Advances in Applied Mechanics, Volume 20

Advances in Applied Mechanics Volume 20

Editorial Board T. BROOKEBENJAMIN

Y. C. FUNG PAULGERMAIN RODNEY HILL L. HOWARTH

T. Y. Wu

Contributors to Volume 20 ERNSTBECKER NEILC. FREEMAN ROTT NIKOLAUS T. TATSUMI

ADVANCES IN

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

VOLUME 20

1980

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London

Toronto Sydney San Francisco

COPYRIGHT @ 1980, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR A N Y 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 Ediiion published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W 1 7 D X

LIBRARY OF

CONGRESS CATALOG CARD

NUMBER:48-8503

ISBN 0-12-002020-3 PRINTED IN THE UNITED STATES OF AMERICA 80818283

9 8 7 6 5 4 3 2 1

Contents vii ix

LISTOF CONTRIBUTORS IN MEMORIUM

Soliton Interactions in Two Dimensions Neil C. Freeman 1

I. Introduction 11. Korteweg-de Vries Equation and Two-Soliton Interactions 111. Inverse Scattering Theory

rV. Multisoliton Solutions V. Positive Dispersion and the Kadomtsev-Petviashvili Equation VI. Cylindrical Korteweg-de Vries Equation VII. Conclusion References

8 14 17

22 30

35 36

Theory of Homogeneous Turbulence T. Tatsumi 39 42 49 65 78 105 127

1. Introduction 11. Mathematical Formulation 111. Statistical State of Turbulence

IV. Cumulant Expansion V. Incompressible Isotropic Turbulence VI. Turbulence of Other Dimensions VII. Concluding Remarks References

130

Thermoacoustics Nikolaus Rott I . Introduction 11. Oscillating Flow over a Nonisothermal Surface

I l l . Damping and Excitation of a Gas Column with Temperature Stratification V

135 138

143

Contents

vi IV. Thermoacoustic Streaming References

168 174

Simple Non-Newtonian Fluid Flows

Ernst Becker 177 179

I. Introduction 11. Non-Newtonian Flow Behavior 111. The Constitutive Equation of Simple Fluids IV. Fully Developed Pipe Flow V. Peristaltic Pumping VI. Viscosity Pumps VII. E f k t i v e Viscosities VIII. Extruder Flow IX. Nearly Viscometric Flow X. Plane Boundary Layer Flow of a Fluid with Short Memory XI. Journal Bearing References

187 192 197 204 210 212 216 219 225

AUTHQR INDEX

227

SUBJECTINDEX

23 1

184

List of Contributors

Numbers in parentheses indicate the pages on whch the authors’ contributions begin.

ERNSTBECKER,Institut fur Mechanik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Federal Republic of Germany (177) NEILC. FREEMAN, School of Mathematics, University of Newcastle upon Tyne, Newcastle upon Tyne, NEl 7RU, England (1)

NIKOLAUS ROTT,Institut fur Aerodynamik, Federal Institute of Technology (ETH), Zurich, Switzerland (1 35) T. TATSUMI, Department of Physics, University of Kyoto, Kyoto 606, Japan (39)

vii

This Page Intentionally Left Blank

In Memorium

While this volume was in press, we received the sad news of the death of Professor William Prager on March 16, 1980, in Switzerland. Professor Prager has been a member of our Editorial Board since 1966. His death has deprived us not only of his valuable service on the Board, but a friend and a counselor as well. The mechanics community has lost an illustrious leader. His presence will be missed, and he will be remembered by many of us with gratitude. Chia-Shun Yih

ix

This Page Intentionally Left Blank

Advances in Applied Mechanics Volume 20

This Page Intentionally Left Blank

ADVANCES IN APPLIED MECHANICS, VOLUME

20

Soliton Interactions in Two Dimensions NEIL C. FREEMAN School of Mathernutics Unioersity of Newcustle upon Tyne Newcastle upon Tyne Eny fand

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Korteweg-de Vries Equation and Two-Soliton Interactions . . . . . . . . . . . . 111. Inverse Scattering Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Multisoliton solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Positive Dispersion and the Kadomtsev-Petviashvili Equation . . . . . . . . . . V1. Cylindrical Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . . . . . VII. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

8

14

17 22 30

35 36

I. Introduction The word “soliton” was coined around 1965 by Zabusky and Kruskal to describe solitary wave pulses, which they observed while numerically integrating a nonlinear partial differential equation-the so-called Kortewegde Vries (K-de V) equation. The solitary wave solution of this equation has been known for many years and its study has its origins in the experimental work described by Scott-Russell in his “Report on Waves” presented to the British Association for the Advancement of Science in 1844. The first analytic representation of the solution as a (sech)2 profile (Fig. 1) was given by Boussinesq in 1870 and independently by Lord Rayleigh (1876). The form of the partial differential equation that describes such waves in one space dimension was obtained by Korteweg and de Vries (1895) and the equation usually bears their name. No further solutions of the equation were obtained until the 1960s when, following their numerical observations, Kruskal and I Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-002020-3

2

Neil C . Freeman

I FIG.1 . The K-de V soliton.

his co-workers (Gardner et al., 1974) developed a technique now referred to as “inverse scattering theory” for the general solution of the equation leading to multisoliton solutions, some of which may have been anticipated (Whitham, 1974). In 1970, Kadomtsev and Petviashvili proposed a generalization of this equation to two space dimensions and Satsuma (1976) showed that this equation has multisoliton solutions. Zhakarov and Shabat (1974) generalized the inverse scattering theory to include such solutions. The interpretation of such solutions in terms of the theory of shallowwater waves was given by Miles (1977a),although experimental observations of the interaction phenomena for such waves were well understood by Scott-Russell (1844). In this chapter a general description of the development of the theory of one-dimensional solitons of the K-de V equation is given. Since this theory has been well reviewed elsewhere (Scott et al., 1973; Miura, 1976), the main interest, however, centers on the generalization of these results to two dimensions. Certain inadequacies of the solutions of the Korteweg de Vries solutions lead us to examine a closely related equation given by Kadomtsev and Petviashvili (1970), which is referred to as the K-P equation. It is, of course, true that the notion of solitons has led to the study of other partial differential equations in a similar way. The development of the theory of such equations, for example the Sine-Gordon equation, parallels that of the K-de V equation in many ways, but with fascinating differences. Two results given by Scott-Russell may be singled out in particular for comment.

Soliton Interactions in Two Dimensions

3

The first concerns the disintegration of an initial disturbance. In ScottRussell’s words: “The existence of a moving heap of water of any arbitrary shape or magnitude is not sufficient to entitle it to the designation of a wave of the first order [solitary wave]. If such a heap be by any means forced into existence, it will rapidly fall to pieces and become disintegrated and resolved into a series of different waves, which do not move forward in company with each other, but move separately, each with a velocity of its own and each of course continuing to depart from the other” (Fig. 2a). This is seen to be a remarkably accurate picture of what was subsequently found theoretically. The second concerns the reflection of a wave from a wall. Scott-Russell writes: “The magnitude of the reflected wave diminishes as the angle of incidence diminishes, until at length, when the angle of the ridge of the wave is within IS” or 20” of being perpendicular to the plane, reflexion ceases, the size of the wave near the point of incidence and its velocity rapidly increases, and it moves forward rapidly with a high crest at right angles to the resisting surface.” Again, this is an acute observation of a phenomenon confirmed later (Fig. 2b). It is doubtful whether a change of name can stimulate new interest in a scientific pheonomenon, but this together with the realization that that new name extends what was a special phenomenon in one field, viz. fluid mechanics, to other wider fields most certainly has done so in this particular case. In this review the progress in understanding the solitary wave of Scott-Russell under the influence of the development of the theory of solitons in other fields is examined. The theoretical form of Scott-Russell’s solitary wave was obtained by Boussinesq (1870) and the profile in more modern notation given by the wave height was shown to be y

(

= qosech2- 3‘~)1’2 (x - rt),

2

h3

where c2 = S ( h + yo)

= &h(l + tro/h),

where h is the water depth and yo the maximum amplitude of the wave. This result confirmed Scott-Russell’s original assertion that the propagation speed was &(h + yo), a point disputed by Airy (1844), Kelland (1839), and Earnshaw ( 1846) later. In 1895 Korteweg and de Vries showed that this was a particular solution of the partial differential equation

.a'

-

4.

.... -..____,, /-~

'-

- .

1 . -

.A

;A

.... ......

.

1

(a1

FIG.2. (a) Detail from Russell's water channel experiments showing the formation of one and two solitary waves. (b) Russell's experiments on reflection at a wall, showing regular and anomalous reflections.

Soliton Interactions in Two Dimensions

5

6

Neil C . Freeman

where T = cot, 5 = x - cot, with co = A h . The solution remained a curiosity in the literature until Zabusky and Kruskal by their numerical studies showed, as Scott-Russell had intimated, that solitary waves were of a more ubiquitous nature. In their investigation Zabusky and Kruskal integrated the above equation with periodic boundary conditions starting with the cosine function as the initial profile. They observed the formation of solitary waves of increasing height, which appeared, disappeared, and reappeared as the motion progressed (Fig. 3). These observations led them to seek a technique of solution that would describe these discrete disturbances. They were eventually characterized as the eigenvalues of an associated Schrodinger equation (Gardner et al., 1974).The construction of the potential of this equation from a knowledge of these eigenvalues and the time evolution of their associated scattering data led to a new wave profile and hence to the solution to the equation. In this way solutions containing many solitary waves could be constructed-the so-called multisoliton solutions. Hirota (1971) and Whitham (1974) showed how such solutions could be obtained by direct substitution in the equations.

NORMALIZED OISTANCE

FIG.3. The formation of solitons from a cosine wave (Zabusky and Kruskal, 1965).

The generalization of these results to more than one space dimension made by Kadomtsev and Petviashvili (1970) sought to extend the dispersion relation of the linear form of Eq. (1.2) to higher dimensions. The term 2q, in the above equation originates from introducing the coordinates 5 = x - cot, z = &cot( E << 1) into the wave operator - c; 2(az/at2)+ (dz/dx2)to approximately at large times. The two-dimensional wave give 2~ d2/& operator - ~ ; ~ ( a ’ / d t ’ ) + ( d 2 / a x 2 ) + (d/ay2) must thus be replaced by e[2(d2/Jt at) + ( a Z / d Y Z ) ]where , Y = ~ ‘ / if~ the y two-dimensional nature of the equation is to be retained. The two-dimensional generalization of the

Soliton Interactions in Two Dimensions

7

K-de V equation therefore becomes (2%+ 3vlrlt; + 4h3tlg&

+ r y y = 0.

(1.3)

It is clear that this equation has solitary wave solutions that are skewed versions of those given by Boussinesq. Written in the same notation they become

where m is a parameter describing the (small) inclination of the wave to the main direction of propagation. Multisoliton interactions of skewed solitary waves could be obtained by direct substitution in Eq. (1.3), as shown by Satsuma (1976). Such waves are not, of course, true localized solutions in two dimensions, but extend to infinity along their length. A novel interpretation of Satsuma’s solution was given by Miles (1977a,b), who derived his results directly from the full equations of motion. Miles considered the problem of the interaction of solitary waves at large angles and showed that the solitary waves moved independently of each other, suffering only a small center shift upon interaction. As observed by Scott-Russell, he noted that an anomalous reflexion took place when the angle between the waves was small and an interaction occurred that could be described by a single soliton that was resonant with the incident and reflected waves. This was indeed the wave described by Scott-Russell and because of the boundary condition at the wall was necessarily normal to it. This type of interaction, familiar in shock wave theory as Mach reflection, thus replaced the regular reflection at small interaction angles. Miles observed that the rnultisoliton solutions due to Satsurna were singular. In this limit the center shift of the post interaction solitons became infinite and a triad of solitons was formed composed of the incident soliton, the reflected soliton, and the resonant soliton. Since any multisoliton interaction is composed of many two-soliton interactions, this means that provided the center shift produced at each individual interaction is large enough the whole interaction can be described in terms of motion of these resonant triads (Anker and Freeman, 1978). Such solutions of the K-de V equation have generated much interest but, fascinating as they are, they are not essentially two dimensional in character, and the search for truly localized two-dimensional disturbances has not been so successful. A closely related equation that we shall refer to as the Kadomtsev and Petviashvili (K-P) equation, which arises in plasma physics and has the opposite sign in front of the dispersion term urrc,has been more fruitful in this respect. Manakov et al. (1977) have shown that truly localized solutions can be obtained for this equation that are purely rational in

8

Neil C. Freeman

character. Unfortunately such solutions are singular and not therefore of immediate physical interest in the case of the K-deV equation. These solutions are obtained as limiting solutions of the multisoliton solutions referred to above and are rational in form. A 2n-soliton solution of exponential type produces an n-soliton solution in the limit. These solutions do, in fact, form part of a much wider class of solutions that may be obtained from the K-de V or K-P equation. Techniques for the construction of such solutions have been suggested by Johnson and Thompson (1978). 11. Korteweg-de Vries Equation and Two-Soliton Interactions The derivation of one of the fundamental equations of soliton theory was given by Korteweg and de Vries in their famous paper of 1895. The equation describes the amplitude of long one-dimensional waves on the surface of a fluid. In physical terms the effects included in the equation are weak nonlinearity and dispersion. The extension of this equation to motions in more than one dimension were given by Kadomtsev and Petviashvili (1970), who generalized the dispersion relation to give an extra term in the equation due to the extra dimension. They also drew attention to the effect of changing the sign of the dispersion term. They noted that the effect of this was to introduce an instability in transverse perturbations and presumably a breakup of the one-dimensional waves. In this section we formally derive the K-de V equation and examine the interaction of waves derived from it in a manner similar to that of Miles (1977a). Waves of amplitude q‘ propagate over the surface of water of uniform depth h, in two dimensions x’, y’. The bottom is denoted by z’ = 0 and the surface by z’ = ho + y’. The equation satisfied within the fluid for an irrotational incompressible motion is then Laplace’s equation

and @’ is the velocity potential of motion defined by q’ = V W , where 4’ is the velocity vector. At the surface two conditions must be satisfied: (1) the kinematic condition that particles on the surface should move with the surface, which may be written

and (2) the surface constant-pressure condition most conveniently written in the form of Bernoulli’s equation

aw + -1 4, 2 + pi

at

2

P

+ g(z’ - h,) = P+!, P

for z’ = ho + q,

(2.3)

Soliton Interactions in Two Dimensions

9

where pm is the surface pressure and p' the (constant) density. A suitable scaling is obtained by using a length scale 1 in the horizontal direction, ho in the vertical direction, a velocity scale c = &ho, and a time scale I c - ' . Thus 6V:@

a2

+ QZz= 0,

+ av2' a2

where V2 - - -ax2

pr+;(of@+ p:)]

+ q = 0,

1

2=1+q

6Crr + Q , ' x l z = l + q ~ x together with the bottom condition Q,zlz=I+q=

@Jz=0

+ Q,oylz=l+g'lyI,

= 0,

where 6 = h$A2 and the unprimed variable have been scaled in the appropriate way. Assuming long waves, then 6 << 1. We write

+ 6@, + 6%,, + . .

Q, = &[Do

*],

(2.5)

where E is an amplitude parameter, and hence obtain the equation

where 40,r$,, 4 2 , .. . are arbitrary functions of x, y, and t . Writing q = &[qO + 61' + . we obtain a]

-(I + ?0)V2fi = ?or + Vl'lOVl40, neglecting terms of order 6. Putting qo = Vl+o we obtain 40*

+ 3 m o + 'lo ?or

= 0,

+ V1Cqo(Vo + 111 = 0,

902

+ 4 o v q o = -V1ro.

(2.7)

These are the equations for large-amplitude, nondispersive waves and take the same form as the compressible flow equations with ratio of specific heats y = 2. The waves described by such equations break and result in the theory of hydraulic jumps and bores. To prevent breaking the steepening effects of nonlinearity must be balanced either by dissipation or, as is the case here, by dispersion. Weak nonlinearity and dispersion require that the amplitude parameter E and the dispersive 6 be small. In fact, we assume that these are of equal magnitude and write 6 = E the proportionality constant being made unity by suitable choice of scale. We write q = E [ ? ~ + 6ql + . . -1and to first approximation obtain

4 o r + ~o

= 0,

?or =

which together require Votr

= VfVO.

-V:fo,

(2.8)

Neil C.Freeman

10

This is the familiar two-dimensional wave equation with plane-wave solutions qo = f(n, x - t), where no is a unit vector in the direction of propagation. Higher order terms will lead to secular terms of the form ct(n, x - t). Such terms grow and will dominate on a time scale of order

-

&-I.

The interaction of two plane waves in the far field after a long time when z = et = O(1) can be considered, and it will be sufficient to discuss the far-

field development of such waves. We assume that the disturbances are localized about some line n r - t = const. Such waves will only need to be considered in the neighborhood of their intersection since far from this interaction the waves are uninfluenced by the presence of each other. Such interactions have been called “weak” by Miles (1977a). In producing two plane-wave coordinates tl and t2 where ti = n, r - t with r = (x, y) and n, = (COSY,, sin”,), we obtain

-

-

(2.9)

Elimination of qo gives 2[1 - cos(Y’, - Yz)]

aZ40 - 0. at1 a t 2

~

(2.10)

Whence for 1 - cos(Yl - Yz) = 0(1), we obtain

a24,/at1 at2 = 0.

(2.11)

This is the two-dimensional wave equation with solution 40

= Fl(t1J)

+ FZ(tZ,T)>

(2.12)

from which

+ Fzr2(tZ,z),

where F, = aF/at. Thus to a first approximation the waves designated by the wave forms F1 and F, do not interact. Following Miles (1977a) we can proceed to a second approximation : q0 = Fl&1,4

+ Fl,,FZ,,[1

+ cos(Y’, - Yz)]

(2.13)

Soliton Interactions in Two Dimensions

11

+ 1/2(F:c1 + Fit,) + F1<,F2,,C O S ( ~ I- Y Z ) ] . (2.14) Elimination of q 1 between (2.13) and (2.14)gives

Thus if F1,F2 satisfy (2.15) then

+

f1

1 2COS(Yl - Y2) (Fl,,F2 + FlF2,2)9 = 2[1 - COS(Yl - Y2)]

plus a sum of arbitrary functions of t1 and z, C2 and z. The surface displacement can then be written most compactly as V = E[Ni(ti

+ XiFzz) + N 2 ( 5 2 + X Z FT)] ~ , + &'ININ2 + . . . ,

(2.16)

where

+

x1

=

1 2COS(Y', - Y2) 2[1 - COS(Yl - Y 2 ) ] '

I=

1

+ 2COS(Y', -

= &2[1-

Y2)

cos(Y1 - Y2] '

+

1 + cos(Y 1 - Y2) COSZ(Y 1 - Y 2) 1 - cos(Y1 - Y2)

Equation (2.15) is the one-dimensional K-de V equation in the variable t normal to the wave front. It is more often written in the form 21,

+ 3qq, + $qrrr = 0,

where 1 = F,.

(2.17)

Its most interesting solutions are the so-called solitary waves or solitons, which may most conveniently be written in the form q = F, = 81' sech2$l(t

- 412r).

(2.18)

Neil C.Freeman

12 We may also note that 112

F = )4;(

[l

+ tanhfiZ(t

- 4Z2r)],

if F

-+

0 as t

--*

- co.

Higher approximations to this result have been given by Laitone (1960) and Fenton (1972) and are necessary for matching this solution at higher orders, but they are not required for our purposes. This technique of expansion obviously fails when 1 - cos(Y - Y ?) = O(E). The waves are then almost aligned and Y, - Yz = O(E'/').The interaction is then no longer weak and, following Miles (1977b),is referred to as strong. Rederiving the previous expansion now gives

,

a290

2lC----at1 at2 =

(& &) +

+

(i$

+

&)

to first order with K = [l - cos(Y - Y 2 ) ] / & . The two phases now differ only by order E and it is convenient to introduce as coordinates and the normal to 5 , with an appropriate scaling, 4'. We write

el

5 = (ycosY, - xsinYl)&,

5 = tl.

(2.20)

Then

a at

a + -,a at2 851

- = (1 - E K ) -

a a5

-= J Z K

a at2

-,

(2.21)

since t2= tl(l - E X ) + 5/2u + O ( E ~ /Equation ~). (2.19) now becomes

Since qo = a+,/ag, we then obtain (2.22) which is the two-dimensional form of the K-de V equation. The linear dispersion relation for this equation can be written 2wk = 3k4 + m2, where the phase function 8 = k t

(2.23)

+ mc - 07.A convenient parameterization

Soliton Interactions in Two Dimensions

13

of this relation is obtained by putting

m = 6(n2 - 12), w = 4&13 + n3). $(l + n), A single skewed soliton solution of this equation thus becomes

k

=

ylo = 2(1+

n)’ sech24$[(1 = 2(1+ n)’ sech2$O(1,n)

+ n)< + @(n2

- 1’)[

- 4(13

(2.24)

+ n’),~] (2.25)

(say), which corresponds to Eq. (2.18) when 1 = n and the wave is propagated in the 5 direction. It can be observed that the wave amplitude is now 2(1+ n)’ and, in general, the wave is characterized by the two parameters 1 and n. Solutions of Eq. (2.22) that describe multisoliton interactions have been given by Satsuma (1976) and a general technique of solution, which is described later, by Zhakarov and Shabat (1974). For our purposes, we examine the two-soliton interaction, which is sufficiently simple to confirm as a solution by direct substitution. The most convenient form for the solution is (2.26) where f = 1 + eel

+ ee2 +

12

ee1+e2 3

with El2

= (11

-m

1 -

nz)/(l1+ n N 2

+

and i = 1, 2, Oi = 0(li,ni), are the phases of the two solitons. For large angles of interaction we expect that this strong solution will correspond to Miles’ weak solution (2.16) with F given by the solitary wave solution of Eq. (2.15). If we align one of the solitons dl with the t1direction, then this requires I1 = nl. Aligning the other with the t2 direction then, by comparison with (2.12), gives K: = 3(12 -

n2)2

or

Y~ - y2= &(12

- n2).

(2.27)

Thus l 2 - n2 -+ co for Y - Y 2 = O(1). Since the amplitude l2 + n2 of the O2 soliton must remain finite, we require l2 -+ co, n2 + - co, with l2 + n, fixed. In this limit we see that

14

Neil C . Freeman

and log(1 + eel)(l + e'z)

3 d5 = 2.41:

sech' $dl

ee1+'Z + 2(1, +1; nZ)ll (1 + ee')(l + e'2)

+ 2(1, + n,)'

sechZ$3,

- 41:

tanhie, sech2@,(1 + tanhie,)

- (1,

+ n2)' tanhie, sech2$Ol(l + tanhfOl)].

(2.29)

Since, in Eq. (2.16), x1

= (24)4,

N,

F1 = (1

= 2(21,)'

x1

= (2/,)4

sech2ff3,,

+ tanh381)4($1/21,,

+

.($,

N,

= 2(1,

F , = (1

I = 2(21,)-2&,

+ n,)'

sechZ&,

+ tanht8,)2

5

this matches exactly the nonuniform behavior associated with the weak interaction when (Y, - Y,)' = O(E).It can be observed that to a first approximation the two solitons do not interact with each other and the phase shift is zero. This is consistent with the limit g 1 2 + 1 as indicated in Eq. (2.28), and f in Eq. (2.26) becomes (1 + eel)(1 + ee2). 111. Inverse Scattering Theory It can be noticed in the analysis of Section I1 that numerical factors such as f i appear throughout. These were not suppressed in the previous sections so that a direct comparison with the physical variables of Section I could be made. Since these factors make the analysis cumbersome, new variables are defined:

t = 61, ?=&, [ = $5, 6 =&lo, (3.1) to remove this behavior. In these variables the K-de V equation becomes

a form that has been used frequently in the literature.

Soliton Interactions in Two Dimensions

15

The general technique of solution for the K-de V equation was given by Gardner et al. (1974).This technique, referred to as inverse scattering theory, was further developed by Zakharov and Shabat (1974) to other equations. The general idea is to relate this nonlinear partial differential equation to a linear eigenvalue problem in the main spatial variable 5 and to consider the development of the scattering data of that equation in time with the eigenvalues fixed. This constancy of 3, is crucial to the subsequent development and is assumed throughout. Formally, therefore, we obtain

where L , is some operator in [, depending on ij. A corresponding operator equation will be valid for the evolution of the eigenfunction $ and eigenvalues ~ ( 3 , ) :

for some operator L, . Provided that l remains constant, then

where [ L , , L 2 ] = L,L, - L,L,. By suitable choice of L , and L,, this equation becomes the original nonlinear partial differential equation (Lax, 1968). For the K-de V equation we write

a2

L - y + q , - a(

A

(3.6)

Then Eq. (3.5) gives

aq -+ a?

aw ap ap3 a t

6 ~a0- +

a30

-=

-,

aw aq = 0,

3 -+

-

a% at

which is Eq. (3.2). If we seek the Fourier transform of I) by writing

A = -k2

p

=

-ik3,

I) = eik2 f

s;" K(p,z, ?, c)eikzdz,

(3.9)

then

(3.10)

16

Neil C. Freeman

and

a

fi = 2-K(e,f;

a4

?,a.

(3.11)

To find K we must solve the linear integral equation

a4 + wt,4 +

K
(3.12)

the so-called Gel’fand and Levitan (1955) equation. The kernel F(s, z ) itself satisfies the linear partial differential equation

a2F

aF a2F -+----o at a t 2

az2 -

(3.13)

’

which has the dispersion relation for the original K-de V equation. The linear nature of the underlying mathematical problem is thus exposed. It is, however, the simple development of the parameters associated with the eigenfunction $ according to linear equations that enables a solution to be constructed by this rather devious route. Seeking a solution of the equations for F of the form F

= Aexp(I4

+ nz + my^- OQ)= -exp

8

(3.14)

(say),we obtain

rn = (n2 - I’),

w = 4(13

+ n3),

(3.15)

which is the parametric form for the dispersion relation already used [cf. (2.24)]. Direct substitution in the Gel’fand-Levitan equation (3.12) gives

K(f,z;

t,t>= ( I + n)ee/(l + ee),

(3.16a)

where A is an arbitrary constant. Thus

a

K(f, ?; t,t )= - log(1

at

+ ee)= $ (I + n)(l + tanh $’),

(3.16b)

and, from (3.11), ij = 2

a f 2 log(1 + ee) = $ ( I + n)’sech2 9. a2

~

This is the single-solition solution [cf. (2.25)] already obtained. Multisoliton

Soliton Interactions in Two Dimensions

17

solutions can now be readily derived by taking m

F

=

C Aiexp(lif + niz + mic - w i t ) i=l m

=

(say). Writing K

C B~exp(@ + niz) i= 1

(3.17)

Kienizthen gives from (3.12)

=

Bie"<

+ nj)5] = 0. + K i - Bi 1Kjexp[(li (li + nj)

(3.18)

j=

Solving for K i by Cramer's rule, we obtain

K k= D k / D

(3.19)

where D = Idij - Bi(li + n j ) - l exp(li + nj)fl and D, is the determinant D with the kth column replaced by - Biefic.Thus (3.20) Thus, from (3.1 l), (3.21) A very important feature of this result is that any exponential factor within D with an argument linear in may be ignored when evaluating q. In Section IV great use is made of this result, and so the notation + or + is used to denote two expressions that are identical or identical in the limit in this sense, i.e., they will give the same fj.

IV. Multisoliton Solutions The general multisoliton solution is given by Eq. (3.21) and the twosoliton solution has already been used in Section 11. Here we examine the structure of the two- and three-soliton solutions in some detail. Putting m = 2 in Eq. (3.19), we obtain I) = 1

+ e@I + e@2+ E

12

where

d i = Bi + log[ -Ai/(li ~ 1 = 2 (11

+ n i ) ] , with Bi = B(li, ni), - 12)(nl - n z ) / ( l l+ n1)(12 + n d .

(4.1)

Neil C . Freeman

18

It is essential that -Ai/(Ii + ni) be positive if singular solutions are to be avoided. This equation has already been examined in the limit I , + nl and, l 2 + n2 fixed, l 2 + a.In this limit c 1 2-+ 1 and D -+ (1 e”)(l + e42) and ‘lois simply the sum of two sech2 solitons. The interaction does not shift the solitons at all and they go on their way unchanged. This zero phase shift limit corresponds in the physical problem to large angles of interaction. The structure of the complete two-soliton solution can best be seen by examining the behavior of D from (4.1) in terms of the phase variables 4 , and c $ ~ . Since these are linearly related to { and c, the orientation of the solitons in the physical plane can readily be obtained after this examination. Considering the limit 42 + - co,we obtain D -+ 1 ebl, which is a soliton aligned along the direction 4 , = 0 [cf. Eq. (3.16)]. The limit 4 , + -co gives the soliton aligned along 42= 0, whereas 42 + 00 gives D 4 1 &12e42rwhich is a soliton aligned with 42 = In&;,’ and 4, + 00 gives D 4 1 + c12e41,which is a soliton aligned along 4 , = In&;,’. Thus, if we regard the incident solitons as those aligned along 4 , = 0 and 42= 0, the post interaction solitions are shifted by an amount In&;,’. In the limit q 2+ 0 this phase shift becomes infinite and we obtain

+

+

+

D

=1

+ e@I+ eb2.

(4.2)

Taking the limits 41-+ - co and 42 -, - co now gives, as before, the incident solitons. The other limits do not give anything of significance, but the limit 4, - 42fixed 4, + co gives

+

This is a soliton aligned along 41= 42.Thus, asymptotically, the infinitephase shift solution (4.2) is itself formed by the two incident solitons and a third one with phase variable 42 - 4 , . The limit e 1 2 + 0 may be achieved by I , + I 2 or n , + n 2 . Taking n , = n2 we obtain

43 = 4 2 - 4, = (13

provided that I 3 we have

=

= (12 -

+ n,)? - ( I ,

r,,g - (if

- If)?

- n:)f - 4(1:

- 4(1; --

I?)?

+ n:)?,

l 2 and n3 = - 1 , . Thus, ifwe impose the condition n , 0 3

=4

1 3 , n3) = 4 4 , n2) - 4 1 1 , n l ) .

(4.4) = n2,

(4.5)

Also

k 3 = I 3 + n3 = l 2 - I , m3 = I: - ns = I f - if

= k2 - kl, = m2 - m ,

(4.6)

Soliton Interactions in Two Dimensions

19

Writing the wavenumber vector k = ( k , m), we have o3= o ( k 2 )- o ( k l ) , k, = k2 - k , and this third soliton is thus resonant with the two incident solitons. The structure of the interaction in the limit of infinite phase shift is therefore asymptotically a triad formed by the two incident solitons and this resonant soliton (Fig. 4). Clearly in the neighborhood of the intersection = 0, the structure is of the three lines # 1 = 0, 42 = 0, and & = 42 more complicated, but as can be seen from Fig. 4, to all intents and purposes the idealized triad is obtained. This representation can be extended more or less exactly to the complete = solution because we see that if + log^,^ and O2 = cP2 + log^,, are

FIG.4. The resonant triad formed by the interaction of two K-de V solitons.

