Advances in Applied Mechanics, Volume 22

Advances in Applied Mechanics Volume 22 Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAUI. GERMAIN RODNEY HILL L. HO...

Author: Chia-Shun Yih

159 downloads 1006 Views 13MB Size Report

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

DOWNLOAD PDF

Advances in Applied Mechanics Volume 22

Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAUI. GERMAIN RODNEY HILL L. HOWARTH T. Y. Wu

Contributors to Volume 22 B. GEBHART BRUCEM.LAKE S. L. LEE R. L. MAHAJAN HENRY C. YUEN

ADVANCES IN

APPLIED MECHANICS Edited by Chiu-Shun Yih DEPARTMENT OF MECHANICAL ENGINEERING AND APPLIED MECHANICS THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

VOLUME 22

1982

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Paris San Diego San Francisco S8o Paul0 Sydney Tokyo Toronto

COPYRIGHT @ 1982, BY ACADEMIC PRESS, 1NC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED O R 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.

ACADEMIC PRESS, INC.

1I 1 Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX

LIBRARY OF

CONGRESS CATALOG CARD NUMBER:

ISBN 0-12-002022-X PRINTED IN THE UNITED STATES OF AMERICA

82 83 84 85

98 765 43 2 1

48-8503

Contents vii

LIST OF CONTRIBUTORS

ix

PREFACE

Aspects of Suspension Shear Flows S. L . Lee 2

I. Introduction 11. Theories on Laminar Shear Flows of a Dilute Suspension Considering

Only Stokes Drag and Neglecting Particulate Volume 111. Theories on Laminar Shear Flows of a Dilute Suspension Considering Both Drag and Lift Forces and Neglecting Particulate Volume IV. Theories on Laminar Shear Flows of a Dilute Suspension Considering Both Drag and Lift Forces, Density Effects, and Finiteness of Particulate Volume V. Experiments on Laminar Shear Flows of a Dilute Suspension by the Use of Laser-Doppler Anemometry Technique VI. Laser-Doppler Anemometry Applied to Turbulent Shear Flows of a Dilute Suspension with a Distribution of Particle Sizes VII. Theories on Particle Deposition in Turbulent Channel Flow of a Dilute Suspension VIII. Concluding Remarks References

3 10

22 30

40 56 61 63

Nonlinear Dynamics of Deep-Water Gravity Waves

Henry C. Yuen and Bruce M . Lake I. Introduction Governing Equations Concept of a Wave Train Properties of Weakly Nonlinear Wave Trains in Two Dimensions Properties of a Weakly Nonlinear Wave Train in Three Dimensions Large-Amplitude Effects Nonlinear Wave Fields Discussion Appendix A. Interaction Coefficients Appendix B. Lorentzian and Bretschneider Spectra References

11. 111. IV. V. VI. VII. VIII.

V

68 69 71 73 96 111 153

214 223 224 225

Contents

vi

Instability and Transition in Buoyancy-Induced Flows B . Gebhart and R . L. Mahajan I . Introduction 11. Initial Instability in Thermally Buoyant Flows 111. The Downstream Growth of Disturbances in a Vertical Flow

IV. V. VI. VII. VIII. IX.

Nonlinear Disturbance Growth Transition and Progression to Developed Turbulence Predictive Parameters for the Events of Transition Plane Plume Instability and Transition Instability of Combined Buoyancy-Mode Flows Higher Order Effects in Linear Stability Analysis List of Symbols References

232 234 237 247 259 214 283 295 305 31 1 312

AUTHORINDEX

317

SUBJECT INDEX

321

-

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

B. GEBHART, Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (23 1)

BRUCEM. LAKE,Fluid Mechanics Department, TRW Defense and Space Systems Group, Redondo Beach, California 90278 (67) S. L. LEE, Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, New York 11794 ( 1 )

R. L. MAHAJAN, Engineering Research Center, Western Electric, Princeton, New Jersey 08540 (231) HENRY C. YUEN,Fluid Mechanics Department, TRW Defense and Space Systems Group, Redondo Beach, California 90278 (67)

vii

This Page Intentionally Left Blank

Preface When one is young one is blessed with boundless energy and an unspoken faith that life will go on forever. Then one day one wakes to suspect the brevity of life and to sense the need to conserve energy and time. Such a morning did dawn upon me some years ago, and the thought, though not pressing, now refuses to be ignored or suppressed. I took on the editorial duties of this serial publication in 1970, and in the more than 10 years since then 12 volumes, Volume I 1 to the present volume, have appeared. It is now time for me to devote more time to my own research. From the point of view of the publication, a change of editors will bring to it the boon of new ideas, new perception, new energy, new enthusiasm, and, not the least among them, untapped acquaintances of the new Editor among the mechanics community-for the most important task of the Editor is to impose successfully upon the right people to write for the publication, and it is ever so much easier to succeed if he has a good number of friends and acquaintances to impose upon. This volume is the last to appear under my editorship. Henceforth the editorial duties will be assumed by Professor John Hutchinson of Harvard University and Professor T. Y. Wu of the California Institute of Technology, distinguished researchers in solid mechanics and fluid mechanics, respectively. My decade with the serial has understandably nourished in me an attachment to it, and knowing that it is now in excellent hands is a great comfort to me. To preserve a measure of continuity, and to assuage aforetime any sadness that might arise, I shall stay on by becoming a member of the Editorial Board. I take this opportunity to thank members of the present Editorial Board for helping me in my work during the years of my tenure as Editor, and Academic Press for its splendid cooperation.

CHIA-SHUN YIH

ix


Advances in Applied Mechanics Volume 22


ADVANCES IN APPLIED MECHANICS, V O L l l M E 22

Aspects of Suspension Shear Flows S. L. LEE Department of Mechanical Engineering State University of New York at Stony Brook Stonv Brook. New York

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . .

2

11. Theories on Laminar Shear Flows of a Dilute Suspension Considering Only

Stokes Drag and Neglecting Particulate Volume . . . . .

. . . , . . . A. Formulation of Governing Conservation Equations . . . . . . . . . . B. Incompressible Boundary-Layer Flow along a Flat Plate . . . . . . . . C. Compressible Laminar Boundary-Layer Flow along a Flat Plate . . . . . .

3

,

D. Incompressible Flow Induced by the Impulsive Motion of an Infinite Flat Plate.. . . . . . . . . . . . . . . . . . . . . . . .

3

5 9

. 10

111. Theories on Laminar Shear Flows of a Dilute Suspension Considering Both

. . . . . . . . . 10 A. Lift Force on a Particle in Shear Flow . . . . . . . . . . . . . . . 10 B. Formulation of Governing Conservation Equations . . . . . . . . . . I2 C. Particle Migration in Laminar Mixing of a Suspension with a Clean Fluid . . . 14 Drag and Lift Forces and Neglecting Particulate Volume

D. Particle Migration in an Incompressible Laminar Boundary-Layer along a Flat Plate . . . . . . . . . . . . . . . . . E. Particle Migration in an Incompressible Laminar Boundary-Layer along a Vertical Flat Wall Including Density Effect . . . . .

Flow

. . . .

. 21

Flow

. . . .

IV. Theories on Laminar Shear Flows of a Dilute Suspension Considering Both Drag and Lift Forces, Density Effects, and Finiteness of Particulate Volume . . A. Formulation of Governing Conservation Equations . . . . . . . . . B. Incompressible Flow Induced by the Impulsive Motion of an Infinite Flat Plate. . . . . . . . . . . . . . . . . . . . . . . . .

V. Experiments on Laminar Shear Flows of a Dilute Suspension by the Use of Laser-Doppler Anemometry Technique . . . . . . . . . . . . . .

.

2I

. 22 . 22 . 25 . 30

A. Application of Laser-Doppler Anemometry Technique to Velocity Measurement of a Suspension of Uniform-Sized Particulates in Laminar Flows . . . . 30 B. Measurements in the Far-Downstream Region of an Incompressible Laminar Boundary-Layer Flow of a Suspension along a Flat Plate . . . . . 33 C. Measurements in the Near-Leading Edge Region of an Incompressible Laminar Boundary-Layer Flow of a Suspension along a Flat Plate . . . . . 37

Copyright 9 1982 hy Academic Press. Inc. All rights of reproduction in any f o r m reserved.

ISBN O-I2-M)2022-X

2

S. L. Lee VI. Laser-Doppler Anemometry Applied to Turbulent Shear Flows of a Dilute Suspension with a Distribution of Particle Sizes . . . . . . . . . . . . . 40 A. Flow Visualization by High-speed Movie Photography . . . . . . . . B. Development of LDA Techniques for Particle Sizing . . . . . . . . . C. Development of LDA Particle Sizing Techniques for Moderately Small Particles in a Nonuniformly Illuminated Measuring Volume . . . . . . D. Simultaneous. in Siru. Local Measurements of Size and Velocity Distributions of Moderately Small Particles and Velocity Distribution of . . . . . . . . . . Fluid by Particle-Path Discrimination Scheme E. Velocity and Concentration Measurements of Turbulent Flow of Dilute Glass Sphere-Air Suspension of Uniformly Large-Sized Particles in a Vertical Pipe by the Scheme of Moving Fringes From Reflected Beams , .

VII. Theories on Particle Deposition in Turbulent Channel Flow of a Dilute Suspension . . . . . . . . . . . . . . . . . . . . . .

. . .

. 40 . 41

. 44

. ,

46

54 56

A. Classical Theories of Deposition Based on Assumed Mechanism of

Eddy Diffusion of Particles in the Conventionally Defined Turbulent Core . . 56 B. Theory of Deposition Based on Particular Dynamical Response Characteristics to Eddy Motion in Surrounding Fluid . . . . . . . . . 57

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

61 63

I. Introduction The flow of a two-phase suspension of particles in a carrier fluid has been the subject of many technical and scientific investigations in a wide range of practical problems. Examples are the transport of firebrands in forest and urban fires, collection of dust and mist from chemical processes, fluidized beds, blood flow, synovial lubrication of bone joints, combustion of droplets and particles, environmental pollution control, sediment transport by water and by air, collection of ice on building and aircraft structures, and centrifugal separation of particulates from fluids. Additional examples include droplet sprays, spray cooling, heat and mass transfer in evaporators and nuclear reactors, evaporation of pesticides, pneumatic transport of particulates such as grains and coal particles, sand blasting, and spray painting, to name a few. Because of the inherent complexity and unique difficulties in each instance and the inadequacy of many standard experimental and analytical tools to deal with them, the dynamics of a particle-fluid system has for a long time been excluded from the general discipline of fluid mechanics. There have been a few detailed reviews on some of the related problems -for instance, the study on a single spherical particle’s motion in a developed laminar pipe flow by Brenner (1966) and the overall viewing of the subject area without the necessity of supplying enough detailed physics

Aspects of Suspension Shear Flows

3

by So0 (1967). It is not the purpose of this article to provide a comprehensive description of the whole field of two-phase suspension flows but rather a systematic review of only the development of the study of particle migration in the shear flows of a two-phase dilute suspension. This phenomenon lies at the center of a class of problems of two-phase suspension flows that are of extreme technical importance. It should be stated that it is not the intention of this article to include all the pertinent publications in the area of dilute suspensions but rather only those that are interrelated well enough to contribute to the central theme of this effort. The coverage will follow a logical, evolutionary sequence of activities that is illustrative of the development of a field of considerable complexity. The task will begin with a theoretical study of relatively simple laminar flow of a two-phase suspension. For this type of flow, a sufficient amount of the analytical formulation of the problem can be readily extended from the established counterpart for conventional single-phase flow, and the all-important linkage of dynamic interaction between the phases can be provided by the considerable preexistent understanding of a particle’s behavior in the flow of viscous fluid. The corresponding experimental efforts will then be presented to provide the necessary comparisons for the theories. Of particular importance is the use of the nonintrusive laserDoppler anemometry optical technique that has been developed for laminar two-phase flows. A natural outgrowth of this is the significant development of a new generation of laser-Doppler anemometry techniques designed specifically for the complex turbulent flow of a two-phase suspension in which the particles are allowed to have a distribution in size. Finally, a presentation will be made of a new theoretical understanding of the migratory behavior of particles in a two-phase suspension in turbulent shear flow by analyzing the dynamical-response characteristics of a particle’s interaction with eddies in the surrounding fluid with prescribed turbulence structure. 11. Theories on Laminar Shear Flows of a Dilute Suspension Considering

Only Stokes Drag and Neglecting Particulate Volume A. FORMULATION OF GOVERNING CONSERVATION EQUATIONS

Conservation equations for nondilute suspensions in laminar flow have been given by Van Deemter and Van der Laan (1961), Hinze (1962), Murray (1965), So0 (1967), and Vasiliev (1960). In the work of Van Deemter and Van der Laan (1961), no discussion took place on the terms containing the stresses or solid-fluid interaction forces. In trying to clarify

4

S. L. Lee

this point, Hinze ( I 962) introduced certain inconsistencies in the interaction forces, as observed by Murray (1965), and made use of an effective composite stress instead of the pure fluid stress, as discussed by DiGiovanni ( 1971). The particular difficulty of properly describing these interaction forces lies in that the effects of flow-field interference within particle clouds are usually very complex. For a suspension of sufficient diluteness, however, these effects are greatly reduced. In this spirit, Marble (1963) derived the conservation equations for dilute suspensions. An outline of this derivation without the inclusion of heat transfer between the phases is given here. The interaction forces between the phases play a pivotal role in the motion of a two-phase suspension flow. In general, these forces depend on the local flow characteristics as well as the interactions between particles. For small particle Reynolds number and sufficiently small molecular mean free path of the fluid, the Stokes drag law can be regarded as an acceptable approximation in the absence of interference from neighboring particles. In this case, the interaction between a spherical particle and the surrounding fluid can be characterized by the velocity equilibrium (or relaxation) time of the particle, or its equivalent, the velocity range (or relaxation distance) of the particle where m is the particle mass, a the particle radius, p the fluid viscosity, ps/pr the ratio of the intrinsic mass density of the particle to that of the

fluid, v the fluid kinematic viscosity of the fluid, and U a characteristic velocity of the flow system. By comparing the particle relaxation distance A, with a characteristic length of the flow system L, one can estimate the dependence of the motion of a particle on the initial and the local fluid flow conditions. For the case in which A,/L. k 0

sw

6

>

. STANDARD

4

D E V I A T I ON

ON A X I A L V E L O C I T Y

\ 2

STANDARD D E V I A T I O N ON L A T E R A L V E L O C I T Y -0-

1-

(

LATfRAL YELOCjTY

0

10

20

30

40

50

60

70

80

90

100

DROPLET SIZE IpmJ

FIG.33. Sample droplet velocity distributions: measuring location was 0.25 rnrn from wall (from Srinivasan and Lee, 1978).

4. Two-Dimensional Measurement in Turbulent Flow of Dilute Water Droplet-Air Suspension of Size Range up to 100 pm in Vertical Channel with Wall Film

Srinivasan and Lee ( 1979) further used the improved particle-path-length discrimination scheme to investigate the turbulent flow of a dilute water droplet-air suspension inside a vertical 10 X 25 mm rectangular channel


53

12

-E AXIAL VELOCITY

10

8

6

4

STANDARD DEVIATION

STANDARD D E V I A T I O N

ON LATERAL VELOCITY

2

0

I 0

1

2

(LATERAL

VELOCITY

3

4

I 5

DISTANCE FROM W A L L (mm)

FIG.34. Air velocity distributions (from Srinivasan and Lee. 1978).

with an established water film on its inside walls. Measurements were similarly made across the channel. An analysis of the experimental data reveals the active re-entrainment of large-sized droplets from the continuous wall film, formed by the accumulated deposition of smaller droplets, and their breakup and subsequent coalescence in the flow, as shown in Fig. 36. These larger droplets from the wall film move into the flow and soon break up into clouds of smaller droplets apparently due to the violent turbulent fluctuations of the air flow. At 1 mm from the wall, a very large number of smaller droplets are found to move at higher lateral velocities pointing away from the wall. These smaller droplets then coalesce to form

S. L. Lee

54

t

P 10pm

0 ,'

1oo

t

t n a w

m

H

3

z

10-2

I-

w b

10-3

1

1

0

1

1 2

1

1

1

3

4

5

DISTANCE FROM WALL (mm)

FIG.35. Migration of droplets in turbulent dilute suspension flow in a vertical rectangular channel (from Srinivasan and Lee. 1978).

large droplets at 2 mm from the wall, which later break up again into medium-sized droplets at 3 mm from the wall.

E. VELOCITY AND CONCENTRATION MEASUREMENTS OF TURBULENT GLASSSPHERE-AIR SUSPENSION OF UNIFORMLY FLOWOF DILUTE IN A VERTICAL PIPEBY THE SCHEME OF LARGE-SIZED PARTICLES MOVINGFRINGES FROM REFLECTED BEAMS Using the scheme of determining the velocity of a large sphere by measuring the beat frequency of the moving fringe pattern formed by the


55

DROPLET SIZE (MICRONS1

FIG.36. Migration of droplets in turbulent dilute suspension flow in a vertical rectangular channel with wall film (from Srinivasan and Lee, 1979).

interference of two beams reflected from the surface of the sphere, which was developed by Durst and Zare (1975), Lee and Durst ( 1979) reported an experimental investigation of turbulent upward flow of a glass partiFle-air suspension with uniform-sized particles in a vertical pipe of inner radius 2.09 cm. The local time-mean axial velocities of the particles and the air were measured at various radial locations for particle sizes of 100, 200, 400, and 800 pm, as shown in Fig. 37. For the most part the particles were found to lag progressively behind the air according to their size, as expected, with the exception of the situation existing in the near-wall region for the 100and 200-pm particles in which the particles were leading the air. The thickness of this reversed slip-velocity region was about 200/0 of the pipe radius for the 100-pm particles and became about 10% of the pipe radius

S. L . Lee

56

IR!

1

0.5

0.5

0

0.5

1

I

0

11 R

1

lb)

fa1

,

II I

I 0.5

,

GIG"

U/G"

1

0.5

0

0.5

ic I

1

0

0.5

1

(dl

FIG. 37. Time-mean velocity of air (0) and glass particles ( 0 ) normalized against maximum air velocity for turbulent upward flow of a suspension in a vertical pipe: (a) 100-pm particles, ii0=5.70 m/sec, +=0.58X I0-j: (b) 200-pm, ii0=5.84 m/sec, +=0.63X lo-': ( c ) (d) 800-pm, ii0=5.66 m/sec, cp= 1.21 X 400-pm, ii0=5.77 m/sec, +=0.72X cp is the ratio of average particle and air fluxes (from Lee and Durst, 1979).

for the 200-pm particles. For the two larger-sized particles (400 and 800 pm), on the other hand, a clearly identifiable particle-free region was found near the wall and the particles lagged behind the air at all radial locations where particles were found.

VII. Theories on Particle Deposition in Turbulent Channel Flow of a Dilute Suspension

A. CLASSICAL THEORIES OF DEPOSITION BASEDON ASSUMED OF EDDYDIFFUSION OF PARTICLES IN THE MECHANISM CONVENTIONALLY DEFINED TURBULENT CORE The deposition of solid particles or droplets from a turbulent particlesuspension flow to channel walls is a problem of fundamental importance

Aspects of Suspension

Shear Flows

57

in a variety of technical areas. Although a large number of articles on this subject have appeared in the literature, reliable results from carefully planned experiments are scarce and mostly relate only to the amount of deposition at the wall without providing an answer to the question of the mechanisms in the flow that are responsible for the motion of the particles toward the wall. Examples are the measurements of deposition of relatively large droplets from a gas by Alexander and Coldren (1951) and Cousins and Hewitt (1968), and of small solid particles from air by Friedlander and Johnstone (1951). Most theoretical treatments of the subject adopt the point of view of a conventional three-layer flow structure in the vicinity of the wall-the viscous sublayer, the buffer zone, and the turbulent core-from studies of single-phase, fully developed turbulent flow. In the turbulent core, particles are assumed to be laterally transported by turbulent diffusion in quite the same way by which scalar quantites such as heat or concentration of species are assumed to be transported in a turbulent stream. Particles reaching the edge of viscous sublayer as a result of this transport are assumed to coast towards the wall across the sublayer to form deposition. A common feature of these treatments is their possession of an adjustable empirical factor that is necessary to achieve a reasonable comparison between the theoretically predicted and experimentally determined amounts of deposition for a particular flow system. Unfortunately this empirical factor is in no case a universal constant for all flow systems. For instance, the factor that is determined for a particular particle size and particle-to-fluid density ratio could make the comparison between theory and experiment on wall deposition as poor as up to four orders of magnitude apart for some other particle size and particle-to-fluid density ratio. Figure 38 shows one such comparison reported recently by J. Wildi (private communication, 1980). This obvious inconsistency leads to the questioning of the correctness of the very physics assumed in these theoretical treatments, particularly the assumed particle transport by turbulent diffusion irrespective of particle size and the ratio of its density to that of the fluid.

B. THEORY OF DEPOSITION BASEDON PARTICULAR DYNAMICAL RESPONSE CHARACTERISTICS TO EDDYMOTION IN SURROUNDING FLUID In a study of a particle’s behavior in turbulent flow, Rouhiainen and Stachiewicz (1970) used the concept of the frequency response of the particle in an oscillating flow field first developed by Hjelmfelt and Mockros (1966). An important consequence of this approach is that the

58

S . L. Lee

Dinenrionks rdaxation time T.

FIG. 38. Comparison of theoretically predicted and experimentally determined rates of deposition from turbulent flow of a dilute suspension: 1. theory of Davies; 11-1V. experiment: (11); Liu and Agarwal, V , =~:/1850 (111): Walls and Kneen and Straws. V , =~:/2530. Chamberlain (IV). Re= 10,000 (0, A), 15.000 (0).dp=0.65(63). 5.0 pm (@). (From Wildi, 1980. private communication.)

validity of the turbulent diffusion assumption for particle transport in a turbulent fluid stream can be characterized by the value of a resultant ratio of the amplitude of oscillation of the particle to that of the surrounding fluid, which is a function of the oscillation frequency of the fluid. On the dynamic behavior of a particle in the viscous sublayer, these authors separately made the observation that the classical concept of the Stokes stopping distance cannot be valid, especially in the case of a dense particle passing through the sublayer, since the effect of the transverse shear-slip lift force first derived by Saffman (1965) is no longer negligible. However, since their basic framework of flow regime classification was still that of the classical three-layer flow structure, these new revelations by themselves could not be expected to make a very significant contribution to a better theoretical understanding of the mechanisms in the flow that were responsible for the motion of the particle toward the wall.


59

In an attempt to bring the concept of frequency response of a particle to the practical problem of particle deposition, Lee and Durst (1979) introduced the simplifying model of particle response characterized by a cutoff frequency below which the particle responds fully to the fluid oscillation and above which the particle is totally insensitive to the fluid oscillation. Therefore for fluid oscillation frequencies smaller than the cutoff frequency, the particle motion is determined by turbulent diffusion, whereas for fluid oscillation frequencies greater than the cutoff frequency, the motion of the particle is controlled by the mean, or quasi-laminar, motion of the fluid. These authors obtained this cutoff frequency as a function of the particle size and the particle-to-fluid density ratio as well as the kinematic viscosity of the fluid. For the same fluid oscillation, a larger and heavier particle will respond only to the low frequencies of oscillation, whereas a smaller and lighter particle will respond to frequencies of oscillation up to a much higher level. This approach was applied to the case of the turbulent pipe flow of a two-phase, sufficiently dilute suspension in which the fluid motion was not seriously distorted by the presence of the particles. A most energetic fluctuating frequency of the fluid was evaluated as a function of radial position from the preexistent information about the turbulent pipe flow of a single-phase fluid. Matching the cutoff frequency from the characteristics of the frequency response of a particle and the most energetic fluctuating frequency from the turbulent motion of the fluid was then executed to produce the cutoff radius for the particle within the pipe, which is a function of the particle size, the physical properties of the particle and of the fluid, and the flow properties. Within the cutoff radius lies the turbulent diffusion core and outside the cutoff radius lies the annular quasi-laminar region for the particle. For the same flow, the cutoff radius decreases with increase of particle size. When the limiting particle size is reached, the turbulent diffusion core for the particle diminishes completely and the particle motion is controlled totally by the mean, or quasi-laminar, motion of the fluid. A sketch of the particle transport flow regime classifications in fully developed turbulent flow in a pipe is shown in Fig. 39. In the theoretical studies of laminar boundary-layer flows of a two-phase suspension of uniform-sized particles by Otterman and Lee ( 1969, 1970), Lee and Chan (1972), and DiGiovanni and Lee (1974), the particles at the edge of the boundary layer have negligible transverse velocity and particles are generally lagging behind the fluid in longitudinal velocity within the layer. The shear-slip lift force helps cause the creation of a low-particlenumber-density region adjacent to the wall. This prediction was virtually verified by the presence of a particle-free region adjacent to the boundary wall by the experiments of Lee and Einav (1972).

S . L. Lee

60

Viscous sublayer

Conventional singlephase fluid-flow regime classification

----------- - -

Viscous sublayer

Turbulent diffusioncontrolled core region Mean fluid motioncontrol led quasi-laminar region

Conventional particletransport flow regime classification

------------

Present frequency response-based particletransport flow regime classification

------ -----

FIG. 39. Particle transport flow regime classifications in fully developed turbulent pipe flow (from Lee and Durst, 1979, 1980).

By including the same shear-slip lift force in the study of the behavior of a particle in the viscous sublayer at the edge of which the particle has a nonnegligible initial transverse velocity, Rouhiainen and Stachiewicz ( 1970) were able to provide a realistic dynamic mechanism for the deposition of particles on the boundary wall across the sublayer. In particular, they pointed out the importance of the directional reversal of this lift force on the two sides of the transverse matching position of the longitudinal velocities of the fluid and the particle. For a given main flow in which the particle is lagging behind the fluid in the longitudinal direction, the particle coming into the sublayer initially experiences a combined resistance of drag and lift forces. If the initial transverse particle velocity is not high enough for it to reach the matching location, the particle will be kept away from the wall in a way similar to the finding of two-phase laminar boundarylayer studies mentioned previously. However, if the initial transverse particle velocity is high enough for it to pass through the matching location, the lift force will thereafter reverse its direction and help propel the particle towards the wall. In the previously mentioned experimental investigation of turbulent upward flow of a glass particle-air suspension in a vertical pipe by Lee and


61

Durst (1979), two inexplicable phenomena emerged from the results, as shown by Fig. 37. For the two smaller-sized particles (100 and 200 pm), there appeared a slip-velocity reversal zone near the wall. For the two larger-sized particles (400 and 800 pm) there appeared a particle-free zone. At first glance, it would seem that both these phenomena could be qualitatively explained by the role played by the shear-slip lift force on a particle along its trajectory in the viscous sublayer in the theory proposed by Rouhiainen and Stachiewicz (1970). However, there is a serious fault in trying to make this connection. Rouhiainen and Stachiewicz’s theory is based on the shear flow within the extremely thin viscous sublayer adjacent to the wall, of a thickness of around 0.018 pipe radius for the present case. The thickness of the apparent shear region for these particle sizes must have been actually many orders of magnitude larger since the radial matching positions for the 100- and 200-pm particles are already approximately 0.8 and 0.9 pipe radius, respectively, and the thicknesses of the particle-free zone for the 400- and 800-pm particles are already of the order of magnitude of 0.10-0.15 pipe radius. According to the analysis by Lee and Durst (1979), which is based on the frequency response of a particle, for each particle size there is a particular value of the cutoff radius that separates the core controlled by turbulent diffusion and the quasi-laminar annular region controlled by the mean fluid motion. For the flow under discussion, this cutoff radius becomes zero at a particle size of only 21 pm and remains zero for larger sizes. In other words, for the particle sizes involved, there is no longer a core controlled by turbulent diffusion. As far as these particles are concerned, the whole flow across the pipe section can be considered quasi-laminar and the motions of the particles are controlled by the mean fluid motion. With this rational justification, the role played by the generalized shear-slip lift force can then be used to provide a qualitative explanation of the behavior of particles in the aforementioned turbulent two-phase suspension flow in a vertical pipe.

VIII. Concluding Remarks The development of the relatively restricted branch of study of two-phase suspension flows-that for a dilute suspension-has been systematically reviewed. Under its own set of restrictions, physically interesting and pottntially useful results have been reported in each of the four selected areas of this field of research, namely theory and experiment for laminar or turbulent flows. However, further progress in the field would depend on the success of efforts to overcome the difficulties resulting from the relaxation of some or all of these restrictions.

62

S.L. Lee

In the area of theory for laminar flows, the most severe restrictions are the assumptions of diluteness and of uniform-sized particles and the ability of most analytical schemes to obtain results only for separated extreme ranges, instead of for the whole continuous range, of a coordinate parameter. For denser suspension, the interactions among particles and, in the case of particles with size distribution, the collisions among particles must be accounted for. Progress in techniques of solving partial differential equations will help construct solutions applicable to the full range of the coordinate parameter. The restriction of the diluteness of the suspension also puts a constraint on the area of experiment on laminar flows as well as the area of experiment on turbulent flows because at higher particle concentration levels the detrimental overblocking of the beams of the optical probing system by the particles will become unavoidable. New experimental techniques will have to be developed to overcome this difficulty. One possibility could be the use of fiber optics to protect the beams from such overblocking. In addition to this, there is always the problem of the need for better and faster electronic and computational facilities to cope with the exigent requirements of data gathering, transmission, and processing. In the area of theory of turbulent flows of two-phase suspensions, because of the apparent difficulties commonly encountered in conventional theory for single-phase turbulent flows together with the difficulties outlined above for the theory of laminar flows of two-phase suspensions, the progress has been slow in coming. Furthermore, unlike the area of laminarflow theory, in which the dynamic feedback to the motion of the fluid from that of the particles has been fully included in the analysis, a similar feedback has not yet been incorporated in the area of turbulent-flow theory. Major and sustained efforts are clearly required for significant future developments in this important technical field.

ACKNOWLEDGMENTS This work has been carried out in the Department of Mechanical Engineering of the State University of New York at Stony Brook, Stony Brook, New York, and the Institute for Hydromechanics of the University of Karlsruhe, Karlsruhe, Federal Republic of Germany. The author gratefully acknowledges the generous support of the U.S. Nuclear Regulatory Commission and the Alexander von Humboldt Foundation. The author would also like to express his thanks to Professor F. Durst for many useful discussions, to Frank Lee and Mr. J. Lisca for proofreading the original draft of the manuscript, to Messrs. J. L. Wang and S . K. Cho for their contribution to the preparation of the final manuscript, and to Mrs. M. Licata For her exemplary effort in the speedy execution of its typing.


63

REFERENCES Alexander, L. G.,and Coldren, C. L. (1951). Droplet transfer from suspending air to duct wall. Ind. Eng. Chem. 45,1325. Beal, S. K. (1968). ‘‘Transport of Particles in Turbulent Flow to Channel or Pipe Walls,” Bettis At. Power Lab. Rep. No. WAPD-TM-765. Westinghouse Electric Corp.. Pittsburgh, Pennsylvania. Brenner, H . (1966). Hydrodynamic resistance of particles. Adv. Chem. Eng. 6, 287-438. Chiu. H. H. (1962). “Boundary Layer Flow with Suspended Particles,” Princeton Univ. Rep. No. 620. Princeton University, Princeton. New Jersey. Corrsin. S., and Lurnley. J. (1956). On the equation of motion for a particle in turbulent fluid. Appl. Sci. Res., Sect. A 6, 114-1 16. Cousins, L. B., and Hewitt, G. F. (1968). Liquid phase mass transfer in annular two-phase flow: Droplet deposition and liquid entrainment. U . K . A t . Energy Auth., Rep. R-5657. Cumo, M., Farello. G. E., and Ferrari. G. (1968). Notes on droplet heat transfer. Prepr. AIChE Pap., Natt. Hear Transfer Con&, 10th. 1968, N o . 21. Cumo, M., Farello, G. E., and Ferrari, G. (1970). A photographic study of two-phase, highly dispersed flows. Comitaro Nazionale Energia Nucleare Rep. RT/ING(71)8. Cumo, M.. Farello, G. E., Ferrari, G., and Palazzi, G. (1973). On two-phase highly dispersed flows. Am. Soc. Mech. Eng. Pap. 73-HT-18. Davies, C . N. (1966). Deposition of aerosols through pipes. Proc. R . Soc. London. Ser. A , 289. 235. DiGiovanni. P. R. (1971). Suspension flow: Impulsive motion over a flat plate and pulsatile tube flow. Ph.D. Thesis, State University of New York at Stony Brook. DiGiovanni, P. R., and Lee, S. L. (1974). Impulsive motion in a particle-fluid suspension including particulate volume, density and migration effects. J . Appl. Mech. 41. No. I. 35-41. Durst. F.. and Umhauer, H. ( 1975). Local measurements of particle velocities. size distribution and concentration with combined laser Doppler. particle sizing system. “The Accuracy of Flow Measurements by Laser Doppler Methods.” Proc. LDA Svmp.. I975 p. 430. Durst, F.. and Zare, M. (1975). Laser Doppler measurements in two-phase flows. “The Accuracy of Flow Measurements by Laser Doppler Methods.” Proc. LDA Symp.. 1975 p. 403.

Dussan. E. B., and Lee, S. L. (1969). Behavior of spherical solid particles released in a laminar boundary layer along a flat plate. Appl. Sci. Rex 20, 465-477. Einav. S., and Lee, S. L. (1973a). Particles migration in laminar boundary layer flow. Int. J . Mulriphase Flow 1, 73-88. Einav. S.. and Lee, S. L. (1973b). Measurement of velocity distributions in two-phase suspension flows by the laser-Doppler technique. Rev. Sci. Instrum. 44, No. 10, 14781480. Farmer, W.M. (1972). Measurement of particle size, number density, and velocity using laser interferometer. AppL Opt. 11, 2603. Friedlander, S. K.. and Johnstone, H. F. (1951). Deposition of suspended particles from turbulent gas streams. Ind. Eng. Chem. 49, 1151. Hinze. J. 0. (1962). Momentum and mechanical-energy balance equations for a flowing homogeneous suspension with slip between the phases. Appl. Sci. Res., Sect. A 11. 33-46. Hjelmfelt. A. T., Jr., and Mockros, L. F. (1966). Motion of discrete particles in a turbulent fluid. Appl. Sci. Res. 16, 149-161. Hutchinson, P., Hewitt. G. F., and Dukler, A. E. (1970). Deposition of liquid or solid dispersions from turbulent gas streams: A stochastic model. U.K . A f . Energy Auth., Rep. AERER-6637.

64

S . L . Lee

Karnis. A., Goldsmith, H. L.. and Mason, S. G. (1966). The flow of suspension through tubes. V. Inertial effects. Can. J. Chem. Eng. 44, 181. Lee, S. L., and Chan, W. K. (1972). Two-phase laminar boundary layer along a vertical flat wall. Hydrotransport 2 A4.45-A4.58. Lee, S. L., and Durst, F. (1979). On the motion of particles in turbulent flows. Sonderforschungsbereich 80 Univ. of Karlsruhe, Karlsruhe, W. Germany. SFB/80/TE/142. Lee, S. L., and Durst, F. (1980). A new analytical approach to deposition from a dispersion in turbulent flows based on particles’ frequency response characteristics. In “Polyphase Flow and Transport Technology” (R.A. Bajura, ed.), pp. 223-231. Am. SOC.Mech. Eng., New York. Lee, S. L., and Einav, S. (1972). Migration in a laminar suspension boundary layer measured by the use of a two-dimensional laser-Doppler anemometer. Prog. Heat Mass Transfer 6, 385-403. Lee, S. L., and Srinivasan, J. (1978a). Measurement of local size and velocity probability density distributions in two-phase suspension flows by laser-Doppler technique. Int. J. Multiphase Flow 4, 141-155. Lee, S. L., and Srinivasan, J. (1978b). An experimental investigation of dilute two-phase dispersed flow using L.D.A. technique. Proc. Heat Transfer Fluid Mech. Inst. pp. 88-102. Lee, S. L.. Srinivasan, J., Cho, S. K., and Malhotra, A. (1979a). A study of droplet hydrodynamics important in upper plenum in LOCA. Proc. Two-Phase Instrum Rev. Group Meet. 1979. p. 111.14-1. Lee, S. L., Srinivasan, J., and Cho, S. K. (1979b). Droplet entrainment studies of dispersed flow through tie plate in LOCA by LDA method. Proc. Water Reactor Sa$ Res. In$ Meet. 7th, 1979. Lee, S . L., Srinivasan, J., and Cho, S. K. (1980). LDA measurement of droplet behavior across tie plate during dispersed flow portion of LOCA reflood. Am. Soc. Mech. Eng. Pap. 80-WA/NE-4. Lin, C. S., Moulton, R. W., and Putman, G. L. (1953). Mass transfer between solid wall and fluid streams. Ind. Eng. Chem. 45, 667. Liska, J. J. (1979). The application of laser doppler anemometry to bubbly two-phase flows. M.A.S. Thesis, University of Toronto, Toronto, Canada. Liu, J. T. C. (1967). Flow induced by the impulsive motion of an infinite flat plate in a dusty gas. Astronaut. Acta 16, 369-376. Marble, F. E. (1963). Dynamics of a gas containing small solid particles. Proc. AGARD Combust. Propuls. Colloq., 5th, 1963 pp. 175-215. Murray, J. D. (1965). On the mathematics of fluidization. I. Fundamental equations and wave propagation. J . Fluid Mech. 21, 465-493. Murray, J. D. (1967). Some basic aspects of one-dimensional incompressible particle-fluid two-phase flows. Astronaut. Acta 13, 417-430. Ogden, D. M., and Stock, D. E. (1978). “Simultaneous Measurement of Particle Size and Velocity via the Scattered Light Intensity of a Real Fringe Laser Anemometer,” Therm. Energy Lab. Rep. No. 78-27. Mech. Eng. Dept., Washington State University, Pullman. Otterman, B., and Lee, S. L. (1969). Particle migrations in laminar mixing of a suspension with a clean fluid. Z. Angew. Math. Phys. 20,730-749. Otterman, B., and Lee, S. L. (1970). Particle velocity and concentration profiles for laminar flow of a suspension over a flat plate. Proc. Heat Transfer Ffuid Mech. Inst. pp. 31 1-322. Rayleigh, Lord (I91 1). On the motion of solid bodies through viscous liquids. Phihs. Mag. [6] 21, 697-711. Rouhiainen, P. O., and Stachiewicz, J. W. (1970). On the deposition of small particles from turbulent streams. J. Heat Transfer 92, 169-177.


