Advances in Applied Mechanics, Volume 31

Advances in Applied Mechanics Volume 31 Editorial Board T. BROOKE BENJAMIN DEPARTMENT OF MATHEMATICS OXFORD UNIVERSXT...

Author: Theodore Y. Wu | John W. Hutchinson

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Advances in Applied Mechanics Volume 31

Editorial Board T. BROOKE BENJAMIN DEPARTMENT OF MATHEMATICS OXFORD UNIVERSXTY

OXFORD, UNITEDKINGDOM Y. C. FUNG

AMES DEPARTMENT OF CALIFORNIA, SAN DEW UNIVERSITY LA JOLLA, CALIFORNIA

PAULGERMAIN ACADEMEDES SCIENCES PARIS,FRANCE RODNEY HILL DEPARTMENT OF APPLEDMATHEMATICS AND THEORETICAL PHYSICS UNIVERSITY OF CAMBRmGE CAMBRIDGE, UNITEDKINGDOM PROFESSOR L. HOWARTH OP MATHEMATICS SCHOOL UNIVERSITY OF BRISTOL BRISTOL, UNITEDKINGDOM C.-S. Ym (Editor, 1971-1982)

Contributors to Volume 31 ROBERTOCAMASSA DARRYL D. HOLM KEH-CHIHHWANG JAMES M. HYMAN R. E. KELLY TURGUT SARPKAYA QING-PING SUN

ADVANCES IN

APPLIED MECHANICS Edited by John W. Hutchinson

Theodore Y. Wu

DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS

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

VOLUME 31

ACADEMIC PRESS, INC.

Boston San Diego New York London Sydney Tokyo Toronto

THISBOOK

IS PRINTED ON ACID-FREE PAPER.

COPYRIOHT

Q

@ 1994 BY ACADEMICPRESS, INC.

ALL RIOHTS RESERVED.

NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MGCHANICAL, INCLUDINO PHOTOCOPY, RECORDINO, OR

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United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1

7DX

LIBRARY OF CONORESS CATALOG CARD

NUMBER: 48-8503

ISBN: 0-12-002031-9

PRINTED IN

THE

UNITEDSTATES OF AMERICA

94 95 96 91 98 99 BC 9 8 7 6 5 4 3 2 I

Contents CONTRIBUTORS

vii

PREFACE

ix

A New Integrable Shallow Water Equation Roberto Camassa, Darryl D. Holm, and James M . Hyman 1

I. Introduction

3

11. The Green-Naghdi Equations

111. 1%'. V. VI. VII.

The Unidirectional Model Solution Dynamics Conservation Laws An lsospectral Problem for the Unidirectional Model Discussion Acknowledgments References

9 15

23 21 32 32 32

The Onset and Development of Thermal Convection in Fully Developed Shear Flows R . E. Kelly I. 11. 111. IV.

Introduction Rayleigh-Bhard Convection in Fully Developed Forced Flows Rayleigh-Blnard Convection in Fully Developed, Thermally Induced Flows Concluding Remarks Acknowledgments References

35 31 100 105

105 106

Vortex Element Methods for Flow Simulation Turgut Sarpkaya 1. Introduction

11. Theoretical Foundations and Numerical Schemes 111. Evolution and Applications of Vortex Element Methods

113 116 184

222 230 230

IV. Concluding Remarks Acknowledgments References V

Contents

vi

Micromechanics Constitutive Description of Thermoelastic Martensitic Transformations Qing-Ping Sun and Keh-Chih Hwang I. Introduction

249

11. Micromechanics and Thermodynamics of Thermoelastic Martensitic

Transformations 111. Energy Changes Accompanying Deformation Processes IV. Constitutive Relations V . Applications to Deformation of Polycrystalline Shape Memory Alloys VI. Concluding Comments Acknowledgments References

25 1 263 268 282 295 295 296

AUTHOR INDEX

299

SUBJECT INDEX

307

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

ROBERTOCAMASSA(l), Theoretical Division and Center for Nonlinear Studies, Los Alamos, New Mexico 87545 DARRYLD. HOLM(l), Theoretical Division and Center for Nonlinear Studies, Los Alamos, New Mexico 87545 KEH-CHJHHWANG(249), Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China JAMESM. HYMAN(l), Theoretical Division and Center for Nonlinear Studies, Los Alamos, New Mexico 87545

R. E. KELLY( 3 9 , Mechanical, Aerospace and Nuclear Engineering Department, University of California, Los Angeles, California 90024-1 597

TURGUT SARPKAYA(1 13), Department of Mechanical Engineering, Naval Postgraduate School, Monterey, California 93943 QING-PINGSUN (249), Department of Engineering Mechanics, Tsinghua University, Bejing 100084, China

vii

This Page Intentionally Left Blank

Preface This volume of the Advances in Applied Mechanics continues the tradition of the series by presenting expository articles on subject areas of fundamental importance in fluid and solid mechanics. The authors of these articles have taken a keen interest in delineating the recent advances in these areas. It is hoped that the elucidations of the theoretical, experimental, and numerical studies they have brought forth in their finest scholarly work will serve handsomely as valuable sources of stimuli, information, and references for years to come. The article by R. Camassa, D. D. Holm, and J. M. Hyman is a preview of a newly discovered model for nonlinear dispersive waves on shallow water. This equation is distinct in being biHamiltonian, possessing an infinite number of conservation laws, and in being completely integrable to form a new family of evolution systems. Of particular interest are its outstanding features in the formation of solitons with pointed peaks, and their retaining certain nonlinear signatures from elastic collisions of the N-soliton solution. R. Kelly’s article is a timely exposition of the basic concepts concerning the instability of Rayleigh-Benard convection in unsteady shear flow of stratified fluids, with possible augmentation by heat and mass transfer. Significant applications cover such diversified interests as cloud rows in the atmosphere, secondary circulation in the earth’s mantle, the technologies of electrochemical plating, production of electronic materials involving phase changes, etc. T. Sarpkaya presents a discussion of the flow phenomena of creation and transport of vorticity in viscous fluids bordered by solid and free boundaries. The article provides a broad coverage of the vortex methods used in current practice of computation for simulating vortical flows in general. Many examples are furnished to exhibit interesting results of vortical dynamics, and in some cases elucidating the underlying basic mechanisms that have become available with the advent of the computer era. ix

X

Preface

The article by Q.-P. Sun and K.-C. Hwang is primarily concerned with a continuum formulation of the macroscopic constitutive laws for phase-transforming materials. The authors’ present approach is based on a micromechanical description of the transformation of the microstructure, crystallography, and thermodynamical processes involved. Applications of the constitutive theory are shown for the individual phenomena of phasetransforming materials and shape-memory alloys. This is a scholarly work discussing the recent advances in this area, especially those in China. Theodore Y. Wu and John W. Hutchinson

.

ADVANCES IN APPLIED MECHANICS VOLUME 31

A New Integrable Shallow Water Equation ROBERTO CAMASSA. DARRYL D . HOLM. and JAMES M . HYMAN Theoretical Division and Center for Nonlinear Studies Los Alamos. New Mexico

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

1

I1. The Green-Naghdi Equations .................................................................. A . Background .................................................................................... B . Green-Naghdi Equations ................................................................... C . Green-Naghdi Equations in One Dimension ..........................................

3 3 6 7

111. The Unidirectional Model .......................................................................

9

IV . Solution Dynamics ................................................................................ A . Steepening Lemma ........................................................................... B . Traveling Wave Solution.................................................................... C . N-Soliton Solutions .......................................................................... D Two-Soliton Dynamics ...................................................................... E Phase Shifts ....................................................................................

15

. .

.

15 16 17 20 22

V Conservation Laws ................................................................................

23

VI . An Isospectral Problem for the Unidirectional Model ................................... A . Spectral Structure ............................................................................. B . The Isospectral Problem for the Extended Dym Equation ......................... C . A Spectral Problem for the N-Soliton Mechanical System.........................

27 28 29 30

.

VII Discussion ...........................................................................................

32

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

32

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

32

.

I Introduction

Completely integrable nonlinear partial differential equations arise at various levels of approximation in shallow water theory . Such equations possess soliton solutions-coherent (spatially localized) structures that interact nonlinearly among themselves and then reemerge. retaining their identity and showing particlelike scattering behavior . In this chapter. we 1 Copyright 0 1994 by Academic Press. Inc . All rights of reproduction in any form reserved.

ISBN 0-12-002031-9

2

Roberto Camassa et al.

discuss a newly discovered completely integrable dispersive shallow-water equation found in Camassa and Holm (1993), U, -k 2KU, - Uxxt -k

3UU, =

2U,U,

-k UU,,

(1.1)

where u is the fluid velocity in the x direction (or equivalently, the height of the water’s free sufrace above a flat bottom), K is a constant related to the critical shallow-water wave speed, and subscripts denote partial derivatives. Camassa and Holm (1993) introduce this equation, discuss its analytical properties, and sketch its derivation. The present chapter shows numerical results for this equation that illustrate the behavior of its solutions, with particular emphasis on the case K = 0. Equation (1.1) is obtained by using a small-wave-amplitude asymptotic expansion directly in the Hamiltonian for the vertically averaged incompressible Euler’s equations, after substituting a solution ansatz of columnar fluid motion and restricting to an invariant manifold for unidirectional motion of waves at the free surface under the influence of gravity. The equation retains higher-order terms in this expansion (the right-hand side) that correspond to higher-order conservation of the fluid energy. Dropping these terms leads to the Benjamin-Bona-Mahoney (BBM) equation, or at the same order, the Korteweg-de Vries (KdV) equation. This extension of the BBM equation possesses soliton solutions whose limiting forms as K + 0 have peaks where first derivatives are discontinuous. These solitons, called peakons because of their shape, dominate the solution of the initial value problem for this equation with K = 0. The evolution of a typical initial condition is shown in Fig. 1. There, an initially parabolic pulse steepens and eventually breaks into a train of peakons. These solitons travel with speed proportional to their height and remain coherent after dozens of collisions in the periodic domain. The way a smooth initial condition breaks up into a train of peakons is by developing a verticality at each inflection point with sufficiently negative slope, from which a derivative discontinuity emerges. Remarkably, the multisoliton solution of (1.1) is obtained by simply superimposing the single peakon solutions and solving for the evolution of their amplitudes and the positions of their peaks as a completely integrable finite-dimensional Hamiltonian system. Equation (1.1) is bi-Hamiltonian; i.e., it can be expressed in Hamiltionian form in two different ways. The sum of its two Hamiltonian operators is again Hamiltonian, and their ratio is a recursion operator that produces an infinite sequence of conservation laws. This bi-Hamiltonian property is useful in recasting the equation as a compatibility condition for a linear isospectral problem, so that the initial

A New Integrable Shallow Water Equation

3

FIG. 1. This space-time plot shows the evolution of the parabolic initial data u(x, 0) = max[O, 1 K

- O.Ol(x - lo)*] as it evolves between t

= 0 and t = 100 by Eq. (1.1) for

= 0 in the periodic domain [0, 1001.

value problem may be solved by the inverse scattering transform method .(Camassa and Holm, 1993). After briefly discussing the Boussinesq class of equations for small amplitude dispersiveshallow water equations, in Section I1 we derive the onedimensional Green-Naghdi equations (Green and Naghdi, 1976). In Section 111, we use Hamiltonian methods to obtain Eq. (1.1) for unidirectional waves. In Section IV, we analyze the behavior of the solutions of (1 .l) and show that certain initial conditions develop a vertical slope in finite time. We also show that there exist stable multisoliton solutions and derive the phase shift that occurs when two of these solitons coIlide. Section V demonstrates the existence of an infinite number of conservation laws for Eq. (1.1) that follow from its bi-Hamiltonian property. Section VI uses this property to derive the isospectral problem for this equation and others in its hierachy. 11. The Green-Naghdi Equations

A. BACKGROUND

Certain small-amplitude fluid flows in thin domains, e.g., internal waves in coastal regions, satisfy the shallow-water approximation, but not necessarily the hydrostatic pressure condition (Wu, 1981). Corrections to account for nonhydrostatic pressure effects have been developed by Peregrine (1967),

4


Green and Naghdi (1976), Wu (1981), and Camassa and Holm (1992). These authors use standard asymptotic perturbation theory to show that nonhydrostatic pressure effects cause additional wave dispersion. The equations they derive fall into the Boussinesq class of approximate dispersive equations for wave elevation q and mean horizontal fluid velocity u. The same Boussinesq tradition of approximations includes the KdV and BBM equations, when restricted to propagation in only one direction by, say, imposing a linear relation between elevation and fluid velocity. The structure of these equations has led to a reasonably complete understanding of the solutions at this level of approximation. In particular, the Kortewegde Vries equation admits solution by the inverse scattering transform method and, thus, allows a complete description of its nonlinear wave interactions. Here we go to higher-order approximations within the Boussinesq class, while retaining the Hamiltonian structure and associated conservation laws inherent in the starting equations, by directly inserting an asymptotic approximation in the Hamiltonian for Euler’s equations in three dimensions. We consider an inviscid incompressible fluid of uniform density with velocity components u = (u, v ) in the horizontal x = (x, y) directions, and w in the vertical (z) direction. The fluid is acted on by gravity and an external pressure and is moving in a domain with an upper free surface at z = [(x, y, t) and a prescribed, possibly time-dependent, bottom boundary at z = -h(x,y, t ) . The dynamics of such a fluid is governed by Euler’s equations, with 3D substantial derivative, d/dt = a/at + u V + w a/&,

-=--( dw dt

) z+pg ’

1 ap p

where p denotes the fluid’s uniform density, g is the constant acceleration due to gravity, and p is the fluid pressure. Incompressibility implies the fluid velocity is divergenceless: aw v‘U += 0. az The kinematic boundary conditions appropriate for such an inviscid fluid are w=[

w = where 4 = dc/dt =

-A

at z = C(x,y, 0 , at z = - h ( x , y , t ) ,

(2.3)

+ u - Vc and u is tangential on any vertical lateral


5

boundaries (free-slip). The dynamic boundary condition is (neglecting surface tension)

P

=P

at z

=

C ( X , Y , 0,

(2.4)

where $(x, y , t ) is the prescribed external pressure. Euler’s equations have several fundamental properties that are worth preserving when making further approximations. First, they are the EulerLagrange equations for a constrained action principle that is stationary under arbitrary variations of the Lagrangian fluid labels. Second, passage from the Euler-Lagrange description in terms of Lagrangian fluid labels to the Hamiltonian description in terms of Eulerian fluid velocity leads to a Lie-Poisson bracket. Third, these equations possess a Kelvin theorem that is related to the advection of potential vorticity and that leads to an infinity of conserved quantities. These conserved quantities are associated with particle relabeling symmetry in the Lagrangian picture and the corresponding degeneracy of the Lie-Poisson bracket in the Eulerian picture. For discussions of the interrelationships among these properties, see Abarbanel and Holm (1983, Holm (1985), and Miles and Salmon (1985). In this section we will discuss how to preserve these three properties-action principle, Hamiltonian structure, and infinity of conservation laws-when making further approximations, particularly when restricting to columnar fluid motion in vertically thin domains. Our approach is to use the principle of generalized coordinates to make approximations directly in an action principle for Euler’s equations, by choosing a simplifying ansatz for the form of the solution before taking variations. Just as in the case of an ordinary constraint, approximations that restrict the form of the solution (and, thus, the class of allowed variations) typically change the equations of motion, and so the accuracy of the approximate dynamics obtained this way must be verified by some other means. In the case at hand, the solution ansatz we choose when substituting the simplified form of the solution into the action principle is obtained from a balance in the Euler equations at first order in an asymptotic expansion of the solution in powers of the thin-domain aspect ratio. (See. e.g., Peregrine (1967) or Camassa and Holm (1992). The solution ansatz arising from this balance corresponds to columnar motion of the fluid in a thin domain. Euler’s equations (2.1) follow from an action principle 6d:= 0, with


6

where D = det(0;"), where 0;" = (NA/ax')is the 3 x 3 Jacobian matrix for the map from Eulerian coordinates to Lagrangian fluid labels, IA(x, z , t ) , A = 1,2,3. These Lagrangian labels specify the fluid particle currently occupying Eulerian position (x, ,x, ,x g ) = (x, z ) . They satisfy the advection law, 0 = dlA/dt = alA/at + uiO;", thereby determining the velocity components ( u l , v z , uj) = (u, w) in the action principle as ui = -(D-'); aiA/at,

i

=

i,2,3.

(2.6)

Variations in (2.5) with respect to IA yield Euler's equations (2.2) with kinematic boundary conditions (2.3). The constraint D = 1 imposed by the Lagrange multiplier p (the pressure) implies incompressibility. For more details, see Abarbanel and Holm (1985), Holm et al. (1988), and Miles and Salmon (1985).

B. GREEN-NAGHDI EQUATIONS By using conservation of energy and invariance under rigid-body transformations, Green and Naghdi (1976) derive an approximate form of Euler's equations appropriate to columnar fluid motion in vertically thin domains. Miles and Salmon (1985) reecover the Green-Naghdi equations from an action principle, by restricting the action principle (2.5) to variations essentially of the form (columnar motion ansatz)

t),

[A

= lA(X,

13

=-

A = 1,2,

z + h c + h'

A = 3,

from which (2.6) implies

u

= U ( X , t ) = i,

w

=

-A

- (Z

+ h)V

U.

The Green-Naghdi equations are all _ - -v

at

qu,

au _ - - u . V u - g V ( t l - h ) + - V 1A at

tl

1 Vh, --B tl

(2.8)


7

where the quantities A and B are given by (2.10)

with, e.g., i j = d 2 q / d t 2 . The Green-Naghdi equations are also rediscovered in Bazdenko et al. (1987) and are derived directly from the Euler equations (2.1) in Wendroff (1992) by substituting the ansatz (2.8) into (2.1) and integrating in z. The Green-Naghdi equations conserve the energy HGN=

.i

d x d y [ t p 2+

t/h2 + q2h(V u ) + ~ g(q - I z ) ~ ] , (2.1 1 )

which may be obtained from the Euler energy,

by substituting the solution form (2.8) for w and explicitly performing the z integration. Holm (1988) observes that the Green-Naghdi equations (2.9) may be expressed in Lie-Poisson Hamiltonian form when the energy HGN is taken as the Hamiltonian.

C. GREEN-NAGHDI EQUATIONS IN ONEDIMENSION We now specialize the Green-Naghdi equations (2.9) to the case of one spatial dimension and constant bottom topography, h = h, = const. Namely, Vr

+ (w),= 0, (2.13)

with conserved energy

H I D= A 2

1

[

dx qu2

I

+ 1 q3u,2 + g(q - hJ2 .

(2.14)

Equations (2.13) are expressible in Lie-Poisson Hamiltonian form in terms of Hamiltonian H , , and dynamical variables q and m, the latter of which is given in one dimension for flat bottom topography by (2.15)


8

In terms of the variables 17 and my the Green-Naghdi equations are expressible in Lie-Poisson Hamiltonian form as

where the variational derivatives are given by the coefficients of 6m and 6q in SHl, =

f

dx [U 6m

+ (-it?

- itj&:+ g(tl - h,)) &].

(2.17)

jch,

Note that had we modeled w 2dz for the kinetic energy due to vertical motion in (2.14) by h,q:/3 = ho(qu):/3 instead of by q3u,2/3,the equations resulting from the Lie-Poisson Hamiltonian form (2.16) would have been the Boussinesq equations (Whitham, 1974): flt

+ (flu),

=

0, (2.18)

It so happens these equations also rise in an asymptotic expansion of the Green-Naghdi equations (2.14) in terms of the small parameters E = ho/L (the thin-domain aspect ratio) and Q in q = h, + CUT (the small wave amplitude), when the balance Q = 0 ( c 2 ) is assumed, and dimensional scales are taken as u + Q ~ U x -+, Lx, and t + tL/&&. From the Boussinesq equations, further asymptotics and restriction to unidirectional propagation in a frame moving near the critical wave speed co = leads to the Korteweg-de Vries (KdV) equation (Whitham, 1974),

a

ut + C,U,

+ $uux + &,hiuxxx

= 0,

(2.19)

or, with the same order of accuracy in the thin-domain expansion, the Benjamin-Bona-Mahoney (1972) (BBM) equation, U,

+ COU, + $uux - ihiuxxt= 0.

(2.20)

In contrast to making asymptotic expansions in the equations of motion, as in the derivations of the KdV and BBM equations, our approach is to make approximations in the Hamiltonian (2.11) that produce unidirectional propagation and preserve the momentum part of the Lie-Poisson structure (2.16).


9

111. The Unidirectional Model

In this section, we make a unidirectional approximation in the GreenNaghdi Hamiltonian system that relates m and q , but preserves the momentum part of the Lie-Poisson structure (2.16). We begin by noticing that I/& is in the kernel of the operator ma + a m :

(ma

+ am) z 1= - a Jmxm + a 6 =

0.

Using this and (2.16),the time evolution of the functional C is given by

=

I?:&&

where we have performed an integration by parts. Thus, if q = const 6, the functional C is a constant of motion, and the integral manifold j?: t] dx = const:!j 6dx is invariant under the motion generated by the Lie-Poisson structure (2.16) for any Hamiltonian. The constant in this relation between m and q is chosen to give the right dimensions. We will set

and because t] ho as 1x1 00, the boundary conditions on m will be assumed to be m hoco,as 1x1 00. The functional C is the Casimir for ma + am and so we will refer to the manifold (3.2)as the Casimir manifold for (2.16). We now restrict to the manifold (3.2). The Hamiltonian energy (2.11) becomes -+

-+

-+

-+

(3.3) The term proportional to the Casimir can be ignored in this expression, because on the Casimir manifold (3.2) only the momentum part of the Hamiltonian operator in (2.16) needs to be considered, and the Casimir C is in the kernel of the Hamiltonian operator ma + am. Rearranging the


10

constant term in order to assure convergence of the integral yields

Expression (3.4) for the Hamiltonian and the relation (2.15) provide an implicit definition of m in terms of u, which we are not able to make explicit. We can, however, find an explicit approximate expression of m when working in the small amplitude regime. We scale u + au and look for m in the form

m = hoco+ aml

+ a 2m2 + a 3 m3 +

(3.5)

Truncating at O(a3),the Hamiltonian becomes

H I D= -1 2

1[ +-

-m

a2(hou’

+7 hi u x ) + -1 a ’(: m1u2 + h: m, u;)] 2

dx

CO

(3.6) By definition, m is the variational derivative of the Hamiltonian with respect to u (as in (2.15)), and so we must have the consistency condition

(3.7) where D: denotes the adjoint of the FrCchet derivative of m with respect to u. Because we seek an evolution equation for m that retains terms up to O(a), we only need to completely determine the form of rn, . Notice that divergence terms (perfect derivatives) in the expressions of m2 and m3 can be ignored, since these terms enter the Hamiltonian (3.6) linearly at O(a2) and O(a3),respectively.

A N e w Integrable Shallow Water Equation

11

The consistency condition at order O(1 ) through @a3) leads to the following expressions for m, , m 2 , and m 3 :

m,

=

m2 = -hou

2h0u - aluxx,

qa22c,

m3 = -7 ho u3 + 3c0 CO

-

2

+ a24,

CO

-

( 3 * 8)

2c, 4

I

where a, and a2 are two undetermined coefficients. With these expressions for m i , i = 1 , 2 , 3 , the Hamiltonian (3.6) (truncated at order O(a3))can be rewritten as HID= r?, + r?2, (3.9) where

(2hou2 + a,u:)dx

(3.10)

and

2

+ a2c0 - a,):

dx

+

-m

(3.11) The equation of motion we are seeking for m up to order a , m = hoco + a m l , can now be determined by r?,. Keeping in mind that when restricting to a submanifold the flow generated by the restricted Hamilitonian rescales time by a factor 2 (Olver, 1988), the Hamiltonian (3.10) must be rewritten as

(3.12) The approximate equation of motion for m on (3.2) is therefore

m,

=

-(ma

dk + am)--. 6m

=

a 2

--(ma

1 + am)u - -cornx.

2

(3.13)

e2

We now fix the coefficients a, and a2 by requiring that is also conserved by the flow (3.13). After some algebra, this leads to the following linear system determining a, and a2: h i + coa, - a, -

3

=

a, --

3'

a1 2a2 = -;

3c0

(3.14)


12

so that 2

a, = j h i ,

a2 =

-.h i

9c0

(3.15)

The final expression for m up to O(a)is therefore

m = h,co

+ am, = h,c, + a(2h0u - $hiu,,),

(3.16)

and in terms of u the equation of motion (3.13) becomes U, -

~ h i u , , , + c0u,

+ $ ~ u u -, ~ h ~ c o ~ =, , ~, a h ~ U , U , +, ~ a h i u u , . (3.17)

Dropping the terms on the right-hand side of this equality gives a BBMlike equation, cf. (2.20). Thus, (3.17) can be seen as the BBM equation corrected by retaining higher-order terms (selected by the Hamiltonian approach) in an asymptotic expansion in terms of the small parameter a. Since the extra terms are quadratic in u, the linearized version of (3.17) has the same dispersion relation w(k) as for a BBM equation written in a frame moving with velocity c0/2. Substitution of the mode eikx-iOc into the linearized equation yields 1 + k2h;/6 (3.18) w=ck 1 + k2hi/3' As argued in Benjamin et ai. (1972), dispersion relations of this kind are preferable to the KdV dispersion w = cok(l - hik2/6), as the large k waves do not propagate with unbounded phase speed. On the other hand, in the long-wave limit hok 4 0, (3.18) coincides with the dispersion relations for KdV, BBM, and Green-Naghdi, as well as for the full linearized shallow-water wave problem, w = dgk tanh hok. Figures 2 and 3 show the comparison among the phase speeds o / k and the group velocities dw/dk, as functions of hok for Eq. (3.18), linearized water waves, BBM, and KdV. The BBM relation and Eq. (3.18) bracket the water wave dispersion relation for all wave numbers. Notice that, unlike the usual derivations of the KdV and BBM models (Whitham, 1974), Eq. (3.17) is obtained through an asymptotic expansion in only one small parameter, a, the amplitude of the wave elevation. Of course, the columnar motion ansatz (2.8) is physically a good approximation for wavelengths that are large compared with the undisturbed water depth, and so a balance between the small (shallow-water) parameter E and the amplitude parameter (Y is implicit throughout the derivation of the GreenNaghdi equations (2.14) as well as in the present derivation of (3.17).


13

Eq 3.18

0 k 2.

ww BBM

1.5.

KdV 1

0.5

w

1.5

2

FIG. 2. Comparison among the phase speeds w / k for Eq. (3.18), linearized water waves (WW), BBM, and KdV.

The restriction to the Casimir submanifold (3.2) is equivalent to the unidirectionality assumption in the usual derivations of the KdV (2.19) and BBM (2.20) models from the Boussinesq system (2.19) (see, e.g., Whitham, 1974; Olver, 1984). In fact, using (3.16) and expanding (3.2) gives

4,

=

fj [ p u-

+ O(a).

(3.19)

Notice that in a long-wave, thin-domain approximation the double derivative term in (3.19) would acquire a factor E* = O(a),and so at leading u. order (3.19) is simply [ =

a

do

dk

31 2. 1. -1.

-2 -3

'

'

FIG. 3. Comparison among the group velocities dw/dk for Eq. (3.18), linearized water waves (WW), BBM, and KdV.


14

Rescaling Eq. (3.17), dropping a, and going to a frame of reference moving with speed 2~ = c0/2 reduces the equation to the form U, -

+ U U , + ~ K U =, - 2 ~ 4 ,+ ~u,u,, + UU,,,,

u,,,

(3.20)

which is the standard form we will use from now on. Notice that (3.20), like BBM, is not Galilean invariant, i.e., not invariant under u + u + K ' , t + t , x -,x + K't. Thus, Eq. (3.20) is best seen as a member of a family of equations parameterized by the speed K ' of the Galilean frame. Equation (3.20) may be rewritten in nonlocal form as u,

+ uu, + K

Sr

~ ' y e - I ~ - ~ l u=, ,-

dye-l"-YI(uu,,+ &,u,,,,),

(3.21)

by using the identity ( 1 - a2)e-Ixl= 24x1.

(3.22)

In this form, dropping the quadratic terms on the right-hand side of the equation gives the equation studied by Fornberg and Whitham (1978).The similarity between the Fornberg and Whitham equation and the present case (3.20) is even more apparent when the Fornberg and Whitham equation is written in the local form u, - u,,,

+ uu, + 2KU, = 3u,u,, + uu,,,.

(3.23)

Fornberg and Whitham show that traveling wave solutions of this equation have a peaked limiting form. Moreover, asymmetric solutions can develop a vertical slope in finite time. Recently, Rosenau and Hyman (1992) have investigated a similar nonlinear dispersion equation, namely, u,

+ uu, = -3u,u,

- uu,., = -+(u2),,,.

(3.24)

This equation has traveling wave solutions that interact almost elastically and have compact support. In what follows, we will concentrate on the scaled form (3.20) of Eq. (3.17).We will consider the initial value problem with u defined on the real line with vanishing boundary conditions at infinity and such that the (rescaled) Hamiltonian +(3.25) is defined (bounded).


15

In accordance with (3.13), H I generates the flow (3.27) through

m The limit K

=u =

m, = - [(m + K ) a

- u,,,

+ a(m + K)] -.au, am

(3.26)

0, u, - u,,, = -3uu, - -a(tU2

+ 2u,u,, + uu,, - 3;.

-

~ ~ x x )

(3.27)

although unphysical (since it corresponds to zero wave speed), is of particular mathematical interest and will be given special attention through the next section.

IV. Solution Dynamics This section discusses the evolution of solutions to (3.20) when K = 0. In this case, an inflection point with sufficiently negative slope will develop verticality in finite time. This singularity leads to a traveling wave with discontinuous derivative at its peak. The traveling wave solution computed for Eq. (3.20) also shows explicitly, in the limit K + 0, that the profile acquires a corner at its peak. We show that the N-soliton solution can be expressed as a superposition of these peaked traveling waves with time-dependent amplitudes and phases. We also give the closed-form solution for the two-soliton dynamics and compute the phase shifts for a binary collision. A. STEEPENING LEMMA

We now show that initial conditions exist for which the solution of Eq. (3.27) develops a vertical slope in finite time. Let us assume that the initial condition is such that it has an inflection point at x = 2,to the right of its maximum, and it decays to zero in each direction sufficiently rapidly for Hl in (3.25) to be finite. Consider the evolution of the slope at the inflection point. Define s as u,(f(t),t ) . Then (3.21) (with K = 0) yields an equation of evolution for s (using uxx(f(t), t ) = 0),

ds dt

+m

2


16

Integrating by parts leads to

I

1

- -s2 2

+ -21 u2(2, t ) .

(4.2)

Then provided u2(2,t ) remains finite, say less than a quantity M, we have ds -5 dt

which implies for s initially

1

--s2

2

+M -,

(4.3)

2

-a,

I

sI acoth(o

+ id%),

(4.4)

where o is a negative constant that determines the initial slope, also negative. Hence, at time 2 = -2ad%, the slope becomes vertical. The assumption that M in (4.3) exists is verified in general by a Sobolev inequality. In fact, M = 2 H , , since max[u2(x, t)] xsR

I

s”

d x ( u 2 + u,”) = 2H, = const.

(4.5)

-

Remark. If the initial condition is antisymmetric, then the inflection point at u = 0 is fixed and d.T/dt = 0, due to the symmetry (u, x) ( - u , -x) admitted by (3.27). In this case, M = 0 and no matter how small Is(0)) (with s(0) < 0) vertically s = -a develops in finite time.

The steepening lemma indicates that traveling wave solutions of (3.27) may not have the usual sech-like shape since inflection points with negative slope lead to unsteady changes in the shape of the solution profile. By a similar argument, the development of verticality in finite time also occurs for K # 0.

B. TRAVELING WAVESOLUTION We seek solutions of (3.27) in the traveling wave form u(x, t) = U(x - c t ) , with a function U that vanishes at infinity along with its first and second derivatives. With these boundary conditions, substituting U in (3.20) and integrating twice yields C-2K-u

(u’)2= u2 c - u

’


17

where primes denote differentiation with respect to x - ct. The usual interpretation of the right-hand side of (4.6) as a potential energy term shows that solitary waves exist only for c 1 2 ~ i.e., ; they travel at supercritical speed, and their amplitude is given by

urn,,= C - 2K.

(4.7)

Integration of (4.6) shows that the function U is defined implicitly by e-(x-ct)

v-c - (v +

v + l

-

(4.8)

J ( i T ) Y

where 7

C C -

(4.9)

2K

and v is related to U by (4.10) In the limit of small-wave amplitudes and so, by (4.7) of near-critical speeds, c - K + 0, so that C -, 00. Equation (4.8) in this limit reduces to

u = (C - K ) sech'[

d

F

(

X -

Et)]

4- o[(C- K ) 2 ] ,

(4.11)

i.e., the same limit form of the solitary wave solution of the Green-Naghdi system (2.13). In the opposite limit of K 0, the curvature of U at its maximum increases and U becomes +

u = ce-lx-ctl + O(Klog K ) .

(4.12)

Indeed, Eq. (4.6) at K = 0 reduces to

(U'-C)[(uy- U2]= 0,

(4.13)

and so the solution (4.12) can be seen as the composition (vanishing at infinity) of the two exponentials that satisfy (4.13). The limiting solution (4.12) travels with speed c and has a corner (that is, a finite jump in its derivative) at its peak of height c.

C . N-SOLITON SOLUTIONS Motivated by the form of the traveling wave solution (4.12), we make the following solution ansatz for N interacting peaked solutions: N

u(x, t ) =

C p i ( t )e-lX-qi@)l. i=

1

(4.14)

Roberto Carnassa et al.

18

Hence, N

m(x, t)

= (1

- a2)u = 2

C p j ( t ) a [ -~ qj(t)I, j=

(4.15)

1

and the peaks in u are delta functions in rn. Substituting (4.14) into (3.27) and using the identity (3.22) yields evolution equations for qj and pi: N

(ii =

C p.e-1qi-qj1, J

j= 1

(4.16)

N

P i = Pi

C Pj sgn(qi - qj) e -1I7i-qj1

j= 1

These equations are Hamilton’s canonical equations, with Hamiltonian HA given by substituting the ansatz (4.14) into the integral of motion H I in (3.25), yielding H~ =

+

N

C

p.p.e-1qi-qjl I f = +HI

IN=sofiton.

(4.17)

i,j= 1

Hamiltonians of this form describe geodesic motion. The peak position qi(t) is governed by geodesic motion of a particle on an N-dimensional surface whose inverse metric tensor is gij(q)

=

e-lqi-qjl,

q

E RN.

(4.18)

The metric tensor gii is singular whenever qi = q j . For the case N = 2, the Gaussian curvature of this surface is (4.19)

This two-dimensional surface is convex (negative curvature) with a peaked ridge along q1 = q2,and it is asymptotically flat. The geodesic dynamics on this surface is completely integrable, since the corresponding two degree of freedom Hamiltonian system (4.16) possesss two functionally independent constants of motion. We will show that the system (4.16) is completely integrable for any N, thereby justifying the term “N-soliton” solutions for (4.14). We integrated equation (3.20) numerically with a variable-order, variabletime-step Adams-Bashford-Moulton method. The spatial derivatives were approximated by a pseudospectral discrete Fourier transform. We monitored the conservation laws and varied the accuracy of the integration method


19

between and lo-’ per unit time and the number of spatial modes was varied between 256 and 1024 to ensure the solutions were well converged. We define the initial conditions for the calculation shown in Fig. 4 to be the sum of solitary waves with velocities 1.O, 0.5, and 0.25 centered at x = -15, 0, and 15 in a periodic domain between -25 and 25. The space-time contour plot illustrates the robust nature of the solitons and the phase shift caused by the collision. Note how the slow soliton (c = 0.25) is shifted more forward in the collision with the fast soliton (c = 1) than it is when colliding with the medium-speed soliton (c = 0.5). Also note that the c = 0.5 soliton is shifted back when it collides with the c = 1 soliton. These phase shifts are calculated explicitly in the next subsection. We have

X

X

FIG.4. The initial conditions are solitary waves with velocities 1.0, 0.5, and 0.25 centered at x = -15, 0, and 15 in a periodic domain between -25 and 25. This space-time plot of the dynamics of the solution demonstrates the robust nature of the solitons. Note the phase shift in the position of the peakon after a collision.


20

performed similar numerical experiments with up to five solitons colliding simultaneously and shown that the solitons remain intact after hundreds of collisions. D. TWO-SOLITON DYNAMICS Consider the scattering of two solitons that are initially well separated and have speeds c1and c2,with c1 > c2 and c1 > 0, so that they collide. The Hamiltonian system (4.16) governing this collision possesses two constants of motion, Ho and H A , expressed in terms of the canonical variables as

Ho = P1 + P2 = c1 + c2,

H,

= +(pf

+ p:) + plp2e-1q1-qzl = ) c,. The position of the peak at intermediate times 1 is +

ql(t) =

CI

t

+

10g[4~(~1 - c , ) ~ ]- log[yC(c'-cl)t + 4~:],

(4.30)


23

for solition 1, and

qZ(t) = czt - flOg[4y(C, -

CZ)’]

+ 10g[ydCL-E2)‘ +

(4.31)

for solition 2. In the limit t + -m these formulas show that the solitons exchange their asymptotic speeds, or equivalently, their momenta and amplitudes, as

Thus, as 1 -P +a the solitions reemerge unscathed, the faster (and larger) soliton ahead of the slower (and smaller) one. The only effect of the interaction is exhibited in the asymptotic positions of the peaks, which are shifted from the positions they would have occupied had no interaction taken place. Defining the phase shift for the fast solition (“1” as t -+ -a)to be A4f

= qz(+m) - q d - m ) ,

and similarly for the slow solition (“2” as t A48

-P

-a),

= % ( + a ) - qd-m),

we then have

These formulas show that when cl/c2 > 2 both solitons experience a forward shift. For 1 < c,/c, < 2, the faster soliton is shifted forward while the slower soliton is shifted backward. The case c,/c, = 2 is the turning point where no shift occurs for the slower soliton (see Fig. 6).

V. Conservation Laws We consider solutions of Eq. (3.27) (K = 0) defined on the real line that The case K Z 0 follows in a similar vanish at infinity with bounded H1. manner. In the case K = 0, (3.27) has a number of extra conservation laws. Because of the conservation form, the total momentum,

s

u, = dxm,

(5.1)

is clearly conserved. Also, by construction (3.27) conserves the Hamiltonian


24

- 20

- 10

0

10

20

X

FIG.6. This space-time contour plot shows the evolution and phase shift when two peakons with speeds 1 and 2 collide. In this situation, the slower soliton does not experience any phase shift.

H I in (3.25), and the Casimir for the Lie-Poisson bracket, H-, =

S

dX&.

The Casimir H - I is distinguished by its property of Poisson commuting under the bracket defined by the Hamiltonian operator ma + dm with any functional of only the momentum density m . In seeking additional conservation laws, it is helpful to notice that Eq. (3.27) follows from an action principle, 66: = 0, with

because variation with respect to 4 produces (3.27) with u replaced by 4,. The Hamiltonian formulation using the momentum canonically conjugate to 4,


25

gives the following Hamiltonian for the 4 dynamics:

The canonical Hamiltonian dynamics is

The last expression defines a second Hamiltonian structure for Eq. (3.27). The two Poisson structures B,

=

a - a3,

B2 = am

+ ma,

(5.7)

with

are compatible. That is, their sum (or any other linear combination) is still a Hamiltonian operator (see Olver, 1986). This means Eq. (3.27) is bi-Hamiltonian and, therefore, has an infinite number of conservation laws. These laws can be obtained by defining the transpose recursion operator (RT = B;'B2, which leads from one conservation law to the next, according to

The operator (RT defined this way recursively takes the variational derivative of H - to that of Ho , to that of H I , and then to H 2 . The next steps are not so easy, however, since each application of the recursion operator introduces an additional convolution integral into the variational derivative of the next constant of motion in the sequence. Correspondingly, the recursion operator (R = B2B;' leads to a hierarchy of commuting flows, defined by Kn+ = (RK, ,

,

n = 0, 1,2, ....

(5.10)


26

The first three flows in the hierarchy are

mj2) = - m,

mi') = 0,

mi3) = -(ma + am)u,

3

(5.1 1)

the third equation being (3.27). The fourth flow in the hierachy, K 4 , is written in terms of u as

mi4) = u, - u,

=

(u - u,)

S" c +m

e-1X-YI[3uuy- 2u,u,

+ +(ux - u,,,)

- uuWy]dy

e-lx-yI[Au2 2 - iu," - uuyy]dy.

J -52

(5.12)

By construction, this cubically nonlinear flow commutes with the other flows in the hierachy, and so it also conserves H - 1 , H,, Hl ,H 2 , and so on. The recursion relation (5.10) can also be continued for negative n. The conservation laws generated this way do not introduce convolutions, but care must be taken to ensure that the conserved densities are integrable. All of the Hamiltonian densities in the negative hierarchy are expressible in terms of m only and do not involve u. Thus, for instance, the second Hamiltonian in the negative hierachy is given by

(5.13) which gives

(5.14) The integral in this expression may not exist for solutions m of (3.27); however, the analog of H - , ,n = 2 , 3 , ..., seems well defined for the family of equations (3.20), obtained by replacing m with m + IC in the integrand and subtracting an appropriate constant for convergence as x +a. +

Remark. The flow defined by (5.13) is very similar to the Dym equation (Ablowitz and Segur, 1981), the only difference being the presence of an extra spatial derivative, (5.15) The consequences of adding the derivative 8 to this known completeiy integrable Dym equation are worth some extra investigation. An indication that this term can be very important will be given in the discussion of the commutator form of (3.27).