Neil C. Freeman

20

kept fixed, then

1

D=-[ c i 2 + e@1+ eo2 + eo1+@2 1 El2

1 -+-e*1[1

+ e@z+ e@2-*1]

as c12 -+ 0

El2

- 1 + e@2+ e@2-@1.

(4.7) For (D1 + + co we obtained the shifted (D2 soliton and O2 -+ co the shifted (D1 soliton. For (D2 - (D1 fixed (D2 -P -co, the resonant soliton (D1 - (D2 is obtained. We thus have two triads with a common resonant soliton. This approach may be extended further (Anker and Freeman, 1978). For three solitons we have

D

= 1+

+ e@2+ e@3+ El2e@1+@2 + 81 3

e@1+@3

+ ~ ~ ~ e 9 2++ @ ~ ,3~ ~ ~ ~ ~ ~ ~ e @ l + @ 2 + @ (4.8) 3. + p42 + loga, where a and p are constants and a is a

Choosing 43= function of t alone, this becomes

D

= 1 + ,@I + e@2 + a & J 1 + 8 @ 2 +

+

13

&(1+a)+#@z

1 2 1 3 23

+

23

2

e@1+@2

ead1+(8+l)d2

(4.9)

eCa+lW1+CS+~)@2.

The function a is obviously of the form of an exponential of time. Thus t -+ 00 implies either a co or a + 0, and t - co implies either a -+ 0 or a -+ 00. Choosing the former we see that -+

D

-+

D

I*

-+

+ e@I+ eb2+ ~ ~ ~ e @as~ t+-+ @-co, ~ 1 + ~13e"+ &23& + ~ 1 2 ~ 1 3 ~ 2 3 e @ ~as + @t ~ + co. 1

-+

The interpretation of this result is now quite clear. The two-soliton interaction of Cpl and #* having interacted with the third soliton # 3 has been shifted a distance log E;: normal to the q51 direction and log E;: normal to the 42direction. The sequence of interactions that take place between these two extremes may be constructed by observing the interaction of 43with each of the solitons-two incident solitons, two postinteraction solitons, and the resonant soliton-which compose the interaction structure of 4l and 42. This can be easily done and two cases are shown in Figs. 5 and 6. The subsequent development in time is shown. In fact, this result is slightly more general than a simple interaction of the third soliton since the form given by Eq. (4.8) allows for different asymptotic behaviors in &, the two ends being 3 E , 3. shifted relatively to each other unless ~ 2 =

Soliton Interactions in Two Dimensions

21

58

FIG.5. The interaction of a third soliton with a two-soliton interaction, case 1.

(4 $6

x-

58

68

(d )

(4

58

58

56

58

58

FIG. 6. The interaction of a third soliton with a two-soliton interaction, case 11.

35

Neil C.Freeman

22

More complicated interactions will obviously occur for larger numbers of solitons interacting together, but the same principles will apply. V. Positive Dispersion and the Kadomtsev-Petviashvili Equation

The K-de V equation (2.22)derived in the previous section has a dispersion relation given by the linear terms, viz.

This equation rises from the linear wave equation with dispersion

as the leading order approximation when the substitutions 5 = X - T, z = ET,and = E ' / ~ are Y introduced. If the sign of the dispersion term is changed, as is the case for wave propagation in a plasma, the equation that corresponds to the K-de V equation (3.2) becomes

This equation is therefore the positive dispersion counterpart of the negative-dispersion K-de V equation. It will be referred to as the Kadomtsev-Petviashvili (K-P) equation, since its properties were first studied by them (1970). It is obvious that a change of variable qo + - vo,t -+ - 5, z + - z converts this equation to

so that it becomes immediately obvious that any solution of the K-de V equation may be converted to a solution of the K-P equation by change of the variable to ill.Introduction of complex numbers, of course, requires also that the parameters in the solution be considered complex also, and these must be chosen so that the resulting solution is real. For the single-soliton solution (2.25) this requires that we choose n = 7, where 7 is the complex conjugate of 1. The wave amplitude then becomes

c

qo = 21' sech2[il<

+ 2pc1 - 44A2 + 3p2)z],

where 1 = 1 + ip with 1 and p real.

(5.3)

Soliton Interactions in Two Dimensions

23

However, Kadomtsev and Petviashvili suggest that such solitons will be unstable since transverse perturbation will grow without bound, causing the soliton to break up. If this is indeed the case, then it is not surprising that localized solitons should be formed in this case. Such solitons have been found by Manakov et al. (1977) for the K-P equation as limiting solutions for large wavelength, i.e., small wavenumber. They are no longer exponential in character, but take the form of rational functions in space and time variables. Similar results have been given by Ablowitz and Satsuma (1978). Such solutions are singular for the K-de V equation. It can also be shown that mixed solutions can also be obtained that are interactions between conventional skewed one-dimensional solitons and these new localized Zakharov-Manakov (Z-M) solitons. Since it takes two conventional solitons to produced a single Z-M soliton, it is appropriate to observe this limiting behavior for the two-soliton solution

(5.4)

where u = 1 for the K-de V and c = i for the K-P equation. The small wavenumber limit I, -+ -n, is sought for each of the two solitary waves i = 1, and i = 2. We write

I,

= L,,

n,

=

-L,

+

1,

v1,

=

L,,

n2

=

-L,

+ v2.

Substitution in (5.4) gives O(v3),

v,,vz = ~ ( v ) as

v

-,0,

where

e;

= x - 2L,yc

- 12L3,

e;

x - 2~~~~ - 1 2 ~ 3 .

(5.5) In general, such a solution will be complex and singular since 0; will have zeros somewhere in the xy plane. However, for the K-P equation, real non singular solutions can be obtained if E , = - L z , whence j =

1 UlU,

( L , + Ed2

+

[X

=

- 2iLly - 12L;tl2

1+

O(v3).

(5.6)

This gives a purely rational soliton, which is localized and decays like r - 2 as r -+ 00. The form of this soliton is shown in Fig. 7. It can be observed that the solution takes both positive and negative values. The soliton as a whole has zero net volume. Since the limit u l , u2 -+ 0 implies that E~ -+ 1,

Neil C . Freeman

24

FIG.7. The 2-M rational soliton.

the two solitons forming the Z-M soliton have zero relative phase shift to first order. In general, Ai

f =

li + nj

”

exp[(li

+ nj)x - a(lZ - n;)y - 4(li3 + nj3)t], for i, j = 1,2k

(say).Putting li

= Li,

ni

=

-Li

+

ui,

A , = ui(1

+ (iui)

then gives f =

Ai [.I

Li - Lj+

uj

exp[(Li - L j

+ uj)x + aui(ui - 213

4

= in =u 1 i[I(x+(i-2Li.-12Li)b,-(l

1

-sij)- Li - Lj

(5.7)

Soliton Interactions in Two Dimensions

Writing Lk= we obtain

25

-L, and t i real makes f real. In particular, when k = 2,

from which we observe that as,

l@’l(2

+ 00

(5.10)

and as

--t

00

(5.1 1)

Thus at large times the single solitons remain unshifted from their preinteraction trajections. Consequently, the interactions result in no center shift of the solitons whatsoever. This is, of course, a consequence of the limiting procedure 1 + - n, which results in zero center shift of the original solitons (as well as zero amplitude). The complete interaction is shown in Fig. 8 for L , and L , real. Such limits are not, however, confined to Z-M solitons alone; mixed interactions can also occur in which Z-M rational solitons interact with conventional exponential solitons. Since two solitons are required to produce one Z-M soliton, these interactions occur for the above limiting procedure to odd multisoliton solutions. The simplest such case is the limit of the threesoliton solution of the K-P equation. Taking the solution f =

1 - eel - ee2

with

1 3ee1+e3

+ ee3 + 1 2

+

-

1 2 2 3 1 3, s I + e 2 + e 39

23

,ez+e3

(5.12)

26

Neil C.Freeman

FIG.8. The interaction of two Z-M rational solitons: (a)-(d) time increasing.

Solution Interactions in Two Dimensions

FIG.8. (Continued.)

27

Neil C. Freeman

28

We put n , = - L 1 + v,, II = L , , 1, = L , , as before and take the limit u l , v2 3 0. Since

n2 = -L2

+

02,

we obtain 1 s=v1u2{(L1d L 2 ) 2 + B ; O ;

(5.13)

For real j,we require Lz = - E , and n3 = T 3 . Thus

(5.14)

Since here O 3 is independent of

el, we observe that as O3 --+

00

(5.15)

which is a Z-M soliton centered on 18;12= 0. However, as O3

s

-

1

U1U," 1

+ Ell2

+ 1s; +

13

(Ll

-

+ 73

23)(73

+ L,)

[I. +

a, then (5.16)

This is a Z-M soliton centered on

Thus the center of the soliton is shifted from the point x = 12(A: y = - 12pc,t,to the point x = 12(A:

+ p?)t + XI),

y = -12p,t

+ yo,

where

xo + i2A,y0

=-

(Ll

13

+7 3

- 13103

+ L)'

L , = A,

+ ip1,

+ p:)t7

Soliton Interactions in Two Dimensions

29

by the passage of the exponential soliton given by f = 1 + eo3.Such an interaction is shown in Fig. 9a. It can be observed that this shift becomes infinite when L , .+ 1, so that as this limit is approached it takes an infinite time for the interaction to be completed. If the phase O3 is shifted initially by 2In[lL, - 13( 113 L 1 [ / ( l + 3 I,)], then in this limit we obtain the mixed soliton given by

+

1

(5.17)

This resonant soliton was obtained by an alternative method by Johnson and Thompson (1978). Its detailed structure is shown in Fig. 9b, but essentially it is an interaction between a Z-M soliton and an exponential soliton that never goes to completion.

FIG.9. (a) The interaction of a rational soliton with an exponential soliton for the K-P equation. The rational solitons are the isolated peaks and the exponential soliton the lower nearly one-dimensional disturbance. The center-shifted rational soliton is created before the original rational soliton is destroyed ! (b) The special case of the rational-exponential interaction given by Eq. (5.17).

Neil C . Freeman

FIG.9. (Continued.)

VI. Cylindrical Korteweg-de Vries Equation

A recent development in the theory of the K-de V equation has been the discovery of an inverse scattering theory for the cylindrically symmetric K-de V equation first studied by Maxon and Viecelli (1974). The cylindricality simply adds an extra term to give

aq + 6q -+ afi a3q q = 0. +at^ a R dR3 2t^

-

This equation is related, as is the K-de V, to a Schrodinger eigenvalue problem. However, the potential is not simply 9, but + R . This result, first noted by Dryuma, has been developed by Calogero and Degasperis (1978). The single-solitonsolution is now oscillatory behind and exponential in front and is derivable in terms of Airy functions. Indeed, the whole development is uncannily close to the derivation of the theory for the normal K-de V

Soliton Interactions in Two Dimensions

31

equation provided exponentials are everywhere replaced by Airy functions. The decay of such functions in the tail of the disturbance is not, however, very rapid and the theory is consequently more difficult to put on a rigorous foundation. For our purposes we can deduce the cylindrical K-de V from the twodimensional K-de V by writing it in terms of the radial coordinate r, provided the derivation is made from the original physical coordinates. The scaling introduced in the previous analysis may be summarized in terms of the physical coordinates f = f i ( x - t), Q = ~,,h and 5 = 6 ~ ” ~ y . Thus, if radial coordinates are chosen in the original formulation of the problem we have r

= (,/-)/’

=

t

126Q

Hence 123+[’

r-t=

= ij(R,?)

f =-+-

J3

12&

Introducing the new coordinate R Assuming that i j

+ 12f? + r^’ + O(E).

= &(r

r^’ 12@’

(6.2)

- t),then

only then Eq. (3.2)becomes

+

If the derivatives of i j vanish at R + 00, this becomes Eq. (6.1).The inverse scattering method is available for this equation. It may be formulated in the Zakharov-Shabat form as outlined in Section I11 for the original K-de V equation. The approach used is to formulate the problem in terms of similarity variables of the equation

u = (12?)2/3ij,

A = R/(12q1’3.

(6.5)

Then

aU a?

12? - + U ~ L L+ ~ U U ,- 4 2 ~ 2 -22.4 = 0.

(6.6)

The equations corresponding to (3.9)-(3.13) are then dK a3K a3K 3?-+7+-+-

a?

aA

az3

3

(

4

dK aA

U-+-

d2K

d2K

aA2

az2

a(Ku)) -A--z--K=O, a2

- (A - z

- u)K,

(6.7)

32 12-

(a)

, 10 -

8-

6-

4-

2-

4

vv\I’v

J

2

4

6

8

I

1

0

Soliton Interactions in Two Dimensions

I:

33

1

-a

-4

-6

-2

2

FIG. 10. The single-soliton solution of the cylindrical K-de V equation as f increases (a)-(c). The horizontal scale is the similarity variable A and thus in terms of R it increases like P3as t increases. Note the asymptotic approach to the similarity form of the oscillatory part of the solution o n the left as t + 0.

with 24 =

a an

2 - K(A,A; q.

The Gelfand-Levitan equation (3.12) has a Kernel function, F , satisfying

aF

3?-+-+-

a3F

a? an3

a3F aF -,I-az3 an

aF

Z--

aZ

F=O

Neil C . Freeman

34

d2F

d2F

aI.2

a22

- (I. - z)F.

(6.10)

The general solution of Eqs. (6.9) and (6.10) can be written F = t-'l3

s"

-m

f(w)Ai(l - ~ t - ' / ~ ) A i (-z wt-'l3)dw,

(6.11)

where Ai(x) is the Airy function (Abramowitz and Stegun, 1965). The single-soliton solution is obtained by putting f ( w ) = 6(w - wl);then (6.12) F = t-1/3Ai(l - ~ , t - ' / ~ ) A i (-z wlt-'l3) and ( zw l t - 1 / 3 ) t-'l3Ai(A - ~ , t - l / ~ ) A i (6.13) K(I.,z) = [1 + t - 1 / 3 a Ai(y - ~ , t - ' / ~ ) d y']

jm

whence

L - ' / ~ [ A ~ ( A- ~ ~ t - ' ~ ~ ) ] ~

d

u = -2-

+ t - ' / 3 :j

dl{1

=2

d2 log(1 dA

[Ai(y - ~ , t - ' / ~ ) ] ~ d y }.

+ t - 1/3[Ai'2(l- w,t-

'j3)

- (I. - ~ , t - l / ~ ) A i ~ -(~I ,. t - ' / ~ ) ] } .

(6.14) This solution, plotted in Fig. 10, is shown for increasing time in terms of the similarity variable A. The solution has also been given by Johnson and Thompson (1978) using different techniques. It can be observed that as t + 0 a solution is obtained in terms of the similarity variable l alone. As t increases, a single solitary wavelike disturbance is absorbed into this oscillatory solution, the whole structure decaying with time and spreading with R. In a similar fashion multisoliton solutions can be obtained by writing F = t- u3

n

Ai(l - wit- ll3) Ai(z - wit- 1/3),

(6.15)

i=1

from which we obtain Ai(y - ~ & - l / ~ ) A i-( yw j t - ' / ' ) d y

I1

Ai'(I. - w i t - lI3)Ai(I.- w j t - ' I 3 ) - Ai(I. - w i t - 'l3)Ai'(A - w j t - 'I3) wi

- wj (6.16)

Soliton Interactions in Two Dimensions

35

where the vertical bars denote the determinant of the matrix with the appropriate (ij)th element. For the diagonal element i = j , the limiting value i + j as given in (6.14) must be taken.

VII. Conclusion Recent advances in the solution of nonlinear partial differential equations by inverse scattering theory have shown that analytic representations of these solutions are available that describe in a strikingly simple way the structure of interaction phenomena. The discovery of the soliton, initiated as it was by the computer, has ironically shown that the modern tendency to reach for a computer to solve all problems is premature to say the least. The full power of such techniques as inverse scattering theory has yet to be realized. The ingenuity of workers in this field as exemplified in Section VI leads to the speculation that at least for nondissipative systems there are many more useful applications yet to be discovered. The main stumbling block to such advances at the present time is a standard technique for constructing the associated eigenvalue problem or even the discovery of a criterion for its existence. As has been observed by Kaup and Newell (1978), the class of solutions made available by inverse scattering theory forms the basis for the study of a wider class of problems, particularly those which include dissipation, by perturbation methods. Such methods are, of course, not dependent on inverse scattering theory and many were developed prior to its discovery. Conceptually, however, the formulation of the problem in terms of the parameters of inverse scattering theory makes the development of such methods easier. A case in point is the K-de V equation for variable depth first treated by Johnson (1972) and placed in the context of perturbation theory by Kaup and Newell (1978).The elegance and beauty of the latter treatment will much commend it to the reader. It also exemplifies the extreme difficulty of guessing a suitable form for the perturbation expansion a priori without the framework of inverse scattering theory. It has become customary in describing soliton theory to quote the remarkable discoveries of Scott-Russell, which have lain buried in the literature so long. The present author offers no apologies for doing this. Had such discoveries been made today there seems little doubt that more notice would have been taken of them. It is a tribute to Scott-Russell's vision that we can still find in his work observations that, considering the inadequacy of his equipment, describe so precisely the theoretical predictions.

36

Neil C . Freeman ACKNOWLEDGMENTS

The author is indebted to Dr. R. S. Johnson, Mr. S. Thompson, and Mr. P. Wilkinson for assistance in the preparation of this chapter. REFERENCES J. (1978). Solitons and rational solutions of nonlinear evolution ABLOWITZ, M. J., and SATSUMA, equations. J. Math. Phys. 19, No. 10,2180-2186. ABRAMOWITZ, M., and STEGUN,I. A. (1965). “Handbook of Mathematical Functions,” p. 33. Dover, New York. AIRY,G. B. (1844). “Encyclopaedic Metropolitana,” Art. 206 (referred to in Scott-Russell, 1844). ANKER,D. A., and FREEMAN, N. C. (1978). Interpretation of three-soliton interactions in terms of resonant triads. J . Fluid Mech. 87, 17. BOUSSINESQ, J. (1870). Theorie de l’intumescence liquide appelee onde solitaire au de translation se propageant dans un canal rectangulaire. C . R . Hebd. Acad. Sci. 72, 755. CALOGERO, F., and DESGASPARIS, A. (1978). Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation. Lett. Nuovo Cimento 23, 150. EARNSHAW, S. (1846). The mathematical theory of the two great solitary waves of first order. Trans. Cambridge Philos. SOC.8, 326-341. FENTON, J. (1972). A ninth order solution for the solitary wave. J . Fluid Mech. 53, 257-272. GARDNER,C. S., GREENE, J. M., KRUSKAL,M. D., and MUIRA,R. M. (1974). Korteweg-de Vries equation and generalisations VI. Methods of exact solution. Commun. Pure Appl. Math. 27,97. GEL’FAND, I., and LEVITAN, B. M. (1955). On the differential equation from its spectral function. Trans. Am. Math. Soc., Ser. 2 1, 253. HIROTA,R. (1971). Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192. JOHNSON, R. S. (1972). Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf). J . Fluid Mech. 54, 81-91.

JOHNSON, R. S., and THOMPSON, S. (1978). A solution of the inverse scattering problem for the Kadomtsev-Petviashvih equation by the method of separation of variables. Phys. Lett. A . 66,279-281. KADOMTSEV, B. B., and PETVIASHVILI, V. I. (1970). The stability of solitary waves in weakly dispersive media. Dokl. Akad. Nauk. SSSR 192, No. 4, 539-541. A. C. (1978). Solitons as particles, oscillators and in slowly changing KAUP,D. J., and NEWELL, media: a singular perturbation theory, Proc. SOC.London, Ser. A 361, 413-446. KELLAND, P. (1839). On waves. Philos. Trans. Roy. SOC.Edinburgh 7,87. KORTEWEG, D. J., and DE VRIES,G. (1895). On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave. Philos. Mag. [5] 39,422-443. LAITONE, E. V. (1960). The second approximation to enoidal and solitary waves. J . Fluid Mech. 9,430-444. LAX.P. (1968). Integrals of nonlinear equations of evaluation and solitary waves. Commun. Pure Appl. Math. 21,467-490. MANAKOV, S. V., ZAKHAROV, V. E., BORDAS,L. A., ITS, A. R.,and MATVEEV, V. B. (1977). Two dimensional solitons of the Kadorntsev-Petviashvili equation and their interaction. Phys. Lett. A . 63, 205-206. MAXON, S., and VIECELLI, J . (1974). Cylindrical solitons. Phys. Fluids 17, 8. MILES,J. W. (1977a).Obliquely interacting solitary waves. J . Fluid Mech. 79, 157-169.

Soliton Interactions in Two Dimensions

31

MILES,J. W. (l977b). Resonantly interacting solitary waves. J . Fluid Mech. 79, 171-179. MIURA,R. M. (1976). The Korteweg-de Vries equation: A survey of results. S I A M Rev. 18, 41 2-459. RAYLEICH, LORD(1 876). On waves. Pliilos. Mug. [I] 4, 257. SATSUMA, J. (1976). N-soliton solution of the two-dimensional Korteweg-de Vries equation. J . Phys. Soc. Jpn. 40, 286-290. SCOTT-RUSSELL, J. (1844). “Report on Waves,” Report of the 14th Meeting, pp. 311-390, Plates XLVII-LVII. Br. Assoc. Adv. Sci., York, England. D. W. (1973). The soliton: A new concept in SCOTT,A. C . , CHU,F. Y. F., and MCLAUGHLIN, applied science. Proc. IEEE 61, 143-1483, G . B. (1974). “Linear and Non-Linear Waves.” Wiley, New York. WHITHAM, ZABUSKY, N. J., and KRUSKAL, M. D. (1965). Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240-243. V. E., and SHABAT, A. B. (1974). A scheme for integrating the nonlinear equations ZHAKAROV, of mathematical physics by the method of the inverse scattering problem. Funct. Anal. Appl. 8,226-235.

This Page Intentionally Left Blank

ADVANCES IN APPLIED MECHANICS. VOLUME

20

Theory of Homogeneous Turbulence T. TATSUMI Department of Physics Unioersity of K y o t o Kyoto. Japan

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

A . Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Characteristic Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Moments and Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 44 46

39

111. Statistical State of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

A . Quasi-Normality of Large-Scale Motions . . . . . . . . . . . . . . . . . . . . B. Quasi-Equilibrium of Small-Scale Motions . . . . . . . . . . . . . . . . . . . Iv. Cumuhnt Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Zero Cumulant Approximation . . . . . . . . . . . . . . . . . . . . . . . . . B. Modified Zero Cumulant Approximation . . . . . . . . . . . . . . . . . . . .

50 61 65

65 74

V . Incompressible Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . .

78

A . EnergySpectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Similarity Laws of Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . C . Energy, Skewness, and Microscale . . . . . . . . . . . . . . . . . . . . . . . .

80 88 101

VI . Turbulence of Other Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . A . Two-Dimensional Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . B. Turbulence of Burgers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

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

130

106 115 127

I . Introduction Homogeneous isotropic turbulence is an idealized concept of turbulence assumed to be governed by a statistical law that is invariant under arbitrary translation (homogeneity).rotation or reflection (isotropy) of the coordinate system. This is an idealization of real turbulent motions. which are observed in nature or produced in a laboratory and generally have much more 39 Copyright @ 1980 by Academic Press. Inc. All rights of reproductionin any form reserved. ISBN 0-12-002020-3

40

T. Tatsumi

complicated structures. This idealization was first introduced by Taylor (1935) to the theory of turbulence and used to reduce the formidable complexity of the statistical expression of turbulence and thus make the subject feasible for theoretical treatment. Up to the present, a large amount of theoretical work has been devoted to this rather restricted kind of turbulence. Remarkable progress has been achieved so far in discovering the various natures of turbulence, but nevertheless our understanding of the fundamental mechanics of turbulence is still partial and unsatisfactory. In fact, turbulence combines the difficulty of a strongly nonlinear dynamical system with that of a nonconservative system. The presence of energy dissipation due to viscosity deprives turbulence of the opportunity of being in a thermal equilibrium state. Instead, what we can expect for this nonconservative system is at most a statistically stable state that all turbulent motions would approach asymptotically. Such a state could be the state of evolution in time governed by a simple similarity law or even a stationary state if the energy is supplied from outside to compensate for the viscous dissipation, but it is definitely not identical with the state of thermal equilibrium. Kolmogorov (1941a) proposed such a similarity state for small-scale components of turbulence irrespective of the variety of its large-scale structure. This state, which is assumed to be stationary, is usually referred to as the universal equilibrium state in the literature of turbulence (see Batchelor, 1953). More recent theoretical works on turbulence cast doubt on the universal validity of Kolmogorov’s equilibrium state for all kinds of turbulence, but whether universal or not this is almost the only simple state known for turbulence. In this sense, this “equilibrium” state provides us with a nice basis for the theoretical approach to turbulence, and indeed all theories of turbulence may be said to have more or less some connection with this state. Physical interest as well as the practical importance of turbulence phenomena are centered on the situation in which the effects of the nonlinear forces dominate that of viscous dissipation, or in other words, the Reynolds number characterizing turbulence is very large. In this sense, turbulence is a strongly nonlinear problem, and it is not surprising that all mathematical methods having the nature ofa formal expansion around a linearized solution do not work well or work only ineffectively for turbulence. The breakdown of the simply curtailed cumulant expansion, which leads to the occurrence of the negative energy spectrum (see Ogura, 1963), and the failure of the Wiener-Hermite expansion with a time-independent base in predicting the correct energy dissipation rate (see Crow and Canavan, 1970) are typical examples illustrating this point. Problems are resolved in some theories of turbulence by utilizing an ad hoc assumption or a model equation to supplement the exact but unclosed equations derived from the basic equations. Unphysical consequences are avoided by introducing an assumption that has some physical ground, and

Theory 05Homogeneous Turbulence

41

moreover the agreement of the theoretical results with the experimental data is attained by choosing a value of an adjustable constant. These theories are, however, not free from the trouble that it is usually difficult to tell if their results have a general validity irrespective of the assumption employed or whether they are totally dependent on a particular choice of ad hoc assumption. For instance, the derivation of Kolmogorov’s spectrum from a theory is in fact a consequence of the expression for an assumed eddy viscosity and thus not to be taken as a proper derivation from an independent source. Likewise, agreement with experiment attained by adjusting an arbitrary constant of a theory cannot be an independent check of either the theory or the experiment. The object of this chapter is not to argue about the merits or demerits of current theories of turbulence [for such a purpose, reference may be made to the reviews by Saffman (1968, 1978), Monin and Yaglom (1975), and Orszag (1977)], but is instead to present an account of a theory of homogeneous turbulence using the method of cumulant expansion. The breakdown of the simply curtailed cumulant expansion has already been mentioned above. This ill behavior of the expansion is rectified by taking into account the difference in the time scales for different length scales of turbulent motion. In practice, a further expansion is made in addition to the usual cumulant expansion assuming that the higher order cumulant has a shorter time scale than the lower order one. Retaining only the lowest order terms of the expansion as the first approximation, we obtain a closed equation for the energy spectrum that is even simpler than that due to the zero fourth-order cumulant approximation. Unlike the counterpart in the latter approximation, this equation is shown to yield positive-definite energy spectra. Statistical information about turbulence such as the energy spectrum, the energy transfer function, the decay of energy, the skewness, and Taylor’s microscale is obtained by solving this equation under appropriate initial conditions. At very large Reynolds numbers, the energy spectrum is found to satisfy different similarity laws at different wavenumber regions, and in particular the similarity in the highest wavenumber region is shown to be identical with Kolmogorov’s similarity law. The presence of the similarity laws in the energy spectrum results in the time similarity of the evolution of the energy, the skewness, and Taylor’s microscale. These similarity laws enable us to evaluate the order of magnitude of the cumulants in each wavenumber region and thus clarify the nature and the limitation of the present approximation scheme. Exactly the same approximation method can be applied to turbulence of any spatial dimension, and in the present work two-dimensional incompressible isotropic turbulence and one-dimensional Burgers turbulence are also dealt with. These non-three-dimensional turbulences have sufficient reality as such and deserve separate treatment on their own merits. They

T. Tatsumi

42

are also interesting in relation to three-dimensional turbulence since they can tell us how far the statistical properties of the latter can be universal and independent of dimensionality. Roughly speaking, two-dimensional turbulence is found to exhibit a qualitatively different nature from its threedimensional counterpart, having a vanishing energy dissipation in the inviscid limit. Burgers turbulence, on the other hand, has a qualitatively similar character and the same similarity laws of the spectrum as threedimensional incompressible turbulence. Before concluding this section a few remarks may be in order on the relevance of studying homogeneous turbulence to an understanding of the mechanics of real turbulent flows, which are usually not isotropic, not homogeneous, and associated with mean shear flows. In such a complicated turbulent flow there are interactions not only between the turbulent fluctuations of different directions and positions but also between the turbulent fluctuations as a whole and the mean flow. These interactions are totally lacking in the framework of homogeneous isotropic turbulence, so that there appears to exist a tremendous gap of tractability between the idealized turbulence and real turbulent flows. The complexity of real flows,however, seems to be somewhat compensated by the relative simplicity of the statistical quantities concerned. For instance, the energy spectrum tensor takes a simple form for isotropic turbulence, being represented by a scalar energy spectrum function, and the same is true for the velocity correlation tensor. For shear flow turbulence, the expressions of these tensors become much more complicated than the corresponding isotropic forms. However, the important quantity for shear flow turbulence is the Reynold stress, which is a limiting form of the velocity correlation tensor for vanishing separation, and the formal complexity of the latter is reduced by taking this limit. Probably a first step toward the complete understanding of turbulent shear flows will be made by filling this theoretical gap between the “energy spectrum” and the “Reynolds stress” and thus clarifying the nature of “turbulent viscosity,” relating turbulent momentum transfer with the mean shear rate. 11. Mathematical Formulation

A. DYNAMICAL EQUATIONS In the major part of this chapter we consider the turbulence that takes place in an incompressible viscous fluid governed by the equation of continuity (2.1) div u = 0,

Theory of Homogeneous Turbulence

43

and the Navier-Stokes equation of motion,

au

1

at

P

- - v Au = -- gradp - (u

- grad)u,

where u = u(x, t ) denotes the velocity, p = p(x, t ) the pressure, x the Cartesian coordinates, t time, and p and v are constants representing the density and the kinematic viscosity of the fluid, respectively. If we eliminate p from (2.1) and (2.2), we obtain a dynamical equation for u, so that the velocity u(x,t) is uniquely determined as a function of x and t if its initial value u(x,O) at t = 0 say, is given as a function of x. In order to investigate the mechanics of turbulence it is convenient to decompose the turbulent velocity field into component motions of different length scales. For this purpose the Fourier transform of the velocity u(x, t ) is considered:

v(k, t ) =

su(x, t )exp( - ik x)dx,

(W3 ~

(2.3)

where k is the wavenumber vector and the integration is taken over the whole space. For homogeneous turbulence the right-hand side of (2.3) represents a divergent integral since u(x, t ) does not vanish for 1x1 + co,so that v(k, t ) should be understood as a generalized function. Since the velocity u(x, t ) is real, v(k, t) is a complex-valued function satisfying the relation

V(- k, t ) = v*(k, t),

(2-4)

where * denotes the complex conjugate. The inverse Fourier transform of (2.3) gives

u(x, t ) = Sv(k, t )exp(ik * x)dk.