65

Rubin. G. (1977). Widerstands-und Auftriebsbeiwerte von rut enden, kugelformigen Partikeln in Stationaren. wandnahen laminaren Grenzschichten. D. Eng. Dissertation, University of Karlsruhe. Karlsruhe, West Germany. Rubinow, S. I., and Keller, J. B. (1961). The transverse force on a spinning sphere moving in a viscous fluid. J . Fluid Merh. 11, 447-459. Saffman, P. G. (1965). The lift on a small sphere in a slow shear flow. J . Fluid Mech. 22, 385-400. Singleton. R. E. (1965). The compressible gas-solid particle flow over a semi-infinite flat plate. Z. Angew. Marh P h p . 16, 421-428. Soo, S. L. (1967). “Fluid Dynamics of Multiphase Systems.’’ Ginn (Blaisdell). Boston, Massachusetts. Srinivasan, J.. and Lee, S. L. (1978). Measurement of turbulent dilute two-phase dispersed flow in a vertical rectangular channel by laser-Doppler anemometry. In “Measurements in Polyphase Flows” (D. E. Stock, ed.), pp. 91-98. Am. Soc. Mech. Eng., New York. Srinivasan, J.. and Lee, S. L. (1979). Application of laser-Doppler anemometry technique to turbulent flow of a two-phase suspension. Int. Symp. Papermach. Headboxes [ P r e p . 1, 1979 pp. 25-30. Stokes. G . G. (1851). On the effect of internal friction of fluids on the motion of pendulums. Trans. Cam. R. Soc. 9. 8-106. Tam, C. K. W. (1969). The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38, 537-546. Ungut, A., Yule, A. J., Taylor, D. S., and Chigier. N. A. (1978). Particle size measurement by laser anernometry. J . Energy 2, No. 6, 330-336. Van Deemter, J. J., and Van der Laan, E. T. (1961). Momentum and energy balances For dispersed two-phase flow. Appl. Sci. Res., Seer. A 10, 102-108. Vasiliev. 0. F. (1960). Problems of two-phase flow theory. froc. Congr. Int. Assoc. Hydraul. Res.. 13th, 1960 Vol. 5-3, pp. 39-84. Wigley. A. (1977). The sizing of large droplets by laser anemometry. U.K.Ar. Energy Auth., Rep. AERER8771. Yeh, Y., and Cummins, H. Z. (1964). Localized fluid flow measurements with a He-Ne laser spectrometer. Appl. Ph,vs. Leli. 4, 176- 178. Yule, A. J., Chigier, N. A., Atakan, S., and Ungut. A. (1977). Particle size and velocity measurement by laser anemometry. J . Energy 1, No. 4, 220-228. Zuber, N. (1964). On the dispersed two-phase flow in a laminar flow region. Chem. Eng. Sci. 19, 897-917.


.

ADVAhUCES IN APPLIED MtC HANIC'S V O l . b M t 22

Nonlinear Dynamics of Deep-Water Gravity Waves HENRY C. YUEN

AND

BRUCE M . LAKE

Fluid Mechanics Department TR W Defense and Space Svstems Group Redondo Beach. California

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . 68 I1. Governing Equations . . . . . . . . . . . . . . . . . . . . . 69 A. Dispersion Relation . . . . . . . . . . . . . . . . . . . . . 70 B . StokesWaves . . . . . . . . . . . . . . . . . . . . . . . 71 111. Concept of a Wave Train . . . . . . . . . . . . . . . . . . . . 71 IV . Properties of Weakly Nonlinear Wave Trains in Two Dimensions . . . . . . 73 A . Nonlinear Schrodinger Equation . . . . . . . . . . . . . . . 74 B. Steady Solutions of Nonlinear Schrodinger Equation . . . . . . . . . . 76 C. Envelope Solitons . . . . . . . . . . . . . . . . . . . . . 77 D . Modulational Instability of the Uniform Wave Train . . . . . . . . . 86 E . Long-Time Evolution of an Unstable Wave Train . . . . . . . . . . . 90 F. Relationship between the Initial Condition and the Long-Time Evolution of an Unstable Wave Train . . . . . . . . . . . . . . . 90 V . Properties of a Weakly Nonlinear Wave Train in Three Dimensions . . . . . . 96 97 A . Steady Solutions . . . . . . . . . . . . . . . . . . . . . . B. Stability of Steady Plane Solutions to Cross-Wave Perturbations . . . . . . 98 C . Recurrence . . . . . . . . . . . . . . . . . . . . . . . . 103 D . Relationship between Initial Instability and Long-Time Evolution in Three Dimensions-Quasi-Recurrence and Energy Leakage . . . . . . 103 E. Summary . . . . . . . . . . . . . . . . . . . . . . . . 111 VI . Large-Amplitude Effects . . . . . . . . . . . . . . . . . . . . 1 11 A. Derivation of Zakharov's Integral Equation . . . . . . : . . . . . . 1 1 1 B. Stability of a Uniform Wave Train . . . . . . . . . . . . . . . . 117 C . Restabilization . . . . . . . . . . . . . . . . . . . . . . 125 D . Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 128 E . A New Type of Three-Dimensional Instability . . . . . . . . . . . .146 V11 . Nonlinear Wave Fields . . . . . . . . . . . . . . . . . . . . . 153 A . Dispersion Relations

. . . . . . . . . . . . . . . . . . . . B . Statistical Properties . . . . . . . . . . . . . . . . . . . . .

C . Properties of the Discretized Zakharov Equation

. . . . . . . . . .

153 180 . 195

67

.

Copyright 1982 bq Academic Prer5 InL All righfs of reproduction in any form reserved ISBN 0-12-002022-X

68

Henry C . Yuen and Bruce M. Lake

VIII. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 214 Appendix A. Interaction Coefficients . . . . . . . . . . . . . . . . 223 Appendix B. Lorentzian and Bretschneider Spectra . . . . . . . . . . . 224 References . . . . . . . . . . . . . . . . . . . . . . . . .225

I. Introduction

This article reviews some recent progress in the nonlinear dynamics of deep-water gravity waves. It attempts to highlight the major developments in theory and experiment commencing with the finding by Lighthill (1965) that a nonlinear, deep-water gravity wave train is unstable to modulational perturbation, up to the present investigations of various aspects of nonlinear phenomena, including three-dimensional instabilities, bifurcations into new steady solutions, statistical properties of random wave fields, and chaotic behavior in time evolution. The governing equations for inviscid, irrotational, incompressible, free surface flows are given in Section 11, together with some basic steady solutions of the system. The concept of a wave train is introduced in Section 111. The stability and evolutionary properties of a weakly nonlinear wave train in two dimensions are considered in Section IV based on the nonlinear Schrodinger equation, which is an equation describing the wave envelope. Some interesting phenomena, such as the existence of envelope solitons, and the Fermi-Pasta-Ulam recurrence in time of an unstable wave train, are examined. Section V extends these results to three dimensions, using the three-dimensional nonlinear Schrodinger equation. The results indicate that whereas the nonlinear Schrodinger equation is remarkably successful in describing the two-dimensional dynamics, it is inadequate for treatment of the three-dimensional case. A more accurate approximation leads to the Zakharov equation, which is derived and studied in Section VI. The Zakharov equation predicts the existence of new types of two- and three-dimensional steady wave patterns. Section VI also reports a new type of three-dimensional instability that becomes dominant for moderate to steep waves, and that is intimately connected with the appearance of bifurcated three-dimensional wave patterns observed experimentally. Application to a wave field is considered in Section VII. A statistical formulation, based on the Zakharov equation, is presented that describes the modulational characteristics of a random nonlinear wave field. The stability of a homogeneous spectrum to modulational perturbations is also discussed. The dispersion properties of a wave field, a subject of considerable recent interest, are examined. Finally, some recent findings on the

Nonlinear Dynamics of Deep- Water Gravity Waves

69

behavior of intermodal interactions of the discretized Zakharov equation are reported. The aforementioned sections (Sections I-VII) concentrate on work either performed by or closely related to that of the staff of the TRW Fluid Mechanics Department. A more comprehensive overview of the various advances in the subject of deep-water gravity waves is given in Section VIII. No details of theory or experiment are given, but through extensive referencing, Section VIII should provide an interested reader with an up-to-date survey of the original contributions that have made the past 15 years such memorable ones for the subject of water waves.

11. Governing Equations

The equations governing the irrotational flow of an incompressible, inviscid fluid with a free surface are known as the Euler equations:

where I# is the velocity potential, q is the free surface, g is the gravitational acceleration, p is an external pressure exerted on the surface of the fluid, the horizontal coordinates are (x, y ) = x, and the vertical coordinate is z, pointing upwards. The Laplacian operator A is a2/ax2+ a2/ay2+a2/az2, and the horizontal gradient operator V is (a/ax, d/dy). Since the fluid is assumed incompressible, the density has been normalized to unity without loss of generality. Unless otherwise specified, the external pressure p will be taken to be a constant and set equal to zero without loss of generality. For a water wave to be considered a deep-water wave, its wavelength X must be small compared with the depth of the water. A gravity wave is one in which gravity is taken as the dominant (and sometimes the only) restoring force. For water waves, this would be a good assumption if the wavelength is substantially longer than 1.7 cm, at which point surface tension and gravitational effects are comparable. Strictly speaking, water waves are not inviscid. However, since viscosity is effective only for small-scale motions, it is negligible for most of the phenomena considered here.

Henry C . Yuen and Bruce M . Lake

10

A. DISPERSION RELATION

If we consider small disturbances on water otherwise at rest, the free surface elevation above mean 1) and the velocity potential cp are both small in the scales of the wavelength and wave period. The free surface conditions can be linearized. In addition, the conditions can be applied on z = 0 rather than z = q ( x , y , t) consistent with the approximation. Thus we have as boundary conditions

Elimination of yields

1)

after differentiating both equations with respect to time t

+,,,+gcpz=O on z=O. (6) This process reduces the nonlinear free-boundary problem to a linear, fixed-boundary problem for +:

A+=O,

-cQ(

k2

1-kz)cod

1

exp(kz)+

. . a ,

where the form for comes from the linear boundary condition, but allowing a to vary slowly with x and t ; the term containing (1 - kz) is required to insure that 9 satisfies Laplace's equation to the order considered. We note that Eqs. (22) and (23) are generalizations of the linear wave train: when the higher-order terms are neglected, and a , w , and k are assumed independent of x and t , they reduce to the linear wave train solutions (10) and ( I 1).


73

Substituting Eqs. (22) and (23) into the expression (21) for the averaged Lagrangian LJ, and retaining terms of order O ( k2a2,e 2 ) , where E characterizes the slowness of the variation: eaa,/ka we obtain a2a,2 aa,,

3% Variation with respect to 6 yields the energy equation (aZ),+(C,a2) .=O,

where C, is the linear group velocity

c,=aw/ak = f(g/k)1/2. Variation with respect to a gives the nonlinear dispersion relation a=( gk)’I2[ 1 + + k 2 a 2 + a , , 1 8 k 2 a ] .

Note that when the wave amplitude is assumed to be a constant, these equations merely state that the wave energy is propagated at the group velocity, and that the wave frequency admits a correction because of the nonlinearity, which are precisely the results of Stokes (1849). Equations (26) and (28) are supplemented by the wave conservation equation, which is obtained by equating 6,, to 6,, : k, w, = 0. (29)

+

This equation states that waves cannot be created or destroyed, and imposes differentiability conditions on k and a,consistent with the assumption of slow variations. Equations (26), (28), and (29) form a complete set of equations for the amplitude a , the wave frequency w , and the wavenumber k.

IV. Properties of Weakly Nonlinear Wave Trains in Two Dimensions In this section, we shall consider the properties of weakly nonlinear deep-water waves in two dimensions. The waves are assumed to propagate in the x direction, with attendant particle motion in the x-z directions (hence two-dimensional). All modulations or variations are in the x direction only, with no )-,-directional dependence. This is clearly an idealization,

74


for the ocean is three-dimensional in nature. However, there are ample justifications for us to consider this idealized case in some detail. First, it avoids the mathematical complexities of an added dimension in space and allows us to concentrate on the important mechanisms associated with nonlinearity. Second, experiments in a wave tank closely approximate unidirectional motion, so that the results obtained can be compared with experimental data. Finally, when waves are generated by a steadily blowing wind, most of the energy is propagated in a single direction-that of the wind, and important information can be obtained by considering unidirectional motion. This restriction will be relaxed in Section V when we consider three-dimensional effects. The second assumption in this section is that the nonlinearity is sufficiently weak so that the leading order approximation in Eqs. (26)-(29) is adequate. It will be seen that even with this assumption, there are some rather surprising phenomena associated with nonlinear effects. Finally, the concept of a wave train requires that the variations of the wave amplitude, wave frequency, and wavenumber be slow compared with the carrier-wave wavelength, frequency and wavenumbers. Note, however, that the total variations over a long time need not be small according to Eqs. (26)-(29). In principle, they are capable of describing wave systems that gradually accumulate a substantial departure from their initial conditions. A. NONLINEAR SCHRODINGER EQUATION

To further simplify the description, we impose the added restriction that the variations must be such that the departures of the frequency o and wavenumber k from their initial conditions are small; we can then write w=w0+(3,

(30)

k = k,+ l, (31) where wo and k , are constants fixed by the initial conditions and G 35.26', however, the relative signs of the dispersive and the nonlinear terms change, and the properties of the solutions also change substantially. We shall confine our attention to the case where (Y < 35.26'. A schematic illustration of an oblique envelope soliton for a = 30' is shown in Fig. 13.

B. STABILITY OF STEADY PLANESOLUTIONS TO CROSS-WAVE PERTURBATIONS We now consider infinitesimal cross-wave perturbations on a steady, plane, envelope soliton represented by P o ( X ) , where X = x -((00/2k0)t. We impose the disturbance given by

P + ( X ) exp( i ( Ky + at))+ P - ( X ) exp( i(Ky - a t ) ) , (78) where K is the transverse wavenumber of the perturbation, the structure functions p + ( X ) and p _ ( X ) are complex, with absolute values small compared with p o ( X ) . Substituting this into Eq. (74), linearizing about P o ( X ) , and equating coefficients of e2'"', we can cast the stability problem into an eigenvalue problem

( L,-

L , - K')

K ~ ) (

P,

=

Q'P,,

(79)

the operators Lo and L , are defined as L,=--.+y I a2 2 ax

-Po(X),

where y 2 is defined in Section III,B as y 2= 4 2 - m 2 ) / 2 .

(82) When 0 < m < 1 the unperturbed solution p 0 ( X ) is periodic. The boundary conditions for Eq. (79) are also chosen to be periodic with the same period. When m = 1, the unperturbed solution represents an envelope soliton, and the boundary condition must be chosen so as to decay with the appropriate exponential rate. When rn = 0, the unperturbed solution is a uniform wave train, the stability of which will be discussed in Section V, D.

Nonlinear Dynamics of Deep- Water Graviv Waves

1

99

(b)

FIG. 14. Sketch of odd (a) and even ( b ) disturbances on an envelope soliton.

When Q2 is real, the perturbations remain fixed with respect to the unperturbed solution in the transverse direction. Complex values of Q 2 represent perturbations that propagate in they direction as well as grow or decay in time. The linear eigenvalue problem, with Q2 as the eigenvalue and p + ( X ) [or p - ( X ) ] as the eigenfunctions, is non-self-adjoint and is most easily approached by numerical techniques. For the purposes of numerical computation and presentation, it is convenient to divide the perturbations into two types: odd and even. The terms apply to the structure of the eigenfunctions p - ( X ) and p : ( X ) with respect to the unperturbed solution. Odd applies to a situation illustrated in Fig. 14a, and even applies to that shown in Fig. 14b. We first present results with Q2 assumed real, i.e., results that represent standing wave instabilities or oscillations in they direction. The stability diagram for even perturbations is shown in Fig. 15. It indicates that the plane dnoidal envelope solutions are unstable to cross-wave disturbances for all values of the modulus in the range 0 < m < 1. In the limiting case of the soliton envelope, where m = I . the solution is stable, and the corresponding diagram for odd perturbations is shown in Fig. 16. This shows that the plane solutions are unstable to odd perturbations for all values of m, including the case m = 1, which is the soliton envelope. The above results for the case m = I were first obtained by Zakharov and Rubenchik (1974) in the long-wave (small K ) limit, and extended to finite


100

LO 0.8

-

0.6

-

0.4

-

0.2

-

i12

0

Y4 -0.2 -0.4

-0.8 -1.0 FIG. 15. Cross-wave instability diagram for stationary, odd perturbations on a plane, periodic solution of the three-dimensional nonlinear Schrodinger equation. [From Martin ei at. ( 1980). Reproduced by permission, Elsevier North Holland, Inc.]

values of K by Saffman and Yuen (1978). The results for other values of m have been obtained by Martin et al. (1980). The stability to traveling disturbances, which is the case when Q2 is complex, was calculated by Martin et a/. (1980). This involves a system double the size of the case when Q2 is real. They found no new instabilities for the odd case, but a new family of traveling instabilities for the even case was found as shown in Fig. 17. Note that these traveling instabilities appear


101

L2 1.0 0.8

0.6

a4 0.2

iE0 Y4

1. i

K

-+T

-0.2 -0.4

-0.6

-0.8 -1.0 FIG. 16. Cross-wave instability diagram for stationary, even perturbations on a plane. periodic solution of the three-dimensional nonlinear Schrodinger equation. [From Martin ei a/. ( 1980). Reproduced by permission. Elsevier North Holland. 1nc.j

to persist even for the soliton envelope case (rn = I), and cover a rather wide range of cross-wave wavenumbers. In fact, the combination of this traveling instability and the odd standing instability makes the envelope soliton unstable to a wide range of perturbation wavenumbers. The fact that the envelope solitons are unstable to cross-wave perturbations casts severe doubt upon the possibility that soliton envelopes can be useful as fundamental entities on which a wave field representation can be based. The next question is whether the concept of recurrence, and the relationship between initial conditions and recurrence, established for twodimensional situations, still hold in three dimensions.


102

a

Re -

Y2

0

0.2

0.4

0.6

0.8 K -

Y

(0)

a

lrn -

Y2

K -

Y

(b) FIG. 17. (a) Frequencies and (b) instability growth rates for traveling cross-wave instability for even perturbatlons of the steady plane solutions of the three-dimensional nonlinear Schr6dinger equation. [From Martin et al. (1980). Reproduced by permission, Elsevier North Holland. Inc.]

103


C. RECURRENCE The phenomenon of Fermi-Pasta-Ulam recurrence observed in the solution of the two-dimensional nonlinear Schrodinger equation can still be found in the three-dimensional equation for certain initial conditions. For example, it has been shown by Yuen and Ferguson ( 1 978b) that at least one set of initial conditions representing two diagonally positioned packets in a periodic box (periodic boundary conditions in both the x and y directions) exhibit recurrence after evolving into a succession of wave forms during evolution. This is shown in Fig. 18. The appearance of many wave groups during the complicated evolutionary process has attracted the attention of many physicists. In other contexts they represent the time-limited creation and destruction of spatially confined entities and are related to what physicists call “instantons.” Despite this interesting behavior, it will be seen that the phenomenon of recurrence in three-dimensional situations is the exception rather than the rule.

BETWEEN INITIAL INSTABILITYAND LONG-TIME D. RELATIONSHIP IN THREEDIMENSIONS-QUASI-RECURRENCE EVOLUTION AND ENERGY LEAKAGE

We recall that there exists a relationship between initial conditions and the eventual evolution of an unstable wave train in the two-dimensional case. Specifically, it has been shown that the long-time evolution of an unstable wave train is composed of the growth and decay of all the harmonics of the initial perturbations that lie within the unstable region. In the two-dimensional case, the region of instability in wavenumber space is finite in extent, implying that the energy of the evolving system will be confined to the same finite band of wavenumbers. The stability of a uniform wave train to three-dimensional perturbations can be analyzed by imposing the disturbances c.exp

(

‘I)

[ ( 2Wk0,) +KVy

i3tki K, x--t

(83)

on the uniform wave train A = a, exp( +ikia&,f)

(84)

Following substantially the same procedures as in the two-dimensional case, we can obtain a dispersion relation for the perturbation frequency

cu II t

II

0 t

II

In t

I'

_I

*-

-1 t = 6.40

,

:1, t = 7

I

t = 8

I

t = 9

.-' I

FIG. 18. Plot of absolute values of the wave envelope of the time evolution of two localized envelope packets. Note that at time I = 12.6. the wave forms return to the initial shapes with a change in relative position. Their relative positions are restored at time f=25.2 (not shown here).

1 06


This shows that the wave train is unstable within the region bounded by the pair of straight lines K,, =

+_-

fiK,,

(86)

and the pair of hyperbolas K:-2K:=8k:a$ Maximum instability occurs along the curves K , f\IT K,. = 2kia,,

and the maximum growth rate is uniformly (Im = +wokia; (89) on the curve (88). This instability region is shown in Fig. 19. The most important observation is that the instability region represented by Eq. ( 8 5 ) is unbounded in the perturbation wave vector space. This is a consequence of the hyperbolic spatial structure of Eq. (74), since the same equation with an elliptic spatial structure would have an instability region bounded by an ellipse in the perturbation wave vector space. If we now extend the relationship between initial conditions and evolution established in the two-dimensional case, we would expect that the long-time evolution of a uniform wave train perturbed by an unstable mode (kx,k,J would be dominated by the harmonics (mk,, nk,,) that lie within the unstable region ( m and n are positive integers). It can be shown that there are infinitely many integer pairs satisfying this condition. Thus, one would expect that the energy initially confined to the uniform wave train and a pair of unstable perturbation modes would eventually spread to arbitrarily high modes. This expectation is verified by the direct computations of Martin and Yuen ( 1980). They calculated the time evolution of a uniform wave train perturbed by a disturbance located at ( k x , k v ) . With their choice of parameters, the first four sets of harmonics of the disturbance lying within the unstable region are (5kx,7kJ (29k,, 41 k,,), (169k,x,239k,),(985k,, 1393k”).This is shown in Fig. 20. The time evolution of the amplitude and the Fourier spectrum is shown in Fig. 21. It can be seen that at ?= 14, the solution has demodulated and the energy has regrouped into the zeroth mode. At t=25, both the fundamental mode and the smallest unstable harmonic (5k,, 7k,) can be detected. The latter dominates at t = 32. The next higher unstable harmonic (29k,v,41 k,,) was found to receive energy at t=40, but the resolution and accuracy-of the calculation becomes suspect at that stage. The results, however, serve to establish the trend. In the three-dimensional modulation of a uniform wave train, the evolution consists of modulation-demodulation cycles that involve ever higher modes with no

Im R 1 2 2 -w k a 2 0 0 0

0

FIG. 19. Three-dimensional instability growth rate of a uniform wave train based on the three-dimensional nonlinear Schrodinger equation.

ioa

Henry C. Yuen and Bruce M . Lake

2kiao

FIG.20. Three-dimensional stability boundary for a uniform wave train based on the three-dimensional nonlinear Schrodinger equation. The dashed line denotes the locus of maximum instability. The two points indicate an unstable mode and its smallest unstable harmonic. The time evolution of a wave train subject to the unstable mode perturbation is given in Fig. 21.

limit as to how high the wavenumber of the energy-receiving mode becomes. Thus, the energy contained in the initial condition is spread to progressively higher modes in a quasi-recurring fashion. This does not conform to the classical concept of thermalization, in which all modes receive equal or near-equal shares of energy. Nevertheless, it is significantly different from the two-dimensional case that exhibits Fermi-Pasta-Ulam recurrence confining energy to a narrow band of wavenumbers. The disturbing part is the fact that these solutions of the three-dimensional nonlinear Schrodinger equation eventually extend beyond the domain of validity of the equation, which results from the assumption that the solution is slowly varying, or that energy is confined to a narrow band around the carrier wave.

t

=o.w

‘1

.-

t = 10.00

t = 0.00

t = 10.00 L -

e=*

6

7 i

t = 14.00

t = 14.00

t = 25.00

t = 25.00

FIG.21. Time evolution of an unstable wave train subject to the perturbation specified in Fig. 20. Plot of absolute value of the wave envelope in ( x . y ) space for various times (left traces) and of Fourier spectrum in ( K L K,,) , for various times (right traces).

t = 32.00

t = 37.00

t = 37.00

c

1/

t =40.00

'?

FIG.21. (Continued)


111

E. SUMMARY We have seen that the inclusion of nonlinear effects, even to first order, produces new and surprising implications for the stability and evolution of deep-water waves. In the two-dimensional context, new phenomena such as the envelope soliton, Fermi-Pasta-Ulam recurrence, and the relationship between instability and recurrence all support the concept of a coherent wave system, adequately described by the nonlinear Schrodinger equation at least in the weakly nonlinear approximation. Unfortunately, whereas most of these phenomena also exist in the three-dimensional context, many of the simple features are lost. The envelope soliton, important for its spatial localization and stability, is no longer spatially localized and is unstable to three-dimensional perturbations. Under certain conditions, recurrence still exists. The relationship between instability and recurrence, however, predicts a leakage of energy to arbitrarily high modes, causing the three-dimensional nonlinear Schrodinger equation to be asymptotically invalid for a wide range of initial conditions. One problem is clear. The three-dimensional nonlinear Schrodinger equation is apparently inadequate for the description of evolution of even weakly nonlinear deep-water waves. A more accurate description is necessary. Crawford et al. (1980) proposed that an integral equation, also first derived by Zakharov ( 1968), be used instead. This integral equation, which we shall call the Zakharov equation, is obtained by performing an expansion in wave steepness in Fourier space to third order. The original derivation by Zakharov (1968) is terse and contains some algebraic errors. In the following section, we outline the steps in the derivation, using a less elegant. but more straightforward approach. VI. Large-Amplitude Effects

In this section we consider properties of deep-water waves that appear as wave amplitude or steepness becomes sufficiently large that analyses of nonlinear wave dynamics must be carried to higher order than is the case in the weakly nonlinear descriptions of the preceding sections. We begin by considering results obtained from the Zakharov equation for a range of values of wave steepness or amplitude. A. DERIVATION OF ZAKHAROV’S INTEGRALEQUATION

We begin with the water wave equations (1)-(3). Following Zakharov, we define the surface velocity potential +’(x, t) as $JS(

x, t ) = +(x, r) (x, t ) ,t ) .

(90)

Henry C. Yuen and Bruce M. Lake

112

Its temporal and horizontal spatial derivatives are

q, I)++z(x,l(x, t ) ,t)%(X9 9 VX+'(x,r ) = V,+(X, ~ ( xr ),, 1 ) + +z (x, q (x, I), t)V,q(x, t ) , +S(% f) =+,(xJI(x,

(91) (92)

where 0,. denotes the horizontal gradient @/ax, a/ay). The free surface boundary conditions can be rewritten in terms of q,+', and the vertical velocity gradient +z as

v,+ ( v x + s ) v x ~ ) - + z1[+ (vxv)2]=a +:+g-rl+ +(VX@)2- ++J I + ( Vxq)2] =o.

(93) (94)

We define the Fourier transform of the velocity potential as

I"

;(k, z, t ) = 1 +(x, z, t ) exp( - ik-x) dx. 2n - m Laplace's equation requires that &k,z, t) be of the form

G(k,r,r)=$(k,r)exp(lk(z).

(95)

(96)

The surface velocity potential is related to $(k, t) by

s"

+'(x, t ) = 1 2n - w @(k,1 ) exp( (k)q(x, t)) exp( ik. x) dk.

(97)

We now assume that lk1q is small and develop a Taylor series expansion of exp((k(q).This gives +'(x, t ) =

Lsw &(k,t)exp(ik.x)dk 2n - m W

+ -Jllkl&(k, 1 2n2

X

where {(k,

1)

t)f(k,, t)exp(i(k+k,)*x)dkdk,

--m

exp(i(k+k,+k,).x)dkdk,dk,

is the Fourier transform of q(x, t):

s"

f (k, r ) = 1 q (x, t ) exp( ik-x) dx. 271 - w In terms of the delta function defined as S (k) =

4 J-1 exp( ik .x) dx (2n)


113

the Fourier transform 4’(k) of +‘(x) can be written as 00

G’(k)=&(k)+ ~ ~ ~ ~ k l ~ 6 ( k l ) $ ( k 2k ,)-8k,)dk, ( k - dk2 2T --m

X

S (k - k, - k2 - k,) dkl dk2 dk3

+ ...,

45,

where the time dependence of 6,and $ is implicit. Inverting Eq. (101) iteratively, we have 00

6(k)=GS(k)- &JJlkll4’(kl)$(k2)8(k-kl

X

-k,)dkI dk,

G5(kI)$(k,)$( k3)S(k - k , - k, - k3) d k , dk,dk3

+ ... .

( 102)

This permits us to express +2 in terms of + 5 a n d 7 through $’, and now the boundary conditions (93) and (94) can be written in terms of &(k) and t‘j(k), which are the Fourier transforms of CpS(x,t ) and ~ ( xt ), , respectively: m

& JJ [ I ~ ikliI

r j , -~ 1 k 1 4 5 + ~) m

- Ik - k I

I

-

- k.k,

] 4 v , ) i ( k 2 ~ w- k , - k2)d k , dk,

-m

Ik - k2l- Ik2 + k, I - Ik I + k3 I ]

X 4’(k1)$ (k2)$(k3)S (k - k , - k2 - k,)

d k , dk, dk, + *

1 .

= 0.

( 104)


114

Following Zakharov, we introduce the complex variable b(k, t) defined

Note that the free surface ~ ( xt), is related to b(k, t) by -q(x, t ) =

Ik( I /" (-)20(k) 27

[ b (k, t) exp( ik-x) + b*(k, t) exp( - ik. x) ] dk.

'I2

('06) Equations (103) and (104) can be combined into a single complex equation for b(k, t): m

b,( k)

+ io(k)b (k) + i s s V' "(k, k , ,k2)b(kl)b(k2) 6(k - k , - k2) d k , d k 2 -m

m

-m m

+ iJ/V')'(k,k,,k2)b*(k,)b*(k2)

6(k+k, +k2)dk,dk2

-m

+i / T /

W' "(k, k, ,k,, k,)b (k,)b (k2)b(k,) 6 (k - k, - k2- k3) d k , dk, dk,

-cQ m

+iJ/J

W2'(k,k , , k,, k,)b*(k,)b(k,)b(k,) S(k+ k, - k2- k3)dk, dk,dk,

-m

m

+ i ~ ~ ~ W ( 3 ' ( k , k l , k 2 , k , ) b * ( k , ) b * ( k 2 ) bS( k( k3+) k , +k2-k,)dk,

dk2dk,

-m m

+iJJJW(4'(Lk,,k2.

k3)b*(kl)b*(k2)b*(k3)6 ( k + k , + k 2 + k 3 ) d k , dk2dk3

-. m

+ ... =o,

(107)

where the interaction coefficients V and W are given in Appendix A. We now assume, as did Zakharov (1968), that the wave field can be divided into a slowly varying (in time) component B and a small, rapidly

Nonlinear Dynamics of Deep- Water Graviiy Waves

1 I5

varying component B ' , and that most of the enrgy in the wave field is contained in B. This assumption permits us to write

b(k, f) = [ d ( k , 7)+ c2B'(k, f')] exp( - io(k)t)

( 108)

where E is a small parameter describing the magnitude of the nonlinearity. The fast time scale is t', and the slow time scale is 7,where the total time . variation a / a t = a / a t ' + E 2 ( a / a 7 ) + Substituting the form of b of Eq. (108) into (107) and collecting terms of o(e2)gives

where we have introduced the compact notation in which the arguments k, in V , B , 8, and w are replaced by subscripts i, with the subscript zero assigned to k. The explicit dependence on time has been suppressed. Thus, for example,

vJ.y!z= V'"'(k, k1, k2), 60-

I

2=

S(k - k I - k2),

B 1 B2 = B (k197)B (k2,7)3 w - w I - ~2 = w(k) - o(k1) - 4 ~ 2 ) .

Since the terms involving Bi in the right-hand side of Eq. (109) do not depend on t', we can integrate with respect to t' to give, to leading order,

+

exp( i(o w I- w z ) t ) + V:.::B:B28o+ (3)

B*B*&

+v0.1.2 I

1-2

2 O+I+Z

W+W, -a2

+

exp( i ( w w I+ 0 2 ) t ) w+ol+02

d k , dk, ('10)

The constant of integration, which corresponds to the phase at an initial time, has been set to zero without loss of generality.


116

We now consider terms of O(c3).We have M

ii!E=-JJ{ aT

V0.1,2( ( 1 ) B I B 2’ + B 2 B’I ) 8o-I-2exP(i(w-ol-w2)t)

-m

+ Vi.t!2(By B; + B2B;*)So+I - 2 exp( i ( w +a,

-w 2 ) t )

+ V[,i!,(By B;* + B; B ; * ) S o + ,+ 2 exp( i ( w + ol+ w 2 ) t ) }d k , dk,

Note that terms involving W “ ) , V 3 )and , W(4) have been dropped, based on the argument that the corresponding time oscillations would be more rapid than the term with W ‘ 2 ’ ,and hence smaller when integrated. Substituting the solution for B’ of Eq. (1 10) into (1 11) and converting into symmetric form, we arrive at Zakharov’s equation for B : m

aB at =

JJT0,I.2,3&

B2B360+

1-2- 3

-m

X

+

exp(i(w w I - w 2 - w3)t) d k , dk, dk,,

(112)

where To,1 , 2 , 3 is expressible in terms of V ( i ) and W(’) and is given in Appendix A. We have also dropped the distinction between the various time scales and used t throughout, since the context is clear. Zakharov’s integral equation (1 12) describes the slow evolution of the dominant components of a weakly nonlinear wave field. The weak nonlinearity requirement is a consequence of expansion to third order in ka. The slow time scale of variation is k2a2w,which is related to the assumption that the frequency difference A o = w , + o , - w 2 - w 3 is small and is of order k2u2.Note that this assumption requires that the dominant modes lie within a band of width O(k2u2)around the resonant curves defined by

b+k,=k2 +k3,

(113)

w(k0) + w(k,) = + @3)7 ( 1 14) but at this stage there is no requirement that the spectrum be narrowbanded. When one insists that the resonant conditions ( 1 13) and (1 14) be satisfied exactly, one recovers the resonant wave theory of Phillips (1960). In order to recover modulational instabilities of the Benjamin-Feir type, however, the nonlinear “detuning” effects [i.e., effects associated with Aw- O(k2u2)] must be included.

Nonlinear Dynamics of Deep- Wafer Gravity Waves

I17

B. STABILITY OF A UNIFORM WAVETRAIN We consider B(k, f ) as a superposition of discrete modes B(k,t)= 2 n B,,(t)S(k-k,).

(115)

Substituting Eq. (115) into (1 12) and evaluating the delta functions, we obtain for each mode k,, the discrete equation

( 1 16)

where w, = o(k,), the generalized Kronecker delta S,,, summation is taken over those subscripts satisfying

+

denotes that

+

kp k, = k, k,,,

(117) and Tp.iJ.mdenotes T(kp,k,,k,,km).At this stage we note that if we specialize to those modes for which wP + o i = w , + w m

( 1 18)

in addition to the constraint ( 1 17), we would arrive at the discrete resonant interaction equation first derived by Benney (1962):

The solution of Eq. (1 16) corresponding to a uniform wave train of wave vector $ = (k,, 0) is

where To= T($, k,, k,,, $)= +wok: and B, is (2u,/k,)'/Za, (where a, is the carrier wave amplitude). We impose perturbations represented by a pair of wavevectors b+K with amplitudes B ? ( f ) . Neglecting the squares of small quantities, it follows from Eq. ( I 16) that B , ( t ) satisfy i -dB -

dr

- T,.=BiB*exp[

- i ( A w + 2 T 0 B ; ) t ] + 2 T ~ ,* B ; B t .

(121)


118

Substituting B + = i?+ exp[ - i ( +Aw+ T oB t )t- iQt], B*=i?*exp[i(iAw+ToB~)t-i91],

( 123)

lk that has no

we obtain a second-order eigenvalue problem in 9, i?+ , and explicit time dependence. The eigenvalues Q are the roots of Q2+2Bi(T-* - T+% +)a+ T + ,- T - . +Bo4

+ (- t Aw-

T0B,2+2T-. - B;)( - f Aw- ToB,2+2T+.+ B i ) = 0 . (124)

The solutions are Q = ( T + .+ - T - . - ) B i +_

( - T+.

-

T - . +B:+

[ - 4 Aw+

2

B i ( T + .+ + T - , - - To)]

]

1/2

.

(125)

The expression for Q given by Eq. (124) gives I m Q correct to O(k,'u$, with no requirement that IKI be small compared with lkol. However, the original approximations contained in the derivation of Eq. ( 1 16) require that AW be of the order of TOE; for the mechanisms described to be dominant. 1. Case of Two-Dimensional Modulation

In this case, we put K = (K,, 0) and define the dimensionless perturbation wavenumber K = K , / k , . From the dispersion relation, it can be seen that for AW to be small, K must be small. In other words, the perturbations that possess the growth rate given by Eq. (125) must be "sidebands" representing long-wave modulations. In the limit of very long perturbational wavelength, i.e., K < < I , we can expand the dispersion relation in powers of K: W,

= a o ?+W"K

-$ W ~ 2 K,

Therefore Au=2w0-w+

-w-

=+J~K~.

Expanding the expressions for T k s and Tt, in powers of retaining only terms to O(kiu;) and O(K') in Eq. (125), we obtain

(127) K

and


I19

Byjamin and Feir (1 967) 1 NO

-0

Y, 3

rw

1

cE

0 c)

2

2 Kx

O.5

c

.-

EE

a

0

0.2

0.4

0.6 0.8 1.0 Dimensionless wavenumber, A

1.2

1.4

FIG. 22. Two-dimensional instability growth rates a s a function of perturbation wavenumber for various values of wave steepness. The Benjamin-Feir result is recovered by taking the limit as wave steepness approaches zero.

This result is identical to the instability first discussed by Benjamin and Feir (1967), and to that which follows from a stability analysis of the nonlinear Schrodinger equation [Section VI, D, Eq. (65)]. We shall now show that Eq. (128) is not a very good approximation of Eq. (125) for moderate but small values of k,a,. To illustrate this, we show in Fig. 22 a plot of the instability growth rate I m Q [as obtained from Eq. (125)] as a function of the normalized perturbation wavenumber

for various values of k,ao. The result of Eq. (128) [which is that of Benjamin and Feir (1967) as labeled] is approached when k,a,+O. For nonzero k,a,, departures arise for larger values of A. These departures become significant with increasing koa,. For koa, = 0.2, the prediction regarding the most unstable wavenumber and the maximum growth rate achieve disagreement as large as 30%. Furthermore, Eq. (125) predicts that the very long waves begin to become stable for koao=0.39. This restabilization of the very long waves agrees qualitatively with the results of Whitham’s

120

Henry C . Yuen and Bruce M . Luke

Longuet-Higgins (1978) I

0.1

0.2 0.3 Wave steepness, koao

0.4

0.5

FIG. 23. Stability diagram for two-dimensional perturbations on a uniform wave train from the Zakharov equation and comparison with results of Longuet-Higgins.

theory, which yield long-wave restabilization at k,a, = 0.34. The quantitative discrepancy of 14% is better than expected, since the present theory is formally accurate only to ~ ( k , j u ; ) . Figure 22 also shows the trend toward restabilization of the entire system for sufficiently large kouo (at about kou,= 0.50). This feature qualitatively agrees with the numerical results obtained from exact water wave equations by Longuet-Higgins (1978). A better illustration of this phenomenon is given in Fig. 23, where we have plotted the stability diagram in the ( K , k,ao) space. The numerical results of Longuet-Higgins (1978), confined to only discrete values of K, are also plotted. It can be seen that the qualitative agreement is satisfactory overall, and quantitative agreement is achieved for small and moderate values of kouo.In Figs. 24 and 25, we plot the real and imaginary parts of the perturbation frequency in the frame of reference moving with the individual wave crests. These plots can be compared with Figs. 5 and 6 of Longuet-Higgins (1978).* Note that no restabilization was

*The explosive instability a t large wave steepness found by Longuet-Higgins (1978) for the case ~ = 0 . 5is not reproduced by Eq. (125). This is to be expected since the Zakharov approximation is correct only to O(k,$i) and the explosive instability is of higher order. See discussion in Section VI, E.

