiently many randomly polarized waves may have zero averaged polarity and zero net handedness.

VorficityDynanzics on Boundaries

141

These QA(k)also form an orthonormal and complete basis in the complex wave-number space, and hence any u(k) can be decomposed to u(k)

uA(k)QA(k), with

=

uA(k)= u ( k ) . QA*(k) (2.31~1)

A = f 1,O

where * means complex conjugate. The components with A = -t 1 and 0 clearly describe the transverse and longitudinal parts, respectively. More remarkably, the transverse components now represent a right-handed or left-handed circularly polarized state, depending on A taking 1 or - 1; and the HWD vorticity components are simply given by wA(k) = AkuA(k),

A

=

+1

(2.31b)

Consequently, by (2.27) and (2.29) and using the continuity equation, for weakly compressible flow, (2.28) can be cast to a very neat form:

where c is the sound speed. Equation (2.32a) is nothing but the Fouriertransformed Liorfex-sound equution at low Mach numbers (Howe, 1975; J. Z. Wu and Wu, 1989a). On the other hand, eq. (2.32b) is both the HWD transverse Navier-Stokes equation and the HWD vorticity equation due to (2.31b). Note that the nonlinear Lamb vector LAin (2.32b) is a convolution integral, which contains the self-coupling among different transverse modes and between the right- and left-handed velocity components. The cross-coupling between the two processes inside a flow field is also evident from (2.32). First, even though an acoustic wave is a longitudinal wave, its fluid dynamic source is the potential part of the Lamb vector, L", which vanishes without shearing process. Second, in the Lamb vector o x u, there can be a contribution of the potential velocity u"(k,t ) . Thus, although a sound wave is not the source of vorticity inside a fluid due to the absence of H in (2.32b), it will affect the advection of a vorticity field. The most typical and important example of such a coupling is the wellknown vortex stretching due to a background irrotational straining flow U = (ax,By, y z ) , which is implicitly contained in the transverse Lamb vector LA(k,t ) of (2.32b) as the counterpart of V x L(x, t) in the physical

142

J. Z. Wu and J. M. WLI

space. This coupling has been the object of extensive studies; for a recent analysis and review of previous works, see Moffatt et al. (1994). We stress that these cross- and self-couplings inside a flow field are essentially due to nonlinear kinematic adcection, which should be distinguished from dynamic couplings discussed before. The polarity of a vorticity field has recently attracted much attention due to its significant effect on vortex evolution and turbulent cascade process (e.g., Waleffe, 1992; Melander and Hussain, 1993; Virk et al., 1994). In a sense, theories on vorticity and vortex dynamics would be incomplete if the polarity effect was ignored. Because this new property contains two independent real scalars (HWD splits a divergenceless vector potential into two), it can be conveniently characterized by the relative amplitude and phase of the right-handed and left-handed components, of which the role in nonlinear interactions should be further explored.

D. SPLITTING

BOUNDARIES: A MODELPROBLEM AND COUPLING ON

Although the kinematic coupling of shearing and compressing inside a flow field due to advection is inherently highly nonlinear, the dynamic coupling on a given boundary 9? is apparently of a linear nature, as exemplified by the Cauchy-Riemann equations (1.2) and (1.3). This apparent linear character makes it possible to develop a general formal theory, which is to be reviewed in this chapter. Here we first clarify some basic facts and concepts. First, eq. (1.2) indicates that the ( w , p ) coupling is necessary for producing vorticity on a rigid wall. The coupling would disappear if the flow were strictly inviscid and if the no-slip condition on .D were removed. Therefore, the dynamic boundary coupling is inherently a iiscous phenomenon. J. Z. Wu and Wu (1993) have stressed that under a pressure gradient the particles of a strictly ideal fluid on a solid wall will only slide over it but never rotate; there is no mechanism to give these particles an angular velocity, and hence no vorticity can be created. This argument applies equally to an interface. Therefore, we shall be confined to viscous fluid exclusively. Even if the Reynolds number approaches infinity, the flow will behave as the Euler limit of a Nauier-Stokes flow (Euler limit, for short) rather than an ideal flow. In this case, the created vortex layer due to the

Vorticity Dynmzics on Boundaries

143

no-slip condition will degenerate to a vortex sheet, which is still essentially different from the pure sliding on a mathematical contact discontinuity. For an elegant exposition of the difference between the Euler limit of a Navier-Stokes flow and the Euler solution of an ideal fluid, see Lagerstrom (1973). Second, in the analysis o f the boundary behavior of vorticity, such as that by Lighthill (1963) and Batchelor (19671, we meet two quantities: boundary vorticity (sometimes denoted o Bfor clarity) and its flux u. As we explained before, o B and u appear in the two types of couplings through the balance of momentum and surface stress, respectively. J. Z. Wu and Wu (1993) showed that on a solid wall the implications and roles of these quantities are very different. This is also true on an interface, as Lugt (1987) stressed. For a fluid element sticking to a material boundary, its vorticity represents twice of the angular velocity of the principal axes of its strain-rate tensor-a physical interpretation of vorticity by Boussinesq (see Truesdell, 1954), and in fact the only consistent interpretation applicable to fluid elements both i n the interior of fluid and on a boundary. Figure 2(a) sketches such a fluid element sticking on a boundary and shows how the principal-axis rotation leads to an o B. Note that, once a material it will vortex line sticks to a boundary (of which every point has an oB), remain there and never go into the interior of fluid; but o B can be diffused into the fluid as shown in the figure. In contrast, as indicated by its definition (1.1), the boundary vorticity flux represents a mechanism that sends vorticity into the fluid from a boundary. Figure 2(b), reproduced from Lighthill (1979), clearly indicates how a fluid

High Pressure Low Pressure (a)

@)

FIG. 2. The physics of boundary vorticity w, and its flux u. (a) The rotation of the principal axes of the strain-rate tensor of a fluid element sticking on a boundary gives w B, which can be diffused into the fluid. (Reproduced from J. Z. Wu and Wu, 1993. Reprinted with the permission of Cambridge University Press.) (b) A fluid element neighboring a stationary boundary is set to rotate due to a tangent pressure gradient and the no-slip condition. (Reproduced from Lighthill, 1991, with permission.)

144

J. Z. Wu and J. M. Wu

element neighboring to a boundary (but not right on the boundary) is forced into rotation by a tangent pressure gradient. As Morton (1984) pointed out, this d p / d x can be replaced by a boundary acceleration; so Figure 203) is a pictorial interpretation of (1.2) and (2.33) later; that is, that of u. An external body force may cause similar effect; for instance, the gravity acted on an inclined boundary (Section I1I.B). Note that, although without a no-slip condition and viscosity the fluid ball in Figure 2(b) would not rotate, the amount of vorticity being sent into the fluid due to (1.2) or (2.33) as well as a body force is independent of the magnitude of v (Lighthill, 1963). We shall also see the explicit viscous effects on u, but except at some special local regions, they are much weaker if v is small. The magnitude of v determines only how deep the vorticity can be diffused into the fluid. In the Euler limit, therefore, the same amount of vorticity is still going into the fluid under the same d p / d x , say, but all confined within a vortex sheet. This sheet exists inside the fluid and is conceptually different from the boundary surface. For example, as a material sheet, it has a definite velocity differing from, say, a stationary solid wall from which it is created (this fact is well known as one approximates a boundary layer by a vortex sheet). To illustrate these basic concepts, we consider a simple model problem taken from J. Z. Wu and Wu (1993) and J. Z. Wu et al. (lYY4b): a unidirectional, incompressible, viscous flow over a flat plate. The fully general case will be treated in the next section. Assume the flow occurs on the half plane y > 0 with u = [ d y ,t > , 0,0], w = [O,O, w ( y , t)] and p = 1. The fluid and boundary are at rest for f < 0, and at t = 0 there suddenly appears a tangent motion of the plate with speed h ( t ) and a uniform, time-dependent pressure gradient d p / d x = P(t). Then, applying (2.23) to y = 0 and imposing the no-slip condition yields the boundary vorticity flux cr(t) =

dh -

dt

+ P(t)

at

y

=

0

(2.33)

which represents the force balance on the plate and in which u is the viscous force. Mathematically, (2.33) gives a Neumann condition for d y , t) and leads to the solution

Vorficit.yDynamics on Boundaries

145

Here, the flux u can be regular or singular. If, at t = 0, there is an impulsive P ( t ) and db/dt, they must cause a suddenly appeared uniform fluid velocity U = ( U , 0,O) and wall velocity b,,, respectively, such that

P ( t ) = -U8(t),

db -

dt

=

b , , 8 ( t ) for 0 - 1 t I 0’

Thus, the vorticity flux will also be singular (still due to a no-slip condition at t = 0):

a ( t >= - ( U

-

h , , ) 6 ( t )= y , , 8 ( t ) for 0 - 1 t 5 0’

(2.35)

where y o = - ( U - 6,) is the strength of the initial vortex sheet. Separating this singular part from (2.34) yields

(2.36) Obviously, the boundary vorticity is given by

indicating clearly that wR is exclusively from u. Moreover, the total amount of the vorticity being scnt into the fluid is (2.38) which is indeed independent of v , as asserted before. In particular, if the pressure gradient and wall acceleration vanish from t = O’, eqs. (2.36) and (2.37) reduce to the Stokes’s first problem (or the Rayleigh problem):

On the other hand, if P ( t ) = 0 for t > 0’ but the wall has a sinusoidal oscillation, say b ( t ) = cos nt, then we have the transient Stokes’s second problem, which has been thoroughly studied by Panton (1968) and of which the boundary vorticity can be integrated from (2.37) analytically:

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J. Z. Wu and J. M. Wu

where S(x) and C(x) are Fresnel's functions. As t classic result

4

=, eq. (2.40) gives the

Stokes's first and second problems have been generalized to include different boundary shapes (J. C. Wu and Wu, 1967). For the flow caused by an impulsively started rotating circular cylinder with constant angular velocity and that by a circular cylinder with rotatory oscillation, the analytical solutions (the latter is confined to the asymptotic steady state t + );. can be found in Lu (1987) and PCpin (19901, respectively. J. Z. Wu et al. (1994b) also gave the computed time evolutions of the vorticity field for these two cases, including a numerical result in the transient period for the generalized Stokes's second problem. The behavior of boundary vorticity and its flux is qualitatively the same as the flat-plate case. From the previous unidirectional-flow solution (2.33)-(2.38), two observations can be made. First, the no-slip condition is the key in deriving (2.33) or (1.21, including the possible singular part of (r, and hence in the creation of vorticity. Therefore, any inviscid interpretation for this creation mechanism should be rejected. Second, it is the boundary vorticity flux u that is directly (locally and simultaneously) coupled with the compressing process ( d p / d x here) through the force balance on the boundary. In contrast, the boundary vorticity o Ris a time-accumulated effect of a.For more general case, the space-time integrated effect of advection and diffusion also contributes to c o n , see Section VILA. Therefore, the boundaly r>orticityflux u, rather than the boundary isorticily c o R , measures the creation rate of iwrticity from a bounduiy. Now, if the rigid boundary becomes an interface S of two immiscible viscous fluids on which the velocity adherence still holds, one would immediately ask if the earlier basic assertion is still true, because (1.4) seems already gives a local and instantaneous relation on the boundary vorticity (which in the present case is zero). The answer is yes. First, eq. (1.4) applies to only a free surface where the fluid motion on one side of S is negligible; otherwise one can obtain a condition for only the vorticity jump (see Section III.A), which is insufficient to determine the vorticities of both sides. Second, as will be seen in Section V, even for a free surface, its geometry and motion are themselves a space-time accumulated effect of the force balance on it, and hence are still an accumulated effect of the boundary vorticity flux.

Vorticity @waniics on Boundaries

147

To support this assertion, J. Z. Wu (1995) extended the above Stokes's first problem to a two-fluid system. Assume that, in addition to an impulsively started rigid wall at y = 0 with U = 1, there is a horizontal flat interface S at y = 1. Thus, the wall drives fluid 1, which then drives fluid 2, which in turn reacts to fluid I . Along S the pressure is constant. Then the dimensionless governing equations and initial-boundary conditions at y = 0 and y = are

db,

dU, - =

V?

dt

~

d y -?

u 2 ( y , 0 ) = 0,

1
for

uz(=,t)

=

0 for

t >0

I

(2.41b)

By applying (2.10) and (2.111, the initial-boundary conditions on S are

Denote a , ( [ )= pressed integral

u 2 at y = 1 by r 4 t ) , and note that, from (2.41a,b,c), there = -dr!/dt. Then, u , ( t ) and u 2 ( t )can be analytically exin terms of the unknown ~ ( t or) cr,(t), for which an additional equation due to (2.41d) holds to close the system: uI

=

-u,(t)

rs(t)

=

F(t)

+ I' 0

i

2G(t

-

7)

+

k

k

=

sE

(2.42)

PI

with F ( t ) and G ( t )being known functions. The second integrand of (2.42) represents the reaction of fluid 2 to fluid 1 that makes the latter rotational on S with an O ( k )vorticity. I t is remarkable that, by comparing (2.42) and (2.37), the vorticity flux on an interface causes a rotutionaZ rdocity there in a way exactly the same as the rwrticity on a rigid wall, but with factor k . As will be addressed later, this is an indication of the basic difference in the mechanisms of vorticity creation from a solid wall and an interface. But on the interface the vorticity creation rate is still u .

148

J. Z. Wu and J. M. Wu

For the water-air interface with k = 3.87 x lop3, the solution of system (2.41) and (2.42) is potted in Figure 3. Figure 3(a) shows the velocity profiles at different times. The shear “layer” of the air is much thicker than that of the water. Figures 3(b) and 3(c) give the computed evolution of o1 and g1 at S, respectively. Indeed, the peak value of lol(l, t)l is about 4.5 X lop3 = O(k). If we replace the interface by a flat free surface with k = 0 in (2.42), then the surface vorticity vanishes (shear-free). From the preceding analysis, it is clear that the shear-j?ee behavior on a flat Pee su$ace comes from the vanishing of k alone, entirely independent of the flow Reynolds number.8 Then, u ( y , t ) and d t ) on S can be easily solved from (2.41a) (Rood, 1994b; J. Z. Wu, 19951, which for the water case are also plotted in Figure 3. Regardless of the missing of the residual interface vorticity, this solution is very close to that of the full water-air system. It was found that, for the generalized Stokes’s second problem of the water-air system, the effect of k # 0 is even smaller than in this example.

111. General Theory of Vorticity Creation at Boundaries

In this section we develop the general theory of vorticity creation from an arbitrary material boundary. This is the major aspect of the boundary vorticity dynamics. As mentioned before, two quantities, the vorticity and its flux on the boundary, are involved in this creation aspect. The other aspect, the dynamic reaction of the created vorticity to the boundary, will be discussed in Section VI. A. BOUNDARY VORTICITY

We first infer the vorticity behavior on a boundary from the surface stress balance (2.11). From now on, we specifically denote the tangential and normal vorticities on 9 by 6 = o, and 5 = o,,= n J , respectively. On a rigid wall one can go no further than (2.22), because the stress experienced by the wall is unknown and, in practice, is to be inferred from the fluid stress-still based on (2.11) with T = 0, a simple use of Newton’s third law. Therefore, we concentrate on an interface S of two Newtonian ‘In particular, we did not invoke a free-slip condition at all; on the contrary, the no-slip condition (2.41~)was used in deriving (2.42).

b

U 0.0045

0.004 0.0035 -

.. .. .,.._..., .. .... -0.0005

c

0

10

20

30

40

50

o -0.004

cwpled

5

-0.012-

>

free-surface

-

-0.006

.e

.-0

80

-

-0.01-

z

70

Time

-0.002 -

2

60

...

-0.008

-0.014. -0.016 -0.018 -

-0.02c

0

10

20

30

40

50

60

70

80

Time

FIG.3. The generalized Stokes’s first problem for the interacting water-air system (from J. Z. Wu, 1995): (a) velocity profiles at different times, (b) the evolution of the water vorticity on the interface, (c) the evolution of the vorticity flux on the interface. Also shown in the figures are the approximate solution with k = 0 (dotted lines). (Reproduced with the permission of the American Institute of Physics.)

J. Z. Wu and J. M. Wu

150

fluids. Except in Section V.D.2, where an elementary discussion is given to the surfactant effect, throughout this chapter we consider only a clean interface with constant surface tension. Thus, the jump of compression variable and tangent vorticity across S directly follows from (2.11), (2.6) and (2.8):

Equation (3.lb) indicates that, although across S the normal vorticity is continuous, the tangent vorticity is generally not (also true for solid wall); the jump of E. j is exclushiely balanced by that of the tangent components of sui$ace-deformation stress. In particular, when t , = 0, as in the case of a flat interface with uniform U , (3.2)

For example, on a flat water-air interface, the tangent vorticity of the water will be about 1.8 X 10- * times smaller than that of the air. The continuity of 5 and discontinuity of US across S implies that a vortex line must be refracted by S, first studied by Lugt (1989b) for the case where (3.2) applies. The general “refraction law” on a curved S can be easily inferred from (3.lb) and (2.12): (3.3) Thus, the refraction occurs in a plane determined by n and 5 , - 2n X ( n . B). Even if 6 , = 0, there is still a refraction, with US2 being 2[ p ] / p 2 times ( n . B), rotating 90” around n. J. Z. Wu (1995) remarked that if a well organized vortex in fluid 1 hits a wavy moving S , its vortex-line refraction is very likely a source of turbulence in fluid 2. Then, what is still continuous across S is the tangent vector (3.4) Note that a shearing near S must cause a rotation of n

Dn ~

Dt

=

1 D -(-dS) dS Dt

= 71

(B. n),

=

W,

X

n

=

-(V,U,

+ U;

K) (3.5)

Vorticity Dynamics on Boundaries

151

Thus, W, is exactly the familiar angular velocity of n. This rotation of n is caused by a nonuniform normal motion and a tangent advection that turns n to a new direction due to the curvature. Although the present theory is developed on a unified basis, the preceding results reveal some interesting differences between a solid wall and an interface regarding the behavior of vorticity on boundaries. First, for an attached flow over a solid wall, the vorticity usually arrives at maximum on the boundary, which is of O ( R e t ) for large Re, as is clear from comparing (2.36) and (2.37). But (3.1) indicates that the interfacial vorticity is of 0(1), of which the physics is explained by Batchelor (1967). Therefore, the maximal vorticity usually occurs in the interior of the fluid. Next, on a solid wall, when its motion and geometry are given, the only thing we know immediately is the normal vorticity, all the rest remains to be solved. In contrast, on an interface the situation seems to be entirely opposite: once the fluid motion on one side of S is given or its effect is negligible, and if the geometry and motion of S are known, the tangent vorticity and compression variable on the other side can be immediately obtained. What remains unknown is the normal vorticity, which does not enter into the stress balance at all. However, the main difficulty is now shifted to the calculation of the geometry and motion of S , which ultimately relies on the space-time accumulated effect of force balance on S. Moreover, because the nonlinear advection is also involved in this force balance, problems with an interface are factually harder to solve than those with a solid wall.

B. BOUNDARY FLUXES OF

VORTTCITY AND

ENSTROPHY

While the velocity and stress conditions are sufficient for dealing with primitive variables, for vorticity dynamics one has to explore further the boundary behavior of w and n. This leads us to the theory of boundary vorticity flux, which stands at the center of entire boundary vorticity dynamics. In short, eq. (1.1) indicates that, if at a boundary 9 the shearing is stronger than that away from .'8 then , the vorticity will be diffused from 9 into interior of the flow, and vice versa (Lighthill, 1963, 1986). In what follows we give a sufficiently general formula, similar to (3.1), that reveals all physical mechanisms causing the boundary vorticity flux. This type of formula, of which the simplest form is (1.2), is obtained by taking the vector product of n and the Navier-Stokes equation applied to 28, along

J. Z. Wu and J. M. Wu

152

with the acceleration adherence condition (2.13). Just as a combination of (2.10) and the boundary stress balance leads to the jump of boundary vorticity and compression variable, a combination of (2.13) and the force balance per unit mass will lead to their respective normal gradients. Once again, the approaches to a rigid wall, a flexible wall, and an interface are unified. 1. The Boundary Vorticig Flux in Arbitrary Continuous Media Since the flux u represents a vorticity source, it is natural to inquire the role of u in the vorticity transport equation; and because in (l.l), u is stated for Newtonian fluids of constant shear viscosity, it is also natural to ask if u can be defined for any continuous media without specifying a constitutive structure. These two questions were addressed by Hornung (1989, 19901, who gave a general definition of u and derived its formula. Hornung’s approach allows for a deeper physical understanding of a,of which a similar version is presented here. Assume that the divergence of the stress tensor T in (2.14) has a formal Stokes-Helmholtz decomposition

V.T

=

p ( a - f)

=

-Vq - Q

x A,

V.A

=

0

(3.6)

where cp and A now represent the compression variable and shear variable, respectively, which become n and ~ D in J (2.1%) for Newtonian fluid with a constant p. Taking the curl of (3.6) and using the continuity equation, we obtain the vorticity equation

Thus, the total vorticity variation in an arbitrary material subdomain B bounded by r ? 9 is given by

D

1

0 . 0 +~ T V p

Dt

X

i

Vcp dV

P-

+

1

nXfdSL23

(3.8)

d

-nX(VXA)dS L3

Here, on the right, the volume integral includes the contribution of vorticity stretching and turning and the baroclinicity (note that the latter is due to a self-interaction of compressing process, as it should be: the density

Vorticity Dynatnics on Boundaries

153

gradient does not align to the compressing force), and the first surface integral is due to nonconservative force. The second surface integral should contain our boundary vorticity flux, which is now defined as the normal gradient of A divided by p (Hornung, 1989): 1 (3 3)

u - -n.VA P

as a generalization of (1.1). Indeed, by using identity (2.7) and noticing that V . A = 0, it easily follows that -

1 - n X ( V X A)

=

u

P

-

1 -(n

X

0) X A

(3.10)

P

This observation also indicates how to obtain the general formula of u for any continuous media on any boundary 9: by (3.6) and (3.10), we simply have u

=

n

X

(a

-

f)

1

+ -{n P

X

Vq

+ (n x

V ) x A)

on boundary 9 (3.11)

It is of which the right-hand side only contains quantities right on 9. worth noticing that in the above procedure u is identified via the vorticity equation (3.7) or (3.8) but its formula is obtained via the momentum equation (2.14). We stress that, if the arbitrary subdomain 9 is entirely inside a homogeneous fluid, then the u on 1?9l becomes an interior flux of A across d 9 , which must be accompanied by an equal but opposite flux on the other side of d i B and hence no net vorticity is created. The flux u represents a vorticity source only when it is on a solid boundary dB or an interface S across which some flow quantities have a jump (see (3.20) below).' Moreover, using (2.8a) as an identity, there is (n X 0 ) X A = VA . n, which disappears in two dimensions. Related to this is the fact that (3.8) seems to suggest that one could take - n x ( V x A)/p, instead of (3.9), as the measure of vorticity-creation rate from 9; since then the explicit dependence of u on (n x V ) x A would disappear in three dimensions as well. That this alterative is physically unacceptable was argued by J. Z. Wu and Wu (1993) for Newtonian fluid, which holds true for the general case as well. Let 9 be a small material volume adjacent to our boundary .i-%9 (a "In other words, inside a homogeneous Navier-Stokes flow, the torque due to surface force causes only vorticity diffusion and dissipation but not creation.

J. Z. Wu and J. M. Wu

154

solid wall dB or an interface S ) . Then, the last term of (3.8) represents the integrated torque over the closed d 9 , of which only a part belongs to 9. In contrast, as a vorticity source strength, u must be defined on any open boundary element of 3, and its formula (3.11) was indeed obtained by applying the tangent components of (2.14) on such an open element. Therefore, using - n X ( V X A)/p to measure the vorticity source would lead to a missing of part of its constituents due to the cancellation during integration. This part is precisely the purely three-dimensional effect (n x 0)X A/p, as is evident by comparing (3.10) and (3.11). The physical correctness of using (3.9) to define the boundary vorticity-creation rate will be further verified when we consider thc reaction of the created vorticity field to the boundary 5 3 ' (Section VI). In fact, as a general rule, defining a boundary source on an open surface based on a close-surface integration always has a risk of some effect being cancelled. This kind of cancellation can be seen more clearly from the integrated effect of u over a closed B' ((3.21) below). Therefore, we use the integrated equation (3.8) only as a clue to identifying the boundary vorticity source, but not as a basis of the entire analysis. Finally, similar to (2.16), we may define a reduced stress tensor

T = -pI+S

(3.12a)

where S - A x I =

-ST

or

S,, =

E,,A ~ , =

-S,,

(3.12b)

is the skew-symmetric tensor associated with the axial vector A. Then by (3.6) there is V . T = V . T. Meanwhile, (3.1 1) can be cast to a neater form

u

=

n x (a

-

f)

-

1 - ( n x v > . T on P

9.

(3.13)

2. Boundary VorticityFlux in Viscous Incompressible Flows We return to Newtonian fluid. For a general compressible viscous flow, the vector potential A in (3.9) will be very complicated due to the variable dynamic viscosity p, as implied by the last term of (2.15a). But, for a

Vorticity Dynamics on Boundaries

155

constant p, (2.1%) indicates that (1.1) is still the proper statement of u, and to obtain its general formula one simply replaces A in (3.13) by po.'" We shall be concerned with this situation exclusively. For simplicity, in most of the rest of the chapter, we further confine ourselves to incompressible flow, governed by a = -gk-Vh-vVxo,

(3.14a, b)

V-u=O

where k is the unit vector vertical up, g is the gravitational acceleration, and h = p / p is the incompressible enthalpy. The gravity effect can be absorbed into a modified enthalpy h = h gz, such that (3.13) is specified as

+

u

= =

+ n x vt; + v(n x V ) x o n x a + (n x V ) . ( h ~ 2 v ~ )on nx a

-

(3.15) c%

where R is the spin tensor first appearing in (2.2). The explicit viscous term exists only in three-dimensional flows. According to the discussion h, coupling is about (2.18) and (2.19), eq. (3.15) clearly shows that the (o, in general inevitable. Like (2.22) and (3.lb), eq. (3.15) can be viewed as a derived boundary condition for vorticity as well. We shall return to this issue in Section VII. Note that, even though on an interface we have only one (vectorial) jump condition for the stress, now (3.15) provides two vector conditions, each for one fluid, but they are related by the continuity of a, gk, and (. We now compare different types of boundaries. For a rigid wall with angular velocity W(t), the flux exists only on one side of dB. Write o = o,+ 2W(t) = 5, 2W(f) due to (2.121, with 5,. being the relative tangential vorticity, in (3.15) we may replace w by 6,. Along with the easily derived normal component of (3.14a), this leads to, on dB (J. Z. Wu, 1986; J. Z. Wu and Wu, 1993).

+

ah _ - -n.(a +gk) dn u I0

=

n x (a

-

v(n x V ) . o

+ vL) + v(n x

V) x

5,

(3.16)

(3.17)

J. Z . Wu and Wu (1993) studied the interaction between a rigid surface and a compressible viscous flow, where the definition of boundary vorticity flux is in terms per unit volume, because in that paper only the Navier-Stokes equation was involved and equations per unit volume are convenient for examining surface forces per unit area (see Section V1.B later) and their relevant integrals. Thc rcsults can be easily re-expressed in terms of per unit mass. But, Wu and Wu defined the normal gradient of pc~w as the flux even for variable viscosity, which is inconsistent with the general definition (3.9).

J. Z. Wu and J. M. Wu

156

For vorticity dynamics, this pair of equations complement the boundary information given by (2.22). Similar to the derivation of (2.8b), eq. (3.17) can be further decomposed to

Thus, the curvature will cause a flux if there exists a relative tangent boundary vorticity or shear stress. The creation of normal vorticity from a rigid wall, which cannot be revealed by the stress balance, is now recognized as due to a nonuniform distribution of relative tangent vorticity. The vorticity fluxes on an interface S can be similarly studied. The fluxes on sides 1 and 2 are a , = v1-

dW I Jn1

=

vl-

I

dn '

(n2 = - n l

a 1 =

=

vz- dw2

an,

-

JW?

-v1-

dn '

-n)

Then, on each side of S, we may decompose (3.15) into (J. Z. Wu, 1995)

This result was also obtained by Yeh (1995) in a different context. Compared with (3.18), both a,, and a,, on S are complicated by an extra term due to the unknown calying normal boundary vorticity. Moreover, the net vorticity-creation rate reads

Although a and g k are part of the vorticity source on a rigid wall or each fluid side, they are no longer so for the net creation rate on S. In all of the previous cases, the tangent and normal components of the boundary vorticity flux, a, and (T,,, represent two different physical patterns by which the vortex lines neighboring a boundary A? enter into the interior of the flow (J. Z. Wu and Wu, 1993). As sketched in Figure 4(a), a, implies a ilortex-line ascending pattern. Due to a a,, vortex lines

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(a)

157

(b)

FIG.4. Two basic patterns for the vorticity to leave a boundary: (a) the ascending of vortex lines close to a boundary due to diffusion, and (b) the turning of tangential vortex lines to the normal direction. The gray vortex lines represent boundary vorticity and always stick to the boundary. (Reproduced from J. 2. Wu and Wu, 1993. Reprinted with the permission of Cambridge University Press.)

infinitesimally close to a wall (but not the boundary vortex lines, which cannot leave 95’) spread from and parallel to the wall (but not necessarily parallel to the boundary vorticity at the foot of n). In two dimensions, ascending is the only possible pattern. But CT, is entirely different. The first equality of (3.19b) clearly shows that the solenoidal feature of the vorticity forces some near-wall vortex lines to turn to the normal direction, provided Vv. 6.1 # 0 (a two-dimensional source or sink). This Llorfex-line turning pattern is shown in Figure 4(b) and is the only mechanism that can form normal vorticity near a nonrotating boundary. 3. Bounduq Enstrophy Flux If we integrate (3.17) over a closed material boundary S’,by (2.9), (3.21 ) with all the rest of the vorticity sources vanishing. This happens because the vorticity lines sent into fluid by those sources are all closed, and their vorticities cancel each other during integration. In fact, eq. (3.21) is a dynamic counterpart of the Fiippl total-vorticity conservation law (e.g., Truesdell, 1954). This observation also tells us that, in defining the boundary vorticity flux or examining its physical implication, one cannot start from this type of integral relation, such as the integral of (3.6) over the fluid domain. To avoid this cancellation, J. Z. Wu (1986; see also J. Z. Wu and Wu, 1993; J. Z. Wu, 1995) invoked the enstrophy consideration to

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J. Z. Wu and J. M. Wu

analyze how much vortical flow, regardless the direction of vorticity, is created from a boundary. For incompressible flow, over the flow domain 9,

in which

and @,=vo:020

in

9

(3.23b)

are the boundary enstrophy flux and the enstrophy dissipation rate, respectively. The first term on the right of (3.22) represents the enstrophy variation due to vortex stretching or shrinking, and the second term is its creation rate on 9. Some remarks are needed here. First, because 77 = 5 . u, + lo;,, on a nonrotating 9 with ( = 0, only uT contributes to the creation of new enstrophy. In other words, a nonrotating boundary does not create new normal covticity dynamically. This is consistent with the kinematic background of a,. Second, eq. (3.23a) implies that the enstrophy, and hence Llorticity, can be diffused across a boundary only if both LIorticity and its flux are nonzero on the boundary. This reflects a fundamental difference between the diffusion of a vector field and a scalar field. Physically, as Lighthill's (1963) pointed out, the vorticity diffusion cannot be viewed as diffusion of angular momentum; rather, it is related to the diffusion of momentum through a momentum gradient, which appears whenever there is a shearing. Therefore, if u # 0 but o = 0 on one side of a boundary 9, say side 1, then it would be imprecise to say that the vorticity is diffused across A?. The correct explanation is that some new vorticity is created on ,Y$ and diffused into fluid 1. We shall return to this point later. Third, in two-dimensional flows whether a piece of boundary is a source or sink of vorticity can be easily identified by the sign of u (Lighthill, 1963, 1986; Morton, 1984; Sarpkaya, 1994). But evidently this criterion is not applicable to three-dimensional flows. The proper source-sink criterion should be the sign of boundary enstrophy flux. The second equality of (3.23a) shows that 77 > 0 on a region of 9 implies that the newly created

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159

vorticity by u enhances the existing one adjacent to the boundary, and hence this region is a vorticity source; inversely, 7 < 0 implies a vorticity sink. For example, in Stokes’s second problem mentioned in Section II.D, as t + M, (3.24) Therefore, during a time period, the flat plate serves as a vorticity source and sink alternatively but not equally: in time averaging, it is always a source. Namely, the wall oscillation makes the near-wall flow field more and more vortical. 4. The Role of Viscosity in Vorticity Creationfront Boundaries In (3.18) and (3.19) the vorticity flux is decomposed with respect to spatial directions. We may alternatively split the terms in u according to whether the viscosity v occurs explicitly, denoted by u,,, and uyIF, respectively. Thus, by (3.19, u,,, = n x (a

I

+~i;>

u,,, = v(n x V ) x

ct)

(3.25a, b)

Obviously, at a large Reynolds number, the apparently inviscid vorticity source (inuiscid, for short) uillV is much stronger than the explicit viscous source uVis. In the Euler limit we simply have u = u We restress that, as argucd in Section II.D, even in the “inviscid” source, the viscosity still plays a key role. More precisely, eq. (3.25a) is a consequence of the no-slip condition, because (2.13) has to be imposed; and if its right-hand side is fixed, the normal gradient of corticity is iniwsely proportional to the kinematic 1-iscosity. But, in the literature, there exists some confusion on this almost trivial fact. For the solid-wall case, some authors favored a completely inviscid interpretation, which has been clarified by J. Z . Wu and Wu (1993). Similarly, for the interface case, some authors attributed the creation mechanism to the baroclinic effect alone, because S is a limiting case o f a density-stratified layer if the surface tension is negligible. We now show that the appropriate explanation for the vorticity creation right on an interface should be a combined baroclinic-L:iscous cffect. Indeed, it was emphasized in Section 1II.A that 5 on S is from the viscous tangential stress balance, and without viscosity, there can by no

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means be a boundary vorticity. But the baroclinicity is also indispensable, because otherwise (3.1) would trivially become 0 = 0. On the other hand, after the preceding discussion, there should be no more doubt about the viscous root for the boundary vorticity flux; but the baroclinic effect does not show up in the momentum equation nor can it be seen in the one-sided u formula (3.13) or (3.17). However, its effect appears in the total flux formula (3.20), where without the density jump (and the jump of u ) u , ( r 2 would be zero, as in the case of inside a homogeneous fluid. The baroclinic effect enters the vorticity equation explicitly; for example, the third term on the right of (3.7). Thus, to further understand how this effect combines with viscosity in the vorticity creation from an interface, we apply (3.7) to a Newtonian fluid with density stratification. The resulting equation has been given by Dahm et al. (1989) in a study of vortexinterface interaction. They considered a finite layer of thickness 6, across which the density changes from p 1 to p z , and then took the limit 6 + 0 to obtain a density interface. The circulation and characteristic length scale were assumed to be r and a , respectively, and the dynamic viscosity p was assumed constant. Using a , a’/T, T/u, I’/u’ and p , + p z to rescale x, t , u, ct), and p , respectively, and writing the density gradient as

+

Dahm et al. obtained the rescaled dimensionless vorticity equation:

where A is the Atwood ratio, and g* = a 3 g / T and E = v / r = R e - ’ are the inverse Froude number and inverse Reynolds number, respectively. The surface tension T is rescaled to yield a Weber number W = ( p 1 + p 2 ) r 2 / a T . Obviously, the second term of (3.26) represents the baroclinic generation of vorticity. Dahm et al. (1989) went on to obtain some further simplifications. Assume W is sufficiently large and 6 / a is sufficiently small, that the surface tension can be ignored and the density gradient can be expressed in terms of a delta function 6 ( n ) . Then, eq. (3.26) reduces to

of which the viscous term was also dropped by Dahm

el

al. (1989) in view

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161

of E < 1. This deletion has led one to explain the vorticity creation on an interface solely by baroclinicity. However, the density-stratified layer is not an immiscible interface; in a strict sense, this baroclinic effect is a body source of vorticity rather than a surface source (hence it is absent from a). Moreover, as remarked before (also will be addressed later), for a fixed circulation density, the thinner is the vortex layer, the stronger would be the normal vorticity gradient and hence the viscous effect. Therefore, the viscous term in (3.27) should be retained in the limiting process. This term contains the mechanism leading to a vorticity-creation rate on the interface, to which the baroclinicity is an indispensable contributor. C. CREATION OF BOUNDARY VORTEXSHEETS

So far the theory applies to flow$ at any Reynolds number, Re. When Re >> 1, the theory can be simplified to boundary-layer approximation. The solid-wall boundary layer has been addressed by Lighthill (1963) in terms of vorticity dynamics, and the interface boundary layer has some special features and will be treated in Section V.C. Here, we jump to a further simplification, to look at the asymptotic form of vorticity creation from boundaries at the Euler limit. In this case, the vortex layer created from a boundary 97 reduces to a vortex sheet of thickness 6 + 0. The jump across 9’ and the attached vortex sheet is different from that only across 9; so we denote the former by a double square bracket . , According to our definition of the direction of n, the vortex sheet strength is given by

(in literature one often writes y = c1 x 1 1 ~ 1 , with i3 = - n pointing from side 2 to side 1). Because the velocity jump ‘[u must be tangent to S, lu = n x y

(3.28b)

On a solid boundary ,jlB we have a single-layer sheet, whereas on an interface S the sheet consists of two shear layers of vanishing thickness with strength y 1 and y 2 , each on one side of S. None of these sheets should be identified as 9 itself, which is a surface in geometric sense.

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J. Z. Wu and J. M. Wu

FIG.5. The sandwich structure of vortex sheets on both sides of an interface.

Thus, for the interface case we have a sandwichlike structure as sketched in Figure 5. The strength of the two sheets are

(3.29)

The total vortex sheet strength is y = y + y ? . For flow over a solid wall the vortex sheet model is certainly oversimplified in modern studies of boundary layer and its separation, but it nevertheless provides overall physical insight and may serve as the basis of perturbation expansion at a finite Re. On the other hand, for simulating an interface the vortex sheet model is still in use. From the viewpoint of three-dimensional boundary vorticity dynamics, two issues addressed by J. Z. Wu (1995) for an interfacial vortex sheet are of interest: the dynamic equation and velocity of a three-dimensional boundary vortex sheet. His results can be applied to a vortex sheet created on a solid wall as well and are reviewed in the following. 1. The Transport Equatiorz of Boundary Vortex Sheets

Unlike a vortex sheet inside an otherwise irrotational incompressible homogeneous fluid, for which all relevant studies are built exclusively on the kinematic Biot-Savart law, the dynamics of vorticity creation plays a key role in the evolution of a boundary i’ortex sheet. To derive the general dynamic equation for an incompressible boundary vortex sheet, it is convenient to start from the Lagrangian description. Let T = t be the time

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163

variable in the Lagrangian description, X be the Lagrangian coordinate of the fluid particles on the sheet. We define fi = w . VX as the incompressible Lugrungiun Llorticity (Casey and Naghdi, 1991; here fl is not to be confused with the spin tensor). Correspondingly, let 6 c = 6 x . 6s = 6 n 6 S ( 6 n = 6x . n 0) be a thin material volume element surrounding a piece of the sheet, then

-

r=y.VX=fl6n

(3.30)

is the Lagrangian vortex sheet strength. Therefore, because lim ( V x a ) 6 n 6n-0

=

- n x ,:a!