(2.5)

Substituting from (2.5) into (2.1) we obtain the orthogonality relation

k * v(k, t) = 0.

(2.4)

Taking the Fourier transform of (2.2) and eliminating the pressure p by making use of (2.6), we obtain the following equation for v(k, t ) : -

Ct

1

+ vk2

v(k, t ) =

-i

J[k

- v(k - k', t)] k [k v(k', t ) ] ]dk', 0

(2.7)

where k = (kl. Thus, like the velocity u(x,t) itself, its Fourier transform v(k, t ) is also uniquely determined as a solution of the dynamical equation (2.7) if its initial value v(k, 0) is given.

T. Tatsumi

44

B. CHARACTERISTIC FUNCTIONAL In the statistical theory of turbulence we consider the ensemble of the velocity fields u(x, t ) each starting from a random initial state u(x,O), say, satisfyinga given probability distribution. Since u(x, t ) is uniquely related to u(x,0) through a dynamical equation, u(x, t ) must have a probability distribution that is uniquely related to that of u(x,O). In this sense the turbulent velocity u(x, t ) is a randomfunction of x, and the same is true for its Fourier transform v(k, t). The complete statistical description of the random variable u(x, t ) or v(k, t ) is given by the probability distribution functional or equivalently by the characteristic functional. The characteristic functional of v(k, t ) is defined by

where the angle brackets indicate the average with respect to the probability distribution of v(k,t) at time t, and z = z(k) is a complex-valued argument function satisfying the condition Z( - k) = z*(k),

(2.9)

and vanishing sufficiently rapidly as k -+ co to make the real-valued integral on the right-hand side of (2.8) convergent. The characteristic functional must satisfy the following consistency conditions :

q o , t ] = 1,

p[Z,

t ] I s 1,

a[ -z,

t ] = @*[Z,t].

(2.10)

For homogeneous turbulence the probability distribution is invariant under an arbitrary translation of the coordinate x, so that the characteristic functional satisfies another condition that follows from (2.3) and (2.8), namely, @[zexp(ia * k), t] = @[z, t ] ,

(2.11)

for an arbitrary real constant vector a. Since the probability distribution of v(k, t ) is uniquely related to that of v(k, 0), the characteristic functional @[z, t ] also is uniquely related to @[z, 01. Actually, it can be shown that O[z, t ] satisfies the following equation, which is derived from Eq. (2.7) and the law of conservation of probability:

= JJ(kk

”@dkdk’, + k;)Aij(k + k’)zi(k + k’) 6zk(k)6zj(k’)

(2.12)

Theory of Homogeneous Turbulence

45

with (2.13) Aij(k) = 6ij - kikj/k2, where 6/6zi(k) represents the functional derivative, 6, is Kronecker's delta, and suffixes i , j , k run over values 1,2,3, the summation convention for repeated suffixes being applied hereafter. Equation (2.12) for the characteristic functional was first obtained by Hopf (1952). It provides us with the basic equation for the theory of turbulence, just like the Liouville equation for statistical mechanics, from which all the statistical information on turbulence can be derived. At the moment, however, the practical merit of the Hopf equation (2.12) is rather limited due to the lack of a mathematical method for solving the functional equation in general. A particular solution of (2.12) is not necessarily a characteristic functional, but in order to be so it must be continuous, positive definite, and satisfy (2.10), and (2.11 ) for the homogeneous case. So far only two such solutions have been obtained by Hopf (1952) and Hopf and Titt (1953) for the limiting cases of weak turbulence and vanishing viscosity. For weak turbulence, the nonlinear terms on the right-hand side of (2.7) can be neglected and hence the second-order functional derivative on the right-hand side of (2.12) also disappears, leaving a0

at

60 + v Jkzzi(k) 6zi(k) dk = O'

(2.14)

~

Hopf( 1952)expressed an exact solution of this equation in the following form:

[ :s

@ [ z , t ] = exp - - aij(k, -k)Ail(k)Ajm(k) 1

x z,(k)z,( - k) exp( - 2vkzt)dk],

(2.15)

where aij is an arbitrary tensorial function satisfying the symmetry condition

aij(k, -k)

= aji( - k, k).

(2.16)

The solution (2.15) corresponds to the normal distribution of v(k, t). In the limit of weak turbulence, (2.2) for the velocity u(x, t ) is also linearized, so that the values of u(x, t ) at different spatial points x become independent of each other. In such a situation, v(k,t), being expressed by (2.3) as an integral of u(x,t), can be considered as the sum of a large number of independent random variables. Then, it follows from the central limit theorem that v(k, t ) is normally distributed. For the limit of vanishing viscosity, on the other hand, we may expect that the viscous dissipation of the turbulent energy also vanishes and the

T. Tatsumi

46

turbulence can be statistically stationary. In such a case, the left-hand side of (2.12) disappears, leaving

ff(k,t

d2@ dkdk‘ = 0. + k;)Aij(k + k’)zi(k + k’) dZ,t(k) dzjw)

(2.17)

It was shown by Hopf and Titt (1953) that the inviscid version (v = 0) of (2.15) gives a solution of (2.17) provided that aij takes the form

afj(k, -k) = adtj, where a is a positive constant. Then, the solution is written as

[ 4f

@[z] = exp --

4,.(k)q(kjz,,,(-k)dk].

(2.18)

This solution gives again a normal distribution for v(k, t ) with the constant energy density a in the wavenumber space. It should be noted, however, that although the characteristic functional (2.18) is an exact solution of (2.17) it has no direct relevance to the real turbulent field. One reason is that the nonzero uniform energy density in the wavenumber space gives an infinite energy density in the physical space, which is of course not physically permissible. Therefore, (2.1 8) should be taken at most as an asymptotic limit of the realistic characteristic functional of the type of (2.15) for vanishing viscosity v + 0. A more serious reason lies, however, in the fundamental assumption of this case that the viscous dissipation of energy vanishes with the viscosity, so that the second term on the left-hand side of (2.12) can be neglected. The real situation is found to be the contrary so far as the three-dimensional turbulence is concerned since the rate of energy dissipation, E say, tends to a nonzero constant, 810

as v-0.

(2.19)

The existence of a finite energy dissipation E in the inviscid limit was assumed by Kolmogorov (1941a) as a premise of his theory of locally isotropic turbulence, and it is shown in this chapter that the relation (2.19) is actually satisfied for three-dimensional turbulence. C. MOMENTS AND CUMULANTS

An alternative way of expressing the turbulent velocity field statistically is to use the moments or the cumulunts of the probability distribution of the random velocity v(k, t).

Theory of Homogeneous Turbulence

47

The moments are defined as the coefficients M(")of the Taylor expansion of the characteristic functional

O[z(k), t ]

=

i" + 17 J. . . JMI:!. .in(kl,. . . , k,; t ) n. x 6(k, + . . . + k,)zl,(kl) . zl,(k,)dkl

1

n=l

* *

dk,,

* * *

(2.20)

where Mi:). . . In(kl, . . . ,k,; t)6(kl

+ . . + k,) *

(2.21) and 6 is Dirac's delta function, whose presence is required by the homogeneity condition (2.11). Likewise, the cumulants are defined as the coefficients Cfn) of the logarithmic Taylor expansion of the characteristic functional

@"z(k),tl

= exp

[,=1 7 J.. . n.

x 6(kl

JCi:).. . ln(kl,. . . ,k,;t)

+ . . + k,)zl,(kl) . . . zln(k,)dkl . . . dk,],

(2.22)

where

(2.23) Obviously, M(")and C(")are invariant under the simultaneous permutation of the suffixes ( I I , . . . ,1,) and the arguments ( k l , . . . ,k,) and, in accordance with (2.9) and (2.10), satisfy the relation

(zy)(-kl,.

. . , -k,;t)

(2.24)

=

As can easily be seen from (2.8) and (2.21), the moments are closely related to the mean velocity products in the following manner:

(ul,(kl,t) . . . ul,(k,,t) = Mi:'*. l,,(kl,.. . ,k,;t)6(kl

+ . . + k,,), (2.25) *

where use has been made of (2.4) and (2.24). Since the moments are related to the cumulants through (2.21) and (2.23), the mean velocity products are also expressed in terms of the cumulants. The first-order term may be taken

48

T. Tatsumi

to be identically zero, i.e., (ui(k1,t))

=

Ci”*(kl;t)6(kl) = 0,

(2.26)

since the mean velocity u(x, t),being constant in the homogeneous turbulence, can be made to vanish by taking a Galilean transformation. Then, the second-, third-, and fourth-order terms may be expressed as

+ k2), k2, k3 ; t)6(k1 + k2 + k3),

(2.28)

+ k2 + k3 + k4) + C‘i,?)*(kl,k2 ; t)Ci:’*(k3, k4; t)d(k1 + k2)6(k3 + k4) + C$)*(ki, k3 ;t)C:?’*(k2, k4; t)6(k1 + k3)6(k2 + k4) + Cf)*(kl,k4; t)C$)*(k,,k3; t)6(kl + k4)8(k2+ k3).

(2.29)

(vi(k1, t)u,(k2, t ) ) = Clf)*(kI,k,; t)6(k1
(2.27)

(’i(kl, t)u,(k2,t)uk(k3,t)u1(k4,t ) > = C$&‘Yki,k2,k,,k4;

t)d(k,

Since the characteristic functional satisfies Eq. (2.12), the cumulants also satisfy the dynamical equations, which are derived by substituting expansion (2.22) into (2.12) and equating coefficients of the same powers of the z,(k) as follows:

(2.30)

+ + +

with k l k2 . . . k, = 0, where k, = lkmland Crl,.. . ,nl means the summation over all permutations of (1,2,. . . ,n).

Theory of Homogeneous Turbulence

49

The first two equations may be written as

+ v(k2 + k”) with k

1

[iJ

1 kpAiq(k)JC$$k

C$’(k,k’; t ) = i

- h,k‘,h; t)dh,

[k,k’l

(2.31)

+ k‘ = 0,

[i+

v(k2 + k”

+ k”’)

[i,j,kl

1

=i

[k,k’,k”]

-

k, A&)

1

C#k, k’,k”; t )

[: s -

C$i,(k

- h, k’, k”, h; t )dh

1

C$)( - k’, k’; t)Cii)(k”, - k”; t ) ,

+

(2.32)

with k k‘ + k ’ = 0, where the summations are taken over all simultaneous permutations of (i,j, k ) and (k, k’, k”). Equations (2.30) constitute an infinite system of equations each relating the cumulant of the nth order with those of lower orders and the (n + 1)th order. Thus the cumulants of all orders are determined as the solutions of (2.30) provided that their initial conditions are specified. In this sense the set of equations (2.30) is equivalent to Eq. (2.12) for the characteristic functional and furnishes us with another complete mathematical formulation of the statistical theory of turbulence. It is, however, not possible to solve this infinite set of equations simultaneously and exactly, and in practice we have to deal with only a finite subset of them. Here arises the difficulty of unclosedness because any subset of equations involves at least one more unknown than the equations due to the presence of the (n + 1)th-order term in the equation for the nth-order cumulant. This difficulty, which is essentially due to the nonlinearity of the equation of motion (2.2), always appears in the statistical treatment of a nonlinear dynamical system by means of the moment or cumulant expansion, so that the unclosedness of equations is the central difficulty in the theory of turbulence. In fact, one of the main tasks of a theory of turbulence is to overcome this difficulty, and there are as many different assumptions for closing the equations as the theories of turbulence proposed so far. 111. Statistical States of Turbulence

The mathematical framework of the theory of turbulence is given by either Eq. (2.12) for the characteristic functional or the infinite set of equations (2.30) for the cumulants. All necessary statistical information on turbulence can be

50

T. Tatsumi

derived by solving these equations under given initial conditions. It is, however, not information about the details of particular solutions that is practically needed but rather the universal nature of solutions possessed more or less in common by all solutions irrespective of their initial conditions. In order to determine such a universal nature of solutions, we look for a physically simple state in which all statistical properties of turbulence are governed by simple laws. It is well known that an isolated dynamical system approaches asymptotically to the thermal equilibrium state, which is governed by simple statistical laws and described in terms of only a few thermodynamical variables. For a closed but nonisolated system such as a viscous fluid in motion, however, the thermal equilibrium cannot be expected to exist since the system continuously loses its kinetic energy by viscous dissipation. Such a system cannot even be stationary in time unless the kinetic energy is constantly supplied from outside in order to compensate for the viscous dissipation. In the present study we do not consider any external energy supply, avoiding an extra complexity due to the presence of an external effect, so that the system is inevitably evolutionary in time. What we can expect for such a system is at most the state of similar evolution in which all statistical quantities change similarly in time and satisfy similarity relations with each other. The similar evolution includes the notion of “self-preservation” of the statistical function, which is sometimes used in the literature of turbulence (see, for instance, Batchelor, 1953, p.148). The former has wider sense than the latter since even when a function is not self-preserving as a whole it can still satisfy a similarity law partly or different laws at different parts. Such a state of turbulence is called quasi-equilibrium, and the stationary state, if possible, is referred to as equilibrium. It should be remembered that in any case the term equilibrium is used in the literature of turbulence in a broader sense that the term thermal equilibrium.

A. QUASI-NORMALITY OF LARGE-SCALE MOTIONS Now let us investigate if there exists any quasi-equilibrium state in the evolution of homogeneous isotropic turbulence and what is the characteristic feature of the state if it exists. It has been known for many years that the turbulent velocity u(x,t) at a fixed point x obeys approximately the normal or Gaussian distribution (see Batchelor, 1953, pp.169-170), which is written for the isotropic case as p ( u l )= (2n(~:>)-’’~ exp[ - ul/@(ul>)],

(3.1) where u1 is a velocity component. The frequency distribution of the velocity in a rigid-generated turbulence measured by Van Atta and Chen (1968)

51

Theory of Homogeneous Turbulence

shows very close agreement with the normal distribution (3.1), as shown in Fig. 1. A sensitive test for normality is provided by the measurement of the moments of the distribution such as the skewness,

s = (u:>/Ku:>)3’2,

(3.2)

and the kurtosis or flatness, K

=

(4>/((u:>)’?

(3.3)

which should be equal to 0 and 3, respectively, for the normal distribution (3.1). Existing experimental results (for instance, Stewart, 1951; Frenkiel and

r

0.40

0.35

0.30

-b

-a

0.25

‘1 b

0.20

0.15

0.10

0.05

0.0 r -3

-2

I

1

1

I

-1

0

1

2

3

U,/c

FIG. I . Measured probability distribution of a velocity component u1 in a grid-generated turbulence (after Van Atta and Chen 1968). Solid line denotes the normal distribution (3.1).

T. Tatsumi

52

Klebanoff, 1967) give S = 0 and K = 2.9-3.0, within the limits of experimental accuracy, thus supporting the normality. It has been proved in Section II,B that, for the case of weak turbulence, not only the one-point distribution but the distribution functional of the velocity u(x, t )at all values of x can be exactly normal. For strong turbulence, however, the normality of the distribution functional as a whole cannot be expected, but there must be some theoretical explantion for the experimentally evident normality of the one-point distribution of the velocity. An approach using the extension of the central limit theorem to the system of weakly dependent random variables has been made by Lumley (1972), Rosenbiatt (1972), and others, but so far no rigorous proof has been obtained for the one-point distribution of the turbulent velocity u(x, t ) itself. For the velocities at two different points, either in space or in time, the joint normal distribution is expressed as P(u1,u;) = [27102(1 - R2)1’*]-1 x exp{ -(u: - 2u1u’,

+ u;’)/[2a2(1

- R2)]},

(3.4)

with u1 = ul(x,t),

u’, = ul(x

+ r, t + s),

where o2 = ( u : ) = (u‘,’),

R(r,s) = (ulu;)/02,

(3.5)

denote the covariance and the correlation, respectively. The measured twopoint distribution are found to show appreciable discrepancy from the joint normal distribution (3.4), as seen from the result obtained by Van Atta and Chen (1968) for the two-time distribution and shown in Fig. 2. Departure from joint normality is more clearly observed in the measurement of the moments of the velocity difference at the two points, such as skewness, S(r, s)

=

((4 - u1I3)/((u;

-

(3.6)

and kurtosis,

((4- u1I4)/<(u;

-

(3.7) Sometimes, the limiting forms of (3.6) and (3.7) for r = Irl + 0, s = 0, or skewness and kurtosis of the velocity derivative, @,s)

=

So = ((au1/ax,)3>/((au1/aX,)”>3/2,

(3.8)

K o = ((au,/a~l)~>/((au,/a~l)~>”,

(3.9)

are used for the same purpose.

Theory of Homogeneous Turbulence

53

0.04

0

ut' FIG. 2. Measured joint probability distribution of two velocity components u , and u', at different times (after Van Atta and Chen, 1968).Solid lines denote the joint-normal distribution (3.4)for R(0,s) = 0.686.

The experimental results of time skewness S(0, s) and kurtosis K(0,s) measured by Frenkiel and Klebanoff (1976) are shown in Figs. 3 and 4, respectively, together with the time correlation R(0,s) for comparison. Apparently, the measured values of skewness and kurtosis are considerably

T. Tatsumi

54 0.4

0.3

-

0.2

LQ

si v)

0.I

0

-0.1

0

0.2

FIG.3.

0.4 0.6 0.8

1.0 1.2

1.4 1.6 Us/M

1.8 2.0

2.2

2.4

2.6

2.8

3.0

Measured time skewness S(0,s) (after Frenkiel and Klebanoff, 1967).

-

Y

0 Y

0

0.2

0.4 0.6

0.8

1.0

1.2

1.4

1.6

1.8 2.0

2.2 2.4

2.6 2.8

Us/M

FIG.4.

Measured time kurtosis K ( 0 ,s) (after Frenkiel and Klebanoff, 1967).

0

Theory of Homogeneous Turbulence

55

different from the values 0 and 3, respectively, corresponding to the joint normal distribution (3.4). It may be observed in these figures that the discrepancy of skewness from its normal value is more persistent than that of kurtosis: S(0,s) does not vanish until R(0, s) vanishes, while K(0,s) becomes approximately 3 at about R(0,s) = 0.2. Thus, the discrepancy from joint normality appears at much shorter distance for kurtosis than for skewness, leaving a larger region of validity of the joint normal distribution for kurtosis than for skewness. Exactly the same trend is observed for the two-point correlations of higher order, RmSn(r,s)= (uyu;") lam+n,

(3.10)

where m and n are positive integers. If the distribution is joint normal, it follows from (3.4) that these higher order correlations of even order are given as functions of the second-order correlation (Fig. 5 ) by the relations R'33

=~

3 . 1= 3R ,

R'*5= R5,' = 15R, R3,3 = 3R(3

+ 2R2),

R2,2 = 1 + 2R2, R2v4 = R4*2= 3(1 R4,4 = 3(3

+ 4R2),

(3.11)

+ 24R2 + 8R4).

On the other hand, the odd-order correlations are all identically zero for the joint-normal distribution (3.4): Rm*"= 0,

for m

+ n = odd.

(3.12)

The time correlations of the fourth, sixth, and eighth order measured by Frenkiel and Klebanoff (1967) are reproduced in Figs. 6,7, and 8, respectively. The experimentally measured higher order correlations are compared with the corresponding normal curves obtained by using relations (3.1 1) and the average curve of the second-order correlation given in Fig. 5. The experimental data agree very well with the curves due to the joint normal distribution, and the deviation, which increases with increasing order of the correlation, is limited to small time intervals even for the eighth-order correlation. The measurement of the same functions made by Van Atta and Chen (1968) is in perfect agreement with the data of Frenkiel and Klebanoff shown in Figs. 6-8. Thus we can conclude that the two-point distribution of the turbulent velocity is approximated quite accurately by the joint normal distribution (3.4) so far as the even-order correlations are concerned. The situation is quite different for the odd-order correlations. The measured results for the third-, fifth-, and seventh-order correlations obtained

-.---FAVRE: X / M

-8

a

40; R ~ M 21,500 =

------STEWART: X/M = 30;Re, = 21,200

ul

e

Us/M

FIG.5. Measured time-correlation R(0, s) (after Frenkiel and Klebanoff, 1967).

Us/M

FIG.6. Measured fourth-order correlations R3*',R',', and R Z s 2 (after Frenkiel and Klebanoff, 1967). Solid lines are obtained by using the joint normal relation (3.1 1). 56

15

$'3

12

15

9

12 GAUSSIAN2'4

3

6

9

3

6

-t 0

0.2 0.4 0.6 0.8

1.2

1.0

2.4 2.6

1.8 2.0 2.2

1.4 1.6 Us/M

2.8 3.0

FIG,7 . Measured sixth-order correlations R4.2,R2.4,and R3s3 (after Frenkiel and KlebanofT, 1967). Solid lines are obtained by using (3.11).

70

I-

--I

I

0

t

0.2

l

I

I

0.4 0.6 0.8

I 1.0

I

1.2

I

'

1.4 1.6 Us/M

I

1

1.8 2,O

I

I

2.2

2.4

I

I

2.6 2.8

! 3.0

FIG.8. Measured eighth-order correlation R4*4(after Frenkiel Klebanoff, 1967). Solid line is obtained by using (3.1 1). 57

T. Tatsumi

58

by Van Atta and Chen (1968)are presented in Figs. 9,10, and 11, respectively, where the plot has been made for the composite correlations defined as

Rms"(r,s) = 3[Rm*"(r,s) - Rm9"(r,s)],

(3.13)

for the purpose of avoiding experimental inaccuracy. The discrepancy of the results with the normality of the distribution is obvious since these correlations must be zero for the normal distribution according to (3.12).

-

0.06

-z --

0.04

096e8QO

-

0

0

0

0

0

a. tJ 8b".,$.

0

N' la

k

0

0.06

0.

-0

o o b ~ ~ ~ c P o ~ 8 000

0

I

-0.20 0

I

I

I

1

I

I

I

I

I

I

i

I

0.4

0.8

1.2

1.6 Us/M

2 .o

I 2.4

I 2.8

3.2

I

I

FIG.10. Measured fifth-order correlations and R3.' (after Van Atta and Chen, 1968). Chain lines are obtained by using the fourth-order Gram-Charlier expansion (3.14).

Theory of Homogeneous Turbulence

0.5

I

0

I

I

0.8

0.4

I

I

1.2

I 2.4

I 2.0

I .6

59

I

2.8

I

3.2

Us/M

FIG. 11. Measured seventh-order correlation R6.', R5.', and R4.3 (after Van Atta and Chen, 1968). Chain lines and broken lines are obtained by using the fourth- and sixth-order Gram-Charlier expansion (3.15) and (3.16), respectively.

It is of interest to investigate if a nonnormal distribution can represent the relations between odd-order correlations as the normal distribution did for even-order correlations. For this purpose the Gram-Charlier expansion of the distribution was employed by Frenkiel and Klebanoff (1967) and also adopted by Van Atta and Chen (1968). The zeroth-order approximation of this expansion is identical to the joint normal distribution (3.4) and leads to the previously obtained relations (3.11) and (3.12). The approximations up to third order give no new relation for the homogeneous field of zero mean velocity, and the fourth-order approximation presents, for the fifth-order correlation, the relations R5.O

= 0,

R4,1 =

6R2.1 7

R3.2 =

3 ( 2~ 1)R2.1,

(3.14)

and, for the seventh-order correlations, the relations R7.0

~

0

R6,1 =

45R2.1,

R4*3= 18(1 - 2R + 2R2)RZ,'.

R5.2 =

15(4~ 1)R2.1,

(3.15)

60

T . Tatswni

Similarly, if we take the sixth-order approximation, we obtain for the seventh-order correlation, the relations R6,1 = 15(W4,1 - 3RZ,l), R 7 . O = 0, ~ 5 =~1 02 ~ ~ 4 + . 11 0 ~ -3 1 ~5 ( 4 ~ 1)jp, (3.16) R4*3= 3R4,' + 6(2R - l)R3,' - 18(1 - 2R + 2R2)RZ,'. It may be noted that, on substitution from (3.14), relations (3.16) are reduced to (3.15) for the fourth-order approximation. The measured data of the fifth- and seventh-order correlations shown in Figs. 10 and 11 are compared with the corresponding curves obtained by using the fourth-order Gram-Charlier approximation relations (3.14) and (3.15) with the curve of R and w 2 > lgiven in Figs. 5 and 9, respectively. Agreement is remarkable for the fifth-order correlations and less satisfactory for the seventh-order ones. For the latter, however, the sixth-order GramCharlier approximation can be applied using the relations (3.16) and the curves of the fifth-order correlations given in Fig. 10, in addition to the data of the lower order correlations. The agreement is significantly improved by the sixth-order approximation for all the seventh-order correlations shown in Fig. 11. It is to be noted that the higher order Gram-Charlier approximation, in addition to accounting for the odd-order correlations, gives still better agreement for the even-order correlations, which were already well represented by the joint normal relations. Frenkiel and Klebanoff (1967) applied the fourth-order Gram-Charlier approximation to the sixth-order correlation R3,3and the fourth- and sixth-order approximations to the eighth-order correlation R494,and found that the small deviations from the direct measurement that existed for the joint normal relations completely disappeared under these approximations. The measurement has been extended to some three- and four-point correlations in a grid-generated turbulence by Van Atta and Yeh (1970). The measured correlations were compared with the corresponding results for the joint normal and the Gram-Charlier distributions, and remarkably good agreement was attained for all cases examined except for small values of the point separations. In view of the remarkable success of the Gram-Charlier approximation for representing the relations between the correlations of all orders, although the test has been made only for the eighth-order and the four-point ones at most, the probability distribution of the turbulent velocity is represented quite well by a finite-order Gram-Charlier approximation. Since the GramCharlier approximation has the nature of the expansion around the joint normal distribution, this type of approximation may be called the quasinormal approximation.

Theory of Homogeneous Turbulence

61

Summarizing the statistical information on turbulent field obtained above, it may be concluded that the large-scale structures of turbulence represented by the correlations at large distances satisfy the normal distribution quite well, while the small-scale structures are governed by a distribution significantly different from the normal distribution but still expressible in terms of a quasi-normal approximation. As stated in the beginning of this section, we are looking for a simple statistical state of the turbulent field. The above conclusion seems to guarantee the existence of such a simple state characterized by the quasi-normal distribution of the turbulent velocity.

B. QUASI-EQUILIBRIUM OF SMALL-SCALE MOTIONS It was seen in Section III,A that while the statistical state of turbulence at large scale is expressed very closely by the normal distribution, the deviation from normality becomes significant at smaller scales. What kind of the simple state can we then imagine for the small-scale structure of turbulence, admitting the existence of such a state? An answer is given by Kolmogorov’s universal equilibrium theory of turbulence (Kolmogorov, 1941a). Kolmogorov deals with turbulent motion in general, which is not necessarily isotropic, not even homogeneous or stationary, and may be associated with a mean shear flow, but instead of dealing with turbulent motion as a whole he restricts consideration to its small-scale components. Usually, turbulence is generated either by the external excitation ofrandom motion at an initial instant or by the growth of small disturbances due to the instability of the shear flow. The former corresponds to the generation of grid turbulence in a wind tunnel, while the latter takes place in practically all shear flows with or without solid boundaries. In any case, the fluctuations first excited have some definite length scale l o , say, comparable to the linear dimension of the turbulence-generating mechanism, but they are in their turn unsteady and the fluctuations of the smaller length scale are generated. Such a process of successive refinement of turbulent fluctuations may be carried out until the length scale of the higher order fluctuations becomes so small that the effect of the viscous damping finally prevents the formation of finer fluctuations. This process may also be accounted for as the successive production of the higher harmonics of the original fluctuations due to the nonlinear inertia terms of the equation of motion (2.2) and the counteraction of the viscous-damping term, which is stronger at smaller length scales. In view of the chaotic mechanism of the modulation of motion from the fluctuations of lower orders to those of higher orders, it may be natural to assume that in the domains of space, whose dimensions are small compared with l o , the fine fluctuations of the higher orders are governed by a statistical

T. Tatsumi

law that is approximately homogeneous and isotropic in space and stationary in time even when the flow as the whole is time varying. This notion of local isotropy is narrower than that of the isotropy introduced by Taylor (1935) in the sense that it requires time stationarity, but wider insofar as it is only concerned with the small-scale fluctuations instead of turbulent fluctuations as a whole. The above situation may be described in the wavenumber space as follows. The energy is supplied to the fluid from outside by exciting the fluctuations of the wavenumbers around k x k , = l/lo. Some portion of the energy contained in this wavenumber region is transferred successively to the fluctuations of higher wavenumbers through the action of inertia forces, which are represented by the right-hand side of (2.7). The effect of viscous damping, which is proportional to vk2 according to (2.7),is not appreciable for the energy transfer at low wavenumbers, but becomes more significant at higher wavenumbers, and eventually the energy transfer is counterbalanced by viscous dissipation at the highest wavenumber range, k 2 k d , say. The hypothesis of local similarity applies to those fluctuations of higher wavenumbers k >> k o , which are not excited directly by the external forces, but owe their existence entirely to the cascadic energy transfer in the wavenumber space. Now, the turbulent fluctuations that already satisfy the condition of the local isotropy are supposed to be in an equilibrium state governed by a similarity law and dependent only upon some external parameters characterizing the system of concern. Kolmogorov takes as such parameters the rate of energy dissipation E and the viscosity v , which determine the input and output of the energy to and from the system, respectively, and makes the hypothesis of local similarity that for the locally isotropic turbulence the distributions are uniquely determined by the parameters E and v. In this context, it should be noted that the existence of a finite energy dissipation, such as in (2.19), is implicitly assumed as a premise to this hypothesis. It then immediately follows from the dimensional analysis that the energy spectrum at the equilibrium range k >> k,, is expressed as

E(k) = ~ ~ / ~ ~ ' / ~ E , ( k / k d ) ,

(3.17)

with

kd = &1/4,,,- 314,

(3.18)

where E , means a nondimensional function. If the hypotheses of the local isotropy and the local similarity are actually observed by the turbulent fluctuations of small scale, the function E , must be of a universal form irrespective of the variety of the large-scale motions. In this sense the spectrum (3.17) is referred to as the universal equilibrium spectrum. So far the functional form of E , is not determined. If the viscosity is very

Theory of Homogeneous Turbulence

63

small, or more adequately, if the Reynolds number R = uo/vk,, uo being the characteristic velocity of turbulence, is extremely large, the distribution at the equilibrium range would become independent of the viscosity v at relatively lower wavenumbers of the range. Thus, Ko!mogorov makes the hypothesis of inviscid similarity that for the larger components of the locally isotropic turbulence the distributions are uniquely determined by the parameter E and not dependent on v. Then the function E , in (3.17) must be proportional to V5I4, so that E(k)= K ~ ~ l ~ k - 5 1 ~ (3.19) in the range (3.20) k o << k << kd, where K denote a nondimensional universal constant that is independent of the viscosity v, or the Reynolds number R, and the details of the largescale structure of turbulence. The wavenumber range (3.20) is usually called the inertial subrange and the spectrum (3.19) is referred to as Kolmogorov’s inertial subrange spectrum. The spectrum (3.19)has been observed by several experimental measurements and now its general consistency with experimental results at very large Reynolds numbers is well established (see Grant et al., 1962; Gibson, 1962,1963; Gibson and Schwarz, 1963; Wyngaard and Pao, 1972; Schedvin et al., 1974; Champagne, 1978). The notion of local isotropy demands the time stationarity of the smallscale structure of turbulence, so that the energy spectrum E(k) and the energy dissipation E are both taken to be stationary in time in the foregoing argument. However, Kolmogorov’s hypotheses of the local similarity and the inviscid similarity can be extended to the nonstationary case by allowing the time dependence of the energy dissipation E. Then the small-scale structure of turbulence is no longer in the equilibrium state, but assumed to be in the quasi-equilibrium state, and the time-dependent energy spectrum E(k, t ) and the energy dissipation ~ ( tsatisfy ) exactly the same relationship as (3.17) and (3.19). Kolmogorov (1941b) used this extended hypotheses for analyzing the decay of the energy of turbulence. Since we are dealing with time-evolutionary turbulence here, Kolmogorov’s similarity law (3.17) and the inertial subrange spectrum (3.19) will provide the description of the quasi-equilibrium state of the small-scale structure of evolutionary turbulence. Before concluding this section it should be noted that Kolmogorov’s inertial subrange spectrum (3.19)has been derived entirely from the following set of hypotheses: (1) independence of E and v, (2) local isotropy, (3) local similarity, (4) inviscid similarity.