27

VI. An Isospectral Problem for the Unidirectional Model This section expresses Eq. (3.27) as the compatibility condition between

a time-independent Sturm-Liouville spectral problem for an eigenfunction v(x, t) and an equation of evolution for this eigenfunction. We seek the spectral problem associated with (3.20) by using the recursion relation of the bi-Hamiltonian structure, following the Gel'fand and Dorfmann (1979) technique. Let us introduce a (spectral) parameter A and multiply by A" the nth step of the recursion relation (5.10). Treating both sides of the recursion relation as terms of a power series and formally summing the series gives

or, by introducing yz(x, t;A)

=

B , #(x, t; A)

=

c

n=-l

A"-

6H"

6m'

we have ABzy/2(~, t ; A).

(6.2)

This equation constitutes a third-order eigenvalue problem for the eigenfunction $, which can be reduced to an ordinary Sturm-Liouville secondorder problem. It is easy to show that if v satisfies

then v/" is a solution of (6.2). Now, assuming A is independent of time, we seek, in analogy with the KdV equation, an evolution equation for r// of the form

vr = avx + bv,

(6.4)

where a and b are functions of u and its derivatives to be determined by the requirement that the compatibility condition vxxt = vtXx between (6.3) and (6.4) implies (3.27). Cross differentiation shows that b = -*ax

and a = -(A

+ u),

(6.5)

and so

vt = -(A + u)vx + +uxv is the desired evolution equation for ty.

(6.6)


28

Remark. The spectral problem for the family of equations (3.20) is simply obtained by replacing m with m + K in (6.3), while the time evolution equation (6.6) remains the same.

A. SPECTRALSTRUCTURE If m vanishes at x = +00 sufficiently fast for H I to be bounded, then the spectral problem (6.3) has a purely discrete spectrum. In fact, the limiting behavior of v/ is w(x)

zm efX?

(6.7)

which implies that the eigenfunctions always decay exponentially at infinity. For instance, if the initial condition u(x, 0) is chosen such that ex - 2 sinh x arctan(ex) - 1

i.e., m(x, 0) = A sech2(x),

(6.8)

for an arbitrary constant A, then it is easy to show (Camassa and Wu, 1991) that the eigenvalues 1 for (6.3) are given by

In =

(2n

+

2A 1)(2n

+ 3)'

n = l,2,

... .

(6.9)

This formula shows explicitly that I = 0 is an accumulation point for the discrete spectrum and the eigenvalues converge to it as l/nz, n + 00, a fact that can be shown to hold in general for an initial condition decaying exponentially fast at infinity. Notice that as soon as K # 0, i.e., for an equation in the family (3.20), the v/ limiting behavior becomes

(6.10) and so a band of continuous spectrum emerges out of the origin in the interval 0 II I2 ~ We . remark that the peculiar feature of the disappearance of continuous spectrum in the limit K --* 0 is essentially caused by the presence of the constant 1/4 in (6.3), which in turn is generated by the first derivative operator in B 1 . In Fig. 7 we see that initial data given by (6.8) breaks into a train of solitons.


29

(a) 0.75 T

i7

0.0 -10

50

X

(b) 0.15

I=

-10

40

X

30

FIG. 7. The initial data u(x, 0) = ( n / 2 ) 6 - 2 sinh x arctan($) - 1 breaks into a train of peakons as it evolves by Eq. (3.26).

B. THEISOSPECTRAL PROBLEM FOR THE EXTENDED DYMEQUATION The eigenvalue problem (6.3) is also isospectral for the extended Dym equation (5.15), since this equation belongs to the same hierarchy (5.10) of flows as Eq. (3.27). The appropriate time evolution law for iy can be found in a similar fashion as for (6.5). We look for iyt defined by (6.4) and notice that in general the evolution equation for rn, produced by the compatibility


30

condition

wxxr = tytxx,is m, = (B2 - LB,)a,

b = -+ax

+ const.

(6.11)

Hence, it is easy to see that the choice (6.12)

reproduces the desired evolution equation. Thus, (5.15) is the compatibility condition between (6.3) and (6.13)

C. A SPECTRAL PROBLEM FOR THE N-SOLITON MECHANICAL SYSTEM

For the N-soliton solution (4.14) of Eq. (3.27), m(x, t ) becomes simply a sum of delta functions: N

mOc, t ) = 2

C pi(t)6[x - qi(t)I* i=l

(6.14)

Rewriting (6.3) in integral form as Ay/(x, t ) =

+S

+m

e-lx-y1/2m(y, t)y/(y,t ) dy

(6.15)

-w

and substituting the expression (6.14) for m yields N

lw(x, t) =

C pje-Ix-qjl’2 j = 1

W(qj, t),

(6.16)

which in particular implies N

dW(qi, t ) =

1

pje-lqi-qjI/z

Y(qj t ) d ~ * 9

(6.17)

j= 1

This expression constitutes an algebraic eigenvalue problem for the eigenvector

i.e., (6.18)


31

with the matrix ~ U. . ( t ) pje-lqi-qi1/2

(6.19)

or L = PQ,

p

diag[pJ,

Q1J. .= -

e-IQi-qh/2.

(6.20)

The evolution equation for \Y can be obtained directly from (6.6): (6.21)

where the matrix A is given by n

A, =

+ C (sgn(qi - qk)e-I~j-9k1/2

P k Lkj

k= 1

+ [sgn(qi - qj) - sgn(qi - q k ) ] e - I q i - Q j 1 / 2 e - I q i - q ~ ~ j p(6.22) k). The evolution equation (6.21) and the eigenvalue problem (6.18) imply that L evolves according to

LtL = AL - LA = [A,L],

(6.23)

which shows that constants of motion can be generated by taking the trace of powers of L : d -(TrL") = 0. dt For instance, the first two constants of motion are Tr L = Ey= pi = Ho and Tr L2 = H A . The eigenvalue problem (6.18) explicitly shows that, in analogy with the two soliton case, soliton overlap (qi = qj at some time t = T)can only occur if the corresponding momenta p i , p j diverge to infinity. In fact, N

det L(t)

=

n&

= const.

(6.24)

k= 1

and the eigenvalues Ak are determined by the asymptotic behavior of the N-soliton solution for t + - 00 : N

N

detL

pk(-a) =

P(-OO)Q(-a) = k= 1

fl c k = k=l

n N

Ak.

(6.25)

k=l

Now, when qi(t) -,qj(t) (for t T), det Q + 0, and so (6.20) and (6.24) imply pk(t) a0 in order to keep det L = const. Time invariance of the Hamiltonian (4.17) then shows that [pi[ + ) p j )+ 00 withp,pj < 0.

nf=,

+

-+

32


VII. Discussion We have derived the model equation (1.1) for dispersive shallow-water motion, under the assumption of unidirectional motion and using an asymptotic expansion directly in the Hamiltonian for Euler’s equations. This model equation has a number of remarkable properties. It is bi-Hamiltonian, and hence it possesses an infinite number of conserved quantities that are in involution and are recursively related. This implies the equation is completely integrable and has other properties (e.g., Lax pair and inverse scattering transform) associated with complete integrability for other soliton equations, such as the Korteweg-de Vries equation. In the present chapter, for the case K = 0, the N-soliton solution for this equation has been introduced, the two-soliton scattering process has been analyzed, and the phase shift for soliton-soliton collisions has been computed. The soliton-antisoliton collision exhibits some interesting behavior, especially its amazing recovery from nearly complete annihilation. The steepening lemma for this equation in the case K = 0 shows that any sufficiently negative slope at an inflection point will reach vertically in a finite time. In particular, a localized initial distribution evolves to develop verticality and then breaks up into a height-ordered train of peaked solitons moving to the right, with the tallest ones ahead. The numerical studies confirmed the central role of these peakons in the dynamics of the solution. Like the solitons for classic integrable equations, these solitons develop from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after colliding with other solitons.

Acknowledgments For their helpful comments during the course of the preparation of this chapter, we thank M. Ablowitz, I. Gabitov, I. M. Gel’fand, B. Fuchssteiner, B. Kupershmidt, P. Lax, C. D. Levermore, F. Magri, S. V. Manakov, L. Margolin, P. Olver, T. Ratiu, P. Rosenau, H. Segur, and T. Y. Wu. This work is partially supported by the U.S. Department of Energy CHAMMP program.

References Abarbanel, H. D. I., and Holm, D. D. (1985). Phys. Fluids 30, 3369-3382. Ablowitz, M. J., and Segur, H. (1981). Solitons and the inverse scattering tramform. SIAM, Philadelphia. Bazdenko, S . V., Morozov, N. N., and Pogutse, 0. P. (1987). Sov. Phys. Dokl. 32,262-266.


33

Benjamin, T. B., Bona, J. L., and Mahoney, J. J. (1972). Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. SOC. Lond. A 227, 47-78. Camassa, R., and Holm, D. D. (1992). Dispersive barotropic equations for stratified mesoscale ocean dynamics. Physica D 60. 1-15. Camassa, R., and Holm, D. D. (1993). A completely integrable dispersive shallow water equation with peaked solitons, Phys. Rev. Lett. 71, 1661-1664. Camassa, R., and Wu, T. Y. (1991). Stability of forced steady solitary waves. Phil. Trans. R. SOC. Lond. A 337, 429-466. Fornberg, B., and Whitham, G. B. (1978). A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. SOC.Lond. A 289, 373-404. Gel’fand, I. M., and Dorfman, I. Ya. R. (1979). Hamiltonian operators and algebraic structures related to them. Func. Anal. Appl. 13, 248-262. Green, A. E., and Naghdi, P. M. (1976). A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237-246. Holm, D. D. (1988). Hamiltonian structure for two dimensional hydrodynamics with nonlinear dispersion. Phys. Fluids 31, 2371-2373. Holm, D. D., Marsden, J. E., and Ratiu, T. S. (1985). Hamiltonian structure ondLyupunov stability f o r ideal continuum dynamics. University of Montreal Press, Montreal. Miles, J. W., and Salmon, R. M. (1985). Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519-531. Olver, P. J. (1984). Hamiltonian perturbation theory and water waves. Contemp. Math. 28, 231.

Olver, P . J. (1986). Applications of Liegroups to differential equations. Springer, New York. Olver, P. J. (1988). Unidirectionalization of Hamiltonian waves. Phys. Lett. A 126, 501-506. Peregrine (1967). Long waves on a beach. J. FIuid Mech. 27. 815-827. Rosenau, P., and Hyman, J. M. (1992). Compactons: Solitons with finite wavelength. Phys. Rev. Lett. 70, 564-567. Wendroff, B. (1992). Private communication. Whitham, G. B. (1974). Linear and nonlinear waves. Wiley, New York. Wu, T. Y. (1981). Long waves in ocean and coastal waters, J. Eng. Mech. Div., Proc. ASCE 107, 502-522.


ADVANCES IN APPLIED MECHANICS, VOLUME 31

The Onset and Development of Thermal Convection in Fully Developed Shear Flows R. E. KELLY Mechanical, Aerospace and Nuclear Engineering Department University of California Los AngeIes, California I. Introduction ........................................................................................ 11. Rayleigh-Btnard Convection in Fully Developed Forced Flows ...................... A. Couette and Poiseuille Flows Contained Between Doubly Infinite Horizontal Isothermal Surfaces.. ........................................................................ B. Couette and Poiseuille Flows with Side Boundaries................................. C. Other Fully Developed Flows ............................................................. D. The Influence of Surface Curvature ..................................................... E. Flow with a Free Surface or Interface ..................................................

35 37 37 85 94 98

99

111. Rayleigh-BCnard Convection in Fully Developed, Thermally Induced Flows ..... A. Case I: Flow in a Tilted Box Heated from Below ................................... B. Case 11: Flow in a Horizontal Box Heated at One End ............................

100 101 104

IV. Concluding Remarks .............................................................................

105

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

105

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

106

I. Introduction The onset and development of Rayleigh-BCnard (R.B.) convection in unstably stratified shear flows has been an active area of research since the early part of this century. During the 1920s, both Idrac (1920, 1921) and Terada (1 928) observed the highly organizing effect that shear has upon thermal convection. Although some theoretical consideration was given to the topic in the same period, e.g., by Jeffreys (1928), most of our quantitative knowledge dates from the mid-1960s and has grown enormously since that time (for further discussion of the early research, see Avsec and Luntz, 1937; BCnard and Avsec, 1938; Brunt, 1951; and Gortler, 1959). 35 Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-002031-9

36

R . E. Kelly

Research on the topic has been motivated by a variety of geophysical and engineering applications. Applications to secondary circulations (e.g .,cloud rows) in the planetary boundary layer have been reviewed by Brown (1980), while a similar concept for secondary circulation in the earth’s mantle was introduced by Richter (1973). Lipps (1971) and Bolton (1984), amongst others, have discussed the dynamics of squall lines by use of unstably stratified shear flow models. Convective motions in lakes have been discussed on the same basis by Farrow and Patterson (1993). Technical applications have concerned the augmentation of heat transfer associated with forced convection due to the onset of Rayleigh-BCnard convection; these combined modes of heat transfer represent one form of “mixed convection” and has been discussed thoroughly for the case of entry flow in a channel by Maughan and Incropera (1990). The importance of such convection in situations involving mass transfer seems to have been recognized first by Tobias and Hickman (1965) in connection with electrochemical plating. Considerable interest in thermal convection in shear flows has also been demonstrated by researchers interested in solidification phenomena associated with the production of electronic materials; see, e.g., Jones (1983), Coriell et al. (1984), Evans and Greif (1989, 1991) and Forth and Wheeler (1992). Some other applications will be mentioned within the body of the chapter. In an earlier review (Kelly, 1977), emphasis was placed on the results of linear stability theory as applied to various fully developed flows, and some of those results will also be represented here. The purpose of this chapter, however, is not to provide a complete review of all the many works that have been devoted to this problem but to present an in-depth account of the basic concepts and of some of the major results that have occurred during the past 15 years. For instance, the interesting consequences of the convective rather than absolute nature of the initial instability have been elucidated recently, and our understanding of the finite amplitude states following the initial instability has been greatly enhanced. For this reason, the chapter will focus primarily on a few rather simple basic states involving a Boussinesq fluid and fully developed forced flows, mainly of the Couette or Poiseuille type, in hopes that these concepts will not be overwhelmed by a plethora of detailed results for other flows. Later sections of the chapter will provide some references to similar phenomena in other basic flows, e.g., when the basic flow itself is due to natural (or “free”) convection (Section 111), but fewer details will be presented. The topic of thermal convection in developing flows, either of the entry flow type or in boundary

Onset and Development of Thermal Convection

31

layers in which turbulent convection can eventually occur, will be discussed only in regard to how phenomena in such flows might affect the results for the fully developed state. A good idea of the current state of knowledge concerning the onset of thermal convection in developing flows can be obtained from the papers by Maughan and Incropera (1990) and Hall and Morris (1992).

11. Rayleigh-BCnard Convection in Fully Developed Forced Flows

In this section, we will consider the stability of unstably stratified parallel shear flows when the shear is established primarily by some mechanical means. Any contribution to the basic velocity distribution due to the variation of temperature acting via the buoyancy term in the equations of motion is assumed to be of secondary importance. We begin by considering flows in which the mean temperature is established by conduction and varies only in the vertical direction. The effects of a horizontal variation of temperature upon the instability will be considered later. The channel will initially also be assumed to have infinite width in both horizontal directions so that the basic state variables depend only upon distance in the vertical direction. The effects of lateral boundaries will then be discussed. A. COUETTE AND POISEUILLE FLOWS CONTAINED BETWEEN DOUBLY INFINITE HORIZONTAL ISOTHERMAL SURFACES Let x and y be horizontal variables and let z measure distance in the vertical direction, antiparallel to the direction of gravity (g). Also, let i,, i,,, and i, denote unit vectors in the x, y, and z directions, respectively. The fluid is assumed to be incompressible, and temperature and density differences are assumed to be sufficiently small that the Boussinesq approximation is valid (for discussion of this approximation, see Section 2.5 of the book by Gebhart et al., 1988). Thus, the density (p) is assumed to be constant except in the buoyancy term where it is taken to depend upon temperature (T) as P

= Po[l -

a(T - 731,

(2.1)

being the coefficient of thermal expansion and the subscript 0 denoting a reference value. The governing equations of motion, often referred to as

a!

R . E. Kelly

38

the Oberbeck-Boussinesq equations, in dimensional form are then

DV 1 - - - --Vvp Dt Po

v * v= 0, DT - = K V2T, Dt

- i,g[l

-

a(T -

&)I + vV2V

(2.2a) (2.2b) (2.2c)

where kinematic viscosity (v) and thermal diffusivity (K)have been assumed to be constant and viscous dissipation has been ignored in the energy equation (2.2~). A characteristic vertical temperature difference AT* is taken to exist across the horizontal fluid layer, with AT* > 0 denoting that the lower surface of the layer is warmer than the upper surface. Each surface is assumed to be isothermal. We use AT* to scale the temperature and let the reference temperature be the temperature of the lower surface, TL;the relative temperature is then

where x , y , and z are nondimensional variables based on the layer thickness H , t is a nondimensional time based on the diffusion time scale H ’ / K , 6 is the horizontally averaged nondimensional temperature, and 8 is the perturbation temperature. The basic flow, which is taken to be both fully developed and unidirectional in the x direction, has the scale U*,say, the maximum velocity for plane Poiseuille flow. We will sometimes discuss how the existence of a shear flow affects known results for the case without shear, and so we first use the diffusive scale ( K / H )to scale the perturbation velocity. The total velocity is then described by

where 0 is the horizontally averaged nondimensional velocity and v is the perturbation velocity. For the linear stability problem, both v and 8 are assumed to be small, and 8 and 0 are the temperature and velocity distributions, respectively, of the basic state. Hence, O(z) corresponds to the laminar flow solution of the momentum equations and 6 to the solution of the conduction equation for the fully developed case. Thus, if we represent


39

pressure by

the equations for the basic state are (2.6a)

o = -dJi2 -

[l -

dz

AT* &)I,

(2.6b)

where d%,/dxis a constant, and, if internal heating is not considered,

o = -d 2 6

(2.6~)

dz2 '

so

e = - z for 0 Iz I 1. The equations required for determining the linear

stability of the system are obtained by substituting (2.3)-(2.5) into (2.2a-c) and retaining only terms involving perturbation terms to the first power. We now write these equations in component form, using

v = i,u

+ iyu + i,w

(2.7)

to define the perturbation velocity components. The linear stability equations are

-1_au

au

an

dO

Pr at

+ v2u,

av an 1 au -+ Re U- = -- + V Pr at

ax

-aw 1 aw -+ ReU-

ax

Pr at

ay

an

= --

az

v,

+ R a e + V2w,

(2.8a) (2.8b) (2.8~)

and

ae

-

ae

- + RePr U- - w at ax

=

V20,

au a0 aw - + - + - = 0, ax au az

(2.8d) (2.8e)

where Re is the Reynolds number (U*H/u), Ra is the Rayleigh number ( g a r A T * H 3 / v ~ )and , Pr is the Prandtl number ( v / K ) . For isothermal

R . E. Kelb

40

no-slip rigid boundaries, the boundary conditions are u=u=w=8=0

atz=O,l.

(2.9)

The set (28a-e) can be reduced to the following two coupled equations for w and 8: d 2 8a w

i a

V2w - Re--

dz2 ax

= RaV:8,

ax

(2.10a) (2.10b)

where

a2

v:=-+-

ax2

a2 au2’

The boundary conditions corresponding to (2.9) are

aw w=-=8=o

az

at z = 0, 1.

(2.10c)

Because the coefficients in (2.10a-c) are at most only functions of z, we can represent w and 8 in a Fourier manner as

w

=

W(z)exp(ik,x

8 = O(z)exp(ik,x

+ ikyy + ot), + ikyy + at),

(2.11a) (2.1 1b)

where k, and k,, are the components of the wavenumber vector k, whose amplitude is k = (k; + k:)”*. After (2.1 la, b) are substituted into (2.10a, b), the resulting 0.D.E.s for W(z)and O(z) are

(2.12b) which, together with the boundary conditions

dW

W=-=@=O dz

at z = 0, 1,

(2.12c)

constitute a sixth-order homogeneous boundary value problem giving rise to eigenvalues for, say, 0 that depend upon Ra, Re, Pr, k, and ky.


41

The situation when Re = 0 is discussed in considerable detail by Chandrasekhar (1961); see also Gershuni and Zhukovitskii (1976) and Platten and Legros (1984). A few pertinent results will now be summarized. (i) For Re = 0, the eigenvalues depend only upon kZ and not k, and k,, separately, due to the occurrence of horizontal isotropy. As a result, no preferred planform can be predicted on the basis of linear theory. (ii) The principal of exchange of stabilities holds; i.e., a disturbance with given k either grows monotonically or decays monotonically, depending upon whether Ra > Ra,(k) or Ra < Ra,(k), where Ra, denotes the Rayleigh number for neutral stability (when CT = 0). Therefore, Ra,(k) is independent of Pr. (iii) As k + 0 or k + 00, Ra, + 00. The neutral stability curve has a minimum at the critical Rayleigh number Ra, a 1707.8 for the problem described. Corresponding to this value of Ra, is a critical wavenumber k, = 3.117, and so the dimensional critical wavelength is almost equal to 2H. (For the case of isothermal, stress-free surfaces, Ra, = 27n4/4 9 657.5 and k, = n / f i = 2.221.) For Re # 0, results (i) and (ii) are no longer valid in general; the mean flow introduces a convective wavespeed, and it is clear from (2.12a, b) that the results depend upon both k and k, (or, equivalently, k, and ky). The exception is the special case when k, = 0, k,, # 0. It is then clear from these two equations that the equations are exactly the same as for the case Re = 0 (note that the influence of Re occurs only via the combination k, Re). For this special case, the disturbance is said to consist of longitudinal convective rolls, which are vortices dependent only upon y and z and with axes in the mean flow direction; they are shown schematically in Fig. 1. As will be discussed in detail shortly, the results of the eigenvalue calculation indicate that all other disturbances are stabilized by the shear at least at relatively low values of Re (so that no hydrodynamic instability due to shear can occur). Point (iii) is therefore still valid if one stipulates that k, = ky and k, = 0 at Ra,. Thus, the critical Rayleigh number is unaffected by shear, but now a preferred pattern consisting of longitudinal rolls is predicted on the basis of linear theory. The preference for longitudinal convective rolls in unstably stratified shear flows was first observed experimentally by Idrac in 1920 (see Fig. 2). The phenomenon has intrigued a number of researchers since then because, for the case of zero mean shear, convection in a Boussinesq fluid develops in the form of locally two-dimensional rolls, but rolls that can have random orientation if a large aspect ratio apparatus

R . E, Kelly

42

DIRECTION OF MEAN SHEAR

yl\

FIG. 1 . Schematic of longitudinal roils; after Benard and Avsec (1938).

with well-insulated sidewalls is used and if convection is allowed to develop from background disturbances (i.e., no controlled initial disturbances are imposed as was done by Busse and Whitehead, 1971). Although in principle the geometry of the apparatus can affect the orientation of the rolls once a steady state occurs, the time scale over which this result might be achieved is very large for a large aspect ratio container (see, e.g., Steinberg et al., 1985). An example of the slowly evolving planform when Re = 0 is given in Fig. 3. In contrast, an early visualization of convection in a shear flow obtained by introducing smoke is shown in Fig. 4. The emergence of a well-ordered motion from a seemingly random state is undeniably striking. Indeed, the effect is so striking that it has led some investigators to use it to explain phenomena in situations where the Rayleigh number is highly supercritical, such as in the atmospheric boundary layer. This result, for instance, has

FIG. 2. Sketch of Idrac’s experiment, after Idrac (1920).


43

FIG. 3. Shadowgraph image of Rayleigh-Btnard convection arising due to natural disturbances in a fluid layer without shear; courtesy of F. H. Busse.

FIG.4. Visualization by use of smoke of spiral motion associated with longitudinal rolls in air. Reprinted with permission of the publisher. D. Avsec and M. Luntz (1937), Meteorologic 3, 180-194.

44

R . E. Kelly

been used to explain the formation of cloud rows (Kuettner, 1971) and of sand dunes in deserts (Hanna, 1969). The occurrence of well-organized cloud rows is common; indeed, it is probably the most commonly observed example of large-scale structure in turbulent flow, and we shall return to the topic later in connection with turbulent convection in shear flows. A schematic of cloud row formation is given in Fig. 5 . The point is that a result based on linear stability theory with the Rayleigh number close to the critical is unlikely to be relevant in any strict sense to such a phenomenon. Longitudinal rolls are three dimensional in that three nonzero disturbance velocity components exist. These components, however, depend upon only two spatial dimensions, and so they are sometimes referred to as being two dimensional. If we now consider only longitudinal rolls, Eqs. (2.8a-e) reduce to 1 au do (2.13a) -- + Re-w = V2u, Pr at dz

a v -- -an _1 _ + V’U,

(2.13b)

1 aw Pr at

(2.13~)

Pr at

ay

--an + R a e + V’w, az

ae - w = v2e, at

(2.13d) (2.13e)

Equations (2.13b-e) are uncoupled from (2.13a) and can be solved by themselves subject to the boundary conditions; indeed, that problem is the same as the R.B. instability problem without shear, as we have already noted. This result means that, v, w, 8, and n can be solved independently of u, or in other words, u is solved from a nonhomogeneous equation (2.13a) once w has been determined. Together with the mean flow, the disturbance velocity components cause a fluid element to follow an open spiral path (cf. Fig. 1) in contrast to the orbital motion in the y-z plane, which would occur for the case without shear when such rolls occur. The x component of disturbance velocity represents the effect of vertical advection of mean flow vorticity by w. From Eq. (2.13a), the magnitude of u is seen to be linearly dependent upon Re, which implies that the characteristic scale for this velocity component should be U*/Pr, not K / H . This means that the x


45

FIG.5. Schematic of cloud band formation in the planetary boundary layer. Reprinted with permission of the publisher. R. Fleagle, Science 176, 1079-1084, 9 June 1972. Copyrighted 1972 by the American Association for the Advancement of Science.

component of disturbance velocity can be quite different in magnitude from u and w , depending upon the value of Re. It also means that significant distortion of the mean velocity profile is more likely to occur for fluids with relatively low values of Pr rather than high values. In order to demonstrate that transverse rolls are more stable than longitudinal rolls, Eqs. (2.12a, b) must be solved subject to the boundary conditions. It turns out, however, that only the results for transverse disturbances need be determined because Squire's theorem can be used to relate, for given Ra and Pr, the general three-dimensional problem to an equivalent twodimensional problem (k, # 0, k,, = 0) holding for a lower value of Re, with the case of longitudinal rolls corresponding to Re = 0. This transformation was used by Gallagher and Mercer (1965) for Couette flow and Gage and Reid (1968) for Poiseuille flow (with Pr = 1) in order to determine the stability characteristics of general 3D disturbances. For Poiseuille flow, the velocity distribution of the basic flow is o(z)= 4(2 - z') if U*is taken to be the maximum laminar flow velocity. The results of Gage and Reid (1968) are shown in Fig. 6. The parameter L in that figure is defined so that I = 90" corresponds to longitudinal rolls and L = 0" to the case of transverse disturbances. It is clear that at low values of the Reynolds number shear has

46

R . E. Kelly

Re12

FIG. 6. Neutral stability boundaries for disturbances at various angles of orientation to the flow direction for plane Poiseuille flow with Pr = 1. Reprinted with the permission of Cambridge University Press. K. S. Gage and W. H. Reid (1968), J. Fluid Mech. 33, 21-32. Copyrighted 1968 by Cambridge University Press.

a stabilizing effect upon all disturbances other than longitudinal rolls, which appear as soon as Ra > Rac,o, the value of Ra, with zero shear. Once the rolls have appeared, the neutral stability curves for the other disturbances lose significance because nonlinear effects associated with the longitudinal rolls must be considered. For plane Poiseuille flow, a hydrodynamic (shear) instability can occur when Ra = 0 if Re is greater than, say, Re,,o. The results indicate that, for Ra < Ra,,o, a transverse disturbance (actually a Tollmien-Schlichting wave) becomes unstable when Re is close to Re,,o. Because the onset of longitudinal rolls is independent of the Prandtl number and buoyancy appears to be negligible as far as the onset of the transverse wave is concerned, the abrupt transition between these two types of instability holds for values of Prandtl number other than unity for the case of plane Poiseuille flow (see Platten, 1971, and Tveitereid, 1974). Pearlstein (1985) has demonstrated by linear analysis that the most critical disturbance to a stratified viscous plane parallel shear flow between rigid horizontal boundaries must be either a transverse (2D) or a longitudinal mode. Thermal convection in a shear flow is interesting in part because two distinct modes of instability, with quite different characteristics, can occur. At this point, we should become more precise about what we mean by the terms “transverse roll” and “transverse wave.” By a transverse roll, we shall be describing a disturbance that exists as Re + 0 and that is either


47

stationary for Re > 0 or propagates for Re > 0 with some constant wave velocity related to the average of 6. Thus, for Couette flow, if 6 = z - 1/2 for 0 Iz I1, the rolls are stationary at low values of Re, but if 6 = z, the rolls propagate in the x direction with a wave velocity equal to 1/2. By transverse waves, we shall mean disturbances that propagate with speeds different from the mean advective speed and that, in general, are dispersive. Such waves occur even for the antisymmetric flow = z - 1/2 if Re is sufficiently large. The terms “rolls” and “waves” will be used from here on with these meanings implied. For transverse rolls in Poiseuille flow at small values of Re, explicit results for Ra,, kN, and the frequency oi,,have been obtained by expanding in terms of powers of Re (Muller, 1990; Muller et af., 1992) as follows:

]

+ 0.02378 Pr2 + 2.194 Pr2 + 0.363 Pr3 + 0.42 Pr4 (0.5117 + Pr)’

0.02156

+ o(R~~), kN = 3.116 -

(:

Re2

(2.14a)

X

r 1.930 + 71.03 Pr - 946.3 Pr2 - 182.8 pr3 - 226.96Pr4i + 565.6 Pr’ + 194.9 Pr6 (0.5117 + Pr)4

+ o(R~~), ci,, =

(??)[

1

Re2

(2.14b)

le409 4*105“](kc Re) 0.5117 + Pr +

+ O(Re3).

(2.14~)

These formulas are claimed by Miiller (1990) to be accurate to within 1Yo as long as Pr Re < 26. Similar results for antisymmetric Couette flow (D= z - 1/2) at small values of Re have been given by Ingersoll (1966a) as follows: RaN = 1707.76

+ [0.06451 + 0.1270 Pr + 0.5598 pr2] Re2 + w e 4 ) , (2.15a)

kN = 3.117 - [1.155

+ 1.503 Pr + 2.325 pr2](10-4)Re2 + o(Re4). (2.15b)

R . E. Kelly

48

The frequency is zero in this limit. Ingersoll also considered the limit Pr -+ 00 with RePr (the Peclet number) held fixed; for instance, Ra, = 27,300 when Pr + 00 with RePr = 200/3. Numerical results for Couette flow at larger values of Re for finite values of Pr have been given by Deardorff (1969, Gallagher and Mercer (1965), and Ingersoll (1966a) and for combined Couette-Poiseuille flow by Fujimura and Kelly (1988) aqd Mohamad and Viskanta (1989). The results indicate that Ra, for transverse disturbances increases both with Re and Pr, and so longitudinal rolls appear in general to be the most unstable buoyantly driven disturbance for a layer infinite in both horizontal directions. It remains to explain now why transverse disturbances are stabilized, at least at low values of the Reynolds number. Both Asai (1964, 1970) and Lipps (1971) have examined the energy transfer mechanisms for the case of Couette flow. Consider a two-dimensional disturbance so that v = 0 in (2.8b, d) and the other disturbance quantities depend only on x, z, and t. An equation for the mean kinetic energy of the disturbance is obtained by multiplying (2.8a) by u, (2.8~)by w, and averaging over a wavelength and the layer depth. If we let (-..)denote such an average, then the rate of change of the mean kinetic energy of the disturbance is 1 d -( u 2 + w2)= Re 2 P r dt

(2.16) where the first term on the right-hand side represents transfer of energy between the mean flow and the disturbance via the Reynolds stress, the second represents the release of potential energy into kinetic energy when coupled with the energy equation, and the last term represents viscous dissipation. For Couette flow du/dz = 1, and so the energy transfer depends largely upon how u and 0 are correlated with w (the dissipation term is always negative). Plots of the perturbation stream function and isotherms for a transverse roll in Couette flow are shown in Fig. 7 when Ra = 12,544, Re = 160, and Pr = 0.7. The critical Rayleigh number for this case is 10,394, and so the flow is unstable. The (+) and (-) signs in the temperature field denote warm and cool cores, respectively. For Re = 0, the dividing boundaries between the cells would be straight, with the region of maximum updraft being located directly above the center of the warm core and the region of maximum downdraft being directly above the cool core. The arrangement


49

, STREAM FUNCTION ~~

TEMPERATURE FIG. 7. Streamlines and isotherms for transverse disturbance in Couette flow with Ra = 12,544, Re = 160, and Pr = 0.7 at a nondimensional time of 33.6 based on a convective time scale. Reprinted with the permission of the American Meteorological Society. F. Lipps (1971), J. Atm. Sci. 28, 3-19.

tends to maximize (Ow). In Fig. 7, we see that the shear leads to a tilt in both the streamlines and isotherms, but more so in the isotherms. Although the region of maximum updraft is still located above the warm core, the ascending motion just to the left of it is advecting upward the relatively cool fluid of the adjoining region as well as warm fluid, due to the tilt. This is bound to decrease (Ow)and so not allow for efficient conversion of buoyant energy into kinetic energy. We also note that u and w have the same sign in most of the ascending and descending regions due to the tilt of the cell. Hence, (uw)is positive, meaning that energy is being transferred from the disturbance to the mean flow. Thus, the Reynolds stress acts to stabilize the flow in this case. We note further that the wavelength of the disturbance is considerably larger than twice the gap height, and so a longer interval of nearly horizontal motion near the boundaries occurs here than in the case of longitudinal rolls. Hence, the buoyancy force must be greater in order to overcome retardation due to viscous effects associated with the boundaries, which means that the Rayleigh number must be correspondingly larger. Also, the minimum distance from, say, the center of a warm core to the isotherm separating the warm core from the cool core is decreased, and

50

R. E. Ketly

so heat conduction is enhanced. While it is not clear which of these effects is the most important in stabilizing the transverse disturbance, it should be clear that the transverse disturbance cannot compete effectively against the longitudinal roll disturbance. Calculations by Vanderborck and Platten (1974) indicate that this result should still be expected when non-Boussinesq effects associated with variable viscosity and expansion coefficient are considered (also see Vasilyev and Paolucci, 1992). An illustration of Idrac’s experimental apparatus, taken from his thesis, is shown in Fig. 2. This diagram along with three brief paragraphs make up his contribution to the subject (he was interested mainly in possible atmospheric applications, including migration patterns of birds from Europe to Africa). On the other hand, Terada (1928) and his second year students of physics at Tokyo University, as well as Terada and Tamano (1929), made detailed observations and photographs. Indeed, Terada made many other interesting observations of thermal convection; for the case without shear, for instance, he observed the now well-known result that rolls line up in a rectangular box so that their axes are parallel to the shorter side of the box. With a shear flow present, Terada observed rolls in both air and various liquids and found, in the case of water, that the wavelength was approximately equal to 2H, at least for the smaller values of gap height considered. He did not attempt, however, to compare his observations with any theoretical prediction; the combination of parameters that we call the Rayleigh number was not mentioned. His attitude is revealed by an introductory statement in his 1929 paper with Tamano. He stated: The aim of our present studies is to observe the actual modes of motion as they occur and not to strive to bring the results of the experiments into agreement or disagreement with such and such mathematical theory based upon some assumption. One may, therefore, regard what follows as a chapter of the “natural history of fluid motion.” Many of the experimentalists of the 1930s who investigated this topic also seem to have been content merely with qualitative descriptions rather than the presentation of data in a systematic manner. Almost all of them used air as a working medium, which is unfortunate because non-Boussinesq effects are then usually more important. Although their observations are interesting from the viewpoint of the nonlinear problem (as we shall discuss later), they are not strictly relevant to the linear problem as we have discussed for a Boussinesq fluid.


51

Avsec (1937) states that Btnard called longitudinal rolls “Idrac-Rayleigh eddies,” after the early experiment by Idrac that led to the first published reports of the rolls (Idrac, 1920, 1921). But apparently Btnard had second thoughts about the matter because, in the article by BCnard and Avsec (1938), the authors refer to this form of convection only as “tourbillions en bandes longitudinales.” Perhaps it is just as well, because Terada (1928) stated that he first observed the rolls more than 10 years before his report was published. Chandra (1938) was the first experimentalist to claim that shear does not affect the critical Rayleigh number for Couette flow, but Ingersoll (1966a) was the first to present definite evidence in the form of heat flux measurements that this conclusion is true (see Fig. 8). In order to minimize end effects, Ingersoll used an apparatus consisting of two horizontal concentric disks with fluid contained between them, as described in more detail in another paper by Ingersoll (1966b). The gap distance was small compared 2 .o

1

I

I

1

I

X

0

+ 1.6 -

Nu 1.4

Ir”

Re

1.8 -

5

0.0

+O

s o

10.5

+g

A

22.6

x

33.4

-

+t

P

-

+O

4

-

3

1.2 -

A

A + + o

+0 1.0

- 8 A+@%+x 1

.5

-

+e I

I .o

1

1

1

1.5

2.0

2.5

3.0

FIG. 8. Ingersoll’s measurement of the Nusselt number as a function of Rayleigh number ( R ) for various Reynolds numbers. Reprinted with the permission of the American Institute of Physics. A. P. Ingersoll (1966), Phys. Fluids 9, 682-689.

52

R . E. Kelly

with the inner radii of the disks. The upper disk rotated at constant angular velocity, while the lower disk was at rest and heated. Hence, the Reynolds number was a function of radius, a fact not accounted for in the theory that has been discussed. Nevertheless, his Nusselt number curve shows that Ra, z 1708 for Reynolds numbers (evaluated at the outer radius) up to at least 33.4. Unfortunately, the apparatus did not allow one to observe the form of convection which occurred for Ra > Ra,. As part of a more general experiment, Akiyama et al. (1971) determined that longitudinal rolls do indeed form in air near Ra = 1708 for the case of fully developed, plane Poiseuille flow (see Fig. 35). The experiment was repeated by Ostrach and Kamotani (1975). Although cellular convection due to non-Boussinesq effects occurred in their experiment when Re = 0, they found that only longitudinal rolls formed for a Reynolds number (based on the volumetrically averaged velocity) in excess of 10. Weak convection in the form of rolls was observed for Ra < 1708 with Re > 0, but this was probably due to subcritical instability arising from non-Boussinesq effects. Ostrach and Kamotani found that the average wavelength of the rolls was approximately equal to twice the gap height for values of the Rayleigh number not only close to Ra, but up to Ra s 6000 for low values of the Reynolds number. Mori and Uchida (1966) had found earlier a similar correlation at larger values of the Reynolds number (Re 300-500), at least for Ra < 6000, approximately. Hence, the linear stability theory for a Boussinesq fluid seems to yield results that are in good agreement with these experimental results. The onset of R.B. convection in most shear flows differs from the zero shear case in a fundamental way that affects how convection might be observed in an experiment. This difference has to do with the concepts of absolute and convective instabilities that arise from the analysis of the initial-value problem. These concepts have been discussed in detail for isothermal shear flows by Heurre and Monkewitz (1990). Absolute instability is pertinent to the zero shear case; i.e., if a localized disturbance to the conduction state is created impulsively for Ra > Ra,, it spreads by diffusion and, as t + m, occurs throughout the fluid. Although the disturbance itself is spatially dependent, clearly its amplitude grows locally with time on a linear basis. Now, say that a mean flow with a small Reynolds number but nonzero net mass flux exists in the (+) x direction. For simplicity, consider first a two-dimensional disturbance. Due to the initial disturbance, a wave packet is generated that propagates in the (+ ) x direction with a group velocity (c,) proportional to the average velocity.