(3.31)

from the incompressible vorticity equation (Truesdell, 1954)

an

-dT

the dynamic equation for

r

-

(V x a ) * VX

reads (3.32a)

or

a-

dr

+ rv,. u

=

- ( n x l L a J .VX

(3.32b)

The corresponding dynamic equation in the Euler description is

where D Dt

=

d

at

+u.v

(3 33b)

The right-hand side of (3.33a) obviously represents the dynamic source of y,which appears only on a boundary <9. Inside a homogeneous fluid (3.33) falls back into pure kinematics. We now show that, on both a solid wall and an interface, the vector - n X a1 is nothing but the “inviscid” part of boundary vorticity flux: -n

X

llal = alnv

(3.34)

which explicitly reveals the dynamic essence of a boundary vortex sheet. Indeed, for a sheet attached to a solid wall dB, there is -all = a,,, - a R ,

J. Z. Wu and J. M. Wu

164

the difference between the acceleration of the inviscid outer flow and that of d B ; but because alnv= - V h , where h has the same value as that on dB, we immediately obtain - n x lla

=

n x (aB + ~i;)

which is precisely the “inviscid” part of (3.18). On the other hand, for an interfacial vortex sheet, the strength y in (3.33) should be the total value y , + y 2 , and Lal means the difference of inviscid accelerations at the outer boundaries of the “sandwich.” Clearly, this simply equals n X V [ h l , and (3.34) becomes the “inviscid” part of (3.20). In practice, on a solid wall, the strength y of the boundary vortex sheet can be directly inferred from the velocity jump l,uI between the outer inviscid flow and the wall. Thus, eq. (3.33) could be used reversely to obtain (T,”~without solving the pressure field. In contrast, because in general the location and shape of an interfacial vortex sheet has to be solved, eq. (3.33) must be included in its governing equations. More specifically, let

be the Atwood ratio and mean acceleration, respectively, then (3.33a) gives (J. Z. Wu, 1995)

DY Dt

--

y.VU

+ yV;U

=

2n

X

i

Aa,, +

TVK ~

PI +

+ Agk]

(3.35)

P2

The two-dimensional version of (3.35) and its Lagrangian counterpart have been derived by Tryggvason (1989) based on the consideration of a pure tangent discontinuity. For instance, eq. (3.32a) reduces to

(3 3 6 ) 2. Bifurcation and Velocity o f a Vortex Sheet in Three Dimensions It was stressed in Section 1I.D that a material vortex sheet is conceptually different from a mathematical contact discontinuity of ideal fluid. The former consists of fluid particles with very high angular velocity, whereas

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165

the latter consists no particles at all (particles o n both sides of the discontinuity only slide over each other). But, in spite of these differences, we saw that the two models lead to the same equation. The reason was explained by J. Z. Wu (199s). He noticed that, however, one difference between these two models may lead to some identifiable effect: the sheet velocity U in (3.33). A material sheet must have a definite velocity, but for a mathematic discontinuity talking about its tangent velocity is meaningless. Wu found that some new aspect of this sheet velocity is closely related to the vortex sheet bifurcation phenomena, which deserve to be briefly mentioned first. It is well known that a boundary layer may bifurcate into several vortex layers when the flow separates from 97, and so too a boundary vortex sheet. For an incompressible and steady flow over a solid boundary d B , by using the Bernoulli integral, it is straightforward to show that a steady, two- or three-dimensional separating vortex sheet must leave the surface tangentially, either from a sharp edge or a smooth surface (Batchelor, 1956; Mangler and Smith, 1970; Smith, 1977; J. Z. Wu et al., 1993a). Following the flow separation theory on a solid wall (Oswatitsch, 1958; see also J. Z. Wu et al., 1988b, and Section I V . 0 , Lugt (1987, 1989a) studied steady separation (or attachment) patterns on an interface by using local Taylor expansions of the Navier-Stokes equation. The separation occurs at stagnation points of S, where U = 0, which, by (3.1), are also the critical points of [ & I (note that Vnq, = 0 for steady flows). The results are in sharp contrast to that of the solid wall case. For example, Lugt found that, in two dimensions, the dividing strcamlines originating from such a critical point can be single, double, or even triple, see Figure 6. Recently, B r m s (1994) revisited Lugt’s work in a more general setting, including a variable curvature and surface tension on S , and obtained a complete description of the bifurcation patterns for two-dimensional steady interfacial flow. Note that, like three-dimensional vortex line refraction (Section IILA), during the bifurcation process dividing streamlines are also refracted (Figure 6(b)). Parallel to the solid wall case, in the Euler limit the appearance of dividing streamlines implies bifurcation of the interfacial vortex sheet. Unfortunately, most of interfacial flows cannot be assumed steady, but the theory of unsteady separation is still not well established even for the solid wall case. For unsteady flows, when and how an interfacial vortex sheet will bifurcate remains unclear. J. Z. Wu (1995) stressed that, in addition to the preceding separationinduced vortex sheet bifurcation, yet another type of bifurcation can

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J. Z. Wu and J. M. Wu

(3) FIG. 6. (a) The single, double, and triple dividing streamlines near an interface. (b) The refraction of dividing streamline across an interface. (Reproduced from Lugt (1987, 1989a), with the permission of the American Institute of Physics.)

happen only in three dimensions: Some cortex lines in a sheet may turn away from the sheet sufuce to form a normal uorticity field. Quantitatively,

in the Euler limit. Here, a nonzero [[‘I does not conflict (2.12); rather, it means that the turning process of Figure 4 occurs inside the vortex sheet. Therefore, in three dimensions, one should not impose an ad hoc condition that the flow is irrotational away from the sheet. In particular, a nonnegligible concentrated “horn vortex” could form by this mechanism, as will be seen in Section 1V.C for a solid boundary and Section V.C for an interface. Equation (3.37) has led J. Z. Wu (1995) to a new vortex sheet velocity formula. It is well known that the normal component of U is always the same as that of u , or u 2 , and the problem is its tangent component. Traditionally, U has been assumed to be the mean velocity 1 U=ii=-(u 2

+u,) l

(3.38)

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167

which is consistent with (2.10a). To support this choice, Milne-Thomson (1967) argued that if S is a mid-surface of a fluid layer of thickness 6, then ul

=

u

-

6 - n - ~+ u 0(6’), 2

u2 =

6

u + -n.Vu + O ( s 2 ) 2

and hence (3.38). But this argument is invalid, because it has nothing to do with the Euler limit of a vortex layer nor with any discontinuity ( S can even be a surface normal to a vortex sheet). In fact, as mentioned before, to a nonmaterial tangent discontinuity no definite tangent velocity n X U can be assigned; choices other than U = ii are allowable; for example, U = ii + ( Y I ’ u I I with (Y being a free parameter (Baker et al., 1982). However, a material vortex sheet must have a definite n X U,which has to be derived in the Euler-limit consideration. We write

U

=

U

+ U’,

U ’ - n= 0

(3.39a)

and determine U’.Based on thc kinematic Kelvin circulation equation

D -Dt $u.dx

=$a.dx

J. Z. Wu (1995) showed that in three dimensions U’ is related to the normal vorticities 6, and t2associated with y :

n x V(U’.’U =~ ) l lull

+ i t ,u’

Let e , and e2 be the unit vectors along the tangent directions of y and u l , respectively, such that U’ = t i ; e , + t i z e , , and s1 and s2 be the corresponding arc lengths measured from a point on S where 2 = 0. Then the solution of the preceding equation reads

An example of using (3.39) to determine U will be given in Section V.C. If =

0 (this includes all two-dimensional sheets and three-dimensional ones

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168

in a potential background flow), we still have (3.38) (Friedrichs, 1966; Saffman, 1992). J. Z . Wu (1995) also noticed a simple relation between U ' and the sandwich structure of an interfacial vortex sheet: 1

U' = - ( y , 2

Hence, if strength.

y2) x n

-

(3.40)

l =0, then the vortex sheets on two sides of S always halie the same

IV. Vorticity Creation from a Solid Wall and Its Control We now take a closer look at various vorticity sources on a solid wall. When the flow is incompressible (or weakly compressible), we take the density p (or the unperturbed density) as unity, so that p = h and p = v. The global Reynolds number is assumed large. We shall consider four constituents of the boundary vorticity flux u, denoted by (see (3.18a, b)) u,] = n x V p , uv13?i = vg,*

K

= -

uvl,,, = -vn(V;Sr)

uo = n

(n

X

a

(4.la, b) (4.1~)

X 7, ) . K

=

-n(n.(V x

7,))

(4.ld)

where T,+= - T = n x v S r is the wall skin friction. Thus, there are contributions from the pressure gradient, wall acceleration, wall curvature (combined with T, or boundary vorticity g r ) , and a rotational T,, distribution. The effect of gravity can be easily added to the pressure force if necessary. The enstrophy fluxes corresponding to (4.la-d) are 1 7p

= -

-T,,

v

.a

(4.2a, b)

where W,,(t,n) is the normal angular velocity of the wall. Note that the first two enstrophy fluxes are of O(Re;), the third is O(1), and the last one is only O(Re-+). Obviously, all these sources are not independent of each other, for instance wall acceleration will certainly affect the ( p , T ) distribution on the wall. We examine them separately merely for convenience.

Vorticity Dynamics on Boundaries A. THE EFFECTOF

TI-IE PRESSURE

169

GRADIENT

As seen from (1.21, up is the most basic vorticity source on a solid wall drB and is the only source in an incompressible flow over a twodimensional, stationary surface. Figure 7 sketches a typical variation of q, along a flat plate, from negative to positive. Also shown are some velocity and vorticity profiles at different x stations, as well as the corresponding distribution of v,] (not to scale). Note that the enstrophy sink 77 < 0 is a sign that the flow may tend to separate downstream, and before and after that we have enstrophy sources. Usually, across a separation point, a;, has a pulse behavior. If multiple separation occurs behind a bluff body, the amplitude of IT, will have violent oscillation, as shown by Koumoutsakos

Y

.................................. (a) Velocity Profile

*

(b) Vonicity Profile

v

(c) Boundary vonicity flux

(d) Boundary enstmphy flux

Flc;. 7. Vorticity source and sink due to pressure gradient (subscript p is omitted): (a) velocity profile, (b) vorticity profile, (c) boundary vorticity flux, (d) boundary enstrophy flux.

J. Z. Wu and J. M. Wu

170

and Leonard’s (1995) computation of an impulsively started flow over a circular cylinder. With this general picture in mind, we now consider a special class of pressure-created vorticity on a solid wall: acoustically generated riorticity waues. The pressure wave is assumed to be harmonic: (4.3) where n is the circular frequency and u u the velocity amplitude. This problem is a slight extension of Stokes’s second problem of Section ILD, simply with u = db,/dt = -nb, sin nf being replaced by Jh

(T

= -=

dX

Re(inu,, e l k ( r - i r} )= - nu,, sin(& - n t )

(4.4)

Let 6 = ( 2 v / n ) i << 1 be the thickness of the Stokes layer excited by (4.4), and [ = y / 6 be the rescaled normal distance. Then the velocity field inside the layer is known as =

u,(l

-

e(’-1)
(4.5a)

indicating that the flow is no longer unidirectional when k vorticity wave produced by the pressure wave is =

(i

-

UO

6

- <ei(kx-flr+c)

+

O(k’6)

f

0. The

(4.6)

where the ignored part is the contribution of d c / d x . The associated enstrophy wave is (here one has to take the real part of (T and wB first then multiply)

which extends (3.24) and also has a positive average. Lin (1956) showed that, even with a mean flow, if the frequency is so high that the Stokes-layer thickness is much smaller than the boundary-layer thickness, then inside the Stokes layer we still have (4.5) and (4.6), with u g being understood as the corresponding inviscid value at y + 0.

Vorticity mnamics on Boundaries

171

More generally, we add a quasi-parallel local mean velocity U ( y ) and assume the flow is weakly compressible with disturbance dilatation 6 = V . u. Then, the disturbance shearing and compressing processes are governed by a pair of linear equations:

The basic boundary conditions are (1.2) and (1.3). Equations (4.8a, b) can be applied to different situations, and we present two examples. 1. Receptiriity of Boundary Layer to Pressure Waivs The first example concerns the amplification of the Tollmien-Schlichting (TS) instability waves by a forcing wave. For spatially developing TS waves we have

where A,, is their wavelength and k , < 0 means growing. Since the classic 1943 experiments of Schubauer and Skramstad (19481, it has been repeatedly found that an acoustic wave can excite TS waves. But A , , is much shorter than the acoustic wavelength A',' under the same frequency; so the question is how such excitation can happen. This is a typical receptkity problem. A heuristic argument was presented by Nishioka and Morkovin (1986), which directly reveals thc creation of vorticity and enstrophy from a solid wall by waves. Assume 6 = 0 and take p = 1. From (4.8a), the unsteady enstrophy equation follows, which, upon being averaged over a time period and integrated from the wall to any y , gives

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J. Z. Wu and J. M. Wu

where

are the mean dissipation and boundary flux of the unsteady enstrophy, respectively. Setting y = 0 in (4.9) gives a mean of (4.2a) and the streaming part of (4.7). Now, the unsteady field contains a TS part and a forcing part, each satisfies (4.8a). The coupling between wTs and wf occurs on the wall only, through the adherence condition. But, in the enstrophy equation, the coupling occurs inside the flow as well. Thus, Nishioka and - identified two inhomogeneous sources in (4.9) for the Morkovin (1986) growing of w & : an interior source ~‘l.wTsU”,which transfers the vorticity of the mean flow to TS waves due to forcing and a boundary source w - , , ( d p / d x ) f , which directly relates the TS-wave vorticity to the forcing pressure gradient. Although the first source belongs to inviscid advection, the second is essentially viscous. More specifically, assume a long-wave ( k + 0) disturbance (T = A ( x ) e ” “ , where the amplitude is x dependent and has a Fourier transform A , ( k ) . Then, if A ( x ) makes a nonzero distribution hA ( k ) at the TS wave length between x - A/2 and x A/2 (quantities without suffix are referred to TS waves), Nishioka and Morkovin found that the mean boundary enstrophy source will yield a net local input into TS enstrophy, with a rate [I,;

+

where w(0) is the value at y = 0. Similarly, the continuity equation implies that uf is proportional to A ’ ( x ) , say, Vf(y){A’(x) ik,A(x)}. This gives a positive contribution to the TS enstrophy from the interior source:

+

where

* stands for complex conjugate.

Vorticity L?ynamics on Boundaries

173

Based on this heuristic argument, Nishioka and Morkovin proposed that the variation of A is a likely effective mechanism for the boundary layer receptivity. The increment A A , ( k ) reflects the commensurability of ATs and the local characteristic length at x introduced by the variableamplitude forcing waves. They carefully analyzed all available experiments, including their own and found that the proposed mechanism works well. 2. Sound- Vortex Irzteractiorz in Duct Shear Flow

In the receptivity problem the pressure wave is given and only (4.8a) was used. Another interesting application of (4.8a, b) is the sound-vortex interaction on a background mean flow U ( y ) in a duct (from y = 0 to y = 2). Because U ( y ) satisfies the no-slip condition and is nonuniform, a sound wave having a plane front at x = 0, say, must be refracted toward the walls, and only some of modes can reach far downstream. In the acoustic community this problem has been known for more than three decades (e.g., Pridmore-Brown, 1958; Shankar, 1971), and the completely inviscid sound-field character has been well clarified by solving (4.8b) alone under homogeneous boundary conditions (an eigenvalue problem). However, through (1.21, the sound must produce a vorticity wave of the same magnitude of order; and once this vorticity wave is formed it will in turn emit sound (Section 1I.C). Consequently, to obtain a full answer one has to solve the coupled system (4.8a, b) under the coupled boundary conditions (1.2) and (1.31, which becomes nonlinear. A simplified approach to this coupling was given by J. Z. Wu et a f . (1994a), who split the process into two subprocesses and solved them sequentially. First, because the viscous effect is ignored in the acoustic equation (4.8b), an inviscid refracted pressure field was computed as Shankar (1971) did. Thus, the vorticity wave followed from (4.8a) and (1.2). Second, eq. (4%) was recasted to a linearized vortex-sound equation in terms of the total enthalpy H. With M = U / c as the Mach number, the dimensionless H-equation reads

from which the pressure field due to the acoustically created vorticity wave can be obtained and added to the initial inviscid solution. Note that, in solving (4.11), a rriscous boundary condition, derived from (1.3), has to be imposed even though the equation is inviscid.

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J. Z. Wu and J. M. Wu

1.0

1.0

08

08

06

06

0.4

04

02

02

00

00 -20

-I0

0

10

lOloSlO(p')

FIG.8. The acoustically created vorticity waves and its effect on the sound pressure level in the duct. M = 0.3. Left: The amplitude of vorticity waves (real part, solid line; imaginary part, dash line) at x = 20 with Re = 1000; top: h = 5, bottom: k = 10. Right: The viscous (solid line) and inviscid (dashed line) sound pressure level at k = 10 and different Y stations; 1994a). top: Re = 1000, bottom: Re = SO00 (from .I. Z. Wu ei d.,

Some results of this approach with a parabolic mean flow M ( y ) = M,(2y - y 2 > are shown in Figure 8. Traditionally, the viscous effect on sound propagation had been thought always as an attenuation, and was approximately treated by an equivalent admittance derived from the high frequency limit (4.5) of the Stokes layer. However, Figure 8 indicates that the viscosity has a nonmonotonic effect on the sound pressure level (SPL). This can be seen more clearly from Figure 9, which gives thc wall SPL variation as k and the Reynolds number Re. When a sound wave excites a vorticity wave through the no-slip condition, it loses some energy. But, as seen in the receptivity example, there is an interior unsteady source U"r' for disturbance vorticity, which makes the acoustically created vorticity

175

VorticityDyrzarnics on Bounduries

I .o

, 0.5

I

35

3.0

.

.

.

40

log io (Re) FIG.9. The wall SPL at x J. 2. Wu el (11.. 1994a).

=

20 for different wavenumbers and Reynolds numbers (from

wave able to absorb enstrophy from the mean flow and become a selfenhanced source of sound.

B. THEEFFECTOF WALLACCELERATION We now turn to (4.1b), the wall acceleration as a vorticity source. Its general feature is similar to cr,' ; here we concentrate on some less familiar issues. A key condition leading to (4.1) is the acceleration adherence (2.13), which was obtained by following fluid particles on a boundary 9. Therefore, the independent spatial variable in (4.1) is in fact the Lagrangian coordinates of these particles, say, X, . Correspondingly, the physics of vorticity creation from 97 should be unambiguously understood in terms of a(x(X,, t ) , t ) . This observation implies that special caution is necessary in dealing with moving boundaries if the Eulerian point of view is to be used. Indeed, once 9 has a normal motion U , , it is meaningless to break a on

176

J. Z. Wu andJ. M. Wu

9 into a local acceleration d u / d t and a convective acceleration u . Vu, because at t + dt each of these Eulerian accelerations will no longer refer to any particle on 93.For example, on a rigid wall d B with velocity b and angular velocity W,

Some confusion would easily occur if i)B has tangential motion because then a h e d spatial point x B , once belonging to JB at t , will always do so. However, this xN is factually sliding along the wall. Consequently, talking about a(x,,t) does not reflect the same physics as a(x(X,, t ) , t >does (J. Z. Wu et al., 1993b). Fortunately, the unidirectional flow model (2.33-2.381, including two Stokes’s problems and their extension to rotating circular cylinder, as well as the examples used by Morton (1984) to illustrate the vorticity creation from moving walls, happen to be free from trouble, because in these situations cr is independent of the location along the wall. Two examples lack of such symmetry are reviewed next. 1. Streaming Effect of Flow oiw a Cylinder with Rotary Oscillation The preceding remark becomes crucially important if wc consider the time-averaged vorticity creation from a periodically moving boundary 9 lack of symmetry. The conventional Eulerian mean (EM) over the period T , C(x)

1a ( x , t ) dt T o I

=

-

T

for fixed x

is in general either meaningless (if has a normal motion) or easily misleading (if 9moves only tangentially). Instead, one should either use a coordinate system comoving with &7 (in that system there is no boundary acceleration) or invoke a Lagrangian-type description, for instance the generalized Lagrangian mean (GLM) developed by Andrews and McIntyre (1978). As illustration, J. Z. Wu et al. (1993b) considered the timeaveraged (T for a uniform flow over a circular cylinder with rotatory oscillation of finite angular amplitude 6. The Reynolds number and Strouhal number were assumed sufficiently high; thus, the pressure gradient on the cylinder can be approximated by that at the outer edge of the boundary layer, and the Stokes layer is well inside the boundary layer so

Vorticity Dynuniics on Boundaries

177

that the outer inviscid flow does not feel the cylinder's rotation. Then, if the flow is attached, it was proven that the E M incorrectly predicts a zero mean vorticity creation. In contrast, for the dimensionless Lagrangian mean G I - , 3'- = 2 ~ ( i ) s i n 2 0

(4.12a)

where 8 is the polar angle and

which has a series of zeros as 6 increases from 0 (the first zero occurs at H^ = 68.9"). This is a typical nonlinear streuming effect (e.g., Stuart, 1963), which in the present case is measured by o,=

cL

-

c r I ~ == ~ ,-2{1 - ~ ( ~ ) ] s i n 2 0

and is strongest when A(G) = 0. At these zeros no mean vorticity is created over the cylinder; thus, thc flow behaves like a mean iiscous potential flow with zero mean pressure drag." Moreover, if the cylinder has a pair of attached wake vortices, IGLI become very small over most of the cylinder surface at 6 = 45", see Figure 10. In this way, Wu et ul. provided the first (though preliminary) explanation for an interesting experiment of Taneda (1978) shown in Figure 11.12 Through a conformal mapping, the flow over a circular cylinder can be cast to that over a flat plate. Thus, J. Z. Wu et al. (1993b) also made a preliminary study on the effect of wing oscillation on the lift generated by a trapped vortex, based on thc inviscid analysis of Saffman and Sheffeld (1977). It was proven that in a certain range of parameters a mean lift increase is possible due to the streaming vorticity flux. 2. Vunishing Mean Drug in Viscous Flow ouer a Flexible Waiy Wall Another interesting example showing the effect of boundary acceleration on vorticity creation can be found from J. M. Wu et ul. (1990) and its "A viscous potential flow is diffusion free due to the absence o f the viscous term in (2.24), hut not dissipation free. Lagerstrom (l9h4) pointed out that this is possible because the dissipation depends on the symmetric part of the velocity gradient only. The Rankine vortex is a simple example of such flow, where h e dissipation occurs in the irrotational flow region outside the vortex core, but the total dissipation can be equally inferred from the core vorticity (J. Z. Wu et al., 1993a). I? So far Taneda's experiment has never been rcproduced numerically for some unknown reason.

J. Z. Wu and J. M. Wid

178

3.0

-

2.0 -

6=0

1.0 -

0.0 -

-1.0 -

I

-4.0

0

60

30

I20

90

150

8

I80

FIG. 10. The distribution of boundary vorticity around a circular cylinder that is in a uniform flow, carrying a pair of FGppl vortices, and performing rotatory oscillation with different angular amplitudes of 0. (Reproduced from J. 2. Wu et ul., 1993b, with the permission of the American Institute of Physics.)

subsequent exploration by the present authors and A. H. Eraslan and K. J. Moore (unpublished, 1992). The problem involves a uniform flow U over a flexible wavy wall y'

=

af(E

-

n t ) = af(k.i

-

nt

+ h),

If1 I 1

(4.13)

which in the laboratory coordinates (2, j ) has an up-down oscillation and thereby forms a train of traveling waves. Here A is the wavelength, k = 27r/h is the wave number, and a the amplitude. In the coordinates

x=P-ct,

y = j

moving with the wave speed c = n / k , the wall motion becomes steady and along its tangent direction, with a variable velocity magnitude

Vorticity Dynamics on Boundaries

179

FIG. 11. Streamlines and streaklines around a circular cylinder performing a rotatory yil l at i o n in a uniform flow: d = 1 cni, U = 0.33 cm/sec, Re, = 35, N = 2 Hz, N d / U = 6 , 0 = 45”, x = U I .(a) x/d = 0; (b) 1.5; (c) 3.4; (d) 12. (Reprinted from Prog. Aerospace Sci. 17, S . Taneda, Visual study of unsteady separated flows around bodies. Pages 287-348. Copyright (19781, with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

and direction toward upstream if c > 0. Therefore, we have a tangentially accelerating boundary. The value of a / A was taken large enough to allow for the generation of separated vortices at the troughs of the wavy wall. A similar problem had been studied by Caponi et al. (1982) with both linearized analysis and numerical computation, the difference being that Caponi et al. imposed a periodic boundary condition to the flow, which was absent in the work of J. M. Wu et al. In this way, they discovered that, for a given wave pattern f(hx),there exists a unique critical waue speed c, with 0 < c,/U < 1, such that at c = c, the r1ortafEow becomes periodic in nature, with both averaged fnction drag and averaged pressure drag being zero. Physically, with c = c, the separated vortices can be well captured and do not grow or break away as one goes downstream. They form a fluid “sheath” to isolate the near-wall shear layer from the main stream. Therefore, for a remote observer who can see only the averaged effect, there is again a viscous potential flow. This type of wavy-wall flow with captured vortices was observed experimentally by Taneda (1978) and

180

J. 2. Wu and J. M. Wu

Vorticity Dynumics on Boundaries

181

Savchenko (1980), and our full Navier-Stokes computation indeed confirmed the existence of a stable zero-mean drag state. Figure 12 shows one with the computed flow pattcrn at the nth and ( n 11th waves from leading edge ( n can be arbitrarily large within the computational ability), with c = c,.. In the computation no periodic boundary condition was imposed, so the flow is naturally periodic at c,. Computations also showed that, under off-design conditions ( c is not close to c,), the flow strongly fluctuated and never reached a steady state or, even worse, vortices broke away. For the present purpose we just remark that, because near the critical state the flow behaves almost inviscid as well as steadily, a simple inviscid analysis similar to that in the earlier cylinder example can predict the critical wave speed c, quite satisfactorily (for one such analysis, see J. M. Wu et al. 19901, and this analysis can well be made based on the requirement that the averaged boundary vorticity flux is zero, exactly the same as in the cylinder case. In fact, to reach the desired critical state one needs two conditions: first, the flow has to be periodic; and second, because the inviscid vortex-induced velocity on the wall is in general different from the wall velocity given by (4.14) (thus, there is a near-wall vortex sheet), the averaged sheet strength has to be zero. This requirement precisely predicts c,/U = 0.414 for the case of Figure 12.” Although the first condition is satisfied by requiring a zero mean u/), the second will be satisfied by requiring a zero mean u 0 .

+

C. THREE-DIMENSIONAL EFFECTS In three dimensions therc are two more vorticity sources uviSii and u V i FCompared n. with cril and wi7, at large Reynolds numbers, they are

small, except for some highly localized regions of the surface where flow separates, which we now focus on. The pressure gradient effect is also involved. The behavior of steady separation in two and three dimensions has been well u n d e r ~ t o o d . ’For ~ three-dimensional steady Navier-Stokes flow, fluid ”In fact, the Navier-Stokes computation was made under the guidance of this inviscid prediction. 14 The concept “separation” can be underhtood in two ways (J. Z. Wu et ul., I%%). One may consider either the boundary layer separation under the boundary layer approximation or the fluid particle separation that may or may not cause the whole boundary layer to separate. The study of the latter is based on the Taylor expansion of the full Navier-Stokes equation and relevant critical point theory. Hcrc we confine ourselves to the latter.

J. 2. Wu and J. M. Wu

182

\ Gc

tedl 1

FIG.13. The separation stream surface and its leading-edge and trailing-edge streamlines.

particle separation initiates from a critical point of the T~ field, say, C , , and terminates at another critical point, say, C,. C, can be a saddle or a saddle node, corresponding to the “closed” or “open” separations, respectively; and C, must be a node, including focus (Zhang, 1985; J. Z. Wu et ul., 1988b). Like a two-dimensional separation streamline, a separation stream suiface grows from the skin friction line between C, and C,, which is the dividing surface of the free vortex layer. This surface has a leadingedge streumline initiated from C, and a trailing-edge streamline initiated from C, (Hornung and Perry, 19841, see Figure 13. To study the local vorticity behavior near these critical points, one may introduce an orthonormal basis (el ,e 2 ,e,) with origin at a C , , say, such that e 3 = ii along the normal and toward the fluid. The directions of e , and e 2 can be chosen along T~ and g , respectively, as one approaches C, from upstream (Figure 13).

1. Vorticity Flux at Wall Critical Points J. Z . Wu et al. (1988b) showed that at C j or C, with T, = 0 there must be vV,. g = - n . ( V x 7,) = 0; so by (4.1), we simply have

u

dP

=

u,, = -e l dX,

dP

-

-e2

(4.15)

Jx,

with q,= 0 there. Away from these points, other sources, cryisnand uvisn, may play some role, as will be treated later. Then, on the (el ,e,) and

Vorticity Dynarnics on Boundaries

183

(e, ,e,) planes, the leading- or trailing-edge streamline has inclination angles's

On the other hand, near critical points we have expansion w(x)

=

x . (V&)O

+ O(lxl')

= -"3U UJO)

+",[:],e*

+ O(lx12)

Here, by (3.18a) and (2.6b), there is d p / d x , = - V T . I,, which at critical points is simply - d ~ , / i l x , in our local frame. Thus, in particular, at a point P on the leading-edge streamline, w(x> =

-

x3 v

2 a,,,(O)e, + -Ir,,(O)e,j 3

+ 0(1xi2)

(4.17)

Similarly, it can be shown that the velocity at P(x) is

(4.18) Therefore, the vorticity at the separation stream surface is fed by a;, at x = 0 and is convected away from the wall by a velocity (4.18), which to O(Ix1) is along the x direction. This vorticity will join those from the boundary layers of both sides of the separation stream surface to form a free vortex layer. For two-dimensional separation, these results reduce to a more familiar form. The separation region has a small streamwise scale of O(Re- ;),thus a,,has a narrow peak of O ( R ef ) .This peak causes the boundary layer to separate; that is, the attached vortex layer bifurcates into a free vortex layer and an attached inverse layer. With the preceding coordinates, we now have a;, = a,,, O 2 = ~ / 2 and , (4.16a) reduces to the result of Oswatitsh (1958). A peak a,, > 0 implies that the separation streamline must be inclined to downstream, and, by (4.171, at least for 1x1 << 1, the [,orticityon separation streamline is dominated by the upstream shear flow. This analysis can be carried to higher orders by a set of recurrence formulas (J. Z . Wu et al., 1088b), but the results will soon become extremely complicated. IF

In J. 2. Wu ef al. (1988b) some relevant formulas were not reduced t o the simplest form.

184

J. Z. Wu and J. M. Wu 2. The Effect of Wall Curi!atureand Rotational r, Field

Because right at the critical points of the T,, field both uvIyn and uv15,, vanish, to examine their effects we need to shift our focus from the neighborhood of these points to the narrow strip neighboring a separation line, which is a T, line connecting the initial and terminal critical points, C, and C,. First, if I K I = O(v-'),the vorticity flux uv1\71 due to the surface curvature is comparable to up.This happens at a sharp edge of d B . Each small piece of the edge can be represented by a straight line, and the T, line must be along the edge. Thus, only the large curvature in the cross section counts. Let e, be the unit vector along the edge, and e, tangent to the sectional curve that has a curvature radius E = O(vi)or even smaller. Then (J. Z . Wu and Wu, 1993), (4.19) indicating a strong vorticity creation along the edge. Note that, by (4.2~1, whether a curved surface is a source or sink of enstrophy depends only on the sign of the sectional curvature of dB in the (e2,n) plane: in terms of our basis, on a nonrotating wall, TV,,,

= V5%2

Therefore, a sharp convex edge is always an enstrophy source. As the edge sweeping angle increases, T~~ will be larger and hence the edge-vorticity creation will be stronger. The total creation from an edge cannot be overestimated, though, because to the leading order the integration of uv,s71 over the edge area is independent of the curvature radius E and hence is still of O(v4).However, this relatively weak vortex layer is a seed of flow separation from edges. In fact, although (4.19) indicates that the created edge vorticity is along the e2 direction (hence, an ascending pattern), after traveling an O ( E )distance it will soon turn to the normal direction simply due to the sharp turning of e2 itself. Then, this vortex layer may become a kernel surface, around which the vortex layers from the upstream of both sides of the edge develop into a strong separating free vortex layer. Therefore, the vortex layer shedding from a sharp edge should have a sandwichlike structure, and further study of the effect of such a structure on the layer's evolution would be desirable. Second, to have an appreciable crvl\,r we need sufficiently large n . ( V x T ~ ~Its ) . peak value appears near a separation line toward which a family of T, lines converge or near an attachment line from which a family of 7,

VorticityDynamics on Boundaries

185

lines diverge. With the local orthonormal frame introduced before, on a separation line (4.ld) reduces to (T3=

1 dr,. -~

h , dx,

+--

T~

dh,

h , h , ax,

(4.20)

where h , ( a = 1,2) are the Lam6 coefficients on the surface. Thus, the faster e , and e 2 change their directions, the stronger will be the normal vorticity. In particular, the magnitude of normal vorticity flux reaches a maximum near a focus of the T~ field, where the separation line rolls up into spirals and brings the created normal vorticity to a local area (a focus of the T, field is always a concentrated sink or source of the 6 field and a source or sink of [ field; see Figure 4(b)). Unlike the vortex layer generated right from a sharp edge, therefore, the spiral structure provides a mechanism to form a concentrated normal vortex, often called tomadolike ilortex (or horn uortex). Such vortices have been observed in many separating flows. As an example of these three-dimensional vorticity sources, Figure 14 sketches a complex vortex system observed in a jet in crossflow (Shi et al., 1991). The jet was issued normally out of an orifice of a flat plate and interacted with the oncoming boundary layer flow. The flow was incompressible, and the vorticity of this vortex system was created entirely from the solid surface, including the flat plate, the inner wall of orifice, and their juncture. From the figure we first see the vorticity in the attached boundary layer (on the left), which was the result of a u,>.Approaching the jet, an inverse pressure gradient similar to that in Figure 7 caused the boundary layer to separate and form a horseshoe vortex. The separated flow reattached in front of the jet and there was a secondary separation and another horseshoe vortex. Similarly, inside the jet pipe a u,] provided the vorticity in the pipe shear flow, that eventually left t h e orifice and became a free vortex layer. Then, because along the side edge of the orifice T~ has a nonzero component, a pair of weak vortex layers was formed near the edge as predicted by (4.191, which was experimentally distinguishable from the orifice vortex layer. Finally, as the horseshoe vortices went downstream of the jet, some horn vortices appeared as the result of u,,,,,.These “microtornados” are somewhat similar to the KBrmrin vortices, as can be seen from Figure 15 (see color plates), a similar experiment on a jet in crossflow (J. M. Wu et a!., 1988. Copyright 0 1988 AIAA. Figure 15 reprinted with permission). However, the fundamental difference is that, unlike a solid

J. Z. Wu and J. M. Wu

186

U

a

U

+

t U-1703

b Z

,Vortex-layer fromJet-pipe Boundary Layer

Tomado-Like Wake Vortices

'Horse-Shoe Vortex due to Primary Separation Oncoming Crossflow Boundary Layer Vonicity C

x

-L

U-16%

FIG.14. The near-field vortex system in jet in crossflow. Sketched based on an experiment: (a) the side view (central plane), (b) the top view, ( c ) a perspective view (from Shi et ul., 1991).

Vorticiry Dynamics on Boundaries

187

cylinder, no new uorticity curl be created from the jet plume boundary, as stressed by Fric and Roshko (1989) and Shi et al. (1991). Unfortunately, to the authors’ knowledge, so far no quantitative results of u distribution came from a three-dimensional flow over a curved surface, and this discussion is of only qualitative feature. The reader may also revisit other available experimental and computational ( p , 7 , ) distribution to obtain a qualitative estimate of the u behavior. Many such examples, among the others, can be found in Doligalski et al. (1994). Finally, it is worth noticing that, although globally u,,,, is much smaller than u p ,they seem to become comparable in the near-wall region of turbulent boundary layer, where the flow is dominated by a much smaller local Reynolds number. Therefore, we may expect that the normal vorticity source, along with the pressure gradient source, could play an important role in understanding the near-wall coherent structures of turbulence, such as those reviewed by Robinson (1991).

D. VORIICITY-CREATION CONTROL Vortex control stands at the ccnter of various flow controls (for an extensive review, see Gad-el-Hak, 1989), including the aeroacoustic noise control, because at low Mach numbers vortices are the only “voice of flow” (Muller and Obermeier, 1988; see the vortex-sound equation (2.32a)). At present time, our knowledge of various vortex controls is going through a transition from an empirical art toward a rational science. Although the process is not yet complete, the general theory of boundary vorticity dynamics nevertheless suggests several guiding principles for achieving a successful uorticity-creation control. Thc principles will be exemplified here by a few typical quantitative results, experimental, analytical, or numerical. Theoretically (and partially practically), vortex control can be achieved at any stage of its life. For a vortex generated from a solid wall, these stages include (1) vorticity creation from the wall; (2) boundary layer evolution; (3) flow separation and the formation of free vortex layer; (4) rolling up into a concentrated vortex, its instability, receptivity, and breakdown; and ( 5 ) dissipation and transition to turbulent eddies (for a systematic exposition of this event sequence, see J. Z . Wu et al., 1993a). However, as stressed by J. Z. Wii and Wu (1991), to control a vortex at different stages is by no means equally easy and effective. The basic guiding principle is, the earlier, the better. For a desired favorable vortex, it is much easier to form an excellent creation circumstance, so that it is

188

J. Z. WuandJ. M. Wu

strong and stable at very beginning, than to enhance it after it has been poorly born. Similarly, it is always less effective to eliminate or alleviate an already formed unfavorable vortex than to prevent its formation. Therefore, whenever possible, the uorticity-creation control is of fundamental importance in various controls of boundary layer and its separation.I6 Typically, we are given a baseline-configuration geometry and a set of flow conditions, and hence also the baseline distribution of various vorticity sources. Therefore, the main objectiiie of vorticity-creation control is to manage a local change of the u distribution to improve the global distribution of vorticity as much as possible. 1. Steady Separation Control by Monitoring Local a,, Usually, the goal of most steady separation controls is to eliminate or alleviate smooth-surface separation, which is almost always unfavorable and uncontrollable. The key local regions for control are the neighborhoods of critical points of the 7,. field; that is, those C j and C, of Section 1V.C. This is because, first, as shown by J. Z. Wu et al. (1988b), the separation stream surface (dividing stream surface of the free vortex layer) consists of only those streamlines initiating from these points, thus, monitoring C, or C, can gain the best effect; and second, due to the topological rules of critical points that the number of nodes minus the number of saddles must equal 2 on a closed, single-connected surface (e.g., Tobak and Peake, 1982; Chapman and Yates, 19911, if one can remove a saddle, say, a node some distance away must disappear simultaneously and hence a local control will lead to a global topological change. In fact, these critical points and their connections are the skeleton of the entire near-surface flow (Hornung and Perry, 1984). Therefore, another guiding principle applied to steady flows is that the local Liorticity-creation control should be focused on critical-point control. In two dimensions, critical points degenerate to straight lines, and as is well known, a control applied to a neighborhood of the separation “point” is indeed most effective. Moreover, it is also well known that more critical points imply more complicated separation pattern and worse flow quality. Thus, the next principle is, for gicen baseline configuration and flow condition, the number of critical points of the T, field should be minimized, the fewer the better. Two remarks are appropriate here. First, controlling the critical points of the T,, field is by no means merely modifying those explicitly viscous vorticity fluxes uvisT and uvin. Rather, the T ,field, and hence these fluxes, Ih

This does not exclude the necessity of later stage control, because the effect of early stage control may not be able to cover the whole working range.

Vorticity Dynunzics on Boundaries

189

is a consequence of the leading-order “inviscid” fluxes ulJand ua (J. Z. Wu and Wu, 1993). Therefore, the focus should be on the latter. Second, for unsteady flow, the separation criteria and a thorough critical-point analysis are not yet available; in that case the principles should be understood from only a time-averaged point of view. An example of truly unsteady control will be reviewed at the end of this section. On a stationary wall, the smooth-surface separation is caused by a local unfavorable up or a sink of enstrophy flux ql, (Figure 71, and any means that can fill up this sink will work. For example, a local suction is an effective means for two-dimensional separation control known since Prandtl. Essentially, associated with sucking out the low-energy retarded fluid in the boundary layer is a local change of pressure gradient, which causes a local change of rlr, from sink to source, see Figure 16. Note that, as pointed out by Reynolds and Carr (19851, the vorticity advection due to a suction with uniform injection velocity is exactly canceled by a corresponding advection term in the vorticity transport equation. Thus, there is

FIG. 16. Enstrophy flux control by local suction near the separation point: (a) suction changes the enstrophy flux by changing the local pressure gradient, (b) sufficient suction removes the separation, (c) insufficient suction moves the separation point to downstream.

190

J. Z. Wu and J. M. Wu

no net gain in removing vorticity from the fluid by suction and the mechanism of suppressing separation by suction is exclusively due to the change of vorticity creation. In three dimensions, the critical points of the 7, field are isolated, which can be recognized from the oil flow pattern with the aid of the aforementioned topological rules and the stability theory of topological structures (e.g., J. Z. Wu et al., 1993a). A local suction is again applicable. This is especially so for removing a horn vortex, because its “root” is highly localized, centered at a focus (a C,). As mentioned already, removing a horn vortex means automatically eliminating a distant saddle point, so a change of the flow behavior in a larger region is achievable. For example, at a high incidence a strong swirling flow may be formed in a S-shaped inlet, resulting in either a sharp reduction of the engine efficiency or a stall. By flow visualization, Guo and Lin (see Lin, 1986) found that the swirl was a horn vortex initiated from a focus at the lip of inlet. Thus, a local suction was applied to an S-duct at an incidence about 60°, which indeed effectively prevented the formation of this swirling flow. The effect of blowing at the same location, in contrast, was found to enhance the horn vortex by increasing its energy, which further deteriorated the inlet behavior.” The same method can be applied to external flow, as confirmed by our water tunnel experiment with a swept-forward wing flow at (Y = 40” and a cylinder-plate juncture flow (J. Z. Wu and Wu, 1991). A single-hole suction with cp = 0.016 could appreciably alleviate an unfavorable horn vortex on the upper surface of the wing and make most part of the outboard flow attached (the inboard flow was massively reversed and could not be controlled by a local suction). Similarly, three suction holes in front of the cylinder (near C , ) with cp = 0.077 greatly suppressed the necklace (horseshoe) vortices. It should be stressed that suction is not the only way to suppress a separation and, in fact, often not a practically adoptable way. Modifying the local configuration within an allowable range, if possible, may be the simplest and most reliable passive control method to achieve a desirable local up distribution. Gupta (1987) reported that, to eliminate the necklace vortex at the juncture of a vertical cylinder and a flat plate, one can install a small delta-wing-like device at a negative angle of attack in front of the juncture. The device produces two effects: first, the ramp formed by the delta wing acts as a barrier to the rolling up of the necklace vortex; 17 Therefore, if one needs to enhance a horn vortex in a duct, blowing near the focus is effective.