64

T , Tatsumi

These hypotheses are considered to represent the essential features of the small-scale structure of real turbulence at large Reynolds numbers. In this sense, Kolmogorov’s conclusion is taken to be universally valid for smallscale components of all turbulent motions irrespective of the details of their large-scale structure. It should be remembered, however, that its universality is solely dependent on the validity of the above hypotheses. For instance, hypothesis (1) is not satisfied for two-dimensioaal incompressible turbulence, so that the spectrum (3.19) is obviously not applicable to this turbulence (see Section VI). On the other hand, hypothesis (1) is satisfied for the one-dimensional turbulence of Burgers, but nevertheless the inertial subrange spectrum of this turbulence is at variance with (3.19) (see Section VI). The reason for this discrepancy should be attributed to the failure of hypothesis (3) for the turbulence of Burgers. A similar discrepancy can occur, to a lesser extent, even for threedimensional turbulence in an incompressible fluid. For instance, for turbulence of enormous linear dimension, such as atmospheric turbulence, the energy dissipation E changes slowly in space and time, reflecting the macroscopic inhomogeneity of the intensity of turbulence (Obukhov, 1962). In such a case, if we assume that Kolmogorov’s hypotheses are valid for the microscopic region at a given dissipation value E , the statistical law for the macroscopic region will show a deviation from the -5/3 power law. Assuming that E has a log-normal distribution, Obukhov obtained a correction to Kolmogorov’s result, and Kolmogorov (1962) made a refinement of his original hypotheses to take into account the variation of E. The fluctuation of the energy dissipation rate in space and time is, in principle, not limited to the rubulence of extremely large linear dimension. In fact, the energy dissipation at the point x and the time t,

ru.auY

8(x,t) = - 2 2 axj

+A axi

,

(3.21)

is a fluctuating quantity like the turbulent velocity u(x, t ) itself, and the quantity E considered so far is actually the average of this fluctuating 2(x, t ) over the space-time region for which the homogeneity of turbulence is assumed: E

= -E = (qx, t)).

(3.22)

As pointed out by Landau (see Kolmogorov, 1962; Landau and Lifshitz, 1959, p. 126), we may have to consider the effect of fluctuations of 2 on the small-scale properties of turbulence. Various attempts have been made so far to refine the theory of the local structure of turbulence along the lines of thought of Kolmogorov and Obukhov mentioned above [a detailed account ofworks ofthis sort is given by Monin and Yaglom (1975),Vol. 2, Section 25)].

Theory of Homogeneous Turbulence

65

A refined expression for the inertial subrange spectrum is given by E ( k ) = K’2Z/3k-5/3(Lk)-p/9,

(3.23)

where K ’ is a nondimensional constant, p the logarithmic dispersion of I, and the average Z is taken over the macroscopic region of the characteristic length L. The experimental value of p is found to be approximately 0.4-0.5, so that the deviation of (3.23) from the -5/3 power law (3.19) is too small to be detected experimentally. Aside from the numerical smallness of the correction, these attempts of modifying Kolmogorov’s hypotheses to take into account the fluctuation of the energy dissipation do not seem to have strong physical grounds unless they are to be applied to a substantially inhomogeneous turbulent field like that of atmospheric turbulence. It is true that there is ample experimental and theoretical evidence indicating the spottiness or the intermittency of the small-scale structure of turbulence (see, for instance, Orszag, 1977, Section 3.3; Frisch et al., 1978). Kolmogorov’s hypotheses require that such a small-scale structure is homogeneous and similar in the statistical sense as to be solely determined by the global energy dissipation E (=Z).If the intermittency of the small-scale structure could invalidate these hypotheses there would be no evident reason to assume that the microstructure of a certain scale is determined by the local average of the energy dissipation E taken over the corresponding microregion.

IV. Cumulant Expansion It has been well established by the investigation of the statistical state of turbulence made in Section I11 that the large-scale components of turbulence obey a quasi-normal distribution that can be approximated by the first few terms of the expansion around the normal distribution, while the small-scale motions are in a quasi-equilibrium state that is almost entirely determined by a few external parameters. These results enable us to construct a rough idea about what we can expect as the universal character of turbulence, and such an idea can be used in turn as a guiding principle for finding a suitable approximation scheme for dealing with the mathematical equations of turbulence.

A. ZEROCUMULANT APPROXIMATION The property of the quasi-normality of the statistical distribution of turbulence can be used for deriving a set of closed equations for the cumulants from the infinite system of the cumulant equations (2.30).If we neglect the

T. Tatsumi

66

fourth-order cumulant C(4)in (2.29), we obtain the following relationship between the mean velocity products of fourth and second orders:

where use has been made of (2.27).Exactly the same relation as (4.1) for the velocity u(x,t) in the physical space is also obtained by assuming a jointnormal distribution for the four velocity components. In this sense this relation may be called the zero fourth-order curnulant relation or the jourthorder normality relation. The zero fourth-order cumulant relation was first introduced to the theory of turbulence by Chou (1940)* and Millionshtchikov (1941) for the purpose of obtaining closed equations for the second- and third-order mean velocity products. In these earlier treatments, however, the rigorous consequence of the relation was not pursued, but different assumptions were made for evaluating the mean pressure-velocity products. The proper mathematical formulation according to this relation and the investigation of the closed equations thus obtained were first made independently by Proudman and Reid (1954) and Tatsumi (1955, 1956, 1957). The closed equations of C'2) and C(3)are immediately derived by neglecting the C4)terms in the pair of equations (2.31) and (2.32) as follows:

(k +

risn

2vk2) CjS)(k, - k ; t ) = i

1

k , Ai,(k)

lk, - kl

x JC',:b(k

[

- h, - k, h; t) dh,

(4.2)

1

+ v(k2 + k" + Idr2) C@(k, k', k"; t ) M.kl

=

-1

'

C

k , Ai,(k)C$)( - k', k'; t)Cii'(k",- k"; t).

(4.3)

[k,k',k"]

The set of equations (4.2)and (4.3)constitutes the cumulant equations based upon the zero-fourth-order-cumulant approximation or the fourth-ordernormality approximation.

* This paper, which recently became available to the author, is probably the earliest work concerning the use of the zero fourth-order cumulant relation in turbulence.

Theory of Homogeneous Turbulence

67

The tensorial complexity of these equations is largely reduced for the case of isotropic turbulence. For isotropic turbulence, C(’) is written as

C$’(k, - k; t ) = 4 ( k ,t )Aij(k), (4.4) where 4 ( k , t ) is a scalar function. The energy of turbulence per unit mass is expressed as

a(t)= 1 / 2 ( ( u ( ~ , t )=( ~1/2JC\:)(k, ) -k;t)dk = 4n JOm

k 2 4 ( k ,t ) d k = JOm E(k, t)dk,

(4.5)

where E(k, t ) = 4nk24(k,t).

(4.6)

The function +(k, t ) represents the energy density at the point in wavenumber space separated from the origin by a distance k, while E(k, t ) gives the surface density of energy on the spherical shell of radius k centered at the origin. The former is called the energy spectrum density and the latter the energy spectrum function. Similarly, the integral of Cf3’on the right-hand side of (4.2)is written as

J

(3) cpjq(

k

1

- h, - k, h; t )dh = - -+(k, t ) [ k , A .(k)

2k

+ k, A Jk)],

(4.7)

where +(k, t ) is a scalar function and the factor - i/2k has been inserted for later convenience. Substituting from (4.4)and (4.7)into (4.2)and (4.3),we obtain the following scalar equations:

k,,2 + p k k k”(1 - p2)dk‘dp, (k2k2 where k”’ = k2 + k” + 2pkk‘, and x

)

W , O )=0

(4.9)

(4.10)

has been assumed as the initial condition. On eliminating +(k, t), the above equations determine the evolution of the energy spectrum density +(k, t ) in time if its initial form 4 ( k ,0) is specified.

T. Tatsumi

68

Equation (4.8) may also be written in terms of the energy spectrum function as

($ + Z v k f ) E ( k ,t)

= T(k,t),

(4.11)

where

T(k,t ) = 4nkZll/(k,t )

(4.12)

is called the energy transfer function. It may be obvious from the antisymmetry of the integrand of (4.9) with respect to k and k' that the present expression of T(k,t) given by (4.9)and (4.12)satisfies the condition

Jr

T(k,t ) d k = 4~

:J

k211/(k,t ) d k = 0,

(4.13)

which must be observed by the energy transfer function. Now Eq. (4.8) or (4.11) for the energy spectrum supplemented by expressions (4.9) and (4.12) for the energy-transfer function furnishes us with the basic equation for the theory of turbulence based upon the zero-fourthorder-cumulant approximation, and various statistical information on turbulence according to this theory can be derived from solutions of this equation. It is easily seen that, for a strictly inviscid case v = 0 or R = 00, the zerofourth-order-cumulant theory conserves energy :

a J" E(k, t)dk = 0,

d dt

- &(t)= -

at

(4.14)

0

which immediately follows from (4.5),(4.11),and (4.13). If we define d as the enstrophy or half the mean square vorticity, so that 2 ( t ) = +(Iw(x, t ) l z ) ,

w ( x , t ) = rot u(x, t),

(4.15)

it is expressed for isotropic turbulence as d(t)= f(lrotu(x,t))2) = $fk2C!T)(k, -k; t ) d k = 47c Jam

k44(k, t )dk = Jam

kZE(k,t ) dk.

(4.16)

Thus, for a viscous case (v > 0 or R < co) it follows from (4.5),(4.11),(4.13), and (4.16) that the rate of energy dissipation is given by E(t) =

d -- &(t) = 2vd(t). dt

(4.17)

Theory of Homogeneous Turbulence

69

Under the zero-fourth-order-cumulant approximation it can be shown for the inviscid case (v = 0) that the enstrophy 2 ( t ) generally increases in time and eventually diverges at a finite time. If v = 0, Eqs. (4.8)and (4.9)reduce to

x

kr,2 + p k k k"(1 - p 2 ) d k dp.

)

(,2k'2

(4.18)

Multiplying Eq. (4.18) by k4 and integrating with respect to k, we obtain after a tedious calculation a remarkably simple equation for 9(t):

d2 dt2

-9 ( t ) = 2/32(t)',

(4.19)

which was first derived by Proudman and Reid (1954). Equation (4.19)is immediately integrated once to give -9(t) d dt

=

2/3[2?(t)3- 9 03 1 112,

(4.20)

where 2l0 is the value of 9 corresponding to the time at which d 9 / d t = 0. The general solution of (4.20)is expressed in terms of the Weierstrass elliptic function (see, for instance, Abramowitz and Stegun, 1964, Section 18) as follows : (4.21) where t , is a positive constant. The B function is a doubly periodic function of the complex variable s, but only its real period 0 I s I 2s0, where so 1.53, is relevant to us. The behavior of B in the real period is depicted in Fig. 12. In particular, B has a double pole at s = 0 and 2s0, and d 9 / d s = 0 at s = so. It follows from (4.11) and (4.16)that, for v = 0,

d 9(t)= dt

-

som

k2 T(k,t ) dk,

(4.22)

so that the initial condition (4.10)with (4.12) gives d 9 / d t = 0 at t = 0. Thus, t = 0 must coincide with s = so, and therefore from (4.20)and (4.21),

20 = 9(0),

t, = (3/21'39;12)so.

(4.23)

Around the double pole at s = 2s0, the function P is expanded as

P(s)= (s - 2s0)-2

+ O[(s - 2s,)4].

(4.24)

T. Tatsumi 14-

B

1

12

-

10

-

8-

\r

6-

4-

2-

I

I

I.o

0.5

0

I

1.5

I

2.0

s/so FIG. 12. Growth of the enstrophy 9(t)(after Proudman and Reid, 1954).

According to solution (4.21), the enstrophy 2(t)increases from its initial value 2!o at t = 0 monotonically in time for t > 0 (s > so) and becomes infinite as t -+ t, (s 2s0) as -+

2(t)z 9(t, - t ) - 2 .

(4.25)

Thus, for the inviscid case v = 0, the enstrophy 2(t)can remain finite only for a finite time period 0 < t t,, and diverges at t = t,. Such catastrophic behavior of the enstrophy 2(t)gives an interesting consequence to the energy ) the relation (4.17),such that dissipation ~ ( tthrough E-+O

as v + O ,

(4.26)

for the period 0 5 t t,, but E is indeterminate for the period t 2 t,. The indeterminacy of E for the latter period makes us suspect the existence of a finite energy dissipation (2.19) in this period, and it is shown in Section IV that this is actually the case. Next, let us examine the asymptotic form of the energy spectrum in the inviscid limit as v -+ 0, or the limit of infinite Reynolds number as R = uo/vk, -+ 00. We assume the existence of a quasi-stationary state in which the energy dissipation is negligible, that is,

a

- E(k, t ) z 2vk2E(k,t ) x 0. at

(4.27)

Theory of Homogeneous Turbulence

71

For such a state, Eq. (4.11) for the energy spectrum becomes T(k,t ) M 0, (4.28) and it was shown by Tatsumi (1960) that (4.28) is asymptotically satisfied by the following power function spectra:

r

(4.29)

E ( k , t ) = Ak-’,

for vk’t << 1,

(4.30)

A k -’ ,

for vk’t >> 1,

(4.31)

where A is a positive constant. The first spectrum (4.29), which gives an equipartition of energy 4(k, t ) = const in the wavenumber space, corresponds to the state governed by the normal distribution (2.18),so that its relevance to the real state of turbulence is exactly the same as that described before for distribution (2.18). The other two spectra (4.30)and (4.31), on the other hand, are shown later to represent the asymptotic forms of the spectrum realized at large Reynolds numbers. It may be interesting to note that neither (4.30) nor (4.31) is exactly in accordance with the Kolmogorov inertial subrange spectrum (3.19), but their exponents - 2 and - 1 bracket Kolmogorov’s value - 5/3. Unfortunately, however, the equations for the energy spectrum (4.8) and (4.9), or (4.11) and (4.12), are found to yield unphysical results, as Ogura (1963)discovered by numerical integration of the equations that the spectrum E(k,t) becomes negative over a finite wavenumber range if the Reynolds number is sufficiently large. Figures 13 and 14 show the numerically calculated spectra for the initial condition E(k, 0) a E0(k/k0)4exp[ -(k/k0)’] and the initial Reynolds number R,(O) = (n1’4/2’/2)(Eo/ko)’/2/v= 7.2 and 14.4, respectively. At the lower Reynolds number, E(k, t ) remains nonnegative at

k/ko

FIG. 13. Evolution of the energy spectrum E ( k , t ) in time (after Ogura, 1963). R,(O)

=

7.2

T. Tatsumi

72 0.6

c

-0.1

L

k/ko

FIG.14. Evolution of the energy spectrum E(k, t ) in time (after Ogura, 1963). R,(O)

=

14.4.

all wavenumbers, but the evolution of E(k, t ) is not very much different from that due to the viscous dissipation alone. At the higher Reynolds number, on the other hand, the solution becomes unphysical due to the appearance of negative values of E(k, t), which must be nonnegative as the energy density in the wavenumber space. The occurrence of the negative spectrum gives rise to oscillation and eventual divergence of the solution, and therefore the asymptotic spectra (4.30)and (4.31) are not attainable at least from the initial condition examined above. Such a consequence of the zero-fourth-ordercumulant approximation seems to cast serious doubt on its validity at large Reynolds numbers. The same kind of failure is found to be shared even by zero-cumulant approximations of higher orders. Kawahara (1968) advanced the approximation a step further by working out the energy spectrum of the turbulence of Burgers under the zero-fifth-order-cumulant approximation. It was found that the spectrum of this approximation still takes negative values at large Reynolds numbers, but it does so at higher wavenumbers and larger Reynolds numbers than those corresponding to the fourth-order approximation. A similar trend in the solution was also observed by Tanaka (1969, 1973), who calculated numerically the energy spectrum of the inviscid turbulence of Burgers using the zero-fifth-order-cumulant approximation and the GramCharlier expansion truncated at various orders. The occurrence of the negative spectrum is not excluded by any approximations examined, but the wavenumber of first appearance of negative values becomes higher for a higher order approximation and the spectrum in the energy-containing range seems to converge to a limiting spectrum as the order of approximation increases. Thus, it may be concluded that a simply truncated cumulant expansion or the Gram-Charlier expansion cannot be free from the occurrence of a negative energy spectrum although it gives a good approximation in the lower wavenumber region.

Theory of Homogeneous Turbulence

73

It is rather difficult to find a good physical explanation for the failure of the zero-cumulant approximation in general. In view of the approximate normality of the large-scale motions of turbance described in detail in Section III,A, the expansion around the normal distribution may be expected to give a good approximation at least for the large-scale motions. Nevertheless, the zero-fourth-order-cumulant approximation leads to the appearance of the negative spectrum in the energy-containing range and not in the higher wavenumber region in which quasi-normality is undoubtedly a poor approximation (see Fig. 14). Such behavior of the energy spectrum reminds us of a kind of nonlinear oscillation, and in fact the set of equations (4.8) and (4.9), or (4.1 1) and (4.12), for the zero-fourth-order-cumulant approximation has the nature of the equation of nonlinear oscillation in the inviscid limit v -+ 0. Orszag (1970) examined the nature of the zero-cumulant approximation applied to a system governed by the inviscid truncated Navier-Stokes equation and concluded that the lack of proper relaxation time in the inviscid form of the zero-cumulant approximation is responsible for the undamped oscillation of the solution around the equilibrium state, and the occurrence of the negative energy spectrum is accounted for as a manifestation of this fundamental weakness of the approximation. In the viscous case v > 0, Eq. (4.9) includes the damping factor which decreases monotonically as time t’ exp[ - v(k2 + k” + / Y 2 ) ( t- t)], goes back from the present t’ = t to the past t’ - t -+ - co, so that the relaxation time for the zero-fourth-order-cumulant theory is of the order of ( v k 2 ) - ’ . For very small v, the damping effect due to this factor becomes too weak and the relaxation time too long at a finite k to suppress the nonlinear oscillation. In order to overcome this weakness of viscous damping it was considered by some authors that some nonlinear effects should be taken into account to cut off the damping factor and thus reduce the relaxation time more efficiently than the viscosity (see Orszag, 1977, Sections 4.6-4.7). The proposed nonlinear effects, or the “nonlinear scrambling of eddies” in a more intuitive expression, should have a relaxation time independent of the viscosity. A relevant choice for this quantity may be to take the time scale of the inertial subrange, which is proportional to ~ - ~ / ~according k - ~ / ~ to dimensional analysis. Various forms of the eddy viscosity were proposed by several authors for expressing the effect of nonlinear scrambling (Edwards, 1964; Kraichnan, 1964, 1971; Herring, 1965, 1966; Leith, 1971). In these theories, damping due to molecular viscosity is assumed to vanish in the inviscid limit and, in order to avoid the lack of relaxation time in this limit, an eddy viscosity supplements the molecular viscosity or, more specifically, the viscous damping of the third-order cumulant. Concerning the consequences of this kind of approximation, reference may be made to the above papers and a numerical study by Herring and Kraichnan (1972).

14

T. Tatsumi

There is, however, a common misunderstanding about the role of viscosity on energy dissipation in all of the above arguments necessitating some kind of eddy viscosity. It is true that viscous damping vanishes with viscosity at a finite wavenumber k, but this by no means implies that the total energy dissipation vanishes with viscosity as in (4.26). In the real situation, the energy transfer function, which itself represents nonlinear effects, transfers energy to a sufficiently high wavenumber region to make the energy dis; k2E(k,t ) d k remain finite in the inviscid limit v --* 0. Here sipation E = 2v 1 lies an essential difference between the argument based upon an identically inviscid system in which v = 0 and one dealing with the inviscid limit v + 0 and keeping a small but nonzero viscosity in mind. The transfer function of the zero-fourth-order-cumulant theory defined by (4.9) or (4.12) is indeed powerful enough to transfer energy to the quasiequilibrium range and thus produce a finite energy dissipation. The actual trouble is instead that the effect of the energy transfer is so strong that it even evacuates the energy from the energy-containing range to give rise to the negative spectrum. It is shown in Section IV,B that such trouble is avoided by taking into account an appropriate scaling for the quasi-equilibrium range. B. MODIFIED ZEROCUMULANT APPROXIMATION

It was seen in Section IV,A that the breakdown of the zero-fourth-ordercumulant approximation is due to an undamped oscillation of the nonlinear system (4.8) and (4.9), or (4.11) and (4.12), having a small damping factor vk2, which is very small at a finite wavenumber k for very small viscosity v. A practical way of suppressing this undesired oscillation may be to introduce some artificial damping effect in the form of an eddy viscosity that is independent of the molecular viscosity. Our concern here is, however, not to try to overcome the difficulty by introducing any artificial device, but to derive a unidirectional approach to a quasi-equilibrium state from the framework of the zero-cumulant approximation by making an additional approximation appropriate to the asymptotic state. As seen in Section III,A, the large-scale structure of turbulence is approximated very well by the normal distribution and the deviation of the real distribution from normality appears only with the small-scale components of turbulence. In terms of the cumulant expansion of the characteristic functional, this situation indicates that the higher order cumulants Cf3), C(4),. . . are generally small in magnitude compared with the second-order cumulant C(2),but they become significant at higher wavenumbers. If we take the above situation as the basis of an approximation we can assume that being significant at wavenumbers higher than the third-order cumulant 03), the characteristic wavenumber of C(’), has a shorter relaxation time (vk2)-

Theory of Homogeneous Turbulence

75

than that of C(’). This assumption may be formulated mathematically as follows. First, expand the product of the 4 on the right-hand side of (4.9) into a Taylor series in time. Then, substituting the series into (4.9) and integrating each term with respect to t’, we obtain the following expression for $ ( k , t ) :

x

1

k,,2 + p k k kI2(1 - p2)dk’dp, L k Z k I 2

where @,(t) =

Ji tlmexp[- v ( k 2 + k’* + k”2)t‘]dt‘.

(4.32)

(4.33)

It may be easily seen that for v ( k 2 + k’2 + k”’)t << I,

(4.34) and for v(k2

+ k 2 + k“2)t >> 1, +

@,(t) z m ! [ v ( k 2

kI2

+ k”2)]-(m+1).

(4.35)

The assumption of a shorter time scale of C3)compared with C(2)amounts to neglecting the change of q5 in (4.9)in time during the time of finite variation of $. Thus, as a first approximation we may take only the first term of the series on the right-hand side of (4.32) and obtain the expression

X

X

[$

+ pkk’] kr2(1- p 2 ) d k ‘ d p .

(4.36)

The same equation was already derived by Tatsumi et al. (1978), using a method of multiple-scale expansion. Since, however, the ordering of the cumulants assumed in this paper is found not necessarily compatible with that of the result obtained, the justification of the approximation based on this idea is not employed here. The approximation used in (4.36)is formally equivalent to updating the time of the functions 4 from the past t’ to the

76

T . Tatsumi

present t, a procedure sometimes referred to as “Markovianization” (see Orszag, 1977, Section 4.7) after the name of the Markov process, whose distribution in the future is entirely determined by the present distribution and is not dependent on its past history. In the eddy viscosity theories, this modification is made a posteriori without fundamental justification, but it may be interesting to observe that these theories owe their physical realizability almost entirely to this Markovian modification and not to a particular choice of the functional form for the eddy viscosity. In this context it should be noted that expression (4.36) has been obtained as the first approximation of the formal Taylor expansion of the 4 terms in (4.321, so that the extent of the validity, or the invalidity, of this approximation can be checked by comparing the order of magnitude of the first approximation solution with those of higher approximations. It may be seen from (4.35) that the approximation of (4.36) is valid in the highest wavenumber region, k >> (vt)-’/’ for finite t, since 0, decreases successively by the factor (vk2)- << 1. This ordering does not apply to other wavenumber regions, but it follows from (4.34)that in the lowest wavenumber for a finite t , the approximation remains good for region, k << (vt)absolutely small time t << 0[4/(d4/at)],since then 0,decreases by the factor t. Thus it may be said that the principle of the present scheme of approximation is to employ the ordering of the cumulants that is valid only in the highest wavenumber region, though having an infinite extension, throughout the whole wavenumber space. In this aspect the present approximation has some analogy to Oseen’s approximation for slow viscous flow (see Lamb, 1932, p. 609). It is shown later that the highest wavenumber region in which this approximation is valid coincides with Kolmogorov’s quasi-equilibrium range described in Section III,B and in this sense the present scheme of approximation may be called the quasi-equilibrium zero-cumulant approximation, It can also be confirmed by the result of the present approximation that the neglected contribution from the fourth-order cumulant C4)to the third-order cumulant C3)is in fact of the same order of magnitude as the contribution from the 0, term to C(3’in the quasi-equilibrium range. Hence, all the contributions from the 0, terms, m 2 1, are found to be minor compared with that from C4)in the same wavenumber range. Such a consideration of the order of magnitude in the quasi-equilibrium range gives some explanation as to why an approximate solution of the zero-fourth-order-cumulant equation should be considered more meaningful than the exact solution of the same equation. The same reasoning, however, does not justify neglecting C4)in the same wavenumber range, and the reason for this approximation should be sought in the phenomenological quasi-normality of the large-scale components of turbulence.

Theory of’Homogeneous Turbulence

77

Now, (4.Q with $ given by (4.36), gives the equation for the energy spectrum due to the quasi-equilibrium zero-fourth-order-cumulant approximation, which may be written for E(k,t) as 1

1 - exp[- v(k’ v(k’ k”

+

+ k” + k”’)t] + k”’)

x [k2E(k’,t ) - k ” E ( k , t)]E(k”, t ) x

(F

k’k’’

+ pkk‘) k“-’(l

- p2)dk’dp.

(4.37)

All statistical information of turbulence derivable from E(k, t ) and T(k,t ) is obtained by solving this equation under appropriate initial conditions. A remarkable property of (4.37) proved by Orszag (1977) is that the spectrum E(k,t) remains positive provided it is so initially. Using the current notation, the proof may be stated as follows. Suppose that E(k, t ) vanished for the first time at a certain wavenumber, k = k , say. Then, (4.37) reduces at k l to

a

Ji

at E ( k 1 7 t, = k: lorn

1

+

+

+ pklk’

1

1 - exp[v(k: k” k“’)t] ( k t + kf2 + klfz)

x E(k’,t)E(k”,t)

(k:k‘2

k”-’(l

- p’)dk’dp.

(4.38)

Changing the integration variables from ( k ,p) to ( k , k”) and using the relation (4.39) we can rewrite (4.38) as

d

-E(k,,t) = dt

~

16k1

so”

kl

+k‘

lk1-fc’i

1 - eXp[ - v(k: v(kf k”

+

x E(k’, t)E(k”, t)[k:(k” x

[(k’ + k”)’

-

+ kI2 + k”’)t]

+ k”’)

+ k”’) + (k” - k”’)’]

k : ] [ k i - (k’ - k”)2](k’k”)-3dk’dk’’, (4.40)

where use has been made of the symmetry of the integrand with respect to k’ and k”. Since the integrand of the right-hand side of (4.40) is positive

definite,

T. Tatsmi

78

so that E(k, t ) cannot become negative at k, and therefore remains positive everywhere. This proof guarantees that the present approximation is actually free from the physical contradiction of the occurrence of the negative energy spectrum accompanying the simple zero-cumulant approximations. It may be obvious from the manner of the above proof that the positiveness of the energy spectrum is shared by all eddy-damping theories that include eddy viscosity terms, q(k), q(k'), and qfk") say, in place of the terms vk', v&', and vk"' on the right-hand side of (4.36), respectively.

V. Incompressible Isotropic Turbulence The energy spectrum equation (4.37) for incompressible isotropic turbulence was solved numerically by Tatsumi et al. (1978)and Tatsumi and Kida (1980)for two typical initial conditions and a wide range of Reynolds number, and the similarity laws of the energy spectrum and other statistical quantities for large Reynolds numbers were derived. In this section, the statistical properties of this turbulence are investigated, mostly by using the results obtained in these works. As the initial condition for (4.37), the following two cases are considered: (1) E(k, 0 ) = Eo(k/ko)2exP[-(k/ko)21,

(5.1)

(11) E(k,0) = E0(k/k0)4exP[-(k/ko)21,

(5.2)

where E, and k, denote the representative values of the energy spectrum function E(k, t) and the wavenumber k, respectively. Cases I and 11, which correspond to a finite and vanishing value of the energy density 4(k, t ) = E(k, t)/(4nkZ)at the origin k = 0, respectively, represent two typical states of the large-scale components of turbulence. The dependence or independence of various statistical quantities of turbulence on the large-scale structure of turbulence in these cases is an interesting subject to be studied in this section. It may be obvious that the initial condition T(k, 0) = 0,

(5.3)

which is equivalent to (4.10), is automatically satisfied by (4.37). For later convenience (4.37) is made nondimensional with respect to E, and k,, and the solutions are expressed as functions of the nondimensional wavenumber, time, and Reynolds number: IC = k/ko,

z = EA/2k:'2t,

R

= E~"/(vkA").