Nonlinear Dynamics of Deep- Water Gravicv Waves

3’ 2 >;

E

-5

0.5

%o

g

s + 3” c 2

121

0.4

fl2

d‘

0.3

g 2 B

4

2 0.2

2

318 .

€ + I

3’ -

0.1

0.1

0.2

03 0.4 Wave steepness. koao

0.5

0.6

0.7

FIG. 24. Normalized frequency of two-dimensional perturbations as a function of wave steepness for various values of perturbation wavenumbers. The coordinate system is traveling with the speed of the carrier wave phase velocity. Shaded region indicates instability.

0.2

Wave steepness, koao

FIG. 25. Two-dimensional instability growth rate as a function of wave steepness for various values of perturbation wavenumbers.

122


Wave steepness, k,a,

FIG. 26. Comparison of calculated instability growth rate with experimental results as a function of wave steepness for two values of perturbation wavenumbers. (O)ti=0.4(Lake er al.. 1977); (O)ti=0.2(Lake er al., 1977): (A)data from Benjamin (1967).

predicted by the analysis of Benjamin and Feir (1967) or that based on the nonlinear Schrodinger equation. For given values of K (0.2 and 0.4), the predicted growth rate as a function of wave steepness has been compared to experimental data of Benjamin (1967) and Lake et at. (1977). The results are shown in Fig. 26. It can be seen that the agreement is quantitatively satisfactory. The Benjamin-Feir result is shown for comparison. In Fig. 27, we compare the predicted results on the most unstable perturbation frequency as a function of wave steepness with experimental data. Again, the agreement between theory and experiment is very good, whereas the result of Benjamin and Feir overpredicts the most unstable perturbation frequency. Lake and Yuen (1977) proposed that the discrepancy between the Benjamin-Feir result and the experimental data is due to the generated waves being in some sense less nonlinear than has been inferred from the measurements. In light of the present results and the results of LonguetHiggins (1978), it appears that this effect is far less significant than was believed and should be disregarded.

123

0. I

0.2 0.3 0.4 Wave steepness, koao Fic,. 27. Plot of most unstable perturbation frequency versus wave steepness and curnparison with experimental data. The data are taken from Fig. 1 [with abscissa corresponding to the scale labeled ( k a ) , ,,I of Lake and Yuen (1977).

0

2 . Case of Three- Dimensional Modulation In the limit of very long modulation wavelength, we can again expand o, and A o about k, in powers of IKI to obtain

Also expanding T ? , .t and T 2 . O(lKI*), we obtain

and retaining terms to O(kiai) and

- t Awk;aiw,+

-

4

where A o is given by Eq. (131). This result agrees with that obtained from

124


FIG.28. Three-dimensional stability boundary from Zakharov equation in strained coor, K v / 2 k ~ a O-,. kouo=O.O1; ---- , koao=O.I; -.-, k 0u0-dinates. A , = K , / 2 k , $ ~ ~A,,= 0.4; ... . k#a, = 0.48.

.

stability analysis based on the three-dimensional nonlir.ear Schrodinger equation as given by Zakharov (1968). Note in Fig. 28 that the instability in the A = (Ax,A,) plane defined by Eq. (1 32) lies in between a pair of straight lines, A x = ?@A,, or Au=O, and a pair of hyperbolas, defined by Au =2k,2a,2u0,whereAu is given by Eq. (131), and is infinite in extent. The general expression for the stability boundary is given by

A w = 2 u o - w + - ~ - = 2 ( T + , + + T - , - - T 0 ~ ( T + , - T - ~ + )(133) ”2)B~, where Au is O(k,$.&,)

with no other restriction on K. This is shown in Fig.

-0.4

FIG.29. Three-dimensional stability boundary from Zakharov equation. ----,kouo= 0: -, ko~o~0.01; k o ~ o = 0 . 1 ;-.-, koUo=0.4; ... ’, k o ~ o 5 0 . 4 8 .

--.


125

28 and 29 in two different sets of scales for the axes. The most important difference between the unstable regions predicted by Eqs. (1 3 1) and (1 3 3 ) is that the latter is now finite in extent. For small values of koao, the instability region lies adjacent to Phillips’ “figure-of-eight’’ diagram (Aw =O), which is valid for weakly nonlinear point spectra. As koao increases, the wave vectors near the edges of the “figure-of-eight” stabilize and the diagram approaches that of a pair of touching “horseshoes.” For sufficiently large koao, the longer waves also begin to stabilize, and the two “horseshoes” split. Just before the total system stabilizes, the instability is concentrated at K= 2 0.78k0,and is strongly two-dimensional. The fact that the instability region is finite in extent, and not infinite as predicted by the three-dimensional nonlinear Schrodinger equation, may have significant consequences in light of our earlier discussions. One of the shortcomings of the three-dimensional nonlinear Schrodinger equation is that its instability region is infinite in extent, leading to the quasi-recurring energy leakage discussed earlier. Now that the predicted instability region is finite in extent, such energy leakage should not occur. Furthermore, the growth rate plots given in Fig. 30 show that the maximum instability always occurs at K,.=O for a given value of k,ao. This suggests that energy spreading t o three-dimensional wave vectors would be relatively slow and weak, in addition to being confined within finite bands. This speculation must be strongly qualified in view of the new results presented in Section VI, E below. C. RESTABILIZATION In addition to the improved agreement between theory and experiment, and the fact that the instability regime is finite in the three-dimensional case, the Zakharov integral equation contains the interesting phenomenon of restabilization at large amplitude. In the two-dimensional case, restabilization of the wave train has been established by Longuet-Higgins (1978) by direct numerical calculations based on the exact equations (1)-(3). Because of the periodicity requirement in his calculations, he examined only those perturbations with wavelengths that are integral multiples or rational multiples of the unperturbed wavelength. In fact, he presented in detail results for six values of the perturbation wavenumber K = K , in the case of subharmonic disturbance. His calculations indicate that for each value of perturbation wavenumber K, there is an amplitude (or wave steepness) at which the system restabilizes against that perturbation. The long-wave perturbations restabilize first at k0a0=0.34. The entire system is then inferred to restabilize at k,a0=o.39. After a given perturbation mode restabilizes, it oscillates with a definite

FIG. 30. Three-dimensional stability growth rates for various values of wave steepness from Zakharov equation. (a) k,a,=O.I, (h) 0.2, ( c ) 0.3, (d) 0.4. I26

FIG.30. (Continued) 127


128

70 80 90 20 30 40 50 60 Direction of modulational wave vector (degrees) FIG.3 I . Stability boundary and neutral stability curves for infinitely long perturbations. Instability boundary is denoted by the dashed lines (ImP=O). Neutral stability boundary is denoted by the solid lines (Im 8 = Re Q = O ) . NLS, Results from three-dimensional nonlinear SchrGdinger equation: 2,results from Zakharov equation: W. results according to Whitham’s theory. 0

10

frequency with respect to the unperturbed wave. I t can be seen from Fig. 24, or from Longuet-Higgins’ (1978) numerical results that this difference in frequency decreases with increasing k,a,. In fact, at a value of koao slightly beyond that for which the mode restabilizes, this difference in frequency drops to zero (see Fig. 31). When this happens, bifurcation of the unperturbed uniform wave into new steady solutions is possible. This is because such a neutral stability point in the linear analysis implies that an infinitesimal disturbance can be superposed on the unperturbed wave with the resulting flow remaining steady. When this infinitesimal disturbance is extended to finite amplitude, new solutions are created. These are the bifurcated solutions, since they coexist in the wave steepness parameter with the uniform waves.

D. BIFURCATION In the two-dimensional case, bifurcated Stokes waves were first discovered by Chen and Saffman (1980). They showed that steady modulated wave patterns with N waves per modulation period can exist. In particular,

Nonlinear Dynamics of Deep- Water Graviiy Waves

129

-0.30L

>

L

-3.00

-2.00

-1.0

-O.3OL

- 0.30L

.-0.3OL

FIG. 32. Wave profile of two-dimensional bifurcated wave resulting in period doubling. The bottom trace is the unbifurcated wave train. Departure from the unbifurcated waves increases as b - b, increases. [For details see Chen and Saffman (1980). copyright 1980 by Elsevier North Holland. Inc.. reprinted with permission.]

they calculated the cases for N = 2 and 3 using the exact water wave equations. Their results are reproduced in Figs. 32 and 33. It is interesting to note that an earlier article by Garabedian (1965) contained a proof that a symmetric, uniform. wave train on deep water is unique provided that its crests and troughs are equal. The findings of Chen and Saffman (1980) do not violate this theorem, but they do show that the requirement of uniform crests and troughs is a crucial assumption for uniqueness, The Zakharov integral equation also predicts restabilization at large amplitude in two dimensions, but the value of wave steepness at which it occurs is about 0.7, which is significantly larger than the Stokes limiting wave value of koa, = 0.448.This unrealistic restabilization value precludes the use of the Zakharov equation to obtain acceptable approximations of bifurcated solutions in two dimensions.


130

b = 0.99023

9(

LL -0.20L

-3.00

-2.00

-1.00

-0.20L

-0.20L

lL:zl -3.00

-2.00

-1.00

b = 0.78023

-0.20L o,20-

-0.20L

FIG.33. Wave profile of two-dimensional bifurcated wave resulting in period tripling. The middle trace is the unbifurcated wave train. The two upper traces represent the two low-one high bifurcation, and the two lower traces the two high-one low bifurcation. [For details see Chen and Saffman (1980). copyright 1980 by Elsevier North Holland, Inc.. reprinted with permission.]

The foregoing discussion can be naturally extended to three dimensions. Calculation of bifurcated solutions in three dimensions using the exact equations is, however, a formidable task and may be impractical even with present-day computational capabilities. It is therefore fortunate that the wave steepness for bifurcation decreases with increasing obliqueness of the perturbations with respect to the unperturbed wave, as noticed by Saffman and Yuen (1980b; in the case of infinitely long disturbances, this fact can be deduced from the results of Peregrine and Thomas, 1979). Furthermore, analytical considerations and actual comparison with exact calculations of three-dimensional instabilities (see Section VI, E) strongly suggest that the Zakharov integral equation yields accurate results for cases with small to

0

h

0

u3

0, 0

0

m

h 0

ID 0

m 0

w 0

c) 0

N 0

0 7

0

c L

_.

W

L

s

L

i m= m

p'- 3 u ?

I32


moderate wave steepness (up to, say, k,u0=0.3). It is therefore possible to use the Zakharov equation to obtain reliable approximations of bifurcated waves provided that the angle between the modulations and the main wave is sufficiently large; in other words, when the steepness of the waves is not excessively large. Plots of the bifurcation wave steepness as a function of the angle between the modulation and the main wave for several perturbation wavenumbers are given in Fig. 34. The case of the infinitely long perturbation found by Peregrine and Thomas (1979), using a modified version of Whitham’s theory that is valid for all wave steepness, is also shown. It can be seen that the agreement between results from the Zakharov equation and those of Peregrine and Thomas is quantitatively satisfactory provided by koa, is less than 0.30. We now highlight the steps in the calculation of steady three-dimensional bifurcated waves. We are interested in solutions corresponding to threedimensional modulations imposed on an otherwise uniform two-dimensional plane wave train that propagates in the x direction with, say, unit wavenumber. Thus we take as solution B(k, t ) =

I

Zl( t)S(k - I) exp( io(k)t),

(134)

where

I = le, + mpe, + nqq,

(135) m and n are positive or negative integers, e, is the unit vector in the x direction, e2 is the unit vector in the y direction, and p = ( p ,q ) is a pair of real numbers. The undisturbed wave corresponds to I = 1, m = 0, and n = 0. The approximations inherent in the derivation of the Zakharov integral equations lead to the neglect of interactions between harmonics of the undisturbed waves and those of the modulations. There is therefore a class of waves in which I= 1 for all modes and for which the mode I is uniquely labeled by an integer pair m=(m,n) for a given fundamental modulation p= ( p ,4). For this class, Eq. (134) can be expressed, for each choice of p, as B(k, t ) =

with

2m (Z,

t)6(k - I)exp( iw( k) i ) ,

+

I = e l + m:p = e , + mpe, nqe2.

(136) (137)

Substitution of expression (136) into Eq. ( 1 12) gives

where g has been taken to be unity, and the simplified notation introduced in Section V1.A has been used for T.


I33

We seek solutions that correspond to free surface displacements of the form V(X, f ) = V( x - Ct, y ) ,

(139) representing patterns propagating without change of shape at a constant speed C in the el direction. The two-dimensional, uniform Stokes solution corresponds to the case with no dependence on y . Fully three-dimensional patterns require solutions of the form Z,=a,,exp( -il(m).Ct)=a,exp( - i ( l + m p ) C t ) ( 140) where C = ( C , O ) and the amplitudes a,, are independent of time i. In this notation, the uniform Stokes wave solution is a,,=a:=a,

m=O,

a,=a;=O,

mfO;

and substitution into Eq. (138) yields

C= I +~ , ~ ~ , ~ , , a * . ( 142) It can be verified that Eqs. (141) and (142) agree with the results of Stokes (1849) since ~,.o,o,o= (4n2) ~

I.

(143)

Substituting Eq. (140) into (138) and its complex conjugate, we obtain the evolution equations, relative to a frame moving with speed C , of modulations associated with a modulation wave vector p,

where physical validity requires the satisfaction of the relation

a to 00. It proves and the members of the integer pairs m, i. j, k run from - c to be convenient, in analyzing the stability of the steady solution to allow (142) to be violated by the infinitesimal perturbations, but it is always satisfied for steady waves. Equations (144) can be written symbolically as iu, = M(u, C ) ,

where M is a nonlinear cubic operator and

( 146)


I34

is a member of a Hilbert space with norm

and an obvious definition of the inner product. Steady solutions satisfy M (u, C ) =O. Suppose that s denotes arc length along a continuous branch of solutions in the infinite-dimensional space ( u , C ) . Along the branch

M ’ = M u d + M,C’=O.

(150)

where the prime denotes d / d s , and the subscripts denote (Frechet) derivatives of the operator M. If the homogeneous equation

Mu+=0

(151)

has a nontrivial solution, then the branch has either a limit point or bifurcates into a new type of steady profile. We shall now show that for arbitrarily given p, Eq. ( 1 5 1) has solutions along the branch of Stokes waves (i.e., solutions in which the only nonzero components of u are those with m = 0 ) , which do not correspond to limit points, at critical wave heights that depend on p. Thus the Zakharov equation predicts that two-dimensional water waves of permanent form are infinitely degenerate to bifurcation into three-dimensional waves of permanent form. It is to be noted that the Zakharov equation is quantitatively accurate when the wave amplitude is small and the modulations are of long wavelength, i.e., p 2 + q2< < 1 . It will be seen that the critical wave height for bifurcation goes to zero as p/q+O. Hence, the results are quantitatively accurate for water waves in the limit of very oblique, long-wave modulations. We can write Eq. ( I 50) in the form

M ‘ Gmak

T,.

1. j. k( i?iUjUk

4- 6,U;Uk -t6iUjU;)

- C’( 1 + mp)am= 0,

A?’

Gm6k-

( 152a)

Tm.i. j. k( aflijcik + u,ci;ci, + uicij6;)

+ C’( 1 + m p p , =0,

(1 52b)

where

C m = [ ( 1 + m p ) 2 + n 2 q 21 /4] - C ( l + m p ) .

(‘53)

Then Eq. ( 1 5 1 ) has a nontrivial solution on the Stokes wave branch if the


135

have nontrivial solutions. The necessary and sufficient condition is

where without loss of generality we can take m = ( T 1, 2 1). With the substitution of (142) for C, Eq. (155) becomes a quadratic for u2 whose solution gives the critical wave height. It will be shown below that this equation has at least one positive root if 4 / p is not loo large. Moreover, the root is the same for each of the four values of m. Then, for given values of p and 4, solutions of the system (152) exist with ak=&=aA,, u-,,,-a-,,,=a, m = ( l , I), ( 156a) a&=a*k=/3Am, u - ~ --u - , = P , m=(l, -I), (156b) *I

I

At

I

and aA,h = 0 otherwise, where

If a and /3 are zero, * Eq. (157) gives the rate of change of a along the branch of Stokes waves, as it is identical with the expression obtained by differentiating Eq. (142). The possibility of nonzero x and /3 indicates the existence of a bifurcation at a critical amplitude a that is a root of (155). One relation between a, /3, and C' is provided by the normalization condition 1111'/12+ c2= 1, (159) but this essentially only fixes the arc length s. To find significant relations, we must consider the higher derivatives of M. From the second derivative evaluated at the critical wave height, we obtain M " G -2C'Uk+2UC {2Tm.i.j.oLifLij+ T,.o.j.kQj'a'k}+M,!nI

a,+n;

(160)

C'+C''

and a similar equation for M: by interchanging a, and 8, and changing *The numbers a and /3 can be supposed real with no loss of generality-this equivalent to an appropriate choice of phase or origin of y .

is just


136

the sign. The last term denotes (152a) evaluated with a: and C" replacing a; and C'.The solubility condition (Fredholm alternative) of the inhomoge-

neous equations for a:, a*:, and C" gives relations between a , b, and C'. There are two equations, one arising from m = ( l , I), and the other from m = ( l , - 1). Substitution of Eqs. (156) and (157) into (160), and elimination of the terms in a:, and C" gives equations that reduce, respectively, to Thus, either

aC'=O,

PC'=O.

a=p=o,

C'=l,

or

C'=O, a2+p2= 1/2(1 +P) (163) [using the normalization condition (159)], where the ratio r = a l p is so far arbitrary. Clearly (162) describes continuation along the Stokes wave branch. But (163) gives a new branch of three-dimensional solutions, in which

c=1 + T0,o.0.oa2,

a0 = a,

a, I .

I)

=€A(

I. I ) ,

a( - I ,

-I)

(164)

=c*

a(I,-l)=r€b a(-l.l)=rc9 correct to order E , These describe three-dimensional waves of permanent form, in which the surface elevation is 1 cos(x - C t ) q ( x , y , t ) = -a - 1 ) 9

TrJZ:

+ €A(

+

€[

(1 + p f + 421 'I8 cos{ ( 1 + p ) ( x - Ct) + qy }

(1 + p y + q2]1/8cos((1 - p ) ( x - C t )- ey}

+ rcA( ,,- ,)[(I + rc[ ( 1 -p)'+

+p12 + 421'I8cos{ (1 + p ) ( x - ~ t- q) y 1 q2]i/8cos{(1 - p ) ( x - C t ) + e y }

+ 0(c2).

(165)

The order e2 terms on a branch can be found by studying M"'=O, and so on, but the algebra quickly becomes too involved to handle easily. It proves easier to follow the branch by solving numerically for steady solutions of a truncated version of Eqs. (144). A truncation in which the integers range from - N to N is equivalent to calculating the ( N - 1)thorder derivative of M(u,C) or finding the coefficient of c N - ' . Note that the coefficients of the waves that start branching as described by Eqs. (164) are all real. However, one condition can be established quickly: that r=O or 1. In


137

other words, the wave is either completely skewed or symmetrical. (The case r = 00 is the same as r = 0 with the y direction reversed.) This follows from noting that the Fredholm alternative takes the form p,,,~',,,+fi~,=~,

m = ( + 1,

I),

(166)

where

Applying the Fredholm alternative to M"' for m = ( I , 1) and m = ( 1, - 1) gives equations for C" that are inconsistent unless r=O or 1. The critical wave heights a at which bifurcations occur depend on p = ( p , q ) . In Fig. 35, we show q as a function of p for various values of wave steepness a / f i T. The case q = 0 corresponds to two-dimensional bifurcation of the type considered by Chen and Saffman (1980) and Saffman (1980), who studied the full water-wave equation. It is interesting to note that the results agree qualitatively even though they lie outside the apparent range of validity of the Zakharov approximation, e.g., the decrease of the critical amplitude with p . The quantitative values of critical wave steepness are about 50% too high for the two-dimensional case. The plot of wave steepness against 8 = tan- ' ( q / p ) which is a measure of the degree of obliqueness of the three-dimensional modulation for various values of p is shown in Fig. 34. Also shown is the curve for neutral stability of a uniform wave train to three-dimensional disturbance of infinite wavelength calculated from Whitham's exact finite-amplitude theory (for details, see Peregrine and Thomas, 1979; Saffman, 1980). It is interesting to note that for steepness less than 0.35, the difference between the curve with p = lo-' and Whitham's result (which is exact in the limit of p,q+O) is practically undetectable. This is a strong suggestion that the present results are quite accurate for wave steepness less than 0.35 (which covers about 80% of the permissible range of water wave steepness) and 8 more than 60". The bifurcation into three-dimensional wave patterns is degenerate because the parameter r can be either 0 or 1. The case r = 1 gives threedimensional waves that are symmetrical about the direction of propagation. For r=O, the pattern is skewed. The calculation of the actual wave patterns is simpler for r=O, since the wavenumbers that are generated by the higher-order interactions are all of the form I=e, + m( I , 1) ( 168) and can therefore be characterized by a single integer m, - 00 Also, the amplitudes are all real. Writing then

L;, = am= b,,,

< m < CQ. ( 169)

I.

1 .I

1

.'

1 ,;

(4

1*I

0.t

0.E

0.4

0.2

P I38

62.83 (a) 0

I-

a > w

-I

w

w

0

a

I

62.83 (b) FIG. 36. Skewed bifurcated wave pattern for three values of a, ,,,,/u0: ( a ) 0. ( b ) 0.2. ( p , q ) = ( O . I , 0.25).

( c ) 0.4. The plots correspond to

FIG. 35. Values of p and q at which bifurcation occurs for various values of wave steepness. Note that for q=O. p is zero until the wave steepness reaches a critical value. after which it increases with wave steepness. These results are in qualitative agreement with the two-dimensional results of Saffman (1980) based on the exact equation. I39

62.83

t c) FIG. 36. (Continued)

FIG. 37. Symmetric bifurcated wave pattern for three values of q i , , > / a w(a-c) Threedimensional wave pattern for ( p . y)-(OS, 1.2). (d-f) Contour plots. 140

1

FIG.37. (Continued) 141

Henry C. Yuen and Bruce M . Luke

142

(d)

FIG.37. (Continued)

the skew waves are given by the solutions of the equations

( [ ( 1 + mu)* + m 2 q 2 ]

-

) + C

C ( 1 + m p ) b,

m=j+k-i

,.

T, ,. rc h,b,bk

= 0.

(170) For symmetric waves ( r = I), a vector index m = ( m , n ) is required and u , satisfies the equation

{ [ (1

mu)*+n 2 q 2 ]

C( 1 + mp))U,+

C

T,,, i,j. kUiUjU,=O.

(171)

m=j+k-i

Equations (170) and (171) can be solved numerically using Newton’s method to give the bifurcated wave solutions. In Fig. 36, we show the three-dimensional plots of the water surface for skewed wave patterns with p=O.1 and q=0.25.

Nonlinear Dynamics of Deep- Wafer Gravity Waves

143

(e)

FIG.37. (Continued)

A plot of the three-dimensional wave pattern for p=0. 5 for three values of ( I , , , I , / a o is given in Fig. 37a-c (pp. 140-141), and contour plots are shown in Fig. 37d-f (pp. 142-144). These calculations were reported in a series of articles on three-dimensional water waves (Saffman and Yuen, 1980a. b, 1982). Independent of the theoretical predictions, M. Y. Su reported experimental observation of three-dimensional steady wave patterns (Su, 1982a, b). His experiments were conducted in an open basin with a test area about 50 f t wide and 1000 f t long at a water depth of about 3 ft. His open-basin observations were later verified by experiments in an indoor wave tank 12 f t wide and 300 f t long. In his experiments, Su generated steep twodimensional waves and observed that after a few wave periods they developed a distinct regular three-dimensional structure that persisted for

144

Henty C. Yuen and Bruce M . Lake

about ten wave periods. This three-dimensional structure of the observed patterns (Fig. 38) is steady and symmetric, and compares well with the symmetric bifurcated solutions shown in Fig. 37. In fact, Su’s observed waves would correspond to a wave steepness of 0.3 1, a value of p = 0.5, and 4= 1.12. From the bifurcation diagram shown in Fig. 35, the experimental observation is in reasonable agreement with the theoretical predictions of the triplet (a,,p,q) for bifurcation to occur. For 90% of the observations, Su observed three-dimensional structures corresponding t o p = 0.5. He also observed p = 0.33 and 0.25 cases for about 1070 and I% of the observations, respectively. The p=O.33 and 0.25 cases do not occur by themselves, however, but mercly coexist with p=O.5 structures. Additionally, the values of the triplet (ao,p , q) do not agree with the theoretical predictions for bifurcation in the q = 0.33 and p = 0.25 cases.

Nonlinear qvnamics of Deep- Water Gravity Waves

145

FIG.38. Photograph of three-dimensional wave patterns from Su (l982a). The waves are generated by a mechanical wave paddle with no prescribed initial perturbations. The threedimensional pattern develops rather suddenly several wavelengths from the wave paddle. The pattern persists for about ten wavelengths. The pattern then subsides and is replaced by two-dimensional instability of the Benjamin-Feir type (Class 1 instability).

146


The preference for p = 0.5 will be understood when we present the results of a recent calculation by McLean et al. (1981) in the following section.

E. A NEWTYPEOF THREE-DIMENSIONAL INSTABILITY Thus far we have presented results for the stability of a uniform wave train to two- and three-dimensional disturbances using the Zakharov integral equation. We have shown that for small and moderate values of wave steepness, the results appear to agree well with the unapproximated results of Peregrine and Thomas (1979) for two- and three-dimensional disturbances in the limit of infinitely long wavelength, and of Longuet-Higgins (1 978) for two-dimensional perturbation with integer or rational wavenumbers. For larger values of wave steepness, however, the results are quantitatively inaccurate, and may be qualitatively unreliable as well. Furthermore, two interesting questions remain outstanding. First, what is the selection mechanism favoring the p = 0 . 5 bifurcated state that is observed for almost all of the experimental cases? Second, what is the three-dimensional extension of the new two-dimensional instability found for very steep waves by Longuet-Higgins (1978)? Considerable light has been shed on both these questions by the results of McLean et al. (1981; see also McLean, 1982), who calculated the stability of a uniform wave train to three-dimensional perturbations using the full water-wave equations. Their results will be summarized here. The exact water-wave equations admit two-dimensional, steady, periodic solutions in the form ~ ( xt ,) =

n

A, cos[ 2nr( x - C r ) / X ] ,

where the Fourier coefficients A,, and the wave speed C are functions of the wave steepness k a = r h / X , h is the peak-to-trough height, and X is the wavelength. The first few terms in the expansion in powers of h/X were calculated by Stokes (1849) and Rayleigh (1917). In fact, the expansion is known as the Stokes expansion, and the solution retaining terms up to second order is the Stokes wave. Stokes (1880) also postulated that there exists a limiting wave steepness beyond which a smooth profile cannot exist. This has been supported by numerical calculations of the steady wave profiles, and various recent calculations have produced solutions up to a wave steepness of about 0.141, which exhibit a very sharp crest. An arbitrary infinitesimal three-dimensional perturbation takes the form 00

$ ( x , y , r ) = exp( i[ p ( x - C t )

+ qy - i l l ] ) - w a; exp(in(x - Ct)),

( 173)

where without loss of generality we have taken X = 2 x and g = 1. The

Nonlinear Dynamics of Deep- Water Gravip Waves

147

perturbation wavenumbersp and q are arbitrary real numbers. I t is obvious that Eq. (173) is an eigenvector of the infinitesimal perturbations to Eq. (172) with 52 the eigenvalue. Instability corresponds to Im52t0, since 52 occurs in complex conjugate pairs. The problem is to determine D and the corresponding a:. This was accomplished numerically by truncating the infinite sum in Eq. (173) to 2N+ 1 modes, substituting q,+q' and the corresponding C+~+C$' into Eq. (171), and satisfying the boundary conditions at 2N 1 points. The resulting homogeneous linear system of order 4N 2 was solved as an eigenvalue problem by standard methods. The accuracy of the solutions was improved by Newton's method when necessary. For small values of h/X (less than O.l), N = 2 0 sufficed to give 52 reliably to three significant figures. As h/X was increased, larger values of N were needed, and for the steepest wave studied (h/X=0.131), N = 5 0 was used.

+

+

1. Results Two distinct regions of instability were identified, denoted as (I) and (11). Plots of instability regions in the p - q plane for various values of h/X are shown in Fig. 39a-g (pp. 147-150). The eigenvectors corresponding to instability region (I) have dominant components n = 1 and - 1 for h / A + 0. For very small values of h / h , the

6 x

0

0.5

1

2

3

FIG. 39. Instability regions in p-q plane for various values of wave steepness: (a) h/h=0.032. (b) 0.064, ( c ) 0.095, (d) 0.111, ( e ) 0.127, (f) 0.131. Shaded regions denote instability. Points of maximum instability are marked by dots -, with the approximate growth rates shown. (g) Resonance curves in p-4 plane for Class I (solid lines) and Class I1 (dashed lines) instabilities for m = 1 and 2.


148

‘r

*r

0

0.5

1

2 P

3

(C) FIG.39. (Continued)

instability band is very narrow and lies near the curve defined by

p - 1+ [ q 2 + ( p - 1)2]’/4=p+1- [ q 2 + ( p + 1)2]”4.

(174)

The band is symmetrical about q = O and p = O (with a;+u’-,J. Near the origin, the instability bandwidth along the p axis is proportional to h / X . Near p = $, the bandwidth is of order (h/A)4. For sufficiently large values of h/h, the band diminishes in size. At h/h=0.108, the instability band


I 9

149

4 x

1

P

0

0.5

2

1

3

P

(4 FIG.39. (Continued)

detaches from the origin ( p = q = 0), indicating that the system is no longer unstable to infinitely long wave perturbations. At h/A=O. 124, this type of instability disappears. The eigenvectors corresponding to instability region (11) have dominant components n = 1 and - 2 for h/A-+O. For small values of h / A , the instability band lies near the curve ~ - 2 + [ q ~ + ( p - 2 I)/ 4~=]p + l - [ q 2 + ( p + 1 ) 2 ] ” 4 . (175)


150

0

0.5

2

1

3

P

(f)

(q) FIG.39. (Continued)

The band is symmetrical about q = O and p=OS (with uA-wL,,). The bandwidth along the p axis is proportional to (h/X)’. Unlike (I), this instability band continues to grow with increasing h / A , being widest at p = 0 . 5 . At h/A=0.13, the instability region touches the p axis a t p = 0 . 5 , indicating the onset of two-dimensional instability of this type. The maximum growth rate of the type (I) instability is proportional to ( h / A ) * for small h / A . For each value of h / A , the maximum instability


151

occurs when q=O, so that type (I) instability is predominantly twodimensional. The maximum growth rate of the type (11) instability is of order (II/A)~. The maximum instability always occurs atp=0.5 and q f O . Thus, type (11) instability is predominantly three-dimensional. For values of h/X>l, the free wave components dominate the system, and the dispersion relation and related properties should be well described by the linear theory. When r (199) where ( ) denotes ensemble averaging. Note that R,, is Hermitian in i and j satisfying R, = RJ. (200) The time evolution of R, is obtained by multiplying Eq. ( I 12) for B(k,,t ) by B*(k,, t), ensemble averaging, and adding the Hermitian transpose:

m

=

111 q. {

I . 2.3'1

+ I -2 - 3

-m

exp( i ( w ; + w I X

-wZ-w3)t)(B7

B:B2B3) -

q.I . 2 . 3 3 + 1 - 2 -

exp( - i ( w j + w , - w 2 - w 3 ) ) ( B , B I B : B ~ ) ) d k , d k 2 d k 3 .

3

(20')

The zero-fourth-order-cumulant hypothesis permits the fourth-order aver-


182

ages to be expressed in terms of second-order averages, retaining only products of terms with opposite-signed phases. Since the subscripts I, 2, 3 denote dummy variables of integration, they can be interchanged and we can write (B:B: B 2 B 3 ) = 2 ( B ~ B 2 ) ( B : B 3 ) = 2 R 2 j R 3 1 ,

(202)

( B , B , B; B f ) = 2( B, B : ) ( B l B ; ) = 2 R , , R I 3 . (203) The steps in arriving at the closure hypothesis have been extensively discussed in articles on random waves (see, for example, Hasselmann, 1962, 1963; Benney and Saffman, 1966), and we shall merely note that they find justification in the Riemann-Lebesgue lemma, which essentially states that nontrivial contributions in the integrals involving random phase functions arise only from terms that satisfy, or nearly satisfy, the stationary phase criterion. Applying the closure hypotheses to Eq. (201), we obtain

-00

-

T/. I . 2.38, + I - 2 -

3 exp( - '(aJ+ W I

- w2 - 0 3 ) t )

x R I 2 R l 3dk, } dk,dk,. (204) This equation governs the evolution of the spectral correlation function R,, as affected by leading-order nonlinear effects described by the Zakharov integral equation. Note that the time scale of evolution is O ( R ) or O(k2a2), in contrast to Hasselmann's mechanism, which has a time scale of O(k4u4). Recall, however, that Hasselmann's theory assumes Q priori that the wave field is homogeneous for all time. This corresponds to writing R, = W , ) S ( k , - kJ) (205) where E(k,) is interpreted as the spectral energy density. If we make this assumption and set i = j in Eq. (204), we find, upon invoking the Hermitian property of R , m . .

-co

,

x Re{ R2iR,,exp( i ( y + w - w2 - w $ ) } d k , dk, dk,. (206) Since R j j = ( B j B : ) is real, both sides of the equation must individually vanish, and we get a E(ki)/a r =0. (207) This result implies that if one adopts the a priori assumption of Hasselmann (1962, 1963, 1968) that the wave field is spatially homogeneous, then


183

the O(k2a2)effects will have zero contribution. In other words, this result is consistent with the findings of Hasselmann and of Zakharov and Filonenko (1967) that homogeneous contributions to the evolution of the random wave field do not enter until the next order, which is O ( R 2 ) . In order to obtain these higher-order terms, the analysis must proceed to O(k6a6),or O ( R ’ ) on the right-hand side. Fortunately, McLean (1982) demonstrated using Feynman diagrams that all contributions to O(k6a6) come from products of 0 ( k 3 a 3 )terms, and that no contributions arise from products O ( k a ) , O(k2a2),O(k4a4),or O ( k 5 a 5 terms. ) This makes it possible to use the “four-plasmon approximation” (see Zakharov, 1966; LonguetHiggins, 1978) and consider the time evolution of the fourth-order correlation function B, BpB,*B:: -(B,BpB,*B,*)= a at

The time derivative terms are given individually by Eq. (1 12), so that we have m

$< B@B,*B:)= ( Ba

Bm B/3B:IJBl B: B ? )

T ~ I,, ,2 . 38r,+

1-2-3

-00

xexp(- i ( w , + w , - w , - w 3 ) t ) d k l d k 2 d k 3 M

m

-00


184

Applied to Eq. (209),we obtain

M

-m

x exp( - i(w,, + w I - w 2 - w3)t)d k , dk2dk3

-w

+

X exp( - i (av w I - w2 - w3)t ) d k , d k , dk3

+

7I

Rf12R I3 T p . 1.2.3'~

+ 1 - 2 -3

-m

x exp( - i(o, + w I - w2 - w3)t ) d k , d k , d k 3 00

+RI,JJIRa2R

I3 ' p , 1.2.3&p

+ I -2 -3

-m

x exp( - i ( w p + w , - w2 - w3)t)d k , dk2dk3 M

-m

+

Xexp(i(wp w l-w2-w3)t)dk, dk2dk3 W

-m

+ wl-w2- w3)t )d k , d k , dk3

X exp( i (up


- RflpJ7j"R21,R31Ta.

185

l.2.3'n+l-2-3

--m

x exp(i(oa +a,-

~ 2 - ~ 3 ) t d ) k ,d k 2 d k 3

-w

Xexp(i(w,+o,

dk2dk,.

--o,--03)t)dk,

(21 1)

We anticipate that the phase term varies far more rapidly than terms containing products of R, so that the latter can be extracted outside the integrals when estimating leading-order contributions. It is now convenient to introduce the slow time T = c2r, where E is a small parameter of order ka measuring the degree of nonlinearity. The rapidly oscillating terms in the integrand can then be evaluated by integrating Eq. (211) from an arbitrary initial time - T ~and letting ~ ~ 3 0 invoking 0 , the "delta calculus" as discussed by Benney and Saffman (1966)

Thus, upon integration of Eq. (210) from - 00 to T , we have

ss 30

1 -( 2i

Ba B f l

B: B:)

=

{ Ra2R,3R

I p T w , I . 2. 3su

+ I - 2 - 3:'+

I

-2 -3

-w

+ Rn2Rf13

- Ral R2p

Iv ' p , 1.2.3'p

R3vT\L

+R"(71Ra2R

I . 2.3'fl+

+ I -2-

w

3'p+

I - 2 -3

I - 2- 3sp"+ I - 2 - 3

+ I - 2 -+:'3

I -2 -3

I . 2. 3Sp+ I - 2 - 3;"'.

I- 2 - 3

I3 Tz*. I . 2,3'v

--m m

+Rfl,~J7sRa2R13T+. -m


186

7

- Ru,J

R 2 v R 3 I [j'.