=


53

At the same time, diffusion occurs in both the (+) and (-) x directions. If the net effect is that the motion decays at a fixed value of x at t + QO, the flow is said to be convectively unstable. The amplitude of the wave packet increases with time because Ra > Ra,, but only the fluid close to the propagating wave packet is disturbed. Miiller (1990) has investigated this problem by solving a linear amplitude equation with the approximate form (2.17) where Q measures the degree of supercriticality and is equal to Ra/Ra, - 1 if Ra, is given by (2.14a) to O(Re2). If A depends only on time, then Q / T , is the temporal growth rate. The group velocity c, is a linear function of Re for small Re. The last term on the right gives the change in Ra, for a spatially periodic disturbance with k slightly different from k,. For a disturbance localized at the origin when t = 0, Miiller determines the Green’s function and for a fixed value of x, say x,, gives the approximate solution for t % 1 by (2.18) Convective instability in the sense explained occurs if

o Rat), one might easily conclude from only the results shown that the onset of convection is x dependent and that the critical value of x depends upon the level of inlet noise. Results for greater values of the Reynolds number would indicate a similar downstream shift in the “origin” of convection. The conclusion is that the development of thermal convection is open-ended flow channels can depend significantly upon the flow environment. For an infinite channel, convection will occur everywhere as t + 00 following an initial localized disturbance to a hypothetical conduction state existing for Ra above the absolute instability curve shown in Fig. 9. The same result will occur presumably as this boundary is crossed from the convective instability side; the wave packet simply begins to spread upstream as well as downstream. Say, however, that convection is suppressed at some location. For instance the flow might go through a porous nozzle prior to heating, as in the experiment by Luijkx et al. (1981). Computations for the case when convection vanishes at both the inlet and outlet are


55

-I

a

0 . -.r

-4

t

I 0

‘

ia

I

I

I x

20

FIG. 10. The development of 2D thermal convection in heated Poiseuille flow in the convective instability regime, due to various levels of inlet noise, increasing from (a) to (c). Reprinted with the permission of the American Institute of Physics. H. W. Miiller et al. (1992), Phys. Rev. A 45, 3114-3126.

shown in Fig. 11. Even though Ra is sufficiently large that absolute instability occurs, steady convection develops only over a certain length that increases with Re. Hence, even for the case of absolute instability, a somewhat subjective assessment about the “onset’’ of instability is possible. In carefully controlled experiments, both the convective and absolute stability boundaries can be determined precisely. For the analogous case concerning the onset of Taylor vortices in an isothermal fluid contained between two concentric cylinders with axial flow when the inner cylinder is rotating about its axis, Babcock et al. (1991) determined the convective instability boundary by introducing controlled disturbances. It was also found that in the convective regime background noise led to the appearance

R . E. Kelly

56

10

20 x (d)

Ro. 11. The development of 2D thermal convection in heated Poiseuille flow in the absolute instability regime when convection is suppressed at the inlet and outlet. Thin lines describe A(x, t ) instantaneously, whereas dark lines denote the envelope. Reprinted with the permission of the American Institute of Physics. H. W. Miiller et al. (1992). Phys. Rev. A 45, 3714-3726.

of traveling vortices in the downstream section with nearly time-independent amplitude but noisy phase. The phase noise was found to disappear once the absolute stability boundary was crossed, which fact was then used to determine accurately the absolute stability boundary. For the three-dimensional case of longitudinal rolls, Brand et al. (1991) have investigated model equations for Ra z Ra, and small values of Re that suggest that convective instability also occurs first. Although their model equations are not derived rigorously from the Navier-Stokes and energy equations for Ra Ra,, H. Miiller, M. Tveitereid, and S. Trainoff (1993) and, independently, P. Hall have demonstrated in as yet unpublished work that the result remains true when an amplitude equation is developed in a systematic manner. The absolute stability boundary, as calculated by H. S . Li based on Hall's amplitude equation, is shown in Fig. 12. For convective instability, the longitudinal rolls develop from a local steady disturbance within a wedge-shaped region, as described for convection in a heated flat plate boundary layer by Hall and Morris (1992). We begin now a discussion of the nonlinear aspects of the stability of unstably stratified Couette and Poiseuille flows and note first that the energy method has been used to obtain bounds on the Rayleigh and Reynolds numbers below which the flow is stable to disturbances of arbitrary amplitude (Le., the flow is globally stable). For instance, such a

=


0.20 h

I

0‘25

I

57

A

1Tm 1.2

Re

FIG. 12. Convective and absolute stability boundaries for longitudinal rolls as calculated from Hall’s amplitude equation. Absolute instability occurs above curves; convective instability occurs in region between curves and mutual stability boundary. (I) Poiseuille flow; (2) Couette flow. (Courtesy of H.-S. Li.)

bound on the Rayleigh number of a Boussinesq fluid in the absence of shear is Ra < 1707.8, meaning that subcritical instability is impossible. Such a bound holds also for the case of longitudinal rolls in a shear flow because I ) , w , 0 are independent of the x component of velocity. For an arbitrary disturbance, however, the Reynolds number appears in the bound. For instance, in the case of plane Couette flow heated from below, Joseph (1966) found that the flow is stable to all disturbances when (+Re* + Ra)

c 1707.8,

(2.20)

when Re is based on the velocity difference between the plates and the gap height. The strong dependence on the Reynolds number reflects the fact that shear flows can often exhibit subcritical instability. The bound obtained by the energy method, however, is usually well below the value of Reynolds number at which subcritical instabilities have been observed in shear flows. The result (2.20) has been extended by Shulze and Carmi (1976) to a combination of Couette and Poiseuille flows, with either heat transfer at the boundaries or internal heat generation via frictional dissipation. It is worthwhile noting that the “most dangerous” disturbance for shear flows as predicted by the energy method (i.e. , the disturbance giving the lowest bound on the Reynolds number) has the form of a longitudinal roll disturbance. However, the physical significance of this result is unclear because, when solving the extremal problem, only kinematically admissible disturbances are considered, not dynamically admissible disturbances.

58

R. E. Kelly

We now consider finite amplitude effects associated with the rolls for the supercritical case when Ra > Ra, . Because the rolls are independent of the x coordinate, the component of the velocity in the streamwise (x) direction cannot affect u, w , and 8, which are determined as for the case without shear by solving numerically the full equations governing two-dimensional convection in the y-z plane. Hence, the Nusselt number (Nu), which is the nondimensional ratio of the horizontally averaged vertical heat flux with convection to that appropriate for pure conduction, would seem to be independent of the Reynolds number as long as only longitudinal rolls exist. The measurements made in air by Ostrach and Kamotani (1975) of the Nusselt number indicate that this result holds, at least for the range of Reynolds numbers considered ( I 100, based on the average velocity). The values of the Nusselt number for their “confined case,” meaning the case of no basic flow, coincide with the earlier measurements by Silveston (1958) and also with their own measurements for the case of Poiseuille flow. More recent experimental results by Fukui et a!. (1983) give slightly higher values than those of Ostrach and Kamotani; in both cases, the ratio of width to height of the channel was about 20. Ingersoll(1966a,b) concluded that Nu is independent of Re in his Couette flow experiment, although a slight increase of Nu with Re seems to be evident in Fig. 8 up to 3 Ra,. The horizontally averaged temperature distribution determined by Ostrach and Kamotani shows the usual development of a nearly isothermal core and pronounced thermal boundary layers near each wall (see their Fig. 11). The fact that they, as well as Mori and Uchida (1966), found that the wavelength was approximately equal to 2H up to at least Ra = 6000 is interesting because, without shear, there is a distinct shift to longer wavelength convection over the same range of Rayleigh number for the case of air (Willis et al., 1972). Presumably the mechanism that allows rolls to adjust to longer wavelengths, as discussed by Busse and Clever (1979) and Busse (1981), is inhibited by the mean shear. Also, although the ratio of channel width to height was sufficiently large (17.8 to 35.6) so as not to be important for the onset of convection, the presence of sidewalls might have been important for the wavenumber adjustment process. In this regard, Cole (1976) observed for Taylor vortex flow that the point of onset of wavy vortices is more sensitive to the value of the ratio of cylinder height to gap width than is the point at which axisymmetric vortices first occur. If the wavelength does indeed remain close to 271/k,, however, the Nusselt number should actually then be greater than for the case without shear because the shift to longer wavelengths in that case tends to decrease Nu. This perhaps explains


59

why Fukui’s results are consistently slightly greater than those of Ostrach and Kamotani. In this sense, the mean shear does affect the value of Nusselt number. Akiyama et al. (1971), Ostrach and Kamotani (1975), and Fukui et al. (1983) all used flow facilities in which a fully developed velocity profile was established prior to the test section at which heating occurred. However, the linear temperature distribution was naturally established only somewhat downstream of the station at which heating first occurred. Ostrach and Kamotani and Fukui et al. therefore made heat transfer measurements sufficiently far downstream that a linear temperature distribution was established prior to the onset of convection. Nonetheless, any instability occcurring in the thermal entrance region could certainly affect the downstream region. In further work, Kamotani and Ostrach (1976) made measurements in the developing region and showed that convection occurs first in the fully developed region; see also Chiu and Rosenberger (1987). This point is important for understanding some experimental results that Platten and Legros (1984) have reported in their book (Chapter VIII) for heated Poiseuille flow in a channel with both aspect ratios large, which conflict with the results already given. They first compare (see their Fig. VIII-6) the critical temperature difference (AT,) with a mean flow to the value for a static layer and find it to be greater, implying that Ra, depends upon Re. The value of Re used, however, is so large that the thermal entrance region is estimated to be more than twice the test section length. Hence, onset of convection in the developing thermal state is being detected, and there is no reason to think that Ra, for this case should be independent of Re (indeed, Kamotani and Ostrach (1976) show in their Fig. 8 that Ra, at a fixed location increases with Re). Although Platten and Legros did not recognize this fact, they did recognize that the mean flow could affect the growth of the longitudinal rolls and therefore did additional measurements at lower values of Re when the basic temperature is fully developed over much of the test section. They then found that AT, agrees with the result for a stagnant layer at the lowest nonzero value of Re used. Many numerical calculations of two-dimensional convection in the absence of shear have now been made that give considerable information concerning u, w , and B for longitudinal rolls; for a survey, see Busse (1978). We will therefore concentrate on calculations that show how the x component of velocity is affected by the rolls for supercritical values of the Rayleigh number. Such calculations have been done for plane Poiseuille

R. E. Keily

60

flow by Ogura and Yagihashi (1969), Hwang and Cheng (1971), Fukui et af. (1983), and Clever and Busse (1991), and for plane Couette flow by Lipps (1971) and Clever et af. (1977). Clever and Busse used a Fourier expansion in they direction combined with a Galerkin expansion in the z direction, whereas the others used finite difference methods. For the case of plane Poiseuille flow, the pressure gradient in the x direction was assumed to be fixed at the laminar value. We noted previously in our discussion of (2.13a) that u is proportional to the Reynolds number for the case of longitudinal rolls. Because the x momentum equation is a linear equation for u, it is desirable to remove this dependence from the governing equation by redefining u ; we now let u = (~*H/K)&(.Y, z).

(2.21)

The x momentum equation for steady flow in the presence of finite amplitude longitudinal rolls is then (2.22) when the mean pressure gradient is maintained constant. This equation indicates that U L R ( y , z) is strongly dependent upon the Prandtl number and vanishes as Pr -, OQ (although u and w depend implicitly upon Pr, Clever and Busse (1974) have shown that the mean kinetic energy is approximately independent of Pr for Pr > 1 and Ra > 3000 when Re = 0). We now let uLR(Y,

2) = &R(z)

+

~ R ( Y z), ,

(2.23)

where ULR(Z) is the horizontal average of U L R and represents the mean distortion to the basic flow, &z). After substitution, rearrangement by use of the continuity equation, and taking the horizontal average, the equation for uLR(z)is found to be

-

-

it is known that w O(Ra - Rae)”’, as For Ra near Ra, and Pr 0(1), is 0 from (2.22); hence, U L R (Ra - Ra,) for Ra E Ra, .The horizontally averaged velocity in the x direction, namely, U(z) + U L R ( z ) , is shown in Fig. 13 for the case of Couette flow with Pr = 0.7 when Ra is substantially above Ra, . The velocity distribution follows the same trend as the horizontally averaged temperature distribution. For larger values of the Prandtl

-


61

1.c

2 0.E

-0.25

0 -%

0.25

0.5

+ z +@z)

Fro. 13. The horizontaly averaged velocity for Couette flow when heated from below with Pr = 0.7. Reprinted with the permission of the publisher. R. M. Clever et ul. (1977), Zeit. angew. Math. Phys. 28, 771-783.

number, however, the distortion of the velocity is much less. Lipps (1971) presents in his Fig. 10 a highly distorted velocity profile for Ra = 20,000 and Pr = 0.7; for Pr = 9.35, however, the distortion of the basic linear profile is nearly imperceptible. Calculations for values of Pr < 0.7 have not been made. The wall shear stress is obviously increased by the boundary layer structure that emerges at the higher values of Rayleigh number. The deviation of the averaged nondimensional shear stress at z = 0 from the

R . E. Kelly

62 10

1

ST

.1

.01 1

10

102

104

Ra - Ra,

FIG. 14. The deviation of the averaged nondimensional wall shear stress from the laminar value. Reprinted with the permission of the publisher. R. M. Clever el ul. (1977), Zeit. ungew. Math. Phys. 28, 771-783.

value appropriate for laminate Couette flow, namely, (2.25)

is plotted in Fig. 14 for Pr = 0.7 and 2.5. The early calculations performed for the case of Poiseuille flow emphasized more the local variations of the x component of velocity because the authors wanted to compare their results with the detailed experimental measurements made by Mori and Uchida (1966). Hwang and Cheng (197 1) obtained qualitative agreement with these results, but sufficient discrepancies existed to cause Fukui et al. (1983) to repeat the experiment and to do their own calculation. As shown in Fig. 15, excellent agreement was found. The distortion of the basic flow velocity is as one might expect at those spanwise stations where the vorticity of the basic flow is being advected upward or downward by the rolls; see also Chiu et al. (1987).


63

Y 1

I-

0.5

0

0

0.5

Y

1

Fro. 15. Distributions of streamwise velocity and temperature at various spanwise stations for Poiseuille flow of air when heated from below. Solid lines correspond to the numerical solution of Fukui et ul. (1983), symbols to their experimental results, and dashed lines to the approximate solution of Mori and Uchida (1966). Reprinted from Int. J. Heat Muss Trumfer 26, K. Fukui e t a / . (1983), “The longitudinal vortex and its effects on the transport process in combined free and forced laminar convection between horizontal and inclined parallel plates,” 109-120, copyright 1983, with kind permission from Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 OBW,UK.

R. E. Kelly

64

Fukui et al. (1983) also calculated the quantity (2.26) where p’ is the nondimensional pressure based on the diffusive scale pv2/h2 and dp’/dxis a constant. The quantity (2.26) is a measure of the increase in pressure gradient required for the same mass flow rate and is a function only of Ra; it is shown in Fig. 16(a). Alternatively, one can determine the ratio of the mass flux with convection to the mass flux of the basic state, say, Sh, when the pressure gradient is assumed to be constant. This ratio has been computed by Clever and Busse (1991) and is shown in Fig. 16(b). In our discussion to this point, we have essentially approached the problem from the viewpoint that the shear is initially established and then the Rayleigh number is increased to a value above Ra, . In such a sequence, longitudinal rolls should always appear first, at least for a Boussinesq fluid, a sufficiently wide channel, and for values of the Reynolds number such that no hydrodynamic instability is possible. We should also consider, however, the situation when convection is established initially and then a mean shear is imposed. It is by no means certain then that longitudinal rolls should necessarily develop for any value of Re > 0 if Ra > Ra,. For Ra close to Ra,, however, one expects that longitudinal rolls should form. In this limit, insight into the development of the longitudinal rolls has been provided by Richter (1973), who was interested in determining whether or not longitudinal rolls might form within the earth’s mantle on a time scale of geological significance. He considered the case of a layer with stress-free surfaces in which an assumed body force drove a basic flow with the velocity distribution

U(z) = Uocosnz,

0s2

I1,

(2.27)

-

where oo O(1). For Ra < Ra,,o, the conduction profile for the temperature was assumed to hold. For Ra > Ra,,o, a solution was developed in terms of a parameter E , defined by Ra = Ra,,,(l

+ E’).

(2.28)

The nondimensional characteristic velocity ([/*H/K) was assumed to be of O(E),and the time scale over which the development of convection evolved was assumed to be 0 ( e 2 ) , leading to the introduction of a slow time variable T = e2t. At lowest order in E , the convection state can then be taken as being composed initially of both transverse and longitudinal rolls, and so the


/Rat

Aa

I

I

l

I

l

l

-................

1.o

65

l

I

l

l

I

J

-....

.. .... ..

f ...

0.8 -

- a

Sm 0.6

-

- 6

r

0.4

-

- 4

0.2 L

0.0

* 2

(b) I

I

l

l

I

l

l

I

l

l

I

0

Ra-Ra, FIG.16. (a) Increase of nondimensional mean pressure gradient for a fvced mass flux due to longitudinal roll convection in Poiseuille flow using air. Reprinted from Int. J. Heat Mass Transfer 26, “The longitudinal vortex and its effects on the transport process in combined free and forced laminar convection between horizontal and inclined parallel plates,’’ 109- 120, copyright 1983, with kind permission from Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 OBW, UK; (b) Decrease in the mass flux for the case when the laminar pressure gradient is maintained. (---): Pr = 7; (---): Pr = 0.71. Reprinted with permission of Cambridge University Press. R. M. Clever and F. H. Busse (1991), J. Fluid Mech. 229, 527-529. Copyrighted 1991 by Cambridge University Press.

R . E. Kelly

66

perturbation temperature was assumed to be of the form 8(x, z, r) = c(k:

+ n2)-'A(t)cos k,x sin 712

+ ~ ( k +i n2)-'B(r)cos kyy sin nz,

(2.29)

where A(r) and B(r) are the amplitudes of the transverse and longitudinal rolls, respectively, and k, = ky = k, = n/\lZ. By expanding consistently in terms of E to O(t3)and applying at that order a solvability condition, the following equations governing A(r) and B(r) were obtained for the case of an infinite Prandtl number fluid: dA dr

1 8

+ B2)- PAB2 - C Y D ~ A , (2.30a)

dB dt

1 8

(2.30b)

- = (kz + n2)A - -A(A2

- = (kz + n2)B - -B(B2 + A 2 ) - BBA2,

where a and 3/ are numerical constants, real and greater than zero. When the convection state was taken initially to be in the form of transverse rolls (B = 0) with Uo = 0, then, from Eq. (2.30a), A(0) = [8(k,2 + n2)]1/2,

(2.31)

coresponding to a supercritical state of convection. For r > 0 when A ( t ) was seen to decay with time to the new amplitude A ( m ) = [8(k,2 + n2) - ( ~ c Y D ~ ) ] ~ / ~ ,

Do # 0, (2.32)

which also follows from Eq. (2.30a). Clearly, convection in the form of transverse rolls is suppressed if

Uo > Do,, = [(kz + Z ~ ) / C Y ] ' P / ~ 5A(O).

(2.33)

Now when B(0) is taken to have a small initial value, say B(0) = 0.01, the longitudinal rolls interact with the transverse rolls so as to cause decay of the transverse rolls even if oo< &, . This interaction is shown in Fig. 17. The curves A3 and A4 correspond to situations where Uo > U0,,, and the amplitudes of the transverse rolls are seen to decay before the longitudinal rolls (B3 and B4) emerge. For the curve A2, however, < oo,,,and so transverse rolls could exist if the longitudinal rolls were not present. As the amplitude of the transverse roll begins to approach the value predicted by (2.28), however, interaction occurs with the growing longitudinal roll B2. The transverse rolls become unstable and decay subsequently to zero,

uo

Onset and Development of Thermal Convection I

I

I

0.2

0.4

0.6

67

1

W

0 3

I

0

0.8

1

NONDIMENSIONAL TI ME FIG. 17. Development of thermal convection in a shear flow based on a weakly nonlinear model allowing for interaction between transverse and longitudinal rolls. Reprinted with the permission of the publisher. F. M. Richter (1973), J. Geophys. Res. 78, 8735-8745. Copyrighted by the American Geophysical Union.

while the longitudinal rolls continue to grow to their steady-state amplitude. Similar instability was stated to occur eventually for the transverse roll A l , at a value of 7 that is off the diagram. Hence, for small values of Reynolds numbers, the transverse rolls always decay and give rise to longitudinal rolls. Richter and Parsons (1975) performed an experiment in which the evolution of longitudinal rolls was observed in a high Prandtl number fluid. They exerted a shear on the surface of an enclosed fluid layer, so that the basic velocity was one corresponding to combined Couette-Poiseuille flow, and observed the decay of transverse rolls and the growth of longitudinal rolls. In fact, they also observed the longitudinal rolls to emerge for a case when the Rayleigh number was sufficiently large that truly threedimensional convection occurred in the absence of shear. However, the functional dependence of Reynolds number upon Rayleigh number required to allow the emergence of longitudinal rolls was not established. For sufficiently large values of Rayleigh number at t = 0, it would seem reasonable that a finite value of shear would be required to destabilize the

68

R. E. Kelly

transverse rolls, say, and allow the longitudinal rolls to emerge (Richter’s analysis holds only for Ra near Ra,). In the experiment by Ostrach and Kamotani (1975), an initial state of steady cellular convection occurred in the absence of shear, but the authors found that longitudinal rolls emerged for values of Ra < 8000 if the Reynolds number was in excess of 10. In this case, Re = 10 serves as a bound. The general dependence of Re upon Ra required to yield longitudinal rolls could be obtained theoretically by examining the stability of the initial state of convection with respect to longitudinal roll disturbances as the Reynolds number increases. The experimental set-up of Richter and Parsons (1975) has been used by Mohamed and Viskanta (1992) but with greater emphasis on endwall effects and boundary layer structure that develops at higher values of Re. To the author’s knowledge, no one has investigated the opposite regime to see whether shear-induced waves emerge from a state consisting initially of longitudinal rolls for slightly supercritical values of Ra as Re Re,,o. In this case, the longitudinal rolls could act as a “thermal roughness” that might trigger the subcritical instability possible for many shear instabilities; if so, the scenario would be dramaically different from the one that has been described. In many of the early experiments concerning Rayleigh-BCnard convection in shear flows, cellular convection in the form of hexagons was usually observed in the absence of shear due to non-Boussinesq effects. A diagram made by Graham (1933) of the evolution of longitudinal rolls in the case of air for Couette flow is shown in Fig. 18. If the hexagonal cells are arranged initially as in Fig. 18(a) and the direction of shear is as shown, the hexagons become distorted by the shear, as shown in Fig. 18(b), (c), but eventually revert to a system of hexagons more suitably aligned with the flow, as shown in Fig. 18(e), so that longitudinal rolls can eventually emerge. Many of the investigators in the 1930s observed this formation of “chains of polygons,” of which an example is shown in Fig. 19. (Phillips and Walker (1932) obtained this pattern by decreasing Re from values at which longitudinal rolls occurred; see also Ma1 (1930).) The next step in the formation of the longitudinal rolls is less documented, but BCnard and Avsec (1938) mention that a roll began to form as the boundary between two cells that was normal to the direction of the flow vanished (see their Fig. 15). An advanced stage of the process is sketched in Fig. 18(f), prior to the emergence of straight rolls as in Fig. 18(g). Avsec (1937) presents a nice photograph of the pattern sketched in Fig. 18(f) and states that the straight boundary between two rolls corresponds to ascending air and the undulatory -+


69

9

f

Dimtion of motion

83 @ a

C

e

FIG. 18. Graham’s sketch of the evolution of longitudinal rolls (g) from an initial state of hexagonal convection (a) as Re increases. Reprinted with permission of the Royal Society of London. A . Graham (1933), Phil. Truns. A . 232, 285-296.

FIG. 19. A chain of polygons in a shear flow. Reprinted with the permission of the publisher. D. Avsec and M. Luntz (1937), Meterorologie 3, 180-194.

70

R . E. Kelly

boundary to descending air. Yoshizaki (1979) has studied the effects of shear upon initially hexagonal convection resulting from a curved mean temperature profile by means of amplitude equations valid near Ra, . For very low values of Re, rolls become a stable form of convection for Ra greater than, say, Ra,. As Re increases, the range of stable hexagons shrinks and vanishes for Re greater than some critical value, say Re,. Unfortunately, numerical values of Re, were not given by Yoshizaki. The problem is being investigated currently by P. Hall and the author. Although the temperature dependence of the viscosity is often the cause for cellular convection, it should be remembered that lack of symmetry in general can give rise to hexagonal convection in fluid layers with zero mean shear. Thus, the chain of events shown in Fig. 18 should be qualitatively representative of other situations involving cellular convection, e.g., convection due to asymmetrical internal heating. Several of the early experimentalists also observed transverse rolls. In the case of Chandra (1938), a boundary vortex was induced at the upstream end of his Couette apparatus, and so the transverse rolls observed at small Re were presumably influenced by this boundary effect. At larger rates of shear, longitudinal rolls were observed. Transverse rolls were observed to exist by themselves in air by BCnard and Avsec (1938) under certain conditions when shear was induced prior to heating. However, they did not observe transverse rolls in the case of a liquid. Although longitudinal rolls are the preferred form of convection for a certain range of Rayleigh and Reynolds numbers, they themselves can become unstable, say, for a fixed value of Reynolds number as the Rayleigh number is increased and so yield a more complicated form of convection. This evolution to three-dimensional and eventually turbulent convection has been surveyed for the case without mean shear by Busse (1978, 1981). For sufficiently small values of the Rayleigh number, two-dimensional rolls are the preferred mode of convection for a Boussinesq fluid. As the Rayleigh number is increased, the rolls become unstable, and either steady or time-dependent three-dimensional convection can develop, depending upon whether the Prandtl number is large or small, respectively. It is natural to ask how roll instability is affected by the presence of a mean shear. Clever et al. (1977) and Clever and Busse (1992) examined the stability of the rolls for the case of Couette flow by performing a linear stability analysis of the three-dimensional flow associated with the rolls and the temperature field appropriate for supercritical values of the Rayleigh


71

number. Thus, they let V(x, y , z, t ) = i,[U*(-+

+ z + U L R ( y , z)l + (dH)li(x, Y , z, t)I

+ ~ ~ ( K / H ) [ V L R ( Y , Z) + +~&C/H)WLR(Y

Z)

+

Y , Z, t)l Y Z, t)I

(2.34)

where T(X,.Y, Z, t ) -

TL

= AT*I-Z

e L R ( y , Z)

&X,J’, 2, t ) ] ,

(2.35)

where ULR, VLR, WLR, and OLR denote properties associated with the rolls (with ky = ky,,= 3.117) when Ra > Ra, and li, a, G, and 8 denote a small perturbation. The governing equations for the perturbation have coefficients that are independent of x and t, and so any disturbance quantity, say li, can be assumed of the form ~ ( xy,, z, t ) = lil(y,z) exp(ikxx + at),

(2.36)

where a , > 0 again denotes instability. The analysis involved a Fourier expansion in y and a Galerkin expansion with respect to the z dependence; see Busse (1991). Modes with two different symmetries in y are possible but, for the most unstable mode, was found to be a symmetric function of y ; i.e., with k, # 0 the instability appears to be “sinuous” in a planview. Calculations were done by Clever et al. (1977) for Pr = 0.71, where for Re = 0 we know that the rolls can become unstable by means of an oscillatory instability with oi # 0, corresponding to waves traveling along the rolls (whether or not they actually do so seems to depend on the wavenumber of the rolls, see Fig. 3 of Busse and Clever (1979)). With nonzero Reynolds number, however, another instability with ai = 0 can occur as the Rayleigh number increases from Ra,. This has been called the wavy instability, although one should remember that it corresponds to a wave of zero temporal frequency for antisymmetric Couette flow. A schematic of the disturbed rolls is given in Fig. 20. The neutral stability boundaries for these two types of disturbance are shown in Fig. 21 for Pr = 0.71. For Re > 39 the wavy instability occurs first at values of the Rayleigh number (Ra,,,) not much beyond the critical value. For instance, Ra,,, - Ra, E 57 for Re = 200. Hence, in order to detect this transition in air at large values of the Reynolds number as the Rayleigh number is increased, very precise control of the Rayleigh number is required in an experiment. Determination of the stability boundary is made even more difficult because the wavy instability first sets in for the

72

R . E. Kelly

FIG.20. Schematic of the disturbed rolls after onset of the wavy instability. Reprinted with permission of Cambridge University Press. R. M. Clever and F. H. Busse (1992), J. Fluid Mech. 234, 51 1-527. Copyrighted 1992 by Cambridge University Press.

unbounded case with an infinite wavelength. As shown in Fig. 23, the value of iX for maximum growth rate changes rapidly as the Rayleigh number increases beyond the critical value corresponding to this secondary instability. Hence, the observed value of kx in an experiment might depend upon the exact manner in which Ra is increased with time to a greater extent than when Re = 0. For Pr = 2.5, Clever and Busse (1992) have found that the minimum value of Re for onset of the wavy instability is about 70 while the minimum value of Ra is slightly above 2000. We note also that if the Rayleigh number is set initially at a value contained on the dashed line in Fig. 21 when the Reynolds number is small, then increase of the Reynolds number might lead to either the wavy or oscillatory instability, depending upon the precise value of the Rayleigh number. Further, if the rolls are established first in region A (bounded by the dashed line and the neutral stability boundaries), then the rolls can in principle be made unstable by actually lowering the Rayleigh number. Determination of similar secondary stability boundaries for longitudinal rolls in flows with variable mean shear is an even more formidable computing problem, but Clever and Busse (1991) have recently succeeded in finding these boundaries for the case of Poiseuille flow. The stability boundaries are shown in Figs. 22(a) and (b) for Pr = 0.7 and 7.0, respectively. For Pr = 0.7, the situation is similar to that shown in Fig. 21 for Couette flow, although the wavy instability sets in at somewhat higher


73

104 L

-

-

OSCILLATORY

103 u 0 K I

--

UNSTABLE ROLLS

2

102

----

WAVY

101

STABLE ROLLS

1

10

1

1

1

1

1

1

1

I

I

I

I 1 1 1 1

103

102

I

1

I 1 " L

104

Re

FIG. 21. Neutral stability boundaries for the wavy and oscillatory instabilities to longitudinal rolls for heated Couette flow with Pr = 0.71. Reprinted with permission of the publisher. R. M. Clever et 01. (1977), Zeit angew. Math. Phys. 28, 771-783.

values of Re and Ra - Ra,. For Pr = 7.0, however, the situation is quite different, because another secondary instability associated with the case Re = 0, namely, the knot instability, occurs at lower values of Ra - Ra, than does the wavy instability even when Re = 100. The variation with Re of the knot instability appears to depend significantly upon the wavenumber of the primary rolls. In order to optimize an experiment aimed at determining the onset of the wavy instability, one needs to use a fluid with a moderate value of Pr so that a critical value of Re, exists as well as having R a , - Ra, large enough so as to be detectable. Another difference between the wavy instability in Couette and Poiseuille flows is that, while oi = 0 for antisymmetric Couette flow (at least when

74

R . E. Kelly

Re F I ~ 22. . Boundaries for instability of longitudinal rolls for Poiseuille flow when heated from below. (a) Pr = 0.7; (b) Pr = 7.0; (-): wavy instability; (--.--): oscillatory; (---): knot. Reprinted with permission of Cambridge University Press. R . M. Clever and F. H. Busse (1991), J. FIuid Mech. 229, 517-529. Copyrighted 1991 by Cambridge University Press.

Ra = RG), q # 0 for Poiseuille flow simply because the mean velocity is nonzero; i.e., a convective wave velocity occurs in a fixed reference system. Clever and Busse (1991) state that the frequency is given within a few percent throughout the parameter space by the formula

(2.37) where (-.-)denotes an average with respect to z. Thus, the wave velocity is simply the average of the mean velocity. Chiu and Rosenberger (1987) observed unsteady longitudinal rolls that gave rise, in their words, to “snaking modulations.” They explained such motions as being due to a

Onset and Development of Thermal Convection 0.4

1

I

1

75

1

0.3

0

0.2

-

0.1

-

0

-0.1

-

-0.2

-

-0.3

-

-0.4 0

0.01

0.02

0.03

~

0.04

0.05

kx

FIG.23. The wavenumber ixfor maximum growth rate of the wavy instability for Couette flow with Re = 1O00, Pr = 0.71. Reprinted with permission of the publisher. R. M . Clever et al. (1977). Zeit. angew. Math. Phys. 28, 771-783.

76

R. E. Kelly

mixture of longitudinal and transverse linear modes and/or due to upstream turbulence. A third reason should be considered in general, namely, the secondary instability of the rolls due either to the oscillatory instability at low values of Re and high values of Ra or the wavy instability when Re z 100 and Ra G Ra,, at least for fluids with low values of Pr (in the experiment of Chiu and Rosenberger (1987), nitrogen was used and so Pr = 0.71). A possible observation of the wavy instability was made by Avsec (1937), who presented a photograph of “les formes onduleks.” Avsec obtained this mode by increasing AT* and stated that further increase in AT* led to “disorganized convection.” Avsec also presents one photograph in which some rolls are sinuous whereas neighboring rolls exhibit a “varicose” type behavior. Unfortunately, Avsec did not present information that can be used to define precisely the conditions under which these forms occurred. Indeed, even the experimental apparatus is not described, although one might assume that it corresponds to the apparatus used by Benard and Avsec (1938) that involved some (undefined) combination of Couette and Poiseuille flows. While Mori and Uchida (1966) apparently did not detect roll instability even for Re = 500 in Poiseuille flow, such instability has been observed and documented for the case of a free convection flow in an inclined channel (see Section 111). In other experiments concerning forced flow in a channel, the maximum reported value of Re appears to be somewhat less than the minimum value of Re,. These experiments should therefore be repeated for higher values of Re. In doing such an experiment, one should keep in mind the possibility that secondary instabilities might occur as convective or absolute instabilities and, in the former case, would be affected by upstream noise in the same way as the initial instability. For Couette flow, Busse and Clever (1992) have also calculated the properties of finite amplitude wavy rolls (i.e., Re > Re, or Ra - Ra, > Ra, - Ra,) in order to determine the corresponding values of the Nusselt number (Nu) and the mean shear parameter (SJ. Some results for Pr = 2.5 are shown in Fig. 24. A somewhat surprising feature is that both Nu and S, are reduced once the fully three-dimensional motion associated with the wavy instability begins. Apparently, the aligned updrafts and downdrafts of the longitudinal rolls provide a more efficient means of heat transport, and the mean flow energy diverted into the fully three-dimensional motion causes S, to be reduced. Further results by Clever and Busse (1992) for different values of Pr and Re reveal that the situation is even more complicated due to the fact that


77

2,

5

2

'01

Ra-Ra,

lo3

FIG.24. Dependence of the Nusselt number (-) and the mean wall shear parameter (. . . .) upon the Rayleigh number for longitudinal rolls in Couette flow with Re = 400, Pr = 2.5, ky = 3.117. Similar values for wavy rolls are given by (---) and (-.-.-), respectively. Reprinted with permission of Cambridge University Press. R. M. Clever and F. H. Busse (1992), J. Fluid Mech. 234, 51 1-527. Copyrighted 1992 by Cambridge University Press.

subcritical instability of the longitudinal rolls can occur, as shown in Fig. 25. Such instability becomes more likely as the Reynolds number is increased and as the Prandtl number is decreased, indicating that it is due to hydrodynamic effects; indeed, three-dimensional steady solutions were obtained even for Ra c Ra, for Pr = 0.71 and Re = 700!

I

I

N

lo2

103

*

5

Ra-Ra,

FIG.25. Dependence of the Nusselt number for Couette flow upon the Rayleigh number for longitudinal rolls with Pr = 0.71 (-) and Pr = 0.3 (-.-.-) and for wavy rolls with Pr = 0.71, Re = 200, k, = 1.3 (---); Pr = 0.71, Re = 400, k, = 0.9 and k, = 1.5 (--- . - - - ) ; P r = 0 . 7 1 , R e = 7 0 0 , k , = 2 . 0 ( - - . - - ) a n d P r = 0 . 3 , R e = 2 0 0 , k , = 1.2(-..-). In all cases, ky = 3.1 17. Reprinted with the permission of Cambridge University Press. R. M. Clever and F. H. Busse (1992), J. FluidMech. 234, 511-527. Copyrighted 1992 by Cambridge University Press. (...a*)

78

R . E. Kelly

It should also be mentioned that some of the steady three-dimensional solutions can themselves be unstable and give rise to time-dependent states; no systematic investigation has yet been made in this regard. However, direct numerical simulation of unsteady, 3D thermal convection with mean shear at large values of Ra has been done by both Hathaway and Somerville (1986) for Pr = 1.0 and Ra = lo4 and Domaradzki and Metcalfe (1988) for Pr = 0.71 and Ra = 35,840 and 150,000; some results were presented by the latter authors for Ra = 630,000, but the resolution was said to have been inadequate at this large value of Ra. For the parameters chosen by Hathaway and Somerville, the convection is both time dependent and three dimensional when Re = 0 according to the well-known diagram of Krishnamurti (1970), but somewhat below the boundary for turbulent convection. The values of Ra used by Domaradzki and Metcalfe are well into the turbulent regime when Re = 0. As far as Reynolds number is concerned, Hathaway and Somerville used Re = 50, 100, and 200, whereas Domaradzki and Metcalfe used values of Re up to about lo3 (they present their results in terms of Ra and Ri, the overall Richardson number, where Ri = -Ra/Re2 Pr is a measure of the effects of buoyancy compared with those of shear; hence, for fixed values of Ri and Pry Re increases with Ra). The computational domain used by Hathaway and Somerville has aspect ratios between 5 and 10, large enough to contain several large-scale eddies, whereas the domain used by Domaradzki and Metcalf was between one and two times the natural wavelength of convection without shear. Hathaway and Somerville used a finite-difference method and imposed boundary conditions corresponding to flow within a channel with isothermal, rigid, noslip walls. Domaradzki and Metcalfe used a Fourier series in the horizontal plane (and therefore used periodic boundary conditions at the side boundaries) and a Chebyshev expansion in the vertical direction (with isothermal, rigid, no slip boundary conditions at the upper and lower surfaces). The most impressive result is the strongly organizing effect of shear upon thermal convection for highly supercritical but still reasonably moderate values of the Rayleigh number, as demonstrated in Fig. 26 (note: Hathaway and Somerville’s paper concerns mainly the case with rotation and was motivated by possible astrophysical and geophysical applications, which explains the notation shown in Fig. 26; “East” is the direction of the mean shear and “North” is in the spanwise direction). At Ra = lO,OOO, the organization shown for Re = 200 is clearly much greater than for Re = 50. As Re increases from 0 to 200, the shear parameter S, increases monotonically, but the Nusselt number decreases initially as Re is increased


79

FtG. 26. Thermal convection in Couette flow at Ra = 10,OOO, Pr = 1.0, and various values of Re at two values of nondimensional time based on a diffusive time scale, with mean shear in the East-West direction. Shading is used to represent temperature, with the lighter shades being hot and the darker shades being cold. The trajectories of markers are plotted on each of the three visible surfaces. (a) Re = 50, t = 1.3; (b) Re = 100, t = 1.3; (c) Re = 200, I = 0.65; reprinted with permission of Cambridge University Press. D. H. Hathaway and R. C. J. Somerville (1986), J. Fluid Mech. 164, 91-105. Copyrighted 1986 by Cambridge University Press.

80

R . E. Kelly

to 50 before beginning to increase for Re > 100. At Ra = 35,840, Domaradzki and Metcalf also found organization in the form of nearly longitudinal rolls for Re s 500 and determined velocity spectra in order to demonstrate the lack of isotropy in the horizontal planes; these are shown in Fig. 27(b). The horizontally averaged terms in the kinetic energy rate equation for the three-dimensional disturbance are shown as a function of z in Fig. 28 for Re = 0 (a) and Re z 500 (b). It is clear that the production term due to buoyancy is not significantly affected as Re increases from zero. The production term due to shear, however, shows that hydrodynamic effects are very important for Re E 500 in regions near each wall. For this

WAVENUMBEA

K

FIG. 27. Velocity spectra in the horizontal directions as a function of wavenumber k: 0,xdirection; +, y direction. (a) Ra = 35,840, Re = 0; (b)Ra = 35,840, Re s 500. Reprinted with permission of Cambridge University Press. J. A. Domaradzki and R. W. Metcalfe (1988), J. Fluid Mech. 193, 499-531. Copyrighted 1988 by Cambridge University Press.


81

WAVENUMBER K FIG. 27-continued.

region, the contour plot of the vertical velocity shown in Fig. 2(c) of Domaradzki and Metcalf at the midplane (z = 1/2) probably exaggerates the amount of organization in the flow. Also, as mentioned by Busse and Clever (1992) in their study of finite-amplitude wavy rolls, isotherms in the midplane show greater distortion than do the contour plots for the vertical velocity. However, their contour plots for w at z = 0.3 and 0.7 certainly still show considerable anisotropy in the x and y directions; see Fig. 29. It would therefore seem that the wavy instability can give rise to a kind of organized convection that can persist to values of Ra well above the value at which the wavy instability begins. At Ra = 35,840, the value of Nu with Re = 500 is only slightly different from the case Re = 0. As Ra increases to 150,000 for the same value of Ri, the degree of organization is much less, based on the contour plot of w ; unfortunately, horizontal velocity spectra for this case

R. E. Kelly

82

were not presented by Domaradzki and Metcalfe. It should be understood that the Reynolds number for Ra = 150,000 and Ri = -0.2 is Re z 1028 and is near the lowest value at which Reichardt (1959) observed turbulent motion in isothermal Couette flow. Hence, the relative lack of organization might signify that shear-induced turbulence is beginning to become dominant over turbulent convection even though the Richardson number is held constant as Ra is increased. The Nusselt number with shear was found to be about 17% below the value without shear, which is significant. Domaradzki and Metcalfe argue that the shear interferes with the plume-type mode of heat transfer that fa)

v 5

.4

.3

.2

.I

N

.o

-.1 -.2 -. 3 -. 4 -.

5

-4

-3

-2

-1

I

I

1

2

E N E R G T I T I M E (x lo5)

3

4

x105

FIG. 28. The vertical structure of the terms in the kinetic-energy-balance equation: 0, dissipation; 0.production due to buoyancy; ,production due to shear. (a) Ra = 35,840, Re = 0; (b) Ra = 35,840, Re 3 500. Reprinted with permission of Cambridge University Press. J. A. Domaradzki and R. W. Metcalf (1988), J. FIuidMech. 193,449-531. Copyrighted 1988 by Cambridge University Press.