(5.4)

Theory of Homogeneous Turbulence

79

Equation (4.37)was solved for the range of Reynolds number R = 5, 10,20, 50,100,200,400, and 800 by Tatsumi et al. (1978) and for extremely large Reynolds numbers R = lo', lo5, and lo6 by Tatsumi and Kida (1980) for both initial conditions (5.1) and (5.2). For the details of the numerical method reference may be made to these papers. Various statistical quantities characterizing turbulence can be derived from the results of the energy spectrum. The kinetic energy of turbulence per unit mass is given by (4.5), or &(t)= JOm E(k, t )dk.

(5.5) The energy &(t)mostly reflects the shape of the energy spectrum E(k, t ) in the energy-containing range of the wavenumber. The skewness of the velocity derivative is defined by (3.8) and for the isotropic turbulence expressed in terms of the energy spectrum and the energy transfer function as

the skewness So(t) is determined by the integrals In contrast to the energy &?(t), of E(k, t ) and T(k,t ) weighted at higher wavenumbers, so that it reflects the shape of the energy spectrum in the higher wavenumber region beyond the energy-containing range. Taylor's microscale or the dissipation length of turbulence 3, is defined and expressed in terms of the energy spectrum as

For nonstationary turbulence it is more convenient to use the Reynolds number based upon the instantaneous state of turbulence rather than that based on the initial state, and for this purpose a Reynolds number is defined in terms of l ( t )and the root-mean-square velocity u(t), where U(t)2 =

( u : ) = f(lul2) =

5 JoW

E(k, t ) d k

[see (4.5)],as follows:

L" "

J

(5.8)

80

T . Tatsumi

Under initial conditions (5.1) and (5.2), the initial value of R1 is related to the corresponding value of R as follows : (I) R = 1.0077RA(0), (5.10) (11) R = 1.0623RA(0). Thus, R and R,(O) are almost identical in either of the above cases.

A. ENERGYSPECTRUM The general appearance of the energy spectrum E ( k , t ) and the energy transfer function T(k,t)may be observed in Figs. 15 ( R = 20) and 16 ( R = 800). The positive definiteness of E ( k , t ) is obvious and the improvement in the behavior of E(k,t ) from that due to the zero- fourth-order-cumulant approximation may clearly be seen by comparison of Fig. 15b with Fig. 14 corresponding to about the same initial Reynolds number [see (5.10)]. The energy initially contained in a narrow wavenumber range around k , is partly dissipated by the viscosity and partly transferred to higher wavenumbers, and the energy transfer becomes more vigorous than the viscous dissipation as the Reynolds number increases. These general trends are in common for cases I and 11. Although it is not evident for smaller Reynolds numbers, there appear more distinctly with increasing Reynolds number two stages of evolution of E(k, t ) and T(k,t ) separated by z = 4 for case I and T = 3 for case 11. In the first stage, 0 I z 5 4 for case I and 0 Iz 5 3 for case 11, both functions change rapidly in time from their initial forms into certain similar forms, which are nearly the same for both cases but apparently differ with Reynolds number. In the latter stage, z > 4 for case I and z > 3 for case 11,on the other hand, they change rather slowly and similarly in time. The presence of these two stages is also observed in the evolution of other statistical quantities as described later, and this seems to be a characteristic feature of the decaying turbulence at large Reynolds numbers. For later convenience, the two stages may be referred to as the initial and the similarity stages." The similarity laws discussed below are all concerned with this similarity stage. The form of the energy spectrum E(k,t), especially its details at higher wavenumbers, is displayed more clearly on a logarithmic scale as shown in * In earlier literature of decaying turbulence, a finite time period after the formation of turbulence in which turbulence is strong enough and its energy decays according to a simple power law is termed the initial period of decay, while the ultimate stage in which the turbulence has become so weak as to make the nonlinear effects negligible is called thefinal period of decay (see Batchelor, 1953, Sections 5.4 and 7.1). The initial and similarity stages considered above are both concerned with strong turbulence of large Reynolds numbers, and in this sense both stages correspond to the initial period of decay.

0.f

0.4

/

0.3

\O

0.4

0.2

0.2 0. I

0

I

3

0

I

2

3

4

0.03

0

-0.1 -0.Or

FIG. 15. Energy spectrum E(k, t ) and energy transfer function T(k,t ) for R (a) Case I, (b) case I1 (after Tatsumi et al., 1978).

= 20.

Numbers by the curves denote the nondimensional time r = Eb/Zki'2t

-

0 0 0

-

FIG. 16. Energy spectrum E(k, r) and energy transfer function

qk, t ) for R = 800. (a) Case I, (b) case I1 (after Tatsumi er ol., 1978).

Theory of Homogeneous Turbulence

83

Figs. 17 and 18. The distinction between the initial and similarity stages is more evident in these figures. Some common features of the spectral curves in the similarity stage are to be noted. First, the positive slope of E(k, t ) at very low wavenumbers, which is initially 2 and 4 for cases I and 11, respectively, is found not to change at all for case I and to change only slightly for case 11. If we express the spectrum at very small wavenumbers as

E(k, t ) / E o = AutiU,

(5.1 1)

a being a positive constant, A , is, in general, a function of time. For case I ( a = 2), however, the invariance of A 2 was proved by Birkhoff (1954) dealing with the velocity correlations and using the relation

nE0 ~

k:

A 2 = u(t)' lim [ r 3 f ( r ,t ) ] ,

(5.12)

r-+ cu

where f ( r , t ) is the longitudinal correlation function, f ( r ,4

=
+ ri, t)>lU(t)2,

(5.13)

i being the unit vector along the x, axis. The same result can be derived more directly as follows. It follows from (4.12) that for very small values of k, T(k,t ) = 4nk31C/(0,t),

(5.14)

where the finiteness of the energy transfer density at k = 0 is assumed. Then, substituting (5.11 ) (a = 2) and (5.14) into (4.11 ) we obtain to the lowest order of k that dAz/dt = 0, (5.15) so that A z is invariant in time. This is in perfect agreement with the behavior of the numerical curves as shown in Figs. 17a and 18a. For case I1 ( a = 4), on the other hand, there is little reason to believe the invariance of A,, which is related to the so-called Loitsiansky integral by

3nE0

7 A,

= u(t)' JOm

r4f(r,t)dr.

(5.16)

k0

In fact, A4 is invariant under the condition that E(k, t ) is regular at k = 0 and A z = 0. Loitsiansky (1939) gave this proof making an equivalent assumption for the correlation function. However, Batchelor and Proudman (1956) showed that even if E(k, t ) is assumed to be regular and expressed by (5.1 1) around k % 0 at a certain time, t = t o say, a logarithmic term of order of k510gk appears for t > t o , so that the invariance of A 4 is lost. The general trend of the numerical curves as shown in Figs. 17b and 18b is in accordance with this analytical conclusion. The positive slope of E(k, t ) at very small k is not exactly constant but the deviation from the constancy is fairly small.

m

\

84

FIG.17. Energy spectrum E(k, t ) for R = 20 on a logarithmic scale. (a) Case I, @) case I1 (after Tatsumi et al., 1978).

h

v

0 0

85 0

FIG.18. Energy spectrum E(k, f ) for R = 800. (a) Case I, (b) case I1 (after Tatsumi et al., 1978).

86

T. Tatswni

Thus, the energy spectrum at very small wavenumbers is entirely or almost determined by the initial condition, reflecting the apparent permanence of the large-scale components of turbulence. Beyond this wavenumber range, the spectrum seems to take nearly the same form for cases I and 11, which changes almost similarly in time in the similarity stage. The existence of such a universality in the spectral form irrespective of the difference in the largescale structure of turbulence strongly suggests the presence of a universal equilibrium in the small-scale components of turbulence. It is shown in Section V,B that this is indeed the case. The form of the energy spectrum in this universal wavenumber range differs considerably with Reynolds number. At small Reynolds numbers such as R = 20, the spectrum preserves its exponential form at all times (see Fig. 17a,b). A closer inspection of the exponential form reveals that its dependence on the wavenumber is not the same as that of the initial conditions (5.1)and (5.2) nor that of the viscous cut off, i.e., exp( -2vk2t), as would be expected from the linearized form of (4.37),but is more closely represented by E(k, t ) K exp( - bk’) (5.17) in the highest wavenumber range, where b is a constant dependent upon v and t, and s is about 1.5 according to the numerical results. I t can be shown, however, that the asymptotic form of the spectrum for k -,00 is expressed by (5.17) with s = 1. This discrepancy therefore indicates that either the asymptotic form is attained beyond the wavenumber range of the numerical curves or the numerical results are not accurate enough. In any case it is to be noted that even at small Reynolds numbers the spectrum does not tend to the purely viscous spectrum characteristic of the final period of decay. This problem is discussed Section V,B. At large Reynolds numbers such as R = 200-800, there appear two intermediate wavenumber ranges in which the spectrum takes the power-functional forms : E(k, t ) cc k-5’3

(5.18)

’,

(5.19)

E(k, t ) K k -

respectively, in the order of increasing wavenumber (see Fig. 18a,b). The spectral form (5.18) has the same exponent as Kolmogorov’s (1941a) inertial subrange spectrum and this conicidence makes us expect that Kolmogorov’s similarity law is actually satisfield in this range. The k-’ spectrum (5.19) is identical to (4.31), which is an asymptotic solution of (4.28)for vkZt >> 1, so that its appearance is not unexpected. At extremely large Reynolds numbers such as R = lo4 to lo6, the energy spectrum takes the forms as shown in Fig. 19a,b ( R = lo6). At such high

L

I

\\'

I

!lo-$

\\\ \\\\ '

01

I

lo-' k/ko

I

\

10

lo3 k/ko

FIG.19. Energy spectrum E(k,t ) for R = lo6. (a) Case I, (b) case I1 (after Tatsumi and Kida, 1980)

lo4

88

T. Tatsumi

Reynolds numbers another wavenumber range appears between the ranges represented by (5.18) and (5.19), in which the spectrum is expressed as E(k, t ) cc k - 2 .

(5.20)

This spectrum is again an asymptotic solution of (4.28) for vk’t << 1. Although the k - 2 spectrum was not noticeable at lower Reynolds numbers R 5 8 0 0 , it appears at larger Reynolds numbers over a finite wavenumber range, which increases steadily with the Reynolds number.

B. SIMILARITY LAWSOF ENERGYSPECTRUM Now let us investigatethe similarity in the evolution of the energy spectrum outlined in Section V,A in more detail. If the energy spectrum satisfies a similarity law with respect to Reynolds number and time in a certain wavenumber range, it must be expressible in the form E(k, t ) / E o = R u z P F [ ~ / ( R y ~ ’ ) ] ,

(5.21)

where E, j?, y, and 6 are real constants and F is a nondimensional function. The similarity is geometrically confirmed by making the spectral curves in the logarithmic scale coincide with each other over a wavenumber range through suitable parallel displacements. In practice the coincidence may not be perfect, and the best agreement of the curves is attained by minimizing the mean-square distance between the curves. Then the values of c1 and y are obtained from the sizes of the vertical and horizontal displacements, respectively,of the curves for different Reynolds numbers, while the values of j? and 6 are determined from the corresponding displacements of the curves for different times. The function F is obtained from the mean of the superimposed curves. The change in pattern of the energy spectrum with Reynolds number indicates that there is no overall similarity law with respect to R that is valid at all wavenumbers, but there are several similarity laws satisfied in different wavenumber ranges. In fact, it was shown by Tatsumi et al. (1978) that the energy spectra for R = 200-800 satisfy two similarity laws for the energycontaining and higher wavenumber ranges, and two sets of numerical values were obtained for the exponents of (5.21) in these ranges (see Tables I and 11). In the energy-containing range, the similarity law is found to be independent of R since c1= y = 0, so that the range is governed by the inviscid similarity law. In the higher wavenumber range, on the other hand, the similarity law has the same dependence on R as Kolmogorov’s scaling, that is, a = - 5/4 and y = 3/4, and the similarity curve shown in Fig. 20 involves the k-’I3 spectrum over a wavenumber range that increases with Reynolds

.-

10-1

FIG.20. Kolmogorov’s similarity of the energy spectrum E(k,t ) for R

= 200, 400, and

800. (a) Case I, (b) case I1 (after Tatsumi et al., 1978).

90

T. Tatsumi

number. Thus, the range may be identified with the universal equilibrium range for which Kolmogorov's similarity law is satisfied. The situation differs at still higher Reynolds numbers since the energy spectra for R = lo4 to lo6 worked out by Tatsumi and Kida (1980) are shown to satisfy three similarity laws for the energy-containing, intermediate, and highest wavenumber ranges, the first and last of which are identical with the two similarity laws for the lower Reynolds numbers. Now let us examine these three similarity laws separately. 1. Energy-Containing Range

First, the curves for a fixed Reynolds number and different times are brought into coincidence with each other over a wavenumber range roughly corresponding to the energy-containing range. Then, the resulting curves for different Reynolds numbers ( R = lo4, los, and lo6) are made to coincide with each other over the same wavenumber range. The numerical values of the exponents thus obtained for the energy-containing range, which are denoted by the suffix 1, are listed in Table I. The vanishing exponents a1 = y1 = 0 indicate that the energy-containing range is governed by the inviscid similarity law. The similarity exponents can be determined analytically by taking account of the behavior of the spectrum at very small wavenumbers. In the limit of vanishing viscosity (v -,0) with finite k and t, (4.37) for the energy spectrum takes the following inviscid form:

a

-

at

E(k, t ) = t

joa ! j [k2E(k',t ) - k"E(k, t)]E(k",t ) (5.22) TABLE I

SIMILARITY EXPONENTS FOR

THE

ENERGY-CONTAINING RANGE

Case I Numerical u1

0.00

81

- 0.79

Y1

0.00

6,

- 0.40

a

Taken from (5.38).

Case I1

Analytical

0

0.00

- 34 --

=

Analytical 0

-0.8

-1.08

- 1.059"

-0.4

0.00 -0.31

0 -0.314"

0

-3

Numerical

Theory of Homogeneous Turbulence

91

Substituting the similarity form (5.21) into (5.22) and equating the powers of R and z on both sides, we obtain the condition a1

+ 3Yl = 0,

pi

-/-

361 = -2.

(5.23)

For case I, the energy spectrum at very small wavenumbers is expressed by (5.1 1) (a = 2) as E(k, t ) / E , = A,!?,

(5.24)

where A , is a constant determined by the initial condition independentIy of the Reynolds number R. Accordingly, the similarity form (5.21) takes the form

~ ( kt ,) l ~=, A ; R ~ ~ Z ~ ~ ( K / R Y ~ T ~ ~ ) ~ ,(5.25) where A; is a nondimensional constant. Comparison of (5.24) and (5.25) and the invariance of A, in time [see (5.15)] give the condition ct1

- 27, = p1 - 26, = 0.

(5.26)

Hence, from (5.23) and (5.26) we have ~ (= 1 ~1

= 0,

=

-415,

61 = -215,

(5.27)

for case I. For case 11, (5.11) is written as E(k, t ) / E , = A,K4,

(5.28)

where again A, is a constant independent of R, and the similarity form (5.21) is written as E(k,t)/E, = AkRa1rPI(K/RY1r61)4.

(5.29)

Comparison of (5.28) and (5.29) gives - 4Y1 = 0,

(5.30)

a1 = y 1 = 0.

(5.31)

ct1

so that from (5.23) and (5.30), If we take A , to be invariant in time, we can proceed as in case I and obtain from (5.28)and (5.29) the relation - 461 = 0,

(5.32)

so that from (5.23)and (5.32), =

-817,

61 = -217

(5.33)

92

T. Tatsumi

for case 11. As stated in Section V,A, however, A4 is generally not invariant, so that the numerical values given by (5.33) are not accurate. In order to evaluate the exponents p1 and 6,, taking into account the time dependence of A4, we substitute (5.28) into (5.22) and obtain the following equation to the lowest order of k: (5.34) Since the right-hand side is a positive-definite integral, A4 always increases in time. Essentially the same equation as (5.34) was obtained by Proudman and Reid (1954) using the zero-fourth-order-cumulant approximation and they concluded from that equation the variation of A4 in time. If we employ the initial form (5.2) as a rough approximation of the similarity form (5.21) at z = 1, we have E(k, t ) / E , = Z@~(IC/Z'~)~ exp[ - ( I C / T ~ ~ ) ~ ] ,

(5.35)

and Ak = 1 for (5.29), where use has been made of (5.31). Substituting (5.35) into Eq. (5.34) and making use of (5.4), we obtain (5.36) On the other hand, from (5.28), (5.29), and (5.31), A , = ,@I

(5.37)

-481.

Then, it follows from comparison of (5.36) and (5.37) and from (5.23) that = -1.059.

(5.38)

These values are in good agreement with the corresponding values obtained numerically by curve fitting. Substituting (5.38) into (5.37) we find that A4(t)= z0.196,

for z 2 1,

(5.39)

which gives a rough estimation of the variation of A4 in time. As a matter of fact, the variation of A , is fairly small.

2. Energy Dissipation Range The similarity law of the spectrum in the highest wavenumber range is obtained in the same manner as in the energy-containing range, and the exponents for this range, which are denoted by the suffix 2, are listed in Table 11. The values of the exponents a2 = - 1.25 and y2 = 0.75 are exactly in agreement with Kolmogorov's similarity law, E(k) oc v5/, and k cc v-~/,,

93

Theory of Homogeneous Turbulence

given by (3.17) and (3.18). Therefore, this wavenumber range is identified with Kolmogorov's universal equilibrium range, or, more adequately, the quasi-equilibrium range, in view of the nonstationarity of the state in this range. According to the numerical values of the exponents in Table 11, the nondimensional wavenumber (vt)l/'k is written ( v t ) W = (R-%)'/2K

= O(R"4),

(5.40)

so that (vt)"'k >> 1 for very large Reynolds number R. For ( v t ) 1 / 2 k>> 1, (4.37) can be written as

J!

2vkZE(k,t ) = JOm

[v(k2

+ k 2 + k'")]-

'[k2E(k', t ) - k"E(k, t ) ] (5.41)

Thus, the energy transfer term is just in balance with the energy dissipation term in this wavenumber range. It should be noted that (5.42) does not involve the time t explicitly, so that the spectrum E(k, t ) depends upon t only as a parameter in this energy dissipation range. The similarity exponents for this range can be determined analytically by the use of (5.41) and the energy-dissipation equation (4.17).Substituting the similarity form (5.21) into (5.41)and equating the powers of R and T on both sides, we obtain the relation a2

- yz =

-2,

p2 - c52 = 0.

(5.42)

In the above derivation it has been assumed tacitly that the integral on the right-hand side of (5.41) is essentially represented by the contribution from the energy dissipation range. Such a localness of the contribution is not evident beforehand, but can be confirmed by the result obtained. In fact it is TABLE I1 SIMILARITY EXPONENTS FOR THE ENERGY DISSIPATION RANGE Case I1

Case I Numerical a2 82

YZ 62

- 1.25 -0.56 0.75 -0.53 Taken from (5.45)

Analytical

-$

=

- 1.25

-M = -0.55 3 = 0.75 -# = -0.55

Numerical - 1.25 -0.61 0.75 -0.58

Analytical

-5

= -1.25

- 0.593"

= - 0.593"

0.75

94

T. Tarsumi

shown later that the contributions from other wavenumber ranges are of minor orders of magnitude compared with the one from the energy dissipation range. Another relation for the exponents, obtained from the equation for the energy dissipation, is d dt

~ ( t=) --

j“E(k, t )dk 0

= 2v

Jo”k2E(k,t )dk,

(5.43)

which follows from (4.9, (4.16), and (4.17).For very large Reynolds numbers, the energy-containing and energy-dissipation ranges are well separated and the energy integral and the dissipation integral in (5.43) are essentially determined by the respective wavenumber ranges. Then, substituting the similarity forms (5.21) with suffixes 1 and 2 into the energy and dissipation integrals in (5.43), respectively, we obtain the relation cI1

-k

)J1

= a2 -k 3y2

- 1,

81

-k 61

- 1 = /?2 -k 362.

(5.44)

Relations (5.42) and (5.44) and the values of the exponents for the energycontaining range, which are given by (5.27) for case I and by (5.31) and (5.38) for case 11, give the following values of the exponents for the energydissipation range: c12

= - 514,

72

(I) 8 2 = 6 2 = -11/20,

(11)

314, /?2

= 62 =

-0.593.

(5.45)

These analytically determined exponents are in excellent agreement with the numerically obtained values in Table 11. The existence of Kolmogorov’s similarity law in the energy dissipation range leads to the appearance of the inertial subrange if the Reynolds number is sufficiently large. For Reynolds numbers R = 200-800, there actually exists a k - 5 / 3 spectrum range as seen from Fig. 20. This spectrum satisfies the similarity laws (I) E ( k , t ) / E o = 1.2R-0.04~-’.44 K -5/3 , (11) E(k, t)/Eo = 1.0R-0.0Z~-’~57~-5’3,

(5.46) (5.47)

which can be written in the form of the inertial subrange spectrum as E(k,t)= K ~ ( t ) ” ~ k - ~ / ~ ,

(5.48)

with Kolmogorov’s constant given by

- 0.62, 0.56 - 0.59.

(I) K = 0.60 (11) K

=

(5.49)

Theory of Homogeneous Turbulence

95

(See Tatsumi et al., 1978, pp. 120, 131.) Thus, the existence of the inertial subrange spectrum seems to be confirmed for these Reynolds numbers. This behavior of the spectrum, however, does not persist for extremely large Reynolds numbers since then the energy dissipation range is connected at the lower wavenumber end with an intermediate range which has a different similarity law from that of the inertial subrange. As seen from Fig. 21, which depicts the similarity forms of the spectrum in the energy dissipation range for R = lo4, lo5, and lo6,the extent of the intermediate range increases with Reynolds number, so that the deviation from the inertial-subrange spectrum also becomes large. In view of the nature of the inertial-subrange spectrum, however, this spectrum is expected to appear in a wavenumber range governed by the inviscid similarity law, that is, the energy-containing range in the present context. Actually, one may notice in Fig. 21 a small subrange of the k - 5’3 spectrum just beyond the maximum of the spectiurn. Although the extent of this subrange is rather small, the spectrum there satisfies the inertialsubrange relation (5.48) with Kolmogorov’s constant given by

(I) K = 0.71,

(11) I( = 0.64.

(5.50)

These values of K are a little larger than those of (5.49) for lower Reynolds numbers but still consistently smaller than experimental values which are reported to be 1.3 1.7 (see references cited on p. 63). It may appear strange that the extent of the wavenumber range identified with the inertial subrange remains small and does not increase with Reynolds number. This is, however, explained by the fact that the present inertial subrange is included in the energy-containing range whose characteristic wavenumber is independent of the Reynolds number. It is shown in Section V,B,3 that there exists a k - 2 spectrum that also satisfies an inviscid similarity law and has a characteristic wavenumber k = O [ ( V ~ ) - ”that ~ ] increases with Reynolds number. This k - spectrum, however, does not satisfy the local similarity relation (5.48) since it also depends upon Eo and ko in addition to c(t). As mentioned in Section IV,B, the present approximation is valid, except for neglecting the fourth-order cumulant, in the energy-dissipation range and partially in the energy containing range, but is not good in the intermediate range. Thus, it is unavoidable that the result is unsatisfactory in the intermediate range. Finally, let us investigate the asymptotic form of the energy spectrum at very large wavenumbers. For vk2 -+ co,the E(k)E(k”)term on the right-hand side of (5.41) becomes of smaller order of magnitude than other terms, so that (5.41) reduces to

-

(k /ko) (R/

Id

( r / 10)o'60

FIG.21. Kolmogorov's similarity of the energy spectrum E(k,I ) for R = lo4, lo5, and lo6.(a) Case I, (b) case I1 (after Tatsumi and Kida, 1980).

97

Theory of Homogeneous Turbulence By applying the change of variables (4.39), (5.51) is written as

x [(k’

+ k’)’

- k 2 ] [ k 2- (k’

- k”)2](k‘k”)-3dk’dk”.

(5.52)

Changing again the variables from (k’,k”) into x

= (k’

+ k”)/k,

y

= (k“ -

k’)/k,

(5.53)

we can rewrite (5.52) as

x (x’

+ y2 + 2x2y2)(x2- 1)(1 - y’)(x’

-~

’ ) - ~ d x d y(5.54) ,

where integration is made on a strip region as shown in Fig. 22. Now, if we assume that E ( k )is a decreasing function of k vanishing as k + 00, the product of E on the right-hand side of (5.54) is a rapidly decreasing function of x for very large values of k and the dominant contribution to the integral comes from the neighborhood of the segment x = 1, - 1 I y I 1. Then, (5.54) reduces for very large k to 2 E(k)N v‘k

f”( x

x (1

1

-

1)dx

+ 3y’)(3 + y

0

k

J:

-q“ - Y)/21E[k(X + Y I P ] (5.55)

y ( 1 - y’)-’dy.

k‘

FIG.22. Domain of integration on the wavenumber plane.

98

T. Tatsurni

In order that (5.55) be satisfied, the product of E, having nonvanishing values only in the narrow domain mentioned above, must satisfy the relation

- Y)/21E[k(X + Y)/2]/E(k)= W) for k

-, co.If

we express E ( k ) as

(5.56)

'

E(k) = exp[g(k)l,

(5.57)

g [ k ( x - YY21 + gCk(x + Y)/21 = g(k).

(5.58)

it follows from (5.56)that Obviously (5.58) is satisfied only by the linear function g ( k ) cc k, and hence

E ( k ) cc exp(- bk),

(5.59)

which corresponds to the case s = 1 in (5.17). If (5.59)is admitted, then it can be shown that

E ( k ) K k3 exp( - bk)

(5.60)

to the next order of magnitude. Substituting the assumed expression

E(k) = Ck" exp( - bk),

(5.61)

C and m being positive constants, into ( 5 . 5 3 , we obtain

2c I=--4"-2 k"-3

Somzexp(-bz)dz JPl-(l1 +3y' 1

- y')"-'dy,

where z = k(x - 1).Hence, it follows that m = 3 and

(5.62)

C = 32v2b2/1= 48.98v2b2,

where (1 - y 2 ) d y = 20 -

32.n ~

3 8 Thus, the asymptotic form of the energy spectrum for k E ( k ) = 48.98v2bZk3exp( - bk).

= 0.6533. -, 00

is expressed as*

(5.63)

According to the similarity law of the energy-dissipation range as given in Table 11, the constant b is expressed as

b = cR-3/4k-1 0

3

(5.64)

* The asymptotic form of (5.60)for k -P co was already mentioned by Kraichnan (1959) and Orszag (1977) without explicit reference to a proof. Recently the author has been informed by Dr. Kraichnan that he has derived this asymptotic form by assuming the general form of(5.61).

Theory of Homogeneous Turbulence

99

where c is a nondimensional constant that may depend upon the nondimensional time z. Then (5.63) is written in nondimensional form as E ( k ) / E , = 4 8 . 9 8 ~ ~ R - ~ ~ ~ exp( ( r c-/ C R K~ /~R~~)/ ~~ ) .

(5.65)

It should be remembered that although the asymptotic form (5.65) is valid in the far-dissipation range characterized by vli2k + co,it still satisfies the similarity law of the energy-dissipation range. As mentioned in Section V,A the asymptotic form of the numerically is represented closely by (5.17) with s = 1.5, obtained spectrum for k + which is at variance with the asymptotic behavior (5.63). It is yet uncertain if this discrepancy is to be attributed to the inaccuracy of numerical calculation, which may have misrepresented the rapid change of the integrand of (5.54), or to the fact that the asymptotic behavior (5.63) is realized at still higher wavenumbers beyond the range of numerical calculation. The asymptotic form (5.63)does not agree with the viscous cut-off spectrum E(k, t ) K exp( - 2vk2t), which is Characteristic of the final period of decay. This disagreement shows that although the intensity of turbulence is very weak in the far-dissipation range, the viscous dissipation is not only a dominant effect in this range but still in balance with the nonlinear inertial effects in the highest wavenumber range. In this sense, all the situations considered in this section are concerned with the initial period of decay. 3. Intermediate Range

The similarity of the spectrum is determined using the same method as in Sections V,B,1 and V,B,2 for the intermediate wavenumber range, which extends from the wavenumbers corresponding to the k - 2 spectrum to those corresponding to the k - spectrum. The numerical values of the exponents for the intermediate range, which are denoted by the suffix 3, are listed in Table 111. A remarkable consequence of the values y2 = 0.51 and h2 = -0.50 is that the nondimensional wavenumber (vt)’i2k = ( R - ‘z)l/’remains finite in the intermediate range. Moreover, this range is bracketed by the k-’ and k - ’ spectra, which are asymptotic solutions of (4.28) for (vt)’/2k << 1 and >>1, respectively, as seen from (4.30) and (4.3 1).Thus, the nondimensional wavenumber (vt)’l2k changes from very small to very large values through the intermediate range, so that we may take (vt)”2k = O(1) (5.66) as the characteristic wavenumber of this range. Using the scaling (5.66) and requiring that the similarity spectrum in the intermediate range be smoothly

T. Tatsumi

100

TABLE 111 SIMILARITY EXPONENTS FOR

THE

INTERMEDIATE RANGE

Case I1

Case I Numerical a3 83

Y3

63 a

Analytical

Numerical

Analytical

- 1.01 -0.60

-1

- 1.02 -0.69

-1

- $ = -0.6

0.51 -0.50

f = 0.5 -f = -0.5

0.51 -0.50

f = 0.5 - f = -0.5

-0.687”

Derived from (5.38).

connected with that of the energy-containing range through the kK2 spectrum, Kida (1980) determined the exponents analytically.The results included in Table 111 are in good agreement with the numerically obtained values. Using these values of the exponents, the k - 2 spectrum and the k-‘ spectrum are expressed in the following similarity forms: (I) E(k, t)/E, = 1 . 7 ~ - ~ ’ ~ ~ : - ~ ,

(5.67)

(11) E(k, t)/Eo = 1 . 7 ~ - ’ . ~ ~ ~ ~ - ’ ,

(5.68)

(I) E ( k , t ) / E , = 4 . 7 R - ” 2 ~ - ’ 1 ~ ’ 0 ~ - 1 , (11) E(k, t)/Eo = 4 . 7 R - ” 2 ~ - ’ . ’ 8 6 ~ - 1 ,

(5.69) (5.70)

where the coefficients have been determined by the mean curves and the analytically determined values have been employed for the exponents. Finally, it is worth noting that, if we ignore the intermediate range, the similarity spectra in the energy containing and energy dissipation ranges can be matched consistently. Requiring that the similarity spectra with the exponents (5.27), (5.31), and (5.38) for the energy containing range and (5.45) for the energy dissipation range be connected smoothly with each other, we obtain the following spectrum in the common wavenumber region:

(I) E ( k ) rx

2-22/15k-5/3,

(5.71)

(11) E ( k ) rx

2-1.582k-5/3.