1.2.3'[1+

I

- 2 - 3'$+

I -2 -3

-aJ M

- R[+J

I

I I R 3 I 'a,

I , 2 . 3 ' ~ ~+ I - 2 - 3':+

I

- 2 -3

-aJ 53

-m M

--M

+

where a,+ I - 2 - = S(w, w I- w2 - w3). Equation (213) gives the next order contribution to the evolution of R,. It would be the leading term if the spatially homogeneous assumption had been made. With this assumption, the term given by Eq. (2 13) greatly simplifies, so that we have 00

5 at =4mJJJT,:

I . 2. 3 a I +

I-2-36,w+I - 2 - 3

-aJ

X

[ E3E,( I' + Ei ) - EIEi( E3 + E , ) ] d k , dk2dk3,

(214)

which can be identified as Hasselmann's equation. In a more general situation, the homogeneous contributions can coexist with the inhomogeneous contributions to yield

-m

where ( B , BflB: B:) is given in terms of R by Eq. ( 2 13). It is therefore clear from the above analysis that the evolution of a nonlinear gravity wave field is composed of modulations with a time scale


I87

of O ( R ) that do not involve net energy exchanges and weak model energy transfer with a time scale of O( R ’). Depending on the particular type of application desired, one or the other of these mechanisms may be more important. For example, in attempts to describe the energetics of a wave spectrum for long-term, large-scale oceanographic forecasting purposes, the energy-transfer term, albeit weaker and slower, is more important. In such an application it would, in fact, be desirable to select an averaging process that would average out the nonhomogeneous contribution. On the other hand, in applications such as interpretation of returns from remote sensors, the description of the modulational characteristics of the ocean surface may be an essential part of the problem and the nonhomogeneous contribution must be taken into account.

2 . Stability of a Homogeneous Spectrum to lnnhomogeneous Disturbances The assumption that the wave field is homogeneous for all time yields the Hasselmann equation, which does not contain information on spatial structures. One may identify conditions under which such an assumption is a good one for describing the dynamics of the wave field by studying the stability of a homogeneous spectrum to inhomogeneous perturbations. In the context of the spectral correlation function R,/, the problem is whether or not the undisturbed, homogeneous spectrum

R(k,Jq= E ( k , ) V , - k , ) is stable to inhomogeneous perturbations of the form

(216)

R’(k,, k,, t ) = e , ( k i ,k,, K)6(ki - k, - K) exp( iat)

+e

(k,, k,, K)6 (k, - k,

+ K) exp( iaf),

(2 17) where K is the modulation wave vector. Substituting Eqs. (216) and (217) into (204) and linearizing, we obtain an eigenvalue problem with S2 as the eigenvalue and e , as the eigenvectors. Stability corresponds to a 2 > 0 , and instability of 8’ < 0. Although no solution of this general eigenvalue problem has yet been obtained, the narrow-band case was analyzed by Alber and Saffman (1978) in the two-dimensional context, and by Crawford et al. (1980) in the three-dimensional context. The narrow-band approximation permits one to replace the interaction coefficient T , , , k . ,by its value at the peak of the spectrum To. This assumption permits T to be extracted from the integral. Correspondingly, the dispersion relation w(k) is expanded about the dominant wave vector k, (these are the same two assumptions used to derive the nonlinear Schro-

-


188

dinger equation-in fact, the analysis of Alber and Saffman was based on the nonlinear Schrodinger equation formulation). This permits cancellation and grouping of a number of terms due to symmetry. We shall briefly highlight the steps here.

3. Narrow-Band Case We consider the limiting case of Eq. (204) to examine the evolution of a narrow-band, inhomogeneous, random wave field. We introduce the average and separation variables n and m of two wave vectors k, and kj:

n = 3(k, +kj), and define

m= k, -kj,

(2’8)

[

F(n, m, t ) = 7 lki’ lkJ’ ]R(k,, k,, t> exp(- i [ w ( k , ) - w ( k , ) ] t ) . (219) ( 2 ~ ) w(ki)4k,) Integrating Eq. (204) with respect to k, and noting

we obtain i-a8 (n, m,2) - [w(ki) - w(kj)]f(n,

m,f )

at

For a narrow-band spectrum, n is close to a constant carrier wave vector k,=(ko,O) and m is small compared with k,. Thus we can expand the frequency w,(k) about wo = a(&):


189

Furthermore, the coupling coefficient can be approximated by its value at k,,, i.e.,

T(k,, k,,k,, k3)= T ( h , $,h,$) = Ikol3/(2~)’. Defining a relative wavenumber p = ( p ,q ) by

(223)

p=n-ko, and evaluating (ao/ak)(k,,), (a2w/3k2)(k,J, we obtain

(224)

An envelope :pectral function F(p,x,t) can be defined as the Fourier transform of F(n, m, t ) with respect to m:

In terms of F(p,x,t) Eq. (225) becomes a transport equation in x, p, and t

We mentioned that Eq. (227) can also be derived by an alternative method based on the nonlinear Schrodinger equation (Alber and Saffman, 1978). This is done by defining the two-point, one-time, envelope correlation function p(r,x,t)=(A(x,,t)A*(x,,r)), (228) where x = ;(x, x2) and r = x2- x, . The equation governing p(r,x,t) can be found by multiplying the threedimensional nonlinear Schrodinger equation for A (x, ,t) by A *(xz,t), adding it to the complex conjugate expression with x2 and x1 exchanged, averaging, and invoking the closure hypothesis that

+

( A (%)A(x,)A*(x,)A*(x,))=2(A (x,P*(x,)) 0 and

W = - i K , u - 2iK, p + i (( K,!- 2 K ; ) { 2 - ( K 2 - ~ K Y ~ ) } ) " ~(252) . We list the results of Eq. (252) for some limiting cases.

Nonlinear Dynamics of Deep- Water Gravicv Waves

195

i . Nonrandom, two-dimensional. The nonrandom assumptions lead to u = ,u = 0. The two-dimensional assumption leads to K,. = 0. The resulting

dispersion relation is

Q =i ( K . f ( 2 - K : ) ) 1 / 2 ,

(253)

which is the Benjamin-Feir stability result for two-dimensional nonlinear wave trains. ii. Nonrandom, three-dimensional. The dispersion relation is

Q = i(( K :

- 2K,'

){ 2 -( K e -

2K:

)})I/',

(254)

which agrees with the stability results for the three-space-dimensional nonlinear Schrodinger equation. iii. Random, two-dimensional. In this case, K,. = p = 0, the dispersion relation becomes Q = - i K , u + i ( K < ( 2 - K : ) ) I/ 2 (255)

.

which is precisely the result given in Eq. (244). The effect of randomness in three-space dimensions is similar to the two-dimensional case in that the instability is diminished with increasing bandwidth due to randomness. One way of geometrically interpreting Eq. (252) is as follows. Start with the deterministic stability result in twodimensions. The effect of randomness with spectral widths u and p is to introduce a plane that makes an angle tan-'u to the K , axis, and tan-'2u to the K, axis, from which the magnitude of the instability is measured. This is illustrated in Fig. 53b.

C. PROPERTIES OF THE DISCRETIZED ZAKHAROV EQUATION Any numerical attempt to solve the Zakharov equation, or any of the statistical equations arising from it, involves some form of discretization. In this section, we present some results on the solution of a simple discretized form of the Zakharov equation, and consider the implications of these results for the long-time dynamics of the water wave system. The simplest form of discretization in the spectral representation is the delta-function expansion. That is, we write the free surface q(x. 1 ) as ~ ( X , I ) = (D(klll,f)exp(ik,,,*x)+ D*(k,,,f)exp( - ik,.x))

(256)

m

This corresponds to a numerical scheme for approximating Eq. ( I 12) that uses a top-hat integration scheme on the integral. D(k,,t) is related to B(k, I) through a scaling factor of [Ik(/2w(k)]'/2. which can be absorbed into a rescaled interaction coefficient T'(k,, k,, k,, kn). Substitution into Eq.


196 ( 1 12) yields

d

i -D(k,,t)+w(k,)D(k,,t) dr

x k,,

c

+ k, = k, + k,

T’(k,,k,,,k,,k,)D*(L,,r)D(k,,r)D(k,,r), (257)

which is a n infinite system of nonlinear, ordinary differential equations in time. There have been many recent reports of chaotic behavior in nonlinear dynamical systems having the form of Eq. (257) (Yamada and Graham, 1980; Holmes, 1979; Greene, 1979; Jorna, 1978). Thus we expect such behavior from Eq. (257) as well. This expectation is verified, as we discuss in the following. 1. Characterization of the System

The energy of the displacement is

It can be seen that the evolution of the energy E ( t ) satisfies

D*(k,, t)D*(k,,i)D(k,, t)D(k,,t). (259) The Zakharov approximation of the water wave is such that the interaction coefficient T‘ does nor possess the symmetry property X

T’(k,, k,, k,, k,)

= T’(k,, k,,

k,, k,,),

(260) so that the right-hand side of Eq. (259) is not zero. This implies that the system does not conserve energy. On the other hand, the system is not dissipative in the normal sense, in that there is no monotonic flow of energy out of the system with time. It will be seen that under many situations, there is an oscillation of E ( t ) with time. The lack of the symmetry noted above also implies that

where we have used the subscript notation. The system is therefore not Hamiltonian and D, and D,*as conjugate variables. It should be noted that the exact equations of motion for water waves (1)-(3) indeed form a Hamiltonian system, as shown by Zakharov (1968) and Miles (1977). We can write Eq. (257) in the form Y =f(y),

(262)


I97

with

Y=

where the subscripts denote dependence on various components of k. It can be seen that the divergence of f, defined as

is identically zero. Thus, the system is volume-preserving in the phase space, in contrast to systems exhibiting strange attractor behavior.

2. Properties of the Solutions We now present a series of solutions to the system that correspond to time evolutions of an unstable wave train in the presence of noise-level modulations. We consider the scalar case with k = ( k , 0). The initial condition for a uniform wave train with wavenumber k , and amplitude a, is Dnl(o)=

(

ra,,

0,

kn,= ko, k,,,# k,.

A small perturbation in the form of a long-wave modulation with wave-

length 277lAk is represented by the presence of nonzero components located at k,+ Ak. The magnitudes of these “sidebands” correspond to the strength of the modulations, and the ratio k , / A k gives the number of carrier waves in one modulation period. In all our computations, we use the normalization that k,= 1. The initial strength of the sidebands is loP6 compared to the primary mode at k, unless otherwise specified. In Fig. 54a-e, we show the results of a series of calculations with seven modes, located at k,, k , , = k , + A k , k r z = ko?2Ak, k , , = k 0 1 3 A k , with Ak =0.2k0. The initial conditions for D, are Do= m,, D, = 1OP%a, ( m = 1, 2,3), and D - r n = - 10-6inu,(m = 1,2,3). In this series the wave amplitude a, is varied with all other parameters kept constant.

koknAk(n =

9

I,

2, 3)

0

"1

NJ

'0

'

.

.

.

.

.

.

.

'

.

,

5 000

.

.

.

.

.

.

10,000

.

.

.

.

.

I5,OOO

.

.

.

.

I

20,000

time, t FIG. 54a. Time evolution for a0=0.05. Top trace: magnitudes (absolute values) of all modes. Second trace: real (-) and imaginary parts)1-( of mode at k 0 - 2 A k . Third trace: real and imaginary parts of mode at k,. Bottom trace: real and imaginary parts of mode at k , + 2 ilk.


199

time, t ( b) Ftcj. 54b. Time evolution for ao=O.l. Top trace: total energy. Second trace: magnitudes of modes k d ~ n n )k,-Ak(-), . k,+Ak( + + +). Third trace: magnitudes of modes koqko-2Ak(-), k 0 + 2 A k ( + +). Fourth trace: magnitude of modes k,. k 0 - 3 A k ( - L k"+ 3 b k ( + ).

++

+

m

2

201 time, t (C'

magnitudes of modes ko(nna), k,+Ak( + + +). Second trace: magnitudes of modes k o , k , - 2 A k ( - ) , k , + Third trace: magnitude of modes k,, k,-3 Ak(-), k,+ 3 A k ( + + +).

FIG. 54c. Time evolution for ao=0.2. Top trace: k,-Ak(-). 2 Ak(

+ + + ).

Nonlinear Dynamics of Deep- Water Gravip Waves

tlmr. t

(d) FIG. 54d. Time evolution for a,=0.45. For legend see Fig. 54c.

20 1


202

k,

I nAk,

n = 1,2,3

0

0

200

600

400

800

1000

time, t

FIG. 54e. Time evolution for a,=0.5. Magnitudes of all modes.

In Fig. 54a, a,=0.05, and the motion is periodic. The sideband components included in the calculations oscillate but their magnitudes remain small. In Fig. 54b, ao=O.lO, and the motion is “recurring.” It can be seen that the sidebands at k,? Ak grow to a significant level (approaching the primary) and then subside, reconstructing the initial condition almost perfectly insofar as the amplitudes of the modes are concerned. The other components in the calculation receive some energy and grow and decay in phase with the sidebands at k,+Ak. The process repeats in time, although not with perfect periodicity. This behavior is normally referred to as the Fermi-Pasta-Ulam recurrence, as opposed to Poincare recurrence, which requires the return of both amplitude and phase to their initial states (see Jackson, 1978). Note that the energy. oscillates in time but does not dissipate on the average. In Fig. 54c, a,=0.2, and the motion is “chaotic.” By “chaotic” we merely mean that no apparent order can be detected in the time history. In Fig. 54d, a0=0.45, and the motion is first apparently recurring, but transits to chaotic behavior at a later time. Finally, in Fig. 54e, a,=0.50, and the system has returned to one with periodic motion. This set of various types of behavior can be related to the stability properties associated with the initial conditions. The stability diagram for infinitesimal perturbations on a uniform wave train is shown again in Fig. 55 with a different scale. Superposed on the diagram are the various cases we have computed, labeled, (a)-(e), It can be seen that for case (a) (corresponding to the calculation shown in Fig. 54a), all the included modes are stable. For case (b), the sidebands at k,+Ak are unstable, but all other components are stable. For case (c), both the pairs at k,?Ak and

ko-3Ak I

ko--2Ak I

ko-Ak I

kI0

ko+ 1 Ak

k o +I2 A k

k o +I3 A k

06

0.5

0.4 0

1 Y

0.3

'\

0

/

0

/

0.2

0.1

0.0 k

FIG. 55. Instability diagram for a uniform wave train with amplitude uo, wavenumber k,). subject to perturbations with wavenumber k - k , ) . The unperturbed system has a single mode at k, (denoted by A). The perturbations for the various cases shown in Fig. 54 are located at points .labeled by 0 or 0. Solid symbols indicate unstable modes. Open symbols indicate stable modes. Dashed line is corresponding result for the nonlinear Schrfidinger equation (see discussion on p. 209).

204


k0+2Ak are unstable. The latter pair can be interpreted as the second harmonic of the modulation perturbation, since it corresponds to modulations with wavelengths 277/2Ak, half of that associated with the sidebands at k,?Ak. For case (d), only the outer pair of components (at ko+3Ak) are unstable. Finally, all components are stable again for case (e). These observations indicate that periodic motion is associated with stable perturbations [cases (a) and (e)]. Indeed, the oscillation frequency is predicted by the stability analysis. When only the inner set of components ( k , , = k,?Ak) is unstable, the motion is recurring. The unstable components grow according to the instability results, return to almost their initial states, and become unstable again. The other stable components behave as forced components of the unstable modes and are more or less phaselocked to them. When more than one set of components are unstable, the motion is apparently chaotic, dominated by the nonlinear interactions of the various unstable modes. In fact, one can detect a tendency to partition the energy of the system among the unstable components and the primary. In this sense, a partial thermalization is achieved. Of most interest is the transitional behavior of case (d). For small times, the behavior appears to be recurring involving only the primary mode at ko?3Ak. However, for a later time, the other modes which are linearly stable participate in the energy-sharing process, and a chaotic behavior follows. The involvement of these linearly stable modes (at k,&Ak and ko?2Ak) is caused by nonlinear instability, which is expected to be effective only when the nonlinearity of the system and the magnitude of the perturbations are sufficiently large. Unlike the linear instability, the growth rate of the nonlinearly unstable mode is dependent on its own magnitude. For small times, the magnitudes of the modes were small enough not to trigger significant growth. However, these modes eventually receive energy from the linearly unstable modes at k,+3Ak during the recurring cycle, and nonlinear instability sets in. Chaotic behavior then appears as a consequence of interactions among all the unstable modes. Evidence of nonlinear instability when a, is large can be seen for a,=0.5, when the system has restabilized against linear instabilities. We have computed the time evolution of initial conditions corresponding to a uniform wave train perturbed by a pair of sidebands located at k - ko+Ak with strengths lo-* and lo-' that of the fundamental mode at k,. In Fig. 56a, we see that a sideband magnitude of lO-*?ra, leads to a nonlinear oscillation that is clearly nonexponential initially. In Fig. 56b, the sideband magnitude was increased to lo-'. The initial growth rate increased correspondingly by an order to magnitude. The oscillation triggers the growth of the other modes and a chaotic state results.

ko

Ak

~~

.w

.

=

k0

x

-- _ v

,

n m

n

a

'

k,?

3Ak

0'

3

FIG. 56. Time evolution for u0=0.5 with initial sideband perturbations (a) D(k,- A k ) = 0.01 nu,, D ( k 0+ Ak) = - 0.01inao; (b) D ( k o--A k ) = 0.1nuo, D( ko Ak)= - 0.1i m , . For legend see Fig. 54c.

+

N

N

time, t ( b) Fra. 56. (Continued)

207


o 0

k o f 3Ak

n I000

R A / L c R h A 2000

3000

A

m

4000

5c

time, t (a)

FIG.57. Time evolution of the nonlinear Schrodinger equation approximation for a,=0.1 0.5 (b). For legend see Fig. 54c.

(a) and

N

ki+P& #

01

n 0

AL

U

0 -

Y

d a

0

N

n

n

At

U

0 -

..... .

Y

time, t (b) FIG.57. (Continued)


209

A wide range of initial conditions covering values of a,, Ak, and perturbation magnitudes have been computed, and the results support the stated relationship between instability and long-time behavior provided that both linear and nonlinear instabilities are taken into account. As we discussed earlier, we can make a further approximation by taking T'(k,,, k,, k;, k,) to be T'(k,,k,, k,, k,) and expanding w(k) in Taylor series about k, to second order. The resulting set of equations is the discretized Fourier transform of the nonlinear Schrodinger equation. As a consequence of the simplifying assumption on T', the system is now conservative and Hamiltonian in the conjugate variables D,,, and D Z . The stability diagram for a uniform wave train subject to infinitesimal perturbations is shown in Fig. 55, where instead of the shaded region, the instability region covers all of the region above the dashed lines. Consequently, we expect from the foregoing discussions to observe periodic, recurring, and then chaotic motions, but there will be no return to periodicity for large a,. This is verified by the results shown in Fig. 57, where we present initial conditions corresponding to cases (c) and (e) according to Fig. 55. 3. Discussion

We have seen that a relationship between the stability of the initial condition and its subsequent evolution exists for the discretized Zakharov equation, which can be considered as a nonlinear, nonconservative, nonHamiltonian, but phase-space volume-preserving system. The evolution may be periodic, recurring, transitional, or chaotic, depending on the initial conditions. In a sufficiently nonlinear system, nonlinear instability must be taken into account in addition to linear instability. It is also interesting to note that this system can be further approximated by the nonlinear Schrodinger equation, which is a conservative, Hamiltonian system, which exhibits part, but not all, of the phenomena observed. Another conclusion that is suggested by these calculations is that the calculated evolution of the wave system may depend crucially on the choice of modes included in the calculation. To illustrate this point, we show in Fig. 58 the evolution of the initial condition D ( k ) = va,,

D ( k ) = 10-2va,,

k = k,, k = k,+ 2 Ak,

D ( k ) = - i 1OW2va,,

k = k, - 2 Ak,

D ( k ) = O(10-6~a,), all other

k,

with a, = 0. I5 and Ak = 0.1k,, using two different distributions of modes (see Fig. 58). Figure 59a shows the results using seven modes located at k,,

h

Q u

210

0.4

0.6

0.8

1.0

1.2

1.4

1.6

(b) FIG. 58. (a) Initial condition corresponding to a uniform wave train at k, perturbed by a 1% sideband component at k , ? 2 4 k , where Ak=O.lk,. for a,=0.15. Mode distribution is such that only one pair of unstable modes is included. (b) Same as (a) except that mode distribution is such that all three pairs of modes included are unstable.

2 12


a

\Ul

FIG. 59. Time evolution of a uniform wave train perturbed by sideband components at k 0 2 2 A k with initial condition and mode distribution given by Fig. 58a (a) and 58b (b).

213


7J-l

(9 0

,,,

time, t ( b)

FIG. 59. (Continued)

.

,, , ,

.,

., . . . .

. .

.., .

..

. .,

214


ko+2Ak, ko?4Ak, k0+6Ak. With this choice, only the k,k2Ak pair is unstable, and the calculated behavior is “recurring.” Figure 59b shows the results using seven modes located a t k,, k,? Ak, k,,? 2 Ak, k,+ 3 Ak. The initial condition remains as in Eqs. (266). The calculated behavior is “chaotic” in the long run, since all the modes included were unstable. Hence, calculations using only the most unstable sidebands and their harmonics that predict recurring motion (Lake et al., 1977) can be misleading, for if more unstable modes are allowed in the calculations, chaotic behavior would result. This seems to suggest that nonlinear water waves form an inherently chaotic system. However, the chaos is confined to a small range of wavenumbers that are initially unstable. We term this phenomenon “confined chaos.” VIII. Discussion In this review, we have attempted to highlight some of the major findings regarding the nonlinear dynamics of deep-water gravity waves, with emphasis on the progress made in the past 15 years. Waves on deep water have always fascinated the curious, for they represent a familiar yet complex phenomenon, easy to observe but difficult to describe. As noted in the address of Sir Horace Lamb to the London Mathematical Society in 1904, the study of waves on deep water “was the first, or all but the first, hydrodynamics question to be attacked systematically from the basis of the general equations, and it offerred ... a field in which (one) could test the efficiency of analytical methods which were ... new and unfamiliar.” Lamb was referring particularly to the problem of the waves produced on deep water by a local disturbance, which was posed as a prize topic by the French Academy in 1816, and solved in the same year by Cauchy and Poisson independently. Apparently motivated by the carefully documented observations of wave motion by Russell (1844), Stokes (1849) published the memorable treatise “On the Theory of Oscillatory Waves,” which not only summarized the state-of-the-art knowledge of water waves at that time, but introduced the Stokes expansion, which formed the cornerstone of modern theories of nonlinear analysis of partial differential equations. He revisited the subject in 1880, when he published a note showing that the waveheight-towavelength ratio of two-dimensional, deep-water gravity waves cannot exceed a finite maximum, at which point the individual wave attains a corner at the crest subtending an angle of 120”. For almost a century after this, attention was focused on the exploration of the steady solutions found by Stokes. Burnside (1916), Nekrasov (1920),


215

and Levi-Civita (1925), among others, examined the question of the existence and convergence of the Stokes expansion for nonlinear corrections to the steady wave profiles. Michell (1 893), Wilton ( I 913), and Rayleigh (1917), on the other hand, were among those who became interested in finding approximate profiles for steep waves. It should be noted that the calculation of steep steady waves is still a problem of great interest, as illustrated by the works of Schwartz (1974), Cokelet (1977), LonguetHiggins and Fox (1977), and Saffman (1980). Until about 15 years ago, the stability of the weakly nonlinear steady wave train on deep water was essentially unquestioned. It therefore came as somewhat of a surprise to workers in the field when Sir James Lighthill (1965) demonstrated, using Whitham’s theory, that such a wave train is unstable to modulational perturbations of long wavelength. The same instability was noted also by Zakharov (1966). A more detailed calculation, with growth rates and ranges of unstable wavenumbers, was given by Benjamin and Feir (1967), (see also Benjamin, 1967), from which the name “Benjamin-Feir instability” was coined. Benjamin (1967) also presented some experimental data that supported the predicted instability, at least qualitatively. The next question that interested workers in the field was what happens to the unstable wave train as it evolves in time. There were three different approaches to the formulation of a time-dependent theory. Whitham ( 1965, 1967, 1970) used a variational principle on the phase-averaged Lagrangian to obtain equations of motion governing the evolution of the amplitude and phase functions of a slowly varying wave train. Benney and Newell (1967) used a multiple-scales approach to obtain slow time dependence of the wave properties, and arrived at the nonlinear Schrodinger equation. The same nonlinear Schrodinger equation was derived by Zakharov ( 1968), using small-amplitude expansion of the Fourier representation. Specific applications to deep-water wave evolution were obtained by Lighthill (1967), Whitham (1967), Hayes (1970a, b, 1973), and Yuen and Lake (1975) via the Whitham theory. Yuen and Lake (1975) derived the nonlinear Schrodinger equation from Whitham’s theory, Chu and Mei (1970, 1971), Hasimoto and Ono (1972), and Davey and Stewartson (1974) via the multiple-scales approach, and Zakharov (1968) (later Crawford ef al., 1980) using the spectral representation. During the mid- 1970s, the nonlinear Schrodinger equation was widely accepted as the equation for description of the evolution of weakly nonlinear wave trains on deep water. The nonlinear Schrodinger equation was not only derivable from all three of these approaches, it also possessed the added attraction that it could be solved exactly for localized initial conditions. This was accomplished by Zakharov and Shabat (1972), using the then newly discovered

216


method of inverse scattering (Gardner et af., 1967; see also Lax, 1968. for generalization of the method, and Scott et al., 1973, for an excellent review). The exact solution predicts that a localized initial condition eventually breaks up into a number of envelope solitons and a tail, and that these envelope solitons are stable both to infinitesimal perturbations (shown by Rowlands, 1974) and to collisions with other wave packets (as predicted by the solution method). Verification of these collision properties was obtained experimentally by Yuen and Lake (1975), whose results explained some of the earlier observations by Feir ( 1967) regarding the disintegration of wave packets. The existence and stability of these envelope solitons (at least in two dimensions) attracted much attention, and models of wave fields based on soliton ensembles were proposed (Cohen et af., 1976; Mollo-Christensen and Ramamonjiarisoa, 1978). The inverse scattering method cannot be easily extended to the case of a wave train, although progress in that direction was made by Lax (1968) and Ma (1977). Numerical solutions of the nonlinear Schrodinger equation were, however, applied to the study of the behavior of wave trains on deep water at large times by Lake et af. (1977). They found that when the most unstable disturbance is imposed, the modulation would grow (as predicted) to a maximum and then subside. This modulation-demodulation cycle would presumably persist indefinitely in the absence of energy dissipation. Such a phenomenon is known as Fermi-Pasta-Ulam recurrence (Fermi et al., 1955), which loosely describes the repeated return of the wave amplitude to its initial condition. This modulation-demodulation tendency was observed experimentally (Lake el al., 1977), but it was noted that because of the presence of surface dissipation and small-scale wavebreaking, this cycle was followed only approximately. Under certain circumstances, it was found that although the wave envelope demodulated, the dominant frequency shifted to a lower value. This frequency-downshifting was not contained in the nonlinear Schrodinger equation. Subsequent numerical calculations by Yuen and Ferguson (1978a) of the nonlinear Schrodinger equation subject to spatially periodic boundary conditions indicated that this simple modulation-demodulation cycle occurs only when the unstable perturbation imposed is such that none of its higher harmonics are unstable. Otherwise, the unstable harmonics actively participate in the evolutionary process, and a more complex evolution pattern follows. Energy still returns repeatedly to the unmodulated state, but each of the unstable modes takes a turn dominating the system. A conclusion that can be drawn is that all unstable modes participate in energy-sharing in the evolutionary process. The stable modes do not receive significant energy, and an analytical bound was established by Thyagaraja (1979) and verified numerically by Martin and Yuen (1980). Since the


217

range of unstable wavenumbers is finite, this implies that the wave train cannot thermalize in the classical sense, since energy is confined to a narrow band of unstable wavenumbers instead of spreading over all modes. At this stage of development, the nonlinear Schrodinger equation appeared to be an extremely attractive model for water wave evolution. It is self-consistent, in the sense that it predicts the confinement of energy within a narrow band, which is necessary for its validity. It predicts coherence of the wave train system, and makes possible the modeling of a more complicated wave system (such as the nonlinear wind-wave system) in terms of wave trains (Lake and Yuen, 1978: Yuen and Lake, 1979). It also dispels the common belief that nonlinearity must act to destroy coherence since it demonstrates that the nonlinearity may actually enhance coherence by counteracting dispersion. Unfortunately, these simple properties were not preserved when extended to three dimensions. The extension of the governing equation was straightforward; it called only for an additional term containing the second derivative in the lateral direction (Zakharov, 1968). All two-dimensional solutions are also plane wave solutions for the three-dimensional equation. The stability of a uniform wave train was studied by Zakharov (1968). Benney and Roskes (1969), Davey and Stewartson (1974), and Martin and Yuen (1980), and it was found that the instability region was not bounded. Therefore, whereas recurrence still occurred for certain choices of initial conditions (Yuen and Ferguson, 1978b), an unstable wave train would exhibit quasi-recurring energy leakage to arbitrarily high modes (Martin and Yuen, 1980) in the sense that there appeared still to be repeated returns to initial conditions, but higher and higher modes (which are unstable harmonics of the initial perturbation) would participate in the evolution as time progressed. Eventually, the solution would march out of the domain of validity of the equation. Finally, it was shown that the plane envelope soliton is unstable to cross-wave perturbations (Zakharov and Rubenchik, 1974), for long-wave perturbations; Saffman and Yuen, 1978, for finitewavelength disturbances; and Martin et al., 1980, for traveling disturbances). The concepts of envelope solitons and recurrence, so prominent and important in the two-dimensional context, appeared to be missing or diminished in importance in three dimensions, at least according to the three-dimensional nonlinear Schrodinger equation. The energy leakage, found by Martin and Yuen (1980), clearly demonstrated that the threedimensional nonlinear Schrodinger equation is not an acceptable equation for describing the evolution of water waves in three dimensions and a better equation is needed. Examination of Zakharov’s ( 1968) approach to derive the nonlinear

218


Schrodinger equation reveals that he made two additional assumptions in order to obtain the nonlinear Schrodinger equation from an integrodifferential equation. These assumptions deal with the requirements that the system under consideration must be narrow-banded, so that the modal interaction coefficient and dispersion relation can both be well approximated by their values at the dominant component. The leakage reported by Martin and Y uen ( 1980) indicated that the narrow-banded assumption in three dimensions is not uniformly valid in time. This observation prompted Crawford el al. (1980) to study the properties of the integrodifferential equation that Zakharov (1968) obtained as an intermediate step. The results were extremely encouraging. First, they found that the instability diagram in two dimensions exhibits a restabilization at large wave steepness that was not present in the earlier approximate results, but was found by Longuet-Higgins ( 1978), who numerically calculated the stability of the wave train to rational wavenumber disturbances using the unapproximated equations. Second, results obtained from the Zakharov equation agreed much better with data regarding the maximum instability wavenumber and instability growth rates when the wave steepness is no longer very small. Last, but not least, the instability diagram based on the Zakharov equation is bounded even in three dimensions. in contrast to instability results obtained from the three-dimensional nonlinear Schrodinger equation. This implies at least that the Zakharov equation does not suffer from the same inconsistency or nonuniform validity as does the Schrodinger equation. Once again, it became reasonable to expect that only certain relatively low modes (and not arbitrarily high modes) would participate in energy-sharing in the evolutionary process. When the wave train restabilizes against two-dimensional disturbances at large wave steepness, the eigenvalues become real, indicating that the disturbances merely travel with respect to the unperturbed waves but do not grow in magnitude. The value of the eigenvalues, of course, continues to depend on the wave steepness. Chen and Saffman (1980) noticed the interesting fact that at still larger wave steepness, the real eigenvalues become zero (this was found for the infinite wavelength limit by Peregrine and Thomas, 1979). This implies that at such wave steepness, the imposed disturbance co-propagates with the undisturbed waves, modifying the undisturbed wave profiles, but imposing no new time dependence. In other words, new steady solutions may come into existence. This possibility was confirmed by the calculations of Chen and Saffman (1980), who found bifurcated solutions representing nonuniform waves with periods N times that of the individual waves. It is interesting to note that this finding did not contradict an earlier proof by Garabedian (1965) that uniform deepwater gravity waves are unique.


219

The bifurcation calculations of Chen and Saffman (1980) were extended to three dimensions by Saffman and Yuen (1980a,b. 1982) using the Zakharov equation. It should be pointed out that in two dimensions, the Zakharov equation cannot be expected to yield satisfactory results since the bifurcations occur at unrealistically large wave steepness. The critical wave steepness. ( k a ) , , at which bifurcation occurs decreases. however, with increasing obliqueness of the pattern (measured by the angle between the normal to the unperturbed wave crest and the normal to the crest of the modulation pattern). In fact, as this angle increases to 90°, ( k a ) , approaches zero. There is a range between 70" to 90" for which ( k a ) , is less than 0.3, so that the Zakharov equation is expected to give good approximations to the exact water wave properties. Saffman and Yuen (1980a,b, 1982) found that there are two classes of bifurcated waves, representing a skewed and a symmetric pattern. Each class is a doubly infinite set in pattern length and pattern aspect ratio (measured by N and the angle in the above notation). They also found that both the unbifurcated and the bifurcated solutions are stable near the bifurcation point for the skewed case. No calculations were made for the symmetric case, but it appeared that such a stability analysis is not likely to generate a selection rule for choosing one particular wave form over another. Totally independently, Su (1982a) reported experimental observations of symmetric and skewed steady wave patterns in an open wave basin, as well as symmetric patterns in an indoor wave tank. Comparison between the observed symmetric pattern and the calculated results is very good (Su, 1982a; Saffman and Yuen, 1982). Because of difficulty of measurement and interpretation, the skewed patterns have not yet been fully analyzed (Su, 1982b). The experiments, however, did show a distinct preference for one set of symmetric patterns: that with wave steepness of about 0.3, transverse wavelength of about 0.8 times that of the unbifurcated wavelength, and longitudinal wavelength of twice that of the unbifurcated wavelength. This set of measurements agrees well with the theoretical values, but they suggest the existence of a strong selection mechanism. The mystery of the missing selection mechanism was finally solved by McLean et a / . (1981) (see also McLean, 1982), who calculated the stability of finite-amplitude deep-water gravity waves to three-dimensional disturbances using the unapproximated equations. Their method also allowed them to deal with arbitrary values of perturbation wavenumbers, and not be restricted to low-order rational values as was required in the work of Longuet-Higgins (1978) in two dimensions. McLean et al. (1981) found a new class of strongly three-dimensional instabilities (whose maximum instability always occurs for three-dimensional disturbances, at least in all the

220


calculated cases) that in fact overtakes the more familiar Benjamin-Feirtype instability for moderate and large wave steepness. They called the new instability the Class I1 instability, and the Benjamin-Feir type the Class I instability. They also found that as wave steepness increases, the Class I instability disappears. The Class 11 instability then connects with the two-dimensional large wave steepness instability reported by LonguetHiggins (1978). Maximum instability remains fully three-dimensional. The results of McLean et al. (1981) also showed the existence of high-wavenumber, two-dimensional instabilities. Their results do not contradict those of Longuet-Higgins (l978), who found no high-wavenumber, two-dimensional instabilities, since Longuet-Higgins’ results were confined to pure or low-order rational harmonics that lie outside the bands of instability. On the other hand, the conclusion by Phillips (1981) that deep-water gravity waves are stable to arbitrary high-wavenumber perturbations is proven erroneous. The error appears to lie in the fact that the WKB method used by Phillips (1981) cannot give higher-order nonlinear instabilities and therefore misses the Class I1 instability entirely, although comparison with Dagan’s (1975) work also casts doubt on the correctness of the kinematic boundary condition used by Phillips (1981). The Class I1 instability is not totally unexpected. Zakharov (1968) hinted at its existence, treating it as a higher-order resonance. A plausibility argument was recently given by Hasselmann (1979) based on the concept of parametric resonance. However, the importance of Class I1 instability lies in its physical implications. The experiments of Su (1982a) generated waves of a given wave steepness using a mechanical wave paddle. It can be verified that at the wave steepness reported by Su, the waves are more unstable to the Class 11, three-dimensional instability than to the Class I instability. Maximum instability occurs at a perturbation with longitudinal wavelength twice, and transverse wavelength 0.8 times, that of the undisturbed waves. Furthermore, these disturbances co-propagate with the undisturbed waves, so that they act as a triggering mechanism for the undisturbed waves to bifurcate into a three-dimensional steady wave pattern. Thus, the Class I1 instability provides the selection rule for three-dimensional bifurcations. These rapid developments in our understanding of the nonlinear dynamics of a single wave train also enhance our knowledge of the properties of a nonlinear wave field. Since linear superposition does not work for nonlinear wave fields, the extension from wave trains is highly nontrivial. The first systematic investigation was made by Hasselmann (1962, 1963, 1968), who studied the nonlinear energy transfer in a homogeneous ocean. He found weakly nonlinear energy transfer occurring at a time scale of (kn)4.The exact effects of this energy transfer mechanism cannot be easily summa-


22 1

rized, but results by Hasselmann et al. (1976), Webb (1978), LonguetHiggins and Fox (1977), and Dungey and Hui (1979) seemed to indicate that it acts to make a broad spectrum narrower, and to shift a narrow spectrum to a lower wavenumber with continuous loss of energy to moderately high wavenumbers. The Zakharov equation, proven successful in describing small to moderate values of wave steepness for a wave train, was used as the basis of a statistical formulation for a nonlinear random deep-water gravity wave field by Crawford et al. (1980). They derived the governing equation for the correlation function for two wave vectors at a given time, and found that it predicts evolutionary processes on a time scale proportional to (ka)’. They further showed that these processes represent large-scale modulations of the wave system and do not result in net energy transfer. In fact, by carrying the analysis to one order higher, they showed that Hasselmann’s equation can be recovered by making the a priori assumption that the wave field is homogeneous for all times (that is, all cross correlations between different wave vectors vanish for all times). In other words, the evolution of a nonlinear random deep-water gravity wave field is composed of a modulational process occurring at a time scale of (ka)’, and a net energy transfer among modes at a time scale of order (ka)4. Crawford et al. (1980) also demonstrated that the homogeneous spectrum is unstable to modulational perturbations if the bandwidth is sufficiently narrow. This implies that the lower-order modulational characteristics are expected to be prominent for narrow-banded ocean wave spectra. In any case, modulational properties should be accounted for properly when dealing with surface sensor returns that may respond sensitively to wave modulations. Recently, there has been a great deal of interest in the phase speeds of individual components in a wave spectrum. Open ocean measurements by von Zweck (1970), Yefimov et al. (1972), Grose et al. (1972), and Ramamonjiarisoa and Giovanangeli (1978). as well as laboratory measurements by Ramamonjiarisoa (1974), Ramamonjiarisoa and Coantic (1976), Lake and Yuen (1978), Rikiishi (1978), and Mitsuyasu et al. (1979) all indicated that there are discrepancies between the measured results and the predictions from the linear dispersion relations. A variety of explanations for these discrepancies have been offered, including drift current effects (Plant and Wright, 1979), directional effects (Huang and Tung, 1977), and nonlinear effects of one form or another (Lake and Yuen, 1978; MolloChristensen and Ramamonjiarisoa, 1978; Yuen and Lake, 1979; Mitsuyasu et al. 1979). Because the measurements quoted to support these theories were all obtained under somewhat different conditions, it is difficult to fully assess at this stage the relative merits of these proposed explanations.