*


83

.4

.3

.2

.I

N

.o

-. 1 -. 2 -. 3

-.4 -.5 -4

-3

-2

-1

ENERGY/TIME

1 (x

2

3

4

lo5)

FIG.28.--rontinued.

exists when Re = 0. Results for the mean velocity and temperature are shown in Fig. 30. The velocity profile is qualitatively very much like the one corresponding to pure longitudinal rolls (Fig. 13) at Ra = lo4, although the flow is drastically different. This fact must be appreciated before future experiments in this area are made. The implications of the preceding results for the explanation of cloud rows are interesting even though Rayleigh and Reynolds numbers for the atmospheric boundary layer are typically much larger than those considered if molecular values for viscosity and conductivity are used. They indicate that organization can exist at large values of the Rayleigh number if the Reynolds number is not too large. Actual pictures of cloud rows (e.g., Fig. 7.2 of the book by Turner, 1973) suggest an alignment of the individual thermals that is characteristic of convection at high values of Ra. Such a

84

R . E. Kelly

X

FIG. 29. Lines of constant vertical velocity in the planes (a) z = 0.8, (b) z = 0.5, and (c) z = 0.2; (b) isotherms and (e) streamlines in the midplane. Also shown are isotherms in the planes (f) x = 0 and (g) x = n / 2 k X . All plots correspond to the case Ra = 3000, Re = 400, Pr = 0.71, k, = 1.5, k,, = 3.1 17. Broken lines indicate negative values; solid lines positive values except for the line adjacent to the broken lines, which indicates zero. Reprinted with permisison of Cambridge University Press. R. M. Clever and F. H. Busse (1992), J. Fluid Mech. 234, 51 1-527. Copyrighted 1992 by Cambridge University Press.

concept was advanced first by Townsend (1972) for the turbulent boundary layer over a heated horizontal plane. It has been shown to be applicable to fully developed, turbulent flow in a heated channel by Fukui et al. (1991). The resulting large-scale structures were found to have an aspect ratio in the spanwise direction of about 1.3 for -0.341 < Ri < -0.111, with about six rolls existing across the span. An individual roll existed over a length in the streamwise direction equal to about 10 to 15 times the depth.


85

B. COUETTEAND POISEUILLE FLOWS WITH SIDE BOUNDARIES Up to this point, fully developed flows have been considered for the case when the fluid layer is effectively infinite in both the x and y directions, so that = @). In laboratory experiments, however, channels with finite lengths and widths are used, so that 0 = u(x, y , z) in general. The variation of with x is a feature of the developing flow, and the onset of R.B. convection in this kind of flow will not be discussed in detail. Suffice it to say at the moment that the developing flow in the case of a channel is more stable than the fully developed flow, and so one can consider the

e

(a)

*’ .4

.3

.2

.1

N

.o

-. 1 -. 2

-. 3 -. 4 -.5 -1.0-.9

-.8

-.7

-.6

-.5

-.4

-.3

-.2

-.1

.O

MEAN TEMPERRTURE FIG. 30. Variation of the mean temperature and mean velocity for Couette flow at Ra = 150,000, Pr = 0.71: (a) mean temperature 0 : Re = 0; 0: Re = 1028; (b) mean velocity, Re = 1028. Reprinted with permission of Cambridge University Press. J . A. Domaradzki and R . W . Metcalfe (1988), J. Fluid Mech. 193, 499-531. Copyrighted 1988 by Cambridge [figure continued over page] University Press.

86

R. E. Kelly

MEAN

VELOCITT

FIG.30-continued.

stability of the fully developed flow alone without being unrealistic. The existence of sidewalls, however, means that 0 = u ( y , z) even for the fully developed flow. The basic temperature can still be assumed to vary linearly with z if, say, the upper and lower walls are each isothermal whereas the sidewalls are insulated (but departures from adiabatic sidewalls can lead to the formation of rolls near these walls even when Ra is subcritical in the normal sense; see, e.g., the paper by Fung el al. (1987) and the references given there). Boundary conditions on the temperature disturbance must, however, be satisfied on the sidewalls as well as on the upper and lower walls. The net result is that two coupled P.D.E.s in y and z must be solved instead of (2.12a, b) for the linear problem. For the case of a finite width channel with an infinite dimension in the x direction, it was mentioned earlier that linear theory predicts that convection will begin for Re = 0


87

in the form of rolls that are periodic in the x direction. Luijkx and Platten (1981), Chana and Daniels (1989), and Daniels and Ong (1990) have = 0 by use of the Galerkin computed such solutions accurately for method and have determined the amount by which the critical Rayleigh number is increased above the result for a doubly infinite layer due to the stabilizing constraint of the sidewalls. When a mean flow is present in the x direction, we know that longitudinal rolls are the most unstable form of disturbance for a doubly infinite layer because the shear stabilizes transverse waves. For flow in a channel with a finite aspect ratio, we therefore have two opposing trends; mean shear in the x direction tends to promote longitudinal rolls, whereas the geometry tends to favor transverse rolls. One might therefore expect that for a given aspect ratio, transverse rolls would be favored at relatively small values of Re whereas longitudinal rolls would be favored at larger values of Re. Thus, a cross-over value of Re would seem to exist, say Re', at which the pattern changes from one of transverse rolls to one of longitudinal rolls, as discussed first by Luijkx et al. (1981); see also Chapter VIII of the book by Platten and Legros (1984). In order to predict the value of Re', the effects of mean shear acting in the y-z plane as well as in the x-z plane must be considered as far as transverse rolls are considered. As mentioned earlier, the onset of longitudinal rolls is not affected by the fact that U varies with y as well as with z, and so the value of (Ra,), is the same as the value when Re = 0, as can be found from one of the papers just cited. The corresponding value of (Ra,.), for transverse rolls is, of course, affected by both mean shear components. In order to gain insight into the novel effects of mean shear in the y direction, we first discuss the result of Davies-Jones (1971) for a basic flow that is still in the x direction but is a linear function of y, not z. Sidewalls located a distance Ly apart are assumed to move oppositely to each other with a velocity difference U*. If we retain the height ( H ) as a length scale, then the basic flow is A A (2.38) - 5 y 5 -, I / * ( y ) = U,*A-'y, 2 2 where A = L y / H is an aspect ratio. If a Reynolds number is defined as Re = U,*L,,/v, then the x momentum equation for a general small disturbance is 1 au (2.39) - - + ReA-'(y? + u ) = --an ax + v2u. Pr at ax

R . E. KeIb

The other disturbance equations are the same as Eqs. (2.8b-e) except that we have ReA-’ substituted for Re. The problem investigated by Davies-Jones (1971) concerns the case when the sidewalls are rigid, nonslip, and perfectly conducting and the top and bottom surfaces at z = 0 and 1 are taken to be nondeformable, stress-free, and perfectly conducting. A linear variation with z of the basic temperature is still possible if it is assumed also to be imposed on the sidewalls. Calculations were made for three values of the Prandtl number, namely, Pr = 0.01, 0.7, and 6.0. The stress-free isothermal boundary conditions allowed Davies-Jones to assume a sinusoidal behavior in the z direction for the disturbance; a Galerkin expansion in the y direction was then used to solve the eigenvalue problem. Davies-Jones found that, as the Reynolds number increased from zero, the critical wavenumber in the x direction (denoted here as k) decreased rapidly, while the critical Rayleigh number increased rapidly. The rate of change is greatest for large values of the Prandtl number and aspect ratio A. For large enough Reynolds numbers, however, the values of critical wavenumber seem to approach limits asymptotically, so that, at least for sufficiently small values of Pr or large values of A, convection can begin in the form of transverse disturbances, albeit with much larger wavelengths in the x direction than for the case Re = 0. The variation of the critical Rayleigh number with Pr = 0.7 is shown in Fig. 31 for various values of aspect ratio and Reynolds number. The values of the critical Rayleigh number for longitudinal rolls for the various aspect ratios is denoted by the “k = 0” values. For A = 4 and 2, Ra, approaches the critical values for longitudinal rolls, indicating that they will be preferred at sufficiently large values of the Reynolds number. For A = 1.0 and 1/2, however, the curves seem to level off before the critical value of Rayleigh number for longitudinal rolls is reached, indicating that transverse disturbances are preferred, at least for Re c 100. Davies-Jones found that the Reynolds stress ( u v ) can be such that transverse disturbances can extract energy from the basic flow for the unstably stratified case and that this energy transfer mechanism can actually be more important than the transfer of potential energy to the disturbance. In their book, Platten and Legros (1984) present some numerical results concerning Re’ for Poiseuille flow in a finite aspect ratio channel. With Pr = 1, two values of A were chosen, namely, A = 2 and 5.2; for A = 2, Re’ = 6.96, whereas for A = 5.2, Ref = 4.83 if Pr = 1. As Pr increases for fixed A , Re’ occurs at smaller values of Reynolds numbers for most


89

Sx 10'

Ru(k=O)

Al=

t

# . j -

2.a

5 x 10' -

rt;

Ra(k=O)

-

2=1.0

D

FIG. 31. Critical Rayleigh number versus Reynolds number for Pr = 0.7 and various aspect ratios where A^ = A-' for horizontal Couette flow. Reprinted with permission of Cambridge University Press. R. P. Davies-Jones (1971). J. Fluid Mech. 49, 193-205. Copyrighted 1971 by Cambridge University Press.

laboratory situations, Some experimental results reported by Platten and Legros (1984) for a silicone oil with Pr = 453 with A = 5.2 verify the numerical result that Re' is indeed very small for this case. More recently, Ouazzani et al. (1990) have presented more detailed data for water in a channel with A = 3.6. Their results are shown in Fig. 32 and certainly indicate that transverse rolls occur for small values of Re (0.6). However, no data are presented in that figure that confirm that a unique value of Re' exists. It should be understood that the notion of Re' is based really on results obtained from the linear stability problem, which concerns infinitesimal disturbances. For thermal convection, the very fact that patterns are observed implies that the disturbance has finite amplitude and that nonlinear effects might be important. Later in their paper, Ouazzani et al. (1990) indicate in their Fig. 18 that a region of mixed-mode convection exists that allows a gradual transition between transverse and longitudinal

R . E. Kelly

90

-1

1qw I

I

Re FIG.32. Observations of laminar flow (O),transverse rolls (A), and longitudinal rolls (X) for Poiseuille flow in a heated channel whose width is less than its length. Reprinted from Int. J. Heat Mass Transfer 33, M. T. Ouazzani et al., “Etude exptrimentale de la convection mixte entre deux plans horizontaux A temptratures difftrentes-11,” copyright 1990, with kind permission from Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 OBW, UK.

rolls. Hence, a unique value of Re+ does not exist, at least in the distinct way suggested by linear theory. Evans and Greif (1991) have obtained solutions of the full governing equations by means of finite-difference calculations that exhibit mixed-mode convection for special choices of the flow parameters. Chiu et al. (1987) conclude that the longitudinal rolls remain steady in channel flow up to Ra = 8300 for Reynolds numbers (based on an average velocity) between about 60 and 120. As seen from Fig. 23(a), this value is above the value of Ra required for secondary instability for Re = 60 for a doubly infinite layer and far above the value required at Re = 120. They claim that the stabilization is due to the presence of a mean shear, but the results shown in Fig. 23(a) cast doubt on this claim. They used an aspect ratio of 10, which might be large enough so as not to influence the onset of the rolls but which might cause the onset of secondary instability to differ substantially from the horizontally unbounded case. It has already been noted that Cole (1976) found that the onset of waviness for Taylor vortices is much more sensitive to changes in aspect ratio than is the point of onset for the vortices themselves. At this time, no theoretical predictions of the effect of sidewalls upon secondary instability of longitudinal rolls in a shear flow exists to the knowledge of the author.


91

The fact that transverse rolls have been both predicted and observed to be more unstable than longitudinal rolls for relatively small values of the aspect ratio and the Reynolds number has stimulated the investigation of such rolls by themselves. Ouazzani et al. (1989, 1990) have both measured and calculated the nondimensional wavespeeds (based on a convective time scale) of finite-amplitude transverse rolls for Re < 9 and found that they are independent of Re, which is confirmation of the idea discussed previously that the wavespeed is related simply to an average nondimensional convection speed. However, they also found that the wavespeeds decrease slightly as the Rayleigh number increases. Evans and Greif (1989) have calculated finite-amplitude 2D solutions of the full non-Boussinesq equations for a gas over a wider range of Ra and Re, but they concentrate on presenting both local and average values of the Nusselt number as a function of the flow parameters. In the experiment by Luijkx et al. (1981), only transverse rolls could be visualized. In one run with Ra > Ra,, such rolls could be seen only in the downstream end of the channel. It was inferred that longitudinal rolls occurred in the upstream region; see also Ouazzani et al. (1990). Thus, there seems to be the possibility of both modes occurring in the channel but at different streamwise locations as well as at the same streamwise location as discussed earlier in connection with mixed-mode convection. These results were explained by Brand et al. (1991) by means of an investigation of amplitude equations for the transverse and longitudinal rolls that allow for their interaction as well as nonlinear equilibration for Ra > Ra, . Although the equations are not derived rigorously and have some arbitrarily chosen coefficients, their solutions exhibit qualitatively the same behavior as seen in the experiments. The explanation is based on the concepts of convective and absolute instabilities, as introduced in Section 1I.A. In Fig. 33, the boundaries for each kind of instability for each mode are shown, first based on linear theory and then allowing for nonlinear interaction. Curves 1 and 2 are convective and absolute boundaries, respectively, for longitudinal rolls based on linear theory, whereas curves 3 and 4 are similar boundaries for the transverse rolls. Coefficients in the amplitude equations were chosen so that the transverse rolls appear first at low values of Re, thereby mimicking one effect associated with the channels’ sidewalls. Say that Re > Re+, however, so that longitudinal rolls occur first. As Ra increases, these rolls acquire a finite amplitude, and so the state existing in the neighborhood of curves 3 and 4 is not the state assumed in their determination. When the effect of the longitudinal rolls is included in the analysis,

R . E. Kelly

92

0

I 2 REYNOLDS NUMBER R

3

FIO.33. Instability boundaries for transverse and longitudinal rolls for flow in a narrow channel, as a function of Reynolds number. See text for explanation of numbers. Reprinted with the permission of the American Institute of Physics. H. R. Brand et 01. (1991), Phys. Rev. A 43, 4262-4268.

the longitudinal rolls are found to become convectively and absolutely unstable to transverse rolls along curves 5 and 6, respectively. These results imply that the actual state existing in the region between curve 3 and curve 6 depends upon how access to that region occurs, i.e., whether Re > Re+ before Ra, is crossed or whether Re increases from small values for Ra above Ra,. This conclusion agrees with the experimental finding of Ouazzani et al. (1990), who found that a region exists in the Ra-Re plane in which the observed structure depends upon the initial conditions. It also implies that hysteresis can exist in going from one mode to the other. By allowing for streamwise dependence as well as temporal dependence in the amplitude equations, Brand et al. (1991) were able to investigate how one state is replaced by the other as, say, Ra changes for a fixed value for Re > Re'; a typical result is shown in Fig. 34. The transverse rolls appear first at the downstream end of the channel and gradually replace the longitudinal rolls as Ra increases. It is clear that states are predicted in which longitudinal rolls occupy the upstream portion of the channel while transverse rolls occupy the downstream portion, in accordance with the conclusion of Luijkx et al. (1 981). Accepting the fact that transverse disturbances can be more unstable for narrow channels, the linear stability results of Fujimura and Kelly (1988)

Onset and Development of Thermal Convection o

.

0

5

30

60

93

s

90

POSITION x Fro. 34. Patterns for fiied Reynolds number Re = 2 at a fixed noise level as a function of (Ra/Ra,,,) - 1. Moduli of the amplitudes of transverse rolls are shown by solid lines, whereas moduli of amplitudes of longitudinal rolls correspond to dashed lines. (a) E = 0.23, (b) E = 0.24, (c) E = 0.25, (d) E = 0.265. Reprinted with permission of the American Institute of Physics. H. R. Brand et al. (1991), Phys. Rev. A 43, 4262-4268. E =

and Mohamed and Viskanta (1989) for the strictly two-dimensional case suggest that interesting nonlinear phenomena might occur for low Pr and moderate values of Re- O(100). In this range, two distinct wavenumbers become unstable at about the same values of the control parameters, and so the corresponding modes can interact when weakly nonlinear effects are considered. Although variation of the mean velocity in the spanwise directions means that these 2D results are not quantitatively applicable, the existence of neutral curves with multiple extrema might still occur when

u = U(y,2). The foregoing results pertain to flow in straight channels. At least one experiment, by Kimura et al. (1971), has concerned flow in a horizontal circular annulus in which the upper wall is moved relative to the lower, and a similar experimental set-up is currently being used by Zhu and Krishnamurti (1992). The advantage to this apparatus is that entrance and exit effects are eliminated. On the other hand, secondary flow effects occur,

R . E. Kelly

94

as discussed by Kimura et al. (1971), that give rise to an even more complicated base flow in general. Pocheau et al. (1987) have also used a circular annulus but one in which fluid is introduced at one station and then withdrawn at the station at the opposite end of the diameter. Hence, the incoming flow divides into two streams that flow through opposite sides of the annulus. They observed transverse rolls whose diameters decreased in the flow direction and demonstrated that the local wavenumbers correspond to points along the Eckhaus marginal stability boundary.

C. OTHERFULLYDEVELOPED FORCED FLOWS In this section, the onset of Rayleigh-BCnard convection will be discussed for other basic flows that are exact solutions of the governing equations, which will serve here as a definition of “fully developed.” For the most part, parallel flows will be discussed, but the case of stagnation point flow will also be mentioned. From here on in this chapter, less detail will be presented. The main idea will be to discuss how the basic concepts and results presented already pertain to other flows and to give a reasonably upto-date, though hardly complete, survey of other investigations. 1. Flows with Directional Change in Shear with Height

Say that we consider a basic flow that is nonplanar of the form (2.40)

so that, in general, the mean shear stress varies in regard to both magnitude and direction when z varies. The disturbance equations analogous to (2.12a,b) are

[a + i Re (k,0 + kyv) - ($- k’)] Pr

($2

-

k2)

(2.41a) and CJ

where k2 = k,’

+ i Re Pr (krU + k y V ) -

+ ky”.

(2.41b)


95

Asai (1972) noted that Eqs. (2.41a, b) can be written in the form (2.12a, b) by defining

I?,o= k , u + kyV,

(2.42)

so that, in accordance with the results of Section II.A, we anticipate that a preferred pattern for convection will be given if we can satisfy the condition d - -((k,U) = 0 dz

=

dU k,dz

+k

dv dz

-

(2.43)

v

for certain k, and ky (for instance, with = 0 we take k, = 0). One can then find a direction such that the shear does not affect the onset of R.B. convection. For the flow

Qz) = az,

Qz)

=

b(1 - z )

(2.44)

such that both the direction and magnitude of the overall shear is independent of z although the direction and magnitude of the velocity varies with z , (2.43) is satisfied if

k,a

=

kyb.

(2.45)

Thus, the usual results are obtained for the limiting cases u = 0 or b = 0 and, if a = b = 1, “longitudinal rolls” will occur for Ra > Rac,Oat an angle of 45” to the x axis. For a # 0 and b # 0, however, the rolls will not be stationary but will propagate in a direction normal to the roll axes with a wavespeed proportional to the corresponding component of the average velocity because the convective term k, 0 + ky V is in general nonzero. Asai (1972) has also calculated the amplification rate as a function of k,/k, for various b/a to show that a preferred direction does indeed exist. For the flow (2.44), convection occurs for all Ra > Ra,,o, because a direction can be defined along which the effects of the mean shear are zero. For general flows, however, Ra, will be greater than Ra,,o because such a direction cannot be found; i.e., disturbances of any orientation will experience the stabilizing effects of shear. This concept forms the basis of an idea advanced by Kelly (1992) and investigated in Kelly and Hu (1993) for stabilizing thermal convection by means of periodic oscillations of one or both surfaces. Stabilization occurs as long as the oscillation produces a nonplanar shear flow regardless of the coordinate system used. For small amplitude oscillation, it was found that Ra, > Rac,Owith the degree of stabilization increasing with Pr. The analysis indicates that a preferred pattern occurs in general, with rolls aligning themselves in the direction of

R . E. Kelly

96

the shear component with greatest amplitude, i.e., in the x direction of lfx,,l,, > / f y z ~ m Whether ~ . this concept might be of practical use depends on whether significant stabilization of R.B. convection can be achieved for finite amplitude oscillations that still have Re sufficiently small that no shear instability occurs. The effect of oscillatory unidirectional shear upon turbulent Rayleigh-BCnard convection has been studied experimentally by Solomon and Gollub (1990). In further work, Solomon and Gollub (1991) examined the effects of steady nonplanar shear upon convection with Ra 0(107) by generating a set of vortices with vertical axes. In contrast to the case of oscillatory unidirectional shear, steady nonplanar shear was found to be capable of increasing the Nusselt number by up to 70%. The most important basic flow in this category of nonplanar flows is Ekman layer flow, which is associated with the viscous region near a no-slip surface in a rotating flow. This topic has already been reviewed by Brown (1980) with an emphasis on the atmospheric boundary layer, however, and the reader is referred to that paper. For results on the interaction of convection, shear, and rotation with relevance to geophysics and astrophysics, see Hathaway and Somerville (1986, 1987) and Busse and Kroop (1992).

-

2. Two-Dimensional Stagnation Point Flow Although stagnation point flow is a valid description of only the region near the nose of a solid body immersed in an airstream, it is a valuable exact solution of the Navier-Stokes solutions and so is discussed here. A boundary layer with constant thickness 6 is possible because the diffusion of vorticity away from the surface is balanced exactly by the advection of vorticity toward the surface. Chen et al. (1983) considered the case of a flow in the - 2 direction toward a heated solid wall located at z = 0, with gravity also pointing in the - 2 direction. They assumed a flow in the x-z plane and considered only disturbances of the longitudinal roll type (periodic in y ) . Due to the fact that the flow accelerates in the x direction, the analysis of transverse disturbances is considerably more complicated. For the case of longitudinal rolls, however, a Fourier representation in the y direction can still be used. By assuming that the x component of the disturbance velocity has the same variation with x as the x component of the basic flow (namely, x'), Chen et al. found that the disturbance equations could be reduced to a set of 0.D.E.s without further approximation (which is not true in general


97

when the base flow is represented by the Falkner-Skan set of solutions to the boundary-layer equations). In contrast to the set (2.13a-e) for a parallel base flow, all three momentum equations are now coupled with the energy and continuity equations, and so the rolls can in principle obtain energy from the base flow as well as via buoyancy. The authors found that the results correlate with the thermal boundary layer thickness S, and that a minimum value of Ra (based on S,) required for the instability of longitudinal rolls varies between approximately 37 and 63 as the Prandtl number varies between 0.7 and infinity. It would seem that an experimental check of this prediction has not yet been attempted, although it would be worthwhile doing. 3 . Channel Flows with Horizontal and Vertical Gradients of Mean Temperature

The experimental investigation of Akiyama et al. (1971) in which Ra, was first determined to be 1708 for Poiseuille flow in a large aspect ratio channel with isothermal upper and lower walls was part of a broader investigation in which a constant streamwise variation of temperature, say 7*, was imposed on both walls. The results of a theoretical investigation by the same group, headed by Professor K. C. Cheng of the University of Alberta, was published by Nakayama et al. (1970). They correlated their results with the nondimensional parameter Re 7,where 7 = r*H/AT*, and found that Ra, could occur well below the nominal value of 1708, especially for large values of Pr; see Fig. 35. Although a free convection component to the basic flow is generated due to 7*, its magnitude is negligible as long as a Rayleigh number based on 7*, say Ra, = 7 Ra, is much less than Re Pr. Only stationary longitudinal rolls were observed. However, they were observed even with stable stratification (Ra < 0) as long as Ra and T had the same sign. If this condition is satisfied, the stability boundary for Ra > 0 merges with that for Ra < 0 for large (71Re, as shown in Fig. 35. This result indicates that a new mechanism for instability has been introduced, which happens to be the same mechanism as governs baroclinic instability of atmospheric flows (see Section 45 of Drazin and Reid (1981) for a succinct discussion of this instability). Essentially, the horizontal temperature gradient means that lines of constant density in the basic state are no longer horizontal as when only a vertical gradient exists but are inclined so that buoyant energy can be released by streamwise advection as well as by vertical advection.

R . E. Kelly

98 10' 8

Jl

I

-

o

-

A

6 -

x

-

L

4

m

-

8

-

0

-

0

c

a

g 2

-

- 8 II

f

1

UNSTABLE UNCERTAIN STABLE -

A

JJ

x

x

z

x

-

4

2

cr ,1

I

1

I

I I I l l

2

FIG.35. Comparison of experimental data with theoretical stabiIity curves for longitudinal vortex rolls in a channel with both streamwise and vertical temperature gradients. p = T Re. Reprinted with the permission of the ASME. M. Akiyama et 01. (1971), J. Heat Transfer 93, 335-341,

The case when the flow is due entirely to T* will be discussed in Section 111. The complexity of the results for that case suggest that more analysis is

desirable for the case when Ra, is comparable with Re Pr.

D. THEINFLUENCE OF SURFACE CURVATURE It is well known that the flow of a constant density fluid past a concave wall can experience a centrifugal instability (see, for instance, Chapter 3 of Drazin and Reid, 1981) that gives rise to vortices similar to the longitudinal vortices discussed in this chapter. Such vortices are called Taylor vortices for flow between differentially rotating cylinders, and Dean vortices for flow in curved channels. In the latter case, Dean (1928) demonstrated that if H is the distance between the walls and R2 is the radius of the outer wall, the pertinent nondimensional parameter (the Dean number) is Re (H/R2)1'2 as long as (H/R,) Q 1.


99

1000 2000 Ra FIG. 36. Critical value of the Dean number as a function of Rayleigh number for flow in a curved channel. Reprinted with the permission of the publisher. K. Kirchgassner (1962), Ing. Archiv. 31, 115-124. 0

The combined effects of wall curvature and heating were discussed first qualitatively by Gortler (1959) and then later in more detail by Kirchgassner (1962) and Kahawita and Meroney (1974). For the case of curved channel flow, the destabilizing effect (as determined by Kirchgassner) of heating upon the Dean number is shown in Fig. 36. It should be pointed out that Kirchgassner considered only the case (H/R,) 4 1, so that the walls appear to be plane as far as the buoyancy force is concerned. This approximation should be improved upon prior to any experiment. It should be mentioned that many interesting results concerning nonlinear effects for isothermal flow in a curved channel have been obtained recently; see, for instance, Guo and Finlay (1991) and Ligrani et al. (1992). E. FLOWWITH

A

FREESURFACE OR INTERFACE

The flow of an isothermal liquid film with a free surface down an inclined wall occupies a unique place in hydrodynamic stability theory. Due to

100

R. E. Kelly

surface deformation, a quite different mode of instability can exist in addition to the usual shearing mode, which is important only at very small angles of inclination. The surface mode has the properties that it is unstable for all values of Reynolds number for the case of a vertical wall and that instability occurs first via a disturbance with infinite wavelength. The physical mechanism for instability of the surface wave mode has been discussed by Kelly et al. (1989) and Smith (1990a). Kirchgaassner (1962) first discussed the case of a heated film but considered only longitudinal roll disturbances. He showed that these become unstable when Ra = Ra,(/3 = O)/cos/3, where /3 is the angle of inclination from the horizontal direction, as one might expect. Kelly and Goussis (1982) showed that transverse rolls are more stable and further that the surface mode of instability is unaffected by heating and is the dominant mode except near zero angle of inclination for Pr O(1). For very large values of the Prandtl number such that Pr O(y-’), where y = y* AT*, y* being the coefficient of volumetric expansion, Smith (1990b) has shown that heating can destabilize the surface mode if the free surface is nearly isothermal. Gummerman and Homsy (1974) investigated interfacial instability in a horizontal Couette flow that is stratified in two distinct ways. The system consists of two immiscible liquids, with the upper one being less dense and so promoting stable stratification. At the same time, the fluid is heated from below, thereby promoting unstable stratification. Without heating, this flow can experience a long-wave interfacial instability if the viscosities of the two layers are unequal, as shown by Yih (1967). Gummerman and Homsy (1974) concluded that with heating longitudinal rolls would be favored if the density jump across the interface is relatively large.

-

-

111. Rayleigh-BCnard Convection in Fully Developed, Thermally Induced Flows

In this section, we will consider the stability of mean flows that are thermally induced; i.e., they are of the “natural” or “free” convection type and occur due to horizontal temperature and density variations. There are two basic configurations that have been studied. In the first (Case I), the fluid is contained in a tilted box. The lower surface of the box is heated whereas the upper surface is cooled. Endwalls are generally assumed to exist, although their effects are assumed to be localized for sufficiently large aspect ratio containers. Thus, a parallel flow is assumed, with flow


101

upwards along the heated wall and downward along the cooled wall, so that there is zero net mass transport (although work has also been done on similar problems involving a nonzero net mass flow). The second configuration (Case 11) consists of a horizontal box heated from below and assumed to have also a horizontal gradient of temperature imposed on one or both walls. So much has been done in connection with both problems that a review comparable in length to Section I1 would be required to cover the results adequately. No such detailed review will be attempted, but it does seem desirable at least to point out how some of the phenomena already discussed also occur for these flows. Such a discussion is also appropriate because interest in nonlinear aspects of thermally stratified shear flows can arguably be said to date from the investigations of John Hart (1971a, b) of Case I. Also, some of the best comparisons between theoretical and numerical results exist for this case.

A. CASEI: FLOW IN

A

TILTEDBox HEATEDFROM BELOW

It is easiest to think about a convection chamber with isothermal walls that is initially horizontal and has a corresponding critical Ra,, say, Rac,O (= 1708 for a large aspect ratio chamber). Now incline the chamber at a small angle fi to the horizontal. For subcritical Ra, an overall Hadley circulation is generated by baroclinic effects. From what has been said already in Section 11, we would expect that as Ra is increased past the critical value, Rayleigh-BCnard convection would take place in the form of longitudinal rolls. This result was predicted by several investigators almost simultaneously: Gershuni and Zhukhovitskii (1969), Liang and Acrivos (1970), Kurzweg (1970), and Hart (1971a). At the other extreme, the stability of the flow in a vertical slot with differentially heated sidewalls has also been studied extensively; see Chapter X of Gershuni and Zhukovitskii (1976). The flow has an inflection point at the midpoint of the slot and becomes unstable to transverse (stationary) rolls at Pr c Pr, , and to transverse waves for Pr > Pr, , where Pr, = 12.45. For a certain range of angles away from the vertical, the 2D shear instability remains the most critical mode. However, this range is only a few degrees for large values of Pr. For Pr = 0.71, longitudinal rolls have been observed for all p c 75" approximately; see Fig. 6 of Hollands and Konicek (1973). Thus, the angle of inclination plays a role somewhat similar to that of the Reynolds number in forced flows, and it is useful to determine the

102

R . E. Kelly

crossover angle when the most critical mode changes in form. Hassab and Ozisik (1979, 1981) have investigated theoretically the influence of varying thermal boundary conditions upon the crossover angle. For a convection box with an aspect ratio of greater than 30 Pr, Hollands and Konicek (1973) state that the assumption of a linear temperature variation across the gap between the heated surfaces for the undisturbed state is valid. Using air, Hollands and Konicek (1973) were able to satisfy this criterion in their apparatus and so found for longitudinal rolls that the theoretical prediction that Ra, = Ra,,o/cos /?was confirmed by the experimental results up to the crossover angle. Hart (1971a) used water and found that the results correlated for angles less than about 80". Experimental verification of the neutral stability curves for transverse disturbances was less satisfactory. For narrow boxes, the presence of sidewalls can favor the onset of thermal convection in the form of transverse rolls, just as for channel flows. The presence of endwalls tends to allow for the formation of a streamwise temperature gradient that can drastically affect the mean flow, as documented by Hart (1971a). Kirchartz and Oertel (1988) have compared fully numerical solutions with their own experimental results for the same configuration. Using a convection box with a length to width to height ratio of 11.7 :25 : 1 or 17.2 : 36.7 : 1, Hart (1971a) observed longitudinal rolls for all angles up to the angle at which the shear instability occurs. In contrast, with a ratio of 10 :4 : 1, Kirchartz and Oertel (1988) found for a silicone oil with Pr = 1780 that transverse rolls precede the onset of longitudinal rolls for small angles of inclination ( p < 25"). For 25" c p < 30", a region of mixed-mode convection occurs, similar to that observed by Ouazzani et al. (1990) for flow in channels with a width much less than the length, at least for moderate values of Re. Pure longitudinal rolls then occur for B > 25", but their amplitude diminishes as increases further, so that they were not experimentally observed for p > 65". Although the rolls are still unstable on the basis of linear theory for sufficiently large Ra, the neutral stability curve rises very steeply, and so Kirchartz and Oertel(l988) found it difficult to determine the stability boundary by their experimental techniques. Of course, the flow becomes stable in respect to R.B. convection for any fixed value of Ra as /Iincreases sufficiently, which is the trend that Yang et al. (1987) found in their numerical calculations for a box with dimensions 7.5 :0.5 : 1. The properties of finite-amplitude longitudinal rolls have been computed by Clever and Busse (1977) for an inclined layer of infinite extent and by


103

Ozoe et al. (1983) and Kirchartz and Oertel (1988) for boxes with finite length and width. For an infinite layer, the tendency to form a nearly isothermal core results in a reduced mean flow, which, after all, is basically driven by the temperature gradient. Clever and Busse (1977) found as a consequence that the mean wall shear stress is reduced for Ra > Ra,, in contrast to the result for forced flows (Figs. 24 and 25). They also found that a preexisting state of longitudinal rolls has a slightly stabilizing influence upon the onset of transverse waves that occur at large angles of attack. Before going on, it should be noted that the stability analysis of the Prandtl buoyancy layer flow (see Section 7.4.2 of Turner (1973) for a discussion of this basic flow) gives rather different results than the results for a tilted slot. For the buoyancy layer, the thickness of the boundary layer depends upon the gradient of the ambient stratification, which is fixed as the boundary is heated. For the tilted box, the streamwise density gradient and the resulting boundary layer structure is dependent upon the wall heating. This difference seems to be crucial because Iyer (1973) found for Pr = 0.72 that transverse waves are preferred for all angles of inclination greater than zero for the buoyancy layer. A weakly nonlinear analysis by Iyer and Kelly (1978) indicates that supercritical finite wave states are possible, and the Nusselt number associated with these finite amplitude solutions was determined. Hart (1971a, b) noticed in his experiment using water that “meanders” developed on the longitudinal vortices at higher Ra, with a typical streamwise wavenumber about equal to 0.57 that of the longitudinal rolls. This observation motivated Clever and Busse (1977) to investigate whether or not the wavy instability might be the cause of the meanders. For this flow, the wavy instability can begin as a disturbance with infinite wavelength (as for forced flows) at relatively small angles of inclination or with finite wavenumber at larger angles of inclination. In Fig. 9 of Clever and Busse (1977), good agreement between the predicted onset of the wavy instability and Hart’s experimental data is obtained at angles of inclination greater than 60°, due partly to this reason. The wavy instability sets in at lower values of Ra for low Pr fluids than for high; for instance, O(10) for Pr = 1 but -O(lOOO) for Pr = 10. Ruth et al. Ra, - Ra, (1980a,b) investigated longitudinal role instability experimentally for air and indicated in their Fig. 10 that for angles greater than 12.5” a direct transition from the base state to wavy vortices occurred. Presumably this result is not strictly true, although the region of stable rolls for slightly supercritical values of Ra is so small for the case of air that the discrepancy

-

R . E. Kelo

104

is a moot point. An important result of this experimental investigation was that the onset of wavy rolls gave rise to a Nusselt number less than the value that would be obtained with straight longitudinal rolls, as mentioned earlier in connection with numerical results for forced flows. Recently, Busse and Clever (1992) have computed finite-amplitude properties of wavy vortices for the inclined layer and obtained a similar result. They also investigated the stability of the wavy rolls and found that their region of stability is also small and that at low angles of inclination the wavy rolls begin to drift in the y direction, presumably corresponding to the meanders of Hart. At higher angles of inclination, a vacillation between a state of longitudinal rolls and a state of nearly transverse oriented vortices is predicted. This vacillation has not yet been observed in an experiment. In a more recent experiment by Shadid and Goldstein (1990) involving a high Prandtl number liquid, visualization of the planform of convection was obtained by means of a temperature-sensitive liquid crystal. At very low angles of inclination, a square-cell pattern was obtained due to boundary effects that, as Ra was increased, gave rise to a superposition of steady cross-rolls and longitudinal rolls and then unsteady longitudinal rolls. At higher angles, steady longitudinal rolls developed that were replaced by transverserolls at the crossover angle. No mention of mixed-mode states near the crossover angle was made. Fujimura and Kelly (1993) have shown that such states are possible in the neighborhood of Ra, , at least for a layer of infinite extent with isothermal boundaries. Shadid and Goldstein (1990) also found that the wavenumber of longitudinal rolls increases with the angle of inclination from the horizontal and described two distinct mechanisms for wavenumber adjustment.

B. CASE 11: FLOW

A

HORIZONTAL BOX HEATED AT ONEEND

In this situation, one considers a shallow horizontal box with differentially heated end walls, so that a horizontal gradient of the temperature occurs and generates an overall circulation. Hart (1972) first studied the stability of this flow. For the case of perfectly conducting boundaries, so that the temperature of each wall is the same linear function of x, the fluid is unstably stratified in the regions near each horizontal wall. Gershuni et al. (1974) presented detailed numerical solutions that predict the occurrence of steady longitudinal rolls for all Pr > 0.24. For Pr < 0.24, steady transverse rolls are predicted that are associated with a hydrodynamic


105

instability. Weber (1978) incorporated the effects of a vertical temperature gradient and found for the case of rigid surfaces that the Rayleigh number based on the vertical temperature difference is increased by the horizontal gradient. Longitudinal rolls are again preferred for large values of Pr. Weber (1978) has also calculated the relative values of terms in the disturbance energy balance for various Pr. For the case of insulating horizontal boundaries, the fluid exhibits stable stratification in the vertical direction but can still be unstable; see Hart (1972, 1983). This problem has a very rich structure. The results have been summarized by Wang and Korpela (1992), who also discuss the nature of the secondary instabilities of transverse rolls. IV. Concluding Remarks

The subject of thermal convection in shear flows is rich in results that have both basic interest and practical importance in science and engineering. Although the field has matured greatly during the part 15 years, many topics still need study. For instance, the effects of channel sidewalls on both the initial instability and the instability of longitudinal rolls need to be clarified both theoretically and experimentally. More effort should be devoted to the use of controlled disturbances in studying the onset of instability due to the convective nature of the instability. Such studies might prove useful also in suggesting how turbulent convection can be promoted or postponed in shear flows. It is hoped that this review of Rayleigh-BCnard convection in fully developed flows will be useful to those investigators interested in convection in non-Newtonian or conducting fluids, in porous media, and in developing flows as well as to those interested in other types of convection, such as thermocapillary and double-diffusive convection. Although various investigations have already been made in these areas, many of the phenomena discussed here for fully developed flows have not yet been studied. Acknowledgments

The modern version of this chapter was started while the author was visting the Japan Atomic Energy Research Institute while on sabbatical leave from UCLA. The author is grateful to Dr. K. Fujimura for organizing this visit. The review was concluded at UCLA with support from the

R . E. Kelly

106

National Science Foundation under Grant CTS-9123553. The author is especially grateful to his editor, Professor T. Y. Wu, for his exceptional patience and continual encouragement during a period of great difficulty for the author. Finally, the author wishes to express his gratitude to Kirsten Maclellan for her excellent assistance in preparing the manuscript. Ref etences Akiyama, M., Hwang, G. J., and Cheng, K. C. (1971). Experiments on the onset of longitudinal vortices in laminar forced convection between horizontal plates. J. Heat Transfer 93, 335-341. Asai, T. (1964). Cumulus convection in the atmosphere with vertical wind shear: Numerical experiment. J. Meteor. SOC. Japan 42, 245-259. Asai, T. (1970). Three-dimensional feature of thermal convection in a plane Couette flow. J. Meteor. SOC. Japan 48, 18-29. Asai, T. (1972). Thermal instability of a shear flow turning the direction with height. J. Meteor. SOC.Japan 50, 525-532. Avsec, D. (1937). Sur les formes ondulees des tourbillons en bandes longitudinales. Comptes Rendus 204, 167-169. Avsec, D., and Luntz, M. (1937). Tourbillons thermoconvectifs et electroconvectifs. Meteorologie 3, 180-194. Babcock, K. L., Ahlers, G., and Cannell, D. S. (1991). Noise-sustained structure in Taylor-Couette flow with through flow. Phys. Rev. Lett. 67, 3388-3391. Benard, H., and Avsec, D. (1938). Travaux rCcents sur les tourbillons cellulaires et les tourbillons en bandes: Application a l’astrophysique et a la meteorologie, J. Physique et Radium 9, 486-500. Bolton, D., (1984). Generation and progagation of African squall lines. Quart. J. Roy. Met. SOC. 110, 695-721. Brand, H. R., Deissler, R. J., and Ahlers, G. (1991). Simple model for the Benard instability with horizontal flow near threshold. Phys. Rev. A 43, 4262-4268. Brown, R. A. (1980). Longitudinal instabilities and secondary flows in the planetary boundary layer: A review. Rev. Geophys. Space Phys. 18, 683-697. Brunt, D. (1951). Experimental cloud formation. In: Compendium of meteorology. American Meteorological Society, Boston, pp. 1255-1262. Busse, F. H. (1978). Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929- 1967.