(5.72)

Taking account of the energy decay laws (5.77) and (5.78) in advance, we find that these spectra are nothing but the inertial-subrange spectrum (1,II) E ( k ) K ~ ( t ) ~ /5/3. ~k-

(5.73)

As mentioned in Section IV,B, the present approximation scheme is well founded in the energy-containing and energy-dissipation ranges but less

Theory of Homogeneous Turbulence

101

satisfactory in the intermediate range. Thus, the above procedure of direct matching of the similarity spectra for the energy-containing and energydissipation ranges is in fact a sensible approximation scheme apart from being the lowest order approximation of the cumulant expansion.

C. ENERGY, SKEWNESS, AND MICROSCALE Various statistical quantities characterizing the turbulence can be derived from the energy spectrum function E(k, t ) and the energy transfer function T(k,t ) obtained and discussed in the foregoing sections. 1. Decay of Energy

The energy of turbulence &(t)is calculated from numerical data on the spectrum E ( k , t ) , using (5.5), and the result is shown graphically in Fig. 23.

I-

t

0.051 I

I

I

I

I

I

I

l

l

2

3

4

5

6

7

8

9

7

FIG. 23. Decay of the energy Bft). (a) Case I, (b) case I1 (after Tatsumi and Kida, 1980).

102

T. Tatsumi

The difference between the initial and similarity stages may be clearly observed in the curves for R 2 100 and more sharply for R 2 lo4. The time, t, say, that separates the two stages is determined as 7, = Eh’2k:/2t, = 2.74 for case I and 1.61 for case 11. At very large Reynolds numbers the energy &(t) does not change at all in the initial stage t < t, and abruptly starts to change in the similarity stage t > t, according to a power law. The data for the largest Reynolds number give the following laws of the energy decay: ) 1.4~-’.~O, (I) 6 ( t ) / ( E o k o= (11) &(t)/(Eoko)= 1.42- 1 . 3 9 .

(5.74) (5.75)

The decay of energy is also determined by the similarity law for the energy spectrum in the energy-containing range. Substitution from (5.24) into (4.5) gives (5.76) &(t) = Eok07”+’~ Joa F(s)ds. By making use of the analytical values of the exponents in Table I, we obtain the following laws: (I) 6(t)cc 7 - 1 . 2 , (5.77) (11) &(t) oc

2-1.373.

(5.78)

The excellent agreement of the exponents with those of (5.74) and (5.75) confirms the accuracy of the similarity law in the energy-containing range. The experimental data on the energy decay law for the grid-generated turbulence are not decisive with respect to the value of the exponent. In early works the exponent was taken to be - 1 (see Batchelor, 1953,Section 7.1), but later experiments gave values ranging between -1.0 and -1.4 (see Table 1 of Comte-Bellot and Corrsin, 1966). In most experiments the exponent cannot be determined uniquely since the energies associated with the streamwise and transverse components of the velocity obey slightly different decay laws. Comte-Bellot and Corrsin (1966) produced almost isotropic turbulence using a weak contraction of the wind tunnel and obtained an exponent of - 1.25, which is fairly close to that in (5.74). On the other hand, a group of more recent measurements by Ling and Wan (1972), Gad-el-Hak and Corrsin (1974), and Tassa and Kamotani (1975) give exponents ranging between -1.30 and -1.35, which are closer to (5.75) than (5.74). 2. Skewness of Velocity Derivative The skewness S(t) of the velocity derivative defined by (5.6)gives a measure of the strength of the vorticity production. S ( t ) is calculated by substituting numerical vahes of E(k,t) and T ( k , t ) into (5.6) and the result is shown graphically in Fig. 24.

Theory of Homogeneous Turbulence

103

FIG.24. Skewness of the velocity derivative S(t). (a) Case I, (b) case I1 (after Tatsumi and Kida, 1980).

The initial condition (5.3)gives S(0) = 0 for both cases. At small Reynolds numbers, S(t) increases gradually in time to an asymptotic value S , = S(o0). At large Reynolds numbers, on the other hand, it overshoots once and then returns rapidly to an asymptotic value. Roughly speaking, this overshooting coincides with the initial stage and S(t) remains nearly constant in the similarity stage. At extremely large Reynolds numbers, the overshooting is largely amplified to the maximum height of about 10 at R = lo6, and S ( t ) is kept exactly constant S(t) = S, throughout the similarity stage. The value of S, increases with Reynolds number and tends to the following limit at infinite Reynolds number: (1,II) S ,

= 0.70.

(5.79)

The perfect agreement of the values of S, for cases I and I1 makes a clear contrast to the discrepancy of the energy decay laws &(t) for the two cases, providing an evidence for the universality of the small-scale structure of turbulence irrespective of its large-scale structure. Earlier measurements of S ( t ) in the grid-generated turbulence made by Batchelor and Townsend (1947, 1949) and Stewart (1951) give values in the range 0.3-0.5, which are compatible with the present results at relatively

104

T. Tatsumi

small Reynolds numbers, but no further comparison is possible owing to the lack of other experimental data. Later measurements by Uberoi (1963) show that S(t) is nearly constant in time with a value of about 0.54. This behavior and value of S(t) are fairly close to the present result for case I at R = 100, which seems to correspond to Uberoi’s measurement in view of the general features of the energy spectrum and the energy decay law. Orszag and Patterson (1972) carried out numerical experiments on decaying isotropic turbulence corresponding to case 11. The numerical curve for R = R,(O) = 21 shows striking agreement with the present curve for R = 20 (see Fig. 8b of Tatsumi et al., 1978). Measurements of stationary atmospheric turbulence made by Wyngaard and Pao (1972) give the values of S ( t ) = S, = 0.70-0.85, the lowest of which is in perfect agreement with the present result (5.79). This agreement seems to be reinforced by the fact that the range of RA= 103-104 for the measurements roughly corresponds to our Reynolds number R = lo6 [see (5.83) and (5.84)]. 3. Microscale

The microscale A(t) defined by (5.7) gives another quantity characterizing the small-scale structure of turbulence. The value of A ( t ) obtained by substituting numerical values of E(k, t ) into (5.7) is plotted in Fig. 25. At small Reynolds numbers A(t) increases monotonically in time, whereas at large Reynolds numbers it decreases rapidly from its initial value to a minimum and then increases almost proportionally with time. Again, this rapid decrease and the linear increase correspond to the initial and similarity stages respectively. By making use of (4.11), (4.13) and ( 5 3 , (5.7) can be written as =

- 10vb(t) db(t)/dt .

(5.80)

Substituting (5.74) and (5.75) into (5.80) we obtain the following similarity laws for the microscale: (I) A(t)’ = 8.33z/(Rkg) = 8.33vt,

(5.81)

(11) A(t)2 = 7.19z/(Rkg) = 7.19vt.

(5.82)

It may be seen in Fig. 25 that these similarity laws are well satisfied by all curves. The microscale Reynolds number defined by (5.9) is easily obtained from the data on b(t)(=fu(t)2)and A(t). In particular, the similarity law for RA(t) at large R and z is immediately derived from the laws (5.74) and (5.75) for

Theory of Homogeneous Turbulence

105

I

I

I

I

I

I

I

l

l

&(t)and (5.81)and (5.82)for A ( t ) as follows:

(I) R,(t) = 2.78R”2~-0“,

(5.83)

(11) R,(t) = 2.59R1’2~-0.2.

(5.84)

These laws, which are satisfied fairly well by the numerical curves for R 2 200, provide us with a useful formula for relating Rn(t)to R. VI. Turbulence of Other Dimensions So far we have been concerned with isotropic turbulence in an incompressible fluid. Although this turbulence is the most familiar type of turbulence having a close connection with real turbulent flows, there are several

106

T. Tatsumi

physical problems in which turbulence of other dimensionality plays an important role. Among many of them, we deal in this section with twodimensional isotropic turbulence in an incompressible fluid and onedimensional turbulence of Burgers, using the same approximation method as for three-dimensional turbulence.

A. TWO-DIMENSIONAL TURBULENCE Two-dimensional turbulence in an incompressible fluid is governed by Eqs (2.1) and (2.2) subject to the condition u(x,t)= (u,,u,,o),

aiax, = 0,

(6.1)

where the x3 axis has been chosen arbitrarily. Under condition (6.1), the vorticity has only the x3 component, w(x, t ) = rot u(x,t ) = (O,O, w),

(6.2)

and eliminating p from (2.1) and (2.2) we obtain the equation of vorticity, am at

-

+ (u

*

grad)w = vA2w,

where A2 = a2/ax: + a2/axi. It follows that the vorticity is conserved for a fluid element in an inviscid fluid. 1. Inviscid Enstrophy Dissipation

For homogeneous turbulence, the rate of change of enstrophy 9(t)defined by (4.15) is found from (6.3) to be d9/dt = -2vS,

(6.4)

where 9 ( t ) = +((au/axi)2)

(6.5)

is called the palinstrophy. Hence, 9(t)is a decreasing function of time and remains finite if it is so initially, and it follows from (4.17) that E+O

as v + O .

(6.6)

This property makes a clear distinction from (2.19) for three-dimensional turbulence, that is, the rate of energy dissipation is nonzero in the inviscid limit. Thus, it may be obvious that Kolmogorov’s similarity law based upon the independence of E and v is not satisfied in two-dimensional turbulence.

Theory of Homogeneous Turbulence

107

On the other hand, the nonzero value of the enstrophy dissipation in the inviscid limit is not excluded. In fact, we see from Eq. (6.3) that

am

-(---)-v((*)z), auj am axi axi axj

dt

axi ax,

(6.7)

in which the first term on the right-hand side represents the rate of amplification of vorticity gradients by the extension of isovorticity lines. If we assume that the material lines are extended on average in two-dimensional as in three-dimensional turbulence, we may expect that the extension of isovorticity lines will amplify s(t) at small viscosity until it makes the right-hand side of (6.4) finite, so that

= -d2/dt

>0 as v -+ 0. (6.8) Batchelor (1969) proposed adopting (6.8) as the hypothesis for twodimensional turbulence and, following the same dimensional argument as Kolmogorov’s for three-dimensional turbulence, derived the similarity form of the energy spectrum as q

E(k) = q’/6~3iZE,(k/kJ, kd = y 1 1 / 6 v - 112,

(6.9) (6.10)

for k >> k , , where E , is a nondimensional function. Provided the Reynolds number is so large that there is a wavenumber range in which the spectrum is independent of the viscosity, (6.9) reduces to

E(k)= C V ” ~ ~ - ~ ,

(6.11)

where C is a nondimensional constant. The same expression for the spectrum was also proposed by Kraichnan (1967) on the basis of similar plausible arguments and by Leith (1968) using a diffusion approximation for the nonlinear energy transfer. There exists, however, counterevidence to hypothesis (6.8). Lilly (1971) derived the following equation for s(t) in an inviscid fluid from the zerofourth-order-cumulant approximation :

$

=2

j: ( k 2 - k‘2)2k’2E(k,t)E(k‘, t )dk dk‘,

(6.12)

where use has been made of the expressions for isotropic turbulence, 9(t)=

jom k Z E ( k ,t ) dk,

(6.13)

Y(t)=

k4E(k, t)dk.

(6.14)

T. Tatsumi

108

Replacing the right-hand side of (6.12) by dominant integrals, we have d 2 P / d t 2 < 22?(t)P(t)I29(0)P(t),

where (6.8) has been used, and hence ~ ( t<)~‘(0)e~p[(29(0))”~t].

(6.15)

Thus, the palinstrophy P(t) remains finite at finite t, and it follows from (6.4) that as v

v-0

-+ 0.

(6.16)

This relation can also be proved using the inequality P(t)< P(0)exp[$d(0)t2],

(6.17)

which was derived by Pouquet et al. (1975) from the “eddy-damped quasinormal approximation,” which includes the present approximation as a limiting case. Conclusion (6.16) is apparently in contradiction to hypothesis (6.8).It is shown below, however, that relations (6.16) and (6.8) represent the states of turbulence in the initial and similarity stages of two-dimensional turbulence, respectively, so that there is no substantial contradiction.

2. Energy Spectrum Now let us examine the consequences of the quasi-equilibrium zerofourth-order-cumulant approximation applied by Tatsumi and Yanase (1980) to two-dimensional turbulence. The energy spectrum equation according to this approximation is written as

(g+

2vk2) E(k, t) = T(k,t),

with T(k,t)=;kJ;J-l

1

1 - exp[ - v(k2 + k” + kr2)t] v(k2 + k 2 + k’2)

x [ k E ( k ,t ) - k E ( k , t)]E(k”, t )

x ($+2p)&(l

-p2)1/2dkdp.

(6.18)

This equation can also be written in the form of the equation for the enstrophy spectrum Q(k, t), where (6.19)

Theory of Homogeneous Turbulence

109

as follows:

($+ Zvk') Q(k,t )

=

W(k,t),

with 4 W(k,t ) = k'T(k, t ) = - k 71

x

jom j-1 - exp[v(k' -+v(k2k'' ++k"k"')+ k"')t] 1

($+ 2.)

$1

- .')"'dk'd.,

(6.20)

where the integrand has been antisymmetrized with respect to k and k'. In view of the invariance of the energy &(t) and the enstrophy 9(t)in an inviscid fluid as required by (6.4) and (6.6),it follows from (6.18)and (6.20) that

lom T(k,t)dk jOw k'T(k,t)dk =

=

W ( k , t ) d k= 0.

(6.21)

These conditions are identically satisfied by the expressions of T(k,t ) and W(k,t ) given in (6.18)and (6.20). Equation (6.18) is solved numerically for the initial conditions (1)

w, 0) (2u$/ko)(k/ko) exp[ =

- (k/kO)'l,

(11) E(k, 0) = ( 2 ~ $ / k O ) ( k / kexp[ o ) ~ - (k/kO)'l, and Reynolds numbers R uo = u(0) and U(t)'

= uo/vko = 20,

(6.22) (6.23)

100, 200,400, lo4, and l o 5 ,where

= ( U 1 ( X ; t ) 2 ) = f(lu(x,t)l')

= &(t).

(6.24)

The energy spectrum thus obtained is shown graphically in Fig. 26. It may be observed that there exists a strong energy transfer toward lower wavenumbers as well as toward higher wavenumbers. This backward energy transfer is necessitated in order to satisfy the two conditions given by (6.21). Probably the presence of the backward energy transfer is the most eminent feature of the energy spectrum of two-dimensional turbulence. (a) Energy-containing range The numerically obtained spectral curves satisfy two different similarity laws in the energy-containing and higher wavenumber ranges. The exponents of the similarity form (5.21)are determined by the same procedure as in Section V,B and the values for the energycontaining range, which are denoted by the suffix 1, are listed in Table IV.

10

I

lo-2 lo-'

I

10-7k o-'

I

lo-'

I

10

lo2

16'

I

10

lo2

k/ko

w

N

r;;

d

FIG.26. Energy spectrum E ( k , t ) of two-dimensional turbulence for R

= u,/uk, =

lo5.(a) Case I, (b) case I1 (after Tatsurni and Yanase, 1980).

111

Theory of Homogeneous Turbulence TABLE IV SIMILARITY EXPONENTS FOR THE ENSTROPHY-CONTAINING RANGE OF TWO-DIMENSIONAL TURBULENCE Case I Numerical u1 B1

Y1

6,

Case 11

Analytical

Numerical

0 1 0 -1

0.00 0.99 0.00 - 0.99

0.00 0.86 0.00 -0.84

Analytical 0 1 0 -1

The vanishing exponents c1, = y, = 0 indicate that this range is governed by the inviscid similarity law. The similarity exponents are also determined analytically using Eq. (6.18) and the invariance of the energy 8.In the limit of v -,0 and finite k, (6.18) reduces to

(6.25) Substituting the similarity form (5.21) into (6.25) and equating powers of R and z on both sides, we obtain tll

+ 361 = -2.

+ 37, = 0,

(6.26)

Two other conditions follow from the independence of the energy d from R and z: p, 6, = 0. (6.27) a1 y1 = 0,

+

+

From (6.26) and (6.27) we have 01, = y1 = 0,

p1 = -6,

=

1.

(6.28)

These values are in good agreement with the numerically obtained values in Table IV for case 11, but less satisfactorily for case I. The similarity relation represented by (6.28) was obtained by Batchelor (1969) using conservation of energy and dimensional analysis. If the enstrophy-containing range is also governed by the same similarity law, the enstrophy and its dissipation rate are expressed as 2(t)=

som

k2E(k,t ) d k

= E,k&X2,

(6.29)

112

T. Tatsumi

where J =

lom sZF(s)ds.

(6.30)

For very large Reynolds numbers such as R 2 lo4, the spectrum takes the k - 3 form over a finite wavenumber range that increases with Reynolds number. The similarity form of this spectrum is given by (6.31)

where the coefficients have been determined by the mean of the superimposed spectral curves and the analytical values have been used for the exponents. By making use of relation (6.29), this spectrum is written in the form of the inertial-subrange spectrum (6.11) with the nondimensional constant C given in cases I and I1 by (6.32)

The closeness of the constants for cases I and I1 shows that the universality of the inertial subrange spectrum is approximately satisfied by the present result. Thus, the present approximation is compatible with the hypothesis of nonzero enstrophy dissipation in the inviscid limit (6.8). If, on the other hand, the enstrophy transfer is absent, we would expect a different spectrum than that of the kK3 form. Saffman (1971) proposed a k P 4 spectrum assuming a piecewise continuous distribution of the vorticity in two-dimensional space, which is analogous to the piecewise continuous velocity distribution of the one-dimensional turbulence of Burgers. Unlike the latter turbulence, however, there is no mechanism to sustain the discontinuities of the vorticity in the two-dimensional turbulence since the crossing of the characteristics is not allowed by the condition of incompressibility, so that the discontinuities are only diffused by the action of viscosity. The data of the numerical simulations of the two-dimensional turbulence are not decisive about the possibility of the inertial subrange spectrum (6.11). Lilly (1971, 1972a,b) reported the k - 3 spectrum while the k - 4 spectrum was obtained by Deem and Zabusky (1971). These earlier results were, however, concluded to be premature by Herring et al. (1974), who obtained spectra of the form k - s - k - 4 for a decade of Reynolds number. More recent numerical study by Fornberg (1977) gives the k - 3 spectrum in the earlier period of evolution (t < 1O00, arb. units) and the k - 4 spectrum in the much later period (t > 3000). A buildup of the phase relations between different wavenumber components is pointed out as a remarkable feature of the latter period. An inspection of the picture display of the isovorticity lines reveals

113

Theory of'Homogeneous Turbulence

that the stretching of isovorticity lines produces regions of high-vorticity gradient in the earlier period while the much later period is characterized by the formation of well-defined vortex regions separated from each other. The generation of high vorticity gradients or large palinstrophy 9 in the earlier period accounts for the appearance of the k - 3 spectrum in this period, and the lack of vorticity transfer in the much later period gives a good reason for the k-4 spectrum in the latter period. (b) Quasi-equilibrium range The similarity exponents in the higher wavenumber range are obtained numerically by the same procedure as before and listed in Table V with the suffix 2. It was not possible to find a constant for p 2 and d2 owing to the lack of time similarity in the spectral curves. The values ct2 = - 1.5 and y 2 = 0.5 for both cases are in accordance with the Reynolds number dependence of the quasi-equilibrium range spectrum (6.9) and (6.10), and therefore this wavenumber range may be identified with the quasi-equilibrium range of two-dimensional turbulence. The similarity exponents are also determined analytically using Eq. (6.18) and the similarity law (6.29) for the enstrophy dissipation 11. For very large wavenumbers (vt)'l2k >> 1, (6.18) reduces to

x [kE(k', t ) - k'E(k, t)]E(k",t )

Substituting (5.21) into (6.33) and equating the powers of R and z on both sides, we obtain a2

- y2

=

-2,

p2 - d2 = 0.

(6.34)

TABLE V SIMILARITY EXPONENTS FOR THE ENSTROPHY DISSIPATION RANGE OF TWO-DIMENSIONAL TURBULENCE Case I1

Case I

a2

Numerical

Analytical

- 1.53

-3 = - 1.5

82

-

0.51

+ = 0.5 -+ -0.5 =

Analytical

1.32

-3

0.45

- f = -0.5 f = 0.5

-f = -0.5

P2

Y2

Numerical

-+

=

=

-1.5

-0.5

114

T. Tatsumi

On the other hand, it follows from (6.4), (6.8), (6.14), and (6.29) that

(6.35) Assuming that the enstrophy dissipation takes place in the higher wavenumber range, we substitute the similarity form (5.21) into (6.35) and obtain c(2

f 5yz =

1,

pz

-k 562 = -3.

(6.36)

From (6.34) and (6.36) we have a2 = - 312,

y z = 112,

pz = Sz = 112.

(6.37)

The similarity law represented by (6.37) is in perfect agreement with that of the spectrum (6.9) and (6.10), with (6.29) taken into account, so that the higher wavenumber range is completely identified with the quasi-equilibrium range. The asymptotic form of the spectrum for very large wavenumbers is obtained by the same procedure as in Section V,B,2 and the result is expressed as

E ( k ) = 128.3vZb3izk5i2 exp( - bk),

(6.38)

or in nondimensional form as

E(k)/Eo = 1 2 8 . 3 ~ ~ ' 3~1R2 (-R - 1 1 2 ~ ) 5exp[ i 2 - c(R- lizic)].

(6.39)

Thus, the asymptotic behavior of the energy spectrum of two-dimensional turbulence for k -+ co has the same exponential form as that of threedimensional turbulence given by (5.63) or (5.65). 3. Enst rophy

The energy 8(t)and the enstrophy 9(t)of turbulence are immediately obtained from the numerical data of the energy spectrum E(k,t) by using relations (5.5) and (6.19). The conservation of energy at large Reynolds numbers is confirmed numerically within an error of a few percent. Enstrophy is not conserved in the present calculation as expected in view of the analytical nature of the governing equation (6.18) examined in Section VI,A,2. In fact, 9(t)is kept nearly constant for a finite time period, t < t, and eventually decays according to a power law, 9(t)K t - ' , 2 , t - 1 . 6 ,

(6.40)

for cases I and 11, respectively. A rather large discrepancy of these values from the analytical value 2 as given by (6.29) is accounted for by the lack of time similarity of the numerically obtained spectrum in the higher wavenumber range.

Theory of Homogeneous Turbulence

115

The time t, increases with Reynolds number unlike the corresponding time t, for the energy decay in three-dimensional turbulence, and its Reynolds number dependence is roughly expressed as t, cc (log R)'/2.

(6.41)

Actually, this relation is consistent with condition (6.17) for making the enstrophy decay (6.4) finite in the similarity stage t > t,. The skewness S ( t ) of the velocity derivative defined by (5.6) is identically zero for the two-dimensional turbulence due to condition (6.21). Sometimes, a two-dimensional skewness defined by

JOm

-

[Jc

k4T(k,t )dk

k2E(k, t)dk]'12

Jr

(6.42)

k4E(k,t)dk'

is employed for dealing with the small-scale structure of two-dimensional turbulence (see Herring et a/.,1974),but we do not discuss this quantity here. Taylor's microscale defined by (5.7) is expressed, on substitution from (5.5) and (6.13), as L(t)2 =

5 &yt)/2(t)

(6.43)

for two-dimensional turbulence. Since &(t)is kept constant and 9(t)decreases in time as t P 2 , the microscale l ( t )increases proportionally to t .

B. TUREKJLENCE OF BURGERS The turbulence of Burgers is a random motion that takes place in the one-dimensional velocity field u(x,t ) governed by the Burgers equation of motion au -

at

d2U + u -du = v--. ax

(6.44)

ax2

It is well known that Eq. (6.44) describes the formation and evolution of weak shock waves in a compressible fluid (see Lighthill, 1956) and that the turbulence of Burgers represents a random series of shock waves each separated by an expansion wave. In fact, it was shown by Tatsumi and Tokunaga (1974) that an arbitrary one-dimensional weak nonlinear wave in a compressible fluid can be decomposed into two independent families of nonlinear waves each satisfying (6.44). In the present context, however,

T. Tatsumi

116

we restrict our consideration to the aspect of (6.44) as a one-dimensional model of the Navier-Stokes equation and investigate the statistical properties of the turbulence of Burgers in comparison with those of threedimensional turbulence. A remarkable feature of turbulence of Burgers compared with other kinds of turbulence is that the general solution of the governing equation (6.44) can be explicitly expressed as u(x, t ) =

j:m[(x - x’)/t] exp[ -(1/2v)U(x, x’; t)] dx‘ m: J

9

exp[ - (1/2v)U(x,x’; t)] dx‘

(6.45)

where U ( X x’; , t) = (x - ”I2 2t

+ Jtu(x”,0) dx”

(see Hopf, 1950; Cole, 1951). The behavior of solution (6.45) for very large Reynolds numbers R = uolo/vand times t, where uo and lo are characteristic velocity and length of turbulence, respectively, was investigated in detail by Tatsumi and Kida (1972)and Burgers (1974).For R >> 1 and z = (uo/lo)t>> 1, the solution takes the form of a random sequence of triangular shock waves, each represented by 1

U. + ui - 2 tanh (6.46) t 2 + xi)/2 c x -= (xi + xi+ 1)/2, where ui = dxi/dt, ui and xi

u ( x , t ) = - (x - xi)

in the region ( x i denote the propagation velocity, the velocity jump, and the coordinate of the ith shock front, i being an integer, respectively. The velocity ui and the length uit are shown to be invariant in time except for the instant of collision of two shocks. When the ith and (i 1)th shocks collide, they are united to a single shock, which is again represented by (6.46),but ui, vi, and xi replaced by u: = (ui ui+J2 = dxi/dt, u: = ui + u i + l , and xi, respectively, and the number i 1 dropped from the sequence (see Fig. 27).

+

+ +

+--

pi

-----I

FIG.27. Random sequence of triangular shock waves.

Theory of’ Homogeneous Turbulence

117

1. Inviscid Energy Dissipation For R >> T >> 1, the velocity field represented by (6.46) becomes almost discontinuous at the shock fronts x = x i , and obviously the energy dissipation takes place mostly in the shock fronts. Consider a large number of shock waves represented by (6.46) with i = 1,2, . . . , N , which occupy a large domain 0 I x I L. If we assume the equivalence of the probability average with the space average (1/L)Jk ( ) dx and denote the arithmetic mean (l/N) by an overbar, we obtain

zF=

e = 2 v ( ( ) 2 ) = - s 2v L 8 N 4 =

Lv

zl);(

m: J

(-)au [2 2

0

dx

ax

sech4

(x - xi)] dx

(6.47)

x?=

where li = xi+ - x i , Ii = L. Thus, the energy dissipation E remains finite in the inviscid limit v + 0, just like (2.19)for three-dimensional turbulence. In this respect, the turbulence of Burgers is expected to have the same similarity law in the energy dissipation range as does three-dimensional turbulence, and in fact this is shown below to be the case. Unlike the similarity in the dissipation range, there exists a big difference between the large-scale structures of these two turbulences. As mentioned before, the velocity field u(x, t ) of turbulence of Burgers is expressed as an integral (6.45) including its initial form u(x,O), and the formation of shock and expansion waves is solely determined by the values of u(x,0) at discrete points on the x axis. Therefore, the system of shock waves inherits its randomness only partially from the initial velocity field. Moreover, the number of shocks composing the system is reduced by successive coalescence of shock fronts, so that the randomness of turbulence of Burgers decreases steadily in time. Such a situation is quite different from that of the threedimensional turbulence governed by the Navier-Stokes equations, in which an infinitesimal randomness in a solution u(x, t ) is enlarged at later times through the action of nonlinear forces so that the randomness is continually produced in time. It is shown later that turbulence of Burgers does not satisfy the inertial-subrange similarity law (3.19), although it satisfies the universal equilibrium similarity laws (3.17)and (3.18)at higher wavenumbers. The lack of the former similarity in the lower wavenumber range is accounted for by the above-mentioned insufficient randomness of turbulence of Burgers.

118

T. Tatsumi

2. Energy Spectrum The equation for the energy spectrum of turbulence of Burgers based on the quasi-equilibrium zero-fourth-order-cumulant approximation is

;(

+ m’)E(k, t ) = T(k,t )

with 1 - exp[-2v(k2 + k 2 + kk’)t] Jrm 2,,(k2 kk’)

T(k7t , =

+k 2 +

x

{ [kE(k‘, t ) + k’E(k,t)]E(k + k‘, t ) - ( k + k’)E(k, t)E(k’,t ) ) dk‘. (6.48)

As in three-dimensional turbulence, the asymptotic solution of this equation for extremely large Reynolds numbers is given by the solution of the equation T ( k ,t ) = 0

as follows :

r

(6.49)

E ( k , t ) = Ak-’,

for vk’t << 1,

(6.50)

Ak-I,

for vk2t >> 1.

(6.51)

Obviously, (6.49) represents the equipartition of energy in the wavenumber space. The other spectra (6.50) and (6.51)have exactly the same forms as the corresponding asymptotic spectra (4.30) and (4.31) for three-dimensional turbulence, respectively. Equation (6.48)was solved numerically by Mizushima and Tatsumi (1980) under the initial conditions (1) E(k, 0)lEO = exp[ - (k/k0)2], (11) E(k, 0)lEO = (k/kO)’exp[ - (klko)’],

(6.52) (6.53)

for a range of Reynolds number R I 200, R = lo4, and the calculation was extended by Mizushima and Segami (1980) to other initial conditions. The energy spectrum at R = uo/(vko)= lo4 is shown graphically in Fig. 28. The general appearance of the spectrum in the similarity stage is quite similar to that of three-dimensional turbulence. (a) Energy-containing range The similarity exponents in the energycontaining range obtained by the curve matching are listed in Table VI with

I

I L

k/ko

FIG.28. Energy spectrum E(k, t ) of turbulence of Burgers for R

k/k, = u,/vk, =

lo4. (a) Case I, (b) case I1 (after Mizushima and Tatsumi, 1980).

T. Tatsumi

120

TABLE VI SIMILARITY EXPONENTS FOR THE ENERGY-CONTAINING RANGE OF TURBULENCE OF BURGERS

Case I1

Case I

Numerical a1 81

Y1 61

Analytical 0 0 0

0.00 0.00 0.00 - 0.66

- 32

Numerical 0.00

- 0.73 0.00

- -0.667

-0.39

-

Analytical 0 - 0.659" 0 - 0.447"

'Taken from (6.71).

the suffix 1. The exponents are also determined analytically by using the asymptotic form of the spectrum for large Reynolds numbers. In the limit of v + 0 and finite k, (6.48) reduces to

a

- E(k, t ) = kt at

:j

m

{ [kE(k',t ) + k'E(k, t ) ] E ( k + k', t )

- (k + k')E(k,t)E(k',t)]dk'.

(6.54)

Substituting the similarity form (5.21) into Eq. (6.54) and equating the powers of R and z on both sides, we have a1

+ 371 = 0,

+ 361 = -2.

(6.55)

Another condition is provided by considering the asymptotic behavior of the spectrum at very small wavenumbers. For case I, (5.11) and (5.21) are written as E(k, t ) / E o = A . = AbRa1rB1,

(6.56)

for k x 0, where A . is free from viscosity but in general a function o f t and Nois a nondimensional constant. Substituting (6.56) into (6.54), we find that (6.57)

dAo/dt = 0,

so that A . is an absolute constant. Hence, it follows from (6.56) and (6.57) that tll

= 81 = 7 1 = 0,

61

= -2/3.