Vorticity Dytiamics on Boundaries

191

second, the wing produces a pair of counterrotating streamwise vortices from the wing tips, of which the sense of rotation is opposite to that of the original necklace vortex and so cancels it. The essence of this device is, in fact, still a local pressure control.18 Even though a qualitative diagnosis by oil-flow patterns, say, may reveal the nature of the problem and what kind of change is desired, a quantitative analysis is necessary to tell the strength of control. To do the latter, one needs a sufficiently accurate computation or measurement of the T,$,field along with the pressure distribution, especially near critical points. This allows one to estimate the force acted on the relevant local region, which is roughly proportional to the strength of control. As illustrated in Figure 15, an insufficient control strength cannot reach the goal. Later we shall see that the theory of boundary vorticity dynamics can provide both a much improved method of computing the T, field and a unique estimate of the control strength directly in terms of boundary vorticity flux. 2. Vorticity-Creation Control by Unsteady Forcing As a further guiding principle, most continuous steady controls that work by imposing a forcing can be replaced by a pulsating control with much less power input yet achieile the same or eivn better effects. Therefore, unsteady controls have recently been a subject of intensive study. The flow to be controlled can be basically stcady or inherently unsteady. The relevant physics, along with many examples, was systematically reviewed by J. Z. Wu et al. (1991). Here we address some topics most directly relevant to the vorticity-creation control. For the unsteady control of a basically steady flow, one’s concern is the mean effect, which is nonzero even if the forcing wave is harmonic. Thc nonlinear interaction and resonance will lead to a net streaming effect. But, unlike the receptivity problem of Section IV.A.l, where one needs to know only if the mean fluctuating enstrophy becomes stronger, for proper control the mean change of iwtorial vorticity field is important. What we require is a favorable vorticity field with certain direction being enhanced. The mean enstrophy-flux consideration cannot distinguish the alternative change of the vorticity direction and hence is not appropriate in studying unsteady controls. 18

We remark that, according to our guiding principles, some configuration modification is better than installing an extra device, which can directly prevent the juncturc separation rather than cancel the separated vortex.

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J. Z. Wu and J. M . Wu

The unsteady means for control can be an acoustic wave, an oscillating flap, periodic blowing or suction, or other devices. The objective of control can be the suppression of separation from a smooth surface and, when the separation is inevitable or desired, achieving a well-organized attachment at a prescribed location (where the flow is usually turbulent even if the separation is laminar). In the latter case, not only vorticity creation but also its advection and instability are involved." So we confine ourselves to the former. Collins and Zelenevitz (1975) first found that an acoustic wave may delay separation. Since then, many experiments have been carried out to clarify relevant mechanisms and improve the control efficiency (see the review of J. Z. Wu et al., 1991). Several mechanisms may be involved in the change of mean profiles by forcing, such as the receptivity of the separated vortex layer and forced transition to turbulence, so for our present purpose of vorticity-creation control, the basic physics should be similar to that discussed in Section 1V.A. But, now the attention must be paid to the time-averaged effect on the pressure gradient and skin friction (or velocity profile). The mean velocity profile, and hence the separation status, can indeed be altered by forcing waves, as seen from Figure 17 from the experiment of Nishioka et ul. (1990). The streaming effect of forcing can be further understood from a two-dimensional perturbation analysis of X. H. Wu et al. (1991). For a steady laminar separation, around the separation point the flow is governed by an interacting triple-deck structure (e.g., Smith, 1982). Let Re = U,L/v >> 1 be the Reynolds number based on the distance L from the leading edge to the separation point, then the basic length scale near the separation point is q,= Re-; << 1. The order of the streamwise length of these decks is E : ; and the normal scales of lower, main, and upper decks are c ( : ,6: (the attached boundary layer thickness), and c i , respectively. Now, upon the basic steady flow we impose a high-frequency forcing wave with circular frequency n , so that a reduced frequency or Strouhal number St = n L / U , >> 1 enters the interaction. Let E = St-f << 1 be the

"To understand such a complicated process, Reynolds and Carr (1985) combined the boundary-vorticity-flux analysis with a qualitative cartoon of unsteady vortex advection driven by oscillating control devices. In this way, they explained some basic physics of many experimental examples of separated flow control.

Vorticity Dynunzics on Boundaries

193

1s

1s

II

x 10

10

5

0

0 0

0

U&

U&

1.0

1.o

0

u/v,

1.o

FIG. 17. Mean velocity profiles with and without acoustic excitation on a flat plate at an angle of attack of 12" and Re,- = 4 x 10'. Curves with triangles show the case of no forcing and thosc with open and black circles show forcing of 200 and 600 Hz, respectively. (Reproduced from Nishioka et ul. Copyright 0 1990 AIAA. Reprinted with pcrmission.)

194

J. Z. Wu and J. M. Wu

normal scale of the viscous oscillating Stokes layer, then the asymptotic structure of the disturbed separating flow naturally depends on the ratio

(4.21) When a >> 1, the Stokes layer is much thinner than the lower deck; when a << 1, the viscous oscillating flow in the Stokes layer enters the main deck and is convected into the upper deck as well due to the up-welling. The frequency between these two opposite extremes, a = 0(1), is known as the Tollmien-Schlichting frequency. This parameter a and the up-welling velocity, along with the x derivative of t h e streamwise velocity (which may be large near separation), jointly determine the flow character and suggest how to introduce an appropriate control for the desired streaming effect. Quantitatively, let (x, y ) be the global coordinates along and normal to the wall, with origin (0,O) at the steady separation point. The steady vorticity w,) vanishes as the skin friction does at the separation point ( x , y ) = (0,O) and becomes positive for x > 0 (Figure 7). Therefore, the goal of control is to create a vorticity wave such that its streaming effect is sufficiently negative to cancel the positive w,):

(4.22)

To this end, X. H. Wu et al. (1991) carried out a perturbation analysis in the triple-deck region. The forcing was assumed to be a tangential oscillation of a small piece of the wall rather than an acoustic wave, because the former directly serves as a vorticity source ua, which creates a transverse vortical wave highly localized in the vicinity of separation point (Figure 1S(a)>.2"Thus, the wall boundary condition is

zu The small oscillating wall could be replaced by a slot from which a forcing sound wave emits.

Vorticity qyniimics on Boundaries

I

a

195

Amplitude

5

b

r

0.50

.~

Numerical

...... Analytical 0.40

-s

..-u

0.30

L

.-c

6

0.20

0.10

0.00

-40

- 30

-20

-10

0

10

20

FIG. 18. Breakaway separation control by a small piece of tangentially oscillating wall: (a) the forcing device, of which the amplitude function is idealized by (4.25) in computations, (b) streaming effect on the skin friction tor Re = 2.0 X lo5 and different forcing frequencies. X is the normalized triple-deck streamwise coordinate. The “original” curve corresponds to no forcing, the dash lines are the closcd-torm asymptotic solution with a >-> I , and the solid lines are numerical results (from X. H. Wu, 1991).

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J. Z. Wu and J. M . Wu

Then, the perturbation expansion shows that, when a >> 1, the classic Rayleigh streaming law (e.g., Stuart, 1963) for an attached Stokes layer,

where 5 = e-leO4y/ is the rescaled normal distance, still holds even for separating flow. This suggests that, to create a negative w,$ on the wall, we need A ’ ( x ) < 0. Figure 1Na) indicates that this corresponds to the right half of the small oscillating wall. The left half is inevitably unfavorable; but, if it is located properly upstream of the strong up-welling region where w o is not too close to zero, it may only reduce the negative vorticity without changing its sign. Thus, not only the amplitude profile 4 x 1 but also its location is important for a successful control. The computation of X. H. Wu (1991) confirmed this observation. In terms of the normalized triple-deck streamwise coordinate X = A - $ e i 3 x with A = 0.332 (Smith, 19821, the forcing amplitude was idealized by the function (4.25)

Here, the free parameters L , and B determine the central location of the oscillating piece of wall and the sharpness of the amplitude slope, respectively. This forcing was applied to both breakaway and bubble-type separating flows. In Figure 18, the basic velocity profile was taken from the triple-deck solution of Smith (1977) for breakaway separation, where the upstream attached boundary layer was assumed to be the Blasius solution. Figure 18(b) clearly shows that a proper forcing of Figure 18(a)’s type may completely eliminate the separation. In Figure 19, the basic flow is a separated bubble similar to that given by Carter (1975). From Figure 19(a) we see that the forcing effect is improved as St decreases for fixed Re. More precisely, the parameter a dominates the control efficiency, as shown in Figure 19(b) for different X stations. Because a = 0 means no forcing and hence w , = ~ 0, there must be an optimal value of a between 0 and O(1). Finally, let us briefly look at the unsteady control of a truly unsteady flow. A typical example is the transition control, where a forcing wave with

.

~Numerical

.....

Analytical

I

-2

0

2

4

6

I

I

10

0

I

I

I

12

X

0.40

0.30

\

a

7. 0.20

X =

X = 0.10

0.00 0.00

0.50

1.00

cL

1.50

2.00

FIG. 19. Bubble-separation control by a small piece of oscillating wall: (a) the streaming effect on the skin friction for Re = 2 x 10' and different frequencies, (b) the effect of a defined by (4.21) at Re = 2 X los (from X. H. Wu, 1991).

198

J. Z. Wu and J. M. Wu

opposite phase and proper frequency and orientation is imposed to cancel the Tollmien-Schlichting wave and thereby retain the laminar state of a boundary layer. The concept of using waue cancellation to eliminate or reduce the TS-wave started with Schilz (1965- 1966), and became popular due to the works of Milling (19811, Liepmann et al. (1982), and Liepmann and Nosenchuck (1982). The control means used by Schilz, Milling, and Liepmann were flexible wall, vibrating wires, and hot films, respectively. The technique can obviously be used to promote transition as well. Liepmann and Nosenchuck (1982) showed that a comparative stabilization by steady heating would require 2 X lo3 times more power than the unsteady wave cancellation. This is a vivid confirmation of our guiding principle about the advantage of using unsteady control. For relevant works see the review of Gad-el-Hak (1989) and a recent paper by J o s h et al. (1994a, b). Our interest here is that the hot films used by Liepmann and Nosenchuck are again a tool of controlling the vorticity creation rate u, but this time through the change of viscosity. A more satisfactory result was obtained by J o s h et al. (1994a,b), who used the direct numerical simulation (DNS) to confirm for the first time that an active periodic blowing and suction may lead to nearly exact cancellation for both smallamplitude (linear) and large-amplitude (nonlinear) disturbances. The computation used two sensors, one actuator, and a spectral analyzer to automate the wave cancellation process. Sensors measured wall pressure for the spectral analyzer, which told the actuator what frequency was evident and at what amplitude to cancel instability. Their results are shown in Figures 20 and 21, where the contours of u velocity, wall pressure, and skin friction with and without control are in a sharp contrast. As in the steady control case, this unsteady blowing-suction control of TS-waves is essentially a up control.

V. Vorticity Creation from an Interface

The general theory of vorticity creation on an interface has been presented in Section 111. In this section we further explore some of its aspects of wide interest, including several examples to illustrate its basic features, especially those that occur in three dimensions only. The flow is assumed incompressible, under a gravitational field.

VorticityDynamics on Boundaries

199

FIG.20. The contours of u velocity without (a) and with (b) control. The TS-wave is in the unstable regime and is intensified downstream for the no-control case. The unsteady control makes the wave approach a steady laminar flow with only slight modulations downstrcam of the controller (from J o s h et a/., 1994. 1995. Copyright 0 1995 AIAA. Reprinted with permission).

A. DIMENSIONLESS PARAMETERS ON

A

VISCOUS INTERFACE

We first cast the general results of Section 111 to dimensionless form as a conventional starting point of our analysis. Let U* be a typical interface velocity and L be a length scale, both common to two fluids. Denote a mean value by an overbar and use U *, L , and 2 p to nondimensionalize all quantities. The Atwood ratio A, Weber number W, and Froude number Fr are defined as

J. Z. Wu and J. M. Wu

200

t

-0.5

"-0

0

Uncontrolled

-1.0

300

1.5

1.0

-1.0

-1.5

350

400

500

450

550

x ~ o - ~

1

I.

C

0

5

Controlled Uncontrolled

0

;

0.8

0.9

1.0

1.1

1.2 x103

X

FIG. 21. The wall pressure (a) and skin friction (b) without and with el a[., 1994, 1995. Copyright 0 1995 A I M . Reprinted with permission).

control (from J d i n

20 1

Vorticity Dynamics on Boundaries respectively. For a viscous interface we have two Reynolds numbers:

both of which are assumed large and from which a single nominal Reynolds number can be introduced: Re

=

=

U*L ~

+ v2

vl

R e , Re,

-

Re,

+ Re,

>> 1

(5.3)

Then, following J. Z. Wu (l995), from A, e l and e2 we construct two constants of O(1) to characterize the jump of dynamic viscosity:

Thus, the dimensionless form of surface stress conditions (3.la, b) become (using the same notations as dimensional ones for relevant variables)

[ p ]= WK

[ 5 ]+ 2Ag =

2EpV

~

-4An

X

'

(V,U,,

Moreover, eqs. (3.19a, b) become, for i

=

u

+ U . K)

€,-

ai; dn

=

€,Il((K

-

In particular, if [ p ] = 0 and hence p = A density jump, eqs. (5.5a, b) degenerate to [PI

=

WK,

(5.5a,b)

S

1,2,

+ E,V,{ + e f t f .K (-l)f-'crrn =

on

o;g,> =

1

on

s

(5.hb)

0, no matter how great the

[g] = 0

(5.7a, b)

That is, the inviscid normal stress balance is recovered and the tangent vorticity is continuous across S, independent of the interface motion and geometry. Note that (5.5a) (along with the motion equation) determines the sueice elevation; thus, for a viscous interface with [ p ] = 0 , the surface

J. Z. Wu and J. M. Wu

202

elevation will remains exactly the same as the corresponding inviscid solution. On the other hand, if S is a free sugace of fluid 1, there must be A=l,

(5.8a, b)

P=A=l

and (5.5a, b) reduce to (we drop the suffix 1)

where

E

is defined by (5.2).For two-dimensional flow, (5.9b) becomes

where s is the arc length along the interface. If, in addition, the flow is steady so that U, = 0, then (1.4) is recovered. The formulas for a are similarly simplified; in particular, because n x (cl, x U) = wU,, - U l , for two-dimensional steady flow we return to (1.5). J. Z. Wu (1995) stressed that, although for inviscid flow the condition A = 1 is sufficient for identifying an interface as a free surface, for viscous flow one should check (5.8b) as well. On a water-air interface A = 0.988, A = 0.964, both being quite close to 1; but p = 0.062 << 1. Therefore, in the free-surface model of a water-air interface, the viscous effect on the normal stress balance is overestimated. This simply favorably ensures that the water-surface elevation is much closer to the inviscid result than the free-surface model predicts. However, if on an interface of two viscous fluids other than pure water and air it happens that A 0.5, say, then the free-surface model has to be given up even if A = 1. Wu also noticed the striking similarity between (5.9b) and the vorticity formed behind a curved shock wave (Hayes, 1957):

-

although their mechanisms are so different. Here the flow in front of the shock is assumed to have a uniform density po and a uniform velocity U and that behind the shock has a variable density p l . This similarity seems to be a hint that extending the present theory to include normai discontinuity might be possible.

Vorticity Dynamics on Boundaries B. FLATINTERFACE

AND

203

FREESURFACE

Based on the preceding dimensionless formulas, we first consider the simplest situation, where Fr << 1 or g" >> 1, so that the interface can be approximated by a flat horizontal surface. 1. General Obsen>ations On a flat interface, the surface tension disappears and so must be V n q l . Thus, because a uniform U,, can be made zero by a coordinate transformation, eqs. (S.Sa, b) and (S.6a, b) reduce to 1 - A (S.lla, b) PI = P2 - 2ePV,. u,, 5, = x 5 *

( - 1)'- 'a,,=

-

e , n ~ , .5,

respectively, where Hi = h , boundary enstrophy flux is

+ U 2 / 2 is the

total enthalpy. Moreover, the

1 , 2 (5.13) If a flat S is a viscous free surface with A = A = P = 1, then 6 , u,,,and q all vanish. This implies that a flat free sugace is free from shearing, where the uorticify can haile only a normal component that must be formed away from the surface and no uorticity clifises across the suface (recall the remark following (3.23)). What remain are a possible pressure variation of O( E ) and a tangent vorticity flux. This flux still creates new tangent vorticity, which, however, entirely enters the interior of the fluid. Moreover, like the case of an interface with [ p ] = 0, there will be no boundary layer near a flat free surface (see Section V.C later). Some further conceptual issues relevant to a flat interface and a flat free surface have been discussed by J. Z. Wu (19951, who stressed that the shear-free condition is not equivalent to and does not need the free-slip condition, as exemplified in Section 1I.D. In addition, when treating a flat water-air interface, say, as a free surface, a residual g on the water side (no matter how small) is necessary to avoid unrealistic physics if the air side is also to be of concern. Nevertheless, the preceding simplifications have been utilized in various

i

=

204

J. Z. Wu and J. M. Wu

approximate analyses and numerical simulations, of which a couple examples are reviewed here. 2. Vortex Pair Rebound from a Flat Interface

As a typical example of using the flat interface or free-surface model in theoretical studies, we consider the vortex pair rebound phenomenon from such a surface. A pair of inviscid vortices of equal and opposite circulation at the same height above a flat boundary, either a solid wall or a free surface, may approach the boundary under their mutual induction. In this process the vortices will separate from each other and never rebound from the boundary (Lamb, 1932; Saffman, 1979, 1991). The observed vortex pair rebounding from a ground has been attributed to the viscous separation induced by the vortex pair and the formation of secondary vortices (Harvey and Perry, 1971; Peace and Riley, 1983; Orlandi, 1990). However, whether or not a viscous vortex pair will rebound from a free surface is a more delicate problem. The experiment of Baker and Crow (1977) and lowReynolds number computation of Peace and Riley (1983) confirmed the rebounding phenomenon; but Orlandi (1990) and Tryggvason et al. (1992) showed numerically that, on a flat, “free-slip’’ (in fact, shear-free) surface, the rebound does not occur at high Reynolds numbers, and the latter attributed the rebounding to the effect of surface contamination. However, it seems that most of these discussions can be settled by the elegant and simple analysis of Saffman (19911, who provided a mathematic proof that on a flat free surface the vorticity centroid does not approach the surface monotonically. We now show that this conclusion can be strengthened by allowing for a residual surface vorticity. Following Saffman (1991), assume the vortex pair initially moves down toward a flat free surface y = 0, with the vorticity antisymmetrical about x = 0. Let

be the total strength and the height of the centroid of the vorticity in the first quadrant, respectively. The boundary conditions are u=w=O

u=0,

on

x=O

o = -

Vorticity Dynamics on Boundaries

205

Note that no free-slip condition is imposed; for a shear-free surface, we simply replace (5.14~)by w = 0 on y = 0. Then, Saffman’s results

(5.16) remain effective no matter if w = 0 on y surface vorticity is that we now have

=

0; the only effect of a residual

where the second term is due to (5.14c), which is always positive. Then, as Saffman argued, as t + m, eq. (5.16) indicates that the vorticity moves asymptotically away from x = 0 and hence the first term of (5.17) will tend to zero. Therefore, without the second term of (5.17) there is Tj

-

const

as

t

.j~0

which, along with (5.151, shows that eventually j will increase, implying a rebound. But, with the extra term we now have

d lim - ( T j ) > 0 1 x dt

-

and hence the rebounding is slightly enhanced (J. Z. Wu, 1995). Even though this analysis proves that the rebounding will happen after a sufficiently long time, it does not tell whether the vorticity will be completely dissipated during this time. If this happens, then the rebounding might not get a chance to occur. Note that the enstrophy dissipation is especially important for a two-dimensional vortical flow bounded only by a flat free surface, because then (3.22) reduces to

Thus, 101 in 9 must monotonically decrease until its distribution is uniform; but = 0 on S implies that the final result can be only a potential flow.

206

J. 2. Wu and J. M. Wu

3. Turbulent Vortices under a Flat Free Sugace Because a flat free surface S can be simply taken as a known shear-free “wall,” using this model in numerical simulations entirely removes the most complicated task of determining the shape and location of S. Therefore, many recent direct numerical simulations (DNS) of free-surface turbulence have adopted this simplified model (e.g., Handler et al., 1993, and references therein). Figure 22 from Handler et al. is a typical DNS result of vortex structures in an open-channel flow with flat free surface, having a Reynolds number of 2340 based on the channel height and mean surface velocity. As expected, the figure clearly shows that the only coherent structure right on S is vertical vortices centered at the spiral

FIG. 22. Turbulent vortex structures in open-channel flow computed by DNS (from Handler et al., 1993): (a) particle paths on flat free surface convected by a frozen velocity field, (b) vortex structure associated with converging spiral in the small rectangle in (a).

VorticityDynumics on Boundaries

207

points of surface pathlines. This kind of pattern has been observed in experiments (e.g., Utami and Ueno, 19871, where the free surface is not strictly flat. Although the numerical picture reminds one of those horn vortices created on a solid wall (Section IV.C), a basic difference is that now since un = 0 the vortices are not creutedfrom S. They must have been turned to the normal direction before reaching S (Figure 21(b)), either by the mutual induction of turbulent eddies or due to a nonzero a, on the bottom wall. Obviously, the flat free-surface model suffers from severe limitation in practical applications, because most rich and colorful vortical structures unique to free-surface flows are missing. These structures appear once the Froude number is no longer small, among which is the free-surface boundary layer to be treated next.

C. FREE-SURFACE BOUNDARY LAYERS As mentioned in Section I, the existence of a boundary layer near a free surface S has been well known. Although the inviscid solution of a free-surface flow satisfies the normal stress balance, in general it does not satisfy the tangent stress balance (5.9b), which forces a vortex layer to form as a correction of inviscid solution. Unlike a solid wall boundary layer, the pressure also needs to be corrected, as first pointed out by Batchelor (Moore, 1963). A theoretical formulation for three-dimensional boundary layer near a free surface has been given by Lundgren (19891, which can be applied to any large Reynolds number free-surface problem with an irrotational global inviscid flow. J. Z. Wu (1995) generalized Lundgren's theory to include rotational outer flow and sharpened its form by using (2.8). This theory and some applications are reviewed here. Extension to an arbitrary interface is straightforward. 1. Linearized Roundary Layer Equation

Denote the inviscid velocity and modified enthalpy by un and respectively. They satisfy the equation

h,,,

Dnun Dn - -v&, +u,.v dt Dt Dt and on the free surface S there is h n = - W K . Thus, in contrast to the solid wall case, the viscous correction of velocity and enthalpy, denoted by u' and h e , must be of o(1). More specifically, we introduce the vector --

J. Z. Wu and J. M. Wu

208

potential A for u', such that u' = V x A and o' = -V2A. Then we find that at large Re = 6 - I the boundary layer thickness is 6 = O ( E ~and ), u;

=

n X dAT - O(S), dn

o; = ~ ( i ) ,

u:,= O W ,

u; = (n

A,

x V).A,

=

=

o(sz>,

O(S2) A,

=0

~

3

\J

(5.18) )

Thus, as remarked before, a free-surface boundary layer is much weaker than its solid wall counterpart. This implies that the boundary layer equation can be linearized. The result is (J. Z . Wu, 1995)

We see that the inviscid rotation enters the equation only through its normal vorticity, consistent with the three-dimensional vortex-sheet theory of Section 1II.C. Note that (5.19a) is a homogeneous equation; it has nontrivial solution only if there exists a forcing mechanism provided by the free-surface condition

d2A,

- - 5 ' = 2n X (V,UO, + U , , - K , , ) + O ( 6 ) on S (5.19b) dn2 That is, the fi-ee-suface boundary layer is drillen by the leading-order sufacedeformation stress. On a flat free surface, this forcing mechanism disappears and hence so does the boundary layer. Moreover, the excess enthalpy he reads --

he = - g ' . A ,

+ O(6')

on

S

(5.20)

which implies only an O(S2) correction to the surface shape. Finally, for the boundary vorticity flux u = uoa+ uh,+ u,,,, + uVisn (J. Z. Wu, 19951,

(5.21a, b, c, d) Clearly, except the first two O(6) terms in u Q sall, the rest are of O( 6 '1. Therefore, the leading-ordersource of vorticity on a free-surj5ace is the viscous correction of the surface acceleration. The weakness of u compared with

Vorticity Dynamics on Boundaries

209

solid wall case has been revealed by the smallness of (T on the flat free surface discussed in Section 1I.D (see Figure 3(c)), where the boundaly layer thickness is 6 = O(lOpl). Note that from a known inviscid solution and uvisn, and the other quantities one may immediately infer uviSr amount to solving the linear problem (5.19a,b) for the tangent vector potential A,. 2. Vorticity Creation Due to Surface Waves The preceding theory covers a wide range of applications. As the first and most classic example, we revisit the viscous two-dimensional linear water wave (Figure 23). This is a slight modification of the flexible wavy-wall case treated in Section IV.A, the main change being the constant-pressure condition on the free water surface S. The problem was solved by Lamb (1932, Section 349) for a freely decaying wave, to which we impose an applied stress to maintain a constant amplitude. Lamb’s result has been extended to three-dimensional flow by Lundgren (1989). Assume the free-surface elevation is y

=f

( x , t ) = a cos(krc - y t ) ,

where T‘

y2

=

gk

+ T‘k3,

ak

2rra

= -< <

h

1

=

T / p . The Cartesian components of the potential velocity are

u

=

ayekY cos(kx - y t ) ,

I)

=

ayeky sin(&

-

yt)

In this problem, (5.19) reduces to a linear equation for the scalar stream function, and the inviscid potential flow satisfies the Bernoulli equation. The resulting linearized vorticity solution is indeed weak: w =

2akyeB’ cos{ku - ( y t + B y ) }

FIG.23. A water wave.

=

O(1)

(5.22)

210

J. 2. Wu and J. M. Wu

where /3 = (y/2v)i. The boundary vorticity flux and enstrophy flux are

respectively, both being of O(vf). The latter has a nonzero positive average, showing that the overall enstrophy in a forced water wave is increasing. As indicated by (5.1c), the Froude number will be larger if the wavelength is smaller. The gravity-capillary wave belongs to this case, where we anticipate a stronger vorticity creation in a thin boundary layer. Figure 24 (see color plates) is an experimental result of a steady near-breaking gravity-capillary wave formed behind a hydrofoil, due to Lin and Rockwell (1995). The measurements were performed using digital particle image velocimeter (DPIV). As a comparison, a DNS of a similar but unsteady wave under slightly different conditions due to Dommermuth and Mui (1995) is shown in Figure 25 (see color plates). The Reynolds number based on wavelength (5 cm) is 3.5 x lo4, and the boundary layer is about 1 mm thick, which was resolved numerically using 4096 X 1025 grid points. The vortical structures can be clearly seen from both experimental and numerical results. Physically, the vorticity inside the flow must come from the u effect governed by the viscous correction of the surface acceleration, a', which creates new vorticity on S and then sends it into the flow. Thus, the location of strongest vorticity should be a mark of the largest la'l on S . Unfortunately, so far no computation has been -made to infer u from experimental or numerical data. Here we note that eqs. (5.5a,b) should be imposed as a primary boundary condition in solving any viscous free-surface flow. In fact, it is a sharpened and physically more appealing version of the stress condition (2.1 11, which usually involves a complicated calculation of all components of the velocity gradient Vu. If (2.11) has been directly used in the common way in a numerical scheme, then (5.5b) can be reversely used to test a posteriori the accuracy of the scheme. Figure 26, again from the DNS by Dommermuth and Mui (19951, gives such an example. The computation used 6.7 X lo7 grid points. The DNS vorticity on S is shown in the figure and compared with (5.10). The numerical error is less than 1%, indicating a very good accuracy of the scheme. Considering the huge number of grid

Vorticity Dynamics on Boundaries

21 1

Surface Vorticity Check 50.0

l

"

'

1

"

'

I

"

'

l

"

'

0.0 .-x

I

.u 5

-50.0

> a

g

-100.0

vl

-150.0

1 -200.0 0.0

1

0.20

,

/

,

1

,

1

/

1

1

1

0.60

0.40

1

1

0.80

1

1

1

1

X-position

FIG. 26. Surface vorticity check of DNS for a two-dimensional unsteady near-breaking gravity-capillary wave with a 16384 x 4097 grid along the length and depth of the wave (from D. G. Dommermuth). At this particular time, the maximum wave slope is 1.22. The solid line is the computed free-surface vorticity, and the dotted line is 10 times the difference of DNS and (5.10).

points, however, this comparison also indicates the difficulty of reaching a high accuracy. It is remarkable to note that, although in general on a free surface 6 = 0(1), Figure 25 indicates that its peak value can be as large as O(Ref), like the solid wall case but confined to very narrow regions of troughs. This is a sign that to capture the detailed structure of stiff short waves the traditional potential wave theory is insufficient, because the created strong vorticity must inversely affect the surface motion.

3. Boundary Layer on u Bubble Surface Moore (1963) studied the boundary layer of a raising spherical gas bubble of radius u in a liquid with constant velocity U at a large Reynolds number Re = U u / v = E - ' = K 2 . The inviscid velocity on S uses the potential-flow solution U, = ( 3 / 2 ) U sin Oe,, where spherical coordinates ( r , 8, 4 ) are assumed, with 0 = 0 along the moving direction. Then, by

.o

J. Z. Wu and J. M. Wu

212

(5.19) and (5.21c, d), we immediately obtain the dimensionless surface vorticity

6’ = 3sin 8e, + O ( 6 )

(5.24)

and its explicit viscous flux:

From the viewpoint of boundary vorticity dynamics, what remain to be solved are the flux due to viscous correction of surface acceleration, u a , , and that due to the excess enthalpy, u h cAlthough . these can be obtained by solving (5.19a), which reduces to a scalar problem for the Stokes stream function, Moore directly solved the velocity components u’ = ( 6 ’ u ’ , 6 u ’ , 0) and excess enthalpy he from the original boundary layer equation. It is evident from (5.18) and (5.21a7b) that for our purpose we need to know only the value of u’,, on S, which is 6 u ‘ ( l , 0)e, and from which A,

=

1”

m

u’,,

X

n dn

=

fi2e,/, u‘dy, on

S,

y

=

F 1 ( 1- r )

=

O(1)

-8

Therefore, from Moore’s (1963) result u’(1,e)

=

-61hsin

0$(8),

2 ~ ( 8 =) -csc4 8(2 9

-

3cos 8 + cos2 0 )

effective for 0 # T (near the rear stagnation point the boundary layer approximation blows up), it follows that (J. Z. Wu, 1995)

the latter diverging as 0 + T . A complete solution of he and u h over the whole bubble surface would be possible if a further singular peiturbation could be introduced near 0 = T as an analogy of the triple-deck structure near the separation from a smooth solid wall. Nevertheless, it can be anticipated that uhewill have a high peak at this stagnation point. A delicate application of the free-surface boundary layer theory was made by Lundgren and Mansour (1988), who considered the oscillating

Vorticity Dynamics on Boundaries

213

drops in zero gravity. After obtaining the inviscid axisymmetric solution where the drop surface was treated as a closed vortex sheet, the authors introduced a high Reynolds number correction to study the viscous damping effect on the oscillation. Because of the complexity of the solution structure, here we shall not go into detail but merely mention that the formulation of this problem, based on Lundgren (19891, can now be sharpened as (5.19a7b) and (5.20); and from the inviscid solution the vorticity creation from the drop surface can be likewise studied. 4. Interaction between a Vertical Vortex and Free Suface

The preceding examples all assume an irrotational outer flow. J. Z. Wu et al. (1995a), considered a situation where the free-surface boundary layer has a rotational outer flow: the interaction between a free surface and a vertical vortex. Assume the vortex is axisymmetric as sketched in Figure 27. Here, we need both the cylindrical coordinates ( r , 0, z ) for describing the basic vortex flow and the coordinates along and normal to S. Let e, and s be the unit tangent vector and arc length along the section curve of S in the ( r , z ) plane, say, z = f ( r ) , and n be the unit normal vector pointing toward air and n the normal distance. Then, as shown in Figure

FIG.27. A vertical vortex interacts with a water-air interface S . In the water, the basic vortex is inviscid and hvo-dimensional (the boundary layer is not shown). In the air, the viscosity is just turned on so that there appears a vortex shcet S + above S. As r + 0, the sheet bifurcates into a thin axial vortex. Both cylindrical and surface coordinates are shown.

J. Z. Wu and J. M. Wu

214

27, the basic geometric relations are de,dS

1 de, r do

dn dS

1

-

4,

- e , cos

r

4 is the angle between

where

-_

K,n,

-

I

- Kse,

1 dn

(5.26a)

--- - ~ # e ~ r do

e, and e , , and 3

K,

=f”

COS3

4 = f ” ( l +f”)-’, (5.26b)

are the principal curvatures along e , and e H . The global inviscid flow, ug, allows for a discontinuity of tangent velocity and normal vorticity on the interface S. The vertical inviscid vortex can be two-dimensional, that is, ug = (0, V,, ,01, and any V,(r) is a solution of the steady Euler equations V:

dh

_ = -

r

dh o=-+g*

dr’

d2

For a given V,,(r) and assuming the surface tension is negligible, the dimensionless shape of S , z = f ( r ) , is determined by f(r)

=

r V,f F r i l -dr

(5.27)

r

-

The prescribed V J r ) should ensure a solid-core behavior V(, r as r + 0, and r - ’ as r + 30 (irrotational). Our interest is the boundary layer structure near S, which amounts to solving (5.19a, b). Following J. Z. Wu et al. (1995a) and in accordance with (5.18), the viscous correction to the velocity can be written as u‘ = ( S U , SV, S 2W ) , with U , V , W = 0(1), and the tangent vector potential is A, = e, A , + e , A , ] . Then (5.19a,b) give a pair of coupled homogeneous equations

-

d2Ao dN’

--

V,I 2-A, r

cos 4

=

0,

dA,

~

I dr,,

+ -r- Adr, dN2

cos 4 = 0,

To = rV, (5.28a, b)

Vorticity Dynamics on Boundaries

215

with N = 6-'n being the stretched normal coordinate, subject to the free-surface boundary conditions

Obviously, eq. (5.29b) is the axisymmetric counterpart of (1.4). Note that V&r) has an associated w O s= w , , sin ~ 4, which would be consistent with (5.29b) only if woz= 2V,/r; that is, the vertical vortex behaves like a solid rotation for all r . But in reality this is not the case, and hence generically there is a viscous correction to l,',,(r), implying the appearance of a boundary layer. Alternatively, one can start from the familiar rotationally symmetric boundary layer equation; then, after subtracting the inviscid solution, the equations for V and the Stokes stream function I) = rA, = 6-'W are factually equivalent to (5.28a,b). J. Z. Wu et al. (1995a) found their analytical solution

C(r)Il -

are two parametric functions. Physically, L ( r ) characterizes the effect of inviscid vortex-core structure, and C ( r ) includes both the effect of L ( r ) and the forcing mechanism (5.29b). The latter dominates the r dependence of the layer. Note that to ensure the effectiveness of the solution, there must be C ( r ) = O(1) for 0 I r <

(5.31)

Two findings of J. Z . Wu et ul. (1995a) based on this solution are of fundamental interest. First, for two most frequently used steady vortex

J. Z. Wu and J. M. Wu

216

models, that is, a normalized “frozen” Taylor vortex

V,= ref(1-r’) and a normalized q vortex

V,(r)

=

4 -(1 - e-”’) r

q

with

=

1.398, s

=

1.256

(the normalization is made to have a unit maximum velocity at r = 11, the C ( r ) behavior does not satisfy (5.31). For the former, d r , / d r changes sign at r = fi and hence C ( r ) has a singularity, but for the latter, C ( r ) grows exponentially. This implies that separation will occur, and the attached boundary layer model, as assumed in deriving (5.191, blows up. Therefore, at least in the case of normal vortices (which often happens in turbulent flows as seen in Section V.B.3 or if a tornadolike vortex hits the surface), thefree-surface boundary layer is very susceptible to separation. In fact, the jet flow shown in Figure 25(b) (see color plates) is nothing but the result of such a separation, even though there is no vertical vortex in that case. J. Z. Wu et al. (1995a) found that the simplest form of C ( r ) satisfying (5.31) and ensuring the correct behavior of V,(r) near r = 0 and at infinity is ar2 which leads to an algebraic vortex model

They set a = 3.5 and r,, = 1.05 times of that of the normalized q vortex, and fixed the constants a and b by normalization. It is remarkable that, as Figure 28 indicates, this model represents merely a very slight modification of the q vortex, but their C ( r ) , and hence their boundary layer structure, are so different. The second finding of J. Z. Wu et al. (1995a) is, on using (5.32) to obtain an attached boundary layer, that away from the layer the axial velocity w = S 2 W does not return to zero. Rather, there is a persistent axial flow

w(r>lN+x=

6’ d C -

--

r dr

VorficityDynamics on Boundaries a

Vo(r)

1.2

217

, Algebraic Vertex Frozen Taylor Vertex ---Q-Vertex - - -

1

0.8

0.6

0.4

0.2

C 1

3

2

4

5

r

I , I I $ , , , /

, , 8 I I II

Algebraic Vertex Frozen Taylor Vertex - - - Q-Vertex

! I

1.5

1 :

-

1 ;

,

,--

I

I ,

,,'

/

I

-0.5-

1

0

1

I

,

1

2

3

4

5

r

FIG.28. The velocity profile V , ( r ) and function C ( r ) of a two-dimensional inviscid vortex, defined by (5.32) with (Y = 3.5. Also shown are the corresponding curves of the normalized q vortex and frozen Taylor vortex. See J . Z . Wu ei al. (1995a).

J. Z. Wu and J. M. Wu

218

with a scale of Fr Re

-=--

vU* gL2

-

viscous force gravitational force

Therefore, the interaction of a vertical vortex with a free surface not only causes a boundary layer, but also alters the vortex structure itself a vertical vortex interacting a free-surface is inherently three dimensional. This phenomenon has been observed in some experiments. Physically, the r dependent axial velocity is induced by a circumferential vorticity component w, that is created from S and sent deeply into the fluid by a a, of O ( S 2 )caused by the interaction, see Figure 29. In this interaction problem, the free surface S can be taken as a water-air interface with small effect on the water motion. If we turn on the viscosity of the air at t = 0, say, to extend the preceding solution to the air side, an unsteady full Navier-Stokes equation has to be solved, of which the initial condition at t = 0' is an air vortex sheet S ' , say, adjacent to S with a nonzero mean normal vorticity 5 = w,,z cos + / 2 . Its

:::: t

-0.3

-0.4

FIG. 29. The circumferential vorticity and associated axial velocity of a vertical vortex interacting a free surface (from J. Z . Wu et al., 1995a). Without interaction, the vortex is two dimensional and inviscid, given by (5.32). In the figure, w 0 ( r ) and w ( r ) are values away from the interacting region (the boundary layer) and hence represent a persistent structural change along the whole vortex.

Vorticity Dynamics on Boundaries

219

u,,

the velocity has to be determined by (3.39). Except the mean velocity normal vorticity causes an additional circumferential velocity (J. Z. Wu, 1995)

which is always an increment to if 2 is single signed. Then, as time goes on, the air vortex sheet evolves to an air vortical flow. D.