Busse, F. H. (1981). Transition to turbulence in Rayleigh-Benard convection. In: Hydrodynamic instabilities and the transition to turbulence (H. L. Swinney and J. P. Gollub, eds.). Springer-Verlag, Berlin, pp. 97-137. Busse, F. H. (1991). Numerical analysis of secondary and tertiary states of fluid flow and their stability properties. Appl. Sci. Res. 48, 341-351. Busse, F. H., and Clever, R. M. (1979). Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319-335. Busse, F. H., and Clever, R. M. (1992). Three-dimensional convection in an inclined layer heated from below, J. Engr. Math. 26, 1-19. Busse, F. H., and Kropp, M. (1992). Buoyancy driven instabilities in rotating layers with parallel axis of rotation. Zeif. angew. Math. Phys. 43, 28-35.


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Busse, F. H., and Whitehead, J. A. (1971). Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 41, 305-320. Chana, M. S., and Daniels, P. G. (1989). Onset of Rayleigh-Btnard convection in a rigid channel. J. Fluid Mech. 199, 257-279. Chandra, K. (1938). Instability of fluids heated from below. Proc. Roy. Soc. London A 164, 231-242.

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R. E. Kelly Fujimura, K., and Kelly, R. E. (1988). Stability of unstably stratified shear flow between parallel plates. Fluid Dyn. Res. 2, 281-292. Fujimura, K., and Kelly, R. E. (1993). Mixed mode convection in an inclined slot. J. Fluid Mech. 246, 545-568. Fukui, K., Nakajima, M., and Ueda, H. (1983). The longitudinal vortex and its effects on the transport processes in combined free and forced laminar convection between horizontal and inclined parallel plates. Int. J. Heut Mass Trumfer 26, 109-120. Fukui, K., Nakajima, M., and Ueda, H. (1991). Coherent structure of turbulent longitudinal vortices in unstably-stratifiedturbulent flow. Int. J. Heat Mass Trunsfer 34, 2372-2385. Fung, L., Nandakumar, K.,and Masliya, J. H. (1987). Bifurcation phenomena and cellularpattern evolution in mixed-convectionheat transfer. J. Fluid Mech. 177, 339-357. Gage, K. S., and Reid, W. H. (1968). The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33, 21-32. Gallagher, A. P., and Mercer, A. McD. (1965). On the behavior of small disturbances in plane Couette flow with a temperature gradient. Proc. Roy. SOC. London A 286, 117-128. Gebhart, B., Jaluria, Y.,Mahahan, R. L., and Sammakia, B. (1988). Buoyancy-inducedflows und transport. Hemisphere Publ. Corp., New York. Gershuni, G. Z., and Zhukhoitskii, E. M. (1969). On the stability of plane-parallel convective motion to three-dimensional disturbances. Prikl. Math. Mekh. 33, 968-000. Gershuni, G. Z., and Zhukhovitskii, E. M. (1976). Convective stability of incompressible fluids. Keter Publ. House, Jerusalem. Gershuni, G. Z., Zhukhovitskii, E. M., and Myznikov, V. M. (1974). Stability of planeparallel convective fluid in a horizontal layer relative to spatial perturbations. J. Appl. Mech. Tech. Phys. 5, 706-708. Gortler, H. (1959). Uber ein Analogie zwischen den instabilitaten laminaren Grenzschichtstromungen an konkaven Wanden und an erwarmter Wanden. Ingen. Arch. 28, 71-78.

Graham, A. (1933). Shear patterns in an unstable layer of air. Phil. Truns. Roy. SOC.A 232, 285-296.

Gummerman, R. J., and Homsy, G. M. (1974). Convectiveinstabilitiesin cocurrent two phase flow: Part I. Linear stability. AIChE J. 20, 981-988. Guo,Y.,and Finlay, N. H. (1991). Splitting, merging and wavelength selection of vortices in curved and/or rotating channel flow due to Eckhaus instability. J. Fluid Mech. 228, 661-691.

Hall, P., and Morris, H. (1992). On the instability of boundary layers on heated flat plates. J. Fluid Mech. 245, 367-400. Hanna, S. R. (1969). The formation of longitudinal sand dunes by large helical eddies in the atmosphere. J. Appl. Meteor. 8, 874-883. Hart, J. E. (1971a). Stability of the flow in a differentially heated inclined box. J. Fluid Mech. 41, 547-576.

Hart, J. E. (1971b). Transition to a wavy vortex regime in convective flow between inclined plates. J. Fluid Mech. 48, 265-271. Hart,J. E. (1972). Stability of thin non-rotating Hadley circulations. J. Atm. Sci. 29,687-697. Hart, J. E. (1983). A note on the stability of low-Prandtl-number Hadley circulations. J. Fluid Mech. 132, 271-281. Hassab, M. A., and Ozisik, M. N. (1979). Effects of radiation and convective boundary conditions on the stability of fluid in an inclined slot. Int. J. Heut Moss Trumfer 22, 1095-1105.

Hassab, M. A., and Ozisik, M. N. (1981). Effects of thermal wall resistance on the stability of the conduction regime in an inclined narrow slot. I n t . J. Hear Moss Transfer 24,739-747.


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Hathaway, D. H., and Somerville, R. C. J. (1986). Nonlinear interactions between convection, rotation and flows with vertical shear. J. Fluid Mech. 164, 91-105. Hathaway, D. H., and Somerville, R. C. J. (1987). Thermal convection in a rotating shear flow. Geophys. Astrophys. Fluid Dyn. 38, 43-68. Hollands, K. G. T., and Konicek, L. (1973). Experimental study of the stability of differentially heated inclined air layers. Int. J. Heat Mass Transfer 16, 1467-1476. Huerre, P., and Monkewitz, P. A. (1990). Local and global instabilities in spatially developing flows. In: Ann. Rev. Fluid Mech. 22, 473-537. Hwang, G. J., and Cheng, K. C. (1971). A boundary vorticity method for finite amplitude convection in plane Poiseuille flow. Dev. Mech. 6, 207-220. Idrac, M. P. (1920). Sur les courants de convection dans I’atmosphhre dans leur rapport avec le vol B voile et certains formes de nuages. Comples Rendus 171, 42-44. Idrac, M. P. (1921). Etudes exphimentales sur le vol a voile. These de Doctorat, Paris. Ingersoll, A. P. (1966a). Convective instabilities in plane Couette flow. Phys. Fluids 9, 682-689.

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ADVANCES IN APPLIED MECHANICS. VOLUME 31

Vortex Element Methods for Flow Simulation TURGUT SARPKAYA Department of Mechanical Engineering Naval Postgraduate School Monterey. California

........................................................................................ I1. Theoretical Foundations and Numerical Schemes ........................................ A . Creation and Transport of Vorticity .................................................... B. Real Vortices and Instabilities ............................................................ C . Biot-Savart Law: The Velocity-Vorticity Relation .................................. D . Computation with Finite-Cored Connected Filaments .............................. E . Cloud-in-Cell (CIC) or Vortex-in-Cell (VIC) Method ............................... F . Body Representation and Classical Discrete Vortex Models ...................... G . Operator Splitting and Random Walk Methods ...................................... H . Diffusion Schemes ........................................................................... 1. Hamiltonian and Contour Dynamics .................................................... 111. Evolution and Applications of Vortex Element Methods ............................... A . Contra Flowing Streams .................................................................... B. Quasi-Two-Dimensional Cross Flows ................................................... C . Flow About a Circular Cylinder .......................................................... D . Flow About Sharp-Edged and Arbitrarily-Shaped Bodies ......................... E . Other Applications ........................................................................... F . General Three-Dimensional Flows ....................................................... G . Miscellanea..................................................................................... IV . Concluding Remarks ............................................................................. Acknowledgments ................................................................................. References........................................................................................... 1. Introduction

113 i16 116 125 129 131 149 155 168 174 181 184 184 190 192 201 205 205 220

222 230 230

.

I Introduction The governing equations of fluid motion are nonlinear and little is known about their solutions at high Reynolds numbers . Thus. it is only natural that there should be an adaptive numerical method that can deal directly with the interaction of parcels of vorticity in any flow situation without the penalties of grids. far-field conditions. and numerical diffusion . 113

.

Copyright 0 1994 by Academic Press Inc. All rights of reproduction in any form reserved. ISBN 0-12-002031-9

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Turgut Sarpkaya

The vortex element methods enable one to recreate the physically relevant dynamics of two- and three-dimensional incompressible flows through the use of the Lagrangian or the Lagrangian-Eulerian description of the evolution of discretized vorticity fields. Helmholtz (1858) was the first to show, in what is now regarded as one of the most important contributions in fluid mechanics, that in an inviscid fluid vortex lines remain continually composed of the same fluid elements and flows with vorticity can be modeled with vortices of appropriate circulation and “infinitesimal cross section”-quantum vortex lines or space curves of infinite vorticity. The dynamics of singular line vortices and filaments’ is a fertile territory for elegant mathematics in dealing with unbounded inviscid flows based on Euler’s equations. The solution of viscous flows based on Navier-Stokes equations requires greater creative imagination in devising numerical models, however, simply because the velocity induced by singular vortices may become unbounded as the vortices approach each other; i.e., the vortices must be desingularized or endowed with a core if they are to be used for flow simulation. Though a vortex with an invariant core is regular and numerically more manageable, it is not a solution of any equation. To produce convergent solutions and to prevent the loss of naturally occurring coherent structures, the cores of the neighboring vortex elements must be made to overlap moderately. The mathematics of the creation, convection, and diffusion of vorticity through the use of such overlapping vortices is a series of elegant approximations buttressed by stability constraints, smoothing functions, ingenious assumptions, and experimental facts. Clearly, the singularity of the vortex filaments, the need to use a large number of vortex patches (currently, with invariant cores of identical shppe and size), and the need to overlap them to preserve convergence are some of the impediments to the effective use of the vortex element methods. These strongly suggest that the core constraints imposed on vortex filaments must be relaxed to make full use of the power of Lagrangian methods, A vortex filament is defined here as a vortex tube whose total vorticity resides in its finite core; i.e., the tube is surrounded by irrotational fluid. When a vortex filament has a vanishing cross section and infinite vorticity, it is referred to as a singular vortexfilament or as a singular curved vortexfilament. A line vortex and a singular line vortex are, respectively, a straight vortex filament and a straight singular vortex filament. Definitions of line vortices, vortex filaments, and vortex tubes or vortices with finite or infinitesimal core are neither uniform nor consistent in the literature.

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particularly when topological changes and close vortex interactions demand core deformation. At present, only the interaction of a limited number of piecewise constant, deformable vorticity distributions has been studied in some detail. Many of the results that established our knowledge about vortex methods can be found in the reviews by Saffman and Baker (1979), Leonard (1980a, 1985), Aref (1983), and Sarpkaya (1989). A conscious effort is made to present an even-handed, comprehensive, and critical account of the mathematical foundations and practical applications of the vortex element methods, highlighting their weaknesses as well as strengths. The reader is frequently reminded that the neighboring vortex elements must be made to overlap moderately to achieve convergence (to sustain the growth of naturally occurring coherent structures and to prevent the birth of unphysical ones) and the cost of the computation must be reduced in accounting for the effects of viscosity, especially at large times. The reader is also reminded that other numerical methods neither enjoy fewer impedients nor have the computational advantages inherent to vortex elements methods: adaptivity, the exact explicit treatment of the outer or far-field boundary conditions, and the ability to provide logical deductions and physical insights by dealing directly with the most fundamental characteristic of the fluid motion: vorticity. As to the future, turbulence has to be the mecca of computational and experimental fluid mechanics, since the Holy Grail of turbulence has not yet been found. The prevailing body of evidence shows that turbulence involves the interaction of many degrees of freedom over a broad range of spatial and temporal scales, and an adaptive numerical method that can deal directly with the interaction of deformable parcels of vorticity should be in a better position, relative to all other methods, to lead the way to the understanding of turbulence. In developing hybrid methods, one can, if necessary, remove the inessential degrees of freedom and simulate the smallest scales of turbulence in some indirect manner, with the encouragement provided by the spectacular advances in computer technology, bearing in mind the fact that the effectiveness of a model does not necessarily increase with its complexity. This chapter is the story of vortex patches and filaments and a critical account of difficulties, limitations, and continuing efforts to improve the simulation of laminar, or preturbulent, flows through the use of vortex element methods.

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11. Theoretical Foundations and Numerical Schemes

A. CREATION AND TRANSPORT OF

VORTICITY

1. Velocity and Vorticity The curl of the velocity vector u(x, t ) is called the vorticity w(x, t); i.e., curl u = V x u = 3 or wi = &i,kUk,j. The significance of vorticity arises because the motion of an incompressible fluid can be represented as the creation and subsequent evolution of a self-interacting vorticity field. In homogeneous fluids, vorticity is produced only at the boundaries of fluid regions. Vorticity can also be generated in the interior of inhomogeneous fluids or at any free surface at which the tangential stress vanishes. Longuet-Higgins (1953, 1992) has shown that in any steady viscous flow in which the tangential stress vanishes, the vorticity at the surface must be given by w, = - 2 ~ ~ where 4 , K, is the curvature of the streamline (positive if the surface is concave) and q is the tangential particle speed. Here, the discussion will be restricted to homogeneous incompressible fluids. The vorticity transport equation for a fluid of uniform density p and viscosity v subjected only to irrotational body forces, is derived from the momentum equation in the form

D3 Dt

-

aiii at

+ u - v 3 = 3 - v u + vv23,

(2.la)

in which use is made of the relations that the divergences of the velocity and the vorticity are zero. In a suitably nondimensionalized form, one has

ai3 DC3 - - - -+ Dt at

1 ". vd = 3 . v u + -vv23,

Re

(2.1 b)

where Re is the Reynolds number. The term u * V 3 represents the rate of change due to convection of fluid. The term i3 Vu represents the rate of deformation of the vortex lines and exists only in a three-dimensional flow. The stretching of the vortex lines concentrates vorticity, increases velocity fluctuations, and decreases the minimum length scale in the flow. The last term represents the rate of change due to molecular diffusion of vorticity. The pressure gradient does not appear in (2.1), and the incompressibility constraint is automatically satisfied. In an inviscid fluid, vorticity is a kinematic property of a given fluid particle, and like matter, it can neither be created nor destroyed. The

-


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circulation of a vortex tube does not change either along the tube or in time. The vortex lines are material lines, and they can undergo only convection and deformation. Consequently, tracking the evolution of vorticity leads immediately to a Lagrangian description. In a viscous fluid, however, the vorticity produced at a boundary is carried away by convection and diffusion. This process determines the entire flow field, which in turn controls the production of vorticity. The discretized representation (approximation) of these processes, particularly in flows where the distribution of vorticity is compact, constitutes the essence of the vortex element methods. The generation of vorticity at rigid boundaries and its subsequent decay have been the subject of much discussion. Lighthill (1963) invoked the existence of vorticity sources in a region of falling pressure along the boundary and vorticity sinks (at which vorticity is abstracted at the surface) in a following region of rising pressure. This interpretation is based, albeit incorrectly, on the fact that, at the wall,

where the velocity vector is zero (see, e.g., Panton, 1984). Equation (2.2) is not apparent from (2.1) since the latter does not include pressure. Batchelor (1967) noted that “vorticity cannot be created or destroyed in the interior of a homogeneous fluid under normal conditions, and is produced only at the boundaries.” Morton (1984) has shown clearly that “vorticity generation results from tangential acceleration of a boundary, from tangential initiation of boundary motion and from tangential pressure gradients acting along the boundary, ” “vorticity once generated cannot subsequently be lost by diffusion to boundaries,” “reversal of the sense of acceleration or of the sense of pressure gradient [in (2.2)] results in reversal of the sense of vorticity generated” (which is interpreted by Lighthill as a vorticity sink), “walls play no direct role in the decay or loss of vorticity,” and “vorticity decay results from cross-diffusion of two fluxes of oppsoite sense and takes place in thefluid interior.” More succinctly, “the acceleration of the surface and the pressure gradient in a plane tangent to the surface, rather than viscosity, are responsible for vorticity production. Viscosity plays only an indirect role in enforcing the no-slip condition” (Lyman, 1990). As noted long ago by Lamb (1932, p. 578), “vortex motion cannot originate in the interior of a viscous liquid, but must be diffused inwards from the boundary. ”

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The solution of real fluid flow problems with vortex models often forces one to think (at times to defend) simultaneously the behavior of vortices in terms of viscous and inviscid concepts. Thus, it is necessary to summarize briefly some of the major differences between the characteristics of vortices in viscous and inviscid fluids. In an inviscid incompressible fluid of uniform density, subjected to irrotational body forces, the circulation around any closed material curve is invariant (Kelvin’s circulation theorem). This is a consequence of the fact that there is no diffusion and the vorticity is transported solely by the convection of the fluid. In a viscous fluid, however, the circulation about a closed contour moving with the fluid depends on the contour of integration. The rate of change of vorticity in a material volume is due solely to diffusion across the boundary of the volume. The appreciation of this difference is of importance in the determination of the vorticity generated by a body and the vorticity found in the wake, and in the estimation of the circulation of a vortex. Sample calculations of the vorticity diffusion in the wake of a cylinder are given by Eaton (1987). 2. Vorticity Transport in Two-Dimensional Flows For a two-dimensional incompressible viscous flow, subjected only to irrotational body forces, (2.1) reduces to

DZ ai3 -=-++.Vd=vV2Z, Dt at

(2.3a)

or to

a,o + a,(uo)

+ a,,(Vw)= v v 2 ~ ,

(2.3b)

and the vorticity, now perpendicular to the plane of flow, is a conservative (and diffusible) scalar quantity, attached to and transported with the fluid. The term Z V u in (2.1), representing the rate of deformation of the vortex lines, vanishes. This prevents the stretching of vorticity and, hence, the energy cascade to smaller scales (a fundamental feature of the three-dimensional turbulence). In fact, it is only in two-dimensional (nonequilibrium) turbulence that energy transfer to larger spatial scales and enstrophy transfer to smaller scales are possible through the capture and retention of vorticity in deformable coherent vortices. The small-scale generation in two-dimensional flows is achieved instead by the stretching of the vorticity gradients.

-


119

The mass conservation equation enables us to define a scalar function (stream function) such that the velocity field is given by u = V w or by

u

=

aw/ay and

v =

-awlax.

w

(2.4)

The stream function may now be related to w to derive the Poisson’s equation v 2y / = - o ,

(2.5)

which enables one to determine w from o.The usual boundary conditions for the foregoing equations are w(x, 0) = o&), no-slip and no penetration on the boundaries, and u + U at infinity. The simplification achieved by the assumption of two-dimensionality is indeed considerable. Other characteristics of the motion, however, such as conservation and diffusion of vorticity, straining and distortion of the vorticity distribution in compact regions, vorticity-gradient production (for the small-scale generation), vorticity merging (for upscale energy transfer), and the mutual-induction instability of vortices (leading to chaos) still offer a number of difficulties and require considerable perseverence for the simulation of planar flows. For the two-dimensional motion of an incompressible inviscid fluid, the right-hand side of (2.3) is zero. The reduced equation describes the evolution of a conservative (and nondiffusible) scalar that is transported solely by the convection of the fluid. The consequence of this fact is that in an inviscid incompressible fluid of uniform density, subjected to irrotational body forces, the circulation around a closed material curve consisting always of the same fluid particles’ is invarient (Kelvin’s circulation theorem) and vortex lines move with the fluid (Helmholtz’s theorem). Thus, the motion of a free3 and singular line vortex (a straight vortex filament of nonzero circulation, vanishing cross section, and infinite vorticity) is determined by the rate of change of its position vector, &(w)/at = q(z) = u

+ iv

with z = x

+ iy.

(2.6)

The same Lagrangian expression is used whether the flow is treated as inviscid or viscous, however, partly because the fluid outside the vortex core ’To emphasize the point that the “closed curve” is a material curve, Lighthill (1986) suggested that one should think of the closed curve “as a necklace of particles of fluid, one whose shape and position change continually as those particles of fluid move.” A bound vortex can sustain a force and occurs only as the limit of a filament consisting of solid matter. The condition that no fluid crosses the sheet holds true for both free and bound sheets, however, and the pressure is the same on both sides of the sheet.

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is assumed to be inviscid and partly because the effects of viscosity (diffusion via the thickening of vortex tubes, generation of vorticity at solid walls, and the reconnection of interacting vortex tubes, leading to major topological changes) are approximated through the use of various schemes: random walk in two orthogonal directions (later extended to three directions) using two independent sets of Gaussian random numbers, diffusion velocity, discrete approximation of the Laplacian using blobs, vortex-core expansion, particle-strength exchange, use of finite-difference methods where topological changes are expected, and their suitable statistical and deterministic combinations, in the context of particle methods. These will be discussed in more detail later. A straight vortex filament at z = zk has a positive counterclockwise circulation of (2.7) where C is the contour surrounding the point zk and q(z) is the complex conjugate of the velocity q(z). The imaginary part of the integral (2.7) is zero when there are no sources or sinks. Throughout this chapter, the circulation will be denoted either by r or by 2 n ~ . The velocity potential for an ideal line vortex is given by

in which the function wR(z) is regular at z = zk and will be ignored throughout the rest of this chapter. The complex velocity q(z) at any point z , except at z = z k , is then given by

The generalization of the foregoing to n line-vortex filaments yields

and n

. (2.11)


121

Unlike a singular curved vortex filament, a free line vortex of vanishing cross section does not give rise to a self-induced ~ e l o c i t y .It~ convects and is convected by what surrounds it. The velocity dz,(t)/dt with which the vortex K~ is convected is given by the regular part of (2.11) plus the components of an external irrotational velocity field evaluated at zk , so that

It is clear that the motion of the system of n vortices is governed by a system of n differential equations of the first order. Using (2.12) and a relatively simple Eulerian integration scheme, one can calculate the new value of zk and, hence, the evolution of a vortex sheet, represented by or discretized into n line vortices,

Zk(t + A t ) = zk + 4(2) At.

(2.13)

Equation (2.13) is first-order accurate in At. If desired and justified, one can achieve greater accuracy by using higher-order schemes, for example, an Adams-Bashforth open integration formula or a Runge-Kutta multistep integration scheme. In doing so, however, one must be aware that the accuracy of the representation depends not only on the integration scheme used but also on a number of other decisions made to perform the simulation, such as the representation of the effects of viscosity, number of vortices, their initial distribution, methods of dealing with difficulties arising from the vortex-vortex proximity, etc. These will be discussed in some detail later. The kinematics of motion possess three integral invariants, in addition to Kelvin’s theorem (often said to be the fourth invariant), similar to the integral invariants of the vorticity distribution [see, e.g., Batchelor (1967) for further details]. These are given by (2.14)

(2.15)

4This follows directly from the Biot-Savart law (2.30), to be discussed later.

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Turgut Sarpkaya

and

That (2.14)-(2.16) are invariants can be easily verified. For example, multiplying both sides of (2.12) by K k and comparing it with the derivative of H with respect to z k , one has (2.17)

since the oppositely signed members of each pair in dH/dzk cancel each other out. Noting that dKk/dt = 0, by virtue of Kelvin’s theorem, one has

d ( c KkZk(f))

=

0,

(2.18)

which, upon integration, yields (2.14). In (2.14), as well as in other invariants, the vortices may have clockwise (- K ~ or ) counterclockwise ( + K ~ )rotations; i.e., the numbers K k need not be positive. If one were to interpret K k as masses, however, M in (2.14) and I in (2.15) would correspond to moment of masses and moment of inertia, respectively. H in (2.16) is known as the Kirchhoff function and could be interpreted as energy. Kirchhoff noted that the equations of motion can be cast in Hamiltonian form and the integrals of motion can be related to kinetic energy (minus “self-energy”), as well as to linear and angular impulse. It is relatively simple to show through the use of (2.12) and (2.16) that (2.19)

which in Cartesian coordinates reduces to (2.20)

The system of Eqs (2.19) or (2.20) is similar in structure to the Hamilton-Jacobi equations in dynamics. The kinematics of the mutual interaction of a number of line vortices in two dimensions through the use of (2.20) has attracted some attention due to its direct connection with “chaos” and conjectured connection with the modeling of turbulence. As an exercise, one may consider the motion of a pair of vortices where in one case K , + K~ = 0 and in another case K~ + K~ # 0. It is easy to show that in the first case the vortices move with


123

a constant velocity along parallel lines (as in the case of a pair of trailing vortices, not subjected to mutual-induction instability), and in the second case, the two vortices move on two circles with center at the origin and with . motion of a pair of vortices radii inversely proportional to K , and K ~ The with arbitrary circulations in the presence of a wall leads to even more complex vortex paths. An interesting historical account of the three-vortex interaction problem is given by Aref et al. (1992). Dritschel(l993) suggested that the three-vortex problem may be germane to two-dimensional turbulence: two vortices cannot merge on their own-they depend on sufficient background strain for this, which means at least one other vortex must be nearby. 3. Evolution of a Vortex Sheet The surfaces on which the vorticity is concentrated are called vortex sheets. In its idealized form, the vortex sheet is a surface of discontinuity across which the velocity exhibits a jump. Strictly speaking, every discontinuity surface cannot be called a vortex sheet. There should be no flux of mass through the surface (no sources or sinks). This condition holds true for free as well as for bound vortex sheets. The velocity acquires a unique value when the vortex sheet is approached from either side (u: and u; in Fig. l(a). The velocity jump Au, = u: - u; in crossing the interface is of special importance in connection with the creation of vorticity on the boundaries. If we consider a directed element 6s along the surface, as shown in Fig. l(b), calculate the circulation r of the flow over a contour abcd, and let, in the limit, bc = du + 0, we find that

r = $ab

u: ds -

$bc

u, ds = Au, ds = y(s) ds.

(2.21)

Thus, Aus may now be regarded as the strength of the vortex sheet per unit length. For this reason, it is often denoted by y(s). It is also known as the vorticity strength per unit length or the circulation per unit length. For flow over a solid boundary, the velocity u; = 0 (Dirichlet condition of zero tangential velocity), and r = u: ds = us ds = y(s) ds. The creation and convection of vorticity generated at the boundaries will be discussed in more detail later. For the time being, it is sufficient to note that the placement of vortex sheets or discrete vortices just outside the boundary, the minimization of the flow leakage across the boundary, and the occasional convection of vortices or vortex-sheet segments into the boundary present complex

Turgut Sarpkaya

124

FIO.1 . (a) A velocity discontinuity or vortex sheet; (b) calculation of the strength of the vortex sheet.

issues to be resolved through suitable assumptions, guided by computational objectives and past experience. The conjugate complex velocity q(z) induced by a two-dimensionalvortex sheet of strength y(s, t ) = W / a s situated on the contour C is given by the Rott-Birkhoff nonlinear integrodifferential equation (Rott, 1956; Birkhoff, 1962)

4(z) = u

-

iv = -

+ U, - iV,,

(2.22a)

where U, and V, are the components of an external irrotational velocity field evaluated at z . The Cauchy principal value is assumedfor the integral to calculate the velocity at points on the sheet. It has been generalized to the case of a vortex sheet with small thickness by Moore (1978). If the circulation r is chosen as the Lagrangian variable to identify points on the sheet, (2.22a) may be written as (Birkhoff, 1962)

Vortex Element Methods f o r Flow Simulation

125

Equations (2.22a) and (2.22b) ensures the continuity of pressure across the sheet and the conservation of circulation of segments lying between any two points moving with the sheet. However, they do not ensure consistent and analytically tractable solutions; one must resort to numerical methods. The standard and perhaps the crudest procedure is to identify the position of the sheet using a finite number of marker points. Then the motion of the sheet is approximated by calculating the trajectories of the marker points. For example, if the trailing-vortex sheet of a wing is represented by an array of n line vortices per half wing, (2.22a) reduces to an initial value problem, consisting of a set of 2n, first-order, ordinary differential equations whose solution requires suitable smoothing schemes. The next level of approximation is to replace the sheet by a large number of segments or panels through the use of piecewise polynomial representations for both y and s and to simulate the highly rolled-up inner region either by an isolated line vortex or by a finite region of distributed vorticity.

B. REALVORTICESAND INSTABILITIES Real vortices are not concentrated singularities of infinite vorticity. The best known among the numerous representations of real vortices are the Rankine and Lamb (Oseen) vortices. The Rankine vortex rotates as a solid body within its core and is characterized by a potential flow outside; i.e., all of the vorticity is confined to the core region. The tangential velocity distribution for an isolated Rankine vortex has the form K Ug = -

r

(r > 6)

and

vo =

r a

K-

(r < a),

(2.23)

with an artificial discontinuity at r = a. In terms of complex variables, the velocity at an arbitrary point zk due to n Rankine vortices at zj may be written as

and as (2.25) The velocity induced at zk by the underlying irrotational flow (e.g., uniform flow, doublet, etc.) will have to be added to that induced by the vortices.

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Turgut Sarpkaya

Evidently, the core shape and the vorticity distribution within the core remain unrealistically invariant even in large strain fields. An axisymmetric version of the Rankine model is discussed by Taylor and Lloyd (1992). The Lamb (1932) model involves a Gaussian vorticity distribution and a circumferential velocity given by

w(r,t) = (rc0/27rvt) exp( - r2/4vt)

(2.26)

u(r, t) = (rco/r)[l - exp(-r2/4vt)].

(2.27)

and

Equation (2.27) is an exact solution of the Navier-Stokes equations for a single viscous vortex in an unbounded incompressible domain and d% = o is the standard deviation of the vorticity distribution. The radius at which the tangential velocity reaches a maximum is r, = 2 . 2 4 f i = 1.5840. Obviously, not even a single vortex whose entire vorticity is confined to an invariant finite core is an exact solution of the Navier-Stokes equations (e.g., a Rankine vortex). A single lamb vortex (which has an infinite support in an unbounded domain) is an exact solution. The velocity field of a multi-Lamb-vortex system is not an exact solution, however, because the nonlinearity of the Navier-Stokes equations does not permit the superposition, without deformation, of a finite number of vortex fields. Interestingly enough, a linear sum of Lamb vortices is an exact solution of the diffusion equation ( a d a t = vV2w) and may be used to simulate viscous effects. Rayleigh (1916) analyzed the stability of single vortices with general distributions of circulation, swirl velocity, and vorticity, but only for axisymmetric perturbations. He showed that stability against such perturbations is assured if r2nowhere decreases with r, i.e., d(r2)/dr= 87r2r2w&> 0

(2.28)

Clearly, a vortex is unstable if it has vorticity whose sign is opposite the swirl velocity. Otherwise, the vortex is stable to axisymmetric perturbations. A vortex with a r2that decreases somewhere is said to have a circulation overshoot (Govindaraju and Saffman, 1971). This somewhat strange result has not yet been observed experimentally, probably due to the fact that numerous tentacle-like vortex sheets of finite length, resulting from helical instabilities, stretch out or are thrown away from the outer edges of a turbulent vortex core, as demonstrated by Sarpkaya (1992b). The vortex peels off randomly and sheds vorticity along its length. In other words, the


127

core of a turbulent vortex is not a benign, smooth, axisymmetric, solid body of rotation. The exchange of momentum between the outer regions and the core leads to the oscillation of the vortex core and the various velocity components. The foregoing is not the only mechanism whereby a single laminar or turbulent vortex is dissipated. Vortex bursting or vortex breakdown has been shown experimentally to play a more dominant and dramatic role in the demise of a vortex (see, e.g., Sarpkaya, 1971, 1983). Axial flow (Bergman, 1969; Widnall and Bliss, 1971; Moore and Saffman, 1972) and ambient turbulence (Sarpkaya and Daly, 1987) are known to affect the stability of a vortex pair and the occurrence of vortex bursting. The stability of vortex sheets and, in particular, the Helmholtz instability have been the subject of intense interest. It is well-known that an infinitesimal disturbance of wavelength A on a plane sheet of strength K grows like exp(nrct/A), according to which the shorter waves grow faster. The stability of unsteady two-dimensional vortex sheets was discussed by Saffman (1974). Moore and Griffith-Jones (1974), and Moore (1976, 1981, 1984). Saffman and Baker (1979) suggested that the spiraling vortex sheets may be stable to Helmholtz instability because their strength decreases at a rate faster than that necessary to stabilize the sheet against a local Helmholtz instability. However, experiments show that the roll-up of both twodimensional as well as conical vortex sheets is accompanied by Helmholtz instability (Pierce, 1961; Gad-el-Hak and Blackwelder, 1985, 1987; Payne et al., 1986; Sarpkaya et al., 1988). Trailing vortices are made of rolled-up vortex sheets. During their formation process, the tightly spiraled regions exhibit velocity jumps between the vortex sheets. They are then liable to helical instabilities, even to Helmholtztype instability. Further downstream, one or both vortices may experience local axisymmetric disturbances (the so-called dotting effect) while the core still remains intact. This is in addition to the inductive instability (Crow, 1970) due to the mutual interaction of the vortex pair. Crow (1970) was the first to show that both symmetric and asymmetric modes of instability will develop on the vortices due to the mutual inductance of the sinusoidally perturbed pair. Crow has also shown that the instability grows exponentially and results either in a linking of the vortex pair into a series of crude vortex rings or in a highly disorganized intermingling of the vortices. Once again, vortex bursting may occur on one or both of the vortices. It is now agreed that sinusoidal instability, vortex breakdown, axial velocity in the vortices, ambient turbulence, and the stratification of the medium govern

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Turgut Sarpkaya

the demise of the trailing vortices and the evolution of internal waves in a stratified medium (Widnall et al., 1971; Moore, 1972; Moore and Saffman, 1972; Widnall, 1975). Sarpkaya (1983) and Sarpkaya and Daly (1987) have shown that any combination of these instabilities might occur, and the nature of their mutual interaction remains unknown. Another form of three-dimensional instability concerns the vortices shed from bluff bodies. Even if the body is two-dimensional and even if the vortices are shed in a two-dimensional manner, three-dimensional vortex instabilities may distort the filament and affect the spanwise and chordwise correlation of pressure on the body. This raises serious questions regarding the applicability of the two-dimensional models, the possible means with which the two-dimensional calculations may be corrected to adequately account for the three-dimensional distortions of vortex filaments (artificial reduction of circulation), and the representation of a three-dimensional vorticity field (containing curved vortex filaments with core and finite selfinduction) by two-dimensional vortices (with or without a core and no selfinduction). In fact, one needs to know the source of the three-dimensionality in order to properly interpret the qualitative nature of two-dimensional simulations. Furthermore, one has to make sure that the mechanism of instability is the same for both the continuous and the discrete systems and that both lead to the same type of large-and small-scale structures. In order to address some of these questions, Widnall (1985a, 1985b) analyzed the three-dimensional instability of two highly idealized cases; a single vortex separating from a cylinder as well as that of the Foppl vortices (two symmetrical stationary vortices behind a circular cylinder; see, e.g., Milne-Thomson, 1960, and Weihs and Boasson, 1979). Her calculations have shown that the most unstable mode of instability for the single vortex separating from a circular cylinder is three dimensional. She reached similar conclusions for various modes (symmetric and asymmetric) for the Foppl vortices. This investigation as well as those of Crow (1970) and Widnall and Sullivan (1 972) suggests that the neglect of three-dimensionality can lead to disagreements with experimental results. This is in addition to two-dimensional instabilities resulting from the discretization of a field of continuous vorticity, which may not be representative of the behavior of the continuous system. The instability of vortex rings has attracted much theoretical and experimental interest following the pioneering work of Krutzsch (1939). A detailed review of the subject is presented by Widnall (1975) (see also Saffman, 1970, 1978, for a succinct critique of earlier stability calculations).


129

The mutual interaction of vortex rings generated, for example, by a bluff body or by a round jet is of importance in the numerical simulation of threedimensional flows (Section II1.F). There is a strong interest in the understanding of the details of the vortex linking and the emergence of new vortex rings (i.e., the cross-linking or the cut-and-connect mechanism) and in the establishment of a possible relationship between these phenomena, noise and, turbulence. The vortex reconnection is prohibited in inviscid flows by Helmholtz’s t h e ~ r e mThus, . ~ viscous effects are necessary for its occurrence (see also Ashurst and Meiron, 1987). A direct numerical simulation of the phenomenon, starting with a closed knotted vortex tube, is given by Kida and Takaoka (1987) through the use of the full Navier-Stokes equations. C. BIOT-SAVARTLAW: THEVELOCITY-VORTICITY RELATION

The velocity induced by the vorticity concentrated in a bounded incompressible region is given by the volume integral (see, e.g., Batchelor, 1967) u(r, t ) =

-L 472

SJ1

(r - r’) x &(r’, t ) du(r’) Ir - rrI3

+ V4,

(2.29)

which was found experimentally by Biot and Savart in 1820 in connection with the determination of the magnetic field intensity (corresponding to u) induced by an element of electric current (corresponding to w ) and was established analytically by Ampere in 1826. In (2.29), the potential 4 only serves to satisfy the boundary conditions. It will be omitted from the subsequent versions of (2.29). For a singular curved vortex filament of vanishing cross section and circulation r (the thin filament approximation), (2.29) reduces to u(r, t ) =

‘s

-4n

[r(s) - r’(s’)]x ar’/dsr ds’, Ir(s) - r’(s’)13

(2.30)

Liquid helium cooled to within 2.172 degrees of absolute zero can flow without viscosity or friction. The quantum vortices in such a flow have extremely small cores and can be modeled as a thin space filament to investigate superfluid turbulence. Feynman suggested that each filament splits in two at the point of closest approach and the lines reconnect (the ends of each line join the appropriate ends of the other). More recent investigations have shown that even this mechanism is not adequate to explain the reconnection mechanism. It seems that when two sections of different vortex filaments come close enough, the circulation decreases or stops in a limited area, for quantum-mechanical reasons, enabling the lines to reconnect without any violation of the laws of hydrodynamics (Donnelly and Swanson, 1986; Donnelly, 1988; Schwarz, 1988).

Turgut Sarpkaya

130

in which r(s) describes the filament centerline in terms of the arclength s, and ar/as’ ds’ is the filament tangent vector. For the one-dimensional continua of N space curves (filaments) ri(

T-30

Yb

-4.3 -3.5 -

. .......

n

..

-6.5 I -2.0

I 0.0

I ~/l,

2.0

FIG.15. The vortex sheet shed by an elliptically loaded wing. The vortex blob positions are plotted on the left, and an interpolating curve is plotted on the right. The value of d (the smoothing parameter) is 0.2, based upon unit half-span. At late times, the vortices are elliptically deformed (Krasny, 1988).

C . FLOWABOUTA CIRCULAR CYLINDER

A great deal of theoretical, experimental, and numerical research has been devoted to the understanding of the near and far wake of bluff bodies, in general, and of a circular cylinder in particular. The impetus for this research comes partly from practical needs and partly from a desire to understand phenomena such as separation, transition, shear-layer evolution, wake instabilities, and fluid-structure interactions. Sarpkaya (1963, 1968b, 1969) was the first to derive general expressions for the lift and drag coefficients for a circular cylinder immersed in a


193

time-dependent flow comprising the ambient flow, a doublet, any number of discrete vortices, and their images. The complex force is given by

D

+ iL = 2npc2U - ip

c q [ ( u j+ hi) - (uji + n

iuji)]

(3.2)

j= 1

or by

in which r j is invariant (i.e., X / a t = 0) and positive in the clockwise direction, uj + iuj is the complex velocity of the j t h real vortex, uji + iuji is the complex velocity of thejth image vortex, and c is the radius of the circle. Equations (3.2) and (3.3) may be easily generalized to cylinders of arbitrary shape summing the impulses of all the real and image vortices. Then one has c d

+ iC, = - i -rtd

c" (qzj

j=l

-

r..z..) - A, JI J1

d(U + iV) dt '

(3-4)

in which c d and C , are the drag and lift coefficients defined by c d = Fd/(pU2&,), C , = FL/(pU2R,), u + iV is the complex time-dependent velocity of the ambient flow, A b is the area of the body, and R , is the characteristic radius of the body. Equation (3.4) may be further generalized to cases where the body is accelerating and/or rotating about its axis. Note that the impulse for o dA, where o is assumed to be uniformly distributed over the eIementa1 area dA, is given by p z o dA or by pzr. A generalized derivation of the force (the inertia due to added mass, the vector sum of the normal surface stresses induced on the body surface by vorticity in the fluid, and the skin friction caused by surface shear stresses) acting on a body in translational motion at velocity U(t) in incompressible, viscous fluid of uniform density is given by Wu (1981), Lighthill (1986), and Howe (1989). Gerrard (1967) was the first to apply the discrete vortex model to the flow about a circular cylinder. He obtained a relatively crude vortex street and lift and drag force traces. Nevertheless, his work has pointed out all the major difficulties to be faced by future investigators. The cylinder problem was taken up by Sarpkaya (1968b), who introduced the nascent vortices at the separation points and a small distance from the cylinder surface. The strength of the nascent vortices was made equal to 0.5(b2- K2)At to account for the oppositely signed vorticity contributed to the shear layer by the reverse flow on the rear face of the cylinder. This was the first time

194

Turgut Sarpkaya

that the effect of rear separation or secondary separation was incorporated into a discrete vortex model. The flow was forced to remain symmetric and represented by fewer than 100 vortices. In spite of these limitations, the results showed the rolling-up of the vortices, the development of the Helmholtz instability (as experimentally observed by Pierce, 1961, and by many others since then), and the rise of the drag coefficient to a maximum at the early stages of the flow due to the rapid accumulation of vorticity in the growing vortices. Sarpkaya’s work was extended by Davis (1969) to include the evolution of asymmetric flow. Subsequently, exploratory models were developed by Chaplin (1973). Chorin (1973) applied his time-splitting method to flow past a circular cylinder. His time-averaged drag coefficients (with expected substantial variance in the instantaneous values) were close to the experimental values. At Re = lo’, the drag coefficient was C, = 0.29. He conjectured “that the rough representation of the boundary layer triggers a premature onset of the drag crisis, analogous to the effect of a rough boundary or a noisy flow.” No Strouhal number was reported. Sarpkaya and Shoaff (1979a, b) have presented a method for determining the flow over a stationary and transversely oscillating circular cylinder using the method of Fink and Soh (1974) to rediscretize the vortices along a vortex sheet at each time. The attachment points of the sheets to the cylinder are the separation points as determined through boundary layer calculations. They have introduced the idea of circulation reduction to account for the effects of three-dimensionality of the flow and were able to obtain results in good agreement with experiments through the use of a suitable circulationreduction scheme. The calculated forces without the circulation reduction were about 25 percent larger. The Strouhal number was essentially unaffected by the circulation reduction over large periods of vortex shedding. Surprisingly enough, 15 years later, more sophisticated and elaborate methods, based on random walk, VIC, or particle-exchange schemes, using as many as 600,000 vortices were unable to reproduce even the early stages of the drag force with comparable accuracy (see Fig. 16). The simulations based on the assumption of two-dimensional flow do not accurately predict the dynamics of the flow (in particular, the lift and drag forces and the pressure distribution). This is partly because they lack the ability to capture small-scale turbulence structures that arise due to vortex stretching and tilting with respect to the plane of flow. This is true whether one uses a traditional vortex model or one with random walks, blobs, etc. Again, as noted, an artificial reduction in circulation is introduced to


195

account for the three-dimensionality effects, assuming that the discrepancy between the calculated and measured quantities is due to the neglect of three-dimensional effects alone. Obviously, it tends to account for all other assumptions regardless of the method used. Cheer (1983) applied the combined sheet-blob algorithm to flow over a circular cylinder for very small times. Stansby and Dixon (1982) used the VIC scheme to emphasize the importance of secondary shedding in twodimensional wake formation at high Reynolds number flow about a cylinder by placing the primary separation point at a judiciously selected point downstream of the velocity maximum. Clearly, the importance of the secondary separation depends on the shape and motion of the afterbody. In some cases the downstream vorticity production is quite small due to the nearly flat pressure distribution there. Koumoutsakos (1993) and Koumoutsakos and Leonard (1992) used the particle-strength-exchange method (see Section II.H.3) and the Biot-Savart law together with fast-summation algorithms to reduce the O(N2)computational cost to O ( N ) for the direct numerical simulation of the initial stages (Ut/R 5 5 ) of the flow development around an impulsively started circular cylinder in the range of Reynolds numbers from 40 to 9500, using as many as 0.6 x lo6 particles. A particle redistribution algorithm was used to overlap the particles in order to avoid the loss of spatial resolution. Except for the case of Re = 9500, the wake remained symmetric. A comparison of their (a) 2.0

-

CD

.

