Advances in Applied Mechanics, Volume 18

Advances in Applied Mechanics Volume 18 Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN RODNEY HILL L. HOWA...

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Advances in Applied Mechanics Volume 18

Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN RODNEY HILL L. HOWARTH WILLIAM PRAGER T. Y. Wu HANSZIEGLER

Contributors to Volume 18 F. H. Buss@

c. K. CHU RODNEYHILL JOHNL. LUMLEY

J. N. NEWMAN SHAN-FUSHEN

ADVANCES I N

APPLIED MECHANICS Edited by Chia-Shun Yih DEPARTMENT OF APPLIED MECHANICS A N D ENGINEERING SCIENCE THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

VOLUME 18

1978

ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT @ 1978, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS,INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W 1

7DX

LIBRARY OF CONGRESS CATALOG

CARD NUMBER: 48-8503

ISBN 0-12-002018-1 PRINTED IN THE UNITED STATES O F AMERICA

Contents vii

LISTOF CONTRIBUTORS

Aspects of Invariance in Solid Mechanics Rodney Hill 1

Introduction I. Preliminary Concepts 11. Constitutive Descriptions 111. Bifurcation Theory References

3 28 50 72

The Optimum Theory of Turbulence F. H . Busse I. Introduction 11. The Optimum Problem for Turbulent Couette Flow 111. Multi-a Solutions IV. Bounds on the Transport of Momentum V. Bounds on the Transport of Mass VI. Bounds on the Transport of Heat VII. General Discussion References

17 80 84 p4

105 110 115 119

Computational Modeling of Turbulent Flows John L. Lumley 124 128 133 143 152

I. Introduction 11. Mathematical Preliminaries

111. The Return to Isotropy IV. The Rapid Terms V. The Dissipation Equations VI. Transport Terms References

160 174 V

vi

Contents

Unsteady Separation According to the BoundaryLayer Equation Shan-Fu Shen I. Introduction 11. Asymptotic Behavior of the Boundary-Layer Solution

Away from the Wall 111. Separation and the Concept of an Unmatchable Boundary Layer IV. The Semisimilar Boundary Layer V. The General Unsteady Boundary Layer VI. Separation in Lagrangian Description References

177 1a2 1a6 192 203 213 218

The Theory of Ship Motions J. N . Newman I. Introduction 11. History 111. The Boundary-Value Problem IV. Fundamental Solutions V. Two-Dimensional Bodies VI. Slender-Body Radiation VII. Slender-Body Diffraction VIII. The Pressure Force References

222 221 235 244 249 258 266 273 280

Numerical Methods in Fluid Dynamics

C.K . Chu I. Introduction 11. Differential Equations and Boundary Conditions 111. Numerical Analysis Background IV. Pseudophysical Effects: Numerical Dissipation and Dispersion V. Gas Dynamics VI. Navier-Stokes Equations VII. Magnetohydrodynamics References

286 287 292 301 309 317 327 329

AUTHOR INDEX

333

SUBJECTINDEX

339

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin

F. H. BUSSE,Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024 (77) C. K. CHU, Department of Mechanical Engineering and Plasma Physics Laboratory, Columbia University, New York, New York 10025 (285)

HILL, Department of Applied Mathematics and Theoretical Physics, RODNEY University of Cambridge, Cambridge CB3 9EW, England (1) JOHNL. LUMLEY, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853 (123) J. N. NEWMAN, Department of Ocean Engmeering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (221) SHAN-FUSHEN,Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853 (177)

This Page Intentionally Left Blank

ADVANCES I N APPLIED MECHANICS, VOLUME

18

Aspects of Invariance in Solid Mechanics RODNEY HILL Department of Applied Mathematics and Theoretical Physics Unioersity of Cambridge Cambridge, England

Introduction . . . . . . . . . . . . . . . . . . . . . . . . I. Preliminary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Tensor Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Geometry of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . C. Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Generalized Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Constitutive Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Measure Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Some Types of Elastic Response . . . . . . . . . . . . . . . . . . . . . . . 111. Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. First-Order Rate Problem . . . . . . . . . . . . . . . . . . . . . . . . . . B. Primary Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bifurcation under Simple Loadings . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

3 3 9 14 17 24 28 28 35 41

50 50 58 66 12

Introduction This article is conceived primarily as an account of certain interrelated ideas that I have contributed over a long period to the basic mechanics of rate-independent solids, and which retrospectively appear to have proved influential. My previous writings on these matters have been purely seriatim and mainly for specialists. It now seems timely to attempt a unified, definitive, presentation which moreover is directed also to nonspecialists.In addition, I have taken the opportunity to incorporate unpublished items from lectures at the University of Cambridge (these are identifiable in the text by an absence of any attribution). The account opens with a recapitulation of properties of second-rank tensors (some standard, some not) that are peculiarly apt to deformation 1

Copyright @ 1978 by Academic Press. Inc. All rights of reprodudon in any form reserved. ISBN 0-12-002018-1

2

Rodney Hill

geometry and kinematics, and to the subsequent formulation of arbitrary measures of strain and work-conjugate stress, together with their objective fluxes. The overall approach differs substantially in spirit and emphasis from the stereotyped expositions in most monographs and treatises; my experience has been that the approach is accepted readily by students, and I myself would have made little headway without it. Clarity at this stage is indispensable for all that follows, so I have not skimped explanations; often, also, more than one route to a formula is indicated to foster helpful crosslinkages. Among other things, some prominence is given to what has come to be known as my “method of principal axes,” which I recommended as a sure way through tensor algebra which can otherwise relapse into labyrinthine complexity. Central to this method is the representation of tensors on trace axes of deformation that sweep through a Lagrangian reference configuration. A Lagrangian viewpoint is also well-suited to handling pure mechanics, as I have long advocated in relation to incremental boundary-value problems. With “nominal” stress as the primary variable, allied to the notion of convexity with respect to functionals,one has an infallible structural guide when investigating uniqueness of solutions and their extremal characters. Failure of incremental uniqueness, or bifurcation, is arguably the phenomenon that is most typical of continuum response at unrestricted levels of deformation. In view of the widespread current interest in its technical implications, I have treated bifurcation theory in detail here. Since the original publications, I have developed and refined the analysis during successive courses of Cambridge lectures, so hopefully the presentation now has a certain freshness, even for specialists. It may be noticed, incidentally, that I abstain from elaborating a collateral theory of instability for inelastic materials, such as I ventured in the past. It is of course in constitutive analyses of material behaviors that the mathematical apparatus assembled at the outset finds its justification. Restriction on space has compelled me to exclude idealized rigid-plastic response, even though I have always been much concerned with it; likewise only brief reference could be made to finite elasticity as such, mainly on the status of constitutive inequalities. Instead, I have elected to concentrate on the elastoplastic response of metal crystals and polycrystals, because it has become possible recently to say much that is new on the subject. Here again, though, I have had to accept constraints for the sake of an overall unity. On completing the text I met an unexpected difficulty in devising a satisfactory title, notwithstanding the initial conception and the ostensible contents. After reflection it became clear that a subconscious kit-motif had operated decisively and selectively throughout. This is the concept of “invariance,” not in the narrow usage of tensor algebra, but in a variety of guises

Aspects of Inuariance in Solid Mechanics

3

and at several levels of sophistication. Rather than enumerate these in advance, I prefer to let the article speak for itself. If the cumulative effect on a reader, whatever his speciality, is to suggest that invariance in its widest sense deserves more attention in every branch of continuum mechanics, the title will have been vindicated. I. Preliminary Concepts

A. TENSORREPRESENTATIONS 1. Decompositions on Reciprocal Bases

In the interests of simplicity, but with no loss of generality, the analytical framework will be expressed in vector and tensor components on a rectangular Cartesian background (curvilinear equivalents can be generated routinely when needed in particular applications). It is nevertheless essential to recognize the contravariant or covariant character of individual quantities that inevitably enter theories of solid continua. In the main these quantities relate to the geometry and loading of an elemental cell which is “embedded” in a continuing deformation. The symbolism to be described was devised to promote this basic viewpoint, and in that respect it differs from conventional notations in abstract tensor analysis. Usage has convinced me that there are consequent gains in clarity, in ease of manipulation, and in feel for structure. It is convenient to begin by recalling some definitions and terminology, and at the same time establish the notation. A pair of vector bases, al, a2,a3 and b’, b2, b3, are said to be reciprocal when

-

ai bj = ,a:

(1.1) where a dot denotes the scalar product and 6: is the Kronecker symbol. Such triads arise naturally in solid mechanics, for instance as edges of elementary cells that were cubes before deformation. Any vector v can be decomposed either as viai or ujbj where

ui = v * b’, uj = v aj. (1.2) The coefficients ui and uj thus have dual interpretations as components on one basis or as resolutes on the other. Let basis a’, a2, a3 be arbitrarily regarded as primary; then the components uj are called couariant and the ui contrauariant. Let T denote the matrix of components of any second-rank tensor on a rectangular background. Since T is expressible as a linear combination of nine independent tensors, which may in particular be dyads, four types of

Rodney Hill

4

tensor decomposition can be proposed: T = tijai@ aJ= tijbi@ bi = ti.jai @Id = t l j b @ aj.

(1.3)

These respectively define contravariant, covariant, and two kinds of mixed components of T. Equivalently, we have the vector decompositions Tbj = tijai, Taj = t i j b , or the vector resolutions tij

=b

. TbJ,

t i j = ai *

Ta,,

Taj = ti.jai,

Tbj = tljb'

(1.4)

ti' = ai . Tbj.

tfj = b' Taj,

(1.5)

To prove equivalence we need only appeal to the operational identity that defines a dyadic product: (u Q v)w = (v * w)u for any u, v, w. Let I denote the unit 3 x 3 matrix. Then the contravariant and covariant components of the unit tensor of second rank are such that I = gijai@ aj = gijbiQ bj. Equivalently, bj = gijai, gij

= b'

bj,

aj = gijbi, 9I.J .= a.I aj .

-

(1.6)

From (1.4) or (1.5) the mixed components are just 6c. Returning with this to (1.3) we obtain the useful identity a, Q bj = I = bj Q aj.

(1.7) This can alternatively be verified directly since it reproduces (1.2) when in product with any v. The arrays gijand g'j are mutual inverses because of the reciprocity of the triads; they act as metrics of the primary and secondary bases, respectively, in that u v = giju'd = g'ju, u j . They have a role also in the linear relations between tensor components of different types; for @ aj by combining (1.3) and (1.6). instance, t ; j = giktiJsince T = tij(gikbk)

-

2. Change of Basis

Let 9, and

6q

be another pair of reciprocal triads. Then i j p ~=

P

Viai + ijp = (

6 *~ai)u',

(1.8)

-

aj + ?q = ( 6 *~a,)(@ aj)ti? (1.9) The transformation rules for the other types of components are similar, and all the coefficients involve products like i, bi and 6 p ai (which incidentally are elements of mutually inverse matrices). Furthermore, 6p * ai ( p = 1,2,3) @iiP

H,

= t'ja,

.

Aspects of Znvariance in Solid Mechanics

5

are components of ai when decomposed on the other primary basis, and likewise b 9, (i = I, 2, 3). Hence, during a continuing homogeneous deformation, in which both primary bases are embedded, all such products are preserved. It follows as a corollary that the linear relations between tensor components of the same type on any pair of embedded bases are invariant under deformation. This observation underpins much recent work on measures of stress and strain. Nevertheless it seems not to have received due mention in the literature.

-

3 . Induced Tensors

Let any three, noncoplanar, fiber segments which are embedded in a homogeneous deformation be regarded as a primary basis. We write A for the matrix whose columns are the rectangular components of a’, a’, a3 in that order; for brevity this triad will be called the A basis. Similarly we write B for the matrix with columns b’, b’, b3.Note that the inverses A - ’ , B-’ are equal to the transposes B’, A’ when the A and B bases are reciprocal. The characteristic context, as we shall see, is where matrix A is the gradient of a deformation that maps a unit cube into a cell with edges a’, a’, a3. Previous formulae can now be rewritten symbolically. Thus from (1.6) ( g i j )= A’A,

(gi’) = B’B,

(1.10)

and from (1.2) (vi)

= B’v,

(Ui)= A’V,

where here v stands specifically for the column of background components. Especially, we draw attention to the replacement of (1.9, namely, (ti’) = B’TB,

(ti’) = A’TA,

(ti.’) = E T A ,

(ti’) = A’TB,

(1.11)

which will be quoted repeatedly. Suppose T is the array of background components of some tensor that is associated with a deformed configuration specified by gradient A. We can define four other tensors such that their background components are given by the arrays in (1.11). Since these components are equal, respectively, to those of the four representations of T on the A basis, we have the situation envisaged in the invariant property just proved. It follows that, if another embedded triad were introduced, the covariant components of tensor A’TA on that basis before deformation would be numerically equal to the covariant components of tensor T on that same basis after deformation. The same is true for the contravariant and mixed components. In other words, if T components of one type on any deformed basis are transferred to that triad as it was initially, another tensor is generated. It is by definition associated

Rodney Hill

6

with the reference configuration, and will be said to be "induced" from T by the deformation. 4. Kinematics of Embedded Bases

A deformation-rate tensor

r is defined by

Pi = rai

(i = 1, 2, 3),

where

b' = -rbj

( j = 1, 2, 3).

(1.12)

Here a dot denotes the rate of change when the primary basis is embedded in a continuing deformation (the reciprocal basis is not similarly embedded, of course). Symbolically, these may be read as the columns of

A=TA,

8= -rB.

(1.13)

From (1.7) and (1.12) we have the explicit formula

r = Pi 0 b'.

(1.14)

Evidently r is independent of the particular triad, again by the invariant property of transformations between embedded bases. Let

r=6+R where

R = )(r- r), 8 = +(r+ r).

(1.15)

By considering, in particular, the principal fibers of a further infinitesimal deformation, and applying (1.12) to them, we easily recognize R as the material spin and 6 as the Eulerian strain-rate. It is not difficult to show that the spin vector associated with the skew tensor R is

R = )b'

A

a',

(1.16)

with a standard notation for the vector product.

5 . Convected Derivatives Let v be any vector. Consider Dv/Dt, where D/Dt follows a material particle and t is a timelike parameter. We can evaluate this from (1.2) and (1.12) by two routes, namely D/Dt(u'ai)= u'ai + &ai,

D/Dt(vjbj)= ujbj - ujT'bj,

Aspects of Invariance in Solid Mechanics

7

where the primary basis is embedded in a deformation whose rate is r. Let these be written as (D/Dt)V = (dC/6t)V+ l 3 = (sc/6t)V - T'v,

(1.17)

so defhng (dc/6t)V= uiai,

(sc/6t)V = ujV.

(1.18)

These are convected derivatives in the full sense, that is, rates of change of representations on an embedded basis (the superscript or subscript index c is intended as a mnemonic notation to distinguish contravariant or covariant, respectively). The right-hand sides in (1.18) are, of course, independent of the basis, either by (1.17) itself or by differentiating (1.8) taking account of the invariance of the coefficients. Parallel definitions of these convected derivatives, in symbolic form, are (s"/ht)V = A(D/Dt)B'v,

(6Jst)V = B(D/Dt)A'v,

(1.19)

and with (1.13) these lead again to (1.17). A special case of (1.17) and (1.19), familiar from classical dynamics, is obtained by setting A = B = Q,proper orthogonal, with Q = RQ. Then (D/Dt)V = ( 9 / 9 t ) V + Qv, where

( ~ / W= QV( W t ) ( Q ' v )

(1.20)

denotes the convected derivative associated with a rotation Q(t).When R is specifically considered to be the spin of a material deformation, 9 / 9 t will be called the Jaumann flux. Then 9 6' -v =-v 9t 6t

6 v -bv + bv =2 6t

(1.21)

from (1.15) and (1.17). For any second-rank T the analog of (1.20) is DT/Dt = ( 9 T / 9 t )+ RT - M , where 9 T / 9 t = Q ( D / D t ) ( QT Q ) Q .

(1.22)

This may be viewed as a statement that the right-hand side of the first equation is spin invariant; its value is simply the flux of components on a fixed frame.

8

Rodney Hill

In the spirit of (1.19) we define background components of four convected derivatives, D Dt

g..

6cc D T = B - (A’TA)B’, 6t Dt

- T = A - (B’TB)A’,

-

6, D - T = B - (A’TB)A’, 6t Dt

64 , D - T = A - (B’TA)B’. 6t Dt

6t

(1.23)

The theorem on embedded bases confirms that D(B’TB)/Dt,etc., transform tensorially (but, of course, they are not associated representations of the same tensor). Then 6“T/6t, etc., are the fluxes of components on the embedded basis that currently coincides with the rectangular background. By evaluating (1.23) with the help of (1.13), we obtain 6“T

DT _

-+

I‘T+T=hT-T‘T-?T 6t

Dt

-T-T‘T+ St

(1.24)

T=> T + I ‘ T - Tl6t

In combination with (1.22) there results

9T -= 9t

dCC St

-T {

+ b T + T b = -a,,6t

S,C

-T

6t

T - b T - T8 (1.25)

6:

- bT + T b=2 T +bT - Tb 6t

in analogy to (1.21). Immediate corollaries are the connections (1.26)

Equivalently, 9T-< -9t

j3

1

D

[A

(B’TB)A’ + B - (A’TA)B’ Dt

{A

D (ETA)# + B - (A’TB)A’ Dt

by reverting to (1.23) and an arbitrary embedded basis.

(1.27)

Aspects of Invariance in Solid Mechanics

9

B. GEOMETRY OF DEFORMATION 1. Deformation Gradient

Let two configurations of a continuum be related by the mapping 5 -+ x in rectangular coordinates on a common fixed background. These configurations will usually be termed reference and current, respectively, or sometimes initial and final. The mapping is required to be differentiable so that (1.28)

Henceforward A symbolizes the Jacobian matrix whose i j element is aij= d x i / d t j and whose columns are denoted alra,, a3 as before. Physically possible mappings have positive determinant, I A 1, since this is the local volume magnification. Matrix A is called the deformation gradient. It suffices to study only first-order relations, on the usual pragmatic view that the local stress is not known to be affected by higher gradients. Since only ratios of differentials matter, such relations at a particle P are formally the same as those representing homogeneous deformation, namely x = AS

(1.29)

where A = (uij) is independent of 5, now drawn from P. For convenience the present discussion will be phrased in this context, and the mapping (1.29) will be regarded as between initial and final positions of either a particle (material point) or a fiber (material line segment). Any result can be transcribed for inhomogeneous mappings by replacing aij by d x i / d y j and prefacing words such as length, area, volume, fiber, etc., by the qualification “infinitesimal.” Incidentally, by writing the mapping as x = ciai,

(1.30)

we recover the notion of an embedded basis, here one that was initially a unit cube. The equation simply states that each particle keeps the same relative position within the deforming cell, and the invariant location is perpetually tagged by the Lagrangian coordinates 5’. Because of the differentiability restriction, A is uniquely determined by given mappings of just three noncoplanar fibers. Indeed, if x, = AS,, x2 = AS2, x3 = A4, where the matrix of columns, (Sl 5, t3),is nonsingular, the unique solution is A = (x1x2x3)(S15, 5 3 ) - As a matter of fact, this is a convenient way to monitor A experimentally, by observing what happens to the edges of a parallelepiped specimen or to lines scribed on three of its faces.

’.

10

Rodney Hill

In particular, the edges of a reference unit cube, with faces parallel to the coordinate planes, are mapped into al, a,, a,. The solution can be expressed more attractively by introducing the triad q', q2, q3 reciprocal to El, k,, 6,. Then A = XI @ q j

(1.31)

by premultiplying the analog of (1.7) by A. So the solution has the form of a sum of three dyads; in verification, operating on ti we obtain (qj ti)xj = 6Cxj = xi as required. To complete the duality, introduce also the triad yl, y2, y3 reciprocal to xl, x2, x,. Then q', q2,q3 are mapped into this triad by the deformation gradient B such that AB' = I, and consequently B = yj @

by analogy with (1.31). In proof, from (1.31) we have AB' = xj @ (Bqj)which is I by definition of B, and so Bqj = yj by (1.7).

2. Deformation of Area

To find how an embedded plane element changes under deformation we can take a face of the cell gl, p,, 5, as typical. Its vector area, say 5, A t3,is expressible as {tlt, t3}q1, where curly brackets signify the scalar triple product. Correspondingly, the deformed vector area is {x1x2x3}y1.But y' = Bq', just proved, and the ratio of the triple products is equal to the volume magnification IA I. Therefore any plane area u is mapped into IAIBu.

(1.32)

Another method is to appeal to the identity ( A t , ) A ( A t , ) = I A I B ( t 2 A 4,). Alternatively, consider a skew cylinder with basal area u and generators of unit length parallel to the jth coordinate axis; under the gradient A the generators become aj and the volume becomes I A I v j , which can be written identically as 1 A I (Bu) aj. Hence (1.32) is the final area of the common base of the cylinders j = 1,2, 3. 3. Principal Fibers The squared final length of a fiber is x * x = 5 * (A'AK.

(1.33)

Note, in passing, that A'A is symmetric and, as this formula shows, also positive definite (since x * x vanishes only when x = 0, which implies 5 = 0 on account of IA I # 0).The ratio (final length)/(initial length) of a fiber is called its stretch. This is therefore a number that represents an extension when in (1, a)and a compression when in (0, 1).

Aspects of Invariance in Solid Mechanics

11

The squared stretch is stationary with respect to varying fiber orientation when 6 = 5, ( I = 1, 2, 3) such that (A‘AK, = Ar26,

(no sum),

(1.34)

where the eigenvalues:A satisfy IA’A - AzI I = 0. Suppose them to be distinct. Since A’A is positive definite, we know from algebra that the eigenvalues are positive and that the associated quadric surface is an ellipsoid. Its principal axes are orthogonal and, being codirectional with their terminal normals, are by (1.34) in the directions +kr. Such fibers are called principal and the 19 are seem from (1.33) to be their stretches. Let (1.34) be premultiplied by A and combined with (1.29). The result is (AA’)x, = Ar2x,

(no sum),

(1.35)

showing that the principal fibers are also finally orthogonal, since AA’ is likewise symmetric and positive definite. The axes of A‘A and AA’ generally have different orientations though, which for brevity will be called Lagrangian and Eulerian, respectively. Of course, if two principal stretches are equal, all fibers in that plane are principal and the triads are to that extent nonunique. 4. Stretch Tensor

The symmetric tensor A, which is coaxial with the Lagrangian triad and has the principal stretches as eigenvalues, is known as the stretch tensor. That is, Ak, = ArL

(no sum).

(1.36)

In other words, the mapping A stretches the principal fibers like A itself but without rotation. So we have the polar decompositions

A = RA,

B = RA-‘,

(1.37)

where R is the proper orthogonal tensor that represents the final rotation needed to turn the principal fibers from the Lagrangian to the Eulerian directions. That AA-’ is automatically proper orthogonal can be checked by algebra : ( A A - ’ ) ’ ( A A - l )= A - l A ’ A A - l = A - l A 2 A - l = I , while I AA-’ I = + 1, since [ A 1 = 1 A 1 by construction. Since R R = I there is also the decomposition A = (RAR’)R.

(1.38)

12

Rodney Hill

This generates A by an initial rotation R followed by a stretch R A R ; this is coaxial with the Eulerian triad and has the principal stretches as eigenvalues. (If T is the array of background components of any tensor, R'TR is the array of its components on a frame rotated by R from the background, cf. (1.11). Equivalently, if the tensor itself is rotated by R, its background components become RTR'.) In practice, given a deformation gradient, the array A of background components of the stretch tensor can be constructed as follows. Calculate from (1.34) the stretches I , (r = 1, 2, 3) and the associated unit eigenvectors, say. Then apply (1.31) in the context of (1.36):

c,

We thereby recover the spectral decomposition formula, from which standpoint each dyadic self-product would be regarded as the background components of a uniaxial tensor with principal values 1, 0, 0. 5 . Kinematics During a continuing deformation, reckoned relative to a fixed initial configuration, the Lagrangian and Eulerian triads generally rotate against the background. Let RL,RE denote their spins and RL, RE their rotations (when these can be uniquely defined). Then

kL= RLRL,

kE= RERE,

by specializing (1.13). Introduce the spin RR of auxiliary axes rotated from the background by R. Then

RE = RRL,

d = RRR.

By eliminating the rates of rotation we obtain

R ( R E- RR)R= RL.

(1.39)

From (1.13), (1.15), and (1.37) there follow

' R'bR = t(hA-' + A- 'A).

R'(R - Ra)R = ~ ( A A - - A- 'A),

(1.4) (1.41)

In passing, it is worth noting that in the initial configuration itself R = RR = R, regardless of the incipient deformation. It is best to deal with such formulae by choosing a special background, namely, the coordinate frame with which the Lagrangian triad coincides momentarily (the principal stretches being supposed distinct). Then we need

Aspects of Invariance in Solid Mechanics

13

the components of tensors A, A, RLon the Lagrangian triad, say, A,,, A,,, a:, and the components of tensors 8, Q RE,RR on the Eulerian triad, say, E,,, orS, OF,, COP,. Equations (1.39)-(1.41) are thereby reduced to 0 , sE

- UP, = o rL, ,

( 1.42) (1.43)

with no summation over repeated subscripts. From the last two equations we have (1.45) Now apply (1.22)with A as the tensor and RLas the spin, again with the special background. Then

still supposing the principal stretches distinct. Combining (1.44)with (1.46) in the case r = s, we have the useful formula err

=X r I L

( 1.47)

In other words, with the Eulerian triad as basis, the normal components of the Eulerian strain-rate are equal to the rates of the logarithms of the principal stretches, regardless of rotation history. (When the same fibers remain principal there is nothing to be proved.) An elementary proof by pure geometry can be grounded on the stationary' property of eigenvalues, here the principal stretches. Take next the case r # s. By combining (1.43)with (1.46) we find (1.48) and then from (1.42)that

( 1.49) Finally, from (1.44)with (1.46)and use of (1.49),explicit formulae for the spins of the Lagrangian and Eulerian triads are obtained in terms of the

14

Rodney Hill

principal stretches and the Eulerian strain-rate:

(1.50) (1.51) Surprisingly, this important pair first entered the mainstream literature fairly recently (Hill, 1969, 1970,where derivations by pure geometry are given); an earlier mention of the second formula by Biot (1965)went unnoticed.

C. STRAINMEASUREB 1. Scale Functions

Suppose that the reference configuration of a material is fully specified; this includes the orientation of its microstructure in the chosen frame. We fix attention on a unit reference cube with edges parallel to the coordinate axes. Any six quantities that together define the shape of the embedded cell mapped by A from this cube can serve as a measure of strain. In crystal lattice theory, for example, the scalar products ai a, have commonly been used. Even more simple, geometrically speaking, are the lengths of the cell edges and their included angles; however, this set does not transform tensorially with the basis. Such variables will be brought within the general formalism later. For the present we restrict attention to tensor measures; these are made coaxial with the Lagrangian triad to obviate dependence on the rotation R. It is sensible to require that a measure vanishes in the reference configuration and agrees with the classical definition when the deformation is first order. The principal values could therefore be any smooth monotone “scale function” of stretch,f(l), subject to f(1)= 0, f’(1) = 1 (Hill, 1968). The correspondence between shape and measure is then one-to-one, but the connection with cell geometry is not simple, as a rule. However, this degree of generality promotes a desirable perspective, without unduly complicating the analysis in many cases. It has also given rise to the notion of measure invariance, which nowadays has a key role in the mathematical description of material response (see later). The array of background components of a measure will be denoted generically by E. Scale functions in the family (A2” - 1)/2n are often considered, where n may have any value. When 2n is a positive or negative integer, the

Aspects of Invariance in Solid Mechanics

15

corresponding array is E(”)= (A’”

(1.52)

- 1)/2n.

In proof, A2gr = Ar2kr, .. ., in ascending powers and A-’$ = A-le,, ..., in descending powers by repeated multiplication by A and A- respectively; so every A’” is coaxial with A and its principal values are A:”. In particular, the scale functions given by n = 1 and - 1 are by custom attributed to Green and Almansi, respectively. Then

’,

E(’) = +(A2 - I ) = +(A’A - I ) ,

(1.53)

E‘-” = ~1 ( -1A-’) = f ( I - BE),

(1.54)

with direct links with the metrics of the embedded cell and its reciprocal. As a further example, E(’’2) = A - 1 is given by the scale function A - 1 (fractional increase in length) and will be called the “stretch measure.” As n + 0, the limiting scale function is log A, which has long been popular in metallurgical testing. We write

(1.55)

E‘O’ = log A

but of course this notation cannot be read componentwise. However, since logA=(A- l ) - + ( A - l ) ’ + - - ,

O
we have the expansion log A = (A

- I ) -+(A

- I)’

+

a * *

(1.56)

when no principal stretch exceeds 2. Likewise, any scale function can be expanded as a convergent series

f(A)=(A-

l)+ff”(l)(A-

1)2+”‘

when IA - 1 1 is sufficiently small. When all principal stretches are in this interval, E = (A - I ) + +f”(l)(A - I)?

+

1 . .

(1.57)

(Although, by the Cayley-Hamilton identity, any E can be written formally as just a quadratic in A, with certain scalar invariants of A as coefficients, this apparently does not help much in practice). Conjoining (1.56) and (1.57), we have the basic connection log A = E - mE’

+ ...,

2m = 1 +f”(l),

(1.58)

Rodney Hill

16

in some E neighborhood of the reference configuration (Hill, 1968). The scalar parameter m is defined so that it vanishes for the logarithm, and the latter is here accorded the status of a norm with which other measures can be compared. Notice that m = n for members of the family (1.52).

2. Rates of Strain Following Hill (1970) let the fixed background be chosen coincident with the current Lagrangian triad. Then explicit components of the rate of change of any tensor measure can be computed by (1.22) and (1.50) with E and RLin place of T and R. Whence, taking account also of (1.47),

1 e, = A,er'Err

(r = s),

(1.59)

where e, =f (A,), e,' =f'(A,), and e,, are the subsequent components of E on the fixed background. It must be kept in mind that E,, are the components of d on the current Eulerian triad. Note that, as A, + A,, the coefficient of E , becomes A, e,' in the limit. With the Green and Almansi scale functions the formulae reduce to e,, = A,A,E,, (Green),

e,,/AJ,

(Almansi).

(1.61)

Alternatively, these representations follow from (1.53) and (1.54) by differentiation : I?(') = A'bA = A(R'bR)A, E(-l) = B'bB = A-l(R'&R)A-I,

(1.62)

with use of (1.13), (1.15), and (1.37). The point of the polar decompositions is that R'bR is the array of components of b on a frame rotated by R from the background; when this frame is the Lagrangian triad, R'bR is just matrix (c,,) and A is diag(A,, A,, 13),leading immediately to (1.61). Incidentally, A'BA and B'dB are two of the tensors induced from d by the deformation. With the scale function I - 1 (1.59) and (1.60) reduce to (1.44), which was derived independently. With the scale function log A the coefficient of E,, is unity while that of E,, is (1.63)

in ascending powers (Hill, 1970). Since the leading term is second order, (d/dt)log A = R'bR

+ O(E2).

(1.64)

Aspects of lnvariance in Solid Mechanics

17

The componentwise approximation by R'bR is close when 2 < &/Is < 2, say. The comparison is unaffected by the dilatational part of the deformation, and there is strict equality whenever the same fibers remain principal ( E , ~= 0, r # s). Rice (1975) has recommended (1.64) with (1.80) as a convenient approximation in constitutive descriptions of plastic response. Because scale functions are normalized in the reference configuration, rates of strain there agree with one another and with 8. Further, in some neighborhood we have from (1.58) the expansion

+

+

e, = {dikdj,-t m(dikej, djkei,) * * . } e ~ ~ )

(1.65)

in powers of E components on an arbitrary background. Fluxes of any two measures thus differ by jrst-order terms in strain unless their m-values are identical, which is to say that their scale functions have equal curvatures at I = 1. Under unrestricted deformation, on the other hand, the connection between a pair of fluxes is best shown in principal representation

.

ez =

der* . err, de,

~

.

e,*,=

e,* - es* ers er - es

(Ir

# Is),

(1.66)

from (1.59) and (1.60) with e = f ( I ) ,e* = f * ( I ) . Higher rates of strain can be investigated along similar lines, but will not be needed here. Authoritative accounts are Chadwick and Ogden (1971) and Ogden (1974).

D. STRESSMEASURES 1. Objective Components

Let Y denote the array of background components of Cauchy stress, where as usual the first index signifies the plane and the second the direction. Body moments are not envisaged here, so Y is symmetric. When the material is rotated under load without deformation, the Cauchy tensor rotates likewise; the array 9'changes accordingly and hence cannot serve as an objective measure of stress. To construct one, consider the cell al, a2, a3 generated by gradient A from the unit reference cube. The jth face of this cell has vector area I A Ibj from (1.32) and, by Cauchy's basic theorem, is subject to the load IAIYbJ= IAIaiiai= IAlaijbj

(1.67)

as in (1.4); the load has here been decomposed on the edges of the A cell and its reciprocal in turn. The coefficients are therefore unaffected by material rotation and are in that sense objective; the arrays of coefficients can be written symbolically as I A I ITYB and 1 A I A ' Y B , respectively, by (1.11).

18

Rodney Hill

The Lagrangian tensors generated by these background components are objective measures of stress. We note that aside from the immaterial factor I A 1, introduced for the sake of the direct mechanical interpretation, these are tensors induced from Y by the deformation A. Their objectivity may be verified in another way by applying polar decomposition to produce IA IA- ‘(R’YR)A- and I A 1 A(R9R)A- l . The tensor R’YR, induced by the components of Y on the R-rotated frame, is, of course, itself objective (hence also suitable as a measure). Dually, considering the reciprocal B cell, the vector load on itsjth face is I B I Y a , = IBlaijbi= IBlai.,ai.

(1.68)

The arrays of coefficients are I B I A’YA and I B I BYA, which involve the two other tensors induced by 9.The explicitly objective versions are 1 B IA(R‘YR)A and I B IA-’(R’YR)A. In the same spirit we can compute the work done, per unit reference volume, in a further differential increment of deformation of the A cell. The loads on the jth pair of opposite faces may be considered to act at points whose vector separation is aj, and so the differential work is altogether dw = IAlai’ai.da,= IAlaiid(hi*aj). Transcribing symbolically, we say that the stress measure

I A I B Y B is conjugate to &4‘A - I ) ,

(1.69)

the strain measure with Green’s scale function. Likewise, the stress measure

I A I A ‘ Y A is conjugate to i(Z - BB),

(1.70)

the strain measure with Almansi’s scale function. These results can also be reached from (1.62) via w = [ AI Tr(YS),

(1.71)

which we quote here as a standard formula in continuum mechanics. On the other hand, although Tr(YS) = Tr{(R’YR)(R’SR)}

(1.72)

trivially, I A 1 R Y R is not conjugate to any strain measure, since RSR is not a direct flux. The tensor 1AIY occurs repeatedly in the foregoing; it will be called “Kirchhoff stress” and its background components will be denoted by r,,.

Aspects of Invariance in Solid Mechanics

19

2. Work Conjugacy

The connections (1.69)and (1.70)motivate a systematic construction of stress measures, namely, as work conjugates to strain measures with arbitrary scale functions (Hill, 1968). Whatever the choice of measure E, the work differential can always be written as a Pfaffian dw = Tr(TdE).

(1.73)

Then the array T of symmetrized coefficients defines the conjugate measure, a function of both the intrinsic loads and the current strain. The array is associated with the rectangular background and so generates an objective tensor. This program is readily executed in principal representation. Let t,, be the coinponents of T on the Lagrangian triad. From (1.71)and (1.73)we have identically

(1.74) rs

rs

where t,, are the components of Kirchhoff stress on the Eulerian triad. Now substitute from (1.59)and (1.60)and compare coefficients on both sides; the result is

where e, =f (A,),e,' =f '(Ar). In passing, we observe that a conjugate measure is coaxial with the Lagrangian triad (trs= 0, r # s ) if and only if the Kirchhoff stress is coaxial with the Eulerian triad (trs= 0, r # s). During an arbitrary loading program this coaxiality is generally no more than momentary; exceptionally, it is perpetual when the material is isotropic elastic and one of its ground states is taken as reference. Noteworthy special cases of these formulae are t,, = trs/ArAs (Green),

A,Astrs (Almansi),

(1.77)

when f ( A ) = +(Az - 1) and i(1 - A-2), respectively. These complement (1.61),from which they are also self-evident via (1.74).Again, t,, = +(l/Ar

whenf(l) = A - 1.

+ l/As)z,,(stretch measure)

(1.78)

20

Rodney Hill

The array of background components of the conjugate of any measure in the family (1.52) will be denoted P).By inspection the representations (1.77) generate T ( ' ) = lAlA-'(R'YR)A-',

T(-"= IAlA(R'YR)A,

as already shown otherwise in (1.69) and (1.70). Similarly,it can be seem that (1.78) generates T"'2' = 2 1 I A I {A- ' ( R Y R ) ( R ' Y R ) A -'}.

+

Or this follows by substituting (1.41) in (1.72). Equivalently, T(1/2) = &(T(')A+ AT(1)).

(1.79)

This measure was introduced by Biot (1965), though in a very different guise and from another standpoint. Certain manipulative advantages in nonlinear isotropic elasticity have been argued by Ogden (1975,1977), more especially in regard to variation principles involving functionals of stress. Further algebraic connections between measures in the family P )are given by Ogden (1974); we do not pursue the matter here. The conjugate to the logarithmic measure (1.55) will be denoted by Fo). In lieu of a closed-form expression, the approximation

I

T'O' = 1 A R ' Y R

+ O(E2)

(1.80)

may be acceptable at moderate deformations, by (1.64) and (1.72). A compact way of stating equations (1.76) is that trs(er- e,) has an invariant value for each pair r # s, which is therefore known from the Green measure and its conjugate. Together these amount to an assertion that the skew tensor TE - ET is measure-invariant.

(1.81)

A direct proof in these terms is instructive (Hill, 1975, unpublished). Consider any incremental deformation in which the Lagrangian triad rotates but the principal stretches stay fixed. Then the components of every E are stationary in a frame whose spin is RL, and so E = RLE- ERLby (1.22). The rate of working is correspondingly

Tr(Tk) = -Tr{(TE - ET)QL}, which is just twice the scalar product of the vectors associated with the skew tensors TE - ET and RL. Since the work rate is independent of the strain measure, and the components of the spin vector can be assigned at will, the components of (1.81) must be separately invariant.

Aspects of lnvariance in Solid Mechanics

21

3. Objective Rates of Stress

With any reference configuration and any strain measure E, the flux T of the work-conjugate stress T is suitable as an objective rate of stress. Other definitions are conceivable, and some can be useful for specific purposes (e.g., Storen and Rice, 1975). Restriction here to the particular class T is motivated by the wish to exploit the measure invariance that arises from a direct link with mechanical work. It is straightforward to calculate the representation of f on a fixed background with which the Lagrangian triad is momentarily coincident. It is the array DT/Dt in (1.22) with T on the right-hand side given by (1.75) and (1.76) and R by (1.50). The resulting expressions are not particularly illuminating. So we concentrate here on evaluating rates of stress in a reference configuration itself, which is to say where E vanishes but T need not. On the other hand, we allow an arbitrary background. With arbitrary f ( I ) we can apply (1.65) near a reference configuration. Then, from (1.73),

+

+

t i j = (6ik6jI - rn(6ikejI 6jkeif) - * * } t i t )

where Po)denotes the exact conjugate to log A. Now differentiate this as it stands, afterwards setting E = 0, fi = 8,T(O)= 9'. The result is precisely

+

T = f ( 0 ) - rn(Y8 89)

(1.82)

in the reference configuration (Hill, 1968). Now, by applying the first of (1.23) to (1.69) and the second to (1.70), there follows

(1.83)

By comparing these, respectively, with the first or second of (1.25), it can be recognized from (1.82) with rn = 1 or - 1 that in the reference configuration

f ( O ) = ( 9 Y / 9 t )+ Y Tr I ,

(1.84)

the Jaumann flux of Kirchhoff stress (Hill, 1968, 1970). This is a result that merits an independent proof. By definition, as in (1.22),

(D/Dt)(I A I R'YR) = R ' ( 9 / 9 t ) ( I A I9')R

where here 9/9t accompanies the R-rotated frame whose spin is RR.But, in the reference configuration, R = 1 and RR = R, the material spin. Then the last right-hand side coincides with the Jaumann flux of Kirchhoff stress, while the left-hand side is Tto)by (1.80).

22

Rodney Hill

Higher rates of stress can also be investigated along these lines; for a systematic account see Chadwick and Ogden (1971) and Ogden (1974). 4. Nominal Stress Objective measures of stress are, of course, primarily intended for intrinsic, rotation-free, formulations of material behavior. In consequence they are not well-suited to matters of pure mechanics such as the balance of forces within a continuum and the specification of tractions over its surface. For that, there are definite advantages in a nonobjective measure which is based partly on an embedded cell and partly on a fixed background. This choice of variable is supplemented by a bridging equation between the material and mechanical descriptions. The whole procedure is standard practice today; it was fist codified by Hill (1957a,b; l958,1959,1961a,b) in a general theory of uniqueness and stability in elastic or plastic continua. Consider the load vector ni on the ith face of an A cell and denote its rectangular components on the background by nij ( j = 1, 2, 3). Under change of background the array N = (nil)generates a tensor; it will be called nominal stress (since reckoned as load divided by reference area, here a face of the original unit cube). N is affected by a final rotation Q of the cell together with its loading (it becomes N Q ) . N is also nonsymmetric: instead the three restrictions demanded by moment balance are ak hnk = O

Or

Or

aiknkj = Ujknki

A N = (AN)'.

(1.85)

Alternatively, these variants follow by setting zero the work-rate expression, in one or other of the forms

w = n, a.I = n.11b.. JI = Tr(NA), when

(1.86)

k is a pure spin, namely &=aha,,

U11. . = Rik Qk j ,

A = RA,

respectively. By the usual tetrahedron construction the current load on a plane element is Nu,where u is its vector area in the reference configuration. The load is likewise calculable from Cauchy's basic theorem as I A I Y'Bu by (1.32), without assuming symmetry of 9. Comparison gives

IA(-'AN

=Y

(1.87)

and we note that this implies (1.85) in the absence of body moments. Correspondingly, by inserting A = TA in (1.86), and cycling the factors in the

Aspects of Znvariance in Solid Mechanics

23

trace, we obtain W = Tr(ANT) =

I A I Tr(YT),

which returns to (1.71) when Y is symmetric. The objective measure of stress with which N is most directly connected is the conjugate of Green strain, F ' )= I A 1 B ' Y B from (1.69). Thus T(')A' = N

(1.88)

by (1.87). With the decompositions (1.37) there are useful equivalents such as T(')A= NR,

T(')A2= NA,

(1.89)

from which it follows that the value of the invariant (1.81) may be put as f ( N A - A'"). It can be seen from (1.79) that the conjugate of the stretch measure is the symmetric part of NR. Contact between N and any other objective measure can be made, in principle, via (1.75) and (1.76). We turn to the rate form of (1.88). Differentiation leads in the first instance to A N = /IT.(')/I' + AT'"A' = (s""/st)( I A 19) + 1 A 1 YT' by (1.13) and (1.23). Written as

+

I A I-'AN

= [(/sCcY/6t) Y Tr 81

IA /-'A&

= [(9Y/9t)

+ Y(8- 0 )

(1.90) by (1.19, this is a form of the bridging equation mentioned previously (Hill, 1959). If the Jaumann flux is preferred as the objective rate of stress, as in Hill (1939, then

+ Y Tr 81 - 89' - 932

(1.91)

from (1.25); here it is worth keeping in mind the connection (1.84). It may be remarked that (1.90) and (1.91) have been arranged in the manner of (1.87), so that each side is independent of the reference configuration. In the sequel this will on occasion be chosen coincident with the current configuration, in which case A is set equal to I and N interpreted correspondingly. For future reference it is convenient to record at this point the formulae Tr(&A) - Tr(f(')l?(')) = Tr(T(')k'k), Tr(NdA) - Tr(T(')dl?')) = d{Tr(+T("k'k)},

( 1.92)

for arbitrary variations in the rates at a given configuration and stress. These formulae are obtained by coupling the derivative of (1.88) with A'k = g(1)+ A'RA; the final quantity is skew and makes no contribution to the trace.

24

Rodney Hill

E. GENERALIZED MODULI 1. Some Transformation Formulae

The notion of conjugacy between measures (T, E) of stress and strain leads naturally to the idea of an associated tensor of moduli. Suppose the response of a material to incremental loading is either linear or piecewise linear. Then a fourth-rank tensor 9 can be defined such that tIJ. . = 9. rjkl i.kl

(1.93)

where 9 has a uniform value over the whole E space or over each of a number of subdomains. The dependence of 9 on the state of deformation will be considered later for various kinds of material; for the moment, however, our concern is with its dependence on the choice of measure. We seek transformation formulae allowing both for a change of scale function and of reference configuration. Such data conversion is a practical necessity since experimentalists and mathematicians often have different preferences. Before undertaking a general analysis we can read off a useful connection from (1.82).We recall that this requires the current configuration to be taken as reference, so let 9 and 9'' specifically ) denote the respective tensors of moduli associated there with E and E(') (the logarithmic measure). By substituting = 9 8 and Po)= Y(')bit is found, as in Hill (1968) that

on an arbitrary background; note the conventional symmetrization of components under the interchange k t-,1. The qualitative conclusion is that, in a material under load, the moduli are not measure invariant even at zero strain; on the contrary, there is a direct dependence on the local curvature of the scale function, represented by the terms in the parameter m. Turning to generalities, we must argue from the invariance of Tr( TdE)/p; this is the work differential per unit muss, p being the material density in the reference configuration. Distinguishing any other choice of variables by stars, we derive ( P / P * ) ~ $= (aemn/ae$)tmn

(1.95)

since each component of E is a determinate function of the components of E*. Now take the differential of (1.99,holding fixed the two reference densities, and introduce (1.93) along with its starred analog. The upshot is (1.96)

Aspects of Invariance in Solid Mechanics

25

Once again the coupling between curvature and load can be observed, here the terms with second derivatives. Note also that the special formula (1.94) could be recovered by explicitly calculating these derivatives at zero strain from (1.58). The transformation rule is particularly simple when the Green scale function is adopted for both measures and only the reference configuration is changed. By combining (1.62) with (1.83) we find

This indicates that the bracketed quantity on the right is invariant under change of reference configuration, and that its value is the tensor of moduli associated with the Green measure based on the current configuration. The transformation rule is implicit in this statement of invariance; it can be brought into line with (1.96), if one wishes, by equating the bracketed quantity to its starred analog, along with A = A*A, where A is the deformation gradient that converts one arbitrary reference configuration into another. Notice that the secondderivative terms have disappeared from (1.96) since the relation between E and E* is linear. Similarly,

implies the transformation rule for the tensor of moduli associated with the Almansi scale function. Because of the central role of nominal stress in the later developments it is necessary to introduce pseudomoduli such that nIJ. . = g. ijkl alk.

( 1.99)

These are not true moduli since they are partly coupled to the material spin. However, they do have a direct link with the Green moduli and this can stand in place of the bridging equation (1.90). Thus, by differentiating (1.88) in the form nij = t$hjk, we obtain

+

Vijk= l a j n u l u 9 ~ ; ~t p, d j , .

(1.100)

The rule of transformation under change of reference configuration is expressible via the invariance of

I A 1-

laimakugmjul

(1.101)

from (1.90) and (1.13). Its invariant value is the tensor of pseudomoduli in (1.99) when the current configuration is taken as reference for both nominal stress and deformation gradient.

26

Rodney Hill

2. Orthotropic Symmetry

It happens not infrequently that a material is isotropic in some ground state and is subsequently deformed so that three fibers are always mutually perpendicular. Then, regardless of the kind of material, the Cauchy stress is coaxial with these fibers and the only nonvanishing moduli are of type Yrrss and Yrsrs referred to the Lagrangian triad. That is,

where the fluxes are of components on a fixed background coinciding with the triad. The rates of strain are already known from (1.59) and (1.60); the rates of siress can be obtained from (1.22) applied to (1.75) and (1.76). This computation involves the flux of components of Kirchhoff stress on the spinning Eulerian triad, which we can write as

where the subscript REemphasizes that 9/9there follows the Eulerian triad (of the deformation referred to the ground state). This defines an objective tensor of moduli X, also having orthotropic symmetry but not associated with any measure in the sense that Y is. A short calculation, taking account of (1.47) and (1.50), leads to (1.102)

with no summation over repeated indices. with the shear modulus It may on occasion be convenient to link Yrws associated with the convected derivative of Kirchhoff stress in a frame rotating with some other spin, R, say. The required transformation is immediate from (1.22) written as

Aspects of Invariance in Solid Mechanics

27

For example, if R is specifically the material spin, the shear modulus associated with the Jaumann flux is (1.104)

by (1.51). As another example, if R is the spin RR of the R-rotated frame, given by (1.42), then

is the associated shear modulus. If we are primarily interested in the dependence of Yrw,on the strain measure, it suffices to cite a less explicit version of (1.103), namely {2%rs(er - es) - (tr - ts)}(er - es)

(1.105)

is measure invariant. This version can be viewed most simply as a consequence of (1.81) in rate form. To illustrate its application, suppose we want to connect the shear moduli associated respectively with the measures E(') and E('l2). The principal values of 7<') and 7<'/2)can be read from (1.77) and (1.78), whence

In principle this and other such relations are obtainable from (1.96), but less conveniently. When, in particular, the material is isotropic elastic the Kirchhoff stress is coaxial with the Eulerian triad. Then the shearing components oft,, are = 0, r # s. By (1.103), therefore, perpetually zero and S,,,

yrsrs = -1- t, - t, 2 er - e,

#

(I s)

(1.106)

and the invariant (1.105) vanishes. Of course, this expression for the shear moduli is also obtainable directly by combining (1.60) with the analogous equation for ir,.In particular, the shear moduli associated with Green's scale function are given by (1.107)

by (1.77). According to (1.97) the right-hand side is the shear modulus when the current configuration is taken as reference (Hill, 1969).

Rodney Hill

28 11. Constitutive Descriptions

A. MEASURE INVARIANCE 1. Generalized Variables and Transformation Rules This account of material response focuses on applications of the preceding mathematical apparatus in constitutive analyses. For the most part the incremental response may be arbitrarily piecewise linear and tensors of moduli may be arbitrarily path dependent. Time dependence and thermal phenomena, however, are excluded for the sake of brevity and a degree of unity. The illustrations touch on the elasticity of ideal monatomic lattices and of polymeric solids, but mainly concentrate on the elasticity and pla-;ticity of metal single crystals and polycrystals. We begin with some elementary observations to do with the work of deformation (Hill, 1972). A simple notation is an advantage here, along with some enlargement of scope. Let the current shape of an embedded cell be specified by any six geometric variables; we can regard them as generalized coordinates and, adopting the symbolism of Lagrangian mechanics, denote them by qa (a = 1, . . ., 6). As usual the cell can be supposed to be a unit cube in the reference configuration. When, in particular, the change of shape is specified by a tensor strain measure E relative to this configuration, we put q1 = e l l , q2 = e22, q3 = e33; q4 = 2e23, q5 = 2e31, 96 = 2e12. Generalized conjugate variables pa are defined by expressing the virtual work rate of the tractions acting over the cell faces as w = p.9,.

(2.1)

In principle, the pa are determinate functions of the tractions and the qa. In particular, p 1 = tll, p 2 = t22, p 3 = h 3 ; p4 = t23, p 5 = t31, p6 = t12, when the generalized coordinates are derived from some E whose conjugate is the stress measure T. Whenever the incremental response is linear within a domain of coordinate space, we can define there a matrix of generalized moduli as the array of coefficients in the differential relations Pa

= cap 4 p .

The notational correspondence between cap and Zjklin (1.93) is analogous to that between pa and tij;that is, a goes with the pair ij,and /?with the pair kl, both in the same manner. The work of deformation, p a d q a , is generally path dependent; here we concern ourselves only with second-order manifestations of the dependence, when the path segment is infinitesimal. Let this be given parametrically by

S

Aspects of Invariance in Solid Mechanics

29

functions qu(8),8 2 0; then the work is expressible as Joe Puq,'dO = ( P a d ) 0 8 + +(Pa4al)o'82+ O(e3),

where differentiation by 8 is denoted by a prime and evaluation at 8 = 0 by a subscript zero. This can be rearranged as

(PAO) + + P a ' ( 0 ) ~ > ~ q A+O+e"(O)e2) )~ and hence written in brief as 6w = ( p a

+ W')

(2.3)

+ +6pa)6q. + ...,

(2.4) correct to second order. This formula will be referred to as the trapezoid rule of quadrature (it generalizes an elementary approximation in the integral calculus, namely mean ordinate times abscissa interval). It will be applied also in the version 6w = pu6q.

+ +cu86qa6q,+ ...

(2.5 )

where cupare the generalized moduli appropriate to the domain of coordinate space in which the segment 6% falls. Suppose that some other set of coordinates qa* is chosen, possibly along with a different reference configuration; let p and p* be the respective reference densities. Since the work rate per unit mass is coordinate invariant, (P/P*)PU* = (aqv/a4a*)Pp (2.6) generalizing (1.95). Moreover 6w*/p* = Sw/p identically in powers of 8, so by (2.5) and its starred analog we have

to second order. The transformation rule for the moduli follows as an immediate corollary:

generalizing (1.96).The final step in (2.7) incidentally avoids a pitfall that has claimed more than one eminent victim: in transforming (2.4) or (2.5) the

30

Rodney Hill

leading terms also must be evaluated to second order in the new coordinates (otherwise the contributions from a2qr/a%* aq,* are lost). 2. Diflerential lnvariants

With the preceding appreciation of the trapezoid rule, it can be used rigorously to examine path dependence. As an elementary illustration, consider two possible routes between nearby initial and final configurations q, and q, dq, + dq,, namely, via (a) q, dq, and (b) q, + dq,. If the responses to dq, and 6q, are governed by the same matrix of moduli, so that dp, = c,, dq, and dp, = cap Sqs in both (a) and (b), the final values of the conjugate variables by either route are pa + dp, dp,, correct to first order. Then, by (2.4), the respective works per unit reference volume are dqa + (pa + d ~ + a 3dPa) h a , (Pa + (a) (Pa + - t a ~ a )h + (pa + dpa + tdpa) &a, (1' correct to second order, and so the work difference per unit mass is

+

+

+

For path independence it is therefore necessary that the matrix of generalized moduli be symmetric (in particular q j k l = Y k l i j for the tensor of moduli associated with any strain measure); conversely, an absence of symmetry presages path dependence. We see also in (2.9)the embryo of another idea (Hill, 1972): a differential form that is bilinear in the generalized coordinates and their conjugates, and whose value is invariant (since shown to be interpretable in terms of work only). The wider significance of this invariance is that it is independent of material response, nor need the routes be first order but only the resultant changes in the variables. The proof is simple: take the d-differential of (2.6) and construct

and similarly with d and 6 interchanged. Subtraction gives

after canceling terms in the second derivatives. An interpretation in terms of virtual work may be had by reference to a triangular cycle of quasilinear arcs from q, to q, + dq, to q, + 6% and then back to q,. Formally associate with these nodes the respective values pa, pa + dp,, pa + 6p,, pa of the conjugate variables, where dp, and dp, are the resultant increments on routes to

Aspects of Invariance in Solid Mechanics q,

31

+ dq, and qa + 6q, from qa. Then, integrating round the cycle,

f

p a 4.1

do = +(dpa dqa - d

~ dqa) a

+ . *

by (2.4), correct to second order. In the context of pathdependent response, such work is indeed only virtual, since in an actual cycle the true nodal values of the conjugate variables (and the tractions they represent) could differ by first order from those assigned. The bilinear differential form will be invoked later in a constitutive analysis of elastoplastic response. Another coordinate-invariant constitutive property derives from the Pfaffian identity 1 dp,*

P*

6(dqa*)- -1 dp,, 6(dqP)= 6 P

a2qfl

* dqa* dql.)

(2.11)

for infinitesimal &variations of the d-differentials (and not of the current state). This is proved similarly to the equation preceding (2.10). Its import stems from the circumstance that the right-hand side is a perfect &variation. Suppose that the constitutive relations between differential increments of the (p, q ) variables admit a potential with one choice of coordinates so that, say, dp,, d(dq,) is a perfect &variation. Then so is dpa* 6(dqa*)and the differential relations in any other coordinates likewise admit a potential. 3. Constitutive Inequalities and Convexity

It is an open question whether a material under load must be in some sense “intrinsically stable” if the conventional kind of constitutive description is to be admissible. Convincing philosophical or experimental grounds for any particular requirement are lacking, in my opinion. The literature on rubberlike elasticity abounds with idiosyncratic formulations of material stability, as can be seen in the accounts by Truesdell and Toupin (1963) and by Truesdell and No11 (1965). Some needed perspective was introduced by Hill (1970) and augmented by Hill (1975) and by Hill and Milstein (1977) in the context of ideal crystals. We follow this approach here, relaxing where possible any restriction to elasticity. Adequate specification of the loading environment is a prerequisite. During any homogeneous virtual deformation of a test specimen, the tractions must be imagined to “follow” the material, since we are here concerned with an intrinsic property (by contrast, the loads in standard laboratory experiments are often frame dependent and their work is affected by rotation of the specimen). The tractions may, in addition, be deformation sensitive and become different in kind from those in the undisturbed configuration. In any event the virtual work of the tractions on a path segment dq, must be

Rodney Hill

32

specified tc; second order, say

+

(2.12) 6u = p , 6q, jk,, 6q, 6q, + per unit reference volume, where the coefficients k,, depend on the current configuration and the choice of generalized coordinates. In a disturbed configuration, q, + 6q,, the actual state of stress under piecewise-linear response is represented by values p, + c,, dq, of the conjugate variables, whereas the virtual state of stress that would be in statical balance with the follower tractions is represented by values pa + k,, dq,. Relative to this environment the material will be called “stable” if

(6~4 k4,

h,)dqa = (cab - kufi) dqa

>0

(2.13)

for every disturbance. This requirement is automatically coordinate invariant since it is equivalent to 6w > 6u for all sufficiently small disturbances. In fact, since Su/p is invariant, the coefficients k,, transform according to (2.14)

by analogy with (2.8) which reflects the invariance of 6w/p; and by subtracting (2.8) from (2.14) the coordinate independence of (2.13) is apparent in detail. For an elastic material with a strain-energy density w(q,) the definition (2.13) requires that the combined incremental potential energy of the specimen and its loading should be positive for arbitrary disturbances (in this case the moduli c,, have the values a2w/aq, 84,). Thus the definition accords with the classical Lagrange-Dirichlet criterion of stability in relation to dynamical systems. For other materials the mechanical significance of (2.13), or of its failure, is less clear. However, the inequality can still be studied in a conjectural constitutive role; henceforth the discussion is confined to this aspect. Loading environments can be envisaged where the coefficients k,, vanish, relative to a particular set of coordinates. In other words, during any disturbance the environment reacts passively so as to hold constant the p , values representing the tractions. In these circumstances the inequality reduces to (2.15) 6Pa h a = cub 4 4 dq, > 0. When converted into other coordinates, however, this returns to the general form in accordance with (2.14). For elastic materials, in particular, (2.15) becomes (2.16) (a2W/aqa aq,) 6% dq, > 0,

which states that the energy density is locally strictly convex in its arguments, or the Hessian matrix of w is positive definite. This, indeed, is the

Aspects of Invariance in Solid Mechanics

33

version of the inequality most frequently encountered in the literature, whether in the context of elastomers or of ideal lattices. Often, though, the special choice of coordinates has been fortuitous and the stability criterion has been routinely quoted as (2.16) without regard to its lack of invariance. Sometimes, on the other hand, a deliberate choice of coordinates has been motivated merely by a priori notions of what constitutes an appropriate passive environment. Full details can be found in the references cited. Leaving this aspect aside, and without commitment to one or any view of intrinsic stability, we can propose a question that has some general interest. Supposing the incrementally linear response of a material to be given, how does the configurational domain defined by (2.15) depend on the choice of coordinates? For the question to be well-posed it seems necessary that the moduli should depend only on the current coordinates. Then the domain can be surveyed by paths leading from a configuration (which would normally be unstressed) where matrix (c,,) is known to be positive definite. The boundary will consequently be encountered where matrix (c,,) first becomes semidefinite, and thereafter indefinite. Necessarily det(c,,) vanishes there in association with an eigenmode dq, such that

s p a = c,, sq, = 0.

So a boundary point is characterized by local stationarity of the conjugate variables along some path through the point (Hill, 1975). Illustrative calculations for cubic lattices, with various sets of coordinates, are reviewed by Hill and Milstein (1977); the types of eigenmode encountered on paths that make the lattice tetragonal are also classified. Still pursuing the same question, we now restrict the choice of generalized coordinates to tensor measures of strain and revert to the T, E notation. Then, in rate form, (2.15) becomes Tr(Tk) > 0,

T = Yk,

(2.17)

where the scale function and reference configuration are both arbitrary. A comparative study of constitutive inequalities in this class for isotropic elasticity was initiated by Hill (1968, 1970). With a ground state for reference and a background frame coincident with the current Lagrangian triad, we can reduce (2.17) to

by (1.59)and (1.60) with tr = aw/der. Necessary and sufficient conditions for the inequality are therefore that the function w(e,, e,, e 3 )is (i) strictly convex locally and (ii) its first derivatives are ordered algebraically as its arguments. Hence the domain boundary will be encountered where either

34

Rodney Hill

det(a2w/ae, ae,) = 0 or t, = t, (r $; s). The respective eigenmodes are coaxial with the principal fibers or are shears of type E, (r # s). If it happens that the domain of (i) in principal strain space is itself convex, one may show that w(e,, e2, e 3 )is strictly convex globally and (ii) is automatically satisfied, by standard theorems, and likewise for any convex part of any domain (i) whatever. On the other hand, when the current configuration is taken as reference, (2.17) is reducible to Tr[(99/9t)8] > 2m Tr(Y@) - Tr(Yb) Tr 8

(2.19)

by (1.82) and (1.84), remembering that every l? = 8 when E = 0. Comparisons of (2.19) with (2.18) for arbitrary scale functions are naturally intricate; further details can be found in the references mentioned. Little eke has been published on the domains of (2.15) or (2.17) and their coordinate relativity for any particular material, but the whole matter certainly merits further study. Leaving it there perforce, we turn to a comparison of one incremental response with another. In mathematical parlance we turn from absolute convexity to relative convexity. This concept was introduced explicitly in continuum mechanics by Hill (1959), to supplement a theory of elastoplastic bifurcation. More will be said about that later; the intention just now is to prove a general theorem (apparently new), namely, that relative convexity is coordinate invariant. In its simplest setting, given a deformed configuration and the current loading, let cap and Zap be any two matrices of moduli (in practice, one matrix often characterizes an actual response and the other a hypothetical one, but this is not the only interpretation). Then the direction of the inequality (6pa - spa) 6 q a = (cap - z a p )

&a

6qp 2 0

(or > 0)

is invariant under transformation of coordinates, for arbitrary increments of deformation. In proof, by (2.8) with p,,, q,,, qa* given by hypothesis,

= (CPV - z p v )

6% 6%. When each response admits a work function, the inequality is a statement that w is locally convex (or strictly convex) relative to iir. The appropriate generalization for any incremental response whatever is A(Spa - &a) A(6qa) 2 0

(or > 0)

(2.20)

where A(6qa) denotes the difference of any two incremental deformations and A(6pa), A(8pa) the differences of the corresponding responses by the

Aspects of Invariance in Solid Mechanics

35

respective materials. In this version the inequality is a statement that one set of differential response relations are convex (or strictly convex) relative to the other set. The proof runs on exactly similar lines, via the Adifferences of the respective 6- and $-differentials of (2.6). B. ELASTOPLASTICITY 1. Phenomenological Framework

Objective rate equations, with a scope fully commensurate with the experimental data, have long been available for metals at unrestricted strains (Hill, 1958,1959). Their basic structure is nowadays agreed by most authorities, though some details are still unsettled (principally concerning the geometry of yield surfaces for polycrystals, and the path dependence of elastic and plastic moduli). The conventional exposition of such rate equations will be taken for granted; the present intention is rather to view this same framework from a radically different standpoint, which is not so well known. A definitive version was given by Hill and Rice (1972, 1973); with minor additions and rearrangements this is followed here. At the outset the material behavior is portrayed only in outline. First, every attainable state in configuration space is supposed to belong to a compact closed domain wherein the response is purely elastic and admits a work function; the boundary of each maximal elastic domain is a so-called “yield surface,” which must be regarded as associated with each and every interior state. Second, any path leaving a domain begins with a segment where the current state remains on the subsequent yield surfaces; the incremental response is now nonelastic overall, and may for instance be piecewise linear. Third, the shape of a yield surface, and the elasticity within it, are bothfinctionals of the entire path of deformation reckoned from some ground state. This behavior will be formalized at whatever observational level the loading and deformation of a representative element of material (whether single crystal or aggregate) can properly be treated as homogeneous overall. Generalized variables p a , qa (a = 1 ... 6) are then appropriate, as defined previously; they will be denoted collectively by p , q when indicating the primary arguments in dual functions of state. Within a yield surface let 4(q,H) be the work function per unit reference volume, where H symbolizes the complete history of nonelastic straining on the path responsible for that domain. Then, at each interior point, 84 = pa 6qa for arbitrary 6qa at fixed H. Whence PA49 H)= a4(e w a q a

(2.21)

36

Rodney Hill

and so the work function acts as an elastic potential in the usual way, but only within the one domain at each H.There, nonetheless, elastic moduli 9as can be defined by = -Zaa aq,,

%p(q, H)= a24(q,H)/aqa 84s.

(2.22)

Proceeding as in the active part of Legendre’s dual transformation, let $(p, H)be the complementary potential such that 4(q, H )+ +(P, H)= P.4,

(2.23)

subject to (2.21). Assuming that det(Y,,,) # 0 within a domain, we can in principle invert (2.21) uniquely and hence express $ as a single-valued function of the conjugate variables. Then S$ = qa Spa at fixed H,or qa(P, H)= a + ( ~ H , ) / ap a *

(2.24)

Elastic compliances A!,@are defined by 6qa =

dP,,

A ~ SH()P =~ a2$(P, H ) / a ~ app, a

(2.25)

and of course matrices (Yap) and (Aaa) are inverses. It should be kept in mind that 4 and $ within any one domain are so far defined only to within an additive constant, which could be any functional of H ;this arbitrariness will be removed later. Now consider a cycle of qa that includes a history increment d H ; physically, this is the only way to vary H but not q in functions of state. In detail, the path in configuration space begins at a point A of the domain H,then proceeds elastically by any route to a point B on the yield surface, then by a specific nonelastic segment (with prescribed injinitesimal length and direction) to a point C on a subsequent yield surface H + dH and finally returns elastically to A by any route within the new domain. Let the resultant “plastic” increment in any quantity be denoted by dP. For instance, d P 4 means 4(q,H + d H ) - $(q, H)and is a function of q, H,and the specific d H . By partial differentiation, (2.26)

dPp, = (a/aqa)(dp4),

showing that the cycle increment in 4 acts as a potential for the increment in pa. Likewise, in the space of the conjugate variables, a cycle of pa via the images of points A, B, C results in (2.27)

dPqa = (a/apa)(dP$)*

More generally, consider any path from A via B and C to a state pa + dp,, qa + dq, in the domain H + d H . Then the increments in the potentials are

d4 = dP4+ p a &a,

d$ = dP$

+

qa

dpu

(2.28)

Aspects of lnvariance in Solid Mechanics

37

by (2.21) and (2.24), respectively. But (2.23) holds also in the domain H dH, and so

+

d p 4 + dP$ = 0,

(2.29)

which is analogous to the passive part of Legendre's dual transformation. For this more general path there are also the differential relations

dPa = dPPa + %p dqp,

dqa = dP% + A

by (2.22) and (2.25). In particular, taking dp,

+

=0

a p

dPp

and dq, = 0 in turn,

dPq, + Aap dPpp= 0,

dPpa gpdPqp= 0,

(2.30)

(2.31)

which are equivalent forms of a connection between the plastic increments of the conjugate variables in the respective cycles. The new constitutive framework is a version of (2.30)that results when the elastic potentials are introduced by (2.26) and (2.27),namely,

d ~ -, y a p dqp = (a/aqa)(dP4), dqa d ~= p (d/dpa)(dP$). (2-32) These are valid throughout a domain, not just at a yield point; the two sets of equations are, of course, equivalent, but the duality is worth preserving. The distinctive feature of this description is the primary role assigned to elasticity, showing how its history dependence gives rise to nonelastic response. In fact, the terms in (2.32) are so arranged that on the left-hand sides there appear, respectively, the plastic increments in pa after cycling q, and in qa after cycling pa, while on the right-hand sides there appear the gradients of the plastic increments in the elastic potentials. To obtain a work interpretation of the latter increments the variations in the values of 4 and $ along nonelastic segments are now made precise. In a cycle of the coordinates the total work is

$

Pa

dqa = [4(41 H)1:

+ J"

C

B

= [+(q,

Pa

dqa + [4(4, H

H + d ~- 4(q, )

+ J"

+ dH)Ic"

C

B

(Pa

dqa - d 4 ) *

It is obviously simplest to choose d 4 = pa dq, along BC, and then automatically d$ = q, dp, since (2.23)is universal. With this choice, the change in 4 is always the work on a possible route between two configurations, whether in the same domain or not; similarly, the change in $ is always the complementary work. In particular,

Rodney Hi 11

38

Further, when a cycle begins on a yield surface, so that A becomes B, these increments vanish to first order since the values of each integrand differ infinitesimally at the same point of segments BC and CB.

2. Invariance and the Normality Rule Within the present framework the constitutive description is completed by specifying the history dependences of the elastic domain and the work potentials. Owing to lack of data at this time, the program can be pursued realistically in certain directions only. One such has to do with the history dependence of 4 and Ic/ near a yield surface. To this end let (2.32) be rewritten as (&a

-3.P

d4,) 84, = 6(dP4),

(AaP

d ~a 4,)8Pa =

- 8(dP$),

(2.34) where the &differentials are at tixed H and so denote purely elastic increments. These expressions are equivalent to one another by (2.29), and to (dpa 84, - 6Pa d4a) = 8

f

Pa

d4a

(2.35)

by (2.22), (2.25),and (2.33),the matrices gap and M a pbeing symmetric. This establishes an affinity with the general bilinear differential form (2.10);in the present context the variations d and 6 have, of course, a special significance. The resulting coordinate invariance of the left-hand side of (2.35), when divided by density p, is confirmed by the connection here with (8 $ dw)/p, which is the difference of the works per unit mass in neighboring cycles of material configuration. According to Ilyushin's axiom (1961) the total work in a cycle of deformation that involves a nonelastic segment is positive. If that were so, even just for evanescent segments (Hill and Rice, 1973), then d P 4 would be positive throughout an elastic domain, by (2.33). However, this carries no implication for 6 ( d P 4 )at interior points. But it does mean that 6 ( d P 4 )2 0

(2.36)

at any point on a yield surface, since dP+ vanishes there by arrangement. Actually, it suffices for (2.36) that Ilyushin's axiom is further restricted to cycles where both elastic segments are evanescent (Hill, 1968). Let it be supposed, then, on whatever grounds, that dP# is positive in some annular neighborhood of a yield surface. Apply (2.34) in a yield-point state; the 8-variations there are directed into the elastic domain but are otherwise arbitrary. We conclude that dpa - YUP dqs, the plastic increment in p., is codirectional with the inward normal to the yield surface in q-space; at a

Aspects of Invariance in Solid Mechanics

39

vertex, if any, it falls within the cone of limiting inward normals. Likewise dq, - &,, dp,, the plastic increment in qa, is codirectional with the outward normal in p-space or else within the cone of limiting outward normals. Moreover, this normality rule is coordinate invariant-always provided that it is expressed in work-conjugate variables. The geometry of the yield surface itself, on the other hand, is obviously affected by the choice of coordinates. In particular, the direction of the normal is not invariant, nor hence the partition of dq, into elastic and plastic components (Hill, 1967a); physically, cycling one set of pa does not automatically cycle any other, even with the current configuration as a common reference. To set these results in perspective it is worth glancing at the situation when potentials 6,II/ do not exist (acknowledging, however, that Cauchy elasticity is judged by many to be unreal). Normality is governed by the definiteness or otherwise of the bilinear forms

z,

-dPPll 6% = %, dPq, &la, dPq, bP, = dPqa 6q, by (2.31) and the first of (2.22), respectively. When Yap # Ppa, however, these two are no longer equivalent in general, and so simultaneous normality in both spaces is not assured. Indeed, coordinate invariance can be proved for the first form only (Hill, 1972; Hill and Rice, 1972). Thus, from (2.6) and (2.8) with the present notation for moduli, (P/P*)(~P,*- YDfSdq,*) = ( d ~ p- q

v dqv)(aqp/dqa*)

which is (l/P*) dPP,*

= (VP) dPP, &I,.

&a*

(2.37)

This is an unpleasing state of affairs and, of course, there is no appeal to observation. So one could abandon conventional normality and propose instead that the bilinear differential invariant should be positive at a yield point (2.38) >0 where dqa and 6% are, respectively, any outward and inward segments. By applying the trapezoid formula (2.4) to the usual cycle of deformation, we find

dPa

- b ~ dqa a

&a

dpa 6 q u - 6Pa d q a = 6

4

Pa dqa + ~ ( Z B - % a ) dqa 6qp

(2.39)

in replacement of (2.35), correct to first order in each variation. This shows incidentally that the right-hand term is coordinate invariant, as seen previously in (2.9) by a purely elastic cycle. Since this term has indefinite sign, Ilyushin’s axiom must also be abandoned; this is no surprise because

40

Rodney Hill

Cauchy elasticity itself is well known to be incompatible with work expenditure in arbitrary cycles (see, for example, Hill, 1968). Pursuing (2.38) by use in turn of the first of (2.22)and (2.25),we find that dp, - 9,, dq, is codirectional with the inward normal in q-space while dq, - A,, dp, is codirectional with the outward normal in p-space (generalizingas usual at a vertex). Thus, an invariant normality rule still holds in each space, but not in relation to plastic increments (since we have here the transposed matrices of elastic moduli and compliances). 3. Normality: The Regular Case In the six-space of the generalized coordinates a yield surface for any actual material may be expected to be smooth, except possibly at isolated vertices (manifolds with dimension 54). Such singularities will be considered later; for the present the implications of the normality rule are analyzed at a regular point. In q-space let I, be chosen codirectional with the unique normal outward from the domain. Define pa in p-space by dpa = Z, dq,, dqa = A a D dp, (2.40) for arbitrary elastic variations, as in (2.22) and (2.25). Then pa is along the outward normal in p-space, and I a = Z,P,, p a = &a, I, (2.41) since the matrices are symmetric. Let pa, q, be a regular yield point at history H, and let dp,, dq, be the nonelastic segment associated with dH, and necessarily such that I, dq, > 0. By the normality rule we have pa dPa = I a

&a

I 0,

dPp, = -I, dy,

dPq, = pa dy,

dy > 0,

(2.42)

where dy is determined by the specific dH. If, correspondingly, we write dP4 = @(q, H ) dy, where @ + Y = 0 by (2.29), then

I, = -aa/aq,,

dp*

= Y(P, H) dy,

pa = aY/ap,

(2.43)

by (2.26) and (2.27), the gradients being evaluated at the yield point. With the assumption that the incremental response is piecewise linear and continuous, d~ = I a dqalg = p a dpalh (2.4) since dy vanishes with I, dq,. Here g and h are scalar functions of state over the yield surface, and g is necessarily positive. Then dPa = Lap dq,,

dqa = Ma, ~ P P , I a dqa > 0

(2.45)

41

Aspects of Invariance in Solid Mechanics where

Map = A a b + ~ a ~ p / h are the matrices of moduli and compliances for the plastic branch of the constitutive law. In retrospect it can be seen that the diagonal symmetry of these matrices is a direct consequence of normality (and conversely implies it, given continuity of response). It remains to relate g and h. By introducing (2.45) in (2.44), one finds that Lap = y a p

- AmAp/g,

g

-h

(2.46)

= Ampb.

It follows that h is not necessarily one-signed and that three types of stress response are conceivable within this framework (Hill, 1967a). These are

h > 0 (strain-hardening),

g > Ampa,

p a dp, > 0;

h < 0 (strain-softening),

g < &pa,

p, dp,

<0;

(2.47)

h = 0 (ideally plastic), = Aapm pa = 0. Thus, starting from the current yield surface in p-space, the yield point moves respectively outward, inward, or tangential (and in this case dq, is not determined uniquely by dp,). With Cauchy elasticity and continuous response, but no normality rule, the dual equations for the plastic branch have the structure (Hill, 1967a) Lap

=yap

- la

Map = A a p

+m,~o

(2.48)

where

1, = =qbaPLg? P, = Jqd, are normals to the yield surfaces as before, and (2.49) are codirectional with the plastic increments in the generalized variables. There is also the consequential connection (1 + Army)(1 - P S l d ) = 1 which parallels (2.46). Finally, since nonelastic segments are defined by " dpa = A, dq, > 0, 1-P d b

the respective criteria for hardening or softening are pdld < 1 or > 1. It should be kept in mind, however, that hardening is a relative term. dependent on the choice of coordinates, like the partition of dq,. In detail, if the magnitude assigned to the normal A, is arbitrarily adjusted under change

Rodney Hill

42

of coordinates in such a way that I , d q a / p is invariant, then I , transforms like pa. Correspondingly, by (2.37) and the first of (2.48), dy and 1, dq, are also invariant; hence so is g / p when 1, = I , / g as in (2.45). Since I , d q , / p = pa9,, dq,/p and Yap transforms as in (2.8), the transformation rule for pa may be stated thus: pad a , / p is invariant when a, transforms like pa in (2.6). So the change in pais not simple. In consequence neither is the change in ma by (2.49)nor in h by (2.46).Explicit formulae are given by Hill (1967a) in the case when the coordinates are varied within the class of tensor measures of strain with the current configuration as reference. The notion of relative convexity, defined in (2.20),can be nicely illustrated with the constitutive relations (2.40) and (2.45). We aim to compare this piecewise-linear response with the hypothetical linear responses defined (over the entire dq-space) by the moduli Y,, and La,, respectively, and we begin by setting up connections between the associated second-degree forms in the A-differences.The piecewise-linear relations can be expressed in terms of the respective moduli as dPa = Y a p dq, - Ia dy = L., dq, - Ia delg, where d o = g dy - I , dq,,

d B 2 0 if d y = 0 ,

de = O

if d y > 0.

Then A(dPa) A(dqm) = g a b A(dqa) A(dq,) - Ja A(&,) A(dy) A\(&) - L a 44,)A(de)/g* Now A(d0) A(dy) 5 0 with equality if and only if the two increments are both elastic or both nonelastic, while I , A(dqa) A(d0) I 0 with equality if and only if they are both nonelastic ( g being positive). So = La,

L., A(dqa) A(dq,) I A(dPa) A(dqa) I ZfiA(dqa) A(@,)

- A ( ~ YA(dy)* )

(2.50) The conclusion is that the actual piecewise-linear response is more convex than a hypothetical linear response with moduli &,, but less convex than one with moduli gap. A prototype theorem of this kind was first obtained by Hill (1958). Note that the continued inequality can be written as 1

A(&) 5 -A(dPpa) 44,)5 IaIp 44,)A(dq,)/g, where every term is individually coordinate invariant, and hence also the absolute convexity of the relation between -dPp, and dq,. Dual inequalities can be proved in identical fashion, starting from

0I

dqa = d

a b

d ~+ p P a dy = Ma, d ~ + p Pa de/h

43

Aspects of Invariance in Solid Mechanics where d 6 = h dy - p a dp,,

d6 2 0 if dy = 0 ,

d6=O

if dy >O.

The outcome is &a*

A(dpu) A ( ~ P ,+ ) h A ( ~ YA) ( ~ Y5) A(dpa) A(dqu) 5 Ma, A(dpa) A ( ~ P * ) , (2.51)

with equality on the left if and only if the two increments are both elastic or both nonelastic; and equality on the right if they are both nonelastic, but otherwise the stated ordering provided h > 0. There are some cross-links between (2.50) and (2.51): the left and right sides of the latter respectively exceed those of the former, the differences being (Ad6)2/g and (Ad6)2/h.Also, the right-hand inequality in (2.50) and the left-hand inequality in (2.51) are directly connected by the identity

when written for the A-differences and the present constitutive law. However, though the relation between dPqa and dp, is convex when h > 0 and by (2.31) implies convexity of the relation between - dPpuand dq, when Tab is positive definite, we recall that neither condition is coordinate invariant. In the context ofinfinitesimaldeformations Hill (1967b)has analyzed the transmission of such constitutive inequalities from micro- to macro-levels of observation (single crystal to aggregate, say). 4. Normality: The Singular Case

A typical vertex will be supposed convex pyramidal with n tangent hyperplanes (n 2 2). Possible elastic increments from the yield point thus satisfy a set of nonredundant inequalities p: dp, = 1,' 64, I 0,

i = 1, .. ., n.

The outward normals to the hyperplanes in the dual spaces are codirectional with AUi,pai, which are related by (2.41) for each i = 1, . . . ,n. By the normality rule, Ya,jdq, - dp, = Lai dy',

dq, - Mas dp, = p: dy',

dy' 2 0,

(2.52)

with the summation convention, where the n scalars dy' are in principle determined by a specific d H . Correspondingly, the plastic increments in the elastic potentials are d p 4 = @(q, H ) dy',

dP$ = '€"(p,H ) dy'

Rodney Hill

44

+

where 0' Y' = 0 by (2.29), and

1;

=

-aa+/aqar

= aYi/ap,

by (2.26) and (2.27). The gradients here are evaluated at the yield point, of course, and 80' = -Aa' dq, 2 0, dY' = pai dp, I0 in its elastic neighborhood. Piecewise linearity and continuity of incremental response are assumed. In contrast to the regular case, however, this does not carry the analysis much farther. The domain external to the yield pyramid in dq-space can still be dissected arbitrarily by hyperplanes into any number of other pyramids; and in each of these a distinct linear response of type dy' = c,' dq, can be assigned, subject only to continuity between branches as across the yield surface. It is usual to prescribe a dissection indirectly, by laying down connections between dy' and dq,. A possible framework considered by Hill and Rice (1972) in relation to slip deformation of single crystals can be adopted also for aggregates, as suggested independently by Sewell (1972). Suppose that 1,' dq, = g'j dy', when dy' # 0, while 1,' dq, 5 g'j dy', when dy' = 0, where the moduli g'j are functions of state at the vertex; symmetry of matrix (g'j) is not assumed. A suggestion by Sewell (1972, 1974) is advantageous here: any such system of constraints can be rephrased as a set of equations, namely, 9" dy' - 1,' dq, = dB',

de' dy' = 0,

(2.53)

where every dy' and do' is nonnegative. With the hardening moduli

hi' = g i j

- &ipaj

(2.54)

defined in the spirit of ( 2 4 3 , the dual set of equations appears as

hi' dy' - p i dp,

= do',

do' dy' = 0

(2.55)

by (2.52). Qualitative features of the dissections of dp-space obtained from various types of matrix (h'j) are reviewed by Hill (1966); for instance, the dissection is by the pyramidal yield-planes themselves only when (h'j) is diagonal. In passing, the formulae -dPp, dq, = 9" dy' dy',

dPq, dp,

= h"

dy' dy'

should be noticed. Before proceeding farther, the effect of transforming the generalized coordinates should be ascertained. Let the arbitrary magnitude of each normal 2,' be so adjusted that 1,'dq,/p is invariant. Then, by (2.37) and the first of (2.52), each dy' is invariant and consequently the moduli g'j simply vary as the reference density p. Hereafter it is stipulated that matrix (g'j) is positive definite, which is therefore a coordinate invariant requirement (with the incidental implication that dPpa dq, 5 0, going beyond the normality rule).

Aspects of Invariance in Solid Mechanics

45

This stipulation has the effect that (2.52) augmented by (2.53) becomes sufficient to determine dp, when a nonelastic dq, is assigned. A short proof by contradiction goes as follows. Supposing two responses possible, let prefix A denote the difference of any pair of like differentials. Thus

de'

+ A(d0') 2 0,

dy'

+ A(dy') 2 0

subject to

{de'

+ A(dO')}{dy'+ A(dy')} = 0,

and so

A(de') A(dy') I 0.