COMPLEX VORTEX-INTERFACE

INTERACTION

AND

SURFACTANT EFFECT

The examples considered thus far have been limited to highly idealized simple circumstances, all assuming a clean interface or free surface. For completeness, before ending this section we briefly exemplify the vorticity creation from a free surface that has a complex interaction with nearby vortices and make some preliminary observations on the effect of surface contamination. 1. Interaction of a Vortex Pair with a Free Sur$ace

A typical complicated vortex-interface interaction occurs when a pair of submerged vortices or a vortex ring moves up to a free surface S , as revealed by the well-known experiments of Sarpkaya and coworkers (for reviews, see Sarpkaya (1992a, b). The experiments showed that as a vortex pair approaches the free surface S under mutual induction, the surface will be humped up to form a Kelvin oval, and at mean time a series of lateral vortices appears, riding on the quasi-cylindrical oval (“striations”), bounded by two rows of “scars” and whirls digging into the water at the roots of the oval (Figure 30). This interesting finding has excited many numerical simulations, such as those based on two-dimensional vortex sheet model for the free surface (e.g., Tryggvason, 1989; Yu and Tryggvason, 1990) and Navier-Stokes solver (Ohring and Lugt, 1991; Lugt and Ohring, 19921, as well as full three-dimensional Navier-Stokes simulation (Dommermuth, 1993). These computations enable us to outline the physics relevant to the vorticity creation in this interacting process. Initially, the interaction of the rising vortex pair with S is apparently a two-dimensional inviscid process and can be mimicked by taking the pair as point vortices and S as a weak boundary vortex sheet. As the Kelvin

220

J. Z. Wu and J. M. Wu /

striations

c7

scars

FIG.30. Schematic of striations and scars (excerpted from Sarpkaya and Suthon, 1991).

oval is formed, the surface tangent vorticity 5 increases to O(1) as indicated by (5.101, or equivalently, the sheet strength y is of O ( 6 ) , 6 = R e - f . Note that the variation of y already contains the vorticity creation process as seen from (3.33) and (3.34). Between the vortex pair and S, the flow can still be irrotational. Then, at a finite Re, new vorticity produced from S will eventually be sent into the fluid; and at a certain stage of the early interaction, the vortex sheet needs to be refined as a free-surface boundary layer. By (5.211, then, the vorticity flux is dominated by the boundary layer correction of surface acceleration; that is, uat= O(6). In two dimensions (5.21a) reduces to

where u: can be solved from (5.18) and (5.19), provided that the elevation of S and its velocity induced by the primary vortices have been known from inviscid calculation. Qualitatively, a, concentrates in the local region of high curvature, where separation may happen at a sufficiently large Froude number (about 0.5 and larger), so that a pair of secondary vortices of opposite sign is formed below S and toward the end of this stage the boundary layer approximation is no longer applicable. This newly produced secondary vortex pair is responsible to the observed scars and possible rebounding of the primary vortices. The preceding two-dimensional picture cannot explain the observed striations, which are related to the vortex instability along the axis. In a I

Vorticity Dynamics on Boundaries

221

three-dimensional Navier-Stokes simulation, Dommermuth (1993) introduced an initial disturbance of the location and vorticity distribution of the primary vortices to observe the effect of instability. It was found that, as a vortex tube interacts itself and its neighbors, sheets of helical vorticity are spiraled off. Due to shortwave inviscid instability, these sheets manifest themselves as braids of cross-axis vorticity, a structure independent of the presence of S . But, as they rotate around and translate with the primary vortices, some braids will approach S and their open ends become normal to S to form the observed whirls as the outer boundary of the scars. This complicated three-dimensional interaction, however, seems not to be accompanied by a strong viscous dynamic process of vorticity creation, although the surface vorticity keeps changing. The main event occurring on S is the formation of normal vortices due to the turning pattern of Figure 4(b), which is essentially a kinematic process.”

2. The Effect of a Suifactunt on Vorticity Creation from a Free Suface In reality, a clean interface can rarely happen. Even a slight surface contamination may significantly alter the interfacial vorticity distribution and hence the surface motion as well as the vorticity-creation rate. The effect of an oil film on calming the interfacial wave has been known for long time. In a broad sense, this calming can be viewed as an early example of intefacial uorticity-creation control, and perhaps introducing proper contamination could be a major means of such control in the future. The appearance of a surface-active material, or a sufactant, will reduce the local surface tension T from its equilibrium value T o , say, and a concentration of surfactant will thereby cause a tangent gradient of T , which in turn drives a motion of both areal and volumetric fluids (the Marangoni effect 1. In addition, the surfactant may have various rheological and chemical properties that can cause additional interactions with the bulk fluids (e.g., Edwards et al., 1991). All these will affect the interfacial vorticity dynamics. Ideally, if the surfactant is also a Newtonian fluid, so that a water-oil-air system, say, forms a sandwich structure with two interfaces, then the clean interface theory developed thus far can still be applied to each interface. However, in many cases, surfactants are non-Newtonian or even not fluids, 21

Viscosity and dynamics would enter if the air vortices were to be studied as well, as illustrated in the previous subsection.

J. 2. Wu and J. M. Wu

222

and their interface with bulk fluids may not be immiscible. On the other hand, it often suffices to take the surfactant as a surface fluid with negligible thickness, density, and bulk motion. As a preliminary discussion, therefore, we confine ourselves to the simplest model. First, we assume the surfactant is Newtonian, of which the motion is governed by a twodimensional analogy of the Navier-Stokes equation (2.15) on a curved surface (Scriven, 1960). This includes introducing a surface shearing viscosity ps and a surface dilatational viscosity A,, the latter being nonzero even if the bulk surfactant is incompressible.22 Second, we assume that the surfactant density is well negligible compared with that of bulk fluids and so is its body force (inertial and external). Consequently, the surfactant simply moves with the interface velocity U. Moreover, the force acted by the surfactant on bulk fluids, say, f , = nf,, f,,, can be only a surface force and hence directly balanced by the surface stress of the bulk fluids. This leads to an extension of the classic interfacial stress condition (2.11):

+

n * [t]

=

T,K + f,,l,

n x [t]

=

n x f,,

(5.33a, b)

Here, f,, and f,, are given by (Scriven, 1960; Edwards et al., 1991) f

,, =

f,,

where

=

- K(AA

+ p,)v,.U

-

I

(5.34a, b)

K E -K:K

(5.34c)

2psk:V,U

+ p.,)V,(V,.U) - P , ( ~ K U , + n x V,{~K.v,u,,) -V,T

-

(A,

-

K= -nXKXn=K-KI,,

with I, = I - nn being the two-dimensional unit tensor on S. For twodimensional flows K = KI, so that K = 0 and K = 0. Note that f, is derived from the two-dimensional analogy of (2.15) in which the divergence of stress tensor, rather than the surface stress itself, is involved. Thus, f, contains derivatives one order higher than those in t. Although (5.34a) implies a modification of the elevation of S, the tangent-vorticity jump [ p.51 across S is modified to

Lp.51= - n x =

([t,,] - f,,)

[ &lo - n X V,T

-(,is

+ p,)n

+ p,V,l

x v,(v,.u)

-

2,411 x (KU,

+ K.v,u,)

(5.35)

"This situation is similar to the two-dimensional divergence of vortex sheet strength; see (3.37).

FIG. 15. Horn vortices in the wake of a jet in crossflow (from the water tunnel viwalization of J. M. Wu c’t d.. 1988).

FIG.24. Experimental result of a wave behind a hydrofoil (from Lin and Rockwell, 1995). The wavelength is about 8.8 cm. The figure shows vorticity contours (upper) and velocity vectors (lower). Red and yellow denote counterclockwise and clockwise vorticities, respectively. (Reprinted with the permission of Cambridge Univer\ity Press.)

FIG.25. DNS result of a 5-cni wave (from Domrncrniuth and Mui, 1995). The vertical scale is stretched to highlight the boundary layer structure. At thi\ particular time tlie maximum wave \lope is 1.07: (a) vorticity contours (the scale is saturated because tlie w in the trough.; are so extreme): ( b ) rotational part of the velocity field. which highlights the newsurface jet that mixes the subcurface flow (a clockwise vortex is clearly seen in the crest of wave where separation occurs): (c) total velocity field (vortical plus potential) viewed in a moving fiame of reference to illustrate the qualitative agreement with experiments (Figure 24). (Reproduced from Dominerwutli and Mui. 1995. with permission from ASME.)

FIG.32. The experimental result of velocity components (a), total pressure (b), and static pressure (c) on a near-wake plane downstream a 76" swept delta wing at a = 20". The wing tip is at y/c = 0.25. Two velocity components in the wake plane are indicated by small arrows, and the chordwise velocity by color contours. M = 0.05, Re = 5 x lo5. The wake plane is normal to the central chord (from Visser and Wdshburn, unpublished).

FIG.33. The vorticity components iii body axe5 a s computed from Figure 32 (from Wu, Ondrusek, and Wu, 1996). In the main vortex core region wv and 0): are roughly antisymmetric with respect to the horizontal and vertical lines thorough the vortex center, rehpectively. Thus, they both I-educe lo zero at the center and then have a negative peak (as one moves upwei-tl and inboard. respectively). which is not clearly shown in the plots.

Vorticity Dynamics on Boundaries

223

where [ pgIOis the value on corresponding clean interface, given by (3.lb). In particular, on a two-dimensional free surface with arc element ds, the dimensionless form of (5.35) reduces to

E = lo+ ReW-

dT* dS

+

(5.36)

where to is given by (5.10), W is the Weber number defined by (5.lb) for equilibrium surface tension T o ,T * = T / T , , and Bo = (As + p s ) / (p L ) is the Boussinesq number, a new parameter. Therefore, the surfactant influences the interfacial tangent vorticity by (1) the gradient of surface tension and (2) the viscous resistance to the strain rate of the surfactant. The first effect is quite strong at a large Re if d T * / d s = 0(1), but not as strong as R e itself; because ReW = T(,/( p U * ) is inversely proportional to the reference momentum. Consequently, a contaminated inteqace may locally behave somewhat in between a clean inteflace and a rigid wall, as confirmed by recent experimental and numerical studies (for a brief review see, e.g., Tsai and Yue, 1995). On the other hand, compared with (3.lb) or (5.10), the additional viscous resistance also tends to increase boundary vorticity. Although usually Bo = 0(1), on a wavy surface the appearance of higher order derivatives may result in a significant local boundary vorticity as well. Interestingly, this resistance includes the gradient of normal vorticity 5, which is exactly of the same form as that in (3.19a) or (3.20a) but now affects [ pg] instead of u. We stress that, under the preceding assumptions, no volumetric force is exerted to the bulk fluids by the surfactant; thus, the force balance that leads to the net-u formulas (3.20a, b) is unaffected. However, the specific level and distribution of u will be indirectly affected by the surfactant, too, mainly via the change of surface acceleration. We return to the interaction of rising vortex pair and free surface, but now let the surface S be contaminated. The experimental measurement and numerical computation with flat free surface by Hirsa et al. (1990; see also Hirsa and Willmarth, 1994; Tryggvason et al., 1992) showed that the presence of surfactant greatly strengthens the formation of secondary vortices and rebounding of primary vortices, so that the contaminated free surface is indeed more like a solid wall. This finding was further confirmed and extended by Tsai and Yue (19951, who made a two-dimensional viscous simulation on the effect of soluble and insoluble surfactant on the

224

J. Z. Wu and J. M. Wu

interaction process. In this study, not only the surface surfactant was introduced as stress conditions (5.33), but also the bulk surfactant with variable concentration and its transport, not reviewed previously, was considered. It was found that the interaction between the surfactant and underlying vortex flows forms a close loop. The primary vortex pair induces gradients of surfactant concentration that leads to Marangoni stresses, strong surface vorticity, a boundary layer, and even separation. These in turn significantly alter the underlying vortex flows. For example, when ReW = O(10) and Bo = 0 0 1 , at a mild Froude number, Fr = 0.15, a free-surface vorticity 6 of O(1) may occur due to two comparable effects in (5.361, ReW-

dT* 6JS

and

Bo-

d’U, ds

*

Owing to the practical importance of surfactant effect on ocean waves and ship wakes, research along this line will certainly be further pursued.

VI. Total Force and Moment Acted on Closed Boundaries by Created Vorticity Fields

So far we have considered various theoretical and applied aspects of vorticity creation from boundaries. This creation process can be viewed as an action of the boundary to the flow field, which is one aspect of the boundary vorticity dynamics. The other aspect is the reaction of created vorticity to the boundary. Locally, the action and reaction have been fully reflected by the stress balance (2.11), where t is a reflection of compressing process, shearing process, and surface deformation process. However, if our concern is only the total force F and moment L acted on a closed boundary, the reaction aspect can be greatly simplified as evidenced by a series of vorticity based formulas of great interest in both fundamental theory of vortex dynamics and applications. For later convenience, we first list the original formulas for F and L. In these formulas, 7 is the total material volume of the fluid surrounding a body with boundary 9, of which the unit normal n points into the body, and I$ is a control volume bounded by 9 from inside and by a control surface Z from outside. The moment is taken about the origin of the coordinates. The flow is assumed incompressible, and the gravitational

Vorticity Dynamics on Boundaries

225

+

force is absorbed into pressure by denoting p - p pgz. Then, in terms of the momentum change or inertial force of the material fluid body, we have D F = --/ppudV= Dt c y D L = -x x p u d ~ - / x x p a d ~ Dt Y

ly

(6.la) (6.lb)

which require knowing the whole flow field and is referred to as the global uiew. Or, in terms of control volume and control surface (only force formula will be considered here for simplicity), F=

--1 pudV+ dt v/ d

/Z(t - p u u . n ) d S

(6.2)

where t is given by (2.6a). For steady flow over a stationary body this requires knowing only the flow on an arbitrary wake plane, say, Y, and is referred to as the near view. When Ymoves sufficiently far downstream (a Trefftz plane), we return to the global view. Finally, in terms of the surface stress on 9, if 9 = dB is rigid, by (2.22) we have

Thus, as noted in Section 11, only the compressing and shearing processes are involved in the rigid-body force and moment analysis. Similarly, if 9 is deformable, say, a flexible solid surface, the surface of an air bubble in water or a water drop in air, we only need to add (2.20) to the moment formula; thus

Equations (6.3) and (6.3') require knowing the stress status on the body surface and are referred to as the close uiew. All these formulas are

226

J. 2. Wu and J. M. Wu

obtained in the framework of primary variables, although the vorticity has entered some of them. A basic observation of (6.11H6.3’) is that the inherent coupling of the fundamental processes, so important in boundary vorticity dynamics, has no reflection at all. In this sense, these formulas are not physically optimized and most appealing. J. Z. Wu and Wu (1993) systematically showed that, for general viscous compressible flow, the total force and moment acted on a closed rigid boundary by the flow field can be attributed exclusively to that by the vorticity field created from the boundary. This result has been extended by J. Z. Wu (1995) to closed fluid interface. Physically, this type of vorticity-based formulas are made possible by the viscous coupling between compressing and shearing processes via the no-slip condition on the boundary, which always enables one to express the force due to the former in terms of the latter even if the former is dominating (e.g., in a supersonic flow). In contrast, J. Z. Wu and Wu (1993) proved that the scalar compressing process alone does not own this nice ability, and hence it is impossible to obtain a complete set of dilatation-based force and moment formulas. In this section, the general vorticity-based force and moment theory are discussed in the order from (6.11, to (6.2), to (6.3), just like following an observer who moves closer and closer to the body from a remote distance. Thus, the observer will get a global view first, then a near view, and finally a close view. The application of the resulting vorticity based formulas will be illustrated by a practical problem, the aerodynamic diagnostics and optimization.

A. THE VORTICITV MOMENTAND KUTTA-JOUKOWSIUFORMULA We start from revisiting the earliest total-force formula in terms of vorticity w or circulation r: the Kutta-Joukowski formula F

=

U x re,

(6.4)

and its three-dimensional counterpart, where e y is the unit vector along the vortex axis. Traditionally, this formula is obtained by using strictly inviscid and irrotational flow models, in which for having a lateral force, a singularity (a point vortex in two-dimensional case) has to be artificially introduced inside the body (Joukowski’s derivation is cited in Batchelor, 1967, pp. 404-406). Although this result was a significant and ingenious

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achievement in early theoretical aerodynamics, today its inviscid derivation is easily misleading and can by no means reveal the physical source of the circulation, which exists only in viscous interaction. To obtain a physically consistent understanding, therefore, we rederive this type of formulas based on viscous consideration. In so doing we shall clarify the conditions for (6.4) to hold. We then make some general remarks on this type of vorticity based formulas. 1. Kutta-Joukowski Formula: Material Volume Derivation It is well known that the total momentum of an incompressible fluid body can be cast to the integral of the vorticity moment x x (I) by integration by parts, via the vector identity (d

-

1)f

=

x x ( V x f)

-

V ( X . f)

+ V . (xf)

in d-dimensional space for any f (Lamb, 1932; Batchelor, 1967). Therefore, by (6.la), we immediately obtain the first general incompressible force formula exclusively in terms of vorticity:

P

F =

d

-

D x x ( I ) ~ V -P D x x (n X d d S (6.5) 1 Dt -(y d-1DEk

+

Lighthill (1979) mentioned that (6.5) has been extensively used in estimating the hydrodynamic loading of offshore structures due to its ability of connecting vorticity and force directly. A systematic exposition of (6.5) and the corresponding moment formula for viscous flow was given by J. C. Wu (1981). Note that this formula gives the force on a fluid bubble or drop 7 as well, as long as the distribution of vorticity in 77 and velocity on 9 are known. Now, assume the viscous fluid is unbounded from outside, with a uniform oncoming velocity U = U e , , and the body surface dB is stationary. Then, the second term of (6.5) vanishes due to a no-slip condition. We consider the Euler limit of the first integral. In this case the boundary layer over dB reduces to a boundary vortex sheet, as discussed in Section III.C, which rolls up as leaving the body and becomes a pair of concentrated trailing vortices. Along with the starting vortex, we have a closed vortex loop, which in the Euler limit does not diffuse and can be represented by a closed vortex filament as viewed by a remote observer. In this case the integrated vorticity moment can be simply expressed as twice the vectorial area S spanned by the loop times its circulation (Batchelor, 1967);

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J. 2. Wu and J. M. Wu

the direction of dS and the vorticity in the loop are defined according to the right-handed rule. Thus,

F=

DS

-pr-Dt

(6.6)

Moreover, if we assume that the wake vortex pair are straight, with constant separation b, and advected downstream by U everywhere, then the increase of S is due simply to the elongation of the trailing vortices as the starting vortex moves downstream with a rate U. Therefore, eq. (6.6) reduces to the simplest lift formula for a finite-span wing:

F, = p b U r

(6.7)

which has been known since the time of Lanchester and Prandtl. Here we used = instead of equality because the trailing vortices have a downward induction, which makes the vortex loop nonplanar and is the source of induced drag. The quantitative determination of downwash depends on the circulation distribution along the span and is not our concern here. However, if we assume that b -+ x such that the downwash approaches zero, and that in any cross-section (x, z ) the flow is the same, then the force per unit span exactly recovers (6.4). 2. Kutta-Joukowski Formula: Control-Volume Denvation

Alternatively, we may obtain (6.6) and (6.7) from (6.2) in a way more closely parallel to the classic approach. Assume C is far away from the body (still the global view), so that we may set u = U + u’ with /u’I2 negligible on C. Note that C must exclude the starting vortices, which is even farther as t + 00, so that the flow inside C can be considered steady. The only deviation from the classic approach, where U‘ = Vcp and is singular inside the body, is that we now consider a uiscous, rotational but regular perturbation u’, with o = V X u‘. The viscous force is still negligible on C; hence, ox

u+v

(

U.U’

3

+-

=

O(IU’I*> on

c

Then, for a two-dimensional steady flow, it is known that in the Euler limit no vortex sheet is shed off into the wake, and in three dimensions, we may again assume the wake vortices are approximately along the direction of U.

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229

Consequently, in both cases w X U = 0 and the Bernoulli integral for steady potential flows still holds on X. This casts (6.2) to

F

=

-

p / {(U u’)n - (U . n)u’ - U(n . u’)}dS

c

5+

by the Gauss theorem and incompressibility, where V = V, is the fluid volume plus solid volume. Because V, is stationary, we finally obtain

F=pUX/wdV=pUX

v,

L

(n.o)xdS

(6.8)

where the second equality of (2.20) was used. We stress that in deriving (6.8) the no-slip condition must be imposed as well, for otherwise a solid sphere, say, can have arbitrary rotation and a nonzero o in V, without affecting the fluid motion. Then, in three dimensions, let x = xe, ye,, we obtain

+

which reduces to (6.7) if wx is concentrated at two points on Yseparated by b. On the other hand, in two dimensions, n * w occurs at the side boundaries of unit separation located at y = k 1/2. Thus, if x = xe, + ye, is the position vector of the centroid of the closed boundary vortex sheet surrounding the body, eq. (6.4) is recovered. Clearly, the conditions for (6.4), (6.6) and (6.9) to hold are (1) the Euler limit of incompressibleviscous flow; and (2) the starting vortex is sufficiently far from the body such that the flow in C is asymptotically steady. We digress to mention that, partially due to the misleading of the inviscid derivation of (6.41, some authors believed that this formula can be used to compute the force on a fluid vortex with circulation r. True, a side force will trivially appear if a vortex can be held by an external force; but it is always incorrect to apply (6.4) to the interaction of any free vortex and cross flow, because its velocity can in no way differ from that of the background flow, which must have included the velocity induced by the vortex. The only nontrivial case is that of a vortex held at t = 0 and then released. Again due to the viscosity and no-slip condition, the sudden appearance of a cross flow at t = 0 will create a vortex sheet “shell”

J. Z. Wu and J. M. Wu

surrounding the “bare” vortex, such that at t = 0’ the sheet induces a velocity inside the bare vortex different from the approaching velocity and therefore hold it in place for 0 < t << 1. However, as Caruthers et al. (1992) showed, in the Euler limit and at t = O+, the total force acted on such an enveloped Rankine vortex is exactly half of the value given by (6.4). This result is also true for a solid rotating cylinder of the same density if it is released at t = 0, as can be inferred from Ting and Klein (1991, Section 2.1). As time goes on, the vortex sheet is rapidly advected downstream of the remaining distorted core with an ever-decreasing influence upon the core motion. The vanishing of the sheet is associated with an acceleration of the core to match the background velocity, so that finally the side force reduces from its maximum value of pU I 7 2 to zero. Therefore, the Kutta-Joukowski formula can n e r w be applied to a free vortex, euen if it is initially bound. This revisit of the so-called vortexcylinder analogy is also helpful for understanding the viscous background of (6.4). 3. General Characters of Vorticity Based Force and Moment Formulas Inspecting the derivation of the preceding results and their structure, some remarks of general significance for the following development can be made. First, comparing (6.5) and (6.la) shows that replacing u by w in the force formula must introduce a position vector x. This is easily understood from the dimensional difference of velocity and vorticity. But the force must be independent of the choice of the origin of x; so there should be a compatibility condition to ensure this physical fact (J. Z. Wu and Wu, 1993). Let I be an integral operator (volume, surface, or line integral or their linear combination), xo be the origin of x, .B any tensor, and 0 any admissible tensor product. Then the general form of the compatibility condition reads I{(x + x , ) o F }

=

Z{xoF}

if and only if

Z{F}

=

0

(6.10)

For example, if we remove the operation x X from the integrand of (6.51, the result is obviously an identity. Similarly, taking off vector x from (6.8) simply tells the solenoidal feature of vorticity. This type of compatibility conditions holds for all vorticity based integral formulas that follow. Second, after integration by parts the new integrand no longer equals the original one; only the integrated result remains the same, as is evident from (6.la) and (6.5). In fact, this difference in integrand gives vorticity

Vorticity Dynuniics on Boundaries

231

based force and moment formulas new but clear physical meaning and significant practical value. The most remarkable point is that the regions in ’27, V,, or on 7with ct) # 0 are much narrower than that with u’ # 0, allowing one to focus on highly localized key regions in force and moment analysis. In addition, a few more features of these derivations are worth mentioning, as they also common to the formulas that follow: 1. Equations (6.4)-(6.9) contain various proofs of the D’Alembert paradox; 2. Unlike the inviscid derivation, where a “phantom vortex” has to be put inside the body, now the vorticity in ‘Y or Vf is continuously created from the body surface due to the no-slip condition; 3. What really counts in F (and L as well) is only the net circulation, part of vorticity with opposite directions having been automatically canceled during integration; and 4. In this type of vorticity based formula, the pressure force is absent, but its effect has been automatically included due to its coupling with vorticity, as explained earlier. B. TOTALFORCE AND CONVECTIVE VORTICITY FLUX O N A WAKEPLANE Equation (6.9), applied as 7 is far downstream, is the prototype of a series of subsequent studies that used vorticity based wake plane data to infer the information of the force status. These include, among the others, Betz (19251, Maskell (19721, J. C. Wu et a/. (19791, Hackett and Wu (19821, Hackett and Sugavanam (19841, Onorato et af. (1984), Yates and Donaldson (1986), Chometon and Laurent (1990), and Brune (1994). The interest in this type of analysis increased recently due to the significant progress of experimental techniques, which enables quantitative survey of flow field over one or multi-wake plane(s) very close to the downstream end of the body, or even cutting across the body. Thus, the wake plane analysis has become a powerful tool for relevant flow diagnostics and optimization. However, most of existing theoretical analyses are confined to leading-order approximation, unable to count thc detailed wake flow structure as one approaches the near wake. The following results (J. Z. Wu et a/., 1987a; J. Z. Wu and Wu, 1989b), again based on recasting (6.2) via an integration by parts, give an accurate version of (6.9) and can be used as the rational

J. Z. Wu and J. M. Wu

232

basis for making any desired approximations regarding the force components acting on the body. 1. Lift and Drag Constituents from the Near- Wake Plane Suruey

We take a uniform flow U over a finite-span wing as illustration. The theory equally applies to analyzing other moving bodies, like automobiles, trains, and ships, as long as the flow can be assumed steady. Let the lateral size of the control surface 2 be sufficiently large, so that except its arbitrary downstream plane 9 ( w e call it the near-wake plane) the flow is undisturbed, see Figure 31. Assume that the flow in the control volume I/ is steady. This implies that the starting vortex is far downstream of F a n d its motion has no effect on the near field. As the mathematic basis of this and next subsections, we introduce a general tensor formula of integration by parts. Let P be any second-rank tensor, x be the position vector, and dS = ndS be a surface element. Then, because in the component form

we have identity

x x (n x V ) . P = -(n x v ) . ( P x

XI'+

~

- - nntrace P

IL-1701

FIG.31. The control volume and near-wake plane ZJ. Z. Wu and Wu, 1989. (Reprinted with permission from SAE paper no. 892346 0 1989 Society of Automotive Engineers, Inc.)

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233

Therefore, from the generalized Stokes theorem (2.9), for any surface S we find X

(n X V ) . P d S = / ( P e n

-

ntrace P)dS

-

S

k

dx.(P x

XIT

(6.11)

Then, following the notation of (3.121, we decompose any vector f into normal and tangent directions, so that formally f = ncp n X A = n . ( P I - S). In (6.10, we set P = cpI and S in turn. This gives a pair of surface integral identities:

+

n X A dS

= -

L

x x ((n x 0 ) x A} dS

+ $ x x (A x dx), dS

d

=

3 only (6.12a, b)

We now use (6.12) to transform (6.2). Assume that the integrand of the surface integral of (6.2) has been decomposed as cpn + n X A,, so that we may apply (6.12) with vanishing line integrals. Then, on the right-hand side cp and A, are under differential operators, so only the near-wake plane 7with n = e, needs to be considered. Denote u = ( u , u , w ) and w = ( w, , m y , wz), one finds

where 0, is the tangent gradient on Zand terms that can be cast to line integrals along the boundary of 9 ( w h e r e the flow is undisturbed) have no contribution. Hence, cp causes only a streamwise force and A, causes only a lateral force. Moreover, the viscous effect in has been accumulated before the fluid arrives at Z which manifests on 7as the variation of stagnation pressure F o = @ + pluI2/2. This effect is much stronger than the surface integral of the viscous term in (2.6a), so only the inviscid stress -pn needs to be retained in (6.2). Thus, cp = @

+ pu2,

A,

=

pu(u X e x )

=

pu(we,

-

ue,)

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J. Z. Wu and J. M. Wu

Therefore, by using (6.13), for the lift and side forces in three-dimensional flow,

and, for the drag in d dimensions,

Now the physical causes of the total force on the wing can be easily identified. For example, for a typical steady attached wing flow with oncoming velocity and root-chord length as reference scales,

Thus, if in (6.14a) u = U + u' with u ' = O ( E ) ,the leading effect is precisely (6.9). But, for a steady separated flow, on a near-wake plane, there can be u' = O(1) and (6.9) is unreliable. Nevertheless, the lift is dominated by the spanwise moment of the adcective streamwise-vorticity flux through a near-wakeplane. Then, the O ( c 2 )term, - p y ( u w , + wwz) ,represents a change of lift due to downwash or upwash and sidewash. Finally, the last term in (6.14a) is O ( e 3 ) ,which reflects a modification due to the curvature of near-wake vortices; it is in general necessary to ensure that the computed F, is independent of the location 5 The side force Fy has exactly the same structure and naturally vanishes if the flow is symmetric with respect to the ( x , 2) plane. Similar identification can be made for the drag F,. But, unlike the lift, all constituents are of the same order of magnitude. The first term of (6.14~1, 1

.

(6.15a) is the pressure drag, which exists as long as f , is not uniformly distributed on X Then, the second term of (6.14~)can be re-expressed as (zwy -yw,)

-

-(x.u,) dX

u,

=

i'eL+ w e ,

Vorticity Dynamics on Boundaries

235

Here, because the viscosity on Y has been ignored, the steady Euler equation V& = p(u X w) applies, yielding

Therefore, we identify a viscous drag m

due to the loss of kinetic energy in boundary layer and wake, which is always associated with the vorticity creation; and an induced drag

yw) dS for d

=

3

only

(6.15~)

caused by the lifting vortex associated with downwash, upwash, and sidewash. The remaining term, 2 FXC",

=

~

d - l b

d

pu-(x. dx

u,) dS

(6.15d)

corresponds to the third term of (6.14a, b) and represents a near-field curvature modification. For example, the two-dimensional viscous drag is directly measured by the vertical moment of advective vorticity flux, 2puzo,, through a near-wake plane, which is equivalent to the well-known wake velocity defect. Note that, through the Bernoulli equation, the pressure drag (6.15a) can also be expressed as the integral of p x * V,lu12/2 over where the tangent variation of the kinetic energy is entirely induced by the wake vortices via the Biot-Savart law. Thus, in three dimensions, even a perfect streamlined wing will experience a pressure drag caused by low pressure in the wake vortex cores, not counted in the induced drag. In fact, all FX,), F,,,,, and Fxind, and even F,,,,, have their roots at the wake vorticity. Compared with those works cited in the beginning of this subsection, this approach has some unique advantages: (1) It is accurate and can cover compressible flows (J. Z. Wu and Wu, 1989b; J. M. Wu et al., 1996); ( 2 ) the plane 7can be shifted upstream and cuts the wing, yielding the force constituents on any desired front portion of the wing to further assess the

236

J. 2. Wu and J. M. Wu

configuration design; and (3) each constituent comes from a physical mechanism, closely related to vorticity and vortex dynamics. Note that the appearance of the x-derivatives of u and w in (6.14a, b) (through wy and w z )and (6.15d) does not necessarily mean that one needs data on at least hvo adjacent near-wake planes to compute the forces, since by using the Euler equation they can be reexpressed in terms of quantities on a single % Alternatively, if the ( u , u, w >data on two adjacent planes are available (e.g., by three-dimensional particle image velocimetry that is being developed), then V T p and V w p o can be expressed in terms of these velocity components and their ( x , y , z ) derivatives, so that without pressure measurement the forces can also be computed.

2. Wake Plane Diagnostics of a Delta Wing As an illustration of the aerodynamic force diagnostics based on the wake plane survey data treated by (6.14) and (6.151, we mention a recent analysis of the experimental data on near-wake planes of a slender delta wing (J. M. Wu et al., 1996). The experimental survey was performed by K. D. Visser and A. E. Washburn (unpublished) at NASA Langley Research Center, with a = 20", M = 0.05, and Re = 5 x lo5. The swept angle was x = 76". The data include velocity components and static and total pressure on a near-wake plane perpendicular to the central chord (using the body axes), as shown in Figure 32 (see color plates). The irregular shape of the total pressure contour around the main vortex indicates some error (the maximum error is about 9%). The data were fitted by spline technique, from which the corresponding vorticity components were computed as shown in Figure 33 (see color plates). The various constituents of the integrand in (6.14a), now the normal force, and that of the chordwise force, (6.15a-d), are plotted in Figure 34, which clearly reveal the key contributors to the normal and chordwise forces and their sources. The survey was made mainly on the upper part of the wing flow, but the effect of model support can be partly seen near the centerline y = 0, which was not excluded in data processing, though this can be easily done if so desired. Another wake-plane data set was used in J. M. Wu et al. (1996), which gives a better result. The normal force is clearly dominated by the spanwise moment of the chordwise vorticity advective flux, the first term of (6.14a). Two peaks of this term appear in Figure 34(a). The higher peak is due to the main leading edge vortex, where the local axial velocity is about 1.8 times of the

VorticifyDynamics on Boundaries a

237

018 0 16

0 14 0 12 01

F

008

E

006

m0

0 04 0 02 0

-0 02 0 04 0

0 05

01

0 15

02

0 25

03

0 35

0.2

0.25

0.3

0.35

Y/C

b

005 0 04 0 03 0 02

0 01

F m

P E

O -001

-0.02 -0.03 -0.04

-0.05 -0.06 0

0.05

0.1

0.15 Y/C

FIG.34. The constituents of the integrand of (6.14a) and (6.15a-d) for normal and axial forces, (a) FN and (b) F A , respectively, and (c) the resulting integrand of lift and drag (from J. M. Wu et al., 1996).

238

J. Z. Wu and J. M. Wu

C

oncoming velocity and hence yields a favorable contribution. This also indicates that beyond small angles of attack applying the classic formula (6.9) to a near-wake plane is indeed inaccurate. The second peak occurs near the edge, a contribution of the feeding vortex layer as it just leaves the edge and before it rolls into the main vortex (behaving as a tip vortex), where the axial velocity is, however, only about half of the undisturbed value [Figures 32(a) and 33(a) (see color plates)]. Then, in the second term of (6.14a), the sidewash and downwash of t h e main vortex causes a downward normal force below its core, but an upwash outside the tip vortex is a favorable contribution (note that the tip vortex has larger y ) . After transforming to wind axes, the total lift coefficient is 0.69, lower than the experimental value 0.744 (no calibration was made in the wake-plane survey). The interference effect from the model support is automatically minimized without additional treatment, because its y = 0. Compared to the normal force F N , the constituents of axial force F A , Figure 34(b), are more complicated. One of the major contributors is the lateral motion of main vortex, which changes sign as crossing the vortex

VorticityDynamics on Boundaries

239

core from inboard to outboard, the latter being larger and beneficial. The same happens for the tip vortex. Note that to a certain extent the contribution of main vortex to the FA is canceled by the static pressure constituent, of which the physics is the balance between centrifugal force and pressure force. Then, a viscous force is manifested by the moment of the tangent gradient of p,, , which has both positive and negative values but the overall effect is a drag. The effect of (6.15d) is not negligible. Adding these together yields the total integrand of (6.15). After transforming the forces to wind axes (Figure 34(c)), we found that the total drag coefficient is 0.333, very close to the experimental value 0.34. The drag is mainly from the tip region, even though the high peak in the FN integrand, due to the main vortex, has a strong streamwise projection-it is largely canceled by the streamwise projection of the beneficial contribution to FA of the same vortex. Therefore, the main vortex has much higher aerodynamic efficiency than the tip vortex. This is a clear indication that the wing-tip shape needs to be modified for higher aerodynamic efficiency. In fact, the main purpose of wake plane survey is precisely discovering what spanwise portion of the wing should be improved. Note that improving the chordwise configuration can be achieved by wake plane analysis as well, provided that the flow is also surveyed on many cutting-in planes. But, in general and particularly for unsteady flows, one has to move to the close view as follows here.

C. FORCEAND MOMENTIN TERMS OF BOUNDARY VORTICITY FLUX We have transferred (6.1) and (6.2) to obtain the vorticity based force formulas on a solid or fluid body, viewed by far-field and near-field observers, respectively. In these results we see that F (L is similar) is related to a continuous shedding of the vorticity from the body. This vorticity can come only from a continuous creation from the body surface 9. This observation brings us back to the boundary vorticity dynamics and suggests that one must be able to express the total force and moment in terms of the boundary vorticity fluxes on 9. Such is achieved by transferring the close-view formulas, (6.3a, b) and (6.3'a, b). The result was first obtained by J. Z. Wu (1987) for incompressible flow over a rigid body, which has been extended to compressible flow (steady or unsteady) and deformable solid body or closed fluid interface by J. Z. Wu et al. (1987a, b),

J. 2. Wu and J. M. Wu

240

J. Z. Wu and Wu (19931, and J. Z . Wu (1995). Here we confine ourselves to incompressible flow.

1. Integrated Moments of Boundary Vorticity Fluxes For recasting (6.3a) and (6.3'a), we invoke the identities (6.12a, b) again. Consider a closed boundary .D in a homogeneous fluid first so that the line integrals are absent. Comparing the left side of (6.12a, b) with (6.3a) and (6.3a') and the right side with (3.17) and (3.15), respectively, it immediately follows that, in d dimensions, j?lnpdS

Lpt

X

n dS

P

x

X

a,;dS,

=

--

=

p / x x u,,,dS,

d-ljhn

d

=

2,3 (6.16a. b)

d

=

3 only

A?

Here, as in (4.1) and (3.251, a h = n X Vh is the vorticity flux due to modified enthalpy (including the gravity effect); and u,,, is the explicit viscous part of vorticity flux, given by the last term of (3.17) and (3.25b) for rigid and deformable boundaries, respectively. Therefore, for threedimensional flow over either type of closed 9, we find an elegant force formula (6.17) Similarly, for the total moment on a rigid body, J. Z. Wu and Wu (1993) showed that the corresponding three-dimensional formula is (6.18) Then, on a deformable body, except for the contribution of a nonuniform normal vorticity 5 , say, uc = v(n x V ) x 5, the remaining part of u is the same as that on a nonrotating rigid surface. But J. Z. Wu (1995) showed that

which is just the last term of (6.3'b). Consequently, eq. (6.18) can also be applied to both rigid and deformable bodies.

Vorticin,Dynamics on Boundaries

24 1

Equations (6.17) and (6.18) are the central result of this section. They reveal that the total force and moment acted on a closed boundary can be solely expressed by the proper moment of i1orticityJiuxesdue to su$ace stresses, which are inherently of viscous origin. In other words, F and L are directly related to the rate of work required to create new vorticity from the wall, that is, that for raising and turning t h e near-boundary vortex lines into the J. Z. Wu and Wu (1993) and J. Z. Wu (1995) took the Stokes flow over a rigid and fluid sphere as a simple example to show how the pressure force on the sphere can be clearly interpreted in terms of up,which is essentially a viscous process, but was explained misleadingly as “the viscous component of the normal stress” (Illingworth, 1963) that should have been referred to ( A 2pM in (2.5). Note that the flux due to acceleration does not appear; as in (6.3), it has been implicitly reflected by the stress status through the Navier-Stokes equation. As noted by J. Z. Wu and Wu (1993), a part of the stress, although exerting a force to the local surface clement, may have no contribution to the total force and moment if it does not send vorticity into the flow. This includes, for example, the hydrostatic pressure and the pressure associated with a potential flow. Even somc viscous stress may have no contribution to the total friction force; this situation occurs on a rigid boundary dB, as seen from (4.1c,d), if on a portion of dB, the boundary vorticity is two-dimensionally divergence free or along its direction dB has no curvature. In this sense (6.17) and (6.18) can be said theoretically irreducible: All local stresses that could cancel each other during integration are automatically excluded. Thus, if only the total force and moment status are to be studied, one can well focus on those local regions where there is a strong boundary vorticity flux-a point of great practical value to be exemplified later. We may also gain a deeper physical understanding of various vorticity based formulas like (6.61, (6.81, (6.14), and (6.15): They all “converge” to (6.17) and (6.18) and become more and more accurate and general. That is, moving toward 9, an observer will pass (6.6) or (6.8) first for the Euler limit of steady flow, then (6.14) and (6.15) for viscous steady flow, and finally, tracing the root of the vorticity or circulation appearing in those formulas, the observer will see the boundary vorticity fluxes and find the most general result (6.17) and (6.18). In short, we arrive at an inherent

+

23

This fact also provides another insightful physical interpretation for the d’Alembert paradox.

242

J. 2. Wu and J. M. Wu

unification of the action and reaction phases between a closed boundary and the vorticity field created thereon, which closes the fundamental theory of boundary vorticity dynamics. 2. Diagnostics and Optimization of an Airfoil

As a simple illustration and complement to the wake plane diagnostics, consider an attached flow over a two-dimensional airfoil in the (x,z ) plane to which (6.16a) can be applied to compute the pressure force. As remarked before, the origin of a Cartesian coordinate system (x,y, z >can be arbitrarily chosen without changing the result of (6.16a); so for convenience let it be so located that for x < 0 there is a favorable pressure gradient, and for x > 0, an adverse gradient, see Figure 35. By (6.16a), the contribution of a unit-length are element ds as x to the lift and pressure drag is given by

where a;, = e y . ur,= - d p / d s . We stress again that, due to integration by parts, these dF, and dF, do not represent the local lift and drag elements at all, but their integrations do give correct Fpz and q,,r. Now, on the upper surface of the airfoil, because up is positive when x < 0 and negative when x > 0, both portions give rise to a lift. The situation is opposite on the lower surface. The regions near the leading and trailing edges (in particular leading edge) are critical, because there we have both a large 1x1 and a sharp pressure gradient. Thus, to increase lift, the airfoil should be shaped so that, on the upper surface, the high-l up/ regions are located as close to the edges as possible, and on the lower surface, they are as close to the origin as possible, provided that the flow remains attached and the off-design performance is not quickly deterio-

FIG.35. The sign and location of vorticity flux due to pressure on airfoils: (a) a traditional airfoil, (b) a Piercey-Whitcomb type of airfoil with higher lift/drag ratio at low speed.