A

-

(Sarpkaya1978 )

A

'

O*OO.'

Experiments

'

'

I

5.

, ' . , , , , . , . . . . I , 10. 15. 20.

,

.

'

25.

ut/c FIG.16. (a) Drag coefficient for a circular cylinder immersed in an impulsively started steady flow,and (b) the evolution of the vortex street (Sarpkaya and Shoaff, 1979a, b). [figure continued over page]

196

Turgut Sarpkaya

0

0

0

FIG. 16-continued.


197

drag coefficients with those obtained by Smith and Stansby (1988), Chang and Chern (1991), and Ta Phuoc LOCand Bouard (1985) shows that there are rather large discrepancies, not surprisingly (see Sarpkaya, 1968b, 1991a), between various results. However, the drag coefficients of Koumoutsakos (1993) and Chang and Chern (1991) agree rather closely for both Re = 500 and lo00 (see Fig. 17). As noted earlier (Section II.H.4), the hybrid scheme used by Chang and Chern is at best a finite-difference/pseudo-Lagrangian scheme and does not take full advantage of the power of the Lagrangian method. As expected, the solutions at small times yield results comparable with those obtained by finite-difference techniques. The solutions at large Reynolds numbers are not too meaningful, however, since the diffusion (aside from that introduced by the numerical scheme itself) is still a viscous diffusion and cannot deal with turbulence. We do not expect the particlestrength-exchange method to yield meaningful results at large times (vortex shedding, etc.) even for relatively small Reynolds numbers partly because the number of particles, and hence the computer time, required will be extremely large and partly because the effort spent to keep the particle demography smooth (reasonable particle overlap) will be excessive, leading (a) 2.0

1.5

u"

1.o

0.5

198

FIG.17-continued.

Turgut Sarpkaya


199

to larger distortions in the velocity, vorticity, and strain fields, contrary to claims regarding the robustness and adaptivity of the scheme and its potential to be a powerful tool for studying unsteady bluff body flows. Smith (1986) and Smith and Stansby (1985, 1987) calculated a number of flows (laminar boundary layer above an infinite plane surface, induced by sinusoidal onset flow and by linear waves; separated laminar flow induced by sinusoidal waves over a rippled bed; and steady flow over cylinders of arbitrary shape for Re = 1OOO) using vortex sheets, random walk, and the VIC scheme. Attached flows were simulated successfully. However, the separated flow cases required the use of a “suitably chosen decay factor” (circluation reduction6) to bring the results into agreement with experiment, even under conditions in which the flow was purely laminar. Applications of discrete vortex models to oscillating flow

( u = urnsin E T) about bluff bodies either have had gross simplifications or have met with various difficulties. In his earlier work, Stansby (1977, 1979, 1981) either determined the separation points on the basis of steady flow calculations or fixed them at k 90 degrees (oscillating ambient flow). Subsequently, Stansby and Dixon (1983) presented a two-dimensional method for calculating laminar flows around cylinders of arbitrary shape in which the vorticity created at the surface at each time step was calculated using a boundary-integral technique. Molecular diffusion even in the highly turbulent wake was simulated by random walks, and the convection was performed with the VIC method. They have obtained agreement with experimental force coefficients and shedding frequencies in their simulation of steady and oscillatory flow about a circular cylinder. Smith and Stansby (1988, 1991) dealt with impulsively started flow around a cylinder (see Fig. 18) and with oscillatory flow about cylindrical bodies using random walk for diffusion and the VIC scheme for convection. In both case, the calculations were restricted to relatively small times and Reynolds numbers. As noted earlier, the existing vortex methods do not deal with turbulence calculations at higher Reynolds numbers (e.g., Re = lo4 or lo5) and are not expected to be very meaningful beyond a small critical time. 6The circulation reduction is an ad hoc scheme used occasionally to account for the differences between the numerical simulation and the flow simulated. It has been introduced in Section II.F.3.

200

FIG. 18.

Turgut Sarpkaya

Comparison at Re = 3000 of (a) the streamlines from the vortex method with

@) experimental visualizations and with (c) streamlines from a finite-difference method

(Ta Phuoc LOCand Bouard, 1985). (Computations and flow visualization by Smith and Stansby, 1988).


20 1

D. FLOW ABOUT SHARP-EDGED AND ARBITRARILY SHAPED BODIES

For sharp-edged bodies, various methods of determining the rate of circulation shedding may be made to produce indistinguishable results through judicious specification of the disposable parameters. Giesing (1969) has shown that the flow must leave the edge tangentially to the windward side provided that the sheet is modeled by a continuous sheet of vorticity in the vicinity of the edge. Similar nascent-vortex introduction schemes were used by Soh and Fink (1971) in their study of flow about bilge keels. When discrete vortices are used to introduce vorticity, however, the correct position of the nascent vortex does not necessarily lie on the tangent. Only the unique positions of the nascent vortex yield separation-velocity profiles that are compatible with the condition that in an inviscid flow the velocity and acceleration extrema occur on the body (Mostafa, 1987). It turns out that the time step used, the separation point velocity, and the point of nascent vortex introduction are intended and none can be assigned arbitrarily or on the basis of trial calculations. One of the earliest applications of the discrete vortex method to sharpedged bodies was made by Ham (1968) in connection with the aerodynamic loading of a two-dimensional airfoil during dynamic stall. His paper contains a number of original ideas, including the use of the amalgamation of vortices to avoid erratic motion. Clements (1973) used the SchwarzChristoffel transformation to map the exterior region of a two-dimensional, square-based, half-infinite body into an upper half plane. The discrete vortices were superimposed on a steady parallel ambient flow. The Kutta condition was not invoked in this version of the model. Instead, the strengths of the nascent vortices were determined from OSU,' At, where Us is the velocity in the plane of the rear face of the body a short distance out from the separation points. Subsequently, the model was modified (Clements, 1973; Clements and Maull, 1975), and the nascent vortices were introduced a short distance downstream of the separation points, in the planes of the body sides, with strengths determined from the Kutta condition. The calculations gave reasonable Strouhal numbers and vortical structures. Sarpkaya (1975b), using variable nascent vortex positions and the Kutta condition, and Kiya and Arie (1977a,b), using fixed nascent vortex positions, have investigated the vortex shedding behind a flat plate at incidence to the flow. Basuki and Graham (1987) used the VIC method to calculate the impulsively started flow past an 11%-thick Joukowski airfoil

202

Turgut Sarpkaya

at 30 degree incidence and concluded that the method predicts too strong a roll-up, an unrealistic suction peak, and excessively large fluctuations in lift. More realistic results were achieved through the use of a vortex decay technique, “provided the circulation removed is transferred so as not to affect the bound circulation.” Sarpkaya and Ihrig (1 986) investigated both experimentally and numerically the impulsively started flow about rectangular prisms. The body and the shear layers were represented by discrete vortices. The condition of zero normal velocity on the body was satisfied by minimizing the error in normal velocity through the use of the method of least squares. The Kutta condition was used to determine the position of the nascent vortices, and the Kelvin condition of zero total circulation is satisfied exactly. The vortices were assumed to be represented by a Lamb vortex. The force exerted on the body was calculated through the use of the generalized Blasius theorem. Comparison of the predicted and measured forces showed reasonably good agreement with respect to the frequency of the oscillations; i.e., the Strouhal number is correctly predicted. However, the amplitudes of the predicted forces are somewhat larger. Vortex shedding from sharp-edged cylinders and plates in steady and oscillating flow was investigated by Graham (1977, 1980, 1985) through the use of three methods (Brown and Michael model (1954), multi-discretevortex model, and the VIC method). Their results have shown that oscillatory flow at low Keulegan-Carpenter numbers can be represented quite accurately. At higher K, the two-dimensional vortex models tend to overestimate the vortex shedding and induced forces without modeling of secondary separation or three-dimensional effects (through the use of an exponential circulation decay “law”). Subsequently, Graham (1986, 1988) and Dolan et al. (1990) have used hybrid mesh techniques (see Section II.H.4) and obtained the results shown in Fig. 19. The streamline plots for an oscillatory flow about a diamond-shaped cylinder (Smith and Stansby, 1991) are shown in Fig. 20. The application of the vortex methods to the prediction of the hydrodynamic damping and the nonlinearities in the response of bargelike bodies in still water is made by a number of investigators (see, e.g., Pate1 and Brown, 1986). Spalart (1982, 1984) and Spalart et al. (1983) described several codes to calculate separated flow about various bluff bodies (squares, airfoils, tiltrotor wing), using the hybrid scheme attributed to him in Section II.H.4. For the circular cylinder, the drag coefficient decreased steadily as the


203

FIG. 19. Vortex distributions for a triangular flowmeter in a duct: (a) symmetric inlet and (b) asymmetric inlet flow (Dolan et al. 1990).

Fro. 20. Streamline plots with zero onset velocity at the end of the half cycle for the Keuegan-Carpenter number (K = U,T / L , U, = amplitude of the oscillating-flow velocity; L , a characteristic body length; and T, the period of oscillation) K = 0.5, 1.0, 1.25, and 1.5 for a diamond-shaped body for fi = L’/vT = 432 (Smith and Stansby, 1991).

Turgut Sarpkaya

204

Reynolds number increased from lo4 to lo7. No drag crisis was observed. Spalart attributed the shortcomings of his models to the “transition” of the separating shear layers, the difficulty of handling the flow around sharp corners, the delicate nature of the coupling algorithm, the three-dimensional character of the real flow, and the interference with the wind-tunnel walls. Nevertheless, his codes have been extensively used (see, e.g., Sisto et al., 1991) for the simulation of more complex flow situations. Sarpkaya et al. (1990) and Sarpkaya and Lindsey (1991) analyzed decelerating flow about cambered plates, with or without porosity, through the use of discrete vortices in order to explain the collapse of large parachutes. Nascent vortices were introduced in such a manner that the conditions of Kutta and the occurrence of velocity and acceleration extrema on the body were satisfied at each time step. No circulation reduction was used in the calculations. The measured and predicted forces were found to be in reasonable agreement, primarily because the calculations were limited to the early stages of the decelerating flow (see Fig. 21).

(a)

(b)

FIG. 21. The velocity field for an accelerating-decelerating ambient flow: (a) about a 120-degree-cambered plate at the early stages of the acceleration phase: and (b) about a 180-degree-camberedplate toward the end of the deceleration phase (Sarpkaya et at., 1990).


205

E . OTHERAPPLICATIONS There are numerous other applications of vortex methods that do not fit conveniently into any of a few broad categories. Inoue (1985b) used the multi-discrete-vortex model to simulate the flow past a porous plate. Peskin (1982) combined a finite-difference method with the vortex blob algorithm to study the flow of a viscous fluid through the mitral valve of the human heart. Special unsteady vortical flows have been investigated in connection with dynamic stall around an oscillating wing (McAlister and Carr, 1979; Sheen, 1986) and in connection with the explanation of extra lift generated by some fish and cetaceans in propulsive movements (Savage, 1979, Maxworthy, 1981). Hsu and Wu (1988) developed a vortex flow model for the two-dimensional blade-vortex interaction, introduced a new trailing-edge flow model (unsteady Kutta condition), and developed closed-form solutions for the vortex-induced unsteady force. Random vortex models have been applied in recent years to “turbulent” combustion in open and closed vessel (Sethian, 1984), to the formation and inflammation ofplanarturbulent jets (Cattolica et al., 1987; Ghoniem et al., 1986) and to turbulence-combustion interactions (Ghoniem et al., 1987b) in a reacting two-dimensional shear layer. In these models, the connection, if any, between the random vortex scheme and turbulence is not made clear. The application of vortex methods to sound generation by nominally steady, low Mach number, mean flow over a cavity has attracted some attention. More recently, geostrophic turbulence has been simulated with two-dimensional vortex methods (see, e.g., McWilliams, 1990). The model determines the vortex characteristics, such as amplitude, size, and shape, and computes the time evolutions and distributions of these characteristics. The results encourage the development of a 2 D-turbulence theory via vortex dynamics. Sarpkaya and Suthon (1991) simulated the temporal growth of V-shaped surface scars, generated by a submerged vortex pair, using randomly distributed vortex patches of varying sizes and strengths (see Fig. 22). F. GENERAL THREE-DIMENSIONAL FLOWS 1. Continuous Filament Applications

The theory of vortex filaments and the methods of regularization through smoothing and cutoff schemes have been discussed in Sections II.D.1-3. Here, only a number of applications of the use of overlapping filaments

206

Turgut Sarpkaya

FIG.22. Streamlinesshowing the temporal growth of V-shaped surface scars above a pair of trailing vortices below the free surface in a translating frame of reference (a) and in a fixed frame of reference (b) (Sarpkaya and Suthon, 1991).

with invariant cores are discussed briefly. These include the interaction of vortex rings (Parakh et al., 1983), interaction of solitons on a rectilinear vortex (Aref and Flinchem, 1984), instability of vortex rings (Ashurst, 1981), and the evolution of the time-developing round jet (Ashurst, 1983). These examples show the difficulties encountered in creating vorticity at the wall, in modeling viscous diffusion, in assessing the effect of the assumptions made, and in carrying out the calculations for sufficiently long times. The growth of a turbulent spot in a laminar boundary layer, as the spot evohes from a localized disturbance in the layer, was simulated


207

numerically by Leonard (1980b) using a vortex filament description of the vorticity field. Each filament is represented by a space curve xi(x, t ) , where x is a parameter along the curve, by a circulation ri,and by an effective core radius oi, which parametrizes the assumed Gaussian vorticity distribution within the filament. The generation of new vorticity at the wall due to the no-slip condition is ignored. Equation (2.37), suitably modified to account for the laminar base flow and the contributions of the images, is used to calculate the motion of the space curves. Leonard found that the gross properties of the spot away from the wall, including the velocities of the leading and trailing edges and the velocity perturbations, are in good agreement with experiment. Leonard’s work may be contrasted with the direct numerical simulation of turbulent spots in plane Poiseuille and boundary layer flow by Henningson et al. (1987). The interaction of vortex filaments with rigid bodies has attracted considerable attention (see, e.g., Pedrizzetti, 1992, and the references cited therein). De Bernardinis et al. (1 98 1) studied the unbounded oscillatory flow around a disk and the bounded oscillatory flow through a sharp-edged orifice. The shed vortex sheet is represented by a sequence of discrete vortex rings, and the solid bodies by a distribution of bound discrete vortex rings whose strengths are chosen to satisfy the Neumann or zero normal velocity boundary condition. In general, the gross properties of the flows are predicted accurately. Pedrizzetti (1992) investigated the evolution of an infinite vortex filament approaching a stationary sphere embedded in a “high-Reynolds-number flow.” He used Rosenhead’s regularization scheme with a constant cutoff distance. The core deformations occurring during close encounters were ignored. It seems that such interactions cannot be modeled adequately by invariant-core-filament models, with or without numerical viscosity. The inclusion of the effects of fluid viscosity and vortex-core deformation on vortex-body and vortex-vortex interactions poses challenging questions and requires new hybrid schemes. Strickland (1992) used discrete ring vortices to simulate the generation and shedding of vorticity from body surfaces (consisting of thin shells such as disks, partial spheres, rings) and a fast multipole method to reduce the computer time. The results are compared favorably with experimental data toward the understanding of the unsteady flow around high-performance parachute systems (see Fig. 23). Ashurt and Meiburg (1988) and Meiburg et al. (1988) presented a numerical study of the evolution of the two- and three-dimensional instabilities in a temporally growing plane shear layer. They included two

Turgut Sarpkaya

208

- *. . ..* . I

FIG. 23. The use of discrete ring vortices in the simulation of the axisymmetric phase of the wake development about an accelerating hemispherical shell at U,,t/R = 6 (a) and at U,t/R = 10 (b) (Strickland, 1992).

signs of vorticity to account for the effect of the weaker boundary layer leaving the splitter plate and used continuous filaments described by cubic splines with second-order integration in space and time. The calculations through the use of (2.38)have only been carried out until the initial filament arc length doubled because of the diminishing time step and the increasing number of node points. These bounds resulted from the need to deal with node depletion due to large strain effects, i.e., from the repeated remeshing of the filaments so as to keep the arc length between the nodes always less than the filament core diameter but more than half of the initial core radius. Ashurst and Meiburg demonstrated the formation of concentrated streamwise vortices in the braids (the two sleeves connecting neighboring spanwise rollers) as observed in the experiments of Lasheras and Choi (1988) with a spatially growing mixing layer with similar perturbations. Figure 24 shows the interface shapes between the free streams forming a plane free shear layer and a plane wake (Meiburg et al., 1988). In order to test the accuracy of the inviscid vortex filament method used by Ashurst and Meiburg (1988),Martin and Meiburg (1992)converted


209

FIG. 24. Topology of the vorticity field in three-dimensional shear layers and wakes: (a) shape of the interface between the free streams forming the shear layer, showing the modulation in the spanwise direction as the interface wraps around the streamwise vortex tubes and (b) the interface between the two sides of a plane wake, showing how the A-shaped vortices affect the shape of the interface: (i) flow visualization and (ii) numerical simulation (Meiburg et al., 1988).

the numerically calculated growth rates to spatial growth rates via a Caster transformation and compared them with those provided by Cohen and Wygnanski (1987) and concluded that the maximum growth rates calculated by the vortex method are too large by about a factor of three or four. They have attributed this discrepancy to the well-known fact that the vortex cores in filament models are not allowed to deform under the influence of the external strain field. However, they have also noted that (i) the existence of a clearly defined wavelength for which the growth rate

210

Turgut Sarpkaya

achieves a maximum, (ii) the decline of the maximum growth rate with increasing smoothness of the velocity profile, and (iii) the amplification of shorter wavelengths with smoother velocity profiles are trends that agree with the results of Cohen and Wygnanski (1987). One may, therefore, assume, as did Martin and Meiburg (1992) that “the vortex filament technique is able to duplicate the physically relevant dynamics, while it may underpredict the length of time it takes for certain events to occur.’’ This conclusion, drawn many times before in connection with numerous other applications, is in conformity with the overall spirit of this chapter that by pointing out the strengths as well as the deficiencies of the vortex element methods we acquire a clearer appreciation of their role in flow simulations relative to all other computational methods. Experience has shown that some appreciation of the incomplete knowledge bases (both numerical and experimental), retrofitting of the data, and the assessment of their consequences are necessary to achieve often qualitative and occasionally quantitative flow simulations. This is a compromise between expectations and achievables and between physically relevant dynamics and specific quantitative results. Knio and Ghoniem (1992) argued that the use of “desingularized vortex sheets to model shear layers, as in the case of Ashurst and Meiburg (1988), may lead to spurious results since the properties of the three-dimensional modes of a vortex structure are strongly dependent on the vorticity distribution within the cross section of the structure.” The major sources of inaccuracies are (i) the cores of neighboring vortex elements cease to overlap (or overlap excessively) after a short time, distorting the vorticity field; and (ii) the use of a limited number of images in enforcing the periodic boundary conditions does not allow one to capture the correct value of the free-stream velocity. In other words, the vorticity stretching and vorticity source terms must be accurately accounted for if one is to prevent the loss of naturally occurring coherent structures and the creation of unphysical ones. Meiburg and Lasheras (1988) and Lasheras and Meiburg (1990) carried out an experimental and numerical investigation of the three-dimensional transition in plane wakes behind flat plates and demonstrated, among other things, that important features of the development of the three-dimensional evolution can be reproduced by vortex methods (identical in most respects to the one used by Ashurst and Meiburg, 1988) even at Reynolds numbers as low as 100. An interesting three-dimensional flow visualization and numerical analysis [through the use of (2.31)] of a coflowing jet is given by Agui and Hesselink (1988), with an eddy captured in a hologram. They have observed


21 1

not only the generation of large-scale vortex rings but also streamwise vorticity in the braids connecting the rings. The method appears to be a useful tool for topological analysis of complex structures. More recently, Martin and Meiburg (1991, 1992) investigated the three-dimensional evolution of a nominally axisymmetric jet subjected to helical perturbations, using the inviscid vortex filament technique. They have found that “for the case of a helical perturbation only, the streamwise vorticity forming in the braid is of the same sign everywhere, with the vortex helix representing streamwise vorticity of opposite sign.’’ In the introduction of an additional periodic perturbation in the azimuthal direction, they have observed “the emergence of concentrated streamwise braid vortices all of the same sign, in contrast to the counter rotating braid vortices of ringdominated jets” (see Fig. 25). The effect of the reconnection of vortex tubes, i.e., the inclusion of the effect of viscosity and core deformation, on the foregoing phenomena remains to be seen. Melander and Zabusky (1988) used high-resolution spectral simulations of the full 3D Navier-Stokes equations to study the interaction and “apparent” reconnection of 3D vortex tubes (see Fig. 26) and have shown that core deformations occuring during close 3D vortex intreractions cannot be ignored and cannot be modeled adequately by invariant-core-filament models; i.e., the assumption of fixed shape is strongly violated.

FIG. 25. The evolution of helically perturbed jets: (a) nodes of the filament centerline with z > 0 are connected to form a surface. The stagnation line is shown in the frame of reference moving with the phase velocity of the evolving helical structure; (b) jet under the additional influence of a periodic perturbation in the azimuthal direction: (i) the streamwise view and (ii) the side view of filaments (Martin and Meiburg, 1992). [figure continued over puge]

Turgut Sarpkay a

212 8.0 6.0

4.0 2.0

2

0.0

- 2.0 - 4.0 - 6.0 - 8.C .O -6.0

- 8.0

-4.0 -2.0

0.0 V

2.0

4.0

6.0

8.0

f l , 12.566

0.000

FIG. 25-continued.

X


t = 2.0

213

I'

CL

(iii) t = 2.5

liu) t = 3.0

FIG.26. Interaction and reconnection of two initially straight vortex tubes with Gaussian vorticity profiles: (a) at f = 0.9 and (b) at t = 2.0, which shows the secondary finger structure and (CL & CR) two perspective views of the evolution of vortex tubes at various times. Left column, CL, is looking down at the x E y plane from z > 0, and the right column, CR, looking in toward the x-z plane from the right (Melander and Zabusky, 1988).

214

Turgut Sarpkaya 2. Disconnected Vortex Particle Methods

An alternative to the continuous-vortex-filaments representation of vorticity in three-dimensional flow is the use of vortexparticles, vortex balls (vortons according to Saffman and Meiron, 1986), and the volume integral given by (2.29). It is because of this reason that this method is called the disconnected-discrete-vortices, or particle method, where the change of magnitude of vorticity requires the determination of the local strain. In some ways it is an extension of the two-dimensional blob scheme. Each ball (vortex segment, arrow, or stick) is characterized by a core radius, a core function, a circulation, and a material vector element that describes the distribution of vorticity along the axis of the element. The core radius and the core function remain invarient with time. The discrete vorticity scheme avoids some of the problems associated with the connectivity of vortex filaments and attracts some new ones. It is difficult to account for the consequences of the nonzero divergence of the field (Saffman and Meiron, 1986; Winckelmans and Leonard, 1988; Greengard and Thomann, 1988), and there is the related problem of formulating an evolution equation for the strength of the vortex particles as they undergo stretching and reorientation. The accurate representation of the continuous vorticity field and the local vorticity intensification require that the disconnected parcels of vorticity must highly overlap and the elements experiencing severe stretch must be split into two in the local direction to preserve vorticity; i.e., subvortons must be created as in Maskew’s (1977) subvortex model (note that the velocity field in the revortoned region is not identical to that of the original vorton field). Thus, it is a method for the motion of disconnected but nevertheless heavily overlapped particles! The calculations may be carried out using singular vortex particles, the inviscid version of regularized vortex particles, or the viscous version of regularized vortex particles. In general, singular vortex particles are not very useful. The inviscid version of regularized vortons leaves the important effect of viscous diffusion out. In many cases, particularly when topological changes (e.g., viscous fusion of two vortex rings) are expected, the introduction of viscous diffusion is mandatory. The vorticity field is discretized onto regularized vortex particles according to

where

ZP and Cp(x)(normally chosen to be radially

symmetric) represent,


215

respectively, the time-dependent strength of the pth particle and the threedimensional regularization function. For rp(x) = 6(x), (3.5) reduces to the singular vortex particle representation. The regularization is often achieved by using a smoothing function similar to that applied in the connected filament scheme. For example, one has

The velocity field is calculated from

d$(t) --

dt

N

-

Z;(x', t ) =

c d[($(t) ZP(t)]x Zp(t), -

p= 1

Xi(O,$O)

=

so,(3.7)

where

is the Biot-Savart kernel for the pure particle representation. For the desingularized particle method, it may be written as

where (T is a smoothing parameter, representative of the particle core size. To proceed with the calculations through the use of the inviscid but regularized version of the vortex particle method, one only needs to update the magnitude and direction of the vorticity of particles as they are subjected to reorientation and stretching. Unfortunately, this is neither straightforward nor unique because of the nonzero divergence of the particle representation. The quality of a circulation-update scheme is measured partly by its ability to keep the vorticity field nearly divergencefree for all times and partly by its ability to conserve, as much as possible, the three linear invariants of the particle motion (vorticity, and linear and angular momentum). Anderson and Greengard (1985) calculate the rate of change of circulation from an explicit differentiation of the velocity field as (3.10)

Turgut Sarpkaya

216

Rechback (1978) and Winckelmans and Leonard (1988) use the transpose formulations of the vorticity transport equation so as to arrive at, respectively,

dZ ( t ) -P (ZJt) * VT)i&(t), t )

dt

and

= dt

1 2

= - [ZJt)

- (V + VT)]J(Tp(t),t ) .

(3.11a)

(3.1 1b)

Choquin and Cottet (1988) have shown that the transpose form of the vorticity transport equation leads to the exact conservation of vorticity, whereas the others do not. In fact, Winckelmans and Leonard (1988) claimed that their transpose equation comes closest to preserving kinetic energy and linear momentum. Although there are a number of proofs (see, e.g., Hou and Lowengrub, 1990) demonstrating the convergence of the various versions of the three-dimensional vortex formulations for the solution of smooth, incompressible, inviscid flow through the use of Euler equations, the nature of the proofs are such that the predicted accuracy is lost rapidly in computations due to the distortion of the Lagrangian grid and the singularity of the Biot-Savart kernel. In order to maintain accuracy in time, one needs to introduce numerical smoothing, local regridding, or desingularization, as noted in Section II.G.2. It is clear from the foregoing that the vortex particle methods are far from developed. In fact, each application is another step in the evolutionary ladder. Each new flow situation requires the invention of a new set of rules and smoothing and relaxation schemes. For example, Winckelmans and Leonard (1992) found, in studying the passing of two vortex rings through each other, that the nonzero divergence of the particle representation can lead to a systematic misalignment of the nieghboring vortons, thereby seriously deteriorating the accuracy of the calculations in a relatively short time. They have devised elaborate schemes (a relaxation scheme to be administered every so often to update the strength and orientation of the particles and a remeshing or revortoning scheme to insert additional vortons in regions of intense stretching) to maintain the vorticity field nearly divergence-free. The undesirable consequences of some of these schemes have been discussed in Section II.D.5. Efforts have been made to account for the effects of viscosity and to bring further realism into viscous flow simulations via regularized


217

particle methods, particularly in unbounded flows. These are, by and large, extensions of the existing two-dimensional diffusion schemes discussed earlier (see Section 11.H):random walk, particle strength exchange, and vortex in cell. Chorin (1980) and Fishelov (19Wa) extended the random walk scheme (Section II.H.l) to three-dimensional flow by giving each particle, at every time step, a random displacement, using independently selected Gaussian random variables (tl,,q,, ,q,), with zero mean and variance 2 At/Re. The shortcomings of this scheme and its slow convergence have already been discussed in detail in Section 1I.G. Winckelmans (1989) redistribiuted the vorticity among particles, adopting a method introduced by Mas-Gallic (1987) and Degond and Mas-Gallic (1989) for solving convection-diffusion equations (see Section II.H.3). The vortex particles that have little or no strength at t = 0 acquire additional strength at the expense of the particles that have some strength, thus imitating viscous diffusion. The method requires that a minimum particle overlap be maintained, which in turn necessitates the use of a large number of particles to ensure quadratic convergence. Even then, in the absence of new or additional passive particles (with little or no strength), the particle strengths tend toward the same constant (except for the effect of vortex stretching) and further diffusion becomes unsustainable. Furthermore, the repetitive convection distorts the vortex locations, and some regions become highly depopulated. These effects, coupled with the poor long-time covergence of the method, limit the calculations to short times and very low Reynolds numbers. Fischelov (199Ob) proposed a similar scheme where the vorticity is tracked along particle line trajectories. The vorticity is approximated by convolving it with a cutoff function. Then, the cutoff function is explicitly differentiated to approximate the second-order spatial derivatives in the viscous term. Recently, Winckelmans and Leonard (1922) considered the “knot” problem (two vortex rings in a “knot” configuration) to examine the consistency of the regularized vortex particles method. The calculations “highlighted some of the serious weaknesses of the method, such as intense stretching, need for the addition of vortex particles during the course of the computation, and non-zero divergence of the particle velocity field over extended times” (Winckelmans, 1989) and have shown that “the computation without remeshing and without relaxation” performs poorly. The reason for this is that the vortex particles become misaligned relative to the direction of the vortex filament they are supposed to discretize.

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Ghoniem et al. (1987a) and Knio and Ghoniem (1988,1991, 1992) applied the vortex-ball and the transport element version of the three-dimensional scheme to the numerical simulation of vortex rings with finite and deformable cores; to the simulation of streamwise vorticity in a periodically excited planar shear layer of finite thickness; and to the evolution of temporal, doubly periodic, uniform- and variable-density shear layers, during the formation and growth of the primary three-dimensional structures. For their latest study, Knio and Ghoniem (1992), rewrote the vorticity transport equation as (3.12)

for a variable density, incompressible, isentropic flow in the low Mach number limit, neglecting buoyancy effects and retaining the solenoidality condition, V * u = 0 (since the pressure and density variations are decoupled) . The presence of the baroclinic term requires the accurate estimation of the density gradient. Thus, taking the gradient of Dp/Dt = 0, one has

D6 + - = - G - V U ’ - G x (3 Dt +

with

2. = V p ,

(3.13)

which helps to circumvent the numerical differentiation of the density field, thereby minimizing a loss of resolution. The tracking of the gradient of a scalar gives the method the name transport element method. Knio and Ghoniem (1992) split (3.12) into two fractional steps (their “baroclinic splitting”) as

Dd -=(3..U’ Dt

(3.14)

and

03 6 DU’ - - - --x-, Dt p Dt

(3.15)

where VU’ is found by analytically differentiating the desingularized Biot-Savart law. The density field is reconstructed through the use of V 2 p = V 6. Thus, the entire method is based on tracking the vorticity and scalar gradients, discretized using a finite number of computational elements. A variable-density temporal vorticity layer with a second-order Gaussian vorticity variation and periodicity lengths Ax and A,, is assumed at t = 0.

-


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The difficulties introduced by the periodic boundary conditions in the evaluation of the flow field and its gradient are dealt with by computing directly the effect of the eight immediate neighbors of the elements and approximating the induced velocity of the images that lie within a square of side 400rl, by interpolation on a fixed grid. The problem of the depletion of computational elements in regions of strong strain, where the separation distance between neighboring elements becomes excessively large, is alleviated by redistributing the vorticity and scalar fields into a larger number of elements. As noted in Section II.D.5, however, the accuracy of the calculations may be sacrificed by remeshing, particularly in regions of high concentrations of vorticity, due to the warping of the velocity and gradient spaces. Shirayama et al. (1987) used disconnected vortex sticks to simulate unsteady flow past a circular disk (a flat disk parachute). Kiya and Ishii (1988) used the inviscid version of the regularised particle method to study the interaction of a vortex ring with an initially straight vortex filament as a model for the effect of free stream turbulence on large-scale structures in free shear flows. Among the various interactions predicted by Kiya and Ishi, the most interesting one is the effective partial cancellation of the vorticities of the ring and the filament. This observation is quite similar to that made in studies on superfluid turbulence, which show that when two sections of different vortex lines come close enough, the circulation stops in a limited area, for quantum-mechanical reasons, enabling the lines to reconnect without any violation of the laws of hydrodynamics (Donnelly and Swanson, 1986; Donnelly, 1988; Schwarz, 1988). The cloud-in-cell methods have been used for inviscid (Zawadzki and Aref, 1991) as well as viscous (Doorly et al., 1992) particle methods. The strengths of the particles are projected onto the nodes of a three-dimensional mesh (e.g., by using a volume interpolation) in a manner very similar to that discussed in Section 1I.E. The velocities at the mesh points are calculated by solving (3.16) v2ii = -v x ih through the use of FFT techniques for the three components of velocity on a regular Cartesian mesh, with w as the mesh vorticity. One may use judiciously the Biot-Savart law and the associated kernel (3.8), together with the mesh vorticity, in dealing with unbounded flows at unspecified Reynolds numbers. As noted earlier, the vortex-in-cell methods do not allow one to prescribe a specific Reynolds number for the flow simulated.

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Doorly et al. (1992) used the three-dimensional vortex-in-cell method to calculate the propagation of a diffusing vortex ring and noted that the discrepancies between theoretical and computed propagation speeds for the inviscid simulation of the ring may be attributed partly to the coarseness of the representation. Sommer (1990) computed the interaction and fusion of two vortex rings, using the same model. Kaykayoglu (1992) used a hybrid method, incorporating the viscous extension of the VIC scheme and the Schwarz-Christoffel transformation, to calculate the vorticity/field-body interactions (e.g., a bluff trailing edge or an elliptical leading edge) (see also Kaykayoglu and Rockwell, 1985). G. MISCELLANEA

A number of interesting studies have appeared in the literature during the past few years on three-dimensional flows, body-vortex interactions, turbulent combustion, and sound generation. Kiya and Ishii (1990) investigated the evolution of square and oval vortex rings, using disconnected vortex filaments with compact support, and found that the total energy serves as the most stringent criterion for the conservation of the dynamical invariants [see (2.14)-(2.16)]. Kamemoto and Nakanishi (1990) combined the boundary element method with the vortex filament method to study the separated flow around a three-dimensional bluff body, moving close to the ground. Marshall (1991) developed a general theory of curved vortices with circular cross-section and variable core area and noted that the assumption of circular compact support would not be expected to remain valid when the magnitude of the external flow is comparable in magnitude with the circumferential velocity of the outer edge of the core. Ashurst (1993) presented two-dimensional simulations (using a single length scale of eddies) of the flame motion through and between swirling eddies and then used the singlescale form as a recursion relation. Recently, Yeung and Vaidhyanathan (1993) investigated both forced and vortex-induced oscillations of a circular cylinder using the random vortex scheme and a complex-variable boundary-integral formulation. They were successful in capturing the well-known “lock-in” phenomenon (Sarpkaya, 1979).

Panaras (1985, 1987) and Poling et al. (1987) used a conformal transformation and discrete vortices to simulate the interaction of a blade and a foil with vortices drifiting with the free stream. Their results have shown that forcing frequencies higher than the frequency of vortex passage can be anticipated (see also Dickinson, 1988, who used blobs). Mathioulakis and


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Telionis (1983) used a combination of conformal mapping, a cutoff length to limit the velocities, and distribution of bound vortices to model the flow through a cascade of blades. The results illustrated how the developing wake of one blade influences the flow over the next blade and eventually induces stall. More recently, Hsu and Wu (1988) developed a vortex flow model for the two-dimensional blade-vortex interaction, introduced a new trailing-edge flow model (unsteady Kutta condition), and developed closedform solutions for the vortex-induced unsteady force. The use of the vortex methods in the analysis of the interaction of strong vortices in the rotor wake with the airframe has attracted considerable interest (see, e.g., Kim and Komerath, 1993; Schreiber, 1990; and Cantaloube and Huberson, 1983). As noted by Kim and Komerath, “The interaction process is rich in detail and complexity, with problems that are at the leading edge of fluid mechanics research. ” Evidently, vortex-vortex and vortex-body interactions require considerable attention to axial velocities and core deformations during close encounters and cannot be modeled adequately by invariant-core-filament models. Random vortex models have been applied in recent years to “turbulent” combustion in open and closed vessels (Sethian, 1984), to the formation and inflammation ofplanarturbulent jets (Cattolica et al., 1987; Ghoniem et al., 1986, who used the so-called dipole-in-cell scheme), and to turbulencecombustion layer through the use of a new transport element method (a generalized Lagrangian particle scheme that is constructed to compute solutions of a convective-diffusive-reactive scalar transport equation). The work of Cattolica et al. is partricularly interesting, for it deals with an axisymmetric flame development and offers a comparison of the predicted flame shapes with those obtained through laser-schlieren visualization. In general, it is assumed that stochastic methods can be used to model the fluid mechanics with the reactive flame front viewed as a flame sheet with infinitely fast chemistry. Even if a connection between the random vortex scheme and turbulence were to be established, it is not applicable to unsteady combustion problems that require a deterministic approach, as noted by Cattolica et al. (1987). Turbulent flow is inherently three dimensional, and the flame speed in such an environment depends strongly on flame stretch. There are a number of other factors that limit the utilization of the vortex methods to study combustion problems. In this connection, the experimental findings of Broadwell and Dimotakis (1986) regarding the modeling of reactions in turbulent flows and the informative review of Spalding (1986) on the application of the two-fluid model of turbulence to combustion phenomena are worth noting.

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The application of vortex methods to sound generation by nominally steady, low Mach number, mean flow over a cavity has attracted some attention (see, e.g., Hardin and Block, 1979; Hardin and Mason, 1977; and Breit et al., 1992). The last investigators have found that the Strouhal number at the peak of the broadband noise spectrum is in the range of the lowest-order edge tones rather than being well above the edge-tone Strouhal numbers as predicted by Hardin and Mason (1977).

IV. Concluding Remarks A numerical model must accurately reproduce a large class of experimental observations and measurements with only a few disposable parameters, and it must make definite predictions about the results of future physical experiments. How well did the vortex element methods fare? To try to answer this question we have given a nearly full and certainly critical account of the theoretical foundations and practical applications of various methods, models, and schemes. Based on this background, partial answers, undoubtedly to be refined in future years, may.be provided as follows. The numerical realization of Helmholtz’s powerful concept, that flows with vorticity could be modeled with line vortices of “infinitely small cross section” (quantum vortex lines), turned out to be anything but simple and revealed the complex nature of the problems to be resolved in dealing with real fluids. These problems are subtle and difficult to assess unambiguously, as evidence from the following digest:

(i) Kelvin-Helmholtz instability, sheet crossings, and chaotic motion of vortices (distinct from that of a few vortex systems) gave rise to various smoothing and cutoff schemes. (ii) Body representation, creating of vorticity, specification of the strength and position of the nascent vortices, determination of separation points, and the asymmetry introduction led to the use of the approximate boundary-layer equations and the Kutta condition in steady and unsteady flows. (iii) Inability to deal with three-dimensional flow effects, vortex stretching, and the annihilation of vorticity in the overlapping regions of oppositely signed vorticity led to the use of ad hoc vorticity-reduction schemes. (iv) Lack of a meaningful definition of the scale effects (Reynolds number) confined the calculations to unknown “high Reynolds number flows,” and the comparisons with experiments to cases where such comparisons appeared to be “reasonable.”