222


The experimental results of Crawford et al. (1981a), however, in which measurements of component phase speeds were made in mechanically generated wave spectra (obtained from records of actual wind-wave measurements) with no wind, show phase speed anomalies similar to those in wind-wave measurements that could be accounted for by effects of nonlinear dynamics using the Zakharov equation. Their results imply therefore that it is the nonlinearity of the wave field that contributes to the observed dispersion properties, and that the correlation of the latter with the ratio of wind speed to dominant wave speed, U / C , in wind-wave measurements enters indirectly through the fact that strong winds generate steep waves rather than as a more direct effect of the wind on wave propagation speeds. While nonlinear wave dynamics may not be the sole cause of the observed dispersion anomalies, the results of Crawford et al. (1981b) have shown that it is at least capable of accounting for such effects and should not be ignored in any proposed theoretical model for ocean wave dynamics. It is impossible to avoid considering the feasibility of constructing an improved theoretical model for ocean wave dynamics after a period of rapid progress in the understanding of the fundamental physics. For predictions of long-time-scale energy transfer, Hasselmann’s theory appeared adequate. Recent advances in remote sensing, however, give rise to a need for models of ocean modulational characteristics. Therefore a solution of the Zakharov-based statistical equation may be very useful. There is one caution: the results of McLean et al. (1981) indicate that even the Zakharov equation is not adequate when the characteristic wave steepness becomes large, since the Class I1 instability becomes dominant. A formulation along the lines of Zakharov’s approach, but including the Class I1 instability, appears rewarding and is indeed under way (Crawford et al., 198la). All numerical attempts at solving nonlinear equations raise the question of adequacy of the discretized representation. It has long been noted that certain analyses based on discretized systems do not extend to the continuous limit (Bretherton, 1964). Some preliminary explorations by Caponi et al. (1982) on this subject confirm this problem. Specifically, Caponi et al. (1982) examined the numerical solutions of the discretized Zakharov equation in two dimensions. They found that when more than one unstable mode is included in the calculations, the solution will exhibit the phenomenon of “confined chaos,” in the sense that the time evolution of these unstable modes appears to be highly irregular, yet such irregularity is confined only to the unstable modes (in contrast to the classical thermalization phenomenon). This implies that the solution can depend sensitively on the number of modes included in the calculation, even though the initial


223

conditions remain practically identical for the different cases. Representation by a finite number of modes is therefore suspect if taken as a source of information on the continuous system. Since the chaos is confined in the above sense, it may mean that solution of the statistical equation (the correlation functions) may be fruitful. Whether this is true or not awaits further investigation. Although much has been learned about the nonlinear dynamics of deep-water gravity waves in recent years, the prospect for the near future is that significant additional progress may be expected, particularly as the numerous unresolved issues related to ocean wave modeling applications are addressed.

Appendix A. Interaction Coefficients The second-order interaction coefficients V ' ' ) appearing in Eq. (107) are given by V ' "(k, k , , k2)= - 2 V ( -k, k , , k2) + V(k1, k2, k), (All V'2'(k,kl,k2)= - 2 V ( - k , k l , -

-k2)+ V(-kl, -k,,k,)

V(k1, -k2,k)-

V(-k,,kI,k),

V'3'(k,kl,k2)=2V(-k, -kl, -k2)+ V(-k,, -k,,k), where

(A2) (A3)

224


where

Appendix B. Lorentzian and Bretschneider Spectra The spectra k ( k / k,) used are nondimensionalized and normalized such that

d m B ( k / k , ) d ( k/ko) = I .

(B1)

The nondimensionalized spectral width 8 = u/k, is defined as the excursion from the peak wavenumber k,, at the level where the spectrum is one half of its peak value: B( 1 ? 6 , ) = @( 1). (B2) If the spectrum is symmetric, 8, = 8 - = 6. Since we expect the asymmetry to be small, we shall characterize spectral width by 6, obtained by Taylor's expansion of the spectral forms 6. The Lorentzian spectrum is defined by

(

k[ ~ ( k / k 0 ) = 2 8 ( ' ) ~k, k,

M

3-82

) + 4 62(4- 382) 2

(2 - 6 2 ) 2

(B3)

Nonlinear Dynamics of Deep- Water Graviy Waves

where the normalizing coefficient $1)

=

482(4-352) (2- 82)2

225

d ’) is

2 - 3e2 28(4-382)

(B4) The Bretschneider (generalized Pierson-Moskowitz) spectrum is defined by

where

p is given by

ACKNOWLEDGMENTS The authors would like to express their deep appreciation to Enrique A. Caponi. Donald R. Crawford, John W. McLean, Y. C. Ma, David U. Martin. Bengt Fornberg. Harry Rungaldier, and Annie Cameron for many stimulating technical discussions. They are especially indebted to Professor P. G. Saffman, who provided invaluable advice and guidance, and to Dr. John H. Chang. whose leadership as manager of the TRW Fluid Mechanics Department created an unmatched atmosphere for research. They also express their gratitude to Miss Janet Nay for her expert preparation of this and many other manuscripts. Part of this work was supported by the Applied Physics Laboratory of The Johns Hopkins University under Navy Contract No. 601038. and by TRW Defense and Space Systems Group Independent Research and Development Projects. REFERENCES Agranovich. Z. S., and Marchenko, V. A. (1963). “The Inverse Problem of Scattering Theory” (translated from the Russian by B. D. Seckler). Gordon and Breach. New York. Alber, 1. E. (1978). The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. SOC.London. Ser. A 363; 525-546. Alber. I. E.. and Saffman, P. G. (1978). “Stability of Random Nonlinear Deep Water Waves with Finite Bandwidth Spectra,” TRW Rep. No. 3 1326-6035-RU-00. Benjamin. T. B. ( 1967). Instability of periodic wavetrains in nonlinear dispersive systems. Pror. R. SOC.London, Ser. A 299: 59-75 Benjamin, T. B., and Feir. J. E. (1967).The disintegration of wavetrains on deep water. Part I . Theory. J. Fluid Mech. 27; 417-430. Benney, D. J. (1962). Non-linear gravity wave interactions. J . Fluid Mech. 14; 577. Benney, D. J., and Newell, A. C. (1967). The propagation of nonlinear wave envelopes. J . Math. Phvs. 46; 133-139. Benney. D. J., and Roskes, G. (1969). Wave instabilities. Stud. Appl. Mafh. 48;377-385. Benney, D. J.. and Saffman, P. G. (1966). Nonlinear interactions of random waves in a dispersive medium. Proc. R. Soc. London, Ser. A 289, 309. Bretherton, F. P. (1964). Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20, 457-479.

226


Burnside, W . (1916). On periodic irrotational waves at the surface of deep water. Proc. London Math. Soc. [2] 15, 26-30. Caponi, E. A,, Saffman. P. G . , and Yuen, H. C. (1982). Instability and confined chaos in a nonlinear dispersion wave system. Phys. Fluids (to appear). Chen, B.. and Saffman, P. G. (1980). Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Stud. Appl. Math. 62, 1-21. Chu. V. H., and Mei. C. C. (1970). On slowly-varying Stokes waves. J . Fluid Mech. 41. 873-887. Chu. V. H.. and Mei, C. C. (1971). The evolution of Stokes waves in deep water. J. Fluid Mech. 47, 337-351. Cohen, B. 1.. Watson, K. M.,and West, B. J. (1976). Some properties of deep water solitons. Phys. Nuid.? 19, 345-350. Cokelet. E. D. (1977). Steep gravity waves in water of arbitrary uniform depth. Philos. Trans. R . Soc. London. Ser. A 286, 183-230. Crawford, D. R.,Saffman, P. G.,and Yuen. H. C:. (1980). Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion 2, 1-16. Crawford. D. R., Lake, 8. M., Saffman, P. G., and Yuen, H . C. (1981a). Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105. 177-191. Crawford, D. R.. Lake, B. M.. Saffman. P. G., and Yuen, H. C. (1981b). Effects of nonlinearity and spectral bandwidth on the dispersion relation and component phase speeds of surface gravity waves. J. Fluid Mech. (in press). Dagan. G . (1975). Taylor instability of a non-uniform free-surface flow. J. Fluid Mech. 67, 113-123. Davey, A. (1972). The propagation of a weak nonlinear wave. J . Fluid Mech. 53, 769-781. Davey, A., and Stewartson, K. (1974). On three-dimensional packets of surface waves. Proc. R. Soc. London 338, 101- 110. Dungey, J. C., and Hui, W. H. (1979). Nonlinear energy transfer in a narrow gravity-wave spectrum. Proc. R . Soc. London, Ser. A 368,239-265. Feir, J. E. (1967). Discussion: Some results from wave pulse experiments. Proc. R . Soc. London, Ser. A 299, 54. Fermi, E., Pasta, J., and Ulam, S. (1955). Studies of nonlinear problems. “Co//ected Papers of Enrico Fermi,” Vol. 2. pp.978-988. Univ. of Chicago Press, Chicago, Illinois. Garabedian. P. R. (1965). Surface waves of finite depth. J. Anal. Math. 14, 161-169. Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M. (1967). Method for solving the Korteweg-deVries equation. Phys. Rev. Lett. 19, 1095-1097. Greene, J. M. (1979). A method for determining a stochastic transition. J. Math. Phys. M, 1183-1201. Grose, P. L., Warsh, K. L.. and Garstang, M. (1972). Dispersion relations and wave shapes. J. Geophy.?.Res. 71, 3902-3906. Hammack, J. L.. and Segur, H. (1974). The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65, 289-314. Hasimoto, H., and Ono, H. (1972). Nonlinear modulation of gravity waves. J . Phys. Soc. Jpn. 33, 805-81 I . Hasselmann. D. E. (1979). The high wavenumber instabilities of a Stokes wave. J. Fluid Mech. 93,491-500. Hasselmann, K. (1962). On the nonlinear energy transfer in gravity-wave spectrum. I . General theory. J. Fluid Mech. 12, 481-500. Hasselmann, K. (1963). On the nonlinear energy transfer in gravity-wave spectrum. 2. Conservation theorems, wave-particle correspondence, irreversibility. J . Fluid Mech. 15, 273-28 1. Hasselmann, K. (1967). Discussion. Proc. R . SOC.London, Ser. A 299. 76. Hasselmann, K. ( 1968). Weak-interaction theory of ocean waves. In “Basic Developments in Fluid Dynamics (Book)” (M.Holt, ed.), Vol. 2, pp. 117-182. Academic Press, New York.


227

Hasselmann. K.. Ross. D. B.. Muller. P.. and Sell, W. (1976). A parametric wave prediction model. J. Plys. Oceanog. 6. 200-228. Hayes. W. D. (1970a). Conservation of action and modal wave action. Proc. R . Soc. London. Ser. A 320, 187-208. Hayes, W. D. (1970b). Kinematic wave theory. Proc. R. SOC.London, Ser. A 320, 209-226. Hayes, W. D. (1973). Group velocity and nonlinear dispersive wave propagation. Proc. R. Sric. London. Ser. A 332, 199-221. Holmes, P. (1979). Philos. Trans. A. Soc. London 292, 419. Huang, N. E.. and Tung. C. C. ( 1977). The influence of the directional energy distribution on the nonlinear dispersion relation in a random gravity wave field. J. Phys. Oceanog. 7. 403 -4 14. Jackson. E. A. (1978). Rocky Mrn. J. Marh 8. 127-196. Jorna. S. (ed.) (1978). "Topics in Nonlinear Dynamics. A Tribute to Sir Edward Bullard" AIP Conf. Proc. No. 46.Am. Inst. Phys.. New York. Kinsman, B. (1965). "Wind Waves: Their Generation and Propagation on the Ocean Surface." Prentice-Hall. Englewood Cliffs, New Jersey. Lake. €4. M., and Yuen, H.C. (1977). "A Note on Some Nonlinear Water Wave Experiments and the Comparison of Data with Theory." J. Fluid Mech. 83. 75. Lake, B. M.. and Yuen, H. C. (1978). A new model for nonlinear wind waves. Part 1. Physical model and experimental evidence. J. Fluid Mech. 88. 33-62. Lake, B. M.. Yuen, H. C.. Rungaldier. H.. and Ferguson, W. E., Jr. (1977). Nonlinear deep-water waves: Theory and experiment. Part 2, Evolution of a continuous wave train. J. Fluid Mech. 83, 49-74. Lamb, H. (1904). On deep-water waves. Proc. London Marh. Soc. 2 , 371-899. Lamb, H. (1930). "Hydrodynamics." Dover, New York. Lax, P. (1968). Integrals of non-linear equations of evolution and solitary waves. Commun. Pure and Appl. Math. 21. 467-483. Levi-Civita. T. (1925). Determination rigoureuse des ordes permanetes d'ampleur finie. Math. Ann. 93. 264-314. Lighthill, M. J. (1965). Contributions to the theory of waves in non-linear dispersive systems. J. Inst. Math. Appl. 1, 269-306. Lighthill, M. J. (1967). Some special cases treated by the Whitham theory. Proc. R. Soc. London, Ser. A 299. 28. Longuet-Higgins, M. S. (1976). Proc. R . Soc. London, Ser. A 347. 311. Longuet-Higgins. M. S. (1978). The instabilities of gravity waves of finite amplitude in deep water. 11. Subharmonics. Proc. R. SOC.London, Ser. A 360. 489-505. Longuet-Higgins. M. S..and Fox, M. J. H. (1977). Theory of the almost-highest wave: The inner solution. J . Fluid Mech. 80, 721-742. Luke, J. C., (1967). A varitional principle for a fluid with a free surface. J. Fluid Mech. 27. 395-397. Ma. Y . C . (1977). Studies of cubic Schrodinger equation. Ph.D. Thesis, Princeton University, Princeton, New Jersey. McLean. J. W. (1982). Instabilities of finite amplitude water waves. J . Fluid Mech. 114. 33 1-341. McLean, J. W.. Ma. Y. C., Martin, D. U., Saffman. P.G., and Yuen, H. C. (1981). A new type of three-dimensional instability of finite amplitude gravity waves. Phys. Rev. Left. (in press). Martin, D. U., and Yuen, H. C. (1980). Quasi-recurring energy leakage in the two-spacedimensional nonlinear Schrodinger equation. Phys. Fluids 23. 881 -883. Martin, D. U.. Saffman, P. G.. and Yuen, H. C. (1980). Stability of plane wave solutions of the two-space-dimensional nonlinear Schrodinger equation. Wave Motion 2. 2 15-229. Masuda. A,. Kuo. Y. Y., and Mitsuyasu, H. (1979). J. Fluid Mech. 92. 717-730. Michell. J . H. (1893). The highest waves in water. Philos. Mag. [5] 36. 430-437.

228


Miles, J. W. (1977). On Hamilton’s principle for surface waves. J. Fluid Mech. 83, 153. Mitsuyasu, H., Kuo, Y. Y.,and Masuda, A. (1979). On the dispersion relation of random gravity waves. Part 2. An experiment. J. Fluid Mech. 92, 731-749. Mollo-Christensen, E., and Ramamonjiarisoa, A. (1978). Modeling the presence of wave groups in a random wave field. J. Geophys. Res. 83, 41 17-4122. Nekrasov, A. I. (1920). On Stokes waves (in Russian). Izv. Ivanovo- Voznesensk. f d i t e k h . Inst. pp. 81-91. Oikawa, M., and Yajima, N. (1974). A perturbation approach to nonlinear systems. 11. Interaction of nonlinear modulated waves. J . fhys. SOC.Jpn. 37, 486-496. Peregrine, D. H., and Thomas G. P. (1979). Finite-amplitude deep-water waves on currents. froc. R . Soc. London, Ser. A 292, 371. Phillips, 0 . M. (1960). On the dynamics of unsteady gravity waves of finite amplitude. J. Fluid Mech. 9 , 193-217. Phillips, 0. M. (1981). The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. 107, 465-485. Pierson, W. J. (1952). “A Unified Mathematical Theory for the Analysis, Propagation and Refraction of Storm Generated Ocean Surface Waves. Parts I and 11,” Dept. Meteorol. Oceanogr., New York University, New York. Plant, W. J., and Wright, J. W. (1979). Spectral decomposition of short gravity wave systems. J. fhys. Oceanog. 9, 621-624. Ramamonjiarisoa, A. (1974). These de Doctorat d’Etat, No.A. 0. 10.023. Universite de Provence, enregistree au C.N. R. S. Ramamonjiarisoa, A., and Coantic, M. (1976). Loi experimentale de dispersion de vagues produites pa le vent sur une faible longueur d’action. C. R . Hehd. Seances Acad. Sci., Ser.

B 282, 111-113. Ramamonjiarisoa, A., and Giovanangeli, J. P. (1978). Observations de la propagation des vagues engendries par le vant au large. C . R . Hehd. Seances Acad. Sci., Ser. B 287, 133-136. Rayleigh, Lord (1917). On periodic irrotational waves at the surface of deep water. fhilos. Mag. [6] 33, 381-389. Rikiishi, K. (1978). A new method for measuring the directional wave spectrum, Part 11. Measurement of the directional spectrum and phase velocity of laboratory wind waves. J. fhys. Oceanog. 8, 518-529. Rowlands, G . (1974). On the stability of solutions of the non-linear Schrodinger equation. J . Inst. Math. Appl. 13, 367-377. Russell, J. Scott (1844). Report on waves. Br. Assoc. Adv. Sci. Rep. 14th Mtg., pp. 311-390. plates XLVII-LVII. Saffman, P. G. (1980). Long wavelength bifurcation of gravity waves on deep water. J. Fluid Mech. 101, 567-587. Saffman, P. G., and Yuen, H. C. (1978). Stability of a plane soliton to infinitesimal two-dimensional perturbations. Phys. Fluids 21, 1450- I45 1. Saffman, P. G., and Yuen, H. C. (1980a). Bifurcation and symmetry breaking in nonlinear dispersive waves. fhys. Rev. Letr. 44, 1097- I 100. Saffman, P. G., and Yuen, H. C. (1980b). A new type of three-dimensional deepwater wave of permanent form. J. Fluid Mech. 101, 797-808. Saffman, P. G.,and Yuen, H. C. (1982). Three-dimensional deep-water waves. 11. Calculation of steady symmetric wave patterns. J . Fluid Mech. (to appear). Schwartz, L. W. (1974). Computer extension and analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62, 553-578,


229

Scott. A. C., Chu, F. Y. F., and McLaughlin, D. W. (1973). The soliton: A new concept in applied science. Proc. IEEE 61, 1443-1491. Segur. H. (1973). The Korteweg-de Vries equation and water waves. Solutions of the equation. Part I. J . Fluid Mech. 59, 721-736. Stokes, G. G . (1849). On the theory of oscillatory waves. Trans. Cambridge Philos. Soc. 8, 4 4 - 4 5 5 . Math. Phys. Pap. 1 : 197-229. Stokes, G. G . (1880). Supplement to a paper on the theory of oscillatory waves. Math. Phys. Pap. 1. 314-326. Su, M. Y. (1982a). Three-dimensional deep-water waves. Part I . Laboratory experiments on spilling breakers. J. Fluid Mech. (submitted for publication). Su, M. Y. (1982b). Three-dimensional deep-water waves. 111. Experimental measurement of skew wave patterns. (In preparation.) Thyagaraja, A. (1979). Recurrent motions in certain physical systems. Phys. F/uid.v 22 (2). von Zweck, 0.H. (1970). Observations of propagation characteristics of a wind driven sea. Doctorate Thesis, Massachusetts lnstitute of Technology, Cambridge. Webb, D. J. (1978). Nonlinear transfers between sea waves. Deep-sea Res. 25. 279-298. Whitham. G. B. (1965). A general approach to linear and nonlinear dispersive waves using a Lagrangian. J . Fluid Mech. 22, 273-283. Whitham. G. B. (1967). Nonlinear dispersion of water waves. J . Fluid Mech. 27, 399-412. Whitham, G . B. (1970). Two-timing. variational principles and waves. J. Fluid Mech. 44, 373-395. Whitham, G . B. (1974). “Linear and Nonlinear Waves.” Wiley, New York. Wilton, J. R. (1913). On the highest wave on deep water. Philos. Mag. [6] 26, 1053. Yamada, T., and Graham. R. (1980). P l y . Rev. Let!. 45. 1322. Yefimov. V. V., Solov’yev, Y. P., and Khristoforov. G . N. (1972). Observational determination of the phase velocities of spectral components of wind waves. Atmos. Oceanic Phys. 8, 435-446. Yuen. H. C.. and Ferguson, W. E., Jr. (l978a). Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrodinger equation. Phys. Fluids 21. 1275- 1278. Yuen, H. C.. and Ferguson. W. E., Jr. (1978b). Fermi-Pasta-Ulam recurrence in the two-space dimensional nonlinear SchrGdinger equation. Phys. Fluids 21. 21 16-21 18. Yuen. H. C., and Lake, B. M. (1975). Nonlinear deep water waves: Theory and experiment. Phys. Fluids 18, 956-960. Yuen. H. C., and Lake, B. M. (1979). A new model for nonlinear wind waves. Part 2. Theoretical model. Srud. Appl. Math. 60.261-270. Yuen, H. C.. and Lake. B. M. (1980). Instabilities of waves on deep water. Annu. Rev. Fluid Mech. 12. 303-334. Zakharov. V. E. (1966). The instability of waves in nonlinear dispersive media. Zh. Eksp. Teor. Fiz. 51; 1107-1 114; Sov. Phys.-JETP (Engl. Trans/.)24. 740-744 (1967). Zakharov. V. E. (1968). Stability of periodic waves of finite amplitude o n the surface of a deep fluid. J. App/. Mech. Tech. Phvs. (Engl. Transl.) 2. 190-194. Zakharov. V. E., and Filonenko. N. N . (1967). Energy spectrum for stochastic oscillations of the surface of a liquid. Sov. Phys.-Dokl. (Engl. Transl.) 11 (10).881. Zakharov. V. E., and Rubenchik, A. M. (1974). Instability of waveguides and solitons in nonlinear media. Sov. Phys.-JETP (Engl. Transl.) 38, 494. Zakharov. V. E.. and Shabat, A. B. (1972). Exact theory of two-dimensional self-focusing and one-dimensional self-modulating waves in nonlinear media. Sov. Phys.-JETP (En#. Transl.) 65, 997-101 1 .


ADVANCES IN APPLIED MECHANICS. V O L U M E 22

Instability and Transition in Buoyancy-Induced Flows B. GEBHART Mechanical Engineering and Applied Mechanics Universiry of Pennsylvania Philadelphia, Pennsylvania

AND

R. L. MAHAJAN Engineering Research Center Western Electric Princeton. New Jersey

. . . . . . 232 . . . . . . 234 111. The Downstream Growth of Disturbances in a Vertical Flow . . . . . . . . 237 IV. Nonlinear Disturbance Growth . . . . . . . . . . . . . . . . . . 247 A. Calculations of Secondary Mean Motions . . . . . . . . . . . . . 247 B. Measurements in Controlled Experiments of Nonlinear Growth . . . . . . 253 C. The Role of Mean Secondary Flows , . . . . . . . . . . . . . . 257 . . . . 258 D. Nonlinear Effects Resulting from Naturally Occurring Disturbances V. Transition and Progression to Developed Turbulence . . . . . . . . . . . 259 A. Transitional Mean Velocity and Temperature Distributions . . . . . . . 263 ,

.

.

.

.

.

.

. . . 11. Initial Instability in Thermally Buoyant Flows . . . 1. Introduction

. .

.

.

.

. .

. .

. . . .

B. Growth of the Boundary Region and the Correlation of Distributions During Transition . . . . . . . . . . . . . . . . . . C. Downstream Velocity and Temperature Distributions . . . . . D. Profiles of Disturbance Fluctuations . . . . . . . . . . . E. Disturbance Frequency during Transition . . . , . . . . . F. Thermal Transport during Transition . . . . . . . . . . . VI. Predictive Parameters for the Events of Transition . . . . A. Correlating Parameter E for the Beginning of Transition B. Predictive Parameter for the End of Transition . . .

. .

. . .

,

. . . . . . . .

. . . . . .

. . . . . .

. 264 . 265 . 267 . 269 . 271

. 274 278 282

23 1 Copyright 01982 by Academic Press. Inc. All rights of reproduction in a n y form reserved. ISBN 0-12-002022-X

B . Gebhart and R . L. Mahajan

232

VII. Plane Plume Instability and Transition . . . . . . . . . . . . . . . . 283 VI11. Instability of Combined Buoyancy-Mode Flows . . . . . . . . . . . . . 295 IX. Higher-Order Effects in Linear Stability Analysis . . . . . . . . . . . . 305

. . . . . . . . . . . . . . . . . . . . . B. Other Vertical Flows . . . . . . . . . . . . . . . . . . . . .

A. Plane Plume Flow

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . References

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

306 310 31 I 3 I2

I. Introduction Density differences in fluids, interacting with the gravitational field, produce an immense diversity of buoyancy forces and flow configurations in our environment, in our enclosures, and in the processes of technology. The atmosphere engine is driven by differences in temperature, water content, and water vapor concentration and by radiation. Circulations in terrestrial bodies of water-oceans, lakes, and ponds-are similarly driven by temperature and concentration differences, as well as by radiation effects and atmospheric interactions. All such motions at large scale are further modified by the earth’s rotational motion. Terrestrial atmosphere and water circulations are a t such large scale that the thermal and concentration transport are often almost entirely by turbulent diffusion. However, in our more immediate environment, laminar and transition flows are more commonly encountered. For example, in a vertical flow in gas or in water, generated adjacent to a vertical surface, transition may not arise or be completed until several meters downstream of flow initiation, depending upon conditions. Many applications arise in which the flow is, and remains, completely laminar. Thus, laminar flows take on first importance in determining transport mechanisms. They also set the eventual stage-provide the flow circumstance-for later breakdown to turbulence. Laminar flows become unstable, to ever-present disturbances, even at the small size scales common in technological and in immediate environmental processes. The difference between the rates of laminar and turbulent transport is very large for most flow configurations. Therefore the questions of how and when or where a flow becomes turbulent have a direct effect on the accuracy and reliability of estimates of transport. Two basically different kinds of instability commonly arise. Thermal instability results from the tendency to motion present in a temperaturestratified medium in which a heavier fluid overlies a lighter fluid. Other density effects, such as salinity gradients, may also result in such tendencies. On the other hand, instability of another kind arises in a laminar flow

Buoyancy-Induced Flows

233

when buoyancy, pressure, and viscous forces in balance contribute net energy to a disturbance, causing it to grow as it is convected along. This is called hydrodynamic instability. This latter mechanism is the one to which the vigorous flows most commonly encountered are subject. The initial instability initiates a train of events that convert a laminar flow to turbulence. In recent years these questions have been studied in considerable intensity and detail. Some of the initial events are now well understood for a number of flow configurations. Later events downstream, all the way to turbulence, have been clarified for several common vertical flow configurations. An earlier review (Gebhart, 1973) summarized the experimental studies up to that date that had guided the linear modeling of initial instability and disturbance growth. That review also presented the accumulation, up to that time, of calculations and experiments concerning the initial instability characteristics of several basically different kinds of steady flow that occur locally in extensive fluid media otherwise essentially at rest. Among partially bounded flows were those arising adjacent to vertical, horizontal and slightly inclined surfaces. This idealization, of an extensive medium, may also be appropriate in some internal flows (completely bounded), if the convection layers are thin compared with the dimensions and spacings of the bounding surfaces. The free boundary flows considered were plumes and nonbuoyant and buoyant jets. For several of these flows, the available information concerning downstream disturbance amplification was also summarized. The numerous further studies of recent years are the subject of the present review. Following the success of linear stability theory in predicting initial disturbance instability and growth, there have been further analyses and measurements concerning downstream growth. These have related principally to the highly selective amplification found in vertical flows adjacent to surfaces. These studies have led to both analysis and extensive measurement concerning nonlinear disturbance growth mechanisms further downstream. Secondary mean flows arise principally in the form of a double longitudinal vortex system. The purposes of these studies, of course, have been to understand the precursors of downstream laminar breakdown, to investigate predictive parameters for such breakdown, and to determine the resulting effects on transport. Recent progress concerning vertical flows generated adjacent to surfaces has determined the conditions for first transition, or turbulent bursts. The downstream progression of transition to full turbulence has been followed in water and in air. The subsequent process whereby a “developed” form of turbulence is achieved has also been followed in water.

234

B . Gebhart and R . L . Mahajan

Experiments have also been carried out specifically to determine the limits of the transition regime, between fully laminar upstream and fully turbulent downstream flow. These results, combined with the other published data concerning these two limits, have suggested general transition regime parameters. Free boundary flows, both plane and axisymmetric, have been the subject of further study. Additional information concerning laminar instability and transition is available. Interesting instability and coupled separation mechanisms have also been determined for horizontal and nearhorizontal flows. Many applications involve multiple flow-inducing effects. The spread of concern generally about instability with buoyancy forces has led to recent studies concerning combined buoyancy-mode flows, as with both thermal and mass transport, These are also often called multiple diffusive flows. Another multiple-effect flow studied is that arising from a combination of thermally caused buoyancy force and an imposed forced-flow condition of the ambient fluid medium, called mixed or combined convection. In the following sections the initial instability characteristics of different types of buoyancy-driven flows are first discussed, followed by the formulation of linear stability theory. Then, the new results are summarized as additions to knowledge in specific areas previously studied and as the results of new research directions. A final section then sets forth the question of improving stability analysis, beyond the assumptions inherent in the boundary-layer and parallel-base-flow formulations. II. Initial Instability in Thermally Buoyant Flows Although turbulence eventually follows the initial instability of a laminar flow to ever-present disturbances, the mechanisms have been found to be different for different flow configurations and bounding conditions. Also, between any initial laminar instability and eventual transition downstream to completely turbulent flow, the particular processes are often different. Such matters are considered in subsequent sections of this review. The first mode in this unstable progression for many flows is the initial growth of very small disturbances. This matter has been extensively studied by using a linear analysis. The usual considerable simplification arises in assuming that disturbance quantities, imposed by the environment on the flow, e.g., u’, o’, and t’, in a two-dimensional plane flow, are very small compared to the basic levels of u ( x , y ) , u ( x , y ) , and t ( x , y ) - t m in the developing laminar flow.

Buovancy- Induced Flows

235

Consider first a vertical laminar flow developing downstream adjacent to a vertical surface. It eventually becomes unstable to some or many of the components of the disturbances imposed upon it by its environment. A periodic component of the disturbance may be isolated. It has been found that a frequent mechanism of such disturbance of growth in buoyancyinduced flow is a propagating downstream periodic wave. Early observations, such as those of Eckert and Soehngen (1951), indicate that the physical process is somewhat like the ocean surface wave motion, which approaches parallel to the shore and eventually becomes breaking surf. Such a motion downstream, in a boundary-region flow, has been called a Tollmien-Schlichting wave. These waves will grow in magnitude if conditions are such that the flow and buoyancy forces channel flow energy into them. However, different flows behave differently with disturbances of different frequencies. They first become unstable to a given frequency at different downstream locations, in terms of the local flow vigor parameter Gr,r or G. A representative stability plane, as would be calculated for small disturbance amplitude, is shown in Fig. 1. The region of this plane of disturbance damping is at low

I

\

Sioble Region

\ \

\

\

\

FIG. I . A typical stability plane for a buoyancy-induced flow. generated adjacent to a vertical surface. for downstream-propagatingdisturbances of the Tollmien-Schlichting type.

236


G. It is separated from the amplifying region, a t high G for a range of frequency f( p), by the condition of neutral stability. A given location on this plane, of /3 and G or of frequency f and downstream location x , is either a stable, neutral, or unstable disturbance environment. The question really is, however, how a given disturbance, of frequency f, is treated by the flow as it is convected and propagates downstream, to increasing G. In Fig. I the dashed curves show the paths of propagation of disturbances of three different given physical frequencies f l > j z>f3. Such paths result, for example, in a flow generated adjacent to a vertical surface dissipating uniform heat flux. This is a frequently convenient experimental arrangement. For the path of fi, the disturbance is seen to first damp and then amplify at an increasing rate downstream, further into the unstable region. The result, for this path and for others in its vicinity not shown, is distubance growth to large amplitude. However, not all disturbance components behave this way, even for the uniform surface heat-flux flow. Note that f l and f3 are damped. This example demonstrates the phenomenon of selective amplification found by Dring and Gebhart (1968). This will be discussed quantitatively in the next section, for this flow. However, here the general way in which disturbances behave will be considered for different flow configurations. Paths of constant disturbance frequency propagation are shown in Fig. 2 for each of five flows. The paths have strikingly different characteristics. For vertical surfaces, they penetrate more deeply into the highly amplified region as they are convected downstream. Detailed behavior in the unstable region will later show, in Figs. 3 and 4, that only a very narrow band of frequencies is highly amplified. Paths of constant physical frequency are the horizontal lines on these figures. Thus such flows selectively amplify only a very narrow band of the frequency components offered by complicated disturbances. However, disturbances in the horizontal and in the plume flows cross the unstable region. The upper branch of the neutral curve is known to be bounded for most circumstances. Therefore any given disturbance, over a very broad band of frequencies, is unstable only over a range of G, i.e., of downstream distance x. The implications of this are very interesting since the disturbances are the origin of the more complicated disturbances later downstream (in x ) that disrupt a laminar flow. The results in Fig. 2 (flows 1-5) suggest that flows adjacent to vertical surfaces are inevitably unstable. However, the free boundary flows 4 and 5, along with flow 3 in which the buoyancy force is principally normal to the flow direction, are eventually stable in the linear range of amplitude to all disturbances. However, all these flows are found to be actually much less stable than flows 1 and 2. Thus, it is known that


237

Neutral Curve

FIG.2. Typical stability plane for buoyancy-induced flows, showing downstream paths of the propagation of a disturbance of a given frequency j' in different kinds of flow. Numbers indicate flow configurations: ( I ) isothermal vertical surface, (2) uniform-heat-flux vertical surface, ( 3 ) horizontal and slightly inclined surfaces. (4) plane plume, and ( 5 ) axisymmetric plume.

other mechanisms become important much more quickly (in x) for flows 3, 4, and 5, presumably because of the absence of a surface to damp disturbances. The explanation for horizontal flow 3 is likely to be the added thermal instability characteristics associated with small disturbances. Sources of detailed instability mechanisms of unstable disturbances are given for flow 3 by Pera and Gebhart (1973), for flow 4 by Pera and Gebhart (1971) and by Haaland and Sparrow (1973), and for flow 5 by Mollendorf and Gebhart (1973). Flow 1 and, in all particulars, flow 2 have been much more intensively studied: to, through, and even beyond transition into a developed turbulence. Therefore, flow 2, a vertical flow generated adjacent to a uniform heat flux surface, is followed through in the next four sections in an account of the detailed mechanisms to full turbulence.

111. The Downstream Growth of Disturbances in a Vertical Flow

Analysis of the stability of laminar fluid motions began over a century ago. The modern linear formulation, which includes a viscous force mechanism of disturbance interaction, has been in use for forced flows for over a

238


half century, Most uses have related to two-dimensional plane flows. The resulting stability equations, or Orr-Sommerfeld equations, are in linearized form, in terms of disturbance quantities. These were generalized to include the effect of a buoyancy force by Plapp (1957), for flow generated adjacent to a vertical surface at a temperature to in a quiescent ambient fluid medium at t , . In this section, the full equations governing transient flow will be set forth, in terms of velocity components u ( x , y , ~ )u,( x , y , r ) , temperature t ( x , y , T), pressure p ( x , y , ~ )and , density p(x, y , ~ )where , T is time. Then disturbances, u’(x, y , r ) , d ( x , y , ~ )etc. , will be postulated for a steady laminar base flow. This flow is steady in the sense that the average values of u , ~etc., , that is, E(x, y ) , Z(x, y ) , etc., are independent of the time interval over which the averages are taken. The quantities u = ti+ u‘, u= ij+ u‘, etc., will then be substituted into the full equations. A series of approximations are then made to reduce the formulation to a set of equations and boundary conditions for the conventional boundary region base flow and the disturbance quantities. After the disturbance forms are postulated, both the base-flow and the disturbance formulations are then converted by the boundary-layer similarity transformation. The last section of this review returns to consider improving these approximations in flow and stability analysis, in a consistent way. For a two-dimensional vertical plane flow, described in x and y , if the pressure and viscous dissipation terms in the energy equation are neglected and if it is also assumed that the molecular viscosity p and thermal conductivity k are constant and uniform in the fluid, the equations are

av + ,av ar

at

+&

ax

at

-+U-+u-=KV

aT

ax

=y v 2 0 +

ay at

ay

2

1.

y- -1 aP P ay ’

(3.3) (3.4)

The positive x direction has been taken in the direction of the gravitational body force F = ( X , Y ) , but opposed to it. Therefore, F = ( X , O ) and X = - gp(x, y , T ) , where g is the gravitational acceleration. The buoyancycaused impetus to motion, or buoyancy force, B ( x , y), is the difference between P and the gradient of the hydrostatic pressure field VP, = - g p ,


239

in the quiescent ambient fluid medium. Thus

-

B( x, y ) = F - V p , = g[ P, - P ( x , JJ, 7 ) ] * (3.5) A motion pressure p m is defined in p ( x , y , T ) = p h ( x )+ p m ( x , y , 7 ) ; that is, p m is the difference between the actual static pressure and the local external hydrostatic. If pressure effects on fluid density are neglected and if temperature effects are closely linear over the range from to to t , , then g(pm - p ) may be taken as g&( t - tm), where pT is the volumetric coefficient of thermal expansion. If, further, Ap r,, the ambient medium being assumed uniform at t,: au au -+-=o, a x ay

ao + uao+& a?

at -

a7

ax

ay

=y

V2U- --, 1 aPm Pr aY

+ u-aaxt +I&= K V2t, a,v U(X,0,“)= U(X,

0,T) = t(X ,

(3.9) CO, 7 )

-f,

= 0.

(3.10)

It is not in general possible at this point to specify other bounding conditions in the ambient medium. The nature of any such applicable conditions depends upon the many additional approximations to be made in (3.7) and (3.8). The four equations above are in terms of the four distributions u, u, t , and p m . They are written below in terms of their mean and disturbance components (the relation between p’ and t‘ is also given): U(X,

y , 7 ) = c ( X , )’)

+ U’(x,)’, T ) , V ( x , y , T ) =

c(x, J’)+v ’ ( x , Y,T) (3.1 1)

t ( x . y , ~ ) =i ( x , y ) + t ’ ( x , y , 7 ) , P m ( X t Y 1 7 ) = ~ m ( X , Y ) + P ~ ( X ,Y’T)?