Vorticity Dynamics on Boundaries

243

rated. On the other hand, to reduce drag, \?,I should be minimized near the maximum thickness point. Remarkably, this simple observation leads to a configuration very like the low-drag airfoil previously proposed by Piercey in 1930s, which evolved to the supercritical airfoil of Whitcomb at a higher speed, with a wide flat midportion on upper surface (Figure 30(b)). The preceding discussion can be quantitizcd by computing the Euler limit of an attached flow (J. Z. Wu et al., 1996). A comparison between a symmetric NACA-0012 airfoil and a lift-optimized (and hence cambered) airfoil of the same maximum thickness is given in Figures 36 and 37 regarding the distribution of pressure and the normal force integrand (6.19a). The Euler limit was obtained by solving the full potential equation, because then imposing the no-slip condition immediately gives the desired u,, and its x moment. The figures indicate clearly that the moment -xu,, is indeed highly localized in the leading- and trailing-edge regions. This is even more so for the optimized airfoil, which by (6.194 explains why it has a higher lift. Note that the optimized airfoil is similar to that of Figure 36(b). Figure 37(b) shows t h e normal force benefit of the optimized airfoil at small angles of attack. The normal force of an airfoil with the same camber distribution as the optimized one but with the thickness distribution of NACA-0012 is also shown (as is well known, changing thickness distribution affects the lift only very slightly). The viscosity of a finite Re flow may cause an early separation if the trailing-edge angle is larger than a critical value, which should be imposed as a constraint that cannot be made by an inviscid code. As indicated by the previous example, the unique advantage of (6.17) and (6.18) over the common formulas (6.3a, b) is that the new formulas automatically focus one’s attention to highly localized key regions in diagnostics and optimization. Further exploration along this line would be of great interest, in particular for the fluid-dynamic optimization of threedimensional configurations.

3. Force on a Body Piercing an Interface The previous formulas can be further generalized to the case of a closed boundary 9 piercing an interface S between fluids 1 and 2. For simplicity we derive only the total force formula, of which the proper basis is obviously the full form of (6.12a, b). Let 9, and be immersed in fluids 1 and 2, respectively, with 9 = .3,+
J. 2. Wu and J. M. Wu

244 a

04

Optimized airfoil NACA0012 - - - ~

03

0.2

01

0

-0 1

-0 2

-0.3

-0 4

0

b

02 I

04

06

I

1

08 I

Optimized airfoil NACA0012 - ~ - ~

-3

-2

-1

0

1

0

0.2

0.4

0.6

0.8

1

FIG. 36. The pressure distribution over a symmetric NACA-0012 airfoil and a liftoptimized and cambered airfoil at M = 0.2 and a = 6". computed by a full potential code. (from J. Z . Wu et al., 1996).

245

Vorticity Llynamics on Boundaries a

150

Upper surface, Optimized airfoil Lpper surface. Optimized airfoil - - - Upper surface, NACAOOI 2 Lower surlace, NACAOOI 2 100

50

-0 5

b

-0.3

-0.4

-0.2

-0.1

0

0.1

03

0.2

0.4

0.5

14

Optimized airfoil -+ Optimized camber. NACA0012 thickness t NACA0012 0 1.2

1

0.8

z

0

0.6

0.4

0.2

0

I

I

1

1

1

1

1

ALPHA (degrees)

FIG. 37. The distribution o f -xu,, on thc two airfoils of Figure 31 and corresponding normal force coefficients at M = 0.2 and small angles of attack. Also shown are the normal force Coefficients of an airfoil with the optimized camber distribution but the NACA-0012 thickness distribution (from J . Z. Wu et d., 1996).

246

J. Z. Wu and J. M. Wu

directions as viewed from two sides of the line. As the intersection of 9 and S, this boundary line is a contacl fine. The total force follows from the procedure leading to (6.19) but retaining the line integrals of (6.12a, b). We denote the contributions to F of the surface and line integrals by F, and F,, respectively. Then,

where [PI and [ p s ] are the jump of stresses given by (3.1a, b). The force F/ can no longer be attributed to boundary vorticity fluxes. Rather, if e is the unit vector along the contact line and dx = eds, then because n . e = 0, by (3.lb), [ pg] X dx = n(e . [t,,]) d ~ ; therefore, ’~ we finally have

Note that U, K, and K measure the motion and shape of interface, of which only the values right of the contact line enter (6.20b). Therefore, F/ strongly depends on the contact line condition, which itself is a subject of much research and will not be pursued here. We remark only that on the contact line some quantities might become singular; but if this does not happen or the singularity is not sufficiently strong, then the line integral (6.20b) is obviously negligible compared with the surface integral (6.20a). Further extension of (6.20) to multi-interface flow over 9 is straightforward. A typical application of (6.20) with great practical interest is the hydrodynamic force of a surface ship, of which a significant part is the resistance to the creation of ship waves. If F, is negligible, it immediately follows that the wave force can come from only the water part of (6.20a). Therefore, associated with the ship-war:e creation there must be a corticity creation. This is another manifestation of the coupling between compressing and shearing processes, of which the mechanism is still the no-slip condition for viscous flow. The vorticity created thereby does not follow the wave 24

If the interface is contaminated, additional terms as in (5.35) should be included.

Vorticity mnamics on Boundaries

247

propagation but is advected downstream and forms a portion of wake vortices, which in turn affect the wave pattern and are the whole source of ship-wake turbulence. Theoretically, the ship wave has been studied under the potential flow assumption, in which the inherent coupling of ship wave and vorticity creation is cut off. However, as in the previous subsection, the coupling can be easily recovered by taking the potential solution as the Euler limit and turning on the no-slip condition. Then, the wave resistance must equal its associated drag component of the water part of (6.20a1, from which the total vorticity flux moment due to the wave can be estimated. It should be stressed that the vorticity source associated with wave creation, although still coming from the no-slip condition, is different from other well-known sources such as those due to flow separation from the hull or a rotating propeller. In fact, the wave resistance and associated u can be clearly identified by its highly nonlinear dependence on the Froude number (e.g., Wehausen, 1973; Newman, 1977). This situation is closely analogous to a supersonic flow over a body surface with shock wave. Although the inviscid theory can well predict the wave pattern and associated wave drag, as the viscosity is turned on the foot of a shock wave on the body surface immediately becomes a singular source of vorticity, because across the shock the pressure gradient is a delta function. For a finite Re, the shock-created vorticity is responsive to the interaction of shock wave and boundary layer. Then, from the compressible counterpart of (6.19) (J. Z . Wu and Wu, 19931, one can equivalently predict the wave drag but in terms of the vorticity creation rate. Further exploration of this interesting aspect of wave-vortex coupling would also be highly desirable.

MI. Application to Vorticity Based Numerical Methods From the discussions in previous sections it is clear that, except for some simple analytical or semianalytical solutions, any further quantitative applications of the theory of boundary vorticity dynamics have to rely on experiment or computation, because the boundary values of those key quantities such as the vorticity and its flux can be only the outcome of the solution. This situation naturally calls for proper numerical methods. Although any existing Navier-Stokes codes may serve the purpose, those vorticity based schemes that have inherent consistency with the theory are

J. Z. Wu and J. M. Wu most favorable. Inversely, the theory also sheds new light onto the resolution of some difficult problems of these methods and hence contributes to their development. In particular, the toughest difficulty is how to impose a proper boundary condition for vorticity on a solid wall, of which extensive studies have been conducted in the past two decades (see Gresho, 1991, 1992; Koumoutsakos et al., 1994; Koumoutsakos and Leonard, 1995; Sarpkaya, 1994). Here, boundary vorticity dynamics can make very unique contribution, which leads to an almost complete resolution of the difficulty. This is the topic to be reviewed here, basically following the recent work of J. Z. Wu et al. (1994b). We shall confine ourselves to the case of an incompressible flow with unit density over a rigid boundary dB, which is simplest and relatively mature. Even in this case, extra complexity will appear if dB has a normal acceleration, as can be felt from the discussion at the beginning of Section 1V.B. Thus, to make the key points stand out, we further assume that the wall has at most a motion along its tangent direction.

OF VORTICITY BASEDMETHODS A. AN ANATOMY

1. Kinematic iiersus Dynamic Vorticity Condition on a Rigid Wall Vorticity based methods solve the vorticity equation, say (3.7), under the uelocity adherence condition (2.10) on d B . For an incompressible flow, such a formulation avoids resolving pressure by keeping the velocity solution divergence free. However, in implementing these methods one has to infer a boundary condition for w from (2.101, either by applying the Biot-Savart law to the solid surface or through a projection theorem (Gresho, 1991). This implies that in its strict nature the vorticity boundary condition is of global type. That is, eq. (2.10) imposes a kinematic integul constraint to the possible w distribution, rather than a local condition for the boundary behavior of w. Such an integral constraint cannot be easily included in a local algorithm (finite difference or finite element). Alternatively, this constraint can be represented as the differential relation between u and w , either by the Poisson equation or a Cauchy-Riemann type of equation. Then w and u must be solved together, which results in a larger coupled system. To avoid this basic difficulty, some well-known fractional-step approaches were proposed, where the near-wall w field is simplified as a

Vorticity Dynatnics on Boundaries

249

vortex sheet y such that the volume integral constraint is reduced to a boundary integral equation or even a local condition for y. These include, among the others, the integral formulation of J. C. Wu and coworkers (e.g., Wang and Wu, 1986, and references cited therein) and the vortex methods of Chorin (1978). A further analysis was recently made by Koumoutsakos et al. (19941, who noted that the boundary vortex-sheet strength y can be manipulated so that a Dirichlet or Neumann type of condition could be modeled. They pointed out that the Neumann type is better suited for vortex methods using the particle strength exchange (PSE) scheme. The error of all these approximations, however, had never been closely analyzed. We stress that any approximation of the near-wall c1) field by a vortex sheet could deteriorate the accuracy of computed boundary vorticity and its flux, which as we saw are of crucial importance in the entire boundary vorticity dynamics. This difficulty reflects a basic fact that neither the local boundary vorticity nor its normal gradient can be rigorously inferred from any kinematic constraint derived from (2. 10). Thus, kinematically, it is impossible to obtain a strict Dirichlet or Neumann condition for the Llorticity equation. Moreover, the vorticity equation is one order higher than the Navier-Stokes equation; hence an additional compatibility condition is necessary to exclude possible spurious solutions due to raising the equation’s order. But again, this condition is not derivable within kinematics. Essentially, a thorough resolution of this boundary conditioning problem lies in the following fundamental observation stated by J. Z. Wu et al. (1990). The Navier-Stokes equation, with primitive variables (u,p ) as unknowns, naturally matches the velocity adherence condition (2.10) (yet special consideration is necessary for the pressure boundary condition). In contrast, the vorticity equation, as a one-order higher equivalence of the Navier-Stokes equation, does not. This mismatch is the root of entire difficulty. It is then clear that the natural boundary condition for (I) should also be one order higher than (2.10); which is nothing but the acceleration adherence (2.131, which in turn leads to the dynamic Neumann conditions (3.17) and (3.16) for second-order vorticity and pressure equations.2s

25Althoughsuch a “derivative argument” is well known in other contexts, Anderson (1989) first applied it to the vorticity boundary conditioning problem. See also Anderson and Reider (I 994).

250

J. Z. W u and J. M. Wu

Symbolically, the incompressible Navier-Stokes and continuity equations can be written as %u)

+ Vp = 0

and

V .u

=

0

(7.la, b)

Then the vorticity and pressure equations, along with their respective dynamic Neumann conditions, are

V

X ~ ( U = )

0,

n

X ~ ( U ) B=

-n x Vp,

(7.2a, b)

where suffix B means values on dB. Note that (7.2b) and (733) are just (3.17) and (3.16), respectively. Suppose the proper initial condition u = uo at t = 0 is always satisfied. Then, guided by the above fundamental observation, J. Z. Wu et al. (1994b) proved the following theorem:

THEOREM 1. Assume u satisfies (7.2a) and ( 7 . 3 ~ )Then . it is the solution of (7.la), ij'either (7.2b) or (7.3b) holds. This theorem can be restated as

COROLLARY 1. Let u be a solution of problem (7.2a, b ) or (7.3a, b). Then it is also a solution of (7.la), if and only if it satisfies (7.3~1,or (7.2a), simultaneously. We have seen the equivalence of (2.10) and (2.13). Now the theorem proves that the compatibility condition mentioned earlier is simultaneously satisfied by the dynamic conditions. Thus, the Neumann problem for a coupled ( w , p ) field is well posed. Note that, once the u field has been solved by whatever method, V p can be inferred from (7.la); but this involves an inconvenient computation of d u/ d t . Solving the second-order pressure equation (7.3a, b) avoids this problem. However, the second form of the theorem shows that this approach needs to satisfy the vorticity equation as a prerequisite. Similarly, solving the vorticity equation (7.2a, b) also requires the second-order pressure equation. The pressure part of this problem has been extensively studied by Gresho and Sani (1987); but the ( w , p ) coupling on dB, and hence an interior ( c o , p ) coupling over the whole flow domain V , should not be ignored or oversimplified. In fact, this

VorticityDynamics on Boundaries

25 1

coupling is precisely the dynamic manifestation of the global kinematic constraint on the o field, and the basic difficulty of vorticity boundary condition would still exist if the effect of this coupling was not further clarified and bypassed. Fortunately, in contrast to the kinematic constraint, the order of magnitude of boundary (w, p ) coupling can be clearly identified and becomes negligible at high Reynolds numbers. This basic estimate leads to a very simple decoupled approximation of the dynamic Neumann condition, which will be addressed next. The most significant advantage of dynamic Neumann condition is that, because it amounts to computing (r (including up and u,,,),an accurate computation of the skin friction 7, on dB becomes possible. For ordinary schemes in terms of primary variables (u,p), as well as some vorticity based methods with kinematic boundary conditions, this is not an easy task because T,, has to be obtained by one-sided differencing along the normal, but the normal derivatives of u and w on dB are extremely large at high Reynolds numbers, O(Re i) and O(Re), respectively. In contrast, because (rl) and cr, are both computed from much milder tangent derivatives of p and T , ,with the knowledge of interior vorticity field and its slope u on ilB, one can use centered diference to obtain a much better estimate of w on dB, and hence 7 , . As will be exemplified in Section VII.B, this unique advantage derived from boundary vorticity dynamics is of great value in practical applications. The use of dynamic condition for vorticity does not eliminate the need for kinematic conditions in solving velocity from vorticity and ensuring the divergence-free property of computed vorticity field. On the contrary, Theorem 1 also sheds new light on the optimal form of such kinematic conditions. Although this chapter focuses on dynamics, this kinematic problem is worth examining briefly. For an incompressible flow with given vorticity w in domain V , the velocity is determined by V x u = w , or more conveniently, by the Poisson equation V2u = - V x w , along with (7.lb) and the normal condition (2.10a). The solution should ensures

V.w=0

in

V

( 7.4)

But, because the scalar condition (2.10) is insufficient for the vector u, additional condition is required that should simultaneously guarantee (7.lb) and (7.4) both in the interior of V and on the boundary d V. Inspired by Theorem 1, the optimal kinematic conditions was found by X. H. Wu

J. 2. Wu and J. M. Wu

252

et al. (1995), who proved the following theorem as a kinematic counterpart of Theorem 1:

THEOREM 2. For any gioen smooth ilector field o,there is a unique solution u of the problem V2u= - V x o

in

(7.5a)

with n x (V

x u)

=

n x o,

n*u

=

n . b on

dB

(7.5b,c)

that satisfies (7.1b) and V X u = cod, where adis an orthogonalprojection of o onto the divergence-free space. Note that (7.5b) is exactly the kinematic counterpart of (7.2b) or (3.17). The significance of this theorem will be mentioned in Section VII.B.2. 2. Solution Structure of the Coupled ( w , p ) Field

To analyze the order of magnitude of the (o, p ) coupling on dB, we first need to understand how this coupling affects the vorticity field in the flow domain V . This can be done by examining the structure of the ( o , p ) solutions without really solving them numerically. Thus, we recast the (o, p ) equations into integral representation by using Green’s functions. Here and below we take p = 1. Let r = x - y, r = IrI, and

be the fundamental solution of the heat equation in free space, with 2 being the step function and d the spatial dimensionality. Then, the integral form of incompressible vorticity equation follows from Green’s second identity:

- c d ~ $ G*n x L, dS

+ /:d.r/vVG* x

L dV

dB

Here the subscript 0 implies values at T = 0. and u is given by (3.17). The first volume integral represents the effect of initial o distribution, and the

Vorticity Dynamics on Boundaries

253

last two, the nonlinear advection effect. As shown in Section II.D, if in an idealized model the flow is started by an impulsive pressure gradient, the boundary vorticity flux u must contain a 6 function at t = 0 and creates an infinitely thin vortex sheet with strength y o , which can be easily separated from its regular part:

c

dr

/

G*u dS

=

[?,G; y o d S

+ /‘ d r / Of

dB

G*u dS

(7.7)

dB

The initial velocity no-slip condition has been implied through (7.7), without which y o would be uncertain and has to be prescribed in advance. Now, a key step is to transfer this integral form of w(x, t ) by using (3.17) and the Stokes theorem (2.9). This finally yields

+cdr/VG*

X

LdV

(7.8)

V

where

t,

=

np,

+nX

VW,

(7.9)

is the wall stress. Similarly, let

be the fundamental solution of the Laplace equation in free space. Then, as the counterpart of (7.81, for stagnation enthalpy H = h + lu12/2 (or stagnation pressure, as we have taken p = 11, H(x)

=

-/T,nVG.t,dS-/VG.LdV

(7.11)

V

Equations (7.8) and (7.11) are a consequence of boundary vorticity dynamics, which have three remarkable features. First, although only the free-space Green’s functions (7.7) and (7.10) are used, the normal derivatives of w and p are absent in the surface integral due to the application of (3.16) and (3.17); that is, the Navier-Stokes equation on dB. Thus, we

254

J. 2. Wu and J. M. Wu

arrive at an innovative integral formulation of a Dirichlet problem with free-space Green’s function. However, as before, this problem is for the coupled (w, p ) . Second, because by (7.7) and (7.10), VG* and VG are along the direction of r = x - y, the boundary stress t, at a point y, E B affects the w field at a point x E V only by its components perpendicular to r, through diffusion; similarly, t, at y, affects the H field at x only by its component along r.26 Third, applying (7.8) to dB implies a global and implicit Dirichlet condition for w in V . Unlike the boundary vorticity flux a,there exists no local equation for w, . This is a general mathematical manifestation of the underlying physics stressed several times before: u rather than w R reflects the vorticity source directly, and w B itself arises through a space-time accumulated effect of u as well as advection and diffusion. The same is true for stagnation enthalpy. We mention that (7.8) and (7.11) are also valuable in various integral formulations of vorticity based methods. These methods are usually more time consuming than finite difference; but, owing to the desire of removing the less accurate random-walk approach from Chorin’s (1978) methods, some authors have recently turned to deterministic diffusion algorithms that are mostly of the integral type, because then a smooth connection between the Lagrangian advection substep and the Eulerian diffusion substep can be achieved (in contrast, to connect a Lagrangian convection and a finite difference diffusion one has to use techniques like vortex in cell, at the expense of artificial viscosity) (see, e.g., Degond and Mas-Gallic, 1989; Cottet, 1990; PCpin, 1990; Lu and Ross, 1991; Koumoutsakos and Leonard 1995; Koumoutsakos et al., 1994). We believe that (7.8) and (7.11) provide a common theoretical foundation to all grid-free deterministic diffusion schemes. 3. The Vorticity-PressureDecoupled Approxima tion

An inspection of (7.8) and (7.11) indicates once again that the boundary ( w , p ) coupling, through the appearance of t, in both equations, is a viscous effect. It does not enter the advection process represented by the volume integrals of the Lamb vector L. Thus, in numerical computations, using fractional-step (or operator splitting) methods can at least greatly 26This is closely similar to the transverse-longitudinal decomposition of shearing and compressing processes made in the wave number space (Section 1I.C).

VorticifyDynamics on Boundaries

255

reduce the ( o , p ) coupling. That is, one solves the Euler and Stokes equations

+ u , . d u , + vpl = 0,

dU1 ~

dt

du2 -

dt

+ Vp2

=

v V 2 u 2 (7.12a,b)

successively for each time step. Here, scalars p1 and p 2 are necessary to guarantee (7.lb). Symbolically, the solution of the Navier-Stokes equation (7.la) can then be written as

where uo is the initial velocity for a time step, and At),9 ( t ) , and E ( t ) stand for the solution operators for the Navier-Stokes, Stokes, and Euler equations, respectively. Ying and coworkers (e.g., Ying, 1987; Zheng and Huang, 1992; Zhang, 1993; Ying and Zhang, 1994) and Beale and Greengard (1992) proved that this scheme is convergent and it is first-order accurate in time (higher order schemes can be designed; e.g., Strang splitting scheme). Corresponding to (7.12a, b), we have dm1

-+ v x L , dt

=

0,

L,

=

w 1 x u,

(7.13a) (7.13b)

vV20, = 0

as the vorticity advection and diffusion equations. Indeed, o1and p1 are fully decoupled in (7.13a) because only (2.10a) is required on dB. In this substep, there is no vorticity boundary condition at all. Here, Lagrangian vortex methods exhibit most of their strength. The coupling between o 2 and p 2 persists in (7.13b) due to (2.10b), which, however, implies a much simpler linearized version of coupled equations (7.8) and (7.11): w,(x, t )

=

/

dB

Ggy,, dS

+ /GXw2,) dV + /‘ d7/ v

0’

VG*

X t,,

dS (7.14)

dB

Here, we have assumed that d B is smooth and the integral in (7.15) should be understood as taking the Cauchy principal value. The velocity does not appear in the coupling; it can be solved separately after o is obtained.

256

J. Z. Wu and J. M. Wu

Our purpose is to obtain a Neumann condition for the vorticity. So we estimate the boundary vorticity flux a after one time step At + 0. First, after solving the substep of inviscid convection (7.13a), a slip velocity us appears on dB, which implies a singular vortex sheet with strength y = n X (b - u,). This vortex sheet is a result of the driving force V p , = O(1) and serves as the initial condition of (7.13b). On the other hand, applying (7.12b) to dB yields du2B

u = n x - dt

+

u 2 p

+

U2"l\

where u2, = n X V p 2 Band uzvl, = v(n x V > x w Z Aare the fluxes due to p I R and respectively. Note that p 2 is the Stokes pressure that, unlike p , , has a viscous origin (Section VI.C.l) and should be classified as an explicit viscous part of a,the same as u2?vlc. Integrating this equation from t = 0 to t = A t , say, and requiring that the resulting u must ensure a no-slip condition at t = A t , there is Y

cr =

-

L At-

+ a,,)+ crzvis

(7.16)

at each x B E dB. Here, the overline means time-averaged values; the first term is the singular part of u caused by advection, and the rest is the regular part caused by pure diffusion that contains the boundary ( w , p ) coupling. The orders of magnitude of a,,, iT2v15, and y / A t are now easily identified. Because ug should satisfy the no-slip condition, lyl = O ( A t ) and hence IyI/At = O(1). On the other hand, by (7.15) and the property of the linear operator acting on p Z B we , have p Z B= O(vlwAI).For a high Reynolds number attached boundary layer lwRl = O(vf), hence lG2pl

-

IG2vlal

-

"llV~GBll

'v

(7.17)

This estimate may have a local change in the neighborhood of separation lines or sharp edges, but because in (7.14) p Z Raffects w through a surface integral, the Re- 4 dependence should be a correct overall estimate. Therefore, for sufficiently high Re, the error due to dropping the last two terms of (7.16) is of O(Re-f), much smaller than the first term. This leads to the decoupled approximation W E - -

Y At

(7.18)

Vorticity Dynamics on Boundaries

257

having been used by several authors based on some plausible arguments (e.g., Hung and Kinney, 1988; P&pin, 1990; Koumoutsakos and Leonard, 1995; Koumoutsakos et al., 1994). Numerical tests have shown that (7.18) is much better than the corresponding local decoupled approximations of the Dirichlet condition, of which one commonly proposed form is

. is where h is a chosen small normal diffusive distance, say d z ~ This not surprising, because no local approximation for o Bcan be rational. Although in the preceding the decoupled approximation was obtained in an operator splitting scheme, a similar approximation can be derived without splitting. X. H. Wu et al. (1994) used a second-order Runge-Kutta method to integrate the vorticity equation in time, where the effect of p H , and hence C T , is updated without really solving it, but is approximated by means of the residual slip velocity u, according to (7.18). It then turned out that at the end of each time step the approximation produces a u, of O ( A t 2 ) (the actual value is much smaller than A t 2 ) . Nevertheless, if necessary, the global (0,p ) coupling can be easily recovered by a simple iteration procedure, which may effectively eliminate the residual slip velocity that appears due to dropping the O(Re-t)-order regular part of (7.16). The analyses of this subsection constitute some basic building blocks in constructing more accurate and efficient schemes. Two typical numerical tests applying these analyses follow. B. NUMERICAL EXAMPLES 1. Impulsively Started Flow over a Circular Cylinder A well-known benchmark test for various numerical schemes is the two-dimensional flow over an impulsively started circular cylinder. Except for the availability of careful experimental results at a Reynolds number up to 9500 (Bouard and Coutaneau, 19801, very accurate numerical data, usually based on vorticity-stream function (w-i,!~) formulation, have also been produced as a standard of comparison. Ta Phuoc LOCand Bouard (1985) used a fourth-order scheme to resolve the Poisson equation for II, and a second-order scheme for w . Anderson and Reider’s (1994) finitedifference scheme is of fourth order in both space and time, which uses

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(2.10) as boundary condition and was performed on a supercomputer with a uniform grid of 2048 x 256 = 0.5 X l o 6 and time step A t = 0.00033. More recently, Koumoutsakos and Leonard (1995) used a grid-free deterministic vortex method to reach about the same accuracy as Anderson and Reider, also performed on a supercomputer. In this scheme the dynamic Neumann condition for vorticity was employed, the time step for Re = 9500 was 0.01, and the particle number increases as time, up to 2 X 10' at t = 3 and more than lo6 at t = 6. All these computations retained the global nature of vorticity boundary condition. J. Z. Wu et al. (1994b) computed the same flow on a workstation by using the w-fi formulation and a finite-difference scheme, of only second order in space and first order splitting in time, with a grid of 301 X 256 (stretched in the radial direction so that it was much denser adjacent to the wall) and time step 0.0025 for Re = 9500. The ( w , p > decoupled approximation of dynamic Neumann condition, eq. (7.18), was imposed. Figure 38 shows the computed flow field using the first order splitting and no iteration, compared with experiment (Bouard and Coutaneau, 1980) and the computation by Ta Phuoc LOCand Bouard (1985). However, this comparison is not critical; many schemes can produce the same flow patterns but not all of them are able to predict the boundary vorticity or skin friction accurately. Figure 39 shows a comparison of computed boundary vorticity w B by J. Z. Wu et al. (1994b) and Anderson and Reider (1994). The violent oscillations of w Roccur due to separated vortices. It is remarkable that the low-order differencing and a simple use of (7.18) can already catch the w B distribution very well, and the small residual difference can be effectively eliminated by only a couple of iterations. Figure 40 shows the computed drag coefficient C, of the cylinder at Re = 1000 by different methods. In this case, the flow does not separate, and hence C, is dominated by skin friction T , ~ Note . that one-sided difference methods cannot predict the asymptotic steady C , satisfactorily, and the third-order method is no better than the second-order one. This test convincingly confirms the ability of boundary vorticity dynamics in improving the skin friction prediction. To check the estimate (7.17) numerically, J. Z. Wu et al. (1994b) carried out a group of tests on the dependence of the residual slip, and hence u Z pon , Re. They define the reduction factor of the slip by

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FIG. 38. The flow patterns of an impulsively started flow over a circular cylinder at Re = 9500. Top: The fractional-step scheme with (7.18) as dynamic boundary condition, J. Z. Wu et a / . (1994b). Middle: Flow visualization of Bouard and Coutaneau (1980). Bottom: Fourth-order simulation of Ta Phuoc Loc and Bouard (1985). (Reproduced from “Dynamic vorticity conditions: Theoretical analysis and numerical implementation,” J. Z. Wu et ul., Copyright 0 1994 (Inter. J . Numericul Mefhods in Fluids). Reprinted by permission of John Wiley & Sons, Ltd.)

150

-

-50

-

-100

-

fourth order no iteration iteration

----

3

-150 -

;

.7nn _”” 0

20

40

60

80

100

120

140

160

180

Angle

lourth order no iteration

-

--- I

t

?

._ ._

8

0.

5 ;-loo

.

0

m

-200 ’ -300 400 -500

I

I

I

I

I

Angle

FIG. 39. The boundary vorticity of impulsively started Row over a circular cylinder at R e = 9500. Comparison between fourth-order result (Anderson and Reider, 1994) and second-order (J. 2. Wu et al., 1994b) computation with and without iteration. (Reproduced from “Dynamic vorticity conditions: Theoretical analysis and numerical implementation,” J. Z. Wu et al., Copyright 0 1994 (Inter. J . Numerical Meihods in Fluids). Reprinted by permission of John Wiley & Sons, Ltd.)

VorticityDynumics on Boundaries I

I

I

I

26 1 I

1.4

1.2 1 c

c

.g

0.8

E 0)

$ 0.6 ol

2

0.4

2nd-order one-side 3rd-order one-side

0.2 0

-0.2 0

1.5

1

0.5

2

2.5

3

Time FIG.40. Comparison of computed transient drag coefficient for flow over an impulsively started circular cylinder at Re = 1000 (from X. H. Wu, 1994).

where u : ’ ; and u:+l are the slip after the convection and diffusion substeps, respectively. Evidently fR also represents the ratio between p , and p 2 , or the inverse of the strength of boundary ( w , p ) coupling. To determine the variation of fR with Re, the index of fR was introduced as

I,

=

In ( fR /fR ,) ln(Re,/Re, 1

TABLE 1 THEREYNOLDS NUMBER DEPENDENCE OF THE ( 0 ,p ) COUPLING STRENGTH AND THE ERROR or: DECOUPLED APPROXIMATION. RESULTS AREOBTAINED WITH At = 0.01 A1 t = 1; j K IS THE REDUCTION FACTOR OF THE AVERAGED SLIP, zR IS ITSINDEX, E AND 1, ARE THE L 2 - N 0 RELATIVE ~~ ERROROF BOUNDARY VORTICITY AND ITS INDEX, RESPECTIVELY. RE 10 100 1000 10000

1.552 4.226 12.94 37.49

0.359 0.475 0.497 0.508

5.093e-3 1.992e-3 7.104e-4 2.659~-4

0.398 - 0.423 - 0.459 - 0.466

(Reproduced from “Dynamic vorticity conditions: Theoretical analysis and numerical implementation,” J. Z. Wu et al., Copyright 01994 (Inter. .I. Numerical Methods in FluidsJ Reprinted by permission of John Wiley & Sons, Ltd.)

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According to (7.17), I , should be about 0.5. Table 1 clearly confirms this especially as R e becomes higher. The table also presents the L,-norm relative error in oB of the decoupled scheme compared to the fully coupled one using iteration. The index of this error, I,, was defined similar to I , and also had a similar trend. Another indication of the strength of (0,p ) coupling is the number of iterations required to achieve convergence in u , . As expected, more iterations were needed for lower R e in the tests. A test with R e = 9500 showed that (7.17) is valid for smoothly separated flow. As the flow becomes more complicated, the estimate becomes invalid and the error of decoupling grows larger. Amazingly, even at Re = 10, about the lowest Reynolds number of most practical interest, the decoupling error might be considered as still reasonably small. 2. Three-Dimensional Lid-Driven Cavity Flow

In three dimensions, extending the dynamic Neumann condition for vorticity is straightforward. The key issue turns to ensuring a divergenceless incompressible (o,u) field. To this problem Theorem 2 provides a theoretical basis. First, owing to the theorem, there is no need to project o prior to solving u; in fact, the projection can be done at no extra cost by taking the curl of u numerically. Second, due to the orthogonality of the projection, the projected od is the optimum approximation of w in the least square sense. Third, because u satisfies (7.lb), in a d-dimensional space one need only solve d - 1 Poisson equations out of (7.5a) for u, inferring the other component from (7. lb). This enables solving velocity 50% and 30% faster than common o-u methods for d = 2 and 3 , respectively. Note that for d = 2 this saving makes o-u methods identical to the vorticity-stream function methods. Based on Theorem 2, as well as a careful treatment of discretization to preserve the kinematic properties of relevant variables inherent in their differential and continuous counterparts, X. H. Wu et al. (1995) arrived at very efficient finite-difference o-u schemes for two- or three-dimensional flows. The schemes were tested by computing the lid-driven flow in a cavity of unit cube. For the case of steady flow at Re = 1000, the result was in good agreement with that by pseudo-spectral computation. For more complicated unsteady flow at a higher Reynolds number, the result with different schemes are compared in Figure 41. It is seen that the nondivergence-free scheme, Figure 41(b), is not acceptable due to the large error of V . o,although it did not bring appreciable difference at lower Reynolds numbers.

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b

a

C

FIG.41. Three-dimensional unsteady lid-driven cavity flow at Re = 3200, T = 51.2: velocity field projected on the symmetry ( x - y ) plane. In the figure, different schemes for computing fluxes due to convection and stretching, respectively, are compared: QUICK and centered difference, denoted Q-C, which is not divergence free; QUICK and QUICK, denoted Q-Q, which is divergence frcc in the interior of the domain but not on the boundary; and Q-C-P, denoting the application of projection to Q-C. (a) Q-C-P, (b) Q-C, (c) Q-Q (from X. H. Wu et al., 1995).

This section did not intend to review various aspects of vorticity based methods. Rather, we have focused on topics relevant to boundary vorticity dynamics and used the examples to show evidence of its significance. In particular, the robustness of the low-order scheme of J. Z . Wu et al. (1994b) indicates that the key issue in vorticity based methods is indeed the proper boundary conditioning, whereas the fast convergence of itera-

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tion confirms the weakness of boundary ( q p ) coupling at high Re. To both of these, the theory of boundary vorticity dynamics made contribution. We may also mention that, because the prediction or measurement of T , at high Re is very often a troublesome engineering task, the excellent ability of computing T , by vorticity based methods with proper boundary conditions has great potential application. For example, based on given data of a pressure distribution over a complicated surface, obtained by either experiment or some numerical scheme (even an Euler code), one could continue the computation with this method and a little additional effort to obtain the best estimate of 7, within the error range of the pressure data.

VIII. Concluding Remarks We have discussed various aspects of the boundary vorticity dynamics, from fundamental theory to applications, from solid boundary to interface, and from the vorticity creation on a boundary to its reaction to the boundary. The main conclusions can be summarized as follows. 1. The boundary vorticity dynamics focuses on the vorticity creation from, and its reaction to, various fluid boundaries. The theory is derived by applying the tangent surface stress balance and force balance (tangent components of the Navier-Stokes equation) to a rigid or deformable boundary 9, along with the no-slip condition. The former gives the tangent vorticity jump [ p.53 across 9, whereas the latter gives the creation rate of vorticity, or boundary vorticity flux u = v d o / d n , from each side of 9. In any case, the boundary vorticity is always a space-time accumulated effect of the boundary vorticity flux, along with that of advection and diffusion inside the flow domain. Therefore, the boundary vorticity flux u is the primary mechanism responsible to the creation of new vorticity from 9. 2. The specific dominating mechanisms of vorticity creation on 9 depend on the coupling situation of shearing process with other surfaceforce driven dynamic processes. Therefore, the splitting and coupling of these dynamic processes on 3 are the key physical basis for understanding the vorticity creation and its reaction. For a Newtonian fluid, the surface force drives three dynamic processes: shearing, compressing, and surface deformation. In general, these processes appear in both stress balance and force balance and are coupled via viscosity and adherence condition. In

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the force balance one also needs to include the effect of external and inertial body forces. A unified theory of boundary vorticity dynamics is thereby developed for any kind of immiscible material boundaries. 3. On a rigid wall, dB, the surface-deformation process is absent and the coupling occurs between the shearing and compressing processes. The local stress balance on dB gives nothing but the ( p , 7 , ) distribution on d B that has to be obtained by solving the entire flow field. Similarly, the corresponding force balance shows that the vorticity creation is dominated by the pressure gradient and tangent wall acceleration, both being of O(1). The explicit viscous effect of O(Re- i ) exists only in three dimensions, which concentrates on highly local regions of dB, where there is a large surface curvature or strongly rotational 7, field. 4. In contrast, a free-surface S always adjusts its motion and shape, which to the leading order is governed by the inviscid normal-stress balance (pressure and surface tension) without coupling with shearing process. Consequently, unlike the solid wall, where the vorticity is of O(Re+),the tangent vorticity on S is only of O(1) and solely balanced by the tangent components of surface-deformation stress t ~,which in turn is dominated by the inviscidly determined velocity and curvature of S. The normal vorticity is free from the stress-balance condition; rather, it may come from the intersection of the free surface and an external vortex or from the kinematic turning mechanism of internal vorticity to the normal direction. Hence, at low Froude numbers the normal vorticity could be the dominating vortex structure near S. The boundary vorticity flux u, then, appears only in the viscous correction to the inviscid motion and hence is of O(Re- i), dominated by the viscous correction of surface acceleration. The coupling between shearing and compressing processes remains merely at a level of O(Re-’). Other types of boundaries, including flexible solid wall, fluid-fluid interface, and contaminated free surfaces, behave in between the two extremes, t h e rigid wall and clean free surface. 5. Because the coupling of shearing processes and other dynamic processes is Reynolds number dependent and weaker as Re increases, at large Re the theory of boundary vorticity dynamics can be simplified. On a solid wall with an attached boundary layer, the pressure is approximately decoupled from and u , in the sense that p can be determined first from inviscid solution and then the latter determined for a given p distribution. On a free surface, the attached boundary layer is weaker and its equation can be linearized. As Re m, that is, in the Euler limit of viscous flows, these attached boundary layers are further simplified to boundary vortex --f

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sheets, of which a basic difference from a free vortex sheet is that they are still continuously created dynamically by a boundary vorticity flux u, manifesting itself as a jump of tangent acceleration across the sheets. 6. The integrated reaction of the created vorticity from a closed boundary 9 to 99 amounts to various vorticity based total force and moment formulas. This is possible again owing to the viscous coupling of the dynamic processes and the no-slip condition, no matter how weak it is at a large Re. In particular, the total force and moment can be expressed solely in terms of proper vector moments of boundary vorticity fluxes, indicating that the rate of work done in creating the vorticity from B' is Compared with conventional faithfully reflected as the reaction to 9. force and moment formulas based on primary variables, the unique feature of vorticity based formulas is the high concentration of their integrand in local regions of S' for typical configurations of engineering interest. The formulas are, therefore, theoretically irreducible, in which those local stresses that could cancel each other during integration are automatically excluded. Hence, this type of formula is of great potential in applications, especially in hydrodynamic-aerodynamic diagnostiq and optimization of the configurations once combined with experimental measurements or numerical computations. Moreover, because the formulas can clearly identify the key regions for vorticity creation, they also provide a clue for optimizing the configurations via various local means of vorticity-creation control. 7. The theory of boundary vorticity dynamics alone is insufficient for solving a bounded vortical flow problem. Rather, it provides natural and optimal boundary conditions for vorticity based numerical methods. On a solid wall, this condition is of a Neumann type, in terms of the boundary vorticity flux that can be efficiently localized at high Reynolds numbers. On a free surface, the Dirichlet condition seems natural for tangent vorticity, if the surface shape and motion are known from inviscid approximation. 8. Although the basic theory of vorticity dynamics is almbst complete, it is highly desired to further explore its various aspects. First, the theory needs to be extended to flows with strong heat conducting with variable shearing viscosity. Essentially, this implies a close study of the boundary coupling between shearing and thermodynamic processes. Next, the potential applications of the theory have never been fully addressed, in particular those relevant to diagnostics and optimization of configurations, to

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near-boundary turbulence structures and modeling, and to developing vorticity based methods for compressible flows and free-surface flows.

Acknowledgments

We owe much to many of our colleagues, friends, and students in preparing this article. Our thanks first go to Professor H. K. Cheng, who recommended writing the chapter for this series. The editor of the series, Professor T. Y. Wu, has been a very strong source of continuous support, understanding, and encouragement; he also carefully read the early drafts of the manuscript and made many valuable suggestions that greatly improved the content of t h e chapter. The helpful discussions with Professors H. Hornung, H. Y. Ma, H. Yeh, and M. Gharib, and Drs. E. Rood and D. G. Dommermuth, are very appreciated. W. L. Sellers, A. E. Washburn, Professor D. Rockwell, Drs. D. G. Domrnermuth, R. D. Juslin, and R. Handler kindly permitted us to cite their work and provided us their experimental and computational results, unpublished or published, to whom we are also very grateful. We also thank our able graduate assistants B. Ondrusek and J. S. Liu, as well as T. G. Zheng, for their help in preparing the manuscript. Our own work reviewed in this article were supported in part by NASA Langley Research Center under the Grant NAG-1-844 J. M. Wu Research Fund, and by National Science Foundation of China during the first author's visit of the Graduate School of Academia Sinica in the summer of 1994. In particular, we are deeply indebted to P. J. Bobbitt and Dr. R. W. Barnwell of NASA Langley Research Center, without whose support and great insight the exploration of relevant theory and applications would have been impossible.