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(v) Computer time and storage demands, though not too excessive relative to those for the operator-splitting methods, forced the development of numerous time-saving schemes (e.g., the CIC scheme, merging of vortices, fast algorithms). Contrary to common belief, however, the long CPU time did not prove to be the greatest impediment to the effectiveness of vortex methods. (vi) The difficulties associated with the assessment or the range of validity of numerous, nonlinearly related, numerical schemes and attempts to imitate high Reynolds number “two-dimensional” flow experiments, by suitably adjusting the disposable parameters until some features of the observed phenomena are mimicked, turned each application into a new model. Consquently, no single line-vortex model emerged that can be applied to a wide variety of flows, However, practically all well-disciplined and well-documented multi-line-vortex models predicted results that are neither too far from those of the “nearly-two-dimensional” experiments nor sufficiently free from a number of assumptions (even if some disposable parameters spanned only over a narrow range). Nevertheless, these models do not, in the strictest sense, fulfill our stated requirements for a satisfactory numerical model. The introduction of vortices with invariant finite cores (blobs and vortons) reduced the propensity of the vortices to sprint and to orbit about each other, but created a whole host of new problems. Finite-cored vortices violate Helmholt’s law that vorticity is a material quantity; i.e., the rigidblob idealization is not dynamically consistent. If the vortices are created, convected, and diffused in a viscous fluid, then one encounters another kind of problem: The nonlinearity of the Navier-Stokes equations does not permit the superposition of finite-cored vortices, not even that of two Lamb vortices in an unbounded domain. This confronted the researchers with a choice between Helmholtz’s infinitesimal-cored vortices and finite-cored, invariant, identical blobs. It seemed that one could smooth the velocity and vorticity distributions and, at the same time, mitigate the nonlinear effects of the Navier-Stokes equations by significantly increasing the number of blobs, by forcing them to overlap, and by judiciously choosing the cutoff radius and the shape of the velocity cutoff function. The use of the Navier-Stokes equations and the creation and diffusion of vorticity in real fluids gave rise to the operator-splitting, random walk, particle-strength-exchange, and hybrid methods. With the operatorsplitting and random walk methods it seemed, at least when they were first

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introduced by Chorin (1973), that the vortex methods have finally attained the mathematical and practical robustness to deal accurately with the convection and diffusion of vorticity in compliance with the laws of motion. Closer scrutiny during the past 15 years, however, has consistently revealed serious problems, and, indeed, the promise of the method outweighed the few results obtained since its inception. The implied explicit link between the blobs, the operator-splitting scheme, and the Navier-Stokes equations became as much obscured as that between the singular line vortices of Helmholtz and the classical discrete vortex models of the past decade because of a number of complex problems, as shown by the following brief summary: (i) None of the applications of the random-walk scheme managed to produce quantitative results at large times in good agreement with analytical solutions or physical experiments without the use of an ample dose of disposable parameters subjected to successive stepwise refinements (e.g. , the number, size, overlap, and maximum circulation of blobs). (ii) The demands for computer storage and time increased by one or two orders of magnitude, confining the calculations to large computers or to shorter times (early stages of flow) or to smaller Reynolds numbers and requiring fast algorithms and vortex-in-cell or hybrid techniques. The use of the CIC or the VIC scheme brought back the grid and, along with it, diffusion and confinement effects. (iii) The statistical nature of the results required the averaging or smoothing of the velocity and pressure distributions and integrated quantities (e.g., lift and drag forces whose instantaneous values often do exhibit unrealistically large variations). (iv) The number of physical parameters, numerical schemes, and convection fixes became very large, making a parametric analysis of their separate as well as combined effects on the predicted results practically impossible, notwithstanding the arguments regarding the robustness of the algorithms devised to account accurately for viscous effects. (v) All of the convergence proofs dealt with laminar flows in the absence of boundaries, assuming a sufficiently smooth initial blob distribution. In order to maintain accuracy in time, one needs to introduce numerical smoothing, local regridding, or desingularization. Excluded from the proofs is the fact that the diffusion of vorticity is affected by the wall proximity and by the boundary layer where the vorticity is not smoothly or uniformly distributed, initially or at any other time (particularly near the


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separation points). In fact, the works of Van der Vegt (1988), Tiemroth (1986), and others cast serious doubt on the ability of the slow-converging random walk scheme to deal with bluff-body flows. Furthermore, the random walk scheme cannot be used to simulate diffusion processes with spatially varying diffusion coefficients and therefore cannot simulate the Reynolds-averaged Navier-Stokes equations. (vi) The blobs must be regarded as mathematical artifices to limit velocity since vorticity carrying, nondeforming, fluid elements cannot occupy the same space at the same time. (vii) The evolution of large local strains increase the blob spacing relative to the core radius (local blob population depletion) and can lead to large errors in the resolution of the vorticity and the velocity field, and the calculations may be carried out only for relatively small fluid displacements. The use of subblobs to repopulate the areas depleted by large strains causes a further blob population explosion. The velocity, vorticity, and strain fields before and after reblobbing are not the same in the vicinity of the new blobs. In summary of the foregoing, the requirements driven by the singularity of the vortex filament cascade down as follows: singularity + smoothing or cutoff + blobs convergence overlap large number of blobs -, fast algorithms and supercomputers. The introduction of diffusion follows a parallel track: singularity + either artificial reduction of the circulation of well-separated singularities, or smoothing or cutoff + blobs + convergence -+ overlap + large number of identical blobs diffusion via core expansion (aging Lamb vortex), or synthetic merger rules and energy cascade, or random walk, or particle strength exchange fast algorithms and hybrid schemes, requiring a grid and a poweful computer. Clearly, the singularity of the vortices, the need to use a large number of vortex patches (currently, with invariant cores of identical shape and size) and the need to overlap them (like stacks of pennies), to preserve convergence, are the root causes of all the impediments to the effectiveness of vortex methods. These impediments strongly suggest that the character constraints imposed on blobs (same invariant size and shape) must be relaxed to make full use of the power of Lagrangian methods. At present, only the interaction of a limited number of piecewise constant, deformable vorticity distributions (contour dynamics of two-dimensional water bags) has been studied in some detail, and it has been shown that the said interactions can and do give rise to smaller as well as larger blobs. +

-+

-+

-+

-+

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The mismatch between model-based predictions and experimental results is not entirely due, or always attributable, to deficiencies in the model, but also lies in the three-dimensional nature of “two-dimensional” experiments. All vortex calculations show that the three-dimensional nature of the flow cannot be ignored, and flows known to be highly threedimensional cannot be modeled with two-dimensional vortex dynamics. The application of vortex methods to three-dimensional flows is limited to relatively simple cases, however, yielding only qualitative information, often in unbounded domains. According to Ashurst (1987), the existing three-dimensional vortex methods suffer either from a short wavelength instability of the connected vortex filaments or from the lack of spatial resolution when disconnected vortons or vortex sticks are used. Recent work (e.g., Martin and Meiburg, 1992) shows that the vortex filament techniques are capable of duplicating the physically relevant dynamics in unbounded domains for relatively short times, while they may underpredict the length and time scales. As noted earlier, Melander and Zabusky (1988) have shown that core deformations occurring during close vortex interactions cannot be ignored and cannot be modeled adequately by invariantcore-filament models; i.e., the assumption of fixed core shape is strongly violated. Experience has shown that some appreciation of the incomplete knowledge bases (both numerical and experimental), retrofitting of the data, and the assessment of their consequences are necessary to achieve often qualitative and occasionally quantitative flow simulations. This is a compromise between expectations and achievables and between physically relevant dynamics and specific quantitative results. As far as the three-dimensional bluff-body flows are concerned, the mostly qualitative laminar-flow simulations have been confined to the early stages of the motion. These remarks are tempered by the fact that the most significant results are often qualitative judgments that provide insights into the real physics of the phenomenon. This is in conformity with the more modest objectives of vortex models: identification of large-scale structures and acquistion of new insights. There seems to be general consensus that large-scale structures rising above (or floating over) the small-scale turbulence can be calculated qualitatively (and sometimes quantitatively, if the disposable parameters are carefully tuned); if the large-scale structures do not become quickly three dimensional (or do not have the propensity to become three dimensional as in the case of turbulent wall-bounded flows); if the large structures do not reside too long in a nearly confined region (e.g., in a recirculation zone), so as not to suffer


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excessive diffusion; and if the flow is in an equilibrium or weakly out-ofequilibrium state. Thus, the early stages of two- and three-dimensional flows without walls (free shear layers, jets, vortex rings and their interaction, two-dimensional combustion based on scalar mixing), the early stages of two-dimensional and axisymmetric flows about bluff bodies (preferably with sharp corners), and the transient flow over airfoils and control surfaces can be simulated with relatively greater confidence and fewer assumptions than the subsequent nonlinear regimes. Some sort of “surgery” on the flow structures is needed to systematically eliminate small scales of motion in order to continue the calculations (e.g., filtering of filaments to remove sections where the radius of curvature is less than the core radius) (see Ashurst and Meiburg, 1988; Dritschel, 1988b, 1993). The relative advantages and disadvantages of the Lagrangian and finitedifference techniques have been pointed out by many investigators and remains a subject for future study and, no doubt, vigorous debate. Often cited among these are the grid-free nature of the vortex methods and the exact treatment of the boundary conditions at infinity (without the use of the VIC scheme); the need to deal only with vorticity, where it exists, rather than with velocities and pressure at every node point on a finite grid (i.e., concentration of the computational resources in a limited spatial domain); the better suitability (with proper smoothing through blob overlap!) of the Lagrangian methods to deal with transient problems where vorticity regions have large deformation with steep gradients (which is hardest for finitedifference methods); the advantages of vortex methods when vorticity fills the computational volume, more or less uniformly; the large CPU times for both methods (Biot-Savart law versus finite differencing over a large number of grid points); the need for a number of suitable assumptions in each method; the artificial diffusion in grid convection schemes versus convection errors and forced diffusion (random walk, particle-strength exchange) in Lagrangian schemes; inability of either technique to deal with turbulence without the use of turbulent diffusion or a subgrid model (e.g., a subgrid-scale eddy-viscosity model); the adaptability and ability of the discrete vortex models to deal with flows of unknown topology (through the use of various smoothing schemes); the advantages of panel methods in dealing with wing vortices when the overall topology is known; and other advantages and disadvantages presented and discussed in this chapter. It must be emphasized that such a comparative listing is somewhat artificial and does not do justice to the subtleties of either method. Only challenging problems and expectations bring out the best and the worst in them. The

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problem and what one expects and wants to do with the answer, rather than exaggerated claims made on behalf of each method regarding its accuracy, novelty, power, and robustness, will determine to a large extent whether one wants to follow the vorticity field or the velocity field. In either case, the code solves only an idealized mathematical problem and the results must be interpreted in view of the real physics. We are reminded by Paul Adrien Maurice Dirac, “A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data.” In the midst of the remarkable progress made so far, it is rather difficult to speculate as to what is likely to happen in the future. Nevetheless, we expect that the demarcation of the areas of application of the existing methods will be further blurred and there will be greater joint use of both methods (Eulerian and Lagrangian) on a given problem (e.g., treating the boundary regions by finite-difference techniques and convecting vorticity away by Lagrangian techniques), so that methods of the future may be hybrids. The Eulerian and Navier-Stokes finite-difference methods may benefit more from the progress to be made on computers, numerical theories (reduction of truncation errors and artificial dissipation), and exciting flow simulations, e.g., computation of flow about geometrically complex configurations, hydro- and aeroelastic response of bodies, various compressible and incompressible unsteady flow phenomena about rigid as well as deformable three-dimensional bodies with massive separation (e.g., the opening of a parachute), dynamic response of lifting bodies, combustion, direct numerical simulation of canonical and noncanonical flows, subgrid-scale models, and, in more general terms, a better assessment of the existing turbulence models. The finite-difference and spectral methods are not expected to eradicate or replace the vortex element methods as a research tool, but they will complement and supplement them invaluably. In spite of their shortcomings, there are a few compelling reasons as to which the vortex element methods will continue to produce remarkable solutions under circumstances where the demands of a particular problem match the character (inherent computational advantages) of the method: better adaptivity (a feature that avoids the loss of spatial resolution and allows the accommodation of high strain rates by judiciously increasing the number of computational elements as the flows evolves), the exact explicit treatment of the far-field boundary conditions, and the ability to provide logical deductions and physical insights by dealing directly with the most fundamental characteristic of the fluid motion-vorticity. The last two advantages are particularly


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important. As noted by Berkooz et al. (1993) in connection with the properorthogonal-decomposition method in the analysis of turbulent flows, “Analytical methods have so far been unable to deal with the interaction of more than a few unstable modes, usually in a weakly nonlinear context, and thus have been restricted to studies of transition or pre-turbulence. Most of the dynamical systems studies have been limited to this area. Computational fluid dynamics bypasses the shortcomings of these methods by offering direct simulation of the Navier-Stokes equations. However, unlike analysis, in which logical deductions lead stepwise to an answer, simulation provides little understanding of the solutions it produces. It is more akin to an experimental method, and no less valuable (or less confusing) for the immense quantity of data it produces, especially at high spatial resolution.” It is our assessment that vortex-element simulations do indeed allow one to make logical deductions by providing an insight into the Lagrangian evolution of the vorticity field. In fact, one cannot help but be impressed by the effort, sophistication, and ingenuity that went into the development of the existing methods and simulations over the past three decades. It is only in recent years that the expectations and limitations of the vortex element modeling came into full focus. It is our hope that the method will sharpen its physics as well as the precision of its predictions. As to the future, the simulation of turbulent flows, in general, and of massively separated bluff-body flows, in particular, will present numerous challenging problems. The prevailing body of evidence shows that turbulence involves the ineraction of many degrees of freedom over a broad range of spatial and temporal scales, and an adaptive numerical method that can deal directly with the interaction of deformable parcels of vorticity should be in a better position, relative to all other methods, to lead the way to the understanding of the nonequilibrium turbulence. It is our belief that the road to the Holy Grail of turbulence is paved with deformable parcels of vorticity, ranging in size from dust to boulders. The understanding of their interaction through the use of two- and three-dimensional deformable vortex patches of different sizes and shapes, faster and more robust algorithms, and larger computers is the most important challenge of vortex element methods. In developing hybrid methods, one can, if necessary, remove the inessential degrees of freedom and simulate the smallest scales of turbulence in some indirect manner, with the encouragement provided by the spectacular advances in computer technology, bearing in mind that the effectiveness of a model does not necessarily increase with its complexity.

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As far as the engineering applications are concerned, the classical vortex methods, preferably f o r bluff-body flows (using separation points, line vortices or blobs, no forced viscous diffusion), the operator-splitting methods, preferably f o r shear layers, laminar flames, and combustion (using blobs, particle-strength-exchange type diffusion, the VIC scheme, and spectral algorithms), and the panel methods, preferably f o r vorlical flows in aerodynamics (in competition with Euler and Navier-Stokes codes), will continue to exist and flourish through complementary numerical and physical experiments as long as they maintain demonstrable advantages over other methods. They may even be able to make definite predictions about the results of future experiments if they are tuned to the physics of the flow. In any case, the glorious vorticity of the fluid motion and the enigmatic smile of vortices will continue to attract new admirers who, like the others before them, will try to minimize the existing deficiencies of these methods, bring new insights, pose new questions, and rediscover the power of the rich and remarkable concepts set forth 135 years ago by Helmholtz. Acknowledgments

The author wishes to express his appreciation to Professor T. Y. Wu of the California Institute of Technology, who has kindly encouraged him to write this chapter. He also wishes to express his gratitude to all those on whose work he has so extensively drawn to produce this review, to many people who have responded generously to author’s requests for advice, references, advance copies of papers, reports, and theses, and/or the originals of some figures, and to the authors and publishers who kindly permitted the reproduction of figures from their publications. The author’s and his students’ projects referred to herein were supported by the National Science Foundation, the Office of Naval Research, and the Defense Advanced Research Projects Agency. References Abernathy, F. H., and Kronauer, R. E. (1962). The formation of vortex streets. J. Fluid Mech. 13, 1-20.

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ADVANCES IN APPLIED MECHANICS. VOLUME 31

Micromechanics Constitutive Description of Thermoelastic Martensitic Transformations QING-PING SUN and KEH-CHIH HWANG Department of Engineering Mechanics Tsinghua University. Beijing. China I . Introduction

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

249

I1 . Micromechanics and Thermodynamics of Thermoelastic Martensitic

Transformations ................................................................................... A . Previous Investigations ..................................................................... B. Thermoelastic Martensitic Transformation ............................................ C . Microstructure and Micromechanics Idealization ....................................

251 251 254 258

I11 Energy Changes Accompanying Deformation Processes ................................ A . Free Energy of the Constitutive Element ............................................... B. Energy Dissipation During Deformation ...............................................

263 263 261

.

IV Constitutive Relations ............................................................................ A . Constitutive Laws for Forward and Reverse Transformations ................... B. Constitutive Laws for Reorientation Process-(1) ................................... C . Constitutive Laws for Reorientation Process-(2) ...................................

268 268 276 280

V . Applications to Deformation of Polycrystalline Shape Memory Alloys ............ A . The High-Temperature (T > Ms) Behavior ............................................ B . The Low-Temperature (T< M,) Behavior ............................................ C . Thermal Cycling ..............................................................................

282 283 287 291

VI . Concluding Comments ...........................................................................

295

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

295

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

296

.

I Introduction

Phase transforming materials. especially those undergoing thermoelastic martensitic transformations. play an important role in many technical applications either as a structural or functional material. such as the quenching of steel components. the toughening of ZrO.. containing ceramics. and the pseudoelastic and shape memory effects displayed by 249 Copyright 0 1994 by Academic Press. Inc . All rights of reproduction in MY form reserved. ISBN 0-12M1u)31-9

250

Qing-Ping Sun and Keh-Chih Hwang

certain alloys. The study of the constitutive relations of these kinds of materials has received greatly increasing attention during the past 10 years. One reason for this is that increasing practical engineering applications of these materials require the establishment of mathematical models that are able to describe the main features of the constitutive behavior under applied mechanical and/or thermal loadings (Bondaryev and Wayman, 1988). Another reason comes from the need of theoretical explanation of the toughening phenomena and further the need for the design of material (Evans, 1900; Hwang and Sun, 1991a,b; Sun et al., 1990~). Generally, under thermomechanical loadings, the thermoelastic phase transforming materials may display various macroscopic constitutive behavior such as superelasticity, shape memory, ferroelasticity, and rubberlike elasticity. Intensive materials science investigations in the past have led to a comprehensive understanding of the physical mechanisms of phase transformation, so that it now seems possible to provide a unified microscopic description of these kinds of behavior. Although the constitutive relation of materials has long been a very important and active research subject in the cross-field of solid mechanics and materials science, the study of the constitutive law for phase transforming materials is much less developed compared with that of the common metals. Even though a lot of work has been done in this area, there has not been a complete constitutive description for the various macroscopic behaviors. This chapter is primarily concerned with the micromechanical description of transformation plasticity, incorporating microstructure, crystallography, thermodynamics, and micromechanics into the continuum formulation of the macroscopic constitutive behavior. An intent of the present chapter is to illustrate that it is indeed possible to describe and to interpret the various phenomena by a unified interrelated micromechanicsmodel, and it is shown that when this is done, the description of rather complex phenomena involved in complex thermomechanical loading paths can be obtained. Examples of such analyses are taken from recent studies of deformation of polycrystalline shape memory alloys (Sun and Hwang, 1991a) and zirconian ceramics. In Section 11, we give a brief review of the previous research activities and recent development in this area. The general characteristics of the thermoelastic martensitic transformations and the most important deformation mechanisms are then outlined. Particular attention is paid to the description of the microstructure and micromechanics idealization of the constitutive element used, the definition of some basic variables, and the

Micromechanics of Thermoelastic Martensitic Transformations 25 1 basic assumptions regarding the deformation process. These are shown to play dominant roles in the derived macroscopic response. In Section 111, the analytical expression of the Gibbs free energy of the material is derived based primarily upon the energy changes during the deformation process. The expression provides clearly the role of macroscopic stress, temperature, and microstructure changes in the deformation history. Also, by the analysis of the energy dissipation during deformation, an expression for the energy dissipation rate is given. In Section IV, the constitutive laws for the forward and reverse transformations and reorientation processes in any thermomechanical loading histories are derived through energy balance analysis and are expressed in an internal variable formalism, by which the inelastic structure rearrangements of the material element on the microscale are related to the corresponding increments of macroscopic plastic strain. The macroscopic yielding characteristics of the material are discussed in detail. In Section V, the individual phenomena of the phase transforming materials are studied and discussed in detail with the present constitutive theory. The theoretical predictions are compared with the available experimental data of shape memory alloys, and some discussions for the complex loading paths are also given. The chapter concludes with some suggestions for further research.

11. Micromechanics and Thermodynamics of Thermoelastic Martensitic Transformations

A. P ~ ~ v r o INVESTIGATIONS us Thermoelastic martensitic materials comprise various kinds of shape memory alloys and zirconia-containing ceramics that are representatives of metallic and nonmetallic thermoelastic martensites, respectively. The study of constitutive relations for thermoelastic martensites is thus mainly concentrated on these two kinds of materials. The shape memory alloys are typical of thermoelastic martensitic transforming materials, early constitutive models of which are limited to the phenomenological description of the uniaxial tension or compression data. Current existing models such as those of Falk (1980) and Achenbach (1986, 1989), which are based upon a phenomenological thermodynamic approach or statistical micromechanics, can only explain the macroscopic behavior

252


qualitatively and are not destined for engineering applications. Tanaka et al. (1986), Tanaka and Fischer (1991) and Patoor et al. (1987, 1988) studied the constitutive behavior of both single and polycrystalline shape memory alloys in the process of stress-induced martensitic transformation from the micromechanical point of view. They connected the microscopic deformation due to transformation to the macroscopic behavior of alloys by introducing two different levels of microstructure, the microregion and the mesodomain, and derived the macroscopic constitutive relation via appropriate averaging procedures. Some novelty may also be found in the work of Patoor et al. (1987, 1988) such as the introduction of an interaction matrix and the self-consistent scheme used for the micro-macro transition. This work, however, is limited to the stress-induced forward transformation and is unable to describe the more complex behavior corresponding to the reverse transformation and reorientation processes, especially under nonproportional loading conditions. Another recently proposed model (Bondaryev and Wayrnan, 1988) that is based on a modified plastic flow theory overcomes some of these limitations but, due to its phenomenological nature, is unable to take account of the microstructural effects such as transformation-induced internal stress both inside the martensite variants and in the surrounding matrix parent phase, the corresponding strain energy stored in the material as well as the energy dissipation during deformation, etc. These effects are shown to have very important contributions to the martensite or twin growth process and to the macroscopic constitutive behavior. Also, the interrelation of elastic modulus variation with temperature with the V-shape of the yielding stress versus temperature curve in the vicinity of transition temperature (Nakanishi et al., 1973) cannot be explained by this model. To fill these gaps, Sun and Hwang (1991a)established a constitutive theory on a more general basis by micromechanics, deformation mechanism, and thermodynamics analysis of the constitutive element. This theory is applied to study the forward and reverse transformations and reorientation processes at high and low temperatures, respectively, and acceptable agreement between theory and experiment is obtained. It should be pointed out that in addition to the foregoing basic modes of deformation, the process due to reorientation of stress-induced martensite at high temperature or of martensite variant produced by stressing the self-accommodating martensite groups at low temperatures is still possible in a nonproportional loading history. This important process was studied in recent work (Sun and Hwang, 1993) and will be described in detail in this chapter.

Micromechanics of Thermoelastic Martensitic Transformations 253 Another important kind of thermoelastic martensitic transformation is that found in Zr0,-containing ceramics, and it has become well known due to the realization of toughening of ceramics by transformation. The study of the constitutive behavior of this nonmetallic material has attracted the strong interest of both material scientists and mechanicians (Evans and Cannon, 1986; Evans, 1990) because a precise constitutive description has essential importance in the explanation and prediction of transformationinduced crack-tip shielding and in stress-strain calculations of ceramics structures. Since the fundamental work of McMeeking and Evans (1982) and Budiansky et al. (1983), most of the research on this subject has been concentrated on taking the transformation shear effect into account properly in the constitutive description (Lambropoulos, 1986a,b; Evans, 1990), and much progress on this subject has been achieved; the most important discoveries are the following. (i) The shear component of transformation has equal importance to volume dilation in the transformation plasticity and the toughness enhancement. The considerable volume strain involved in transformation makes the transformation stress have strong features of pressure sensitivity. This fact has been firmly demonstrated by experiments of Chen and Reyes-Morel (1986, 1987), Sun et af. (1990b, 1991c), and recently Qing et al. (1992). (ii) The experiments of Reyes-Morel et al. (1988), Reyes-Morel and Chen (1988), Sun et al. (199Ob), and Dauskardt et al. (1989, 1990) revealed that the tetragonal to monoclinic transformation in Zr0,-based ceramics, macroscopically as a whole, is occurring in a gradual process. That is, when transformation proceeds under applied stress the increase in macroscopic plastic strain is due to a gradual increase in volume fraction of transformed ZrO, grains in the material sample, which experiences microstructural evolution similar to that of the metallic shape memory alloys. Hence, the previous critical transformation assumption (Budiansky et al., 1983) in the macroscopic transformation behavior description and toughening calculation is unrealistic and needs further investigation (Chen, 1991; Dadkhah et al., 1991; Qing et al., 1992). (iii) The martensitic transformations in partially stabilized zirconia (PSZ) and tetragonal zirconia polycrystal (TZP) ceramics are thermoelastic in nature (Lee and Heuer, 1988), and the pseudoelastic and shape memory behaviors typical of thermoelastic martensitic transformations have been demonstrated to exist in experiments by Reyes-Morel et al. (1988), Huang et al. (1990), and Sun et al. (1991b).

254


Based upon these scientific findings, Sun et al. (1991a) reconsidered the description of the transformation plasticity in a macro-micro combined approach and established the constitutive law in the framework of the internal variable constitutive theory (Rice, 1971, 1975). This description is still not complete, however, because other deformation mechanisms of thermoelastic martensite such as the reorientation process of martensite variants under applied stress were not included and the reverse transformation condition proposed is correct only under proportional loadings. So, it is clear that further work is necessary to overcome these deficiencies. This objective will be achieved in the unified description of this chapter.

B . THERMOELASTIC MARTENSITIC TRANSFORMATION 1. General Thermodynamic Characteristics

Martensitic transformations can be induced by the application of stress as well as by changes in temperature. Both temperature and stress as macroscopic variables affect the transformation because they influence the thermodynamics and kinetics of the transformation. Moreover, the thermodynamic and kinetic effects are strongly dependent on the direction of stress with respect to lattice orientation. During transformation, both a parent phase and one or more product phases (martensite),are present, which may share the same chemical composition but differ in their microstructure, volume, and shape. The interaction between the microregions occupied by these phases gives rise to microstresses that can influence the macrobehavior of the material. In the theory of martensitic transformation, the thermoelastic transformation is realized if the martensite forms and grows continuously as the temperature is decreased (or the applied stress is increased), and shrinks and vanishes continuously as the temperature is raised (or the stress is decreased). The transformation proceeds essentially in equilibrium between the chemical driving energy and the resistive energy whose dominating component is the stored elastic energy. In general, the material undergoing thermoelastic martensitic transformation may display two major kinds of mechanical behavior, i.e., pseudoelasticity (PE) and shape memory effect (SME) at high and low temperatures where austenite and martensite, respectively, are stable or metastable (Delaey el al., 1974; Wayman, 1983). The pseudoelastic behavior is a complete mechanical analog to the thermoelastic transformation; the transformation or reorientation proceeds continuously with increasing applied stress and is reversed continuously

Micromechanics of Thermoelastic Martensitic Transformations 255

(e)

(9)

(f)

(h)

FIG. 1. Schematic illustrations of the mechanisms of PE by transformation (a, b, c, d) and reorientation (e. f, g, h), respectively.

when the stress is decreased below a definite value. The shape memory effect arises if the macroscopic deformation of these processes remains after unloading and then disappears by heating to reverse the transformation. It must be recognized that the pseudoelastic behavior and shape memory effect are interrelated; i.e., if the hysteresis in the case of PE is so large that the reverse transformation or reorientation is incomplete when the applied stress is removed, the residual martensite can be reverted by heating, i.e., by employing SME. The schematic illustrations of the basic mechanisms of PE and SME in a crystal of shape memory alloys are shown in Figs. 1 and 2, respectively. M,

< T < A,

t

(f)

(9)

(h)

(i)

FIG.2. Schematic illustrations of the mechanisms of SME by transformation (a, b, c, d, e) and reorientation (f, g, h. i , e), respectively.


256

Extensive microscopic studies have revealed that the various macroscopic behaviors under thermomechanical loadings are caused microscopically by one or more of the following metallurgical elementary processes (the detailed description and physical interpretation can be found in various reviews; see, for example, Delaey et al., 1974; Perkins, 1975; and Wayman, 1983): (i) thermoelastic parent to martensite (p + m) transformation and its reverse transformation (m -,p); (ii) reorientation of the self-accommodating martensite groups under applied stress (m + m) and its reversal, i.e., the reversible movement of the twin or intervariant boundaries; (iii) reorientation of one martensite variant to some other martensite variant that has a different crystallographic orientation from the first. Because the volume and shape change of the crystal is elastically accommodated by the surrounding elastic matrix and the growth or shrinkage of martensite in the process of the forward-reverse transformation or reorientation are accomplished by the reversible motion of the p-m interfaces or m-m intervariant boundaries, the transformation-induced elastic energy stored in the material and the property of the interface have very important influence on the thermodynamics and morphology of transformation. 2. Brief Descrbtion of the Constitutive Behavior

The macroscopic constitutive behavior is characterized by a strong temperature dependence of the stress-strain diagrams. It has become customary to classify the experimental phenomena of thermoelastic martensite under the following major headings: the shape memory effect, the two-way (or reversible) shape memory effect, the ferroelasticity, the superelasticity, and the rubberlike elasticity. The last two effects may also be classed together as pseudoelasticity effect (see Christian, 1982; Otsuka and Shimizu, 1986, for the terminology). As a schematic illustration of the various effects, the stress-strain (S-S) curves as a function of temperature for a typical thermoelastic martensite are plotted in Fig. 3. The S-S curves change markedly with test temperature relative to the characteristic temperatures (material constants), M, (martensite-start temperature on cooling), Mf (martensite-finish temperature), A s (reverse transformationstart temperature on heating), and Af (austenite-finish temperature). At temperatures above A f , an apparent inelastic strain due to stressinduced p m transformation occurs after linear elastic deformation, but this strain recovers completely through a hysteresis on unloading by the reverse m p transformation as shown in Figs. 3(a),(b). This -+

-+


r-

Slrain

(1)

(9)

(h)

(0

(i)

FIG.3. Schematic illustrations of the various S-S curves as a function of temperature for a typical SMA > & > T, > ... > > lj).

(c

pseudoelasticity by forward and reverse transformations at high temperatures is called superelasticity. The inelastic strain in this case recovers on unloading since the martensite produced are completely unstable in the absence of stress at temperatures above A , . If the test is conducted at temperatures lower than A , , some strains due to p + m transformation remain after unloading, but these strains can be recovered by heating the specimen to a temperature above A,, as indicated by broken lines in Figs. 3(c),(d). This phenomenon is the so-called shape memory effect (SME). If the temperature is lower than M,, the SME can also occur by reorientation; the specimen is originally in a self-accommodating martensite state, and in this temperature range the inelastic deformation caused by the intervariant boundary movement upon stressing will remain after unloading. On heating the specimen above A,, the martensite reverts directly to the parent phase in the original orientation by virtue of the crystallographic reversibility of the thermoelastic martensitic transformation, as shown in Figs 3(e),(f). If the remanent strain on unloading is removed by opposite stressing, such as in Fig. 3(g), then the behavior is called ferroelasticity. If the temperature or the resistance to the twin boundary displacement is low enough, the reverse motion of twin boundary can happen during unloading producing pseudoelastic loop, as shown in Figs. 3(h)-(j). In this case, the phenomenon is called rubberlike elasticity. Another important feature shown in Fig. 3 is that the yielding stresses in the S-S curves have a minimum value in the vicinity of M , temperature; such a V-shape phenomenon is caused by the anomalous decrease of the elastic

25 8


moduli in the vicinity of the transition temperature, i.e., lattice softening (Hasiguti and Iwasaki, 1968; Nakanishi et al., 1973). In the following, we describe the microstructure change and the micromechanics idealization of the foregoing processes.

C. MICROSTRUCTURE AND MICROMECHANICS IDEALIZATION 1. Microstructure of the Constitutive Element

In establishing the micromechanics constitutive law of material, a constitutive element (a very small representative material sample) is usually taken as the subject of study, which contains enough microscale deformation information. The macroscopic continuous medium such as the stress-induced transformation zone near crack tip and the uniaxial tensile specimen is considered to be piled up by a large number of such elements as shown in Fig. 4. For the thermoelastic martensitic transformations of polycrystalline SMA and ceramics considered here, the constitutive element consists of a great number of untransformed grains of parent phase (as the matrix) and the transformed grains as dispersed second phase martensitic inclusions (with volume fraction f) embedded coherently in the elastic matrix. A temperature T is uniformly distributed everywhere in the element, and the external macroscopic stress ( C ) or strain ( E ) is applied on the boundary. Microscopic observation reveals that under a critical thermomechanical condition the martensite/parent interface propagates with a velocity that is much larger than the speed of the applied thermomechanical load in general, which implies that only two states exist for a grain (or at

Bulk structure

The constitutive element

The single

grain

FIG.4. Microstructure and micromechanics model of the constitutive eiement.

Micromechanics of Thermoelastic Martensitic Transformations 259 the level of microregion): the untransformed state and transformed state. The partial transformation or the intermediate state of transformation (i.e., 0 c f c 1) is only meaningful in the sense of the macroscopic volume average over the constitutive element. In order to arrive at a macroscopic material description from the microstructure considerations, some microscopic variables must be introduced. For convenience, we denote the microscopic stress and strain in the elements by 0 and E and the volume of the element, matrix, and martensitic inclusions by V, V,, and V,, respectively; and V = V, + V,. Then 0 , E , C, and E can be further decomposed into volumetric (superscript “v”), deviator (superscript “d”), elastic and plastic parts (Leblond et al., 1986): 0?I. .=

s.. u +

C.. IJ =

s.. + Crn 6..

(2.2)

&e. + &P1J ’ Y

(2.3)

&.. 1J

=

0m 6.. u’

ij

IJ

(2.1) 3

&?. IJ = E??d U + & q U ,

(2.4)

where the thermal strain is neglected for simplicity. Principally, the microstructural evolution of the constitutive element accompanying transformation or reorientation can be summed up as the process where the volume fraction of p + m transformation or m + m reorientation changes and each transformed grain (or region) experiences a chemical free energy change, interfacial (surface) energy change, and a plastic strain ep (stress-free transformation lattice strain or eigenstrain, Eshelby, 1957; Mura, 1987) that, once transformation is completed, does not change any more until m + p reverse transformation or m -+ m reorientation happens. 2. Some Basic Assumptions

In the process of stress-induced transformation at high temperatures (T > M s ) , the volumetric part cPVof transformation strain is the constant lattice volume dilation and independent of stress state. The deviatoric part cPd,however, is stress state-dependent because of the directionality in the crystallographic orientation and its shear nature. Among the 24 possible variants of martensite in shape memory alloys, only a particular one that

260


is energetically most favorable can be preferentially formed and grow up to the grain boundary in a polycrystal (Christian, 1982; Delaey et al., 1974; Olson and Cohen, 1981). The ~5~ may be considered to be uniform in a grain (inclusion) in the sense of long-range effect, and the magnitude of ~8~is limited by the lattice shear. In the deformation history, the &Ed in a grain may vary within itself or may vary from inclusion to inclusion and have different direction due to the stress-biased favorable orientation of the variant. Because a transforming grain is embedded in the matrix, we assume that at any instant of transformation in the deformation history the &Zd in a grain is always along the direction of or parallel to the average stress s r o f the matrix where this grain is located at that instant, i.e.,

where u," = ($S?S?)'/~, g = fie!: =fi($~~ representing ~&~ the~ ) " ~ . be emphasized again that the &Ed in a given grain, intensity of ~ 8 It ~must once transformation is finished, does not change any more until reverse transformation or reorientation occurs. The s? is the measure of the combined effect of external macroscopic stress and the transformation internal stress in the matrix. The detailed calculation of s y is given in Section 111. Generally the magnitude of g in Eq. (2.7) should depend on the magnitude of a? (or matrix strain), and there should exist a critical value ocrbelow which the value of g increases with the magnitude of o," and further increases with the applied macroscopic stress, as suggested in the experiments of Hsu and Wechsler (1981) and recently of Wang (1992). When 0," = 0, g must be zero, because there is no stress bias and the transformation lattice shear in a grain is averaged out to be zero due to the self-accommodating twins. When u," > u,,, g is independent of 0," and is limited by the lattice shear. The relation between g and u," is schematically shown in Fig. 5 . In this chapter, the g will be taken as a material constant for most cases. The basic assumption in Eq. (2.7) is consistent with the experimental facts that the stress field causes the appearance only of variants that provide maximum work done with respect to the inelastic strain (Otsuka and Shimizu, 1986). The preceding matrix stress bias effect upon &$* is most obvious and plays a dominant role in the two-way, or reversible, SME (Christian, 1982; Wayman, 1983). At low temperatures (T < Mf),the material is originally in a fully twinned martensitic state. The differently oriented martensite variants are spatially


.____

0cr

0 :

Equivalent stress of matrix FIG. 5. Schematic illustration of the relation between g and u," at a given temperature.

arranged in so-called self-accommodating groups such that the shape changes of those variants compensate each other, the diviatoric internal stresses are very low (so = 0, f = 1 at the beginning), and there is no macroscopic shape change. Upon stressing, the twin or variant boundaries are displaced in such a way as to cause the growth of those variants which are most favorably oriented with respect to the average matrix stress sf (equal to the applied stress So at the beginning of the process). Microscopically, this process in a grain is accomplished by the variant-variant coalescence and group-group coalescence until a single variant of martensite is eventually formed (Christian, 1982; Wayman, 1983). When the applied stress is removed in the case of rubberlike elasticity, the structure returns to its original "self-accommodating" state with minimum strain energy again as schematically shown in Fig. 1(e)-(h). For convenience, the preceding process is termed as the reorientation process (l), as distinguished from the reorientation process (2) to be defined later. The maximum shape change that can be achieved by reorientation is also limited by the lattice shear as in the case of p + m transformation. Thus, the relation between deviatoric and the matrix stress sf still obeys Eq. (2.7). reorientation strain &id' Hereafter, we shall not distinguish and &id. In view of the foregoing, we can logically conclude that, at the limiting strain of a pseudoelastic loop, the structure, orientation, and shape change from the original parent phase is now equivalent to that of stress-induced martensite, apart from trivial changes arising from the different temperatures of the two processes (Christian, 1982). This equivalence was confirmed by Cook (1981) in his experiment of polycrystalline specimen. Another difference from the

&id'

262


p + m transformation is that the displacement of existing twin boundaries dominates the process, and so comparatively the surface energy term is negligible and the major driving force is from the external applied stress because there is no chemical free energy change. The stored elastic energy and the twin boundary friction are the main terms of the resistive energy. Finally, for the process of the reorientation from one martensite variant to the other martensite variant that has a different crystallographic orientation from the first, we assume that for a grain that experienced a transformation (or reorientation) strain t$d(t') at past time t', the reorientation strain Eid(t)with respect to the original parent phase of this grain under the applied stress at current time t still obeys Eq. (2.7), i.e.,

For convenience, this process is termed as the reorientation process (2). We use eQ to represent Eid in the following derivation.

3 . Some Relations Between Macro- and Microquantities In the complex loading history, the previously described processes may all be involved in the deformation of the constitutive element. If we denote the volume average by ( ) and neglect the thermal expansion term and the small difference in elastic constants between parent and martensite phases at a given temperature, then by taking volume average of Q and E over the constitutive element it is easy to prove the relations

z = ( d v = fW",+ (1 - f )(a)", , E = ( E ) V = Ee

+ E P = M(T):C+ f(Ep)6,

(2.9) (2.10)

and the macroscopic strain increment corresponding to an incremental thermomechanical loading dZ and dT (Sun and Hwang, 1991a, 1993):

+ dEp = M ( T ) :dX + &pv df + (&Pd)dVrdf

dE = dE'

(2.11) Here M ( T ) is the macroscopic elastic compliance tensor, V, (=fV)and V , ( =f ' V ) are the volumes occupied by the transformed martensite and the

variant characterized by Eq. (2.7) (or Eq. (2.8)) respectively; dV,, df and

Micromechanics of Thermoelastic Martensitic Transformations 263 d K , df' are the corresponding increments in their volumes and volume fractions; dR and dr are the volume and volume fraction of reorientation process (2) corresponding to an incremental loading. Because under an incremental loading these possible processes are happening in a common matrix stress environment, Eq. (2.11) can be expressed as dE = M ( T ) :dZ

+ epvdf + E~~ df + [cPd- cPd(t')]dr.

(2.12)

The evolution law of the preceding microvariables will be given in Section IV.