(2.56)

g" A(dy') A(dy') > 0

(2.57)

By hypothesis, for distinct responses, regardless of the signs of A(dy'). On the other hand, A(dq,) = 0 in (2.53)implies g'j A(dyj) = A(dOi), with resulting incompatibility of (2.56) and (2.57).Hence the set of dy' is unique and so is dp, given by (2.52). When (gij)is not positive definite, examples can easily be constructed where uniqueness fails. In just the same way, positive definite (h'j) is required for unique dy' and dq, when dp, is given. However, the latter requirement is not coordinate invariant; indeed, as was seen even in the regular case, the transformation of h'j is quite complex. But by (2.54) we do have a simple connection between the respective pair of quadratic forms in n variables:

(g" - hi') A(dy') A(d#) = YaD(A: Ady')(A#j Ad#).

(2.58)

Their difference is thus another quadratic form which can be arranged in the six variables :A A(dy'), with matrix (YUs). With this arrangement it must be kept in mind that the variables can vanish collectively for certain nonzero A(dy'), namely those that correspond to any linear dependences between the vertex normals. A necessary and sufficient condition for dependence is the vanishing of the n x n Gram determinant, det(AJr2,'); of course, this happens automatically when n > 6 (the dimensionality of coordinate space). That being said, it obviously suffices for (2.57)that matrix ( V j )be positive definite when, as we shall suppose, the matrix (YUs) of elastic moduli is positive definite (for metal crystals and polycrystals this will ordinarily be the case with conventional choices of coordinates). However, it can then happen in practice that matrix (h'j) is not positive definite, but semidelinite or indefinite, as for instance in crystals deforming by slip when self-hardening of the slip systems is less than their latent hardening. It can happen, too, with artificial though seemingly plausible hardening rules; for example, both det(h'j) and det(g'j) vanish identically when n > 6 if h" = cup&'A$ with any

Rodney Hill

46

coefficients c,, (since h'jqj = 0 for qj such that A,'q' = 0). So the question arises: how can (2.57) be prearranged when matrix (h'j) is indefinite in coordinates for which matrix (Y,,) is positive definite? This is crucial for theories of texture development under progressive deformation, where lack of uniqueness in response has been a serious drawback in the past. In partial answer, one can note that when the vertex normals are independent it suffices for (2.57) that the now-positive Y,, form should dominate the sometimes negative h'j form. Put otherwise, the all-around elastic stiffness of the material must be sufficiently pronounced; in theoretical calculations this can easily be arranged by proportionately increasing the moduli gap. A continued inequality analogous to (2.50) is available under the condition (2.57) guaranteeing uniqueness. For this purpose we need only the matrix of moduli for the particular plastic branch where every dy' > 0 ; by (2.53) this corresponds to the domain of dq-space such that g! A,' d4, > 0 ( i = 1, . . ., n), where g! are elements of the matrix inverse to (g'j). From (2.52) the associated moduli are

hp= Yap - g'l1 I&j.

(2.59)

Note that this matrix is not symmetric unless g'j = SJ'. The derivation now runs on similar lines, beginning with W

P

U

)

A(d4a) = -%a A(d4u) A(d4,) - L,i A(d4a) A(dy') = &p A(d4,) A(d4,) - g! 1 I,' A(d4,) A(dfP).

Appealing next to (2.56) and to g! I,' A(d4,) A(d@ = A(d0') A(dy') - g!

A(dB') A(d@) 5 0,

we obtain Lap

A(d4.z) A(d4,) 5 A(@,) A(d4u) 5 Y a p A(d4a) A(d4,) - 9" A(dy') A(d?).

(2.60) The left-hand inequality is due to Sewell (1972, 1974); it states the (coordinate invariant) relative convexity of the actual piecewise-linear response compared to the hypothetical linear response with moduli (2.59). The righthand inequality is due to Hill and Rice (1972); in view of (2.57) it implies convexity of the relations between -dPpa and d4,. The dual continued inequality, analogous to (2.51), is A(dpu) A ( ~ P @ + )hi' A(dy') A(&') 5 A(dpu) A(d4a) 5 M a p A(dpa) A ( d ~ p ) * (2.61) Here the matrix (2.62) Ma, = Jab + h! lpa'p,' Ja,

Aspects of Invariance in Solid Mechanics

47

is inverse to (L,#)in (2.59) while matrix (h! is inverse to ( h i j ) , which must be supposed positive definite. The left-hand inequality is due to Hill (1966); the right-hand inequality has not been found in the literature. As in the regular case, there are cross-links between (2.60) and (2.61).For instance, the left and right sides of the latter, respectively, exceed those of the former; the differences are g!! A(de') A(d@), as noted by Havner (1977a,b), and h! A(d@)A(d6'). A final remark is relevant to the subsequent account of boundary-value problems for an elastoplastic continuum, in particular the self-adjointnessof the differential system and the existence of a variation principle. We have already noted that matrices (Las)and (Mu&are symmetric if and only if (g'j) and (hi') are, and this can be said of every branch. Equivalently, as may be shown directly from (2.53) and the first of (2.52), dP, Wq,) = dq, WP,) = W P , dq,)

(2.63)

for nonelastic d-increments and arbitrary infinitesimal &variations (of course, the last equality is trivially consequent on the first). Whence (Hill, 1966) the constitutive relations admit a potential function, in that

(2.64) where 2U = dp, dq,. In principle this is expressible as a piecewise quadratic form in dq, or dp,, respectively, via and the branch relations for dy'

C. SOME TYPES OF ELASTIC RESPONSE It is not in my terms of reference to review the present status of theoretical elasticity. Purely within the confines of constitutive analyses, and in the spirit of this article, I aim only to glance at a few developments and to highlight one or two outstanding problems. It is convenient to begin with macro-isotropic artificial rubbers. In the matter of constructing phenomenological work-functions, Treloar (1973, 1976) is a valuable source of all-round information. The unstressed ground state will be taken as reference, and as usual w denotes the free energy per unit reference volume. Until fairly recently, following the precedent set by Mooney and other pioneers, it was customary to fit empirical data only by functions w of polynomial invariants of the stretch tensor, and often the proposed functions were disagreeably complicated. In departing from this tradition, my 1969 paper was (if I am not mistaken) the first (i) to emphasize

48

Rodney Hill

that any symmetricfunction of the stretches is admissible in principle as a candidate w , and (ii) to add point to that remark by showing that the resulting mathematics can actually be made simpler than with invariants as arguments. Elsewhere I noted correspondingly (1969 unpublished, except for a footnote in Ogden 1972a) that empirical data for quasi-incompressible rubbers at very large stretches could be well represented by a simple function, namely w = (p/n)Tr E(")

(2.65)

where E(") is the strain measure defined in (1.52), but 2n need not be an integer. The principal Cauchy stresses satisfy a, - 6,= 2p(e? - ep)),

r Z s,

so p is the classical shear modulus at zero strain. Ogden (1972a)carried this line of enquiry much farther and was notably successful in fitting all the data available from the standard tests on continuum rubbers, just by additive combinations of such functions: (c(")Tr

w= n

E'")},

(nc'")},

p=

(2.66)

n

where the paired values of n and c(") are disposable (only three being needed in general). Changes of volume in both continuum and foam rubbers can be accounted for by an extra additive function of the volume ratio, I, A,& (Ogden, 1972b, 1976). A mathematically attractive proposal of this type, which I have not seen in the literature except for a hint by'Blatz and KO (1962), is

where the parameter v is a generalized Poisson's ratio. The principal Kirchhoff stresses are

If, now, the path of deformation is such that I z = I 3 = A T v it can be seen in the formulae for z2 = z3 that each summed term vanishes separately. This path is therefore produced by uniaxial stress zl, and the ratio of logarithmic transverse contraction to longitudinal extension has the constant value v. Some support for this distinctive property of (2.67) was reported for a polyurethane foam rubber (50% voids by volume) by Blatz and KO, who favored just the single term n = - 1 with v = 2. Whether (2.67) itself has any

Aspects of Inuariance in Solid Mechanics

49

future remains to be seen, but the general notion of conjugate measures of stress and strain does seem to offer a suggestive framework that ought to be explored. Just to mention one example,

F")= Iz Tr E(")I + 2pE'")

(2.69)

where I , p, n are all disposable, admits an isotropic work function and might be regarded as a natural generalization of Hooke's law. Elasticity of a completely different kind is exhibited by a monatomic crystal. Regardless of the interaction "forces," geometric symmetry by itself ensures that any primitive (Bravais) lattice of atoms can be equilibrated by surface loading ("at infinity") alone. Our standpoint will be that every primitive lattice can be regarded, quite simply, as a homogeneous deformation A of a simple cubic array; this is a unifying view not found in conventional texts on crystallography. Then, adopting this array as a convenient reference configuration (with the nearest-neighbor spacing as the unit of length and the cubic axes as background coordinates), the problem is construct appropriate work-functions w ( A ) .As observed by Ericksen (1970), aside from the trivial requirement w(QA) = w ( A ) for every proper orthogonal Q , it is necessary that w ( A q fW(A)

(2.70)

for every unimodular r with integer coefficients. Such deformation gradients r map the simple cubic array onto itself. For, if 6 is a node, so obviously is any l-6 since the components of 5 are integers. Conversely, if q is any node, a 5 can always be found with integer coefficients such that l-6 = q, because the inverse matrix r-' also has f integer coefficients when det r = 1. The group of all such r has just two generators; they can be chosen, for instance, as a rotation of 120" about the direction ( 1 11) and a unit shear over a cube face parallel to a cubic axis. If, now, we consider a deformed configuration A, the new associated group i= of self-mappings is easily identified. Thus, the configuration r becomes Ar under A, while A itself is converted into AT by

r = ArB:

(2.71)

r

It is of course the rotation subgroups of that give rise to the crystallographic classification of lattice symmetries. From our standpoint it is more interesting to remark that (2.71) can be put as

B T A = r,

(2.72)

r

which by (1.11) expresses the invariance of the mixed representation y!, of on the A cell. To date, the construction of closed-form work functions subject to (2.70)

50

Rodney Hill

has not been solved satisfactorily. The exceptional difficulty of the task can be understood from the investigations by Parry (1976, 1977). Of course, if the atomic interactions are modeled by pairwise forces, there is then the classical formula (2.73) w= +(r2)

+1

where the summation extends over all bonds on one atom in the infinite cubic array; r is a typical internodal distance after deformation, and +(r2) is the potential energy of a single pair of atoms. Such infinite sums automatically have the property (2.70), but they are inconvenient in applications. Also, the pairwise model is not too realistic in its macropredictions, except for nickel and a few other substances. For example, regardless of what may be, an inevitable consequence is that the Green moduli have "Cauchy symmetry," viz.

+

g$ = ggjI

(2.74)

not just in the ground state (as is well known) but also under load (Hill, 1975). Specifically,since r2 = (Sij 2eIf))tit j ,

+

It can be recognized from (1.96) that Cauchy symmetry under load does not hold for any other scale function. 111. Bifurcation Theory A. FIRST-ORDER RATEPROBLEM

1. Field Equations

From now on our concern is with the inhomogeneous response of a finite continuum to a quasi-static loading program, and especially with whether or not the response is unique. Constitutive laws will be applied pointwise and are taken to be incrementally linear or piecewise linear, in the sense we have described ; more generally, whenever the analysis allows, the relation between rates of stress and strain can be nonlinear homogeneous of degree one. Various types of incremental loading are envisaged and will be detailed, in particular some that are deformation sensitive; for the moment, though, it is enough to say that all types impose some local condition on the surface velocity and/or its gradient, separately or conjointly. Within the continuum the quasi-static velocity must satisfy second-order equations of continuing

Aspects of Inuariance in Solid Mechanics

51

equilibrium, which will be obtained presently. Supposing that the current geometry, material properties, and state of stress are known, these field equations together with the surface data pose a well-set boundary-value problem for the velocity distribution at each stage of the loading program. This is the first-order “rate problem,” and suitably analytic solutions are called deformation modes. A critical stage may be reached where there exists more than one mode, usually infinitelymany, which is a phenomenon customarily referred to as bifurcation. This strictly means a simple branching of the path of deformation, and in fact the nonuniqueness is probably always reducible to that by reference to rate problems of higher order (for the quasi-static acceleration, etc.); naturally, this involves a sufficiently close specification of the subsequent loading. We shall not venture into this territory here; practically all that is known is limited to continua with simple geometries and to general systems with finite freedoms (e.g., Hill and Sewell, 1960, 1962; Budiansky, 1974; Hutchinson, 1974; Thompson and Hunt, 1973). The field equations will be written from a Lagrangian standpoint, in terms of the tensor of nominal stress nij (Section I,D,4) based on a past or present configuration of the continuum. The reference coordinates of a typical particle are denoted by &, a volume element by d l , and a surface element by d&. Body forces are admitted and are written as y j per unit reference volume. Then, for any part of the continuum, the global equations of linear equilibrium are

jnij dXi + 5 y j d< = 0 which, by the divergence theorem and the usual argument, imply (an,j/aei)

+

=0

(34 locally. Differentiationby a timelike parameter is of course immediate in this formulation, so the first-order equations of continuing equilibrium are just yj

+

(ahij/agi) f j = 0 (34 where nij and f j are rates of change following a particle. From the global rate equation, applied in the standard manner to a disklike volume at an interior jump surface of tiij, it can be deduced that the nominal traction-rate is continuous. The rate of deformation, previously denoted hij, where aij is the deformation gradient a x i / a c i , will henceforth be written as d u i / d r j in terms of the velocity ui. Correspondingly, the rate equations (1.99) for an incrementally linear material are nij

= wijkdt)(aut/atrJ

(3.3)

Rodney Hill

52

where possible spatial variations of the pseudomoduli are indicated by the bracketed symbol 5. Any branch of a piecewise-linear material can be written similarly, provided one keeps in mind that the moduli depend on the particular domain of velocity-gradient space. More generally (Hill, 1959), we could envisage a thoroughly nonlinear response of type

where the potential U is a function of the velocity gradient, necessarily homogeneous of degree two. U is a functional also of the deformation history, while under change of reference configuration it varies simply as the reference density. Relations (3.3) admit as potential the quadratic form

au. aU, u = -1 vijkl(<)

when

J

2

a
%?ijkl

= gklij.

(3.5)

According to (1.100) this symmetry required of the pseudomoduli implies the same symmetry for the objective moduli associated with Green’s scale function, and hence with any other, since diagonal symmetry of the generalized moduli is coordinate invariant by (2.8). As we saw in connection with (2.9), this characterizes any incrementally linear response that is, in a minimal sense, conservative; in particular, therefore, it is automatic for an elastic material that admits a strain-energy potential. For elastoplastic materials we noted in (2.45) its presence at a regular yield point under the normality rule; and, in (2.64),at a singular yield point also when the hardening moduli (2.54) are symmetric. The existence of U is associated with a “reciprocal relation” i i js(aU,/a<,)= (auj/at,)hitij

(3.6) where in general the &variation must be infinitesimal. Exceptionally, when U is the simple quadratic form ( 3 . 9 6uj can be replaced by the finite difference Auj = uj* - uj and we can write equivalently

riij(aUj*/agi)= ir;(auj/ati). From the reciprocal relation there follows the identity

a

(0. J

air..- n.. 60.) J 1J

1J

which establishes the self-adjointness of the field equations and leads immediately by integration to a reciprocal theorem

juj 6yj d< +

uj 6nij d X i =

s

y j 6uj d t

+ 5 tiij 6uj dZi,

(3.7)

Aspects of Invariance in Solid Mechanics

53

within the class of fields satisfying (3.2), (3.4), and the interior jump condition. In particular, for incrementally-linear response, Avj can replace 60, in (3.7), with the result

which is analogous to Betti's theorem in classical elasticity.

2. Surface Data The main categories of loading will now be formulated analytically. The intention is to illustrate the range of possibilities, not to produce an exhaustive classification. Several categories will subsequently be brought within an overall notation, which is a useful shorthand in the analysis of bifurcation. Those that cannot will be treated separately, as we explain by an example. However, once the theoretical guidelines are understood, the procedure for any type of loading is virtually routine. Some standard kinematical formulae are assembled for convenience at the outset. They relate to an embedded surface element whose vector area is dCi = vi dC in the reference configuration, when t = 0 say, and is dSi = ni dS when t 2 0 (both normals are unit and outward from the material). Then at t = 0 we have the convected derivatives

d dS dt -

=

auj

( 6 , - njnl) - dS, 3x1

d dSi = (ni djk - nj 8 i k ) dt

-

where on the right-hand sides d C , v i , tiare replaced by dS, ni, x i to indicate that the current configuration is being taken as reference. An immediate application is to loading by a uniformly pressurized fluid. The vector load on a surface element is

nij(t)d Z i = - p ( t ) dSj(t), where p is the Cauchy pressure. So tiij

and hence, by (3.8),

dSi = - p dSj - p ( d / d t ) dSj,

t 2 0, t

= 0,

Rodney Hill

54

in the current configuration taken as reference. This is a case of “deformation sensitive” (configuration-dependent)loading, which typically is characterized by linear terms in the velocity or its gradient. Their presence is due to factors beyond the control of the experimenter, who for example can here only prescribe p and @.It may be remarked that, with any kind ofexternally applied loading, the net contribution from the gradient terms can arise only from in-surface deformations. Since what happens in the body of the material makes dui/dxj at the surface uncertain to the extent of an additive term qi nj, where qi is an undetermined vector function over S, the net contribution of such dyadic products to the load rate expression is necessarily nil. This is a useful check on the validity of any proposed formula; thus (qlnj q k nk bjI)nI= 0 in (3.9). Loading via contact with a tool is the most common arrangement in engineering practice. The tool may be a movable die or punch, or a fixed foundation or container, and in the simplest case it can be supposed smooth and rigid. Then, when the motion of the tool is predetermined, the vector velocity of each particle of its surface is known, but only the normal component of velocity of the contiguous particle of material, and that only if contact is maintained. So, during a continuous process, we have the following conditional data, with p in (3.9) now standing for the local contact pressure (which in principle is known at t = 0) and uiT denoting the given local velocity of the tool. If p > 0 then ni(ui - u,’) = 0; if p = 0 then either ni(ui- ui’) = 0 with i, > 0, or ni(u, - u,’) I 0 with p = 0.

(3.10)

This data must be complemented by the no-friction condition, and here one must beware of a too-facile analogy with the classical mixed data in infinitesimal elasticity. The nominal traction-rate over the surface is not necessarily purely normal (contrary to statements in the literature), but its tangential component must actually be calculated. Either formally via (3.8) and (3.9), or by a simple ad hoc argument, we have

(aij - ninj)tikjdSk = -pii dS.

(3.1 1)

On the right-hand side the variation of n, follows a material particle of S, and comes partly from any curvature of the tool surface and partly from any rigid-body rotation: )li

= Kij(Uj

- Uj’)

+ eijkoj‘nk

(3.12)

where eijk is the alternating tensor, ojTis the spin of the tool, and K i j is the local curvature tensor of S. If, as is feasible, we arbitrarily choose its com-

Aspects of Invariance in Solid Mechanics ponents so that

icij nj

55

= 0, then

icij =

(6ik - nink)(6, - njn,)

where F ( x , , x2, x3) = 0 is the equation of the surface. A quite different kind of deformation-sensitive loading, discussed by Nemat-Nasser (1968), among others, involves tangential “follower forces.” Here the local traction is perpetually tangent to an embedded fiber element of the surface. It is hard to imagine how this could be realized in practice, except in mean as a resultant over a small but finite area; an example might be a slender column compressed by a terminal force, whose line of action is tied to the centroidal axis. We can write nij(t) dZi = q ( t ) m j ( t )dS(t),

t 2 0,

where m j is the running unit tangent to the fiber element, and qmj = ni aij.By another standard formula, one has

(d/dt)mi= (6ik - mimk)mj(avk/8xj)

(3.13)

at t = 0. If the force magnitude qdS is held fixed, for instance, then tiij

dSi = q(hj, - m,m,)mk(do,/dxk) dS

(3.14)

in the current configuration taken as reference. If, alternatively, the force magnitude is maintained proportional to the fiber length, the nominal traction vector varies as if it were itself embedded in the deformation, so that tiij dSi = qmk(dvj/dxk) dS.

(3.15)

Albeit highly artificial, this case features prominently in the well-known book on elastic stability by Bolotin (1963). These examples, along with others, suggest that all worthwhile cases of nonconditional data can be treated collectively under a unified formalism (Hill, 1962): (3.16) tiij dSi = (Cj dS

+A)

where cj is the controllable part of the incremental loading (independent of material response) and f, is the deformation-sensitive part (a linear homogeneous expression in the velocity and its gradient, allowing for deformability of both tool and material). For instance in (3.9) we have c j = -pn’

j,

f’= P(6iI J

6jk

-

6ij hkl)ni(avl/axk),

(3.17)

while in (3.14) and (3.15) cj is zero andfi is the coefficient of dS as stated. In general the velocity gradient contributions can be written as niejk.(av, /&k) ifdesired, where &jk[nink = 0 (it can happen that e j k t = - q,ilbut this iS

Rodney Hill

56

not necessary).Then, for example, when cj = 0 and there is no direct velocity contribution tofi, the surface data can be put as

fli(Wijk1- q j k l ) aul/axk = 0, from (3.3) with the current configuration as reference. Now consider the following mixed data over the surface S of a continuum. A part S’ is fully constrained so that the vector velocity is prescribed; over the remainder S - S’ the loading is of type (3.16) with at least piecewise continuous cj and These data are called “homogeneous” when S’ is fixed and cj vanishes over S - S’; otherwise they are called “inhomogeneous.” Homogeneous data are said to be “self-adjoint” when

fi.

(fiuj*

- ujfi*) dS

=

I (s Auj

- uj

Afi) dS = 0

(3.18)

where uj and uj* are any pair of continuous and piecewise-continuously differentiable fields whose difference Auj = uj* - uj vanishes over S‘. Sincefi is linear homogeneous, equivalent definitions are

5 (fi6uj - u j S f j ) dS

=0

and

jfi 6uj = iS

u j f j dS

(3.19)

where, within the admissible class of continuous and piecewise-continuously differentiable fields, uj is arbitrary while Suj is any infinitesimal variation that vanishes over S . A work interpretation can be had via (1.86) and (2.4), from which we obtain the second-order formula

5 (nij + ;anij) 6uj dSi

(3.20)

for the virtual work of surface tractions over S in an incremental displacement Suj. Apply this to each of two routes to xi + uj* 6t, namely, direct or via xi uj 6t. The difference in the contributions by the deformationsensitive part of the loading is f(6t)2 x (3.18), which must vanish if this part is conservative in a minimal global sense. To illustrate the self-adjointness criterion we can apply it to fluid-pressure loading. Suppose (3.17) is given over a part S” of the surface. The contribution to (3.18) is reducible to

+

p

curl(v A 6v) d S .

Thus, if the integral of v A 6v round the perimeter of S” is to vanish, S must adjoin only S and not another part of S - S’. Correspondingly, the virtual work of a constant pressure is - p times volume swept out by S”, which is unique only when the motion of its perimeter is prescribed.

Aspects of Inuariance in Solid Mechanics

57

A final remark on pressure loading may be of interest to amplify the preceding statement. The volume swept out by a nonuniform finite displacement u in taking an open surface C into another surface S can be expressed as

3

1

u . (dS

+ dI;) + $

I

u . {(Tr A)I - A’} dC

where dS = I A 1 B dZ as in (1.32). An equivalent formula, here less attractive, was given by Pearson (1956).The first variation of this volume, namely its difference from that swept out by u + 6u, is expressible as

j6u

*

dS

+ f (dur\u).(I + 2 A ) d 5

where the line integral is round the perimeter of C.Since the first term is the volume generated by 6u from S , the geometric interpretation of the second term is obvious; moreover, its structure links it directly with the line integral in the preceding criterion for self-adjointness. Further information on deformation sensitivity in general, and on conservative loadings in particular, can be found in Sewell (1965, 1967). 3. Solution Properties Before embarking on a bifurcation analysis we mention some elementary properties of any solution of the rate problem, whether unique or not. We begin with a statement of the principle of virtual work in terms of the present variables. Let uj be a continuous and piecewise-continuously differentiable field of velocity such that the constitutively associated tiij satisfies (3.2) and applies a continuous traction over any interior jump surface of duj/axi and tiij itself. Let 6uj be any continuous and piecewise-continuously differentiable field whatever, not necessarily infinitesimal. Then (3.21) by the divergence transformation. Now assume the existence of a velocity-gradient potential as in (3.4).By the choice 6oj a v j in (3.21),or by direct transformation, 2

I

1

U d( = nijvj d &

+j

i)jvj

dt,

(3.22)

which stands in analogy to Clapeyron’s formula in linearized elasticity. However, the surface data here are much more diverse, as we have seen. Suppose, for example, that the data are homogeneous and that j J j = 0 within

Rodney Hi 11

58

S, as when the body forces are solely due to the Earth’s gravitational field. The whole rate problem is then homogeneous and in general can have only a null solution, but there may be special configurations, or eigenstates, that admit nontrivial modes uj = wj,say. For these (3.22) reduces to

1U(W)d t

=f

~S,(W)W,dS

(3.23)

when S‘ is fixed and S - S is subject to (3.16) with c j = 0. (For clarity, function dependence on wjor its gradient is made explicit by the bracketed symbol w.)Note, incidentally, that in (3.22) the quantities U d t , ij d t , and nij dXi are invariant for the same material particles under change of reference configuration. So, if preferred, the volume integral in (3.23) could be over the current configuration provided the meaning of symbol U is mentally adjusted. Returning to (3.21) we can rearrange it as

if 6uj is infinitesimal and vanishes over C’.A true variation principle is obtained when the data over S - S’ is self-adjoint as in (3.16)with (3.19), and then (Hill, 1962)

{V(v) - i;uj} d t

-

j

s-S’

{cj

+ fh(v)}uj dS

1

=0

(3.24)

and conversely (by reversing the analytical steps and applying the standard argument of the calculus of variations). In the homogeneous problem corresponding to (3.24) an eigenmode is characterized by

6

[ U(w)

dt -3

1

1

f;(w)wjdS = 0

s-S’

(3.25)

in addition to (3.23), and conversely. When converted into finite-element procedures, of one kind or another, the two principles are currently the preferred basis for computations of deformation paths and primary bifurcations.

B. PRIMARY BIFURCATION 1. Linear Case

With incrementally linear response the inhomogeneous rate problem is linear, and so its solution is unique when and only when the current configuration is not an eigenstate for the associated homogeneous rate prob-

Aspects of Invariance in Solid Mechanics

59

lem. In fact, the difference of distinct velocity solutions would be an eigenmode, while, conversely, arbitrary multiples of an eigenmode could be added to one solution to generate others. Thus, to guarantee uniqueness it is enough to exclude the possibility of eigenmodes. By reference to (3.23) a clearly sufficient condition is

I

4

U(w*) d l - Ifi(w*)wj* dS > 0

(3.26)

for all analytically admissible fields wj* that vanish over S’ but are not identically zero. This expression will be called the “exclusion functional.” To obtain a work interpretation we recall from (3.20) that the secondorder work done by the surface loads and body forces in a virtual displacement 6uj is

I (nij+ 46nij)

6uj dSi

+ I y j 6uj dx

with the current configuration as reference. Likewise, the second-order virtual work of internal deformation is

I

+

(nij f6nij)

a axi (6uj) dx.

-

The energy deficit is therefore

a

f j6nij - (6uj) dx - f j anij 6uj dSi axi

since the first-order terms cancel (virtual work principle) because the current distributions of stress and body forces are in equilibrium. Here bnijwithin S must be calculated from U(6u), with the current configuration as reference, while 6nij over S - S’ must be calculated fromfj(6u). So the deficit is

under fixed constraint over S’, from which we recognize that the exclusion condition prevents eigenmodes simply by ensuring an insufficiency of energy available for eoery admissible displacement. In particular, therefore, for an elastic continuum in a conservative environment, the condition coincides with the classical criterion of stability; this connection was originally made explicit by Hill (1957b), at the same time emphasizing the simplifications resulting from the present Lagrangian formulation. We must now assess the criticality of the exclusion condition, by examining the circumstances of its incipient failure. For this purpose consider a

Rodney Hill

60

path of deformation (actual or conceptual) which is generated by some monotonically varying parameter (a load or a modulus or a geometric dimension, etc.), beginning where the exclusion functional is positive. Suppose that a stage is eventually reached beyond which the functional becomes indefinite. In general the functional at this stage is semidefinite and vanishes for some admissible field. It is evident that such a field necessarily renders the functional stationary. In an eigenstate, by comparison, a mode wj makes the functional vanish (3.23),but need not make it stationary, as we saw in the rearrangement of (3.21). But when the deformation-sensitive data are selfadjoint, the exclusion condition is critical in that either a primary eigenstate does occur where it first fails, by the converse to (3.25), or (for instance) the configuration beyond which it ceases to hold is a point of accumulation for the spectrum of eigenstates. This latter phenomenon has been encountered in particular cases by Biot (1965), Hill and Hutchinson (1975), and Young (1976); there may, of course, be other kinds of pathological behavior. In passing, it may be remarked that inhomogeneous data cannot be prescribed freely in an eigenstate if the rate problem is to admit a solution (cf. an algebraic system of linear equations whose determinant is zero). For example, when there are fixed constraints over C' and the deformationsensitive data over C - C' are self-adjoint, we see by choosing uj* = w j in the reciprocal theorem (3.7) and using (3.18) that the inhomogeneous data must comply with

jyjwj d< + jc j w j dCi = 0 for every distinct eigenmode. This may be regarded as a generalized orthogonality, requiring that the controllable part of the incremental loading should have no component in the eigendirection, so to say. When the inhomogeneous rate problem is self-adjoint and the exclusion condition holds, it can be shown that the guaranteed unique solution makes the functional in (3.24) an absolute extremum over the entire class of admissible velocity fields. We start from (3.26), putting uj* - uj in place of wj*, where uj and vj* are any distinct fields taking the values given over C'. After rearrangement and use of (3.18), we obtain

Now take uj to be the actual solution; the expression on the right transforms to cj(uj* - u j ) dS. yj(uj* - u j ) d< +

j

j

s-S'

Aspects of Invariance in Solid Mechanics

61

Hence the functional in (3.24), evaluated with any vj*, exceeds its solution value. This extremum principle, and its dual with the nominal stress-rate as the primary variable, are due to Hill (1962) with forerunners (1958, 1959, 196la,b). 2. Nonlinear Case The incremental response can now be thoroughly nonlinear as in (3.4). With homogeneous data the condition (3.26) excluding eigenstates is still relevant. Exactly as before this condition is critical under self-adjoint data and an eigenmode is characterized by stationarity of the exclusion functional. It goes without saying, however, that there is no longer an automatic connection between the homogeneous and inhomogeneous problems. So in regard to the latter we must proceed along different lines. Following Hill (1959) the starting point is taken to be the virtual work identity

J” Anij a

-

ati

Avj d t = J” Anij Avj d X i

in the difference Av, = vj* - v j of any presumed pair of distinct solutions to a particular inhomogeneous problem. For prescribed velocities over S’ and the data (3.16) over S - S’, the identity becomes

J” Anij a

-

J”

Avj d t = Af;. Avj dS.

ati

Bifurcation is thus excluded by the requirement

J” Anij a

-

ati

I

Avj d t > AA Avj dS

(3.27)

for every pair of analytically admissible fields taking the values prescribed over S’. Note that this condition takes no account of the particular cj values given over S - S’. It is therefore likely to be oversufficient. For instance, suppose the part S’ is either fixed or absent. Then the null field is admissible in (3.27) and can be paired with any wj* as previously defined, so reproducing (3.26). It follows that (3.27) in this case excludes bifurcations under homogeneous data also and is therefore stronger than (3.26). So typically we have the following sequence of events along a path of deformation: first a segment where (3.27) holds, guaranteeing universal uniqueness; then a segment where (3.27) fails but uniqueness persists; then a critical stage at which the first primary bifurcation occurs under some particular dj distribution; then a succession of stages where primary bifurcations occur under other ij distributions, somewhere among them being the homogeneous data dj E 0.

Rodney Hill

62

Before proceeding, it is perhaps worth mentioning that the condition excluding bifurcations can be weakened whenever one solution is known. In that event the inequality need only be stipulated for pairs having that field as one member. While this situation is common, it appears that the weakened condition has not been noticed or put to any use. Guided by the notion of function convexity,extended to functionals (Hill, 1959, 1961a,b), and by an associated “automatic” procedure for generating extremum principles from uniqueness conditions (Hill, 1956), it will be shown that (3.27) is equivalent to (3.28) for every pair of admissible fields (Hill, 1962). Assuming this result for the moment, choose uj to be the guaranteed unique solution. Then (3.28) transforms to

j AU d t >

(cj

+fi + $A&) Avj dS +

yj Avj d t .

Now

j (fi+ *A&) Au, dS = 4A under the self-adjointness condition (3.18). Hence the solution value of the functional in (3.24) is an absolute minimum within the entire class of admissible fields (Hill, 1962). In proof of the stated equivalence, it is obvious in the first place that (3.27) is implied by (3.28), simply by interchanging v j and vj* in the latter and adding the two inequalities. To proceed in the reverse direction we consider any pair v j and vj* and introduce a continuous sequence of fields v j + AAv j such that A I u j = A Avj, 0 5 1 I 1, where 1is uniform throughout the continuum. Such fields satisfy the constraint data over s’when v j and uI* do, and are therefore admissible in (3.27). Define

j

+(A) = ( A A U - n, where AAoperating on any function means the difference between its values at uj + A Auj and v j . It is required to prove that +(1) > 0. We have

A

4 a a = j AAnijag AAvl d< - j A,& AAvjdS i

Aspects of Invariance in Solid Mechanics

63

and so d 4 / d 1 > 0 when 0 < A 5 1 by (3.27). Since 4(0)= 0 it follows that

#(A) > 0 when 0 < 1I 1, as required. To date, direct application of the uniqueness test (3.27) or (3.28) has been discouraged by the volume integrand not being a single-valued function of Avj, but of v j and vj* jointly. However, the following indirect approach has been found effective. This is based on the notion of an incrementally linear comparison material that is in a sense “less stiff (Hill 1958, 1959, 1961a, 1967~).Let ULdenote its velocity gradient potential at the same local Cauchy stress. Suppose that the exclusion condition (3.26) is provable for material UL in the current configuration under some particular homogeneous data (disregarding trivial rigid-body motions where necessary). Then (3.26) obviously holds also for material U if function U - UL is nonnegative at each point. If, more strongly, function U - UL is convex at each point, then bifurcation is ruled out for material U under any associated inhomogeneous data. In proof we need only observe that (3.27) is identical to (3.26) for material UL,and that then the property of relative convexity, uiz. (3.29) ensures (3.27) for material U . Now, in (2.20), relative convexity was defined in regard to rate equations expressed objectively in work-conjugate generalized variables, and the property was shown to be coordinate invariant. And, in fact, it is equivalent to relative convexity in regard to rate equations expressed nonobjectively in nominal stress and deformation gradient. Equivalence follows from a finite-difference analog of the first of (1.92), namely,

Tr(AN AA) - Tr(Ap’) AE(’)) = Tr(F1)Ak’ Ak),

(3.30)

which can be proved in identical fashion via AN = Ai<”A’

+ F” AA’,

A’ AA

= AE“”

+ A’ AOA.

The right-hand term in (3.30) makes for a nonsimple connection between absolute convexities in the two formulations, but disappears from the connection between relative convexities, viz. (3.29) and A?ltijAe,, 2 At; Aeij,

(3.31)

when this is written in the Green measure and relative to the incrementally linear comparison material in particular. We see further from (1.92) that existence of a velocity-gradient potential U implies, and is implied by, existence of a rate-of-strain potential W such that (when symmetrized in its arguments) $)

=

aw/aq,

2 w = i!!)e!!) V IJ

(3.32)

Rodney Hill

64 where

Moreover, by (2.1I), this existence is a coordinate-invariant property. Function W was introduced by Hill (1959) with the current configuration as reference, in which case the components cij of Eulerian strain-rate are its arguments. With this choice of reference the strain-rate potential associated with any other scale function follows at once from (1.82), namely,

‘i“&” 2 EJ

V =

w + (1 - m ) o i j & i k & j k .

In particular from (1.84), or even (1.25) by going back to first principles, f[(goij/gt)

+ aij&k.]&ij = w +

CijEikEjk.

(3.33)

By introducing (3.32) into (3.27) we both connect the sufficiency condition directly with an objective constitutive description and partially split the volume integrand into terms in rate-of-strain and spin:

It can then be seen qualitatively that the exclusion of bifurcations depends on contributions of type stress times (spin)2 being dominated by those of type modulus times (strain rate)2. This is even more apparent in the exclusion condition for incrementally linear materials when (3.26) is written as

(3.35) Here the current configuration is taken as reference, 2’pIjiIis the corresponding tensor of Green moduli, u j denotes any admissible field vanishing over s’, and cij is its strain rate. When the geometry is such that the admissible class includes fields in which large spins are compatible with small strain rates, as in slender beams and columns, then bifurcation can occur while the stress is still only a fraction of the relevant modulus. For elastic continua (3.35) with fj = 0 coincides with the classical condition for stability under dead loading over S - S’; for such materials and in that particular context, essentially the same inequality was first given by Pearson (1956). For elastoplastic continua, with piecewise-linear response subject to the normality rule, the optimum choice of comparison material at each plastically stressed point is decided by (2.50) or (2.60), respectively, for the regular

Aspects of lnvariance in Solid Mechanics

65

or singular cases (Hill, 1958; Sewell, 1972). According to each left-hand inequality the matrix of moduli defining the material UL should be Lap.in (2.45) or (2.59), corresponding to the “fully” plastic branch. Transliteratmg this into the present notation, when the generalized coordinates are components eij of some tensor measure of strain, we have to set Lijk,

= q ’ j k 1 -!!g

A;jAi,;

r, s = 1, . .., n,

(3.36)

where Aij heij = 0 are the n tangent hyperplanes in strain space at the yield point ( n = 1 in the regular case). Elsewhere in the continuum the tensor of moduli is, of course, taken as the local gjk,. The bifurcation problem is thus reduced, in the first instance, to determining primary eigenstates for this incrementally-linear material. Under compatible inhomogeneous data in such a configuration it can happen that, among the infinity of deformation modes for the UL material, there are those for which the strain rate at every plastically stressed point of the continuum falls in the fully plastic domain of the U material itself. Then, clearly, such modes are also solutions of the given inhomogeneous rate problem for the U material, so a primary bifurcation has been identified. In short, there are circumstances where the whole procedure is critical and not just conservative. The classic example is Shanley buckling, as analyzed by Hill and Sewell (1960,1962)for columns and by Sewell (1963, 1964, 1973) for plates. Further applications are treated by Miles (1971, 1973, 1975), Hutchinson and Miles (1974), Storikers (1975). In recent years the outstanding development is the devising of effective computational techniques (e.g., Hibbitt et al., 1970, Nagtegaal et al., 1974), which have brought within range problems of a complexity that could not be attempted even a decade ago. Representative references in the bifurcation context are Needleman (1972a,b, 1973, 1975a,b, 1976, 1977), McMeeking and Rice (1975), Needleman and Tvergaard (1976, 1977), and Tvergaard (1976a,b, 1977). These references contain a wealth of details that illuminate many aspects of the general theory, but an adequate account could not be attempted in this survey. Instead, we describe in Section II1,C two simple rate problems which, while solvable by pure analysis, are sufficiently nontrivial to underscore important points of principle. 3. Modijcations for Conditional Data Active or passive constraint by smooth rigid tools was instanced as the archetype of data which is conditional in the sense that only the solution as a whole determines which of two mutually exclusive loadings shall apply locally (Section III,A,2). Short of an unwieldy symbolism, data of this kind cannot be handled on a common footing like that in the preceding analysis of bifurcation. It seems more sensible to treat each case separately on its

Rodney Hill

66

merits, and here we follow through the archetypal example to show precisely what is involved. With the total surface now denoted by S + S”, suppose that the vector velocity is prescribed over S‘, self-adjoint nonconditional data (3.16)over S - S‘, and conditional data (3.10)with (3.11) and (3.12)over S . In retracing the steps leading to (3.27)we have to evaluate Anij Avj over S”. Where p > 0 we have from (3.11) and (3.12)successively, An, Avj dSi = - p Ani Avi dS = -picij Avi ADj dS,

remembering that nj Avj = 0 by (3.10).Where p = 0, on the other hand, no more can be concluded than that Anij Avj dS, 5 0,

since the inequality would prevail were contact to be lost in one field but maintained in the other (i, > 0). On the right-hand side of (3.27)we must therefore add the term

-I

picij Avi Avj d S ,

and require the admissible fields to satisfy (3.10).Likewise, on the right of the equivalent uniqueness condition (3.28)we must add one-half of this same term. The proof of equivalence goes as before; the sequence of fields v j + AAv j automatically satisfies (3.10);and to the definition of 4(A)must be added

4

picij

AAviA A v jdS”.

Finally, in the absolute minimum principle characterizing the guaranteed unique solution, we must add to the functional in (3.24)the expression

+

p{+cijvivj - KijviTvj eijkviojTnk} dS“.

We omit details of the algebra, from which it could be seen that the extremum is not analytic, in that the functional has no true stationary value.

c. BIFURCATIONS UNDER SIMPLE LOADINGS 1. All-Around Dead b a d

Consider a continuum whose entire surface is unconstrained and subject only to “dead” loads (ie., iijdSi = 0 with the current configuration as reference). The intention of the following analysis is to locate any primary eigen-

Aspects of Znvariance in Solid Mechanics

67

states. Rigid translations are, of course, formally possible but are of no interest as eigenmodes. So we modify the exclusion condition (3.26) insubstantially to U(v*) dx > 0 for nontranslational fields v*; the material need not be incrementally linear. Correspondingly, we look for configurations where J U(v*) dx 2 0 with equality for some nontrivial field v. Suppose, for simplicity, that the continuum was originally homogeneous and that subsequently it has been uniformly strained. Then U is currently the same function of velocity gradient at every point. By choosing fields v* with arbitrary uniform gradient, we deduce first that the exclusion condition holds when and only when the function U is positivedefinite, and second that U is positive semidefinite in a primary eigenstate and is therefore made zero and stationary by the gradient value of a primary eigenmode. That is, no matter what the surface geometry, a primary eigenmode necessarily has a uniform gradient given by the eigenequations (3.37)

which express the stationarity of the nominal stress within the continuum. Conversely, any field with a uniform gradient satisfying (3.37) is obviously an eigenmode since it keeps the loading dead, but it is not primary when U is only indefinite in that configuration. In terms of the strain-rate potential W, citing (3.32) written with the current configuration as reference, primary eigenmodes are given by 2W(v*) + oij

auk* avk* __ 2 0,

~

axi axj

(3.38)

It is seen that the exclusion condition implies oija:ia;j = (ol

+ 02)wp+ +

*-.

>O

for arbitrary spins with vector components (a1*, a2*,a3*) on the axes of the principal stresses (ol,02, 03).Thus the condition requires the current stress to be within the domain 01

+ 6 2 > 0,

6 2

+ 6 3 > 0,

03

+

01

> 0.

(3.39)

Inequality (3.38), of course, also imposes requirements on the incremental response, which may indirectly restrict the domain even more. The significanceof (3.39), according to the work interpretation of (3.26), is simply that the opposite inequalities would allow the dead loads to do work in certain (virtual) rotations of the continuum.

Rodney Hill

68

It is seen also from (3.38) that an eigenmode must comply with (3.40) These are just the three independent equations that enforce continuing balance of moments of the tractions on a deforming cell; cf. the rate version of (1.85) when nil is stationary, namely, u i k n k j = u j k n k i evaluated in the reference configuration. When referred to a coordinate frame coaxial with the stress, (3.40) reduces to aik(dvj/dxk)

= ojk(dvi/dxk)

(i # j ) .

+ O 2 b 3 = -(61

(3.41) - 62)E12 and two similar. In particular, when o1 + o2 = 0, a rigid rotation about axis x3 is statically possible under the dead loads (which is to say without expen(O1

diture of work), but it may be debarred constitutively; in any event the configuration is said to have an “axis of equilibrium” (trivially if o1 = a2 = 0). As an alternative to (3.40) one may profitably derive (3.41) in elementary fashion, by considering an embedded cell that is currently a principal unit cube and then calculating the incipient lever arms of the dead loads ol,0 2 ,o3 acting at the centroids of opposite faces. Henceforward we take the linear case. Then the first group of eigenequations in (3.38) is y(l1 lk’ lEkl

+ ulE1l

=

(3.42)

and two similar. When there is no axis of equilibrium, the spin components can be eliminated from the second group by means of (3.41), with the result (3.43) and two similar. Then we have six eigenequations altogether in the strainrate components alone; the spin needed to maintain balance of moments can be calculated afterward from (3.41). If, however, an axis of equilibrium is present, say when o1 + o2 = 0, then E~~ = 0 and

+0103 =0

(3.44) replaces the corresponding member of the group (3.43). Suppose that the tensor of moduli has orthotropic symmetry with respect to the axes of stress. Then the six eigenequations decouple: y\:)kl&kl

etc. When n1+ o2 = 0 then o3= 0 (unless o1 = o2 = 0), which points up the previous remark that rotations may sometimes be statically but not

Aspects of Invariance in Solid Mechanics

69

constitutively possible. According to (3.45) there are two distinct types of eigenmode, namely, coaxial with the stress and shearing in a principal plane. To settle which is primary we would need to resort to the inequality in (3.38). When the material is isotropic elastic with work function w ( I l ,I , , 13),the standard elementary formulae for the moduli 9:jisin terms of the principal stretches can be substituted in the first of (3.45). This leads to the eigenequations (3.46) for coaxial modes. Actually, this could have been foreseen ab initio since the conjugate loads are just aw/i31r and coaxiality automatically maintains their alignment with the principal fibers. By (1.107) and the second of (3.45) shearing modes such as c12 # 0 are possible when

a1lI1

or

= a,/&

(3.47)

~ l / &= + 2 / & )

provided I 1 # I,. This can also be established in elementary fashion by requiring that the mode be such as to keep the varying stress axes (determined by the dead tractions on the deforming cell) aligned with the Eulerian triad (determined by the incremental deformation, relative to the ground state). When ,I1= I , , shearing can be examined as a coaxial mode, and det(a2w/aI, an,) factorizes correspondingly. Up to this point the analysis is taken from Hill (1967c, 1975). For isotrop ic elastic materials, an instructive alternative route to eigenmodes keeping the nominal stress stationary has been mentioned recently by Ogden (1977). This centers on using the measure F 1 / 2defined ) in (1.79), which here reduces to NR, where A = RA relative to a ground state. Then N = 0 is equivalent to fC'lz) = T(1'2)(R'12RR) where RR = RR'. With the Lagrangian triad as basis the component eigenequations are

t'y) =0

9

( t y ) - t ( ' / z ) ) W ; s +t!'12'W,R,

= 0,

(3.48)

as in (1.46). Since t!'/') = t , / I , = aw/aI, by (1.78), the first group returns to (3.46). In the second group each pair r # s jointly implies (t!'/,) + tS.'/2))oE= 0, so the group as a whole is equivalent to

6) or (ii)

c!'/~) + tj'/') = 0 t!1/2)

with wP, + 2w),

- &'Iz) =0

with

= 0,

0 : = 0.

These agree with (3.47) and further, by (1.45) and (1.50), w,, =

4+ Is I, -I, ~

70

Rodney Hill

or (ii) which both return to (3.41). 2. All-Around Fluid Pressure

The surface S of a continuum is completely immersed in a uniformly pressurized fluid. We consider the homogeneous self-adjoint rate problem dS over the whole of S, where set by nij dSi

=fi

] . = p ( - au. - - Sauk ij)dSi J a x j axk

from (3.9). Rigid rotations and translations are possible but will not be counted as eigenmodes, so the exclusion condition (3.26) is modified to (3.49) for all straining fields uj*. That this would indeed be an equality for pure spins ui* = o $ x j is easily verified: on the left we would have aij dx)wk*, by (3.38) and on the right, - p ( J x i dSj)W$iiOlr*/,and these are equal by the mean-value theorem of stress analysis, which in this case reads

(I

gij

s

dx = xiajk dSk = - p

s

xi dSj.

Since the surface integral in (3.49) extends over all of S, the divergence theorem can be called on, with the result ~ [ U ( V *+)

fp(yavi* - av.* 1 auj* __ axi axj

axj axi

Suppose that the current state of stress within S is uniform, namely aij = - p dij. Then, on introducing the strain-rate potential defined in (3.32), the inequality becomes

5 { W(v*) + p(&$~j:.-

E$E$)}

dx > 0,

with an integrand involving strain rate alone. By (3.33) this can be reduced finally to (3.50)

71

Aspects of Invariance in Solid Mechanics

where goij

a

-

a aEij

(W(V)

+

P(4EiiEjj

- EijEij)}

is the constitutive law under hydrostatic stress. An equivalent conclusion, but expressed in Kirchhoff stress, was reached by Miles (1973). In retrospect it is possible to convince oneself that the simple inequality (3.50) is “obvious,” when considered in the light of the energetic gloss on the exclusion condition (3.26). To begin with, in any finite deformation of S the work supplied by the constant pressure in the fluid is exactly p times the decrease of volume bounded by S. Also, ifthe Cauchy stress remained equal to - p hij within S, the state of equilibrium would continue and precisely the same work would be absorbed in internal deformation. So the actual shortfall in the supply of energy must be the work absorbed by reason of the constitutive change in Cauchy stress, and this work is +(St)’ times the volume integral in (3.50),to second order. Suppose, henceforward, that the material response is also uniform in the current state. Then, by the line of argument leading to (3.37), though now with strain rate and not velocity gradient as the independent variable, it can be deduced that a primary eigenmode is generally homogeneous. It is in fact any eigensolution of 90ij/9t = 0

(3.51)

that makes the potential function in (3.50) positive semidefinite. Conversely, any other solution of (3.51) is an eigenmode, though not primary, since it evidently keeps the stress hydrostatic, uniform, and compatible with the pressure maintained in the fluid. Note, incidentally, that the Jaumann flux is identical with Daij/Dt,the rate of change of background components, when the stress is hydrostatic. As an illustration consider a material which is homogeneous at zero pressure; then the fundamental path of deformation under varying uniform pressure or tension is such that it remains so. If, further, the microstructure initially has cubic symmetry the ensuing incremental response to prescribed dp is purely volumetric. Hence the cubic symmetry is not disturbed and by induction is retained along the fundamental path. So at any later stage, with coordinates along the cubic axes, we can write Q‘kk/gt

= 31CEkk,

(g/gl)(‘ll- .22)

= 2p(E11

- & 22) ,

9 o , 2 / 9 t = 2pIE12

and two similar pairs. It can be recognized that K is the instantaneous bulk modulus, p the modulus for shearing at 45” to the cubic axes, and p’ the modulus for shearing parallel to the cubic axes. The exclusion condition

Rodney Hill

72

requires that these three moduli are all positive; for isotropic elastic materials a more elaborate analysis with consonant conclusions was given by Miles (1973). From (3.51) the corresponding types of primary eigenmode are (i) purely volumetric: E ~ ~ = E ~ ~ = E ~ q ~ 2# =O0 , ... whenK=O;

(ii) volume-preserving coaxial: &kk = 0,

&12 =

0,... when P = 0;

(iii) volume-preserving shearing: q1= E~~ = E~~ = 0,

c12

# 0, . . . when p‘ = 0.

The first type is trivially coincident with the fundamental path at a stage where the pressure is stationary. Implications for other sets of moduli follow from relations such as y(1) 1111

=K

+ 4P + P,

y C1122 1)

= K -3P

-P,

y(1) 1212=

P’

+ P.

For pairwise interaction models of monatomic lattices the implications of this theory have been reviewed by Hill and Milstein (1977), and extensive computations of the pressure dependence of the moduli for Morse potential models are presented by Milstein and Hill (1977).

REFERENCES BIOT,M. A. (1965). “Mechanics of Incremental Deformations.” Wiley, New York. BLATZ,P. J., and KO, W.L. (1962). Application of finite elastic theory to the deformation of rubbery materials. Trans. SOC. Rheol. 6, 223-251. BOLOTIN,V. V. (1963). “Non-Conservative Problems of the Theory of Elastic Stability” (G. Herrmann, ed.). Pergamon, Oxford. BUDIANSKY, B. (1974). Theory of buckling and postbuckling behavior of elastic structures. In “Advances in Applied Mechanics,” Vol. 14, pp. 2-65. Academic Press, New York. CHADWICK, P., and OGDEN,R. W. (1971). On the definition of elastic moduli. Arch. Rational Mech. Anal. 44,41-53; 54-68. J. L. (1970). Nonlinear elasticity of diatomic crystals. Int. J. Solids Structures 6, ERICKSEN, 951-958. HAVNER, K. S. (1977a). On unification, uniqueness, and numerical analysis in plasticity. Int. J . Solids Structures 13, 625-635. HAVNER, K. S. (1977b). On uniqueness criteria and minimum principles for crystalline solids at finite strain. Acta Mechanica 28, to appear. HIBBITT,H. D., MARCAL, P. V., and RICE,J. R. (1970). A finite element formulation for problems of large strain and large displacement. Int. J . Solids Structures 6, 1069-1086. HILL,R. (1956). New horizons in the mechanics of solids. J. Mech. Phys. Solids 5, 66-74. HILL,R. (1957a). On the problem of uniqueness in the theory of rigid-plastic solids. J. Mech. Phys. Solids 5, 153-161; 302-307.

Aspects of Invariance in Solid Mechanics

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HILL,R. (1957b). On uniqueness and stability in the theory of finite elastic strain. J. Mech. Phys. Solids 5, 229-241. HILL,R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J . Mech. Phys. Solids 6, 236-249. HILL,R. (1959). Some basic principles in the mechanics of solids without a natural time. J . Mech. Phys. Solids 7 , 209-225. HILL,R. (1961a). Bifurcation and uniqueness in nonlinear mechanics of continua. In “Problems of Continuum Mechanics,” N. I. Muskhelishvili Volume, pp. 155-164. SOC.Ind. Appl. Math., Philadelphia. HILL, R. (1961b). Uniqueness in general boundary-value problems for elastic and inelastic solids. J . Mech. Phys. Solids 9, 114-130. HILL,R. (1962). Uniqueness and extremum principles in self-adjoint boundary-value problems in continuum mechanics. J. Mech. Phys. Solids 10, 185-194. HILL,R.(1966). Generalized constitutive relations for incremental deformation of metal crystals by multi-slip. J . Mech. Phys. Solids 14, 95-102. HILL, R. (1967a). On the classical constitutive laws for elastic-plastic solids. In “Recent Progress in Applied Mechanics,” F. K. G. Odqvist Volume, pp. 241-249. Wiley, New York. HILL,R. (1967b). The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 79-95. HILL,R. (1967~).Eigenmodal deformations in elastic-plastic continua. J . Mech. Phys. Solids 15, 371-386. HILL, R. (1968). On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229-242; 3 15-322. HILL,R.(1969). Some aspects of the incremental behaviour of isotropic elastic solids after finite strain. In “Problems in Mechanics: Deformation of Solid Bodies,” V. V. Novozhilov Volume, pp. 459-466. Leningrad. HILL,R. (1970). Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. SOC.London Ser. A 314, 457-472. HILL,R. (1972). On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. SOC.London Ser. A 326, 131-147. HILL, R. (1975). On the elasticity and stability of perfect crystals at finite strain. Math. Proc. Cambridge Phil. SOC. 75, 225-240. HILL,R.,and HUTCHINSON, J. W. (1975). Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids 23, 239-264. HILL,R., and MILSTEIN, F. (1977). Principles of stability analysis of ideal crystals. Phys. Rev. B 15, 3087-3096. HILL, R., and RICE,J. R. (1972). Constitutive analysis of elastic-plastic crystals at arbitrary strain. J. Mech. Phys. Solids 20, 401-413. HILL,R., and RICE,J. R. (1973). Elastic potentials and the structure of inelastic constitutive laws. S.I.A.M. J. Appl. Math. 25, 448-461. HILL,R., and SEWELL,M. J. (1960). A general theory of inelastic column failure. J. Mech. Phys. Solids 8, 105-111; 112-118. HILL,R., and SEWELL, M. J. (1962). A general theory of inelastic column failure. J . Mech. Phys. Solids 10, 285-300. HUTCHINSON,J. W. (1974). Plastic buckling. I n “Advances in Applied Mechanics,” Vol. 14, pp. 67-144. Academic Press, New York. HUTCHINSON, J. W., and MILES,J. P. (1974). Bifurcation analysis of the onset of necking in an elastic-plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22, 61-7 1. ILYUSHIN, A. A. (1961). On a postulate of plasticity. Prikl. M a t . Mekh. 25, 503-507. MCMEEKING, R. M., and RICE,J. R. (1975). Finite-element formulations for problems of large elastic-plastic deformation. Int. J. Solids Structures 11, 601-616.

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MILES,J. P. (1971). Bifurcation in plastic flow under uniaxial tension. J. Mech. Phys. Solids 19, 89-102. MILES,J. P. (1973). Fluid-pressure eigenstates and bifurcation in tension specimens under lateral pressure. J. Mech. Phys. Solids 21, 145-162. MILES,J. P. (1975). The initiation of necking in rectangular elastic-plastic specimens under uniaxial and biaxial tension. J. Mech. Phys. Solids 23, 197-213. F., and HILL,R. (1977). Theoretical properties of cubic crystals at arbitrary pressure. MILSTEIN. I. Density and bulk modulus. J. Mech. Phys. Solids 25, 457477. J. C., PARKS,D. M.,and RICE,J. R. (1974). On numerically accurate finite-element NAGTEGAAL, solutions in the fully plastic range. Comput. Methods Appl. Mech. Eng. 4, 153-177. A. (1972a). Void growth in an elastic-plastic medium. Trans. Am. SOC.Mech. Eng. NEEDLEMAN, (December) 964-970. NEEDLEMAN, A. (1972b). A numerical study of necking in circular cylindrical bars. J. Mech. Phys. Solids 20, 111-127. NEEDLEMAN, A. (1973). A numerical study of uniaxial compression in circular elastic-plastic columns. Int. J. Solids Structures 9, 981-998. A. (1975a). Post-bifurcation behaviour and imperfection sensitivity of elasticNEEDLEMAN, plastic circular plates. Int. J. Mech. Sci. 17, 1-14. A. (1975b). Bifurcation of elastic-plastic spherical shells subject to internal presNEEDLEMAN, sure. J. Mech. Phys. Solids 23, 357-367. A. (1976). Necking of pressurized spherical membranes. J . Mech. Phys. Solids 24, NEEDLEMAN, 339-359. NEEDLEMAN, A. (1977). Inflation of spherical rubber balloons. Int. J. Solids Structures 13, 49-42 1. NEEDLEMAN, A., and TVERGAARD, V. (1976). An analysis of the imperfection sensitivity of square elastic-plastic plates under axial compression. Int. J. Solids Structures 12, 185-201. NEEDLEMAN, A., and TVERGAARD, V. (1977).Necking of biaxially-stretched elastic-plastic circular plates. J. Mech. Phys. Solids 25, 159-183. NEMAT-NASSER, S. (1968). On local stability of a finitely deformed solid subjected to followertype loads. Quart. Appl. Math. 26, 119-129. OGDEN,R. W.(1972a). Large deformation isotropic elasticity. 1. On the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. SOC. London Ser. A 326, 565-584. OGDEN,R. W.(1972b). Large deformation isotropic elasticity. 11. On the correlation of theory and experiment for compressible rubberlike solids. Proc. R. SOC. London Ser. A 328, 567-583. OGDEN,R. W.(1974). On stress rates in solid mechanics with applications to elasticity theory. Proc. Cambridge Phil. SOC. 75, 303-319. OGDEN,R. W.(1975). A note on variational theorems in non-linear elastostatics. Math. Proc. Cambridge Phil. SOC.77, 609-615. OGDEN,R. W . (1976). Volume changes associated with the deformation of rubberlike solids. J. Mech. Phys. Solids 24, 323-338. OGDEN,R. W.(1977). Inequalities associated with the inversion of elastic stress-deformation relations and their implications. Math. Proc. Cambridge Phil. SOC.81, 313-324. G. P. (1976). On the elasticity of monatomic crystals. Math. Proc. Cambridge Phil. PARRY, SOC. 80, 189-211. PARRY, G. P.(1977). On the crystallographic point groups and on Cauchy symmetry. Math. Proc. Cambridge Phil. SOC.82, 165-175. PEARSON, C. E. (1956). General theory of elastic stability. Quart. Appl. Math. 14, 133-144.

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RICE, J. R. (1975). Continuum mechanics and thermodynamics of plasticity in relation to micro-scale deformation mechanisms. I n “Constitutive Equations in Plasticity” (A. S. Argon, ed.), pp. 23-75. MIT Press, Cambridge, Mass. SEWELL, M. J. (1963). A general theory of elastic and inelastic plate failure. J. Mech. Phys. Solids 11, 377-393. SEWELL, M. J. (1964). A general theory ofelastic and inelastic plate failure. J. Mech. Phys. Solids 12, 279-297. SEWELL, M. J. (1965). On the calculation of potential functions defined on curved boundaries. Proc. R. SOC.London Ser. A 286, 402-411. SEWELL,M. J. (1967). On configuration-dependent loading. Arch. Rational Mech. Anal. 23, 327-35 1. SEWELL, M. J. (1972). A survey of plastic buckling. In “Stability” (H. Leipholz, ed.), pp. 85-197. Univ. of Waterloo Press, Ontario. SEWELL, M. J. (1973). A yield surface corner lowers the buckling stress of an elastic-plastic plate under compression. J. Mech. Phys. Solids 21, 19-43. SEWELL, M. J. (1974). On applications of saddle-shaped and convex generating functionals. I n “Physical Structures in Systems Theory” (J. J. van Dixhoorn and F. J. Evans, eds.), pp. 219-245. Academic Press, New York. STOR~ERS, B. (1975). On buckling of axisyrnmetric thin elastic-plastic shells. Int. J. Solids Structures 11, 1329-1346. STOREN,S., and RICE,J. R. (1975). Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421-441. THOMPSON, J. M. T., and HUNT,G. W. (1973). “A General Theory of Elastic Stability.” Wiley, London. TRELOAR, L. R. G. (1973). The elasticity and related properties of rubbers. Rep. Progr. Phys. 36, 755-826. TRELOAR, L. R. G. (1976). The mechanics of rubber elasticity. Proc. R. SOC.London Ser. A 351, 301-322. TRUESDELL, C., and NOLL, W. (1965). “The Non-Linear Field Theories of Mechanics,” (S. Fliigge, ed.), Section 87, Handbuch der Physik Band II1/3. Springer, Berlin. TRUESDELL, C., and TOUPIN,R. (1963). Static grounds for inequalities in finite strain ofelastic materials. Arch. Rational Mech. Anal. 12, 1-33. TVERGAARD, V. (1976a). Effect of thickness inhomogeneities in internally pressurized elasticplastic spherical shells. J . Mech. Phys. Solids 24, 291-304. TVERGAARD, V. (1976b). Buckling behavior of plate and shell structures. Proc. 14th Int. Cong. Theor. and Appl. Mech. (W. T. Koiter, ed.), pp. 233-247. North-Holland, Publ., Amsterdam. TVERGAARD, V. (1977). On the numerical analysis of necking instabilities. I n “Proceedings of an International Conference on Finite Elements in Nonlinear Solid and Structural Mechanics,” to appear. YOUNG,N. J. B. (1976). Bifurcation phenomena in the plane compression test. J. Mech. Phys. Solids 24, 77-9 1.

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ADVANCES I N APPLIED MECHANICS VOLUME

18

The Optimum Theory of Turbulence F. H. BUSSE Institute of Geophysics and Planetary Physics University of California Los Angeles. California

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Optimum Problem for Turbulent Couette Flow . . . . . . . . . . . . A . Basic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The Energy Stability Limit . . . . . . . . . . . . . . . . . . . . . . . . . I11. Multi-a Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Convection in a Porous Medium . . . . . . . . . . . . . . . . . . . . . . B. General Properties of the Extremalizing Solutions . . . . . . . . . . . . C . Boundary-Layer Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Bounds on the Transport of Momentum . . . . . . . . . . . . . . . . . . . A . Extremalizing Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . B. Comparison with Observations . . . . . . . . . . . . . . . . . . . . . . . C . Momentum Transport in a Rotating System . . . . . . . . . . . . . . . V . Bounds on the Transport of Mass . . . . . . . . . . . . . . . . . . . . . . . A . Turbulent Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Turbulent Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Bounds on the Transport of Heat . . . . . . . . . . . . . . . . . . . . . . . A . Convection in a Layer Heated from Below . . . . . . . . . . . . . . . B. The Infinite Prandtl Number Limit . . . . . . . . . . . . . . . . . . . . VII . General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Further Applications of the Optimum Theory . . . . . . . . . . . . . . B. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 80 80 82 83 84 84 86 89 94 94 98 101 105 105 108 110 110 113 115 115 117 119

I . Introduction The optimum theory represents an approach toward the understanding of turbulent flows without the introduction of heuristic assumptions which are commonly used in other theories of turbulence .Since there does not seem to be a generally accepted definition of turbulence we shall regard for the 77

Copyright Q 1978 by Academic Press, Inc. All rights of reproduction in any form reserved . ISBN 0-12.M)2018-1

78

F. H.Busse

present purpose a fluid flow as turbulent if the details of the velocity field are too complex to be of interest and only the information about certain average properties is needed. The optimum theory is hardly appropriate for all turbulent systems of this general nature and a more suitable mathematical definition of turbulence will be required later. The main point is that turbulent fluid systems are characterized by an information gap which is neither possible nor desirable to fill. Because of the nonlinearity of the Navier-Stokes equations lack of information about a particular aspect of a turbulent velocity field affects in general the knowledge about all other aspects. In other words, the theoretical description of average quantities cannot be separated from the behavior of the fluctuating velocity field. The attempt to derive equations for average quantities yields an infinite hierarchy of moment equations which can be analyzed only if suitably chosen assumptions are introduced. Highly sophisticated schemes have been developed for this purpose in statistical theories of turbulence, but the heuristic character of the assumptions cannot be eliminated and the theoretical predictions have been successful only in idealized cases such as isotropic homogeneous turbulence. The optimum theory represents an alternative approach in which the lack of information about properties of the fluctuating velocity field is reflected by the theoretical results. Instead of a theoretical prediction of the physically realized average properties, bounds on those properties are obtained. No assumptions are introduced in the theory, with the exception of technical assumptions such as those involved in boundary-layer methods. The optimum theory has thus been capable of producing mathematically rigorous quantitative results in contrast to other theories of turbulence. The idea of the optimum theory is to consider not the manifold of solutions of the basic equations for a particular problem but a larger manifold of vector fields which includes the actual solutions. This manifold of fields shares with the solutions kinematic relationships such as boundary conditions and the energy balance. By generalizing a certain average physical property such as the momentum transport as a functional of the manifold of fields, extremal values of the functional can be determined by variational methods. The extremal values provide the bounds for the values which the property assumes for the subset of the actual solutions of the Navier-Stokes equations. The bounds can be improved, naturally, by using a narrower definition for the manifold of vector fields. In addition to the kinematic and energetic constraints other constraints may be introduced, and in principle it is possible to conceive a hierarchy of constraints such that the bounds approach the extremal values assumed by the actual solutions of the equations of motion. Little work has yet been done on the development of the optimum theory in this direction.

Optimum Theory of Turbulence

19

The optimum theory would be primarily of mathematical interest if there were not strong indications that the physically realized solutions of turbulent fluid flow often tend to optimize transport. Malkus’ (1954b) suggestion that the convection solution of maximum heat transport is the most stable one provided the stimulation for the first development of the theory of bounds by Howard (1963). When the theory was extended and applied to turbulent shear flow (Busse, l968,1969a,b, 1970b) further evidence appeared for the tendency of the realized solution to maximize the transport of momentum or mass. It must be emphasized that there is only a tendency and that in general the realized flow in a turbulent system does not represent the solution of maximum transport. Even in the case of thermal convection the Malkus hypothesis is strictly correct only in the limit of small amplitudes. The tendency toward optimal transport properties is sufficiently strong, however, to cause similarities between the observed structure of turbulent flow and the properties of the extremalizing vector field. The physical insights provided by these similarities are probably the most fascinating results of the optimum theory. Related to those similarities is the model character of the Euler equations determining the extremalizing vector field. The optimum theory would hardly be useful if the variational problems for the extrema of the functionals mentioned above would not be simpler to solve than the basic NavierStokes equations. Even though the Euler equations are highly nonlinear they can be solved analytically in certain cases where a solution of the Navier-Stokes equations would be an impossible task even with the help of modern computers. Because the Euler equations resemble the equations of motion and incorporate some of their nonlinear features while neglecting others, they can be regarded as model equations. The investigation of model equations has had a long history in the theory of turbulence and much of the theoretical understanding of turbulence has been derived from model equations rather than from the basic Navier-Stokes equations. The Euler equations share with the so-called mean field equations the property of optimally combining a reasonably close description of the physics and a mathematical simplicity which often permits analytical solutions. The three main aspects of the optimum theory-the model character of the equations, the similarities between the extremalizing solutions and the structure of the observed turbulence, and the rigorous character of the bounds-provide an unusual variety of facts and insights which should appeal to a wide spectrum of scientists from mathematicians to experimental fluid dynamicists. In this article we attempt to keep the mathematical treatment sufficiently simple that it can be followed by fluid mechanicians with a modest theoretical background. On the other hand, open mathematical problems are pointed out and the limitations of proven mathematical

F. H.Busse

80

properties are mentioned. A more mathematically oriented introduction to the theory of bounds is given by Howard (1972) and in Chapter XI1 of Joseph's (1976) book. 11. 'Ihe Optimum Problem for Turbulent Couette Flow A. BASICRELATIONSHIPS In order to introduce the basic concepts of the optimum theory we consider the simplest physical problem of fluid dynamics, the transport of momentum between two parallel rigid plates moving relative to each other with a constant speed V,. The laminar solution for this problem is plane Couette flow and the corresponding turbulent state is sometimes called turbulent Couette flow. We assume that the fluid is homogeneous and incompressible and that the plates are infinitely extended. For the dimensionless description of the problem the distance d between the plates is the length scale and d2/v is the time scale, where v is the kinematic viscosity of the fluid. The dimensionless Navier-Stokes equations for the velocity vector V are

v x (V x V) + v p + v - vv + ata v = 0,

(2.la)

v.v=o.

(2.lb)

-

Using a Cartesian system of coordinates with the origin halfway between the plates and the z-direction normal to the plates we can write the boundary condition for the velocity vector V in the form a t z = ki,

V = TjRei

(2.2)

where i is the unit vector in the x-direction and the Reynolds number is defined by Re = V, d/v. Assuming that the velocity vector and the pressure are bounded everywhere we introduce two kinds of averages. The average over planes z = const. will be indicated by a bar, the average over the entire fluid layer will be indicated by angle brackets (...), i.e., UWE lim L+oo

j L j L uwdxdy 4E

-

-L

and

(uw)=j

112

Wdz.

(2.3)

- 112

-L

Separating the velocity vector into its mean and fluctuating parts, so that V =U +5

with U

= v,

81

Optimum Theory of Turbulence

we obtain the following equations from (2.1) by first taking the average over planes z = const. and then subtracting it from the original equations: a2 -

az2

u - ata u = aaZ -at+, -

~

a p = a -$?, aZ aZ

-_

-

v x (V x 5) + V(p - p ) + ata 5 + 5 - VQ - 5 -

*

VS + u . v5 + ~.

a u = 0,

-

az

V-i=O. We have used the fact that the component of U normal to the boundary vanishes because of the boundary condition. lii denotes the component of 5 parallel to the plates while is the normal component. By multiplying Eq. (2.6) with i and averaging it over the entire fluid layer we obtain

+

I d - - ( 1512) 2 dt

+ (Iv x 512) +

where the property has been used that some integrals can be transformed into surface integrals which vanish because 5 vanishes at z = k+and because the contributions from the remaining surface become negligible as the volume over which the average is extended tends to infinity. We are interested in turbulent flow under stationary conditions long after any change in the relative motion of the plates has occurred. We define this are indepencase by the assumption that all mean quantities like U or dent of time. It is possible that certain solutions of the Navier-Stokes equations corresponding to coherent oscillations of the fluid layer do not satisfy this property. In that case the above averages could be defined as ensemble averages. With the exception of some singular cases there appears to be no need for such a distinction and we shall not introduce it here. The time independence of mean quantities allows the integration of Eq. (2.4) d -U = - (a&) - Re i, dz

a

where the constant vector of integration has been determined by the boundary condition (2.2). Expression (2.9) can be used to eliminate the mean flow from the energy balance (2.8):

( I V x 5 1’)

+ ( lz - (iM)1’ )

- Re(6,fi)

= 0,

(2.10)

82

F. H. Busse

where fix is the x-component of Q. In accordance with the definition of turbulence under stationary conditions we have neglected the time rate of change of the kinetic energy. The use of the identity

(G(G- (ha)))= ( IG - (a*) 1)’ in the derivation of relationship (2.10) has made it obvious that the average Reynolds stress (6, a ) must be a positive quantity. B. THEVARIATIONALPROBLEM The dimensionless momentum transport in the case of the laminar solution of the above problem, i = 0, U = -zi Re, is given by Re. The xmomentum transport in the z-direction in the case of turbulent flow is given by (2.11) M = Re + (fix$). Since (.*,a) is a positive quantity the momentum transport is bounded from below by the value of the laminar solution. The goal of the optimum theory is to derive an upper bound for (fi,a) at a given value of Re. A particular method of deriving such an upper bound is the determination of the maximum p ( R ) of the functional (u, w ) among all vector fields v that satisfy (~vxV~’)+(~uw-(UW)J2)-R(u,w)=0,

v-v=o, v=O

atz=+&,

(2.12a) (2.12b) (2.12c)

and are bounded everywhere. As before u and w denote tangential and normal components of v. Obviously, the solutions of the Navier-Stokes equations with time-independent average properties are part of the manifold of vector fields v. Accordingly, p ( R ) provides indeed an upper bound for (&a) at a given value Re of R. The advantage of the manifold defined by conditions (2.12)is that the maximum p ( R )can be determined by variational methods, which in general is not possible for the submanifold of the solutions of the Navier-Stokes equations. The upper bounds can be improved when more constraints derived from the Navier-Stokes equations are taken into account, in addition to those given by (2.12). We shall return to this point in Section VI1,B. On the other hand, weaker but more easily accessible results may be obtained if the constraint of V * v = 0 is removed. An exact analytical solution of the variational problem for Couette flow in this case has been obtained by Howard (1972).

83

Optimum Theory of Turbulence

It is inconvenient to solve the variational problem in the formulation given above. Instead the following variational problem is considered. Given p > 0, find the minimum R ( p ) of the functional

among all bounded vector fields v which satisfy (2.12b)and (2.12~)and have (ux w ) > 0. Because the functional (2.13) is homogeneous the amplitude of the minimizing solution v remains undetermined. If the amplitude is chosen such that ( U X d

(2.14)

=P

is satisfied, condition (2.12a) is fulfilled. To complete the proof of the equivalence of the two variational problems it must be shown that R ( p ) is a monotonous function. Following Howard (1963) this can be easily seen from the following sequence of inequalities which hold for p* > p’ with v* and v’ representing the extremalizing solutions for p* and p’,

( 1 wllll - (W’U’) I(p* - p’)

(w’u,’)’

1)’

.

(2.15)

In the limit when p* approaches p’, (2.16) can be deduced whenever v* approaches v’ at the same time. Relationship (2.16) can be used to show that dR(,u)/dp is a decreasing function of p. This follows when relationship (2.16) is used in the first and last term of (2.15).

C. THEENERGYSTABILITYLIMIT The minimum R ( p ) of the functional (2.13) provides a lower bound for the Reynolds number Re at a given value p of (d, ~ )the, Reynolds stress part of the momentum transport. Since R ( p ) is a monotonically increasing function of p and since (dx&) must be positive according to (2.10), no fluctuating flow can exist for Reynolds numbers less than R(0).Indeed, in the limit p = 0

F. H.Busse

84

the variational problem (2.13) becomes identical to the energy stability problem which was originally posed by Reynolds (1895) and Orr (1907). The energy stability limit RE = R(0)determines the region 0 < Re < RE in which laminar Couette flow is absolutely stable such that any arbitrary disturbance of the flow decays exponentially. For Re > RE laminar Couette flow may still be stable but disturbances exist which grow in time, at least initially. For a detailed exposition of the energy stability problem in more general cases, see Joseph's (1976) book. The energy stability problem is always included in the optimum problem as the special limit when the Reynolds stress or an equivalent quantity vanishes. On the other hand, the energy stability problem can be posed in more general cases, since the optimum problem depends on the separation of a mean and a fluctuating part of the velocity field. The energy stability problem is essentially a linear problem giving rise to linear Euler equations even though it provides a general sufficient condition for stability. The optimum problem may be regarded as the nonlinear extension of the energy stability problem in those cases in which the laminar solution is invariant with respect to the averaging procedure. Although the possibilities for the most general applications of the optimum problem have not yet been explored, a straightforward mathematical analysis appears to be possible only in those flow problems which are characterized by laminar solutions that are iddependent of the Reynolds number except for the amplitude factor. 111. Multi-a Solutions

A.

CONVECTION IN A POROUS

MEDIUM

Before discussing the variational problem (2.13) in more detail, we shall consider a simpler problem in order to demonstrate the boundary-layer methods for the solution. The simplest variational problem which retains all relevant mathematical properties of the problem (2.13) is posed by the task to find the minimum F ( p ) of the functional 8 ; p) ={( 1vl2)(

Ive12) + p((We-

(we))2)}(we)-2 (3.1) among all fields v, 6' that satisfy the constraint V * v = 0 and the boundary condition w=e=O a t z = ++. (3.2) As before, w is the z-component of the vector v. This variational problem provides an upper bound for the heat transport F(V,

Optimum Theory of Turbulence

85

in a horizontal porous layer heated from below. Although the equations from which (3.1) has been derived are of doubtful validity in the turbulent regime of convection in a porous medium, we shall consider them nevertheless in order to provide a physical background for the discussion of the variational problem (3.1). The physical parameters of convection in a porous medium are the thickness d of the layer, the applied temperature difference T2 - T,, the viscosity v of the fluid, the Darcy permeability coefficient K, and the thermal diffusivityK of the porous medium. For the nondimensional description of the problem we use d, d2/rc,and (T, - T,) Ra- as units for length, time, and temperature, respectively, where Ra is the Rayleigh number defined below. The Darcy-Boussinesq equations for convection in a porous layer are

'

B(ai/at

+ 5 . vi)= -vp + k(T - T' + i, v-i=o,

aTpt

+i

'

(3.3a) (3.3b)

VT = V ~ T ,

(3.4)

and the boundary conditions are given by

T = Ra(T2 + Tl)/2(T,- T,) T Ra/2

k.i=$=O a t z = ki, (3.5) where k represents the unit vector in the z-direction. The nondimensional parameters of the problem are defined by B

E uK/dZv,

Ra

= ygKd(T2 - T,)/vK,

where g denotes the gravitational acceleration. By taking the horizontal average of Eq. (3.4)and subtracting it from (3.4) we obtain two equations for the mean temperature T and the fluctuating temperature 8 = T - F, i.e.,

(3.7) In accordance with our discussion in the preceding chapter we assume a statistically stationary state and neglect the time dependence of averaged quantities. This allows us to integrate (3.6) and obtain

aaZ T = $9

-

-

($8)

- Ra,

where the constant of integration has been determined by applying boundary condition (3.5). By multiplying (3.3a) by i and (3.7) by 8 and taking the

F. H.Busse

86

average over the porous layer we obtain two dissipation integral relationships: (p12)= (s@, (3-9) ( I V6 )1’ + ((3 - ($6))2) = R a ( ~ 6 ) . (3.10) It is evident that the last equation has been obtained in the same way as relationship (2.10) by eliminating the mean field T with the help of expression (3.8). The dimensionless heat transport H across the porous layer can be obtained from its value at the boundary:

H

= -dT/dz

lz=1,2

=

(&) + Ra.

(3.1 1)

From (3.9) it follows that the heat transport with convection is always larger than the static heat transport and that the latter provides a lower bound for the heat transport of all convective motions. An upper bound p for the convective part ($6) of the heat transport at a given value of Ra can be obtained by finding the maximum p of (we) among all vector fields v, 8 that satisfy Eqs. (3.3b), (3.9), and (3.10) and the boundary condition 8 = w = 0 at z = i~$.Instead of solving this variational problem it is convenient to solve the equivalent problem stated in the beginning of this section. Since the functional (3.1) is homogeneous of degree zero with respect to v as well as with respect to 8 the extremalizing solution can be normalized to satisfy ( IvI’) = ( w e ) and p = ( w e ) . In this case relationship (3.10) is satisfied with F replacing Ra. Accordingly F provides a lower bound for the Rayleigh number Ra at a given value of the convective heat transport p. Since F ( p ) is a monotonic function of p in the same way as discussed in the case of R(p),the two variational problems are indeed equivalent. We have already mentioned that the range of validity of Eq. (3.3a) for a porous medium is doubtful. Since we are more concerned with the mathematical properties of the variational problem than its practical applications we shall not discuss this point in detail and refer the interested reader to Busse and Joseph (1972) and Joseph (1976). B. GENERAL PROPERTIES OF THE EXTREMALIZING SOLUTIONS The Euler-Lagrange equations for an extremum, or more exactly, a stationary value of the functional (3.1), are given by

+ [ ( F + p ) ( w e ) - pG]ek - vz = 0, ( ( V I2)v2e- [(F + p ) ( w e ) - ~ G ]=w 0,

( I ve 12)v

v-v=o,

(3.12a) (3.12b) (3.12~)

Optimum Theory of Turbulence

87

where IL represents the Lagrange multiplier by which the constraint V . v = 0 has been taken into account. Equations (3.12) are clearly similar to Eqs. (3.3) (3.7) if expression (3.8) is inserted in (3.7). The main difference is that the time derivative has disappeared from Eqs. (3.12) and the nonlinear term in (3.12) is solely dependent on the z-coordinate. The constants in front of v and 8 in Eqs. (3.12) can be eliminated if suitable normalization conditions are used. In contrast to Eqs. (3.3) and (3.7) the nonlinearity appears symmetrically in (3.12) since the former equations cannot be derived from a variational principle. As a first step of the solution of problem (3.12) we eliminate (3.12~)by introducing the general representation

v = V x (V x k x ) + V x k+ for the solenoidal vector field v. By taking the z-component of the curl of (3.12a) we find (3.13) This means that the toroidal part V x k+ of the velocity either vanishes or is restricted to horizontal translations or rigid rotations about a vertical axis. We shall neglect these possibilities and assume z 0 since the toroidal component must vanish for the minimizing solution because it makes only a positive contribution to the first term in the wavy bracket of (3.1). In the second step toward a solution of problem (3.12) we introduce a Fourier series representation for the functions x and 8 :

+

N

N

8 = fYN)E

1

@,(Z)~,(X,

y)a; 1/2,

(3.14)

n= 1

where 4, satisfies the equation A, 4, = -an24,,and the orthonormalization condition 4,,4,,,= 6,. In general, the number of modes is assumed to be finite, but the possibility that N tends to infinity is not excluded. The functions W,,On,and 4, and the wavenumbers an depend on the cutoff parameter N of the series, but this dependence will be indicated only where necessary. By taking the z-component of the curl-curl of Eq. (3.12a) an equation for x is obtained. This partial differential equation can be reduced to a system of N ordinary differential equations by multiplying it with 4, and taking the horizontal average after the representation (3.14) has been inserted. After dealing with Eq. (3.12b) analogously we obtain the following

F. H.Busse

88

set of 2N equations

+ +

(d2/dz2- a,2)Wn a,@@, = 0, (d2/dzZ - an2)@, cr,@W, = 0,

n = 1, ..., N ,

(3.15)

where

1

@ ~ F C ( W m 0 , ) + p~ ( W m O , , , ) - ~ W m O m . m

(m

m

Unless indicated otherwise the summation is extended from 1 to N. In addition, we have used our freedom of choice of two normalization conditions by assuming

( Ive(2)= ( I v I ~ ) = 1.

(3.16)

The following general properties of the solutions of (3.15) can be demonstrated : (i) The symmetry of Eqs. (3.15) suggests W, = 0,. This property can be For a comproven by considering the equations for W, + 0,and W, - 0,. plete proof see Busse and Joseph (1972) or Joseph (1976). The problem is thus reduced to

+

(d2/dz2 - an2)@, a,@@, = 0

for n = 1, . .., N .