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ADVANCES I N APPLIED MH'HANICS. VOLUME 32

Some Applications of the Homogenization Theory CHIANG C. ME1 Massachuseti.5 Institute of Technology

JEAN-LOUIS AURIAULT Insiitul de Mecanique de Grenoble. France

and

CHIU-ON NG Massachuscm Institute of Technology

I . Introduction

.......................................

278

I1 . One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . A Differential Equation with Oscillating Coefficients . . . . . . . . . . . B. One-Dimensional Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . C. Typical Procedure of Homogenization Analysis . . . . . . . . . . . . . . .

279 279 283 287

111. Seepage Flow in Rigid Porous Media . . . . . . . . . . . . . . . . . . . . . . . A. Darcy'sLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Uniqueness of the Cell Boundary-Value Problem . . . . . . . . . . . . . C. Properties of Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . D . Numerical Solution of thc Cell Problem . . . . . . . . . . . . . . . . . . . E. Effects of Weak Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. A Spatial Averaging Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . G . Porous Media with Three or More Scales . . . . . . . . . . . . . . . . . .

287 288

IV . Diffusion and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Heat Conduction in a Composite . . . . . . . . . . . . . . . . . . . . . . . . B. Dispersion of Solutc in a Channel Flow . . . . . . . . . . . . . . . . . . . . C. Dispersion in a Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . D . Porous Media with Disparate Diffusivities . . . . . . . . . . . . . . . . . . E . Dispersion in Wave Boundary Layers . . . . . . . . . . . . . . . . . . . . .

309 309 313 319 328 335

277

292 293

295 297

300 303

.

Copyright Q 1990 by Academic I'rcs? I i i c . All right\ of rrproductirm i n any f o r m rescrved . IS13N 0-12-002032.7

278

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

V. Other Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

I. Introduction In engineering mechanics and applied physics one often must know the effective properties of inhomogeneous or composite materials. When the heterogeneity is localized in a few regions, such as a plate with one or a few holes or inclusions, the mathematical problem can often be solved by analytical or numerical means. When the heterogeneity spreads over a large number of regions, a detailed analytical or even numerical approach becomes infeasible. A natural idea is to gloss over the rapid variations of the heterogeneities and replace the composite by an equivalent homogeneous medium whose behavior over a macroscopic scale represents the averaged behavior of the composite. There are two ways to achieve this goal. One is the phenomenological approach, in which one establishes directly the equations governing the macroscopic behavior without inquiring about the physics on the microscale. The form and coefficients of the constitutive laws must be obtained by experiment. Another more basic route is to start from the microscale of heterogeneities and deduce the effective equations on the microscale by a rational process of averaging. The effective constitutive laws are derived and the material coefficients are calculated. The continuum theories for fluids or solids belong to the first kind, while the kinetic theory of gases to the second. There are several types of averaging in applied mechanics: the statistical averaging (Kroner, 19861, the self-consistent method (Zaoui, 19871, all methods using the average theorem (Nigmalutin, 1981; Bedford and Druhmeller, 1983; Howes and Whitaker, 1985; Gilbert, 1990), and the asymptotic method of homogenization. In this review we discuss the last type, which is characterized by the mathematical techniques of multiple scales and is especially useful for materials with a periodic microstructure. The use of the multiple-scale expansions as a systematic tool of averaging for problems other than wave propagation can be traced to the earlier works by Sanchez-Palencia (1974) in France, Keller (1980) in the United States and Bakhvalov (1975) in Russia. There now exist several mathemati-

Some Applicaiions of the Homogenization Theoiy

279

cal treatises of the method (Bensoussan et al., 1978; Sanchez-Palencia, 1980; Ene and PoliSevski, 1987; Bakhvalov and Panasenko, 1989). In our opinion the level of mathematics used in these books may appear forbidding to many engineering readers. In this review we shall explain the idea of this powerful method in more physical and less abstract terms, discuss its advantages and present several recent applications. Although differing in technical details, the basic idea of the theory of homogenization has been employed for a long time. In the theory of wave propagation over slowly varying media, the familiar ray theory (geometrical optics approximation) is one such example. There one employs the method of multiple scales to average over the locally periodic waves and find the slow variation of the wave envelope. The procedure is rarely known as homogenization, however. This vast topic, which covers both linear and nonlinear waves, is excluded from the present survey; only a simple example in Section I1 is given to show the affinity here.

11. One-Dimensional Examples

We begin our introduction of the basic idea of homogenization by one-dimensional examples described by differential equations with fast oscillating coefficients.

A. A DIFFERENTIAL EQUArlON

WITH OSCILLATING COEFFICIENTS

Consider an ordinary differential equation: d

du

=O

O<X
(2.1)

where E varies periodically in x, with possible discontinuities. Across a discontinuity, the following jump conditions are imposed: du [u]

=

0,

[Ez]

=

0

(2.2)

where the brackets signify the jump. Equation (2.1) may describe the static deformation of an elastic laminate consisting of density layers of different materials, where u represents the displacement and E represents Young’s modulus. The boundary conditions (2.2) represent the continuity of displacement and stress, respectively. Alternatively, (2.1) may also describe

280

Chiang C. Mei,Jean-Louis Auriault, and Chiu-on Ng

the seepage flow in periodically layered porous medium. Then u stands for the pore pressure and E the hydraulic conductivity, while E(du/du) stands for the rate of fluid flux. For this simple differential equation, an exact solution is, of course, possible. Suppose that one is more interested in the averaged variation over a region much greater than the typical period and less with the detailed variation in each layer. Can one bypass the details to find an equation governing the variations on the global scale L? The important characteristics of this problem is the existence of two vastly different length scales: the microscale I, which characterizes the typical layer thickness, and the macroscale L , which characterizes the global variations of external forcing or boundary data. The perturbation method of multiple scales is particularly suited for problems involving contrasting scales. Let E = 1/L with E << 1. We introduce two coordinates defined by and x' = E X (2.3) which will be treated as two independent variables. The unknown displacement is now represented in the form of a power series in E : x

u

=

uo

+ E U , + E * U 2 + ...

(2.4)

where u l , i = 0,1,2,. . . , are functions of both x and x ' . The original derivative then becomes, according to the chain rule, d

a

- = -

du

a

+€Y dx dx

It follows from (2.1) that

Equating the coefficient of each power of E to 0 leads to a sequence of perturbation equations. At the order O ( E ( ' we ) , get

which governs the microscale variation of u,), subject to the jump conditions

Some Applications o i the Homogenization Theory

28 1

Because the microstructure is assumed to be periodic on the 1-scale, uo should be likewise. The general solution to the homogeneous equation ( 5 ) is

where A , ( x ’ ) and A 2 ( x ‘ )are integration constants. To ensure periodicity over the distance I, A , ( x ’ ) must vanish, implying that the leading-order displacement depends only on t h e macroscale uo = u&‘)

=

(2.7)

A,(x’)

At the next order O ( E ) the , perturbation equation is d

(2.8) with the jump conditions (2.9) where use has been made of (2.7). In addition, u I must also be 1-periodic. Equation (2.8) is an inhomogeneous equation for u , . Linearity suggests that u1 may be sought in the form (2.10) Q ( x , x ‘ ) must then satisfy

(2.11) because of (2.8). From the jump conditions we get

[Ql

=

0,

[E(l+

g)] o

(2.12)

=

In addition, Q must be 1-periodic. Thus, Q is the solution of a canonical problem in a typical period. Successive integrations yield JQ

dx

-

Di

E

1

Q = -x+D,

(2.13)

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Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

where D , ( x ' ) and D,(x') are constants of integration. Because Q must be I-periodic, it is necessary that

Mathematically, this equation is the solvability condition for the inhomogeneous problem for Q or u , , given that the homogeneous problem has a nontrivial solution u,. Let us denote the harmonic mean of E by E, (2.14) It follows from (2.10) and (2.14) that dU,

- - -

du,,

E, du,

+7 E dx

-~

dx'

dX

(2.15)

Next we proceed to the perturbation equation at the order O ( E * ) :

du, +-d x i E dx y d

)

i

d du, + y Edx dx

(2.16)

)

=O

which is again an inhomogeneous equation for u 2 , similar to (2.8) for u1, with the jump conditions

[ u 2 ]= 0 ,

2 + 2)]

[E(x)(

=

Furthermore, u2 is 1-periodic. Because from (2.15)

(2.16) may be written

d

du,

+-d x ( E-d x

)

=O

0

(2.17)

Some Applications of the Homogenization Theory

283

By virtue of periodicity, the average of the preceding equation over an 1 period is (2.18)

This equation governs the macroscale variation of the mean displacement, while E, is the effective Young’s modulus. Subject to further boundary conditions on the macroscale, u o can be solved. Afterward one may find the microscale variation from (2.10) and (2.14). Let the 1 period be composed of two distinct layers. Young’s moduli and thicknesses are, respectively, ( E ,, E 2 ) and (1 - nl, nl). From (2.15)

In particular, if n or 1 - n is not too small compared to unity, then

These results can easily be verified by solving the problem in each layer and matching u and E(du/du) on the interface. B. ONE-DIMENSIONAL ELASTODYNAMICS Let us consider some variations of (2.1) and examine the following partial differential equation, which describes one-dimensional oscillations of an elastic body: d d,(

rlu

E-

dx)

=p-

d *U

dt2

(2.19)

Let E and p be periodic in x with the same periodic 1. Let the time scale be 2m-/w, where o is the characteristic frequency. On the microscale the importance of inertia relative to stress force is measured by

(2.20)

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Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

where A=

Ei

is the length of the elastic wave of frequency w . There are several physical possibilities.

1. Lowfrequency, or (1 / A)’

=

O(e2)

The motion is quasi-static over a microscale. It was shown in the previous section that the effective equation of the leading-order average displacement is derived by considering the stress term alone up to O(E’). Hence, the inertial term is of importance to the macroscale motion only if the preceding length ratio is on the order of O(E*>.In this case, we rewrite the governing equation as d2U

(2.21) By following the two-scale analysis we can readily show that the effective equation is (2.22) In the limiting case of l / A << 0 ( e 2 ) the quasi-static result (18) is recovered. For slightly higher frequency, say (l/A)’ = O(E), straightforward multiple-scale analysis shows that the U ( ~ ) ( X ‘ )must vanish identically. Therefore, at the leading order, the short and long scales cannot be separated and one cannot get an equation describing the long-scale motion. Therefore, (2.19) is said to be nonhomogenizable. Physically, the phenomenon is complicated by wave scattering on the short scale. However, if the magnitude of inhomogeneity is sufficiently small, then scale separation is still possible, as shown in the next example of Bragg resonance. 2. ffighfiequency,or I / A

=

O(1)

Let us consider the accumulated effects of small and periodic inhomogeneities by taking (2.19) with

E

=

E,(1

+ ED cos K x )

(2.23)

285

Some Applications of the Homogenization Theory

where K1 and D are of order unity. Diffraction of shallow water waves by periodic sandbars can be described by this equation. The three-dimensional counterpart of this problem occurs in X-ray diffraction of crystals. If we let u = u g + E U ~ ... , the crudest solution is easily found to be

+

A ug

where

+*

= - eikx-iwt

2

(2.24)

* signifies the complex conjugate of the preceding term, and (2.25)

At the next order the governing equation is d

(2.26)

-E,D 2

d

( e ~ K+~e - ~ K t

elkl-lwr

ikA* -

ikx+ iwt

2

dx

Clearly, when (2.27)

K = k2k some of the forcing terms on the right will be of the form i(kxe ? i(kx+ w l ) OJt)

These terms are homogeneous solutions to the wave equation and represent incident and reflected waves, which would force unbounded resonance of u l . In particular, the reflected wave in E U ~would grow as E X grows and become comparable to u g after a distance E X = 0(1), rendering the perturbation series invalid. Equation (2.27) is the well-known condition for Bragg resonance. To render the solution uniformly valid for all x, we introduce slow variables x' = E X , t' = c t and assume that u = U " ( X , x';t , t ' )

+ E U , ( X , x';t , t ' ) + ...

(2.28)

After making the changes d

d

d

d

d

d

-+-+€I ---++El dx dt dt dt dx dx

(2.29)

Chiang C. Mei, Jean-Louis Aunault, and Chiu-on Ng

286

and substituting (2.28) and (2.29) into (2.21), we get (2.30) at O(1). Anticipating strong but finite reflection, we take the solution to be A

u o = -e i k x - i w t

+

*

B

+ -e-~kx-iwr

2

2

+*

(2.31)

At the order O ( E ) we , have d

dZU, =

-2E0-

d2U"

d x dx'

+ 2p- dt dt'

To avoid unbounded resonance of u l , that is, to make sure that the problem for u1 is solvable, we equate to 0 the coefficients of forcing terms proportional to e i k x i w t . The following equations are obtained: dt'

dA c 7 dx

dB

dB

dA

-+

ikcD = --

4

i kcD

--c-=-dt' dx'

4

B

(2.33)

A

(2.34)

where c = w / k denotes the phase speed. These equations govern the macroscale variation of the envelopes of the incident and reflected waves and can be combined to give the Hein-Gordon equation: d 2A ~dtI2

d2A kcD c 2d X-f 2+ ( ~ )

A=O

(2.35)

Some Applications of the Homogenization Theoly

287

With suitable initial and boundary conditions one finds the slow variation of these wave envelopes, hence the global features of wave motion. For applications of homogenization theory to Bragg resonance, see Mei (1985) for water waves over periodic sandbars and Naciri and Mei (1993) for sound waves beneath surface waves over a shallow sea.

c. TYPICAL

PROCEDURE OF

HOMOGENIZATION ANALYSIS

These elementary examples demonstrate the typical steps of the homogenization theory that can be extended to many problems with a periodic structure on the microscale. Let us summarize these steps as follows: 1. Identify the micro- and macroscales. 2. Introduce multiple-scale variables and expansions and deduce cell boundary-value problems at successive orders. The leading-order cell problem is homogeneous; either the solution itself or the coefficient of the homogeneous solution are indeterminate and independent of the microscale coordinates. 3. Use linearity and express the next-order solution in terms of the leading-order solution and deduce an inhomogeneous canonical cell problem. 4. Require the solvability of the inhomogeneous cell problem. 5. Get the equation governing the evolution of the leading-order solution (or the coefficient of the homogeneous solution) and calculate the constitutive coefficients from the solution of a canonical cell problem.

It should be emphasized that the identification of scales is a consideration of physics and is crucial to the success of the mathematical theory.

111. Seepage Flow in Rigid Porous Media

Seepage through a porous media is one of the first examples to which the method of homogenization was applied (Sanchez-Palencia, 1974; Keller, 1980). It is a good example to explain the role of physical scales and the mathematical procedure for three-dimensional problems. Furthermore, it can be used to illustrate the derivation of many properties of the constitutive coefficients. In this section we begin to establish the classical results for two-scale media and then to extend to three scales. Numerical compu-

288

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

tations based on the solution of canonical cell problems will also be discussed.

A. DARCY'S LAW Let a rigid porous medium be saturated by an incompressible Newtonian fluid of constant density. Driven by a steady ambient pressure gradient, the steady flow velocity ui and pressure p in the pores are governed by the Navier-Stokes equations

-=o

(3.1)

ax1

(3.2)

On the wetted surface of the solid matrix uj=o

r, there is no slip,

xj€r

(3.3)

For slow flows, the two terms on the right-hand side of (3.2), representing the pressure gradient and the viscous force, must be dominant. Then both the pore pressure and the flow velocity vary according to two scales: the local or microscale 1 characteristic of the size of pores and grains, and the global or macroscale L imposed by the global pressure gradient. We assume that l / L << 1. Equating the order of magnitudes of the global pressure gradient P / L to the local viscous stress, we get

which defines the velocity scale U. Let us normalize the space coordinates by the local scales and the unknowns according to the estimates just found, (3.4) where the primed quantities are dimensionless and p here stands for the change in pressure, because it appears only in differential form. Equation (3.2) becomes, formally,

Some Applications of the Homogenization Theory

289

where Re =

p12P E-

(3.6)

ru2

is just the Reynolds number. The dimensionless continuity equation remains in the form of (3.1) and need not be repeated. We assume the Reynolds number to be no greater than O ( E ) that ; is, (3.7) Note that the pressure gradient term appears formally dominant in (3.9, because xi is normalized by the micro-length scale I in every term. From here on we shall return to dimensional variables but retain the ordering symbol E (3.8) The boundary condition on the wetted surface r of the pores remains (3.3). Let us assume that the geometry of the porous matrix is periodic on the microscale, as depicted in Figure 1, although the structure may still change slowly over the macroscale L. Each periodic cell 1R is a rectangular box of dimension O ( 0 . We then expect u , and p to be spatially periodic from cell to cell. I I

I

I I

I I

-

I I I I I

I

I I I

FIG.1. A typical cell on the microscale.

290

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

We now introduce the multiple-scale coordinates x i ,x ; perturbation expansions

= EX,,

and the

where d J )p,( J )are functions of x , and x i . From (3.1) we get at the first two orders O ( e O )and O ( E ) : duj"'

-=o

(3.11)

dx,

dujU

du(W

- d+x -,

ax;

dUj2'

du'l'

dx,

ax;

-+-=o

=o

(3.12)

(3.13)

Similarly, we get from (3.8) (3.14)

(3.15)

(3.16)

r, the velocity vanishes; hence, u(0) = u (1) , = u (2) , = ... = o X , ~ r

On the wetted interfaces I

(3.17)

In a typical R cell the flow must be periodic, up,

u p , u y ,. . .

p(o)

p(l)

,

p'2'

)...

are

R-periodic (3.18)

From (3.14) it is clear that p(")= p(x:)

(3.19)

Some Applications of the Homogenization Theoiy

291

Because of the linearity of (3.15), u!”’ and p(’) can be formally represented by

where j7(’)(x;) is independent of x , . It then follows from (3.111, (3.151, (3.17), and (3.18) that K , , ( x , ,x i ) and A , ( x , ,x : ) must satisfy d Kii -- - 0 (3.21) axi

(3.22) where

K,,

=

0 on

K , , ,A, are

62-periodic

(3.23) (3.24)

( A , ) can be set to 0. Equations (3.21) to (3.24) define a canonical Stokes’s flow boundary-value problem in an R cell, which can, in principle, be solved numerically for any prescribed microstructure. Defining the average over an R cell by

(3.25) where R, is the fluid volume inside the R cell, we get from (3.20) (3.26) (3.27) where n denotes the porosity that is the ratio of fluid volume in the cell to the total cell volume (p(l))

= nij(l)

(3.28) Equation (3.26) is just the celebrated law of Darcy with ( K i , ) being the hydraulic conductivity. The R-average of (3.13) gives

292

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

The interchange of volume integration with respect to x i and differentiation with respect to x' is allowable when n = constant. Otherwise the same is justifiable by virtue of the spatial averaging theorem to be discussed in Section 1II.F. By using the Gauss theorem and the boundary conditions, we see that the volume integral vanishes; hence, (3.29) This result implies, in turn, that (3.30) Equations (3.26) and (3.29) or (3.30) govern the seepage flow in a rigid porous medium on the macroscale. Theoretical derivation in the present manner was first given by Sanchez-Palencia (19741, Bensoussan ef al. (19781, and Keller (1980). If the medium is isotropic and homogeneous on the L scale, we have

(Kf,) = K6,,

(3.31)

where K is a scalar constant. It follows from (3.30) that

a2

p

--

ax;

ax;

-0

(3.32)

With proper boundary conditions on the macroscale, p"') can then be found. It may be pointed out that assumption (3.7) renders the perturbations equations linear at each order. Had the Reynolds number been assumed to be finite, that is, Re = 0(1), then the convective inertia would be on the order of O ( E )and (3.15) would be replaced by (3.33) Together with (3.11) the cell problem is fully nonlinear (Ene and SanchezPalencia, 1975). Equations (3.30) and Darcy's law (3.26) no longer hold. B. UNIQUENESS OF

THE

CELL BOUNDARY-VALUE PROBLEM

We first settle the mathematical issue that the solution to the cell problem is unique; that is, if ( K : ] ,A',) and (K:, , A:) are solutions to the

Some Applications of the Homogenization The0y

293

same cell boundary problem, then the values of K ' must be the same and the two expressions A, can differ by only a constant. Let us consider their differences. k,, = K:] - KYl and 2I = A 'I -A" I ' which must be the solution to the homogeneous cell problem, defined by (3.21) to (3.24) with 0 on the right of (3.22); that is, (3.34) Taking the scalar product of K,] and (3.34)' we get, after partial integration,

If the entire equation is integrated over a,, the first two volume integrals can be turned to surface integrals by Gauss's theorem and must vanish because of the boundary conditions; therefore, do= 0

-

This condition is possible if and only if

K,] = 0

vx,

E

a,

can at most be a function of xi. The From (3.34), it is obvious that uniqueness of K,, and A, is proven. C . PROPERTIES OF HYDRAULIC CONDUCTIVITY One of the advantages of the homogenization theory is that certain important physical properties of the effective coefficients can be proven from the cell boundary-value problems. Two properties will be shown in this section: symmetry and positiveness. Let us form the scalar product of K;,, and (3.22)

294

Chiang C. Mei,Jean-Louis Auriault, and Chiu-on Ng

Taking the volume average over R and integrating by parts, we get

By Gauss's theorem and the boundary conditions, the first two integrals vanish; hence,

Since the left-hand side is unaltered if the subscripts j and p are interchanged, we must have ( K c p >= ( K , , , )

(3.35)

which states that the permeability tensor must be symmetric. Next let us consider the balance of mechanical energy by taking the scalar product of u!') and (3.15):

Taking the R average and integrating by parts, we obtain

The first two integrals on the right vanish after using Gauss's theorem and the boundary conditions, although the last integral is positive-definite. Applying Darcy's law we get (3.36)

Because the pressure gradient can be arbitrary, the matrix ( Ki,)must be

295

Some Applications of the Homogenization Theory

positive-definite. In particular, each diagonal term of the conductivity matrix must be positive-definite.

D. NUMERICAL SOLUTION

OF THE CELL PROBLEM

There are many more theoretical papers applying the method of homogenization to a large variety of physical problems than there are quantitative solutions of the cell problems. The latter, however, is ultimately necessary to complete the scientific task. A numerical solution of the Stokes’s flow through a bed of uniform spheres in cubic packing has been described by Snyder and Stewart (1966) using a Galerkin method with trigonometric series as trial functions (see also Sorensen and Stewart, 1974). Later, Zick and Homsy (1980) applied the technique of integral equations and obtained more accurate solutions for six different arrays of uniform spheres. The cases of contacting spheres are taken as special limits, corresponding to just six different values of porosity. As an alternative three-dimensional microstructure, Lee, Sun and Mei (1995) have chosen the Wigner-Seitz grain, which is a polyhedron with 14 sides, shown in Figure 2(a). For a cubically packed array, each grain is in contact with six neighbors and the porosity may take any value from 1/6 to 5/6 by varying a length parameter. At the lower limit ( n = 1/6), communication between adjacent pores is no longer possible (Figure 2(b)); beyond the higher limit ( n = 5/61, the grains lose contact and are suspended in fluid (Figure 2(c)). The cell is a unit cube containing just one grain, and the cell problem is solved by the finite-element method. First let us recast the Stokes’s problem in the cell as a variational principle; that is, its solution is

i c

a n

Y

b

C

FIG.2. (a) A Wigner-Seitz grain in cubic array; (b) 1/8 of a grain at the smallest porosity f; ( c ) 1/8 of a grain at the largest porosity n = i.

=

296

Chiang C. Mei, Jean-Louis Aunault, and Chiu-on Ng

equivalent to the extremization of the following functional:

We can verify the stationarity by taking the first variation and partial integration to get

By Gauss's theorem, the first two volume integrals can be transformed to surface integrals

where dR, is the boundary on R,. Both integrals vanish by virtue of the boundary conditions and the periodicity. If K , , and A , are the solution of the cell boundary-value problem, the remaining volume integrals also vanish. Hence, the first variation of J is 0 and J is stationary. Conversely, because 6K,, and 6A, are arbitrary, the vanishing of 6 J also implies the boundary-value problem. Thus the Stokes's problem in the R cell is equivalent to the stationarity of the functional J. Because of the geometrical symmetry of the Wigner-Seitz grain, it is necessary to consider only ( K , , ) induced by the unit pressure gradient in the direction of xl. Finite elements are used to discretize 0,. A matrix equation for the nodal unknowns is obtained by extremizing J and is solved on a computer. Many one-dimensional experiments have been performed in the past for nearly uniform spheres, powders, and natural sand with a considerable size variation and for a wide range of Reynolds numbers (0 < Re < O(10)). Based on measured data, empirical formulas have been proposed. The best known of such formulas is by Kozeny and Carman (Carman, 1937),

(3.38)

Some Applications of the Homogenization Theory

'

.

.

'

'

Wigner-Seitz

'

297

"

'

'

"

'

.

..."I

.

.

.

. , .

.

.

Grain

K11

lo-=-. 0.2

.

.

I

0.4

.

.

, 0.6

0.8

1.o

Porosity n FIG.3. Comparison between theory and the empirical Kozeny-Carman formula.

where 2 is the ratio of volume to surface area of the grain, a length characterizing both the size and shape of the grain. For the Wigner-Seitz particle, the value of 3 can be calculated and put in the Kozeny-Carman formula; the empirical result is compared with the finite-element calculations in Figure 3. Despite the geometrical idealization in the theory, the agreement is excellent over the porosity range of 0.35 < n < 0.66, where most of the measurements were recorded. Outside of this range, the empirical formula is only an extrapolation of measured data. It should be pointed out that the experimental data may deviate from the empirical formula by 10 to 2096, due likely to the irregularity in shape and size variation of the specimen tested (Carman, 1937). The variation of the normalized conductivity, defined by ( K , , >p/12with respect to porosity n, is plotted in Figure 3.

E. EFFECTS OF WEAKINERTIA Under the assumption that Re = O ( E )<< 1, the effect of weak inertia can be found at the next order of perturbation analysis (Mei and Auriault, 1991). Let us combine (3.20) and (3.12) with (3.16) to get

298

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

and

Because the operators on the left are linear, we may substitute

and

into (3.39) and (3.40). The new coefficient tensors must then satisfy the following governing equations: (3.43) (3.44)

(3.45)

(3.47) (3.48)

Some Applications of the Homogenization Theory

299

Similar substitution into the boundary conditions yields

L r.l.k

=

0,

L c J kB,, , ;

L‘,,~= Kl!j = 0 on

L’,,, , B;, ;

l7

K:J, A’, R-periodic

(3.49) (3.50)

Without loss of generality, we may take

which can be simplified to ( P ( ~ )=) 0, since these inhomogeneous cell problems for L L l k , Blk , and B;k can be solved numerically as in Section 1II.D. Afterward the f l averages may be taken to give

(3.51)

( p‘2’)

= np(2’

(3.52)

The R average of (3.39) gives (3.53) which implies the governing equation for the first-order correction to the mean pore pressure p“’:

Once p(O) is known, p ( l ) is found as the correction to the mean pressure due to inertia. This inhomogeneous equation must be solved under appropriate global boundary conditions. In general, (3.51) implies that the correction of Darcy’s law for weak nonlinearity is not only quadratic in the velocity but also dependent on the velocity gradient.

300

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

In the special case where the medium, though not the flow, is isotropic and homogeneous with respect to the macroscale, the forcing terms in (3.44) to (3.48) do not depend on x:. It follows that the effective coefficients are independent of x:. However, according to the theory of Cartesian tensors, ( L l , k ) and (L'[]k) must be linear combinations of the permutation tensor which has the property that

Similarly, the second forcing term in (3.54) also vanishes; hence, p"' may be absorbed by p") and set to 0. This implies, finally, that

(up)=0

(3 .57)

Therefore, the nonlinear corrections for convective inertia to Darcy's law are at most a third-order effect. This result is indeed consistent with existing experiments, all of which show that the linear law of Darcy can be good even up to Re = O(1) (Kovacs, 1981). This theoretical deduction has been experimentally confirmed for a special microstructure by Rasoloarijaona and Auriault (1994).

F. A SPATIALAVERAGING THEOREM In the derivation of the effective equations, a frequent operation is to interchange differentiation with respect to the macroscale variable and volume averaging over a unit cell; hence, one needs to know the relation between

Some Applications of the Homogenization Theory

301

The interchange of differentiation and integration is a trivial matter if R, is uniform with respect to the macroscale coordinates. However, when R, or, equivalently, the porosity n = lClfl/lCll varies in xi, the two operations do not commute. Greater care is needed and a theorem for spatial averaging is derived next, in the manner of Gray and Lee (1977). In a unit cell R, we let the fluid-solid interface r be described by the equation F ( x , x ' ) = 0, the fluid domain R, by F > 0, and the solid domain by F < 0. The cell volume of R is assumed to be spatially invariant but the fluid volume may vary with respect to x'. An integral of any vector f(x,x') over fl,(x'> can be replaced by an integral over R with the help of the Heaviside step function H ( F ) :

f, diZ

=

/fiH(F)d R n

(3.58)

where

H(F)=

( 01

if if

F < 0 in solid F > 0 in fluid

(3.59)

After changing the integration domain to R, we may now proceed as follows:

(3.60)

where a, is a thin shell enclosing the interface r with a thickness 2.5, where E << 1, and R; = 0, n RE.Although the first RE integral in the square brackets vanishes as E -, 0, the second one requires further manipulation. First note that

302

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

where 6 ( F ) is the Dirac delta function of F. Hence,

(3.62)

where 5 is the normal distance from dent variable from f to F by dF

=

(or F

=

0). Changing the indepen-

1VF1 d (

(3.63)

we can write the last integral as

Putting this result into (3.62) and then (3.60) and letting

E +

0, we obtain

(3.65) or, in terms of the averaging symbols, (3.66) This result is a variant of the averaging theorem of Slattery (1967), Howes and Whitaker (1985), and Whitaker (1985, 1986), now expressed specially for two-scale problems. As an immediate application, we let f, = u , , then the surface integral vanishes on account of the no-slip boundary condition on r. In this case, volume-averaging and macroscale differentiation commute. The result ensures that Darcy’s law and the effective equation for seepage flow in Section III.A, that is, (3.26) and (3.30), remain valid in an inhomogeneous medium where the porosity, hence, ( K , , ) , varies on the macroscale. The significance of the theorem is less trivial as will be illustrated later in Section IV.

Some Applications of the Homogenization The0y G.

POROUS

303

MEDIAWITH THREEOR MORE SCALES

Stimulated by environmental concerns of solute transport, much recent attention in ground water mechanics has been focused on the effects of soil heterogeneity over spatial scales much greater than the dimensions in the laboratory. Specifically, the aquifer depth O(10 m) is usually much smaller than the geological formation depth O(100 m), which is in turn much smaller than the regional size 0 ( 1 10 km). Because it is impractical to collect comprehensive data on all scales, statistical models have been constructed to schematize the seepage problem, so that, with a finite number of parameters (to be supplied by field measurements) certain statistical information on the largest (regional) scale can be predicted (Dagan, 1987). In these stochastic theories, Darcy’s law is taken as the starting basis, and permeability is regarded as a given random quantity. Assumptions such as stationarity and small departure from the statistical mean are often added to enable analytical progress. To model natural materials as periodically structured porous media is, of course, crude. Nevertheless, one may avoid empiricism by constructing a theory from the microscale. Indeed, the formalism of multiple scales is equally powerful for problems with three or more contrasting scales. An idealized model of three-scale soil is a stratum consisting of periodically alternating layers of sand and clay. Then I corresponds to the granular size, I‘ to the depth of a sand-clay layer period, and I” to the global scale, with 1 << I‘ -=c1”. The homogenization theory can be achieved in two equivalent ways. One way is to break the homogenization process into two steps: obtain the mesoscale equations from the microscale and then obtain the macroscale equations from the mesoscale results. Another way is to apply the method of multiple scales with three levels of fast and slow variables to the basic equations valid for all scales (Mei and Auriault, 1989). Let us demonstrate the second alternative. The key is to recognize that the driving pressure gradient must be O(P/I”); that is, the characteristic pressure gradient over the global scale. The proper normalization is as follows:

-

(3.67)

304

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

The dimensionless Navier-Stokes equations are (3.68) and

J P*

d2UT

ax:

ax; ax;

- - - E2

- E*

Reu*-

JUT

ax;

(3.69)

where c 2 = 1/1"

and

Re

=

pP12/p2.

(3.70)

We shall assume the Reynolds number to be small: Re = O ( E ) .The results to obtained are valid for R 5 O ( E ) . Let us return to physical variables and retain the ordering symbols. Thus the dimensional Navier-Stokes equations are written as (3.71) and JU;

- E 3 u--

' ax,

(3.72)

x:'

€2X,

(3.73)

The following multiple-scale vectors x,,

x; = EX,,

=

and the perturbation expansions

are introduced, where p ' k ) and

depend on x,, x:, and x:' . Substituting

Some Applications of the Homogenization Theory

305

(3.74) and (3.75) into (3.71) and (3.72), we find at successive orders (3.76)

(3.77)

(3.78) and (3.79) dp'O'

cjrp'l,

-+ax;

dp'0'

dp('1

ax,

The boundary conditions are U(o) = U ( l ) 1

u! '),

u:')

I

=

ap@'

+ - -ax,

ax:l+ ax:

(2' =

=o -

(330)

pV2ujO)

... = 0 on

r

... p ( ( ) ) , p ( ' ) are R- and n'-periodic

Equations (3.79) and (3.80) arc satisfied by p(o) = p'o'(x" ) p"' =p("(x',x'')

(3.81)

(3.82) (3.83) (3.84)

whose macroscale variations we now seek. Linearity of (3.81) suggests the formal solution: (3.85)

(3.86) Then the second-order tensor - K l , ( x , , x ; , x ; ) and the vector A , ( x , , xi, x;) must satisfy (3.21) to (3.24) in an (1 cell. Once K,, and A , are solved, we find the R average of (3.85):

306

Chiang C. Mei, Jean-Louis Aunault, and Chiu-on Ng

(3.88) is the permeability tensor on the 1' scale. Of course Kil is symmetric and positive-definite, as shown before (Ene and Sanchez-Palencia, 1975). By taking the fl average of (3.77) and using Gauss's theorem, the spatial averaging theorem (3.66), and the boundary conditions, we find

This equation immediately suggests a representation of p ( ' ) by (3.90) where A J ( x ' ,x " ) is another unknown vector that is governed by (3.91) and must be a'-periodic. Equation (3.91) is the mesoscale canonical boundary-value problem for AJ in an fl' cell, which can also be solved numerically in general. In the particular case of periodical layers the mesoscale cell problem is one-dimensional and can be solved analytically (see Mei and Auriault, 1989). Afterward we get from (3.85) the microscale velocity (3.92) and the mesoscale velocity (3.93) Defining the meso-cell average by

we find the macroscale velocity (3.94)

Some Applications o f the Homogenization Theory

307

where (3.95) Finally, the twofold equation for p"':

(a,fl') average of (3.78) gives the effective governing (3.96)

on the macroscale. Note that the macroscale equation is of the same form as the mesoscale equation, which is, however, different from the microscale equation. Clearly, the theory can be extended in principle to a medium with n-scale periodicity. We shall now prove that the macroscale conductivity tensor K:; is symmetric and positive-definite.

1. Symmetry Because K:, is symmetric and positive-definite, so is ( K : , ) n , .One need only to prove that (3.97) After changing k to m, let us multiply both sides of (3.91) by A; and then integrate over an 0' cell. By partial integration, the left side gives (3.98) and the right side becomes (3.99) after invoking the symmetry of K L l j . Because (3.98) is symmetric with respect to the interchange of m , n , then (3.99) is symmetric likewise. Hence, (3.97) is verified and

K k n = K:m

(3.100)

308

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng 2. Positir ieness

Let us examine the energy dissipation rate in a mesoscale cell and rewrite (3.81) as

where (3.102) Multiplying both sides of (3.101) by u~"'and taking the twofold average of the result, we obtain, after partial integration and using the symmetry of elk and the boundary condition,

Because of (3.89) we have

after using (3.91). It follows that the second line on the right of (3.103) vanishes. Because the left-hand side of (3.103) is the positive rate of energy

Some Applications of the Homogenization Theory

309

dissipation, we conclude that (3.106) is positive-definite.

IV. Diffusion and Dispersion In this section we shall begin with the classical problem of predicting the heat diffusivity of a composite medium. Then the physically more complex problem of dispersion (flow-enhanced diffusion) in porous media will be discussed. A model problem of environmental interest is the theory of vapor extraction for removing volatile organic compounds from the pores of unsaturated soil. Homogenization provides a sound theoretical basis for making reasonable predictions without excessive empirical elements. Finally, we shall give a recent example of dispersion in wave boundary layers where homogenization is carried out with respect to time instead of space. A. HEAT C0NI)UCTION

IN A COMPOSITE

Consider heat conduction in a composite medium such as fiber-reinforced plastic. Let the subscripts a = 1,2 distinguish the two components and T,, p , , c,, and K , denote, respectively, the temperature, density, specific heat, and heat conductivity in the a component of the composite. In each component, the heat equation applies (4.1) For simplicity, we assume the two component materials to be isotropic and their corresponding coefficients p,, c,, and K , to be comparable. We allow K , to vary slowly over the global scale 1'. On the interface r, the temperatures and the rates of heat flux must match,

T,

=

T2

(4.2)

(4.3)

310

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

where n = Ini) denotes the unit normal vector pointing from phase 1 to phase 2. In principle, there are two time scales

l2 K/

and

PC

L2

K/W

Let the transfer of heat be caused either by the initial condition or by boundary constraints characterized by the much larger scale 1'. Then only the latter is relevant I 2

L

When the variables are normalized as follows X*

=

x/l,

T*

=

T/AT

t*

=t/T

(4.5)

the heat equation becomes (4.6) Let us assume that the contrast of length scales is large so that 1/L = E < < 1. The left-hand side is formally smaller than the right by a factor O(E').The boundary conditions (4.2) and (4.3) preserve their forms under the normalization. Returning to physical coordinates but preserving the order symbols, we write dT,

d

at

axi

(4.7)

Upon introducing the fast and slow coordinates xi and xi = e x i and the usual series expansions for T,, we get the following sequence of problems in an Cl cell. At the order O ( E " ) , (4.8)

(4.10)

T, : R-periodic

(4.11)

Some Applications o j the Homogenization Theory

311

It is evident that this homogeneous boundary-value problem in the R cell admits a simple solution that is independent of the fast coordinates

T,'lj)= T")) 2 = T'yx' t) I )

(4.12)

To prove this obvious result, we multiply (4.8) by T'()) and integrate the result over a,to get

Because of R-periodicity, only the interface r contributes to this surface integral. Now we add the integrals for both components and apply the boundary conditions on r. If use is made of the fact that n1 = - n 2 = n, we get (4.13)

Now for this equality to hold, each gradient in the preceding integrand must vanish; hence, (4.12) is proven. With the last result, the governing equations at the order O ( E )can be simplified to the following linear system: (4.14)

T,!') : &periodic

(4.17)

The boundary-value problem for T ' ' ) is linear and inhomogeneous; hence, the following representation can be proposed: (4.18)

312

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

Then b, must satisfy (4.19) where ajj is the identity tensor of the second rank. The implied boundary conditions are bl, = b2,

x, E

r

b, : LR-periodic

(4.20)

(4.22)

This problem must, in general, be solved numerically. At the next order O ( E * )the , governing equations are

TL2): R-periodic

(4.25)

We now integrate (4.23) for the (Y phase over the partial volume R, and add the results for both phases. Defining the two-phase average by (4.26) and the effective conductivity by (4.27)

Some Applications of the Homogenization Theory

313

we see it follows straightforwardly from (4.23) to (4.25) that (4.28)

which is the macroscale diffusion equation for the composite. It is easy to show that ( K I , ) is symmetric and positive-definite. In summary, we have obtained the effective diffusion equation for the composite and defined the effective constitutive coefficients ( p c ) and ( K l , ) . In particular, the effective diffusivity ( K I , ) must be obtained by solving the canonical cell problem defined by (4.19) to (4.22). These formal results have been obtained in this way by Bensoussan et al. (1978). For the special case of a laminated composite, the cell problem is one-dimensional and has been solved by Auriault (1983) to get the effective heat conductivity. The heat conductivity of fiber-reinforced composites has been computed by Bourgat (1978), who solved the two-dimensional cell problem by employing finite elements to discretize a unit cell. Perrins et al. (1979) also treated circular cylinders in a matrix by a similar numerical method. Many papers have been devoted to the calculation of heat conductivity of packed uniform spheres (see, for example, Keller, 1963; McPhedran and McKenzie, 1978; McKenzie et al., 1978; Sangani and Acrivos, 1983). Auriault (1983) further pointed out that, if the conductivities of the two media are sharply different, the effective equation governing the macroscale diffusion is an integro-differential equation. With some modifications to account for fluid flow, the extension of such a theory is relevant to soil vapor extraction, an important technology for cleansing soils contaminated by gasoline leakage. This extension will be discussed in Section 1V.C. B. DISPERSION OF SOLUTE

IN A

CHANNEL FLOW

In 1953, G. I. Taylor discovered for pipe flows that the cross-sectionally averaged concentration of a dye cloud is not simply convected by the mean velocity and diffused by molecular or eddy viscosity. Instead, velocity shear across the pipe greatly enhances the diffusion in the direction of flow, which he called dispersion. Because the knowledge of contaminant transport dilution in rivers and estuaries is the key to proper control of water

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

314

quality, Taylor's theory of dispersion and its extensions have become the scientific cornerstone for predicting surface-water and groundwater pollution. For the one-dimensional case, Taylor's original analysis was heuristic. A more formal procedure called the method of moments was introduced later by Aris (1956), who also studied the case of pulsating flow in a pipe. Here we demonstrate the use of the homogenization theory. Consider the uniform flow in a channel of constant width a. The one-dimensional velocity flow is given and consists of a steady part Us and an oscillatory part with amplitude U, : u

=

~ , ( y +) !R [ ~ , . ( y >e - f w t ]

(4.29)

The concentration C is governed by dC

-at+ -

d(uC)

(

d2C

d2C

=D -

dX

dX2

+

(4.30)

7)

with the boundary conditions dC

-_

-

y

0,

=

O,a

(4.31)

dY

where D denotes the molecular diffusivity. Assume the channel to be so narrow that diffusion affects the whole width within a few periods of oscillation; that is,

a2

277

(4.32)

- u -

W

D

We now consider the diffusion across a horizontal distance L much greater than a. Let U, be the scale of U, and U, and x

=

Lw',

y

=

ay',

u

=

t

Uout,

U2

=

WU2

-t' D '

a = -D

(4.33)

Equation (4.30) is normalized to

dC'

-d+t '- -

U,a a d ( u ' C ' ) D L

dx'

-

u 2 d2C'

--

L' a x t 2

d2C' +,2

dy

(4.34)

Let the P6clet number Pe = U,a/D = U(1) be on the order of unity and denote a / L = E . Equation (4.34) becomes dC' -+ePe dt '

d(L4'C') dX'

= c2-

d2C' dXt2

+

d2C'

-

dy'2

(4.35)

Some Applications of the Homogenization Theory

315

where u'

=

u7'+ >yiude-lil''

(4.36)

with the boundary conditions dC'

- -

dy'

-

0,

yl

=

0,l

(4.37)

For brevity the primes are dropped from now on. There are three time scales: diffusion time across a , convection time across L , and diffusion time across L. Their ratios are

(4.38) Therefore, we introduce the multiple time coordinates t,t,

= Et,

t,

(4.39)

= E2t

and multiple-scale expansions

C' where C"' O(€O),

=

=

C'"' + EC'I' +

E2C(2)+

...