(6.58)

For case 11, (5.11) and (5.21) are written as

E(k, t ) / E o = A& = A;R"1-2y1z~1-261~2,

(6.59)

Theory of Homogeneous Turbulence

121

for k N 0, where again A 2 is a function of time and A; is a nondimensional constant. Substituting (6.59) into (6.54), we obtain an equation for A,: dA2 -dt- - k: t

Eo j:m E(k', t)2 dk'.

(6.60)

If we employ, as a rough approximation, the initial form (6.53) for the similarity form (5.21) at 7 = 1, we have

E(k, t)/Eo = Ro'1~P1[~/(RY1~61)]2 exp{ - [ K / ( R ~ ~ T ~ ~ ) ] ~(6.61) }, and A; = 1. On substitution from (6.61), (6.60) assumes the form (6.62) Comparing (6.59) and (6.62), we obtain

The values of the similarity exponents given by (6.58) and (6.63) for cases I and 11, respectively, are in perfect agreement with the numerically obtained values in Table VI except for PI and 6, for case 11, for which the error is not negligible but still fairly small in view of the roughness of the approximation employed. At higher wavenumbers in the energy-containing range, the spectrum takes the k-' form. The similarity law for this spectrum is given by

(I) E ( k , t ) / E o = 0 . 2 8 ~ - ~ ' ~ ~ - ~ ,

(6.64)

(11) E ( k , t ) / E o = 0 . 1 5 ~ - ' . ~ ~ ~ - ~ ,

(6.65)

where the numerically obtained values have been used for the exponents. The similarity law corresponding to (6.64) and (6.65) is also derived from the asymptotic expression of the velocity field (6.46) for R >> 1, z >> 1. If we take a pair of spatial points x and x + r that are so close to each other that both are included in a triangular shock region represented by (6.46), u(x

+ r,t) - u ( x , t ) = t

2 (6.66)

122

T. Tatsumi

If we employ the same assumption as that used for obtaining (6.47), the velocity covariance is expressed as

( [ u ( x + r, t) - u ( x , t)]')

r 1

= - v2 coth

(6.67)

for r << 7. The energy spectrum is related to the velocity correlation by 1 E(k, t) = 27c

-

s"

(u(x

0

--J"1

+ r, t)u(x,t ) ) cos kr dr a2

27ck2

-( u ( x 0

dr2

+ r, t)u(x, t ) ) cos kr dr.

(6.68)

Upon subsitution from (6.67), (6.68) takes the form 7cV2

E(k, t) = -cosech2

7

(6.69)

For small wavenumbers k << ij/v, (6.69) becomes

E(k, t) = (U2/4~7)k-~.

(6.70)

According to the shock statistics made by Tatsumi and Kida (1972),

(6.71) and a = 2/3 and 1/2 for cases I and 11, respectively. Thus, (6.70) can be written as

(6.72) (11) E(k, t) = ___ '0 2nty

t-3/2k-2

(6.73)

The asymptotic forms (6.72) and (6.73) of the spectrum are in excellent agreement with the numerical results (6.64)and (6.65), respectively, although comparison of the numerical factors is not possible owing to the difference in the time and length scales employed. In this context, it should be noted that the similarity laws (6.72) and (6.73) are confirmed, again but for the numerical factors, by a rigorous treatment of shock statistics made by Kida (1979). (b) Energy-dissipation range In the energy-dissipation range characterized by the condition (vt)'I2k >> 1, energy spectrum equation (6.48)

Theory of Homogeneous Turbulence reduces to

k 2vk2E(k,t)= 2v

s"

--co

123

(k2 + k 2 + kk')-'

x ([kE(k', t )

+ k'E(k, t)]E(k",t ) - ( k + k')E(k,t)E(k', t ) }dk'. (6.74)

Substituting the similarity form (5.21)into (6.74) and equating the powers of R and z on both sides, we obtain the relation a2

- y2 =

-2,

p 2

-

a2 = 0.

(6.75)

Another relation is derived from consideration of the energy dissipation, which, by virtue of (44,(4.16), and (4.17) can be written as

d dt

~ ( t=) --

s" 0

E(k, t )d k = 2v

k2E(k,t )dk.

(6.76)

At very large Reynolds numbers, the energy-containing and energy-dissipation ranges are well separated in the wavenumber space, and the energy integral and the dissipation integral in (6.76) are solely determined by the respective wavenumber ranges. Then substituting the similarity forms (5.21) with suffixes 1 and 2 into the energy and dissipation integrals in (6.76), respectively, we find the relation

+

71

= Ct2

+ 372 - 1 ,

+ 61 - I =

P2

4- 362.

(6.77)

Thus, from (6.75), (6.77), and the numerical values of the exponents in Table VI we obtain

(;I)

P 2 = 62 =

(:go).

(6.78)

The values of a2 and y2 given by (6.78) are identical with those of (5.45) for three-dimensional turbulence, and therefore turbulence of Burgers is governed by the same Kolmogorov similarity law as three-dimensional turbulence in the energy dissipation range, which is characterized by kd = p V - 3 / 4 [see (3.18)].Nevertheless, there is no evidence of the inertial subrange spectrum (3.19) for turbulence of Burgers, but another inviscid similarity spectrum proportional to k - is realized instead. Thus, it may be concluded that the inertial subrange spectrum should not be taken as a unique consequence of Kolmogorov's similarity in the dissipation range, but it is only one of possible inviscid spectra that are compatible with the latter.

T. Tatsumi

124

In this sense, Kolmogorov’s second hypothesis of inviscid similarity is not a corollary of the first hypothesis of local similarity but an independent assumption. A turbulent motion that gives an inviscid energy dissipation is likely to satisfy the first hypothesis in the dissipation range, but not necessarily the second hypothesis at lower wavenumbers. In order to satisfy the latter as well, the turbulence must have a detailed similarity of largescale components, but such a similarity is easily broken by the interrnittency associated with large-scale motions. In this sense, it may be rather natural that Kolmogorov’s inviscid similarity is not realized in turbulence of Burgers, whose intermittent structure is clearly observed in the succession of triangular shock waves. In the far dissipation range k >> k , = ~ ~ ~ the~ energy v -spectrum ~ ~ de-~ creases exponentially in k as seen in Fig. 28. The asymptotic behavior of the spectrum for k + 00 is obtained by using essentially the same method as in Section V,B,2 as follows:

E ( k ) = (4/I)vzk exp( - bk) = 19.12v2kexp( -bk),

(6.79)

where 1

Y(1 - Y )

271 3$ - 1 = 0.2092,

or in nondimensional form, E(k)/Eo = 19.12R-s/4(~/R3/4) exp( - ctc/R3I4).

(6.80)

This behavior of the spectrum is in qualitative agreement with that expected from (6.69) for k >> i+. (c) Intermediate range As the Reynolds number increases, the energycontaining range and the dissipation range become more and more separated from each other, and there appears between them an intermediate wavenumber range that satisfies a different similarity law from those of neighboring ranges. The similarity exponents of this range are determined analytically in the same way as in Section V,B,3 (see also Kida, 1980), and are cI3

= - 1,

(1)

P3

= - 1,

(11)

P3

=

-0.553,

y3 =

1/2,

(r3

=

63 =

1/6,

(6.8 1)

-0.500,

where the analytical values have been used for P1 and d1 for case 11. These values of a3 and y3 are equal to the corresponding values for three-dimensional turbulence given in Table 111. Thus, it is concluded from this result

,

125

Theory of Homogeneous Turbulence

and those for the energy-containing and dissipation ranges that the turbulence of Burgers satisfies exactly the same similarity law with respect to the Reynolds number as does three-dimensional turbulence in each of the three wavenumber ranges.

3. Energy, Skewness, and Microscale (a) Decay of energy The energy &(t) calculated from the numerical data of the energy spectrum E(k, t ) decays in time according to a power law, as follows:

(I) & ( c ) / &=~1 . 0 7 ~ - O . ~ ~ ~ ,

(11)

(6.82)

&(t)/&, = 0.925~-'.'~.

(6.83)

On the other hand, Tatsumi and Kida (1972) obtained the following energy decay laws using statistics of shocks: (6.84)

(I) &(t)= 33,4'3t-2'3, (11)

&(t) = $l&

(6.85)

' t - 1.

The numerically obtained exponent for case I is in perfect agreement with the analytical value - 2/3, which also follows from dimensional analysis, but the agreement is less satisfactory for case 11. The numerical simulation of turbulence of Burgers was carried out by several authors. Crow and Canavan (1970) obtained &(t) a t - ' for case I1 as the limit of infinite Reynolds number, but Walton (1970) reported &(t)a t - ' . 2 5 for case 11. Yamamoto and Hosokawa (1976) made a Monte Carlo simulation for various initial conditions and obtained &(t) cc t - 0 . 7 3 for case I and 8(t)a t-'.I7 for case 11. More recent numerical experiments t - ' . 0 5 for by Kida (1979) give 8(t)cc t C 2 j 3 for case 1 and &(t)cc t-"" case 11, the exponent varying slightly due to the choice of the initial distribution.

-

(b) Skewness of oelocity derivative The skewness of the velocity derivative is expressed for turbulence of Burgers as

The evolution of S(t) calculated from the numerically obtained E(k, t ) and T(k,t ) has a similar appearance to that of three-dimensional turbulence shown in Fig. 24. The numerical value of S ( t ) is, however, consistently larger than that of three-dimensional turbulence, and the asymptotic value

T. Tatswni

126 S , = S(o0) tends to

(I) S , = 2.0,

(11) S , = 1.9

(6.87)

for infinite Reynolds number. According to the statistics of shocks, S ( t ) is expressed as (6.88)

Using relation (6.71), we write (6.88) as (I) S(t) = (2$/5)R:/2(t/t,)'/6,

(11) S ( t ) = (2a/5)R:I2,

(6.89) (6.90)

where R1 = l:/vto. The almost invariance of S(t) in time is compatible with the behavior of the numerical results, but the Reynolds number dependence of the former is at variance with the trend of the latter, which seems to approach an inviscid limit (6.87). The large-valuedness of S(t) for turbulence of Burgers compared with that of three-dimensional turbulence reflects the highly asymmetric structure of turbuIence of Burgers consisting of regions of very strong compression and weak expansion. (c) Microscale The microscale is written for turbulence of Burgers as

The evolution of the numerically obtained A@) is again similar to that of three-dimensional turbulence as shown in Fig. 25. In the similarity stage it satisfies the similarity law (I) A ( t ) = 1.92k,'R-0.5'~0.46, (11) A(t) = 1.31k~1R-0.50~0.50.

(6.92) (6.93)

The similarity law can also be derived from the relation A(t)2

2vb(t) d&(t)/dt'

= -~

(6.94)

and is, for cases I and 11, (I) A(t) = 1.73k; 'R-

1/2~1/2,

(11) A ( t ) = 1.33k~'R-'/2~'/2.

(6.95) (6.96)

Theory of Homogeneous Turbulence

127

The agreement of these two sets of numerical results proves the consistency of the numerical calculation. On the other hand, if we substitute (6.84) and (6.85) into (6.95) we obtain the similarity law according to the shock statistics :

(I) A(t) = J ? v W / 2 , (11) A ( t ) = $ 2 V 1 / 2 t ?

(6.97) (6.98)

Agreement with the above numerical results is again satisfactory.

VII. Concluding Remarks In concluding this chapter it may be appropriate to review the nature of the approximation employed in this work. The present approximation scheme consists of two superimposed expansions, one being the ordinary cumulant expansion and the other the Taylor expansion of the nonlinear terms in each cumulant equation in time. The cumulant expansion gives the system of equations (2.30),which may be written schematically as

the first two equations being identical with (2.31) and (2.32). The physical arguments can be made more conveniently by means of the averaged cumulants defined by

F3' = JJO3)(k, k')k2K2 do do', C4)= JJJC(4'(k, k', k " ) k 2 k , 2 k r ' 2 do do' do",

(7.4)

T. Tatsumi

128

where c denotes the solid angles in the wavenumber space. Then, the cumulant equations are written as

If we neglect C(4)in Eq. (7.6), its solution is immediately expressed as

F 3 )=

Ji exp[ - vk2(t - t')][kC(2)C(2)],. dt'.

Equations (7.5) and (7.8) constitute the dynamical equation for F2), or equivalently the energy spectrum E(k, t ) = * C f ) ( k ,t), under the zero-fourthorder-cumulant approximation. Furthermore, if we expand the product of C(') on the right-hand side of (7.8) in time around time t, (7.8) can be written as

where 0, is defined by (4.33). Taking only the first term of this expansion, we have c(3) = @o(t)[kC(2)C(2)],. (7.10) The set of equations (7.5) and (7.10) is the energy spectrum equation used in this chapter. In the framework of this approximation, it has been established that the energy spectrum E(k, t ) satisfies different similarity laws in different wavenumber ranges. For three-dimensional turbulence and turbulence of Burgers, there exist three wavenumber ranges: '

energy-containing range: intermediate range: energy-dissipation range:

k x ko = O(1),

(7.11)

k = ki = O(v-'l2),

(7.12)

k x k, = O ( V - ~ / ~ ) .

(7.13)

For two-dimensional turbulence, on the other hand, there exist only two wavenumber ranges: enstrophy-containing range:

k w ko = O(l),

(7.14)

enstrophy-dissipation range:

k x k, = 0(v-'l2).

(7.15)

Theory of' Homogeneous Turbulence

129

According to the evaluation of the order of magnitude of 0, given by (4.34)and (4.35),the 0 expansion is asymptotically good for v + 0 in the energy-dissipation range and for t -+ 0 in the energy-containing and enstrophy-containing ranges, but rapid convergence is not generally guaranteed in other ranges. As the viscosity v decreases or the Reynolds number increases, the width of the intermediate range in three-dimensional turbulence and turbulence of Burgers increases like Y 3 I 4 , so that the wavenumber range in which the expansion is not good increases with the Reynolds number. Such a trend seems to be unavoidable in view of the basic character of the cumulant expansion as an ascending power series of Reynolds number. This drawback of the present approximation, however, does not essentially affect the behavior of the energy since, as seen in Section V,B,2 and VI,B,2, the decay of energy is determined by the energy spectrum in the energycontaining range and the energy-dissipation range, while the intermediate range acts merely as a lossless transmitter of the energy. The same argument does not apply to the enstrophy decay of two-dimensional turbulence, owing to its different similarity character, but the argument is equally valid for the period t >> ( v k i ) - ' . Finally, let us consider the higher order approximation of the cumulant expansion for three-dimensional turbulence. In the energy-dissipation range k z kd, where the 0 expansion is well founded, the order of magnitude of and C ( 3 ) is determined from (7.5) and (7.6),with C ( 4 ) omitted, as follows:

e(')

C(2)= 0 ( , , 5 / 4 ) ,

C(3)

= o(v9/4).

(7.16)

Likewise, (7.6) and (7.7) are consistently satisfied by C(4)

= 0(~13/4),

C(5)= o(,,17/4

1,

(7.17)

together with (7.16). Similarly, it can be shown, in general, that

C(n) = O(,,n-(3/4)).

(7.18)

Thus, although the cumulant itself decreases monotonically with increasing order, the effect of the higher order cumulant in a cumulant equation remains exactly of the same order of magnitude as that of the other terms. In this sense, the zero-cumulant approximation of any order is not well founded in the energy-dissipation range. On the other hand, the simple regularity of the situation in the energydissipation range as exemplified by (7.18) makes it possible to deal with this wavenumber range exactly by taking the whole system of cumulants into consideration. In fact, it can be shown that the quasi-equilibrium similarity law (5.45) for c1 and y in the energy-dissipation range and the asymptotic similarity form (5.59) of the spectrum in the far dissipation range are exact analytical results uninfluenced by taking account of higher order cumulants (see Tatsumi and Kida, 1980). It is hoped that these exact results of the

130

T. Tatsumi

cumulant expansion can be used as a step to a more rigorous treatment of turbulence than that provided by successive approximations of the cumulant expansion. ACKNOWLEDGMENTS The author wishes to express his hearty thanks to Dr. Shigeo Kida, Dr. Jiro Mizhushima, and Mr. Shin-ichiro Yanase for their collaboration in preparing this chapter, which is essentially based on the results of my joint work with them. The author’s thanks are also due to Dr. Takuji Kawahara, who kindly read the manuscript and gave useful suggestions for improvement. The author retains, however, full responsibility for any errors that may remain. Finally, the author wishes to record his sincere gratitude to Professor Uriel Frish for his stimulating and suggestive discussions, Professor Chia-Shun Yih for inviting him to write a chapter in this volume, and to those authors and publishers who kindly permitted to reproduce figures from their publications. During the course of this work the author has been in receipt of a grant-in-aid for scientific research from the Ministry of Education of Japan. REFERENCES ~ R A M O W I M., T Z ,and STEGUN,I. A. (1964). “Handbook of Mathematical Function.” U.S. Department of Commerce, Washington, D.C. BATCHELOR,G . K. (1953).“The Theory of Homogeneous Turbulence.” Cambridge Univ. Press, London and New York. G. K. (1969). Computation of the energy spectrum in homogeneous twoBATCHELOR, dimensional turbulence. Phys. FluidF 12, Suppl. 11,233-239. I. (1956). The large-scale structure of homogeneous BATCHELOR, G. K., and PROUDMAN, turbulence. Philos. Trans. R. SOC.London, Ser. A 248, 369-405. BATCHELOR, G. K., and TOWNSEND, A. A. (1947). Decay of vorticity in isotropic turbulence. Proc. R. SOC.London, Ser. A . 191, 534-550. A. A. (1949).The nature of turbulent motion at large waveBATCHELOR, G. K., and TOWNSEND, numbers. Proc. R. SOC.London, Ser. A 199, 238-255. BIRKHOFF,G. (1954). Fourier synthesis of homogeneous turbulence. Comrnun. Pure Appl. Math. 7, 19-44. BURGERS, J. M. (1974). “The Nonlinear Diffusion Equation.” Reidel Publ., Dordrecht, The Netherlands. CHAMPAGNE, F. H. (1978). The fine-scale structure of the turbulent velocity field. J . Fluid Mech. 86,67-108. CHOU,P. Y. (1940). On an extension of Reynolds’ method of finding apparent stress and the nature of turbulence. Chin.J . Phys. 4, 1-33. COLE,J. D. (1951). On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225-236. COMTE-BELLOT, G . , and CORRSIN, S. (1966). The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25,657-682. G. H. (1970). Relationship between a Wiener-Hermite expansion CROW,S. C., and CANAVAN, and an energy cascade. J . Fluid Mech. 41, 387-403. N. J. (1971). Ergodic boundary in numerical simulations of twoDEEM,G. S . , and ZABUSKY, dimensional turbulence. Phys. Rev. Lett. 27, 396-399. S. F. (1964). The statistical dynamics of homogeneous turbulence. J . Fluid Mech. EDWARDS, 18,239-273. FORNBERG, B. (1977). A numerical study of 2-D turbulence. J. Comp. Phys. 25, 1-31.

Theory of Homogeneous Turbulence

131

FRENKIEL, F. N., and KLEBANOFF, P. S. (1967). Higher-order correlations in a turbulent field. Phys. Fluids 10, 507-520. FRISCH,U., SULEM,P. L., and NELKIN,M. (1978). A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87, 719-736. GAD-EL-HAK, M., and CORRSIN,S. (1974). Measurements of the nearly isotropic turbulence behind a uniform jet grid. J . Fluid Mech. 62, 115-143. GIBSON,C. H., and SCHWARZ,W. H. (1963). The universal equilibrium spectra of turbulent velocity and scalar fields. J. Fluid Mech. 16, 365-384. GIBSON,M. M. (1962). Spectra of turbulence at high Reynolds number. Nature (London) 195, 1281-1283. GIBSON,M. M. (1963). Spectra of turbulence in a round jet. J . Fluid Mech. 15, 161-173. GRANT,H. L., STEWART,R. W., and MOILLIET, A. (1962). Turbulence spectra from a tidal channel. J. Fluid Mech. 12, 241 -268. HERRING,J. R. (1965). Self-consistent-field approach to turbulent theory. Phys. Fiuids 8, 22 19-2225. HERRING, J. R. (1966). Self-consistent-field approach to nonstationary turbulence. Phys. Fluids 9,2106-2110. HERRING,J. R., and KRAICHNAN, R. H. (1972). Comparison of some approximations for isotropic turbulence. Lecture Notes Phys. 12, 148- 194. HERRING, J. R., ORSZAG,S. A,, KRAICHNAN, R. H., and Fox, D. G. (1974). Decay of twodimensional homogeneous turbulence. J. Fluid Mech. 66, 417444. HOPF,E. (1950). The partial differential equation u, + uu, = uxx.Commun. Pure Appl. Math. 3, 201 -230. HOPF,E. (1952). Statistical hydromechanics and functional calculus. J. Ration. Mech. Anal. 1, 87-123. HOPF,E., and Tim, E. W. (1953). On certain special solutions of the @-equation of statistical hydrodynamics. J . Ration. Mech. Anal. 2, 587-592. KAWAHARA, T. (1968). A successive approximation for turbulence in the Burgers model fluid. J. Phys. Soc. Jpn. 25, 892-900. KIDA,S. (1979), Asymptotic properties of Burgers’ turbulence. J. Fluid Mech. 93, 337-377. KIDA,S. (1980).To be published. KOLMOGOROV, A. N. (1941a). The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nuuk SSSR 30,301 -305. KOLMOCOROV, A. N . (1941b). On degeneration of isotropic turbulence in an incompressible viscous liquid. Dokl. Akud. Nuuk SSSR 31, 538-540. KOLMOGOROV, A. N. (1962). A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J . Fluid Mech. 12, 82-85. KRAICHNAN, R. H. (1959).The structure of isotropic turbulence at very large Reynolds numbers. J. Fluid Mech. 5, 497-543. KRAICHNAN, R. H. (1964). Approximations for steady-state isotropic turbulence. Phys. Fluids 7, 1163-1 168. KRAICHNAN, R. H. (1967). Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417- 1423. KRAICHNAN, R. H. (1971). An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513-524. LAMB,H. (1932), “Hydrodynamics.” Cambridge Univ. Press, London and New York. LANDAU,L. D., and LIFSCHITZ, E. M. (1959). “Fluid Mechanics.” Pergamon, Oxford. LEITH,C. E. (1968). Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11,671 -673. LEITH,C. E. (1971). Atmospheric predictability and two-dimensional turbulence. J . Atmos. Sci. 28, 145-161.

T. Tatsumi LIGHTHILL, M. J. (1956). Viscosity effects in sound waves of finite amplitude. In “Surveys in Mechanics” (G. K. Batchelor and R. M. Davies eds.), pp. 250-351. Cambridge Univ. Press, London and New York. LILLY,D. K. (1971). Numerical simulation of developing and decaying two-dimensional turbulence. J. Fluid Mech. 45, 395-415. LILLY,D. K. (1972a). Numerical simulation studies of two-dimensional turbulence: I. Models of statistically steady turbulence. Geophys. Fluid Dyn. 3, 289-319. LILLY,D. K. (1972b). Numerical simulation studies of two-dimensional turbulence: 11. Stability and predictability studies. Geophys. Fluid Dyn. 4, 1-28. LING,S. C., and WAN,C. A. (1972). Decay of isotropic turbulence generated by a mechanically agitated grid. Phys. Fluids 15, 1363-1369. LOITSIANSKY, L. G. (1939). Some basic laws of isotropic turbulent flow. Rep. Cent. Aero. Hydrodyn. Inst. (Moscow) No. 440; translated as NACA Tech. Memo. 1079 (1945). LUMLEY,J. L. (1972). Application of central limit theorems to turbulence problems. Lect. Notes Phys. 12, 1-26. MILLIONSHTCHIKOV, M. (1941). On the theory of homogeneous isotropic turbulence. Dokl. Akad. Nauk SSSR 32,615-618. J., and TATSUMI, T. (1980). To be published. MIZUSHIMA, A. M. (1975). “Statistical Fluid Mechanics,” Vol. 2. MIT Press, MONIN,A. S., and YAGLOM, Cambridge, Massachusetts. OBUKHOV, A. M. (1962). Some specific features of atmospheric turbulence. J. Fluid Mech. 12, 77-81. OGURA,Y. (1963). A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence. J. Fluid Mech. 16, 38-41. ORSZAG,S . A. (1970). Analytical theories of turbulence. J. Fluid Mech. 41, 363-386. ORSZAG,S. A. (1977). Lectures on the statistical theory of turbulence. In “Fluid Dynamics,” Les Houches, 1973 (R. Balian and P. L. Peube ed.), pp. 235-374. Gordon and Breach, New York. ORSZAG,S. A., and PATTERSON, G. S. (1972). Numerical simulation of turbulence. Lect. Notes Phys. 12, 127-147. J. C., and BASDEVANT, C. (1975). Evolution of high Reynolds POUQUET, A., LESIEUR, M., ANDR~, number two-dimensional turbulence. J. Fluid Mech. 72, 305-319. I., and REID,W. H. (1954). On the decay of normally distributed and homogeneous PROUDMAN, turbulent velocity fields. Philos. Trans. R . SOC.London, A 247, 163-189. ROSENBLA~, M. (1972). Probability limit theorems and some questions in fluid mechanics. Lect. Notes Phys. 12, 27-40. SAFFMAN, P. G. (1968). Lectures on homogeneous turbulence. In “Topics in Nonlinear Physics” (N. J. Zabusky ed.), pp. 485-614. Springer-Verlag, Berlin and New York. SAFFMAN, P. G. (1971). On the spectrum and decay of random two-dimensional vorticity distributions at large Reynolds number. Stud. Appl. Math. 50, 377-383. SAFFMAN, P. G. (1978). Problems and progress in the theory of turbulence. Lect. Notes Phys. 76, 273-306. J., STEGEN,G . R., and GIBSON,C. H. (1974). Universal similarity at high grid SCHEDVIN, Reynolds numbers. J. Fluid Mech. 56, 561-579. STEWART, R. W. (1951). Triple velocity correlations in isotropic turbulence. Proc. Cambridge Phibs. SOC.47, 146-147. TANAKA, H. (1969).0-5th cumulant approximation of inviscid Burgers turbulence. J. Meteorol. SOC.Jpn. 47, 373-383. TANAKA, H. (1973). Higher order successive expansion of inviscid Burgers turbulence. J. Phys. SOC.Jpn. 34, 1390-1395 Y., and KAMOTANI, Y. (1975). Experiments on turbulence behind a grid with jet injection TASSA, in downstream and upstream direction. Phys. Fluids 18,411-414.

Theory of Homogeneous Turbulence

133

TATSUMI, T. (1955).Theory of isotropic turbulence with the normal joint-probability distribution of velocity. Proc. Jpn. Natl. Congr. Appl. Mech., 4th, 1954 pp. 307-31 I . TATSUMI, T. (1956). The energy spectrum of incompressible isotropic turbulence. Acres Congr. h i . Mecan. Appl., 9th, 1956 Vol. 3, pp. 396-404. TATSUMI, T. (1957). The theory of decay process of incompressible isotropic turbulence. Proc. R . Soc. London, Ser. A 239, 16-45. TATSUMI, T. (1960). Energy spectra in magneto-fluid dynamic turbulence. Rev. Mod. Phys. 32, 807-812. TATSUMI, T., and KIDA,S. (1972). Statistical mechanics of the Burgers model of turbulence. J. Fluid Mech. 55,659-675. TATSUMI, T., and KIDA,S. (1980). To be published. TATSUMI, T., KIDA.S., and MIZUSHIMA, J. (1978). The multiple-scale cumulant expansion for isotropic turbulence. J. Fluid Mech. 85, 97-142. TATSUMI, T., and TOKUNAGA, H. (1974). One dimensional shock turbulence in a compressible fluid. J. Fluid Mech. 65, 581-601. TATSUMI, T., and YANASE, S. (1980). To be published. G. I. (1935). Statistical theory of turbulence. I-IV. Proc. R. Soc. London, Ser. A 151, TAYLOR, 421 -478. UBEROI,M. S. (1963). Energy transfer in isotropic turbulence. Phys. Fluids 6, 1048-1056. VANATTA,C. W., and CHEN,W. Y. (1968). Correlation measurements in grid turbulence using digital harmonic analysis. J. Fluid Mech. 34, 497-515. VANATTA,C. W., and YEH,T. T. (1970). Some measurements of multipoint time correlations in grid turbulence. J . FIuid Mech. 41, 169-178. J. J. (1970). Integration of the Lagrangian-history approximation to Burgers’ equation. WALTON, Phys. Fluids 13, 1634-1635. WYNGAARD, J. C., and PAO,Y. H. (1972). Some measurements of the fine structure of large Reynolds number turbulence. Lect. Notes Phys. 12, 384-401. K., and HOSOKAWA, I. (1976). Energy decay of Burgers’ model of turbulence. YAMAMOTO, Phys. Fluids 19, 1423-1424.

This Page Intentionally Left Blank

ADVANCES IN APPLIED MECHANICS, VOLUME

20

Thermoacoustics NIKOLAUS ROTT Instirut fur Aerodynamik Federal Institute of Technology ( E T H ) Zurich, Switzerland

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

135

11. Oscillating Flow over a Nonisothermal Surface . . . . . . . . . . . . . . . . . . .

138

A. Velocity Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Temperature Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The Velocity Perpendicular to the Wall . . . . . . . . . . . . . . . . . . . . .