(3.17)

(ii) The functions 0,are either symmetric or antisymmetric in z. This property originates from the similarity of the problem (3.17) with linear eigenvalue problems. For a proof see the above mentioned references. (iii) It can be assumed that all a, are different. If two alphas are equal, say a, = a,, then O,/O, = const. follows from the following consideration. By multiplying the equation for 0,with 0,and vice versa and subtracting the result we obtain 0,”0, - 02”01 = 0. Integration of this expression and application of the boundary condition 0,= 0 at z = +fleads to

0,’0, - 0,’0, =0

or

(@,/@,)‘

= 0.

Accordingly 0,= c 0 , and the solution of the form (3.14) can be rewritten with N - 1 replacing N and @‘few) = 0 , ( 1 + c2)’/’, a?“)= On+ for n > 1. Using the property a, # a, for n # rn we shall assume a, > a,,, for n > m in the following. (iv) A most important property is expressed by

,

(o,,,’o~) = a,a,(O,,,O,).

(3.18)

Optimum Theory of Turbulence

89

This property can be proven for rn # n by multiplying the nth equation of (3.17) by a; '0, and the rnth equation by a; 'On,subtracting the result and averaging it, which, after partial integration, yields the desired relationship (a,

- a,)/a,a,{(O,'O,')

= 0.

- a,a,(O,O,))

For rn = n (3.18) holds as well but does not follow from Eqs. (3.17). Instead the functional (3.1) must be considered as a functional of the functions 0, and the wavenumbers a,, so that 9(0,,..., 0,; a l , ..., a,; p )

i

]'I[ C I - ' (0,')

J 2 + p COm'-~(OmZ) [m

nt

where J is defined by J

= 1((@:)a;' m

,

(3.19)

m

+ a,(@,')).

(3.20)

Then 9 is minimized with respect to a,. Since the minimizations of F and J with respect to a, are equivalent, relationship (3.18) for rn = n is obtained readily by differentiating (3.20) with respect to a,. The similarity of the solutions 0, with the eigenfunctions of a linear eigenvalue problem suggests that 0, should be a symmetric function of z for odd n and an antisymmetric function for even n. Although this property appears to be borne out by numerical computations it has not yet been proven in general. The asymptotic analysis to be discussed next is not dependent on this property. C. BOUNDARY-LAYER THEORY

In minimizing the functional (3.19) we shall focus our attention on the case of asymptotically high values of p. We regard N as a parameter of the problem and consider the minimum of expression (3.19) with respect to N only after the minimization with respect to all other dependences has been carried out. For simplicity we use as normalization condition for the discussion of this section (3.21) (0,2) = 1.

c

To minimize the functional 9 in the limit of large p, Em0,' must be as close to its average value as is possible in view of the boundary conditions 0,= 0 at z = +f.The drop of Cm0,' toward the boundary value is moderated by the fact that the expression (3.20) for J tends to infinity when ([Em0,' - (Om2)]') tends to zero because ofthe required high values

F . H.Busse

90

- 1/2

-2

FIG. 1. Boundary-layer structure of multi-a-solutions. In the case of convection in a porous medium, W, = Q,.

of one or more of the functions em'. A balance between the two terms within the braces of (3.19) obviously determines the minimum F")(p). In the case N = 1 it is easy to visualize the minimizing solution O1.The dependence of el2 is similar to that shown by the thick line in Fig. 1. Indeed the problem can be solved exactly in this case in terms of elliptic integrals (Busse and Joseph, 1972). The case of N > 1 offers richer possibilities for the solution. Since large derivatives 0,'correspond to high values a,, expression (3.20)can be minimized by using 0,to describe the boundary layer. In order to keep (0,')small, 0,must decay rapidly toward the interior, although not as rapidly as it rises from the boundary value 0,= 0. At the same time 0,- rises from the boundary value 0,- = 0 to ensure 0 , '+ z 1. This process is repeated, as sketched in Fig. 1, until O1 covers the interior of the interval -4 I z I 4. In the following we demonstrate that this heuristic picture does indeed correspond to the asymptotic solutions of the Euler equations of the variational problem. We start by introducing different symbols for the rising and decaying part of the function 0,in the neighborhood of z = -4, for ( z + 3) z O ( P - ~ ~ ) for ( z + 4) = o(p-'*-') 6,([,,)

for n = 1, . . . , N . (3.22)

The procedure at the upper boundary z = 4 is analogous and does not have to be considered separately. It is assumed that the boundary layers scale with powers of p as p tends to infinity. Accordingly the boundary layer coordinates c,, are defined by (3.23)

Optimum Theory of Turbulence

91

We shall assume that 0, differs from zero essentially only in the nth and (n - 1)th boundary layers such that

Gn2+ 6,2+zz 1 for (z + 3) z The functions

for n = 1, . . ., N - 1.

6, and 6, satisfy the boundary @,(O)=O

conditions

an(m)=O

and

(3.24)

(3.25)

and must be matched at their maximum value 1. From the scaling (3.23),

an2= ( O ; ~ ) / ( O , , ~ >

-

m

6;' dC,,//

pr"+r"-l jom

= Prn+rn-l bn2

(1 - 6;- 1) dr,-

0

for n = 2, . . ., N ,

and m

j

a 1 2= pr12

6;' dr, = pr'b12,

(3.26)

0

where (3.21)and (3.24) have been used. Neglecting terms of higher order the boundary-layer approximation of the functional (3.19) can be written in the form

+ 2p1-" jOm (I - 6~')' dcN.

(3.27)

Differentiation of this expression with respect to the exponents r, yields the result that all powers of p in (3.27) must be equal, so that 1 - rN = rN - rN- = ... = r2 - r l

=

rl,

which yields the solution r, = n/(N

+ 1)

for n = 1, . . ., N .

(3.28)

F. H.Busse

92

Accordingly (3.27) can be rewritten in the somewhat simpler form

where

The Euler equations corresponding to a stationary value of the functional (3.29) can be written in the form

6,''+ b,b,+16,, = 0 16~" + bN( 1 - 6 ~ ' ) =60.~

for n = 1, ..., N - 1,

(3.30a) (3.30b)

Equivalently, these equations could be derived by introducing the boundary-layer approximation in Eq. (3.15). The solutions of Eq. (3.30) satisfying condition (3.25) are given by

6, = +sin(b,b,+

l)l/zcn

for 0 I (, I 7~/2(b,b,+~)'/~, (3.31a)

6,= f t a n h [ ( i b ~ / J ) ' / ~ [ ~ ]

for ( N 2 0.

(3.31b)

The endpoint of the interval of c, is the appropriate matching point with the function 6,= f(1 - 6;- ')lj2. We have allowed for both signs of the solutions since only quadratic expressions enter the functional. Antisymmetric functions 0,have opposite signs in the two boundary layers while symmetric functions 0,have the same sign. Using (3.31) in the definition (3.26), the values of b, can be computed. Thus, for n = 1, ..., N - 1, b, = (b,+lb,-l)llz bl = 7~(b~/b,)'/~/2,

(3.32)

bN= 8(bN- /2S)'/'/3.lr which together yield the general expression b, = (7~/2)(8/3~2N1/2)(2"i)/(N+

From the results (3.32) and (3.33) we find

J = 2Nb1,

1)

(3.33)

Optimum Theory of Turbulence

93

and finally

PN)'(p) = p " / ( N + ' ) N (+ N 1)4b12 = x Z N ( N+ 1 ) ( 6 4 , ~ / 9 ~ ~ N ) ' / " + ~(3.34) ). So far we have minimized the functional (3.1) with respect to fields of the form (3.14)with given value of the parameter N . The result (3.34)shows that the minimum F ( p ) among the class of functions {F")(p)) is assumed successively by N = 1, 2, .. ., as p increases. The corresponding upper bound for the convective part of the heat transport in a porous medium is shown in Fig. 2. At finite values of p the conclusions drawn from the boundary-layer analysis are at best tentative and we have distinguished the results for this reason by the superscript from the exact expressions. However, the direct numerical solutions of the Euler equations (3.17) confirm the basic conclusions of the boundary-layer theory as shown by the comparison in Fig. 3 and indicates that the asymptotic theory provides a fair description at finite values of p. The "kinks" in the upper bound curve are also confirmed by a more detailed analysis except that the first derivative is continuous at the transition from F")(,u) to P N + ' ) ( p ) and the discontinuity occurs in the second derivative. The ( N + 1) - a solution branches from the N - a solution at this point in very much the same way as solutions resulting from hydrodynamic instabilities bifurcate from a stationary basic flow. The similarities between the branchings of the optimizing vector fields and the transitions observed in convection are particularly striking. The reader is referred to the discussion in Section VI. A

0

FIG.2. Upper bound for the Nusselt number Nu (= heat transport with convection/heat transport by conduction only) in the case of convection in a porous medium.

94

F. H. Busse

-0.5

-0.4

ZFIG.3. The 2-a-solution at R = 5 0 d : The graph shows numerical computations (solid line) and boundary-layer theory (dashed line) of 0 , and 0,. This figure differs from Fig. 1 of Busse and Joseph (1972) in that an improved numerical approximation has been used.

IV. Bounds on the Transport of Momentum A. EXTREMALIZING VECTORFIELDS The multiple boundary-layer technique developed in the preceding chapter will now be used for the solution of variational problem (2.13). We start by considering the Euler equations for the extremalizing vector field of (2.13), i.e.,

V x (V x V)

+ V7t + w dzd U + ku -

d - U = 0, dz

(4.la)

v.v=o,

(4.lb)

where the expressions d dz

-B

= UW-

(uw) - Ri

and

(4.1~)

R = R ( p ) - ( IV x v I’)/2(UXW)

Optimum Theory of Turbulence

95

have been introduced to demonstrate the similarity of the Euler equations with the basic Navier-Stokes equations (2.6), (2.7), and (2.9) of the problem. The normalization condition (2.14) has been used and IT is the Lagrange multiplier by which the continuity equation has been taken into account. Equation (4.la) represents a “symmetrized” version of Eq. (2.6), d o / & appears in all components of Eq. (4.la), while dU/dz does not appear in the z-component of Eq. (2.6). For this reason the definition (4.1~)is not directly analogous to Eq. (2.9) and l? equals R(p)/2 in the limit p + 0. The properties that time does not enter the variational problem and that nonlinear terms depend only on the z-coordinate are unrealistic features of the Euler equations. The same features, however, allow an analytic solution of the equations for large values of p which is impossible in the case of the equations of motion. In solving Eq. (4.1) we introduce the hypothesis that the minimizing vector field v is independent of the x-coordinate. This hypothesis has been proven in the limit of small p (Busse, 1972c) and the fact that y-independent vector fields lead to much higher values of the functional (2.13) provides additional support for the hypothesis (Busse, 1970b). In view of its importance for the optimum theory of turbulent shear flows it is highly desirable to replace the hypothesis by a rigorous proof. The x-independence of the optimum vector field may be seen in contrast with the fact that the equations of motion (2.6), (2.7), and (2.9) d o not permit a nondecaying x-independent velocity field v. This impression is misleading, however, as a much closer relationship between the optimum vector field and the physically realized velocity field is found in a rotating system (see Section IV,C). An x-independent solenoidal vector field v can be represented in the form u,

= u, = e,

uy

= uy

=

-a+/az,

u,

=

a$/ay.

(4.2)

In the following we shall neglect the term Uyw in the functional (2.13). This is justified because this term makes only a positive contribution to the functional and u i i vanishes for the solutions of the Euler equations derived from the reduced functional. Using the representation (4.2) the dissipation term ( I V x v 1)’ can be written in the form ( I V x v 1 2 ) = (IV~l”+((V2$)’).

(4.31

When the ratio D of the amplitudes of 8 and $ is varied while the product of the amplitudes is kept constant, all terms of the functional (2.13) (in which uyw = 0 is assumed) remain unchanged with the exception of the term (4.3). This term reaches a minimum as a function of D when

( 1 ve 1)’

+ ( ( v ~ $ )=~2() I ve 12)1/2((v2$)2)1/2

(4.4)

96

F . H. Busse

holds. We may thus assume that the minimizing solution of the variational satisfies the property (4.4) and consider the following variational problem in place of (2.13): given p > 0, find the minimum R ( p ) of the functional

among all fields 8(y, z), $(y, z ) that satisfy the boundary conditions

e=$=a$laz=o

a t z = &+

(4.6)

and the condition ( w 8 h ) > 0. We have introduced the function h ( z ) to treat a slightly more general variational problem which can be applied later to the case of channel flow. We assume ((h(z))’) = 1,

= ho.

h(4) = h ( - + )

(4.7)

The case of Couette flow is recovered by specifying h ( z ) = h, = 1. Because the functional (4.5) is homogeneous and of degree zero with respect to 8 and I) we can impose two normalization conditions. We choose

(w eh) = 1, ( w z ) = (ez). (4.8) The form of the functional (4.5) suggests a close similarity of the present variational problem to the problem treated in Section II1,A. The fact that a higher order differential operator occurs in (4.5) and that four boundary conditions must be fulfilled by $ instead of two destroys the symmetry of 8 and w. But the extremalizing fields 8, w are rather similar in all other respects. We consider solutions of the variational problem of the form n= 1

n=l

with +,,(y) satisfying the relationships dZ - + n ( Y ) = -an24n(Y),

(4n4m)

= anm*

dY2 In order to minimize the functional (4.5) in the limit p + 00, must approach the function h ( z ) throughout the interior of the layer. Near the boundary a sharp drop must occur such that the boundary conditions (4.6) become satisfied. The “sharpness” of the drop depends on the balance between the two terms of functional (4.5). The availability of components with different length scales in the Fourier-series representation (4.9)allows one to minimize the functional (4.5) by fields w, 8 with a multiple boundary-layer

Optimum Theory of Turbulence

97

structure. In analogy to (3.221, we assume at the boundary z =

-4

where the boundary layer coordinates c,, are defined by (3.23). Throughout each boundary layer only two components of the representation (4.9) are essentially different from zero such that w,@,

+ w,- ,0,- ,

%

- + iu,- ,0,. ,

it,,@,

%

ho,

for n = 2, . . . , N , (4.11)

is satisfied. In the interior w,0,

= h(z)

(4.12)

must be fulfilled. Because of conditions (4.8) w1 and 0 , have the same order of magnitude in the interior, i.e., s1 = 0, while in the boundary layers different dependences on p must be admitted as shown in expressions (4.10). After taking into account the dependence of the wavenumbers a,, on p, i.e., an2= pqnbn2,

(4.13)

the functional (4.5)can be minimized as a function of the exponents r,,, 4,,,pn, and s, with the result (Busse, 1970b) 1 - 4-" r, = 2 - 4 - N ) ~

qfl=

2 - 4 ' 4-" 2-4-N

9

4-" 2pn=2-4-N'

%=o.

The corresponding p-dependence of the asymptotic minimum l@N)(p) of the functional (4.5) can be written in the form P"(p)

= F ( N ) pl i ( 2 - 4 - N ) .

(4.14)

To obtain an expression for F ( N ) the function R must be minimized with respect to the functions G,,,6,, it,, 6,. Instead of listing the complete set of Euler equations for those functions, we restrict our attention to the equations for the interior dependence, i t , , 6,,which will be needed for a later discussion. These are D - ' b 1 2 i t ,=

with

and

D b 1 2 6 ,= "it,,

(4.15a)

98

F. H.Busse

D denotes the ratio of the second to the first square root in the first term of the right-hand side of (4.5). From Eqs. (4.8) and (4.15a) we conclude D = 1. After the Euler equations are solved in terms of the parameters b,, the boundary-layer approximation of expression (4.5) can be minimized with respect to b,, yielding F ( N ) = 2h0h1(2- 4-N)4Nb12,

(4.16)

where h, is defined by h , z (Ih(z)()/h, = (612)/ho = ( i c 1 2 ) / h o .

The minimizing values of the wavenumber parameters b, are

b,,, = (blh1/44/3/3)1-4-"b 14"

for n = 1, ..., N

-

1,

(4.17)

where 0 = 0.337,

/3 = 0.624

are constants arising from the boundary-layer solutions which have been computed by Howard (1963) and Busse (1969b). We note that the value of /3 is slightly different from that calculated by Busse (1969b), because the integral constraint 3

jm ii11'

dt

+ jw

Qt2

0

0

W

dt

=2

j

(1 -

iis)dt

0

had not been taken into account in that work and its consideration leads to the improved value for 8. Asymptotically, relationship (4.16) yields R(m)(p)

= h;/247/3 (03 /3)1/4p1/2,

(4.18)

which shows the same dependence on p as the solution of the variational problem without the constraint of the continuity equation (Busse, 1969a;see also Howard, 1972). The coefficient of pl/' is lower by a factor 0.38 in the latter case. This close correspondence is another indication that a value lower than (4.18) can hardly be expected if the assumption of x-independent extremalizing vector fields is relaxed. WITH OBSERVATIONS B. COMPARISON

The functions ff("(p) are asymptotic expressions which have been derived under the assumption that p tends to infinity. As in the case of convection in a porous layer discussed earlier, it is reasonable to expect that the expressions (4.14), (4.16) give a fair representation of the exact function R")(p) at

Optimum Theory of Turbulence

99

finite values of p. The absolute minimum among the functions R")(p) exhibits the characteristic property that it is assumed by R"), R(2),and so on, as p increases. Detailed calculations are likely to show that the kinks in minimum R ( p ) correspond actually to discontinuities in the second derivative instead of the first derivative as a literal interpretation of the asymptotic boundary layer results would suggest. But the general character of the curve R ( p ) is not influenced by this distinction. To obtain an upper bound for the momentum transport (2.11) in turbulent Couette flow, the inverse functions jYN)(R)of f i ( N ) ( pmust ) be considered, M

= Re + (fixfi)

IRe

+ ma~,@(~)(Re)].

(4.19)

The comparison of the experimental data with the upper bound (4.19) in Fig. 4 indicates that the upper bound exceeds the measured values by a factor of about 10. It is noteworthy, however, that the measured data parallel the slope of the upper bound. The fact that the momentum transport tends to approach a Re2-law can

FIG.4. The upper bound for the momentum transport M by turbulent Couette flow. The bound for (M/Re) - 1 has been plotted in comparison with the experimental values by Reichardt (1959) for water ( x ) and for air (+ ). The line labeled I indicates the asymptotic bound derived without the constraint of the equation of continuity. (After Busse, 1970b) (Copyright by Cambridge University Press.Reprinted with permission.)

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F. H. Busse

be understood from the following simple argument. The Reynolds number based on the thickness 6d of the viscous sublayer adjacent to the wall in which the momentum is transported by viscous stresses is Re 6. Re/6 is a measure of the dimensionless momentum transport M since the velocity change across the sublayer is of the order Re. Applying a criterion for the instability of laminar flow, we find that the viscous sublayer becomes unstable if Re 6 2 R,,

(4.20)

where R, is a constant of the order lo3.By expressing the sublayer thickness 6 in terms of the momentum transport, 6 z Re/M this criterion can be rewritten in the form Re2 2 R, M.

(4.21)

.Accordingly, the laminar sublayer will be unstable unless the momentum transport grows like Re'. On the other hand, it follows from (4.19)that M cannot grow stronger than Re2 asymptotically. The momentum transport is thus forced to grow like Re2 which appears to be in approximate agreement with the experimental observations in the Couette case. The similarity between the extremalizing vector field and the observed turbulent flow is more pronounced in the comparison of the mean flow. Using the relationship

(4.22) we find that the right-hand side approaches -R(")(p)/4 as p tends to infinity. The mean shear of the extremalizing vector field thus amounts to 4 of the shear of the laminar solution. Surprisingly, this $law appears to be borne out by the measurements shown in Fig. 5. The persistence of a finite mean shear in a fully developed turbulent flow seems to contradict the intuition that the Reynolds stresses wipe out any mean shear except near boundaries. The logarithmic law of the wall which should be particularly appropriate for the constant stress layer realized in turbulent Couette flow is in obvious disagreement with this result. Since the effects of advection and time dependence are neglected in the optimum theory, a much lower level of correspondence with observations must be expected for the fluctuating vector field. Perhaps the most interesting aspect is the dependence of the characteristic length scale on the distance from the wall. The value of N for which expression (4.14)becomes a minimum at a given value of p is asymptotically determined by

101

Optimum Theory of Turbulence

-0.8

. .

.

2 a.

-1

FIG. 5. The mean velocity in plane Couette flow measured by Reichardt (1959) at Re = 1200 (O), Re = 2900 ( x ), Re = 5900 (+), and Re = 34,000 (A). The straight line describes the asymptotic profile corresponding to the extremalizing vector field.

The boundary-layer scales dn and the associated wavenumbers of the extremalizing vector field approach, therefore, asymptotically the relationships d n -= p-'"(b n bZ n + l )-ll3

+ hl/fl4"-'

for n = 1, ..., N - 1,

dN G p - r N( b ~ h ~ ) - -+l ' h1(CJ/fl)''2/f14N+ ~ ',

(4.23)

i.e., the thickness of subsequent boundary layers differ by a factor of 4. Although for a factor of 4 the boundary-layer assumption made in the analysis appears to be not well satisfied, the rather good agreement with numerical results in similar cases suggests that the boundary-layer description remains essentially correct. Figure 6 shows the boundary-layer structure of the extremalizing vector field. As in Prandtl's mixing-length theory, the characteristic scale is proportional to the distance from the wall. But instead of a continuous spectrum of wavelengths, the length scale changes in steps in accordance with the discrete wavenumber spectrum of the extremalizing vector field. C. MOMENTUM TRANSPORT IN A ROTATINGSYSTEM A basic reason for the differences between the fluctuating component of the velocity field in turbulent shear flow and the extremalizingvector fields is the fact that x-independent fluctuating velocity fields always decay in the

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F . H.Busse

s

FIG.6. Sketch of the boundary-layer region of the vector field maximizing the momentum transport.

presence of a mean shear flow in the x-direction (Joseph and Tao, 1963). Moreover, linear stability theory for plane parallel shear flow predicts [Squire’s (1933) theorem] that y-independent rather than x-independent disturbances are the most unstable ones. But these properties are restricted to a nonrotating system. The Coriolis force in a rotating system can release the dynamical constraint which prevents the realization of x-independent fluctuating velocity fields. Taylor vortices between differentially rotating coaxial cylinders are the best-known example for such a release of a dynam-

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ical constraint by the Coriolis force. In the following, this example will be studied in the small gap limit where curvature effects can be neglected. The optimum theory formulated in Section II,A holds in rotating systems as well as in a nonrotating system, since the effects of rotation enter the dynamics of an incompressible fluid only in the form of the Coriolis force, which does not contribute to the energy relationship (2.8). f i a t the upper bound is independent of the rotation rate implies, on the one hand, a general applicability to a wide variety of fluid systems. On the other hand, it emphasizes a shortcoming of the optimum theory-at least in the form in which it has been considered up to this point-in that important dynamical effects like those of the Coriolis force are not taken into account as constraints for the optimizing vector fields. In a particular case mentioned below, however, it is obvious that an additional constraint involving the Coriolis force cannot improve the upper bound. We consider the equivalent of Eqs. (2.6) and (2.7) in a rotating system. In the limit when ir is sufficiently small such that nonlinear terms can be neglected, we find

VzS - V(p - p) = 2S2 x v

+U

*

a

a

az

at

VS + $ - U + - S,

V.S=O.

(4.24) (4.25)

In order to eliminate the continuity equation (4.25),we introduce the general representation for the solenoidal field v:

S=V

x

(V x k 4 ) + V

x

k$,

(4.26)

where k is the unit vector in the z-direction normal to the boundaries. The laminar solution of the problem,

(4.27)

U = -Re iz,

is the same as in a nonrotating system if the z-component of &2vanishes, so that fi = n,i

+ 0,j.

Restricting our attention to the case of x-independent stationary solutions of Eq. (4.24), we obtain, after operating with k . V x (V x ) and k V x on (4.24),

az4 - m v -aYz 4

a3 -*=o,

y ay3

a2

V2 7t+b - (Re - 2Qy) aY

a3 ~

aY3

4 = 0.

(4.28)

F. H.Busse

104 The elimination of $ ytelds

V6 - 2R,(Re The solution of this equation, together with the corresponding boundary conditions a 4=-+=V4+=0 atz=

*+,

a2

has first been obtained in the context of the onset of convection in a layer heated from below (Pellew and Southwell, 1940). The minimum Reynolds number for the onset of x-independent disturbances is given by

+ --1708

(4.29) 2% It corresponds to a wavenumber a, = 3.117 of the y-dependence exp{iay}.As a function of R,, the lowest value of expression (4.29) is reached for Re, = 2R,

2R, = (1708)”,.

(4.30)

At this point the value of Re, becomes equal with the minimum value R ( p ) for which a solution of the problem (4.1)exists. Indeed, when the representation (4.26) for the vector field v is used, it can easily be verified (Busse, 1970a) that the problem (4.7)becomes identical to problem (4.28)in the limit p = 0. This implies that in the case (4.30)the instability must occur in the form of infinitesimal x-independent disturbances and that the possibility of subcritical finite-amplitude instability can be excluded. This particular case in which the stability problem is completely solved and in which the optimum problem coincides with the physical state in the limit p = 0, is realized experimentally between two coaxial cylinders with radii R 1 and R,, and angular velocities R1,R, if the small gap limited ( R , - R 1 ) / ( R 2 R , ) 4 1 is approached such that

+

-4- R , - R1 +R, R,+Rl is satisfied. Since the x-component of does not enter the relationship, other experimental realizations can be obtained by an additional relative motion of the cylinders in the axial direction, as in the experiments by Ludwieg (1964) and Nagib (1972). These experiments clearly demonstrate the destabilizing role of the Coriolis force and the Occurrence of the longitudinal vortex in a wide variety of flow configurations. The optimum theory for turbulent flow between coaxial differentially rotating cylinders has been considered for arbitrary radius ratios in the papers

a1 R1

Optimum Theory of Turbulence

105

by Nickerson (1969) and Busse (1972a). The optimum theory indicates a significant variation of the mean flow profile as a function of the parameters of the problem. Because of the scarcity of experimental measurements of the turbulent mean flow, it has not yet been possible to realize the interesting possibilities for a comparison between optimum theory and observations in this case.

V. Bounds on the Transport of Mass A. TURBULENT CHANNEL FLOW Next to Couette flow, the flow driven by a constant pressure gradient in a channel defined by two infinite parallel plates represents the simplest case of fluid motion. The application of the optimum theory to this system can be accomplished by a slight modification of the discussion in Section 11. Assuming that the imposed pressure gradient is directed in the negative x-direction the energy balance becomes

( I v x v* 1)’

+ ( I ii3 - (a*) 1)’

- A,( ri,$12) = 0,

(54 in place of relationship (2.10). Using the average flow in the x-direction for the definition of the Reynolds number Re, the relationship between Re and the magnitude A, of the gradient can be written in the form Re = (U

(

) 1:

i) = - zz : U i = - A, - (li,3z).

(5-2)

The following variational problem provides a lower bound for Re(12)”’ at a given value p of ( ~ ~ , I X J Z ( I ~ ) ~ / ’ ) : Given p 2 0, find the minimum R ( p ) of the functional

among all bounded vector fields v that satisfy (2.12b) and (2.12~)and have (ii, 3z(12)’/’) > 0. In anticipation that (uw) vanishes for the extremalizing solution, this term has been neglected in the definition of the functional. Since R ( p ) is a monotonically increasing function of p, it is equivalent to say that R ( p ) provides a lower bound for Re(12)”’ at a given value p of ( ~ ~ w z ( l 2 ) ’ ~or’ )at a given value ( p + ~(p))(12)’/’of A,. The transport of mass through a channel at given pressure head is thus bounded from below by R ( p ) , while an upper bound is given by A,/12 since (u, w z ) is positive according to the energy balance (5.1).

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F. H. Busse

The factor 12l” has been introduced in the definition (5.3) in order to apply the analysis of Section IV directly. Following through the steps described by (4.2)and (4.4),the functional (5.3) can be-transformed into the form (4.5) with h ( z ) = z(12)’/2,

and the results (4.16), (4.17),and (4.18)can be applied directly to the present case by using

h , = 4. A more detailed discussion is given by Busse (1970b). Asymptotically the optimizing solution satisfies Prandtl’s wall proximity law which states that the structure of the turbulent flow becomes independent of the Reynolds number if the friction velocity U , and the length U , /v are used as scales for the velocity and the distance from the wall. In the dimensionless units of this paper, the friction velocity Ur is defined by ho = 3112,

where the last equality is approached asymptotically for fully developed turbulence when the transport of momentum is carried almost entirely by the Reynolds stresses. Among the features which have been found to be independent of the Reynolds number in experimental observations of turbulent channel flow is the ratio between the rms value of the x-component of the fluctuating velocity field and the mean flow close to the boundary. Laufer (1951) found 0.18 for this ratio, a more recent value given by Eckelmann (1974) is 0.24. In the optimum theory, this ratio becomes asymptotically

/(3)1’2p% 0.49, C=O

(5.4)

where the value of a@/dtI,=, has been taken from Howard’s (1963) boundary-layer calculation. The result that the value (5.4)is more than twice the observed value is not surprising if it is remembered that the optimum vector field is much more efficient in transporting momentum than the physically realized turbulent velocity field and accordingly grows more rapidly away from this boundary. That the structure of the optimizing vector field is approached qualitatively by the realized turbulent flow is evident from the observations of the longitudinal eddies in the viscous sublayer by

Optimum Theory of Turbulence

107

Kline et al. (1967)and Gupta et al. (1971).Again, the observed eddies are not as close to the wall as the Nth eddy in optimizing solution and thus the observed wavelength A+ =loo based on the friction velocity scale is more than twice as large as the wavelength

The mean profile corresponding to the optimizing vector field exhibits a finite shear as in the Couette case. The measured profiles are slightly flatter toward the center of the channel than the optimal profile shown in Fig. 7. The traditional theoretical description of the mean-flow profile of turbulent shear flow is based on the assumption that the turbulent flow outside the viscous sublayer becomes independent of the viscosity and that the only relevant length scale is the distance from the wall. This leads to the universal logarithmic velocity profile connecting the viscously dominated shear region near the wall and the region outside where the assumption of a constant momentum transport is no longer valid. The optimizing vector field does not conform to the assumption of a single length scale determining the structure of the profile. The presence of the other wall is always felt and the boundarylayer part of the mean profile (not shown in Fig. 7) is proportional to the

FIG. 7. Asymptotic profiles of the mean flow corresponding to the extremalizing vector field in the case of channel flow (a) and pipe flow (b).

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F. H. Busse

inverse of the distance from the wall rather than depending on it logarithmically. The optimal profile obviously does not fit the observations as well as the suitably adjusted logarithmic profile, although some of the discrepancy may disappear if the boundary-layer theory is replaced by a more accurate numerical solution. The fact, however, that the optimizing vector field does not obey the scaling assumption, together with the evidence that the logarithmic distribution seems to fail in the Couette case, which represents its most appropriate application, casts some doubts on the universal validity of the scaling assumption. B. TURBULENT PIPEFLOW Turbulent flow in circular pipes is easier to realize experimentally than turbulent channel flow. The optimum theory on the other hand becomes somewhat more difficult for a cylindrical geometry because only a discrete spectrum of wavenumbers in the azimuthal direction is available instead of the continuously varying wavenumbers in the case of the channel. A consequence of this property is that the extremalizing solution is not quite independent of the x-coordinate along the axis of the pipe, as is evident from the solution by Joseph and Carmi (1969) in the energy stability limit p = 0. Because the&-dependence is very small, we shall neglect it in the discussion of the extremalizing vector field. Since the results of the optimum theory depend primarily on the boundary-layer structure which is characterized by relatively high values of the wavenumbers, the fact that the parameters @n assume only integer values should not have an appreciable effect. We thus proceed with the application of the theory developed in Section IV. A more exact treatment of the problem has been given by Busse (1972b) in which the property has been taken into account that the interior wavenumber a1 assumes only integer values. The equations corresponding to (5.1) and (5.2) in the case of channel flow are

where a cylindrical system of coordinates (r, 4, x ) has been assumed. The bar indicates the average over surfaces r = const. and the angular brackets denote the average over the infinite volume of the cylindrical pipe whose radius has been set equal to 1. A, is the component of the mean pressure gradient in the negative x-direction. The following variational problem provides a lower bound for 81/2 Re at a given value p of ( 1 i 5 ~ 4 ) 2 ~ / ’ .

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109

Given p 2 0, find the minimum R ( p ) of the functional

among all bounded vector fields v that vanish at r = 1, satisfy V . v = 0 and have ( u , u, r(2)’/’) > 0. As in the case of channel flow, R ( p )provides a lower bound for 8’12 Re at a given value of A, = 8’/’(p R ( p ) ) as well as at a given value of p. Using the definitions

+

the functional (5.8) can be transformed into the form (4.5) by replacing h(z) with r(2)’/’ and the results (4.16),(4.17),and (4.18)can be applied to the case of pipe flow by specifying

ho = 2’12,

h , = 2.

The relationship between the optimizing vector field and the observed turbulence in pipe flow is essentially the same as in the case of channel flow. The discrepancy between the lower bound R ( p ) and the realized value of Re(8)’12 increases as Re increases. Measurements by Nikuradse (1932) give (see Hinze, 1959)

while the asymptotic result (4.18) yields the lower bound

Because of the circular cross section of the pipe the mean profile of the optimizing vector field differs slightly from the profile for channel flow as shown in Fig. 7. The property that the curvature of the mean profile in the case of pipe flow is higher than in the case of a channel flow at the same maximum velocity agrees with the experimental findings. For the comparison with the observed profile, a more accurate description of the optimal profile is desirable which would take into account the boundary-layer contributions. But this will require a numerical solution of the problem. The structure of optimizing vector field is in many respects similar to the observed structure of turbulent pipe flow. In particular, the observation made by Laufer (1954) that the rate of energy production at a point is approximately balanced by the rate of energy dissipation is reflected by the Euler equations because the advection of energy does not enter the optimum

F. H.Busse

110

problem. The shape of the observed rms velocity distributions is similar to that exhibited by the optimum theory (Busse, 1970b).Another feature of the optimum theory, namely, the equality of the energies dissipated by the mean flow and by the fluctuating motion, is also approximately shown by the observed turbulence. The maximum of the energy production was found by Laufer at a distance (1 - r ) o , zz 11.5 from the wall. Because of its higher transport efficiency the optimizing vector field exhibits this maximum at (1 - r ) o , z 1.70 * (/3/0)’/~ * 4ll3 = 3.15. The number 1.70 represents the value of 5 where QW(l - O W ) reaches its maximum in Howard’s (1963) boundary-layer analysis. Asymptotically this point represents the distance at which the laminar shearing stress is equal to the turbulent shearing stress. We conclude that mainly because of the low correlation coefficient ~ / ( ~ ) 1the” realized , turbulent flow differs in all quantitative aspects from the optimizing vector field. The qualitative similarity between the two fields suggests, however, that the realized velocity field tends to approach optimal transport properties. VI. Bounds on the Transport of Heat

A.

CONVECTION IN A

LAYERHEATED FROM BELOW

The closest correspondence between the optimum theory and the physically realized turbulent flow has been found in the case of thermal convection. In contrast to shear flow turbulence in a nonrotating system the onset of convection is not delayed by dynamical constraints and a gradual development of turbulence characterizes convective systems. In the problem of convection in a layer heated from below the energy-stability limit coincides with the actual onset of convection, and in the special case of stress-free boundary conditions the upper bound for the heat transport equals the physically realized heat transport in the limit of small convection amplitudes. The nondimensional Boussinesq equations for convection in a layer heated from below are identical with Eqs. (3.3) and (3.4) i f f in (3.3a) is replaced by -V2f and the Darcy permeability coefficient K is set equal to d2. In this case the parameter B equals the inverse of the Prandtl number Pr = V / K Accordingly the integral dissipation relationships

( (VfIZ) = ($0)

(6.la)

($8))’)

(6.1b)

and

(lV812)

+ (($8-

= Ra(48)

Optimum Theory of Turbulence

111

are obtained in place of expressions (3.9) and (3.10). Based on relationships (6.1), the variational problem can be formulated which provides an upper bound p on the heat transport by convection ($8) at a given value R ( p ) of the Rayleigh number or, equivalently, a lower bound R ( p ) for the Rayleigh number at a given value p of the heat transport ($8). Since the latter possibility is mathematically more convenient, it will be used here: Given p 2 0, find R ( p ) , the minimum of the functional

among all bounded fields v, 0 that vanish at z = *iand satisfy the equation of continuity, V . v = 0. We have assumed the case of rigid boundaries which is realized in most experimental studies. The case of stress-free boundaries has been considered by Straus (1973, 1976a). The solution of the variational problem (6.2) without the constraint of the continuity equation has been derived by Howard (1963) in analytical form. In the same paper the problem is also solved under the assumption of a single horizontal wavenumber. The technique of multiple boundary layers was developed in connection with problem (6.2) by Busse (1969b). Since the analysis is analogous to those described in Sections II1,B and IV,A, it will not be repeated here. The main result is the asymptotic minimum for the N-a-solution of the Euler equations, where b, is given by 1 4-~)-2. b4(3 - 4 - N , = 4 - 6N(./p)3 (8443)4( 1- 4 - ~ ) ( 1

(6.4)

As in previously discussed cases the absolute minimum R ( p ) is given, one after another, by the functions R")(p) starting with N = 1. The predictions of the boundary-layer theory have been verified by the numerical computations of Straus (1976b). Since some higher order terms have been neglected in the boundary-layer approximation R")(p) for R("(p), the exact upper bound p ( R )shown in Fig. 8 is lower than predicted by the asymptotic theory. The transitions from N to N + 1 in the bounding curve are in good agreement, however. The transitions or "kinks" in the function p ( R ) are a prominent feature of the optimum theory which has a direct physical counterpart in turbulent convection. Schmidt and Saunders (1938) were the first to report a transition in the heat-transport curve beyond the initial transition corresponding to the onset of convection. Malkus (1954a) found six transitions in a thermal decay experiment which is known for its low noise. Since then, transitions

112

F . H.Busse

Ra FIG.8. Upper bound for the Nusselt number Nu for convection in a layer heated from below. The graph shows numerical computations (solid, after Straus, 1976b), and asymptotic results (dashed). (a) and (b) are single-a bounds, (c) and (d) are two-a bounds.

have been found by many experimenters, although the reality of some transitions is doubtful, while others may depend on the Prandtl number of the convection fluid (Willis and Deardorff, 1967; Chu and Goldstein, 1973). The combination of visual observations and heat-flow measurements employed by Krishnamurti (1970) has shown clearly that the transition in the Rayleigh-number range of 1.7 x lo4 to 2.2 x lo4 is caused by the onset of bimodal convection (Busse, 1967; Busse and Whitehead, 1971). Since, in addition to the basic convection rolls, a second system of rolls of higher wavenumber occurs in this case in the thermal boundary layers, bimodal convection is directly analogous to the 2-a-solution of the optimum theory. In the case of the next higher transition observed at a Rayleigh number of about 6 x lo4, a direct analogy is not easily seen, even though the upperbound transition from N = 2 to N = 3 occurs nearly at the same Rayleigh number, namely at R = 6.3 x lo4. Experimenters identify this transition usually with the onset of time-dependent convection; but the Prandtl number dependence expected for such a transition has not been observed. According to the optimum theory a velocity field with a discrete spectrum of the horizontal wavenumber is more efficient in transporting heat than a velocity field with a continuous spectrum. Indications of a discrete spectrum in turbulent convection were found by Deardoff and Willis (1967), but their measurements could not be reproduced in the more recent study of Fitzjarrald (1976). It is difficult to find evidence for a discrete wavenumber spectrum because the normal experimental procedure determines the spectrum of the wavenumber component in a single horizontal direction, while discrete values can be expected only for the total horizontal wavenumber.

Optimum Theory of Turbulence

113

New evidence for several discrete wavenumbers present in the spectrum of convection at Rayleigh numbers of the order of lo5 has been obtained with an optical method by Parsapour (1977). As predicted by the optimum theory the spectrum changes abruptly at the transition points of the heat flux curve, but between these points the wavenumbers tend to decrease rather than increase with increasing Rayleigh number as suggested by the theory. It should be kept in mind that the optimum theory is not capable of giving a detailed picture of the physically realized turbulence especially since a constraint involving the time dependence has not yet been taken into account. But the remarkable similarity between the optimizing vector field and the realized turbulent convection must be interpreted as the tendency of the latter to approach a maximum heat transport. The discrete wavenumbers and discrete transitions observed in turbulent convection provide an important reminder that the commonly acknowledged tendency toward randomness represents only one aspect of fully developed turbulence. On the smooth background discrete peaks seem to persist in the wavenumber spectrum of turbulent transport processes even though they may not always be easily recognizable. Statistical theories of turbulence emphasize the continuous properties of the spectrum, while the importance of discrete wavenumbers is stressed in the optimum theory. B. THEINFINITEPRANDTL NUMBER LIMIT

In the limit when p tends to infinity, the minimum among the expressions (6.3) approaches R‘“’(p) = 3 c ~ ( p ’ f l 4 ~ ~ ’= ~ ) 1’ ’0~. 1 6 ~ ” ~ , (6.5) which implies that the upper bound for the Nusselt number (= total heat transport/heat transport in the case of conduction alone) grows as Ra’” Experimental determinations of the Nusselt number give results scattered around 0.2 . RaO.’* for fluids of moderate Prandtl numbers. Simple dimensional arguments suggest 4 for the exponent. This result is convenient for practical applications, but the dimensional argument are not reliable since they give the wrong answer when applied to the optimum problem. The &power law is in rough agreement with Kraichnan’s (1962) prediction for the heat transport by turbulent convection based on mixing-length arguments. Since the expression [Ra/(ln Ra)3]’’Z derived by Kraichnan for the Nusselt number depends on the hydrodynamic instability of the thermal boundary layer, the expression becomes valid only at Rayleigh numbers in excess of lozo,which is far beyond the present experimental capabilities. The discrepancy between the bound (6.5)and the observed power law has led to a search for additional constraints by which the upper bound for the

114

F. H. Busse

heat transport could be improved. Such a constraint is readily available if the limit of infinite Prandtl number is considered. Since the nonlinear terms in the Boussinesq equation of motion are multiplied by Pr-', the equation of motion becomes linear for infinite Prandtl numbers 0 = -VR

+ k0 + V'V.

(6.6) Solutions of the variational problem (6.2) incorporating the additional constraint (6.6) have been obtained by Chan (1971). In spite of the increased complexity of the problem, the multi-a-solutions of the Euler equations can still be obtained by the multiple boundary-layer method described in Section II1,B. As asymptotic upper bound for the convective heat transport at infinite Prandtl number, Chan obtained p ( R ) 0.152R4I3. (6.7) This expression is remarkably close to the experimentally measured dependence, but it must be remembered that all experiments are carried out at a finite Prandtl number. From numerical solutions of two-dimensional convection (Moore and Weiss, 1973), it is known that the heat transport tends to reach its maximum at Prandtl numbers of the order unity and that the infinite Prandtl-number limit is approached only in the regime Pr > (Ra/RaJ2I3. The result (6.7) is also close to the prediction of the meanfield theory of convection (Herring, 1964) in which all nonlinear terms are neglected in the Boussinesq equations with the exception of those affecting the horizontal mean of the temperature field. The mean-field approximation thus becomes similar to the optimum problem in the limit of infinite Prandtl number and Chan (1971) has shown that the heat transport becomes actually identical in the asymptotic limit of R tending to infinity if the maximizing values of the wavenumbers a, are used. A numerical single-a solution for Chan's problem has been obtained by Straus (1976b). Both numerical and asymptotic solutions have been obtained by Gupta and Joseph (1973) in the analogous optimum problem for convection in porous medium. The infinite Prandtl-number limit has the advantage that the effects of rotation on the upper bound for the heat transport can be considered. The rotation rate R enters the nondimensional description of the problem in the form of the Taylor number T = 4R2d4/v2. Since in physical situations the infinite Prandtl number limit is approached in the form v -+ 00 rather than K -+ 0, a finite and possibly large Taylor number is difficult to achieve in the limit when V/K tends to infinity. The optimum problem in this limit is of mathematical interest, however, because it can no longer be assumed that the vertical vorticity vanishes for the optimizing solution. The Taylor number parameter and the toroidal component of the trial vector field induced by the finite vertical vorticity increase the complexity of the vari-

Optimum Theory of Turbulence

115

ational problem significantly. We refer to the original papers on this subject by Chan (1974), Hunter and Riahi (1975), and Riahi (1977). Another problem for which the infinite Prandtl-number limit is of interest is the problem of doubly diffusive convection, which also is known under the name thermohaline convection after its major physical application. The addition of a second diffusiveprocess affectingthe density can lead to interesting phenomena as for example salt-finger convection which is caused by a destabilizing salt concentration gradient in the presence of a stable thermal stratification. In applying the optimum theory to the problem of doubly diffusive convection, two cases may be distinguished. The salt-finger phenomenon is an example of convection in the case when the low diffusivity is associated with the destabilizing component of the density variation. Straus (1974) analyzed this problem in the infinite Prandtl-number limit. The case when the less diffusive component of the density variation has a stabilizing influence has been considered by Lindberg (1971), who used inequalities to obtain upper bounds on the heat and solute fluxes without actually solving a variational problem.

VII. General Discussion

A. FURTHER APPLICATIONSOF THE OPTIMUM THEORY In the problems considered in the preceding sections a straightforward physical interpretation can be given for the various terms of the energyintegral relationship. In particular, it has been possible to derive bounds for transport quantities from variational problems that arise naturally from the energy balances. In general, intermediate steps will be required. The difficulty of the interpretation of the energy-production term becomes apparent when the case of combined Couette and Poiseuille flow is considered (Busse, 1969a) Neither an upper bound for the momentum transport nor a lower bound for the mean flow can be obtained directly in this case. Instead, a combination of the applied pressure gradient and the applied shear becomes extremalized when a variational problem is formulated in analogy to (2.13). More useful results can be obtained only if additional inequalities are applied. The problem of interpretation becomes amplified when more complicated flow processes are considered. We have already emphasized that the application of the optimum theory in its simplest form is limited to those cases of fluid flow for which a laminar solution exists that depends on a single coordinate only. Turbulent jets and turbulent boundary layers cannot be

F. H.Busse

116

easily attacked by variational methods unless restrictive assumptions are introduced. It is of interest to consider at this point an example of a flow between parallel planes which is quite different from Poiseuille or Couette flow. The application of the optimum theory in this case suggests a new way in which the realized turbulence may tend towards an extremum and a comparison with experimental observations will eventually be of considerable importance. Using the nondimensional description introduced in Section II,A, the flow given by UI = iG(z/4 - z3)/6 (74 is physically realizable in a channel inclined with the angle 4 with respect to the horizontal plane when two different temperatures, TI and T,, are prescribed on the bounding plates. Assuming the limit of vanishing Prandtl number, the problem can be considered on the basis of the hydrodynamic equations (2.1) alone if the effects of gravity are taken into account. The deviations of the temperature from the basic linear profile are vanishingly small as long as thermal conduction dominates over convection effects. The solution (7.1) is stable when the Grasshof number G = yg sin 4 I T, Tl ld3/vZ is sufficiently small (Ayyaswamy, 1974). For large values of G, a turbulent flow must be expected. In order to avoid the distinction of whether the upper or lower plate is heated, the convention is used that the unit vector i indicating the x-axis points upward in the latter and downward in the former case. The mean shear in the case of statistically stationary turbulence is determined by dU/dz = iG( 1/24 - z2/2) + - (a+), (74 and the energy balance for the fluctuating velocity field can be written in analogy to (2.10)

a

0 = (IV x SI')

+ (1113 - (a+) 1)'

- G(u*,+(1/24 - z2/2)).

(7.3) A physical interpretation of the last term in this equation is obtained when the y-component of the mean angular-momentum density of the layer with respect to a point on the median plane is considered. This component is

where h ( z ) = (180)'/2(1/12 - z 2 ) .

(7.5)

Since the last term in (7.4) is positive according to relationship'(7.3), the angular-momentum density of the system in the turbulent state is smaller than in the laminar state at a given value of G, i.e., at a given torque exerted

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by gravity. A lower bound for M y may be obtained from the variational problem: Given p > 0, find the minimum R(p) of the functional

(IV x vl’)

+

4 = ~(u,wh)

( I u w - ( u w ) - ih(u,wh)I’) (u, wh)’

(7.6)

among all vector fields v that satisfy (2.12b), (2.12~)and have ( u , wh) > 0. When the normalization condition (7.7) and the property ( h 2 ) = 1 are used, the extremalizing vector field satisfies relationship (7.3) if G is eliminated by means of Eq. (7.4) and M yis replaced by (720)’”R. Thus R(p) provides a lower bound for MY/(720)”’ at a given value of p or, equivalently, at a given value of G. As in the case of the variational problem (2.13), the extremalizing vector field of the functional (7.6) has the property Uyw = 0. In addition, the property (u, w) = 0 can be anticipated, in which case the functional becomes identical to expression (4.5), after the representation (4.2) has been introduced for v and relationship (4.4) has been used. Accordingly, the analysis of Section IV,A is directly applicable and the results (4.16), (4.17), and (4.18) hold in the present case with P = (uxwh)

There appear to be no experimental data available for the turbulent flow in an inclined heated layer at a sufficiently low Prandtl number. The hypothesis that the realized turbulence tends to minimize the angular momentum density thus has not yet gained observational support. Because of its unusual properties, an experimental investigation of the problem is desirable. There are numerous possibilities for turbulent systems with physical properties depending on a single coordinate like those considered in this paper. The effects of a homogeneous magnetic field have not yet been considered and the possibility of the generation of magnetic fields by turbulent motions leads to other interesting applications of the optimum theory. A first step in the latter direction has been made by Kennett (1974). But the scarcity of observational data in those cases restricts the possibilities for the interpretation of the realized turbulence in terms of the optimum theory. B. EXTENSIONS The major task in the further development of the optimum theory of turbulence is the introduction of additional constraints in order to improve the bounds. Although it is in principle possible to approach the actual solution of the Navier-Stokes equations with the optimum transport by

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adding the hierarchy of moments of the equations as constraints, in practical applications it has proven to be rather difficult to proceed beyond the energetic constraints considered in this article. An attempt has been made in the case of Couette flow (Ayyaswamy and Busse, 1978),but the results have not been very encouraging. By separating the fluctuating velocity field v into its “toroidal” and “poloidal” parts, so that v = V x k+

+ V x (V x k4) = E+ + 154,

where k is the unit vector normal to the plates, and by multiplying the equation of motion (2.6) by V x k+ and by V x (V x k#), two separate energy-integral equations are obtained in place of the single relationship (2.10):

Because of the triple product terms in these relationships, the Euler equations of the corresponding variational problem can no longer be solved by analytical methods. Numerical methods on the other hand are limited to a small range of Reynolds numbers because of the computational expense associated with the three-dimensional representation of the extremalizing vector field. A different aspect of the optimum theory that appears to be promising for future applications is evident from the fact that the variational problem and the extremalizing vector fields are surprisingly similar for different physical situations,’if the appropriate variables are compared. The analogous role of the fluctuating temperature 8 in the problem of convection and of the fluctuating velocity component u, in shear flow turbulence appears to be well confirmed by observations. In Fig. 9 these two variables and the corresponding normal velocity components are shown as functions of the distance from the walls without the use of an adjustable parameter. The similarity suggested by the optimum problem is quite strikingly borne out within the scatter of the measured data. Deardorff (1970) has made use of this property for the scaling of the unstable planetary boundary layer. Similar analogies can be expected in other problems of turbulence and the formulation of a general similarity law for the turbulent transport from rigid walls appears to be feasible. It is appropriate to close this article with the preceding remark since it illustrates the power of the optimum theory as an analytical tool in the interpretation of the structure of turbulent flow. In this respect the know-

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FIG.9. Root mean square (rms.) values of the fluctuating component of the velocity in streamwise direction, ijx/Ur, and normal to the wall, measured by Laufer (1954) at Re = 2.5 x lo4 (+ )and Re = 2.5 x lo5 ( x ). For comparison the rrns values of the temperature fluctuations 0 and of the vertical velocity component w, which were measured in turbulent thermal convection by Deardorff and Willis (1970) at Ra = 2.5 x lo6 ( 0 )and Ra = 1.0 x lo7 (O), are plotted in units resulting from the correspondence of the variational problems. (After Busse, 1970b). (Copyright by Cambridge University Press. Reprinted with permission.)

w/oz,

ledge gained from the optimum theory is complementary to the results of statistical theories of turbulence. Since the latter are developed in the ideal limit of homogeneous isotropic turbulence in which spatial structures disappear, they tend to emphasize random properties of turbulence. The combination of the two approaches may ultimately provide a satisfactory description of physically realized turbulent flows. REFERENCES AYYASWAMY, P. S. (1974). On the stability of plane parallel flow between differentially heated tilted planes. J. Appl. Mech. 41, 554-556. AYYASWAMY, P. S., and BUSSE,F. H.(1978). Improved bounds for turbulent Couette flow. To be submitted. BUSSE,F. H.(1967). On the stability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46, 140-150. BUSSE,F. H.(1968). Eine neuartige Methode zur theoretischen Behandlung turbulenter Transportvorgange. Z . Angew. Math. Mech. 48,T187-Tl90. BUSSE, F. H. (1969a). Bounds on the transport of mass and momentum by turbulent flow between parallel plates. J. Appl. Math. Phys. ( Z A M P ) 20, 1-14.

F. H . Busse BUSSE,F. H. (1969b). On Howard’s upper bound for heat transport by turbulent convection. J. Fluid Mech. 37,457-477. BUSSE,F. H. (1970a). Uber notwendige und hinreichende Kriterien fur die Stabilitat von S t r a mungen. 2.Angew. Math. Mech. 50, T173-T174. B u w , F. H. (1970b). Bounds for turbulent shear flow. J. Fluid Mech. 41,219-240. BUSSE,F. H. (1972a). The bounding theory of turbulence and its physical significancein the case of turbulent Couette flow. Springer Lect. Notes Phys. 12, 103-126. B u w , F. H. (1972b).The bounding theory of turbulence and its physical significancein the case of pipe flow. Symp. Math. 9,493-505. BUSSE,F. H. (1972~).A property of the energy stability limit for plane parallel shear flow. Arch. Ration. Mech. Anal. 47,28-35. BUSSE,F. H., and JOSEPH,D. D. (1972). Bounds for heat transport in a porous layer. J. Fluid Mech. 54, 521-543. BUSSE,F. H., and WHITEHEAD, J. A. (1971). Instab es of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305-320. CHAN,S.-K. (1971). Infinite Prandtl number turbulent convection. Stud. Appl. Math. 9413-49. CHAN,S.-K. (1974).Investigation of turbulent convection under a rotational constraint. 1.Fluid Mech. 64,477-506. CHU,T. Y., and GOLDSTEIN, R. J. (1973).Turbulent convection in a horizontal layer of water. J. Fluid Mech. 60, 141-159. DEARDORFF, J. W. (1970).Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211-1213. DEARWRFF, J. W., and W ~ L I SG. , E. (1967). Investigations of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28, 675-704. ECKELMANN, H. (1972). The structure of the viscous sublayer and the adjacent wall regon in a turbulent channel flow. J. Fluid Mech. 65,439-459. FITZJARRALD, D. E. (1976).An experimental study of turbulent convection in air. J. Fluid Mech. 13,693-719. GUPTA,V. P., and JOSEPH,D. D. (1973). Bounds for heat transport in a porous layer. J. Fluid Mech. 57,491-514. GUPTA,A. H., LAUFER, J., and KAPLAN,R. E.(1971). Spatial structure in the viscous sublayer. J. Fluid Mech. 50, 493-512. HERRING,J. R. (1964). Investigation of problems in thermal convection: Rigid boundaries. J. Atmos. Sci. 21,277-290. HINZE,J. 0. (1959). “Turbulence.” McGraw-Hill, New York. HOWARD,L. N. (1963). Heat transport by turbulent convection. J. Fluid Mech. 17,405-432. HOWARD,L. N. (1972). Bounds on flow quantities. Ann. Rev. Fluid Mech. 4, 473-494. HUNTER,C., and RIAHI,N. (1975). Nonlinear convection in a rotating fluid. J. Fluid Mech. 72, 433-454. JOSEPH,D. D. (1976). “Stability of Fluid Motions.” 2 vols. Springer, Berlin, Heidelberg, New

York. JOSEPH, D. D., and CMI, S. (1969). Stability of Poiseuille flow in pipes, annuli, and channels. Quart. Appl. Math. 26, 575. JOSEPH,D. D., and TAO, L. N. (1963). Transverse velocity components in fully developed unsteady flows. J. Appl. Mech. 30, 147-148. KENNETT,R. G. (1974). Convectively-driven dynamos. I n “Notes of the Geophysical Fluid Dynamics Summer Program.” Woods Hole Oceanogr. Inst. Rep. 74-63,94117. KLINE,S. J., REYNOLDS, W. C., SCHRAUB, F. A., and RUNSTADLER,P. W. (1967).The structure of turbulent boundary layers. J. Fluid Mech. 30,741-773. KRAICHNAN, R. H. (1962). Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 13741389,

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KRISHNAMURTI, R. (1970). On the transition to turbulent convection. I. The transition from two to threedimensional flow. J. Fluid Mech. 42, 295-307. LAUFER,J. (1951). Investigation of turbulent flow in a two-dimensional channel. N A C A Rep. 1053.

LAUFER, J . (1954). The structure of turbulence in fully developed pipe flow. N A C A Rep. I 1 74. LINDBERG, W. R. (1971). An upper bound on transport processes in turbulent thermohaline convection. J. Phys. Oceanogr. 1, 187-195. LUDWIEG,H. (1964). Experimentelle Nachpriifung der Stabilitats-Theorien fur reibungsfreie Stromungen mit schrauben-linienformigen Stromlinien. Proc. 4th Int. Congr. Appl. Mech., 1045- 105 1. MALKUS, W. V. R. (1954a). Discrete transitions in turbulent convection. Proc. R. SOC.London, Ser. A 225, 185-195. MALKUS,W. V. R. (1954b). The heat transport and spectrum of thermal turbulence. Proc. R. SOC.London, Ser. A 225, 196-212. MOORE,D. R., and WEISS,N. 0.(1973). Two-dimensional Rayleigh-Binard convection. J. Fluid Mech. % ! , 289-312. NAGIB,H. M. (1972). On instabilities and secondary motions in swirling flows through annuli. Ph.D. Dissertation, 225 pp., Illinois Institute of Technology, Chicago. NICKERSON, E. C. (1969). Upper bounds on the torque in cylindrical Couette flow. J . Fluid Mech. 38, 807-815. NIKURADSE, J. (1932). Gesetzmassigkeiten der turbulenten Stromung in glatten Rohren. Forsch. Arb. 1ng.-Wes. No. 356; see also Forsch. Geb. Ing. W e s . 3, 260. ORR,W. McF. (1907). The stability or instability of the steady motions of a liquid. 11. A viscous liquid. Proc. R. Ir. Acad., Sect. A 27, 69-138. PARSAPOUR, H. (1977). A new approach to the detection of Malkus transitions in a horizontal layer of fluid heated from below. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York. A., and SOUTHWELL, R. V. ( 1 9 4 ) . On maintained convective motion in a fluid heated PELLEW, from below. Proc. R. SOC.London, Ser. A 176, 3 12-343. REICHARDT,H. (1959). Gesetzmassigkeiten der geradlinigen turbulenten Couettestromung. Mitteilungen Max-Planck-Institut fur Stromungsforschung, Gottingen, No. 22. REYNOLDS,D. (1895). On the dynamical theory of incompressible viscous fluid and the determination of the criteria. Philos. Trans. R. Soc., Ser. A 186, 123-164. RIAHI,N. (1977). Upper-bound problem for a rotating system. J . Fluid Mech. 81, 523-528. SCHMIDT, R. J., and SAUNDERS, 0.A. (1938). On the motion of a fluid heated from below. Proc. R. SOC.London, Ser. A 165, 216-228. SQUIRE, H. B. (1933). On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. London Ser. A 142, 621-628. STRAUS,J. M. (1973). On the upper bounding approach to thermal convection at moderate Rayleigh numbers. Geophys. Fluid Dyn. 5, 261-28 1. STRAUS,J. M. (1974). Upper bound on the solute flux in doubly diffusive convection. Phys. Fluids 17, 52&527. STRAUS, J. M. (1976a). A note on the multi-a solutions of the upper bounding problem for thermal convection. Dyn. Atmos. Oceans I, 7 1-76. J. M. (l976b). On the upper bounding approach to thermal convection at moderate STRAUS, Rayleigh numbers. II. Rigid boundaries. Dyn. Atmos. Oceans I, 77-90. WILLIS,G. E., and DEARDORFF, J. W. (1967). Confirmation and renumbering of the discrete heat flux transitions of Malkus. Phys. Fluids 10, 1861-1866.

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ADVANCES IN APPLIED MECHANICS VOLUME

18

Computational Modeling of Turbulent Flows?. JOHN L. LUMLEY Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca. New York

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. History and Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . B. General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Realizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. The Return to Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Reynolds Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . The Rapid Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Heat Flux Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The Temperature Variance Integral . . . . . . . . . . . . . . . . . . . . . D. The Reynolds Stress Integral . . . . . . . . . . . . . . . . . . . . . . . . . V . The Dissipation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Mechanical Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . B. The Thermal Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The Transport Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . The Transport Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. A Gaussian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Order of Magnitude Analysis . . . . . . . . . . . . . . . . . . . . . . . . D . The Pressure Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Zeroth-Order Transport Terms . . . . . . . . . . . . . . . . . . . . . . . F. Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 124 127 128 128 131 133 133 137 142 143 143 147 147 150 152 152 156

159 160 160 162 165 168 170 171 174

t This work supported in part by the U S. National Science Foundation. Meteorology Program under Grant Number ATM77.22903. and in part by the U . S. Office of Naval Research. Fluid Dynamics Branch. It is a pleasure to acknowledge fruitful discussion with B. Brumley. and the computational help of D. Hatziavromidis . 123

Copyright @ 1778 by Academic Press. Inc All rights of reproduction in any form reserved. ISBN 0-12-002018-1

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I. Introduction

A. HISTORY AND GENERALITIES At the 1968 AFOSR-IFP-Stanford Conference on Computation of Turbulent Boundary Layers (Kline et al., 1969),all the methods formally presented were oriented specifically toward the turbulent boundary layer, and most involved some variant of the mixing length formulation. During the Monday afternoon discussion, however, C. Donaldson made an informal presentation (reported on pp. 114-118) in which he strongly supported an idea he was then developing, which he called “invariant modeling.” He said “. ..it has been my feeling that one should keep track of the dynamics of all the second-order correlations of importance.. . . one seeks to express those terms which are unknown ... in terms of the second-order correlations themselves.. . . In making these choices one is constrained only by the requirements of symmetry and the general conservation laws.” And in another place, “Using general tensor notation . .. one.. . seeks.. .the simple invariant form that will reduce to one of the forms generally seen in traditional boundary-layer studies.. . .” Later in the conference, an ad hoc committee was formed to deal with the importance of invariance (consisting of Bradshaw, Donaldson, and Mellor). This committee’s brief report (p. 426) states in part: “Invariance of a particular formulation to coordinate transformation is important . .. particularly .. . if one is attempting to describe threedimensional flows.. . . If one chooses models which are of invariant form, it should be found that these models have the greatest generality; they should have the highest probability of describing turbulent flows which depart from the particular geometry for which the parameters in the model were adjusted to agree with experimental data.” This was the birth of the technique which has become known as “secondorder modeling” (as well as invariant modeling, and which the French school refers to as the one-point closure). During the last decade it has undergone very rapid development in the hands of numerous authors whose work will be mentioned explicitly in the appropriate sections below. The development at the present time is by no means complete, and the present work must be regarded as an interim report. One thing is already clear, however; in many situations of practical importance this technique makes possible computations which often agree with what data are available. Inevitably, the technique is also being applied in many situations in which data do not exist, which must be regarded as a dangerous practice, since the limitations of the technique are not known with any precision. It is primarily the possibility of practical computation which has been responsible for the great interest in this method.

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The method is not without historical antecedents. A good description of the early development of these ideas is contained in Monin and Yaglom (1971, p. 318). Specifically, Kolmogorov (1942) appears to have been the first to suggest characterizing turbulence entirely by its intensity and scale and using this to simplify the equations, an idea which is also used by all authors. Chou (1945a,b, 1947) suggested a number of closure schemes, and in particular was the first to use the equations for the third moments, eliminating the fourth moments by various hypotheses, something which we shall discuss later. Finally, the suggestions of Rotta (1951a,b) and Davidov (1958, 1959a,b, 1961) for the modeling of the pressure-strain correlations and related matters have had extensive influence on the development. of this technique. It is not an exaggeration to say that there is little in use at the present time that was not suggested by these authors. These authors all predated the easy availability of large-scale computers, so that their suggestions, for the most part, could not be explored extensively. Second-order modeling, even in its most stripped-down form, results in general in the simultaneous solution of four partial differential equations in the domain of interest; more elaborate models in a three-dimensional situation might require the simultaneous solution of as many as 36 partial differential equations to obtain the mechanical field only. Fortunately, this is within the capabilities of present computers at a reasonable price, which cannot be said of any other technique. Direct simulation (Orszag and Patterson, 1972) is limited to relatively low Reynolds numbers, but this is not serious, since the large scales (which are responsible for transport) are dominated by inertia, and thus are essentially independent of Reynolds number; however, if there is no homogeneous direction in which averages may be taken, several hundred realizations must be generated to obtain stable statistics, which is prohibitively expensive. There are also problems with initial conditions (Lumley and Newman, 1977): problems of differencing errors in current codes restrict the initial conditions on turbulent structure to fairly unrealistic ones. Almost the entire computational time is used in setting up a realistic turbulence, by which time the mean initial conditions have already changed. Direct simulation is thus not an alternative for practical computation. The various sophisticated closures (Leslie, 1973) suffer from essentially the same problems as the direct simulations and hence are also limited to homogeneous situations. Thus, the second-order modeling is at present the only possibility for practical computation. Second-order modeling may also be said to have as antecedent the work of the school of Rational Mechanics, which had its roots in the work of Stokes (and to a lesser extent Navier), and the modem development of which is associated with the name of Truesdell and his co-workers (with, of course, many others unnamed in between). In early work on non-

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Newtonian fluids, closures were developed for flows in particular geometries, relating the stress to the deformation state. Typifyingsuch closures is the power-law fluid, in which the shear stress is assumed proportional to a power of the local strain rate. This is not an invariant formulation and cannot be easily generalized to other geometries and flows. The school of Rational Mechanics, leading to the work of Coleman and No11 (1961), adopted the point of view that it was necessary to discover first the general form for a general geometry and flow, which the dependence of stress on deformation could take, considering the various mathematical and physical restrictions to which it was subject; this would permit the identification of a relatively small number of invariant functions, which could then be determined by experiment in particular geometries, the results being applicable to flows in all geometries. This is practically a statement of Donaldson’s point of view, quoted at the beginning. In fact, the techniques used for the computation of turbulent boundary layers at the Stanford Conference are reminiscent of the situation in continuum mechanics before the advent of rational mechanics; in just a few decades this approach has revolutionized nonNewtonian fluid mechanics, and we may hope that a similar approach will do the same for the computation of turbulent flows. Regarding rational mechanics, it is fair to say that there are those who feel that it hrts been philosophically productive but has not been greatly useful for computations; that in its general form it is too complex, and there are many more fluids in nature than can be easily encompassed by it; and that it is consequently necessary to fall back on the old empiricism for many practical computations. Just so, in turbulence computations, there are those, typified by Bradshaw (personal communication), who feel that the behavior of turbulence is so complex that the search for general closures is probably futile and that practical computations will require empirical techniques developed for the specific geometry. There are also those who feel that the general second-order modeling produces forms too complex to be of use in practical situations. My position is not diametrically opposed to these. Rather, I would say that I believe in the ultimate possibility of developing general computation procedures based on first principles; and under certain circumstances I believe that it is possible to do this rationally by the techniques of second-order modeling. While it may be necessary to fall back on empiricism for computations in complex situations, I believe that rational second-order modeling can at least provide a guide for the construction of the more empirical models and can certainly serve in general to indicate the range of applicability of these techniques. Quite apart from its utility as a practical computational tool, I have found that the attempt to devise models of this kind which behave like turbulence brings to light aspects of turbulent behavior that might never have been noticed. That is, one often finds that experiments do qualitative things which

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models cannot be made to do, no matter how the constants are adjusted. This indicates that a basic physical mechanism has been omitted from the model; if it can be identified and included (often difficult), something fundamental will have been learned about turbulence. This, incidentally, shows that Gauss was not quite correct when he remarked that, given seven constants, he could produce an elephant on a tightrope, and that with nine, he could make it dance.

B. GENERAL ASSUMPTIONS The development of second-order models has proceeded on a somewhat ad hoc basis, the degree depending on the predelictions of the particular author. Although nearly everyone recognizes the desirability of having all terms be tensors of the appropriate rank, with the correct symmetry and other properties (such as the vanishing of traces where appropriate), there has been little consideration of turbulence dynamics beyond this, and almost any convenient quantity with the proper tensor properties has been fair game, without consideration for its behavior at large or small Reynolds numbers, large or small anisotropy, etc. We will try to develop here fragments of a rational approach, from which we will see that the second-order models appear to be an orderly expansion about a homogeneous, stationary turbulence, the large scales of which have a Gaussian distribution. In fact, homogeneous turbulence is observed to be approximately Gaussian in the large scales (Frenkiel and Klebanoff, 1967a,b),even in the presence of homogeneous distortion (Marechal, 1972); what is envisioned here is a turbulence which is made non-Gaussian in the large scales by inhomogeneity, but which would, on the removal of the inhomogeneity, relax to a Gaussian state. The expansion about the homogeneous, stationary state suggests that the ratios of the turbulence length scale to the length scale of the mean flow inhomogeneities and of the turbulence time scale to the time scale of the mean flow evolution are both small. This is essentially a kinetic theory type of approximation. It is known experimentally that these ratios of scales are not small in real turbulence, being in general of order unity (since the turbulence structure and the mean flow inhomogeneity are generally produced by the same mechanism, unlike the artificial situation in a wind tunnel, where a homogeneous turbulence and the associated homogeneous mean shear may be carefully produced by different mechanisms). It is legitimate to ask why one might expect second-order modeling to resemble real turbulence; that is, why one has a right to expect the first term in an expansion in a small parameter to be applicable when the parameter is of order unity. There are many other examples of this phenomenon: for instance, the first two terms in an expansion in small Reynolds number for laminar flow around a cylinder work well up to Reynolds numbers of order 10 (Van

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Dyke, 1964). That, of course, is not an explanation. The explanation here probably is this: by following a rational procedure we have created a physically possible phenomenon, not quite real turbulence perhaps, but one which conserves momentum and energy; transports the right amount of everything budgeted (momentum, energy, Reynolds stress, heat flux, etc.), although not by quite the right mechanism; satisfies realizability (so that nonnegative quantities are never negative, Schwarz’ inequality is always satisfied, etc.); behaves correctly for both large and small Reynolds numbers; and reduces to real turbulence in one limit (weak inhomogeneity and unsteadiness). Probably any mechanism that satisfied all of these restrictions would behave about the same. The physical input from experiment is essentially used to fix the amount of transport. Of course, it is also possible, as Donaldson suggests (personal communication), that there is some other basis on which these equations can be derived, under which they have broader applicability. A given set of equations can often be derived from different sets of hypotheses of different degrees of generality. An example is the equations for global energy of a disturbance to Couette flow, which can be derived both from the exact equations and from the equations in the small disturbance approximation (Lumley, 1972).We only claim here to have found a consistent basis. We have already seen that we are making a specific assumption about the probability density of the turbulence; we are also making an assumption about the spectrum: it can be parametrized by the large scales. That is, if we know the characteristics of the large scales and the Reynolds number, then the shape of the rest of the spectrum follows; in fact, all of the statistics of the turbulence are determined by the large scales and the Reynolds number. This suggests two things: the turbulence has had time to come to equilibrium spectrally with the large scales and hence changes in the large scales are slow enough for the small scales to follow, and boundaries and initial conditions are far enough removed to have no direct influence on the present state. Otherwise, these would, in general, introduce another parameter (or several) which should be included.

11. Mathematical Preliminaries

A. REPRESENTATIONS In what follows, we will use a number of ideas familiar to workers in continuum mechanics, but perhaps not so familiar to workers in other areas. The principal concept that we will need is that of the form which an isotropic

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tensor function of other tensors may assume. Many references exist (e.g., Lumley, 1970a), but none covers the subject in quite the way we will need it. Thus, we will give a brief introduction here. Suppose that we have an isotropic symmetric second-rank tensor function of a symmetric second-rank tensor, both of zero trace c$ij(bpp), say. This form will occur when we consider the return to isotropy. We may consider that bij and 4ijare average second-order turbulence moments. First, by an isotropic function we mean that the functional relation is isotropic; that is, it is not dependent on any other suppressed variable, such as the direction of a magnetic field or the axis of a wind tunnel. The values of 4ijare not isotrop ic, but any anisotropy in 4ijis induced by anisotropy of bij;if bij is isotrop ic, 4i, is isotropic. A general technique for determining the dependence of 4ijon bij is to select two arbitrary vectors A , and Bj and form an invariant 4 i j A iBj. This is now a tensor of zeroth rank, having no free indices, and must consequently be the same in any coordinate system, hence, an invariant. As an invariant, it must be a function only of invariants, those that can be made from A , , Bj, and bij. We now must ask what are the independent invariants of this collection of quantities. From the Cayley-Hamilton theorem (Lumley, 1970a, Section A2.6) we know that in three dimensions only the Oth, lst, and 2nd powers of bij are linearly independent since bij must satisfy its own secular equation:

b;

- lb$

+ IIbij - I l l b ; = 0,

(2.1) where b$ = 6,, Kronecker’s delta, and b$ = bikbkj,etc. The quantities I, 11, and 111 are, respectively,

+

I1 = (biibjj - b,Z,)/2, 111 = (biibjjbkk - 3biib;j 2b;)/3! (2.2) Note that bif.is the trace of b$ and not biib,. These quantities (2.2) are the only independent invariants of bij. In addition tofhese, we have the invariants of A , and B,, which are the lengths and included angle of this vector pair, or equivalently A i A i ,BiBi,and A i B i . We have in addition, the invariants that can be constructed between the vectors and the tensor; what is invariant here is essentially the orientation of the vectors relative to the principal axes of the tensor, which should give no more than six quantities. These can conveniently be gotten from the collection A i b i j A j ,A,b$A,, Ai bijB,, A , b$ Bj, BibijBj, B, b: Bj. Higher powers of bijare not independent by the Cayley-Hamilton theorem. We are assuming, incidentally, invariance under the full rotation group; that is, we include invariance under improper rotations, or reflections, presuming that there is no preferred spin to the turbulence relations (not to the turbulence!). We are also working in Cartesian coordinates; the generalization to non-Cartesian systems is relatively straightforward but adds complexity. Note that if bij were not symmetric

I = bii,

130

John L Lumley

there would be other independent invariants, since b , b,, # b, b,, and A, bijBj # Ai bji Bj. Now, 4ijAi Bj is bilinear in Ai Bj, and thus the right-hand side must also be since these vectors are arbitrary. We may thus exclude all forms quadratic in Ai and Bj, and set 4 i j A iBj = a(I, II, III)b$ Ai B, + @(I, II, IIl)b, Ai Bj + y(I, II, III)bi Ai Bj. (2.31 We may now remove the A, and Bj and obtain

4ij= adij + @bij+ y b i ,

(2.4)

where a, @, and y are unknown scalar (invariant) functions of the invariants I, II, and III. If we now include the condition that b, has zero trace, I = 0, I1 = - b i / 2 , and I I I = b i / 3 ; if we take account of the fact that 4ii= 0, then a = 211~13so that

4ij= fibij + y(b$ + 211dij/3).

(2.5)

This form is essentially completely general, granting only that there are no suppressed variables. We have now reduced the determination of the dependence to the determination of two unknown scalar functions of scalar arguments, independent of flow geometry. These considerations can be extended to more complex situations. For example, suppose that chi, were a function of b , and a vector, ci. The invariants of the vectors with each other and with the tensor would now be AB, Ac, Bc, cc, AbB, Abc, Bbc, cbc, Ab’B, Ab’c, Bb’c, cb’c in an obvious notation. The general expression would not only have terms proportional to AB, AbB, Ab’B, but also proportional to AcBc, AcBbc, AcBb’c, BcAbc, BcAb’c, etc., making all possible pairs bilinear in A and B leading to 12 terms in all. The invariant functions would depend on cc, cbc, and cb’c in addition to the previous terms. It is clear that we are including in this way more invariants than we need (more than are independent) since we begin with 18 (two from the set A, B, and c, and one from the set bo, b, and b’), while only 15 appear to be independent (from a consideration of which ones are necessary to determine the projections of the three vectors on the principal axes of bij).It is possible to construct a reduced basis (Lumley, 1970b)which considerably simplifiesthe forms; however, in most cases we will consider if the turbulence is isotropic, b will be proportional to Kronecker’s delta, and the reduced basis will no longer span the space. Thus, it is better to be somewhat redundant, and include extra invariants, in order to have a basis which is valid in all cases.

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131

B. REALIZABILITY Recently, Schumann (1977) introduced the concept of realizability. He pointed out that no matter what equation is used to predict values o f w , it must have the property that does not allow correlations to exceed the limiting values dictated by Schwarz’ inequality and does not permit negative component energies. One might argue that this is a mathematical nicety so far as the component energies are concerned (although it is clearly important in regard to the correlations); that is, one might feel that the occurrence of vanishingly small component energies is so rare in nature that requiring correct behavior of model equations in that situation is unnecessary. However, anyone who has had experience computing with second-order models knows that a frequent cause of aborted calculations is the occurrence of negative component energies, often associated with recovery from poor initial conditions. There are, in addition, real physical situations in which one component is strongly suppressed, notably in stably stratified buoyant turbulence. Hence, it is physically important and computationally convenient, as well as mathematically nice, to require realizability. The most convenient way of satisfying the requirement is to transform the equation to principal axes of uiuj; now there are no off-diagonal terms; we may designate the component energies in this coordinate system (the eigenvalues) by We now require that if were to vanish, its time derivative too would vanish, so that it arrives at zero with zero slope and cannot cross to negative values. Just how this is implemented, we will see in Section I11 when we discuss the return to isotropy. Note that it was erroneously stated in Lumley and Newman (1977) that the method just presented, satisfying realizability for ,is foolproof only if the direction of the principal axes is not changing with time. That this is not true can be seen by differentiating the eigenvalue relation. That is, if we write at every instant

z.

v,

then differentiation of the second relation and substitution of the first gives: BijX$k)+ ~ X k f=)U:k,Xlk)+ %Xik).

(2.7) If (2.7) is now multiplied by XIk)and if the second equation of (2.6) is used, the second and fourth terms, containing the time derivative of the eigenvector, cancel, leaving -

,i(k) - X !I k ’ B . . X y ,

(2.8) and if it is required that the right side of (2.8) is to vanish when the eigenvalue vanishes, then the condition is satisfied. IJ

132

John L h m l e y

This general concept of realizability has many other applications, which have not been explored before the present work. For example, the equations for intrinsically positive quantities such as the dissipation of energy F must also have the property that the time derivative will vanish if the quantity vanishes. Further, we look at the equation for ?,the total turbulent energy; if ;TI vanishes, the time derivative of must vanish. This is a slightly different requirement from Schumann’s realizability in practice, although included in it in principle. That is, we will implement Schumann’s realizability by requiring that the time derivatives of u& vanish if the component vanishes, even if the other components do not. The sum of the three time derivatives, of course, must be equal to production minus dissipation; if vanishes, the requirement that correlation coefficients not exceed unity will assure the vanishing of the production. However, we must separately assure vanishes to guarantee that the time that the dissipation vanishes when derivative vanishes. Roughly speaking, this means that, as vanishes and if cdoes not, the time derivative of B must go to - 00 to assure that it will also vanish at the same point. The same consideration applies to the temperature variance. Finally, we must consider correlations between unlike quantities, like velocity and temperature, which are also constrained by Schwarz’inequality. These must ke treated by a slightly different technique, because the two do not form a tensor. However, we may still examine the eigenvalues of the correlation matrix. These eigenvalues are nonnegative and vanish in only three cases: if the variance of either variable is zero, or if the correlation coefficient becomes unity in absolute value. If we write the secular equation for the matrix and require that the time derivative of the eigenvalue vanish if the eigenvalue vanishes, we obtain as a condition that the time derivative of the determinant will vanish if any of these conditions is met:

a

a

a

__

-2abub

a

+ > p + 23 = 0

(2.9) for two arbitrary correlated quantities u and b. If we arrange to have 2 vanish when 2 vanishes and the same for b, Eq. (2.9)will be satisfied if ab also vanishes; if the correlation coefficient cannot exceed unity (in absolute value), this will be assured. Hence, we must consider the case of the correlation coefficient approaching unity. If p is the correlation coefficient, we have A _

*

p / p = ab/ab - 2 / 2 2 - $/2p.

(2.10)

Substitution in (2.9) permits us to write --

2pp = ( 1 - p2)(ue2/a2 + b2/b2). -L-

(2.11 )

Hence, if the variances are vanishing, but are not yet zero, and the correlation coefficient rises to unity (i.e., the correlation is not vanishing fast enough), the time derivative of the correlation coefficient must vanish, and

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133

the equation for the correlation must be constructed in such a way as to assure this. This type of realizability, with regard to velocity-temperature correlations, has been completely neglected until now. In the case of the correlation between a scalar and a vector, such as temperature and velocity, in addition to the constraint on the heat flux along each axis, we may obtain a more general condition:

__

( G j P- e u i e U j ) A i A2j o

(2.12)

for an arbitrary vector A i . The conditions on the individual components are included in this and can be obtained by taking Ai successively parallel to the three axes. However, (2.12) is more general. The easiest way to implement the condition is to transform to principal axes of the tensor in parentheses in (2.12); then a statement equivalent to (2.12) is that all eigenvalues of that tensor must be nonnegative. In these coordinates then, we may write equations like (2.9) in each coordinate direction and assure ourselves of satisfying the general condition (2.12), since each eigenvalue is in the form of the usual Schwarz’ inequality. Again, the development of (2.6-2.8) is applicable, and the fact that the directions of the principal axes are changing in time is irrelevant.

111. The Return to Isotropy A. INTRODUCTION It is part of the folklore in the field that anisotropic turbulence tends to become more isotropic; that is, in the absence of other influences the components interchange energy so as to become more nearly equal, and this exchange process proceeds faster than the decay. The experimental evidence for this is individually not very strong; Fig. I, from Lumley and Newman (1977) is typical. Nevertheless, the evidence taken together is incontrovert-

10

15

20

25

jn

35

4n

45

50

55

i M

FIG.1. The return to isotropy of homogeneous turbulence following a contraction. Measurements of Uberoi. (Lumley and Newman, 1977).

John L Lumley

134

ible: such a return to isotropy does exist, but it appears to be very slow for weak anisotropy. This return to isotropy is produced by the pressure terms. Let us consider a homogeneous flow, without mean velocity gradients. The equation for the Reynolds stress is:

(m+ m)/p- 2VUi.k

uj,k*

-(m+ p,iui)/p = -[(puj).i + (pui),jl/~+

J4Uj.i

(34 Although it makes no difference in this flow, it is customary to remove the transport effects (Tennekes and Lumley, 1972,Section 3.2) from the pressure term, writing =-

+ ui,j)/p*

(3.2) Although Lumley (1975b) has pointed out that this separation is not unique, and (3.2) may not be the most appropriate choice, there are reasons of convenience (which will become apparent in Section VI) for making the separation as in (3.2). We concern ourselves with this matter here because we intend to use the modeling which we devise for this pressure term also in inhomogeneous situations. It is evident from the presence of the strain rate ~ ) the trace vanishes; that is, the term serves only to interin ~ ( u+~u ,~ ~, that change energy among the components, not to create or destroy energy. The viscous term in (3.1), although its primary function is to dissipate uiuj stuff to heat, can also cause interchange of energy among the components at any Reynolds number. Although the term is observed (Monin and Yaglom, 1975, p. 453) to become more isotropic with increasing Reynolds number, in agreement with Kolmogorov’s (1941) hypothesis of local isotropy, if u1 = 0 (which can occur conceptually or computationally, and even approximately in reality) then U1.k U1.k = 0 also. We will add and subtract the trace, which is twice the dissipation of energy: 2 v u x = [ 2 v w k - 2Edij/3] + 2E6ij/3.