(4.40)

C ( ' ) ( xy, , t , t , , t 2 ) . Then the perturbation problems are, for

(4.41) with the boundary conditions

(4.42)

(4.43)

(4.44 1

316

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

and for O ( E ' ) ,

with tlC(2)

dY

=o,

y=o,1

(4.46)

The solution at O ( E ( )is)

If C'") is independent of y initially, then C!") = 0 for n = 1,2,3, ... . If C!') # 0 for some integer n , the series terms die out in the range t , = O(1) and are unimportant if we are interested in the long-term behavior. We therefore take the solution at O ( E ( )to) be

Taking the time average over a period, (4.50)

with --

dY

where

-

0,

y

=

0,l

(4.51)

Some Applications of the Homogenization Theory

317

Integrating (or averaging) across the channel, we get dC"" at I

+ Pe(q)-

d C'O' =

dX

0

(4.52)

where

( h ) = /'hdy 0

Now subtract (4.52) from (4.49)

(4.54) In view of the linearity of (4.531, we assume the solution for C(') to be (4.55) Then (4.56)

(4.57) with the boundary conditions (4.58) These two boundary-value problems for B, and B, are straightforward. After solving for B,, B,, we go to 0 ( e 2 ) ;that is, (4.45):

318

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

From (4.55) and (4.52) we find

It follows that

Taking the time average over a period, we get

with (4.63)

Integrating (4.62) with respect to y from 0 to 1, we get (4.64) with

E

=

i

1 - Pe2

1

(Q.B,) + - % ( u ~ B ; ) 2

I

(4.65)

which is called the effective diffusion coefficient or the dispersion coefficient. Finally, we add (4.52) and (4.62) to get

After the perturbation analysis is complete, there is no need to use

Some Applications of the Homogenization Theoly

319

multiple scales; we may now write (4.67) Thus the long-time evolution of the sectional-averaged concentration obeys this one-dimensional convective-diffusion equation, which can be , solved for any given initial data C ( " ) ( x0). The case for a circular pipe can be similarly treated. In particular, the contribution to E by the steady shear flow is = 1/192, due to Taylor (1953). The contribution by the oscillatory part requires the solution for the canonical cell problem for B , , which has been worked out by Aris (1960). We now turn to three-dimensional problems.

(q,Bs)

c. DISPERSIONIN

A

POROUS MEDIUM

Given the complexity of soils and rocks in nature, the theory of dispersion of a solute in porous media inevitably involves mathematical simplifications in which natural soils are modeled only incompletely. Some theories bypass the microscale details and begin from Darcy's law for the seepage flow on a scale much greater than the pore size. A convectivediffusion equation is either assumed to govern the solute transport on a similarly large scale or formally justified through a Reynolds averaging formalism to which closure hypotheses must be added. The effective diffusion coefficients accounting for the Taylor dispersion are usually based on empirical relations between the effective longitudinal and transverse dispersivities and the local seepage velocity (Bear, 1969; Fried and Combarnous, 1971). This approach has been the basis of statistical theories for large regions where soil heterogeneity over different length scales are important (see Dagan, 1989). For fully three-dimensional models of microstructure, Brenner (1980) developed the basis of a rigorous theory for strictly periodic porous media. Starting from the Brownian theory for the probability density of suspended particles, he has extended Aris's method of moments (Aris, 1956) and shown that the dispersion tensor should be calculated by first numerically solving a convective-diffusion problem in a periodic cell on the microscale. Extensions of Brenner's approach, which is now called the generalized Taylor dispersion theoly, have been made in recent years by Brenner and

320

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

collaborators. For a recent account, see Dill and Brenner (1982) and Brenner and Edwards (1993). As will be described here, Brenner’s theory can be derived and extended straightforwardly by the homogenization theory without resorting to the Brownian theory (Rubinstein and Mauri, 1986; Mauri, 1991; Mei, 1991; Auriault and Adler, 1995). In particular, we shall consider the medium to be inhomogeneous on the macroscale and demonstrate the use of the spatial averaging theorem in Section 1II.F. The volume concentration C(x,, t ) of the solute is governed everywhere in the pore fluid by dC ~

dt

dC

+u = DV2C ax,

in

RJ

(4.68)

where D denotes the molecular mass diffusivity. The pore fluid is incompressible, (4.69) In this section we allow the porosity to vary slowly over the global scale and demonstrate the use of the averaging theorem of Section 1II.F. Accordingly, we let the granular boundary r be denoted by F(x,) = 0 and be impermeable, then dC d F

ax, dx;

-0

r

on

(4.70)

because the unit normal n , is proportional to d S / d x , . Introducing the normalization C

=

(AC>C*

t

=

Tt*

where AC and T are, respectively, t h e scales of concentration variation and time, we get l 2 dC* ~~

TD d t *

dC

+ Peu*-ax,. I

=

V$C*

Some Applications of the Homogenization Theory

321

where Pe is the PCclet number (4.71) We are interested in the physical processes over much longer time than the scale for microscale diffusion f 2 / D .For many solutes in groundwater flows, the Prandtl number Pr = v / D is large though Re is small (e.g., for salt in water D = cm2/sec, whereas v = cm2/sec so that Pr = lo3), therefore we shall allow Pe = O(1) for generality.' Two other time scales can be expected to be physically important, times for convection and diffusion over L:

Their ratios to the diffusion time over 1 are

These estimates suggest that in a steady flow C depends only on the slow variables Et* and E 2 t * . Let us now return to (4.68) in physical variables and introduce the slow time coordinates tl

=

Et,

t'

=

(4.72)

E2t

and the multiple-scale expansion for C

c = c'"'+ E C ' l )+ E2c'2'+ ...

u,

)'(I.

+

+

=

E2uj2)

+ ... (4.73)

where C""), m = 0,1,2,. .. depends on x k , x;, t l , and t,. It follows from (4.73) that at the orders O ( E " )to O(E'> (4.74) JC'O'

dtl

duCl)C'O'

du'O"'''

f

J

dx, =

+

I

ax,

du(0)C'0) +

I

JX;

(4.75)

D(V'C(1) + V . V'C(())+ V' . VC")))

'If the pore fluid is air, as in unsaturated soils, these estimates can be greatly different, see Section 1V.D.

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

322

where (4.77) As in Section III.F, we allow the granular surface r : F = 0 to change slowly in space, hence F depends on x, and x:. The boundary conditions on the grain-fluid interface r and the outer boundary d R of an R cell are dC(O' J F dx,

-

0

(4.78)

=

0

(4.79)

ax,

and

C'") : R-periodic (4.81) for m = 0,1,2, . ), by (4.74), (4.78), and (4.81) The cell problem at the order O ( E ' ) defined with m = 0, has the formal solution

c'"'= C'W(

I 7

t I ) t2 )

(4.82)

which is independent of the local spatial coordinates. Taking the R average of (4.751, we obtain by Gauss's theorem and the boundary conditions d C'O)

n-

dtl

Clearly, the right-hand side vanishes because of the boundary condition (4.79). Hence, (4.83)

Some Applications of the Homogenization Theoly

323

in view of (3.29). Therefore, at the leading order, the concentration is simply convected by the mean flow, as anticipated. Subtracting (4.83) from (4.79, we get (4.84) where $ )I) =

uI( 0 ) - (ujo))/n.

(4.85)

To solve the linear equation (4.84) for C(’),we let (4.86) and define N l ( x i ,xi) to be the solution to the following canonical boundary-value problem in an Q cell: (4.87)

n

dN1

-=nl dx,

N I Q-periodic

on

r

(4.88) (4.89)

The vector function Nl can be made unique by adding the condition that

( N / >= 0

(4.90)

Without loss of generality, we take c ( ’ ) ( x :t,, ,t,) = 0, which amounts to defining C(O) to be the cell average with an error no greater than O ( E ~ ) . ~ The uniqueness of the cell problem can be proven as in Section 1II.C (Brenner, 1980, Appendix A). This linear cell problem can, in principle, be solved numerically for N, for any cell geometry. * l f Cc’’is not taken to be 0, we would have to add

to the left-hand sides of (4.91) and (4.93). The result would lead to the same (4.101) if we redefine C = C“” + eC(’’ as the average in (4,101).

324

Chiang C. Mei,Jean-Louis Auriault, and Chiu-on Ng

Last, we average (4.76) over the R cell. By using Gauss's theorem and boundary conditions, we get

dC'"](

+-

ax;

zz),til

(4.91)

+ D ( V ' . VC'") + D(V'2C'"')

Invoking the averaging theorem (3.661, the last two terms on the right become

which reduces (4.91) to

because of (4.80), where

which is nonzero when either the medium or the flow is inhdmogeneous on the macroscale. The dispersion tensor is given by

and depends on the flow in the pores. The form of Djl is arranged to be symmetric, resulting in the pseudo velocity defined by (4.94). Equations (4.83) and (4.93) jointly govern the convection and diffusion of C'O'.

Some Applicatioris of the Homogenization Theory

325

From (4.87) and the boundary conditions (4.88) and (4.89), it can be shown that

-

D ( q V 2 N , + NIV”,)

dNj dN, =2D-(dxk dxk)

(4.96)

(z: : )

-D-+-

after using continuity and (4.75). Hence, (4.94) may be written in terms of N/ only, (4.97)

The combined dispersion-diffusion tensor may now be written

(4.99) where 6 , = N, - x,. The last form (4.99) was first derived by Brenner (1980) via the Brownian motion theory. Brenner further showed that for any 4 that is independent of x k , the bilinear

is clearly positive-definite. Hence the total dispersion-diffusion tensor Dj, + nD6,, is positive-definite. The same argument applies to the part

(4.100) which also must be positive-definite. Finally, we may add (4.83) and (4.93) to get the convective dispersion-diffusion equation for C(’). Because the derivation has been

326

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

completed, the superscripts and the ordering parameter carded and the distinction between xi and xi. removed:

ax;

at

ax;

(D,,

+ nD6. )-ax,.

E

may be dis-

(4.101)

+

where (a,) = (u{O)) ( u ; ' ) ) . The preceding theory has been extended by Mei (1992) to the diffusion of heat in a fluid-saturated porous medium. To calculate the dispersion tensor, one must solve the canonical cell problem defined by (4.87) to (4.89). Lee et al. (1995) have shown that the cell boundary-value problem for convective diffusion of heat can be recast as a variational principle. The case for solute transport being a special case, its variational principle is

6J

=

0

(4.102)

where the functional J is

(4.103) in which A, is a Lagrange multiplier. For a cubic array of Wigner-Seitz grains, the dispersivity tensor depends, however, on the direction of the global flow. In the computations of Lee et af., the mean flow is directed along the x axis ( 6 = 0"). D,, is diagonal with two independent components that are the longitudinal D, and transverse D , dispersivities: D,,(O = 0) = D, and D,,(O = 0) = D,,(O = 0) = D,. For any other flow direction in the q plane, there are only four nonzero independent dispersivity coefficients: D , , , D,, , D,, , and D,, = D 2 1 ;D,, = D,, = 0. Computed values of longitudinal and transverse dispersivity coefficients D , and D , are shown in Figures 4 and 5, respectively, for Pe up to 300 for D , and 200 for D,, and for two porosities n = 0.38 and 0.5. To conform with experimental literature the abscissas in both figures are the Pkclet numbers defined in terms of the mean-flow velocity averaged over R, only; that is, ( u ) l / n D = Pe/n. In Figure 4, the longitudinal dispersivity is also compared with the measured data for simple cubic packing of uniform

327

Some Applications of the Homogenization Theory

10000.0~

. "."" '

- ee 00 .. 35 8 ......

c c

1000.0 r

..

'

'"''1

' " " ' ' 1

'

""".?

'

0

e 0 . 4 8 (Koch eta/.. 1989) Salles eta/.. (1 993) ( m 0 48)

' ~ - T

'

'

9.. -

-2.-

-

-

10.0 r

-

0.1

10000.0~

,

I

.

'

"""'

-e0.38 ......

.

,

'

'

, , , , ,

, , ,

,

- " " "

'

,,.,I

""""

.

'

fa) I , , ,

.

,.,,I

' ' ' . ' ' I

, , ,

,

.

'

'"77

e0.5 m0.48 (Koch e t a / . ,1989)

1000.0 F

-

100.0 5

-

-

-

0.1

I

, , . , . , , , I

........ I

(b)

, . , , , , , , I

I

........

spheres and the calculations (Zick and Homsy, 1980) for a simple cubic lattice of uniform spheres with n = 0.48, 0.74 and 0.82. The results for n = 0.48 by Koch et al. (19891, based on an approximate analysis for dilute concentration, are also included. All are in qualitative agreement for D, .

328

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

Natural soils are composed of grains of a wide range of shapes and sizes, and D, is known empirically to be linear in the PCclet number. Such media still remain a challenge for the theorist.

D. POROUSMEDIA WITH DISPARATE DIFFUSIVITIES Gasoline spills from storage tanks have contaminated soils from filling stations, airports, petroleum refineries, and oil fields. One effective method for removing gasoline vapor from the unsaturated soils is called air rlenting or soil-vapor extraction. The idea is to pump air from the pores through a well piercing through the contaminated zone. The air flow also enhances diffusion from the soil aggregates that contain volatile organic compounds either as a solute in the water phase or as sorbate on the surface of soil grains. Once diffused in the pore air, the chemicals are again more easily removed by the moving air. For soils composed of aggregates the micropores within each aggregate are generally much finer than the macropores between the aggregates. Water captured inside the aggregates is nearly immobile so as to be approximately stationary. The physical problem is, therefore, dominated by the air flow through the macropores and convective diffusion, with chemical partition on the aggregate scale. A common feature is that the diffusivity in air is usually much greater than that in the aggregate, so that the time scale for diffusion over a macroscale length of soil is comparable to the diffusion scale through the small aggregates. In this diffusion problem, the time contrast results in some distinguishing features. A slightly simpler case is the problem of diffusion through a flowing groundwater in a saturated soil, where the diffusivities of a solute in the water far exceeds that in the aggregates. The macroscale equation of diffusion law has been derived by heuristic arguments in the chemical engineering literature of packed beds. For periodic aggregates Hornung (1991) has used the theory of homogenization to get the same governing equation. Here we describe a recent work of Ng and Mei (199.9, which accounts for chemical partitioning for modeling air venting of volatile organic compounds. The soil is modeled here as a periodic array of packed spherical aggregates of uniform size. Water with dissolved and sorbed chemicals is held immobilized in the aggregates, and air with chemical vapors can flow in the pore space between aggregates. Thus a unit micro-cell Cl consists of

Some Applications of the Homogenization Theory

329

a,

a spherical aggregate of radius a, surrounded by an air-filled void a,; the surface of the aggregate is denoted by rwg. The volume ratios are denoted by ng = I.R,I/I.RI and n , = lflwl/lRl, and the volume averages of a quantity f over an R cell are denoted by

For air-venting problems, the pressure variation can be large enough that air compressibility is important. Let us begin with the Navier-Stokes equations for air in the macropores:

(4.104) p u -d U 1

' dx,

dP + p - d2U,

= -~

dx,

ax;

1

+ -p3

d2U, a x , ax,

(4.105)

Assuming the ideal gas law, we get the equation of state

PM -=P RT

(4.106)

Let 1 = 2 a and L be the micro- and macro-length scales, respectively. Their small ratio f / L = E << 1 will be used as an ordering parameter. The local coordinates are x , and the global coordinates are x: = EX,. Under the usual assumption that the Reynolds number is small R e = U g l / v g= O ( E ) we , introduce two-scale expansions for u , and p . It then follows that

(4.108) Locally, the pressure is constant and the air flow is incompressible. At the next order, it is easily found that

(4.109)

330

Chiang C. Mei,Jean-Louis Auriault, and Chiu-on Ng

where (4.110) The permeability tensor K,, is found from the same canonical cell problem defined in Section 1II.A.Thus, the air-flow equation (4.109) is nonlinear in general. Let cg and c , be the concentrations of the chemical compound in the pore air and in the immobile water, respectively. Their variations are governed by the laws of diffusion (4.1 11) in the pore air, and (4.112) in the immobile water, D, represents the effective diffusivity accounting for adsorption of the chemical on the surface of the solid

with 4 representing the porosity in the aggregate, D, the effective diffusivity of water-soil mixture in the aggregate, K , the sorption coefficient, and ps the density of the solid in the aggregate. In addition, we require the continuity of flux across the air-water rgwinterface of the aggregate; that is, (4.113) where ni is the unit normal vector directing into the aggregate. Also, Henry’s law of concentration partition holds along rgw

K,

=

cg

(4.1 14)

W

where K , denotes Henry’s law constant. Along the cell boundaries, c, and c, are R-periodic. Using the microlength 1 to normalize the spatial

331

Some Applications of the Homogenization Theory

coordinates, the following dimensionless parameters appear. Based on typical data for air-venting operations, their magnitudes are estimated as follows: 1. Pkclet number Pe

=

U,l/D,

2. Ratio of diffusivities D,/D,

Re(v,/D,)

= =

=

O(Re)

=

O(E).

0(e2).

3. Time scale T is controlled by the diffusion time in the immobile aggregate: 12/TDe = 0(1), implying that 1 / TD = O ( E2 ).

Returning to physical variables but retaining the order symbols according to the estimates, we get the vapor transport equation: (4.115) Introducing two-scale expansions c,

=

+ E Cx( ~ +) ... , we obtain

)c:

(4.116)

(4.118)

Similarly, diffusion in the immobile water 2

E

dc, -=

dt

a,

d2C,

e2De-

in the aggregates obeys (4.119)

dX'

where c , is the chemical concentration in the immobile water inside an aggregate. Expanding c, = c:)) + ... , we get the leading-order equation: (4.120)

332

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

By expanding the boundary conditions similarly, we obtain, for the continuity of flux along r,,, (4.121)

(4.122)

where ni is the unit normal vector directing into the aggregate. From Henry’s law of chemical partition, c(nl)

Ktl

=

$

m

=

0,1,2, ...

(4.124)

W

Along the cell boundaries, c p ) and ckm), m = 0, 1,. . . are cR-periodic. From (4.116) and (4.1211, it is obvious that

, now reduces to At the order O ( E ) (4.117) d 2 C p

O=D-

ax,’

In view of (4.122) we let (4.126) where an arbitrary function $’)(xi) has been discarded by redefining C(‘)). A’/ must then be the solution of a canonical boundary-value cell problem defined by (4.127)

Some Applications of the Homogenization Theoly

333

and the interface condition (4.128) In addition, N/ must be R-periodic and have 0 mean over the cell, ( N / ) = 0. At order 0 ( e 2 ) ,we average (4.123) and (4.120) over an 0 cell and add the results to get

(4.129)

where nK= IKL,I/lfil

is the volume fraction of the air phase, and (4.130)

Note the following symmetry

):=():(

(4.13 1)

which can be proven as in Section III.C, and ensures the symmetry of D , / . The source term a( c$'))w/dt, which represents the exchange of chemical between air and immobile water, is evaluated as follows. From (4.125) c:) is uniform in a local cell, hence, cc') is a function of local radial position r and time only; that is, (4.132) with the boundary condition (4.133) and the initial condition c;?(r,t

=

0)

=

c,,,<x~>

(4.134)

334

Chiang C, Mei, Jean-Louis Auriault, and Chiu-on Ng

The solution to this classical diffusion problems is known (Carslaw and Jaeger, 1959, p. 233):

where A,

Den2n2 =

~

a'

(4.136)

Upon averaging over fl and differentiating with respect to time, the source term is calculated

(4.137)

where cgo = cf) ( t = 0). If initially the partition between air and immobile water is in equilibrium, the two terms inside the parentheses cancel out each other. Substitution of (4.137) into (4.129) leads to

Equation (4.138) is an integral-differential equation governing the macroscale transport of the vapor phase solute cg. It must be solved under appropriate boundary conditions on the ground surface, at the well casing, at the water table below, if any, and at infinity. Afterward c, can be computed from (4.135). Laboratory experiments have been performed by Gierke et al. (1990, 1992) for a column of soil partially saturated with water and contaminated with volatile organic compounds such as toluene and benzene. Air is pushed in from one end of a pipe and out the other at the speed ug. Using

Some Applications of the Homogenization Theory 1.1

1

I

,

,

,

40

60

'

335

,

.c. C

.-0

c

C

0 a C

0

0

.c

-ma

a

a .

0

20

80

100

120

140

Gas pore volumes

FIG.6 . Comparison between theory and measurement of toluene concentration at the exit of a contaminated soil column.

parameters measured by Gierke et al., the theory here agrees well with their experiments, as shown in Figure 6 for the exit concentration cg of toluene as a function of time.

E. DISPERSION IN

WAVE

BOUNDARY LAYERS

Dispersion of suspended particles near the seabed by waves and currents is of interest not only to the transport of sand or mud but also of planktonic larvae (Denny, 1988). For example, when sewage or fine silt dredged from an estuary is deposited somewhere offshore, one wishes to know whether it might drift back in a storm. In coastal regions, diffusion of suspended particles by waves is most effective in the bottom boundary layer where there is dispersion of the Taylor-type due to the presence of strong shear. It is well-known that the flow field therein has not only an oscillatory component but also a steady streaming induced by Reynolds stresses (see, e.g., Longuet-Higgins, 1953; Hunt and Johns, 1963; Mei, 1983). How do steady streaming and enhanced diffusion contribute to the spreading and transport of suspended particles? In a recent paper, Mei and Chian (1994) assume the particles to be small enough so that their inertia is negligible, and the particle velocity

336

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

due to fluid oscillation to be essentially equal to the ensemble mean velocity of the local fluid. The fall velocity, however, is not ignored, as it is unrelated to wave motion and nonzero even if the ambient fluid is completely calm. Consider particles of a certain size whose fall velocity is - w, . Let C denote the volume concentration of these particles. The diffusion equation for the concentration C of a very dilute sediment cloud can be approximated by

where i = 1,2 with ( x , , x , ) = (x,y ) and ( u ,, u z >= ( u , representing, respectively, the horizontal coordinates and the fluid velocity components, and D representing the eddy diffusivity of mass. At the sea bottom, the boundary condition for C is the least certain if the bed surface is erodible. We shall take the simplifying assumption for very small particles that they are kept in suspension by turbulence and that the bed is not erodible ~

dC Ddz

+ w,C=0,

z=O

1

)

(4.140)

Outside the boundary layer, we assume

c=o,

z + x

(4.141)

Initially, the horizontal distribution of the depth-averaged concentration is prescribed in some source area. The complexity of this problem stems from the presence of several characteristic length scales. The first is the thickness of a steady concentration layer due to the balance of downward sedimentation by gravity and vertical diffusion d--

D

(4.142)

wo

Associated with fluid oscillations with frequency w are two additional vertical length scales; that is, the oscillatory boundary-layer thicknesses 6

- J2.e/w

and

6,

- 42D/w

(4.143)

Some Applications ofthe Homogenization Theory

337

corresponding, respectively, to oscillatory momentum and mass diffusion. Finally, there are the wavelength (2n-/k) and the typical dimension of a topography. For generality, all three vertical scales are assumed to be comparable; that is, 6

< O(1) d -

(4.144)

-

and (4.145) where Sc is the Schmidt number. Similarly, both horizontal lengths are assumed to be comparable, too. Invoking Reynolds analogy, D = v,, and using v, K U * & as an estimate, where K = 0.4 is KArmAn's constant (Kajiura, 1968), we find d K U . 6/w,,; (4.145) is seen to be consistent with (4.144). It will be assumed that the boundary-layer thickness is far smaller than the wavelength, and the wave slope is also very mild. Let A denote the typical wave amplitude. We assume

-

-

E

= k A << I

and

p

=

kS << 1

(4.146)

In most natural flows of interest p is much smaller than E . We shall, however, make a generous assumption that O ( E )= O( p ) . A consequence is that the horizontal diffusion is retained in the final diffusion equation, as will be shown. Introducing the following normalization

t h e diffusion equation (4.139) is rescaled to become

dC* -

dt*

au;c*

i)

7 +( [ ;d Z -Pe + E W * ) C * ] +€ dX. (4.148)

where Pe = w,S/D is a PCclet number based on the fall velocity of a particle. Having identified the orders, we return to the dimensional form

338

Chiung C. Mei, Jean-Louis Auriuult, and Chiu-on Ng

(4.139) by preserving the ordering parameter

dC -

dt

du,C

d

+ E - a x i + -d[z( - w ,

+ E W ) C =] D

E

as follows:

ax, ax,

+

dZ2

(4.149)

The two sharply contrasting length scales imply two distinct time scales: one for vertical diffusion across the boundary layer, O ( w - ' ) = O ( S 2 / D ) , which is the same as a wave period; and one for horizontal diffusion across a wavelength, O(l/k2D). The ratio between the two is O(k2S2)= O( p2). Under the assumption E = O( p), we shall introduce multiple-scale coordinates for time: t and t' = ~ ' t The . velocity and concentration are expanded as follows:

+ EUj') + 0 ( c 2 ) w = wC1) + E d 2 ) + O ( E 3 ) c c(o)+ € c ( I+)E ~ ~ ( +2 0) ( ~ 3 )

u,

=

uy

=

(4.150) (4.151) (4.152)

where u)") and w(") are functions of x,, z , and t and 0 " = ) C ( " ) ( x ,z,, t , t '1. Ignoring initial transients that diminish after a few wave periods, we take C'") = C(o)(x,,z , t ' ) to represent the period average and depends only on t ' . The diffusion equation at O(1) is an homogeneous ordinary differential equation in z

dC'Q -wo-

d 2C'"' -

dZ

D

T

0
(4.153)

The homogeneous boundary conditions are (4.154)

C'"'

=

0,

-3

m

(4.155)

Thus, a nontrivial solution exists

where (4.157) and d is the concentration at the seabed and represents the horizontal variation of C to the leading order. The exponential vertical variation is the consequence of constant eddy diffusivity assumption.

Some Applications of the Homogenization Theory

339

At O ( E ) ,C'I) represents the concentration fluctuations due to the oscillating velocity field and satisfies

a c(')

dC'" --

at

' w

dz

d2C'"

-D-=

d(w'"C'o')

d(U!l)C(Q) -

dx,

dz?

-

dZ

(4.158)

and the boundary conditions (4.159) (4.160)

(4.161)

dx; dx,

dZ

and the boundary conditions

C'2'

=o,

z=o

(4.162)

=o,

z-w

(4.163)

Our interest is in the slow diffusion at the leading order, and attention will be focused on the governing equation for the factor &xi, t ' ) , defined by (4.156). Let us assume that the velocity field at the leading-order ( u ! ' )w(')) , is simple harmonic in time with the frequency w . All the forcing terms on the right of (4.158) are then simple harmonic in the fast time t . Let the period-average (time average with respect to a wave period) of C(l) be denoted by C(l).Then C(') satisfies the homogeneous equation (4.153) also and the homogeneous boundary conditions (4.154) and (4.155). Without loss of generality we shall take = 0 so that C(l),which corresponds to the departure from the zeroth order mean C("),consists only of first-

cc')

340

Chiung C. Mei, Jean-Louis Auriauft, und Chiu-on Ng

harmonic fluctuations: C'"

=

% ( C , ,e p l w f )

(4.164)

where C,, = C , , ( x , ,z , t ' ) and % denotes the real part of its argument. (For later use, 3 denotes the imaginary part.) Averaging (4.161) to (4.164) over the wave period, we find that the time average of C'2) satisfies an inhomogeneous differential equation similar to (4.153) and the boundary conditions (4.154) and (4.155). Solvability of the inhomogeneous boundary-value problem for leads to

c(2)

c(')

Again, this is just the Fredholm alternative for the inhomogeneous boundary-value problem, which possesses a nontrivial homogeneous solution (here C"')). It follows by using (4.157) that

a -

- -

(4.166) d'

- ( U : ~ ) C ( ~+ ) )D [e(F)l ax, dx,ax,

where ( f ) represents depth integration of f from z = 0 to z = m. This gives the effective convection-diffusion equation for 2. In the preceding equation, U!') represents the second-order Eulerian streaming induced by Reynolds stresses. The first term on the right-hand side of (4.166) is the correlation of velocity and concentration fluctuations. The last term represents horizontal diffusion due directly to turbulence. At the first order, the concentration fluctuation, C"), formally given by (4.164), must satisfy (4.158). By substituting the known result of Eulerian streaming (see Hunt and Johns, 1963; Mei, 1983) in terms of the oscillatory velocity components q,and V, just above the boundary layer and (4.157) into the right-hand side of (4.158) and extracting the coefficient of the first harmonic part, we get

d 2 C , , + -1 dC,, dz2 d dz with

(4.167)

Some Ayplicutions of the Homogenization Theory

34 1

where F , ( z ) and F,(z) are explicit functions of z with parametric dependence on Pe and Sc. These results arc given in Mei and Chian (1994). From (4.167) and (4.1681, the forcing terms of C , , involve both d e / d x , and therefore, the solution for C , , must be a linear combination of 6 and its horizontal gradient. Using the solutions for C , , , we get, after some algebra, the effective diffusion equation for 6:

c;

d e -

dt’

d

+ -(Z,?) ax,

=

D-

ae

(4.169)

where

Z! and Y”are the components of the weighted depth-average of Eulerian

streaming and Eli arises from the correlation tensor ( u ~ ” C ( ’ ) )The . formulas deduced here are general for any small amplitude wave field as long as the first-order inviscid velocity (U, , 5 )in the tangential direction is known at the upper edge of the boundary layer. Expressions of the complex coefficients H , , H,, and H4 and the real coefficient H , are functions of Pe and Sc (see Mei and Chian, 1994). Using this theory, Mei and Chian have calculated the streaming velocity, dispersion coefficients, and the transport of an initially gaussian cloud in several types of progressive and standing waves. Figures 7 and 8 display the effective streaming velocity and dispersion coefficients under a partially reflected plane wave. For an initially line cloud of suspended particles released at t = 0, the subsequent dispersion and convection is shown in Figure 9 for several values of the reflection coefficients R.

342

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

0.10 I

I

X

FIG.8. Dispersion coefficient in the boundary layer under a partially reflected plane wave.

Some Applications of the Homogenization Theory

343

FIG.9. Evolution of concentration in the boundary layer under a partially reflected plane wave.

These results highlight the fact that the phenomenon of dispersion is the consequence of flow; therefore, numerical modeling based on constant dispersion coefficients calibrated with a few field measurements cannot be satisfactory. On the other hand, further improvement of the theory is needed by including a more realistic model for the eddy viscosities and a model accounting for erosion and deposition on the seabed.

V. Other Applications As mentioned in the introduction, we have not included the vast literature on many linear and nonlinear wave propagation problems involving slow modulation of near periodic waves. Many of these problems can be and have been studied by what is essentially the method of homogenization (see, e.g., Mei, 1983, 1989). The modulation can be caused by the narrow width of the frequency band or by the slow variation of the medium properties, or by both. Examples of applications are particularly rich in the mechanics of composite media. The obvious example is to determine the relations between stresses and strains in a fiber-reinforced material (Tong and Mei,

344

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

1992). The deformation of a saturated porous medium is of fundamental importance to the mechanics of ground consolidation due either to surface loading or pumping of groundwater. By modeling the soil as a periodic cell of aggregates, the work of Auriault and Sanchez-Palencia (1977) puts the phenomenological theory of Biot on a micro-mechanical foundation. In this theory the motion in both the fluid and solid phases is quasi-static. Propagation of sound in a saturated porous media has been treated independently by Levy (1979) and Auriault (1980). Because the interstitial flow is dominated by viscosity, the dynamical problem is memory dependent and the final equation is of the integral-differentia1 type. Extension from two to three contrasting scales such as soil stratum consisting of periodic layers of sand and clay has been made by Mei and Auriault (1989). Fractured porous media have been modeled by assuming double porosity and has been treated by Auriault and Boutin (1992, 1993, 1994) and Royer and Auriault (1994). The problem of sound propagation through a liquid populated sparcely by bubbles is another interesting application treated by Caflisch et al. (1985) and Miksis and Ting (1986, 1987, 1994). The main objective is to find an effective equation for the propagation of sound whose wavelength is much greater than the bubble spacing, which is, in turn, much greater than the bubble radius. In particular, it is known that the presence of a small volume fraction of bubbles in water can drastically reduce the speed of sound. For dense concentration of bubbles the work by Boutin and Auriault (1993) is germane. Although only a few selected examples of homogenization theory have been discussed here, we hope to have demonstrated the point that the key to a successful application to a new physical problem depends on the a priori identification of spatial or temporal scales based on physical considerations. Then the asymptotic analysis can be straightforwardly carried out by the introduction of multiple-scale coordinates and the perturbation series. It is worth emphasizing that the theoretical results can be of practical use only if the canonical cell problems are solved quantitatively, which usually entails considerable numerical work; much hard work remains to be done. One of the major challenges for future years is to extend the homogenization method to natural media, which are random. Some progress is being made by choosing periodic cells, each of which contains a large number of grains of varying sizes (see Cruz et al., 1995).

Some Applicutions of thc Homogenization Theoiy

345

Acknowledgments

CCM and CON wish to acknowledge the financial support for many years by the U.S. National Science Foundation (Fluid Mechanics and Hydraulics Program, and Geomechanics, Geotechnical and Geoenvironmental Program), the U.S. Office of Naval Research (Ocean Engineering Program) and by the U.S. Air Force Office of Scientific Research. JLA thanks the continued support of the French Centre National de la Recherche Scientifique. References Ark, R. (1956). On the dispersion o f a solute in a fluid flowing through a tube. Proc. R. Soc. London, Ser. A 235, 67. Ark, R. (1960). On the dispersion of a solute in pulsating flow through a tube. Proc. R . Soc. London, Ser. A 259, 370-376. Auriault, J.-L. (1980). Dynamic behaviour o f a porous medium saturated by a Newtonian fluid. Int. J . Eng. Sci. 18, 775-785. Auriault, J.-L. (1983). Effective macroscopic description for heat conduction in periodic composites. J . Heat Mass Transfer 26(6), 86 1-869. Auriault, J.-L., and Adler, P. M. (1995). Taylor dispersion in porous media. Ado. Water Res. 18(4), 217-266. Auriault, J.-L., and Boutin, C. (1992). Deformable media with double porosity, quasi statics. I. Coupling effects. Transp. Porou.s Mediu 7, 63-82. Auriault, J.-L., and Boutin, C. (1993). Deformable media with double porosity, quasi statics. 11. Memory effects. Transp. Porous Mediu 10, 153-169. Auriault, J.-L., and Boutin, C. (1994). Deformable media with double porosity, quasi statics. 111. Acoustics. Transp. Porous Me& 14, 143- 162. Auriault, J.-L., and Sanchez-Palencia, E. (1977). Etude du comportement macroscopiquc d’un milieu poreux saturk dCformable. J . M6c. 16(4), 575-603. Bakhvalov, N. S. (1975). Averaged characteristics of bodies with periodic structures. Dokl. Akad. Nauk SSSR 221, 516-519. Bakhvalov, N. S., and Panasenko, G . ( 1989). flomogenisation: Ar3eraging processes in periodic media. Kluwer Academic Publishers, Dordrecht, The Netherlands. Bear, J. (1969). Dynamics of fluids in porous media. Elsevier, Amsterdam. Bedford, A,, and Druhmeller, D. S. (1983). Recent advances in the theory of immiscible and structured mixtures. Int. J . Eng. Sci. 21, 863-960. Bensoussan, A,, Lions, J. L., and Papanicolaou, G . (1978). Asymptotic analysis ,for periodic structures. North-Holland Publ., Amsterdam. Bourgat, J. F. (1978). Numerical experiments of the homogenization method for operators with periodic coeficients, Rapp. Rech. No. 277. I.R.I.A. Rocquencourt, France. Boutin, C., and Auriault, J.-L. (1993). Acoustics of Newtonian fluid with large hubble concentration. Eur. J . Mech. B/Fluids 12, 367-399. Brenner, H. (1980). Dispersion resulting from flow through spatially periodic porous media. Philos. Trans. R. SOC. London, Ser. A 208, 81-133.

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Brenner, H., and Edwards, D. A. (1993). Macrotransport processes. Butternorth-Heinemann, London. Caflisch, R. E., Miksis, M. J., Papanicolaou, C. C., and Ting, L. (1985). Effective equations for wave propagation in bubbly liquids. J . Fluid Mech. 153, 259-273. Carman, P. C. (1937). Fluid flow through granular beds. Truns. Inst. Chem. Eng. 15, 150-166. Carslaw, H. S., and Jaeger, J. C. (1959). Conduction qf heat in solids. Oxford Univ. Press, Oxford. Cruz, M. E., Ghaddar, C. K., and Patera, A. T. (1995). A variational-bound nip-element method for geometrically stiff problems; application to thermal composites and porous media. Proc. R . Soc. London, Ser. A 449, 93-122. Dagan, G. (1987). Theory of solute transport by groundwater. Annu. Rer>.Fluid Mech. 19, 183-215. Dagan, G. (1989). Flow and transport in porous formuhons. Springer-Verlag, Berlin. Denny, M . W. (1988). Biology and mechanics of the wur~e-.sweptencironment. Princeton Univ. Press, Princeton, NJ. Dill; L. H., and Brenner, H. (1982). A general theory of Taylor dispersion phenomena. V. Time-periodic convection. Phys. Chem. Hydrodyn. 3, 267-292. Ene, H. I., and PoliSevski, D. (1987). Thermal flow in porous media. Keidel, Dordrecht, The Netherlands. Ene, H. I., and Sanchez-Palencia, E. (1975). Equations et phtnomkne de surface pour I’tcoulement dan un modkle de milieu poreux. J . Mic. 4, 73- 108. Fried, J. J., and Combarnous, M. A. (1971). Dispersion in porous media. Ad(%.Hydrosci. 7, 169-282. Gierke, J. S., Hutzler, N. J., and Crittenden, J. C. (1990). Modelling the movement of volatile organic chemicals in columns of unsaturated soil. Water Resour. Res. 26, 1529-1547. Gierke, J. S., Hutzler, N. J., and McKenzie, D. B. (l9Y2). Vapor transport in unsaturated soil columns: implications for vapor extraction. Waler Resour. Res. 28, 323-335. Gilbert, F. (1990). Change of scale in multi phase media: The use of structured soils. In: Geornatenal.r, constitutir,e equations and niodelling (F. Darve, ed.). Elsevier, London. Gray, W. G., and Lee, P. C. Y. (1977). On the theorems for local volume averaging of multiphase systems. Int. J . Multiphase Flow 3, 333-340. Gunn, D. J., and Pryce, C. (1969). Dispersion in packed beds. Trans. Inst. Chern. Eng. 47, T341-T350. Hornung, U. (1991). Miscible displacement in porous media influenced by mobile and immobile water. Rocky Mt. J . Math. 21, 645-669. Howes, F., and Whitaker, S. (1085). The spacial averaging thcorem revisited. Chem. Eng. Sci. 40(8), 1387-1392. Hunt, J. N., and Johns, B. (1963). Currents induced by tides and gravity waves. Tellus 15, 4. Kajiura, K. (1968). A model o f the bottom boundary layer in water waves. Bull. Earthquake Res. Inst., Unii.. Tokyo 146, 75-123. Keller. J. B. (1963). Conductivity of a dense medium containing a dense array of perfectly conducting spheres or cylinders. J . Appl. Phys. 34, 991. Keller, J. B. (1980). Darcy’s law for flow in porous media by the two space method. In: Nonlinear partial differentid equations in engineering and applied sciences (R. L. Sternhcrg, A. J. Kdlinowski, and J. S. Papadakis, eds.). Dekker. New York. pp. 429-443. Koch, D. L., Cox, R. G., Brenner, H., and Brady, J. F. (1989). The effect o f order on dispersion in porous media. J . Fluid Mech. 200, 173-188. Kovacs, G. (1981). Seepage hydraulics. Elsevier, Amsterdam. Kriiner, E. (1986). Statistical modelling. In: Modelling snzull deformations of polycysystals (J. Gittus and J. Zarka, eds.), pp. 229-291. Elsevier, London.