P( X, y , 7 ) = ii( x, y ) + P’( x , y , 7 ) = P( X, y ) - Pr a t ’ .

(3.12) (3.13) (3.14)

240


When these are introduced into (3.6)-(3.9), we obtain (3.15)

(3.17)

=K

v2i+K V2t’.

(3.18)

These results are written in expanded form to make clear the nature and impact of the approximations that follow. These approximations will yield the Orr-Sommerfeld equations, in terms of a conventional boundary-layer solution. The collection of approximations may be classified as follows: 1. The base flow is taken as that resulting from the first-order boundarylayer theory formulation. 2. Only first-order or linear terms in the disturbance quantities u’, u’, t‘ are retained. 3. After approximations 1 and 2 are made, the base-flow quantities U and the x derivatives of ii and i are taken as zero, sometimes called a parallelflow approximation. 4. In the postulated disturbances, x independence is assumed.

Applying assumption 1, the following collection of equations obtained from the above four equations are the equations governing the base flow:

-+-=o, az ac ax

ay

(3.19) (3.20)

(3.21) These equalities eliminate all these terms from (3.15)-(3.18). Also eliminated there, as a result of conventional boundary-layer approximations, are all other effects expressed purely in base-flow quantities. These consist of pm,all purely base-flow terms in (3.171, and stream-wise second derivatives.

24 1


Approximation 2 amounts to a small disturbance magnitude or linear approximation. This in turn means, for example, that periodic disturbances will not generate either harmonics or mean flow components. By assumption 2, all terms involving any product of u’, d,and t’ and/or their derivatives are eliminated. Approximation 3 is an ad hoc one. A base flow and transport, ii, 5, and i, were postulated in (3.19)-(3.21). The value of G in the resulting solution is not zero. Nevertheless, 5 and the x derivatives of both E and i are next taken equal to zero in the disturbance equations that remain after approximations 1 and 2 are made. The principal result is that all remaining terms in 5 disappear. The residual equations then become as follows: (3.22)

+ ii- au‘ + 0’- a ij = u v 2u’+ g & f ax ay

au’ -

aT

1 -, ap:, Pr ax

(3.23) ( 3.24)

(3.25)

The internal consistency of this collection of approximations is considered in the general discussion of such approximations, which is the last section of this work. Equations (3.19) and (3.22) suggest a stream function $(x, y , T)=&(x, y)++’(x, y , ~ such ) that

E=+,,

-

-5=+,,

u’=+’L ’

-V’=#;.

(3.26)

The conventional base-flow similarity transformations, the governing equations, and the boundary conditions are given below for Pr (Prandtl number) of order 1 or less: &=YC(x)F(q),

(3.27)

q=yb(x),

+ ( q ) = ( f - t m ) / ( i o - t w ) = ( f - t,)/d(x),

~(x)=4(Gr,/4)”~= G,

d(x)=N,x”,

b ( x ) = G/4x,

G r , = ( gx3/.’)PT(i0-

tm),

F ” ’ + ( n +3)FF” - ( 2 n +2)( F’)*++=O,

+”+Pr[(n+3)F+’-4nF’+]=O, F’(O)= F(O)= I -+(O)=

6(x)=4x/G,

F’(00)=+(00)=0,

Ucx/v= G2/4.

(3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34)

242


The listed boundary conditions are for an impervious vertical surface beginning at x=O, in an isothermal and quiescent ambient medium. The characteristic transport distance and tangential velocity are S(x) and U,( x), respectively. The numerical solutions of these equations for various values of n have been obtained for a wide range of Prandtl numbers (see, for example, Ostrach, 1964; Gebhart, 1971). The form of disturbance equations (3.23)(3.24), along with (3.26), suggests forming a disturbance vorticity equation in 4’. Then, the remaining disturbance functions, 4‘ and t’, are similarly postulated, for any given disturbance mode, as follows: 4’(x, y , 7 ) = 61/,@exp[i(c+x- j7)],

(3.35)

t’(x,y,7)=d(x)sexp[i(&x-87)1,

(3.36)

a=

c+S=c+,S+ic+,S=2nS/X+ iai,S=a,+ia,,

p = j S / u, = j , S / uc.

(3.37) (3.38)

The dimensionless amplitude functions across the transport region, @(q) and s(v), are in general complex. The two disturbances, linked through their residence in mass, are taken to be of similar form. Disturbance behavior is assumed to be periodic in x at any give? time. The wavenumber is 6,. They are periodic in time at all x, since fl=2nf, where f is the frequency. Amplitude damping or growth is assumed spatial and exponential, in x, in terms of a,. Neutral stability is a,=O and downstream amplification occurs for a, < 0. At this point, approximation 4 is invoked, to yield the usual OrrSommerfeld equations. These equations are in terms of the disturbance amplitude functions @ and s. The first part of approximation 4 is that the amplitude functions depend only on 77, that is, that @=@(q) and s = s ( q ) . This is analogous to the base-flow approximation that )C and F depend only on 9. The second part of approximation 4 is that the x derivatives of each disturbance spatial amplification rate &, and the disturbance wave length A, or G,, are taken as zero. The first three of these measures, concerning @, s, and &,, are supported quite well by numerous accurate measurements, in several different vertical buoyancy-induced flows. However, these same data, along with optical visualizations, indicate that X varies appreciably downstream. In fact it must, since frequency f is constant in an analysis by frequency modes, whereas the vigor of the base flow is x dependent. These downstream effects may be much more severe in more complicated flows of different geometric and bounding conditions. The improvement of the analysis, in any of the four categories of approximation previously set forth, must be consistently done. Any improvement effort


243

must be cognizant of the total body of approximations and not merely an ad hoc or selective treatment. The formulation of a consistent scheme of improved approximation is set forth in Section IX for several vertical flows. Substitution of (3.35)-(3.38) into (3.22)-(3.26) yields the Orr-Sommerfeld equations, in terms of disturbance amplitude functions @(q) and s( q). These are given together with the boundary conditions characteristic of a quiescent ambient medium:

( ~ ' - p / ~ ) ( w ' - ~ *~q" -' @ = ( @ ' ' " - 2 ~ 2 ~ " + ~ 4 @ + ~ (3.39) 1)/i~c, (F'- /3/a)s -+'@=(s"

- a2s)/icuPr G ,

@( 00)=@'(00)=s(00)=0.

(3.40) (3.41)

Three other boundary conditions, at 9 =0, depend upon the particular flow under consideration. The equations contain five independent parameters a r ,a , ,p, Pr, and G. Also, F ( q ) and +(q) are dependent on Pr. Solutions amount to finding G( p ) for a ,= O as the neutral stability condition, or curve, on /3, G coordinates. Contours of constant amplification rate - aiare then found in the unstable region. For constant physical PUJS a PG 3 / x 2 . Connecting x to G , for the to frequency we have f a variation with x that is appropriate to any particular flow, we have a relation between /3 and G at constant physical disturbance frequency. Thus we may follow a given disturbance on p, G coordinates and find the change in its amplitude, in the manner of Dring and Gebhart (1968), as it is convected along in the flow to larger x (that is, G). This is the way the constant-frequency paths in Fig. 2 were constructed for different flows, even for a horizontal flow. A simpler interpretation of downstream amplification results if /3 is generalized differently as follows:

8=

Q=j&?Gm/Uc.

(3.42)

For an isothermal condition, n -0 in (3.28) and m = f. The cumulative downstream amplitude growth, for the neutral condition at x, or G,, for a given value off, is calculated as follows. If A , is the disturbance amplitude of the periodic two-dimensional disturbance as it reaches the x location of neutral stability and A, is the amplitude farther downstream, then A is defined by A,/A,=eA,

4A=-

lnaidG.

(3.43)

The neutral curve is A =O. Neutral curves and constant-amplification contours have been calculated for many buoyancy-generating processes, since the first calculations of neutral stability by Plapp (1957), Szewcyzk (1962), Kurtz and Crandall (1962), and Nachtsheim (1963). The kinds of stability planes reproduced

244


here are for a vertical surface dissipating a uniform heat flux q“, n = 4 in (3.28). Figures 3 and 4 are stability planes for Pr=0.733 and 6.7, respectively. However, the generalization in these figures is in terms of a flux Grashof number G* and a generalized frequency &?*defined as follows:

G* =5(Gr:/5)’I5,

Gr: =g&x4q”/kv2.

(3.44) (3.45)

Q* = p G * ‘ / 2 ,

Note that the characteristic transport distance 6 and tangential velocity U, for the uniform flux surface are 6=5x/G*,

(3.46)

U , x/ v= G*2/ 5,

Similar stability planes, for plane plumes and for combined buoyancymode flows, are given in subsequent sections. For the surface-generated flow upon which Figs. 3 and 4 are based, the apparent additional boundary conditions were given by Knowles and

4.0 3.5

t

0.5 I”j

1 0

I

I

3.00

1

I

400

I

I

600

I

I

800

1

I

1000

ti* FIG.3. Stability plane for Pr=0.733 showing measured disturbance frequency for the unstable laminar flow (open symbols) and for locally laminar portions of the flow in the transition region (partially or fully shaded symbols). The dashed line is a constant-physicalfrequency path. Data are from Mahajan and Gebhart (1979).

s

245 A

1.6

-

7

/

/7

I

0

=O

200

I

I

I

400

600

800

I

1000

I

I

1200

1400

G+ FIG.4. Stability plane for Pr=6.7 showing amplitude curves in the unstable region. The dashed line represents the path of a rapidly amplifying frequency. The measured disturbance frequency data are from Qureshi and Gebhart (1978).

Gebhart (1968) as

@(O)

= W ( 0 )= 0,

(3.47) s(0) = i s ’ ( O ) / p , Q( G*)3’4, where

0, the relative thermal capacity parameter, is Q= ( ~ r c ’ ~ / p C ~ ) ( g p ~ q ’ ’ / k v ~ ) ’ ’ ~ .

(3.48)

Thus s(0) is not taken as zero unless the surface is massive, that is, very large. Since this is usually the practical circumstance in air, Fig. 3 was calculated with s(O)=O. However, Fig. 4 is based on s’(O)=O. It is seen in Figs. 3 and 4, from the trajectories of the constant-physicalfrequency paths shown, in 52* that these flows are very sharply selective in their amplification characteristics. The disturbances are filtered for essentially a single frequency (characteristic frequency) as they are convected downstream. The experimental data, like the points seen on Figs. 3 and 4, strongly substantiate this prediction. Comparison of the A contours in Figs. 3 and 4 indicates that the selected value of Q* is Pr dependent. The collection of available calculated and measured behavior was assembled as in Fig. 5 by Gebhart and Mahajan


246

I

I

1

I

10-2

I

102

Pr

I

104

Flc. 5. Characteristic Frequency data for vertical natural convection flows. Large and small Prandtl number asymptotes and + data points are from Hieber and Gebhart (1971a, b). Other data: 0, Polymeropoulous and Gebhart (1967). 8. Eckert and Soehngen (1951); 0, Knowles and Gebhart (1969); v, Shaukatullah (1974); Godaux and Gebhart (1974): a. Jaluria and Gebhart (1974).

m,

(1975). The right-hand ordinate in D corresponds to the isothermal surface condition, n=O. The !eft-hand one, in 9*,is for uniform surface flux and is defined in (3.45). In Fig. 5 the asymptotic dependences shown at large and small Pr were inferred from Hieber and Gebhart (1971a, b). The crosses at intermediate values of Pr are derived from detailed stability planes for each specific Pr value. The data points shown in this range are seen to be in very good agreement with the calculations. The kind of remarkable agreement seen above, between linear stability theory and experimental results, has been found to extend also to other and much more subtle aspects of unstable flows and transport. Such success is not common across the broad range of such research in fluid flow. Perhaps the reason for such close agreement in these flows is that the disturbance amplification is so highly selective and thus is not heavily dependent on particular aspects of each specific physical situation. Subsequent sections will indicate other very interesting downstream consequences of this initial characteristic.

Buoyancy- Induced Flows

247

IV. Nonlinear Disturbance Growth The disturbances, when sufficiently weak, grow downstream as predicted by the linear stability theory. However, as the disturbances become large, nonlinear mechanisms arise and their development begins to deviate from these predictions. These mechanisms are of two kinds: the possibility of generating higher harmonics and the generation of secondary mean flow due to nonlinear interaction of two-dimensional and transverse disturbances. The possible effects of higher harmonics on the disturbance growth are difficult to conjecture. They are perhaps not even significant in the later stages preceding transition, since the available experimental data for flows subject to both natural and controlled disturbances strongly indicate that a simple sinusoidal form of the highly amplified disturbances is retained during these later stages. The generation of a secondary mean flow, on the other hand, has been found to play an important role in the breakdown of the flow from laminar to turbulence. It has been studied both analytically and experimentally. OF SECONDARY MEANMOTIONS A. CALCULATIONS

The theoretical treatment is due to Audunson and Gebhart (1976). Their analysis postulates a two-dimensional disturbance modulated by a standing transverse disturbance. The relative amplitudes of the two disturbance components are allowed to vary while their phase velocities and wavelengths are assumed equal. These last two assumptions are similar to those used by Benny and Lin (1960) and by Benny (1961) in the nonlinear analysis of Blasius flow. However, Stuart (1965) has shown that two- and three-dimensional waves are unequal near the neutral curve, and therefore presumably are not in the amplified region downstream. Similar objections to the assumption of synchronization of two- and three-dimensional waves in forced flow have been raised by Hocking et al. (1972). An uncertainty regarding this synchronization also exists in natural convection flows. On the other hand, measurements of Jaluria and Gebhart (1973), of controlled disturbance propagation in a buoyancy-induced flow, indicate that the velocities of the two waves are nearly the same (see Fig. 6). Their phases are, however, about one-quarter of a period apart. Nevertheless, the great simplicity of the analysis following the assumption of equality of phases, along with a posteriori good agreement between the predictions of the analysis and the experimental results, justifies the simplifying assumptions in the analysis. Retaining these postulated nonlinear interactions in the disturbance equations, finite amplitude effects were calculated by a systematic perturba-


248

(b)

(d)

FIG.6 . Two-dimensional and transverse velocity disturbance versus time as measured at various downstream locations: G' =350 (a). 400 (b). 500 (c). 545 (d). Upper signal for each G* is the transverse one (from Audunson and Gebhart 1976).

tion of linear stability theory. The nonlinear interaction of the solutions to the homogeneous Orr-Sommerfeld equations provides the driving functions for the first perturbation from linearized analysis. For details, the reader is referred to Audunson (1971). Numerical solutions were obtained for four base-flow conditions having very different linear stability characteristics, for Pr = 0.733. The flow conditions chosen are shown on the two-dimensional disturbance stability plane in Fig. 3. Points B, C, and D lie close to the path of most amplified frequency. D is at the neutral condition, whereas C and B are in the amplified domain. Point A is at the same value of G* as B but lies on the neutral curve. The results indicate a strong dependence of the resulting secondary flows on both G* and frequency Q*. For point B lying in the highly unstable region, a double longitudinal mean secondary vortex system is indicated (see Fig. 7). In Fig. 7 X,/X, is a measure of the relative strength of two-dimensional and transverse disturbances, B is the transverse wavenumber, and z is the coordinate in the transverse direction. The streamlines shown are for conditions of A,/& = 100 (highly two-dimensional disturbance) to A,/h,=O (a purely transverse oscillation). For h,/h,>> I (Fig. 7a) the streamlines indicate mean cellular vortex structure motion with spanwise periodicity of 2n/fl. With increasing three-dimensionality of the flow, i.e., decreasing A,/&, the centers of the outer rolls move toward spanwise locations 8z = 2ns, whereas the centers of the inner rolls are pushed towards spanwise locations 8z = ( 2 n 1)a (see Figs. 7b, c). For the extreme case of purely transverse primary oscillation, the vortex structure is shown in Fig. 7d. The spanwise period is a/O.

+


249

FIG. 7. The calculated streamlines of the mean secondary flow for point B in Fig. 3. Stream function value 0, O.oooO5: I , 0.0001; 2, 0.0002: 3. 0.0005; 4. 0.001: 5, 0.005: 6.0.01: 7.0.05; 8,O.l. G*=700;X,/h,= 100 (a). 2.0 (b). 0.2 (c), 0.0 (d) (from Audunson and Gebhart. 1976).

Such secondary motions imply a large momentum transport across the boundary region and result in important modifications of the mean flow. The result in Fig. 7a is of primary interest, since it closely represents the mainly two-dimensional flow preceding transition. At 8z = (2n + I)n the inner roll carries primarily high-momentum fluid from the inner part of the boundary layer to the outer slower-moving region. On the other hand, the counter-rotating outer vortex brings low-momentum fluid from the far field into the boundary region at this same z location. These cross flows result in steepening of the outer part of the mean velocity profile at locations 8z =(2n 1)s and flattening of it at locations Bz =2nn. Since energy transfer to a disturbance is at least approximately proportional to the velocity gradient, or shear, of the mean flow, the disturbance

+

250


growth rate is strongly augmented at 6z = (2n + 1)a locations, the regions of high shear. At Bz=2na locations, reduction in disturbance growth is suggested. Note that the calculated locations of high shear are just the opposite of those found in analogous Blasius forced flow. The quiescent far field, rather than the region near the surface, is the source of low momentum that causes the high-shear region. Further, in forced flows the presence of only a single-longitudinal-vortex system was detected (Klebanoff et al. 1962). These vortices occupied only the inner half of the boundary region. In the buoyancy-induced flows considered here, however, a double-vortex system is indicated and the outer vortices stretch across the boundary region, out into the quiescent fluid. These vortices may therefore be expected to cause a great distortion in the longitudinal mean velocity profile. Associated with these secondary flows is also an alternating spanwise modification of the mean temperature distribution across the boundary layer. At spanwise locations Bz = 2na the local heat transfer is augmented, whereas at 6z = (2n + 1)s a decrease occurs. The computed results for point C were shown to be in complete agreement with those discussed above for point B. This is not surprising since both points lie in the highly unstable regions and have the same physical frequency. However, it is of interest to see if the streamlines calculated for point D at the neutral curve, but along the same high-amplification path, show similar secondary flow characteristics. The results are shown in Fig. 8. For X,/X,= 10 one dominating outer roll is seen, but a weak inner circulation also begins. The general behavior at other values of A,/h, is the same as for the highly amplified flow at point B. Thus, along the highamplification path, a double-longitudinal-vortex system is produced at the earliest stages of instability and continues to be found in the region of highly amplified disturbances. Consider now the streamlines shown in Fig. 9 for point A, which is at the same value of G* as B but does not lie along the path of amplifying disturbances. The results are very different from those at point B. Only a single longitudinal roll results. Increase in three-dimensionality of the flow produces similar changes in the streamline pattern as for point B (Figs. 9b,c). The resulting modifications of the mean flow caused by highly two-dimensional oscillation (Fig. 9a) are also compatible with former results. Spanwise locations Bz = (2n + l)n experience a momentum defect in the inner and outer parts of the boundary layer, and at Oz=2nv the situation is reversed. Thus, there is again an alternating spanwise thinning and thickening of the boundary layer. However, this single-roll system does not appear to produce any significant steepening of the outer part of the velocity profile. The profile merely shifts in and out from the surface while

Buoyancy-lnduced Flows

25 1

FIG. 8. The calculated streamlines of the mean secondary flow for point D. Stream function value: I . O.ooOo5; 2. 0.0005: 3, 0.005: 4. 0.05: 5. 0.2: G*= 160: ?,,/A,= 10 (a), 0.2 (b). 0 (c) (from Audunson and Gebhart, 1976).

retaining its original form. These changes would not be expected to augment disturbance growth. From the above results, it appears that a double-vortex system is predicted at points D, C, and B, as the most highly amplified disturbance is convected downstream. Thus, the secondary mean flow configuration need


252

(b)

(C)

FIG. 9. The calculated streamlines of the mean secondary flow for point A. Stream function value: 0. O.OOO1; 1, 0.OOOS:2. 0.001; 3, 0.005; 4, 0.01; 5 , 0.05; 6 , 0.1; 7, 0.5; 8. I. G*=700; &/A,= 10.0 (a), 0.2 (b), 0.0 (c) (from Audunson and Gebhart, 1976).

not appreciably change as this disturbance is convected downstream. It merely enhances itself. This probably occurs simultaneously with the continued concentration of disturbance energy into the filtered twodimensional primary wave. In other words, linear and nonlinear mechanisms appear to proceed hand in hand in a highly filtered way. This important result is very different from what is observed in forced flows. It


253

would be interesting to calculate the integrated downstream effect of an initially three-dimensional disturbance along this filtered path and compare the results with the experimental measurements of the disturbance form and of transition.

B. MEASUREMENTS IN CONTROLLED EXPERIMENTS OF NONLINEAR GROWTH Excellent corroboration to these analytical results has been provided by the detailed experimental studies of Jaluria and Gebhart (1973). The measurements were in the flow adjacent to a vertical uniform flux surface, in water. Controlled two-dimensional disturbances, with a superimposed transverse variation, were introduced in the flow by a vibrating ribbon (see Fig. 1Oc). The input disturbances were introduced at location G*= 140. 0.06-

I

I

I

l

l

0.04 0.02

-w K C,

-

0-

-0.02 -

-0.04-

- 0.06 L

“I

0.8 0.6 1.0

“&ax

0.4

-

0.2

-

I

I

w: I

“

r -

I

1

I

I

I

I l

I

I I

I

I

1

l2.7mm

b-4

r n

l

l

I 1

MAX MIN MAX MIN 1 I , I

I

I

VIBRATOR

FOIL

Fw. 10. Configuration of vibrating ribbon and measured downstream spanwise distribution of u‘ and W. The measurements are at a single value o f each x and I’.

254

B . Gebhart and R. L. Mahajan

This lies well in the unstable portion of the stability plane (see Fig. 4). The resulting behavior of the disturbances, and of the mean flow downstream, was studied in detail. Recall that the actual local mean flow is the sum of the base flow ‘ii and E and any secondary mean flow that might arise through interaction among the disturbances and this base flow. Denoting the resulting mean flow components as U , V , and W, the resulting components of the secondary mean flow are U - U, V - V , and W. The periodic parts of the velocity components, as before, are u’, u’, and w’. In the first few experiments of Jaluria and Gebhart (1973), the frequency of the vibrating ribbon was varied at constant disturbance amplitude and measurements were made of the amplitude velocity disturbance u’, downstream at different x and z locations. The data indicated that the most rapidly amplifying frequency was almost exactly equal to that predicted by linear stability theory. The conclusion was that the frequency filtering mechanism discussed in Section I11 for boundary-layer flows subject to two-dimensional disturbances is not affected by a spanwise variation in the input disturbance. In subsequent experiments this frequency was chosen for the input disturbance. The data are in the range G* =400-600. The spanwise distribution of u’ normalized by the maximum value measured in the transverse direction z across the surface, is shown in Fig. lob. The initial spanwise positions of maxima and minima in u’ were found to be preserved downstream in a given flow, indicating more or less vertical propagation of the disturbance pattern. However, peaks and valleys are seen to be very sharp, quite unlike the input disturbance. This indicates that these measurements were taken in the region of amplifying transverse effects, which, through linear and/or nonlinear interactions, accentuate the ribbon input spanwise variation of u’. Other data indicated that, for smaller disturbance amplitudes, linear effects dominate. Nonlinear interactions arose at larger amplitudes. Associated with this spanwise variation of u’ is the variation of the transverse component W of the secondary mean flow across the boundary region (shown in Fig. 10a). Evidence of longitudinal rolls is seen in Fig. 11. The measured y distribution of W , normalized by Emax,the measured maximum velocity in the base profile without disturbances, is plotted against 7. These distributions indicate the changing form of W ( 7 )with G* (Fig. l l a ) and with z (Figs. I la, b). In each distribution, W changes sign twice across the boundary region. In Fig. l l a , W starts with negative values at low 9 , rises to a peak at positive W, and returns to negative values again before gradually dying out at large 9. The two sign reversals suggest the centers of two rolls, since W is in one direction on one side of each center and in the opposite


255

0.04

0.02 x

E 0

3 113

-0.02

-0.02t

1

- 0.04 -0.06

k

-0.061 ,

0

l

I

I

2

1

1

4

1

1

6

1

I

8

7 (b)

Fw. 1 I . Distribution of the transverse component W across the boundary region. ( a ) 0. G*=400 at z=86.36 rnm: 0. G*=460 at z=86.36 m m ( b ) A, G*=460 at r = 7 6 . 2 mm: A. G*=460 at z=78.74 rnm.

direction on the other. Assuming two rolls, the location of their interface is where W attains the highest absolute values. Since this occurs around TJ = 1.7, the inner roll extends from q = 0 to TJ = 1.7. The outer roll, on the other hand, stretches from q = 1.7 to q = 7 for G* = 400 and beyond for G*=460. The extent of the inner roll for these two G* values appears to be the same. The distribution in Fig. 1 Ib at z = 78.74 mm is just the opposite of that in Fig. 1la, at z = 86.36 mm. It starts at a positive value of W at low q, goes through two sign reversals, and dies out at small positive values at large TJ. This comparison, at two z locations at the same G*, indicates a plane of demarcation between these transverse locations. On the opposite sides of this plane, the vortices rotate in opposite directions. The minimum in the input disturbance, at z = 83.32 mm, is the most probable site of the location of this plane. The maxima and minima in the input disturbance are locations of symmetry. Zero mean transverse flow is expected at these locations. Similar reversal in form of these curves from z=73.66 to z = 69.45 mm (not shown in the figure) indicated t = 7 1.12 mm, the location of input disturbance maximum, as the plane of symmetry for the two counterrotating vortices. Thus, each pair of longitudinal vortices stretches in the transverse direction from positions of a maximum to a minimum in the input disturbance.


256

This is further confirmed by the measurements of W in the transverse direction (see Fig. 10). The component W reaches zero value at z=71.63, 83.31, and 96.27 mm, indicating that one vortex pair extends from z = 71.63 to z = 83.3 1 mm and that the adjacent vortex pair extends from z = 83.3 1 to approximately ~ 1 9 6 . 2 7 mm. Each vortex pair thus lies between a maximum and a minimum, located at z=83.32 and z=71.12 mm for the input disturbance. The slight difference is probably due to a shift of the vortex pattern caused by the hot wire sensors used to measure W. The total picture of the longitudinal vortex system that emerges from these measurements is shown in Fig. 12. The vortex pair lies between adjacent maxima and minima in the input disturbance. The inner roll is close to the wall, whereas the other stretches out across the boundary layer into the ambient medium. This double longitudinal vortex is in excellent agreement with the analytical predictions of Audunson and Gebhart ( 1976) (see, for example, Fig. 7a). Since the Prandtl number (Pr=6.7) in these measurements is different from that of the analysis (Pr=0.733), a quantitative comparison between the two is not relevant. However, the overall features are the same. At high Prandtl number one would expect the vortex system to be closer to the wall. These measurements in water, when compared with the theoretical vortex pattern in air, indicate that this

I I I I

a,

1 I I

I

I I

I I

I

I

I

I I

0

min

I

I I

1 I

I

I

I I

I I I

I I

l

max

- 2

FIG. 12. Sketch of the double mean longitudinal vortex system.


257

C. THEROLEOF MEANSECONDARY FLOWS

Such a double-longitudinal-vortex system results in a large momentum transport of downstream momentum across the boundary layer. At the location of a maximum in the input disturbance, the outer vortex tends to convect higher-velocity fluid outward in the boundary region. However, at a minimum, it brings fluid into the boundary region from the quiescent medium. The inner roll does just the opposite. The resulting mean flow velocity induced by these momentum transports is steepened around the inflection point at the spanwise position corresponding to a minimum. It is flattened at a maximum (see Fig. 13). Thus, there is an alternate spanwise steepening and flattening of the mean velocity profile, accompanied by an alternate thinning and thickening of the boundary layer. In the regions of

0

1

2

3

4

5

6

7) FIG. 13. Measured longitudinal mean-flow profiles. compared with that for undistributed flow. Data: 0 . at spanwise location of primary disturbance minimurn: 0. at spanwise location of primary disturbance maximum: B. undisturbed flow.

258


local steepening, the disturbance growth is augmented. The measurements indicate that with increasing G*,the spanwise distortion of the mean velocity profile increases. Consequently, augmentation of the disturbance growth becomes increasingly intense. These results are again in good agreement with the analysis of Audunson and Gebhart (1976). Prior to this analysis and to the conclusive corroborative experimental results of Jaluria and Gebhart (1973), there was some controversy concerning the role and the form of three-dimensional disturbances and of nonlinear mechanisms in natural convection flows. Colak-Antic (1962) had suspended highly reflective aluminum particles in water and observed their behavior during transition in the convective layer formed adjacent to a heated vertical flat surface. Two longitudinal vortices, similar to those predicted by the theory of Audunson and Gebhart, were seen. However, by dye visualization Szewczk (1962) observed vortices whose axes were transverse. The formation of a vortex loop was postulated, in which the vorticity field is considered to be concentrated. Although this hypothesis appeared to provide a plausible explanation for observed phenomena, there is some ambiguity in relating dye injection to vorticity concentration. Interestingly, a similar difference of opinion regarding such mechanisms arose in forced flows until Klebanoff et al. (1962) established, from detailed experiments with controlled three-dimensional disturbances, that secondary mean-flow longitudinal vortices occur as a consequence of nonlinear and threedimensional interactions. Additional explanations that have been offered are as follows: the generation of higher harmonics and the effects of the concave streamline curvature associated with the wave motion, in addition to the vortex loop formation. However, it is now reasonable to conclude that these effects do not dominate the overall behavior of transition and that regions of high shear, along with other consequences of the longitudinal vortex system, are associated with the onset of transition to turbulence.

D. NONLINEAR EFFECTS RESULTINGFROM NATURALLY OCCURRING DISTURBANCES Although the nonlinear growth mechanisms discussed above arose from artificially introduced disturbances, there is ample reason to believe that these effects also arise in flows subject only to naturally occurring disturbances. A frequent source of such disturbances, which arise from ambient disturbance sources, is through external vibration (see Tani, 1969, concerning forced flow). This mode of origin is very similar to that imposed by a vibrating ribbon in controlled experiments. There are, however, important differences. The artificial disturbances were introduced across the boundary region at one streamwise location x , beyond the neutral curve, in the


259

unstable region of the stability plane. Natural disturbances, on the other hand, may be fed in over the entire boundary region. Further, these disturbances contribute a spectrum of frequencies, as indicated by hot-wire measurements of background disturbances by Jaluria and Gebhart ( 1977). No dominant frequency was found. However, for any natural disturbances of a given amplitude, introduced at many different downstream locations, the one introduced.at the neutral stability condition would reach the highest amplitude downstream, since this disturbance would have had the longest path of amplification. Further, any such disturbance. with a broad range of component frequencies, is subject to selective amplification, or filtering, downstream. Therefore in the amplified domain the disturbance components that really count are those that are impressed at a location corresponding to the early stages of initial instability and that are at the most rapidly amplifying frequency. Thus, the end result for the two circumstances-of the introduction of disturbance through a vibrating ribbon at the most amplifying frequency near the neutral curve, or through a spectrum of naturally occurring disturbances across the entire boundary region-is expected to be the same. Available experimental data confirm this. Jaluria and Gebhart ( 1974) measured W in their investigation, in water, of transition mechanisms in natural convection flow, subject to naturally occurring disturbances. Its small magnitude and the random noise level made complete measurements impossible. However, the measured variation of W over a portion of the boundary layer is in good agreement with the distribution already shown in Fig. 11. The presence of longitudinal rolls is thus implied. The double longitudinal vortex observed by Colak-Antic (1964) is also for a flow subject only to naturally occurring disturbances, again confirming that the nonlinear interactions cause such mean flow modifications.

V. Transition and Progression to Developed Turbulence The previous sections discussed the sequence of events leading toward eventual transition. The laminar boundary layer becomes unstable to ever-present disturbances at some distance downstream. These first amplify in excellent agreement with the linear stability theory. Later the consequences of nonlinear interactions arise. A double longitudinal mean vortex is generated that, through the formation of regions of higher shear, contributes to rapid disturbance growth. These events, in broad outline, are similar to those observed in forced convection flows. However, the analogy ends here. In forced flows it is thought that the high-shear region, acting as a secondary instability, generates rapidly oscillating “hairpin eddies.” These


260

are produced intermittently in the boundary layer and immediately precede the formation of turbulent spots. On the other hand, the process of transition in natural convection flows is long drawn out and also more complicated. The velocity and temperature fields in natural convection flows are coupled together, and this has been found to cause significant additional effects on transition mechanisms in such flows. The way the two fields interact and influence each other is also Pr dependent, which in turn, then, becomes an additional parameter. Most studies of transition in natural convection flows have been in the two most common fluids in nature: water (Pr=i6.7) and air (Pr-0.7). Although the overall transition mechanisms in these fluids appear quite similar, some important differences are apparent. A brief review of what is known of transition in these fluids follows.

Developed Turbulence Laminar 1.2 BoundaryRegion Thickness

Turbulent Cascade to Smaller Scoies BL -I I

1.0

!

x(m1 0.81

0.6

I ,

Transition Proceeds

L

T, . / !

0.4

, Y ~ T ,Mean Flow Deviation Begins

FIG. 14. The sequence of downstream events during transition in water from a stable laminar flow to full turbulence. The spatial extent of each regime is shown to scale for a uniform flux surface condition of q ” = 1000 W/m2. G:, neutral stability for most rapidly amplified disturbance; GtT and G k , beginning of velocity and thermal transition, respectively; G&, end of transition.

Buoyancy-Znduced Flows

26 1

In water, to date, the most detailed investigations of transition mechanisms are by Godaux and Gebhart (1974), Jaluria and Gebhart (1974), and Qureshi and Gebhart (1978). Prior investigations (e.g., VIiet and Liu, 1969; Lock and Trotter, 1968, deal primarily with turbulent flows, although a few measurements during transition are also reported. An overall picture of transition that emerges from these studies is shown in Fig. 14. The turbulent disturbances first occur in the thicker velocity boundary layer. These apparently then cause the first signs of turbulence inside the thermal boundary layer. Further downstream, the maximum value of the base-flow velocity begins to decrease from its laminar trend. Simultaneously, the mean velocity profile also deviates from its laminar form. This is the beginning of velocity transition. The velocity disturbances then become strong enough to diffuse the thermal layer material into the outer velocity layer. This begins a change in the mean temperature profile-the beginning of thermal transition. As transition progresses, the velocity and thermal boundary layers mix and thicken, and the mean profiles deviate progressively from the corresponding laminar ones. The end of transition is simultaneously marked by no appreciable further change in the distributions of local velocity and temperature intermittency factors 1, and Z,. These factors, I, and 1,,are defined as the fraction of the time the flow at any point is turbulent, in the velocity and temperature values, respectively. The end of transition (see Bill and Gebhart, 1979) is followed by a regime of spectral and transport development. The spectrum of velocity fluctuations broadens and temperature fluctuations decrease in magnitude. Also, the turbulent heat-transfer mechanisms become more effective, despite the leveling of the growth of velocity disturbances. This development process continues downstream, until the distributions, scales, intensities, and other turbulent parameters largely adjust to the final characteristics of the turbulent flow. In gases, some data in the transition regime are available from the experimental studies of turbulent flow by Regnier and Kaplan (1963), Cheesewright (1968), Warner (1966), Warner and Arpaci (1968), and Smith (1972). A detailed determination of transition mechanisms in gases is provided by experiments of Mahajan and Gebhart (1979), in a vertical natural convection flow adjacent to a uniform heat-flux surface in pressurized nitrogen. These measurements indicate that the overall mechanisms are similar to those in water. The thermal transition follows velocity transition. However, since the two boundary layers are of comparable thickness, as opposed to water in which the thermal boundary layer is well contained in the velocity boundary layer (Prandtl number effect), the disturbance fluctuations in velocity more quickly affect the temperature boundary region. Both the velocity and thermal transitions are thus triggered almost simultaneously.

262


The region of spectral development beyond the end of transition, to the achievement of developed turbulence, has not been systematically studied in air. However, the experimental data reported by Smith (1972) in his study of turbulent flow adjacent to an isothermal plate in air indicate, unlike in water, no well-defined region of adjustment of turbulent-flow parameters. The reason for this difference can probably again be traced to Prandtl number effect. In water, the lag in the development of temperature disturbance levels observed in the early stages of transition continues until the end of transition. Thus, the spectral region may be looked upon as the regime in which temperature disturbances must catch up with the velocity disturbances. In air, however, the velocity and temperature disturbances develop equally side by side, from the beginning to the end of transition. Before beginning a detailed description of the stages of transition, in both air and water, a few remarks about the definition of the beginning and end of transition are in order. Although the criterion described above, to define end of transition, seems to have been successfully used for both liquids and gases, a multiplicity of criteria have been used in the past to mark the beginning of transition. Further, excepting the recent studies, no distinction seems to have been made between a velocity and a thermal transition. The beginning has been said to be signaled by the presence of significant temperature fluctuations, an increase in heat transfer effectiveness from the laminar trends, a decrease in temperature difference across the boundary layer from its laminar value for a uniform flux condition, and a deviation from laminar mean temperature profile. These are the most commonly used criteria to indicate the beginning of what is now known as thermal transition. Mahajan and Gebhart (1979) found that the events upon which the last three criteria are based occur almost simultaneously. Here, these criteria have been used interchangeably, although Mahajan and Gebhart found that a decrease in AT, the temperature difference across the boundary layer, from the calculated laminar value is a sharper indicator than others to define the beginning of thermal transition. The beginning of velocity transition, first studied in detail by Jaluria and Gebhart ( 1 974) in water, was indicated by the presence of a higher-frequency component superimposed on the single laminar filtered frequency. However, in gases, Mahajan and Gebhart (1979) found that this criterion could not be used unambiguously. Instead, deviation of U,,, , the observed local maximum value of the tangential flow velocity across the boundary region, from its laminar trend downstream was used to mark the beginning of velocity transition. In the experiment of Jaluria and Gebhart (1974) in water, this event occurs downstream of their designated location of the beginning of transition. The criterion of Mahajan and Gebhart (1979) is now used as the indicator of beginning of transition because of its precision and applicability to both liquids and gases.