(3.3) The deviatoric part (in square brackets) now acts to interchange energy among components, but neither creates nor destroys total energy. If we define -E4ij = p(ui,j uj,i)/p - 2~2Zdij/3 (3.4) then Eq. (3.1) may be written as u,ul = -E4ij - 2Edij/3 and 4ijis dimensionless, has zero trace, and is solely responsible and responsible only for the return to isotropy. The tendency toward equipartition which returns the turbulence to (or toward) isotropy can also be regarded as a decay of Reynolds stress, since it is only a question of being, or not being, in principal axes of the Reynolds stress tensor. It seems natural to consider also the decay of other correlations, notably the heat flux. There is very little evidence for this decay; Fig. 2 is the only known data, reproduced from Warhaft and Lumley (1978a).

+

+

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135

XlM

FIG.2. Decay of temperature-velocity autocorrelation coefficient in grid-produced turbulence after Warhaft and Lumley (1978a; see also Newman et a/., 1978a). Curves were produced by the analysis of Section 1II.C.

Fortunately, the data show incontrovertible decay of the correlation coefficient. In the same homogeneous flow, without mean velocity or temperature gradients, the equation for the heat flux may be written as

iiJ

= --/p

- (v

+ y)B,iui.i,

(3.5) where y is the thermometric diffusivity. Although it is not at all customary, we will separate the first term on the right-hand side just as we did (3.2):

-DIP = -@),i/p

+ KIP.

(3.6) This is not ordinarily done, since the term obtained (the first on the right) is not in the form of a transport term, although it is a divergence (a transport term should be of the form (q),i for the turbulent transport of a quantity A). However, realizability requires that this term be separately modeled. To see this, suppose that all other terms in the equations are realizable. Then, in

John L Lumley

136 --

--

principal axes of f12uiuj - flui flu,, applying Eq. (2.9), realizability requires

-

--

when Pu,ui = flu, flui. This is clearly true for the unmodeled terms (in their natural form as given in (3.7)). More to the point, however, since we will be modeling the term on the left for inclusion in the Reynolds stress equation, we must include a model of the term on the right in the heat flux equation, and the model must satisfy (3.7). Let us define -

_-

-4!jflujElq

2 - 3 -

- P ,iIp - ( V

-

+ Y)fl,jui.j

(3.8)

so that Eq. (3.5) becomes simply -

~ i =8

-- &,flu, E/q2.

(3.9)

This is not a completely general form, but is sufficiently general for our purposes. The molecular transport part of (3.8) will become essentially zero for high Reynolds-Peclet numbers under local isotropy, but that need not concern us. We also need the equation for the temperature variance:

f12 = -2E0.

(3.10)

We can now apply realizability to these equations. First, if we transform and take = 0, then the right-hand side of (3.4) to principal axes of ~(iu~, must vanish, so that we require 411= -4 (since the dissipation and -other _ components _ are notzero). Next, -~ we transform to principal axes of f12uiu, flui fluj and take pu,uj = 8 ~ fluj. 1 Then, applying Eq. (2.9), we require

Note that, if in addition the one-heat flux vanishes, all components of the Reynolds stress containing u1 vanish; this is equivalent to the vanishing of the one-eigenvalue of the Reynolds stress tensor, and hence by the realizability requirement for the Reynolds stress, the second term on the left of (3.11) must also vanish. From this, if the one-heat flux vanishes, the right -side of (3.11) must vanish. So @ must , have the same principal axes as f12uiuj 0%. BU, . Thus, it must be a function only of that tensor, although it may be a function of invariants of other tensors, parameters, etc.; for, if it were a function of any other vectors or tensors, the principal axes would not be the same.

Computational Modeling of Turbulent Flows

137

B. THEREYNOLDS STRESS Now let us consider dij.If Eq. (3.4) is considered as an equation for q5ij, it is clear that it is determined by the history of the Reynolds stress and the present value of the dissipation. We could write in general

4ij = 4ij(uVj, c v},

(3.12)

that is, a functional of the past values. The kinematic viscosity has been included for generality, although it is not explicitly present in the equation. Time does not appear explicitly, since we anticipate that the relationship will not change with time (although the quantities involved will). The functional will involve integrals over time, however, and the most satisfactory normalization_is_to define a new time d.r = dte/?; this is equivalent to T - to= f In (q2/qo2).This new dimensionless time is monotone in the true time since the turbulence is decaying. The argument of (3.12) contains eight quantities involving two dimensions; hence it is possible to form six independent dimensionless groups. We choose to do this in such a way as to make the role of anisotropy most evident; we will define an anisotropy tensor (3.13) This vanishes identically if the turbulence is isotropic, it is dimensionless, symmetric, and has zero trace; hence, it has five independent components. For the sixth group, we take a Reynolds number R, = (?)'/~Ev.

(3.14)

The factor of nine has been included so that, if we take ? = 3u2, B = u3/l, (3.14) becomes the traditional Reynolds number based on this length and velocity. Hence, (3.12) can be written as a functional of (3.13) and (3.14). Now, we may presume that the turbulence has a fading memory and will ultimately forget its initial state; if changes in the mean state are sufficiently slow relative to the turbulence time scale, (3.12) can be expanded in a functional Taylor series about the present state (see Lumley and Newman, 1977, for details); keeping only the first term, (3.12) reduces to a function of the present values of (3.13) and (3.14). The ratio of the turbulence time scale to the time scale of change of the mean field is almost always of order unity in real turbulence, so such an expansion is not formally justified; however, we will find that the resulting expressions work remarkably well. Thus, (3.15)

John L h m l e y

138

This is now of the form discussed in Section II,A, and we can immediately write the form (2.5)

dij = B(11, Ill, R,)bij+ ~ ( 1 1I,l l , RJ(b$ + 211dij/3).

(3.16)

We must now determine the forms of /3 and y in their dependency on the invariants and on the Reynolds number. (This y is, of course, not to be confused with the thermometric diffusivity.) First, it is instructive to consider how turbulence can be characterized in terms of the invariants. A plot of all turbulence on axes of 11 and I l l is shown in Fig. 3. (Note that our definitions (1.2) are the classical ones and differ from those used in Lumley and Newman (1977) by simple factors: 11 = - 11'12, 111 = 111'13, where the prime indicates the expressions used in that paper.) In Fig. 4, the possible region of variation of the eigenvalues of bij, say bl and b, (since the trace is zero, b3 = -b, - b,) is shown. Each eigenvaluecan be no smaller than - 3, corresponding to the vanishing of that component, and can be no larger than +$, corresponding to the vanishing of the other two. Thus, b , and b, are constrained to lie within the triangle in

-II

(2/27.1/3) . 1D

AXISYHMETRIC

ISOTROPIC 2D

AXISYMMETRIC

I -0.05

0

0.05

01

m FIG.3. Possible states of turbulence parametrized by the independent invariants of the anisotropy tensor after Lumley and Newman (1977). Turbulence must occur within the region (or on its boundaries) delimited by the axisymmetric and two-dimensional states.

Computational Modeling of Turbulent Flows

139

FIG.4. The possible range of variation of the two independent eigenvalues of the anisotropy tensor. Turbulence must occur within the triangular region, or on its edges, which correspond to the two-dimensional state. The maxima and minima of 111 on the curve of constant I1 are indicated, where this curve crosses the lines corresponding to axisymmetric states.

Fig. 4. A constant value of I1 corresponds to the ellipse; if I1 is small in absolute value, the ellipse lies entirely within the triangle; if 1 I1 I is larger, it lies partly outside, and the parts outside are excluded (that is, they do not correspond to possible values of bl and b2).At the largest possible value of -11 = 3, the ellipse just touches the corners, corresponding to the three possible one-dimensional turbulences. On the segments of the ellipse lying within the triangle it is possible to determine the total derivative of 111 with respect to b, or b2 for fixed 11; it is readily found that relative maxima occur where the ellipse crosses the lines joining the apexes to the origin, corresponding to axisymmetric turbulence. Relative minima occur where these lines cross the other sides of the ellipse, when the ellipse is small enough to lie entirely within the triangle, also corresponding to axisymmetric turbulence. The relative maxima correspond in each case to two components being equal and the third being larger than the average, while the minima are the opposite. The minima disappear when the component that is smaller than the average reaches zero. Moving away from the maxima, the value of I11 decreases monotonically until an edge of the triangle is reached, corresponding to two-dimensional turbulence. In the two-dimensional and axisymmetric cases, there are simple relations between I1 and 111 which may readily be found. In the axisymmetric case, if

bij = ( 0b -b/2 0 0

8)

-b/2

, (3.17)

John L Lumley

140

then I1 = -3b2/4 and 111 = b3J4; hence 111 = +2(-11/3)3/2. In the twodimensional case, if bij is placed in principal axes and bl = -f the expressions for I1 and 111 can be readily reduced to give

+ 3111 + I1 = 0.

(3.18)

These lines are shown on Fig. 3. By inspection it is clear that a stronger statement can be made relative to the quantity in Eq. (3.18):

6 + 3111 + I1 2 0

(3.19)

everywhere. Hence, the vanishing of (3.19) can be used as an indicator of two-dimensionality of the turbulence, and we will avail ourselves of that possibility. First, Lumley and Newman (1977) have shown from the data of ComteBellot and Corrsin (1966) that for axisymmetric turbulence in vanishingly small anisotropy,

41 Jb = 2.0 + 8.0/R,?12.

(3.20)

The value of 2 corresponds to no return to isotropy. This can be seen by forming the equation for bij: (3.21) dbijldr = - (4ij - 2bij) so that if q511 = 2b in the axisymmetric case, the value of bij remains unchanged. Hehce, (3.20) suggests that there is no linear return to isotropy at an infinite Reynolds number; the effect is thus either a viscous or a nonlinear effect. By forming the equations for I1 and I l l , and considering very small anisotropy, Lumley and Newman (1977) have shown that y must vanish as the anisotropy vanishes at infinite Reynolds number. We have an additional requirement. The so-called final period of decay (Batchelor, 1956) describes turbulence at a Reynolds number so low that the nonlinear terms may be neglected. Since they are responsible for the return to isotropy, there is consequently no longer any return to isotropy. Consequently, the right-hand side of (3.21) must be zero for a zero Reynolds number. We may now attempt to satisfy these various requirements in the simplest way possible. We will be able to satisfy the conditions with y = 0, giving a relatively simple expression. [Lumley and Newman (1977) give a complete expression, but the form is cumbersome.] If 4ij= /3bij, then by (3.21) we must have /3 2 2 always, since f l < 2 would correspond to the spontaneous increase of anisotropy in the absence of any external agency. There is no proof that this should not happen, but it seems unlikely. Since the expression (3.19) is always positive and vanishes if one of the eigenvalues of bij takes the (i.e., the turbulence becomes two-dimensional), it is tempting to value

-4

141

Computational Modeling of Turbulent Flows

take fl = 2 + F(R,, ZI, Ill)($+ 3111 + 11).This would automatically satisfy realizability for the Reynolds stress. It has the disadvantage that, if the turbulence became two-dimensional, there would be no return to isotropy of the remaining two components. However, it is not our intent here to make a model that behaves properly for two-dimensional turbulence, which is fundamentally different in so many respects. We are interested only in a simple, workable form which satisfies realizability, and since a form & j = C1bij is widely used in the literature (see, e.g., &man and Lumley, 1978), the form proposed is likely to be satisfactory. The unknown function F must be determined so that it vanishes at a zero Reynolds number, takes on the value 8.O/R:l2 for small anisotropy and a large Reynolds number, is always positive, and otherwise fits the existing data. The form

fl = 2 + e~p[-D/R;/~](72/Rj"~

+ A h[l + B(-ZZ + CZZZ)])($ + 3111 + ZZ), A = 80.1,

B = 62.4,

C = 2.3,

(3.22)

D = 7.77

fits the data given in Lumley and Newman (1977) nearly as well as the more complicated expression given there (Fig. 5 ) ; the agreement with the form

1

2

3

4

5

6

7

8

-ux10*

FIG. 5. The return to isotropy + , , / b (the function from the analysis of Section II1,B) for an axisymmetric flow with negative 111. The light curves are the more complete description of Lumley and Newman (1977), while the heavy curves are the analysis of Section II1,B. The experimental points are those reproduced in Lumley and Newman (1977).

142

John L Lumley

given by Lumley and Newman (1977) is equally good for the axisymmetric case with 111 > 0, for which no data exist. The behavior in Fig. 5 displays the various characteristics we have discussed. Note particularly the fact that the curve arrives at the twodimensional state (- 211 = $) with finite slope which indicates that this state is unstable; if perturbed, there will be a finite rate of return to isotropy. C. THEHEATFLUX

We may now consider +!, beginning from the realizability condition (3.11). Let us introduce a tensor

__

(3.23) D,, = iqqa - eu, euj/P?. This has nonnegative eigenvalues that vanish only if the correlation is perfect in that component. From the realizability condition, we have already , be a function of Dij. It is straightforward but tedious to found that &must show that (in principal axes of D i j ) if D l l = 0, then 2

4 + 3111 + 11 = 3DzzD3,8U1 /@?.

(3.24)

Substituting this in (3.22), and that in (3.11), we obtain for realizability in principal axes of Dij with D l l = 0: =

1

+ r + (F/2)($ + 3111 + 11 -

D22D33),

(3.25)

where F is the expression in (3.22) which multiplies the last set of parentheses. We may identify D Z 2D3, as ll,, the second invariant of the tensor Dij under these conditions. r is the time scale ratio, r = (~/P)(42@). Thus, to satisfy realizability, we could assume = [I

+ Y + (F/2)(3 + 3111 + I I -

llD)]dij

+ gDij + hD$,

(3.26)

where g and h are functions of the invariants of D,,, I,, ll,, and 111,. If we consider decay of the correlation coefficient in an isotropic turbulence, we find, if p is the correlation coefficient,

I, = -p(f$&

- r - l)E/?.

(3.27)

We find 11, = 2( 1 - p2)/9 + f and

- 1 - r = (F/2)(3- 11,)

+ gD11 + hD:l.

(3.28)

With D1 = ( 1 - p2)/3 finally we have d In p/dz = -(8/3 - F/9)(1 - p’) - (h/9)(1 - p2)’.

(3.29)

Excellent agreement with the data of Warhaft and Lumley (1978a) is obtained by taking h = 0, g/3 - F/9 = 1.6 (the curves in Fig. 3 were obtained

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143

using these values). Until more data are available indicating variation with the Reynolds number, etc., we will use these values. We have a choice; the F appearing in (3.29) is F with I I = I I I = 0, say F,. In defining g in general, we could use either the general F, or just F,. Lacking any information, we will take F because it produces the simplest equation. If we form the equation for the decay of the correlation coefficient of an axial heat flux in an axisymmetric turbulence, we obtain with this choice d In p/dr = -(B - 2)/2 - 3bll/(l

- (1

+ 3b11)

+ 3bll)(l - p2)(1.6+

b11 F/6)

(3.30)

so that the decay rate is increased if b l l is positive (and vice versa). This must await experimental confirmation but has a certain appeal. The axisymmetric form (3.30) is applicable to the conditions of &man and Lumley (1976), where a buoyancy-driven atmospheric surface mixed layer was considered. The heat transfer is entirely vertical, and the turbulence is axisymmetric about the vertical axis. Using Zeman and Lumley’s (1976)fixed value for B = 3.25, we find for the right-hand side of (3.30) a value of -2.44, whereas the value using their equations, which assume a fixed value for 4; = 7.0, is - 4.79; the value we obtain for 4; = 4.65 under their conditions. It seems very likely that the larger value was necessitated by the failure of their rapid terms to satisfy realizability conditions, driving the heat flux to impossibly high values, and requiring a larger return to isotropy term to kekp the heat flux under control. IV. The Rapid Terms A. INTRODUCTION

In what we have done so far, we have considered only homogeneous situations without mean velocity gradients or buoyancy. If we begin from the Navier-Stokes equations for the fluctuating velocity in an incompressible situation (in the Boussinesq approximation, if buoyancy is importantsee Lumley and Panofsky, 1964, Section 2.1),

+ ui,juj+ ui,juj + ui,juj -

+

+

- p , ~ p pie vui,jj and take the divergence, we obtain an expression for the pressure: tii

=

+

(4.1)

-v2pip = 2ui,juj,i ui,juj,i- pie,i. (4.2) This contains two types of terms on the right-hand side. The first and third are linear in the fluctuating quantities, while the second is quadratic. The second term is the nonlinear scrambling (Hanjalic and Launder, 1972) or the

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John L Lumley

return-to-isotropy term; it is the pressure field associated with the mixing of the turbulence by itself generated by this term which is responsible for the gradual equalization of the energy in the various components of the turbulence. It is the other terms which we wish to examine here. We may conveniently split the pressure into two parts, defining

The correlations with p") are those which have already been modeled in Section 111. The correlations with p") and its gradients are those which we refer to as the "rapid" terms. The term "rapid" is used for two reasons. First, there is a classical problem in turbulence called the rapid distortion problem (Batchelor and Proudman, 1954), in which one imagines that the turbulence is subjected to a velocity gradient so intense that for some little time one may neglect the transport and distortion of the turbulence by itself, and hence neglect p(') and the nonlinear and viscous terms in Eq. (4.1). Since the pressure p") is the only pressure present during this rapid distortion, it is natural to refer to it as the rapid pressure. Second and more important for us, however, it is not difficult to show that, if an initially isotropic turbulence is subjected to a sudden distortion, not necessarily rapid, all correlations in the equations not involving p(l)begin from isotropy and gradually develop anisotropy in response to the distortion; the same is true for the sudden application of a gravitational field. The terms involving p(l), however, are instantly anisotropic. This also justifies the use of the term rapid, and in addition makes clear why it is necessary to model these terms with considerable care; in nearly every situation, they exert a very strong influence on the structure of the anisotropy of the velocity field (see, e.g., Townsend, 1970). Incidentally, the concept of rapid distortion has been extended to the case of the sudden application of a very strong gravitational field (Gence, 1977)within the same approximation. The first of Eq. (4.3) may readily be solved by standard techniques to give p") explicitly in terms of the right-hand side. Of the many solution techniques available, we will use Fourier transforms, which are appropriate to a homogeneous field. Although there are, in general, terms in the various correlations with p") arising from inhomogeneity, consideration of which would require a solution technique for p(') that did not suppose homogeneity, these terms are never considered, and there is no indication that the homogeneous forms are insufficient in any practical situation. For an introduction to the use of Fourier transforms for homogeneous stochastic fields, see Lumley (1970a). Indicating the Fourier transform off by [f], we may write

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145

To obtain the various correlations that appear in the equations, we must now multiply by the Fourier transform of the appropriate quantity, average, and integrate over the Fourier space. Taking the simplest case first, let us multiply by [el* (the complex conjugate), average, and integrate over Fourier space, designating S the spectrum of temperature variance, s, the spectrum of the heat flux, and s,,, the spectrum of the Reynolds stress for future reference. In all cases the integral of the appropriate spectrum over the entire Fourier space gives the quantity in question. We will indicate an integral over three-dimensional Fourier space by 1dK.

Taking the second part of the integral only, without the buoyancy parameter, we will consider

j” (KqKi/K2)SdK= Iqi,

say.

Now, Iqi must clearly be symmetric. If i is summed on q, the term in parentheses goes to unity and the integral is, by definition, the temperature variance, so that we have as a condition I,, = p.In addition, if the turbulence is isotropic, S is a function only of the magnitude of the wavenumber vector, and hence is spherically symmetric. If the integral is carried out in two steps, first over spherical shells, and second, over the radius, the term in parentheses can be integrated immediately to give

on a spherical shell. If the integral is now carried out over the radius, we have for the isotropic case

We must develop a model for Iqi which satisfies these restrictions. Most authors use the isotropic value for Iqi for all situations. However, we must also satisfy the realizability conditions. Writing Iqij and Iqijk for quantities constructed like Zqi in (4.6),but using S j and S , and applying the realizability condition (2.9)to the equations for the heat flux, taking in account the fact that the realizability conditions must be separately satisfied by the terms multiplying the buoyancy parameter since it may be given any value and orientation (the same being true of the terms multiplying the mean velocity gradient), and using the fact that the spectra and hence Iqi,etc., may be

John L Lumley

146

(say, by unfelicitous initial_conditionsJ-from _ fl, and U i S jwe , obtain (in principal axes of the tensor p u i u j - 8ui 8 u j )

manipulated separately

&I,,

=

F Z , ~ , , GI,,= PI,,,,

G I =, ~ FPz,~~ (4.9)

as the conditions which must be satisfied if either the corresponding velocity component vanishes, or the corresponding correlation coefficient achieves the (absolute) value of unity, or the temperature variance vanishes. If we can assure ourselves that the corresponding condition (4.9) is satisfied when the correlation coefficient achieves unity in absolute value, then by (2.11) the correlation coefficient will be bounded. Hence, if either the temperature variance or the corresponding velocity variance vanishes, the corresponding heat flux will vanish. If we arrange that Iql vanishes when the one-heat flux vanishes, and similarly for the other components, then (4.9) will be satisfied under all conditions. Thus, we must (1) arrange for I , , , to vanish if vanishes, etc., and (2) must satisfy the corresponding member of (4.9) if the correlation coefficient in question becomes f 1. Wyngaard (1975) and &man and Lumley (1978) looked at the horizontally homogeneous case of a vertical temperature and velocity gradient and considered a strong stable stratification which annihilated the vertical velocity and heat flux. The equation for the vertical heat flux in these circumstances is -

ew =

-83

+ p3P.

(4.10)

Since it seems likely that the vertical heat flux should remain zero under these circumstances, they were led to require that 1 3 3 = F as the vertical velocity vanishes. As a general prescription this is, however, not consistent with (4.9), if we assume that the value of lqiis determined entirely by the temperature and velocity fields as we shall. For then, if the velocity and heat flux vanish in any direction, the diagonal component of Iqi in that direction must take on the value p,while the other two must vanish, since lii= p. The condition is immediately violated if two components of the velocity vanish. In fact, an exact condition can be obtained from (4.9). If the horizontal velocity and heat flux vanishes, then the first two of (4.9) are satisfied; if the correlation coefficient in the vertical is unity, then 1 3 3 must vanish, since Ipii= 0 by continuity, and hence 1,33 = 0 since I,, and I,,, are zero (since 8ul and = 0) (see,for example, Lumley, 1970a, Section 4.7). Hence, 1 and I , , cannot both be zero. This exact condition was also invoked by Wyngaard (1975) and &man and Lumley (1978). We may add one final condition: the diagonal components of lqimust be nonnegative, since the temperature spectrum is nonnegative. In the case when the correlation coefficient achieves the value of 1, the conditions (4.9) have a simple interpretation. If u , and 8 are perfectly cor-

,

,,

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147

related, they are proportional, and consequently the spectra S, S , , and S I 1 are all proportional. Thus, we see that, not only must we construct a model for Iqi consistent with all these conditions, but we must model I,, at the same time, since their limiting values must be coupled. The equivalent conditions on Iqij are I4V. . = I.W J..’ I44J. = 8u.; J I W.. = 0 ; and Inij -,0 as -,0. These have all been discussed before or are completely analogous to conditions on Iqi. We have, in addition, the conditions on Iqij which arise from realizability of the Reynolds stress. We may obtain these conditions by placing the equations for the Reynolds stress in principal axes; if the intensity in, say, the onedirection vanishes, then for the vanishing of the time derivative we require that I,, vanish also (since the one-heat flux vanishes) for arbitrary p . Thus this is not an independent condition.

6

B. THEHEATFLUX INTEGRAL In the past it has been customary to take for Iqij a simple linear combination of G.For example, &man and Lumley (1978) and nearly all other authors use

Iqij = (3)[Si,euj

-

(+)(SijK + Sqj&)].

(4.11 )

This satisfies the first, second, and third of the conditions in Iqij, but it does not satisfy the important condition that it vanish when the j-heat flux vanishes, both required by realizability and by the fact that S j vanishes when the j-heat flux vanishes. This almost surely causes computational difficulties (Zeman and Lumley, 1978). We could attempt to remedy the situation by constructing a very general formalism, expressing Iqij as a function of bijand the heat flux, forming invariant functions, etc. Fortunately, it is possible to construct by inspection a simple expression that at least satisfies all the necessary conditions : 1,ij

= (6,i

-- _ _ -8~,0~i/tl~,8~,)8~j/2.

(4.12)

C. THETEMPERATURE VARIANCE INTEGRAL

The construction of an expression for Iqi which satisfies all requirements of its own and is properly related to the expression (4.12) by realizability conditions is not quite so simple, and we have been unable to find a completely general expression. The following expression has nonnegative eigenvalues, has the proper trace and the proper value under isotropy, but

148

John L Lumley

satisfies realizability exactly only on the total heat flux, not on the components: I 1.4 = eZsi,/3

+ Phi, - (3/2a)(G&,

-

__

-.e~,e~,s,/3).

(4.13)

To see in what senserealizabilityis satisfied, suppose we are in principal axes of the tensor @uiuj - flui 8 u j , and suppose that the correlation is perfect in the one-direction. Then, because off-diagonal terms of the tensor and because -- the leading diagonal term also vanvanish in principal axes, ishes, we may write @u,u, = Bu, Ou, ;substituted in the expression for bqi, this may then be combined with the other terms in (4.13), producing

__

-- _ _

I,, = (eu, eu,/2q2)(s,1 - eu, eu,leu, eu,).

(4.14)

At the same time we have from (4.12) the expression

-- _ _

I,, = (8U,/2)(sq1- eu, eu, /eu,eu,).

(4.15)

These have the same dependence on q ; the realizability condition produces --

eu,eu,

=

(4.16)

which is true only if all components are perfectly correlated (in which case all three realizability conditions will be satisfied exactly). This can occur approximately in free convection, in which the horizontal velocity essentially vanishes; htnce, the expressions (4.12) and (4.13) may be useful in that situation. When 24, = 0, I , , = Ou, 8u,,/2q2; if the horizontal heat flux is small, this will be essentially zero, as is desirable (see Eq. (4.10)).No computations have been done yet with (4.12)-(4.13), and we must regard them as an interim solution since it seems likely that additional terms may be required. Lumley (1975a) suggested that perhaps these rapid terms should be functions of the mean field parameters, that is, the buoyancy parameter and the mean velocity gradient. However, Reynolds (1976) quite correctly pointed out that the Iqi,etc., are determined by the form of the spectra; if the spectra are set by initial conditions in a particular state, the Iqi, etc., will be completely determined independent of the buoyancy parameter and the mean velocity gradient. These will exert their effect as the field evolves, of course, but only by their influence on the form of the spectra. Hence, in determining forms for these terms, we exclude all quantities other than those characterizing the state of the turbulent field, the second-order correlations. Lumley (1975a) also suggested that the expressions for Iqi, etc., must be linear in the second-order quantities. The reasoning was as follows: the rapid terms are those which would appear in rapid distortion theory. Rapid distortion theory is a linear calculation, and therefore results are superposable. We may model rapid distortion theory by using our rapid terms, neglecting return to isotropy and dissipation; hence, our modeling results must also be

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149

superposable. Hence, the expressions for lqimust be linear in the secondorder quantities. This position seems to have found acceptance (Reynolds, 1976). However, it must be incorrect in some sense, since it is impossible to satisfy realizability with linear expressions. This is a perplexing problem, which deserves further thought. The answer could, of course, be that it is not possible to parametrize even approximately the terms lqiin terms of the second-order quantities; however, I feel that that is unlikely. A more appealing explanation is the following: in rapid distortion theory the solutions are dependent on the time multiplied by the mean velocity gradient due to the linearity; since geometrically different mean velocity fields produce different solutions, the solutions can be expressed as functions of dimensionless structural parameters, constructed from the mean velocity gradient (Lumley, 1975a); due to the way in which time appears, these structural parameters may, in each solution, be replaced post hoc by expressions constructed from the solution variables themselves, producing an apparent nonlinearity. Hence, a general expression, parametrized in terms of the second-order quantities, in the course of a rapid distortion problem, would have certain combinations of these quantities which remained constant, playing the role of structural parameters. It has also been suggested (Lumley, 1975a)that the rapid distortion problems could be used to calibrate the modeling of the rapid terms. While it is possible to construct rapid terms which model the rapid distortion problem quite successfully (Gence, 1977), it now seems likely that this is not a useful procedure. The problem is, that the spectra, for the same values of the large scale parameters (the second order quantities) will, during rapid distortion, have quite different forms from their usual ones, since the strain rate, or buoyancy, is supposed to be sufficiently strong to dominate the nonlinear effects. Thus, the strain rate, or buoyancy, will be felt directly in the small scales of the spectra, which will not have an equilibrium form. It is conceivable that the rapid terms could be modeled incorporating a parameter representing the ratio of the time scale of the distortion to the time scale of the dissipative eddies; during the distortion this is essentially zero, while for the cases we are interested in modeling, it is essentially infinite. One could envision calibrating the model against rapid distortion with the parameter set at one value, and afterwards setting it at the other value. However, this would be a complicated and delicate procedure and seems unlikely to be worth the effort. Since even the extremely crude models for the rapid terms currently in use (which do not satisfy realizability and sometimes not even incompressibility) give results which are in many ways satisfactory, it appears probable that we can, by devising models which do satisfy these conditions, produce results which are completely satisfactory without going to extremes.

John L. Lumley

150

D. THEREYNOLDSSTRESS INTEGRAL We may now consider the rapid terms which multiply the velocity gradient. From Eq. (4.5) we may evidently write the appropriate terms in the heat flux equation as h i

-

=***

- Ui,kUkd + 2Up,,1piq,

(4.17)

and in the Reynolds stress equation as 2up,q1piqj+ 2upsqIpjqi - ui,km - U j , k m ~ i . (4.18) Again, if we rotate to principal axes of the Reynolds stress, and consider the vanishing of, say, the one-component, then for the time derivative of the one-component to vanish we require lplql a dpq. We have used the fact that the velocity gradient U i , j is arbitrary relative to the principal axes of -the Reynolds __ stress and that Ui,i= 0. If we now rotate to principal axes of 02uiuj - 8ui Buj , and apply Eq. (2.9), supposing that the one-eigenvalue, say, vanishes, i.e., that the correlation is perfect in the one-direction, we obtain

uiuj =

* * *

-

-

~21p,q1 = 8% 1,1,

+ Adpq,

(4.19) ~

--

where A is arbitrary, and use was made of the fact that P u i u, = Bui Bu, ,the mean velocity gradient is arbitrary, and Ui,i = 0. We have in addition the lpiqj = lipqj; various symmetry and other conditions on lPw:lpiei= lpijq; lppqj = ii&(which we will term normalization); lpiij = 0 (incompressibility). We can add, finally, that if the turbulence is isotropic, we must have (Crow, 1968; Rotta, 1951a) -

1P4V.. = (4dijdpq - dpid, - dpjdqi)q2/30

(4.20)

obtained by integrating the isotropic form of the spectrum (Lumley, 1970a). The realizability conditions above are new, but all other conditions have been known for some time. The conditions other than realizability can all be satisfied by a linear combination of terms in bij: Iijpq

+ (APijbpq - (i+)(aipbjq + diqbjp + djpbiq + djqbip)l + (?/11)[5dijbpq - (dipbjq + d i q b j p + d j p b i q + djqbip)I

= C[bijapq

+ (?/30)[46ijdpq - dipdjq- diqdjp).

(4.21)

This form was independently devised in different ways by a number of different workers (for a full discussion, see Launder, 1975). The constant C is not determined by the various conditions, and is usually determined empirically. However, different workers obtain different values, when different flows are used for calibration. This is probably because the form (4.21) does not satisfy realizability for any value of C, as can be easily checked. If we

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151

,

transform to principal axes of b,, suppose that b , = -4,and form Iilplrwe find that it is zero for i # p. However, for i = p # 1 we find ZPlp1 = -bp,(3C

+ a ) / l l + C/11 - ?/330

(no sum)

(4.22)

while for i = p = 1 we find

l l l I l= -C/11

+ 2?/55.

(4.23)

Thus, there is no value of C which will satisfy realizability. Depending on the value which is chosen, this term will either oppose the growth of the offdiagonal terms of the Reynolds stress, not permitting them to grow large enough in some geometries, or permit them to grow too large. In case the other two components of the velocity are equal, we could take C = 2?/5, which would then satisfy realizability, since the diagonal components would then be equal. Launder (1975) suggests a value of 0 . 2 7 5 a (our C = c2 2 / 2 in Launder’s notation). The situation with regard to realizability relative to the heat flux is even worse. There is essentially no relation between IP,,,and It is evident that the failure of realizability is due primarily to the terms of the form 6 , bip, etc., in (4.21), since on setting j = q = 1, we are left with bip,which cannot be made to behave properly. Hence, the entire group of four terms (required by symmetry) must be excluded. It is possible to satisfy all symmetry, incompressibility, normalization, and isotropy conditions with the following expression :

This does not satisfy realizability, however (relative to the Reynolds stress), since in principal axes of bij, if b , , = -$, we have Iilpl

= (?/30)(36i,6p,

- dip).

(4.25)

This, at least, has only delta functions, which could be canceled by the addition of another term. We must create a function to add to (4.24)which is symmetric in i - j and p - q, which has zero trace when i is summed on j and when j is summed on p, which vanishes under isotropy, and which, in principal axes of b,,, if b , = -4, will produce only terms like those of (4.25) so that a coefficient can be chosen to cancel them. Note that the d i p can remain (since the whole expression will be multiplied by U i , ,which is incompressible). It is necessary to go to third order in order to obtain such a term (note that, to be sure of having only terms like those of (4.25)we must never have bi, appearing):

,

bib,,

+ (11/3)bijhPq+ (211/3)6ijb,, + (2Z11/10)6ij6pq- (3111/10)(6ip6jq+ 6jp6iq).

(4.26)

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John L Lumley

This vanishes under isotropy, has both traces zero, and in principal axes of bij, if b,, = -4, reduces to (taking the ilpl component) dil dP1(-& - 11/3 - 111/10) - (3111/10)dip.

(4.27)

We can satisfy realizability relative to the Reynolds stress by adding (4.26) to (4.24) with a coefficient 27?/( 10 + 9011 + 27111). If we examine the realizability relations relative to the heat flux, we again find that it is not possible the_ general relation, but it is possible to satisfy the relation for the to satisfy_ case of 8ui 8ui = pa,when all components are perfectly correlated (or only one component exists) by the combination of (4.24) and (4.26). Note that there are eight independent expressions which all satisfy the requirements of symmetry, incompressibility, etc., any of which can be multiplied by (4 + 3111 + 11) and added to the combination of (4.24) and (4.26) without condition _ on either heat flux (approximately) or affecting the realizability_ then 4 + 3111 + 11 = 0. Reynolds stress, since if 8ui 8ui = These expressions are forbiddingly complicated, but where is it written that turbulence must be simple? The principle is simple; the result is an odd tensor polynomial on a bounded domain. The complexity is primarily so that the expression will say the same thing regardless of the geometry. It should be emphasized that these expressions are by no means the last word, especially since they have not been tested by computation. They are predicated on the simple form for Zijk, which is, in some respects, the easiest to model. A more complex form could be constructed for Zijkr with constants which could be adjusted to conform to experiment, and which would then lead to more complex forms for Z i j and Zijkl. The failure to satisfy realizability exactly relative to the heat flux may be serious, and it may be necessary to construct these more complicated forms for this reason. These interim forms at least point the way toward improvements on the forms in current use, which do not satisfy realizability in any sense.

e'z

V. The Dissipation Equations A. THEMECHANICAL DISSIPATION The equations for the mechanical and thermal dissipation are in the sorriest state. Whereas realizability and considerable physical reasoning has given substantial form to the equations for the second and third moments, when the equations for the dissipations are reduced by high-ReynoldsPeclet number assumptions (Lumley and Khajeh-Nouri, 1974) what is left is

B + E , i ui +

io+ Eo,i ui+

= - (2/?)+, = - (EoZ/P)Jlg,

(5.1)

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153

where the dimensionless invariant functions on the right-hand side contain the entire mechanism for the production and destruction of the dissipations. The situation is particularly difficult, because while the dissipation occurs at small scales, it is fixed by large-scale mechanisms; what we are really trying to determine is the rate at which energy or temperature variance is passed along the spectral pipeline to scales at which it can be attacked by molecular transport (see Tennekes and Lumley, 1972, Section 3.2). If the molecular diffusivity is changed in a turbulent flow, it simply changes the scale at which the dissipation occurs, but does not change the level of the dissipation. Since the spectral transport is related to the skewness of the velocity differences at two points (Monin and Yaglom, 1975, Section 21.4) for intermediate separation distances, it would probably make sense to try to model the equation for this quantity, as suggested by D. C. Leslie (personal communication). In the meantime, we must do what we can with what we have. If the Reynolds stress equation is regarded as an equation for the dissipation, we may probably conclude that I(/ is a functional of the history of the Reynolds stress, the heat flux, and the dissipation, as well as the mean velocity gradient, the gravity vector, and possibly the viscosity. Making the same assumption regarding slow changes, we can write II/ as a function of these quantities:

*

*(w,G,E, ui,j,

(54 We cannot exclude the mean velocity gradient and buoyancy vector as we did previously (in other equations) because it is not at all clear that (5.2) is determined solely by the state of the turbulence at a given instant. It is certain, for example, that if the buoyancy vector is zero, a heat flux cannot affect the mechanical dissipation. It also seems clear that the sudden imposition of a mean velocity gradient cannot have any immediate effect on the rate of change of dissipation if the turbulence is isotropic; on the other hand, additional sources of production in the energy equation must cause changes in the level of dissipation. We can thus conclude that the heat flux and buoyancy vector must occur as a product and the mean velocity gradient probably cannot occur without the anisotropy tensor. We may select a collection of nondimensional groupings in (5.2), as =

pi,

-

v).

$ = $(blj, q2ui,jfGGPjh RI)* (5.3) This must now be a function of the invariants that can be constructed from these quantities. Unfortunately, they are numerous, particularly because the mean velocity gradient is not symmetric. Pope (1974; see also Spencer and Rivlin, 1959, 1960) has examined a related problem. A complete list includes 22 invariants, even with the restrictions we have placed on the problem. It is clearly impossible to determine the dependence of a function of 22 variables by any imaginable set of experiments. The list is usually restricted to include only quantities of first degree in the mean velocity gradient and the buoy-

John L Lumley

154

ancy vector on the principle that these may be regarded in the first instance as weak. Such a restricted list is: -

$ = $(ll, I l l , bijq2Ui,j/E,b;TUi,j/E, & f i i / E ,

Gbijfij/E, &bifij/E, R,). (5.4)

It is also consistent with the position that the mean velocity gradient and the buoyancy are weak to expand (5.4) in a series in these quantities, keeping only linear terms: $ = $0

+

$1

+

bijq2Ui,j/E $2b;TUi,j/E

+ 4b3&Bi/Z + $4&bijfij/E + t,bsGbifij/E,

(5.5)

where the coefficients are functions of I I , I l l , and the Reynolds number. Even these are not all used. For example, Zeman and Lumley (1978) use $0

$3

= 3.8

+ 6011/(1 + 3( -211)1'2),

=

+0.95,

=

-3.8,

= 20/( 1

+ 3( -211)'12).

(5.6) These forms are determined on a largely empirical basis; the forms in the denominator of $o and t+b4 are found necessary in computation of the buoyancy-driven atmospheric surface mixed layer, which becomes very anisotropic near the inversion base. The leading constant is the only one that can be determined cleanly: the data of Comte-Bellot and Corrsin (1966) give $4

= 3.78

- 2.77R;

'I2

+ 18.1811 +

(5.7) for large Reynolds number and very small anisotropy (Lumley and Newman, 1977). The correlation coefficient for the Reynolds number variation is0.8, while that for the variation with anisotropy is 0.33, and hence this coefficient should not be taken too seriously. If we consider turbulence without production from either mean velocity gradient or buoyancy, then there are two limiting cases for which values of the leading coefficient can be determined: in the final period of decay, for very small Reynolds number and all anisotropies, Lumley and Newman (1977) determine that $o = they also obtain the same result for one-dimensional turbulence and arbitrary Reynolds number, i.e., for If = -3. In the absence of any other information, they suggest a simple interpolation formula relating these values: $0

v;

+ 0.980 exp[ -2.83R; '/'I[

1 - 0.33

In( 1 - 5511)l.

(5.8) At observed values of anisotropy, this does not differ much numerically from the form used by Zeman and Lumley (1978). $o =

Computational Modeling of Turbulent Flows

155

It has been known for some time that computation schemes of the type discussed here, when calibrated against plane flows, will not predict the development of axisymmetric flows properly, as discussed by Pope (1977). Specifically, the spreading rate of a round jet is overestimated by about 40% (Pope, 1977). Pope suggests that this is due to the omission in (5.5) of a term of the form (Ui,j - Uj,i)(Uj,k- Uk,j)(Ui,k+ Uk,i). The spectral transfer is produced by the stretching of large-scale vorticity (Tennekes and Lumley, 1972, Section 8.2), and this term is intended to represent this process; in particular, the term vanishes in two-dimensional mean flow, when the mean vorticity is normal to the plane of the mean strain rate. In axisymmetric flows, however, this is not true, and the term is nonzero. The inclusion of such a term restores the correct spreading rate for the axisymmetricjet. I feel, however, that the term suggested cannot be quite right, because it is independent of the anisotropy of the turbulence. That is, if the turbulence is isotropic, it should make no initial difference what the geometry of the mean velocity field is. Another possibility presents itself immediately: in developed turbulence, to a zeroth-order approximation, one can set approximately bij oc (Ui,j+ U j , i )(Lumley, 1967), and this can be substituted in the expression proposed by Pope. Evaluating the expression in a plane shear flow, we fmd that it will vanish if b,, (normal to the plane of the shear) vanishes, which it approximately does in plane deformations. This suggests other possibilities that have the same property, in particular 111; it seems more rational that +o should depend on 111 than on Pope’s (1977) suggestion, but this must be determined from a comparison of the decay of isothermal anisotropic turbulence without mean velocity gradients in the axisymmetric case, when 111 # 0, and in the case of plane deformation when it is approximately zero. It is unfortunate that the data are so poor; the determination of t,b from experimental data is equivalent to determining the second derivative of the data, and very little data are sufficiently accurate to permit this. We can adduce one rigorous condition, which is unfortunately of no help in determining the form of our equation. For realizability, it is necessary that e vanish if 7 vanishes. Hence, the F equation must be arranged so that, if vanishes without F, will become infinite so as to drive B to zero. As it stands (5.9, or even (5.3), has this property. We may introduce one further possible criterion. At the edge of a wake or jet, or thermal plume, the production closely balances the dissipation as both vanish (Tennekes and Lumley, 1972, Section 4.2). We have already considered the necessity for assuring that the dissipation vanishes when the velocity vanishes; here, however, we must say something stronger to assure that they vanish together at the proper rate. If they are to vanish at such a rate that (?)”/E is bounded for some n, and production and dissipation are of

+

156

John L Lamley

the same order, then we should have

zz = nz?

= 2nE(P, - e),

(5.9) where we have written P , for the total production, and (5.9) should be valid as both energy and dissipation vanish and production is of the order of dissipation. All the terms in Eq. (5.5) will be of the same order; hence, this suggests that $?, $4, $5 should not be present, and that -4b3 = = $o = 2n. Since +~I! is at least 3.78, n = 1.89 or larger so that the dissipation will go to zero faster than the energy as observed (Tennekes and Lumley, 1972; Section 4.2). Elimination of 4b4 would have caused difficulty in the calculation of Zeman and Lumley (1978); however, what was needed was a reduction of the dissipation production in the case of nearly onedimensional turbulence; this could be achieved in other ways, say, by the introduction of a factor of 4 + 3111 + 11. The equality between and -$3 we will see again when we discuss the equation for the dissipation of temperature variance. B. THETHERMAL DISSIPATION We can now consider the second part of (5.1), the equation for the dissipation of temperature variance. Proceeding in the same way, we obtain the These allow the same set of variables, with the addition of Ee, p,and addition of several invariants and a ratio, r = ie;fz/esE, the time scale ratio, which can appear as a variable in all the coefficients. If we consider the case of decaying anisotropic turbulence without production of any kind, we have

_ -

---

I+P= $ o e ( ~ 111, ~ , r, R,, 8U,i&$FTTI euibijeuj/iPq2,euieujb;/iPz), (5.10) Some authors avoid the entire problem by assigning a constant value to r (Spalding, 1971). However, it seems likely that r may vary depending on local conditions of transport and production. We may form an equation for r, obtaining d In r/dr = r(2 - i,hoe) - (2 - $o). (5.11) Extensive measurements by Warhaft and Lumley (1978b) indicated that, when the turbulence was isotropic, there was absolutely no tendency for r to change during the decay; it could be set at any value initially and would maintain that value until the end of the tunnel. This suggests taking, at least for the isotropic case, (5.12) i,ho0 = 2 - (2 - i,b0)/r. It also suggests that any tendency for the ratio r to tend toward an equilibrium value must be due to anisotropy and/or the production terms, which

Computational Modeling of Turbulent Flows

157

are producing temperature and velocity fluctuations from gradients by the same mechanism. We can write by analogy with (5.5) $* = *oe

+

(5.13)

$18zi&3,i/Eg,

where we have suppressed the other terms, taking the position (at least temporarily) that the temperature equation should not depend directly on the buoyant production or mechanical production (Newman et al., 1978a). If we form the equation for r, we find (designating the mechanical production by P,, the buoyant production by P b and the thermal production by P,) d In r/dz = i - ( t / ~ ~-* 2jP,/&

+ (2 - $l)Pm/E+ (2 + $3)Pb/E- r$$ + (5.14)

where we have designated by $: and the departures of $oe and $o from their isotropic values. Let us consider first an anisotropic decaying turbulence without production. If r has reached its equilibrium value re, which must be determined later, then we expect the time derivative to vanish, so that we must have re $$ =

(5.15)

If it is assumed that $$ is not a function of r, (5.15) fixes its value. Now consider an equilibrium situation with production, but without buoyancy so that P, = 0. We expect that P , = E, and P, = E ~ If. r has reached its equilibrium value re, we must again have the time derivative vanish, which gives (5.16) re($1*- 2 ) = $ l - 2. If we now introduce buoyancy, and let P , + P b = i, we find - $ 1 = $ 3 . Although their forms were somewhat different from those we have assumed, the above relations (with the exception of - $ 1 = t+b3) were approximately satisfied in the work of Zeman and Lumley (1978). These relations provide for an orderly relaxation of the time scale ratio whenever there is anisotropy or production. We obtain, making use of these various relations, d In r/dr = (2 - $,)[(P,

+ P&

-

+

(P,/EB)r/re] $o'(l - r/re).

(5.17)

The fact that the equation for dissipation of temperature variance does not contain, with these assumptions, terms quadratic in the heat flux, is not serious. Newman et al. (1978a) found from an analysis of the data of Alexopoulos and Keffer (1971) that the inclusion of such a term was not warranted by the data. We must now determine a value for re. For vanishing anisotropy and infinite Reynolds-Peclet number it is clear that the number should be near

158

John L. Lumley

unity. A test field model simulation (Newman et al., 1978a; Newman and Herring, 1978) suggests the value 1.0. However, the measurements of Warhaft (Warhaft and Lumley, 1978b) suggest a value of r = 1.34 when the three-dimensional velocity and temperature spectra peak at the same wavenumber, which one would expect in a real equilibrium situation. For a vanishing Reynolds number and all anisotropy, we have the final period of decay. Corrsin (1951)has shown that then re = 0.6. For large anisotropy we may consider one-dimensionalturbulence. Newman et al. (1978a)consider a somewhat special type of one-dimensional thermal turbulence which is isothermal in the direction of the unique velocity component, and take a limit from the not-quite-one-dimensionalcase (because the limit is singular); they find that re has the same value for all Reynolds numbers as it does in the final period of decay. Having only three limiting values of re,it is reasonable to construct an interpolation formula among them, the simplest possible. Newman et al. (1978a) take the variation to be of the same type as was assumed in Eq. (5.8). If the value of 1.34 is taken as the value for re at infinite Reynolds number and vanishing anisotropy, then we have (5.18)

i+bo - 2 = 4r,/3

for all Reynolds numbers and anisotropies, since the limiting values of the two quantities are then related. Putting these various relations together, we obtain

he = 2 - (2 - h 0 ) / 1 + 4 ( h - h 0 ) / 3 ( h - 2), = 2 + 4(Ic11 - 2)/3(h - 21,

(5.19)

where $oo is the isotropic value of @o. The following forms were used by Zeman and Lumley (1978): t,hoe = 3.0(1 =

__

__

+ r/4) - 3ohi&~/r2e2 q2,

+0.97.

(5.20)

Distinction among these various forms will require more experiments on isotropic decay with heat flux and anisotropic decay with heat flux, followed by the same flows with mean velocity and temperature gradients. From the work done so far, it is certain that terms involving the mean gradients of temperature and velocity must be present in the equations, as well as the terms involving anisotropy. Note that we have a realizability requirement on the dissipation of temperature variance: if the temperature variance vanishes without the dissipation, V ,I must become infinite to force the dissipation to zero. However, this condition is satisfied.

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159

C. THETRANSPORT TERMS In the following section we will discuss the transport terms in the equations for the Reynolds stress and the heat flux. However, the transport terms in the dissipation equations d o not fit in the same picture, because it is not possible to write exact equations for the transport of dissipation, as it is for the transport of velocity and temperature variance, etc. Hence, we will discuss these terms here. We may again apply the somewhat tentative condition that as the dissipation and the velocity variance vanish, they should d o so in such a way that the left member of (5.9) is satisfied. When we wrote (5.9) we had in mind a homogeneous situation, but now the transport terms must also be included. Thus, we must require

+ 2p/p)uk],k = ($b(q?u,),k

=~[(qz

?(Ek),k

(5.21)

as the velocity variance and the dissipation vanish, where we have made use of the homogeneous approximation for the pressure transport (see Section VI). We may satisfy this requirement by replacing throughout in the expression for &k (see Section VI) ( w j ) , k

(5.22)

=wjE,k/M

42

which supposes that b,, remains finite as vanishes; if we then write where q2ukis the expression with the replacement (5.22), the condition (5.21)will be automatically satisfied. This also has the advantage of being independent of the value of n. For example, in the nonbuoyant case we obtain from (6.54)

mk = (3%/5?)&k,

m k

= -(+)(?/&)[3/(10 -(:)Eipu,/u,/q2

+ 4C,)][&,p(u,p + 2 W k w / ? ) -(

~ ~ , ~ p u p / q z ] . (5.23)

Exactly the same consideration can be applied to the transport of Ee. If we and Ee vanish together so that (e')n/Ee remains bounded, presume that then one should require

n(82u,),kEe= @(m),k. This may be satisfied by replacing in the expression for

(eUJj =eu,EB,i/2nZe,

(5.24)

6

= e'ze,j/nEe

(5.25)

and calling the resulting expression (p/rEe)EE. In the nonbuoyant case this results in ___ --= - (e'/~~)[j( 1 C ~ / ~ ) ] [ E ~ , ~ euj ( We u k / e z )- ( + ) ~ ~ . ~ e ~ ~ (5.26)

+

+

e~~/e~

John L Lamley

160 VI. Transport Terms

A. INTRODUC~ION In the equations for second-order quantities, in inhomogeneous flows transport terms appear which are of third order. For example, in the equafor the transport term is ),k; in the equation for it is tion (ui0uk),k. These represent the divergence of the flux of the quantity in question, produced by the fluctuating velocity. In many flows, these terms represent the principal source of energy, heat flux, etc.: for example, in the atmospheric surface layer driven primarily by surface heating, the erosion of the inversion base, and gradual thickening of the surface mixed layer, is due entirely to these terms. To close the equations for second-order quantities, we must somehow express these terms as functions of the second-order quantities. If we approach this problem from the point of view of invariant modeling (Lumley and Khajeh-Nouri, 1974), we would say that, in the nonbuoyant (ignoring the Reynolds case, u,ulu, must be a functional of B and number dependence). Since the third moment must vanish in the case of homogeneity, it must depend on gradients of the arguments of various orders. It is possible to write an expansion, supposing weak anisotropy and inhomogeneity; to first order this gives:

(w

w,

q,

where a and b are unknown constants. This is clearly unsatisfactory,since it makes no distinction among the components. To second order, six more constants are introduced. Each of the fluxes (say of temperature variance or heat flux) will contain as many more undetermined constants. From a practical point of view, there are not enough well-documented experiments to unambiguously determine all these constants. In addition, it is not clear that second order is high enough. We have already seen, in connection with realizability, that it is not generally sufficient to consider anisotropy small in any sense. What is needed is a physical model which will produce a form for the fluxes, consistent with our general form, but with all the constants determined. Classically (see, e.g., Tennekes and Lumley, 1972) a simple mixing length or gradient transport argument was used in a simple shear: (q2/2 + P/p)O

- vT a(?/2)/ay

(6.2) with no attempt made to consider tensor properties. The eddy viscosity used is the same as that obtained (not unambiguously) from the ratio =

Computational Modeling of Turbulent Flows

161

-iiB/(dU/dy). This type of model is currently used in invariant form (Lewellen, 1975):

6= -c(F),jqEjqz/E,

(6.31

where the constant still must be determined from comparison of predictions with some class of experiments. The ?/E may be replaced by some other scalar combination of the correct dimensions, depending on the author’s preferences. With regard to uiuiu,,it was recognized early on (Hanjalic and Launder, 1972) that there was no justification for writing -

- ( ~ ) , ~ q 2 / ~

uiujuk

(6.4)

since the three velocities have equal standing and should be treated equally. Hence, we should write something like uiujuk

(mu,),* +

-(T/c)[(wj),lm +

(vk),m]. (6.5)

The constant of proportionality was optimized by Launder e f al. (1975) at 0.055. Many authors find this forbiddingly complicated, particularly in more complex flows. Since there is not much theoretical justification for it, and since the constant still must be determined from experiment, many prefer to use a simpler form such as (6.4). For example, Daly and Harlow (1970) use (6.4); Launder (1975) recommends a value of the coefficient of 0.125. Hanjalic and Launder (1972) tried to provide justification for the form (6.5). If the equation for ~ i ~ isj written, ~ k it consists of the substantial derivative; production terms of the form U i , , , T and its permutations; and its permutations; dissipressure gradient terms of the form - p . i / u , l p pation terms of the form - vui,,,uj,,, uk - vui,,,uj uk,,, and its permutations; and finally, on the left-hand side a collection of terms of the form:

(m),, -

-

wj(mU, ),l wk(m),I u j ( q I ) , I -

(6.6)

Hanjalic and Launder (1972) suggested the following set of assumptions: neglect all substantial derivatives, production, and dissipation terms; since any rapid terms arising from the pressure gradient correlations will be of the same form as the production terms, neglect them also; replace the pressure gradient correlation with a relaxation term proportional to -

- ui ujuk q2/c

(6.7)

with an undetermined coefficient; in the expression (6.6) introduce a quasiGaussian assumption (Lumley, 1970a): ___. -- -~~

UiUjUkUl

= uiujukul

+

uiukujul

+ ujukuiul.

This collection of assumptions leads to the form (6.5).

(6.8 1

162

John L Lumley

The same set of assumptions was applied to the buoyant case by Zeman and Lumley (1976) with excellent results. Forms for the fluxes were produced which can be identified with definite physical mechanisms (Lumley et al., 1978). The relative success of this technique suggests that the assumptions may be justifiable in some way; that is, there may be some single guiding principle from which they all follow naturally. We can ask a number of questions and make several comments which may help to shed light on this: homogeneous turbulence is observed to be Gaussian in the energy containing range (Frenkiel and Klebanoff 1967a,b) even in the presence of nonzero velocity gradients (Marechal, 1972); departure from Gaussian behavior, therefore, is associated with inhomogeneity (which is clear because nonzero values o f i p j i i k are non-Gaussian and are fluxes); hence, (6.8) is presumably correct only in a homogeneous situation and should carry a correction for inhomogeneity; it would be possible to justify neglecting fourth cumulants (recall that all cumulants of second order or higher vanish for a Gaussian) in an equation for third cumulants if cumulants of successive orders had relaxation times which were successively shorter; if inhomogeneity is responsible for nonzero third moments, a term like U i , , , m is of order &/El relative to (6.7), and if this quantity is small, we could justify neglecting these terms (we have taken Ui,,,to be of order u/l where u and 1 are the scales of the energy containing eddies); finally, how d o we know that (6.7) is the right form? Other linear combinations of third moments are equally attractive. It seems clear that what is needed is a model of turbulence that will relax to Gaussian behavior in the absence of inhomogeneity, successive cumulants relaxing faster, and from the equation for the density (or equivalent quantity) equations for the moments may be obtained. The model must be constructed in such a way that our second moment equations are reproduced unchanged, and the third moment equations are reproduced with as few changes as possible; that is, there are presumably constraints on possible forms, consistent with all moments being derivable from a single density which will relax to a Gaussian distribution in the absence of inhomogeneity. This is a type of realizability although somewhat more subtle. B. A GAUSSIAN MODEL In order to examine these possibilities further, it is simpler to work at first with a passive scalar quantity; the added complexity of a vector quantity such as ui is a nuisance. Consider a quantity 8:

B + eViui + oSiui+ e,iui- 6=

(6.9)

Computational Modeling of Turbulent Flows

163

Since both 8 and ui appear in this equation, we will have to generate an equation equivalent to that for the joint density of 8 and ui. It is convenient to work in terms of the moment generating function, since its derivatives at the origin are the cumulants and it is quadratic for a Gaussian distribution: F = In If the equation for ui iri

G,

G = exp[ik,u, + i ~ ] .

+ ui,juj+ ui,,uj+ ui,juj -a= - p , , / p + pie + vui,jj

(6.10) (6.11)

is multiplied by iki that for 8 is multiplied by il, the two are added, and if the resulting equation is multiplied through by G, averaged and the resulting equation divided by G, we obtain (with a little rearrangement)

F

+ aF/aXj uj + (ik, U,,, + ilO,,) aF/aik, - ik,& aF/ail

+ a2F/axj aik, + (aF/axj)(aF/aikj)- (u,In G),, ~= -ik,p,,G/pG + vik,Gu,,/G + yilG8,jj/G. -

-

--

(6.12)

We have used the fact that G-I

a2G/axjdikj = a2F/axj aik, + (dF/axj)(aF/aikj)

(6.13)

from the definition. Now, the terms on the right-hand side of (6.12) may be rearranged in suggestive forms, and physically appealing assumptions may be made directly regarding the terms. However, one has relatively little feeling for the moment generating function and how it may interact with other quantities. It is more productive to adopt the position that we know what equation we wish to obtain for second order (specifically, the terms on the right of (6.12) must give rise to the rapid terms, the return to isotropy, the pressure transport, and the dissipation) and select for the right-hand side of (6.12) the simplest form that will (a) relax to Gaussian in the absence of inhomogeneity, and (b) will produce the correct second-order terms. Let us consider a joint Gaussian distribution. The moment generating function is given by:

F = - & [ k i k j m + 2 k i l d + l2P].

(6.14)

Let us consider only the return to isotropy terms and the dissipation terms; we will deal with the rapid and pressure transport terms later. Using the simplest forms, we have (with C1 and C, as defined by Zeman and Lumley, 1978) u:u. 1 1 + ... = -C1(E/&(iZpj - 6,,7/3) - 6,,2-?/3, ui +

$+

* a *

=

- Ce(E/z)Q,

*.. = -2$.

(6.15)

John L Lumley

164

The ellipses indicate the omitted terms: production, transport, pressure transport, rapid, etc. We can form the time derivative of F :

p = ... = -i{ki k j [ - Cl(C/z)(ii& - Sij?/3) - 6i,Z/3]

+ 2ki I[ - C , ( E / ~ ) Q +] lz[-2(G/?P)F]},

(6.16)

where the ellipses indicate the other terms omitted on the left side of (6.21)as well as those arising from the rapid and pressure transport terms. Now, in a Gaussian distribution iipq = a2F/aikidikj = F,ij, -

uie = a2F/aikiail = F,il,

(6.17)

-

e2 = a2F/aii ail = F , ~ ~ ,

and we propose as the simplest possible generalization of (6.16)

p + ...= - f { k i

k j [ - ~1 ( E / Z M F , i j

- dij ~ , p p / 3 ) S i j ~ . p p W @ I

+ 2ki I[ - C,j(E/z)F,iJ+ 12[ -2(F,j/P)F,11]}-

(6.18)

In the absence of inhomogeneity, buoyancy, and mean gradients, the terms not shown explicitly in (6.18)vanish. The equation then has solutions other than Gaussian, but which all relax to Gaussian. This is easiest to see in the scalar case, setting k = 0. Then

p = 12 ( E- e /7e

)F,ii*

(6.19)

Indicating the nth cumulant by C,, we have (differentiating n times with respect to i l ) (6.20) C, = -n(n - I)(C,j/eT)C,. If we normalize by Cyz, designating P, = C,/C;'2, we have P, = - n(n - ~)(E,/P)P,.

(6.21)

Hence, all normalized cumulants above the second decay to zero, no matter what the initial values, leaving a Gaussian distribution. The choice of right-hand side in (6.18)is not as arbitrary as it seems. To second order, the - lz?P in (6.16)can be replaced only by - 12F,11,ilFv1,or 2F. The second does not converge to Gaussian, producing constant P,; the third diverges, the coefficient in (6.21)being positive. Hence, the only further generalization would be the inclusion of third and higher derivatives. Equation (6.18)appears to be adequate, however, since it produces terms at third order which are also obtained by simple physical reasoning. For example, in the equation for e", the dissipation term may be written as -

e3/3

-~

+ -.. = y e z e , j j = y ( e 2 e , j ) , , - 2ye,je,je 1:

-22z -2(~,j/?P)8"

(6.22)

Computational Modeling of Turbulent Flows

165

(Zeman and Lumley 1976), where we have used the usual high-Reynolds number approximation, and the idea that fluctuations of O2 and of Eg should be well correlated, so that regions of high O2 dissipate most of it locally. From (6.18) we have (6.23)

which is the same as (6.22). The same is true of the dissipation terms produced at third order from the mechanical part of (6.18Fthey are identical with those obtained from physical reasoning [like that leading to (6.2211.

C. ORDEROF MAGNITUDE ANALYSIS Let us consider now the full equation for the nth cumulant of 8. This equation contains no terms arising from pressure transport or rapid terms, or buoyancy, so that we may obtain it directly from (6.12), using (6.18) as a right-hand side. Cumulants involving velocity also appear, and we will need a designation for them; let us say C . for the cumulant involving 8 n times - 38 uj8. We will consider n greater than 2, and uj once. That is, C3j = so that the last term on the left-hand side of (6.12), being linear in the independent variables, will not contribute. We obtain

& Y-

(6.24)

We wish now to carry out an order of magnitude analysis on this equation, in the case of weak inhomogeneity,in an attempt to identify the more important terms. Let us designate by 8 and u the rms values of 8 and ui, respectively. We will write C , = One,,

Cnj= Pue,,

(6.25)

and we will make the assumption that enj= O(en+l).That is, we presume that since both variables share a distribution they will approach Gaussian together. To fix ideas, let us assume that we are in a nearly parallel, weakly inhomogeneous, steady, two-dimensional flow such as a late wake. Then, from the wake scaling relations, if L and 1 are, respectively, the downstream and cross-stream length scales, U1/L= O(u/l) (Tennekes and Lumley, 1972, p. 109). We also have @ , j = 0(8/l).Such terms as CnjSj are dominated by the

John L..hmley

166

cross-stream gradients, and are hence of order Cnj/l.The orders of magnitude become

c , ,uj: ~ endnull= n(n - l)(Eg/P)dnen[uP/GIn(n- I)], nO,jCn-lj:ne,Pu/I = n(n - l)(Eg/P)Pen[uF/coIn(n- I)],

c , , ~en+ , ~pull : = n(n - l)(G/eZ)Pen[(en+l/en)uP/EgIn(n- 111,

n(n - l)(Eg/@)Cn:n(n

-

l)&/e’)Pe, = n(n - l)(~/@)O”en[l]. (6.26)

Let us designate by 4 = up/&.Of course, in a real wake, or in any naturally occurring turbulent flow, q = O(1). However, we must consider an artificial situation, in which q is small compared to unity, a weakly inhomogeneous situation which could be produced in the laboratory. As q goes to zero, some term in (6.26)must balance the last term. Clearly the first two cannot, since they both vanish. The third is a possibility; this would give qen+l/enn(n - 1) = O(1)

(6.27)

but this would give e3 = O(2/q), e4 = O(12/q2),etc. Here we have used the fact that ez = 1 by definition. Hence, as q goes to zero, all normalized cumulants go to infinity. This is clearly contrary to observation since (as we have already commented) homogeneous turbulence is observed to be Gaussian. The only other possibility is that the quadratic terms balance the last term. Thus, for example, qezen- l/en = O(1)

(6.28)

(ignoring for a moment the various factors of n). This gives en = O(qe,- 1) (again using e2 = 1) and e3 = O(q)so that all normalized cumulants vanish as they ought to, and higher orders vanish faster than the lower orders do. All these terms are of the same order, the general term being given by qek en- k + 1 /en*

(6.29)

Computational Modeling of Turbulent Flows

167

If (6.28) holds, en = O(q'-2), (6.29) is also of order 1 . Hence, the entire collection of quadratic terms must be of order unity. Since there are n - 2 of them, each must be approximately of order ( n - 2 ) - :

'

(J

qe,e,-,+,/e,n(n - 1 ) = O(n - 2 ) - ' .

(6.30)

e, = O(k!(q/2)k-2),

(6.31 )

If

+

then the left-hand side of (6.30)is given by 2(n - k l)/n(n- 1). If we sum this from 2 to n - 1, we have 1 - 2/n(n - I ) , which approaches unity for large n. Hence, (6.31)must be approximately true for large k. Note that (6.31)does not imply that ek+ /ek+ 0 as k + 00 for fixed q, but rather that e,, /ek 0 for fixed k as q + 0. In fact, ek/ek- = kq/2; by picking a small enough q, we can make this ratio as small as we like for the first few cumulants; eventually, however, for k > 2/q, the cumulants will begin to increase. This is to say that the distribution can be made to look Gaussian in as large a neighborhood of the origin as we like, but that far enough out in the tails it will never look Gaussian. This is not a problem, however, since our concern was simply to be able to neglect fourth cumulants in the equation for third cumulants, and this is justified if q is small enough. Using (6.31),the third term becomes --+

qen+ 1 /enn(n - 1 ) = (q2/2)(n+ l)/n(n- 1 )

(6.32)

so that this is a second-order term in q. Hence, to the zeroth order we can write

(4)

C,,jC,-

,j

+ ... +

in

C,- I,jClj= - n(n - l)(Eo/p)Cn.

(6.33)

The neglect of the second-order term (6.32)corresponds to neglecting cumulants of order n + 1 in the equation for cumulants of order n. The neglect of the first-order term corresponds to the neglect of the substantial derivative and the production terms. Note that, in the case of cumulants involving velocity, which can be handled in exactly the same way, the rapid velocity gradient terms are of the same form as the production terms, and hence will also be of first order. We will deal separately with the problem of the buoyant terms, which requires special treatment. Thus, to zerdh order we have

__

30: Buj = - 6(%/8')8"

(6.34)

John L Lumley

168

and 6 E K j + 4e3eUj = - 12(G/e’)(F

- 3822).

(6.35)

Equation (6.34) is the standard mixing length approximation; if we took

o2 a - q u , ( i P / ~ J

(6.36)

we would obtain (6.34) on multiplying by 0, but with an unknown coefficient. Equation (6.34) involves the neglect of the fourth cumulant. To first order in (6.34), we should keep the substantial derivative and production terms (presuming that the time derivative is of the same order as the advective term):

8 + q U j + 30,,-

+ 3cBuj= -6(FB/ p”)e .

(6.37)

To second order we should include a zeroth-order approximation for the neglected term C,,,,; this order approximation for C,, will presumably be of the form (6.35), but we must derive it explicitly, since there may be other complications. Deardorff (1978) has pointed out that, if an equation such as (6.37) is carried for the third-order quantities, then diffusion terms (which will be provided by C , , , ) are essential to stabilize the solutions and prevent the development of spurious peaks. It should be noted that when we made our assumption (6.18), we were essentially neglecting derivatives of order 3 and higher, which we now see will be of first order in q. Thus, while (6.34) and (6.35) are consistent approximations, it is possible that (6.37) neglects terms of the order of those retained. TRANSPORT D. THEPRESSURE We must now turn to consideration of cumulants involving velocity, dealing first with the nonbuoyant case. The rapid terms, as already noted, will cause no difficulty, since they are of the same form as the production terms. The only problem is pressure transport. Traditionally, p,ij is split into a deviatoric part and an isotropic part, which is a transport term (though the split is not unambiguous: Lumley, 1975b). There appears to be no reason to split m k in a similar manner, particularly since it leaves as remainder Pi,ik + which one might expect to be poorly correlated (although one has little intuition for such correlations; in fact ui,,is evidently well correlated with p , and perhaps the two have a common large-scale part which is well correlated with uk). On the other hand, the entire pressure gradient correlation in the second-order equation consists of the rapid and

m,

Computational Modeling of Turbulent Flows

169

the return-to-isotropy parts and the pressure transport; if the pressure gradient terms in higher order equations are obtained from an equation such as (6.18), all must be present, their combination representing the entire term; the term obtained by generalizing the return-to-isotropy part alone being only a piece of the total. The pressure term in Eq. (6.12) may be written as - -

~-

+

-ikip,,G/pG = -iki(pG),i/pG i k i z / p c . The second term on the right may be rewritten as - -

- -

-kikjpui3jG/pG- kilpO,iG/pG

(6.38) (6.39)

-

which is of the form of (6.18). In the equation for ui uj , this produces P(uj,i + ui.j)/P while the first term on the right of (6.38) gives

(6.4)

Thus, the separation (6.38) represents the traditional separation into pressure transport and pressure-strain correlation. Since we are carrying to higher orders the generalization of the second term on the right of (6.38) (from Eq. (6.18)), we must also carry to higher orders the generalization of the first term on the right, so that we will have the entire term. Beginning from Eq. (4.3) in a homogeneous flow we may write

- [p'2']/p = ( K i K j / K 2 ) [ U i U j ] .

(6.42)

Multiplying by [uk]* and averaging and integrating, we obtain (6.43) where Sijk is the spectrum of w k . Although it proved to be dangerous with regard to realizability in Section IV, we will attempt here to express the integral as a linear combination of this triple correlation. If we define

we have the following conditions: Iijpqr = Zjipqr; Iijpqr = Iijqpr; Iijpqj =0; IiiPq = the last but one resulting from incompressibility.Realizability only requires that the term vanish if uk vanishes. The most general linear k contains five coefficients. The application of these conditions form in w determines all the coefficients, resulting in the form :

w,, Iijpqr

= ( f ) G i j u p u q u , - (&)(GirJqq&

+

Gj,ii&&).

(6.45)

John L Lamley

170 Finally, we obtain Iijijr

=

, = (314 1 ur/P - P(2

ur2

(6.46)

and since this does vanish if ur vanishes, it satisfies realizability. Just as our viscous terms-could be obtained through third order by generally replacing E by q2c/q2, the form of (6.46) suggests replacing everywhere (6.47) -PIP = (4*- ?)/5 so that the first term on the right-hand side of (6.38) becomes

- ?)I, (6.48) and this term could be added to the right-hand side of the equation for F. (+)iki[(a/axi

+aF/axi)(F,pp + ~

, p ~ , p

This cannot be correct for all orders, however, since for fifth order and above the cubic term in (6.48)introduces terms of lower degree in q than those we have kept, that is, terms of order q-'. This seems unlikely to be true and suggests that there is more wrong with (6.48) than meets the eye. In fact, there is the same ambiguity regarding the correct form as there was relative to the right-hand side of (6.18). We know the form we wish to produce at second order, (6.46), but several possible general expressions will produce this form. In particular, we could eliminate from (6.48) any term which does not contribute to second order, such as the troublesome cubic term; this could be done by eliminating either the second term in the first parenthesis of (6.48), or the second term in the second, or both. The1 first possibility leaves quadratic terms which contribute to third-order cumulants to zeroth order; computation with these forms shows that the contributions from the pressure terms very much over-correct, leading to negative diffusivities. Hence, we must discard this alternative. The second and third possibilities give no zeroth-order contributions to third-order quantities, but the second will produce zeroth-order contributions to fourth-order quantities. This seems unlikely, and lacking better information we select simply (f)iki(a/aXi)(F,pp

- q2).

(6.49)

Note, incidentally, that the inclusion of this expression produces in the equation for the heat flux a term of the form

-( e p ) , k / P = ( 8 q z ) , k / 5

(6.50)

(see Eqs. (3.6) and (3.7)). This term does satisfy realizability as required. E. ZEROTH-ORDER TRANSPORT TERMS We may now proceed to consider the equation for cumulants including velocity. Let us take first C , j , since it appears in the equation for the temper-

Computational Modeling of Turbulent Flows

171

ature skewness. We know from our discussion above that we may ignore substantial derivatives, production and rapid terms, and higher order cumulants, since our reasoning is equally applicable to equations for cumulants involving velocity. Expression (6.49) may be neglected like the third term in (6.26). Using the notation c,jk for the cumulant involving uj, uk, and temperature n times, we obtain for C,,

qG + 3 ( G ) , k G + 330% + 3(G),k82U, =

+ C & / ? ) ( P / G ) ] ( G - 3eZeUj).

-3@,/F)[1

(6.51)

We have neglected C 3 p pIf. this were inserted in Eq. (6.37), we would have the diffusion terms which stability requires. DifferentiatReturning now to third-order quantities, let us consider ing, we obtain to zeroth order

6.

This is close to the form assumed by previous authors, but here we have a definite value for the coefficient. The zeroth-order approximation for Bui uj is (8ui),kvj

=

Finally, for m

+

+

(%j),k%&

(mj),kak

-[c1(@)(G - 428dij/3) + (2E/3z)fiSij _+ 2c&//q2)8ui~j]. k

(6.53)

we obtain

(uij),pulrup + (u,u,),puiup + (uju,)*puiu, =

-3Cl(E/?)[wk

- ($)(dijfi

- (2E/32)(dij&

+

dikfi

+

+ 6ikfi + djkfi)] djkfi).

(6.54)

Note that if i f j # k # i, then the right-hand side of (6.54) reduces to the form (6.5), but with an explicit coefficient. However, the coefficients are quite different in the different directions. F. MODIFICATIONS In deriving Eqs. (6.15) and the following, we used very simple return-toisotropy forms. If we wish to be more elegant and make use of the material in Section 111, we must consider what is the proper generalization of Eq. (6.18).

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John L Lumley

In fact, there are many generalizations possible; probably the most convenient is to replace Cl by fi and Ce by &,, leaving the arguments of these functions unchanged; then, when the differentiations are carried out with respect to iki, etc., these quantities act just like C1 and C,, so that the expressions we have derived (6.51)-(6.54) will remain unchanged, except for the simple substitution. The next level of complexity, the substitution of expressions (6.17) in the arguments of /3 and 6fj,would result in the appearance of additional terms in (6.51)-(6.54) corresponding to the differentiation of these coefficients with respect to iki,etc. The additional complexity does not appear to be worth the effort, unless it proves to be necessary. That is, suppose that on computation the coefficients in (6.51)-(6.54)prove to be too large or small; they could be modified by including these terms and adjusted by adjusting the dependency of the coefficients in the Reynolds stress and heat flux equations. We must now consider the effect of buoyancy. If the equation for Cnjis written, a term of the form flj C,, appears on the ri t-hand side. Relative to the return to isotropy term, this term is of order POq /z&n(n - 1). We must make a decision relative to the magnitude of this quantity. One possibility is to say that the influence of buoyancy is weak, that is, the turbulence is primarily shear produced, and the buoyant influence is secondary. PO?/@ is jOlt/u2 where It is a turbulent integral length scale; in supposing this small we are s'aying that the buoyant acceleration PO is small relative to the turbulent acceleration u2/lt. If it is of order q, we can neglect it and the associated rapid terms just like the mean velocity gradient terms. On the other hand, if we are considering a flow that is produced by buoyancy, we must have jOlt/u2 = O(1) (Tennekes and Lumley, 1972, Section 4.6). The buoyancy terms will then be of the same order as the return to isotropy terms, and must be kept to zeroth order, that is, in the expressions (6.51)-(6.54). This, of course, increases the complexity considerably, since the rapid terms in buoyancy must also be kept. These rapid terms may be obtained by the substitution of the expressions (6.17) in the various rapid terms (see Section IV) followed by appropriate differentiation. A simplified version of this gives excellent predictions of the evolution of the buoyancydriven atmospheric surface mixed layer (Zeman and Lumley, 1976, 1978). In the order of magnitude analysis which we carried out on Eq. (6.24), other choices are possible, corresponding to other physical situations. For example, if we had said that &,/eZ = u/l, which is realistic, but that C,j,j = O(Cnj/L),while U i , j= O(u/l)and a,, = O(O/l),which would correspond to a near-equilibrium situation of a turbulence maintained by almost uniform gradients, with weak inhomogeneity, then our small parameter becomes 1/L and the order of magnitude analysis remains the same with the exception of the terms involving the mean velocity and mean temperature

e

Computational Modeling of Turbulent Flows

173

gradients, which become of order one; thus we would have to keep the mean velocity and mean temperature gradient terms, as well as the rapid terms in the mean velocity and mean temperature gradients in the expressions (6.51)-(6.54). Each situation should be considered on its merits. For example, at the inversion base over the atmospheric surface mixed layer, both mean velocity and mean temperature gradients are strong, while the turbulence has been produced primarily by buoyancy. In this situation we may expect that it will be necessary to keep both the buoyancy terms and the terms in the mean gradients in the expressions (6.51)-(6.54). An approximate form of this, keeping the buoyancy terms and the mean temperature gradients, has been worked out in Lumley et al. (1978). Finally, a word of caution is in order. Some situations are not in any sense almost Gaussian. The temperature fluctuations in a heated wake are a case in point. In the intermittent region, the probability density for the temperature fluctuations will have a spike, corresponding to the free-stream temperature, lower than the-local mean (Antonia and Sreenivasan, 1977). This will be combined with an almost Gaussian distribution corresponding to the temperature fluctuations in the turbulent portion of the wake with the average offset to higher temperatures. As one moves toward the axis of the wake, the magnitude of the spike decreases, but never completely disappears, even on the axis, since there is still a small but finite probability of finding oneself in a tongue of nonturbulent fluid engulfed by the wake. This combined density cannot be approximated in any sense by expansions about the Gaussian state. The part of the density corresponding to the fluctuations within the turbulent fluid probably can; the spike, however, must be treated separately. Moments of any order can be written explicitly as moments of the continuous distribution plus terms involving the displacement of the spike from the mean of the continuous distribution and the intermittency (the time spent in the turbulent fluid). A rational approach probably would involve dealing separately with the two parts, using our technique for the continuous part and handling the spike explicitly (Libby, 1975). These considerations apply to any intermittent situation, such as the edge of a wake or jet or boundary layer, the entrainment region at the inversion base in the atmospheric surface rnixed layer, etc. They apply, in addition, to the modeling of chemical reactions. Initially, the reactants are not mixed on a molecular level but may be macroscopically mixed. The density will consist of two spikes, if we imagine reactants such as acid and base that can be identified on a single numerical scale. As the reaction progresses, the spikes will disappear, and a continuous distribution of reactant will appear. Again, we may be able to apply our ideas to the continuous part, but the spikes should be dealt with separately.

174

John L Lumley REFERENCES

ALEXOPOULOS, C. C., and KEFFER, J. F. (1971). Turbulent wake in a passively stratified field. Phys. Fluids 14, 216224. ANMNIA, R. A., and SREENIVASAN, K. R. (1977). Conditional probability densities in a turbulent heated round jet. Australas. Hydraul. Fluid Mech. Con$, 6th, Inst. Eng. Aust. Nat. Conf. Publ. NO. 77/12, pp. 411-414. BATCHELOR, G. K. (1956). “The Theory of Homogeneous Turbulence.” Cambridge Univ. Press, London and New York. BATCHELOR, G. K.,and PROUDMAN, R. 1. (1954). The effect of rapid distortion of a fluid in turbulent motion. Q.J. Mech. Appl. Math. 7, 83-103. CHOU,P.-Y. (1945a). On velocity correlation and the solution of the equation of turbulent fluctuation. Q. Appl. Math. 3, 38-54. CHOU,P.-Y. (1945b). Pressure flow of a turbulent fluid between parallel planes. Q. Appl. Math. 3, 198-209. CHOU,P.-Y. (1947). Turbulent flow along a semi-infinite plane. Q. Appl. Math. 5, 346-353. COLEMAN, B. D., and NOLL,W.(1961). Recent results in the continuum theory of viscoelastic fluids. Ann. N.Y. Acad. Sci. 89,672-714. G., and CORRSIN, S. (1966). The use of a contraction to improve the isotropy of COMTE-BELLOT, grid-generated turbulence. J. Fluid Mech. 25, 657-682. CORRSIN, S. (1951). The decay of isotropic temperature fluctuations in an isotropic turbulence. J . Aeronaut. Sci. 18, 417-423. CROW,S. C. (1968). The viscoelastic properties of he-grained incompressible turbulence. J. Fluid Mech. 33, 1-200. DALY,B. J., and HARLOW, F. H. (1970). Transport equations in turbulence. Phys. Fluids 13, 2634-2649. DAVIDOV, B. I. (1958). Phenomenological equation of statistical dynamics of an incompressible turbulent fluid. Zh. Eksp. Teor. Fiz. 35, 527-529. DAVIDOV, B. I. (1959a). Statistical dynamics of an incompressible turbulent fluid. Dokl. Akad. Nauk SSSR 127, 768-771. DAVIDOV, B. I. (1959b). Statistical theory of turbulence. Dokl. Akad. Nauk SSSR 127,98&982. DAVWV,B. I. (1961). Statistical dynamics of an incompressible turbulent fluid. Dokl. Akad. Nauk SSSR 136,47-50. DEARDORFF, J. W.(1978). Closure of second- and third-moment rate equations for diffusion in homogeneous turbulence. Phys. Fluids 21, 525-530. F.N., and KLEBANOFF, P. S. (1967a). Higher order correlations in a turbulent field. FRENKIEL, Phys. Fluids 10, 507-520. FRENKIEL, F. N., and KLEBANOFF, P. S. (1967b).Correlation measurements in a turbulent flow using high speed computing methods. Phys. Fluids 10, 1737-1747. GENCE,J. N. (1977). Turbulence homogdne associee A des effets de gravite. Thdse, Docteur Ingenieur, Universite Claude Bernard, Lyon. K.,and LAUNDER, B. E. (1972). A Reynolds stress model of turbulence and its HANJALIC, application to thin shear flows. J . Fluid Mech. 52, 609638. KLME,S. J., MORKOVIN, M.J., SOVRAN, G., and COCKRELL, J. J. (4s.) (1969) Proc. Cornput. Turbulent Boundary Layers-1 968 AFOSR-IFP-Stanford. Thermosci. Div., Stanford University, Stanford, California. A. N. (1941). Local structure of turbulence in an incompressible fluid at very KOLMOOOROV, high Reynolds numbers. Dokl. Akad. Nauk SSSR 30,299-303. KOLMOOOROV, A. N. (1942) Equation of turbulent motion of an incompressiblefluid. Izu. Akad. Nauk SSSR, Ser. Fiz. 6, 56-58.