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Lee, C. K., Sun, C. C., and Mei, C. C. (1995). Computation of permeability and dispersivities, of solute or heat in periodic porous media. Inf. J . Heat Mass Transfer (to be published). Levy, T. (1979). Propagation of waves in a fluid saturated porous elastic solid. Inf. J . Eng. Sci. 17, 1005-1014. Lnnguet-Higgins, M. S. (1953). Mass transport in water waves. Philos. Trans. R . Soc. London, Ser. A 245, 535-581. Mauri, R. (1991). Dispersion convection and reaction in porous media. Phys. Fluids A 3, 743-756. McKenzie, D. R., McPhedran, R. C., and Derrick, G. H. (1978). The conductivity of lattices of spheres. 11. The body-centered and face-centered cubic lattices. Proc. R. Soc. London, Ser. A 362, 211. McPhedran, R. C., and McKenzie, D. R. (1978). The conductivity of lattices of spheres. 1. The simple cubic lattice. Proc. R. Soc. London, Ser. A 359, 45. Mei, C. C. (1983). The applied dynamics of ocean sufuce waivs. Wiley (Interscience), New York. (1989), World Scientific. Mei, C. C. (1985). Resonant reflection of surface water waves from periodic sandbars. J . Fluid Meell. 152, 315-335. Mei, C. C. (1991). Heat dispersion in periodic porous media by homogenization method. In: Symposium on multiphase transport (R. Eaton, ed.), ASME, New York, pp. 11-16, Mei, C . C. (1992). Method of homogenization applied to dispersion in porous media. Trump. Porous Media 9, 261-274. Mei, C. C., and Auriault, J.-L. (1989). Mechanics of heterogeneous porous media with several spatial scales. Proc. R . Soc. London, Ser. A 426, 391-423. Mei, C. C., and Auriault, J.-L. (1991). The effects of weak inertia on the flow through a porous medium. J . Fluid Mech. 222, 647-663. Mei, C. C., and Chian, C. (1994). Dispersion of small suspended particles in a wave boundary layer. Am. Meteorol. SOC. 24(12), 1-17. Miksis, M. J., and Ting, L. (1986). Wave propagation in a bubbly liquid with finite amplitude asymmetric huhhlc oscillations. Phys. F h d s 29, 603-618. Miksis, M. J., and Ting, L. (1987). Viscous effects on wave propagation in a huhhly liquid. Phys. Fluids 30, 1683-1689. Miksis, M. J., and Ting, L. (1994). Effcctivc equations for multiphasc flows-waves in a bubbly liquid. Adi'. Appl. Mech. 31, 142-256. Naciri, M., and Mei, C. C. (1993). Strong Bragg reflection of low-frequency sound from a line source in shallow water. I : One uniform surface wave. J . Acousr. Soc. Am. 94, 1598-1608. Ng, C. O., and Mei, C. C. (1995). Application of spherical diffusion model for soil vapor extraction in aggregated soils. Suhniitted for publication. Nigmatulin, R. I. (1981). Three-dimensional averaging in the mechanics of heterogeneous media. J . Fluid Mech. 10(4), 72- 107. Perrins, W. T., McKenzie, D. R., and McPhedran, R. C. (1979). Transport properties of regular arrays of cylinders. Proc. R . Soc. London, Ser. A 369. 207. Rasoloarijaona, M., and Auriault, J.-L. ( 1994). Nonlinear seepage flow through a rigid porous medium. Eur. J . Mech. B/Fhiids 13, 177-195. Roycr, P., and Auriault, J.-L. (1994). Transient quasi-static gas flow through a rigid porous medium with double porosity. Trun.sp. Porous Media 17, 33-57. Ruhinstein, J., and Mauri, R. (19%). Dispersion and convection in periodic porous media. SIAMJ. Appl. Math. 46, 1018-1023. Salles, J., Thovert, J.-F., Delannay. R., Prevors. L., Auriault, J.-L., and Adler, P. M. (1993). Taylor dispersion in porous media. Determination of the dispersion tensor. P11,ys.Fluids 5,2348-2376.

348

Chiang C. Mei, Jean-Louis Auriault, and Chiu-on Ng

Sanchez-Palencia, E. (1974). Comportement local et macroscopique d'un type de milieux physiques hCtCrogknes. Inf. J . Eng. Sci. 12, 331-351. Sanchez-Palencia, E. (1980). Nonhomogeneous mediu and ~~ibrabion theory. Springer-Verlag, Berlin. Sangani, A. S., and Acrivos, A. (1983). The effective conductivity of a periodic array of spheres. Proc. R. Soc. London, Ser. A 386, 262-275. Skittery, J. C. (1967). Flow of viscoelastic fluids through porous mcdia. AIChE. J . 13, 1066-1071. Snyder, L. J., and Stewart, W. E. (1966). Velocity and pressure profiles for Newtonian. creeping flow in regular packed beds of spheres. AlChE J . 12(1), 167-173. Sorensen, J. P., and Stewart, W. E. (1974). Computation of forced convection in slow flow through ducts and packed beds. 11. Velocity profile in a simple cubic array of spheres. Chem. Eng. Sci. 29, 819-825. Taylor, G. I. (1953). Dispersion of solute matter in solvent flowing slowly through a tube. Proc. K. Soc. London, Ser. A 219, 186-203. Tong, P., and Mei, C. C. (1992). Mechanics of composites of multiple scales. Comput. Mech. 9, 195-210. Whitaker, S. (1985). A simple derivation of the spatial averaging theorem. Chem. Eng. Educ. 19, 18-21,50-52. Whitaker, S. (1986). Flow in porous media. I: A theoretical derivation of Darcy's law. Trump. Porous Mediu 1, 3-25. Zaoui, A. (1987). Approximate statistical modelling and applications. In Homogenization techniques for composite media (E. Sanchez-Palcncia and A. Zaoui, eds.), pp. 338-397. Springer-Verlag, Berlin. Zick, A. A,, and Homsy, G. M. (1980). Stokes flow through periodic arrays of spheres. J . Fluid Me&. 115. 13-26.

Author Index

Numbers in italics refer 10pages on which the complete references are cited A

Abdollahi-Alibeik, J., 204, 223, 26Y, 273 Ablowitz, M. J., 3, 56, 61, 115 Acrivos, A,, 313, 348 Adler, P. M., 320, 345, 347 Aki, K., 65, 115 Alekseenko, S. V., 29, 32, 34, 45, 56 Anderson, C. R., 249, 257-258, 260, 267 Anderson, K. E., 2, 56 Andrews, D. G., 176, 267 Ark, R., 314, 319, 345 Armstrong, R. C., 12, 31, 58 Asai, M., 192-193, 271 Aubry, I.V., 2, 56 Auriault, J.-L., 277, 297, 300, 303, 306, 313, 320, 344, 345, 347

Bensoussan, A,, 278, 292, 313, 345 Bergdniasco, L., 61, 117 Berry, M. V.,101, 116 Betz, A., 231, 268 Bliven, L. F., 63, 116 Bouard, R., 257-259, 268, 272 Bourgat, J . F., 313, 345 Boutin, C., 344, 345 Brady, J . F., 327, 346 Brennen, C., 128, 274 Brenner, H., 221-222, 268, 319-320, 323, 325, 327, 345-346 Brokaw, C. J., 128, 274 Br@ns,M., 165, 268 Browand, F. K., 83, 111, 116 Brown, G. L., 198, 270 Brown, R. A,, 12, 31, 58 Brune, G . W., 231, 268

B Baker, G. R., 167, 267 Baker, S. J., 204, 268 Bakhvalov, N. S., 278, 345 Balmforth, N. J., 30, 40, 42, 56 Banilower, E., 13, 58 Bankoff, S. G., 25, 57 Batchelor, G. K., 122-124, 129, 132, 143, 151, 165, 226-227, 268 Beale, J. T., 255, 268 Bear, J., 319, 345 Bedford, A,, 278, 345 Behringer, R. P., 2, 56 Bendat, J. S., 65, 67, 115 Benjamin, T. B., 13, 56, 63, 115 Benny, D. J., 60, 115

C

Caflisch, R. E., 344, 346 Caponi, E. A,, 179, 268 Carman, P. C., 296-297, 346 Carr, L. W., 123, 189, 192, 272 Carslaw, H. S., 334, 346 Carter, J . E., 196, 268 Caruthers, J., 230, 268 Casey, J., 163, 268 Caswell, B., 129, 268 Chang, H.-C., 1 , 1, 7, 11-13, 15, 20, 22-23, 25, 27-28,30-31,40-41,43, 54, 56-58 Chapman, G. T., 188, 268 Chen, H. H., 3, 58 349

350

Author Index

Cheng, M., 15, 20, 22-23, 54, 56, 58 Chereskin, T . K., 72-73, 116 Chian, C., 335, 341, 347 Childress, S., 128, 268 Chomenton, F., 231, 268 Chorin, A. J., 249, 254, 268 Chu, V. C., 61, 116 Cohen, L., 65, 67-68, 116 Collins, F. G., 192, 268 Combarnous, M. A,, 319, 346 Costelli, A. F., 231, 271 Cottct, G. H., 254, 268 Coullet, P., 2, 56 Coutaneau, M., 257-259, 268 Cox, R. G., 327, 346 Crittenden, J. C., 334, 346 Crow, S. C., 204, 268 Cruz, M. E., 344, 346

D Dagan, G., 303, 319, 346 Dahm, W.J.A., 160, 268 Davcy, A., 61, 116 Davis, S. H., 13, 25, 57 Degond, P., 254, 268 Deissler, R. J., 13, 56 Delannay, R., 347 Demay, Y., 17, 19, 57 Demekhin, E. A,, 1, 7, 11, 13, 15, 22-23, 30-31, 40-41, 43, 54, 56 Denny, M. W., 335, 346 Derrick, G. H., 313, 347 Dill, L. H., 320, 346 Dishington, R. H., 129, 268 Dofigalski, T. L., 127, 187, 268 Dommermuth, D. G., 210-211,219,221, 268 Donaldson, C. duP., 231, 274 Donelan, M. A., 71, 98, 101, 105, 113, 116 Drazin, P. G., 2, 57 Druhmeller, D. S., 278, 345

E Edwards, D. A., 221-222, 268, 320, 346 Elphick, C., 2, 34, 36, 56-57 Ene, H. I., 278, 292, 306, 346 Eraslan, A. H., 135, 177-178, 268 Erlebacher, G., 198-200, 269 Evans, J. W., 2, 57

F Feir, J . E., 63, 115 Ferguson, W. E., 61, 79, 93, 113, 116 Fornberg, B., 179, 268 Fric, T. F., 185, 269 Fried, J. J., 319, 346 Friedrichs, K. O., 168, 269 Frisch, U., 41, 57 Fung, Y . C., 128, 269

G

Gad-el-Hak, M., 187, 198, 269 Gaponov-Grekhov, A. V., 11 1, 116 Garrone, A,, 231, 271 Ghaddar, C. K., 344, 346 Cierke, J. S., 334, 346 Gilbert, F., 278, 346 Glendinning, P.. 30, 39-40, 57 Goldstein, R. E., 2, 56 Gollub, J. P., 13, 20-21, 23, 25, 29, 32, 41, 45, 50, 57-58 Gray, W. G., 301, 346 Greengard, C., 255, 268 Gresho, P. M., 121, 248, 250, 269 Gu, J. W., 165, 181-183, 188, 274 Gunaratne, G. H., 2, 56 Gunn, D. J., 346 Gunzburger, M. D., 198-200, 269 Gupta, A. K., 190, 269

H

Hackett, J. E., 231, 269, 273 Hald, 0. H., 121, 269 Hancock, D. W., 62, 117 Handler, R. A., 206, 269 Hara, T., 113, I16 Harvey. J. K., 204, 269 Hashimoto, H., 61, 716 Hasselmann, K., 63, 116 Hayes, W. D., 202, 269 Hayot, F., 41, 57 Hines, D. E., 62, 117 Hirsa, A,, 204, 223, 269, 273 Ho, C . M., 83, 111, I16 Holmcs, P., 2, 56 Homsy, C . M., 295, 327, 348

35 1

Author Index Hornung, H., 124, 126, 152-153, 182, 188, 269 Hornung, U., 328, 346 Howe, M. S., 141, 269 Howes, F., 278, 302, 346 Huang, M., 255, 275 Huang, N. E., 59, 60, 63, 68-76, 79, 81-83, 92, 98, 101, 105, 111, 116-117 Huerre, P., 15, 57 Hung, S. C., 257, 269 Hunt, J. N . , 335, 340, 346 Hussain, F., 142, 271, 273 Hussaini, M. Y., 198-200, 269 Hutzler, N. J., 334, 346 I Ierley, G. R., 30, 34, 36, 40, 56-57 Illingworth, C. R., 241, 269 Infeld, E., 114, 116 looss, G., 17, 19, 57

J Jaeger, J. C., 334, 346 Janssen, P.A.E.M., 23, 57 Jayaprakash, C., 41, 57 Joets, A,, 2, 57 Johns, B., 335, 340, 346 Joo, S. W., 13, 25, 57 J o s h , R. D., 198-200, 269 K Kachanov, Y. S., 2, 26, 57 Kajiura, K., 337, 346 Kalaidin, E., 23, 40, 43, 56 Kalliadasis, S., 27, 57 Kapitza, P. L., 32, 57 Kapitza, S. P., 32, 57 Kawahara, T., 28, 30, 42, 57 Keller, J. B., 2, 58, 278, 287, 292, 313, 346 Keolian, R., 2, 58 Kida, S., 142, 271 Kinney, R. B., 257, 269 Kirschgassner, J., 17, 19, 57 Klein, R., 230, 273 Knight, D. D., 179, 268 Koch, D. L., 327, 346 Koehler, F., 65, 117

Kopelevich, D. I., 7, 11, 13, 15, 22, 30-3 I , 40, 41, 54, 56 Koumoutsakos, P., 169-170, 248-249, 254, 257-258, 270 Kovacs, G., 300, 346 Kroner, E., 278, 346 Kruskal, M. D., 3, 57 Kuznetsov, E. A., 61, 117

L Lagerstrom, P. A,, 143, 177, 270 Lai, R. J., 71, 98, 101, 105, 116 Laitone, E. V., 124, 132, 273 Lake, B. M., 60-63,79,88,93, 113, 116-117 Lamb, H., 124, 204, 209, 227, 270 Larson, J. W., 61, 117 Laurent, J., 231, 268 Leconinte, Y., 123, 270 Lee, C. K., 295, 326, 346 Lee, P. C. Y., 301, 346 Lee, Y. C., 3, 58 Leighton, R. I., 206, 269 Leonard, A., 123,170,248-249,254,257-258, 2 70 Lesieur, M., 140, 270 Levy, T., 344, 347 Lewis, Z . V., 101, 116 Liepmann, H. W., 198, 270 Lighthill, M. J., 122-123, 128, 143-144, 151, 158, 161, 227, 270 Lillcy, D. E., 231, 273 Lin, C. C., 170, 270 Lin, J. C., 210, 270 Lin, Q., 190, 270 Lin, R. Q., 60, 68-76, 81-83, 116-117 Lin, S. P., 13, 28, 57 Lions, J. L., 278, 292, 313, 345 Liu, J., 13, 20-21, 23, 25, 29, 32, 41, 45, SO, 57-58 Liu, J. S., 213-218, 274 Lo, J. M., 72, 117 Long, S. R., 59, 60, 63, 68-76, 79, 81-83, 92, 98, 101, 105, 116-117 Longuet-Higgens, M. S., 124. 270 Longuet-Higgins, M. S., 9.5, 113, 116-117, 335, 347 Lu, Z. Y., 146, 254, 270 Lugt, H. J., 125, 143, 150, 365-166, 219, 270-271

352

Author Index

Lumley, J. L., 2, 56 Lundgren, T. S., 124, 207, 209, 212-213, 271 M Ma, H. Y., 124, 144, 146, 165, 173-175, 177, 187,190,213-218,230,248,250, 258-261, 263, 268, 274 Majda, A,, 121, 271 Mangler, K. W., 165, 271 Manneville, P., 28, 32, 58 Mansour, N. N., 212, 271 Mas-Gallic, S., 254, 268 Maskell, E. C., 231, 271 Mauri, R., 320, 347 McIntyre, M. E., 176, 267 McKenzie, D. B., 334, 346 McKenzie, D. R., 313, 347 McLean, J. W., 179, 268 McPhedran, R. C., 313, 347 Mei, C. C., 61, 113, 116, 277, 286, 295, 297, 303, 306, 320, 326, 328, 335, 340-341, 343-344, 346-348 Mei, L., 110-111, 117 Meiron, D. I., 167, 267 Melander, M. V., 142, 271, 273 Melville, W. K., 71-73, 81, 88, 90, 111, 113, 117 Meron, E., 2, 57 Mielke, A,, 17, 19, 57 Miksis, M. J., 344, 346-347 Milling, R. W., 198, 271 Milne-Thornson, L. M., 167, 271 Moffat, H. K., 2, 58 Moffatt, H. K., 142, 271 Mollo-Christensen, E., 60, 63-64, 71-73, 88, 116-117 Monkewitz, P. A., 15, 57 Moore, D. W., 124, 207, 211-212, 271 Moore, K. J., 178 Morkovin, M. V., 171-172, 271 Morton, B. R., 123, 144, 158, 176, 271 Moses, H. E., 140, 271 Mui, R.C.Y., 210-211, 268 MGller, E. A,, 187, 271 Muskhelishvili, N. I., 65, 67, 117 N

Naciri, M., 286, 347 Naghdi, P. M., 163, 268

Nakaya, C., 28, 32, 58 Nakoryakov, V. E., 29, 32, 34, 45, 56, 58 Newell, A. C., 60, 115 Newrnan, J. N., 247, 271 Ng, C.-O., 277, 328, 347 Nicolaides, R. A., 198-200, 269 Nigmatulin, R. I., 278, 347 Nishioka. M., 171-172, 192-193, 271 Nosenchuck, D. M., 198, 270

0 Oberrneier, F., 187, 271 Ohkitani, K., 142, 271 Ohring, S., 219, 271 Ondrusek, B., 235-237, 273 Ono, H., 61, 116 Onorato, M., 231, 271 Oppenheirn, A. V., 65, 117 Orlandi, P., 204, 271 Orszag, S. A,, 167, 267 Osborne, A. R., 61, 114, 117 Oswatitsch, K., 165, 183, 271

P

Panasenko, G., 278, 345 Pandit, R., 41, 57 Panton, R. L., 123, 145, 271 Papanicolaou, C. C., 344, 346 Papanicolaou, G., 278, 292, 313, 345 Patera, A. T., 344, 346 Paul, J. D., I?, 58 Peace, A. J., 204, 271 Peake, D. J., 188, 273 Pearson, C. E., 123, 271 Pipin, F., 146, 248-249, 254, 257, 270-271 Perrins, W. T., 313, 347 Perry, A. E., 182, 188, 269 Perry, F. J.. 204, 269 Petti, M., 114, 117 Phillips, 0. M., 60-61, I f 7 Piersol, A. G., 65, 67, 115 Piquet, J., 123, 270 Pokusaer, B. G., 29, 32, 34, 45, 56 Pokusaev, B. G., 32, 58 PoliSevski, D., 278, 346 Pomeau, Y., 28, 32, 58 Prevors, L., 347

Author Index Pridmore-Brown, D. C., 173, 272 Prokopiou, T., 20, 58 Pryce, C., 346 Purnir, A., 28, 32, 58

0 Qian, S., 3, 58

R Rabinovich, M. I., 111, 116 Radev, K. B., 32, 58 Ramarnonjiarisoa, A., 60, 63-64, 83, 88, I17 Rasoloarijaona, M., 300, 347 Regev, O., 34, 36, 57 Reider, M. B., 249, 257-258, 260, 267 Reynolds, R. C., 123, 189, 192, 272 Ribotta, R., 2, 57 Richards, P. G., 65, 115 Riley, N., 204, 271 Rizk, Y. M., 123, 273 Roache, P. J., 123, 272 Roberts, A. J., 17, 58 Robinson, S. K., 121, 187, 272 Rockwell, D., 210, 270 Rood, E. P., 125, 148, 272 Roshko, A,, 185, 269 Ross, T. J., 254, 270 Rowlands, G., 114, I16 Royer, P., 344, 347 Rubinstein, J., 2, 58, 320, 347 Rudnick, I., 2, 58 Rundgaldier, H., 61, 79, 93, 113, 116 S Saffman, P. G., 168, 177, 179, 204, 268. 272 Salarnon, T. R., 12, 31, 58 Salles, J., 347 Sanchez-Palencia, E., 278, 287, 292, 306, 344, 345-346, 348 Sangalli, M., 3, 58 Sangani, A. S., 313, 348 Sani, R. L., 250, 269 Sankar, N. L., 123, 273 Sarpkaya, T., 124, 127, 158, 219-220, 248, 2 72 Savchenko, Y. N., 179, 272 Schafer, R. W., 65, 117

353

Scheil, C. M., 160, 268 Schilz, W., 198, 272 Schubauer, G. B., 171, 272 Scott, J. F., 62, 117 Scriven, L. E., 222, 272 Sefik, B., 28, 30, 58 Segr6, E., 114, 117 Segur, H., 3, 56, 61, 115 Shankar, P. N., 173, 272 Sheffeld, J. S., 177, 272 Shen, Z . , 59, 60, 68-76, 81-83, 110-111. 116-117 Sheriff, R. E., 65, 117 Shi, Z., 185-186, 272 Shkadov, W. Ya., 12, 58 Shum, K. T., 71-72, 117 Sirovich, L., 2, 58 Skramstad, H. K., 171, 272 Slattery, J. C., 302, 348 Smith, C. R., 127, 187, 268 Smith, F. T., 165, 192, 196, 272 Smith, J. H., 165, 271 Snyder, L. J., 295, 348 Sorensen, J. P., 295, 348 Sparrow, C. T., 30, 39-40, 57 Spiegel, E. A., 2, 30, 34, 36, 40, 56-57 Sternberg, P., 2, 58 Stewart, W. E., 295, 348 Stewartson, K., 61, 116 Stone, E., 2, 56 Stuart, J. T., 177, 196, 272 Sugavanam, A., 231, 269 Sun, C. C., 295, 326, 346 Suryadevara, O., 28, 57 Suthon, P., 220, 272 Swean, T. F., Jr., 206, 269 Swearingen, J. D., 206, 269 Swift, R. N., 62, 117

T

Taneda, S., 177, 179, 272 Tanner, M. T., 65, 117 Ta Phuoc LOC,257-259, 272 Tayfun, A,, 72, 117 Taylor, G. I., 313, 319, 348 Thovert, J.-F., 347 Thual, O., 41, 57 Ting, L., 230, 273, 344, 346-347

Author Index

354

Titchmarsh, E. C., 6.5, 67, 117 Tobak, M., 188, 273 Toh, S., 28, 30, 41-42, 57-58 Tong, P., 343, 348 Tracy, E. R., 61, 117 Truesdell, C., 129, 135, 137, 143. 157, 163, 2 73 Tryggvason, G., 160, 164, 204,219, 223, 268-269, 273, 275 Tsdi, W.-T., 223, 273 Tsvelodub, 0. Yu., 28, 42, 58 Tung, C. C., 71,79,92,98, 101, 105,116-117 Turner, J. S., 113, 116

U

Ueno, T., 207, 273 Unal, G., 28, 30, 58 Utarni, T., 207, 273

WU, J . M., 119, 124, 129, 133-136, 138-139, 141-144, 146-147, 153, 155-157, 159, 165, 173-178, 181-192, 194,226, 230-231, 235-237, 239-241, 243-245, 247-252, 257-263, 272-274 WU, J . Z., 119, 123-125, 129, 131, 133-136, 138-139, 141-144,146-150, 153, 155-157, 159, 162, 164-168, 173-178, 181-192, 194, 201-203, 205, 207-208, 212-219,226,230-231,235-237, 239-241,243-245,247-252,257-263, 268, 272-274 Wu, M. L., 71, I17 WU, T. Y., 128, 146, 273-274 WU, T. Y.-T., 2, 58 WU, X. H., 144, 146, 176-178, 194~197,248, 250-252, 257-263, 274 Y

V

Vakili, A. D., 181, 185, 191-192, 273 Virk, D., 142, 273 Visser, K. V., 236

W

Waleffe, F., 142, 273 Walker, J.D.A., 127, 187, 268 Walsh, E. J., 62, 117 Wang, C. M., 249, 273 Wang, C. Y., 13, 57 Wasan, D. T., 221-222, 268 Washburn, A. E., 236 Wehausen, J. V., 124, 132, 247, 273 Whitaker, S., 278, 302, 346, 348 Willrnarth, W. W., 204, 223, 269, 273 Wu, C. J., 124, 133, 177, 181, 231, 239, 249, 273-2 74 Wu, J., 2, 58 Wu, J. C., 123, 146, 227, 231, 249, 269, 27.3

Yates. L. A., 188, 268 Yeh, H., 133, 156, 274 Yetes, J. E., 231, 274 Yih, C.-S., 13, 58 Ying, L.-A,, 255, 275 Yoshida, S., 192-193, 271 Yu. D., 219, 275 Yu, F. M., 185, 27.3 Yuan, Y., 71, 98, 101, 105, 116-117 Yue. D.K.P., 223, 273 Yuen, H . C., 60-63, 79, 88,93, 113, 116-117, 179, 268 Z Zakharov, V. E., 60-61, 63, 117 Zaoui, A,, 278, 348 Zelenevitz, J., 192, 268 Zhang, H. X., 182, 275 Zhang, P.-W., 255, 275 Zhen, S. S., 41, 57 Zheng, Q., 255, 275 Zheng, T. G., 243-245, 274 Zhou, M. D., 124, 165, 177, 187, 190, 274 Zhou, Y., 136, 274 Zick, A. A., 295, 327, 348

Subject Index

three-dimensional effects, 181- 187 wall acceleration, 175- 181 splitting and coupling, 134-148 total force and moment, 224-247 triple decomposition of stress tensor, 128- 131 vorticity creation, 148-168, 198-224 Boundary vorticity flux, 122, 239-240 in arbitrary continuous media, 152-154 integrated moments, 240-242 in viscous incompressible flows, 154-157 Bounded states, 28 Bow waves, 29, 42-53 Bubblcs boundary layer on surface, 211-213 sound propagation through liquid with bubbles, 344

A Aeroacoustic control, 187 Airfoil, diagnostics and organization, 242-245 Air venting, gasoline spills, 328 Atwood ratio, 164, 199 B

Benjamin-Feir instability, 62 Benjamin-Ono equation, 3 Benney’s equation, 11-12, 13, 28, 32 Binary interaction theory, 53 Biot-Savart law, 162 Boundary, vorticity flux, 143 Boundary conditions, 128-13 1 Boundary enstrophy flux,157-159 Boundary flux, vorticity, 151-157 Boundary layer on a bubblc surface, 211-213 cell boundary-value problem, 292-293, 295-297 free-surface boundary layers, 207-219 wave boundary layers, dispersion, 335-343 Boundary layer (BL) equation, 7-8, 11-12, 13, 3 1-32 linearized, 207-209 Boundary vortex sheets creation, 161- 162 transport equation, 162-164 Boundary vorticity dynamics, 11Y- 128, 224-264, 265 boundary conditions, 131-134 solid boundary, 168-198 controlling vorticity creation, 187- 187 pressure gradient, 169-175

c Cauchy motion equation, 133, 134 Cauchy-Riemann equations, 142, 248 Cell boundary-value problem, 292-293, 295-297 Center manifold theory, 17, 19 Channel flow, dispersion of solute, 313-319 Characteristic scale decomposition method (CSDM), 69, 71, 74 Chordwise vorticity advective flux, 236 Coalescence, stationary waves, 42-53 Coastal regions, dispersion in wave boundary layers, 335 Coherent structure theory, 2-3, 32-42 Composite media heat conduction, 309-313 mechanics, 343-344 355

356

Subject Index

Compression equation, 139 Compression variable, 129 Conduction, heat conduction in compositc, 309-313 Conductivity, hydraulic, 293-295 Continuity of acceleration, 132-133 Continuity of normal vorticity, 132 Convective instability, 13, 15 Convective vorticity flux, on a wake plane, 23 1-239 Coupling boundaries, 142-148 cross-coupling, 141- 142 inside a flow field, 139-142 momentum balance, 134-138 stress balance, 138 Crest loss, 73 Crest pairing, 64, 73, 88, 90 Cross-coupling, 141 142 Curvature tensor, 131 -

Dynamics boundary vorticity dynamics, 119-267 falling-film wave dynamics, 27-55

E

Effective diffusion coefficient. 318 Effective dispersion coefficient, 318 Elastodynamics, one-dimensional, 283-286 Enstrophy boundary flux, 157-159 in a forced water wave, 210 Enstrophy dissipation rate, 158 Enstrophy flux, 189 Envelope decomposition method, 107 Eulerian mean, vorticity creation, 176-177 Euler limit, 143, 243

F D

Darcy’s law, 286-292, 302, 303 Data processing, Hilbert transform, 65 Decoupled approximation, 254-257 Delta wing, wakc plane diagnostics, 236-239 Derived conditions, 132 Differential equation, with oscillating coefficients, 279-383 Diffusion coefficient, 3 18 Diffusivity molecular mass diffusivity, 320 porous media, 328-335 Dirichlet condition, 249 Dispersion in a porous medium, 319-328 of solute in a channel flow, 313-319 Taylor’s theory, 313-314 in wave boundary layers, 335-343 Dispersion coefficient, 318, 341 Dispersion-diffusion equation, 325-326 Dispersivity coefficients, 326-327 Drag, near-wake plane, 232-236 Duct shear flow, sound-vortex interaction, 173 Dynamic Neumann conditions, 249, 250 Dynamic processes Newtonian fluid, 128-134 splitting and coupling of, 134-148

Falling-film dynamics, model equations, 5-12 Fiber-reinforced material, stresses and strains, 343 Film thickness, falling-film dynamics, 9 Flat films, linear stability, 13 Flow coupling inside a flow field, 139-142 dispersion of solute in channel flow, 313-319 duct shcar flow, sound-vortex interaction, 173 impulsively started flow over a circular cylinder, 257-262 Navier--Stokes flow, 143, 152, 181-182 seepage flow, 286-304 splitting inside a flow field, 139-142 three-dimensional cavity flow, 262-264 viscous flow over a flexible wavy wall, 177-181 Force, on a body piercing an interface, 243, 246-247 Fourier transform, nonlinear waves, 66-70 Free-surface boundary layers, 207-219 Frequency downshift, nonlincar waves, 59-115 Froude number, 199 Fundamental frequency, 18

357

Subject Index G

I

Gallilean symmetry, 34, 35 Gasoline spills, air venting, 328 Gaster transformation, 17 Generalized Taylor dispersion method, 3 19 General jump condition, 133 Geophysics, Hilbert transform, 65 Ginzburg-Landau equation, 3, 26-27 Ground water mechanics, seepage models, 303

IBL equation, 17, 19, 32, 42, 45 Inertia, homogenization theory, 297 Interfacial velocity, falling-film dynamics, 9 Inverse scattering transform, 3

H Heat conduction in a composite, 309-3 13 convectivc diffusion, 326 Helical wave decomposition (HWD), 140-142 Hilbert transform nonlinear waves, 65-75, 93, 1 I2 uses, 65 Homogenization theory, 277-278 diffusion and dispersion, 309-343 dispersion porous media with disparate diffusivities, 328-335 in a porous medium, 319-328 of solute in a channel flow, 313-319 in wave boundary layers, 335-343 heat conduction in a composite, 309-313 one-dimensional examples, 278 differential equation with oscillating coefficients, 279-283 elastodynamics, 283-286 typical procedure, 286 other applications, 343-344 seepage flow in rigid porous media, 286 cell boundary-value problem, 292-293, 295-297 Darcy’s law, 286-292 hydraulic conductivity, 293-295 porous media with three or more scales, 303-304 spatial averaging theorem, 300-302 weak inertia, 297-300 Horn vortex, 185, 190 Horseshoe vortex, 185, 186 Hydraulic conductivity, properties, 293-295

K

Kapitza number, 7 Kinematic condition, vorticity, 248-25 1 Korteweg-de Vries (KdV) equation, 26 Korteweg-de Vries soliton (KdV soliton), 2 Kuramoto-Sivashinsley (KS) equation, 11, 12, 13, 17, 26, 28-32, 34-42 Kutta-Joukowski formula, 226-231

L

Lagrangian vortex sheet strength, 163 Lagrangian vorticity, 163 Lift, near-wake plane, 232-236 Linear theory, at wave inception, 13 Localized structures, 2-3 Longitudinal dispersivity, 326-327 Lost crest, 60, 63, 88 Low-drag airfoil, 243

M Macroscale conductivity tensor, 307-309 Marangoni effect, 221 Maximum growing frequency, 17, 18 Mean curvature, 131 Melnikov functions, 45 Method of moments, 314, 319 Molecular mass diffusivity, 320 Momentum balance, splitting and coupling, 134-1 38 Momentum gradient, 158

N

Navier-Stokes (NS) equations, 31-32 dimensional, 304 falling-film dynamics, 5-7 linearized, 14 triple decomposition, 134 vorticity, 249-250

358

Subject Index

Navier-Stokes flow, 181-182 Euler limit, 143 homogenous, 152 Near-wake plane, lift and drag constituents, 232-236 Negative interfacial gradient, 9, 10 Neumann condition, 249, 250 Newtonian fluid dynamic processes, 128-134 vector potential, 154-155 n-hump solitary waves, 28 Nonlinear kinematic advection, 142 Nonlinear waves, 60-64, 111- 115 frequency downshift, 59-1 15 Hilbert transform, 65-75, 93, 112 laboratory experiment, 75-98 surface wave dynamics experiment, 98-1 11 Nonlinear wave-wave interaction, 59, 60 Normal form theory, 28 No-slip condition, 132 No-through condition, 132

0 Ocean waves, frequency downshift in nonlinear wave evolution, 59- 115 One-dimensional elastodynamics, 283-286 One-hump waves, 28, 30, 32, 39 Orr-Sommerfeld equation, 14 Oscillating coefficients, differential equation with, 279-383 Overtone triad, 19

P Particle strength exchange, 249 Phase reversal, wave group evolution, 73, 81-82 Poincare map technique, 30 Poisson equation, 248 Porous media with disparate diffusivities, 328-335 dispersion, 319-328 modeling, 344 seepage flow, 286 cell boundary-value problem, 292-293, 295-297 Darcy’s law, 286-292 hydraulic conductivity, 293-295 spatial averaging theorem, 300-302

three or more scales, 303-309 weak inertia, 297-300 Pressurc gradient, vorticity creation on a solid wall, 169-175 Primary boundary conditions, 132 Progressive waves, dispersion coefficient, 341

R Rayleigh streaming law, 196 Reduced stress tensor, 135 Refraction law, vorticity, 150 Relative vorticity, 129

S

Saddle point, 13 Scallop waves, 54-55 Schrodinger equation, nonlinear, 61, 79 Seepage flow porous media, 286 cell boundary-value problem, 292-293, 295-297 Darcy’s law, 286-292 hydraulic conductivity, 293-295 media with three or more scales, 303-304 spatial averaging theorem, 300-302 weak inertia, 297-300 Self-coupling, 141 Separation streamline, 182-183 Shearing, 131, 134, 138, 140, 150 Shilnikov theory, 30 Sideband triad, 19 Soil, modeling, 344 Soil-vapor extraction, 328 Solitary waves, 3-5, 53-55 coalescence, 42-53 construction, 27-32 dynamics, 27-55 formation, 13-27 linear theory at wave inception, 13-18 model equations, 5-12 quasi-stationary stages, 25 speed, 35 symmetries, 32-42 synchronization, 25-26 transition state, 42-53 Soliton interfacial waves, 2

Subject Index Sound propagation, through liquid with bubbles, 344 Sound-vortex interaction, duct shear flow, 173 Source-sink criterion, 158 Spatial averaging theorem, 300-302 Splitting boundaries, 142-148 inside a flow field, 139-142 momentum balance, 134-138 stress balance, 138 Squires theorem, 13 Standing waves, dispersion coefficient, 341 Stokes-Helmholtz decomposition, 135, 139 Stokes-Helmholtz potentials, 135-136 Stokes layer, 194 Streaming velocity, 341 Stress, splitting and coupling, 138 Stress tensor reduced, 135 triple decomposition, 128-131 Stuart-Landau equation, 20 Subharmonic secondary instability, 2G24 Subharmonic triad, 19 Supercritical airfoil, 243 Surface-strain-rate tensor, 129 Surface wave dynamics experiment (SWADE), 98- 111 Surfactant, vorticity creation from a free surface, 221-224 Symmetries, solitary waves, 32-42 Synchronization, solitary waves, 25-26

T Tangent discontinuity, 164 Theoretical mechanics, Hilbert transform, 65 Thin-film dynamics, 3 Three-dimensional cavity flow, 262-264 Three-dimensional effects, vorticity creation, 181- 187 Tollmien-Schlichting wave, 171, 198 Tornado-like vortex, 185 Total-vorticity conservation law, 157 Transition, stationary waves, 42-53 Transverse dispersivity, 326-327 Transverse shearing process, 128 Turbulent vortices, 206 Two-hump waves, 28, 30

359 U

Unstable frequencies, linear filtering, 18

V

Vanishing mean drag, viscous flow over a flexible wavy wall, 177-181 Vector potential, Newtonian fluid, 154-155 Vertical vortex, interaction with free surface, 213-219 Viscosity, in vorticity creation from boundaries, 159-161 Viscous stress tensor, 128 Vortex-interface interaction, complex, 219-224 Vortex pair rebound, from a flat interface, 204 Vortex sheet, bifurcation, 164-168 Vortex-sound equation, 141 Vorticity boundary conditions, 131-134 boundary flux, 151-157 boundary vorticity dynamics, 119-128, 224-264, 265 creation, 148-168, 198-204 kinematic condition, 248-25 1 solid boundary, 168-198 controlling vorticity creation, 187-198 pressure gradient, 169-175 three-dimensional effects, 181-187 wall acceleration, 175-181 splitting and coupling, 134-148 triple decomposition of stress tensor, 128- 131 Vorticity flux at wall critical points, 182-184 boundary, 143 Vorticity moment, 226-231 Vorticity-pressure decoupled approximation, 254-257 Vorticity source strength, 122 Vorticity waves, acoustically generated, 170

W

Wake plane, 236-239 Wall acceleration, as vorticity source, 175-181

360

Subject Index

Wave boundary layers, dispersion, 335-343 Wave breaking, 112 Wave cancellation, 198 Wave evolution nonlinear waves, 60-61 solitary waves, 27-53 Wave fusion, 60, 74, 90, 111, 112 Waves bow waves, 29 n-hump solitary waves, 28 nonlinear, 60-64, 111-115 frequency downshift, 59- 115 Hilbert transform, 65-75, 93, 112 laboratory experiment, 75-98 surface wave dynamic experiment, 98-111 one-hump waves, 28, 30, 32, 39 progressive waves, dispersion coefficient, 34 1 scallop waves, 54-55 solitary waves, 3-5, 53-55 construction, 27-32 dynamics, 27-55 formation, 13-27 linear theory at wave inception, 13-18

model equations, 5-12 quasi-stationary stages, 25 speed, 35 symmetries, 32-42 synchronization, 25-26 transition state, 42-53 soliton interfacial waves, 2 standing waves, dispersion coefficient, 341 stationary waves, coalescence, 42-53 Tollmien-Schlichting instability waves, 171 two-hump waves, 28, 30 vorticity waves, acoustically generated, 170 Wave studies boundary vorticity dynamics, 119-265 dispersion in wave boundary layers, 335-343 frequency downshift in nonlinear wave evolution, 59-1 15 frequency upshift, 113 homogenization theory, 335-343 Weak inertia, homogenization theory, 297 Weber number, 199 Wigner-Seitz grain, 295 Wind wave field, frequency downshift, 59- 1 15

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