263

A. TRANSITIONAL MEANVELOCITYAND TEMPERATURE DISTRIBUTIONS

Figure 15 shows the velocity measurements of Mahajan and Gebhart (l979), in nitrogen, at several values of q", in laminar flow and in transition. The downstream distance x was 22 cm and the ambient pressure of N, was 8.36 atm. From these data, the values of G* at the beginning of velocity transition, the beginning of thermal transition, and the end of transition (G&, G&, and Gg,) are 450, 480, and 61 1, respectively. In Fig. 15a, measured distributions of U , normalized by the maximum value U,,, found across the boundary layer, are plotted versus the laminar similarity variable 7). The measured values of U,,, normalized by local calculated laminar values, Urnax,at the same conditions are plotted in Fig. 15b. At G* =434, the measured value of U,,, and the mean velocity profile are in good agreement with the calculated laminar values. However, at G*= 470, which is downstream of the beginning of velocity transition, U,,, is about 4% below the laminar value and the mean profile has begun to deviate. As transition progresses, the deviations in form increase. The flow penetrates deeper into ambient medium as a consequence of growing turbulence in the boundary region. As the flow region thickens, the profile in the outer region is progressively flattened. This trend decreases as the end of transition is approached at around G* = 61 1. Measured mean temperature distributions for the same conditions as in Fig. 15 are plotted in Fig. 16a, in terms of nondimensional temperature + = ( t - ?,)/(to- t,) versus y , where t , to, and t , are the local fluid, (b)

(a1

G* I .o

0.8 - 0.6

0.6

II

3 3

-

0.4

0.4

0.2

0

0

2

6

4

8

10

77

Flci. 15. (a) Development of mean velocity profiles. (b) Variation of Urnax/iinlax during transition for experimental conditions of x=22 cm and p=8.36 atm (from Mahajan and Gebhart. 1979). Data: X , G* =434; 0 . 4 7 0 : 0.503; @, 543; 0,579; 0 . 6 1 1; 8, 648.

264


10

I

06

' .--.A

-

10

/O"

/a

- 08

Laminar theory

06

+

4 U

-06

$

4

G

04

-04

02

-02

5

surface, and ambient fluid temperatures, respectively. At G* = 470, the temperature profile is in agreement with the laminar profile. Note that at this value of G*, velocity transition had already begun. At G*=503, immediately downstream of the beginning of thermal transition at GFT= 480, the measured AT has already decreased from the laminar value (see Fig. 16b). The mean temperature profile has also started to deviate from the laminar trend. As thermal transition progresses, the thermal boundary layer thickens. The profiles steepen close to the wall and flatten in the outer region. These deviations follow those of velocity field, as expected, since they are initially almost completely coupled. As the flow penetrates into the ambient, it diffuses warm fluid outward, thereby thickening the thermal boundary layer, At the end of transition, at G* = 61 1, this modification is almost complete. Further variation with increasing G* is small. The mean velocity and temperature distribution modifications during transition in water show similar trends (see Jaluria and Gebhart, 1974). B. GROWTH OF THE BOUNDARY REGIONAND THE CORRELATION OF DISTRIBUTIONS DURING TRANSITION The increasing penetration of the upward flow into the ambient flow, with increasing transition, amounts to a more rapid growth in boundary region thickness. In air, the thermal and velocity boundary layers grow together. In water, the velocity boundary layer starts increasing in thickness


265

from its laminar value earlier due to delayed thermal transition. Although the thermal boundary-layer thickness grows more rapidly after the onset of transition, it does not catch up with the velocity region thickness either by the end of transition or even further downstream in fully turbulent flow. It is of interest to see if mean velocity (U/UmaX)and temperature (G) distributions can be correlated as a function of some dimensionless distance away from the wall. The laminar similarity variable q = y / 6 can not accomplish this (see Fig. 15a). When the temperature distribution in Fig. 16 is plotted in terms of 11, the same conclusion is indicated. Godaux and Gebhart (1974) showed that a modified q , defined as ,Y/&, or ,~/6,,, where aVMand ,a-, are, respectively, the measured local values of velocity and temperature boundary-layer thicknesses in transition is also not successful. Other attempts to correlate fully turbulent flows by a single variable, e.g., 77 = (y/x)Gr:' by Cheesewright (1968), have not been successful. For fully turbulent flows, however, George and Capp (1977) have recently shown that two different scaling parameters, one close to the wall and the other further out, correlate reasonably well the mean velocity and temperature distributions. Very near the wall there is a conduction viscous sublayer in which the mean temperature and velocity profiles are linear. In the other layer, called buoyant sublayer, the mean temperature and velocity profiles show, respectively, a cube root dependence and an inverse cube root dependence on distance from the wall. These predictions have found corroboration from the experimental work of Qureshi and Gebhart (1978). The data of Mahajan and Gebhart (1979) from the end of transition to early turbulence (in Fig. 16), when plotted in terms of these scaling parameters, also support these findings. However, examination of the data indicates that this scheme of scaling does not correlate distributions during transition. The two layers, inner and outer, are not clearly differentiated and the temperature in the outer layer decays faster than the indicated y ' I 3 . ~

C. DOWNSTREAM VELOCITY AND TEMPERATURE DISTRIBUTIONS

As the boundary layer undergoes transition, disturbances are found to amplify as they are convected downstream (see Fig. 17). The disturbance data of Mahajan and Gebhart (1979) in air, and of Jaluria and Gebhart (1974) in water, are plotted in the bottom part of Fig. 17 in terms of measured values of the maximum temperature and velocity fluctuation levels t' and u', respectively, normalized by the measured local mean-flow maximum values A t = 1,- t , and U,,,. Following the beginning of transition there is a rapid increase in disturbance magnitudes. The disturbances then grow less rapidly during later stages of transition. After reaching


POINT NO.

400

I

I

600

800

I

loo0

G*

I

I

I

1400

I

1 2 000

--.

velocity data: temperature FIG. 17. Downstreams growth of disturbance level. -, data. Data in water: A, Jaluria and Gebhart (1974); Bill and Gebhart (1979). Data in air are from Mahajan and Gebhart (1979) (for legend of their data see Fig. 15).

m,

maximum values the ratios tend to decrease as the end of transition is approached. In air, the velocity and temperature fluctuations reach their maximum almost simultaneously. In water, on the other hand, the velocity fluctuation ratio reaches its maximum by the end of transition, whereas the temperature fluctuation ratio continues to grow. This lag in growth characteristic is a continuation of the lag in the earlier stages of transition. Bill and Gebhart (1979) examined this point in detail, taking measurements in water during the later stages of transition and further on, to fully developed turbulence. Their data are also shown in Fig. 17, plotted as (?2)'/2/(f'2)',/,2, and (li'2)'/2/(Z'2),!,(~ versus G*. The data point 3 corresponds to the end of transition. Heat-transfer measurements suggest that points 1 and 2 are in fully developed turbulent flow. Thus, although transition ends at point 3,

Buoyancy-lnduced Flows

267

turbulent heat-transfer mechanisms are not fully developed until the maximum temperature fluctuation level is reached (point 2). The region between points 3 and 2 is the zone of rapid spectral change, defined earlier, in which the turbulent parameters adjust themselves from the end of transition to fully developed turbulent values.

D. PROFILES OF DISTURBANCE FLUCTUATIONS Distributions of temperature and velocity fluctuation levels across the boundary region, from laminar flow, through transition, to early turbulence, in air, are shown in Fig. 18. The disturbance data during transition

7l

(b)

FIG.18. Velocity (a) and temperature (b) disturbance distributions across the boundary region in air. -, theoretical curve for G* =4W,11*= 1.23. The data of Mahajan and data points in early Gebhart (1979) are during transition (for legend see Fig. 15). The turbulence from Smith (1972).

m,

B . Gebhart and R. L . Mahajan

268

from Mahajan (1977) are plotted in terms of instantaneous values, whereas the data of Smith (1972), in early turbulence, are expressed in terms of rms values. In laminar flow, the disturbance profiles are in good agreement with the calculated laminar curves of Mahajan (1977) computed from linear stability theory for two-dimensional disturbances. However, as transition progresses, the distributions extend further out, with the thickening of the boundary layer. They deviate progressively from the laminar calculations. In particular, the position of the inner peak moves closer to the wall while the peak in the outer region broadens and finally disappears with the development of turbulence. As the end of transition is approached, the disturbance profiles adjust themselves to a fully developed turbulent distribution. Compare the profile of Mahajan (1977) at the end of transition to that of Smith (1972) in early turbulence. Data of Jaluria and Gebhart (1974), in water, show similar trends except for a persistence of the outer peak in distributions of velocity fluctuations, even at the end of transition. However, the data of Bill and Gebhart (1979), in early stages of fully developed turbulent flow, based on rms values of the disturbance fluctuations, show no such peaks. Nor is such a peak expected, since the turbulent diffusion would tend to smooth out sharp gradients in the boundary layer. Noting that the distributions of Jaluria and Gebhart are based on the maximum values of the fluctuation, as seen in an analog

1.0

1

0.8 . 0.6 . Iv

0.4.

0.2’ I

I

0

,

I

I

2

\, 3

~

4

5

6

7

8

~

9

7 FIG. 19. Velocity intermittency distributions across the boundary layer during transition (for legend, see Fig. 15).


269

record, it is quite possible, as pointed out by Bill and Gebhart, that the peaks observed might represent the passage of particular bursts of turbulence. Progressive penetration of disturbance fluctuations into the thickening boundary layer as the flow progresses from laminar to turbulent can also be expressed in terms of distributions of intermittency factors I , and I,. I, distributions during transition from the experiment of Mahajan and Gebhart (1979) in pressurized nitrogen are shown in Fig. 19. Corresponding I, distributions, although not shown, are identical and follow the same developments. In early stages of transition, the region of maximum turbulence is located around the inflection point and flow is not fully turbulent ( I , = 1) anywhere. However, as transition progresses, the intermittency rapidly increases and spreads in both directions from the inflection point. The process continues until, at some value of G*, both I, and I, distributions change little further downstream. This by definition is the end of transition. In water, although the temperature disturbance layer is smaller in thickness than the velocity layer (the Prandtl number effect), the data of Jaluria and Gebhart (1974) indicate that as the end of transition is approached, both I, and I, distributions reach their final shape almost simultaneously and thereafter change insignificantly further downstream.

E. DISTURBANCE FREQUENCY DURING TRANSITION The selective amplification or filtering effect predicted by linear stability analysis has been found not only in the earlier stages of laminar instability, where linear processes dominate, but also in the region of nonlinear and three-dimensional disturbances, as discussed in Section IV, B. It is of interest to see if this filtering mechanism is modified during transition. An analog record of naturally occurring disturbances during transition in water, taken from the experimental investigation of Qureshi and Gebhart (1978), is shown in Fig. 20. In locally laminar portions of the flow, the disturbance frequency remains essentially unaltered during different stages of transition. Further, this frequency is the same as the filtered frequency predicted by linear stability theory (see the data at large G* in Fig. 4). The data of other investigators had shown this same result (see also Fig. 3, for comparable data in air). The process of selective disturbance amplification seems to extend far beyond the range of linear development and downstream well into transition. In Fig. 20, it is seen that other and higher frequencies also arise, later in transition. These disturbances begin to occupy a larger part of the record, that is, extent of the flow field, at the expense of the characteristic frequency. The disturbances become distributed over a much broader

270

B . Gebhart and R. L . Mahajan

(el

(fl

FIG. 20. Analog record of amplification of natural oscillations as the laminar flow undergoes thermal transition. (a) G*=682, (b) 889, (c) 962, ( d ) 1155, (e) 1375, ( f ) 1561 (Qureshi and Gebhart, 1978).

frequency range. This is the beginning of the broad range of length scales or eddy sizes that characterizes the eventual completely turbulent flow. An experimental determination of this frequency broadening during transition, and after, is provided by the s p e c t r y analysis of Bill and Gebhart (1979) (see Fig. 21). Cumulative spectra J$+df for the disturbance energy ii’*are presented for the flows a t the beginning of transition, the end of transition, and in full turbulence. The cumulative spectrum covers only the frequency range 0.006-10 Hz, since above 10 Hz the low levels of the spectral density approached the noise levels of the anemometer unit. Arrows in Fig. 21 correspond to the characteristic frequencies of the local flow. Near the beginning of transition, only about 5% of the disturbance energy lies above the characteristic frequency. About 45% of the disturbance energy is concentrated in a smdl frequency range containing the characteristic frequency. By the end of transition turbulent energy has been extracted from the mean flow through nonlinear processes and distributed more evenly across the spectrum. Approximately 14% of the energy is now distributed in the frequency range above the filtered frequency. This value is somewhat conservative, since the energy spectrum was cut off at 10 Hz, although some energy may be contained in the range above 10 Hz. Broadening of the spectrum continues beyond the end of transition. For the data in turbulence, energy above the filtered frequency increases to about 27% of the total disturbance energy. Further downstream, the spectrum of turbulence continues to develop until a condition is reached in which regions of local isotropic turbulence exist at “inertial sub range.” (For further details, refer to Bill and Gebhart (1979).) It is apparent from


27 1

1.07

-

0.8

J

'$df 0.008

0.6

-

0.4

-

a2

-

/'%if

0.006

Frequency (Hz) FIG.21. Cumulative energy distribution versus frequency: 0. the beginning of transition: transition: W, in turbulence. Data are taken From Bill and Gebhart (1979).

0;end of

Fig. 21 that the process of breaking up of large-scale eddies into smaller scales begins during transition. Taking disturbance frequency as a measure of turbulent length scales, the length scales during the early stages of transition are large since the observed laminar filtered frequencies are very low. However, as transition progresses, energy is transferred from the narrow band of frequencies centered on the characteristic frequency, to higher frequencies that indicate smaller-scale eddies.

F. THERMAL TRANSPORT DURING TRANSITION The most important practical aspect of transition is the improvement in heat-transfer mechanisms, compared to steady laminar flow. The progression of local heat transfer according to the laminar mechanism apd after is shown in Fig. 22. These data are from the experiments of Qureshi and Gebhart (1978), taken in flow induced adjacent to a vertical uniform flux surface in water. Accompanying the increase seen in local coefficient of


212

4

0

'

lo3

to2

10

10"

I 0l4

I 0''

FIG.22. Variation of local heat transfer from laminar through transition to turbulence. Data from Qureshi and Gebhart (1978): 0 , q=583.80; 0,1323.75; 2326.4; 0,3714.45; 0,4488.60 W/rnZ.

a,

heat transfer, h,, is a corresponding decrease in local surface temperature from its laminar value. The deviation from the laminar trend, the Jj root variation shown at smaller Gr:Pr, increases with the progression of transition, for each of the five heat flux levels. Eventually a fully developed turbulent heat-transfer trend is established further downstream. Clearly, the additional turbulent transport modes account for this more effective transport. The two possible modes are turbulent convection of heat downstream, p C F n , and increased transport of heat from the inner wall region to the outer boundary layer, p C F n . The downstream development of these modes, from the later stages of transition to turbulent flow, is shown in Fig. 23. Clearly there is an increase in both, with the maximum occurring beyond the location of end of transition. The increase in mode p C ' n is at the expense of mean thermal transport. As the flow deviates from the laminar condition, the streamwise velocity component U decreases progressively from its laminar value (see Fig. 15b). Since this decrease takes place in the region of steep temperature gradients, near the waii, there is a reduction of mean thermal transport. Since the downstream convection of heat is the aggregate of the convection by the mean base flow, JZpC, Ut dy, and the turbulent convection downstream,

273


0.0

‘

600

1

800

I 1200

It

1000

I

1400

I

1600

G*

POINT NO. 4

3

2

--

I

u‘t’lu’t’max; FIG.23. Downstream growth of turbulent heat transport 0,

.,

_u’t‘/u’t’max

at 7-6; q ” = 1920 W/m*. Data are taken from Bill and Gebhart (1979); vertical line is the location of end of transition from Jaluria and Gebhart (1974).

J , P p C p n d y ,a decrease in the former with progression of transition is overcompensated by an increase of the latter. As the flow undergoes transition, it penetrates further into the ambient and diffuses warm fluid outward. This results in the observed increase of the second turbulent transport mode p C p a (see Fig. 23). This increase is seen to lag the streamwise turbulent convective transport. The lag is similar to the delay observed in the development of temperature fluctuations (see Fig. 17). This suggests that the development of the normal heat-transport mode is more closely coupled to the temperature fluctuation levels in the wall region. Beyond the end of transition, the increase in both transport modes continues, which is to be expected. It is now known that fully developed turbulent heat-transfer mechanisms set in further downstream. In the intervening region of spectral transition, these transport modes therefore increase to adjust to the fully turbulent values. Thus, the modification of the laminar heat-transfer rate starting at the beginning of transition continues beyond the completion of transition, through the spectra! region, until the turbulent value is achieved.

214


V1. Predictive Parameters for the Events of Transition Perhaps the most important aspect of instability and transition studies is the establishment of predictive parameters for the two ends of transition. Both tradition and the success of linear stability theory in correctly predicting the dependence of the growth rate of two-dimensional disturbances, solely on the basis of the local Grashof number, led earlier investigators to attempt to use the Grashof number as the correlator of the beginning of transition. However, an examination of data (for example, as collected in Tables I and 11, indicates that each end of transition occurs over a broad range of Grashof number. Some of the spread may be attributed to different criteria. However, it is important to remember that there is no strong reason to believe that the prediction of transition events should depend solely on a parameter whose primary importance first arises in laminar transport and in the linear theory of stability. Strong nonlinear interactions precede transition. There may be many other important considerations. Vliet and Liu (1969) concluded from their experimental study of turbulent natural convection boundary layers that Ra: alone did not correlate the beginning of transition. No other parameter was proposed. The first experimental study to investigate this particular matter was that of Godaux and Gebhart (1974), in a flow induced adjacent to a uniform flux vertical surface, in water. The beginning of thermal transition was judged by traverses with a thermocouple probe at various downstream locations x, at different surface heat flux levels q” for each location. The data unequivocally proved that the beginning of transition was not correlated by local Grashof number alone. An additional dependence on at least x or q” was indicated. It was found that thermal transition, defined by the change in mean temperature profile from its laminar shape, began at an approximately constant value of G * / x 3 / 5 c c ( q ” x ) ’ / 5 aQ(x>’I5;that is, transition began when the local thermal energy Q(x), convected in the boundary layer at a downstream location x, had reached a certain value. The measurements of the mean and disturbance quantities during transition also confirmed the failure of G* alone to correlate transition (see, for example, Fig. 24). The thermal intermittency factor Z, for each flow condition is listed in the figure. At x- 100 cm, for G* =948 the mean temperature distribution has just deviated from the laminar. However, at x=36.2cm, it has already changed considerably at a much lower value of G*= 625. A similar conclusion follows from an examination of the disturbance data (not shown in Fig. 24). An additional downstream parameter must arise.

OF TRANSITION TABLE 1. DATA FOR THE BEGINNING

Transition Fluid

Pr"

Water Water N,

6.7 6.25-6.81 0.71

Air

0.71

Water

6.7

Water

5.05-6.4

Air Air Water

0.71 0.71 11.0

Water

11.4 6.7

G

G*

580- 1030 749-950 380-610 400-645

504-802 563-802 855-960

x

P

(cm)

(atm)

36.2-100 43.3-78.6 13.2-33 13.2-33 91.5 580 466 65 38.1 - 121.9 38.1-121.9 28-79

Instrument used to detect transition

I

Thermocouple Thermocouple Hot wire 1-15.92 Thermocouple I Thermocouple Thermocouple I 1 Hot wire Thermocouple I Thermocouple I

1 1- 15.92

E

EIE,h

Transition criterion

16.8-21.6 18.6- 19.4 16.8-23 17.6-24 24 23 13.6' 15.2 22.2"

0.82-1.05 0.9 1-0.95 0.86-1.17 0.86-1.17 1.17 1.12 0.70 0.74 I .08

25 30 16.3 20.5 17.2

I .28 I .46 0.8 I .03 0.88

As proposed in Section V" As proposed in Section V' As proposed in Section V" As proposed in Section VK AS proposed in Section V" As proposed in Section V" Jump in the frequency of disturbance above laminar frequency' Deviation of wall temperature from its maximum value During transition' During transition' During transition'?' During transition"' Beginning of vortex formation"

20

0.98

First appearance of Tollmien Schlichting waves" First appearance of turbulent burst'' First appearance of turbulent burstr

Air

0.70

400

61

1

Hot wire Thermocouple Thermocouple Hot wire Flow visualization using dye Interferometer

Air

0.71

622

92

I

Interferometer

26.4

1.29

CO,

0.77

460-547 645-702 54 I 378 605

12.5 25 17 9 25

Interferometer

23.8-24.9 23.2-26.4 24.4 18.3 19

1.16-1.21 1.13-1.29 1.19 0.89 0.93

572 713

84 97 24 27.4 60

1

485 665 665

I I I I

4-9 4-9 5 10 11

Proposed values of E for beginning of transition velocity transition. 19.5: thermal transition. 20.5. Prandtl number of ambient medium. Best fit value. dAverage value. 'Godaux and Gebhart (1974). 'Qureshi and Gebhart (1978). cMahajan and Gebhart ( 1979). Warner and Arpaci (1968). 'Jaluria and Gebhart (1974). /Vliet and Liu (1969). 'Colak-Antic (1964). 'Cheesewright (1968). "' Lock and Trotter (1968). "Szewczyk (1962). "Eckert and Soehngen (1951). [ ' R e p i e r and Kaplan (1963). "

TABLE I1

DATAFOR Transition G* G

X

P

(cm)

(atm) 4.18-15.92

Fluid

Pf

N,

0.71 495-980

13.2-33

Water

6.7

61

870

ENDOF TRANSITION

Instrument used to detect transition Hot wire and thermocouple Hot wire and thermocouple

990 I140 1320

Air Water

a

0.72 6.2 6.4 6.4 5.05

83.8 100.7 121.9 845 115 1140 113 1195 80.7 60.5 1385 1615 53.6

1

THE

1 1

Prandtl number of ambient medium. Q E T = I 1.4 for end of transition. Best fit value.

’Proposed value of

Thermocouple Thermocouple

QET

QET/QkT

Reference

11.4‘

1 .o

Mahajan and Gebhart (1979)

10.94

0.96

Jaluria and Gebhart (1974)

10.62 10.84 11.75 11.89 10.18 12.67 16.96 18.73

0.93 0.95

tb

scz

a-

F: “L

a

i3.

1.03

1.04 0.89 1.11

1.49 1.64

Cheesewright (1968) Vliet and Liu (1969)

% a a-

e. a 3

Buoyancy-fnduced Flows

277

-

&\\

0

Lam inar theory

3

2

I

4

rl Symbol

x(cm)

G*

0 0

36.2 36.2 36.2

485 608 625

I

r(cm)

G*

I,(7=2.5)

+

100

X

100

0.05 0.6

0

loo

948 1031 1131

I,(?J-2.5) Symbol 0 0.18 0.68

0.9

These striking results of Godaux and Gebhart (1974) generated a much more detailed investigation of transition by Jaluria and Gebhart (1974), also in water. This study confirmed that neither the beginning nor the end of transition is a function of only the local Grashof number. A function of the plate heat flux q” or, equivalently, of the streamwise coordinate x must also arise. Using both thermocouples and hot wires, it was found that velocity transition preceded thermal transition and that each began at a

278


particular value of G * / x 2 l 5 . The mean flow and disturbance quantities during the progression of transition strongly confirmed the above additional dependence on x . The end of transition was found to be approximately correlated by G * / x ’ . ~ ~ . A. CORRELATING PARAMETER E FOR THE BEGINNING OF TRANSITION The quantity G * / x 2 l 5 has a direct physical significance. It is proportional to the fifth root of the boundary-layer kinetic energy flux e , defined by

or e / p v 3=(G * 5 / x 2 ) F ( P r ) ,

where F(Pr) is a nondimensional function that may be calculated from the laminar similarity solution. Nondimensionalization of e / p v 3 in terms of g and v results in 2/3

e;/’=[$($)

or

] =(-$)

i!;:, ( i’3)

2/15

G*F(Pr)

(6.2)

2/15

--

-

-

G*= E,

where g x 3 / v 2 is the unit Grashof number. Jaluria and Gebhart (1974) found that velocity transition, indicated by the presence of higherfrequency disturbance components, began at E = 13.6. Thermal transition occurred later, at E = 15.2. An implication of these results is that the kinetic energy flux of the mean flow is the energy available for disturbance growth and this, in turn, determines the onset of transition. To test the validity of E as a general correlator of the beginning of transition, Jaluria and Gebhart (1974) calculated the values of E from the transition data of other investigators in air and in water, which are summarized in Table 1 of their article. A 70% spread in E was found, over a range of Pr=0.7-11.85. This spread is clearly due, in part, to the use of different criteria to identify the beginning of transition. This effect was demonstrated from the detailed experiments of Qureshi and Gebhart (l978), in water. The most commonly used methods of specifying transition were reviewed. They were as follows: (i) the appearance of significant temperature fluctuations, (ii) the deviation of the mean temperature profile from the laminar trend,


279

*0003 2000

G*

1000

-

500

-

,001 100

6

,

,

I

1000

1

I

I

I

1

1

1

1

1

1

10,000

q" (W/m2)

FIG. 25. E as a correlator of different criteria of beginning of thermal transition. ( I ) E = 17.5. appearance of significant temperature fluctuations in boundary layer, f ' = 0.05. (11) E = 19.2. deviation of mean temperature profile from laminar trend. (111) E=22.7. maximum surface temperature. Data are taken from Qureshi and Gebhart (1978).

and (iii) the attainment of maximum surface temperature for a uniform heat flux condition, the equivalent of the minimum heat transfer condition. It was found that each of these events is correlated approximately by a different unique value of E. Values of E for each of these different transition criteria, over a range of heat flux levels, are determined in Fig. 25. Constant-E parameter loci correlate each event. The value of E thus determined for the beginning of thermal transition in water, as defined by the deviation of mean temperature profile from the laminar trend is 19.2. This criterion is the one used in the correlation next discussed. A more rigorous test of E as a correlator of beginning of transition is provided by the experiments of Mahajan and Gebhart (1979), in pressurized nitrogen. Recall that the factor ( ~ * / g ) ~was / ' ~ introduced in the definition of E for nondimensionalization only. However, this imposes a particular dependence of G* on v, at the beginning of transition [see (6.2)]. Since v is inversely dependent on density in gases, this dependence was tested in transition experiments by changing the pressure level of nitrogen from 1 to 15.92 atm.

280


Measurements taken at a fixed level of pressure confirmed the dependence of the beginning of transition on G * / x ~ / ~However, . at a given downstream location, the values of G* (both G$T and G&) at which transition began showed a systematic dependence on pressure given closely by G * a p 2 / l 5 .Recalling that, for an ideal gas, v a l/p, these data indicate that G* a v - ~ / ' ' This . is different from the dependence G* a K4/I5implied by the definition of E in (6.2). Based on these dependences of the transition value G* on x and Y , the following new correlating parameter QeT for the beginning of transition in gases was formulated:

Here q is the fifth root of nondimensional local heat flux to the boundary region flow. Fixed values of QsT=290 and 315 were found to characterize the beginning of velocity and thermal transition, respectively. When calculated in terms of parameter E, transition data in nitrogen gas of Mahajan and Gebhart (1979) (see Table I) indicate an average value of 20.5 for beginning of thermal transition. Since this is very close to the value E = 19.2 suggested for the beginning of thermal transition in water, it is of interest to see if a single value of E can, within reasonable accuracy, predict the beginning of transition in both liquids and gases. To determine this unique value, the values of E were calculated from the data of various investigators concerning the beginning of transition. The results are summarized in Table I. The downstream locations, where not stated in some of the studies, have been estimated as accurately as the descriptions of the experiments permitted. Data for an isothermal surface condition was reduced using the relation

G = G*[ (0.q4/ - +'(O)] ' I 5 , where +'(O) is a Pr-dependent constant that is given, for example, in Table 8.1 of Gebhart (1971). For some of the data entries in Table I, the transition criterion was different from that proposed in Section V, as enumerated in the last column. Considering first the condition for the beginning of thermal transition, the data in the first four entries in Table I correspond to the same criterion, the response of a thermocouple probe. From this data, an average value of E=20.5 is indicated. The maximum difference from this value is 17%. Since the data correspond to the wide ranges G or C* =400-1030, x = 13.2100 cm, and p = 1-15.92 atm, this deviation in E is not surprising. The resulting estimates of G b or GTT,obtained using this single value of E to detect the beginning of thermal transition in both gases and water, should

Buoyancy-Induced Flows be sufficiently accurate for engineering calculations. Thus, G&=20.5( ~ . x ’ / Y - ) 2/15 .

28 1

(6.5) On the other hand, the beginning of thermal transition, as defined by the downstream location where the wall temperature begins to decrease from a maximum value (Vliet and Liu, 1969), appears to begin at an average value of E2522.2. This is close to the value of E=22.7 found by Qureshi and Gebhart (1978) for the same criterion. The observations in which an interferometer detected the beginning of transition resulted in a number of substantially higher values of E . This discrepancy is thought to be due to the insensitivity of the interferometer to small and/or concentrated turbulence. This is analogous to the differences noted in data for transition in supersonic boundary layers between Schlieren measurements and thermocouple measurements, as noted by Schubauer and Klebanoff (1956). Bill and Gebhart (1975) reached the same conclusion concerning interferometry, from their study of transition in plane plumes. The lower values of E for two data points of Regnier and Kaplan (1963) could also be due in part to inaccurate resolution of interferometer output for the low values of AT for these experiments. Earlier transition is suggested by the lower values of E for the data of Eckert and Soehngen (1951). This is possibly due to the particular nature of their experiment. Very high disturbance levels were present and the flow was actually a transient cooling of a surface. From the interferometric data in Table I, a representative value of E to mark the first appearance of turbulent burst seems to be about 25. For the beginning of velocity transition, as marked by deviation of U,,, from its laminar value (see Section V), the value of E in gases from the data of Mahajan and Gebhart (1979) appears to center around an average value of 19.5. This corresponds to a value of x for the beginning of velocity transition (xVT)to be 5% upstream of that for the beginning of thermal . this criterion, no such precise information is available transition ( x ~ ) For in water. In the detailed transition experiments of Jaluria and Gebhart (1974), in water, a different criterion was used. This criterion was the presence of a higher-frequency component, superimposed on a single laminar filtered frequency, as the indication of the beginning of velocity transition. This was found to occur at a location G* about 12% upstream of the location of the beginning of thermal transition. The deviation of U,,, from its laminar trend occurs in between the locations of these two events. Thus, it is not unreasonable to assume that the beginning of velocity transition, according to the criterion used here, takes place in water also at about 5% upstream of xn. Indeed, examination of the data in Fig. 15 of Jaluria and Gebhart (1974) also suggests this assumption. Thus, it appears


282

that in both gases and water, velocity transition occurs about 5% upstream of thermal transition. Using the proposed value of Em= 20.5, this amounts to a value of EvT=19.5. In other words,

B. PREDICTIVE PARAMETER FOR THE END OF TRANSITION As for the beginning of transition, the recent investigations have conclusively shown that the end of transition is also not correlated by G* aIone. Additional dependence on x, q", and/or v must be considered. Jaluria and Gebhart, in the investigation of water, found that G*/x", where n--0.54, approximately correlated the end of transition. No nondimensional parameter was proposed. The more detailed study by Mahajan and Gebhart (1979), in pressurized nitrogen, indicated n-0.50 and an additional depen1000

-

7

-000 900

700 600

G~~ 500 400

i // 0

-

0/

O

-

- 20

- -d

r+

- 15 m-

- 10 Q~~

- 5

Buoyancy-Induced Ffows

283

dence of G* on v as G* av-’’’(see Fig. 26). Based on these results, the following correlating parameter was proposed:

(6.7 1 where Ra* is the Rayleigh number for a uniform flux surface condition. The value Q E T from these data in gases was found to be 11.4 (see Table 11). The Prandtl number dependence as Pr’/’ in the definition of QET above was included to account for any independent effect of Prandtl number on QET.

The data of other investigators for the end of transition in air and water, converted to QET form, are-also listed in Table TI. As for the beginning of transition, there is much scatter in the values of G& or GET. However, when converted to QET, the data seem to collapse around the value of Q E T = 1 1.4. For the transition data of Cheesewright (1968), in air, the value of QET is 11.89. Note that his data is for an isothermal surface and was converted to the form of Q E T using (6.4) with +(O) taken at the value estimated from the data at the end of transition. The average value of Q E T calculated from the data of Jaluria and Gebhart (1974), in water, is about 11, with the maximum difference from the suggested value of QET being only 7%. The QET values for the first two data points of Vliet and Liu (1969) in Table I1 are close to those predicted. For the last two entries, however, the Q E T values are much higher. The values of A t across the boundary layer for these two data points were much larger (15-22OC>, leading to the uncertainty in property evaluation. The above observations suggest that a given value of Q E T is a consistent indicator of the end of transition. The Pr dependence included in the definition seems to be adequate at least for the data in air and in water. Based on the value of Q E T = 11.4, the following relation may thus be used to determine the location for the end of transition, using local sensors, to a reasonable degree of accuracy both in water and gases: Ra* = 308( ~

x ~ / P ~ ) ’ / ~ .

(6.8)

VII. Plane Plume Instability and Transition Plume flows are very different from those adjacent to a surface. The surface damps disturbances. They are also very different in that the two mirror-image flows of a plume may freely interact in disturbance mechanisms across the midplane and even by fluid motion across the midplane.

284


FIG.27. Plumes perturbed with sinusoidal disturbances at several frequencies for air at atmospheric pressure: f=2.4 (a); 3.6 (b); 5.1 (c), 7.0 Hz(d). Q’=56.3 W/m, wire length= 15.5 cm, wire diarneter=0.013 cm (Pera and Gebhart, 1971).

As a result, free boundary flows are much less stable (in G ) than those

adjacent to surfaces. Also, disturbance mechanisms that are asymmetric, with respect to the midplane, are found to be much more unstable than those that are symmetric. The plane plume may be thought of as arising in a wake above a horizontal concentrated source of energy, like an electrically heated wire.


285

The flow results entirely from thermal buoyancy. A plane plume in air, subject to controlled disturbances of several frequencies, is seen in Fig. 27. The midplane temperature to decreases approximately as ( t o- t,)a x i.e., n = -0.6 in (3.28). The velocity increases as x0,*. The local flow parameter is again G in (3.29), as for the vertical surface. The plane plume flow is discussed in detail in Section IX. Two-dimensional disturbances were postulated by Pera and Gebhart (1971) as in (3.35) and (3.36). The stability equations in terms of and s are again (3.39) and (3.40). The coupled base-flow temperature and stream functions (+ and F) are found from (3.31)-(3.33). The remote boundary conditions in Q, and s are still the same43.41). The other three conditions admit the possibilities of disturbance motion at 17 = 0 and of disturbances being symmetric or nonsymmetric about 17 = 0. An extreme of nonsymmetry is complete asymmetry. The asymmetric mode was found to be less stable and neutral stability curves were determined. The result for Pr=0.7 is the neutral curve of Fig. 28. The first values of G for instability were found to be very low, an order of magnitude less than those shown in Figs. 3 and 4 for flows adjacent to surfaces. Haaland and Sparrow (1973) repeated the stability analysis, retaining two of the several terms excluded in the conventional approximations, as set forth in Section 111. Similar results are obtained. 'The question of the consistency of higher-order approximations in stability analysis for vertical buoyancy-induced flows is considered in Section IX. The paths that disturbances follow as they are convected along at constant frequency are also indicated in Fig. 28. The particular frequencies shown are those relevant to experiments in atmospheric air at a source strength Q'=56.3 W/m. These paths show a very different behavior than for vertical flows adjacent to surfaces (recall Fig. 2). Also, the base flow amplifies all frequencies below a certain limit. However, all frequencies are eventually stable. Of course, this doesn't happen in an actual plume. Other and nonlinear mechanisms intervene for some of the amplified frequencies. Experiments by Pera and Gebhart (1971) tested these stability predictions. A 15.3-cm-long horizontal wire of 0.0127-cm diameter was electrically heated in atmospheric air. The interferograms of Fig. 27 show the extent of the thermal boundary region. Since, for Pr = 0.7 the velocity and thermal boundary regions are of almost equal extent, the region shown is essentially the whole plume. Controlled disturbances were introduced with the vibrator shown near the plume source. Disturbances of lower frequency are very strongly amplified. These observations are in very good agreement with the predictions of Fig. 28. We may conclude that the predictions of instability are again in close agreement with experimental observations, although the extent of the comparison concerns only the ranges of frequency and of G in which disturbances amplify.


286

-

QOl-

1

I 1 l l l l

I

I

I

I

I

I

I

I

I

I

I

FIG. 28. Computed neutral stability curve (Pera and Gebhart, 1971). Constant frequency contours for air for Q'=56.3 W/m. Data of highest frequency velocity disturbances: 0, Q'=65.9 W / m ; M, Q'=65.9 W/m (turbulent flow); v , Q'=3.1 W/m. Data are taken from Bill and Gebhart (1975).


287

Some calculations of amplification rate - a i have been made and the values are very large compared to those found for flows adjacent to surfaces. The low levels of G for instability and the rapid amplification rates, both from theory and as found in experiment, suggested that other nonlinear effects must very quickly (in x ) become important in such flows. Since the calculated downstream range of instability is very short (in G), even the relatively high values of - a i that were calculated result in maximum values of A of less than about 2.0 for the disturbance frequencies that appear to be sufficient to disrupt such plumes. Thus, such plumes are relatively much less stable than vertical flows adjacent to surfaces. A considerable amount of study has been devoted to plume transport, beyond the simple initial laminar flow that is amenable to similarity analysis. Some of these studies relate to transition, as do the foregoing stability analyses. Forstrom and Sparrow (1967) generated flows sufficiently vigorous to disrupt the laminar patterns. Thermocouple measurements indicated turbulent bursts and their first appearance was taken as the beginning of transition. Characterizing the local vigor of the flow by a local Grashof number based on the heat input rate, these occurred at the flow Grashof number, GrQ,,I , of 5 X lo8, where Q’ is the line source strength per unit length. This Grashof number is defined by GrQ,, =g&x3Q’/pCpv3. (7.1) At the highest heating rate used, and at the most distant location downstream, turbulent bursts were observed with great frequency. Furthermore, under this condition, a time-averaged temperature profile showed a thickening of the flow region, with respect to laminar flow. Based on this observation, Forstrom and Sparrow concluded that full turbulence occurred at Gr,.-.=5x lo9. Only this single data point was measured in what was taken to be the turbulent region. No comparisons could be made with the temperature decay in x for turbulent plumes predicted by Zel’dovich (1937). Even though flow at the highest Grashof number was found to be primarily turbulent, the maximum centerline temperature of the plume still was found to follow the functional dependence on heat flux predicted by laminar theory. It is noted that temperature levels for laminar flow depend on Q‘4/5, whereas in turbulent flow (either plane or axisymmetric) the dependence is indicated to be at least approximately Q’2/3. The turbulent data taken by Rouse et al. (1952) and by Lee and Emmons (1961) are difficult to interpret as thermal buoyancy-induced line source plumes since in the former study the plume flow was generated by a row of gas burners and in the latter by burning liquid fuel in a channel burner. In both experiments the plume source was of appreciable size and introduced initial momentum flux, diffusing chemical species of different molecular ~

288


weight, initial disturbances, and nonuniformity of energy production rate. Results of Lee and Emmons are tabulated only in terms of fuel consumption rate per unit length of burner. Perhaps related to possible additional mechanisms arising in such measurements are observations by Miyabe and Katsuhara (1972). Another mode of instability was seen in spindle oil. Transverse sinusoidal oscillations were reported, i.e., across the span of the plume. An experiment by Bill and Gebhart (1975), in atmospheric air, studied transition as it related to instability and to eventual turbulence. Using fine thermocouple and hot-wire anemometer probes, along with an interferometer, disturbances and turbulence conditions were studied in a plane plume subject to naturally occurring disturbances. The measured disturbance frequencies were in accord with the predictions of linear stability theory. Increasing local Grashof numbers were obtained either by increasing the heat input or by moving the probes further downstream. Velocity disturbance signals were decomposed with a spectrum analyzer. Somewhat surprisingly, it was found that all frequency components of appreciable amplitude fell in the amplified region of the stability plane, even to the end of transition. This indicates that linear stability considerations are important even in regions of large disturbance amplitude. This same characteristic was found during transition in the flows generated adjacent to a vertical surface, as discussed in Section V. The plumes were visualized with a 20-cm aperture Mach-Zehnder interferometer. Thereby, the locations of the thermocouple and hot-wire probes were known, in relation to the general flow configuration. The region considered to have turbulent bursts by Forstrom and Sparrow was found to consist of a flow in which two-dimensional sinusoidal-like disturbances had reached large amplitudes and higher-frequency disturbances had begun to appear. These large disturbances were seen to disappear downstream and leave a well-ordered and completely laminar boundary region. With increasing local Grashof number such disturbances became more frequent and eventually the laminar boundary layer broke down completely. This corresponds to the condition for which Forstrom and Sparrow reported a thickening of the profile and concluded the presence of complete turbulence. However, upstream of this last condition, the maximum instantaneous midplane temperature was still accurately predicted by the laminar theory. Further downstream, the changing centerline temperature finally came to depend on turbulent field parameters. In these observations, above 25.4-cm-long electrically heated wire at different downstream locations and heat input levels, these events occurred at roughly the same values of the local Grashof number. Thus, the beginning and end of transition approximately correlated in terms of this parameter.