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LAUNDER, B. E. (1975). Progress in the modeling of turbulent transport. In “Lecture Series 76: Prediction Methods for Turbulent Flows.” von Karman Inst. Fluid Dyn., Rhode-St.Genese, Belgium. LAUNDER, B. E., REECE,G. J., and RODI, W. (1975). Progress in the development of a Reynolds stress turbulent closure. J. Fluid Mech. 68, 537-566. LESLIE,D. C. (1973).“Developments in the Theory of Turbulence.” Oxford Univ. Press, London and New York. LEWELLEN, W. S. (1975). “Use of Invariant Modeling,” ARAP Rep. No. 243. Princeton, New Jersey. LIBBY,P. A. (1975). On the prediction of intermittent turbulent flows. J. Fluid Mech. 68, 273-295. LUMLEY, J. L. (1967). The applicability of turbulence research to the solution of internal flow problems. In “Fluid Mechanics of Internal Flow” (G. Sovran, ed.), pp. 152-169. Elsevier, Amsterdam. LUMLEY, J. L. (1970a). “Stochastic Tools in Turbulence.” Academic Press, New York. LUMLEY, J. L. (1970b). Toward a turbulent constitutive relation. J. Fluid Mech. 41, 413-434. LUMLEY, J. L. (1972). Some comments on the energy method. In “Developments in Mechanics” (L. H. N. Lee and A. H. Szewczyk, eds.), Vol. 6, pp. 63-88. Notre Dame University Press, Notre Dame, Indiana. LUMLEY, J. L. (1975a). Introduction. In “Lecture Series 76: Prediction Methods for Turbulent Flows.” von Karman Inst. Fluid Dyn., Rhode-St.-Genese, Belgium. LUMLEY, J. L. (1975b). Pressure strain correlation. Phys. Fluids 18, 750. LUMLEY, J. L., and KHAJEH-NOURI, B. (1974).Computational modeling of turbulent transport. Adv. Geophys. 18, 169. LUMLEY, J. L., and NEWMAN, G. R. (1977). The return to isotropy of homogeneous turbulence. J. Fluid Mech. 82, 161-178. H. A. (1964). “The structure of Atmospheric Turbulence” (R. E. LUMLEY, J. L., and PANOFSKY, Marshak, ed.), Interscience Monographs and Texts in Physics and Astronomy, Vol. 12. Wiley (Interscience), New York. O., and SIES, J. (1978). The influence of buoyancy on turbulent transLUMLEY, J. L., ZEMAN, port. J. Fluid Mech. 84, 581-597. MARBCHAL,J. (1972). Etude experimentale de la deformation plane d’une turbulence homogene. J. Mec. 11, 263-294. MONIN,A. S., and YAGLOM, A. M. (1971).“Statistical Fluid Mechanics” (J. L. Lumley, ed.), Vol. 1. MIT Press, Cambridge, Massachusetts. A. M. (1975).“Statistical Fluid Mechanics” (J. L. Lumley, ed.), Vol. MONIN,A. S., and YAGLOM, 2. MIT Press, Cambridge, Massachusetts. J. (1978). Second order modeling and statistical theory modelNEWMAN, G. R., and HERRING, ing of a homogeneous turbulence. J. Fluid Mech. (to be submitted). NEWMAN, G. R., LAUNDER, B., and LUMLEY, J. L. (1978a). Modeling the behavior of homogeneous scalar turbulence. J. Fluid Mech. (to be submitted). NEWMAN, G. R., WARHAFT, Z., and LUMLEY, J. L. (1978b). The decay of heat flux in gridgenerated turbulence. J. Fluid Mech. (to be submitted). G. S. (1972). Numerical simulation of turbulence. In “Statistical ORSZAG,S. A., and PAITERSON, Models and Turbulence,” Lecture Notes in Physics, Vol. 12, pp. 127-147. Springer-Verlag, Berlin and New York. POPE,S. B. (1974). A more general eNective viscosity hypothesis. J . Fluid Mech. 72, 331-340. POPE,S. B. (1977). “An Explanation of the Turbulent Round Jet/Plane Jet Anomaly,” Rep. No. FS/77/12. Imperial College, London. REYNOLDS,W.C. (1976). Computation of turbulent flows. Annu. Reo. Fluid Mech. 8, 183-208.

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ROITA,J. C. (1951a). Statistische Theorie Nichthomogener Turbulenz. 1.2. Phys. 129,547-572. ROITA, J. C. (1951b). Statistiche Theorie Nichthomogener Turbulenz. 2. 2.Phys. 132, 51-77. U. (1977). Realizability of Reynolds stress turbulence models. Phys. Fluids 20, SCHUMANN, 721-725. SPALDING,D. B. (1971). Concentration fluctuations in a round turbulent free jet. Chem. Eng. Sci. 26, 95-107. SPENCER, A. J. M., and RIVLIN,R. S. (1959). The theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch. Ration. Mech. Anal. 2, 309-336. SPENCER, A. J. M., and RIVLIN,R. S.(1960). Further results in the theory of matrix polynomials. Arch. Ration. Mech. Anal. 4, 214-230. TENNEKES, H., and LUMLEY,J. L. (1972). “A First Course in Turbulence.” MIT Press, Cambridge, Massachusetts. TOWNSEND, A. A. (1970). Entrainment and the structure of turbulent flow. J. Fluid Mech. 41, 13-46. VAN DYKE,M. (1964). “Perturbation Methods in Fluid Mechanics.” Academic Press, New York. J. L. (1978a).The decay of temperature fluctuations and heat flux in WARHAFT, Z., and LUMLEY, grid-generated turbulence. In “Lecture Notes in Physics,” Vol. 76, pp. 113-123. SpringerVerlag, Berlin and New York. J. L. (1978b). An experimental study of the decay of temperature WARHAFT, Z., and LUMLEY, fluctuations in grid-generated turbulence. J. Fluid Mech. 88,659-684. WYNGAARD, J. C. (1975). Modeling the planetary boundary layer-extension to the stable case. Boundary-Layer Meteorol. 9, 441-460. ZEMAN, 0.. and LUMLEY, J. L. (1976). Modeling buoyancy driven mixed layers. J. A m o s . Sci. 33, 1974-1988. O., and LUMLEY, J. L. (1978).Buoyancy effects in entraining turbulent boundary layers. ZEMAN, A second order closure study. In “Lecture Notes in Physics.” Springer-Verlag, Berlin and New York. To be published.

ADVANCES IN APPLIED MECHANICS. VOLUME

18

Unsteady Separation According to the Boundary-Layer Equation SHAN-FU SHEN Sibley School of Mechanical and Aerospace Engineering Cornell Uniuersity lthaca, New York

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

177

11. Asymptotic Behavior of the Boundary-Layer Solution Away from the Wall A. The Steady Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Weakly Unsteady Case . . . . . . . . . . . . . . . . . . . . . . . . .

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

182 182 185

111. Separation and the Concept of an Unmatchable Boundary Layer . . . . . A. The Case of Steady Separation . . . . . . . . . . . . . . . . . . . . . . . . B. The Case of Unsteady Separation . . . . . . . . . . . . . . . . . . . . . .

186 187 189

........ . ......... ......... ......... V. The General Unsteady Boundary Layer , . . . . . . . . . . . . . . . . . . . A. Finite-Difference Schemes and Local Flow Reversal . . . . . . . . . . . B. The Numerical Example of Telionis, Tsahalis, and Werle . . . . . . . . C. Criterion for the Separation Singularity . . . . . . . . . . . . . . . . . . VI. Separation in Lagrangian Description . . . . . . . . . . . . . . . . . . . . .

192 192 195 199

,

.

IV. The Semisimilar Boundary Layer . . . , , . . . . . . . . . . A. Definition and Results of Computation . . . . . . . . . . B. Reformulation of the Semisimilar Problem . . . . . . . C. Separation Criterion and the Goldstein Singularity . .

203 203 207 208 213

A. The Two-Dimensional Boundary-Layer Equation in Lagrangian

Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Genesis of the Separation Singularity . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

214 215 218

I. Introduction It was Prandtl who founded the boundary-layer approximation and, at the same time, introduced the term “separation” to designate the phenomenon that the boundary layer would leave the body surface under certain circumstances. His physical argument was that as the pressure is essentially constant across the thin boundary layer, the slow-moving fluid particles near an immobile wall would eventually lose their forward momentum under 177

Copyright 0 1978 by Academic Press. Inc. All rights of reproduction in any form rmrved. ISBN 0-12-002018-1

178

Shun-Fu Shen

prolonged adverse pressure, and thus act as a barrier to force the flow away from the body surface. Separation in a steady boundary layer was then directly associated with flow reversal, which in turn implies vanishing shear at the wall. Observations seem to support that an adverse pressure gradient is indeed a prerequisite to separation. For the steady two-dimensional boundary-layer equation, a mathematical analysis with an adverse pressure gradient over fixed wall and in the vicinity of the point of flow reversal is contained in the classical paper of Goldstein (1948). It clearly shows that the solution develops a local singularity such that T,

a (x,

- x)’”,

where T, is the wall shear, x is the downstream distance, and x, marks the position of flow reversal along the wall. Furthermore, the displacement thickness tends to infinity as x + x, and no real solution is possible for x > x,. All numerical calculations have confirmed these features, and it has become a tradition to identify the point of vanishing wall shear as the “separation point.” A number of questions of course may be raised, and much recent activity can be found in two different directions. First, if the boundary-layer equation admits no continuation downstream of x,, what must be the different structure in the vicinity of x, such that the physical flow can be properly described? Second, if the wall is moving or the outside flow is unsteady, how does the boundary equation behave when actual separation, in the original Prandtl sense of the breaking away of the boundary layer from the wall, again takes place? The problem of the flow structure near the steady separation point, capable of getting around the Goldstein singularity and penetrating further downstream into the wake region, has attracted many investigators. Since the singularity is apparently forced to occur with a prescribed adverse pressure gradient, allowing for a pressure readjustment due to viscous-inviscid interaction near the separation point holds promise for its removal. In an inverse formulation Catherall and Mangler (1966) propose to prescribe the displacement thickness growth, and Horton (1974) chooses the alternative of specifying the wall-shear variation. By insisting on a “regular” separation, the prevailing pressure gradient can be determined as a consequence. The validity of the boundary-layer approximation up to and beyond the separation point is implied, however. The direct approach of unraveling the flow structure including the unknown pressure modification near the separation point requires a more refined boundary-layer theory. The way is paved after the important discovery of the triple-deck description near the trailing edge of a flat plate, due to Stewartson (1969) and also Messiter (1970). Applying the same technique to the case of the steady separation point, Stewartson (1970) concludes that if the upstream pressure gradient is indeed adverse, the

Boundary-Layer Equation

179

refined structure still is incapable of matching with an ordinary boundary layer downstream. In short, the Goldstein singularity is not removable, and must be considered as “a real phenomenon terminating the flow which, at high Reynolds number, exists upstream.” Conceivably the dilemma is caused by the assumed adverse pressure gradient. As a different starting point it may be postulated instead that the limiting flow as the Reynolds number tends to infinity should be described by the free-streamline theory for inviscid fluids. For the example of a circular cylinder, Sychev (1972) and Messiter and Enslow (1973) thus succeed remarkably to fit in a triple-deck local-interaction model, where the boundary layer has a faoorable pressure gradient upstream and a constant pressure plateau downstream. There is still an adverse pressure gradient locally, but of vanishing strength and over a vanishingly small distance as the Reynolds number tends to infinity. It seems not yet clear how this picture can be reconciled with the observed fact that, even for the circular cylinder, separation seems to have never occurred without an existing adverse pressure gradient upstream. In fact, the recent experimental data of Dobbinga et al. (1972) show a rearward movement of the separation point as the Reynolds number (based on the momentum thickness) is increased. The trend is contrary to the expectation that eventually the limit flow should separate under favorable pressure gradient. Besides, there is also the famous Schubauer experiment over an ellipse, where separation is found to arise after an extended region of adverse pressure gradient, already pointed out in the Addendum of Stewartson (1974). Perhaps the explanation lies in the unavoidable free-stream turbulence, in all experiments, which tend to delay separation. There will undoubtedly be fascinating further development to clarify the flow structure near the separation point. In a detailed study of the flow over the trailing edge of a flat plate, Veldmann (1976) suggests that the triple-deck forms “only the beginning of an infinite series of such regimes.” The introduction of time dependence for unsteady boundary layers cannot help but add greatly to the complexities. But if we restrict ourselves only to the global phenomenon of separation as the breakaway of the boundary layer from the body surface, the primary attention is focused on whether the outer edge of the viscous region no longer lies within a distance of O(v”*), v being the kinematic viscosity, from the wall. Even for boundary layers within which a bubble of reversed flow exists, if the overall thickness remains to be vanishingly small for large Reynolds numbers, clearly the limiting inviscid flow does not separate. We should still have a prescribed pressure distribution for the boundary layer, with possible local refinements near the points of flow reversal. The parabolic nature of the boundary-layer approximation should allow us to continue marching downstream. If the flow is, in fact, already separated, and the prescribed pressure distribution is taken to be that which actually exists just

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Shan-Fu Shen

outside the free-shear layer, the thickness of the boundary layer (between the body surface and the outer edge of the shear layer) is O( 1) even as v -,0, thus becoming infinite in the boundary-layer scaling. In other words, the naive boundary-layer approximation must exhibit a singularity which terminates its applicability in the presence of separation. It is on these grounds that we concur with Sears and Telionis (1975) in maintaining the traditional boundary-layer equation as a diagnostic tool for unsteady separation. For simplicity, only two-dimensional incompressible laminar boundary layers will be examined. The governing equations are well known, au au au au -+u-+u-=-+u-+at ay at ax

au

a2u

ax

ay2’

(1.1)

au a0 -+-=o, ax ay

where (x, y) denote the Cartesian coordinates along and normal to the wall, respectively, t is the time, (u, u ) are the velocity components in the respective x and y directions, and U = U ( x , t ) is the free-stream velocity at the outer edge of the boundary layer. The kinematic viscosity v is assumed to have been absorbed in the scaling and does not appear. The boundary conditions are, if the wall is fixed, y=o, u=u=o, y -, 00, u -,u(x, t). (1.3) For the initial condition, properly specified velocity distributions are needed, and, in general, there should also be data along x = 0. The emphasis is on the behavior of the solution which may suggest separation through the development of a singularity, with or without the presence of flow reversal (u < 0) in some part of the domain. In contrast to the steady case, our knowledge about such boundary layer in detail, either experimentally or numerically, is still very limited. Particularly important is the distinction between flow reversal and separation (as breakaway) for unsteady flows. Though recognized in the literature, it has been most emphatically stressed by Sears and Telionis (1971). They take the position that, instead of vanishing wall shear, the appearance of a Goldstein singularity, or its like, in the boundary-layer solution is a more universal feature of separation. In the earlier studies of the steady flow over a rotating circular cylinder by Moore (1958) and Ludwig (1964), the vanishing wall shear is shown to be clearly irrelevant for separation and the separation criterion has been replaced by

au/ay = o

at u = 0,

(1.4)

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181

referred to as the M-R-S (Moore-Rott-Sears) condition by Telionis (1970). The separation over a rotating cylinder becomes an unsteady phenomenon when viewed from a set of coordinates corotating with the cylinder. Consequently, they are led to propose that the M-R-S condition might prevail for all unsteady boundary layers at separation, when viewed from a suitable set of mooing coordinates. As a concept the moving separation singularity is physically very descriptive, but its genesis and demise in arbitrary unsteady boundary layers need to be clarified by further analysis. The study by Buckmaster (1973), for instance, suggests the possibility of situations where an initial singularity may in fact first disappear before resurfacing again later somehow. But supporting evidences have come forward, mostly from detailed numerical case studies as described in Sears and Telionis (1975). In this paper, we shall review mostly our recent studies of the unsteady boundary-layer equation relevant to the separation problem: the asymptotic approach to separation, outlined in Shen and Nenni (1975), is aimed at the prediction of separation without the boundary-layer details which must be obtained by accurate stepwise integration. Attention is only focused on the behavior of the u-component boundary-layer velocity at the outer edge of the boundary layer. This requires an improved analysis of the asymptotic behavior for steady flow due to Tollmien (1946), and the results are generalized to a class of weakly unsteady flows. It appears that the wall shear can sometimes be retained as a useful indicator when the boundary-layer singularity, hence separation, is encountered. Aside from their basic interest, these findings open up the possibility of practical methods of predicting unsteady separation through approximate boundary-layer calculations. Meanwhile the semisimilar boundary-layer equations are reexamined. The numerical examples provided by Williams and Johnson (1974qb) for these special unsteady boundary layers confirm the details of the M-R-S condition. Our analysis shows that the validity holds for all semisimilar boundary layers, for which the M-R-S condition may be rephrased without the notion of suitable moving coordinates. For the general case, the separation singularity is interpretable as due to the coalescence of the characteristics in the (x, t)-plane, or the crossing of path lines at the same instant. The boundary-layer equation is next formulated in Lagrangian coordinates for a unified treatment of the separation in both steady and unsteady flows. The evolution resulting in separation is followed numerically in a model problem, and singular behavior of the solution is found to occur when at a given time the Lagrangian variable x, the downstream penetration distance, develops a stationary point. This condition formalizes Prandtl's intuitive statement that the piling up of fluid particles is responsible for the boundary layer to break away from the wall. Without the longitudinal diffusion term, the boundary layer equation can predict the piling up only as a singularity, analogous to the shock

182

Shan-Fu Shen

discontinuity in gas dynamics. The crucial mechanism responsible for such an event is, likewise, none other than the wave-steepening in decelerating flows. 11. Asymptotic Behavior of the Boundary-Layer Solution Away

from the Wall A. THESTEADY CASE

Following a straightforward transformation, in the steady case Eqs. (1.1) and (1.2) may be rewritten as

aE- u a2E -

a4-Eag2’

where 4 and I) are basically the von Mises coordinates .x

.Y

4= J Udx, 0

+ =J

udy,

0

and E is the modified dependent variable representing the energy deficit,

E = U 2- u2. The asymptotic behavior of E as I) -+ 00 is obtained by letting u/U 4 1 and reducing (2.1) to the heat equation. It has been given by Tollmien (1946) in terms of an initial profile Eo(+) at 4 = 0 and arbitrary data Eo(4) along = 0. Since ulU -+ 1 is invalid near I) = 0, the condition E = U 2at the wall must be relaxed and 8,(4)may be regarded as resulting from the introduction of an effective slipping velocity. We shall refrain from a display of the Tollmien solution, except to point out that the contribution to the asymptotic solution E at a given station 4 comes mainly from the form of E0(4)near the leading edge 4 = 0; see Fig. 1. This Tollmien layer is therefore passive in nature, wrapping around the boundary-layer core within which are manifested the effects of the true boundary condition at the wall. Near the separation singularity, I) is expected to bifurcate and its use as one of the coordinates becomes obviously undesirable. In order to bring out the proper asymptotic behavior, particularly for the u-component velocity, Nenni (1976; also Shen and Ninni, 1975) uses the dependent variable 2 = U(U - U ) (2.2) and replaces the independent variable I) by

+

q = uy.

(2.3)

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183

Tollmien layer, /’\,/-

JI

//

//

/ ,

-

, ’ /- I , , ’ I

A;<;

Equation (2.1) then takes the form

where all the nonlinear terms are gathered in N:

N=

(zU ,

uz u

- 2,)

+ [$

2, dq - 2

u3 u, j”

Z dq\Z,,.

0

(2.5)

The boundary conditions on Z ( 4 , q) are z(430) = - u2,

z(4,m) 0, +

Z(0, q) = Z o ( q ) ,

(2.6)

Zo(q) reflecting the initial velocity uo(q) at x = 0. If the right-hand side is

treated as given, the inhomogeneous heat equation (2.6) has the formal solution, satisfying the exact boundary and initial conditions,

It is now straightforward to evaluate the asymptotic behavior of Z as q -,00. Only the double integral of N in (2.7) is not considered in Tollmien’s linearization, and its contribution to the asymptotic behavior is to be found from

184

Shan-Fu Shen

the behavior of N(&, q’) for q’ small. It becomes necessary to describe Z(4, q) for q small, that is, the velocity profile near the wall. So as long as the boundary layer has yet to reach the singularity, it is expected that for q small, OD

Z

=

C ai(4)qi. i=O

The coefficients ai must satisfy the compatibility relations of the boundarylayer equations through successive differentiation of the latter with respect to q. By substitution of (2.8) into (2.5), it follows OD

where the coefficients ai turn out to be ao= -UU,, a1 = air, a2=

alal’ -2v2

1 (UU, 2

--

+ U,Z),

(2.10)

with a,‘ = dal/d+.

In terms of these coefficients, it follows for q -,co,

(2.11) where b = q/24’/’ and Zo*(q)is the asymptotic representation of the initial profile Zo(q).It is noted that the wall condition enters only through ao, al, ..., and in the higher order terms of lib.

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Boundary-Layer Equation

B. THEWEAKLY UNSTEADY CASE The above analysis is not difficult to extend to the weakly unsteady case where the free-stream velocity is of the form U(X, t ) =

UO(X)

uo > 0,

+

-

Ul(X, t),

IU,I/V, G 1.

We define again the same variables that lead to (2.4) except replacing U everywhere by U o . For the asymptotic behavior, Z = ( U o - u)Uo < O(1). Now, corresponding to Tollmien’s linearized approximation, the unsteady boundary-layer equation in (x, y, t ) simplifies to

--+--oy-=u 1 az az u ’ az uo at ax uo ay

1*

1 azz + (U,U,),+ -. uo ayz

(2.12)

It indicates that for given initial data on t = 0 and boundary conditions on x = 0, the integration should be along the characteristics d x l U o = dt.

(2.13)

Hence the solution in the (x, t)-plane (Fig. 2) may be constructed separately depending on t 2 t* = 6 d x / U o . In either case, after reinserting the nonlinear terms omitted in (2.12), the governing equation can be reduced to the same form, (2.14) where, for the “large-time” solution, t > t*(x), ,x

(2.15) but for the “small-time” solution, t < t*(x),

5=

c‘

UoZ(x*)dt,

q

=

Voy,

T = t*

‘0

x * = x*(t

+ T),

the inverse of t * ( x * ) = t

- t,

+ r.

(2.16)

The solution of Eq. (2.14) and its asymptotic behavior as -, co can be determined from the same equation (2.11) as previously done for equation (2.4), except for an additive term to Z equal to U o U1. The coefficients ai(#) and a@) become ai(t, t),respectively, satisfying the compatibility relations

Shan-Fu Shen

186 +

t=t“x1

Large-llme

X

FIG.2. Large- and small-time domains and the characteristics as path of integration

following the unsteady boundary-layer equation. Their contributions to the asymptotic behavior of 2 again are exponentially small and of higher order than those from the linear theory. [For details, see Nenni (1976).]

111. Separation and the Concept of an Unmatchable Boundary Layer It has been repeatedly mentioned in the literature that all numerical integrations of the boundary-layer equations break down when the normal velocity and the displacement thickness start to increase their magnitudes precipitously. Thus to locate separation there is no need to examine the boundary-layer profile in detail. In Shen and Nenni (1975), the specific proposal is made to associate separation directly with the condition that the boundary layer should become “unmatchable,” in the sense that the induced normal velocity at the outer edge of the boundary layer attains such a magnitude as to invalidate the basic assumption of u ( R ) ’ / ~ 0(1), R being the Reynolds number. It remains to search for the circumstances under which unmatchability takes place, but frees us at least from preconceived notions about the differences between steady and unsteady separation. Although the asymptotic behavior of 2,essentially the u-component velocity, is found to be insensitive to the local wall conditions, the same cannot be said for the u-component velocity. From continuity, Eq. (1.2), we have

-

As q + 00, the first term on the right-hand side is easily seen to be nothing but the inner behavior of the outer (inviscid) solution, in the language of the method of matched asymptotic expansions. The last term of Eq. (3.1) is due to the boundary layer solution 2. As v -+ 00, it can be identified with

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Boundary-Layer Equation

a( U6*)/ax,6* being the usual displacement thickness: 6* =

1”

(1

-e

1 ) d y = --

”

v2 0

2 dq.

Thus the “unmatchability condition” amounts to the statement: u’

a (ua*)> O(l), =ax

(3.3)

u‘ denoting the perturbation normal velocity on the inviscid flow. Separation can happen, consequently, if 6* or its x-derivative becomes unbounded. Note that 6* depends on the entire boundary-layer velocity profile, but as an integral is not very sensitive to local details.

A. THECASEOF STEADY SEPARATION It is necessary now to apply Eq. (3.3) first to steady boundary layers and show its equivalence to the criterion of vanishing wall shear and the concomitant Goldstein singularity. What is needed is only a sufficiently accurate representation of 6*. For instance, the Pohlhausen momentum-integral method provides an approximate equation for 6*(x), although it is notoriously unreliable for predicting separation. Furthermore, in general it does not exhibit the Goldstein singularity when the wall shear vanishes. The reason of such failures can only be that the assumed quartic for the velocity profile is too crude. This observation is dramatically demonstrated when, as in Shen and Nenni (1975), an approximation to the wall shear is made by patching together the asymptotic expansion equation (2.11) with a polynomial of the form equation (2.8). When truncated to four terms, Eq. (2.8) becomes explicitly

Z

I-

U 2+ 2(4)”*ba,

- 24b2UU,

+ 23

- +2

b4 a,a,’. u

(3.4)

On the other hand, Eq. (2.11) to O(exp( - b2)/b3) is unaffected by a,. If Eq. (3.8) is used for b i b* and Eq. (2.11) for b 2 b* to form a composite approximation with continuous first derivative (Fig. 3) two conditions are obtained from which b* can be eliminated. We obtain in principle an equation for the wall shear a, : %a,’ = %,4,

u, q.

(3.5) The coefficient a, of course is proportional to the wall shear. Although Eq. (3.5) may be only a very rough description of the shear along the wall, it

Shun-Fu Shen

188

2

FIG.3. Patching of asymptotic solution with polynomial approximation near wall

clearly exhibits the Goldstein square-root singularity at a, = 0 unless the right-hand side vanishes simultaneously, i.e., for special types of the pressure distribution. The displacement thickness 6* contains the integral of Eq. (3.4) from b = 0 to b*, and becomes singular together with al'. Unmatchability thus is seen to lead to the correct features of steady separation as long as the approximate profile near the wall includes the a,a,' term which may be traced to the coefficient a4 as a result of the compatibility conditions. The Pohlhausen quartic is constrained by only the compatibility conditions on a, and az, hence unable to reproduce the proper singularity. The physical significance of the compatibility condition on a4 is pointed out by Nenni (1976) as being the first contribution due to the convective terms in the boundary-layer equation. Its omission is tantamount to replacing the boundary-layer equation with a Stokes approximation, which is unacceptable beyond an extremely thin sublayer. We believe this observation opens up the possibility of improved momentum-integral-type methods for the prediction of separation. Preliminary investigations along this line by S. K. Kim (unpublished) at Cornell University produce encouraging results, but so far appear to require a heavy amount of algebra. The role played by the compatibility relation on a4 as illustrated in the following crude example may be of some interest. Instead of Eq. (6.4) let us return to the physical (x, y)plane and use near the fixed wall y = 0, u =a,y

+ a,yz + a3y3 + a4y4,

(3.6) the coefficients ai, i = 1, 2, 3, 4, being now functions of x only. Then the compatibility conditions, up to a4, are az =

-$UU',

a3

= 0,

a4 = (1/12)a1a1'.

(3.7)

Boundary-Layer Equation

189

We take the asymptotic behavior at large y as simply the free-stream velocity U ( x ) and patch it with Eq. (3.6) by requiring u(y*)= U ,

fil aY

= 0.

y*

The procedure leads to two algebraic equations in terms of y* and a4. Near a , = 0 it is easily found

Goldstein (1948) has quoted Jones (1948) that for the Howarth problem U = 1 - (x/8), separation occurs at x, = 0.959 and the wall shear obeys U,

= 23’2(0.47)(~,- x ) ~ ”= 1.33(~,- x ) ~ ” .

The prediction of Eq. (3.9) is seen to be U,

= [3(1 - &X,)]”~(X, - x)’” = 1.62(~,- x)’”.

Not only is it the proper singularity, but the strength is surprisingly close in this particular case, by means of the crude patching.

B. THECASEOF UNSTEADY SEPARATION Extending the concept of unmatchability to the unsteady case, we should be again looking for the singular behavior of the displacement thickness 6* with respect to either x or t. The patching described above is one way to assume a boundary-layer profile of minimum acceptability for this purpose, and the wall shear is seen to be capable of reflecting the singular behavior. It does not seem unreasonable again to focus on the wall shear. The wall shear in addition is an easily observable quantity in experiments. In real flows, the wall-shear singularity with respect to x or t at the separation point of course will be smoothed out locally, but the smoothing presumably is limited within a vanishingly small distance along the wall for large Reynolds numbers. We may mimic Eq. (2.8) and pose a polynomial approximation of degree rn near the wall as m

1 ai(t, r)qi9

(3.10)

i=O

and require that the compatibility at least up to a4 should be observed. But since the asymptotic behavior of U - u remains exponentiallysmall, any such

Shun-Fu Shen

190

approximation must be sufficiently flexible to represent the large variety of possible unsteady boundary-layer profiles. Otherwise, the displacement thickness might be grossly in error and the deductions therefrom would be totally misleading. For rapid oscillatory changes of the freestream, for instance, the boundary-layer profile may develop “wrinkles” and the needed number of terms in Eq. (3.10) becomes necessarily large. This difficulty is well recognized in the momentum-integral methods for unsteady flows, and the patching procedure, considered as a rough shortcut, must suffer similarly. However, the wall shear cannot escape being the solution of an initial- and boundary-value problem in the (x, tFplane. At least for simpler cases, the way in which a singular behavior develops may be qualitatively examined by the patching procedure. The investigations of Nenni (1976) include both the patching of Eq. (3.10)to his asymptotic solutions for large q, as well as the use of a composite rational-fraction approximation for Z:

where the ps and fs are functions of 5 and z defined by Eq. (2.15) or Eq. (2.16).These coefficients are required to satisfy the compatibility conditions near the wall, up to a4, and to agree with the first two terms of the asymptotic solution for q large. His results are illuminating but the details are cumbersome. Instead of quoting Nenni (1976) further, we shall present a more direct demonstration along the lines of Eqs. (3.6) and (3.8). For more generality than Eq. (3.6), let us allow an extra term for possibly a moving wall, = a0

+ a , y + a2yZ+ a3y3 + a4y4 +

***,

(3.12)

the coefficientsai now depending on both x and t. By direct substitution into Eqs. (1.1) and (1.2), the compatibility conditions are found to be uo

+ a0 ao’ = -p, + 2a2, (3.13)

... where (’) is the time derivative, and ( )’ the x-derivative, and -p, = U,+ UU,.Again truncating Eq. (3.12) as written and applying the crudest patching condition equation (3.8), we get

+ a4y*4 = u - a,y* - a2y*2, 3a3y*2 + 4a4y*4 = -a, - 2a,y*. a3y*3

(3.14)

Boundary-Layer Equation

191

The elimination of y* from the pair in Eq. (3.14) leads to an implicit equation: (3.15) q = 0. = 0, a, must equal ipx, therefore prescribed as

qao, a,, 4, a39 a4,

By Eq. (3.13), for fixed wall a. usual, a3 represents the time rate of change of a, (the wall shear), and a4 depends on the change of a, along the wall. Equation (3.15) then can in principle be cast as, similar to Eq. (3.5), a4 = G(hl, a,,

. . .),

(3.16)

the dots in the parenthesis referring to quantities obtainable from the prescribed free stream. A closer examination of (3.14) shows that y* may be solved from a cubic equation in terms of the coefficients ao, a,, a,, and a3. Equation (3.16) is only an alternate form of either of the pair in Eq. (3.14) with y* so defined. Together with the third line of Eq. (3.13), it may be regarded as the equation for a,. The nature of possible behaviors, even for cases where the profile is well represented by Eq. (3.12), is evidently highly complex. We can nevertheless draw certain qualitative conclusions. If the free stream changes sufficiently slowly with time, it should be possible to expand Eq. (3.16) for b small, a4 = Go(al, ...)

+ G,(a,, ...)bl + G2(al, ...)b12+ ...,

(3.17)

where G o , G,, G,, etc., are obviously the appropriate derivatives of G with respect to a,. When on,ly the linear term in a, is kept, the result may be rewritten as hlhl

+ alal‘ = k,,

(3.18)

with h , and k , depending on a, and other quantities defined by the free stream. An equation of the form of (3.18) has already made its appearance in Shen and Nenni (1975). It is noted to resemble the Burgers equation in gas dynamics; thus the wall shear can develop a singularity analogous to the shock formation through the coalescence of the characteristics of Eq. (3.18). The “shock path” in the (x, t)plane gives the movement of the separation point. As a gas-dynamic shock sometimes starts after a finite time after the disturbance, Eq. (3.18) is seen to be capable of explaining the emergence of separation at finite time under suitable conditions. By retaining the quadratic term in Eq. (3.17), Eq. (3.18) is replaced schematically by

hlhl

+ h Z h l 2+ alal’ = k z ,

(3.19)

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Shan-Fu Shen

where again, h,, h,, k2 depend on a , and the free stream. An equation similar to Eq. (3.19) can be found in Nenni (1976),whose derivation is based on Eq. (3.11). Nenni’s coefficients corresponding to h , , h,, and k,, however, contain complicated dependences on ti1 and a,. At any rate, Eq. (3.19) suggests that the separation singularity as manifested by the wall-shear behavior is actually considerably more complex than the gas-dynamic analogy of Eq. (3.18). We caution that the patching procedure as described is not to be taken seriously as providing a quantitative answer for the wall shear. The construction is based upon only local properties and must have severe limitations. The next step for reasonable accuracy and more general applicability might be to employ a momentum integral formulation in conjunction with such a patched profile that incorporates the right ingredients to turn properly singular. Our experience with preliminary studies of this type shows that hidden difficulties are abundantly strewn along the path and must be overcome very likely on a case-to-case basis. The major consideration should be the adequate representation of the velocity profile where it contributes most to the displacement thickness. This is most difficult to achieve when the velocity profile ceases to be monotonic. Much experimentation is definitely needed before the practical usefulness of such procedures can be fairly assessed.

IV. The Semisimilar Boundary Layer A. DEFINITION AND RESULTSOF COMPUTATION As an attempt to gain some information for the unsteady flow associated with a diffuser when the angle of divergence varies with time, or an airfoil with time-dependent angle of attack, Tani (1958) has considered a generalization of the Howarth problem with the free-stream velocity

u = 1 - 5,

5 = x/(l

- At),

(4.1) A being a constant. With the usual boundary-layer variable r , ~= y/2(x)’I2 and a series expansion of the stream function $, $ = (x)”2[fo(?) + tfl(r,I)+ t 2 f 2 ( r , I ) + . . .I, (44 a sequence of ordinary differential equations is obtained forfo,fl, . . . . The functions upf5 are solved for the appropriate boundary conditions for fixed wall, ,f;:(O) =h‘(O)= 0,

i = 0,1, 2, . . .,

Boundary-Layer Equation

193

and the conditions at the outer edge,

j,’(co= ) 2,

fl’(co) = $,

$(a)= 0, i

2

2.

5. The calculation for 5 > 0.05 is carried out by the momentum-integral method with an approximate profile

As in the steady case, the series expansion converges slowly with

u = W ( q )- tfl’(?)- -..-
(4.3)

where G ( q )is determined according to Howarth (1938) and given in a tabulated form. He finds thus the flow reversal to occur at = 0.15 for A = 1. In Tani’s problem, $ / ( x ) l i Z is only a function of 5 and q. Following an earlier analysis of Hayasi (1962), Williams and Johnson (1974a) search for the general conditions for the class of “semisimilar” unsteady boundary layers. These have the property that the governing equation is reducible to a form involving only two suitable independent variables t ( x , t) and q(x, y, t ) , and a properly transformed stream function. The standard numerical techniques developed for the steady boundary layers may then be directly borrowed to provide nontrivial examples of unsteady boundary layers, in particular with flow reversal as well as separation. However, the multitude of requirements force them to resort to an “indirect method of solution” by assigning certain a priori conditions, following which a class of admissible free streams is found to be

<

U = U(<),

5 = ( x + kt)/(l - At),

(4.4) A and k being constants. Numerical examples are provided again for the generalized Howarth problem U = 1 - 5, with k = 0 in Williams and Johnson (1974a) and A = 0 in Williams and Johnson (1974b). The former, of course, is just the Tani problem, and the flow reversal point is found to occur also at 5 = 0.15 for A = 1. The latter corresponds to the steady flow over a moving wall. In the Tani problem, the integration is carried out further beyond the reversal point until separation. It is, however, necessary to avoid reverse flows in standard implicit finite-difference schemes, e.g., Blottner (1970), developed for the steady boundary layer. They make the observation that if 5, is the separation point, then the actual separation point x, obeys X,

= <,(1 - At),

dx,/dt =

-At,.

In other words, when viewed from a moving coordinate system with k = At, in Eq. (4.4), the semisimilar solution should resemble a steady boundarylayer solution with standing separation. In the new coordinates, the free stream becomes U = 1 - 5 + A<, and the wall is moving at velocity

194

Shan-Fu Shen

FIG.4. Velocity profile for U = 1 - {, fixed wall, k = 0, according to Williams and Johnson (1974a).

u(<, 0) = At,. We display here only the final equation:

(

f ” ‘ + 1- (1 )ff“- 1 + x 2 1 + At, - 5

s

-t

)(I9 ‘ 7

where f =f (t,q), q = y [ V / ( x - At, t ) ] ” 2 , f = 8f8q, etc., and the boundary conditions are

v o t e that f (5, q ) in Eq. (4.5) is different from Tani’s definition Eq. (4.2).] But, of course 5, is unknown beforehand. An iteration scheme is necessary, by starting with an assumed value 5, and finding out whether the solution does blow up at the same value. A typical solution, for A = 1.0, transformed back to the fixed coordinates is shown as Fig. 4. Note the presence of reverse flow for 5 > 0.15, and the rapid increase of the boundary-layer thickness downstream once flow reversal is encountered. The large flat portion of the velocity profile for u/U < 0 near separation contributes most heavily to the displacement thickness and obviously is difficult to represent with the simple polynomial approximations such as Eq. (3.12). It is also found that, in the moving coordinate system, the solution cannot be continued further downstream as soon ass‘ = 0 is reached somewhere within the boundary layer. Geometrically this point, say (&, q,), is a minimum of the profile f’&, q), hence f”(<,,q.) = &it corresponds to the point of zero shear. These features are in complete agreement with the M-R-Scondition, and

Boundary-Layer Equation

195

I

[,=3.16

t

FIG.5. u-component velocity for U = 1 - t, moving wall, I = 0, according to Williams and Johnson (1974b).

emerge again in the companion moving-wall calculations of Williams and Johnson (1974b). Furthermore, the minimum of the boundary-layer velocity (in moving coordinates) turns out to vary like (t, - t)'/', i.e., [(x, - x)/(l - It)]'/'. Thus the singularity is actually of Goldstein type. These calculations, from a different viewpoint, naturally lend support to the unmatchability condition requiring merely that the o-component velocity at the outer edge of the boundary layer or, equivalently, the displacement thickness should develop a singularity at <,. This is evident in Figs. 5 and 6. As for the calculated wall shear, the trend toward a singularity is progressively more obscure as I becomes larger. The result indicates a loss of sensitivity of the wall shear as a good indicator of separation when the reverse-flow region becomes extensive. It explains the highly complicated form possible for the equation governing the wall shear in Section 111. B. REFORMULATION OF THE SEMISIMILAR PROBLEM In the procedure described above, to avoid the numerical difficulty of the reversed flow, the unknown moving separation point has to be introduced and solved by iteration. An alternative formulation without introduction of the stream function, proposed in Wang and Shen (1977), turns out to be both easier to analyze and simpler to integrate. In addition the physical nature of the separation singularity is also less obscure.

c FIG.6. Displacement thickness for U = 1 - <,fixed wall, k = 0, according to Williams and Johnson (1974a).

196

Shan-Fu Shen

The main feature of the semisimilar boundary layers is that the governing equations may be written in two independent variables [(x, t ) and q(x, y, t). Without loss of generality, let us define u = u(5, q),

u = u(5, q),

q = Y/6(X, t ) ,

(4.7)

4 5 , 00)- W ) , and 6(x, t ) as well as ((x, t ) are yet to be specified. The continuity equation (1.2) becomes 6, 1 u t t x (-qu,) - u - = 0, (4.8) 6

+

+

while the momentum equation (1.1) takes the form

With ii = u, U = u6, two equations are obtained from (4.8) and (4.9) for these variables in terms of ( and q, provided

5 :&:-6x.6,. - -1 = a1:a2:a3:a4:a5, 6’6‘62

(4.10)

where a,, . . . , a,, being dependent on x and t , can at most be functions of 5. The relation (4.10) of course is equivalent to four conditions: (4.11)

Instead of looking for <(x, t ) and 6(x, t), it is convenient to interchange the roles of the independent and dependent variables. The first line of (4.11)may be then recast as XC F2t5 = 0, xg

+ +F

1 tg

= 0,

(4.12)

which, after eliminating x by cross-differentiation, is reduced to (F2

- F1)tac - Fl’tg = 0.

(4.13)

Hence, direct integration gives the general solution

(4.14)

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197

A,, B,, and B, being arbitrary functions. It is now straightforward to go

back to the second line of (4.11).We first deduce

h2 = (B2’- F,’t)F4,

(4.15)

and find next, besides the trivial case F , = const.,

Bz = a, F1 + a2,

(4.16)

a, and a, being constants. Hence the combination of (4.14) and (4.16) leads

to (x - az)/(a1 -

=

W).

(4.17)

It appears, consequently, that Williams and Johnson’s solution of admissible [(x, t), Eq. (4.4), is in fact the most general one for semisimilar boundary layers. As for 6(x, t ) , following (4.15) and (4.16),

6,

= (a1 - t)F4(<),

or equivalently, (4.18) 6’ = (x - aZ)(F4/FI). There is therefore some flexibility in the choice of F4(5)to suit the problem. In contrast to Williams and Johnson (1974a,b),Wang and Shen (1977)have simply taken F4 F , , and 6 independent of t. For the Tani problem defined by (4.1), we take a, = l/A, a, = 0, and F , = A< in (4.17) and F4 = 45 in (4.18). The set (4.8) and (4.9) become

-

25n,

- qu,

+ +B,

(4.19)

= 0,

4<(A<+ ilp, - (2qn - B)ci, = 4t(ng + U ) U ,

+ n,,,

(4.20)

where q = y / 2 ( ~ ) ’ /a~= , u, B = 2u(x)’/’. For fixed wall, the boundary conditions are U(<,0) = v(<, 0) = 0,

q 5 , 0 0 ) = U(5), ~ ( 0q,) = u&),

(4.21) gven.

Note that there is no intrinsic advantage to use the normalized velocity ii/U(<)when the solution is sought by numerical integration. It is evident that standard finite-difference schemes developed for steady boundary-layer equations in terms of the continuity and momentum equations should be again directly adaptable. Furthermore, (4.20)is parabolic and of the general form K ( < , up, + ... = u,,,

(4.22)

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Shun-Fu Shen

with K ( < , ii) = 45(A< + a) in the present case. The coefficient K plays the role of the inverse of thermal diffusivity in the analog heat equation. We have a well-posed problem to continue Do for > 0, so long as K > 0. Thus, for decelerating flows I > 0, local flow reversal will not disrupt the numerical integration until it is sufficiently negative, at least not before K changes sign. It turns out K starts to change sign only at t = t,, as will be shown. There is no need to find the separation position 5, by trial and error. Other than the generalized Howarth flow U = 1 - 5 , two additional examples of semisimilar solutions have been calculated by Wang and Shen (1977) with the formulation (4.19) to (4.21), using the Crank-Nicholson scheme outlined in Blottner (1970).With 5 = x/l - It, the free-stream velocity distributions are prescribed to be:

+

(ii) U = 1 - A , t/(1 T2t), A , , A,, T,, & being positive constants. Figure 7 shows a plot of the modified streamfunction II/ in the (<, q)plane for case (i), $ being defined by $ = 1,h/2(x)”~,where $ is the true stream function according to the usual definition. Although II/ = constant no longer represents the streamline, the bubble beneath II/ = 0 nevertheless indicates the region of reverse flow. In the example K > 0 is maintained in the bubble region and no apparent

7r

7)

IF 0 005

0.01 0.62 0.03 064 0105 006 007 0.08 OD9 010 011 012

t FIG.7. An example of unsteady boundary layer containing a recirculating bubble without separation.

Boundary-Layer Equation

199

instability has arisen due to the presence of reverse flow. As a precaution, the steps in the [-direction are carefully refined to insure that any indication of instability is not suppressed.

C. SEPARATION CRITERION AND THE GOLDSTEIN SINGULARITY We return to (4.22)and observe that in its numerical integration by marching downstream, instability is certain to occur beyond a certain [ when K < 0 for all q. The locus of the condition K = 0 may be thought of as a neutral curve in the (5, q)-plane, and defines a critical point (tC,q c ) when jointly (4.23)

the subscript c denoting the value at (lc,qc). If there are several such points, we take the one with the smallest value of [, so that tCis the minimum critical value beyond which K < 0 in some interval of q. It remains to examine whether the solution necessarily breaks down as the critical point is approached from upstream. In the degenerate case of I = 0, i.e., steady flows, K = u and (4.23) becomes identical to the M-R-Scondition for the separation singularity ([, q,), where both u and the shear vanish. Now for the generalized Howarth flow U = 1 - [ and I > 0, the following results are stated in Williams and Johnson (1974a): I

0

5, 0.117 Ii,

0

0.5 1.0 0.161 0.22 1 - 0.0805 - 0.221

The data are in full agreement with (4.23), if indeed 5, = 5,. These correlations are highly suggestive and have led to the conjecture in Wang and Shen (1977) that the critical point might actually be a barrier of the solution, indicating separation. Because K = 4[(I[ + a) for these semisimilar boundary layers, (4.23) and the M-R-Scondition coincide. The latter is phrased in terms of a moving observer at an unspecified “suitable speed,” which is now specified by satisfying K = 0. By retracing the steps back to (4.9), it is apparent that K is proportional to the forward velocity component of the fluid particle. When in addition dK/dq = 0, physically in an infinitesimally small neighborhood of the critical point all particles cease to advance downstream. From this viewpoint, the breakdown of the boundary layer solution is seen to reflect

200

Shun-Fu Shen

the incipient turning back of these slow-moving fluid particles, as originally pictured by Prandtl in the steady case. Goldstein’s classical analysis of the separation singularity in 1948 was motivated by Hartree’s numerical study of the case U = 1 - x. We now have the computational results of U = 1 - 5 following (4.4) and (4.9, which apparently exhibit the same behavior characterizing the Goldstein singularity. To our knowledge, these represent the only evidence from detailed study in support of the emphatic conviction of Sears and Telionis (1975) that at separation there should be even in unsteady cases “the appearance of the Goldstein singularity . . ., modified us necessary, in the solution of the boundary-layer equations.” (Italics are by the present author.) Other numerical solutions of the unsteady boundary layer are cited as showing also singular behavior, but definitive evidence of the square-root nature of the classical Goldstein singularity is lacking. Their statement presumably should be understood as strongly anticipating a singularity with an unbounded u-component velocity and possibly the Goldstein (square-root) behavior. The formal resemblance of (4.19) and (4.20), for semisimilar boundary layers, to the governing equations for steady flows points to the probable validity of their conjecture. However, there seems to be no previous analysis in open literature. we use (4.11) to rewrite Without specializing the choices of 5 and U(<), (4.8) and (4.9) as (4.24)

with F* = F4(F1 + U ) for brevity. Corresponding to (4.22), it is seen that, for the general case, (4.26) K(u, 5 ) = F4(F1 + u), and the critical point according to (4.23) remains always the same as obtained by the M-R-S conditions of aupq = 0 and specifying u = - F, for the moving coordinates. To study the behavior of the solution upstream and in the vicinity of (t,, q,), let

5 = 5,

- 51,

u = u,

+

Ul,

v=

vc

+ v1, (4.27)

Boundary-Layer Equation

201

where the subscript “c” denotes the value at the critical point and F*‘ = dF*/dS, etc. Equations (4.24) and (4.25) become

= Fc*- F r ’ 5 ,

,

aZu + .. . + -

a1112’

(4.29)

As suggested by the numerical evidence, we pattern after Goldstein (1948) and look for a local solution of the type (4.30) Thus, the terms of O(5,) in the coefficientsof (4.28)and (4.29)are O(x4), and those from q1 are O(x). For the expansion (4.30) up tofl it is sufficient to consider the simplified set: -F4,c

aa

au,

au 1

+ = 0, a111 a111

-- F 3 , c V c -

851 8%

(4.31)

-

- F 4, c u -+[U-11c~cF3,c]-=Fc*+--

at,

aZu

1

a111

(4.32)

By redefining 52

= 51/F4,c>

01 =

112 = 1119

(4.33)

- F3.c11cUc,

these are further reduced to au, av, -_ +--0, a52 a112 au,

-u1 - + U 1 a52

au

(4.34) a2u

-=Fc*+-8112 a1122.

(4.35)

In other words, precisely the same equations as considered in Goldstein

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Shan-Fu Shen

(1948) govern the behavior offo andf,. But of course the boundary conditions are different, as now the critical point does not lie on the fixed wall. The consequences of this modification have already been formally discussed by Brown (1965) in a different context. Prior to the critical station t,, the velocity profile u,(& q) for 5 given is well defined. As in the steady case, the numerical data for U = 1 - g suggest that there is also no anomaly of u1 as t + 5, from upstream. Assuming this to hold in general, we may therefore write for t2 and q2 small, ui = ao(52) + a1(52)t/2 81 = bO(t2)

+ b,(t,)t/2

+ az(tzhz2 + %(tz)r/z3 +

‘**,

(4.36)

**..

At the critical station 5, = 0, ao(0)= q ( 0 )= 0 by definition, and the compatibility conditions, from substitution of (4.36)into (4.34)and (4.35),yield a2(0)= - F,*,

a3(0)= - bo(0)F,*, etc.

(4.37)

The implication of (4.37) is that, for q, # 0, lim (-00

f:o rr+, = ((2)1’2~a,+~(0), r = 0 , 1 .

(4.38)

It is now possible to borrow directly from Goldstein (1948).A rescaling of the velocity and length suffices to reduce F,* to - 1 and will be omitted here, and (4.30) is understood as having been thus properly scaled. Then the equations forfo andf, are known to be: fo”’ - 3f0 fo”

f,‘”-

+ 2f0’~= 1,

+c3f1” + $cvl’- 4rlfi = 0.

(4.39) (4.40)

The boundary conditions for fo are unchanged from Goldstein’s case, fo(0)=fo’(0)= 0, The solution is still

fo’

-c

as c + 00.

f o = ic3.

(4.41)

Forf,, however, since a3(0)# 0, it is necessary to consider all three independent complementary functions of (4.40). These are c2, hl(c), and gl(c), the details of which are fully described in Goldstein’s paper. The most important property of h , and g1 is that asymptotically both grow exponentially large with c. On the other hand, the exponential growth can be eliminated by a suitable linear combination of h, and 6,: Let w,(c) =

(-a)!

(-2)!

g1

+ 2-(1’4)(-$)! (-2)! hl,

(4.42)

Boundary-Layer Equation

203

-

then w1(Q l4 as ( -,00. Therefore, to satisfy (4.38) we must have in the present case the solution (4.43)

where a1 and /I1are two constants. It goes without saying that B1 is completely determined by a3(0)because of (4.38). As for a l we find it proportional to b,(O). To summarize the solution (4.30) now may explicitly be written as, for fured q2 and T2 + 0, 2 1 2

u1 = 2% [d = [q2'

+ x(2a1C + 4B1r3) + -1

+ 8B1q23 +

* * a ]

+ 4a1q22(<2)1/2+

* . a .

(4.44)

Note that the variables have been rescaled to make Fo* = - 1, and are no longer those originally introduced in (4.30). The terms in the bracket represent the profile at t2= 0, or 4 = tc.The deviation of the velocity u1 (= u - u,) from the profile at tC still varies as ( C 2 ) l I 2 for given q2 # 0 and as the critical 5, is approached from upstream. The critical point is therefore coincident to the separation singularity postulated by Sears and Telionis (1975). Further, we have indeed the true Goldstein singularity as in the steady case, for all semisimilar boundary layers.

V. 'Ihe General Unsteady Boundary Layer A. FINITE-DIFFERENCE SCHEMES AND LQCAL FLOWREVERSAL

For the general unsteady boundary layer, the equations are directly integrated often with a suitable finite-difference scheme. We focus once more on the breakdown of the solution as it progresses in the (x, t)-plane, which signals separation in the present viewpoint. There is no need to reiterate that separation frequently is further downstream from the occurrence of a local flow reversal. Typically the numerical integration is performed as an evolution in time. Let us first discretize time into the set of points t,,, n = 0, 1,2, . . . ,and write U(X,

Y , t n ) = un(x, Y ) ,

~ ( x Y, , t n ) = un(x, Y).

It is obvious that the one-sided difference for the time derivative should be used :

,

At, = t, - t,-

1.

Shan-Fu Shen

204

At each t = t,, the momentum equation (1.1) becomes

au, ax

u,, -

au, u, u,-l a2u, + 0, + -= ( U , + UU,)” + - + ~, a Y At, At, ay2

(5.2)

but the continuity equation (1.2) remains to be au, av, -+-=o.

ax

(5.3)

ay

The boundary conditions (1.3) for fixed wall are un(x, 0) = un(x, 0) = 0,

un(x, 00) = U(X,tn),

(5.4)

and data along x = 0 at t, = 0 are also specified. We have to solve thus a succession of quasi-steady boundary layer equations carrying t, as a parameter, provided u,- l(x, y) at the previous time step is known. The integration of (5.2) is typically by advancing in the x-direction and numerical difficulties arise as soon as u, < 0 for all y at a given x. Because of the boundary condition as y oo,and usually U > 0, negative values of u, may occur over only part of the boundary-layer thickness. A limited amount of local flow reversal is often tolerable, but special handling is required in the finite-difference procedure. The key notion is that, as a convective derivative, u,(au,,/ax) in (5.2) should be approximated with the proper amount of “upstream” influence, the upstream direction depending on the sign of u,,. Let the domain of interest in the (x, y)-plane be now discretized into the set of points (xl, ym),I, m = 0, 1,2, . ..,at which the velocity components are expressed by

-

Un(x1, Ym) = Ul.m,n,

un(xl, Ym) = Ul.m.n-

(5.5)

Two of the successful current schemes in coping with local flow reversal are, with .hx, = XI - XI- 1, (i) the upwind scheme (Phillips and Ackerberg, 1973)

and (ii) the zigzag scheme (Telionis et al., 1973)

Both are first-order accurate, and contain information beyond the station of interest x1 at the previous time tn- 1.

Boundary-Layer Equation

205

The stabilizing effect of (5.6) and (5.7) in the numerical integration is given some insight by Wang and Shen (1977). The implications of a difference formula are made more transparent with the diagrams in Fig. 8, where the grid work is drawn only in the (I, nbplane because all quantities in (5.2) are evaluated at the same level in. The circles denote the points at which the variable u appears in the difference formula. A line connecting two circles denotes that the differenceis taken between them, the arrowhead pointing to the one from which the value at the other end is subtracted. In Fig. 8a, the solid path from (I, n) to (I + 1, n - 1) represents the numerator of (5.6), but the broken-line path through an intermediate point (I, n - l), marked by a cross, is its equivalent, since UI+l.n-l

- U1.n

= ( U I + l . n - l - U1.n-1)

+ (U1.n-1 - U1.n).

It is clear that replacing the path between any two end points in the diagram by any other path through intermediate points does not alter the result. Figure 8a shows that (5.6) is equivalent to

By expanding the variable u of the right-hand side in a Taylor series, it follows that to first order of At and Ax, (5.8) implies the approximation

or

where A1+lJl

= A.xl+l/Atn.

Consequently, the original momentum equation (1.1) is approximated by

=

(V, + UK)I,n + A+1,n

(5.10)

in a quasi-steady form, which may be immediately compared with (4.22). (Note that au/ax In- is assumed known, hence put to the right-hand side.) Similarly, in Fig. 8b the zigzag scheme (5.7) is depicted and the direct path

206

Shun-Fu Shen

(C)

FIG. 8. Diagrams For interpretation OF difference schemes: (a) upwind, (b) zigzag, and (c) time derivative.

from (I - 1, n ) to (I, n) is detoured via the broken-line path in three segments. It may be easily verified that (5.7) is tantamount to replacing (1.1) by

again to first-order accuracy. In both (5.10) and (5.11), the coefficient of au/ax does not change sign unless u ~is, sufficiently ~ negative, since I as the

ratio of the step sizes is always positive. From its appearance, (5.11) is apparently more stable than (5.10) if the same sizes are employed. The diagramming above invites a reinterpretation of (5.1). In Fig. 8c the direct path connecting (I, n - 1) to (I, n ) is detoured via two intermediate points as shown. Therefore (5.1) is seen to imply the approximation: (5.12)

Putting (5.12) into (l.l),we obtain a different picture of (5.2) before taking any specific scheme for au/ax:

(5.13) The structure is similar to that of (5.10).The upwind scheme (5.6) may be regarded as consistent with the onasided time difference (5.1),but the zigzag scheme is apparently not.

207

Boundary-Layer Equation

If we start from (5.13) instead of (5.2), there should be no numerical instability before, in this case, K = (ulVn+ &) changes sign. Although a critical point can be again defined according to (4.23),it in general does not occur at the separation singularity. After all, the choice of the step sizes Ax, and At,, is largely arbitrary, so that unlike for the semisimilar flows of Section IV the stabilization due to unsteadiness is not inherent but artificially introduced. This state of affairs should not be surprising if we recall that in (5.2) K = u,, and the critical point is reached as soon as flow reversal takes place. Nevertheless, either (5.6) or (5.7) enables further integration downstream. The distinction is that (4.22) is an exact equation and all other terms are of O(l), while now the approximation is achieved through the addition and subtraction of the &/ax terms at adjacent points with a resulting stabilization effect. Near the singularity au/ax tends to become unbounded. The approximation must deteriorate since it replaces terms of O(1) by the difference between two large numbers unless, e.g., the step size Ax, is properly adjusted. Apparently the practice is to use increasingly small Ax to avoid the artificial overstabilization in the search of the separation location xs, as mentioned in Telionis et al. (1973). B. THENUMERICAL EXAMPLE OF TELIONIS, TSAHALIS, A N D WERLE In the numerical investigation of unsteady boundary layer separation involving other than semisimilar flows, Telionis et al. (1973) introduce first the coordinate transformation:

x = X(X,

t )=

jxU(x, t ) dx, 0

p = p(x, t ) = yU/(2x)'/2,

(5.14)

-

t = t,

analogous to those adopted by Gortler (1957) in the steady case. The dependent variables (u, u ) are rewritten as (5.15) Omitting the straightforward details, we quote only their final set of equations: aF dG (5.16) 2x - + F +- = 0,

ax

@

Shan-Fu Shen

208

and the boundary conditions:

B=O, F = G = O ; J-Go, F+l. (5.18) The coefficients A iare generally functions of x and Fonly, except that A , is also linear in y. Aside from the a F / a i term, (5.17) and (5.18) may be compared with (4.24) and (4.25) in the semisimilar case, but of course the variables are defined differently. The numerical solution is by the usual approach of approximating the time derivative at r= 6 according to (5.1),

ar ,= F n -A
_

_

At, = t,

-

- t,-

1,

(5.19)

and reducing (5.17) to

(5.20)

Equation (5.20) is of course analogous to (5.2), and together with (5.16) and (5.18) may be solved as again a quasi-steady problem. The special case actually treated by these authors has U = 1 - T(t)x,as a further generalization of the semisimilar Howarth problem U = 1 - of Section IV. The function T ( t )is chosen to be a polynomial representing an increasingly adverse pressure gradient in time, inducing a forward movement of the separation point. The numerical solution is achieved by using the subroutine due to Werle and Davis (1972). The separation point X , ( o is located from the manifestation of lack of convergence of the iteration procedure. The u-component velocity profile remains without anomaly up to the separation point. On the other hand, near separation the u-component velocity rises rapidly and becomes apparently unbounded in the limit, as shown in Fig. 9. (The numbers in the parenthesis denote the number of iterations required to satisfy their convergence criterion.) This feature of an unbounded u-component velocity alone implies a singularity. Without further details, however, it may be referred to as of the Goldstein type only in a vague sense. But we have no qualms to accept the results as substantiating the “unmatchability condition” mentioned in Section 111.

<

C. CRITERION FOR THE SEPARATION SINGULARITY Based on the specific finite-differenceschemes, their inadequacy near separation obviously makes further general discussions difficult. An alternative route might be followed which would be free from any approximation of the singular behavior. As in Goldstein (1948) for the steady case, it may not be

Boundary-Layer Equation

209

X

FIG.9. Typical o-component velocity versus x at p = 0.5 in the example of Telionis et al. (1973).

unreasonable to accept the Limited evidences of numerical results and postulate that the velocity profile u(x, y, t) remains regular in the y-direction prior to and even at separation. We assume, in short, that du/ay and a2u/ay2exist and can be represented by suitable finite-difference formulas. Once the ycoordinate is discretized into the points m = 0,1,2,.. .,with yo = 0 denoting the wall, du/ay(, and d2u/ay2Im are but Linear combinations of u, = u(x, y,, t). For instance, the usual central difference formulas give

if the step size Ay is uniform. (A slight modification is of course needed for

rn = 0 and 1.) At each level y,, the basic equations (1.1) and (1.2) may be written as (5.22) (5.23) The boundary and initial conditions are understood as properly prescribed and play no role in the following discussion. From continuity

(5.24)

2 10

Shan-Fu Shen

where Ay, = y , - y,- and a, are constants depending on the quadrature formula. Hence, (5.21) is recast into

(5.25)

Because of the coupling between different levels m arisen from the difference formulas for au/ay I, and a2u/ay2I, the solution of (5.25) must be carried out as a simultaneous system with one such equation for each m. Solution of the system of equations means that the data prescribed along x = 0 and t = 0 are to be continued according to (5.25) into the first quadrant of the ( x , t)-plane, and separation is said to occur when such a continuation becomes impossible. We assume there is a barrier of the solution in the (x, t)-plane represented by the curve, Fig. 10, (5.26)

f ( x , , t,) = 0,

to the left of which, starting from sufficiently small x and t, all derivatives aum/at and au,,,/ax exist. The solution for semisimilar flows shows that a singular point can be present at the critical point with coordinates (<,, qc) according to (4.23). The curve (5.26) in this case is none other than the locus of 5 = <,, or ( x , - az)/(al - t,) = <,,constant. This relation describes the path of the “moving separation singularity” as conceived by Sears and Telionis (1975). Furthermore, as qc is fixed the singularity occurs at a value of y , = yc(x,) along the curvef= 0. It seems not unreasonable to assume that for the general case we may also associate a specific value y , at each point alongf= 0. Suppose the discretization is arranged to have y , coincide with one of the ym’s, say y,, = y,. For

x,(m“)

x.

-

X

FIG.10. Barrier to flow and formation of singularity due to crossing of characteristics.

211

Boundary-Layer Equation

m # m*, the solution is regular and we imagine it as given. Then our attention may be fooused on Eq. (5.25) at m = m* by itself. For illustration, let I, and azu/ay2I, be evaluated according to (5.21), and collect all terms which are definitely 0(1)into a single expression F* on the right-hand side. The result is

As a single equation, there is again the obvious gas-dynamic analogy similar to (3.18). Its integration may be performed along the characteristics defined bY

which, after integration, may be written schematically as

(5.29)

x = s ( t , xo),

xo being the integration constant determined from the prescribed urn,(xo,O), the velocity at level m* when t = 0. Along each characteristic, (5.27) becomes durn*Jdt = F*,

(5.30)

and F* is the sum of finite quantities, hence finite. Thus urn*can always be obtained from integrating (5.30) with u,,,+(xo,0) known, and remains bounded at finite time. A singularity arises only when adjacent characteristics coalesce. For fixed m*, the singular point (xs, t s ) is then obtainable from solving simultaneously

(5.31) The resulting x,(m*) and ts(m*) define parametrically the curvef= 0 (Fig. 10). As urn* is finite, the singular behavior has to be connected with its derivatives. By differentiating (1.1) with respect to y and using (1.2), it is obtained:

a -

at

uy

a az + u axa u, + u uy = - u ay ay2 -

(5.32)

y’

The operator on the left-hand side is obviously identical to that of (1.1). Proceeding as in the derivation of (5.22) and what follows, we find easily that for the discretized uy I,* the characteristics are still defined by (5.28). It is thus concluded that uyI,* turns singular at exactly the same point (xs,t s ) as

212

Shun-Fu Shen

for u,.. Also, since uyis finite, only its derivativescan become unbounded at the singular point. Let us now reexamine the above analysis for the steady case, for which (5.27) degenerates into

i

urn+ - Cr,*

It is evident that the singularity occurs when (5.33) or, as Ay + 0, u,. 0, i.e., at flow reversal. As in the derivation of (4.23), by the definition of m*,Eq. (5.33) cannot be satisfied at m # m*.There must be in particular K,,, > 0 and K,.- > 0, or as Ay + 0, urn++ > 0 and urn.- > 0. In other words .,u is a local minimum, hence au/ay. , 1 = 0. We thus completely recover the M-R-S condition for steady flows. It may likewise be of some interest to take up the semisimilar case of Section IV, where u = u(t, q), ( = <(x, t) and q = y/(x)lI2. Let us first change the independent variables from (x, y, t) to (x, q, t), so that the left-hand side of ( 1.1 ) becomes

(

-au + u - + au at ax

-qu+-

2x

(x)l’Z

)= ... .

With the assumption that the u is regular in q, essentially the same equation (5.25) will result except to replace y by q and slight modifications of the y-derivative that do not change their orders of magnitude. If we shortcut the arguments by going to Ay + 0 directly, the characteristics for the integration of urn,,at q = q,,,., are now defined by dx/dt = u,,,.(x, t ) = urn+(().

(5.34)

For the semisimilar case where ( = x/( 1 - At),

(5.34) is reduced to (1 - At)(d(/dt) = u,+

+ A<.

(5.35)

Integration of (5.35) yields explicitly

i,.

d5 U,*

+A(

1

+ -A h(1- At) = 0.

(5.36)

Boundary-Layer Equation

213

Since go = xo in this case, it can be verified that the condition for forming an envelope is Km*(<s)

= urn*(ts)

+

%s

= 0,

in agreement with the critical point condition (4.23). It follows that u,. is again a local minimum for the semisimilar boundary layers, according to the formulation of (4.23). We have here, of course, no more than a qualitative picture. The emphasis is on how the solution may develop a singular behavior. Like the intuitive M-R-S condition suggesting au/ay = 0 at the singularity, the verifications must come after the solution has been determined. Nevertheless, (5.31) results from a straightforward deduction, and clearly allows the possibility that the “moving separation singularity” may suddenly appear under suitable conditions. We merely have to recall that in gas dynamics there is no mystery in the formation of a shock, again due to the coalescence of characteristics, away from the body.

VI. Separation in Lagrangian Description

In the semisimilar case (Section IV) we have mentioned that the crucial coefficient K in (4.23) is proportional to the forward velocity component following a fluid particle. It is further recalled that in Section 11, for the asymptotic behavior away from the wall, the integration is carried out along the characteristics defined by dx/dt = Uo(x).Since u = Uo in the asymptotic region, physically the fluid motion has been followed again along the path lines in the (x, t)-plane. In fact, the real content of (1.1) is but Newton’s law applied to a fluid particle. The thought naturally arises that some of the pitfalls near separation in the formal discretization of the boundary-layer equations in Eulerian description perhaps are avoidable if we revert to the Lagrangian description. Furthermore, in Lagrangian description there is no distinction between steady and unsteady flows. The vexing question of the Goldstein singularity in unsteady boundary layers, perhaps only of academic interest anyway, no longer arises. We have thus a unified approach to the separation problem. In general the Lagrangian description of fluid motion results in equations much more intractable than those from the Eulerian description. The boundary-layer equations are relatively simple, and have been studied recently by van Dommelen and Shen (1977). A summary of this work is presented below.

214

Shun-Fu Shen

A. THETW~DIMENSIONAL BOUNDARY-LAYER EQUATION IN LAGRANGIAN COORDINATES

As is customary, let the particle path be designated as, in Cartesian coordinates, x = 4 5 , q, t ) ,

Y = Y(5, q, t ) , (6.11 where 5 and q are constants for a given particle, typically chosen as its location at t = 0, i.e., 5 = x(5, q, 0),q = y(5, q, 0). We seek to transform the boundary-layer equations (1.1) and (1.2) into the coordinate system 5, q and time t = t . The velocity components are, by definition, u = x,, u = yr. (6.2) The derivatives in the two sets of coordinate systems ( x , y, t ) and (5, q, t )are formally related by

[;: ; [;; 'I: Jj -[; ; 5, 5, 5,

X<

X,

Xr

-1

-xq

Y,

=f

=

OX,

- UY,

uY<;ux<]'

(6.3)

where J is the Jacobian: J =X
- Y+q.

(6.4)

In place of the continuity equation, it is well known that (1.2) implies

J = 1. The momentum equation (1.1) is then turned into U, =

U,+ UU, + D2u,

(6.5) (6.6)

where the operator D is the Eulerian operator a/ay but explicitly evaluated in the Lagrangian description. To interpret the Eulerian derivatives,we note, following (6.2)

a

a

Thus, written out in full, (6.6) becomes

a

215

Boundary-Layer Equation

We shall refer to (6.8) as the Lagrangian boundary-layer equation, fully equivalent to (1.1) and (1.2). Basically, the integration of (6.8) is to be effected jointly with (6.2) as an initial value problem for the set of trajectories for all 5, q. At t = 0, the conditions are

x

= (9

Y = q,

u = U o ( L q),

0

(6.9)

= h(<, q).

At subsequent times, we have x = x(<, q, T ) from (6.8) and obtain y(& q, z) from (6.4) and (6.5),

XCYS - X,Y<

= 1,

(6.10)

which may be written in characteristic form for numerical integration,

d(:dq:dT:dy

=

-x,,:X<:O:l.

(6.11)

With known x(5, q, T) and y(& q, T), the velocity components u and u are given by (6.2). Certain simple cases such as the Blasius flow and the similarity solution over a wedge are discussed in the original reference, but they have no bearing on separation. The details of a computational scheme described therein will also be omitted here.

B. GENESIS OF THE SEPARATION SINGULARITY

A special case has been numerically integrated to elucidate the physical process of how an initially smooth boundary layer finally may be expected to separate through the development of a singularity. It is assumed that at t=O (6.12) uo = U ( t ) f ' ( q ) , 00 = - U'(5)f(rl), where f is the exact Hiemenz solution for the plane stagnation point flow, and U = sin 5. At t > 0, the outer flow is kept unchanged: U = sin x. Thus we have a problem which models the separation over a circular cylinder where the outer flow is potential and steady, but begins with an artificial boundary layer other than that follows from an impulsive start previously calculated in the Eulerian description by Telionis and Tsahalis (1974). In the latter, there seem to be some difficulty, due to the numerical procedure, in pinpointing the first appearance of singular behavior in the sense of an unbounded u-component velocity. Nevertheless, that with only smooth initial data the singular behavior does make its entrance at some later time is beyond reasonable doubt. Eventually the boundary layer should become steady and feature a Goldstein singularity at x = 1.82. We thus anticipate our example also to develop a singularity after sufficient time, noting that in Langrangian description u, blows up at the singularity.

216

Shan-Fu Shen

Figure 11 shows the resulting time-dependent velocity profiles converted to the Eulerian description. At small values of x, the profiles are seen to become quickly settled and converge to that from the Eulerian calculation of the steady boundary layer under the given free stream. At larger values of x, reverse flow is present but has caused no computational difficulty. By examining the results up to t = 2.4, both x and u, as well as their first derivatives, are found to remain bounded, but yt, y,,, and u, show a tendency to blow up at the same value of From (6.7),if u, becomes infinite while both ut and u,, are finite, either or both of yr and y, should be expected to become unbounded. It also implies that the mapping between (x, y) and (t,q) turns singular. This is illustrated in Fig. 12, which shows the Lagrangian grid in the Eulerian (x, y)-plane. (The actual 33 x 73 grid is plotted at only selected points to form a 5 x 7 grid for clarity.) The gross distortion of the rectangular elements near x = 2, at t = 2.4, is evident. In accordance with (6.11),the displacement y can be integrated along the curve x = constant in the (t,q)-plane. If s is the arc length along x = constant, with s = 0 at the wall q = 0,

r.

(6.13) In Fig. 13 the contours of both x and (xt2 + xq2)'/' at t = 2.4, are plotted in the (t,/l)-plane, where /l = (2/11) tan-l(q/lS), its use permitting us to cover 0 < q < co in a finite plot. There is indeed a minimum of (xr2 + x , , ' ) ~ ' ~ marked by 0,and it lies close to the contour for x = 2. The peaking of y at t = 2.4 near x = 2 in Fig. 12 reflects this behavior. To save computing time, ~ ~ ~with coarser grids. The miniafter t = 2.4, a study of (xc2+ x , , ~ is) made mum is observed to move closer to zero with increasing time, and a systemxh.0.2

Y

I

Y

0.6

/ P

U

U

U

FIG. 11. Evolution of local velocity profile in the example of van Dommelen and Shen (1977).

Boundary-Layer Equation

1.0

217

1'12

FIG. 12. Distortion of the Lagrangian grid in the example of van Dommelen and Shen (1977).

atic trend showingthe effect of grid size is also obtained. By extrapolation the estimate is reached that (xr2 + xs2)112= 0 at t = 2.75, at which time, therefore, the solution becomes singular in the sense that y -, 00. On hindsight, the vanishing of (x5* + xs2)1/2means of course xy = 0,

xs=0,

(6.14)

which may be identified as the conditions for the formation of an envelope of x((, q, t), the forward penetration along the path line for different particles (t,q). This event takes place for a certain particle (<,, qc),say, at a certain time given by (6.14). Physically, (6.14) expresses the fact that the fluid particles in a small volume near (tC, t , ~ ~arrive ) at the same point in the xdirection simultaneously. Because of continuity, the original small volume is

< FIG.13. Contours of x and (x; (1977).

+ x:)'~~

in the example of van Dommelen and Shen

218

Shun-Fu Shen

forced to stretch tremendously in the y-direction.The path lines literally run up against a virtual barrier and further forward penetration becomes impossible. This phenomenon clearly should correspond to separation. Since (6.14) originates from (6.13), it is a constraint due to continuity on the physical realizability of the flow. The same constraint must therefore apply to all flows, steady or unsteady, regardless of the dynamic forces which produce the motion. To demonstrate its consequences in the steady case is particularly simple. The path lines now coincide with the streamlines. Introducing the stream function $, we can always write the u-component velocity in terms of the von Mises coordinates as u = u(x, $). Thus, along the path line. (6.15)

where $ = I)(<, q ) represents the path line followed by the fluid particle with the location (l,q ) at t = 0. Hence, t,X<

+ ts*< + t<= 0, t, xq + tJI$, = 0.

(6.16)

Barring the exceptional cases, we see immediately that (6.14) is satisfied when l/t, = 0, or u(x,, $) = 0. If u(x, $) > 0 for x < x,, u(xs, $) must be a local minimum for the velocity profile at x = x,. The M-R-Sversion of the Prandtl condition for separation is therefore derivable from (6.14). Within the boundary-layer description, incidentally, the separated region should be depicted as a vertical barrier rather than the customary sketch of an embedded blunt body. In essence, (6.14) is no more than a formalized expression of the Prandtl concept-that the boundary layer must break away when a packet of fluid particles are stopped in their forward advance along the wall. In the present form, however, it may serve as the stepping-stone for further explorations. ACKNOWLEDGMENT This work has been partially supported by the Air Force Office of Scientific Research Grant AFOSR-74-2659.To W. R. Sears, friend and long time colleague, I owe my initial interest of the problem and many conceptual discussions. REFERENCES BLOTTNER, F. G. (1970) Finite-difference methods of solution of the boundary-layer equations. A I A A J . 8, 193-205. BROWN,S. N. (1965). Singularities associated with separating boundary layers. Phil. Trans. R. SOC.London, Ser. A 257,409444.

Boundary-Layer Equation

219

BUCKMASTER, J. (1973). The unsteady evolution of the singularity at separation. J . Eng. Math. 7 , 223-230.

CATHFRALL, D., and MANGLER, K. W. (1966). The integration of the two-dimensional laminar boundary-layer equation past the point of vanishing skin-friction. J . Fluid Mech. 26, 163-182.

E., VAN INGEN,J. L., and Koor, J. W. (1972). Some research on two-dimensional DOBBINGA, laminar separation bubbles. A G A R D CPP-102, 2-1-2-9. GORTLER, H. (1957). A new series for the calculation of steady laminar boundary-layer flows. J. Math. Mech. 6. GOLDSTEIN, S. (1948). On laminar boundary-layer flow near a point of separation. Q.J. Mech. Appl. Math. 1, 43-69. HAYASI, N. (1962). On semi-similar solution of the unsteady quasi-two-dimensional incompressible laminar boundary layer equations. J . Phys. SOC.Jpn. 17, 194-203. HORTON,H. P.(1974). Separating laminar boundary layers with prescribed wall shear. A I A A J . 12, 1772-1774. HOWARTH, L. (1938). On the solution of the laminar boundary layer equations. Proc. R . SOC. London, Ser. A 164, 547-579. JONES,C. W. (1948). On a solution of the laminar boundary-layer equation near a position of separation. Q. J . Mech. Appl. Math. 1, 385-407. LUDWIG,G. R. (1964). An experimental investigation of laminar separation from a moving wall. A I A A , N e w York, Pap. 64-6. MESSITER, A. F. (1970). Boundary layer flow near the trailing edge of a flat plate. S I A M (Soc. Ind. Appl. Math.) J . Appl. Math. 18, 241-257. MESSITER, A. F., and ENSLOW,R. L. (1973). A model for laminar boundary-layer flow near a separation point. S I A M (Soc. Ind. Appl. Math.) J. Appl. Math. 25, 655-670. MOORE,F. K. (1958). On the separation of the unsteady boundary layer. In “Boundary Layer Research” (H. Gortler, ed.), pp. 296310. Springer-Verlag, Berlin and New York. NENNI,J. P. (1976). An asymptotic approach to the separation of two-dimensional laminar boundary layers. Ph.D. Thesis, Cornell University, Ithaca, New York. PHILLIPS, J. H.,and ACKERBERG, R. C. (1973). A numerical method for integrating the unsteady boundary-layer equations when there are regions of back-flow. J. Fluid Mech. 58,561-579. SEARS,W. R., and TELIONIS,D. P. (1971). Unsteady boundary-layer separation. In “Recent Research on Unsteady Boundary Layers” (E. A. Eichelbrenner, ed.), pp. 404-447. Lava1 Univ. Press, Quebec. SEARS,W. R., and TELIONIS, D. P. (1975). Boundary layer separation in unsteady flow. S I A M (Soc. Ind. Appl. Math.) J . Appl. Math. 28, 215-235. SHEN,S. F., and NENNI,J. P. (1975). Asymptotic solution of the unsteady two-dimensional incompressible boundary layer and its implications on separation. In “Unsteady Aerodynamics” (R. B. Kinney, ed.), Vol. 1, pp. 245-259. Univ. of Arizona Press, Tucson. STEWARTSON, K. (1969). On the flow near the trailing edge of a flat plate 11. Mathematika 16, 106121.

STEWARTSON, K. (1970). Is the singularity at separation removable? J . Fluid Mech. 44,247-364. STEWARTSON, K. (1974). Multistructured boundary layers on flat plates and related bodies. Adu. Appl. Mech. 14, 145-239. SYCHEV,V. Y. (1972). Concerning laminar separation. Izo. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 3,47-59. TANI,I. (1958). An example of unsteady laminar boundary layer flow. Aero-Res. Inst., Univ. Tokyo, Rep. No. 33 1. TELIONIS, D. P. (1970). Boundary layer separation. Ph.D. Thesis, Cornell Univ., Ithaca, New York.

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TELIONIS, D. P., and TSAHALIS, D. TH. (1974). Unsteady laminar separation over impulsively moved cylinders. Acta Astronautica 1, 1487-1505. TELIONIS, D. P., TSAHALIS, D. T., and WERLE,M.J. (1973). Numerical investigation of unsteady boundary layer separation. Phy. Fluids 16, 968-973. TOLLMIEN, W. (1946). On the behavior of a flow along a wall at the outer edge of its boundary layer. Br. A R C Rep. No. 9739. V A N DOMMELEN, L. L., and SHEN,S. F. (1977). The laminar boundary layer in Lagrangian description. Bienn. Fluid Dyn. Symp., f 3 t h , Olsztyn, Poland (to be published). VELDMAN,A. E. P. (1976). Boundary layer flow past a finite flat plate. Ph.D. Thesis, Groningen Univ., Groningen, The Netherlands. WANG,J. C. T., and SHEN,S. F. (1977). On the closed recirculating bubble inside an unsteady boundary layer and its effects on heat transfer calculation. A I A A , New York, Pap. 77-684; retitled as: Unsteady boundary layers with flow reversal and the associated heat transfer problem. A I A A . J., in press. WERLE,M.J., and DAVIS,R. T. (1972). Incompressible laminar boundary layers on a parabola at angle of attack: a study of separation point. J. Appl. Mech. 39, 7-12. J. C., and JOHNSON, W. D. (1974a). Semi-similar solutions to unsteady boundaryWILLIAMS, layer flows including separation. A I A A J. 12, 1388-1393. WILLIAMS, J. C., and JOHNSON, W. D. (1974b). A note on unsteady boundary layer separation. A I A A J . 12, 1427-1429.

ADVANCES IN APPLWD MECHANICS. VOLUME

18

The Theory of Ship Motions? J . N . NEWMAN Department of Ocean Engineering Massachusetts Institute of Technology Cambridge. Massachusetts

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

222 227

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

235 236 237 240 241 242

111. The Boundary-Value Problem A . Exact Formulation . . . . .

B. The Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . C. Linear Decomposition of the Unsteady Potential . . . . . . . . . . . D . Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Slender Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Two-Dimensional Green Function . . . . . . . . . . . . . . . . . . B. The Three-Dimensional Green Function . . . . . . . . . . . . . . . . . V. Two-Dimensional Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Radiation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The DiNraction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . C. Applications of Green's Theorem . . . . . . . . . . . . . . . . . . . . . D . LongWavelength Approximations . . . . . . . . . . . . . . . . . . . . VI . Slender-Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Outer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Inner Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The Inner Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Slender-Body Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Outer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Inner Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The Inner Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. The Long-Wavelength Solution . . . . . . . . . . . . . . . . . . . . . . VIII . The Pressure Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Added Mass and Damping . . . . . . . . . . . . . . . . . . . . . . . . . B. The Exciting Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

244 245 246 249 250 252 253 256 258 258 259 262 265 266 267 268 269 271 272 273 275 277 280

t Preparation of this article was supported by the National Science Foundation and by the Office of Naval Research. 22 1

Copyright 0 1978 by Academic Press. Inc All rights of reproduction in any form reserved. ISBN 0-12-002018-1

222

J . N . Newman

I. Introduction Oceangoing ships are designed to operate in a wave environment that is frequently uncomfortable, and sometimes hostile. Unsteady motions, and structural loading of the ship hull, are two of the principal engineering problems which result. Early research in ship hydrodynamics was devoted primarily to operations in calm water, but a landmark paper “On the Motions of Ships at Sea” by Weinblum and St. Denis (1950) focused attention on this subject, and extensive work has followed. Ships generally move with a mean forward velocity, and their oscillatory motions in waves are superposed upon a steady flow field. The solution of the steady-state problem is itself of interest, particularly with regard to the calculation of wave resistance in calm water; a comprehensive survey has been given by Wehausen (1973). The opposite special case is that of wave interactions with a vessel which has no mean velocity; this topic is reviewed by Wehausen (1971), and more recent numerical solutions are described by Mei (1977). The problem of ship motions in waves can be regarded as a superposition of these two special cases, but interactions between the steady and oscillatory flow fields complicate the more general problem. All three topics are discussed by Ogilvie (1977). In order to predict its motions in waves, a ship may be regarded as an unrestrained rigid body with six degrees of freedom as defined in Fig. 1. The three components of translation are surge parallel to the longitudinal axis, heave in the vertical direction, and sway in the lateral direction orthogonal to surge and heave. Rotational motions about the same axes are roll, yaw, and pitch, respectively. If the unsteady motions of the ship and the waves are of small amplitude, systematic perturbation procedures can be justified, with the leading-order solutions linear in these small amplitudes. Furthermore, the ambient seaway

FIG.1. The coordinate system and six modes of ship motion.