111. Energy Changes Accompanying Deformation Processes

In this section, we first derive the elastic strain energy of the constitutive element and calculate the changes in chemical free energy and surface energy, and then we formulate the expressions of the complementary and Helmholtz free energy of the constitutive element. Finally, the analytical expression for the energy dissipation during transformation and reorientation are given. Some thermodynamic foundations of this analysis have been given by Kato and Pak (1984, 1985) and more recently by Ortin and Planes (1988, 1989). A. FREEENERGY OF THE CONSTITUTWE ELEMENT 1. Elastic Strain Energy

As shown in Fig. 4, under the action of external stress C and temperature T, the corresponding volume fractions are f and 1' ( f sf).The total elastic strain energy of the constitutive element is the sum of W,and W,, where W,is caused directly by the external macroscopic stress,

W, = i Z : M ( T ) : Z= i E e : M - ' ( T ) : E e ,

(3.1)

and W, is caused by the internal stresses produced by transformation or reorientation. In the derivation of W, , two effects should be considered: (i) the size of the constitutive element is finite, and so Eshelby's solution (Eshelby, 1957) for an inclusion in an infinitely extended elastic body cannot be used directly; (ii) for the polycrystalline ceramics or shape memory alloys, the spatial density of the transformed grains (inclusions) increases with their

264


volume fraction, and so the interaction between the stress fields produced by individual transformed inclusions must be taken into account. This implies the use of micromechanics that accounts for explicit interaction at the level of a continuous matrix phase and one or more inclusion phases. To achieve this, among a class of micromechanics models, the self-consistent Mori-Tanaka’s theory is adopted (Mori and Tanaka, 1973; Mura, 1987). A much more direct and simplified derivation of the theory has been given by Benveniste (1987). For simplicity, the inclusions here are taken as equalsized spheroids. It can be extended to the general case of ellipsoids. Denoting the internal stress or eigenstress (see Mura, 1987) in the element by 6, and the Eshelby’s solution by o;, we then have

where ’ ( T ) , K ( T ) , v are the elastic shear modulus, bulk modulus, and Poisson’s ratio, respectively, s; = 0; - a,“6,, 0; = +o:, Mori and Tanaka (1973) proved that -f(@v,,

(3.4)

-fe,(T ) ( E ~ ”- ) V ~

(3.6)

(6,)VM

=

which can be further written as

(6m)vM

=

Thus, the internal stress in any inclusion can be approximately expressed as (Mura, 1987)

6.. U = a m IJ + (6..) 1J V M ‘

(3.7)

By superposing Eqs. (3.5) and (3.6) with the macroscopic applied deviatoric and volumetric stresses Sij and C,, the average deviatoric and volumetric stresses in the matrix are, respectively, sM= U S 1.J. - f ‘ B 1(T)(&Cd)F,

0,” =

Cm -fBz(T)&”.

(3.8)

From these equations, the elastic strain energy W,per unit volume of the

Micromechanics of Thermoelastic Martensitic Transformations 265 constitutive element can be directly calculated by (Mura, 1987; Sun et al., 1991a)

-

3 2

3

- Bz(T)(&'")2f+ 2 B2( T)(Epv)2f '.

(3.9)

2. Chemical Free Energy and Interfacial Energy Accompanying a p + m transformation, the total change in chemical free energy per unit volume of the constitutive element is

AG,h,,(T, f) = [G"(T) - G P ( T )f] = AGP'"(T) f,

(3.10)

where G m ( T )and G p ( T )are chemical free energy of m and p phases, respectively, depending on temperature as shown in Fig. 6, in which & is the stress-free equilibrium temperature of two phases (AGP-'"(&) = 0). For equal-sized spherical particles, the total change of interfacial energy Kntcan be simply expressed as Knt

=

6Ysf/do

9

(3.11)

where y s = ym - yp , and ym , yp , do are the surface energy of martensite and parent phases per unit area and the diameter of the particle, respectively.

To Temperature

FIG. 6. Chemical free energy versus temperature diagram.


266

3. Helmholtz and Complementary Free Energy

If we denote the Helmholtz free energy by @ and the complementary free energy (minus Gibbs free energy) by Y,then from the results derived in Eqs. (3.1)-(3.11) the analytical expressions of @ and Y for a unit volume of the constitutive element in the deformation history can be formulated as @@,

T , f , f ( E $ ~ ) v ,=) W, + W, + Writ + 9‘

AGchem

= +(E - f (eP)5):M - ’ ( T ) :(E - f (E’)v,)

- +B1(T)g2f1+ tB,(T)(Egd)v$E$d)V,fI2 -

Y(C, T ,f , f

I,

+B2(T)(&Py2f+ $B2( T)(&PV)2f2

+ 6y, f / d o + AGP””(T)f , ( E $ ~ ) K ) = -(Wl+ W2 + qnt + AGchem- X :E ) = +z:M ( T ) : c + c :(f&PV + f ‘(&Pd)Vr)

(3.12)

+ f B , ( T ) s Y ‘ - + B , ( T ) f12(E$d)v$E8d)y,

+ +B2(T)(&pv)2f- +B2(T)(&pv)2f -

6y, f / d o - AGP’”(T)f.

(3.13)

In Eq. (3.13), C :ePvmeans the double dot of the tensors Zij and Thus far, the free energy changes for the three major processes as operative on the microscale in the deformation of a typical thermoelastic martensitic material (see Section I1.B) have been incorporated into the preceding unified expression. From the point of view of the thermodynamic internal variable constitutive theory (Rice, 1971, 1975), it is clear that the thermodynamic state of the material is completely defined by the variables E (or Z), T, f , f I, and in Eqs. (3.12) and (3.13), where E (or X) and T are obviously the internal are macroscopic variables and f , f and variables describing the microstructural change of the material during deformation; that is, the dependence of the material response on loading history can be replaced by a dependence on what it has produced, namely, the current pattern of structural arrangement on the microscale of material element that is represented by the current values of internal variables. When the internal variables are fixed, the material response is elastic. For example, the monotonic uniaxial tensile stress-induced transformation is

Micromechanics of Thermoelastic Martensitic Transformations 261 characterized by df = df' > 0; the reverse transformation in superelasticity or in shape memory effect is by df = df ' < 0; the monotonic reorientation of twinned martensite at low temperature (reorientation process (1)) is characterized by df = 0 but df ' > 0; whereas the reorientation from one martensite variant to the other (reorientation process (2)) is characterized by df = df' = 0 but dr > 0 and d(e$d)K# 0.

B. ENERGY DISSIPATION DURINGDEFORMATION In the case of forward and reverse transformation, because they are realized by the forward and reverse motion of p-m interfaces, the essential contributions to the energy dissipation are associated with the p-m interfacial friction and irreversible defects production that, macroscopically, will result in stress hysteresis in the superelasticity or temperature hysteresis in thermal cycling. In addition, the conversion of elastic waves to heat at the microlevel during transformation (as in the acoustic emission measurement) would also cause energy dissipation. Through the analysis of existing experimental data and for simplicity of mathematical treatment, we may reasonably assume that the total energy dissipation W, of unit volume constitutive element is proportional to the accumulated volume fraction of transformation f , , ( = j{ Idf I), i.e.,

w, = Dofcu,

(3.14)

where Do is a positive material constant or generally a material function. More detailed investigation showed that Do is dependent on the test temperature, the kind of variant produced and the strain rate (Otsuka and Shimizu, 1986; Christian, 1982). The rate form of Eq. (3.14) is

% = Dolfl,

(3.15)

where ( ) = d( ) / d t . Similarly, for the reorientation process (1) and the process (2) the corresponding energy dissipation rate can be approximately expressed as

Wpl= DF'lf'l

(3.16) (3.17)

respectively, where i = dr/dt > 0, and DF' and OF2 are positive material constants or functions.


268

IV. Constitutive Relations In this section, by the energy balance analysis, the constitutive laws corresponding to the three major processes described in Section I1 under general nonproportional loading conditions are derived, and it is shown that the procedure used here coincides with those of the internal variable constitutive theory (Rice, 1971, 1975). A. CONSTITUTIVE LAWSFOR FORWARD AND REVERSE TRANSFORMATIONS 1. Constitutive Law for Forward Transformation in Stress Space

a. Derivation of General Relations When transformation proceeds dynamics requires

(f# 0),

the second law of thermo-

where (> = d( ) / d t . The thermodynamics energy balance requires

*I,=

(4.2)

wd.

Substituting Eqs. (4.1), (3.13), and (3.15) into Eq. (4.2) and using Eqs. (2.12) and (2.7) to eliminatef andf' (noting that f = f' > 0 and i = 0 in stressinduced transformation), the forward transformation condition (yielding condition) in stress space can be finally derived:

+ 3&PV(Z, - fB2(T)&PV)- C,(T) = 0

(4.3)

where J@ij

-f'Bm(Egd)v) = [+(Sij - frBI(T)(&fid)vr)(Su -f '

Co(T) = Do

+ 6ys/do + AGP"'l(T)

-

~ 1 ( ~ & 8 ~ ) v ~ ) 1 ~ (4.4) ' ~ s

i B , ( T ) g 2 - +BZ(T)(E~')~. (4.5)

Because Bl(T) and B,(T) are negative, the material will exhibit a softening response when Co(T)is independent o f f (Eqs. (4.3) and (4.5)) due to the assumption of equally sized particles. For example, in the case of

Micromechanics of Thermoelastic Martensitic Transformations 269 pure volume dilation transformation (cPd= 0). we have, by (2.11) and (4.3) with g = 0, c'rn = - qAT)Eii. (4.6)

-

This negative slope prediction of the Zm Eji curve during transformation (Sun et al., 1990a) coincides with the critical transformation of Budiansky et al. (1983), although it is derived in quite different manner. It is easy to prove that Eq. (4.3) represents the balance between the average of all the local energetic forces on all the moving interfaces and the frictional resistance and that Eq. (4.3) has similar physical meaning to the energy balance during crack growth in fracture mechanics (Eshelby, 195 1, 1956, 1957, 1961, 1970; Sun et al., 1990a) Microscopically, the average stress in the matrix caused by transformation (Eqs. (3.5) and (3.6), which is negative of the so-called back stress, is conducive to the autocatalysis of further transformation of the remaining grains of parent phase in the matrix (Cohen and Wayman, 1981). This effect is most obvious in the case of such an equal-sized grain assumption and the resulting equal potency of nucleating sites in each grain (Chen and Chiao, 1983, 1985; Chen et al., 1985). This chain reaction, once initiated, is expected to require a somewhat lower driving force to sustain, hence resulting in an externally applied stress decrease. The softening dilatancy stress-strain relation in Eq. (4.6) is consistent with the foregoing interpretation and shows the importance of the microscopic internal stress. On the other hand, however, the published experimental macroscopic stress-strain curve during transformation shows an overall trend of increasing resistance to transformation with increasing volume fraction transformed, i.e., hardening rather than softening (though a serrated stressstrain curve during transformation may sometimes be observed). The microstructural reason for such a hardening effect is due to mechanisms such as the particle size distribution (for example, the compressive yield stress of ZrOz ceramics increases from 750 to 900 MPa when the grain size decreased from 2 to 1 pm (Reyes-Morel et al., 1988)), the crystallographic orientation effect, and the mutual interference of transformed regions, etc., in reality. Thus, the hardening contributions must be included in the model or, alternatively, must be treated simply by assuming that C,( T) increases linearly with f , i.e., replacing Co(T)by Co(T,f ) , (4.7)


270

where a ( T ) is an experimentally identified function. Equation (4.3) can also be expressed in terms of average matrix stress a?: g M be

d3

+ 3ePVaF- C,(T,f) = 0 ,

(4.8)

where it is clearly seen that the matrix stress required to transform the parent matrix into martensite increases with volume fraction transformed. The incremental stress-strain relations for stress-induced forward transformation can be readily obtained by the usual routine of internal variable constitutive theory (Rice, 1971, 1975): E.. = Ee. + E!? u u u

where j ( = j ' )can be determined by the consistency condition with the aid of Eqs. (2.7) and (2.12):

a qf, + F. - -aX qU . + - axQ af giving

a q f'+ . -a Tq +.

af

= 0, (4.10)

aT

[

d(AGP'"(T) dT

[

+ iB1(T)g2 1 + ~Bz(T)(&")~ + a(T)].

+ a(T))

(4.11)

b. Discussion From Eqs. (4.3) and (4.8), it is easy to prove that (4.12)

Micromechanics of Thermoelastic Martensitic Transformations 27 1 i.e., the inelastic strain rates are normal to the yielding surface (Eq. (4.3)) in stress space. The normality rule is naturally satisfied, and there is no need to assume the normality a priori, as was done in conventional phenomenological treatment. When transformation proceeds, the center of the yielding surface, as shown in Eq. (4.3), moves with the average internal stress in the matrix: ",=

-f 'B1(T)(€;d>"r- fB*(T)&PVSu;

(4.13)

and at the same time the yielding surface expands in stress space as well (Eqs. (4.3) and (4.7)), having the characteristic of both kinematic and isotropic hardening, i.e., mixed hardening. In terms of classical plasticity, the back stress is equal to -(8jj)vM, which can be directly calculated from Eq. (4.13) and need not be assumed, as was done by Prager (1956) and Ziegler (1959) in the theory of metal plasticity. The yielding surface in stress space is schemetically shown in Fig. 7. The transformation yielding stress (Eq. (4.3)), through dependence on material functions AGP'm(T), B , ( T ) ,B,(T), etc., depends on the temperature T. This character of transformation yielding dependence on temperature and stress clearly reflects the interchangeability of the two basic state variables affecting transformation, and from the properties of AGP'm(T), B , ( T ) , and B,(T) it is seen that the forward transformation stress increases with temperature, which is consistent with all the available experimental data (see, for example, Reyes-Morel and Chen, 1988; Schroeder and Wayman, 1979).

'Proportional loading path

(i > 0 )

Fro. 7 . Three-dimensional view of forward and reverse transformation yielding surfaces in stress space.

272


Because both the shear and volume strain of transformation are taken into account, the derived yielding condition has the most general yielding characteristic of pressure sensitivity that is common in geomaterials and porous ductile metals (Rudnicki and Rice, 1975; Gurson, 1977) and is consistent with the experimental data of stress-induced transformation of zirconia ceramics, where a considerable volume expansion (about 4.5%) is involved (Chen and Reyes-Morel, 1986; Reyes-Morel and Chen, 1988). Furthermore, if shear effect is neglected, as in the cases of transformations under hydrostatic tension and stress-free thermal cycling, Eq. (4.3) is consistent with the results of Budiansky et a[. (1983) and Sun ei a[. (1990a). In most of thermoelastic shape memory alloys, the transformation volume change is relatively small (usually less than 1% (Wayman, 1976)) and can be neglected in case of stress-induced transformation. Then according to Eq. (4.3) the transformation condition has the Von-Mises form. It must be pointed out that the yielding condition of Eq. (4.3) is derived under the assumption that g is a stress state-independent material constant; this assumption is only correct when the a," is larger than a critical value, as pointed out in Section II.C, and may be unrealistic if:a is small, In the latter case, Eq. (4.3) will lead to discontinuous behavior for polycrystalline materials in the following sense. If hydrostatic tensile stress is applied together with an arbitrarily small positive af"z, then when transformation proceeds the constitutive law predicts a volumetric expansion plus a finite positive shear transformation strain. Whereas if ! a is arbitrarily small but negative, there would be a finite negative transformation shear strain. This physically unsound discontinuous behavior can be avoided by a physically based assumption that g is proportional to a," and inversely proportional to p ( T ) , i.e., (4.14)

where (>0) is a material constant and reflects the constraint of the elastic matrix to the transforming grains. The maximum value of g in Eq. (4.14) equals the lattice shear. By a similar derivation to the foregoing and in Section 111, the transformation yield condition can be approximately expressed in the form of matrix stress a,M:

Z(T)(a,")'

+ 3a,M~~' - C'o(T,f) = 0;

(4.15)

Micromechanics of Thermoelastic Martensitic Transformations 273 or in the form of macroscopic stress Xu:

Z(T)[J(SU- f ' B I ( T ) ( E ~ ~+) ~E"[X, ~ ) ] ~ - ~ B Z ( T ) E- ~Eo(T, ~ ] f) = 0, (4.16) where

Eo(T,f) = Do

3 + 6y,/do + AGP'"l(T) - -BZ(T)(E~')~ + a(T)f. 2

Equations (4.15) and (4.16) have a similar form to the pressure sensitive criterion proposed by Lambropoulos (1986) for ceramics and to that of Gurson (1977) for porous ductile metals. The corresponding incremental stress-strain relation can be similarly obtained and is omitted here. Finally, we mention that the expression of Co(T)in Eq. (4.5) can in some cases, such as at very low temperatures ( T 4 &), have a value of zero or even a negative value. Physically, this implies that the thermal driving force (-AGP+"l(T))for forward transformation is large enough so that the stress-free transformation can occur automatically. In this case, the application of external stress will cause reorientation or reverse transformation processes to happen in such thermal-induced self-accommodating martensites, the constitutive response of which will be described in the following subsections. 2. Constitutive Law for Reverse Transformation in Stress Space a. Forward TransformationMemory Function It is easy to recognize that the foregoing monotonic forward transformation is ideal and does not always happen in an actual nonproportional loading history since the reverse transformation often gets involved during deformation. Because the reverse transformation is essentially the crystallographic recovery of the lattice deformation formed in the previous forward transformation, the reverse transformation plasticity must be restricted by the forward transformation history, having a strong memory character. According to Eqs. (2.7), (2.1 l), and (2.12), the existing deformation state (E!, f , f r, etc.) of the material element at the present time t can


274

be regarded as an integral to the loading history; i.e.,

[

f(t) =

j$')

f '(t) =

dt',

5:

f'(t') dt',

(4.17)

where f(t') = df(t')/dt', i(t') = dr(t')/dt', f'(t') = df'(t')/dt', t' E [0, t ] , t" E [0, t ' ] , t" E [0, t'], and t"' i t" = t' in the case of forward transformation. For convenience, we introduce forward transformation memory functions f ( t ' ) and f'(t'), which are defined in the forward transformation interval (E [0, t ] ) ,not recovered by reverse transformation, and are equal to f(t') and f'(t'), respectively in their intervals of definition. For a given loading history, &') andf'(t') are known functions. If the lattice deformation caused by the forward transformation at time t' is recovered at present time t , t' being a function of t, then according to Eqs. (2.7) and (2.12) the plastic strain rate at t is (4.18)

where f = df(t)/dt and f' = df '(t)/dt.It is clear that there are only two possibilities: ( 1 ) j < 0 andf' = 0, (2)f = f' < 0. In the following, we only discuss case (2); case (1) will emerge in Section V. b. Constitutive Law In a similar way to the derivation in forward transformation, by using Eqs. (3.13), (3.15), (4.1), (4.2), and (4.18), we obtain the reverse transformation condition of the martensites that are forward transformed at t' (letting$ = f' < 0):

FAL TYfYf

'Y

(&;*)KY

t')

= 3&PV(Z,- fB*(T)&P')

+g

a [SU - f 'B,(T)(&8dP)"JSu(t')- f'(t')B,(T(t'))(&~d)~(tl,l 2J[Su(t') - f'(t')B,(T(t'))(&~dP)Y,(tl,l

- CdTYf)

=

0,

(4.19)

Micromechanics of Thermoelastic Martensitic Transformations 215 or

F I G ,T , f , fI, (&Ed)vr,t‘) = 3&’’(Z,

- fB2(T)EPv)+ [Sjj - f ‘B1(T)(t$d)vJ&$d(t’)

- C , ( T , f ) = 0,

(4.20)

where C,(T,f) = Co(T ,f) - 2Do.For a given t , t’ (I t ) is determined from (4.19) or (4.20). The incremental stress-strain relations during reverse transformation are

(4.21) where f ( = f r < 0) can be calculated from the consistency condition

where

i’ ( = d t ’ / d t )is determined by the condition (4.23)

i.e.,

t’ =

f df(t’)/dt’‘

(4.24)

Thus, using Eqs. (4.18), (4.22), and (4.24),fcan be finally calculated from and T. c. Discussion In a real deformation process involving both forward transformation and its reversal, the combined use of Eqs. (4.9, (4.9), (4.19), and (4.21) is necessary in constitutive calculation, which is very complicated for the case of nonproportional loadings. If the martensite is transformed under proportional loadings, however, the reverse transformation condition in stress space is simply represented by a hyperplane defined by Eq. (4.20) with outward normal of ( f l / 2 ) g s 3 t ‘ ) / a r ( t ’ )+ ~ ~ “ c 5 ~ the , , three-dimensional view of which is shown in Fig. 7.

276


Because aF,/at’ = 0 for proportional loading history (g is assumed to be a constant), the “loading criterion” (in terms of classical plastic flow theory) for reverse transformation of a strain hardening material

( a ( T )+ iB,(T)g2+ ~B,(T)(E~’)~ > 0, see Sun and Hwang, 1991a, 1993) is simplified as

aF, . -tij Tc0 axij + aT

(f< 0).

(4.25)

In contrast with classical plasticity, there are two sets of yielding surfaces in stress space for superelastic behavior, which microscopically correspond to the forward and reverse transformation processes, respectively. This means that before the load is completely removed, the elastic unloading from the p -, m forward transformation yielding surface will proceed only until the stress state of the material satisfies another yielding criterion, which is associated with the reverse transformation. Detailed discussions can be found in recent work of Sun and Hwang (1993).

B. CONSTITUT~VE LAWSFOR REORIENTATION PROCESS-(1) In addition to the forward (p -, m) and reverse (m -, p) transformation, the reorientation process under the applied loading is another dominant deformation mechanism of thermoelastic martensitic materials. This process involves (1) reorientation of a fully twinned martensitic structure at low temperature (T < Mf), which is responsible for the shape memory and the pseudoelastic rubberlike behaviors in this temperature range; (2) reorientation from one martensite variant to another under applied thermomechanical loadings, which may operate at both high and low temperatures and usually happens under nonproportional loading conditions. In this subsection we give the constitutive description for the first kind of process. 1. General Characteristics Because the deformation under applied stress in this case is by the twin or variant boundary displacement in such a way as to cause the growth of those martensite variants which are most favorably oriented with respect to the average deviatoric matrix stress sf, it is clear that this process does not

Micromechanics of Thermoelastic Martensitic Transformations 277 involve lattice volume change and that the volume fractionf of martensite does not change in an incremental loading ( f = 0). Another important characteristic is that because the interfacial energy change is negligible and there is no chemical free energy change in this process ( f is fixed), the major driving force is from the external applied stress, and the stored elastic energy and the twin boundary interfacial friction are respectively the main parts of the resistive energy and energy dissipation. When the applied stress is removed, the microstructure returns to its original "self-accommodating" stable state with minimum strain energy again. Due to the equivalence of transformation-induced and reorientation-induced martensite variant (Sun and Hwang, 1991a; Christian, 1982; Cook, 1981), the free energy for the foregoing forward and reverse processes can still be expressed by Eqs. (3.12) and (3.13). 2. Constitutive Relations

When reorientation under the applied stress proceeds, the second law of thermodynamics requires (f= 0, f' # 0) (4.26)

and the energy dissipation rate

Wd, from Eq.

(3.16), is

W . l = 0;'If' l .

(4.27)

Thermodynamic energy balance requires YIZ,J =

W.1.

(4.28)

Substituting Eqs. (3.13), (4.26), and (4.27) into Eq. (4.28) and by the similar derivations in Section A, the yielding condition for the forward reorientation process (f' > 0 and f = 0) can be proved to be Fr&,

g T , f ', (~f3P)v) = 5J(Sij - f 'B1(T)(&Cd)~) - CZ(T,f '1 = 0, (4.29)

where Cz(T,f ') = OF' - $B,(T)g2 + a , ( T )f ', a , ( T ) is the function reflecting the hardening effect (the corresponding mechanism is not well understood so far) in the reorientation process. From Eq. (4.29) it is seen that the reorientation stress depends on temperature through the temperature dependence of elastic shear modulus

(BI(T) = 2 p ( T ) ( 5 ~ - 7)/15(1

-

v));


278

i.e., p ( T ) decreasing below T, and increasing above T, with increasing temperature (Hasiguti and Iwasaki, 1968; Zirinsky, 1956). The important contribution of this elastic anomaly to the phenomenon of the “V-shape” of the yielding stress-temperature curve of shape memory alloys in the vicinity of equilibrium temperature T, was first quantitatively incorporated into the constitutive description by Sun and Hwang (1991a, 1993) and will be discussed in detail later. Similarly, we can derive the yielding condition for the reverse reorientation process (j-‘ < 0,f = o),

FrerG, T,

1 ‘ 3

(&Ed>K, t’)

and the incremental stress-strain relation during the reverse process,

where C3(T,f‘) = C2(T, f‘) - 2Ll;’ ;f‘ ( G) due to the increase in AGP'"'(T). Within the range of reorientation at low temperature (T < M, or T < M s ) ,the temperature dependence of reorientation stress is determined by that of the elastic shear modulus of the martensite or parent phase, as indicated in Eq. (4.29). This means that, for a given value of DF', the required stress for twin boundary motion increases with decreasing temperature, thus leading to the transition from ferroelasticity or shape memory to rubberlike elasticity as shown in Fig. 3(f)-(j). Within the range of stress-induced transformation at temperature T > M, , the temperature dependence of initial transformation stress ( f = 0) in Eq. (4.3) is determined by that of the chemical driving energy (the term AGP'"'( T ) ) and the positive temperature dependence of the elastic in shear modulus of the parent phase (the term -4B,(T)g2 - %B2(T)(ep')2 Eq. (4.5)). Therefore, the smaller the magnitude of the elastic modulus, the smaller the elastic strain energy (Eq. (3.9)) that must be stored in the particles and matrix to maintain thermoelastic balance. The values of p ( T ) (parent) and p ( T ) (martensite) correspond to the resistance against the transformation shear to the m-phase and twinning shear in the m-phase, respectively. So, it is clear that the yielding stress in both p and m phases increases when the deformation temperature departs from T, (Fig. 11). But the minimum of yielding stress does not strictly coincide with T, because of interfacial frictional resistance, elastic strain energy and interfacial energy. The comparison between the preceding theoretical prediction and the experimental data for Ti-Ni polycrystalline shape memory alloys will be given in Section V.

280


It must be pointed out that an elastic isotropy is assumed in the present theory, whereas during the actual lattice softening the elastic anisotropic factor C,,/C’ is as high as 10, where C’= (CI1- C,#2 is the elastic shear constant corresponding to the resistance for the lattice shear (Nakanishi et ul., 1973; Zirinsky, 1956). Therefore, the foregoing theoretical explanation for the V-shape phenomenon, in the strict sense, is only an approximation, and in further work the anisotropic effect must be taken into account.

c. CONSTITUTIVE LAWSFOR REORIENTATION PROCESS-(2) In this subsection, we turn to the constitutive description of the second kind of reorientation process, i.e., the reorientation under the applied stress from one existing martensite variant (either previously transformationinduced or reorientation-induced at time t’) possessing a deviatoric eigenstrain Esd(t’) to another martensite variant characterized by deviatoric eigenstrain &$d(t). As already stated in the basic assumptions in Section 11, the &gd(t)is along the direction of the average deviatoric stress of the matrix at the instant of reorientation, so that the maximum work is done with respect to the inelastic strain, i.e.,

d3

& f ( t )= -g2

sM(t)

a,M(t)‘

(4.35)

An important feature of this process is that corresponding to an incremental loading dE and dT there are no changes in volume fractions f and f caused by the variation of eld (df = df = 0) and only changes in in a small reoriented region dR within V,. Thus, there is only elastic strain energy change accompanying deformation. Also, the energy dissipation is mainly related to the frictional resistance to the “switching” (Nakanishi et al., 1973), and no lattice volume change is involved in this process. 1. The Constitutive Relations

The energy balance relation during the process is (4.36)

Micromechanics of Thermoelastic Martensitic Transformations 28 1

By substituting Eqs. (3.13) (and (3.17) into the preceding and using the relation (from Eq. (2.12))

q =f

=

[&td(t)- &Ed(t‘)]i

(4.37)

to eliminate i (>0), we get the reorientation condition Freof the martensite variant formed at t’ in stress space:

Ee(x,T , f,f r, =

g m i j

-

[Sij

(&fid))yr,

t’)

- f rB1(T)(E;d)vrl

- frB1(T)(~fid)v,]&td(t’) - OF2=

(4.38)

0,

from which t’ is determined. The incremental stress-strain relations are

where i is determined from the consistancy condition

aEe . + aPreT. + aFre axij aT

-Eij

*

(Eij )

. + -aFre t’ at’

= 0.

(4.40)

p T V

Similarly as before, the reorientation memory function f ‘ ( t ’ ) is used to determine t ’ ( = d t ’ / d t , noting that f ‘ ( t ‘ ) and eEd(t’) are all loading history-dependent known functions):

t’

=

i/(df”‘(t’)/dt’).

(4.41)

Substituting Eq. (4.41) into Eq. (4.40) and using Eq. (4.37), i can be finally calculated from and T as @e

+aT f)/[ B1(T)e(t,t’) -

““;-I at

1

(df‘(t’)/dt’) , (4.42)

-

where e(t, t’) = [Etd(t)- &td(t’)][&td(t) - czd(t’)].It is easy to show that the inelastic strain increments are parallel to the normal of yieldng surface Fre= o in stress space.

282


-c

(I'a-0) Fra'=o

'L

(4

(b)

FIG. 8. Two-dimensional view of reorientation surfaces (Eq. (4.38)) in Z,,-ZI2 plane for shape memory alloy. (a) Initial yielding surface for a previously elongated specimen. (b) Initial yielding surface for a previously sheared specimen.

2 . Discussion It is easy to prove that the reorientation condition of Eq. (4.38) in stress space represents a parabola. The two-dimensional image of this reorientation surface in the Xl1-X,, plane for a shape memory alloy at a given temperature T is shown in Fig. 8. It is obvious that the surface = 0 is activated only for complex loading paths (at least in opposite stressing). More detailed discussions about this process under complex loading conditions are given in recent work of Sun and Hwang (1993). The foregoing theoretical predictions are subject to experimental verification.

V. Applications to Deformation of Polycrystalline Shape Memory Alloys

This section describes the various typical responses of the thermoelastic martensitic shape memory alloys under different thermomechanical loadings by applying the constitutive models previously developed. The theoretical results are compared with the available experimental data for Ti-Ni alloy in which the transformation lattice volume change is negligible. For the results of transformation plasticity in ZrO, ceramics with considerable

Micromechanics of Thermoelastic Martensitic Transformations 283 volume dilation, the reader is referred to previous work (Sun et al., 1991a; Hwang and Sun, 1991a; Chen and Reyes-Morel, 1986; Reyes-Morel et al., 1988; Reyes-Morel and Chen, 1988). For convenience we divide the macroscopic behavior into three categories, i.e., high temperature ( T > M s ) , low temperature ( T < Mf), and thermal cycling. It should be recognized that this division is not complete because, for example, in the temperature range Mf < T < M , both transformation and reorientation processes are possible and they are not strictly separated from each other. Thus, such a division serves mainly as a starting point for the analysis (Sun and Hwang, 1993). A. THEHIGH-TEMPERATURE ( T > M,) BEHAVIOR In the high-temperature range ( T > M,), shape memory alloys (SMA) may exhibit two major kinds of behavior, i.e., pseudoelasticity (PE, also called superelasticity) and shape memory effect (SME), both caused by forward and reverse transformations. So far, only experimental data under uniaxial tension and compression are available. In this section, by applying the theoretical model established in Section IV, we study the PE and SME under proportional loadings, and comparisons between theory and experiments in uniaxial stress state are made. For simplicity, we neglect the contribution of cPVbecause the lattice shear is dominant in stress-induced transformations of SMA. Finally, a discussion concerning the hightemperature behavior of SMA under opposite stressing is given. 1. PE by p

+

m

+

p Transformation (Superelasticity)

Under proportional loading and unloading conditions (C = A X o , L > 0), it is easy to prove that (within the stress range where g is constant (see Fig. 4))

and the forward and reverse transformation yielding conditions of Eqs. (4.3) and (4.20) are simplified as g F'(z9 T , f ) = -&

-

1 -B1(T)g2f - CO(T9.f)= 0, 2

(5.2)

g F,(C, T , f ) = -Xe

-

1 -BI(T)g2f 2

(5.3)

0

0

-

C,(T,f) = 0 .


284

a. Uniaxial Tension From Eq. (5.2) the forward transformation stress X i l in uniaxial tension is

where, as defined in Section IV, Co(T) = Do

+ 6y,/do + A C P d m ( T )- )B1(T)gZ,

~ 7)/15(1 - v ) . Bl(T) = 2 p ( T ) ( 5It is clear that for the hardening material the function a ( T ) should be chosen so that a ( T ) + )Bl(T)g2> 0 . The function AGP'm(T) has the properties that AGP'"'(T) = 0 at the equilibrium or transition temperature & and AGP'm(T) > 0 (or & (or < &). From the measured experimental data, A G P d m ( T )can be approximately assumed to be a linear function of temperature, i.e., AGP'm(T) = k(T - &),

(5.4)'

where k (>0) is a constant representing the entropy of transformation. The function p ( T ) , from the experimental data of Hasiguti and Iwasaki (1968) and Zirinsky (1956), can also be approximately assumed to be a piecewise linear function of temperature: P(T) =

[

T > & a n d & >o, T c &andqzcO.

qi(T- &)+Po, q2(T- & ) + p o ,

(5.4)"

The experimental data shows that at high temperature range, the initial forward transformation stress (f= 0) increases almost linearly with temperature (Sun and Hwang, 1993), being consistent with the foregoing prediction: d%l = dT

g

[

"1

k - g2q1(5v 30(1 - V )

=

const.,

from Eq. (5.4).

Finally, from the measured M, value, the constant term Do + 6y,/d0 in Co(T ) should be determined by the minimum transformation stress condition Eil = 0 (when f = 0 and T = M,) in Eq. (4.16) instead of Eq. (5.4) (noting that C , + 0, g + 0), or vice versa.

Micromechanics of Thermoelastic Martensitic Transformations 285 When unloading, the reverse transformation stress Cyl, from Eq. (5.3), is

where (2fi/g)D0is the stress hysteresis ( A x i l = Cil- Cyl) in the PE cycle, with Dothe constant defined in Section 111. A more general form of the stress hysteresis should be A x i l = ( 2 f i / g ) D 0 ( T , k )for a given martensite, as mentioned previously (Otsuka and Shimizu, 1986). Finally, the incremental form of stress-strain ( S - S ) relation at a given temperature during both forward and reversetransformations is

exhibiting linear hardening. b . Uniaxial Compression

In a similar way to the preceding, the forward and reverse transformation stresses in compression X i l , C;\ can be derived:

C;;

=

-e[ g

C,,(T)

)

+ ( a ( T ) + ?B1(T)g2f

].

- Zoo

(5.8)

Comparing Eqs. (5.7) and (5.8) with Eqs. (5.4) and ( 5 . 9 , we can see = - X;l , Xyl = - Xi\ , AE; = -AX; . Experimental data has already demonstrated such a symmetry of S-S curves with respect to the origin in the tension-compression PE cycle. It should be noted, however, that if a different martensitic transformation is induced in compression or a considerable volume change is involved in transformation such as in ceramics, the stress hysteresis is accordingly different, and there will be an asymmetric S-S curve (see Christian, 1982; Sun, 1989; Sun et af., 1991a).

C:

2. SME by p -+ m + p Transformation a. Heating Process From Eqs. (5.4), ( 5 . 9 , (5.7), and (5.8), it is seen that the absolute values of both forward and reverse transformation stresses decrease with decreasing temperature. If T is low enough or Dois large enough so that


286

t

FIG.9. S-S-T curve in a typical SME cycle by p -, m -, p transformation.

reverse transformation no longer occurs during unloading (i.e., Xyl I0 in Eq. (5.5)), then the remanent strain can be recovered by heating, i.e., by SME. The temperature required to induce the m + p reverse transformation of a specimen previously loaded in tension or compression must satisfy the equation (from Eqs. (5.5) and (5.8)) C,(T) + [cY(T)+ +B1(T)g2] f-

mo = 0.

(5.9)

The inelastic strain Efl (EE, = 0) of a previously tensioned specimen during heating is thus

(5.10)

(i< 0) is solved from Eq. (5.9). The typical S-S-T

where f in Fig. 9.

curve is shown

b . Opposite Stressing It should be noted that Eq. (5.5) implies that instead of heating, the remanent strain of a previously tensile loaded specimen can be reverted by opposite stressing, i.e., by compression. The compressive stress fyl (< 0) needed to induce the m p reverse transformation, from Eq. (4.20), is -+

ftr 11

[

(i

fi C,(T) + - B , ( T ) g 2 + a ( T ) -g

=

2d3

c;, - - Dg o , (5.11)

which has the same form as Eq. (5.5). The incremental form of S-S relation during reverse transformation in compression is the same as Eq. (5.6).


I

-

.:, --n l > O

I

;>o

I C O

Ls I > O

(a)

( b)

(c)

Fro. 10. The temperature dependence of S-S curves in reverse transformation caused by opposite stressing (T,> > T, > &).

It must be pointed out that if the forward transformation ( f > 0) at opposite stressing is to occur, then the required stress g l ,from Eqs. (4.3) and (4.7), is 1 (5.12) - C o ( T ) - a r ( T ) f + -2B , ( T ) g ' f ] . Comparing Eq. (5.11)with Eq. (5.12), we can see that in opposite stressing at T > T,, the reverse transformation always happens prior to the forward transformation, because Co(T)> Do (as T > To) and a ( T ) > 0 lead to 0 > Eyl > G1.The stress at which the reverse transformation finishes (f= 0), from Eq. (5.11), is

If we continue the opposite stressing after this, then the forward transformation will happ,en still according to Eq. (5.7). Figure 10 shows the typical S-S curves for the foregoing processes of reverse transformation by opposite stressing under different constant temperatures. At still lower temperatures, the self-accommodating martensites will preexist, and the deformation process under applied load is dominated by the twin boundary motion of martensite, which will be studied in the following section.

B. THELOW-TEMPERATURE ( T c Mf) BEHAVIOR At low temperatures, the microstructure of the stress-free specimen is in the state of a fully self-accommodating martensite group. For proportional loading, the yielding condition for the forward reorientation process,


288 \

\

Ti-51 Ni

\ \ \

a

GOO

u)

400

-

200

-

m

z

In

-

.-

0

,)

\

-

1.4

L

N

-

1.2

-

1.0

k-0.92 MPai'C g-26%

-

V-0.25

;3

- - - Experimenl (p)

c v)

r-"

\

1.6

N

q1-5.8 MPal"C q2--17.4 MPa/"C

t0

L

u) 0,

0

\

------

-

o

I

0'

Experiment (z:,.x

,'~,~~)

-Equat ions(S.4,h nd (516) I

I

I

FIG. 11. The temperature variation of elastic shear modulus p (dotted line) for a TiNi specimen ( p is related to the period of vibration A t of a torsion pendulum by p a (At)-*) (redrawn with permission from the publisher; Husiguti and Iwaskai (1968), J. Appl. Phys. 39, 2182) and the corresponding temperature variation of yielding stress above and below T, (redrawn with permission granted by Gordon and Breach Science Publishers; Funakubo (1987).

from Eq. (4.29), is

and the reverse reorientation yielding condition corresponding to the recovery of the forward process during unloading is (from Eq. (4.30))

where

C , ( T, f ' ) = q1 - iB1(T)g2- cr'(T)f', and C3(Tf') = C,(T,f') - w : ' . Because the elastic shear modulus p decreases abruptly with increasing T on approaching T, (dotted line in Fig. l l ) , contrary to the case of stressinduced transformation, the initial yielding stress (f' = 0) required for reorientation process decreases with increasing temperature, as reflected by the term - i B l ( T ) g 2 in C,(T, f') of Eq. (5.14).

Micromechanics of Thermoelastic Martensitic Transformations 289 1. PE by Reversible Twin Boundary Displacement

If the hysteresis in the forward and reverse motion of the intervariant boundary is small enough, the pseudoelastic stress-strain behavior can also appear at low temperature, which is usually called the rubberlike elasticity. Under uniaxial tension and uniaxial compression, the reorientation stresses Ze,ll %e,ll are (from Eq. (5.14))

G , l l

=

-Ze,11

(5.16)'

*

Upon unloading, the stresses for the reverse motion of the intervariants boundaries are (from Eq. (5.15))

(5.17) (5.17)'

where (2fi/g)Df;' = Zte,ll - Z:,ll represents the stress hysteresis. The plastic strain rate during the forward and reverse motion of the intervariant boundary, according to Eqs. (5.16) and (5.17), will be

or, from Eq. (5.16) ( f o r j ' > 0),

(5.18)'

2. SME by Reorientation and Transformation (m

+

m -+ p)

If reorientation of a fully martensitic structure by an applied stress is associated with a sufficiently high frictional resistance D r l , the remanent strain after unloading then can be recovered by heating, i.e., to cause the

290


martensites to transform directly to the parent phase. So in addition to the yielding condition for the reorientation process shown in Eq. (5.16), we have the reverse transformation condition upon heating (f' = f < 0, from Eq. (4.19)):

Co(T) + [cY(T)+ $B1(T)gz]f' -

200

=

0.

(5.19)

The inelastic strain rate Ef, during heating (f > 0) is

(5.20)

or in finite form, by using Eq. (5.19)

(5.21) which has the same form as Eq. (5.10).The corresponding S-S-T curve of the foregoing SME process is similar to Fig. 9 .

3 . Ferroelasticity and Elastic Anomaly (Lattice Softening) Since ferroelasticity and elastic anomaly are the most peculiar properties of thermoelastic martensitic transformations, they must be given special attention. Instead of heating the specimen, the residual strain in the reorientation process after unloading from uniaxial tension can also be recovered by opposite stressing, i.e. , by compression, which is commonly known as ferroelasticity. The required stress E:,l, (

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