289

Some of the patterns established for transition adjacent to a vertical surface, Jaluria and Gebhart (1974), are paralleled by this data. Twodimensional disturbances were found to amplify selectively. At increasing amplitude, three-dimensional effects became apparent. However, a delayed thermal transition effect was not observed. This might be expected in air (Pr = 0.7) since velocity and temperature disturbances appear to grow together and remain coupled. After a period of intermittency the flow adjusts to turbulent parameters. In the experiment, plumes were generated in an enclosure, from horizontal electrically heated wires of length L=25.4, 15.5, 5.1, and 2.5 cm and values of L/dw=741. 445, 400, and 400, where d,,, is the wire diameter. These plume spans, along with different levels of energy input, resulted in downstream plume behavior that varied from that of a plane plume toward that of an axisymmetric one. A measure of the downstream (in x) transport in a concentrated source plume is the nature of the decay of the temperature field, due to entrainment. This is expressed in terms of the actual midplane or axis fluid temperature io(x) as i,,(x)- t , =d(x) analogous to (3.28) in the similarity solution for plane laminar plumes. Here, io(x) is the time-averaged value. Each of the two plumes, plane (P) or axisymmetric (A), may be either laminar (L) or turbulent (T). The four variations are written below in terms of a general variable T , where I=g(Pr) is an integral of +F’:

T=4JZ( fo- t,)( p C p I ) / Q ’ . LP: T = Gr,; TP: T a IGr;’l2, LA: T=K,,(Q/Gr,)”*,

(7.5)

TA: T a K,, Q ’’rGr;5/4.

(7.6)

The value of the Grashof number Gr, in (7.3) and (7.4) is based on to- t , resulting from Q’, as calculated for steady laminar boundary-layer flow. A similar procedure is used in (7.5) and (7.6), using Q’ to calculate i o - t , from laminar theory. Comparisons of actual data with these trends were used to infer downstream plume transport. Typical examples of such transport are seen in Fig. 29 at two heat input conditions and at differing locations downstream. In Fig. 30 are shown the time variations of midplane velocity and temperature, at three downstream locations, in a given plume. The records in (a) and (b) clearly indicate that the velocity and temperature disturbances are strongly coupled, as would be surmised from the interferograms.

290

B . Gebhart and R . L. Mahujan

FIG. 29. lnterferograms for L=25.4 cm. G=68.8 (a, b). 186.0 (c). and 228.0 (d) at arrow locations. which are .x=7.1 and 30.5 cm. respectively. Q ’ = 5 0 (a, b). 98.1 W/m ( c , d ) (from Bill and Gebhart, 1975).

In (a) the temperature disturbance frequencies were found to be 0.2 Hz or less. These large fluctuations resulted from the swaying of the plume perpendicular to its average midplane, as well as from oscillations along its span. Despite these disturbances, the plume appeared to remain primarily laminar. This would be expected from Fig. 29a, where the listed local value of G applies at the level of the arrow. In Fig. 29b the flow is apparently still laminar, despite the large spanwise distortion seen. With increasing Grashof number, small-amplitude, higher-frequency disturbances begin to appear (Fig. 30b.) These are superimposed on the higher-amplitude, low-frequency disturbances seen at lower Grashof number. Simultaneously, large unsteady and wavelike disturbances were visible in the interferometer (as seen in Fig. 29b immediately downstream of the arrow marking the local value G = 186. These very complicated occurrences are taken as the first signs of local turbulence and are defined as the beginning of transition. In the following transition region, the laminar flow was intermittently disturbed by the passage of disturbances, followed by relaminarization. Yet further downstream, disturbance frequency and amplitude increased. The end of transition was taken as the location in x at which a thickening of the mean-flow boundary layer and no relaminarization occurred. Beyond the end of transition, the thermocouple and hot-wire outputs were dominated by high-frequency components (as seen in Fig. 30c.) The


I cm

29 1

(C)

FIG.30. Time variation of hot wire and thermocouple outputs. respectively, at G=68.8 (a) and 160.3 (b) and thermocouple output at G=228 (c). ---,laminar theory: L=25.4 cm,

Q'=34.4 W/m; chart speed=] crn/sec (From Bill and Gebhart, 1975).

292


turbulent condition shown in Fig. 29d is characterized by a thickened boundary layer and a chaotic temperature field. Conclusions were not based primarily on a statistical study of the disturbances that arose under various conditions. 7 he flow had been found to be alternately laminar and turbulent early in transition. This suggested characterizing the flow locally in terms of the maximum measured instantaneous local temperature tb<x). These temperature trends downstream were then compared with (7.2)-(7.6), to indicate the kind of transport the flow was approaching downstream. It had been noted from experimental measurements in laminar flows that the measured T is about 15% below (7.3) (see Brodowicz and Kierkus, 1966; Forstrom and Sparrow, 1967; Schorr and Gebhart, 1970). Lyakhov (1970) demonstrated that this arose from flow generated upstream of the flow-generating electrically heated wire. Such on-flow is not included in the boundary-layer formulation, which restricts the domain to x > 0 for Q' > 0. Therefore this effect on T will again show up in subsequent comparisons. However, in Fig. 31 is 'shown the very interesting consequence already seen during the downstream transition processes of several other kinds of

LP; ----, FIG. 31. T versus Gr, data for nonturbulent plane plumes, in air: --, experimental correction; 0 , Forstrom and Sparrow (1967); Brodowicz and Kierkus (1966); 0 , , X , Bill and Gebhart (1975).

+

+,


293

0.01

\ 0.001I I 0’

I

1

1

1

1

I

I

I

I

1

I

108

rx FIG. 32. T versus Gr, data for turbulent plumes taken from Bill and Gebhart (1975). -, LP: ----, TP: ---, TA; experimental correction. 0. Q ‘ = 5 0 W/m; m. 63.3: A, 75: 0, 84.6; v,98.1.

----.

vertical flows; that is, laminar transport continues to penetrate often, completely unchanged, far downstream into the transition region. Figure 3 I shows that measured instantaneous temperature maxima during transition are in excellent agreement with corrected laminar theory. The extent of the transition region is indicated on the figure. The first comprehensive data comparison (shown in Fig. 32) is for a plume initially of 25.4-cm span, for a range of energy input. The LP, TP, TA, and “corrected” LP (15% low) downstream trends are shown. Maximum measured instantaneous midplane temperatures are shown for plumes generated by five different levels of Q‘. Only the first two measurements at the lowest Q’ are seen to be still in the transition region. All the others are in completely turbulent flow; that is, they diverge downward from the LP corrected trend. Only the variation of T may be shown for the TP and TA plume, since the constants of proportionality in (7.4) and (7.6) are not

1

294


known in general. These curves were placed to best agree with the data. For the TP plume, T is only a function of Grashof number. However, in the axisymmetric flow, there is also a further dependence on Q ” / * . The turbulent region data does not correlate well with either theory alone. Immediately after the end of transition, at perhaps Gr,=3X lo’, it conforms most closely to TP plume behavior. Further downstream the slope of the data decreases further and then conforms most closely to the trend for a TA plume. However, the data does not show systematic Q‘ dependence. The theories for turbulent plumes have been derived for mean-flow values and neither their accuracy nor their applicability to temperature maxima are established. Similar measurements for L = 15.3 cm indicated that transition is complete after Gr, = 4 x 10’. However, a similar plot does not show the succession of later trends seen in Fig. 32. For L = 5.1 cm the deviation, likely the achievement of full turbulence, did not occur until about Gr, = 2 x 10’. Thereafter the TP plume trend was followed. Surprisingly, for L = 2.5 cm, first deviation appeared to occur at a decreased value, at Gr, = 1.5 X 10’. The other important aspect of such a transition is how the predictions of stability theory relate to actual disturbance growth and transition. For example, it is seen in Fig. 28 that the theory prediction is for disturbance amplification only below a certain cutoff frequency. Disturbance velocity spectra were determined down to 2.5 Hz during the above measurements. Comparison of spectra at different values of Gr, for a given plume indicated which disturbance components had been amplified. The resulting points, in p and G , are shown in Fig. 28. These data are for L=25.4 cm at Q’=65.9 and 3.1 W/m. The trend of the data with G indicates that disturbance energy is found at increasingly higher frequencies further downstream. We also see that all disturbances detected up to G = 194 have, with one exception, traversed the amplified region of the stability plane. This is almost to the end of transition, which was about at G = 208. Beyond the end of transition, energy is found in higher frequencies, clearly indicating the kind of nonlinear disturbance growth and propagation mechanisms found by Jaluria and Gebhart (1974) in flows adjacent to a vertical surface. The spread of observed frequencies for G < 208 does not indicate a narrow-band filtering process, but a one-sided process. The much higher frequencies beyond G = 208 represent the conversion of disturbance energy by turbulent processes. This is a broadenidk of the spectrum, through turbulence. Finally, these estimates of transition limits are compared with those of Forstrom and Sparrow (1967). For the beginning of transition, the values are GrQ,*= 1 1.2 X 10’ and 5.0 X lo’, respectively. This discrepancy may in


295

part be due to the insensitivity of the integrated interferometer output to small local disturbances. For the end of transition, the value of Bill and Gebhart (1975) is Grq..=7.9x lo9, compared to a single data point at 5 X lo9. The criterion for this latter value was a thickened temperature profile. This does not define a precise point for the completion of transition, and, from the later data, it appears that such a local flow was still within the transition region. In summary, the above measurements indicated that, for both L = 25.4 and 15.3 cm sources, the beginning of transition was at Gr, =6.4X 10’. It ended at approximately the single value Gr, = 2.95 x lo8. The correlation of centerline temperature with laminar theory provided a strong and unequivocal standard for the determination of the end of transition. After the complete disruption of the boundary layer, the flow begins to adjust to turbulent parameters, and the laminar centerline temperature is no longer achieved. Nonlinear effects have become important and spread disturbance energy to frequencies above those of the initially amplified disturbances. Turbulence intensity and scale then begin to decrease.

VIII. Instability of Combined Buoyancy-Mode Flows Mass transfer occurring in a fluid also gives rise to a buoyancy force if the molecular concentration gradient causes density differences. If the concentration of the diffusing chemical species is sufficiently small, the equations governing the phenomenon are identical to those governing a thermally induced flow. A frequently occurring circumstance in our environment and in technological applications is the simultaneous transport of both thermal energy and of one or more chemical constituents. The occurrence of a second buoyancy-inducing transport process may be expected to cause major alterations of the stability characteristics of the resulting flow. There are very complex interactions between disturbances in velocity, temperature, and concentration. The additional complexity may arise from two separate aspects. One is the possibility of opposing buoyancy force components. Then, the chemical species transport layers may be of different spatial extent. The measure of this is the Lewis number, Le= Sc/Pr = K / D,where Sc = Y / D is the Schmidt number. This additional characteristic is similar to the role of the Prandtl number in expressing the relative extents of the velocity and thermal effects. These effects may be seen in Fig. 33, in which buoyancy force distributions-f?,. thermal; B,, chemical; and B = B,+ &-are sketched for Le< I for both aiding and opposing buoyancy force components. Although combined buoyancy-mode flows appear to be very compli-

296


3t

b

Y

FIG.33. Aiding and opposing buoyancy force components E , and E,. For Sc > Pr, the variation of the local buoyancy force B = ET+ B, for lo> r, and BT > 0. Gr, = Grx,I + Grr,c.N=Grf.,,./Gr,,l. (a) Local buoyancy force variation B for ) & ( f o - t , ) I > I,&(C,CJ. (b) Local buoyancy for reversal for I &.(Co- C,I > I &( t o - t J . Ea++ C, where C = ( C - Cm)/(C0- Cm).

cated, a very fortunate characteristic arises in a large proportion of the combined buoyancy-mode flows in our environment and in technology. First, the chemical species diffusion-caused density differences are often very small. For example, ordinary humidity levels in air are only a few percent of the total density. The discharge of solvents and other agents from many surfacing materials is often accomplished at very low gas-phase concentrations. Even for seawater, the salinity level is only about 3.5%. Yet

Buoyancy - Induced Flows

297

the buoyancy effect resulting from the species diffusion is comparable to thermal buoyancy for a wide range of processes and conditions that very commonly arise (see Gebhart and Pera, 1971). In one of the first investigations of the flow resulting from combined thermal and mass transport, by Somers (1956), the dual buoyancy force variables were employed in an integral analysis. Gill er al. (1965) assessed the effects of multiple and comparable concentration levels. Possible similarity solutions were examined by Lowell and Adams (1967) and numerical results were presented by Adams and Lowell (1968). Among numerous experimental studies, the results of Bottemanne (1972) show close agreement with calculations. In formulating the instability of such transport at low level concentration, the similar methodologies in Gebhart and Pera (1971) and in Bottemanne (1971) will be followed. With small concentration differences, the buoyancy force contribution B , may be calculated in terms of the concentration differences as €3, =p&( C- Cm).This is analogous to the Boussinesq approximation in B,, in that the density is assumed to vary linearly with concentration C , where BC-is the equivalent volumetric coefficient of expansion. Then B = B,+ B , = g d PT( t - t m) + PCC c - C d l . Since the gradients of concentration are also small, both the Soret and the Dufour diffusion effects are negligible in a convective circumstance. Then the equations governing thermal transport and chemical species diffusion are equally simple and of identical form, except that the parameter of the first is the Prandtl number, Pr= Y / K , and of the second is the Schmidt number, Sc= v / D , where D is the Fickian molecular diffusion coefficient . Another fortunate characteristic is that the species diffusion rate is, relatively, very small. In particular, when the mass diffusion rate is converted to an equivalent velocity of the medium at density p, this velocity is usually very small compared to the flow velocities generated by buoyancy (see Gebhart and Pera, 1971). The resulting equation for convection and Fickian diffusion of a specie of local concentration C ( x , y , T ) is

aT

+ u-ac + 0-ac = D V’C, ax

ay

where C satisfies the general boundary condition

c(x,m,7)-cm=o.

(8.2)

+

The base-flow and disturbance levels are related by C(x, y , T ) = F(x, y ) C’(x, y , 7). The additional buoyancy force associated with C’ and the boundary region base-flow and disturbance equations are written, using the

298


three approximations set forth in Section 111, as follows:

into The transformations in (3.26)-(3.30) may again be used to cast boundary-layer similarity form, with the following additions and modifications: 6=(c-C,)/(co-C,)=(C-C,)/e(x), e ( x ) = N , x ” , (8.6) Gr, = PGr,. ,+ MGr,,

[ PP,(I,-

=( g x 3 / v 2 )

G- cm)].

tm) + M P ~ (

(8.7) N =Grx A = P,(ioGrx.I

t,)lP,(

co- Cm),

(8.8)

where P and M are convenient constants and N is positive for aiding effects and negative for opposed. The functions b ( x ) and c ( x ) in (3.29) remain the same, as do the energy equation (3.32) and the conditions (3.33). The transformation of (8.4) and of the base-flow force momentum balance, with the additional buoyancy force term in 6, yields the following equations:

C‘” + Sc[ ( n + 3) F c ’ - 4 n F ’ 6 ] = 0, F”’ + ( n + 3 )FF’ - (2n + 2) ++ + NC“= 0. FI2

(8.9) (8.10)

Here P is taken as 1.0 and M as 0, following the analysis of Boura and Gebhart (1976). Then Gr, again becomes simply as given in (3.30). The characteristic length and velocity are again defined as in (3.34). These equations coupled with energy equation (3.32) have been solved for air and water for various practical values of Schmidt number and for multiple buoyancy effects aiding (positive N ) and opposing (negative N ) (see Gebhart and Pera, 1971). The disturbance stream and temperature functions J/’ and t’ remain as defined in (3.35) and (3.36). The disturbance C’is similarly defined in terms of an amplitude function a ( q ) : ~ ( xy , 7 ) = (

Co-

c,)a(q)exp[ i ( d x - /%)I.

(8.1 1)

The equation for a(?) from (8.5) becomes identical in form to that for s(q), (3.40), namely, ( F ’ - P / a ) u - era=(a”- a2u)/iaSc G.

(8.12)


299

The previous disturbance force-momentum balance (3.23) must now be augmented with the added buoyancy force component due to C ’ , as seen in (8.3). The result is that the disturbance velocity equation (3.39) is the same, except that 3‘ is replaced by s’+ Nu‘. The additional boundary conditions, on a, are analogous to those on s given in (3.41) and (3.47) for an assigned surface condition at q = 0: a(oo)=O,

a(O)=Ka’(O)

(8.13)

The above formulation for the amplitude functions @, s, and a, in terms of parameters cu,, q,p, Pr, Sc, N and G, is of seventh order. This, combined with the much greater complexity that may arise in the base flow with opposed effects when Pr # Sc, makes calculations much more difficult. Nevertheless, calculations have been made for the surface conditions t (x, 0, T ) = t,=constant and C(x,0,7)= C,=constant. Therefore, n=O and the full equations and boundary conditions, where G is as defined in (8.7) but with P = 1 and M = 0, are as follows: ( F ’ - @ / a ) ( @ ”-a2@)- F’”@=(~’”-2a2~”+cw4@+s‘+ Na‘)/iaG. (8.14)

(F’-/3/a)s-cp’@=(s’’-a2s)/iaPrG,

( F ’ - / 3 / a ) a- c“@= (a” - a a ) / i a Sc G,

9(0)= cp’(0)- s(O)= a(0)= @( 00)’

@‘(OO)=S(W)

= a(00)=0.

(8.15) (8.16)

(8.17)

It is apparent from these relations that a = d if Pr = Sc. Then the only effect of mass diffusion is through the coefficient of its buoyancy force term ( I N ) s in (8.14). Opposing buoyancy effects mean only that N is negative and (1 + N ) is reduced. Should ( I f N ) be found to be negative, it is replaced by -(I N ) and the assumed positive direction of x is reversed. In any event, the proper stability plane is that which applies for thermally caused buoyancy above. However, the interpretation is now different. The coordinates previously generalized in terms of Grx., must now be interpreted in terms of Grx,,( 1 N). However, the interesting question here concerns what additional effects arise when Pr # Sc, that is, for Le # 1. Then the buoyancy force variation is different and buoyancy force reversal may arise across the boundary region. The result may be large effects on the velocity distribution, which is usually the prime determinant of instability characteristics. For the most common fluids, air and water, the Lewis number for ordinarily occurring diffusing chemical species is usually different from 1 .O. For air, Pr-0.7, Sc ranges from 0.22 for hydrogen gas to around 2.5 for a hydrocarbon vapor, and Le ranges from 0.3 1 to 3.6. For CO,, Sc = 0.94 and

+

+

+


300

0.I4 -

0.10-

P

-

0.06-

0.02I

100

I

1

200

150

1

250

G FIG. 34. Neutral curves for Pr-0.7 and Sc=0.94 (carbon dioxide in air) in terms of thermal Grashof number.

Le= 1.34. For water at around 20°C, Pr-7.0, and Sc and Le range from 152 and 22 for hydrogen gas to about 1700 and 240 for sugar, respectively. For salt, Sc = 840 and Le = 120. Extensive stability calcuIations have been made for Pr = 0.7 and Sc = 0.2, 0.94, and 2.0. The remaining parameter to define a stability plane is N in (8.8). This is the ratio of the units of thermally and mass transfer-caused buoyancy. Figure 34 shows the effect of N on neutral stability for Sc=O.94 in terms of p and G as defined in (3.38) and (8.7),with P = 1 and M = O . Neutral curves for Pr=0.7, Sc=0.94, and six values of N, from -0.8 to +0.5, are seen. Increasing mass transfer buoyancy upward, that is, N >0, strongly destabilizes the flow in terms of G. This is to be expected, since G is not a reliable measure of the actual total buoyancy force. However, each of these neutral curves strongly suggests the very sharp selective disturbance amplification first found in a purely thermally driven flow. It amounts to selectively amplifying only certain components of a more complicated naturally occurring disturbance. A more realistic plot of these effects is seen in Fig. 35. There the


30 1

t y

0.14

b.2

FIG. 35. Neutral curves for Pr=0.7 and Sc==0.94 (carbon dioxide in air) in terms of a combined Grashof number.

coordinates are P I , and G , , instead. These are based on (8.7) for P- M = 1 as follows: (8.18)

pI= p( 1 + N ) - 3/4.

(8.19)

These coordinates are much more appropriate since Le= 1.34. The small effect on stability in the N range from -0.2 to +0.2 agrees with the conclusions of Gebhart and Pera (1971). A single neutral curve would result for Pr=Sc, for all N, when GI is used. The small difference in Pr and Sc is first strongly felt for N = - 0.5 and very strongly at N = - 0.8. The effect for N = -0.8 is seen here as formally due to the singularity of the transformation of ,l3 into p, at N = 1.0. This singularity does not actually occur for Le # 1.O; that is, we should not take P = M for Lef I .O because the two transport processes have different spatial extents, and their simple sum does not properly represent the actual buoyancy effect, or whatever

302


GI -,

FIG. 36. Downstream disturbance amplification for Sc=0.94 in terms of G , . ----,N = 0 . 5 ; N s O . 0 : ...., N = -0.5.

else is appropriate as their combined effect. We recall that for Le = 1.0 there is no flow for N = 1.0, no matter what values are assigned to P and M. The downstream [ G ( x ) ] amplification rates are given by - ai. The ratio of the amplitude of any particular sinusoidal disturbance component downstream, at G, to the amplitude it had on crossing the neutral curve, at G,, is given by (3.43), where 4A there for the flux condition is here replaced by 3 A for the uniform surface condition of n = O . This amplitude growth calculation is approximate to the extent that the form of the disturbance amplitude distributions across the boundary region changes downstream with G and is also subject to all other approximations already made. The above integration is performed in the p,, G, plane along paths of constant physical frequency f. This path is PIG ; l 3= constant. Contours of constant A downstream have been calculated for Pr=0.7, Sc =0.94, 2.0, and 0.2, and for several values of N for each value of Sc. The contours have been determined across the band of frequency that experiences most rapid amplification in each circumstance. Again the frequencies most rapidly amplified are not the earliest in G to be unstable; that is, the notion of a critical G was again found not to be important. The results for Sc=O.94 and N = 0 . 5 , 0 and -0.5 are shown in Fig. 36 in terms of G I . The contours of A again show the sharp downstream frequency filtering found for a purely thermally driven flow and since abundantly corroborated by experiment. With the combined buoyancy modes,

Buoyancy- induced Flows

303

we see that disturbances are amplified less rapidly for increasing N, in terms of G I . N = 0.5 and N = - 0.5 appear to cause opposite effects of comparable amount. Increasing N appears to stabilize the flow and also to reduce the most highly amplified frequencies. However, the location of the filtered band seems largely independent of N. Such inferences are not purely quantitative, since P = M = 1 is still somewhat arbitrary, even for Pr = 0.7 and Sc = 0.94. However, these results for Sc = 0.94 do not amount to a demanding test of the effects of combined buoyancy modes on stability and disturbance growth mechanisms. The transport effects are only approximately included by the parameter (1 + N). Recall that for Le= 1.32 the boundary layers are of comparable extent. For Sc = 2 (Le= 2.9), the concentration boundary layer is relatively thin. Stability results for Sc = 2.0 and N = - 0.5, 0, and + 0.5 are seen in Fig. 37 in terms of G I ,The effects of N on stability are much greater. An opposing buoyancy effect destabilizes the flow and an aiding one stabilizes. These effects are large. The destabilization for negative N is consistent with that found for Sc=O.94. We note that for values of Sc>Pr the concentration boundary layer is always thinner than the thermal one. Some results are also available for Sc=O.2, at N=O and 0.2. The

+

0.10-

PI

-

0.06-

I

I

I

I

200

100

I

I

I

G,

-.

I

400

300

I

I

I

500

FIG.37. Downstream disturbance amplification for Sc=0.2, in terms of G , . ----.N = 0 . 5 ; N=0.0;

. . . . , A'=

-0.5.


304 0.121

0.08

I t

c

0.04

I

100

I

I

200

1

I

300

I

I

1

400

FIG.38. Downstream disturbance amplification for Sc =0.2, in terms of G, . ----,N = 0.2; -,

N = 0.0.

calculations were limited to the A =0, 0.5, 1, and 2 contours because of long computing times. They are shown in Fig. 38. The curves indicate that a positive N again stabilizes. These results indicate stabilization with increasing N for all three values of the Schmidt number. This is rather surprising since this range of Sc spans the condition of Le= 1; that is, the concentration gradient layer is both thinner and thicker than the thermal layer, over the range. The explanation for this is not now clear. There have been no experiments for comparison, even to the extent of measured favored frequencies. Nevertheless, these results might be expected to be realistic estimates of the stability characteristics of actual flows because of the detailed past successes of linear stability theory, compared to many experiments, in its predictions of disturbance growth rates, filtered frequency, etc. There is the remaining question of how one may most accurately estimate the effective local vigor of a flow induced by combined buoyancy modes. The effect of Lewis number on the respective spatial extents of the diffusion layers, and the differing resultant modification of the form of the velocity field, makes this a very complicated and still unanswered question. The methods used here may be directly applied to other important combined buoyancy-mode flows. Both bounded and unbounded flows of great practical importance arise through combined modes and in many different fluids.


305

IX. Higher-Order Effects in Linear Stability Analysis The instability characteristics of the different types of buoyancy-induced flows discussed in Sections 111, VII, and VIII were obtained using analysis based on parallel flow and conventional boundary-layer approximations. Using these approximations (see Section 111) Orr-Sommerfeld equations (3.39) and (3.40) were obtained from a complete set of stability equations (3.16)-(3.18) by omitting some terms of O(I / G ) from the latter eqwtions. These terms can be shown to involve derivatives of lower order than appear on the right-hand side of (3.39) and (3.40) and therefore may be neglected. For many flow circumstances of interest, this procedure is justified. For example, for a flow adjacent to a vertical surface, the location of the first instability occurs around G*-I00 (see Figs. 3 and 4), so that the error introduced due to neglecting terms of order O(I/G*) is small. Further downstream (higher G*), in the region of highly amplified disturbances the error is even smaller. This region is of interest, since it immediately precedes transition. However, for a plume generated by a line heat source, unstable conditions extend to very low values of G (see Pera and Gebhart, 1971). Using parallel-flow analysis, the lowest value of G, is calculated to be approximately 3. At such low values of G, it is to be suspected that effects arising from O(I / G ) terms neglected in the parallel-flow analysis may be nonnegligible. An improved stability analysis to better predict the instability characteristics of such flows and other flows in general at lower values of G must consider “nonparallel” and other associated effects in a consistent manner. Such effects have not been adequately investigated. Haaland and Sparrow (1973) considered this matter for a plane plume flow. In the analysis, the Orr-Sommerfeld equations were extended to include the effects of streamwise dependence of base-flow quantities. These terms are shown underlined in the following equations:

at.+,K+u’ar+,-+o‘-=K at) ai a7 ax ax ay ax

(9.2)

Equation (9.1) is the vorticity equation derived from Eqs. (3.15)-(3.17) linearized in disturbance quantities and by following the standard procedure to obtain the vorticity equation from the continuity and momentum equations. Here vorticity l=aG/ax - aii/ay and disturbance vorticity {’ = au‘/ax - au’/ay. Equation (9.2) is the energy equation derived from (3.18) linearized in disturbance quantities. It can be shown that these

306


underlined terms are of O( 1 / G) and lower. However, as first pointed out by Hieber and Nash ( 1975), Haaland and Sparrow (1973) neglected other terms that are of the same order as those retained. There are higher boundary-layer effects of O( 1/ G) in the base flow. These, through interaction with the disturbance quantities, give rise to terms of the same order in the stability equations. Further, the terms arising from the x dependence in S' and from the x dependence of (Y for fixed physical frequency are also of O( 1 / C). An internally consistent higher-order stability analysis must also include all these terms. Such an analysis for the plane plume is given by Hieber and Nash (1 975). The higher-order boundary-layer plume solution is calculated and the stability of the resulting flow has been analyzed by a systematic expansion in the disturbance equations. This procedure and the results are outlined next. A. PLANEPLUMEFLOW

Using the asymptotic matching technique, higher-order boundary-layer effects were first calculated in terms of perturbation parameter E = Gr; 'I4, with the leading terms (zeroth order) satisfying the following governing equations: F0"'+ 1F F " - IF'F'+@,=O, 5 0 0 5 0 0 (9.3)

+," + $Pr( For& + F6+o) =0,

(9.4)

F{(O) = Fo(0)= 1 - +o(0)= F6( 00) = (Po( 00 ) = 0. (9.51 Note that these equations are exactly the same as (3.31)-(3.33) with n = - $ except that the coefficients of some of the terms in (9.3) and (9.4) are different. This difference is due to the different definitions of similarity variables used. In this work, q = ( y / f i ~ ) ( G r , ) 'whereas /~, Hieber and Nash use 1 = ( y / x ) ( G r X ) ' l One 4 . set of equations may be transformed into the other. The first-order equations in F , and ( P I were determined to be governed by F,"'+ 25 F0 F1" I5 F'F' 0 1 + I= 0, (9.6)

+

+

+

+

Pr( For$', + 2 F&,) $PrF;@,= 0, Fl(0)= 0 = F;'(O)=+',(O) a), F;( 00)' $ cot yFo(00).

(9.7) (9.8)

The zeroth-order plane plume solutions have been obtained by several investigators. Closed-form solutions for Pr = 2 and 2 were presented by Yih (1956), and later rediscovered by Brand and Lahey (1967). The most


307

comprehensive numerical solutions for a range of Pr = 0.01 - 100 were presented by Gebhart et al. (1970). The numerical solution to the first-order equations above is given by Hieber and Nash (1975). To assess the higher-order stability effects, more general forms of equations (3.35) and (3.36) were assumed: +b'(x,y , z) = 6U,@ exp[ i ( A ( x )-

/%)I,

('f x, y , t)= d ( x)s exp[ i ( A( x) -

where

A=

/&)I,

(9.9) (9.10)

S,'ac x ) dx.

As for the base flow, the disturbance quantities were also then expanded in terms of z, as @=@o(q)+E@l(q)+ * . . , s = So(?) ff

+ zsl(q) +

= &6= f f , +

Eff,

+

, ,

**.

(9.1 1 ) (9.12) (9.13) (9.14)

c=co+€cl+~~'.

where c=p/ff.

The linearized vorticity disturbance equation (9. I ) was then evaluated in terms of the higher-order base-flow and disturbance quantities defined above. Taking terms in different powers of E resulted in the following equations: ~ ~ : L ( @ o ) = FAi ~ o c,)(@; ( €'

- &Do)-

iaoFl'@o=O

:L(@,)=5,+ff,52

(9.15) (9.16)

(9.18) Here (9.19)

308


arises from the streamwise dependence of a. and the property that a. depends only upon P, being the eigenvalue of the inviscid problem. The leading term from the disturbance energy equation (9.2) is (9.20) so = %@o/ ( G - CO). At large distances from the plume, the disturbances vanish so that +(0 0 ) = 0.

(9.21)

The boundary condition at the center of the plume depends on the nature of disturbances, i.e., @(O) =0 For symmetric disturbances, (9.22) W(0) = 0 for asymmetric disturbances.

(9.23)

It was shown by Pera and Gebhart (197 1) that the asymmetric mode is less stable than the symmetric mode so that the appropriate boundary conditions are (9.21) and (9.23), i.e., @;(o)=o=@0(00),

(9.24)

0;(0)= 0 = 0 ( 00 ),

(9.25)

The equations (9.15) and (9.24) define the inviscid problem and can be easily solved to calculate a. and Q0 for a given value of P. To determine a , , first note that the homogeneous problem for is the same as that for !Do. It is therefore required that

(9.26) where x is a nontrivial solution of the adjoint homogeneous problem:

+

(FG - cO)(x”- (~lix) 2 F{x’= 0,

(9.27) x’(O)=O=x(00).

With a. and @, known, x is determined from (9.27) and t2 is evaluated from (9.18). To determine t , , @ ’; and at’’ are obtained by successive differentiation of (9.24); s’, is similarly derived from (9.20) and y is obtained from knowing a. at neighboring values of P, namely

F P

(Yo(

P + AP ) -

(Yo(

2 AP

P - AP ) .

,

(9.28)

aI is now obtained from (9.26), i.e.

(9.29)

Buoyancy-Induced Flows 0.14

r

309

--

=\

-*

1

40

80

I20

160

1

200

c 39. Constant-amplitude-ratio curves for plane plume flow. Pr=0.7. -, Hieber and Nash (1975): -.-..Haaland and Sparrow (1973); - x - x , Pera and Gebhart (1971). FIG.

Calculating the values of a. and a , for a range of fi, the neutral stability and amplification curves can be obtained. Using the series in (9.13), truncated to two terms, - a i = -(aOi+€ali), (9.30) where - a i is the desired amplification rate. In particular, for the neutral stability curve ( - ai= 0), l / ~ = G r - : / ~ G/2fi =

= -a,,/aOi.

(9.31)

As in Section 111, constant-amplitude curves can be obtained along paths of

constant physical frequency given by fJ;2/jG-1/3,

(9.32)

i.e., m = - f in (3.42). Figure 39 shows these curves for different values of A . They have been replotted here from the curves /3 versus G of Hieber and Nash (1975) in (52, G ) coordinates. Horizontal lines are again the constantphysical-frequency disturbance trajectories. For comparison, the neutral curve from parallel-flow results of Pera and Gebhart (197 1) is also shown. Clearly, higher-order effects have significant effect on the instability characteristics of plane plume flows. With the inclusion of these effects, the neutral curve exhibits both minimum Grashof number and a lower branch. Also shown in the Fig. 39 is the neutral stability curve obtained by nonparallel analysis of Haaland and Sparrow (1973). The apparent discrepancy in their results is due to the inconsistent approximations used in that analysis, as discussed before.


310

B. OTHERVERTICALFLOWS Higher-order stability effects for flows adjacent to a vertical flow have not been evaluated. An approach to solve the problem is outlined below. As for a plume, the analysis must include the terms contributed by the higher-order boundary-layer effects, along with those arising from streamwise dependence of both base-flow and disturbance quantities. First consider the higher-order boundary-layer effects. For an isothermal surface, these have been most recently analyzed by Hieber ( I 974), with perturbation parameter E =Gr; 'I4. Mahajan and Gebhart (1978) analyzed these effects for a uniform flux surface, c being equal to 5 / G * . To proceed with stability analysis, one first notes that, unlike in plume flow, there is a predominantly viscous inner layer next to the surface at q = 0, where the no-slip condition must be satisfied. Away from the surface in the outer layer, the viscous forces are negligible. For these two regions (inner and outer), two separate solutions (inner and outer) arise that must match in the region q = O ( l ) . These solutions may be obtained by the method of matched asymptotic expansions. Appropriate outer expansions for CP and s are for fixed q and r,+O,

@-a&)+ e,@?(q)+ €;@;(TI)+ s-sg(

q)

+

€,ST(

q)

+ +z"(

**.

11) + * ' *

,

.

(9.33) (9.34)

Appropriate inner expansions for fixed y and c,+O are

,

@.-YEl@;(q)+E;@1;(q)+ S-€,S;(q)+E:S;(q)+

where

-

0

.

.

(9.35) (9.36)

may be related to the base-flow perturbation parameter E , and is the inner variable. Also, as before, we introduce the following asymptotic expansions: E,

{=?/el

cr--cw,+Eiai+E:a2+ . . * ) c-c,+

E,Cl

+ EfCZ+

* * *

.

(9.37) (9.38)

Using these expansions in conjunction with (9.1), (9.2), (9.9), and (9.10), and collecting terms of like powers of c l , the governing equations for inner and outer vorticity (a:, CP;, . ..) and temperature disturbance (sg,s;, . . . ) functions can be obtained. These equations can then be solved for a0, a,,... for fixed values of p, from which the amplification curves can be obtained.

Buoyancy-Induced Flows LIST OF SYMBOLS a

A

E C C, D E

f R G GI

G*

Gr, Gr: h k

Le N N,

Nu,

P Pr 4,'

Q Ra Ra* P

sc I

T AT ii, L?

U V. W u'. c'. w'

Uc s y i

Species disturbance amplitude function. equation (8.1 I ) Disturbance amplitude. A = - fj:,,., dG Buoyancy force. see equation (3.5) Concentration Constant-pressure specific heat Species diffusion coefficient Transition parameter, E = G*( ~ ~ / g . x - ' ) ~ / ' ' Disturbance frequency Gravitational acceleration Modified Grashof number, 4(Gr,/4)'/4 G(I N)'I4 Modified flux Grashof number, 5(Gr;/5)'lq Local Grashof number. $,(lot,)x3/ju2 Local flux Grashof number. g/jTq"x4/kv' Heat-transfer coefficient Thermal conductivity Lewis number, K I D GrV,

Advances in Applied Mechanics, Volume 22

Advances in applied mechanics

Advances in Applied Mechanics