Theory of Ship Motions

223

can be decomposed into individual components which are unidirectional and sinusoidal. In the jargon of this field, spectral analysis makes the study of ship motions in regular waves applicable ultimately to an irregular seaway. This synthesis is well known in other fields of applied mechanics, particularly in random vibrations and acoustics where the analogies are obvious. The first application of spectral analysis to ship motions was made by St. Denis and Pierson (1953). More recent developments are described by Price and Bishop (1974), and in the proceedings of two symposia: Society of Naval Architects and Marine Engineers (1974), and Bishop and Price (1975). The present article is restricted to ship motions in regular waves. The dynamics of ship motions are governed by equations of motion which balance the external forces and moments acting upon the ship, with the internal force and moment due to gravity and inertia. (Hereafter the term “force” will be used in a generalized sense to include the moment.) Assuming the ship to be in stable equilibrium in calm water, its weight is balanced by the force of hydrostatic pressure. Similarly, the steady drag and propulsive force are balanced. These steady forces may be neglected, and our attention is focused on the unsteady perturbations. The principal unsteady force acting upon an unrestrained vessel is due to the hydrostatic and hydrodynamic components of the normal pressure acting on the submerged surface. Additional force components which generally are neglected include the force on the ship’s propeller, viscous forces acting on the submerged surface of the hull, and aerodynamic forces acting upon the ship above the free surface. With the assumption of small unsteady motions, of the ship and surrounding fluid, linear superposition can be applied. Thus we consider separately the radiation problem, where the ship undergoes prescribed oscillatory motions in otherwise calm water, and the diflraction problem, where incident waves act upon the ship in its equilibrium position. Interactions between these two first-order problems are of second order in the oscillatory amplitudes, and may be neglected in the linear theory. The radiation problem may be decomposed further, by considering separately the six degrees of freedom defined above. In each of these modes, outgoing radiated waves will exist on the free surface. The existence of radiated waves implies a complicated time dependence of the fluid motion, and hence the pressure force. Waves generated by the body at time t will persist, in principle for an infinite time thereafter, and the resultant pressure force on the body will act similarly. This situation can be described mathematically by a convolution integral, with the fluid motion and pressure force at a given time dependent upon the previous history of the ship’s motion. In this respect the irrotational flow due

J . N . Newman

224

to unsteady motions of a floating body is analogous to unsteady liftingsurface theory, where in the latter problem the source of “memory” is the shed vorticity in the wake. This aspect of ship motions is emphasized by Cummins (1962) and Ogilvie (1964). Similar effects due to viscous separation are analyzed by Brard (1973). Experimental techniques which account for the convolution effect are discussed by Bishop et a!. (1973) and by Wehausen (1978). For steady-state oscillatory motion a simpler description can be utilized: since the linear pressure field is oscillatory in time, and proportional to the amplitude of forced motion in each mode, the resulting force on the ship hull must be of the form

1

Fi(t)= Re eio‘

6 j= 1

tjtij/,

(i = 1, 2, ..., 6).

(1.1)

Here t j is the complex amplitude of the ship’s oscillatory displacement in each mode of motion, o is the frequency, and t i j is a complex transfer function which depends on the geometry of the ship hull, the frequency w, and the forward velocity U. Re denotes the real part, which is implied hereafter when the time dependence eiof is displayed. The complex transfer function t i j can be expressed in terms of its real and imaginary parts. From the physical standpoint a more useful decomposition is t I.J . = w2a.. - job..- c.. (1.2) IJ IJ IJ‘ The coefficientsaij, bij,cijare real and correspond, respectively, to the force components due to acceleration, velocity, and static displacement of the ship. It is important to emphasize that the coefficients in (1.2) are not constant, but depend on the same parameters as do t i j . In particular, the coefficientsaij, bij, and cij will depend in general on the frequency w, and the representation (1.2) is not meant to imply that t i j is a second-order polynomial in o. The damping codjcients bij in (1.2)are specified uniquely by the imaginary part of t i j . The diagonal elements bii and suitable combinations of offdiagonal elements, can be associated with the work done to oscillate the ship. These can be related to the energy flux in the radiated waves, under the usual assumption of an inviscid fluid. The separation of the added-mass coefficients aijand restoring coefficients cij is somewhat arbitrary. A physically appropriate subdivision can be affected by defining cij = - lim t i j . 0-0

Theory of Ship Motions

225

With this definition, the restoring coefficients cij can be associated with the hydrostatic pressure gradient and, to a lesser extent, with the steady-state dynamic pressure field. In the diffraction problem the ship moves in its steady-state orientation, in the presence of incident plane progressive waves of prescribed amplitude, wavelength, and direction. The resulting oscillatory exciting force on the ship hull is proportional to the wave amplitude A, and can be expressed in the form Fi(t)= AXieimf. (1.4) Here Xi is a complex force coefficient which depends on the wavelength and direction of the incident waves and on the ship geometry and forward speed. In a fixed frame of reference the incident-wave frequency oois related to the wavelength A and wavenumber K O = 2n/A by the dispersion relation. For water of constant depth h, wo = (gKo tanh K0h)”’,

(1.5)

and in the deep-water limit (h -+ 0 0 ) o0= (gK0)”’. In the moving reference frame of the ship, the incident waves arrive with the frequency of encounter o = 1010 - U K o cos

/3I.

(1.7) Here /3 is the angle of incidence, between the phase velocity of the waves and the forward velocity of the ship. In effect, (1.7) introduces a Doppler shift between the wave frequency and the frequency of encounter. The frequency of encounter is reduced infollowing seus (/3 = 0),whereas o is a maximum in head seus (/3 = n). The total oscillatory pressure force acting on the ship hull is the sum of (1.1) and (1.4). The equations for unrestrained motion of the ship in a prescribed incident-wave system follow from Newton’s equations. Since these equations of motion are linear and algebraic in ti,the only nontrivial task is to predict the coefficients in (1.2) and (1.4). The six modes of ship motions can be categorized in terms of the magnitudes of the corresponding restoring coefficients cii. These determine the scale of the natural frequency in each mode and the resulting response characteristics. For surface ships (as opposed to submarines), small vertical motions are opposed by a hydrostatic restoring force proportional to the waterplane area S. The resonant frequency can be estimated by neglecting hydrodynamic forces and equating the restoring force to the product of the ship’s mass

226

J . N . Newman

and acceleration. Since the mass is proportional to the displaced volume V, the heave natural frequency is of order (1.8) where T is the draft. The principal hydrodynamic effect is to increase the effective mass of the ship and to reduce the natural frequency, but the order of magnitude is not changed. The same estimate applies to pitch, since the relevant moments of inertia of the waterplane area and ship’s mass are both proportional to L?.In this respect, heave and pitch are dynamically similar. The existence of a substantial static restoring force ensures that the amplitude of the motions will be relatively small, except near the resonant frequencies. The response at resonance depends on the magnitude of the damping forces and the degree of tuning with respect to the exciting forces. The damping of pitch and heave motions is due principally to the radiation of wave energy. In these modes the damping is generally subcritical but sufficient to prevent highly tuned resonant response. The exciting forces in pitch and heave are significant only if the wavelength is comparable to or greater than the ship length. From the dispersion relation it follows that excitation will occur principally from waves with a characteristic frequency ooIO(g/L)’”. For conventional ships L >> T, and the estimate (1.8) indicates that resonance will occur in combination with significant wave excitation only if the frequency of encounter is substantially greater than the wave frequency. When the Doppler shift (1.7)is considered, it follows that heave and pitch are most severe in head seas. Under these circumstances the acceleration and structural loading on the ship hull are most severe. In most cases, heave and pitch are the modes of greatest practical importance. Moreover, the small amplitudes and large inertial effects of these modes can be described by a linearized theory which assumes that the unsteady motions are of small amplitude and that the fluid is inviscid. Thus, theoretical approaches based on the methodology of applied mechanics have been most successful in predicting these modes. The static restoring moment in roll is small for conventional ships. This is due in part to the narrow beam. In addition, the vertical position of the center of gravity may be positioned to reduce this restoring moment, lengthen the natural period, and hence to reduce the angular acceleration. The radiation damping is weak in roll, especially at the low frequencies near resonance. Thus resonant motions occur with a large amplitude and with significant nonlinear and viscous effects. No satisfactory method exists for predicting the rolling motion of ships with engineering accuracy. The remaining motions are in the horizontal plane, unopposed by (gS/V)”2 = 0(g/T)’/2,

Theory of Ship Motions

227

hydrostatic restoring forces. The response is nonresonant, but motions of large amplitude may occur at low frequencies in following waves. Particularly serious in this context is broaching due to the combined effects of dynamic instability and prolonged unidirectional excitation. As in the case of roll, the modes of surge, sway, and yaw may be influenced significantly by nonlinear and viscous effects. A comprehensive discussion of ship motions in following waves is given by Oakley et al. (1974). Following a brief historical outline in Section 11, the remainder of this article is devoted to a linearized analysis of ship motions in incident waves. Slender-body approximations are applied to the ship’s hull form, without restricting the scale of the wavelength or frequency. All six modes of ship motion are considered. In our analysis the fluid is assumed ideal, with irrotational motion, and unbounded except for the submerged portion of the ship hull and the free surface. Surface tension is neglected. The ship hull is assumed symmetrical, in the “port-and-starboard” sense, about a vertical centerplane which contains the longitudinal axis. The unsteady motions are assumed sinusoidal in time, and they are of small amplitude. These assumptions, and the discussion above, imply that the results may be of the greatest practical value for predicting heave and pitch motions. The exact and linearized boundary-value problems are formulated in Section 111. Fundamental solutions are summarized in Section IV, including the two- and three-dimensional source potentials or Green functions which satisfy the linear free-surface boundary condition. Two-dimensional solutions of the boundary-value problem adjacent to each section of the ship hull are outlined in Section V. The method of matched asymptotic expansions is used with these twoand three-dimensional solutions, to derive a slender-body approximation for the three-dimensional oscillatory flow field. This analysis is carried out for the radiation problems in Section VI, and for the diffraction problem in Section VII. In Section VIII these solutions are used to determine the hydrodynamic pressure force acting on the ship.

11. History Early sailing vessels favored trade-wind routes with following seas and were unable to move with great speed to windward. For this reason, heave and pitch were not of great importance whereas rolling motions in waves were reduced by the stabilizing influence of the sails. The first steamships could not attain high speed in head seas. However,

228

J. N. Newman

the absence of aerodynamic damping increased the importance of roll. This may explain the initial study of rolling motions by William Froude (1861). Subsequently, the importance of pitch and heave motions increased with the power and speed of ships, attracting attention and study by the Russian naval officer Krylov (Kriloff, 1896). Froude and Krylov derived differential equations of motion for the inertial and restoring forces of the ship. No attempt was made to analyze the hydrodynamic disturbance associated with the presence of the ship hull. Only the pressure field of the undisturbed incident waves was considered, and the resultant force on the ship has become known as the Froude-Krylov exciting force. The first significant step to account for the hydrodynamic disturbance due to a ship hull of realistic form was the steady-state wave-resistance theory of Michell (1898). Michell assumed the ship to be thin, with small beams (B) compared to the ship length, draft, and wavelength. In this respect, his approach is related to the solution of the thickness problem in thin-wing theory, which followed subsequently. Michell recognized the possibility of extending his theory to include unsteady motions, but a promised sequel on heave and pitch was never published. Another major advance in accounting for the ship’s hydrodynamic disturbance followed in a study by Lewis (1929), of the added mass associated with hull vibrations in structural modes. In this problem the characteristic frequency is sufficiently large such that inertial effects are dominant, and gravitational forces can be neglected. Thus wave effects are ignored, greatly simplifying the analysis. Lewis assumed the ship hull to be slender, and used a striptheory approach to integrate the hydrodynamic force longitudinally in terms of the two-dimensional characteristics of each transverse section. Not content with this simplified approach, Lewis derived three-dimensional correction factors by reference to the exact solutions for a prolate spheroid. This appears to be the first development of a strip theory in ship hydrodynamics. In an inviscid fluid, the vertical motion of a thin ship is equivalent mathematically to the “wave-maker problem” of a prescribed normal velocity on a vertical plane. The latter problem was solved by Havelock (1929),during a prolific career devoted primarily to the theory of wave resistance. It was not until his last published paper, however, that Havelock (1958) explicitly studied the oscillatory motions of a thin ship. A comprehensive analysis of pitch and heave motions was made in two papers by Haskind (1946a,b). Green’s theorem was used to construct the velocity potential due to the presence of the ship hull, and the necessary Green’s function or source potential was derived. The thin-ship approximation was invoked to solve the resulting integral equation. Haskind treated

Theory of Ship Motions

229

the initial-value problem for arbitrary time dependence, and the special case of sinusoidal motion. A notable feature of Haskind’s work is the decomposition of the velocity potential into a canonical form which includes separately the solution of the diffraction problem and solutions of the radiation problem for each mode of oscillatory ship motion. The thin-ship approximation was reexamined in a critical fashion by Peters and Stoker (1957). A systematic perturbation procedure was adopted with the ship’s beam and the unsteady motions assumed to be of the same small order of magnitude. On this basis the Froude-Krylov exciting force is the only first-order hydrodynamic force. This rather trivial first-order theory essentially confirmed the approach of Froude and Krylov, but cast doubt upon the value of Haskind’s more extensive work. The thin-ship approximation was refined by Newman (1961), with a more accurate statement of the boundary condition on the oscillatory ship hull. A systematic expansion in multiple small parameters was used to avoid the results of Peters and Stoker (1957). Computations, however, of the damping coefficients presented by Gerritsma et al. (1962) did not correlate well with experiments. It is obvious that the (inviscid) hydrodynamic disturbance due to vertical motions of a thin ship is small in proportion to the beam. To avoid this situation, Peters and Stoker (1957) advocated a complementary “flat-ship” approximation with the draft small compared to the beam and length. This leads to an integral equation similar in form to that of lifting-surface theory, but with a more complicated kernel. Moreover, the intersection of the free surface and the ship is a singular region which must be examined with great care. Steady-state solutions have been derived for planing boats, as described by Ogilvie (1977), but unsteady solutions are restricted to the case of zero forward speed. Typical ship hulls are elongated, with the beam and draft of the same small order of magnitude compared to the length. Thus it is logical to develop a three-dimensional approximation analogous to the slender-body theory of aerodynamics. This was done initially for the steady-state waveresistance problem by Cummins (1956); subsequent references are given by Ogilvie (1977). An important restriction resulted from the assumption, not always explicitly stated, that the ship is slender relative to the characteristic wavelength. Thus, the beam and draft were assumed small compared to the wavelength scale U 2 / g as well as the ship length L. Equivalently, the Froude number U(gL)-‘’’*’ was assumed to be of order one. Unsteady solutions based on similar slender-body assumptions and applicable to the prediction of ship motions in waves were derived by Ursell (1962), Joosen (1964), Newman (1964), Newman and Tuck (1964), and Maruo (1967). Here the long-wavelength assumption rZ = O(L)seemed phys-

J . N . Newman

230

ically appropriate and mathematically convenient. In this case, however, as in the thin-ship approach, most nontrivial hydrodynamic effects are of higher order by comparison to the hydrostatic restoring force and the Froude-Krylov exciting force. Moreover, the inertial force due to the body mass is of higher order, with the result that the leading-order equations of motion are nonresonant. Nevertheless, for shiplike vessels without forward speed, this simple longwavelength theory gives reasonable predictions of the pitch and heave motions, as illustrated in Fig. 2. An explanation of this fortunate situation is the disparity between the natural frequencies and the wave frequencies where the exciting force and moment are significant. A different situation results if the ship moves in head seas with forward velocity. Here the domains of resonance and significant excitation overlap, 1.25 X

F = 0.00

0 X

X

F = 0.14

1.00

0.75

l€Sl A

0.50

0

i

0

0.25

I

I

0.00

I

%&I

a

I

5.0

I

2.0

I

1.0

x

I

I

0.5

0.3

c-

L

FIG.2. Amplitude of heave motion, per unit wave height, predicted from the leading-order low-frequency slender-body theory, and compared with experimental measurements for zero forward velocity and a Froude number F = 0.14. This figure is reproduced from Newman (1977), where a similar plot of the pitch motion is included.

23 1

Theory of Ship Motions

due to the Doppler shift toward higher frequencies of encounter. The effect of this shift is indicated by the experimental data with forward velocity in Fig. 2. Naval architects have not awaited a three-dimensional theory of ship motions which is both rigorous and practical. Instead the numerical solutions of a simpler class of two-dimensional problems have been utilized, where the body floats on the free surface and performs small oscillatory motions without forward speed. The wavelength and body dimensions are not restricted, but the solution is most useful, and nontrivial, when both length scales are of the same order of magnitude. This type of problem was first solved rigorously by Ursell (1949) for the heaving motions of a halfimmersed circular cylinder. Various extensions and generalizations have been carried out subsequently, as described in the survey of Wehausen ,(1971). A compendium of two-dimensional results is given by Vugts (1968), with more recent numerical techniques described by Chapman (1977) and Mei (1977). Typical results for the heave added mass and damping of a rectangular cylinder are shown in Figs. 3 and 4. The first utilization of two-dimensional results in a three-dimensional striptheory approximation for ship motions was made by KorvinKroukovsky (1955). This work was self-contained in the sense that all of the

I

0.0 0.0

I

I

0.5

u4-

1.0

I

1.5

FIG. 3. Added-mass coefficients for a family of two-dimensional rectangular cylinders, based on the computations of Vugts (1968). With the normalization shown, the added-mass coefficient is logarithmically infinite for w 0, and also for B/T -,0.

-.

J . N . Newman

232

1

I.o

t b33

pe'w

0.5

0.a

FIG.4. Damping coefficients for a family of two-dimensional rectangular cylinders, based on the computationsof Vugts (1968).The scale on the right gives the exciting-forcecoefficient, in accordance with the Haskind relations (5.30).

relevant forces were considered in a linearized analysis of pitch and heave motions in head seas. Use was made of concepts from the slender-body theory in aerodynamics, supplemented by shrewd physical insight, to account for the effects of forward speed. Some refinements and extensive experimental comparisons were provided in a sequel by KorvinKroukovsky and Jacobs (1957). Theoretical workers were slow to accept the striptheory approach of Korvin-Kroukovsky, due to the lack of a systematic and rational derivation. This defect was of less concern to practical naval architects, who recognized the computational simplicity of the strip theory and the generally satisfactory agreement with experiments. In both respects, this nonrigorous approach compared favorably with the thin-ship theory. Typical calculations for heave and pitch are compared with experimental data in Fig. 5. More comprehensive evaluations of the strip theory are given by Salvesen et al. (1970) and Gerritsma (1976). Grim (1960) provided some of the first systematic calculations of the two-dimensional added-mass and damping coefficients for shiplike forms, suitable for use in the strip theory. Not content with this approximation, Grim also proposed a heuristic correction based on the distribution of threedimensional sources on the surface of the ship, with the source strength

Theory of Ship Motions

233

FIG. 5. Heave and pitch motions predicted from strip theory and compared with experiments for a Froude number F = 0.3. (From Newman, 1977.)

determined from the two-dimensional solution at each section. Further corrections were introduced to account for the induced normal velocity of these sources at other longitudinal positions along the ship hull. By comparison to the slender-body results illustrated in Fig. 2, the superiority of the strip theory for predicting pitch and heave motions in head waves can be explained by the practical importance of the resonant frequency regime. From the estimate (1.8) this is precisely the shortwavelength regime where o z T / g = 0(1) and thus, for a slender ship, o24g 9 1. A rational foundation for the strip theory was suggested by Vossers (1962) and Joosen (1964), based on the slender-body assumptions B/L = E << 1, BIT = O(1). These authors showed that in the short-wavelength regime the solution of the three-dimensional radiation problem in the near field adjacent to the ship hull is identical to the strip-theory solution. In the absence of forward speed, this conclusion is fairly obvious: if a slender ship is radiating short waves with slowly varying phase along the length, these waves will be locally two-dimensional, and three-dimensional interference effects are absent from the inner region. The principal effect of forward speed is to introduce the convective derivative Ud/ax, but this is dominated by the time derivative in the high-frequency regime. A systematic analysis of the short-wavelength slender-body problem for heave and pitch was carried out by Ogilvie and Tuck (1969). In addition to

234

J . N . Newman

the leading-order zero-speed striptheory results, Ogilvie and Tuck conThese sistently retained the higher-order terms of relative magnitude E higher-order corrections are linearly proportional to U and thus provide a rational approximation for the effects of the ship’s forward speed. In spite of the work of Ogilvie and Tuck (1969), certain aspects of the strip theory remain unsatisfactory from the rational viewpoint. The principal questions concern the validity of the solution at lower frequencies, the emergence of forward-speed effects only as higher-order corrections, and the intractable nature of the diffraction problem in short incident waves. From the underlying assumptions, it is clear that the strip theory is invalid at low frequencies of encounter. An example is that the (two-dimensional) added-mass coefficient for vertical motions is logarithmically infinite as the frequency tends to zero, as shown in Fig. 3. This is of little importance for predictions of ship motions in head seas since it is the product w’aij which occurs in the equations of motion. Since the strip theory correctly predicts the hydrostatic effects which are dominant as w -+ 0, the resulting equations of motion are also correct for w -+ 0. (However,-this is a consequence of the fact that the Froude-Krylov force is left in its original form; if it is expanded consistently for large frequencies, the result is divergent for w -,0.) In the strip theory, forward speed affects the hydrodynamic force simply by introducing terms proportional to (U/w) and (Ulw)’. By assumption, U / w = O(E’’’),and thus the effects of forward speed are higher-order corrections to the zero-speed leading-order theory. Ogilvie and Tuck (1969) retain terms of relative order E’/’ while neglecting higher-order terms O(E).As such, their theory is mathematically consistent, but at variance with most of the intuitive versions of the theory which retain some additional corrections proportional to (U/w)’. The diffraction problem is complicated in the strip-theory synthesis by the assumption of high-frequency, short-wavelength incident waves. These are rapidly oscillatory along the ship length, and the scattering potential must be sought as the product of a longitudinal oscillatory function times a slowly varying solution of the Helmholtz equation. For head seas the latter problem is singular and a special analysis is required. Various schemes have been devised to circumvent this difficulty, with Green’s theorem used to replace the diffraction problem by a simpler radiation problem. An alternative approach outlined by Newman (1977) restricts the incident wavelength to be intermediate between the ship’s length and beam; the resulting formula for the exciting force is used in Fig. 5. The conventional strip theory is deficient not only for low frequencies, but also for high speeds. A complementary approach initiated by Chapman (1975) uses a high-Froude-number approximation suggested by Ogilvie (1967). The flow at each section along the ship is analyzed in a quasi-two-

’”.

Theory of Ship Motions

235

dimensional manner, with interactions propagated downstream by the freesurface condition. This technique is discussed further by Ogilvie (1977). Chapman’s computations for the sway and yaw response of a thin ship are supported by impressive agreement with experiments. Ship motions are of primary interest offshore in deep water. However, the operation in coastal waters of supertankers, and other ships of deep draft, has increased the importance of the shaliow-water regime. In the slenderbody theory, the effects of finite depth are confined to the outer field for h/L = 0(1),but for depths comparable to the beam and draft the near field is affected and the order of magnitude of the hydrodynamic forces is increased. Beck and Tuck (1972) have studied the latter regime, using shallow-water approximations to simplify the results. The necessity of analytic approximations may be questioned in the present era of numerical fluid mechanics. Only Chang (1977) has reported success with the direct numerical solution of the linearized threedimensional ship-motion problem. Calculations of the added-mass and damping coefficients are presented for a realistic ship hull in all modes except surge. The results show reasonable agreement with experiments, except in the roll mode, and confirm the limitations of the striptheory approximations.

111. ‘Ihe Boundary-Value Problem It is helpful to define three Cartesian coordinate systems, with xo = (xo, yo, zo) fixed in space, x’ = (x’, y’, z’) fixed with respect to the ship, and x = (x, y, z) moving in steady translation with the mean forward velocity of the ship. The space-fixed system xo is the simplest in which to express the free-surface boundary condition, whereas the shipfixed system x’ is the best in which to derive the boundary condition on the ship’s wetted surface. The steadymoving coordinate system x is an inertial reference frame in which the motions are periodic. We take z, = 0 as the plane of the undisturbed free surface, the xo-axis positive in the direction of the ship’s forward velocity, and the zo-axis positive upward. The steady-moving coordinate system is defined by the transformation

x = (xo - Ut, yo, zo),

(3.1)

with U the mean forward velocity of the ship. The shipfixed coordinate system is defined such that x’ = x in steady-state equilibrium.

236

J . N . Newman

A. EXACTFORMULATION With the assumptions noted in Section I, the fluid velocity vector V(x,, t ) is equal to VCP, with the velocity potential CP(xo, t ) governed by Laplace’s equation V’CP = 0 throughout the fluid domain. The fluid pressure p(x,, t ) is given by Bernoulli’s equation p = -p(CPt

+ +vz + gz,) + pa.

(3.2) Here p is the fluid density, g is the gravitational acceleration, and pa is the atmospheric pressure which is assumed constant. In (3.2) and hereafter, when the independent variables ( x , t ) appear as subscripts partial differentiation is indicated. On the submerged portion of the ship’s surface S, the normal velocity is equal to that of the adjacent fluid. The appropriate boundary condition is

-

(Vs - V ) n = 0 on S, (3.3) where V, is the local velocity of the ship’s wetted surface. The unit normal vector n is defined to point out of the fluid domain. The free surface is defined by its elevation zo = c(xo, y o , t). On this surface, the kinematic boundary condition is expressed by means of the substantial derivative DfDt 5 afat V . V, in the form

+

c.

(D/Dt)(C - zo) = 0 on z, = (3.4) Since the position of the free surface is unknown, an additional dynamic boundary condition is imposed, that the pressure on the free surface is atmospheric. From Bernoulli’s equation (3.2) it follows that

+ &ifz+ gz,

=0

on z , = c.

(3.5) This boundary condition can be used to determine the free-surface elevation from the implicit equation

c = - (1/9)(@t+ w2)zo=[.

(3-6) Since (3.5) holds on the free surface for all time, its substantial derivative can be set equal to zero. This gives an alternative boundary condition for the velocity potential, CPtt

+ 2VCP

V a t + $VCP * V(V@ . V@)

+ gCPz, = 0

on zo = [.

(3.7) Equations (3.3) and (3.7) are the principal boundary conditions of the problem, valid on the ship hull and on the free surface, respectively. If the fluid domain contains no other boundary surfaces, the additional requirements are imposed such that V -+ 0 as z, + - 00, and such that the energy *

23 7

Theory of Ship Motions

flux of waves associated with the disturbance of the ship is directed away from the ship at infinity. The latter is the radiation condition. The problem stated above is exact within the limitations of an ideal incompressible fluid. However, the nonlinear free-surface condition precludes solutions without further simplification. Moreover, there are unresolved questions regarding the explicit form of the radiation condition, and the singularities at the intersection of the ship hull and free surface. Further progress requires the fluid motion to be small in some sense. B. THELINEARIZED PROBLEM In the theory of water waves it is customary to assume the amplitude of the oscillatory wave motion to be small by comparison to the wavelength. Equivalently, the slope of the free surface is assumed small compared to unity. With the neglect of second-order terms quadratic in derivatives of @, Eq. (3.7) can be replaced by the linearized free-surface boundary condition

+

Ort gOZ,= 0

on zo = 0. (3.8) Note that this condition is imposed on the mean position of the free surface, since the difference between the value of @ or its derivatives on zo = [ and zo = 0 is a second-order quantity. There are extensive solutions of Laplace's equation and the linearized free-surface condition (3.8), as described especially by Wehausen and Laitone (1960). Of particular importance is the plane-progressivewave of constant amplitude A and sinusoidal profile, for which the velocity potential in deep water is given by @ = ( i g A / o , ) exp[K,(z,

- ix, cos fl

- iy,

sin fl) + io,t].

(3.9)

Here K O is the wavenumber, with I = 27r/K0 the wavelength, oo is the radian frequency in the space-fixed reference frame, and fl denotes the angle of wave propagation relative to the x,-axis. Substituting (3.9) into (3.8) gives the dispersion relation (1.6). In the steady-moving reference frame the velocity potential can be redefined in the form

a

YO7

'07

l) = @(.

+ U t , y, z, l)

+(x3 y7 Thus, in accordance with the Lorentz transformation, '('07

a,= 4(xo - U t , y o , z,, at

t) =

-

(:t

z7

l).

- u - +(x, y, z, t).

(3.10)

(3.1 1)

J . N . Newman

238

Transforming the linear free-surface condition in this manner, it follows that

4,, - 2U4,,

+ U2&, + g& = 0

on z = 0.

(3.12)

In the moving reference frame the plane-wave potential (3.9) takes the form

4 = (igA/oo) exp[Ko(z - ix cos /3 - iy sin /3)

+ iot],

(3.13)

where the frequency of encounter is defined by (1.7). If the ship is stable, and if the amplitude A of the incident wave system is small, the oscillatory motions of the ship and surrounding fluid will be proportional to A. Linearization of the unsteady problem can be justified on this basis. However, suitable geometric restrictions must be placed on the ship hull to ensure that the steady-state disturbance is small. Before restricting the geometry for this purpose, we first consider the general case where only the oscillatory flow is linearized. An overbar will be used to denote the velocity potential due to the steady forward motion of the ship, q x 0 , t) = UI$(x).

(3.14)

The velocity vector of the steady flow relative to the moving reference frame is

w = UV(8 - x).

(3.15)

The boundary condition on the hull surface in its steady-state position S takes the form onS.

W.n=O

(3.16)

In the moving reference frame the nonlinear free-surface condition is

+ w .v(w’) + gI$), = o

on z =

c,

(3.17)

with the steady free-surface elevation given by the implicit formula

e = -(1/2g)(W2

- v”),,,.

(3.18)

The total potential can be written in the form q x , , t ) = 4(xy t ) = UI$(X)

+ cp(x, t),

(3.19)

where the unsteady component cp is assumed small. Neglecting second-order terms in cp, the free-surface boundary condition is

4w

*

+ d), + cp,, + 2 w . vcp, + w V ( W - Vcp) + # V q . v(W) + gqz = 0 on z = [. V(W2)

*

(3.20)

Theory of Ship Motions

239

The corresponding expression for the free-surface elevation is

Here the error is O(cpz), and the last form of (3.21) follows from (3.18) and a Taylor-series expansion. Using this formula to solve for the difference (C - r) gives

c = e - “4% + w . Vcp)/(g + w - W,)l,=i. (3.22) The contribution from the steady terms in (3.20) can be evaluated by expanding from c to c and by using (3.17). Thus, the unsteady velocity potential is governed by the first-order free-surface condition

+ cp,,+

2 w . vcp,

+ w . V(W . Vcp)

+ +Vcp . V ( W ) + gcp, = 0

on z =

c.

(3.23)

If the perturbation of the steady flow due to the ship is neglected, W = - Ui and (3.23) reduces to (3.12). The linearized boundary condition on the hull requires a similar analysis. We begin by decomposing the velocity V, in the form V, = Ui

+ a,

(3.24)

-

where a = x x’ is the local oscillatory displacement of the ship’s surface and the overdot denotes time differentiation in the reference frame of the ship. Since a is a small oscillatory quantity, this vector displacement can be expressed as a =6 +R x (3.25) XI.

Here 6 and R denote the unsteady translation and rotation of the ship, relative to the origin x’ = 0. Substitution of (3.15), (3.19), and (3.24) in (3.3) gives the unsteady boundary condition cp,=a*n-W*n

onS.

(3.26)

The last term must be retained, in spite of (3.16), since W = O(1) and the difference in the boundary values of W n on S and S is O(a). The two first-order contributions to W n on S are from the rotation of the shipfixed coordinate system, and from the gradient of the steady flow

-

J. N . Newman

240

field. After accounting for both effects,

(W * n), 2 ([W

- R x W + (a

*

-

V)W] n)s.

(3.27)

Invoking the steady boundary condition (3.16), and substituting (3.27) in (3.26), it follows that (p,, = [a + R x W - (a V)W] * n (3.28) on S , S. 3

Since each member of (3.28) is O(a),this boundary condition can be applied either on S or S with the difference O(a2). An alternative to (3.28) can be derived using (3.25), in the form q,, = [a

+ (W . V)a - ( a . V)W] - n

on S , S.

(3.29)

The first two terms in brackets give the rate of change of a in a frame of reference moving with the steady flow. Finally, since a and W have zero divergence, a vector identity gives the more compact expression (p,,

= [a

+ V x (a x W)] - n

on S , S.

(3.30)

The boundary condition (3.30) was derived by Timman and Newman (1962) to account in a consistent manner for the interaction between the steady and oscillatory flow fields. In most prior ship-motion analyses an incomplete form of (3.30) was used, which led to an erroneous asymmetry of the coupling coefficients between heave and pitch. If the perturbation of the steady flow field due to the ship is neglected in (3.29), W = -Ui, and thus qn=

[(;

= [a -

-

Ug)a]*n

U(R x i)] * n.

(3.31)

The term proportional to U can be interpreted as the product of the ship's forward velocity and the angle of attack due to pitch and yaw.

c. LINEARDECOMPOSITION OF THE UNSTEADY POTENTIAL Since the unsteady motions are assumed small, the potential (p in (3.19) can be decomposed linearly into separate components due to the incident wave, each of the six rigid-body motions, and the scattered disturbance of the incident wave. With the restriction that the unsteady motions are sinusoidal in time with the frequency of encounter w, the motions of the ship are denoted by (3.32) 5 = (51, 5 2 , 53)eiOt, R = (al,R2, 03)ei"'

= (54, 5 5 , 56)ei0'.

(3.33)

24 1

Theory of Ship Motions

With this notation, the unsteady component of the velocity potential can be expressed as (3.34) Here cpo is the incident-wave potential of unit amplitude, cpo = (ig/o,) exp[Ko(z - ix cos fl - iy sin

p)],

(3.35)

and cp, is the scattered potential such that

(d/dn)(cpo

+ cp7) = 0

on S.

(3.36)

The components cpj ( j = 1, 2, . . . , 6) in (3.34) are the radiation potentials due to motions of the ship with unit amplitude in each of six degrees of freedom. From (3.25) and (3.29), these are governed by the hull boundary conditions cpj, = ionj

+ Umj

on S.

(3.37)

Here the components n j are defined as (n1,

n2, n 3 )

(3.38)

= n,

(n4, n 5 , H 6 ) = (x x n),

(3.39)

and, following Ogilvie and Tuck (1969), (ml, m2, m 3 )= rn = - (n . V)W,

(m4, m,, m6) = -(n

*

V)(x x W).

(3.40) (3.41)

On the free surface, the radiation potentials satisfy the boundary condition (3.23). The same condition applies to the sum (cpo + q7),but since the incident wave satisfies (3.12), the scattered potential cp7 satisfies an inhomogeneous form of (3.23) with derivatives of cpo on the right-hand side. D. SPECIAL CASES

The steady flow field W is a major complexity in the free-surface and hull boundary conditions. Moreover, while it is implicit that W is known in the boundary conditions for cp, the solution of the nonlinear steady-flow problem is beyond the present state of this field. Thus, regardless of whether the steady-flow problem is of direct interest, it must be simplified in order to solve for the unsteady flow. The simplest case of a moving ship is obtained when the hull shape is restricted to be a small perturbation from a plane which contains the x-axis.

242

J . N . Newman

As in the analogous case of thin-wing theory, perturbations of the steadystreaming flow - Ui are small, in proportion to the thickness of the hull, and to leading order the boundary conditions (3.12) and (3.31) apply. Examples are the thin ship and the flat ship, mentioned in Section 11. While it is more appropriate to the study of submarines than it is to surface ships, the assumption that the ship hull is “deeply submerged” should be noted as an alternative to restrictions on the shape of the hull surface. The leading-order steady flow near the body is that for motion in an infinite fluid, and on the free surface the steady disturbance from the body can be linearized. Thus the unsteady problem can be treated with relative ease. This assumption was utilized extensively in the collected works of Havelock (1963), although studies there of the unsteady problem generally were carried out with the incomplete hull boundary condition (3.31) instead of (3.28-3.30). The steady-flow problem also can be simplified if one adopts the “slowship” assumption that the forward velocity is small in some sense. This is a topic of contemporary importance in wave-resistance theory (Baba and Hara, 1977; Keller, 1978). The implications for unsteady motions in waves have not been explored. From the boundary conditions it is clear that as W + 0, the zero-forward-speed case will result, except possibly for a regime where the frequency of encounter is small. Singularities in the steady-state solution may be expected to occur at the bow and stern stagnation points. E. SLENDER SHIPS Finally we consider the consequence of restricting the geometry of the ship hull such that its beam (B) and draft (T) are small compared to the length (L) by factors of order E G 1. It is convenient to choose the unit of length such that L = O(l), whereas (B, T) = O(E).To leading order the components of the factors in (3.38-3.39) reduce to

n = (n1, n 2 , 4 ) , (x x n) = (yn, - zn,, - x n 3 , xn2).

(3.42) (3.43)

The first components of these two vectors are O(E),whereas the remaining contributions are O(1). We shall retain the first components in order to derive nontrivial results for ship motions in surge and roll. In the inner region where (y, z) = O(E),a coordinate stretching argument . can be used to show that gradients in the y-z plane are of order 1 / ~ Except for the diffraction potential rpo + rp,, gradients in the longitudinal direction are 0(1),and the velocity potential is governed to leading order by the

Theory of Ship Motions

243

two-dimensional Laplace equation 4yy

+

422

= 0.

(3.44)

Equation (3.44) applies in particular to the steady potential 0, and from the hull boundary condition (3.16) 6 = O(E2).tThus, to leading order in E, the components in (3.40-3.41) reduce to

m = - ( n 2 a/ay + n3 a/az)v$, -

m4 = - n 2 4 *

m, = -xm3

+ n36y + ym3 - zm2, + n3,

(3.45) (3.46a) (3.46b)

m6 = xm2 - n 2 .

(3.464

The factors rn, and m4 are O(E),whereas the remaining components of (3.45-3.46) are 0(1), as in (3.42-3.43). The free-surface boundary condition must be dealt with separately in the inner and outer regions. In the outer region (y, z) = 0(1), gradients of the potential are 0(1) in all three directions and, since the steady potential is O(E’), the leading-order free-surface condition is (3.12). In the inner region, the steady free-surfacecondition (3.17) is dominated by the second term, and the elevation (3.18) is O(E’).The leading-order free-surface condition for the steady problem is the “rigid-wall” boundary condition

$,=O

on Z = O .

(3.47)

The resulting inner solution for 6 and the factors mj in (3.45-3.46) are independent of the forward velocity U. The free-surface condition for the unsteady potential rp can be derived in the inner region from (3.23). Using the above results for I$, it follows that

+ u2(Px* + 26,rp,t + 6yyrpt - U 6 y y r p x - 2u6x,(P, - 2U6YrpXY + 4 y 2 ( P y y + 3 6 y 6 y y ( P , + 9rpz

(Pa - 2urpxt

=0

+ O(~rp,~ V r p )

on z = 0.

(3.48)

Since d/az = O ( E - ’ ) ,the last term on the left side of (3.48) is O(V/E).The remaining terms are O(rp), if the time derivatives and gravity are assumed to be O(1). Thus, with the assumption that the frequency of encounter is 0(1), the rigid-wall boundary condition (3.47) applies also to the unsteady potential. I n this case there are no wave effects in the inner region, to leading order in the slenderness parameter E. t Strictly, 6 = O(E’ log essential.

E).

Logarithmic error factors will be deleted unless their display is

244

J . N . Newman

Wave effects can be introduced in the inner problem to leading order if or if o = O(&-(l/’)). In this circumstance the leading-order terms of (3.48)are qtt g q z = 0 on z = 0, (3.49)

a2/atz = O ( E -l),

+

corresponding to the zero-forward-speed condition (3.8). Justification for assuming that o = O(E-(”’)),or that o’B/g = 0(1),can be based on the fact that this is the resonant frequency domain (1.8) for heave and pitch. As noted in the Introduction, this high-frequency regime is of the greatest practical importance for ship motions in head seas. The fact that it leads to a relatively simple free-surface condition and, ultimately, to a correspondingly simple strip theory, is an additional reason for its study, but not the only one. ) . terms of that order are The error in (3.49) is a factor 1 + O ( E ~ /If~ the retained a more accurate free-surface condition results, in the form

+

qff g q Z - 2Uq,,

+ 2$,,qyf+ $Yyqt= 0

on z = 0.

(3.50)

The free-surface boundary conditions (3.47) and (3.49-3.50) lead to two separate theories for ship motions, applicable respectively for long and short wavelengths or low and high frequencies. These are discussed in the Introduction, and in greater detail by Ogilvie (1977). In Sections VI-VII we shall develop a more general approach, which seeks to unify the two separate theories. The velocity potentials q j in (3.34)will be derived from an asymptotic analysis which is valid to leading order in the body slenderness, for all values of o I O(E”(’/’)).The free-surface conditions to be satisfied in the outer and inner regions are (3.12) and (3.49),respectively. With the sinusoidal time dependence indicated in (3.34), it follows that - 0 2 q j - 2ioUqjx

+ U’q,, + g q j z = 0

on z = 0

(3.51)

in the outer region, and -w2qj

+ gqjz = 0

on z = o

(3.52)

in the inner region. The first term of (3.52) is of higher order for o = 0(1), but not for the extended frequency regime o < O(&-(’/’)). IV. Fundamental Solutions

The interaction of a body with an exterior flow field can be represented by suitable distributions of sources, dipoles and higher order multipoles. In the simplest example, a point dipole may be used to represent the uniform flow past a circle or sphere. More complete multipole expansions are applicable for other body shapes, with the singularities situated at a point or on a line

Theory of Ship Motions

245

within the body. In the most general case, sources and normal dipoles may be distributed in a continuous manner on the body surface, with Green’s theorem used to derive integral equations for the unknown source strength or dipole moment. The source potential is known also as Green’sfunction and will be denoted here by the symbol G. This is the fundamental singularity, since dipoles and higher-order multipoles can be derived from the source by differentiation. The elementary three-dimensional source potential is G = - (47cr)-’ with r = 16 - x I the distance between the source and field points. An analogous result holds in two dimensions, with G = (27c-l log r. In both cases the normalizing factor is such that the source generates a unit rate of flux. It is possible to use the elementary source potential for free-surface problems, as shown by Yeung (see Bai and Yeung, 1974).In linearized problems, however, it is common to use a modified source potential satisfying the free-surface condition, radiation condition, and (for infinite depth) the condition of vanishing at z = - co.When expressed in terms of this singularity, the velocity potential will satisfy all of the boundary conditions except that on the body. For a particular body geometry and normal velocity one then seeks appropriate distributions of surface sources and/or dipoles, or interior multipole expansions, so as to satisfy the body boundary condition. Source and multipole solutions which satisfy the linearized free-surface condition are described systematically by Wehausen and Laitone (1960, Section 13). In slender-body theory the source and dipole potentials are particularly useful for the outer solution, where the body boundary condition is absent. Typically, the outer solution consists of sources and transverse dipoles, distributed on the longitudinal axis of the body. For a slender ship these singularities are on the free surface, and the net flux associated with vertical motions implies the need for sources, whereas lateral motions of the ship hull in relation to the surrounding flow are represented by transverse dipoles. Before considering the three-dimensional source potential we first discuss the simpler result for two dimensions, which will be used subsequently for the inner solution. A. THETWO-DIMENSIONAL GREENFUNCTION

For two-dimensional flow in the y-z plane, the free-surface condition (3.52) is applicable. The corresponding source potential is given by Wehausen and Laitone (1960,Eq. 13.31). In the present notation, with the source point at the origin in the free surface,

1 GZD(y,z) = -- lim 2R p + o +

2‘”cos(k’y)

IOm dk’ k’

- (w - ip)2/g *

(44

J. N. Newman

246

The parameter p can be interpreted as a Rayleigh viscosity coefficient, representing a fictitious dissipation which suppresses incoming waves at infinity. Alternatively, this parameter can be associated with a complex frequency such that the factor ei(a-ip)fgrows slowly from a state of rest at t = - GO.In either case, the limit p -,O + determines the contour of integration in the complex k'-plane, such that the radiation condition is satisfied. Since the imaginary part of the pole k' = (a- ip)2/g is negative, p may be set to zero in (4.1)if the contour of integration is deformed to pass above the pole at k' = w2/g = K . An alternative form for (4.1)is in terms of the exponential integral

defined such that the complex parameter u is exterior to a branch cut along the negative real axis. After a reduction it follows that G,,(y,

1 1 Re{eK('+'Y)El(Kz+ iKy)} + - ieK(z-ilyl). (4.31 2n 2

Z) = --

The residue term in (4.3)results from deforming the contour in (4.1)to avoid the branch cut. The asymptotic properties of the two-dimensional wave source can be obtained from the corresponding approximations of the exponential integral (Abramowitz and Stegun, 1964,Eqs. 5.1.11 and 5.1.51). For small values of Kr the sourcelike logarithmic singularity is displayed in the approximation

Here y = 0.577. . . is Euler's constant, and (r, 8) are polar coordinates such that y = r sin 8, z = - r cos 8.The error in (4.4)is a factor 1 + O(K2r2).For large values of K Iy I the asymptotic approximation of the exponential integral confirms the outgoing two-dimensional plane waves in the form G z D &eK('-'IYI) for K l y l 9 1. (4.51 B. THETHREE-DIMENSIONAL GREENFUNCTION

The three-dimensional source potential which satisfies the outer freesurface condition (3.51) corresponds physically to a source of oscillatory strength, moving with constant velocity U.This source potential was derived initially by Haskind (1946a),and subsequently by Brard (1948).The solution is described by Wehausen and Laitone (1960,Eq. 13.52)and by Lighthill

247

Theory of Ship Motions

(1967). With the source point at the origin on the free surface, this source potential is given by 1

G(x, y, z ) = -2lim 8ff p + o +

j

m

2n

k‘dk‘

0

d8

0

exp[k‘z + ik’(x cos 8 + y sin 8)] k‘ - (o- ip + Uk’ cos 8)’/g ’

(4.6) Ultimately we shall distribute these sources along the longitudinal axis, and an inner approximation of this distribution will be required. For this purpose Fourier transforms are particularly helpful. Thus we shall analyze the Fourier transform of (4.6), in the form G*(y, z ; k) =

[

m

dxeikXG(x,y,

‘-m

(4.7)

2).

After substituting the solution (4.6),and using generalized harmonic analysis to perform the integral in (4.7), it follows that G*(y, z; k,

1 . lim 4n p + o +

K) = - -

j-,

du

exp[z(k2 + u’)”’

(k’

+ iyu]

+ uz)1/2- (o- ip - Uk)’/g’

where u = k‘ sin 8. Here, for future convenience, we define = (a - Uk)’/g.

(4.9) The parameter p may be set to zero in (4.8),if the contour of integration is deformed to avoid the poles at f(K’ - k’)’/’. For k < K the poles are symmetrically situated on the real axis, and the sign of the imaginary part as p + O + is such that the contour should pass above or below the pole according as u(w - U k ) is positive or negative, respectively. For k > K the poles are imaginary, and no deformation of the contour is necessary in (4.8). The value of G* for k = 0 is the longitudinal integral of the threedimensional source potential, which reduces t o the two-dimensional source potential (4.1). To confirm this we note that K

1 G*(y, z ; 0,K ) = -411

exp[zlul + iyu] du j-, (uI - ( o - i p ) ’ / g

= GzLdY,

4.

(4.10)

Approximations similar to (4.4) and (4.5)can be derived for the transform G*(y, z ; k, K). For this purpose we shall assume that k = O(1). An asymptotic expansion of (4.8) for K r -g 1 is derived by Ursell (1962, Eq. 2-19), in the special case U = 0. In the present notation, Ursell’s result

J . N . Newman

248

can be expressed in the form G*(y, z ; k, K ) E (1/2n)(1

+ Kz)[log(* Ik I I ) + y + ( I 1 - k Z / K ZI ) - ( ' I 2 )

1

( K / l k l )- II - K Z + KyO , )cosh-' ( K / l k l )+ ni COS-'

(4.11)

where the upper or lower expression in braces is applicable according as K/ I k I 5 1, respectively. The error in (4.11) is a factor 1 + O ( K 2 r Zk2r2). , This approximation is analogous to (4.4),except for an additional homogeneous solution of the form (1 + K z ) F ( K / l k ( ) . Generalization of (4.11) for U # 0 simply requires that the parameter K is replaced by K and also requires that the conjugate contour of integration is used if w - Uk < 0. After a straightforward reduction it follows that G*(y, z ; k ,

K)=

G*(y, z; 0, K ) - (1/2n)(1 + K z ) f * ( k , K , K )

+ O ( ( K - K ) r , K2r2,k z r z ) ,

(4.12)

where G*(y, z ; 0, K) is defined by (4.10),and f * ( k , K , K) = log(2K/ I k I ) + ai - ( I 1 - k Z / K Z I ) - ( I i z )

1

.\

cos-'(K/lkl) - a \cosh-'(K/ I k I ) + ni sgn(w - U k ) '

(4.13)

The asymptotic approximations (4.11) and (4.12) are valid for K r < 1, irrespective of the magnitude of K. This can be confirmed by writing (4.8)in terms of the nondimensional variables Ky and Kz before deriving (4.11).By the same argument, the requirement that k = O(1) can be replaced by the less restrictive assumption that k/K = O(1). The functionf* defined by (4.13)tends to zero for K % 1, with the limiting behavior

f* = log(K/K) + O ( K - ' ) = O(K-('/')).

(4.14)

A complementary approach is required for the short-wavelength regime K B 1. First we deform the contour of integration in the expression (4.8)for G*, into the upper or lower half of the complex plane u + iu, according as y 2 0. The contour can be deformed ultimately to a large semicircle I u + iu I = 00, except for a branch cut I u I > I k I along the imaginary axis. There is no contribution to the integral from the large semicircle, but one must include the residue from the pole situated in the appropriate half-plane

Theory of Ship Motions

249

for p > 0. For short wavelengths such that o - Uk > 0 and final result of this procedure is the expression G*(y, Z ; k,

K)

= )i(l - k’/K’)“’’’

K

> 1 k 1, the

exp[Kz - i Iy I (K’ - k’)”’] exp[-u lyl

+ iz(u’ + iK

k2)”21.

(u’ - k2)”’

(4.15)

For large K z K, the last term can be approximated by

where K l is the modified Bessel function, and the error is O(Ky)-’. After expanding the modified Bessel function for small kr and substituting the result in (4.16), we obtain the approximation 1 2

G* = - i(1 - k’/K’)-(”’’ exp[icz

-i

1 y I (K’

- k’)”’]

cos 9 ++ O(KY)-’. 211Kr (4.17)

Comparison with the limiting value of (4.17) for k = 0 gives the result G* z GZD,

(4.18)

with the error a factor 1 + O(k’y/K, K1’’y, ( K y ) - ’ ) .

V. TweDirnensional Bodies In the inner region close to a slender ship hull, its boundary surface is approximated by a long horizontal cylinder having a two-dimensional profile defined by the local cross section of the ship. We shall refer to such a cylinder as a “two-dimensional body.” In the radiation problem of forced oscillatory motions, in the y-z plane, the inner flow is governed by the two-dimensional Laplace equation (3.44). The resulting velocity potential is denoted by 4 ( y , z ) to distinguish this from the outer three-dimensional potential q ( x , y , z). In both cases the complex timedependent factor eio‘ is implied. For oblique waves incident upon a two-dimensional body, the threedimensional diffraction potential can be expressed as the product of a sinusoidal function of x, and a two-dimensional function @(y, z). The latter is governed by the Helmholtz equation, reducing to Laplace’s equation in the special case of beam seas. There is an extensive literature on wave radiation and diffraction by two-

250

J . N . Newman

dimensional bodies. Numerical results have been obtained for a variety of body profiles, using several different methods of solution. These are described in the surveys of Wehausen (1971) and Mei (1977).Our discussion in this section is limited to the derivation of analytic properties which are needed subsequently in the three-dimensional slender-body analysis. A. RADIATIONPROBLEMS

Forced motions of the three-dimensional ship hull in sway (j= 2), heave (j= 3), and roll (j= 4) can be related directly to the same motions of the two-dimensional body in the y-z plane. Pitch and yaw motions will be related ultimately to appropriate translations of the two-dimensional body in heave and sway, respectively. The remaining surge mode (j= 1) corresponds to a dilation of the two-dimensional body, with normal velocity proportional to the longitudinal component of the unit normal vector on the ship hull. Thus we must consider the four radiation problems of surge, sway, heave, and roll. In each case the two-dimensional Laplace equation (3.44) and free-surfacecondition (3.52) apply. With the normalization (3.34), the corresponding potentials satisfy the boundary condition

4Jn. = iconj

(j= 1, 2, 3, 4)

(5.1) on the body profile. Restricting the body to be symmetrical about y = 0, the surge and heave potentials are even functions of y, whereas the sway and roll potentials are odd. Each boundary-value problem is completed by imposing a radiation condition of outgoing plane waves at y = k 00 and by requiring that the motion vanish for z + - 00. Following an approach introduced by Ursell(1949), we shall express the solutions for surge and heave in the form m

+j=ajGzD+

c

m=l

cos 2mO a j m ( 7

K

+ (2m-

1)

cos(2m - l)e rZm-l

(j= 1, 3).

(54 In this expansion G,, is the two-dimensional source potential (4.1), and the higher order multipoles have been combined to form the “wave-free potentials” in braces. The source strength aj and coefficients aimare unknowns which must be determined from the boundary condition (5.1) on the body surface. In practice this leads to an infinite system of simultaneous equations, which can be truncated and solved by numerical methods. Ursell (1949) proves that this process is convergent for a circular body profile. For more general body profiles the expansion (5.2) is valid for symmetric mo-

25 1

Theory of Ship Motions

tions exterior to a circle of constant radius which encloses the body, as shown by Ursell (1968a). Since the wave-free potentials in (5.2) vanish at infinity, the radiated waves are associated only with the source term. Using (4.5) with (5.2) it follows that

4J. N- 2 l j a . eJ K ' z - ' l Y l )

( i g / o ) A eK(Z-'IyI)

as lyl

+GO.

(5.3)

Here Aj = t(o/g)aj (j= 193) (5.4) is the complex amplitude of the radiated wave. Since the potential & j is normalized for a unit amplitude of motion, (5.4) is nondimensional. Similar results hold for sway and roll. The appropriate singularities can be obtained by differentiation of (5.2). After redefining the unknown coefficients it follows that

(5.5)

Here the unknown coefficients p j , ah are determined from the boundary condition (5.1). Since the radiated waves are associated only with the wave dipole, c$j

z

(ig/o)A eK('

'

iY)

as y - ,

+GO,

(5.6)

where A = -+ i ( o K / g ) p j

( j= 2,4).

(5.7)

Hereafter we shall assume that these two-dimensional radiation potentials are known. The hydrodynamic pressure then may be determined from the linearized form of Bernoulli's equation (3.2), p = - iop4e'"'. (5.8) Ignoring the restoring force cij in (1.2), due to the hydrostatic pressure, the remaining components of the pressure force are associated with the addedmass and damping coefficients. These can be computed by integration of (5.8) over the submerged portion of the body profile, o Z A i j- iwBij = - i o p

jp n i 4 j dl.

(5.9)

Here A i j and Bij denote the two-dimensional values of the added-mass and damping coefficients, and P denotes the submerged portion of the body profile.

252

J . N . Newman

Other properties of the added-mass and damping coefficients are derived by Wehausen (1971). Included in that survey are the computed results for a family of rectangular cylinders, including the coefficients for heave results reproduced here in Figs. 3 and 4, as well as similar results for sway and roll. These calculations and additional results for other body profiles are due to Vugts (1968).

B. THEDIFFRACTION PROBLEM If a progressive wave system of the form (3.35) is incident upon a twodimensional body, the fluid motion will be periodic along the body axis. The effect of forward speed along this axis can be ignored, since the body is two-dimensional. Therefore the analysis can be performed in a frame of reference fixed with respect to the fluid, and there is no need to distinguish the wave frequency oofrom the frequency of encounter o. With the incident-wave potential defined by cpo = ( i g / o ) exp[K(z - ix cos

/I - iy sin /I)],

(5.10)

the scattered potential cp7 is subject to the condition that the total potential cpo + ip7 has zero normal velocity on the body. Since the flow is periodic in the x-direction, we can write cpj(x, Y , z ) = Oj(y,z)e-"",

j = 0, 7,

(5.11)

where

1 = K cos /I

(5.12)

is the longitudinal component of the wavenumber. With these definitions, the boundary condition (3.36) takes the form 07,, = io(n, - in, sin j?)exp[K(z - iy sin /I)],

(5.13)

on the body profile. After substituting (5.11) in the three-dimensional Laplace equation, the two-dimensional functions Oj satisfy the Helmholtz equation

+

Ojyy Ojzz- PO,= 0.

(5.14)

Both Oo and O7 vanish for z -,- 00 and satisfy the free-surface condition KOj - Ojz= 0

on z = 0.

(5.15)

The scattering function O7 satisfies a radiation condition of outgoing waves for y + +a. Expansions similar to (5.2) and (5.5) can be applied to the symmetric and antisymmetric portions of 07,except for the singular case sin /I = 0, where

253

Theory of Ship Motions

I

the two-dimensional solution is unbounded for I y --* 00. The proof is given by Ursell (1968a). The appropriate wave-free singularities which satisfy the Helmholtz equation involve the modified Bessel functions K,,,(lr);these are exponentially small at infinity. The corresponding source function which satisfies (5.14) can be derived from the Fourier-transformed three-dimensional Green's function G*, by setting U = 0 and k = 1 in (4.8). With these substitutions, and ignoring the wave-free functions, it follows that (5.16)

where Z, denotes the source strength and M , denotes the dipole moment. The radiated waves in the far field can be derived by substituting (4.17) in (5.16), with the result

I

CD, 2 *(iC, csc

p I f K M , ) exp[K(z - i I y sin /3 I )I,

(5.17)

for y -, f00. Since the incident wave amplitude is unity, the radiated wave amplitude at y sin /3= - 00 is equal to the rejection coejjcient R = ( 0 / 2 g ) ( C ,csc /3+ i K M , ) sgn(/3).

Similarly, the total wave amplitude at y sin mission coeficient T=1

/3 = + 00

(5.18)

is equal to the trans-

+ ( 0 / 2 g ) ( C , csc /3 - X M , ) sgn(/3).

(5.19)

In (5.18) and (5.19), --R < /3 < A. If the body profile is symmetric about y = 0, the dipole moment M , is an odd function of b, whereas the source strength C,, and hence R and T, are even functions of /3.

C. APPLICATIONSOF GREEN'S THEOREM Green's theorem may be applied to the pair of functions (Yl, Y 2 )in the plane x = constant, in the form

f (YlY2n- Y2Yln)dl = C

(Y1V2Y2 - Y2V2Yl)dS.

(5.20)

S

The closed contour C is the boundary of the simply connected domain S . The surface integral vanishes if, in this domain, both functions satisfy the same governing equation (3.44) or (5.14). We shall apply (5.20) to the fluid domain in the y-z plane, between the body profile P and a pair of vertical contours at y = k 00. These are connected by horizontal lines at z = 0 and z = - 00 to form a closed contour.

J . N . Newman

254

There is no contribution from the horizontal lines to the contour integral in (5.20),provided that the functions Y i satisfy the linearized free-surfacecondition (5.15) and vanish for large depths. Thus (5.20) gives the result (Y1Yzn - Y2Yln)dl =

-I

0

dz[YIYz, - Y z Y l y ~ Z ? m . (5.21)

-m

In this form, Green’s theorem can be used to derive various relations for the forces acting on the body and for the characteristics of the radiated waves at infinity. In the simplest example, we apply (5.21) to two solutions of the radiation problem +i and dj. Since these are subject to the radiation condition at infinity, the right side of (5.21)vanishes. After using the boundary conditions (5.1) and comparing the left side with the pressure force (5.9), it follows that the added-mass and damping coefficients are symmetric, i.e., A, = Aji and BII. . = BI..t ’ If (5.21) is applied to the radiation potential +i and its conjugate $i,the left side of (5.21) is proportional to the damping coefficient Bii. The contribution from the right side is nonzero. After a reduction using (5.3)or (5.6),it follows that Bii

I

= (pg2/co(’3) Ai 12

(i = 1, 2, 3, 4).

(5.22)

Alternatively, this relation can be derived from energy conservation. If (5.21) is applied to the diffraction solution Oo + (P, and its conjugate, there is no contribution from the left side due to the body boundary condition. The contribution from the integration at infinity gives the familiar result IRIZ+ ITl2=1. (5.23) This relation also can be derived from energy conservation in an obvious manner. Alternatively, if (5.21) is applied to the diffraction potential with an angle of incidence 8, and the conjugate of the diffraction potential with angle of incidence R + 8, it follows that

RT

+ RT = 0.

(5.24)

Here the symmetries noted after (5.19) are used, and (5.24) holds only for a body profile symmetrical about y = 0. Further relations can be obtained by applying Green’s theorem to suitable combinations of the radiation and scattering problems. TO preserve the Helmholtz equation for both functions, we define a class of radiation problems where the normal velocity on the two-dimensional body surface is periodic in the same manner as (5.11). This corresponds physically to a

Theory of Ship Motions

255

forced sinuous motion which propagates along the cylinder with phase velocity w/l.To be more specific, “generalized radiation functions” are defined to satisfy the same conditions as 0,, except on the body profile where

ajn= i o n j

( j = 1, 2, 3, 4).

(5.25)

These functions can be expanded in the same manner as the scattering potential, with source strength C j for the symmetric modes ( j= 1, 3) and dipole moment M j for the antisymmetric modes (j= 2, 4). The radiated waves in the far field may be defined in a similar form to (5.17). For y - , +a, Q j z (ig/w)Ajexp[K(z - iy I sin B I )I, (5.26) where Aj

= 9(0/g)cjlcsc

BI

Aj = -*i(oK/g)Mj

( j = 1, 3),

(5.27)

( j = 2, 4).

(5.28)

The same results hold for y + - 00, provided the sign of (5.26)is reversed for = 0, the wave amplitudes (5.27-5.28) reduce to the corresponding values defined by (5.4) and (5.7). If Green’s theorem (5.23) is applied to the diffraction function a0 and the generalized radiation function Oj, one obtains the result

j = 2,4. When cos

iw

’

sin /3( ) jp nj(Oo+ 0,)dl = -i(g/o)’Jj ( -sin.

c

+

for j =-l , 3 - 2, 4

).

(5.29)

The left side of (5.29) is proportional to the linearized pressure force on the body profile, due to the interaction of the fixed body with the incident waves. With the notation (1.4) it follows that the two-dimensional exciting force X j can be related to the wave amplitude of the generalized radiation function, in the jth mode, by means of the formula

(5.30) This is the two-dimensional form of a more general result known in ship hydrodynamics as the Haskind relations. Equation (5.22)can be combined with (5.30) to relate the damping coefficient and the magnitude of the exciting force, as shown in Fig. 4. A different result follows if the generalized radiation function is replaced by its conjugate Gj. In this case - io

jp nj(mo + (D7)dl = i ( g / o ) * J j ( R_+ T )( -Isinsin. BIj?)

for

0 1. j=l 3 ’ = 2, 4

(5.3 1)

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J . N . Newman

Adding (5.29) to (5.31) gives the relation (5.32) These linear equations can be solved for the reJection and transmission coefficients R and T in terms of the ratios Aj/Aj.A corollary is that the phase of the radiated waves (5.27-5.28), and hence the phase of the exciting force (5.30), is equal to the argument of -h(R f T) for symmetric or antisymmetric modes, respectively. Since R and T are independent of the particular modes, the two phase angles are likewise invariant with respect to the distribution of normal velocity on the body profile. Finally, we combine (5.32) with (5.18-5.19),and obtain relations for the source strength X, and dipole moment M , ,

C, = -(g/u)~sin~ l ( + 1 Aj/jj) M , = -i(g2/03) s g n ( ~ ) ( l -Zj/Zj)

( j = 1,3),

(5.33)

( j = 2,4).

(5.34)

The relations derived in this section from Green’s theorem are special cases of more general results which are summarized by Newman (1976).

D. LONGWAVELENGTH APPROXIMATIONS When the wavelength is long compared to the dimensions of the body profile, the free-surface condition can be replaced to leading order by the “rigid-wall” boundary condition (3.47). In this limit the source strength for surge and heave can be determined from a continuity argument,

I,ah

dl = -4Zj

( j = 1, 3).

(5.35)

Substituting the boundary condition (5.l), it follows from geometrical considerations that

XI = -2ios’(x),

(5.36)

-2ioB(x).

(5.37)

x 3

=

Here S(x) is the submerged area of the body profile; S’(x) = dS/dx; and B(x) is the beam, or width of the body profile at the waterplane. Derivations of these results are given by Newman and Tuck (1964, Appendix 2). The dipole moment for sway can be determined by considering the body profile plus its image above the free surface, in an infinite fluid. For this

257

Theory of Ship Motions “double-body” the dipole moment is given by the formula M , = -2io(S

+ m,,/p),

(5.38)

where 2m,, is the added mass of the double body. Equation (5.38) results from a theorem due to G. I. Taylor, which is rederived and extended by Landweber and Yih (1956).The dipole moment due to rolling motion can be derived in a similar manner, noting that on the double body the normal velocity for roll is an even function of z. The corresponding dipole moment is given by M4 = 2iw(szB

+ B3/i2 - m,4/p).

(5.39)

Here zB < 0 is the vertical coordinate of the center of buoyancy, or the centroid of the submerged profile, and 2mZ4 is the added-mass coupling coefficient for the double body, between roll and sway. The same estimates apply for the source strength and dipole moment of the generalized functions Oj.The boundary condition (5.13) for the scattering potential implies the long-wavelength approximation O, z -03

+ i sin POz

(5.40)

and, hence, from (5.36-5.37),

x, z 2ioB(x), M , z 2 4 s + m z z / p )sin /3.

(5.41) (5.42)

The amplitudes of the radiated waves can be found using (5.27-5.28). These results are summarized as follows:

A, z -iKS’Icsc 81, A, z - K 2 ( S + m,,/p), A, z -iKBIcsc 81,

(5.43)

,T4 z K’(SZ, + ~ ~ / -1 m,4/p). 2

(5.46)

(5.44)

(5.45)

More accurate approximations for the phase angles of the radiated waves follow from (5.33-5.34) and (5.41-5.42) in the form arg(Aj) z

IL --

2

+ K B I csc fl I

arg(d,) z IL - K 2 ( S + m z , / p ) I sin /3 I

( j = 1, 3),

(5.47)

( j = 2,4).

(5.48)

These can be used with the Haskind relations (5.30) to approximate the exciting forces, and with (5.32) to obtain approximations for the reflection and transmission coefficients.

J . N. Newman

258

VI. Slender-Body Radiation Hereafter the slender-body assumption is invoked for the ship hull with the slenderness parameter E defined as the ratio of the beam (B) or draft (T), divided by the length (L). It is convenient to presume a length scale such that L = O(l), and thus (B, T) = O(E).Following the method of matched asymptotic expansions, approximate solutions for E 4 1 are derived separately in the outer region (y, z) = 0 ( 1 ) and in the inner region (y, z) = O(E).These two separate solutions then are required to match in a suitable overlap domain E 6 (y, z) 4 1. In this section we analyze the radiation problem for each mode of rigidbody motion. The corresponding velocity potential 'piis defined? by (3.34). The three-dimensional Laplace equation and linearized free-surface condition (3.51) apply in the outer region, together with a radiation condition and the requirement that the solution vanishes for z + - 00. The inner solution is governed by the two-dimensional Laplace equation (3.44), the linearized free-surface condition (3.52), and the boundary condition (3.37) on the ship hull. The method of matched asymptotic expansions has been applied to the radiation problem for a slender ship by Newman and Tuck (1964), for the case where the characteristic wavelength I = O(1), and by Ogilvie and Tuck (1969) for I = O(E).Here we seek a more general "unified" approach, which is valid for all wavelengths 1 IO(E). The objective of a unified theory requires a careful analysis of the matching error. For this reason the errors in the inner and outer solutions will be estimated, and the overlap region will be chosen to minimize the largest of these. The accuracy of each solution will be indicated by an error factor 8, with logarithmic factors neglected in these estimates. A. THEOUTER PROBLEM

Since the ship hull is collapsed onto the longitudinal x-axis as E + 0, the outer solution can be constructed from a suitable distribution of singularities on this axis. For the symmetric modes (j = 1,3,5) the three-dimensional source potential (4.6) is the appropriate singularity, whereas the antisymmetric modes ( j = 2, 4, 6) require axial distributions of transverse dipoles. (These statements are somewhat intuitive, but they will be confirmed ultimately by matching the results with the inner solution.) t Throughout this section the integer j takes all values from one to six, unless otherwise noted.

Theory of Ship Motions

259

The strength of the source and dipole distributions will be denoted by 4 j ( x ) and d j ( x ) , respectively. The potential for a source at the point x = l is obtained by shifting the longitudinal coordinate in (4.6), and the transverse

dipole potential can be derived by differentiating with respect to y . Thus the outer solution is expressed in the form

with the convention that 4 j = 0 if j is even, and d j = 0 if j is odd. The Fourier transform of (6.1) is derived from the convolution theorem in the form

(

)

+d.* a G*(Y, 2 ; k, K), (6.2) ’aY where G* is defined by (4.74.8). Assuming the source and dipole distributions vary slowly along the hull, the Fourier transforms 4j* and dj* will tend to zero if k 9 1. Thus it is reasonable to assume that k = O(1), in seeking asymptotic approximations of (6.2). Inner approximations of (6.2) for small values of the coordinates ( y , z) can be constructed from the results of Section IV. For small values of Kr, the approximation (4.12) is used to give ‘pi*

= qj*

Here G z D andf* are defined by (4.1) and (4.13), and the error in (6.3) is the factor d = 1 O ( ( K - K ) r , K 2 r Z ,k2r2). (6.4) Alternatively, for K 1 y ) 9 1, (4.18) gives the approximation

+

with the error factor

d = 1 + O ( k Z y / K ,K’”y, ( K y ) - ’ ) .

(6.6)

B. THEINNER PROBLEM In constructing the inner solution we shall ignore temporarily the matching requirement, replacing this by the two-dimensional radiation condition. The result can be identified with the strip-theory synthesis, and a superscript (s) is used to distinguish this solution.

260

J . N . Newman

In view of the boundary condition (3.37) on the body profile, the striptheory solution can be expressed in the form

cpp = + u$j $Ij

(6.7)

= iconj,

(6.8)

where, on the body profile, +j,,

$.Jn = m i .

(6.9) The factors n j and mj are defined by (3.42-3.43) and (3.45-3.46), respectively. The two-dimensional potentials 4j and $ j are governed by the Laplace equation (3.44),satisfy the free-surface condition (3.52), and vanish as 2 - -aoO. These boundary-value problems and their solutions do not involve the forward velocity U. The two-dimensional potentials $ j ( j = 1,2,3,4) are analyzed in Section V,A. Analogous results can be derived for d j if the factors mj are known. In practice a numerical solution is required not only for the two-dimensional potentials, but also for the factors mj.Hereafter, we shall assume that these are known. The remaining potentials for pitch ( j = 5 ) and yaw ( j = 6) follow from the definitions of n j and mjr

cpp = - xcpy + (U/iw)&, cpt)= xcp$’ - (U/iw)&.

(6.10) (6.11)

In general the matching requirement with the outer solution will differ from the condition of outgoing radiated waves satisfied by the strip-theory potentials. Moreover, the potentials (6.7)are unique (assuming unique solutions of the two-dimensional radiation problems), and depend only on the local cross-sectional geometry of the hull. Since the outer solution includes a longitudinal function of x which depends on the three-dimensional shape of the ship hull, a more general solution is required in the inner region. In the classical slender-body theory of aerodynamics, and also in the long-wavelength case where the rigid free-surface condition (3.47) holds in the inner region, the inner solution is generalized simply by adding an arbitrary “constant” C, which may depend on x. In the present case, the free-surface condition (3.52) requires a nontrivial homogeneous solution. Since the homogeneous solution satisfies a boundary condition of zero normal velocity on the hull, it can be identified physically with the scattering of incident waves by the fixed body. A solution which is symmetric about y = 0 can be derived by combining two waves of equal phase incident from opposite sides of the body. An antisymmetric solution can be derived similarly, with incident waves of opposite phase. In either case standing waves will exist in the far field of the two-dimensional problem. An alternative derivation of the homogeneous solutions follows by

261

Theory of Ship Motions

observing that the boundary conditions (6.8) are purely imaginary, hence is a homogeneous solution. Similarly, from (6.9), Im($j) is a homogeneous solution which will differ from Re(4j) by a multiplicative constant. With an arbitrary multiplicative factor, only one homogeneous solution is required in each mode, and we shall take this to be (4j+ Bj). With this choice, the general form of the inner solution is given by 'pj

= Cpy

+ Cj(X)(4j+

(6.12)

Bj).

Here C j ( x ) is a function to be determined by matching with the outer solution. The outer approximation of the inner solution (6.12) can be derived from the expansions (5.2) and (5.5). Since the radial derivatives of these expansions are O(1) in the inner region, as r + O(E), (6.13)

ajm= O(E'"'O~, &'"pj).

Thus, the wave-free potentials can be neglected for r B E , and the only contribution to the outer approximation of the inner solution is from the source or dipole. For the potential 4 j ( j = 1, 2, 3, 4) the source strength 01,3or dipole moment pz,4is defined by the two-dimensional solution. These definitions are readily extended to the potentials for yaw and pitch where, in accordance with (6.10-6.11), os = - x o 3 and p6 = x p z . With the same convention as in (6.1), the outer approximation of the potential 4j is

(6.14)

A similar representation can be applied to the potentials

G

$j,

-

DjGzD.

(6.15)

For convenience in subsequent expressions we also define the differential operators

+ UDj, R j = Dj + Dj. S j = Dj

(6.16) (6.17)

With these definitions, the outer approximation of (6.12) is q j

z SjGzD + Cj(DjGzD + D j G z D ) = (Sj

+ C j R j ) G z D- iCjDjeKzcos K y .

(6.18)

J . N . Newman

262

The last result follows from (4.3). The Fourier transform of (6.18)is given by 'pj* E [S?

+ (C,Rj)*]G2D- i(CjD,l)*eKzcos K y .

(6.19)

The errors in the inner solution may be summarized in the following manner for r & E. The two-dimensional Laplace equation and body boundary condition involve second-order errors in the ratio of longitudinal to transverse gradients, or a factor of 1 + O(kr)'. The two-dimensional freesurface condition (3.52) contains an error factor 1 + O(K'I2kr). Neglect of the wave-free potentials involves the error factor 1 + O(EZ/r2). Thus the cumulative error in (6.19) is the factor

d = 1 + O(k2r2, K1I2kr,E2/r2).

(6.20)

C. MATCHING

The inner and outer solutions are matched in a suitable overlap domain 6 r 4 1 to determine the unknown source strength and dipole moment of the outer solution and the coefficients C , in the inner solution. Initially it will be assumed that the overlap region is close to the ship, in terms of the wavelength, and thus Kr 4 1. This assumption will become invalid for short wavelengths, at which point a separate approach will be adopted. Matching of the inner and outer solutions is carried out in the Fourier domain. Thus we equate (6.3) to (6.19), and obtain the relation E

= - (S,+ CjRj)*G2D- i(CjBj)*eKzcos K y .

(6.21)

The dominant terms in this matching relation are the antisymmetric contributions associated with the dipole potential, of order l/r, and symmetric contributions from the source potential, of order log r. First we consider the antisymmetric terms in (6.21), corresponding to the modes j = 2, 4, 6. Equating the factors of the dipole terms gives dj*

=

[pj

+ Ufij + C j ( p j+ pi)]*

( j = 2,4, 6).

(6.22)

The long-wavelength approximation (5.48) can be used with (5.7) to show that the dipole moment p, is imaginary with the error a factor 1 + O(K2c2). Thus the interaction term proportional to C , may be neglected in (6.22). After inverting the Fourier transforms, d, = p j

+ Ufi,

( j = 2,4, 6),

(6.23)

Theory of Ship Motions

263

and the dipole moments are identical in the inner and outer solutions. The remaining antisymmetric part of (6.21) is a higher order contribution from the last term, proportional to sin Ky. To leading order the antisymmetric solution is strictly two-dimensional in the inner region and given correctly by the striptheory approach. This is a familiar situation in slender-body theory, where lateral body motions without a net source strength contain no longitudinal interactions. Ogilvie (1977) refers to this as a “primitive” strip theory. Next we consider the symmetric modes, which are dominated in (6.21)by the source potential. Equating the factors of G z Dgives a relation for the source strength qj* = ( S j

+ CjRj)*

( j = 1, 3, 5).

(6.24)

Equating the remaining terms in (6.21) of order one gives qj*f* = 2ni(CjDj)*

( j = 1, 3, 5).

(6.25)

A comparison of (6.4) and (6.20) indicates the cumulative error in (6.24) and (6.25) to be the factor

d=1

+ O(k2r2,K112r,K 2 r 2 ,c2/r2).

(6.26)

The optimum location of the overlap region may be selected to minimize this error. For sufficiently small values of K, the dominant contributions to (6.26)are O(K”*r, E2/r2).The optimum value of r is such that these two errors are equal and thus

,.=

0(&2/3~-(1/6)

).

(6.27)

With this choice, the maximum error in (6.26) is O ( E ” ~ K ” ~provided ), K &-(li2). For K > E - ( ’ / ~ ) , the dominant contributions to (6.26) are O ( K Z r 2E2/r2). , Equating these defines the optimum overlap region r =O(E/K)”~.

(6.28)

The corresponding error in (6.26) is O(EK). As K + O(E-l ) a separate matching must be constructed from the shortwavelength inner approximation of the outer solution. Proceeding on this basis with (4.18), (6.21) is replaced by

G z D2 ( S j + C j R j ) * G z D- i(CjDj)*eKzcos Ky.

(6.29)

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J. N . Newman

The solution of (6.29) is the strip-theory result (6.30)

cj*= 0,

(6.31)

in accordance with the short-wavelength analysis of Ogilvie and Tuck (1969). The error factor in (6.29-6.31) is 8 = 1 + O(kzrz,K”*r, c2/r2, ( K y ) - ’ ) .

(6.32)

The maximum error in (6.32)is O(K-(1’3’)if the overlap region is defined by r = O(K-(5/6)). Since the alternative error in the long-wavelength matching is O(EK),the short-wavelength matching should be adopted if K 2 E - ( ~ / ~ ) . At this stage we observe that the long-wavelength results (6.23-6.25) are consistent with the striptheory results (6.29-6.31) for K 9 1, sincef* vanishes in accordance with (4.14). For this reason (6.23-6.25) are valid in general, for all wavelengths such that K IO(E-’), and will be used exclusively hereafter. The inverse transforms of (6.24-6.25) can be expressed in the form qj = Sj

2nicjDj =

+ CjRj,

J”L qj(<)f(x - <) d<,

(6.33) (6.34)

f o r j = 1,3,5. The kernelf(x) is the inverse transform of (4.13).Elimination of C, gives an integral equation for the outer source strength:

= g,(x)

+ Ukj(X)

( j = 1, 3, 5).

(6.35)

Here the two-dimensional source strengths have been substituted for the differential operators using (6.14-6.17). The relations (5.4) and (5.32) can be used to replace the factor ( 0 , / 8 ~ + 1) by (1 - R - T),where R and Tare the two-dimensional beam-sea reflection and transmission coefficients. The kernel in (6.35) is identical to that which accounts for longitudinal interactions in the long-wavelength slender-body theory (Newman and Tuck, 1964). On the other hand, the source strengths 0, and 6, are derived from the striptheory solution. Thus the integral equation (6.35) provides a blend between the two limiting theories, such that the result is valid for all intermediate wavelengths.

Theory of Ship Motions

265

The term containing the integral in (6.35)tends to zero for the two limiting regimes K = 0(1) and K = O ( E - ’ ) . For long wavelengths the factor ( a j / S j 1 ) = O ( K E )from (5.4) and (5.43).For short wavelengths the kernel vanishes, in accordance with (4.14).To leading order it follows that

+

qj

E bj

+ u&j.

(6.36)

This approximation can be refined by iteration. D. THEINNERSOLUTION

At this stage the inner solution (6.12) is determined. The two-dimensional potentials 4j and t$j ( j = 1, 2, 3, 4) must be obtained numerically at each section along the hull. The coefficients C j in (6.12) vanish for ( j = 2, 4, 6), and are otherwise determined from (6.34) in terms of the outer source strength qj. The latter is determined from the integral equation (6.35) or by approximation from (6.36). The resulting inner solution can be expressed in the form

The “unified” inner solution (6.37) is the principal result of our analysis. This velocity potential is a linear superposition of the strip-theory solution (6.7),and the homogeneous solution (4j + $ j ) . The homogeneous solution is multiplied by the same longitudinal integral of the outer source strength which appears in the long-wavelength slender-body theory. In the long-wavelength regime K = 0(1),the inner free-surface condition is the “rigid-wall” boundary condition (3.47). This governs the twodimensional potentials 4j and $, and thus qy).The homogeneous inner solution is a constant and the last term in (6.37) is a function only of the longitudinal coordinate x. The outer source strength qj is given explicitly in terms of the net flux at each section, in accordance with (5.36) and (5.37). Longitudinal interference is accounted for by the kernel f (x - t),but transverse interactions are neglected. This is the “ordinary slender-body theory” described by Newman and Tuck (1964) and by Ogilvie (1977). In the short-wavelength regime K = O ( E - ~ the ) , kernel is of higher order and (6.37) is dominated by the potential cpy). Transverse interference is accounted for in this two-dimensional potential, but longitudinal interactions are negligible. This is the striptheory solution. The unified potential (6.37) is valid for all wavenumbers K I O(E-’ ) . In the special case of zero forward velocity this solution reduces to that derived

J . N . Newman

266

by Mays (1978) and outlined by Newman (1978). The latter reference also treats an analogous problem in acoustic radiation. The unified result may be compared with an “interpolation solution” derived by Maruo (1970) for the case U = 0. Maruo’s approach is rather different, but the only change in the final result is that the homogeneous solution in (6.37) is replaced by (1 + K z ) , and the amplitude of the twodimensional striptheory potential is modified accordingly to satisfy the boundary condition on the body. Maruo’s source strength is governed by an integral equation similar to (6.35), with the kernel simplified by the restriction to zero forward velocity. VII. Slender-Body Diffraction

In this section we consider the diffraction problem for a slender ship, in the presence of incident waves. With an incident wave of unit amplitude defined by ( 3 . 3 9 the scattered potential q7 must be determined such that the total potential ‘po + q 7 satisfies the boundary condition of zero normal velocity on the ship hull. Our approach here is similar to that used for the radiation problems in Section VI. The conditions in the opening paragraph of that section apply. The principal difference is with respect to the boundary condition on the ship hull, which for the scattered potential is given by ‘pTn =

cos p - in, sin

p) exp[K,(z - ix cos p - iy sin p)]

-iw,(n,

- in,

on S.

(74 Here oo is the incident wave frequency, in a fixed reference frame, and K O = o o 2 / g is the corresponding wavenumber. The diffraction problem is simplified by the absence of forward-speed effects in (7.1), by comparison to the factors mjand the corresponding solutions $ j in the radiation problem. On the other hand, the normal velocity (7.1) is oscillatory along the ship’s hull, at a rate which is proportional to the longitudinal wavenumber component K O cos p. In the beam-sea case (cos b = 0),the right side of (7.1) is slowly varying for all values of K O I O(E-’). Thus the beam-sea diffraction problem can be treated in an identical manner to the radiation problems, except that the striptheory solution in the inner region corresponds to the two-dimensional scattering problem. The boundary condition (7.1) is simplified also for head or following seas (sin = 0),but this simplification is deceptive. Head seas propagating along

Theory of Ship Motions

267

the two-dimensional body in the inner problem are diffracted over a transverse width that increases without limit. Thus it is not possible to derive a conventional striptheory solution in the inner region. The singular nature of the head-sea diffraction problem was established by Ursell (1968a,b). Ursell’s proof states that head seas cannot propagate along an infinitely long cylinder in a periodic manner, unless the diffraction potential is unbounded at large distances from the cylinder axis. A detailed analysis of the head-sea diffraction problem has been carried out by Faltinsen (1971) for the case where the incident wavelength is O(E). To leading order the incident wave is canceled in the near field by an equal and opposite longitudinal wave. Ursell’s unbounded solution is utilized in a higher order inner solution, and matched with the outer solution in a consistent manner. A singularity is encountered at the ship’s bow, of the sort which generally occurs in short-wavelength scattering problems. Maruo and Sasaki (1974) present a modified approach intended to remove this singularity. Both solutions are discussed further by Ogilvie (1977, 1978). For a ship moving in head seas with U = O(l), the frequency of encounter o is increased by the Doppler shift (1.7). The regime of resonant pitch and heave motions (1.8) coincides with incident wavelengths of order cl/’, intermediate in scale between the ship’s length and transverse dimensions. In this regime the head-sea problem can be analyzed in a relatively simple manner from the long-wavelength slender-body theory. A solution of the diffraction problem will be derived from the unified slender-body approach which was developed for the radiation problems in Section VI. This theory is intended to apply for incident wavenumbers K O IO ( E -’), but we shall concentrate on the regime where K O I O(&-(”’)). Beam seas will emerge as a relatively simple limit, but for head seas the unified solution is singular for all wavenumbers. The alternative longwavelength assumption will be used to provide a simple remedy for this defect. A more fundamental extension of the unified theory is warranted to include head seas, but this task is left for future research.

A. THEOUTERPROBLEM Since the scattering potential (p7 differs from the radiation potentials of Section VI only with respect to the body boundary condition, the boundaryvalue problems in the outer region are identical. The outer solution (6.1)is applicable directly to the potential (p7, with unknown source strength q , and dipole moment d 7 . The Fourier transform of the outer scattering solution is given by (6.2), with j = 7.

J . N . Newman

268

B. THEINNER PROBLEM

The oscillatory longitudinal factor of the boundary condition (7.1) suggests expressing the inner solution for the diffraction potential in the form (p7 = Q, exp( - iKox

cos fl) = Q exp( - dox).

(74

After substituting in (7.1), and neglecting the longitudinal component n,, the function Q, satisfies the boundary condition On= -iwo(n, - in, sin

fl) exp[Ko(z - iy sin fl)]

on S.

(7.3)

Since (7.3) is slowly varying in the x-direction, the same behavior is expected of Q,. Substituting (7.2) in the three-dimensional Laplace equation and neglecting longitudinal derivatives of Q then gives

+

a,,

-

lo% = 0.

(7.4)

Thus Q, is governed by the two-dimensional Helmholtz equation. The leading-order free-surface equation can be derived from (3.49) or (3.52), KoQ,-Q,=O

onz=O.

(7.5)

Here (1.7) has been used to replace the frequency of encounter by the incident-wave frequency oo, and K O = wo2/g. The problem defined by (7.3-7.5) is identical to the two-dimensional diffraction potential Q7 derived in Section V,B, except for the appearance of the frequency oo in place of o.Thus the striptheory solution of the inner problem is the solution of the two-dimensional diffraction problem, which will be denoted here by Q7. The general solution of the inner problem is given in a form analogous to (6.12): @ = @7

+

(74

Since the real and imaginary parts of the boundary condition (7.3) are antisymmetric and symmetric, respectively, the general homogeneous solution is given by @(h)

= C,(@,

+ as)+ C,(@, - a,).

Here

and (Cs, C,) are arbitrary constants.

(7.7)

Theory of Ship Motions

269

The outer approximation of this inner solution can be obtained from (5.16) in the form

G*(y, Z, lo, KO)

+

CsE7 - C , R 7

(C*- G*).

(7.9)

After using (4.15) to evaluate the last factor, and taking the Fourier transform, it follows that

C s z 7- C a M ,

-)aYa * eKoz cos(Koy sin B).

(7.10)

This outer approximation of the inner scattering potential is analogous to (6.19) for the radiation problem. The error in (7.10) is the factor (6.20), with K O substituted for K , and with the Fourier parameter k replaced by lo if lo > k. Since (D is slowly varying with respect to x, the significant domain for (7.10) is k = O(1).

C. MATCHING The Fourier transform (7.10) of the inner solution is to be matched with the transform (6.2) of the outer solution. Recalling the oscillatory factor in (7.2), it is necessary to shift the transform parameter from k to k - lo in (7.10), or alternatively to shift from k to k + lo in the outer solution. Adopting the latter approach, (6.2) is rewritten in the form (P7*(Y,z ; Q = [q7*(jt)+ d,*(Q(a/ay)lG*(y,

2;

IT, 3,

(7.11)

where K=k+lO

(7.12)

and K‘ = (0 - UE)’/q = (00 - Uk)’/g.

(7.13)

Matching of the inner and outer solutions is performed by equating (7.10) and (7.11) in a suitable overlap region. Since the transforms G* in these two

J . N . Newman

270

solutions contain different arguments, it is not possible to match their factors directly. Instead, the long-wavelength inner approximation (4.12) of G* is used in the solutions, with (4.4), to derive the relation 1 - q,*(E)(l + Kz)[log(Kr) + y + ni -f*(& II-, K) - Kz + KyO] 2n 1 +d,*(E)(l + Kz)[(sin O)/r + KO] 271 1 z - [C, + Cs(C7+ X,)]*(l + K,Z) 2n [log(Kor) + y + ni - f * ( l o , K O ,K O )- Koz + KoyO] 1 +[ M , + C,(M7 - &f7)]*(1 + Koz)[(sin O)/r + KoO] - ilcsc flleKoZ 2n

. [(Cs&)* cos(Koy sin fi) + (C,l\si,)*(Ko sin fi) sin(Koy sin /I)]. (7.14)

Proceeding as in Section VI,C, the antisymmetric terms in (7.14) are matched by neglecting the higher-order interactions, proportional to C,, with the result d7*(E) = M7*(k). (7.15) After inverting this Foilrier transform the outer dipole moment is given by d 7 ( x ) = M 7 ( x )exp( - ilox).

(7.16)

Once again there is no interaction in the antisymmetric solution, and the only difference between the inner and outer dipole moments is the oscillatory factor in the definition (7.2) of the inner solution. The dominant symmetric terms in (7.14) are proportional to log r. Equating these, (7.17) q7*(Q = [C7 + CS(C7 + %)I* Using this result, the remaining symmetric terms in (7.14) can be equated to give the relation 274CsC7)* = 47*(l;)f7*(k9 lo, 4, (7.18) where from (4.13), f,*(k, lo, k) = 1 sin fl I log( I l o / E l ) - ( 1 1 - R / i 2I ) - ( ' I 2 )

[

cos-'(i?/~E~) - II + xi sgn(oo- U k ) (cosh-l(i/ I

+ [cosh-'(sec

/3)

+ nil.

(7.19)

Theory of Ship Motions

27 1

Equations (7.17-7.18) are analogous to (6.24-6.25) in the radiation solutions. The error factor in (7.14-7.18) can be deduced in a similar manner to (6.26). With k = O(l), the error is a factor

I = 1 + O(KO2r2,&'/r2).

(7.20)

~ ' ~this ; choice The optimum overlap region is defined by r = O ( E / K ~ )with the error factor is 1 + O(&KO).A separate analysis must be performed for K O-+ O ( E - ' ) ,using (4.17) to approximate G* in (7.10) and (7.11). We shall not repeat that analysis here, since the results are similar to the radiation solutions, and serve only to confirm the validity of (7.15-7.18) for the extended regime KO < O ( E - ' ) . The Fourier transforms (7.17) and (7.18) can be inverted, after noting the shift in the parameter (7.12). The result is a pair of simultaneous equations, q7 = [C,

+ C,(C7 + C,)] exp(-il,.x),

(7.21) (7.22)

wheref7(x) is defined by its Fourier transform (7.19). From (7.21), the outer source strength can be expressed in a similar form to (7.16), q7(x) = Q7(x) ex~(-ilox),

(7.23)

with Q7 a slowly varying unknown function of x. Elimination of C , from (7.21) and (7.22) gives an integral equation for Q 7 ,

Q ~ ( x-) ( 2 4 - ' ( & / x 7

+ 1) '[L Q 7 ( 5 ) f 7 ( ~ - 4 ) d4 = C7(x).

(7.24)

This integral equation is similar in form to (6.35). Once again the integral term in (7.24) vanishes in the limits K OE 4 1 and K O B 1 so that to leading order (7.24) is approximated by Q7b)

= C7(x).

(7.25)

D. THEINNERSOLUTION The solution in the inner region follows from (7.6-7.8), with the coefficient C, determined from (7.22) and the (antisymmetric) coefficient C , = 0. The

result is of the form

272

J . N . Newman

Here @, is the solution of the two-dimensional diffraction problem outlined in Section V,B, X7 is the corresponding source strength, and QS is the symmetric part of @., In all cases the incident-wave frequency wo and wavenumber K O apply to the two-dimensional solution. In view of the definitions (7.12-7.13) and (7.19), this inner solution is not dependent on the frequency of encounter o,or the corresponding wavenumber K , but depends instead on the frequency wo and wavenumber KO.A physical interpretation for this distinction is given by Newman (1977). For short wavelengths the kernel in (7.26) vanishes, and the striptheory result follows. This two-dimensional solution has been studied by Bolton and Ursell (1973), Choo (1975), and Troesch (1976). E. THELONGWAVELENGTH SOLUTION Since the solution @, of the two-dimensional diffraction problem, is singular in head or following seas (fl= 180" or 0", respectively) the unified solution (7.26) is not well behaved in these limits. In the long-wavelength regime this difficulty does not arise. The essential change is to simplify the inner boundary-value problem by assuming that lo and K O are much less than E - ' in (7.4) and (7.5). The two-dimensional Laplace equation (3.44) and rigid free-surface condition (3.47) follow. The source potential for tbe inner solution is (271)- log r, and the homogeneous solution is a constant. With these simplifications the outer approximation of the inner solution (7.9) is replaced by 1 (7.27) @ 2 - (Z, M, log r C,(x), 271

$)

+

+

with the source strength C7 and dipole moment M , given by(5.41)and (5.42). Matching follows by equating the Fourier transform of (7.27) to the left side of (7.14). The relation for the outer dipole moment (7.15-7.16) is unchanged, but (7.17-7.18) are replaced by (7.28)

47*(lt) = Z 7 * ( k ) ,

2nC,*(k) = q,*(Q[log(K)

+ y + xi - f * ( &

2, K)].

(7.29)

After inverting these transforms and substituting the source strength from (5.41), it follows that q7(x)= - 2iw0 B ( x ) exp( - ilox),

2nC7(x)exp( - ilox) = q7(x)[log K

+ y + nil L

(7.30)

q7(5)f(x - t) d5. (7.31)

Theory of Ship Motions

273

Here the kernelfis the inverse transform of (4.13)' as in (6.34-6.35). As intended, the source strength q , and constant C, are well behaved in (7.30-7.31) as sin fi + 0, and these results can be utilized for head and following seas. On the other hand, the factor log K in (7.31) suggests difficulties for large wavenumber. This is a defect of the long-wavelength slender-body theory which is absent from the unified results.

VIII. The Pressure Force The six components of the pressure force (Fl, F,, F 3 )and moment (F4, F s , F 6 ) can be expressed in the form

F, =

ljpn, dS

(i = 1, 2,

. . . , 6).

(8.1)

S

Here S is the submerged portion of the ship hull, and the factors n, are defined by (3.38-3.39). The pressure p is determined from Bernoulli's equation (3.2), and can be separated into hydrodynamic and hydrostatic components. The hydrostatic pressure force is analyzed by substituting p = -pgz in (8.1). The nonvanishing components (i = 3 , 4 , 5) are analyzed by Wehausen (1971) and Newman (1977). The results are simplified for linearized motions of a ship hull which is symmetrical about y = 0. Further simplification follows by adding the gravitational force and moment due to the ship's mass, and assuming equilibrium when the unsteady motions vanish. The latter assumption implies that the ship's weight is balanced by the buoyancy force pgV and that the horizontal coordinates of the center of gravity and center of buoyancy are coincident. With these restrictions, the force components due to the sum of the hydrostatic pressure and the ship's weight are given by F3 = -pg(t3SOO - tSSlO)e'"',

(8.2)

+ ~ ( Z B- Z G ) ] C ~ ~ ' ~ ' , = pg{s10t3 + v(zB -

F4 = -Pg[So, FS

[sZO

zG)]<5}ei0'.

(8.3) (8.4)

Here zB and zG denote the vertical coordinates of the centers of buoyancy and gravity, and

sij = ljX'yJ dx dy, where the integral is over the plane z = 0, interior to the ship hull.

J . N . Newman

274

The steady component of the hydrodynamic pressure follows by substituting the velocity field W(x) in Bernoulli’s equation. From symmetry, the only contributions to (8.1) are for (i = 1, 3, 5). The steady portion of Fl is the (negative) wave resistance, which is balanced by the propulsive force. The steady vertical force and pitch moment are balanced by static “sinkage and trim,” which modify the equilibrium position of the ship hull but d o not contribute directly to the unsteady force or motions. Hereafter we consider the unsteady component of the hydrodynamic pressure force, with the usual assumption that the oscillatory motions of the ship and the fluid are small. Neglecting second-order terms in Bernoulli’s equation (3.2), the pressure is given by

An additional contribution to (8.1) results from the oscillatory position of the ship’s surface S with respect to the mean surface S. From a Taylor-series expansion the total oscillatory pressure in (8.1) is p = -p(q,

+ w . vq + +x . vwz)s.

(8.7)

The last term in (8.7) gives a force proportional to the unsteady displacement of the ship, and hence an additional contribution to the restoring coefficients cij in (1.2-1.3). This force is due to the unsteady motion of the ship within the steady pressure field. A similar contribution from the oscillatory change in the upper boundary of S is noted by Timman and Newman (1962). For a slender ship, the last term of (8.7) is approximated by

The resultant force from this change in pressure is O ( E )relative to the restoring-force components (8.2-8.4). For horizontal translation the contribution from (8.8) is zero and the only first-order effect from (8.8) is a sway force, and moment, due to a static yaw angle. The latter are analogous to the aerodynamic lift force and moment on a slender body, with important effects on the low-frequency steering maneuvers of ships. For oscillatory motions in waves these generally are neglected. The remaining contributions to the linearized unsteady pressure force are due to the added-mass and damping coefficients defined in (1.2) and the exciting force defined by (1.4). These are discussed separately below, using the solutions for the radiation and diffraction potentials derived in Sections VI-VII.

Theory of Ship Motions

A. ADDED MASSAND

275

DAMPiNG

With the notation of (l.l), our task is to evaluate the transfer function t . .= - p

jj(ioqj+ W . Vqj)nidS.

(8.9) Here Bernoulli’s equation has been used in the form (8.7), together with (3.34) and (8.1). The real and imaginary parts of (8.9) give the added-mass and damping coefficients defined by (1.2). The term in (8.9) proportional to the steady velocity field W can be transformed by means of a theorem due to Tuck (Ogilvie and Tuck, 1969, Appendix A), IJ

1’1

1s q j m i d S

S

S

(W . V q j ) n ,dS = - U

-

U

fqj&ni

dl.

(8.10)

C

-

This result follows from Stokes’ theorem, and the fact that V W = 0. The last integral in (8.10) is over the boundary of S, i.e., the intersection of the ship hull with the plane z = 0. Since the rigid free-surface condition (3.47)l applies to &, the line integral is of higher order and can be ignored. Substituting (8.10) in (8.9) and using (6.7) and (6.12) for the unsteady potential q j , it follows that t I.J. = - p

jj(ion, - Umi)[c$j+ U J j + C j ( 4 j+ Bj)] dS.

(8.11)

S

The two-dimensional potentials 4j and $ j are defined in Section VI,B, and the interaction coefficient Cj( x)is determined from the source strength of the outer solution by (6.34). Equation (8.11) may be rewritten in the form

where

(8.13)

- mi4j) dl,

T!;)= - p U P

mi$j dl,

TI;’ = pU2

(8.14) (8.15)

P

T $ ) = -pCj

1 (ioni P

-

Umi)(+j+ $ j ) dl.

(8.16)

276

J . N . Newman

From Green's theorem (5.21), and the boundary conditions (6.8-6.9), TIIO.') = T$)g2),

(8.17) (8.18)

(8.19) TI;) = (- 2ipgAjCj/K)(Ai + UA,). Equation (5.32) can be used with (6.34) and (6.37) to show that the contribution from (8.19) to the integrated force (8.12) is symmetric with respect to the indices i and j. Thus the total three-dimensional force (8.12) satisfies the reverse-flow theorem of Timman and Newman (1962), i.e., ti;) = t);), where the superscript denotes the direction of the forward velocity. The local force (8.13) can be related to the two-dimensional added-mass and damping coefficients (5.9), and the contribution to (8.12) is the zerospeed striptheory result. The contribution from (8.15) is similar, but requires a separate analysis of the potentials 4j.There is no contribution from (8.16) to the fluxless modes (j = 2, 4, 6) where C j = 0. From (8.18) the integrals defining TI;) vanish for i = j , and give nonzero contributions only for the cross-coupling coefficients. For coupling between heave ( j = 3) and pitch ( j= 5), we note from(3.43) and (3.46b) that 4, = -x&, and m, = -xm3 + n3. From (8.18) it follows that 7''') - - " ( 15 )3 (8.20) 35 - (W4TS"J. By a similar argument, for sway and yaw, T':Q =

- Thy = - (U/io)T$"J.

(8.21)

Thus these contributions to the cross-coupling coefficients can be expressed in terms of the "zero-speed" potentials 4j. In the short-wavelengthstriptheory regime, the interaction coefficientsC j vanish, in accordance with (6.31). This leaves only the integrals (8.13-8.15) to be considered. For heave and pitch there are two variations of the resulting formulas. In the intuitive approach, gradients of the steady-state velocity field are neglected with the result that the only nonzero elements of mi are m, = n3 and m6 = -nz. Thus the two-dimensional potentials 4, vanish and = -&. With these simplifications in except for iw4, = (8.12-8.15),

(8.22) (8.23)

277

Theory of Ship Motions

It’

I- J;. -

-

T&”Jxdx & ( U / i o ) t 3 , ,

(8.24)

t55

=

S, T&’Jx’ dx + p(U/o)’t33,

(8.25)

t66

=

jL T$’iX2 dx -t

(8.26)

b53

p(U/W)2t22.

These equations are essentially? identical to the striptheory results for heave and pitch derived by Salvesen et al. (1970) and by Newman (1977). Somewhat different results are obtained in the systematic approach of Ogilvie and Tuck (1969),as noted in the Introduction. With o = O(&-(”’)), the moment proportional to (U/w)’ is discarded as a higher-order effect. Additional cross-coupling terms are added to (8.24) from the free-surface boundary condition (3.50). The heave force t 3 , is unchanged from (8.22). The comparative merits of these different approaches to strip theory have not been resolved, although Faltinsen (1974) has shown that the comparison with experiments is improved by using the Ogilvie-Tuck cross-coupling coefficients. The results (8.12-8.16) are not limited by the strip-theory assumptions and apply more generally for all frequencies and wavelengths. The principal complication is that the forward-speed potentials and the factors mjmust be determined, as well as the interaction coefficients Cj. Such calculations have not been performed in the general case U # 0. For zero forward velocity (V = 0), calculations of the heave and pitch coefficients by Mays (1978) show good agreement with “exact” threedimensional computations for slender spheroids with E < $. Computations of the added mass and damping at zero velocity also have been performed by Maruo and Tokura (1978), based on the “interpolation solution” of Maruo (1970) which is described in the closing paragraph of Section VI. Maruo and Tokura show good agreement of their results with experimental data. The difference between Maruo’s theory and the unified solution is of little practical significance for U = 0, and both offer substantial improvement relative to the ordinary slender-body theory and the strip theory.

4,

B. THEEXCITING FORCE

The exciting force (1.4) is the result of the pressure associated with the diffraction potential. The coefficient X i in (1.4) can be interpreted as the t In Salvesen et al. (1970), a “transom-stem correction” is introduced for ships where the after end of the hull is not pointed; the validity of this correction is questionable. In Newman (1977) a different coordinate system is used, and the expression for the pitch moment contains an error in the sign of the term proportional to (V/o)*.

J . N. Newman

278

complex amplitude of the exciting force due to an incident-wave system of unit amplitude. With the diffraction potential substituted in (8.9), this coefficient is given by

xi= - P

\jni(io + W V)(cpo + jjs (ioni - Umi)(cpo + *

(p7) dS

‘s = -P

( ~ 7d )s,

(8.27)

where the last form follows from (8.10). A direct evaluation of the exciting-force coefficient follows by substituting the inner solution (7.26) for the diffraction potential: Xi = -p

j exp(il,x) dx L

(iconi - Urni)[@,

+ O7+ C,(x)(Os + a,)]dl.

P

(8.28) The contributions proportional to the factors ni can be evaluated directly from the two-dimensional zero-forward-speed exciting force (5.30). The contributions from the factors mi depend on the steady-state solution, and the interaction coefficient C , is dependent on the forward speed. For the fluxless modes (i = 2, 4, 6), and also in the short-wavelength regime, C , does not contribute to (8.28) and the exciting force coefficients are linear functions of the forward velocity. Since Stokes’ theorem has been used in the last form of (8.27), the integrand in (8.28) cannot be interpreted as the local force. To emphasize this distinction we note the special case of a long “parallel middle body” where the ship hull is cylindrical and, in the inner region, W = - Ui. Neglecting the interaction coefficient, the only effect of the gradient operator in (8.27) is on the oscillatory factor exp(ilox ) , with the result that (iw + W - V) = io,,and the local exciting force is independent of the forward speed. This is confirmed physically by the fact that a long cylinder may be moved axially (in an inviscid fluid) without affecting the local pressure field except near the ends. In this connection we recall the discussion following (7.26). Green’s theorem can be applied to the three-dimensional potentials in (8.27), or alternatively to the two-dimensional functions in (8.28). In both cases the result is a form of Haskind’s relations, with the solution of the diffraction problem replaced by an appropriate radiation solution. To derive Haskind’s relations in the three-dimensional form, following Newman (1965), the boundary conditions (3.37) are combined with (8.27) to give

xi = - P

jf s

CPG)((P~

+ ( ~ 7 d) s .

(8.29)

Theory of Ship Motions

279

Here the superscript (- ) denotes the reverse-flow solution of the radiation problem, with negative forward velocity. This radiation potential satisfies the free-surface condition (3.51) with U replaced by - U . After an application of Green’s theorem to (8.29) with the boundary condition (3.36) imposed,

Jj (cp{,’cpo

- cp,cp~-’) dS. (8.30) s Here a line integral similar to the last term in (8.10) is neglected. In this form, the total exciting force on the ship is expressed in terms of the solution of the radiation problem. The inner solution for cpj-’ may be substituted, with the integral performed over the mean surface of the ship hull. Alternatively, this integral can be performed over a closed surface at large distance from the ship, where the far-field asymptotic form of the radiated waves may be used. McCreight (1973) has extended the Haskind relation (8.30) to the striptheory regime of Ogilvie and Tuck (1969), including the higher-order terms in the free-surface condition (3.50). Equation (8.30) is unchanged, but the integral cannot be evaluated at infinity due to the inhomogeneous freesurface condition in the higher-order solution. McCreight notes that the leading-order contribution to (8.30) is the Froude-Krylov force. The next term in a systematic perturbation expansion can be expressed in terms of the zero-speed two-dimensional potential 4. If Green’s theorem (5.21) is applied to the two-dimensional functions in (8.28), we find after a reduction using (5.33) that

Xi = - p

Xi = i ( p g / K , ) I sin and

. [l

fl I

exp(il,x)(& - U j i ) L

+ C,(AIi/zi - Ji/Ji)]dx

(i = 1, 3, 5),

(8.31)

I

X i = -i(pg/K,) sin

B

exp(il,x)(& - U A , ) dx

( i = 2, 4, 6).

(8.32)

JL

Here the local exciting force at each section is expressed in terms of the wave amplitude of the generalized radiation function &i and the corresponding forward-speed function &. $or the antisymmetric modes the exciting force follows directly from (8.32) but for the symmetric modes (i = 1, 3, 5) one must determine the interaction coefficient C , of the diffraction problem, by solution of (7.21-7.22). If the inner solution of the three-dimensional radiation problem is substituted in (8.30), the result is an integral along the length involving twodimensional solutions of Laplace’s equation. By comparison, the integrands

280

J. N. Newman

of (8.31) and (8.32) contain solutions of the two-dimensional Helmholtz equation. It is not obvious that these alternative integrals for the total exciting force will be equivalent, even in the asymptotic sense for small values of the slenderness parameter. Computations of the heave and sway exciting forces have been made by Troesch (1976) for the case U = 0, using these two alternative expressions; the results are in satisfactory agreement with each other and with experiments. In the intuitive strip theory initiated by Korvin-Kroukovsky (1955), thZ local exciting force at each section is derived from a “relative-motion” assumption. This states that the exciting force can be expressed in the same form as the pressure force of the radiation problem, but with the ship’s velocity and acceleration replaced by the relative motions of the incident wave, at a suitable mean depth. (In accordance with G. I. Taylor’s theorem, there is an additional component of the exciting force, equal to the product of the incident-wave acceleration and the mass of fluid displaced by the hull.) In this relative-motion approach the pressure force of the radiation problem is expressed in terms of the added-mass and damping coefficients, evaluated at the frequency of encounter w. Similar expressions for the exciting forces are derived by Salvesen et al. (1970), using the two-dimensional Haskind relations. In modified results derived by Newman (1977) the added-mass and damping coefficients are evaluated at the incident-wave frequency wo. An argument in favor of this modified approach is that, along a parallel middle body, the exciting force is independent of the forward velocity of the ship. REFERENCES ABRAMOWITZ, M., and STEGUN,I., eds. (1964) “Handbook of Mathematical Functions.” U.S. Gov. Print. ON., Washington, D.C. BABA,E., and HARA,M. (1977). Numerical evaluation of a wave-resistance theory for slow ships. Proc. Int. Con$ Numer. Ship Hydrodyn., 2nd. pp. 17-29. Univ. California, Berkeley. BAI,K. J., and YEUNG,R. W.(1974). Numerical solutions to free-surface flow problems. Proc. Symp. Nav. Hydrodyn., 10th ACR-204. pp. 609-647. ON. Nav. Res., Washington, D.C. BECK,R. F., and TUCK,E. 0. (1972). Computation of shallow water ship motions. Proc. Symp. Nav. Hydrodyn., 9th ACR-203, pp. 1543-1587. ON. Nav. Res., Washington, D.C. BISHOP,R. E. D., and -Ice, W. G., ads. (1975). Proc. lnt. Symp. Dyn. Mar. Vehicles Struct. Waoes. Inst. Mech. Eng., London. BISHOP,R. E. D., BURCHER, R. K., and PRICE,W. G. (1973). The uses of functional analysis in ship dynamics. Proc. R.SOC.London, Ser. A 332,23-35. BOLTON, W. E., and URSELL, F.(1973). The wave force on an infinitely long circular cylinder in an oblique sea. J . Fluid Mech. 57, 241-256. BRARD,R. (1948). Introduction a l’etude theorique du tangage en marche. Bull. Assoc. Tech. Marit. Aeronaut. 47, 455479. BRARD,R. (1973). “A Mathematical Introduction to Ship Maneuverability,” Rep. No. 4331. Nav. Ship Res. Dev. Cent., Bethesda, Maryland.

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28 1

CHANG,M.4. (1977). Computations of three-dimensional ship motions with forward speed. Proc. Int. Con/: Numer. Ship Hydrodyn., 2nd, pp. 124-135. Univ. California, Berkeley. CHAPMAN, R. B. (1975). Numerical solution for hydrodynamic forces on a surface-piercing plate oscillating in yaw and sway. Proc. Int. Con/: Numer. Ship Hydrodyn., l s t , pp. 333-350. David W. Taylor Nav. Ship R & D Cent., Bethesda, Maryland. CHAPMAN, R. B. (1977).Survey of numerical solutions for free-surface problems. Proc. Int. Cot$ Numer. Ship Hydrodyn., Znd, pp. 5-16. Univ. California, Berkeley. CHOO,K.Y. (1975). Exciting forces and pressure distribution on a ship in oblique waves. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts. CUMMINS, W. E. (1956).The wave resistance of a floating slender body. Ph.D. Thesis, American University, Washington, D.C. CUMMINS, W. E. (1962). The impulse response function and ship motions. Schiffstechnik 9, 101- 109. FALTINSEN, 0. (1971). Wave forces on a restrained ship in head-sea waves. Ph.D. Thesis, University of Michigan, Ann Arbor. FALTINSEN, 0. (1974). A numerical evaluation of the Ogilvie-Tuck formulas for added mass and damping coefficients. J. Ship Res. 18, 73-85. FROUDE,W. (1861). On the rolling of ships. Inst. N a n Archit., Trans. 2, 180-229. GERRITSMA, J. (1976). A note on the application of ship motion theory. Schiffstechnik 23, 181-185. GERRITSMA, J., KERWIN,J. E., and NEWMAN, J. N. (1962). Polynomial representation and damping of Series 60 hull forms. Int. Shipbuilding Prog. 9, 295-304. GRIM,0. (1960). A method for a more precise computation of heaving and pitching motions both in calm water and in waves. Proc. Symp. Nav. Hydrodyn., 3rd ACR-65, pp. 483-524. Off. Nav. Res., Washington, D.C. HASKIND, M. D. (1946a). The hydrodynamic theory of ship oscillations in rolling and pitching. Prikl. Mat. Mekh. 10, 33-66. (Engl. transl., Tech. Res. Bull. No. 1-12, pp. 3-43. SOC.Nav. Archit. Mar. Eng., New York, 1953.) HASKIND, M. D. (1946b). The oscillation of a ship in still water. Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk 1,23-34. (Engl. transl., Tech. Res. Bull. No. 1-12, pp. 45-60. SOC.Nav. Archit. Mar. Eng., New York, 1953.) T. H. (1929). Forced surface-waves on water. Philos. Mag. [7] 8, 569-576. HAVELOCK, HAVELOCK, T. H. (1958). The effect of speed of advance upon the damping of heave and pitch. Inst. Nav. Archit., Trans. 100, 131-135. HAVELOCK, T. H. (1963). “Collected Works,” ACR-103. Off. Nav. Res., Washington, D.C. JWEN, W. P. A. (1964). “Oscillating Slender Ships at Forward Speed,” Publ. No. 268. Neth. Ship Model Basin, Wageningen. KELLER, J. B. (1978). The ray theory of ship waves and the class of streamlined ships. J. Fluid Mech. (in press). KORVIN-KROUKOVSKY, B. V. (1955). Investigation of ship motions in regular waves. SOC.Nav. Archit. Mar. Eng., Trans. 63, 386-435. KORVIN-KROUKOVSKY, B. V., and JACOBS, W. R. (1957). Pitching and heaving motions of a ship in regular waves. SOC. Nav. Archit. Mar. Eng., Trans. 65, 590-632. A. (1896). A new theory of the pitching motion of ships on waves, and of the stresses KRILOFP, produced by this motion. Inst. Nav. Archit., Trans. 37, 326-368. LANDwEBeR, L., and YIH,C. S. (1956). Forces, moments and added masses for Rankine bodies. J. Fluid Mech. 1, 319-336. LEWIS,F. M. (1929). The inertia of water surrounding a vibrating ship. SOC.Nav. Archit. Mar. Eng., Trans. 37, 1-20. LIGHTHILL, M. J. (1967). On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27, 725-752.

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MCCREIGHT, W. R. (1973). Exciting forces on a moving ship in waves. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts. MARUO,H. (1967). Application of the slender body theory to the longitudinal motion of ships among waves. Bull. Fac. Eng., Yokohama Natl. Uniu. 16, 29-61. MARUO,H. (1970). An improvement of the slender body theory for oscillating ships with zero forward speed. Bull. Fac. Eng., Yokohama Natl. Uniu. 19, 45-56. MARUO, H., and SASAKI, N. (1974). On the wave pressure acting on the surface of an elongated body fixed in head seas. J . SOC. Nov. Archit. Jpn. 136,3442. MARUO,H., and TOKURA, J. (1978). Prediction of hydrodynamic forces and moments acting on ships in heaving and pitching oscillations by means of an improvement of the slender ship theory. J . SOC.Nau. Archit. Jpn. 143, 111-120. MAYS,J. H. (1978). Wave radiation and diffraction by a floating slender body. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts. MEI,C. C. (1977). Numerical methods in water-wave diffraction and radiation. Annu. Rev. Fluid Mech. 10, 393-416. MICHELL, J. H. (1898). The wave resistance of a ship. Philos. Mag. [5] 45, 106-123. NEWMAN, J. N. (1961). A linearized theory for the motion of a thin ship in regular waves. J . Ship Res. 3:(1), 1-19. NEWMAN, J. N. (1964). A slender-body theory for ship oscillations in waves. J . Fluid Mech. 18, 602-618.

NEWMAN, J. N. (1965). The exciting forces on a moving body in waves. J . Ship Res. 9, 190-199. NEWMAN, J. N. (1976). The interaction of stationary vessels with regular waves. Proc. Symp. N a n Hydrodyn., 11th pp. 491-501. Mech. Eng. Publ., London. NEWMAN, J. N. (1977). “Marine Hydrodynamics.” MIT Press, Cambridge, Massachusetts. NEWMAN, J. N. (1978). Wave radiation from slender bodies. Proc. Symp. Appl. Math. Dedicated to the Late Prof: Dr. R . Timman pp. 101-115. Sijthoff & Nordhoff, Groningen. NEWMAN, J. N., and TUCK,E. 0. (1964). Current progress in the slender-body theory of ship motions. Proc. Symp. Nao. Hydrodyn., 5th ACR-I 12, pp. 129-167. Off. Nav. Res., Washington, D.C. OAKLEY, 0. H., Jr., PAULLING, J. R., and WOOD,P. D. (1974). Ship motions and capsizing in astern seas. Proc. Symp. Nau. Hydrodyn., 10th ACR-204, pp. 297-350. Off. Nav. Res., Washington, D.C. OGILVIE, T. F. (1964). Recent progress toward the understanding and prediction of ship motions. Proc. Symp. Nau. Hydrodyn., 5th ACR-112, pp. 3-128. Off. Nav. Res., Washington, D.C. OGILVIE, T. F. (1967). Nonlinear high-Froude-number free-surface problems. J . Eng. Math. 1, 2 15-235.

OGILVIE, T. F. (1977). Singular-perturbation problems in ship hydrodynamics. Adu. Appl. Mech. 17, 91-188.

OGILVIE, T. F. (1978). End effects in slender-ship theory. Proc. Symp. Appl. Math. Dedicated to the Late Prof: Dr. R . Timman pp. 119-139. Sijthoff & Nordhoff, Groningen. OGILvIE, T. F., and TUCK,E. 0. (1969). “A Rational Strip Theory for Ship Motions,” Part 1, Rep. No. 013. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. PETERS, A. S., and STOKER, J. J. (1957). The motion of a ship, as a floating rigid body, in a seaway. Commun. Pure Appl. Math. 10, 399490. PRICE,W. G., and BISHOP, R. E. D. (1974). “Probabilistic Theory of Ship Dynamics.” Chapman & Hall, London; Wiley (Halsted), New York. W. J. (1953). On the motion of ships in confused seas. SOC.Nau. ST. DENIS,M., and PIERSON, Archit. Mar. Eng., Trans. 61, 280-354. SALVESEN, N., TUCK,E. O., and FALTINSEN, 0. (1970). Ship motions and sea loads. SOC. Nau. Archit. M a r . Eng., Trans. 78, 250-287.

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SOCIETY OF NAVAL ARCHITECTSAND MARINE ENGINEERS (1974). Seakeeping 1953-1973 : Tech. Res. Symp. S-3. SOC.Nav. Archit. Mar. En&, New York. TIMMAN, R., and NEWMAN, J. N. (1962). The coupled damping coefficientsof symmetric ships. J. Ship Res. 5(4), 34-55. TROESCH, A. W. (1976). The diffraction potential for a slender ship moving through oblique waves. Ph.D. Thesis, University of Michigan, Ann Arbor. URSELL,F. (1949). On the heaving motion of a circular cylinder on the surface of a fluid. Q.J. Mech. Appl. Math. 2, 218-231. URSELL,F. (1962). Slender oscillating ships at zero forward speed. J . Fluid Mech. 19,496-516. URSELL, F. (1968a). The expansion of water-wave potentials at great distances. Proc. Cambridge Philos. SOC.64, 811-826. URSELL, F. (1968b). On head seas travelling along a horizontal cylinder. J. Inst. Math. Its Appl. 4,414427.

VOSSERS,G . (1962). Some applications of the slender-body theory in ship hydrodynamics. Ph.D. Thesis, Delft University of Technology, Delft. VUGTS,J. H. (1968). “The Hydrodynamic Coefficientsfor Swaying, Heaving and Rolling Cylinders in a Free Surface,’’ Rep. No. 194. Shipbuilding Lab., Delft University of Technology, Delft. WEHAUSEN, J. V. (1971). The motion of floating bodies. Annu. Rev. Fluid Mech. 3, 237-268. WEHAUSEN, J. V. (1973). The wave resistance of ships. Adu. Appl. Mech. 13, 93-245. WEHAUSEN, J. V. (1978). Some aspects of maneuverability theory. Proc. Symp. Appl. Math. Dedicated t o the Late Prof Dr. R . Timman pp. 203-214. Sijthoff & Nordhoff, Groningen. WEHAUSEN, J. V.,and LAITONE, E. V. (1960). Surface waves. In “Handbuch der Physik” (S. Flugge, ed.),Vol. 9, pp. 446-778. Springer-Verlag, Berlin and New York. WEINBLUM, G. P., and ST. DENIS,M. (1950). On the motions of ships at sea. SOC.Nau. Archit. M a r . Eng., Trans. 58, 184-248.

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ADVANCES I N APPLIED MECHANICS. VOLUME

18

Numerical Methods in Fluid Dynamics C. K . CHU Department of Mechanical Engineering and Plasma Physics Laboratory Columbia University New York. New York

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Differential Equations and Boundary Conditions . . . . . . . . . . . . . . A . Differential Equations for Compressible Flow . . . . . . . . . . . . . . B. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Differential Equations and Boundary Conditions: Incompressible Flow 111. Numerical Analysis Background . . . . . . . . . . . . . . . . . . . . . . . . A . Pure Initial-Value Problems: Convergence, Stability, Truncation Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Examples of Useful Difference Schemes . . . . . . . . . . . . . . . . . . C. Solution of Implicit Schemes: Fractional-Step Methods and AlternatingMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV. Pseudophysical Effects: Numerical Dissipation and Dispersion . . . . . . A . Dissipation and Dispersion of Waves . . . . . . . . . . . . . . . . . . . B. Numerical Dissipation and Dispersion from Fourier Components, and Order of Dissipation and Dispersion . . . . . . . . . . . . . . . . . . . C. Modified Equation of Truncation Errors . . . . . . . . . . . . . . . . . D . Relation to Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Implicit Schemes and Large Time Steps . . . . . . . . . . . . . . . . . . F. Mixed Initial Boundary Value Problems . . . . . . . . . . . . . . . . . V. Gas Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Flows with Shocks and Contact Discontinuities: One-Dimensional Unsteady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Lagrangian versus Eulerian Methods . . . . . . . . . . . . . . . . . . . C. Implicit Methods for Wave Filtering . . . . . . . . . . . . . . . . . . . . VI. Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Stream-Function and Vorticity Methods . . . . . . . . . . . . . . . . . B. Navier-Stokes Equations in Primitive Variables for Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Large-Reynolds-Number Limitation . . . . . . . . . . . . . . . . . . . . D. Compressible Flow Navier-Stokes Equations . . . . . . . . . . . . . . E. Steady-State Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

286 287 287 288 29 1 292 293 295 297 301 301 302 304 305 306 307 309 309 313 315 317 317 319 321 322 325 321 329

Copyright 0 1978 by Academic Press. Inc All rights of reproduction in any form reserved. ISBN 0-12-002018-1

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C. K. Chu

I. Introduction The advent of the high-speed computer has produced a profound and lasting change in scientific research. While up to this time scientific research was carried out by theory or by experiments, today it can be carried out by computation as well. It is a fortunate fact of nature that problems most amenable to theoretical analyses, to experimentation, and to Computation or simulation do not coincide, so that an adroit choice of problems in each category permits useful cross-fertilization and rapid building of physical intuition. Intelligent use of computation not only permits understanding and interpretation of experiments, but has given effective anticipation and extension of theory. Fluid dynamics happens to be a field which has strongly benefited from computation, since it is a field in which the physical laws are relatively simple, the mathematics is nonlinear and analytically difficult, and the physical phenomena contained are rich and often dramatic. Such a field is ideal for exploitation by computation. Over the past several decades, while the practice of computation is used in ever-widening areas, there has been an inadequate appreciation of the science and art of computation, at least in the view of this author. Regarding computing as a straightforward routine, some theoreticians still tend to underestimate its intellectual value and challenge, while practitioners often ignore its accuracy and overrate its validity. Rather than giving a catalog of techniques and results, we intend as the theme of this paper to present the overall spirit of modern computing and its philosophy. We shall study the numerical-analytic properties and difficulties, accuracy and pseudophysical phenomena, and some techniques and results that illustrate and emphasize these features. In Sections I1 and I11 we review and summarize the salient and relevant features of partial differential equations and numerical analysis which are directly applicable to fluid-dynamic computing. In Section IV we study the pseudophysical effects, namely, dissipation and dispersion, that arise from computing. These properties will be fully analyzed on some numerical schemes applied to simplified model equations. Sections V and VI will describe typical practical applications to compressible flows and to NavierStokes equations. We close the paper with a short section (VI) on magnetohydrodynamics. Because of space limitations and of matters of taste, many important topics have not been included. For example, special techniques for transonic flow-boundary layer calculations, turbulence, lifting surfaces, and rarefied gas dynamics-have all been omitted.

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11. Differential Equations and Boundary Conditions A. DIFFERENTIAL EQUATIONS FOR COMPRESSIBLE FLOW

The differential equations of motion for the time-dependent flow of a compressible fluid form a set of five nonlinear equations: conservations of mass, the three momentum components, and energy for the five variables: density p, velocity u (three components) and temperature T. Other thermodynamic variables, such as pressure p , internal energy e, enthalpy h, entropy s, etc., can be introduced at will, each additional variable requiring an additional equation of state, which being an algebraic equation, need not be counted among the five basic differential equations. If viscosity and thermal conductivity are included, this set of equations (Navier-Stokes) is parabolic in nature; if they are excluded, the set (Euler equations) is hyperbolic in nature. Both sets have standard mathematical properties, and we recall that parabolic equations have infinite signal propagation speeds while hyperbolic equations have finite propagation speeds. The Navier-Stokes equations for a gas are aP

at

+v

vp

1

*

(pu) = 0,

+vp=pv’u

1

+ (5 +/? V ( V . u),

The inviscid or Euler equations for a gas are correspondingly aP at

p

:i-+

+ v . (pu) = 0,

1

(u.V)u + v p = o ,

as

at

+ (u . v)s = 0.

Here ,u and fl are the shear and bulk viscosity coefficients and K is the thermal conductivity. Mixed vector-tensor notation is used for simplicity.

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288

Often, one prefers or needs to use systems (2.1) and (2.2) in conservation (or divergence) form. Then (2.1) becomes aP at

-

a

- (pu) at

+ax,a

+v

*

(PUiUj)

(pu) = 0,

+ vp = pv2u +

(:+ 1 -

fi V(V

*

u),

(2.1Y

where n’is the viscous stress tensor

and the corresponding Euler equations in conservation form are the same as with the right-hand side set to zero. In conservation-law form, the energy equation uses the total energy density and not entropy. When shocks (discontinuous or weak solution) form, entropy will not be conserved, but energy will still be conserved. B. BOUNDARY CONDITIONS Standard finite-difference or finite element methods can be applied for pure initial-value problems. Both parabolic and hyperbolic systems require initial values on all variables. For practical purposes in solving pure initialvalue problems, the fact that hyperbolic equations have characteristics or finite propagation speeds and parabolic equations have infinite propagation speeds almost need not be considered. However, all the interesting problems in fluid dynamics are mixed initial and boundary-value problems (flow past a body, flow in a channel, etc.). Then, the parabolic or hyperbolic nature of the equations becomes crucial in determining the type of boundary conditions to pose. Let us deal with hyperbolic equations first, and consider (2.2) in one dimension. Then we can write (2.2) in matrix form as

au + A ( U )au -=

at

ax

a

-

at

(:i p

+

ax A[!] =0

(2.3)

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(the velocities perpendicular to the x-direction drop out trivially so that u denotes the x-component of u). The characteristics of the system are given by the eigenvalues of the matrix A; in this case, they are curves in the x - t plane with dx/dt = u + a, u - a, u. We refer to the three characteristics as C', C - , Co. If we multiply (2.3) by the left eigenvectors corresponding to each eigenvalue, we get the differential equations in characteristic form, which in this case represent directional derivatives in the three directions, respectively. Now consider Fig. 1, which represents a region in the x-t space, and let us require the solution be found in the regions to the right of the curve S. The number of boundary conditions (and initial conditions) required at any given point of the boundary is exactly equal to the number of characteristics pointing into the domain of interest. Thus, at points typified by A , B, C , and D, the number of boundary conditions required are, respectively, 3,2, 1, and 0. Moreover, no boundary conditions are needed as x + 00. Nonuniqueness of the solution results when too few boundary conditions are given, while the solution will in general not exist when too many boundary conditions are given. A complete discussion of these questions can be found in Courant and Friedrichs (1948). Of course, one-dimensional gas dynamics is extremely simple, and often explicit solutions can even be obtained through Riemann invariants. In higher dimensions the problem becomes much more complicated. Nevert

a-

oo

FIG.1. Schematic diagram of boundary curve and characteristics at the boundary determining the number of boundary conditions needed.

290

C. K . Chu

theless, the determination of the number of boundary conditions to be prescribed at each point on the boundary can still be achieved by the same procedure as in one dimension. For example, consider the system (2.2) in two dimensions which becomes

a

-u

at

+ A(U) a u + B(U) a u -

-

aY

ax

\ o

\o

0

0

0

0

Ul

U I

= 0.

It is well known that the characteristics are the sonic cone and the particle trajectory counted twice. But in determining the boundary conditions needed at a boundary surface (boundary curve), one only has to consider the projection of the characteristic cone onto the plane spanned by the normal to the boundary curve and the t-axis. Thus, for instance, if x = 0 (or y = 0) is such a boundary, it suffices to find the eigenvalues of the matrix A (or matrix B) and ignore matrix B (or matrix A), and count the characteristics pointing inward into the domain, exactly as in one dimension. For a more general boundary one introduces an x'-axis locally in the direction of the normal to the curve, and a y'-axis, say, perpendicular to it; one then transforms the system to these new independent variables locally at each boundary point, and considers the coefficient matrix of the a/ax' term alone. In this way, the characteristic cone never has to be considered in the determination of the number of boundary conditions. One must be cautioned that, while counting the number of boundary conditions needed is simple, choice of the permissible variables (or combination of variables) that may be prescribed is not always obvious. In one dimension, the situation is simple for Eq. (2.3). For instance, at a boundary where one boundary condition is required, one may prescribe p or u, but not s; where two boundary conditions are required, one may prescribe p or u

Numerical Methods in Fluid Dynamics

29 1

and s, but not both p and u. In two or three dimensions, the situation is more subtle; the reader is referred to Kreiss and Oliger (1973) for such a discussion. For parabolic equations, the boundary conditions are readily applied: either each variable or its normal derivative must be prescribed at each boundary; and boundedness must be prescribed at infinity, as is discussed in standard texts on partial differential equations. For the Navier-Stokes system (2.1), which is a parabolic system, the mass-conservation equation contains no second-derivative terms for the density p while the other equations contain second derivatives of T and u. Thus, all components of u and the temperature T are prescribed on all boundaries, whereas the density p is only to be prescribed at injlow boundaries and not at outflow boundaries. In other words, that equation alone has the form of a hyperbolic equation (even though the entire system is parabolic), and the particle path dx/dt = u plays the role of a characteristic, insofar as specifying boundary conditions is concerned.

c. DIFFERENTIAL EQUATIONS AND BOUNDARYCONDITIONS: INCOMPRESSIBLEFLOW The Navier-Stokes equations for incompressible fluids look considerably simpler than for compressible fluids, but are in fact less standard in type: P

(a. at+

(U.V)u

1

+Vp=pVZu,

The system is still parabolic, but we have lost the time derivatives which advance p and T (and thereby p). The continuity equation div u = 0 now plays the role of a constraint: the velocity u advanced by the momentum equation must satisfy it at all times; this determines the pressure in an indirect manner. The energy equation is not needed if one is interested in the flow only. If one is studying heat convection, it is needed, but it is not coupled with the mass- and momentum-conservation equations, and may be treated as a standard parabolic equation after the flow field u is first found. This is not the case, however, if buoyancy is important. Then, the energy equation, typically written for temperature, must be solved jointly with the system (as in compressible flow), but it is still a standard parabolic equation. Initial and boundary conditions for the Navier-Stokes equations will be

C.K. Chu

292

discussed first. Initial conditions are u at all points, satisfying the constraint div u = 0. Boundary conditions are for u (or normal derivatives) at all fixed boundaries, including inlet and outlet and at infinity, and pressure p at a single point at all time. [See Ladyzhenskaya (1963) for proofs of wellposedness of these problems.] In contrast to inviscid fluids, there is no additional complication with boundary conditions for Navier-Stokes equations in doubly and multiply connected domains, although, when the stream function is used instead of u and p , there is an added difficulty; see Section V1,A. On a free surface, the correct boundary conditions should be the kinematic boundary condition, normal stress balance (in the absence of surface tension and small viscosity, this reduces to the standard condition of constant pressure), and vanishing of tangential stresses. The author is unaware of proofs of well-posedness of free-surface viscous flow problems. For inviscid flows, the differential equations reduce to the standard Euler equations for incompressible fluids: (2.5) with the right-hand side set to zero. Initial conditions are the same as for the Navier-Stokes equations, u prescribed at all points and satisfying div u = 0. Boundary conditions are different from those for the Navier-Stokes equations : now only the normal component of the velocity u * n (or its normal derivative) are prescribed at fixed boundaries, including all inflow and outflow sections, and the pressure is given at a single point. Boundary conditions for the tangential components are lost with the dropping of the second derivatives in the momentum equations, and the tangential components may not be prescribed. For multiply connected domains, one must give in addition the circulations T = u ds around each of the internal boundaries as functions of time, or else one must determine them by a condition of the Kutta-Joukowski type. On a free surface, we now revert to the classical water-wave boundary conditions: kinematic condition for the free surface and constant pressure. The vanishing of the shear stress is not prescribed as this is the boundary condition that is lost when we dropped the higher derivatives in the NavierStokes equations.

-

111. Numerical Analysis Background

We summarize very concisely those notions and results of numerical analysis which are particularly important to fluid-dynamic computing. Whenever possible and convenient, model equations will be used to illustrate these ideas.

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A. PUREINITIAL-VALUE PROBLEMS:CONVERGENCE, STABILITY, TRUNCATTON ERRORS

We recall some well-known definitions. For a set of differential equations, which we denote by Lu=O (3.1) and appropriate initial and boundary conditions, the problem is said to be well-posed if a solution exists, is unique, and depends continuously on the data (i.e., initial conditions, boundary conditions, coefficients, forcing functions). We approximate the differential equation by some finite difference equation (or finite element equation)

&I u h = 0,

(3.2) where the subscript h represents the space steps Ax, Ay and time step At and h + 0 denotes Ax, Ay, At + 0 in some prescribed manner. We apply appropriate initial and boundary conditions, and then ask that the approximate solution u h converge to the exact solution u in some precise sense. To this end, we adopt some norm, denoted by I( 1 , (for example, maximum or square integral norm), such that the norm of the difference of two functions measures the “distance” between them. For practical purposes, the maximum norm is preferred, although for theoretical proofs, the square integral (or Hilbert space) norm is often more convenient. The approximate solutions u h are said to conuerge to the exact solution u if lluh - uI1 -+ 0 as h + 0, that is, as At, Ax, Ay + 0. For sufficiently small h, if I ( & , - uI I = O(Atpl, Axpz,etc.), this quantity is called the error of the solution, and the p’s are called the rates ofconvergence. The difference operator L h is said to be a consistent approximation to the differential operator Lif l l L h u Loll + 0 as h -,0 for all smooth functions v to which L can be applied. For sufficiently small h, if 114,u - LOII = O(Atql, Axqz, etc.), the right-hand side is called the truncation error, and the q’s are called the (formal) orders of accuracy of the scheme. In this context, when we say h -,0, we must specify how At, Ax, Ay, etc. approach zero in relation to each other. For example, U(X, t

+ At) - )[u(x + AX, t ) + U ( X - A X ,t)] At

+ U ( X + A X ,t2) -Ax

U(X

is a consistent approximation to

-au+ - = au o at

ax

- A X ,t )

=O

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C. K . Chu

if At -+ 0 with Ax = O(At), but is not a consistent approximation if At -,0 with At = O(Ax2).Rather, it would in the latter case approximate

The stability of a numerical scheme is the property that the solution to the approximate equations at any fixed t be bounded as the number of steps increases to infinity and h -+ 0. This property of the numerical scheme is not to be construed to mean that the solution cannot grow as t increases, since the exact solution u can very well grow with t. Thus, a difference approximation is stable if, as h + 0, there exist constants C1 and C2 such that (3.3) where C1 and C2are independent of h, and C2is not greater than the growth rate of the exact solution. With these definitions, there holds for linear equations the well-known (but sometimes misinterpreted) Lax Equivalence Theorem. Theorem. For a well-posed initial-value problem for linear differential equations, and for a consistent difference approximation, stability is necessary and sufficient for convergence (of u,, t o u). The reader is referred to Richtmyer and Morton (1967) or Kreiss and Oliger (1973) for the proof of this theorem. It should be emphasized that, first, the theorem holds for linear equations only. Second, it can easily be seen that the rates of convergence are the same as the formal orders of accuracy of the scheme, provided that the latter is stable, and the exact solution is sufficiently smooth. Third, the theorem applies to pure initial-value problems, and extends to mixed initialboundary-value problems only under added assumptions. The stability of a scheme is investigated principally by two methods: the energy method (in which the norm is usually the square integral norm), or the von Neumann method (Fourier components). The former applies only to some cases, but when applicable, it is very convenient to use. The latter is familiar to most readers, and is a straightforward procedure. For nonlinear equations, and for linear equations with variable coefficients, the method is also applicable if the coefficients (which include derivatives in the nonlinear case) are frozen at local values. The procedure then consists of Fourieranalyzing the solution at the new and old time steps, and expressing the former as an operator (amplifcation matrix) multiplying the latter. The von Neumann criterion is that the norm of this operator (e.g., the maximum eigenvalue) should not exceed 1 + O(At) for the scheme to be stable. The

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O(At) term permits the approximate solution to have the exponential time growth, which the exact solution might possess. This criterion is necessary for stability, but is in general not sufficient. Sufficiency criteria are usually rather technical (see Richtmyer and Morton, 1967, for details). There are various proposed stability criteria based on examining truncation errors. These are in general rather weak criteria, but are very convenient to use in eliminating unstable schemes. More will be said about them in Section IV,D.

B.. EXAMPLES OF USEFUL DIFFERENCE SCHEMES Let us illustrate the previous ideas on some useful schemes applied to the simple model equation

au -

at

with

au +a ax = 0

(a = constant),

u ( x , 0) = uo(x)

(3.4)

on the entire x-axis. We first consider the “upwind differencing” scheme: U(X, t

+ A t ) - U(X, t ) + a U(X, t ) - U(X

- A X ,t )

Ax

At

or u;+1

= uj”

+ al(I.&,

=O

(3.5)

- uj”),

where x = j Ax, t = n At, u(x, t ) = u ( j Ax, n A t ) = uj”, l = At/Ax. To apply the von Neumann criterion, one Fourier-analyzes in x. Thus, with

one obtains

q+1= a;(l + aA(e-ikh - 1)). The amplification matrix is just a single complex number, which we shall call the complex amplification factor r, and the magnitude of it is the amplification factor defined in the previous section. Thus r ( l , k ) = $+l/ii;

+ al(cos k Ax - 1) - ial sin k Ax, (3.7) 1 r I > 1 + const. At if a l < 0 (downwind) or a l > 1 =1

from which we see that (At too large). This pair of conditions is often called the CourantFriedrichs-Lewy condition (Courant et al., 1928).

C . K. Chu

296

A simple geometric interpretation is the following (Fig. 2). The difference solution at the point P is completely governed by the initial data in the segment AB, and the slope of the line P A is At/&. On the other hand, the exact solution at Pis uo(x - at, 0).Surely, this point Q must lie within segment AB, for otherwise one can alter the exact solution by changing the initial data without the approximate solution even recognizing this change, so that convergence cannot possibly occur. This requirement is identical to the pair of inequalities given in the previous paragraph. Unfortunately, this argument does not always work, as seen in the following example of a well-known unstable scheme, with forward-time differencing and centered-space differencing: $+I

- ur

At

+a

u;+i 2 Ax

o,

By the same algebraic manipulations as before, r = 1 - i d sin k Ax,

(3.9) i.e., r > 1 for all I and k Ax, and the scheme is never stable. The domain of determinancy argument does not give the correct result in this case, for it would indicate stability for a l < 1. In Table I, we summarize the most commonly used schemes for the model equation (3.3). For each scheme, we give a figure indicating which are the points in the x - t plane that the difference equation involves, the difference equation, the amplification factor, the stability condition, and the truncation error. We comment on this table. The Lax-Friedrichs scheme stabilizes the unstable forward-time centered-space scheme shown earlier, and introduces enough dissipation to calculate shock waves (see Sections IV and V). The

FIG.2. Geometric interpretation of the Courant-Friedrichs-Lewy condition: the domain of determinancy of the approximate solution (A-B) must contain that of the exact solution (point Q).

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297

fully implicit scheme has the advantage of being unconditionally stable, but is also only accurate to the first order. The Lax-Wendroff scheme is accurate to the second order, but has dispersion (Section IV) and therefore produces oscillations and wiggles near sharp transitions. The same is true of the leapfrog and the Crank-Nicolson schemes, both of which are conservative or reversible schemes (I 11 = I), so that there is no numerical dissipation, as will be discussed in Section IV. The leapfrog scheme is a three-level scheme, and the equation for the amplification factor contains an extraneous root which corresponds to a wave in the wrong direction. This wave must be filtered out by proper choice of initial conditions at the first two time levels. The Lax-Wendroff scheme has had various modifications that are different in the nonlinear case, but identical in the present model equation. Hence they are not separately listed. We have also omitted third-order schemes because of their enormous complexity. The fourth-order leapfrog scheme has been included in the table because of its simplicity, and it is obviously possible to get high accuracies in those cases where At can be taken small and Ax cannot be taken as small. C. SOLUTION OF IMPLICITSCHEMES. FRACTIONAL AND ALTERNATING DIRECTION METHODS STEPMETHODS

It is amply clear that from the viewpoint of stability, implicit schemes (i.e., new time variables at more than one point enter each difference equation) are superior to explicit schemes (i.e., new time variables at only one point enter each difference equation) in that they have no stability limit. Their disadvantage is also well known: a large number of algebraic equations have to be solved simultaneously instead of successively, resulting in an immense increase in labor in general. This situation is relieved in one dimension, when each implicit difference equation only involves the values at the two immediate neighboring points. In other words, the difference equation Au"+'

= Bu",

U"

{uj"}, j = 1, ..., N ,

(3.10)

has a tridiagonal coefficient matrix A . For such matrices, one can use standard inversion schemes, variously called Gauss elimination, double-sweep, matrix factorizations, which consist of successively solving linear algebraic equations first in one direction and then backwards. The total computational labor is only about double that of an explicit scheme (see Richtmyer and Morton, 1967, p. 198, or other standard texts on numerical analysis). In more than one dimension, the situation appears much worse, since the matrix A will involve five diagonal rows in two dimensions, seven diagonal

2

f

J

3

L:

I

- I N

2 Y

.-C I

.-

*

2

3

Y

I

1 298

Q

*

Im

+

I

.cI

.-0

a

-

+

-i

u

+I

=

n

*

C .-VI

Q n

Iro

+

2 k?

Q

.-s I

.-

*

+

la

5 IW

;%-I

+

H 299

II

0 "I

C . K . Chu

300

rows in three dimensions, etc., even if the difference equation still only involves new time variables from its immediate neighbors. However, this difficulty can be overcome by the class of methods known variously as alternating-direction implicit methods, splitting, or fractional-step methods. For example, to solve the equation au

au

au

at

ax

ay

-+a--+b-=O

in two dimensions plus time, one could use two half-steps: during the first half-step the scheme is implicit in x and explicit in y, and conversely so during the second half-step. Thus, instead of inverting a five-diagonal matrix, we invert two tridiagonal matrices successively by the method already developed for one dimension. When the implicit and explicit schemes are properly chosen, e.g., by Crank-Nicolson, then one even gets second-order accuracy. This scheme is usually called the alternating direction implicit method (ADI), and it is due to Peaceman and Rachford (1955) and Douglas (1955). The related splitting or fractional-step methods do not aim for secondorder accuracy in general, and thus they give more freedom to splitting the operator. The original idea, due to a series of Soviet authors (see, e.g., Yanenko, 1971), is to solve 1 au au =a-

_-

2

at

ax

during the first half-step, and to solve 1 au - -- b - au 2 at

ax

during the second half-step. The approximation is consistent only at the end of the two-step cycle, and the accuracy is only first order. Fractional-step or splitting methods, however, carry further implications. The operator need not only be split in two directions, but can be split in numerous other ways. Thus, in solving for example

au

--

at

au azu - a - - + v 7 , ax

ax

one could split the right-hand side, if desired, to

I au au --a--, 2 at ax

~-

and

1 au aZu = v 7, 2 at ax --

in two half-steps. In fact, most of the modern methods in solving fluiddynamic problems, including particle-in-cell, Lagragan-Eulerian methods, etc., all utilize the idea of splitting operators at various stages.

Numerical Methods in Fluid Dynamics

30 1

IV. Pseudophysical Effects: Numerical Dissipation and Dispersion A. DISSIPATION AND DISPERSION OF WAVES

It is well known by now that numerical schemes introduce numerical dissipation and dispersion, in roughly the same way as physical dissipation and dispersion occur in phenomena of fluid flow. It is important to have some precise notions of these effects, so that in calculating or modzling actual flows, one may minimize, ignore, or even exploit these effects, depending on the nature of the problem. For example, if we are solving the NavierStokes equations, we must minimize the numerical dissipation arising from differencing of the convective terms; otherwise the physical dissipation may well be swamped by the numerical dissipation, and the entire calculation may be meaningless. To see these effects clearly, we consider the model equations

au

au

-+a-=O at ax

au

au

-+a-=v,, at ax

au

au

-+a-=&at ax

’ a2U

(4.21

ax a3v

(4.3)

ax3’

which model ideal fluid flow, the Navier-Stokes equations, and dispersive waves (Korteweg-deVries equation), respectively. Plane waves of the form G~ eik(x -c(k)r) - a(k)r 7

when substituted into (4.1) and (4.3), will propagate undissipated and undispersed according to (4.1), with

c(k)= a = const.,

u = 0,

or dissipate according to (4.2)’ with c ( k ) = a = const.,

a(k)= vk2,

or else disperse according to (4.3), with c ( k )= a

+ &k2,

u = 0.

Precisely, dissipation means the damping of Fourier components or plane waves, and dispersion means the propagation of plane waves of different k at different speeds c(k).

302

C. K. Chu

B. NUMERICAL DISSIPATION AND DISPERSION FROM FOURIER COMPONENTS AND ORDER OF DISSIPATION AND DISPERSION

Now let us solve (4.1) by a typical difference scheme. Then the solution u(x, t) will naturally be a function of Ax and At as well. More precisely, each plane wave of wavenumber k will have c and a dependent on the two Similarly, if we solve dimensionless parameters k Ax = and a At/Ax = d. (4.2) or (4.3), c and c1 will not just depend on k, v, and E, as in the case of the exact solution, but will depend in addition on k Ax, al,v At/(Ax)2 = v7, and E A ~ / ( A x ) ~(These . dimensionless ratios are, of course, not all independent, but it is convenient to introduce all of them at times.) To be definite, let us again consider the Lax-Friedrichs scheme and the leapfrog scheme to Solve (4.1), as detailed in Section II1,B. The complex amplification factors for the two schemes are respectively, from Table I,

<

r = cos k Ax - ial sin k Ax

(4.4)

and

r=

- ia3, sin

k Ax

+ (1 - a’l’

sin’ k Ax)”’.

(4.5) From this, it is easy to see that the former scheme is dissipative, while the latter is nondissipative, or conservative, since the amplitude Irl is given respectively by

Irl = (1 - (1 - a’1’) sin’ k Ax)’/’

(4.6a)

Irl = 1

(4.6b)

and At the same time, both schemes are dispersive, as one can readily see by examining the phase angle after time At. The exact solution has changed in phase by an angle cp = - ak At, while the solutions from these two schemes have changed respectively by the angles cp = arc tan( - a l tan k Ax)

(4.7a)

and

(

cp = arc tan -

a l sin k Ax (1 - a’1’ sin’ k

(4.7b)

The wave speeds c(k) are simply obtained as c(k) = I cp 1 / k At. Figure 3 shows the relative errors of each scheme as a function of k Ax and a1. In general, dissipative schemes smear out sharp discontinuities, but the dispersiveness comes out to be fairly harmless, as the smallest wavelengths are damped out the fastest. On the other hand, conservative

303

Numerical Methods in Fluid Dynamics

I

I

0

*/2

w

kAa

FIG.3. Dissipation and dispersion for the Lax-Friedrichs scheme and for the leapfrog scheme for various values of a l . Curve 1, exact solution and leapfrog scheme; curves 2-4, Lax-Friedrichs scheme with a l = $, i. $, resp.;curve 5, leapfrog for a l = t.f (changesslight).

a,

schemes invariably produce a large number of spurious oscillations near sharp transitions, as is familiar to most readers. Roberts and Weiss (1966) appear to have been among the earliest workers to examine dispersion of schemes in this fashion. Since, in a numerical calculation, it is the long wavelengths (relative to the grid size) that should be made accurate-the short wavelengths can never be accurate anyway-it is meaningful to expand the expressions (4.6) and (4.7) in powers of small k Ax. Thus, from (4.6a), we see that the Lax-Friedrichs scheme has

Ir I

-

1 - *(l - a’A2)(k Ax)’,

(4.8) while Irl = 1 for the leapfrog scheme. The phases at the end of At are expanded from (4.7a) and (4.7b) into q

- -ak At

aA 3

- - (k

AX)^(^

- a2A’)

(4.9a)

C. K. Chu

304 and cp

-

-ak At

+ U6 l (k -

AX)^(^

(4.9b)

- a’l’),

where the correct phase of the exact solution is in the first term, and the rest is dispersion. Following Kreiss, we define the lowest power of k Ax in each case as the order of the dissipation or dispersion. Thus, the Lax-Friedrichs scheme has second-order dissipation and third-order dispersion, while the leapfrog scheme has no dissipation and third-order dispersion.

c. MODIFIEDEQUATIONOR TRUNCATION ERROR The other way to examine dissipation and dispersion is to expand the difference equation in Taylor series, to get the original differential equation plus higher order terms. Thus, for example,

If we substitute these into the difference equation, we recover the differential equation, plus higher order terms in At and Ax, with higher derivatives u,,, uxx,etc., as coefficients. To see the dissipation and dispersion, we eliminate all higher-order t-derivatives in favor of x-derivatives by repeated differentiation of this expression (i.e., the entire expression as an infinite series and not just the original differential equation). The resulting equation after elimination and substitution is called the modijied equation, after Warming and Hyett (1974). Applying this procedure to our examples, we get the modified equation for the Lax-Friedrichs scheme U,

+ UU,

1 (Ax)’ 2 At

= - -(1 - ~’l’)u,, +$a(Ax)’(l

- a212)u,,,

+ ..*, (4.11a)

and the modified equation for the leapfrog scheme U,

+ UU,

= -&(Ax)*a(l - ~~l’)u,,,.

(4.1l b )

The quantities in (4.11a) multiplying u,, and in (4.11a) and (4.11b) multiplying u,,, are, respectively, the numerical dissipation and dispersion coefficients, by comparison to Eqs. (4.2) and (4.3). The order of the dissipation and dispersion as previously defined refers here to the order of the

Numerical Methods in Fluid Dynamics

305

derivative. Thus for example, a scheme with coefficient zero in front of u,, but a nonzero coefficient in front of uxxxxwill have fourth-order dissipation. It is important to recognize that this agreement between dissipation coefficients viewed from Fourier components and viewed from the modified equation is a direct consequence that both procedures represent expansion in small k Ax, hence both agree for long wavelengths. How waves dissipate and disperse at small wavelengths (large k Ax) cannot be seen by the truncation error procedure, and one must look at the full expressions such as (4.7). Dissipation and dispersion coefficients for small k Ax for all the schemes in Table I can be read off from the truncation errors; one merely has to substitute the appropriate x-derivatives for the t-derivatives. D. RELATIONTO STABILITY If we apply the modified equation procedure to the unstable scheme (3.9), i.e., forward-time difference plus centered-space difference, we obtain u,

+ au, = - At2 a%,, + ...

(4.12)

-

The instability of the scheme is now seen in yet another way: the dissipation coefficient is now negative, which corresponds to instability. For higher order terms, similarly, for stability to occur, it is necessary that the coefficient of u,,,, be negative, that for ubXbe positive, etc. This fact is easily established by considering the total energy integral (letting v and v’ denote the coefficients of the second and fourth derivatives):

-at j a

m

-m

u2dx=v

“!

m

MU,,

dx

+ v’

-m m

(u,)’ dx

= -v -m

+ v’

m

( u , , ) ~dx.

(4.13)

-m

Note that the same holds for higher order terms of even order, which must alternate in sign for stability, while the odd-order terms all integrate out into boundary terms at infinity. Hirt (1968) first proposed using this “heuristic” stability criterion by examining the coefficient of the second derivative. Certainly, this is a very simple necessary condition, and very useful in excluding obviously unstable schemes or regimes. But it is rather weak in that many unstable schemes need not show up that way, particularly if the instability comes from the short wavelengths, which this procedure does not account for at all and which are often crucial in causing instability.

306

C . K . Chu

An important stability criterion was given by Kreiss (1964). For hyperbolic systems of linear differential equations (variable or constant coefficients), with coefficient matrices which are symmetric and sufficiently smooth, and for explicit difference approximations of the system, if the difference scheme is dissipative of order 2r and accurate of order 2r - 1 or 2r - 2, the scheme is stable. The order of the dissipation is defined in the lines following Eqs. (4.8) and (4.9), and, in addition, the scheme is defined to be dissipative if Ir I < 1 for all k Ax. Thus, the heuristic stability criterion of Hirt, as described in the previous paragraph, would be rigorously correct (since it applies only to first-order schemes with second-order dissipation) if dissipation at large k Ax is also present. More recently, Warming and Hyett (1974) proposed the strengthening of this criterion by including terms of all orders in the modified equation. This clarifies the situation, but is not necessarily easy to use. Yanenko and Shokin (1969) obtained necessary and sufficient criteria based on this approach by restricting the class of schemes considered, but these become rather specialized. In short, when compared with the von Neumann criterion, this approach is simpler in those cases where instability is found outright, particularly for nonlinear equations or variable coefficients. On the other hand, strengthening this criterion often leads to long algebraic manipulations, and whether it is still simpler than the von Neumann procedure becomes debatable. AND LARGE TIMESTEPS E. IMPLICIT SCHEMES

It has been seen in Section 111 that implicit schemes have the advantage of being unconditionally stable, so that one is often tempted to take very large time steps to economize computing. That the accuracy will suffer is intuitively clear. But, in addition, we show rather easily here that large At may produce spatial oscillations or wiggles in some schemes and will invariably slow down the wave speeds. The latter may even be desirable, as we shall see in Section V,C, but one should certainly be aware of this fact. Unwanted oscillations are most easily seen on a Crank-Nicolson scheme, which gives an amplification factor (Table I) ial 1 - - sin k Ax r=

2 i d

(4.14)

l+-sinkAx 2 Here Irl = 1 and hence the scheme is unconditionally stable, and the scheme is accurate to the second order in addition (for small a l and k Ax).

Numerical Methods in Fluid Dynamics

307

However, if one becomes greedy and takes a large time step, thereby making a l 9 1, then for shorter waves, r -+ - 1. Thus, the short wavelength modes would “flip-flop” and produce spurious oscillations in space. This phenomenon is more striking when one solves the heat conduction equation, since there the correct solution is monotonic, but these oscillations appear anyway when At is taken too large. If we use the fully implicit scheme (Table I)

r=

1+

1 ial sin k Ax ’

(4.15)

then for large a l , I r I + 0, which results in decay due to large dissipation, but no oscillations are expected. On the other hand, the wave speed has been much reduced. For small k Ax, the phase lag is cp = -arc tan a l sin k Ax

- -arc tan a l k

Ax

in contrast to the exact phase lag c p = -ak A t = - a l k Ax.

Now for a l very large, the speed of the wave is approximated by a’ =

arc tan a l k Ax k At

(4.16)

which is a small fraction of the correct value a.

F. MIXEDINITIAL-BOUNDARY-VALUE PROBLEMS Since all the interesting time-dependent fluid-dynamic problems are mixed initial-boundary-value problems (rather than pure initial-value problems), it would be particularly desirable to have some understanding of the numerical questions associated with them. Unfortunately, while pure initialvalue problems have been studied by numerical analysts for some time, mixed initial-boundary-value problems have been treated only very recently. Thus, our state of knowledge is still comparatively sketchy, and the technical details are rather formidable, so that we shall just content ourselves with a brief discussion. The general notion of numerical stability is still the same, so that at a fixed t , the solution should not go to infinity as the time step is refined. A precise definition to replace (3.3) is more complicated, since the boundary condition

308

C. K . Chu

must now be incorporated. We leave the readers to read Kreiss and Oliger (1973) for this definition and further details. To prove stability, the von Neumann procedure can no longer be used. One must now use a procedure, due originally to Godunov and Ryabenkii (see, e.g., Godunov and Ryabenkii, 1964) and extended and strengthened by Kreiss and co-workers, often called mode analysis, which is closely related to the Laplace transform. The method is difficult to use and is rather technical. [See Kreiss and Oliger (1973) for the procedure, and see Gustafsson et al. (1972) for greater details and many examples.] There are two interesting questions that appear because of the boundary. The first is that the difference equation near the boundary will in general have a different form from the difference equation at interior points, particularly if the latter involves many space points. In general, since one tends to “run out of” points as one gets near the boundary, there is a temptation to use a lower order accuracy scheme near the boundary than in the interior. This is usually permissible. In fact, Gustafsson et al. (1972) showed that for reasonably general cases, the rate of convergence of the solution remains the same as the formal accuracy of the scheme in the interior (as for pure initial-value problems) if the boundary difference scheme is one order less accurate formally. The second interesting question is what to d o with those variables which are not specified for the differential equation problem, but which may be called for in the difference scheme. For example, in terms of the discussion following Fig. 1 in Section II,B, suppose that the differential equation problem demands one boundary condition on u, and the difference equation requires the value of p as well, how d o we treat this problem? Many workers use the ad hoc method of reflection or extension (setting the boundary value equal to some combination of the interior values) which is equivalent to adding a condition on the derivative of the unspecified variable. This overspecifies the problem, but in many cases, one gets away with this simply because the overspecification is a rather “gentle” one. If one specifies the variable instead of the derivative, a numerical boundary layer will form, and if there are any characteristics pointing inward into the domain, the entire solution will be affected by the convecting inward of this boundary layer. A more natural method is to write some kind of one-sided difference equation near the boundary and solve for the unspecified variable. Gustafsson et al. (1972) discussed the theoretical aspects of this question in some detail. Chu and Sereny (1974) made several test calculations on simple fluid-dynamic problems, using various one-sided boundary difference equations, as well as extrapolation (overspecification). In these tests, extrapolation turns out to give rather good results, but we feel it is a fortuitous result (see also Sundstrom, 1975). Chu and Sereny also proposed a

Numerical Methods in Fluid Dynamics

309

procedure, similar in spirit to studying dissipation and dispersion in pure initial-value schemes, to investigate the amplification of reflected waves at boundaries, as a means of characterizing the relative accuracy of various boundary schemes. Further work needs to be done in this direction. V. GasDynamics This section is concerned mainly with inviscid flows, in which shock waves, contact discontinuities, propagation of sound waves, etc., play the main roles. Problems in which viscous effects are important are left to the next section. By far the best-known, best-developed, and most accurate method for solving inviscid gas dynamics problems in one dimension plus time or in steady two-dimensional supersonic flow, is the method of characteristics. Its main drawback, however, is its difficulty for extension to higher dimensions; this is done by some workers, but it is an art difficult to use in general. The basic method for hyperbolic equations in two dimensions ( x - t or x-y), however, is well treated in almost all texts in gas dynamics, partial differential equations, etc., so that it is unnecessary to repeat a discussion here. Rather, we shall concentrate on difference methods, and study the special features involved in solving problems of inviscid gas dynamics. We shall deal with the treatment of discontinuities with some care, mainly in one dimension, but the ideas generalize to higher dimensions rather easily. We shall then treat partly Lagrangian and partly Eulerian methods (including the particle-in-cell methods), which are particularly desirable with interfaces that convect with the fluid. Finally, we shall deal with the problem of filtering out the fast wave speeds, in order to compute ior phenomena on a slower time scale. A. FLOWS WITH SHOCKS A N D CONTACT DISCONTINUITIES. ONE-DIMENSIONAL UNSTEADY FLOW

To be specific, we shall limit our discussion here to time-dependent onedimensional flows, since generalization to higher dimensions is straightforward. Writing the differential equations (2.2) in difference form and proceeding to solve them in a routine way will yield a usable answer, except when there are shocks, boundaries, and contact discontinuities or interfaces. The question of boundaries has been discussed, albeit briefly, in Section II,B, so that we shall concentrate here only on the internal discontinuities: shocks and contact discontinuities. A large proportion of workers in the field

C . K . Chu

3 10

(see, e.g., the works of Moretti, 1969) believe that the best way to solve such problems is to write the difference equations on different sides of the discontinuities and use Hugoniot relations (for shocks) or various continuity conditions (for interfaces) as interior boundary conditions. We grant that this is the most accurate procedure, since discontinuities then remain sharp discontinuities. Nevertheless, except in simple geometries and flows, the fact that the locations of these discontinuities are not known a priori can result in great complications. Hence, the so-called shock-smearing methods have justifiably gained wide acceptance, particularly in recent years. Shock smearing was first used by von Neumann and Richtmyer (1950), in which a viscous term (i.e.,2nd derivative in velocity) with an artificial viscosity is introduced. Shocks then appear as smooth S-shaped transitions. Lax (1954) appears to be the first person to exploit the inherent numerical dissipation of a finite-difference scheme (in this case, the Lax-Friedrichs scheme) to structure such shock waves. Because the dissipation is of the second order and proportional to k Ax, these shocks come out rather thick, some 10 to 20 grid points in thickness. Subsequent introduction of the Lax-Wendroff scheme (Lax and Wendroff, 1960), resulted in thinner shocks (since the dissipation is now of the fourth order), but spurious oscillations also resulted from the third-order dispersion. A question commonly asked by classical fluid dynamicists is: why not use the real viscosity instead of an artificial numerical viscosity? The answer is that a real viscosity shock is too thin to be calculated on a grid compatible with most practical problems. Thus, in shock-smearing methods, one aims to find correctly both the inviscid flows outside the shock transition and the position and speed of the shock, but not the details of the transition. This implies that the computed shock widths must still be small when compared with characteristic lengths of the problem being solved, even though they cannot be as thin as that governed by the real viscosity. In calculating flows with shock-smearing methods, it is desirable (almost necessary) to write the differential equations in conservation-law forms, i.e., in the forms of equations (2.1’) with the right-hand side set to zero. The reason is that the shock speed, and hence position, are directly governed by the “jump” of the (integrated) variables, and not by the derivatives, and hence it is essential that they be integrated accurately. We illustrate this by a simplified example: suppose we wish to solve the simple model equation au at+u

-au= - +auax

subject to the initial conditions t=O:u=a u=b

at

axa ( u3 3 ) = o -

in-co<x
9

Numerical Methods in Fluid Dynamics The solution is clearly ’=

I

a b

311

i n x - Ut<0, inx-Ut>O

with a discontinuity speed U to be determined. Choose L to be large, so then the discontinuity is well inside it. Then

But obviously,

1

a L

-

at

u dx = U(a - b),

-L

so that U=

(b3 - a3)/3 =-[u3/3] b-a [MI ’

where [ 3 denotes the jump of a quantity across the discontinuity. If we solve the equation in conservation form by finite differences, using, say, added viscosity, we have 4+1-uj”

Ac

1

+ 2 A x [+($+ ~

- +(ti- 1)3]

+ 2nd differences = 0.

Then, denoting - N to N as the total number of x, we obtain

since the differences all “telescope” out, except possibly for boundary terms multiplied by Ax. Hence the integrals and the shock speed are correctly reproduced by the difference scheme to order Ax. O n the other hand, if we had done the same to the equation not in conservation-law form, we would have gotten

The right-hand side evidently does not “telescope” out to the quantity ( a 3 / 3 )- (b3/3),so that there is now an error of zeroth order in the shock speed.

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C. K . Chu

The original Lax-Wendroff second-order scheme has been modified by Richtmyer (1962) into a centered two-step scheme, and by MacCormack (1969) into a directional two-step scheme. For linear equations with constant coefficients, these schemes all coincide, but for nonlinear equations, the accuracies of the modified schemes are better than in the original scheme. Rusanov (1970) and Burstein and Mirin (1970) introduced third-order schemes for flow calculations with shocks. Because the dissipation is now of the fourth order, and the dispersion of the fifth order, the resulting shocks are indeed much thinner and the spurious oscillations much less. Figure 4 shows a comparison between three shocks, calculated with a first-order scheme, second-order Lax-Wendroff scheme, and third-order Rusanov scheme, respectively. The main drawback of third-order schemes is that they are quite complicated to use. For meteorological problems, which contain no shocks, Kreiss and Oliger (1973) recommend the use of the leapfrog scheme, second order in time and fourth order in space, which is convenient to use and gives high accuracy. For flows with shocks, dissipation (of either second or fourth order) can be added locally at the shock transition. Even a finer grid can be used locally together with dissipation to ensure a thin shock. Only slight difficulties are present since the location of the shock has to be determined. A different approach, known as flux-corrected transport (FCT), has been given by Boris and Book (1973). Carving away the details, their procedure is

1.0

u

-

u, = ou, At

a-;;t

= 0.2

FIG.4. Comparison of a discontinuity calculated with a first-order Godunov scheme (I), Lax-Wendroff scheme (2), and Rusanov’s third-order scheme (3). The abscissa is in number of grid points.

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essentially the following: split the second-order Lax-Wendroff scheme into two steps, a first-order half-step followed by an antidiffusion half-step, with just enough negative dissipation to eliminate the dissipation introduced in the first-order half-step. (Instability does not result from the negative diffusion since it is interlaced with a first-order half-step.) If the dissipation is entirely killed off, the scheme returns to a Lax-Wendroff scheme, or modification of it in nonlinear cases. But they now use an artifice, in which the negative diffusion is restricted (flux-limiting) to an amount just below that which would cause the appearance of the spurious oscillations, i.e., the antidiffusion half-step is not permitted to produce any new maxima or minima. In these regions, then, the accuracy remains of first order. The sharp gradients in shock transitions are identical to those given by a LaxWendroff scheme, while the spurious oscillations are lopped off, albeit by somewhat ad hoc means. Certainly one can construct many examples for which this method would routinely fail, but in many practical problems, the method does work very well, with a considerable degree of convenience in programming.

B. LAGRANGIAN VERSUS EULERIAN METHODS In many practically interesting problems, it would be most convenient to use a coordinate system that moves with the fluid, i.e., a Lagrangian coordinate system. In as simple a problem as the classical piston problem, with the piston moving nonuniformly in time, the advantages are already obvious : the piston is always a fixed coordinate point in a Lagrangian system, while the piston path in the x-t plane will thread through many grid points (and in general not coincide with all of them) in an Eulerian system. In onedimensional problems, Lagrangian schemes are indeed easy to use and to program. The procedure is straightforward. For example, one divides the interval into segments or cells, as in regular finite differencing index each by i = 1, . . . , n. The velocities of the cell boundaries and positions, denoted by and x i + are variables to be recorded and calculated for subsequent ui+ times, and the temperature T , pressure p i , and density pi are defined at cell centers. (This is one way to proceed-many other ways are possible as well.) When quantities defined at cell centers are needed at boundaries, one takes an average, and this is similarly done for quantities defined at cell boundaries. Subsequently, accelerations are determined by pressure differences and Newton’s law, velocities and positions are determined by appropriate time integrations of the accelerations, densities are determined from the change in cell volume and mass conservation, temperatures are determined from the change in internal energy as cells are compressed or ex-

3 14

C. K . Chu

panded, and pressures are determined from densities and temperatures by the equation of state. Higher order schemes, implicit schemes, etc., are of course usable at each of the steps. Although this is very convenient in one dimension, it is usually very troublesome in higher dimensions. The reason is that in more than one dimension, a cell of reasonable shape can distort into a very complicated shape (for example, a rectangle into a very narrow ribbon at similar volumes), and the calculation of cell volumes and cell areas become prohibitively difficult and inaccurate. Thus, pure Lagrangian schemes in more than one dimension are often impractical. Nowadays, it is very common to use a mixed Lagrangian-Eulerian grid, in which the grid moves, but not necessarily with the velocity of the fluid. There are several practical instances in which such a scheme would be useful, even though it is obviously complicated. One instance is that of two different fluids separated by an interface, each having velocities normal and tangential to the interface. In order to keep the interface sharp, it is important to have a Lagrangian grid perpendicular to the interface, but the distortions inherent that would appear in a Lagrangian grid due to large fluid distortions can be avoided by allowing the fluid and the grid to have relative motion in the direction parallel to the interface, the parallel velocity of the grid being so chosen as to maintain reasonable grid shapes. Another instance is in magnetohydrodynamics, where one often needs to use flux coordinates to follow the magnetic-flux surfaces accurately, and the motion of these flux surfaces will not coincide with the fluid motion if there is any dissipation present. In all such cases, one deals with the grid motion and the fluid motion relative to the grid at the same time. In such arbitrary grids, then, the central features are the following: (1) The differential equations must first be cast in moving curvilinear coordinates, corresponding to the velocity of the moving grid, which may be known or unknown (i.e., to be computed together with the solution). (2) The difference equation must be chosen to be stable. (3) The positions of the grid points with respect to a fixed grid must be recorded accurately. (4) An accurate interpolation scheme, for example, using splines, must be used to transfer results on the moving grid to a stationary frame.

A well-known fluid-dynamic code with arbitrary grid motion is the ALE (arbitrary Langrangian-Eulerian) program, due to Hirt (1971). When combined with an implicit differencing procedure, it is called ICE-ALE (Harlow and Amsden, 1971; Hirt and Amsden, 1972). The same procedure applied to magnetohydrodynamics is due to Brackbill (1976). An earlier mixed Eulerian-Lagrangian method that has been used with

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wide success is the particle-in-cell method (Harlow, 1964). It is not as commonly used nowadays as the newer methods because of its inherent fluctuations and hence lower accuracy, but it is still useful when one deals with multiple fluid media, large fluid distortions, etc., when other mixed schemes become inconvenient to use. In the particle-in-cell method, density, pressure, temperature, and velocity are assigned to each cell, as in a regular Eulerian grid. Together with the cells is a system of particles (fluid elements), which are endowed with position, mass, velocity, and internal energy. These particles move according to Newton’s laws, the forces being determined by the pressure gradients in the particular cell in which the particles reside. As the particles cross cell boundaries, detailed accounting of the masses, momenta, and energy is carried out, and the cell densities, velocities, and temperatures are recalculated. Cell quantities are then reassigned to the particles in such a way as to ensure overall conservation, improve numerical stability, and reduce fluctuations. C. IMPLICIT METHODSFOR WAVEFILTERING The Courant-Friedrichs-Lewy condition for gas dynamics is the inequality I u f a 1 (At/Ax) 5 1, where again u is the flow speed and a is the sound speed. In many problems of interest, IuI 6 la1 and the phenomena associated with sound wave p r o p agation may not be of much interest; in that case, it is uneconomical to let the time step be dictated by the sound speed a, when in fact it should be dictated by the flow speed u. Obviously, implicit methods are precisely what is needed, as described in Section IV,E. One such scheme is the ICE (implicit, compressible, Eulerian) scheme (Harlow and Amsden 1971). Stripped of details, the procedure is essentially the following, which we describe in one dimension for the inviscid conservation laws, i.e., Eqs. (2.1’) with dissipative terms set to zero: (5.3a) (5.3b) gat( l 2p u ’ + p e ) =

-&(ipuz+ph).

(5.3c)

The quantities on the right-hand side are all treated explicitly, while the quantities in the left-hand side of the first two equations are treated either

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C . K . Chu

fully implicitly, or by the Crank-Nicolson scheme, with the provision that the pressure p is represented by a predicted value fi obtained from (5.4)

This closes the first two equations, and one solves for p n + l , (pu)"" (and thereby u""), and jj by, for example, iteration. The energy equation is completely explicit, and the final value of p n + l is obtained, after both p"+' and e"+' are obtained, using the equation of state. If viscous terms are present in the momentum equation, they are thrown to the right-hand side and made explicit. If Crank-Nicolson differencing is used, and the numerical dissipation from the left-hand side is small, numerical dissipation can also be added explicitly to the right-hand sides of all three equations. The reasons for this particular choice of implicit variables are amply clear. In the first place, the eigenvalues u f a come from the first two equations, while the eigenvalue u comes from the third equation; thus the first two equations should be made implicit in some way, while the third can be left fully explicit. Second, fi had to be introduced in the momentum equation, in order to close the two equations as a subset, otherwise p"' will have to await the solution of the third equation as well. Third, the momentum pu can be eliminated from the first two equations, and a second-order equation for p"+ or 6 can be solved if more convenient to d o so, instead of iterating. Because of the complications and nonlinearities of the fluid-dynamic equations, and particularly of magnetohydrodynamic equations, there is a great temptation to use partly implicit and partly explicit schemes successively, as seen in the following example. While the scheme may appear reasonable, it may have very troublesome stability properties. The following example of simple isentropic flow, linearized to a constant background, is due to M. F. Reusch. For aP au ap 5 po -k uo - = 0, ax ax

+

(5.5)

the difference approximations are py+' -

At ";+I

pj"

- Ujn

At

n+l-

n+l

n+l-

n+l

+ uo Pj+l2 AxP j - 1

uj-1 2 Ax

uj+l

+ uo

u;+l - UY-1 - -Po 2Ax ' 2

n+l

(54

n+l

-5 P j + i -Pj-1 po

2 AX

'

which appear both reasonable and convenient. One solves the first equation (tridiagonal matrix), and uses the updated p in the second (another tridi-

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317

agonal matrix). Nevertheless, the eigenvalues of the amplification matrix come out to be r=

1 + iuoA sin k Ax

(1-

ao2A2sin2 k Ax 2(1 + iuoA sin k Ax)

’

aO2A2sin’ k Ax ( ( l - 2(1 + iuoA sin k Ax)

jz j””, -

1

so that the scheme is stable only if luol 2 laol.

The conclusion is quite surprising, and the scheme is rather useless. The explanation is that for very small u, this partially implicit scheme is practically two explicit steps, each using forward-time and centered-space differences, hence it would be unstable. The use of adroitly chosen implicit methods to slow down the sound-wave speeds, as typified by the ICE procedure described, is one such approach to deal with the time-step problem. Recently, other approaches have been successively made. MacCormack (1976) used a scheme, in which by skillfully splitting the equations into fractional steps, he is able to integrate the sound waves by a characteristic method. Kreiss (1977) has proposed a method in which the initial data are particularly tailored such that the sound waves, or other fast time-scale phenomena, are filtered out.

VI. NavierStokes Equations We discuss here numerical methods for (a) two-dimensional flows of a viscous incompressible fluid, for which the stream function and the vorticity are the dependent variables, (b) flows of a viscous incompressible fluid in two or three dimensions, for which the velocity and the pressure are the dependent variables, and (c) flows of a viscous compressible fluid. We also discuss their inherent limitations and some remedies for treating flows at large Reynolds numbers and attempt to clarify the relationship between a time-dependent calculation and its limiting steady state. A. STREAM-FUNCTION AND VORTICITYMETHODS

The use of the stream function t,b and the vorticity ( in place of the primitive variables u and p is the simplest method for treating the NavierStokes equations for two-dimensional or axisymmetric flows of an incompressible fluid. For simplicity, we describe only the plane case, with the

C. K. Chu

318

velocity variable u = (u, 0). Equations (2.5) are readily replaced by the equations for $ and l = curl u = (au/ay) - (av/ax): (allat)

+ v . (uc) = vvzc,

(6.la)

-l,

(6.lb)

vz* = u = a$/ay,

v = -a$/ax.

(6.1~)

These equations were first solved numerically by Thom (1933), and with great success by Fromm (1964) in his studies of Karman vortices. The advantages of the method lie in the standard form of the equations: a parabolic equation for the vorticity, a Poisson equation for the stream function, and first derivatives for the velocity components. The vorticity equation is solved by standard numerical techniques: in the case of Fromm, by a leapfrog scheme for the convective operator, and a Dufort-Frankel scheme for the diffusion operator. Thus,

The Poisson equation is again solved by standard techniques: in the original work of Fromm, by successive overrelaxation. Nowadays, with sufficiently simple boundaries, one can use fast Fourier transforms, or direct methods for the Poisson equation. The disadvantages of the method are twofold. First, it is (more or less) limited to two-dimensional (plane and axisymmetric) problems, since the stream function $ exists only in two-dimensional problems. Second, the stream-function vorticity method usually results in complicated and clumsy treatment of the boundary conditions resulting in inaccuracy. We illustrate this by the simple case of a solid wall, moving with velocity U parallel to itself. The boundary conditions for the primitive variables are u(x, 0) = U , u(x, 0 ) = 0, which translate into $(x, 0) = constant and $,,(x, 0) = U . This gives one boundary condition too many for the Poisson equation, and one boundary condition too few for the vorticity equation. In Fromm's original procedure, the old values $(x, y ) and r ( x , y ) in the interior, together with the old boundary values [ ( x , 0), are used in (6.la) to calculate the new vorticity values in the interior, [ ( x , y ) . Together with the boundary value $ ( x , 0), the

Numerical Methods in Fluid Dynamics

3 19

Poisson equation is solved for new values of $ ( x , y) in the interior, and the velocity u(x, y) and u(x, y) everywhere in the interior. The wall vorticity [ ( x , 0) is then found from

au

[ ( x , 0 ) = - ( x , 0 )=

u(x, AY) AY

u

dY The method is indirect and error-prone. An improvement of this procedure is used by Sood and Elrod (1974). They use a nine-point difference formula for the Poisson equation and a consistent linear combination of [ from the nine points for the inhomogeneous term. In this fashion, in the difference equation for $ at the points next to the wall, the boundary vorticity [ ( x , 0) appears as unknown and the stream function $(x, Ay) as known (i.e., found from U = $ y ) . This is neater and more accurate. Because of these difficulties, and particularly for problems with more complicated boundary conditions, such as free-surface conditions, it is often better to keep the primitive variables u and p. B. NAVIER-STOKES EQUATIONS IN PRIMITIVE VARIABLES FOR INCOMPRESSIBLE FLUIDS More recently, methods have been devised for solving the Navier-Stokes equations for incompressible fluids in the primitive variables p and u. The advantages of these methods lie in the simplicity of applying boundary conditions (essentially u on solid walls and p on free surfaces) and in their applicability to three dimensions without drastic changes needed from a two-dimensional procedure. Two very similar methods are Harlow’s marker-and-cell method (MAC) (Harlow and Welch, 1965) and Chorin’s (1968) method. The name MAC is somewhat misleading since markers are used to delineate free surfaces, and are not essential to the method. (When free surfaces are absent, markers d o not appear at all.) As mentioned in Section 11, Eqs. (2.5) form an unconventional set: the momentum equation must advance the velocity u in time with just the right pressure p such that the constraint div u = 0 holds at all times. Upon setting

F(u) (2.5) simply becomes

VV’U - (l/p)(u ’ V)U,

C . K . Chu

320 and its divergence gives

a v2p = -- div u + div F(u). P at By virtue of div u = 0, the first term on the right side of (6.4) is of course zero. But in the difference formulation this will not be exactly true, and this fact is utilized as an additional condition. The difference scheme is

where Vh and div, denote the finite-difference approximations to the V and div operators. At each time step, (6.6) is first solved for the new pressure p"", and (6.5) then advances the velocity u"+'. Note that div u"" is set to zero in (6.6)to determine the pressure, but because of finite-differenceerrors, the resulting u"" from (6.5)is never exactly divergence-free. Hence the term div d is retained in (6.6) to correct for this small error from the previous time step. A similar procedure was proposed by Chorin (1968), who splits (6.5) into two steps, and replaces (6.6) by an iteration procedure for p"" and u " + l simultaneously. Let p"' I,'", u"+ be the mth iteration for p"+' and u"+' The scheme is written as u - u" = F(u"), At '9'"

pn+l,m+l = n+l,m -

P

1 divh Un+l.m+l 9

(6.9)

where I. is a convergence factor chosen to improve the efficiency of the iterative process. Since in most cases (6.6) is solved by iteration anyway, Chorin's method provides a greater freedom in designing the iteration process. This feature has indeed been adopted in the newer version of the MAC code (simplified MAC or SMAC). However, in simpler problems (6.6) permits a direct solution of the Poisson equation-for example, by fast Fourier transform, which is faster than iteration. Both the MAC and SMAC codes have been applied to a host of problems involving free surfaces (e.g., breaking of a dam), with rather dramatic results. While the precise correctness of the free-surface boundary conditions in their studies may be debated, they have produced interesting and important qualitative insight for fluid dynamicists.

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321

C. LARGE-REYNOLDS-NUMBER LIMITATION Most finite-difference methods for solving the Navier-Stokes equations, including all the ones described in the previous two sections, suffer from a severe limitation: the methods become useless at slightly large Reynolds numbers (far below the Reynolds numbers corresponding to transition to turbulence). One common reason is that viscosity is often relied on either to stabilize or at least to damp out the oscillations due to the differencing of the convection operator; this effect decreases as the Reynolds number increases, and the calculation becomes either unstable or oscillatory. We shall discuss this aspect in the next section. The other reason is that the boundary-layer thickness, which decreases with an increase in the Reynolds number, soon becomes smaller than a grid width, and all resolution is lost. This limiting Reynolds number can be estimated very readily. Consider the flow around a body of characteristic length L, with the Reynolds number defined as Re = U q v , where U is some characteristic velocity (e.g., free stream), and v is the kinematic viscosity. At a distance x, a few grid widths from the leading edge (x a Ax, a a small integer), the thickness of the boundary layer is

-

The limiting Reynolds number corresponds to '(x) < Ax, or

or Re > a(L/Ax).

(6.10)

Such a number is typically about 100. Recently, a method has been proposed by Chorin (1972) to overcome this difficulty. N o grid is used at all. For the flow around a body, an inviscid-flow problem is solved in the first step, in which the vanishing of the normal velocity is satisfied on the body, but not the vanishing of the tangential component. Discrete vortices are then introduced on the boundary, of just the right strength to satisfy the latter condition. These vortices are then convected with their instantaneous local velocities. In the next time step, they also contribute to the inviscid velocity field, and the cycle continues. The preliminary results obtained from this method appear very encouraging, and the method appears to be indeed effective in dealing with highReynolds-number problems. Another partial remedy to this phenomenon is provided by nonuniform grid sizes (small grid near the boundaries). On boundaries where conditions

322

C . K . Chu

are not crucial, another standard device is to make the flow locally irrotational, that is, replace the boundary condition u = 0 by the conditions (6.11)

On important boundaries, this device can also be used and a boundary-layer calculation matched to it, provided the boundary layer does not have a significant influence on the rest of the main flow, such as in wakes.

D. COMPRESSIBLE-FLOW NAVIER-STOKES EQUATIONS The equations for viscous compressible flow (2.1) are standard parabolic equations, except for the slight difference in the mass-conservation equation, which was discussed in Section I,B. To a large extent, therefore, we have already gained much understanding by studying the simpler model equation (4.2), often called the Burgers equation. We shall consider time-dependent problems in this section, and postpone our discussion of steady-state questions to the next section. There is considerable feeling in the field that the Navier-Stokes equations should be written in conservation law form before any finite-differencing is performed, i.e., having the Navier-Stokes equations in the form (2.1')instead of (2.1).The argument is never completely convincing. Indeed, the equations in conservation law form ensure overall conservation of mass, momentum, and energy, but d o not ensure accuracy beyond that. It is our belief that in problems involving shock waves (see Section V,A), or problems involving a fixed mass of fluid in a closed domain, overall conservation is important; then the equations should be in conservation law form, and the difference schemes should be conservative as well. For flows through tubes, flows past bodies, etc., an accurate scheme does not have to ensure overall conservation, and in fact, overall conservation is often used as a check for the accuracy of the method as a whole. If the dissipation parameter is very small (or the Reynolds number is very large), then it is essential that the finite-difference scheme be of high-order accuracy, so that not only the dissipation of the scheme but also its dispersion should be small, as the physical dissipation is not adequate to dampen the small wavelengths if the scheme is too dispersive. This can be a very difficult task, and often limits the Reynolds number to considerably below that of transition to turbulence, as mentioned in the previous section. In the easier cases, when the dissipation is not too small, the main concern is that the numerical dissipation be not so large as to dominate the physical

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323

dissipation. We illustrate this important point by examining several popular schemes. It is quite obvious that any first-order (e.g., Lax-Friedrichs) differencing of the convective operator will be quite poor when used on the Burger’s equation, since the large numerical dissipation will dominate the physical even at fairly small Reynolds numbers. We can estimate that very easily. The numerical dissipation coefficient of the Lax-Friedrichs scheme was found in (4.1la) and it would exceed the physical dissipation when 1 (Ax)’ VI-(1 - a’n’), 2 At or

a Ax v

2

2a3,

(6.12)

(1 - a2A2)’

-

The left side is the Reynolds number based on the grid size, while the right side is a number 1, since a 1 I 1 is required for stability. Thus, when the grid Reynolds number becomes of order of 1 , the numerical dissipation swamps the physical dissipation. Hence, the scheme cannot be used at all. Next, we use a higher order (i.e., dissipation of order higher than two) scheme, e.g., the Lax-Wendroff scheme, to difference the convection operator, and we difference the diffusion operator in three different ways: fully explicit, full implicit, and Crank-Nicholson. The difference equations are, respectively,

$+’- uj” At

+a

$+1

a’ At $+1 - u ; - ~- _ _ 2 Ax

+ u’jV1- 2uj”

2

(Ax)’ (6.13a) (6.13b)

=v[

ui”+1

+AX)' u;- - 2uj” + u;:: + un+’

3- 1

1

- 2u;+’

2(Ax1’

1

.

(6.13~)

The complex amplification factors for the three schemes are, respectively, (an = a(At/Ax), vt = [v At/(Ax)]’) re.+, = 1 - iaL sin k Ax - 4 (a’;”” ~

+ vt

) sin’ 2k ,Ax

(6.14a)

C. K . Chu

324

1 - ial sin k Ax - 2a212sin’ rimp

=

kAx ~

2

(6.14b)

k Ax 1 + 4vz sin’ 2 1 - ial sin k A x - 2(a2A2+ vz) sin’

k Ax

Since the Lax-Wendroff operator is accurate to the second order (ie., with third-order dispersion and fourth-order dissipation), the second-order dissipation for all three schemes are correctly given by the physical dissipation for each case.

Ir I = 1

-

pT(k AX)’

= 1 - pk’ At

+ ...

+ ....

(6.15)

The dispersion for the three schemes are different, although of the same order:

( k AX)^

qexp = -ak At - a l qimp = -ak At - a l (pcN =

-ak At - a l

1 ( -: + $1

:

--

+

~

a:2

( k AX)^

+

+ ..-,

+ ...,

”;) ( k AX)^ + ....

-

(6.16a) (6.16b) (6.16~)

The dimensionless quantity al/vz = (a Ax/v) is again the Reynolds number based on the grid size, Re,. The stability of the schemes deserves some comment. The explicit scheme will have the standard stability limits a l I 1 and vt I4,respectively, only in the limiting cases of v = 0 or a = 0. Otherwise, (6.14a)shows clearly that for fixed a l , the presence of v decreases the amplification factor I r I, but for fixed vz, the effect of a is less obvious. Thus the allowable time step At is changed from the value of min(Ax/a, Ax2/2v), and must be calculated for given values of a, v, and Ax. For the implicit scheme, on the other hand, (6.14b) shows that the presence of v in the denominator reduces I r I for given values of a l . Thus, the permissible time step At will always be increased over the value of Axla by the presence of the viscosity v. Similar investigations must be made for other schemes for the convective operator, particularly if the convective operator is taken implicitly.

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325

A rather popular scheme due to Brailovskaya (1965), and subsequently adapted by Cheng (1969) and his co-workers is the following two-step procedure, in which an intermediate u is calculated:

ii - u; At

+a

-

u7-1

2 Ax

= v u;+1

-

u;+1 - u;

At

u;+l

+U

iij+ 1 - u j 2 Ax

+q - 1 ,

1

=v

u;+1

-

2u; 3

(Ax)2 I

+ t4-1

(6.17) - 2u;

(Ax)’

For this scheme, we have

a2A2 sin2 k Ax

+ 4vz sin’

1 - 4vz sin2 and IrI

-

1 - (a’l’

~

~

2

2

+ vz)(k Ax)’.

(6.18) (6.19)

Here the ratio of the numerical dissipation to the physical dissipation is a2A2/vz = aA Re,.

(6.20)

In other words, it is in fact a bit worse than even the Lax-Friedrichs scheme. Hence the method is of little value for transient flow calculations. What is rather amazing, as we shall see in the next section, is that this is a very good scheme for calculating the steady state as a long-time limit of a transient flow.

E. STEADY-STATE CALCULATION We briefly discuss the steady-state solution obtained as the long-time asymptotic limit of time-dependent solutions. We shall not consider in this paper methods specifically designed for calculating the steady state by other techniques. The most surprising feature is that the orders of the dissipation, dispersion, and accuracy of a time-dependent scheme do not immediately ensure the same orders for the steady-state solution. Thus, a superior timedependent scheme may give rather poor steady-state results, while an inferior time-dependent scheme may in fact do better. A separate investigation is needed before any such conclusions can be made. We shall continue with the several examples given in the previous section. We saw that for time-dependent solutions the leapfrog and Lax-Wendroff schemes (plus explicit or implicit or Crank-Nicholson diffusion term) were

326

C. K. Chu

comparable in accuracy, while the Brailovskaya scheme was decidedly inferior. Now we shall see that for the steady state, the leapfrog and the Brailovskaya schemes are comparable in accuracy, while the Lax-Wendroff scheme is poorer, unless it is properly retailored. But since the Brailovskaya scheme is highly dissipative in time-dependent calculations, it is in fact an efficient method of obtaining the steady state, as it will converge faster than the less dissipative schemes. To look at the steady-state behavior, we should of course introduce boundary considerations. As this is difficult to handle, a simpler device is to consider an inhomogeneous forcing term F in the equation au at

au ax

a2u +F, ax2

-+U--=V-

(6.21)

in which the steady-state solution satisfies

au

aZu

--v-= ax ax2

F.

(6.22)

Fourier analyzing as before, with u = iieikx,F = peikx,etc., we have

ii = F/D, where

D = vk’

+ iak

(6.23) (6.24)

for the exact solution. At steady state, u“” = u“, the leapfrog convection term plus (whether explicit or implicit) the diffusion term gives 4v D = -(Ax)’

-

k Ax

a +i sin k Ax Ax [vk’ + O(k Ax)’] + i[ak + O(k Ax)’], 2

(6.25)

with second-order error in k Ax in both the real and imaginary parts. In fact, a fully implicit space-centered scheme for the convection operator will give the same result. For the Lax-Wendroff scheme for the convection operator, if we proceed in a straightforward manner by just adding F to the right-hand side of the equation as in (6.21), we end up with 4 a’ At kAx a D = -( v +T) sin’ - + i - sin k Ax (Ax)’ 2 Ax

- ( “p’) v+-

(k’

+ O(k Ax)’) + i(ak + O(k Ax)’).

(6.26)

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327

A first-order error has now crept into the real part! This defect, however, can be removed, if we add to the difference equation not just F on the right side, but a correction term proportional to At. Thus

4+l- ujn + a At

= Fj

$+1

u ; + ~+ti-! - 2uj”

- $-1

2 Ax

(AxY

+ a2 At Fj+ 12 -AxFj-

1

(6.27)

(v+v)

Now at steady state, we have

(k’ + O ( k Ax)’)

F D z T

N

ak At l+i2

U

-

(vk’

+ i(ak + O(k Ax)’)

+ O(k Ax)’) + i(ak + O(k Ax)’).

(6.28)

These correction procedures, however, may be very troublesome to apply in real calculations, where there is no forcing function F but there are boundary conditions. And finally, for the Brailovskaya scheme, we can proceed in a straightforward manner by adding F to each equation, without adding a correction term. At steady state, u”+l = u“, but neither is equal to ii. By a simple calculation

a2L2 sin’ k Ax

+ 4 v t sin’

D=

-

(vk’

k Ax 2 1 + ial sin k Ax ~

+ O ( k Ax)’) + i(ak + O ( k Ax)’).

1 - 4vr sin’

~

2 (6.29)

Hence, the steady-state results of this method has second-order accuracy in both the real and imaginary parts, and the excessive dissipation in the timedependent calculation has disappeared in the steady state.

VII. Magnetohydrodynamics Even though magnetohydrodynamics may be considered as a rather special topic to fluid dynamics, we have included a short section at the end of this paper for two reasons. First, it has been the field of research of this author for the past several years, so that he has a particular perspective of

328

C . K . Chu

the subject, and second, there are some very exciting applications of numerical calculations to problems of enormous practical importance in fusion research, using rather ingenious techniques. In line with the spirit of this paper, we include for discussion only time-dependent calculations. The late 1950s saw a large group of fluid dynamicists engaging in research in magnetohydrodynamics, but the activity soon faded on the ground that little application could be found for the theory. In the meantime, fusion physicists were concentrating on kinetic theory. In the past decade, however, with the construction of large toroidal experiments (on tokamaks, pinches, etc.), magnetohydrodynamics in toroidal systems became a main tool of research. Kinetic theory is now used mainly to supply coefficients and transport data to magnetohydrodynamics. Currently, there are three or four main directions of computational magnetohydrodynamics, related to toroidal experiments, which are highly active. They are one- and two-dimensional transport, stability, and initialvalue or implosion simulations. One-dimensional transport calculations (see, e.g., Hogan, 1976) are characterized by pressure equilibrium and time-dependent equations for current, energy, and mass. A vast amount of physics is included, and it is the main tool for comparison with experiments and diagnostics used by experimental groups. Two-dimensional transport is quite subtle (see, e.g., Grad et al., 1975). The plasma is assumed to be in pressure equilibrium, i.e., no accelerations, but velocities appear due to diffusion. The resulting differential equations become completely nonstandard in form, and appear as a hybrid between a partial differential equation and an ordinary differential equation. Both the mathematical theory and the numerical analysis are interesting in their own right. Stability calculations using the initial-value problem approach have been carried out by Wesson and Sykes (1974) and by Bateman et al. (1974). The method consists of taking a plasma in equilibrium, perturbing it, and solving the linearized equations until the most unstable mode dominates, thereby generating an eigenfunction for the most unstable mode. The numerical procedure is necessarily highly nondissipative. Typically, leapfrog schemes are used. Implosion calculations are full dynamic simulations of a toroidal plasma from rest to equilibrium (Lui and Chu, 1975, 1976). The purpose of such calculations is to simulate an experiment and to interpret the dynamic phase of the plasma behavior. The plasma is quite far from equilibrium, thus the accuracy used is less than that required for stability calculations. Nevertheless, very good agreement has been obtained between computed results and several experiments.

Numerical Methods in Fluid Dynamics

329

ACKNOWLEDGMENTS Over the years, the author has benefited from many highly valuable discussions with his friends and colleagues. He feels particularly grateful to K. 0. Friedrichs, H. 0. Kreiss (with whom the author is preparing a forthcoming monograph on the subject), P. D. Lax, K. W. Morton, K. V. Roberts, and F. H. Harlow. He has also benefited much from his students at Columbia, notably, H. C. Lui, M. F. Reusch, A. Sereny, W. Park, N. Sharky, S. H. Schneider, and A. M.M. Todd. The writing of this paper has been supported by the U.S. E.R.D.A. under Contract EY-76-S-02-2456.A large part of the writing was completed while the author was visiting Science Applications, Inc., and he acknowledges with pleasure the hospitality of Drs. N. Byrne and N. Krall. BIBLIOGRAPHY S., and ROTENBERG, M., eds. (1962) “Methods in Computational PhysALDER,B., FERNBACH, ics,” Vol. 2, “Fundamental Methods in Hydrodynamics.” Academic Press, New York. S., and ROTENBERG, M., eds. (1963). “Methods in Computational PhysALDER, B., FERNBACH, ics,” Vol. 3, “Applications in Hydrodynamics.” Academic Press, New York. S., and ROTENBERG, M., eds. (1976) “Methods in Computational PhysALDER,B., FERNBACH, ics,” Vol. 16, “Physics of Fusion.” Academic Press, New York. CHU,C. K., eds. (1968) “Computational Fluid Dynamics,” AIAA Reprint Series, Vol. 4. Am. Inst. Aeronaut. Astronaut., New York. F. H., ed. (1973). “Computer Fluid Dynamics-Recent Advances,” AIAA Reprint HARLOW, Series, Vol. 15. Am. Inst. Aeronaut. Astronaut., New York. ROACHE,P. (1974). “Computational Fluid Dynamics.” Hermosa Press, Albuquerque, New Mexico. (Text with very complete bibliography.)

REFERENCES G., ScHNtmER,W., and GROSSMANN, W. (1974).MHD instabilities as an initial value BATEMAN, problem. Nucl. Fusion 14, 669. BORIS,J. P., and BOOK,D. (1973). Flux corrected transport I-Shasta. A fluid transport algorithm that works. J. Comput. Phys. 11,38. BRACKBILL, J. U. (1976). Numerical magnetohydrodynamics for high-beta plasmas. In “Computer Applications to Controlled Fusion Research” (J. Killeen, vol. ed.), Methods in Computational Physics, Vol. 16, p. 1. Academic Press,New York. BRAILOVSKAYA, I. Y. (1965).A difference scheme for numerical solution of the two-dimensional nonstationary Navier-Stokes equations for a compressible gas. Sou. Phys.-Dokl. 10, 107. BURSTEIN, S. Z., and MIRIN,A. A. (1970).Third-order difference methods for hyperbolic equations. J. Comput. P hys. 5, 547. CHENG,S. I. (1969). Accuracy of difference formulation of Navier-Stokes equations. Phys. Fluids 12, Suppl. 11, p. 34. CHORIN,A. J. (1968). Numerical solution of the Navier-Stokes equations. Math. Comput. 22, 745. CHORIN,A. J. (1972).A vortex method for the study of rapid flow. Proc. 3rd Int. ConJ Numer. Methods Fluid Mech. Springer-Verlag Lect. Notes Phys., No. 19, Vol. 11, p. 100. CHU, C. K., and SERENY,A. (1974). Boundary conditions in finite difference fluid dynamic codes. J. Cornput. Phys. 15, 476.

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COURANT, R., and FRIEDRICHS, K. 0. (1948). “Supersonic Flow and Shock Waves.” Wiley (Interscience), New York. COURANT, R., FRIEDRICHS, K. O., and LEWY,H. (1928) Ober die partielle differenzengleichungen der mathematische physik. Math. Ann. 100, 32. DOUGLAS, J. (1955). Oh the numerical integration of u, = u,, by implicit methods. J. SOC. Ind. Appl. Math. 3.42. FROMM, J. E. (1964). The time dependent flow of an incompressible fluid. In “Fundamental Methods in Hydrodynamics” (B. Alder, S. Fernbach, and M. Rotenberg, eds.), Methods in Computational Physics, Vol. 3, p. 345. Academic Press, New York. GODUNOV,S. K., and RYABENKII, V. S. (1964). “Introduction to the Theory of Difference Schemes.” Wiley (Interscience), New York. GRAD,H., Hu, P. N., and STEVENS, D. P. (1975). Adiabatic evolution of plasma equilibrium. Proc. Nat. Acad. Sci. U S A 72, 3789. GUSTAFSSON, B., KREISS,H. O., and SUNDSTROM, A. (1972). Stability theory of difference approximations for mixed initial-boundary-value problems. Math. Comput. 26, 649. HARLOW, F. H. (1964). The particle-in-cell computing method in fluid dynamics. In “Fundamental Methods in Hydrodynamics” (B. Alder, s. Fernbach, and M. Rotenberg, eds.), Methods in Computational Physics, Vol. 3, p. 319. Academic Press, New York. HARLOW, F. H., and AMSDEN,A. A. (1971). A numerical fluid dynamics calculation method for all speeds. J. Comput. Phys. 8, 197. HARLOW, F. H., and WELCH,J. E. (1965) Numerical calculation of time-dependent incompressible flow of a fluid with free surface. Phys. Fluids 8, 2182. HIRT,C. W. (1968). Heuristic stability theory for finite difference equations. J . Comput. Phys. 2, 339. HIRT,C. W. (1971). Arbitrary Lagrangian-Eulerian computing technique. Proc. 2nd Int. C o n t Numer. Methods Fluid Dyn. Springer-Verlag Lect. Notes Phys., Vol. 8, p. 350. HIRT,C. W., and AMSDEN, A. A. (1972). “Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds,” Rep. LA-DC-72-1287. Los Alamos Sci. Lab., Los Alamos, New Mexico. HOGAN,J. (1976). Multifluid tokamak transport models. In “Computer Applications to Controlled Fusion Research” (John Killeen, vol. ed.), Methods in Computational Physics, Vol. 16, p. 131. Academic Press, New York. KREISS,H. 0. (1964). On difference approximations of the dissipative type for hyperbolic differential equations. Commun. Pure Appl. Math. 17, 335. KREISS, H. 0. (1977). “Problems with Different Time Scales for Differential Equations,” Report No. 68. Uppsala Univ. Dept. of Computer Sci. KREISS,H. O., and OLIGER,J. (1973). “Methods for the Approximate Solution of TimeDependent Problems,” CARP Publ. No. 10. World Meteorol. Organ., Geneva. 0.(1963). “Mathematical Theory of Viscous Flows.” Gordon & Breach, New LADYZHENSKAYA, York. LAX,P. D. (1954). Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159. LAX,P. D., and WENDROFF, B. (1960).Systems ofconservation laws. Commun.Pure Appl. Math. 13, 217. LUI, H. C., and CHU, C. K. (1975). Two-dimensional magnetohydrodynamic simulation of toroidal pinches. Phys. Fluids 18, 1277. LUI, H. C., and CHU,C. K. (1976). Two-dimensional magnetohydrodynamic simulation of toroidal pinches 11: belt pinches. Phys. Fluids 19, 1947. MACCORMACK, R. W. (1969). The effect of viscosity in hypervelocity impact cratering. A I A A , New York Pap. 69-354.

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MACCORMACK, R. W. (1976). An efficient numerical method for solving time-dependent compressible Navier-Stokes equations at high Reynolds numbers. N A S A Tech. Memo NASA

TM X-73, 129. MORETIT, G. (1969). Importance of boundary conditions in the numerical treatment of hyperbolic equations. Phys. Fluids 12, Suppl. 11, p. 13. PEACEMAN, D. W., and RACHFORD,H. H., JR. (1955). The numerical solution of parabolic and elliptic differential equations. J. SOC.Ind. Appl. Math. 3, 28. RICHTMYER, R. D. (1962). A survey of difference methods for nonsteady fluid dynamics. Natl. Cent. Atmos. Res. Tech. Note 63-2. RICHTMYER, R. D., and MORTON,K. W. (1967). “Difference Methods for Initial-Value Problems.” Wiley (Interscience), New York. K. V., and WEISS,N. 0. (1966). Convective difference schemes. Math. Comp. 20, 272. ROBERTS, RUSANOV, V. V. (1970). On difference schemes of third-order accuracy for nonlinear hyperbolic systems. J. Comput. Phys. 5, 505. SOOD,D. R., and ELROD,H. E. (1974). On the flow between two long eccentric cylinders. J . Am. Inst. Aero. Astro. 12, 636. SUNDSTROM, A. (1975). Notes on the paper “Boundary conditions in finite difference fluid dynamic codes.” J. Comput. Phys. 17, 450. THOM,A. (1933). The flow past circular cylinders at low speeds. Proc. R. SOC.London, Ser. A 141, 651. VON NEUMANN, J., and RICHTMYER, R. D. (1950). A method for the numerical calculation of hydrodynamical shocks. J . Appl. Phys. 21,232. WARMING, R. F., and HYETT,B. J. (1974). The modified equation approach to the stability and accuracy analysis of finite difference methods. J. Comput. Phys. 14, 159. A. (1974). Two-dimensional calculation of tokamak stability. Nucl. Fus. WESON,J., and SYKES, 14, 645. YANENKO, N. N. (1971). “The Method of Fractional Steps.” Springer-Verlag. Berlin and New York. YANENKO,N. N., and SHOKIN, Y. I. (1969). First differential approximation method and approximate viscosity of difference schemes. Phys. Fluids 12, Suppl. 11, p. 28.

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Author Index Numbers in italic refer to the pages on which the complete references are listed.

A

Coleman, B. D.. 126, 174 Cockrell, J. J . , 124, 174 Chadwjck, P., 17, 22, 72 Chan, S. K . , 114, 115, 120 Chang, M.-S., 235, 281 Chapman, R. B., 231. 234, 281 Cheng, S. I.. 325, 329 Choo, K. Y., 272, 281 Chorin, A. J., 319, 320, 321, 329 Chou, P.-Y., 125, 174 Chu, C. K., 308, 328, 329,329. 330 Chu, T. Y., 112, I20 Cornte-Bellof, G., 140, 154, 174 Corrsin, S.. 140, 154, 158, 174 Courant, R., 289, 295. 330 Crow, S. C., 150, 174 Curnrnins, W. E . , 224, 229, 281

Abramowitz, M., 246, 280 Ackerberg, R. C., 204, 219 Alder, B., 329 Alexopoulos, C. D., 157, 174 Amsden. A. A., 314, 315, 330 Antonia, R. A,, 173, 174 Ayyaswarny, P. S., 116, 118, 119 B Baba, E.. 242, 280 Bai, K. J., 245. 280 Baternan, G., 328, 329 Beck, R. F., 235, 280 Biott, M. A.. 14, 20, 60, 72 Bishop, E. E. D., 223, 224,280, 282 Blatz, P. J., 48, 72 Blottner, F. G . , 193, 198, 218 Bolotin, V. V . , 5 5 , 72 Bolton, W. E., 272, 280 Book, D., 312,329 Boris, J. P., 312, 329 Brackbill, J . U.. 314, 329 Brailovskaya. I . Y., 325, 329 Brard, R., 224, 246. 280 Brown, S. N., 202, 218 Buckmaster, J., 181. 219 Budiansky, B., 51, 72 Burcher, R. K., 224, 280 Burstein, S. Z . , 312, 329 Busse, F. H., 79, 86, 88, 90, 94, 95, 97, 98, 104, 105, 106, 108, 110, 1 1 1 , 112, 115, 119, 119, 120

D Daly, B. J . , 161. 174 Davidov, B. I . , 125, 174 Davis, R. T., 208. 220 Deardorff, J. W., 112, 118,120, 121. 168.174 Dobbinga, E.. 179, 219 Douglas, J., 300, 330

E Eckelmann, H.. 106, I20 Elrod, H. E., 319, 331 Enslow, R. L., 179, 219 Ericksen, J . L., 49, 72

C

Carmi, S., 108, I20 Catherall, D., 178, 219

F Faltinsen, O., 232, 267, 277, 280, 281, 283 Fernach, S., 329 333

334

Author Index

Fitzjarrald, D. E., 112, 120 Frenkiel, F. N., 127, 162, 174 Friedrichs, K. 0.. 289, 295, 330 Fromm. J. E., 318, 330 Froude, W., 228,281

I Ilyushin, A. A,, 38, 73

J G Gence. J. N . , 144, 149, 174 Gerritsma, J., 229, 232. 281 Goldstein, R. J., 112, I20 Goldstein, S., 178, 189, 201, 202, 208, 219 Godunov, S. K., 308,330 Gortler, H., 207,219 Grad, H., 328, 330 Grim. 0.. 238,281 Grossmann, W., 328,329 Gupta, A. H., 107, 120 Gupta, V. P., 114, 120 Gustafsson, B., 308, 330

H Hanjalic, K., 143, 161, 174 Hara, M., 242, 280 Harlow, F. H.. 161, 174. 314, 315, 319,329, 330 Haskind, M. D., 228, 246, 281 Havelock, T. H., 228, 242, 281 Havner, K. S., 47, 72 Hayasi, N., 193, 219 Herring, J . R., 114, 158, f20, 175 Hibbitt, H. D., 65, 72 Hill, R., 14, 16, 19, 20, 21,22, 23,24,27,28, 30,31,33,34,35,38,39.40,41,42,43,44,

Jacobs, W. R., 232, 281 Johnson, W. D., 181, 193, 194, 195, 197, 199, 220 Jones, C. W., 189, 219 Joseph, D. D., 80, 84, 86, 88, 90, 94, 102, 108, 114, I20

K Kaplan, R. E., 107, 120 Keffer, J. F., 157, 174 Keller, J. B., 242, 281 Kennett, R. G., 117, 120 Kerwin, J. E., 229, 281 Khajeh-Nouri, B., 152, 160, 175 Klebanoff, P. S., 127, 162, 174 Kline, S. J., 107, 120, 124, 174 KO, W. L., 48, 72 Kolmogorov, A. N., 125, 134, 174 Kooi, J. W., 179, 219 Korvin-Kroukovsky, B. V., 231, 232, 280, 281 Kraichnah, R. H., 113, 120 Kreiss, H. O., 291, 294, 306, 308, 312, 317, 330 Kriloff, A., 228, 281 Krishnamurti, R., 112, 121

46,47.50,51,52,55,58,60,61,62,63,64,

65, 72, 72, 73, 74 Hinze, J. 0.. 109, 120 Hirt, C. W., 305, 314,330 Hogan, J., 328, 330 Horton, H. P., 178, 219 Howard, L. N., 79, 80, 82, 83, 98, 110, 111, 120 Howartn, L., 193, 219 Hu, P. N., 328,330 Hunt, G. W., 51, 75 Hunter, C., 115, I20 Hutchinson, J. W., 51, 60,65, 73 Hyett, B. J., 304, 306, 331

L Ladyzhenskaya, 0.. 292,330 Laitone, E. V., 237, 245, 246, 283 Landweber, L., 257, 281 Laufer, J., 106, 107, 109, 119, 120, I21 Launder, B. E., 135, 143. 150, 151, 157, 158, 161, 174, 175 Lax, P. D., 310, 330 Leslie, D. C., 125, 175 Lewis, F. M., 228, 281 Lewellen, W. S., 161, 175

Author Index Lewy, H., 295, 330 Libby, P. A., 173, 175 Lighthill, M. J., 246, 281 Lindberg, W. R., 115, 121 Ludwieg, H., 104, 121 Ludwig, G. R.. 180,219 Lui, H. C., 328,330 Lumley, J. L., 125, 128, 129, 130, 131, 133, 134, 135, 137, 138, 140, 141, 142, 143, 144, 146, 147, 148, 149, 150, 152, 153. 154, 155, 156, 157, 158, 160, 161, 162, 163, 165, 168, 172, 173, 175, 176

M MacCormack, R. W., 312, 317, 331 Malkus, W. V. R., 79, 1 1 1 , 121 Mangler, K. W., 178, 219 Marcial, P. V., 65, 72 Marechal, J., 127, 162, 175 Maruo, H., 229, 266, 267, 277, 282 Mays, J. H., 266, 277, 282 McCreight, W. R., 279, 282 McMeeking, R. M., 65, 73 Mei, C. C., 222, 231, 250, 282 Messiter, A. F., 178, 179, 219 Miles, J. P., 65, 71, 72, 73. 74 Milstein, F., 31, 33, 72, 73, 74 Mirin, A. A., 312,329 Mitchell, J . H., 228, 282 Monin, A. S., 125, 134, 153, 175 Moore, D. R., 114, 121 Moore, F. K., 180, 219 Moretti, G., 310, 331 Morkovin, M. J., 124, 174 Morton, K. W.. 294, 295, 297,331

335

Newman, J. N., 229,230,233,234,240,256, 258, 264,265, 266,272,273,274,276,277, 278, 280, 281. 282. 283 Nickerson, E. C., 105, I21 Nikuradse, J., 109, I21 NOH, W., 31, 75, 126, 174

0

Oakley, 0. H. Jr., 227, 282 Ogden, R. W.. 17, 20, 22, 48, 72, 74 Ogilvie, T. F., 222, 224, 229, 233, 234, 235, 241,244,258, 263,264, 265,267, 275,277, 279, 282 Oliger, J., 291, 294, 308, 312, 330 Om. W. McF., 84, I21 Orzag, S. A,, 125, 175

P Panofsky, H. A,, 143, 175 Parks, D. M., 65, 74 Parry, G. P., 50, 74 Parsapour, H., 113, I21 Paulling, J. R., 227, 282 Patterson, G. S., 125, 175 Peaceman, D. W., 300, 331 Pearson, C. E., 57, 64, 71 Pellew, A,, 104, 121 Peters, A. S., 229, 282 Phillips, J . H., 204, 219 Pierson. W. J., 223, 282 Pope, S. B., 153, 155, 175 Price, W. G., 223, 224, 280, 282 Proudman, R. I.. 144, 174

N

R

Nagiv, H. M., 104, 121 Nagtegaal, J. C., 65, 74 Needleman, A,, 65. 74 Nemat-Nasser, S., 5 5 , 74 Nenni, J. P., 182, 186, 187, 188, 190, 191, 192,219 Newman, G. R., 125, 131, 133, 135, 137, 138, 140, 141. 142, 154, 157, 158, 175

Rachford, H. H. Jr., 300, 331 Reece, G. J., 161, 175 Reichardt, H., 99, 101, 121 Reynolds, D., 84, I21 Reynolds, W. C., 107, 120. 148, 149, 175 Riahi, N., 115, 120, 121 Richtmyer, R. D., 294, 295, 297, 310, 312, 33I

Author Index

336

Rice. J. R., 17, 21, 35, 38, 44, 46, 65, 72.73 , 74, 75

Rivlin, R. S., 153, 176 Roache, P., 329 Roberts, K . V., 303, 331 Rodi, W., 161, 175 Rotenberg, M., 329 Rotta, J . C., 125, 150, 176 Rusanov, V. V., 312, 331 Runstadler, P. W., 107, 120 Ryabenkil, V. S., 308, 330

S Salvesen, N., 232, 277, 280, 283 Sasaki, N., 267, 282 Saunders. 0. A , , 111, I21 Schmidt, R. J., 111, I21 Schneider, W., 328, 329 Schraub, F. A , , 107, I20 Schumann, U., 131. 176 Sears, W. R., 180. 181, 200, 203, 210, 219 Sereny. A , , 308, 329 Sewell, M. J., 44, 46, 51, 57, 65, 73. 75 Shen. S. F., 181. 182, 186, 187, 191, 195, 197, 198, 199, 205, 213, 216, 217,219. 220 Shokin, Y. I . , 306, 331 Siess, J., 162. 173, 175 Sood, D. R., 319.331 Southwell, R. V., 104, I2I Sovran, G., 124, 174 Spalding, D. B., 156, 176 Spencer, A. J. M..153, 176 Squire, H. B., 102, 121 Sreenivasan, K. R., 173, 174 St. Denis, M., 222, 223, 282. 283 Stegun, I . , 246, 280 Stevens, D. P., 328, 330 Stewartson, K., 178, 179, 219 Stocker, J . J., 229, 282 Storikers, B., 65, 75 Storen. S., 21, 75 S t r a w J. M.. 1 1 1 , 112, 114, 1 1 5 , 121 Sundstrom, A., 308, 330, 331 Sychev, V. Y., 179,219 Sykes. A., 328,331

T Tani, I., 198, 219 Tao, L. N., 102, 120

Telionis. D. P., 180, 181, 200, 203, 204. 207, 209, 210, 215,219, 220 Tennekes, H., 134, 153, 155, 156, 160, 165, 172, 176 Thorn, A., 318,331 Thompson, J . M. T., 51, 75 Timman, R., 240, 274, 276, 283 Tokura, J., 277, 282 Tollmien, W., 181, 182, 220 Toupin, R., 31, 75 Townsend, A. A., 144, 176 Treloar, L. R. G., 47, 75 Troesch, A. W., 272, 280, 283 Truesdell, C., 31, 75 Tsahalis, D. T., 204, 207, 209, 215, 220 Tuck, E. 0.. 229, 232, 233, 234. 235, 241, 256, 258, 264, 265.275,277,279,280,280. 282, 283 Tvergaard, V., 65, 74. 75

U Ursell, F., 229, 23 1 , 247, 250, 25 1 , 253, 267, 272, 280, 283 V Van Dornmelen, L. L., 213, 216, 217, 20 Van Dyke, M.,126, 176 Van Ingen, J. L., 179, 219 Veldmann, A. E. P., 179, 220 Von Neumann, J., 310, 331 Vossers, G., 233, 283 Vugts, J. H., 231, 232, 250, 283

W Wang, J. C. T., 195, 197, 198, 199, 205,220 Warhaft, 2.. 134, 135, 142, 156, 157, 158, 175, 176

Warming, R. F., 304, 306.331 Wehausen, J. V., 222, 224, 231, 237, 245, 246, 250, 252, 273, 283 Weinblum, G. P., 222, 283 Weiss, N . O., 114, 121, 303, 331 Welch, J. E., 319, 330 Wendroff, B., 310, 330 Werle, M. J., 204, 207, 208, 209,220 Wesson, J., 328, 331 Whitehead, J. A., 112, 120

Author Index Williams, J. C., 181, 193, 194, 195, 197, 199, 220 Willis, G . E . , 112, 120. I21 Wood, P. D., 227,282 Wyngaard, J. C., 146, 176

Y Yaglom, A. M . , 125, 134, 153, 175 Yanenko, N . N . , 300, 306,331

337

Yeung, R. W., 245, 280 Yih, C. S . , 257, 281 Young, N. J. B., 60,75

Z

Zeman, O., 141, 143, 146, 147, 154, 156, 157, 158, 162, 163, 165, 172, 173, 175. 176

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Subject Index A

Absolute convexity, nonlinear bifurcation and, 63 Actual state of stress, loading environment and, 32 Added mass coefficient exciting force and, 280 pressure and, 275-277 in radiation problems, 251-252 for steady-state oscillatory motion, 224 strip theory and, 235 Adverse pressure gradient, boundary layer and, 178-179 ALE program, see Arbitrary LagrangianEulerian program Almansi scale function, transformation formulae and, 25 Alternating direction methods, in fluid dynamics, 297-300 see also Fractional step methods Anisotropic turbulence, return to isotropy and, 133-143 Anisotropy distortion and, 144 mechanical dissipation and, 154-155 return to isotropy and, 133-143 Reynolds stress and, 137-143 in second-order modeling, 129 Anisotropy tensor, defined, 137 Arbitrary Lagrangian-Eulerian program, in fluid dynamics, 314 Axisymmetric turbulence mechanical dissipation and, 155 heat flux in, 143 Reynolds stress and, 139-142

B Beam seas, diffraction problem in, 266-267

Bifurcation defined, 51 deformation and, 2 first-order rate problem and, 50-58 under fluid pressure, 70-72 invariance and, 50-72 linear case, 58-61 nonlinear case, 61-65 primary, 58-66 under simple loading, 66-72 Boundary condition diffraction problem in, 252-253 in fluid dynamics, 287-292 for incompressible flow, 291-292 Green’s theorem and, 254 long-wavelength approximation, 256-257 radiation problems, 250-252 slender-body diffraction, 266-273 Boundary layer, unmatchable, 186-192 Boundary layer equations asymptotic behavior of, 182-186 two-dimensional, 2 14-21 5 unsteady separation in, 177-218 Boundary layer theory heat transport and, 110- 113 turbulence theory and, 78, 84-86, 89-93 turbulent channel flow and, 105-108 turbulent pipe flow and, 108-110 Boundary layer velocity, variation of, 195 Boundary-value problem elastoplasticity and, 35-47 field equations for, 51-58 in ship motion theory, 235-244 Boussinesq equations, heat transport and, 110, 114 Brailovskaya scheme, Navier-Stokes equations and, 325-327 Bravais lattice, elastic response in, 49 Broaching, defined, 227 Buoyancy effect of. 172-173

339

340

Subject Index

Convected derivatives, concept of, 6-8 Convection see also Thermal convection in layer heated from below, 110- I13 in porous medium, 84-86 Convergence, numerical analysis of, 293295 Convergence rate, defined, 293 Convexity C absolute, 34 constitutive inequalities and, 31-35 Cauchy elasticity, normality rule and, 39-41 nonlinear bifurcation and, 62-63 Cauchy stress relative, 34 elastic response and, 48 Coordinate invariance nominal stress and, 22 normality and, 39, 44-47 orthotropic symmetry and, 26-27 relative convexity and, 34 stress measures and, 17-23 Coriolis force, in a rotating system, 102-104 Cauchy symmetry, elastic response, SO Couette flow Cayley-Hamilton identity disturbances to, 128 scale functions and, 15 energy production term, 115-1 16 second-order modeling and, 129 Navier-Stokes equations and, I18 Closure, turbulence behavior and, 125- I26 optimum problems for, 80-84 Compatibility relation, in steady separation, turbulent channel flow and, Courant-Friedrichs-Lewey condition, up188- 189 Component energies, negative, 131-133 wind differencing scheme and, 295 Compressible flow Covariant bases, decomposition and, 3-4 differential equations for, 287-288 Crank-Nicolson scheme Navier-Stokes equations for, 317-327 for initial-value problems, 297, 300 Computer Navier-Stokes equations for, 323-324 effect on fluid dynamics, 286-328 stability and, 306-307 effect on scientific research, 286 for wave filtering, 316 Conjugacy, stress and strain measures of, Crystal, monatomic, 49-50 19-24 Crystal lattice theory see also Work conjugacy elastic response, 49-50 Conservation-law form scale functions and, 14 of differential equations, 310-31 1 Cubic lattices, boundary point and, 33 of Navier-Stokes equations, 322 Current configuration Conservative schemes, dissipative schemes defined, and, 302-304 inequality and, 33-34 Constitutive description loading environment and, 31-32 bifurcation and, SO-72 rates of strain and, 16 invariance and, 28-49 transformation formulae and, 24-25 Constitutive inequalities, convexity and, 31-35 Contact discontinuities, numerical methods in 309-313 D Continuum configuration, types of, 9 Continuum response, deformation and, 2, Damping coefficient exciting force and, 280 9- 10 Contravariant basis, decomposition and, pressure and, 275-277 3-4 in radiation problems, 251-252

in mechanical dissipation, 153-154 in rapid distortion, 148-149 scale ratio and, 157 thermal dissipation and, 157 Burgers equation, Navier-Stokes equations and, 322-323

Subject Index resonant response and, 226 for steady-state oscillatory motion, 224 strip theory and, 235 Darcy-Boussinesq equations, convection and, 85 Darcy permeability coefficient convection and, 85 heat transport and, 110 Dead load, bifurcation under, 66-70 Decaying anisotropic turbulence see also Return to isotropy mechanical dissipation and, 154-155 process of, 133-143 thermal dissipation and, 156- 158 Decomposition principal fibers and, 10-1 1 on reciprocal bases, 3-4 stretch tensor and, 11-12 Deformation of area, 10 bifurcation theory and, 50-72 change of basis and, 4-5 convected derivatives and, 6-8 defined, 51 elastoplastic, 35-47 geometry of, 9-14 induced tensors, 5-6 orthotropic symmetry and, 26-27 rates of strain and, 16-17 reciprocal bases and, 3-4 spin, see Material spin stability tests and, 31-35 tensor of moduli and, 24-27 Deformation gradient defined, 9 deformation geometry and, 9-10 stretch tensor and, 12 transformation formulae and, 25 Deformation-rate tensor, defined, 6 Deformation-sensitive loading exclusion functional and, 60 surface data on, 54-57 Differential equations boundary conditions and, 287-292 for compressible flow, 287-288 conservative-law forms of, 3 10-3 1 1 in fluid dynamics, 287-292 for incompressible flow, 291-292 Differential invariants, measure invariance and, 30-31

34 1

Diffraction potential exciting force and, 277-278 Green’s theorem and, 254-255 three-dimensional, 249 Diffraction problem defined, 223 description of, 225 in strip theory, 234 in two-dimensional bodies, 252-253 Dipole in outer problem, 258-259 in ship motion theory, 244-245 Dipole moment Green’s theorem and, 245, 256 inner problem and, 261 long-wavelength approximation and, 256-257 of outer solution, 262 Discontinuities, numerical methods in, 309-313 Dispersion steady-state calculation and, 325 of waves, 301 Displacement thickness growth in separation, 178 in semisimilar boundary layer, 195 in steady separation, 187-188 in unsteady separation, 189-192 Dissipation physical, 322-325 steady-state calculation and, 325 of waves, 301 Dissipation equations, in second-order modeling, 152-159 Domain boundary, inequality and, 33-34 Doppler shift frequency of encounter and, 225, 226 in head seas, 231 Downstream penetration distance, boundary layer equations and, 181 Dufort-Frankel scheme, for vorticity, 318

E Eigenmode boundary point and, 33-34 in dead loading, 67-70 under fluid pressure, 70-72

342

Subject Index

linear response and, 59 rate problems and, 58 Eigenstate in dead loading, 66-70 exclusion functional and,60 linear response and, 58-60 nonlinear response and, 61 rate problems and, 58 Elasticity bifurcation and, 64 material stability and, 31-35 measure invariance and, 28 Elastic response, types of, 47-50 Elastic stiffness, normality and, 46 Elastomer, inequality and, 33 Elastoplasticity bifurcation and, 34, 52 comparison materials for, 64-65 convexity and, 34 invariance and, 35-47 of metal crystals and polycrystals, 2 phenomenological framework for, 35-38 Embedded bases deformation of, 3-5, 9 kinematics of, 6 measure invariance and, 28 scale functions and, 14-16 Energy balance for statistically stationary turbulence, 115 turbulence theory applications, I15 for turbulent channel flow, 105 Energy density, stability and, 32-33 Energy-stability limit in convection, 110 momentum transport and, 83-84 turbulent pipe flow and, 108 Equipartition, return to isotropy and, 134 Euler equations for compressible flow, 287-288 for Couette flow, 118 for extremalizing vector field, 94-98 for heat transport, 114 for incompressible flow, 292 optimum theory of turbulence and, 79 Eulerian methods, Lagrangian methods and, 213-215, 3 13-3 15 Eulerian strain-rate defined, 6 kinematics and, 13- 14 Eulerian triad kinematics and, 12-14

orthotropic symmetry and, 26-27 principal fibers and, 11 rates of strain and, 16 stretch tensor and, 12 work conjugacy and, 19 Euler-Lagrange equations, for extremalizing solutions, 86-89 Exciting force see also Froude-Krylov exciting force expression of, 225 in head seas, 230-231 in heave, 226 in pitch, 226 pressure and, 277-280 strip theory and, 234 Exclusion functional, bifurcation and, 59-60 Explicit schemes implicit schemes and, 297 Navier-Stokes equations and, 323-324 Extremalizing solutions, general properties of, 86-89 Extremalizing vector field momentum transport and, 94-98 turbulent pipe flow and, 108-1 10 F

FCT, see Flux-corrected transport Fiber, principal, see Principal fiber Finite-difference methods for boundary value problems, 288 for general unsteady boundary layer, 203-207 for Navier-Stokes equations, 317-322 shock waves and, 310 Finite element methods for boundary value problems, 288 for initial-value problems, 293 First-order rate problems field equations for, 50-53 solution properties, 57-58 surface data for, 53-58 Flat-ship approximation, hydrodynamic disturbance and, 229 Flow reversal local, 203-207 separation and, 177 for unsteady flows, 180 Flow structure, near steady separation point, 178

343

Subject Index Fluid-dynamic problems, time-dependent, 307-309 see also Initial boundary value problem Fluid dynamics numerical methods in, 285-328 optimal theory of turbulance and, 77-1 19 Fluid flow, turbulent, 77-1 19 Fluid mechanics, numerical methods in, 285-328 Fluid pressure bifurcation under, 70-72 self-adjointness and, 56-57 Flux-corrected transport, shock waves and, 3 12-31 3 Follower forces, deformation-sensitive loading and, 55 Following seas defined, 225 diffraction problem in, 266-267 long-wavelength solution in, 272-273 Force, defined, 223 Fourier components numerical dispersion from, 302-304 Fourier transform in outer problem, 259 rapid terms and, 144-145 Fourth-order leapfrog schemes, for initial value problems, 297 Fractional step methods, in fluid dynamics 297-300 see also Alternating direction methods Free-stream turbulence, separation and, 179 Free-stream velocity, in weakly unsteady case, 185 Free-surface boundary coi.Ation elementary source potential in, 245 radiation problems and, 250 in slender ships, 243-244 steady flow field in, 241-242 in three-dimensional flow, 246-249 in two-dimensional flow, 245-246 Frequency of encounter strip theory and, 225, 234 wave frequency and, 226 Froude-Krylov exciting force defined, 228 long-wavelength assumption and, 230 strip theory and, 234 thin-ship approximation and, 229 Froude number slender ship approximation and, 229

strip theory and, 234-235 Fundamental singularity, in ship motion theory, 244-249 Fundamental solutions, in ship motion theory, 244-249

G Gas dynamics boundary value problems in, 289-290 Courant-Friedrichs-Lewy condition for, 315 differential equations for, 287 numerical methods for, 309-317 Gaussian distribution homogeneous turbulence and, 127 in second-order modeling, 127, 163- 165, 173 Gaussian model, of turbulence, 162-165, I73 Goldstein singularity genesis of, 215 in semisimilar boundary layer, 195 separation and, 178-180, 199-203 in steady separation, 187-188 in unsteady separation, 208 Grasshof number, turbulence theory approximation and I16 Green measure, work conjugacy and, 20 Green modul, Cauchy symmetry and, 50 Green’s scale function orthotropic symmetry and, 27 transformation formulae and, 25 Green’s theorem see also Source potential applications of, 253-256 exciting force and, 278-279 strip theory and, 228, 234 three-dimensional, 246-249 two-dimensional, 245-246 Green strain, conjugate of, 23 Haskind relations exciting force and, 278-280 Green’s theorem and, 255 Head seas defined, 225 diffraction problems in, 266-267 heave in, 226 long-wavelength solution in, 272-273 pitch in, 226 slender ships in, 244

344

Subject Index

ship movement in, 230-231 geometry of, 242-244 strip theory and, 234 outer problem and, 258-259 Heat flux pressure force on, 273-280 in mechanical dissipation, 153 radiation problems and, 250-252 return to isotropy and, 142-143 slender-body diffraction and, 266-273 Reynolds stress integral and, 150-152 slender-body radiation and, 258-266 transport terms and, 160 structural loading of, 222 vertical, 146 Hull boundary conditions Heat flux integral, construction of, 147 in slender ships, 242-244 Heat transport steady flow field in, 241-242 see also Thermal convection in unsteady motion, 241 bounds on, 110-115 Hydrodynamic disturbance convection solution of, 79 attempt to account for, 228 optimal theory of turbulence and, 79 flat-ship approximation and, 229 variational problems and, 84-86 ship hull and, 228 Heave Hydrodynamic pressure defined, 222 analysis of, 273-274 in early sailing ships, 227 in radiation problems, 251 exciting force and, 280 strip theory and, 234 historical interest in, 228 unsteady motion and, 223 linearized problem and, 240 Hydrostatic pressure long-wavelength approximations and, 256 analysis of, 273 prediction of, 227 strip theory and, 234 pressure and, 276-277 unsteady motion and, 223 radiation problems of, 250-252 Hydrostatic restoring force, effect on ships, restoring force and, 226 225-226 in slender ships, 244 Hyperbolic equations in strip theory, 232-234 for boundary value problems, 288-291 Helmholtz equation for compressible flow, 287 diffraction problem and, 252-253 Green’s theorem and, 254 strip theory and, 234 I Heuristic stability criterion, instability and, 305-306 Homogeneous data, loading and, 56 ICE scheme, see Implicit compressible Homogeneous flow, return to isotropy and, Eulerian scheme 134- 136 Ideal crystal, material stability and, 31 Homogeneous mean shear, production of, Ideal lattice, inequality and, 33 127 Ideally plastic stress, normality and, 41 Homogeneous turbulence Implicit compressible Eulerian scheme, for Gaussian behavior of, 162-165 wave filtering, 315-317 second-order modeling and, 127 Implicit differencing procedure, in fluid Homogeneous virtual deformation, testing dynamics, 314 of, 31-32 Implicit schemes Howarth problem computing of, 306-307 generalization of, 208 Navier-Stokes equations and, 323-324 unsteady flow and, 192-195 solution of, 297-300 Hugoniot relations, shock waves and, 310 for wave filtering, 315-317 Hull Implosion calculations, in magnethohydrodeeply submerged, 242 dynamics, 328

Subject Index Incident wave diffraction problem and, 252-253 slender ship diffraction and, 266-273 Incompressible flow differential equations for, 291-292 Navier-Stokes equations for, 317-327 Reynolds stress integral and, 150-152 Incremental loading, types of, 50-72 Induced tensors, concept of, 5-6 Inequality, stability and, 31-35 Inertial forces, of ship, 228 Infinite Prandtl number limit, heat transport and, 113-115 Inflow boundary, equations for, 291 Inhomogeneous data, loading and, 56 Initial boundary value problem see also Time-dependent fluid problems in magnetohydrodynamics, 328 mixed, 307-309 numerical solutions for, 288-289 pure, 293-295, 307-309 well-posed, 293 Initial conditions, for incompressible flow, 291-292 Inner problem in slender-body diffraction, 268-269 in slender-body radiation, 259-262 Inner region, in slender ship motion, 242244 Inner solution matching and, 262-265 in slender-body diffraction, 271-272 in slender-body radiation, 265-266 Invariance bifurcation theory and, 50-72 constitutive descriptions of, 28-50 normality rule and, 38-47 preliminary concepts of, 1-27 in solid mechanics, 1-72 Inversion schemes, in fluid dynamics, 297300 Inviscid flow differential equations for, 292 free streamline theory for, 179 numerical methods in, 309-317 separation and, 179 Irregular seaway, defined, 223 Irrotational flow, unsteady motion and, 223-224 Isotropic elasticity

345

constructive inequalities for, 33 orthotropic symmetry and, 27 work conjugacy and, 19-20 Isotropic tensor function, mathematical representation of, 128-130 Isotropic turbulence effect of distortion, 144 homogeneous, 78 time scale ratio for, 156-157 Isotropy see also Return to isotropy orthotropic symmetry and, 26-27 Reynolds stress and, 150-152 thermal dissipation and, 156-158

J Jacobian matrix, deformation gradient and, 9 Jaumann flux defined, 7 nominal stress and, 23 orthotropic symmetry and, 27 rates of stress and, 21

K Kinematics, deformation geometry and, 12-14 Kinematic viscosity, separation and, 179180 Kirchhoff stress defined, 18 elastic response and, 48 Eulerian triad and, 19 orthotropic symmetry and, 26-27 rates of stress and, 21 Korteweg-deVries equation, dispersive waves and. 301

L Lagrange-Dirichlet criterion, stability and, 32 Lagrangian-Eulerian methods, in fluid dynamics, 300

346

Subject Index

Lagrangian methods, Lagrangian methods and, 313-315 Lagrangian tensor, objective stress measures and, 18 Lagrangian triad bifurcation theory and, 51-53 deformation gradient and, 9 kinematics and, 12-14 orthotropic symmetry and, 26-27 principal fibers and, I 1 rates of strain and, 16 scale functions and, 14 separation and, 213-218 stretch tensor and, 1 1 tensor representation and, 2 work conjugacy and, 19-21 Laminar flow momentum transport in, 80-84 Reynolds number for, 127-128 Laplace equation diffraction problem and, 252 inner problem and, 262 radiation problems of, 250 in slender-body radiation, 258 two-dimensional, 249-250 Lax equivalence theorem, defined, 294 Lax-Friedrichs scheme for initial-value problems, 296-297 modified, 304-305 Navier-Stokes equations and, 323-325 numerical dispersion and, 302-304 numerical dissipation and, 302-340 Lax- Wendroff scheme for initial-value problems, 297 Navier-Stokes equations and, 323-324 shock waves and, 312-313 in steady-state behavior, 326-327 Leapfrog scheme for initial-value problems, 297 modified, 304-305 numerical dispersion and, 302-304 numerical dissipation and, 302-304 shock waves and, 312 for vorticity, 318 Legendre’s dual transformation, elastoplasticity and, 36-37 Linear decomposition, of unsteady potential, 240-241 Linearized boundary-value problem in ship motion theory, 237-240

source potential in, 245 Linearized free-surface condition, defined, 237-240 Linear response, primary bifurcation and, 58-61 Loading analytical formulations for, 53-58 bifurcations under, 66-72 deformation-sensitive, 54-57 elastoplasticity, 35-47 stability and, 31 Loading environment, loading tests and, 31-32 Local flow reversal, for general unsteady boundary layer, 203-207 Local isotropy, hypothesis of, 134 Long-wavelength approximation inner problem and, 260 inner solution and, 265 matching and, 262-265 of ship motion, 256-257 slender-body approximation and, 229-230 in slender-body diffraction, 272-273

M MAC, see Marker-and-cell method Macroisotropic artificial rubbers, elastic response in, 47-49 Magnetohydrodynamics, numerical methods in, 327-328 Marker-and-cell method, for incompressible fluids, 319-320 Mass transport, bounds on, 105-1 10 Material deformation, Jaumann flux and, 7 Material spin defined, 6 kinematics and, 13-14 Matching in slender-body diffraction, 269-271 in slender-body radiation, 262-265 Mean flow inhomogeneity, production of, 127 Mean velocity gradient in mechanical dissipation, 153- 154 in rapid distortion, 148-149 Measure invariance, in deformation, 28-35 Mechanical dissipation, equations for, 152-156

347

Subject Index Mechanical production, scale ratio and, 157 Metal single crystal, elasticity and plasticity of, 28 Mode analysis, in mixed-initial-value problems, 308 Modified equation, defined, 304-305 Momentum transport bounds on, 94-105 in a rotating system, 101-105 turbulent Couette flow and, 80-84 Monatomic crystal, elastic response, 49-50 Moore-Rott-Sears condition semisimilar boundary layer and, 194 for separation, 181, 199-200 separation singularity and, 212-213, 218 M-R-S condition, see Moore-Rott-Sears condition Multiple boundary-layer technique turbulence theory and, 84-93 for variational problem soulution, 94-105 Multipole, in ship motion theory, 244-245

N Navier-Stokes equations for boundary-value problems, 291 for compressible flow, 287-291 in fluid dynamics, 317-327 for incompressible flow, 291-292 numerical dissipation and, 301 optimum theory of turbulence and, 79 solution of, I 17- 118 turbulence and, 78-82 turbulent Couette flow and, 80-82 Nominal stress stress measures and, 22-23 transformation formulae and, 25 Nonelastic response, elastoplasticity and, 35-47 Nonlinear response, bifurcation theory and, 52 Normality rule invariance and, 38-47 regular case, 40-47 Reynolds stress integral and, 150-151 singular case, 43-47 Numerical dispersion, of waves, 301-309 Numerical dissipation

Navier-Stokes equations and, 322-325 of waves, 301-309 Numerical methods, in fluid dynamics, 286-328 Nusselt number, upper bound for, 113

0 One-point closure, see Second-order modeling Optimum turbulence theory advantages of, 78-80 applications of, 94-1 19 boundary-layer solution methods in, 84-93 for Couette flow, 80-84 heat transport in, 110-115 mass transport in, 105-110 momentum transport in, 94- 105 use of, 77-119 Order of accuracy defined, 293 steady-state calculation and, 325 Order of magnitude analysis, of transport terms, 165-168, 172-173 Orthotropic symmetry, in tensor of moduli, 26-27 Oscillatory motion, steady state, 224-225 Oscillatory pressure, analysis of, 274 Oscillatory pressure force, defined, 225 Outer problem in slender-body diffraction, 267 in slender-body radiation, 258-259 Outer region, in slender ship motion, 243244 Outer solution, matching and, 262-265 Outflow boundary, equations for, 291

P Parabolic equations for boundary value problems, 288-291 for compressible flow, 287 Particle-in-cell method, in fluid dynamics, 315 Path dependence, symmetry and, 30

Subject Index Pfaffian identity, differential invariants and, 31 Piecewise-linear response bifurcation theory and, 52 normality and, 42, 44 Pitch defined, 222 in early sailing ships, 227 historical interest in, 228 inner problem in, 260 linearized problem and, 240 prediction of, 227 pressure and, 276-277 radiation problems of, 250 restoring force and, 226 in slender ships, 244 in strip theory, 232-234 Plane Couette flow, turbulent Couette flow and, 80 Plane waves dispersion of, 301 dissipation of, 301 Plasticity, measure invariance in, 28 Pohlhausen quartic, in steady separation, 188

Poiseuille flow, energy production term and, 115-1 16 Poisson equation, vorticity equation and, 318-319 Polycrystals elasticity and plasticity of, 28 structure of, 35 Polyurethane rubber, elastic response of, 48-49 Porous medium, convection in, 84-86 Prandtl condition, separation singularity and, 218 Prandtl number see also Infinite Prandtl number limit heat transport and, 110, 112 turbulence theory applications and, 116117 Prandtl’s wall proximity law, turbulent flow and, 106 Pressure field, turbulence and, 143-152 Pressure force, in slip motion theory, 273280 Pressure gradient, separation and, 178 Pressure loading, self-adjointness and, 56-57

Pressure transport, in second-order modeling, 168-170 Primary basis, decomposition and, 3-4 Primitive variables, Navier-Stokes equations for, 319-320 Principal fibers, defined, 10-1 I Pseudomoduli, transformation formulae and, 25 Pseudophysical effects, in fluid dynamics, 301-309

Q Quasi-static loading, inhomogeneous response to, 50-72 Quasi-static velocity, equilibrium and, 50-5 1

R Radiated waves, pressure force and, 223 Radiation condition defined, 237 in inner problem, 259-262 Radiation damping, in roll, 226 Radiation potential, in unsteady motion, 241 Radiation problem added-mass coeficient, 251-252 damping coefficient and, 251-252 defined, 223 exciting force and, 279-280 Green’s theorem and, 254-256 slender-body, 258-266 in two-dimensional bodies, 249-252 Rapid distortion problem, in turbulence, 144 Rapid pressure, defined, 144 Rapid terms, in second-order modeling, 143-152 Rate equation, elastoplasticity and, 35-38 Rate-independent solids, mechanics of, 1-72 Rate-of-strain potential, nonlinear bifurcation and, 63-65 Rate problem first-order, 51-58 homogeneous, 56-61 inhomogeneous, 56-65

349

Subject Index Rational Mechanics, school of, 125-126 Rayleigh number convection and, 85-86 heat transport and, 111-113 Rayleigh viscosity coefficient, defined, 246 Realizability in Gaussian method, 162 heat flux and, 142-143 rapid terms and, 145-147 Reynolds stress integral and, 150-152 in second-order modeling, 131-133 temperature variance integral and, 147149 Reference configuration defined, 9 rates of stress in, 21 transformation formulae and, 24-25 Reflection coefficient, defined, 253 Regular wave, defined, 223 Relative convexity nonlinear bifurcation and, 63 normality and, 42 Resonance in head seas, 230-231 in ship motion, 225-226 Restoring coefficients, for steady-state oscillatory motion, 224-225 Restoring forces, equations for, 228 Return-to-isotropy pressure field and, 144 in second-order modeling, 128-130, 133143 Reynolds number for Couette flow, 118 Goldstein singularity and, 179 isotropy and, 134-136 limitation on large, 321-322 mass transport and, 106 Navier-Stokes equations and, 322-325 return-to-isotropy and, 137-141 in a rotating system, 104 second-order modeling and, 125, 127-128 in thermal dissipation, 158 for turbulent Couette flow, 80-84 for turbulent pipe flow, 108-109 unmatchable boundary layer and, 186 in unsteady separation, 189 Reynolds-Peclet number mechanical dissipation and, 152 in thermal dissipation, 157

Reynolds stress decay of, 134-136 energy stability limit and, 83-84 mass transport and, 106 mechanical dissipation and,153- 154 return to isotropy and, 137-142 Reynolds stress integral, construction of, 150-152 Rigid-wall boundary condition, wavelength and, 256-257 Roll defined, 222 dipole moment for, 257 prediction of, 226 radiation problems of, 250-252 in slender ship motion, 242 Rotating system, momentum transport in, 101-105 Rotational motions, defined, 222 Rubber, artificial, 47-49 Rusanov third-order scheme, shock waves and, 312

S

Scale function strain measures and, 14-16 transformation formulae and, 24 Scattered potential, in unsteady motion, 241 Schwarz’ inequality, realizability and, 132133 Seaway, defined, 223 Second-order modeling assumptions concerning, 127-128 development of, 124-127 mathematics of, 128-133 rapid terms in, 143-152 of turbulent flow, 124-173 Second-order rate problem, linear case, 59-60 Self-adjointness exclusion functional and, 60 under fluid pressure, 70-72 loading and, 56 nonlinear bifurcation and, 62 Semisimilar boundary layer, separation in, 181, 192-203, 212 Separation

350

Subject Index

see also Steady separation; Unsteady sep-

aration analysis of, 177- I82 asymptotic approach to, 182-186 boundary-layer approximation of, 177218 defined, 177 general unsteady boundary layer and, 203-2 13 in Lagrangian description, 213-218 semisimilar boundary layer and, 181, 192-203, 212 unmatchable boundary layer concept and, 186-192 Separation criterion, Goldstein singularity and, 199-203 Separation point defined, 178 flow structure near, 178-181 in unsteady boundary layer separation, 208 Separation singularity criterion for, 208-213 genesis of, 215-218 in outer problem, 258-259 separation criterion and, 199-203 Shanley buckling, bifurcation and, 65 Shear moduli, orthotropic symmetry and, 26-27 Ship hull, see Hull Ship motion boundary-value problem in, 235-244 fundamental solutions to, 244-249 historical study of, 222-235 pressure force in, 273-280 slender-body diffraction in, 266-273 slender-body radiation in, 258-266 theory of, 221-280 three-dimensional, 23 1-235 two-dimensional, 249-257 Shock-smearing methods, use of, 310 Shock waves, numerical methods in, 309313 Singularity, see Fundamental singularity; Goldstein singularity; Separation singularity Skew tensor defined, 6 work conjugacy and. 20 Slender-body approximation fundamental solutions to, 245

ordinary, 265 ship hull and, 227-229 strip theory and, 235 theory of, 228-230 Slender-body diffraction inner problem in, 268-269 inner solution in, 271-272 long-wavelength solution in, 272-273 matching in, 269-271 outer problem in, 267 in ship motion theory, 266-273 Slender-body radiation inner problem in, 259-262 inner solution to, 265-266 matching in, 262-265 outer problem in, 258-259 ship motion theory and, 258-266 Slenderness parameter defined, 258 exciting force and, 280 in slender ships, 243 Slender ship defined, 242-244 diffraction problem in, 266-273 pressure analysis of, 274 radiation problem in, 258-266 two-dimensional approximation of, 249257 Slip deformation, normality and, 44 Slow-ship assumption, steady-flow field and, 242 Solid mechanics, invariance in, 1-72 Source distribution, in outer problem, 258259 Source potential see also Green’s theorem in outer problem, 258-259 in ship motion theory, 244-249 three-dimensional, 245-249 two-dimensional, 245-246 Source strength Green’s theorem and, 256 inner problem and, 261 long wavelength and, 256 of outer solution, 262-265 Spectral analysis, of ship motion, 223 Splitting methods, see Fractional-step methods Stability constitutive descriptions, 31 criteria for, 305-306

Subject Index defined, 294 in magnetohydrodynamics, 328 in mixed initial-boundary-value problems, 308-309 modified equation for, 305-306 numerical analysis of, 293-295 Static restoring moment, in conventional ships, 226 Statistically stationary turbulence, mean shear in, 116 Steady boundary condition asymptotic behavior of, 184 separation in, 178, 181. 182-184, 187-189, 193, 202-203, 218 in ship motion, 238-240 Steady flow field, description of, 241-242 Steady free-surface condition, in slender ships, 243-244 Steady separation asymptotic approach to, 181. 182-184 flow structure and, 178-179 in Lagrangian description, 218 semisimilar boundary layer and, 193,202203 unmatchable boundary layer concept and, 187- I89 Steady-state oscillatory motion, description of, 224-225 Steady-state problem, in ship motion, 222 Steady-state solution. for Navier-Stokes equations, 325-327 Steady-state wave resistance theory, hydrodynamic disturbance and, 228 Steamship, motion of, 227-228 Stokes theorem, exciting force and, 278 Strain-energy density, elasticity and, 32 Strain-hardening stress bifurcation theory and. 52 normality and, 44-45 Strain measures invariance and, 14-17 orthotropic symmetry and, 27 Strain rate in rapid distortion, 149 strain measures and, 16-17 Strain-rate potential, see Rate-of-strain potential Strain-softening stress, normality rule and, 41 Stream function for incompressible flow, 292

351

in Navier-Stokes equations, 317-319 Stress measures, invariance and, 17-23 Stress rates, objective, 2 1-22 Stress response, normality rule and, 41-43 Stretch tensor defined, 11-12 elastic response and, 47-50 scale functions and, 14-16 Strip theory exciting force and, 280 inner problem and, 259-260 inner solution and, 265-266 matching and, 263-264 pressure and, 276-277 three-dimensional, 23 1-235 Structural loading, in ships, 222 Submarines, steady-flow field and, 242 Surge defined, 222 influences on, 227 long-wavelength approximations and, 256 radiation problems of, 250-25 I in slender ship motion, 242 Sway defined, 222 dipole moment for, 257 exciting force and, 280 influences on, 227 long-wavelength approximation and, 256 pressure and, 274,276 radiation problems of, 250-252 in strip theory, 235

T Taylor vortice, Coriolis force and, 102-103 Temperature variance, dissipation of, 156158 Temperature variance integral, construction of, 147-149 Tensor see also Induced tensors; Rate tensor; Skew tensor; Stretch tensor deformation and, 1-2 of moduli, 24-27 second-order modeling and, 127- 128 turbulence dynamics and, 124-173 Tensor representativeness, concept of, 3-8 Tensor strain measure, measure invariance and, 28

352

Subject Index

Thermal convection see also Heat transport optimum turbulence theory and, 110-113 transport properties and, 79 Thermal dissipation, equations for, 152, 156-158 Thermal production scale ratio and, 157 thermal dissipation and, 157 Thin ship, see Slender ship Thin-wing theory steady-flow field and, 241-242 thickness problem and, 228 Three-dimensional flow, Green's function and, 246-249 Time-dependent flow differential equations for, 287-288 one-dimensional, 309-3 13 Time-dependent problems pure initial-value problems and, 307-309 Navier-Stokes equations and, 322-325 Time-dependent solutions, for NavierStokes equations, 325-327 Time-scale ratio, in thermal dissipation, 156- 158 Time steps, large, 306-307 Tollmien asymptotic behavior, 182 Tollmien's linearization, asymtotic behavior and, 183-185 Toroidal systems, magnetohydrodynamics in, 328 Transformation formulae generalized moduli and, 24-25 measure invariance and, 28-30 Translational motion, defined, 222 Transmission coefficient, defined, 253 Transport see also Heat transport; Mass transport; Momentum transport one-dimensional, 328 second-order modeling and, 128 two-dimensional, 328 Transport terms for dissipation, 159 in second-order modeling, 160- 173 third-order, 160 Trapezoid rule of quadrature differential invariants and, 30 measure invariance and, 29 Triple-deck local-interaction model, boundary layer and, 178-179

Truncation error defined, 304-305 numerical analysis of, 293-295 Turbulence boundary-layer theory of, 89-93, 124 defined, 77-78 Gaussian model of, 162-165 heuristic assumptions concerning, 77-78 invariant characterization of, 138-142 loading environment and, 31-32 measure invariance and, 28 principle of, 57-58 V

Viscosity Navier-Stokes equations for, 317-327 real vs. numerical, 310 Viscous-inviscid interaction, separation point and, 178 Von Neumann criterion defined, 294-295 stability criteria and, 306 in upwind differencing scheme, 295 Vorticity, in Navier-Stokes equations, 3 17-3 19

W Wall shear in semisimilar boundary layer, 195 separation and, 178, 181 in unsteady separation, 189-192 vanishing, see Vanishing wall shear Water-wave boundary conditions, for incompressible flow, 292 Wave effects, in slender ship theory, 244 Wave filtering, implicit methods for, 315317 Wave-free potential, in radiation problems, 250-251 Wavelength, ship length and, 226 Wave-maker problem, thin ship and, 228 Wave resistance calculation of, 222 slow-ship assumption and, 242 Waves, effect on ship motions, 222-280 Weakly unsteady case, of boundary layer solution, 185-186

Subject Index Work conjugacy measure invariance and, 28 as objective rate of stress, 21-22 stress measures and, 19-20 Work differential, transformation formulae and, 24 Work function, elastoplasticity and, 35-47 Work rate, measure invariance and, 28-29

353

inner problem and, 260 linearized problem and, 240 pressure and, 274, 276 radiation problems of, 250 in strip theory, 235 Yield surface, elastoplasticity and, 35-47

Z

Y Yaw defined, 222 influences on, 227

A 6

8

c D

9 O

E l F

2

G 3 H 4 1 5 J 6

Zeroth-order transport terms, in secondorder modeling, 170-171 Zigzag scheme, for local flow reversal, 204-207

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