138 139 141 143

111. Damping and Excitation of a Gas Column with Temperature Stratification. . . .

A. Preliminary Calculations for Thin Boundary Layers . . . . . . . . . . . . . . B. The History of the Stability Problem. . . . . . . . . . . . . . . . . . . . . . . C. The Complete Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . D. Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Thermoacoustic Streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Isothermal Walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Nonisothermal Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 146 148 153 168 168 170 173

174

I. Introduction The use of the word “thermoacoustics” is not very widespread; however, its meaning is rather self-explanatory. Nevertheless, the limits of the subject matter treated in this chapter must be carefully defined, in view of the possible most general interpretation of the word thermoacoustics, which would include all effects in acoustics in which heat conduction and entropy variations of the (gaseous) medium play a role. Considering first the effect of heat conduction, it is known that whenever friction is taken into account, heat conduction cannot be neglected. Thus all acoustics in gases in which diffusive effects are considered belongs rightfully to the field of thermoacoustics. 135 Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-002020-3

136

Nikolaus Rott

However, a practical classification offers itself since the creation of the theory of thermoacoustics in this general sense by Kirchhoff in 1868. As is well known, the two important results deduced by Kirchhoff from his theory were corrections to the theory of friction; one result corrected Stokes’ formula for the sound attenuation of plane waves in an unlimited medium; the second gave a modified result of the Helmholtz-Rayleigh theory of the attenuation in a duct. Nevertheless, the great significance of the Kirchhoff theory was immediately manifest; Rayleigh fully incorporated it in his book, in the chapter on friction and heat conduction. In a different chapter of his “Theory of Sound” [in Vol. 2, Sections 322f-i of the (reprinted) 1896 edition], Rayleigh (1896) describes several examples of “maintenance by heat of aerial vibrations.” These phenomena are called thermoacoustic effects in the more restricted sense of the word, which is accepted here. The full theoretical treatment of certain phenomena described by Rayleigh, in which sound is directly produced by heat, occupies the central portion of this chapter. The theory is based on a generalization of Kirchhoffs work, which is extended to include the effect of the temperature stratification of the medium, i.e., the effect of nonconstant entropy. Now the most general theory of acoustics in a nonisentropic medium, which logically also includes the effect of heat sources (including combustion), covers a multitude of phenomena that transcend the limits of one monograph. Therefore, for instance, combustion effects are excluded: all heat sources are assumed to be located at solid boundaries. Also, a considerable literature has developed recently on acoustic effectsconnected with nonisentropic flow in ducts of different kinds. These flow-dominated effects certainly require a special survey paper and are not covered here. A recent paper that is concerned with thermoacoustic effects and is included in this survey was published by Kempton (1976), who investigated a number of examples of “heat diffusion as a source of aerodynamic sound.” Part of his examples are treated (in modified form) in the present work. A particular question discussed first is the effect of heat (entropy) spots on sound generation in turbulence, an essential problem in Lighthill’s theory. The character of the most important acoustic source term was clarified by Kempton (1976) and Morfey (1976); the following discussion follows the arguments given by Kempton. The fundamental question is whether the mixing of several “lumps” of gas with different temperatures can yield an acoustic source term. For an answer using only elementary considerations, it is noted that the equation of state of an ideal gas can be written in the form p = (y - l)ep, where e is the internal energy per unit mass, and ep the internal energy per unit volume, which is thus proportional to the pressure p . It follows that a mixing at a

Thermoacoustics

137

constant pressure p preserves the total volume V of the components, as the total energy E=- P Y-1

XI.:=Y-1

is constant together with p = const, provided that y is the same for each of the components with the volume K.However, when different gases are mixed (Morfey) or in the presence of relaxation effects (Kempton), the specific heats and therefore y are not necessarily the same for the mixing components, so that volume changes could occur, leading to acoustic monopoles. The practical significance of this interesting mechanism is not yet clarified experimentally. For constant y, however, only the possibility of a (weak) dipole source exists. In a monograph in the series “Reports on Progress in Physics” of the year 1956, Mawardi published a report on “Aero-Thermoacoustics.”The part on aeroacoustics is essentially a theory of sound generated by turbulence, albeit without the effect of the heat sources mentioned above. The thermoacoustics part is mainly concerned, as is this chapter, with what Rayleigh has called the “maintenance” of sound by heat. Mawardi gives an account of the theory of the Rijke phenomenon (1859), first treated theoretically by Carrier (1954). This theory is not included here, as it has already been adequately covered in the literature; furthermore, as is well known, a steady (DC) flow component through the Rijke tube is an essential ingredient for the functioning of this device. In this work, a different sound-producing tube without throughflow, also described in Rayleigh’s book, is treated. In summary, then, Section 11 is concerned with thermoacoustic effects caused by heated surfaces, with particular emphasis on situations in which large amplitude acoustic oscillations are maintained. These are caused, as is well known, by configurations that are unstable. The details of the stability calculations in these cases are very lengthy, and it is therefore intended to give an outline only, with algebraic details (which in some cases can be rather involved) to be filled in. Finally, this chapter also includes a section on a thermoacoustic effect that was apparently never treated under this special heading: the production of heat by sound. Actually, in every theory of diffusive effects in acoustics, the overall heat production easily follows from simple energy considerations. However, when this acoustic heat production is observed in a duct, its distribution within the duct is by no means readily explainable. Key to the understanding of these effects is the realization of the existence of “thermoacoustic streaming” (Merkli and Thomann, 1975), a phenomenon that is completely analogous to the acoustic streaming of mass discovered by Rayleigh. Therrnoacoustic streaming, a second-order effect, describes the

Nikolaus Rott

138

time-averaged overall heat flux associated with a linear solution of the acoustic equations. In case that the amplitude of the acoustic solution is too high (as manifested by the occurrence of shocks), the linear solution is not adequate. In such cases, the heat effects produced by the gas oscillations are particularly prominent. However, the theory in the cases in which nonlinear acoustics is needed for the description of the basic oscillations is not included in this report. 11. Oscillating Flow over a Nonisothermal Surface

A. VELOCITY OSCILLATIONS This fundamental problem is solved here in the case that the nonoscillating equilibrium situation involves a temperature stratification of an infinite plane in a direction x (say),and the gas in contact with this plane (but otherwise unbounded) has the same equilibrium stratification T,(x) as the wall. Now the half-infinite gas region is oscillated parallel to the plate in the x direction with velocity u,e'"' everywhere except in a thin region near the wall, where the velocity u = ii(y)e'"' is given as the solution of the equation a u p t = va2u/aY2,

(2.1)

so that, with the condition u = 0 at y = 0,

ii = u m { l - e~p[-(iw/v)'/~y]}.

(2.2)

Equation (2.2) is the classical Stokes solution, except that the boundary layer thickness is now a function of x, because of the variation of the kinematic viscosity v with T,(x). For a viscosity law p TB,and with pm = const, pm T i the boundary layer thickness varies as

-

-

',

6

- ( v / ~ ) ' / ~- T g

+n)/*.

(2.3)

As a consequence of (2.3), the velocity component u in the y direction is no longer 0, and Eqs. (2.1) and (2.2) are not exact any more, as they were in the isothermal case. However, v would only enter the nonlinear inertia terms that are smaller compared to the one retained in (2.1)by a factor of the order

u dlogT, dlogT, -sw dx dx ' where s is the amplitude of the motion. Thus, the solution (2.2) is valid only as long as the percentage change of the absolute temperature over a distance s

Thermoacoustics

139

is small: s d log T J d x << 1,

(2.4) a restriction that is accepted henceforth. Consistent with this degree of approximation is the application of the solution (2.2) to cases where u,eiW‘ is replaced by u,, which is a function of x and t, whereby the oscillations of the gas are caused by acoustic waves, standing or propagating in the x direction. In a stratified medium they obey the equations

au, -+--

ax

1

ap

p,a2 at

= 0,

where a’ = yRT, is the local speed of sound. (Note that pma2 = y p , is a constant.)

B. TEMPERATURE OSCILLATIONS Now the temperature fluctuations are calculated from the energy equation. The boundary layer approximations certainly apply; the temperature distribution T,(x) is thought to be imposed (by means to be discussed later) and conduction in the x direction is actually ignored as in the isothermal case. The appropriate nonlinear equation for the boundary layer temperature T is

Linearization is justified by known arguments together with the use of the restriction (2.4). Also, as p is independent of y , the linear term d p / a t can be inserted from (2.6).The result is

By setting T

=

T,

+ $eiwtand u, = u,(x)eiO‘, one obtains

whereby ii has to be inserted from (2.2) and CJ is the Prandtl number: r~ = pCp/k.

(2.9)

Nikolaus Rott

140

The appropriate boundary condition at y = 0 is in most cases 9 = 0 (its justification and possible exceptions are discussed presently). The condition at y = co is implied in the inhomogeneous equation (2.8); the solution is, corresponding to the two inhomogeneous terms in (2.8),split in two parts : 9 = 91 + 9,. With the notation ( i ~ / v ) ”y~ = 11 one obtains i durn 9,= (y - 1)T, __ (1 - e-qJz),

dx

0

(2.10)

in case that 0 # 1; in the special case that the Prandtl number equals 1, the solution for QZ is i dT, 9, = - -U, [I - (1 + ~ ) e - ~ ] . (2.12) o dx The term 9, expresses the effect of the heat conduction from the gas, whose temperature changes adiabatically, to the wall, which has a temperature T, independent of time. In the second term Q,, the “driving” temperature gradient in the y direction is produced by oscillating the stratified body of gas over the plate; in first approximation, i.e., with the restriction (2.4), this effect is proportional to dT,/dx (see Fig. 1). Now the question of the boundary condition for 9 at y = 0 is reconsidered. The value 9 = 0 is appropriate when the heat capacity per unit volume is much higher in the wall than in the gas; this is usually the case and the simple boundary condition 9 = 0 is valid. For a more accurate condition, the heat conduction problem in the solid wall has to be solved. Under the assumption (usually amply fulfilled) that the wall is thick enough for the fluctuating temperature to decay to zero within the solid, the relation between q,eio‘,

+

+ Z ’ , / , /

,,,/,//

,,/,

/,,/,,/,,,,,./,,,,,,~

x, FIG. 1 . Visualization of the first-order heat flux effect produced by gas oscillations along a nonisothermal wall.

Thermoacoustics

141

the common heat flux in the gas and the solid at the wall y = 0 (positive into the wall), and 9,eiU1,the common contact temperature at y = 0, is found from the elementary solution in the solid: qw = (iwp,C,k,)1'2 9,,

(2.13)

where the quantities with the index s are the density, the specific heat, and the heat conductivity in the solid, respectively. The same relationship holds for the heat flowing out on the gas side, so that the boundary condition becomes = (iwpsc,k,)1/2 9

(2.14)

This condition together with the inhomogeneous equation (2.8) can change only the general homogeneous solutions in (2.10) and (2.1l), which is the last term in both equations. The result is expressed with the help of the quantity (2.15) n = 1 + (pCpk/PsC,k,)1~2, as follows: the homogeneous solution exp( - yl&) is multiplied by a factor c1 = l/n, in (2.10), and by a factor 1 - Jo n

c,=-+&

in (2.11). For the wall temperature, the final result is durn 1 (7- l)T,-+-u,dx 1 +&

w

dTm]. dx

(2.16)

c. THEVELOCITY PERPENDICULAR TO THE WALL The velocity component in the y direction, which has been neglected thus far, can now be calculated a posteriori, in an approximation consistent with the previous results, by use of the (linearized) continuity equation. From this, v is given by

In linear approximation, one replaces the second term in the integral as follows [by use of (2.6)] : 1 ap

1 Sp

1 dT iit

---- - _ - - _ pm at pm at T,

du,

iw9

142

Nikolaus Rott

Writing u = 13eio', the following split of 13 into three terms offers itself, with some obvious rearrangements motivated by the physical interpretation of the results :

-

yo=

(2.17a)

--

(2.17b) (2.17~) as d log pm/dx =

- d log T,/dx.

By use of (2.2),(2.10),and (2.11)one obtains (2.18a) (2.18b)

1

A

v2=--1-

1 dT, urn T , dx

(&)

1/2

[(I - e-.&) - &(l - e-.)].

(2.18~)

For a = 1,6, becomes

(The results are shown in the special case n = 1, 9, = 0 only.) The main result is the value of u for y -+ GO, which is reached for y >> 6, i.e., outside the Stokes boundary layer. The total amplitude u, is given by the following formula, which includes the effect 9, # 0, not shown in Eqs. (2.18):

-

d dx

(

+ n(o

(;)'I2 +

(2.19)

~)

(with the abbreviation 8 = d log TJdx). By use of (2.3), the first term in (2.10) can be split into two parts and the results rearranged as

)

(2.20) urnB]

More convenient for the later applications is the form v , = (1

;I'");(

+ e) [urn

nfi + -1 2 1 n 1+Ba+&

(-

~

-

&

$

(;)lI2

(2.21)

Thermoacoustics

143

Now, u , is the amplitude of “pistons” to be arranged in the plane y = 0 to represent the effect of the boundary layer on the main flow; the first term in (2.20) or (2.21) is found in isothermal flow; the second term gives the effect of the variable wall temperature and includes the effect of the variation of the viscosity with temperature. The influence of urn on the flow is found either by summing the effect of the acoustic sources imbedded in the plane, with the amplitude u,, or alternatively (and equivalently) by taking u, as a modification of the boundary conditions at y = 0, for the next approximation. This is the approach taken in the next sections. 111. Damping and Excitation of a Gas Column with Temperature Stratification A. PRELIMINARY CALCULATIONS FOR THINBOUNDARYLAYERS The results obtained thus far are applied now to the acoustics of a gas column with a temperature stratification along its axis x, enclosed by a cylindrical wall with generators parallel to x. In equilibrium, wall and gas have the common temperature T,,,(x),imposed by sources and sinks outside the wall. In the gas, the radial variation of the equilibrium temperature is ignored. This implies that the tube is long compared to a typical crosssectional dimension (e.g., the hydraulic radius) of the cylinder. Damping and excitation of the acoustic oscillations of such a gas column can be treated by use of the results obtained in Section 11, provided that the Stokes boundary layer thickness is small compared to the hydraulic radius (v/w)”2 6 << Yh. (3.1) A detailed discussion of the significance of this important restriction is given later. The plan for the analysis, under assumption (3.1), is as follows: first all diffusive effects are neglected, and the acoustic oscillations are investigated for the appropriate boundary conditions. Then the velocity u, at the nonisothermal wall is determined from the results of Section 11, with u given from the inviscid solution. Finally, the energy exchange between the inviscid core and the boundary layer is calculated ; its sign determines whether damping or excitation is found. The separate and sequential calculation of the nondiffusive and the boundary layer solutions is only justified when condition (3.1) is fulfilled; urn in this limit is not affected by the finite cross section of the gas-filled domain or by the curvature of the wall. The limit between damped and excited oscillations, i.e., the stability limit, is found when the time-averaged energy exchange between the inviscid core and the boundary layer, integrated over the whole wall surface, is zero.

-

144

Nikolaus Rott

To find the stability limit, it is assumed that the pressure paeiorof the inviscid core oscillation is given; according to (2.6), it is related to u, by the formula pa = (i/o)a2p, du,/dx. (3.2) The instantaneous power exchanged between the core and the boundary layer is the product of the acoustic pressure and the velocity u, at the outer edge of the boundary layer. To calculate this product and its time average, using the complex notation, it is recalled that the physical p is Re p (the real part of p ) , etc. Then, the time average in question is iRejjav,(=$Repav“,)

=p

x ,

(3.3)

where a tilde indicates the complex conjugate. The mean power exchange between the core and the boundary layer is

9=

C L --[ 2 0 Rejjau,dx

(3.4)

for a tube of length L and circumference C; the sign makes 9 positive if energy flows from the core to the boundary layer, as u, as defined in Section I1 is positive when directed away from the wall. Now Pais introduced from (3.2) and u , from (2.21). The first term of u , in the form (2.21) is a perfect differential, and use is made of this fact to perform a partial integration. The integrated part is set equal to zero, as either u, = 0 or du,/dx = 0 is supposed to hold at the ends of the tube (which are either “closed” or “open”). In the remaining integral d2u,/dx2 appears. From the fundamental Eqs. (2.5) and (2.6), this is replaced by (3.5) d2u,/dx2 = - (Cu2/a2)u,. In this way the first term in u,, which is the only one found for an isothermal wall, leads to an integral over urn& with a unique sign. The final form is written making use of the fact that a2pmis a constant along the tube; the integral corresponding to the second term of u, is left unchanged. The result is

The necessary condition for the stability limit is 9 = 0, and it is possible that 9 vanishes, as the second integral has no definite sign. The next problem is the evaluation of u, from (2.5) and (2.6); this is, in principle, the basic step in the present scheme of successive approximations.

Thermoacoustics

145

For a given T,(x), the numerical determination of u, presents no problem; but only very special distributions T,(x) are suitable for analytic treatment. One that is of practical importance is the “discontinuous model,” in which T , is piecewise constant in two regions of the tube, which are joined by a jump at x = 1 (say): TI, O I x < I, (3.7) ‘“‘={T2, l<x
a2

which determines the frequency. It remains to introduce (3.8) in the integrals appearing in (3.6). When evaluating the second integral, one realizes that, because of ( 3 3 , the kinematic viscosity does not change except in a small vicinity of x = 1. One finds, therefore,

(3.10) The remaining routine calculations are not carried out here, because the stability conditions obtained by this method are in poor agreement with

146

Nikolaus Rott

experience. The theory is presented, nevertheless, as an example for the applications of the boundary layer calculations given in Section 11, which puts the essential mechanism-the energy exchange between the inviscid and the viscous regions-in evidence. The failure of the theory, as shown by the author (1969), is a consequence of assumption (3.1). It is not permissible to assume that the viscous layer is very thin compared to the tube radius, if one wants to discuss the stability problem. A fresh start without assumption (3.1) is made, but first a historical introduction is given. B. THEHISTORY OF THE STABILITY PROBLEM The preliminary calculations of Section III,A facilitate the historical and the qualitative discussion of the phenomenon under consideration. Spontaneous oscillations of a gas column with temperature stratification were first investigated experimentally in a systematic way by Sondhauss (1 850), whose paper is quoted in Rayleigh’s book; therefore, occasionally the device is called a Sondhauss tube. Sondhauss himself, however, gives several references to earlier observers; the earliest published record is apparently a “Letter to the Editor”, dated April 25, 1804, by a Dr. Castberg in Vienna, in Gilbert’s Annralen der Physik Vol. 17. He described the device as a “glowing glassharmonica”; actually, it was proposed later by Marx (1841) to use the effect for the design of a musical instrument, a totally impractical idea (on which Marx had a correspondence with Goethe in 1827). The configuration is extremely simple: the part near the closed end of a glass tube is heated, while the open-end part is left cold. Sound is produced when the temperature ratio between the hot and the cold section is sufficiently high. From the melting point of glass, the ratio of absolute temperatures is estimated to lie between 2.5 and 3 for the early experiments.The hot part was mostly formed by a portion of the tube with an increased diameter (a “bulb”) at the closed end. Systematicexperiments by Sondhauss have shown that this configuration is superior in producing sound effects, provided that the hot part extends to the beginning of the section with a small cross section (the “neck”)of the tube. The theory presented later gives a full explanation of these effects. A necessary condition for the instability was given by Rayleigh (1896) in his book, based on energy considerations. It states that in a phase of the acoustic oscillation in which the pressure is high, the displacement has to be directed toward the hot end (and vice versa) for instability. The heat addition at high pressure (and vice versa) means energy addition for the oscillations. For the basic configuration of a half-open tube, it is the closed end that must be hot. A new era for the observation ofthese phenomena was opened after helium was liquefied (in 1908).Helium vapor has a temperature of4.2 K at 1bar, and

Thermoacoustics

147

tremendous temperature ratios were produced in ducts leading out from helium dewars to regions of room temperature. Under these circumstances, unstable configurations were extremely hard to avoid, while in the early experiments with modest temperature ratios they were hard to produce. Qualitative explanations of the effect were given by Taconis (1949), and Kramers (1949)gave the first stability theory, which contained all elements of the theory to be presented in Section IKC, with one exception: Kramers evaluated the results of his analysis only in the limit of a very thin boundary layer. Therefore, the results of Section II,A suffice for the discussion of Kramers’ paper. Kramers found, in (the equivalent of) expression (3.6),the possibility of excitation; this effect is represented by the second term on the right-hand side. The first factor in this term depends only on the constants of the gas [and has been named “Kramers’ constant” by the author (1969)] : (3.11) Now for helium with y = 4, the Prandtl number c = +,and the exponent for the viscosity-temperature power law P = 0.647 (which is very accurate over a wide temperature range, from 4 to 300 K), one finds that dK = 0.002. A positive value of dKis necessary for possible excitation in the framework of this theory [see (3.19)], but the extremely small value of Kramers’ constant for helium leads to very high temperature ratios (of the order lo3)as a necessary condition to reach the stability limit. There is no indication of any agreement with experience, and Kramers conjectured that the linear theory is not adequate. Actually, the Kramers theory can be brought to perfect agreement with experience when extended to arbitrary values of the parameter that gives the ratio of the width of the viscous region to the tube radius. With the provision that the parameter is determined at the cold end (index c), it is defined as Y, = Yw(0/V,)1’2

(3.12)

(for round tubes). The theory that the stability limit is given by 9’= 0 in (3.6) yields only the asymptotic value of the critical temperature ratio between the hot and the cold part for stability in the limit Y,+ 00,provided that Kramers’ constant is positive. This is the case for all diatomic gases (also for air), but even then the information on the stability curve given by the theory for Y, + cc is not adequate for the explanation of all observed phenomena. In case Kramers’ constant is zero (a condition almost exactly fulfilled for helium, and very nearly for other monatomic gases), it has been shown by the author (1969) that the asymptotic value of the critical temperature ratio for

148

Nikolaus Rott

’

stability varies as Ydifl +B1 for 5 00. A second-order theory in Y , is needed to obtain this result. Finally, in case that Kramers’ constant is negative, there is an upper limit for 5 beyond which instability is impossible, as was shown by the author (1973).Actually there are no simple gases (with a constitution that remains unchanged over a wide temperature range) known to have a value d , < 0. Several asymptotic properties of the stability curve can be discussed analytically; here, however, the point of view is taken that a numerical determination of the stability limit for arbitrary Y, poses no problem and is accepted as the “best” method. A common property of all stability curves is the existence of an “optimal” (or rather a “most dangerous”) value of Y,, for which the critical temperature ratio for stability has a minimum. This can be understood physically, as it is the boundary layer that does the driving, and it cannot be too thin if it has to be effective. On the other hand, if it is too thick, viscosity prevails. It is also clear that optimal conditions are only required where the wall has a temperature gradient. In the isothermal sections, the boundary layer has only damping effects. It follows that an isothermal section should have a diameter that is as big as practically feasible. The condition is most important in the hot section, where the boundary layer has its maximal thickness. Thus, the “bulb” at the hot end of the original Sondhauss tubes reduces the critical temperature ratio very effectively. The discussion of optimal conditions can only be complete when the theory covers the case of tubes with variable cross section along the axis. This generalization of the theory was given by Rott and Zouzoulas (1976), and is presented in Section II1,C. C. THECOMPLETE STABILITY THEORY The basic equations of the complete stability theory are obtained first by simplification and then by generalizations of the Kirchhoff theory (1868), i.e., the exact linear acoustic equations with friction and heat conduction in a duct. The simplification involves the following assumptions: (i) the radial variation of the acoustic pressure is neglected, and (ii) friction and heat con‘ductiondue to axial gradients are ignored. These simplificationsare the same as those of the Prandtl boundary layer theory, except that here they are only applied in the “dynamical” sense: the “geometrical”restriction that there has to be a thin viscous layer is not applied. These assumptions have been first introduced by Iberall (1950) and give results in excellent agreement with the exact Kirchhoff theory provided that the acoustic wavelength is sufficiently bigger than the tube radius (which in turn has to be much bigger than the

Thermoacoustics

149

mean free path). These assumptions are amply fulfilled in the examples considered. In a first generalization to the case of the axial temperature stratification by the author (1969),the radial variation of the mean temperature was neglected, together with the attendant radial variations of the coefficients of viscosity and heat conduction. Implied is a restriction given by (2.4).A corresponding type of restriction is needed for the generalization to the case of a variable tube radius along the axis (Rott and Zouzoulas, 1976),namely, s d log rw/dx << 1.

(3.13)

Nevertheless, for the temperature variation (as already discussed in Section II1,A) and also for the cross-sectional change, solutions with a discontinuous distribution will be admitted. For a jump in the tube radius, the same arguments for the validity of the theory can be given as before: the effects typical for a discontinuity are not covered, but the mode of oscillation shows only insignificant changes when the limit of a discontinuous solution is reached. Actually, a discontinuity in the cross section can lead to flow separation, an important effect not covered by any linear theory. On the other hand, the stability limit problem is linear and should not be affected by separation, which, however, is expected to have a dominant effect on amplitude limitation. The following theory is restricted to the case of circular sections. All quantities without subscript in the subsequent equations are understood to represent the (complex) amplitudes of the acoustic quantities; the time variation with the factor eiW* is taken into account by setting d/at = io. The mean state variables have a subscript m as before. All equations are linearized. The continuity equation is in case of cylindrical symmetry:

(3.14) and the axial momentum equation is, neglecting the effects of compressive friction,

(3.15) The radial momentum equation is ignored, or rather replaced by the simplifying assumption ap/& = 0. Finally, the energy equation is written again, this time using the axial symmetry of the problem:

Nikolaus Rot2

150

It is advantageous to eliminate T (a small disturbance) by the relation (3.17)

The result is, as d p p r

= 0 and

pm = const, (3.18)

The boundary condition at r = r, has been discussed by the author (1973); its derivation is completely analogous to the one given in Section ILB, leading to (2.14). The only difference is due to the (circumferential)curvature of the temperature boundary layer in the solid. The result is, after eliminating again T by (3.17) (and noting that p = pw),the following relation at r = r,: (3.19)

where 'pl = n - 1with n given by (2.15) and cp2 expresses the curvature effect mentioned above. For thermal boundary layers in the solid that are thin compared to r,, the author has found (1973) that

In most cases cp2 = 1 is a very good approximation. The equations are now complete; the following combination of (3.14) and (3.18) is useful for the subsequent operations: (3.20)

Now let this equation be multiplied by the factor 2a'rdrlr; and integrated between 0 and the tube radius r,. By introduction of the quantity (3.21)

the first term in (3.20) becomes, after performing the integration, (3.22)

whereby use was made of pma2= ypm = co.nst and of u = 0 for r = r,. The full equation (3.20) becomes, as the radial velocity v = 0 for r = r,, (3.23)

Thermoacoustics

151

The same operation applied to (3.15) gives iw$

+ a'

- = ___

dx

rw

(3.24)

Elimination of the quantity (&$) between (3.23) and (3.24) leads to

-

r:

(3.25)

dx

The diffusive effects are represented by the right-hand side terms of (3.25). It remains to solve (3.15) for u and subsequently (3.18) for p. In (3.15), p and v are functions of x, but as no derivatives of u with respect to x occur, the solution is (with u = 0 at Y = r,) given by the Bessel function J o : (3.26) where (3.27) Similarly, the inhomogeneous equation (3.18) gives

-

-

(1 - a)w2 dx

(3.28)

This solution is only given here for o # 1, and with the boundary condition (3.19) simplified to the case when only the first term is present ( q l= 0). For the subsequent application only (dp/dr), is needed; this quantity has to be inserted in (3.25). It has been shown by the author (1973) that (dpldr), for the general boundary condition (3.19) is related to (dp/dr)wsfor the special boundary condition with q 1 = 0 (index s) by the relation (3.29) where

152

Nikolaus Rott

When lqw( -, co, one finds that (p3 = 1 ; when the thermal boundary layer in the wall is also thin, then N = n. With (du/dr), from (3.26) and (dpldr), from (3.28) and (3.29) inserted in (3.25), the final equation for the acoustic pressure p in function of x is

r t a’ f* -f dp _ _ _ _ _ _ 0-=0, No2 1-a

dx

(3.31)

where (3.32)

and, as before, 0 = d log T,,,/dx. For the subsequent calculations, whether analytical or numerical, it is advantageous to change the second-order equation (3.3 1) into two firstorder equations : 1 d* p = ----’ (3.33) H dx (3.34)

where a

(3.35)

a

and (3.36) (3.37)

The form of Eqs. (3.33) and (3.34) makes it evident that p and 1(1 must be continuous whenever a discontinuity in rw(x)or T,(x) causes a jump in H or k. The significance of $ can be traced back by comparison of (3.34) to (3.24)’ to give, after some calculations,

*

=

-(i/0)E&p.

(3.38)

It is seen that the flux (~$4)has to be multiplied by the factor E given by (3.37)’ to give the quantity that remains continuous across a temperature jump, a fact that follows from the basic equation (3.31) when brought to the

Thermoacoustics

153

form (3.33) and (3.34). For a closed end, where u = 0 and thus 4 = 0, we have t,b = 0. The appropriate boundary condition for the open end is, for a linear theory, p = 0.

D. DISCUSSION OF THE RESULTS I. Remarks on the Method

of

the Solution

In general, Eq. (3.31) or the equivalent pair (3.33) and (3.34) must be solved by a step-by-step integration. A “case” is defined by two functions T,(x) and r,(x), and by the gas properties, which are ideally fixed by three constants y, 0,and B. [It is possible to let y and 0 be a function of the temperature, and to prescribe an arbitrary function p( T,); these generalizations do not affect the essence ofthe method.] The unknown parameter (eigenvalue) is the complex frequency o,and a search procedure has to be provided for this quantity, for which a guess is needed at the very beginning of the calculations. With w one finds first q,, Eq. (3.27), as a function of x, then f and f* from (3.32), and E is determined by a step-by-step integration of (3.37), with N given by (3.30). Then both k and H , (3.35) and (3.36),are known, and the integration can proceed. Beginning at one end (x = 0), the known boundary condition ( p = 0 or 3/ = 0) is used, and at the end of the integration, at x = L (say),some complex values p L and t,bL are found. One of these, depending on the boundary condition at x = L, must be zero. The vanishing of such a complex quantity is the determining condition for the complex eigenvalue o,which has to be found by the search procedure. Depending on the sign of the imaginary part of o,one has damped or excited oscillations; a second search has to be initiated for cases for which the imaginary part of w vanishes, that is, for the stability conditions. It is convenient to investigate an affine family of temperature distributions, best characterized by the ratio of the highest to the lowest value of T,; this parameter a = Tm,ot/Tmco,, (3.39) is used for the discussion of all the results in this section. The multitude of possible cases with two arbitrary functions is staggering, and a reasonable choice of typical special cases is imperative. First the group with a constant tube radius r , is discussed. For r , = const, I], becomes a function of the temperature only [see (2.3)], and the integral (3.37) for E can be evaluated as a function of T,, noting that 0 d x = dT,/T,,

154

Nikolaus Rott

and that the integrand too depends only on T,. (Constant thermal properties of the tube wall along x are also necessary for a solution of this type.) The function E can be computed (for N = 1) for different gases in advance; an arbitrary multiplicative factor, arising from one free limit of integration, does not affect the system (3.33)-(3.34). A constant value of N , appropriate for thin boundary layers, can be accounted for analytically. [For the special case CT = 1, N = 1, one finds E = (1 - f ) l i c l + @1) . The absolute value of qw at a fixed position of the tube is the suitable parameter, as discussed in Section III,B to express the ratio of r, to the thickness of the viscous region. The quantity Y,, given by (3.12), defined at the coldest spot of the tube, is a useful parameter, but it has the disadvantage that w is not known (exactly) in advance: it depends on Y,. The way out is to use a fixed w at a well-defined state for the reference parameter. The state chosen is the inviscid case (Y,= co), for which the frequency w o is real and depends only on T,(x) (for a given tube length and gas). The temperature distribution, in turn, is chosen in such a way that it corresponds to the asymptote Y, + co of the stability curve; this point is discussed later in detail. The parameter thus defined,

Y,, = r,(wo/v,)1’2,

(3.40)

is used throughout this section. Finally, the function T,(x) must be fixed. Kramers (1949) assumed a discontinuous distribution, which has been also used by the author previously (1969, 1973), and in the present work [Section III,A, Eq. (3.7)]. The method given in Section III,A for the inviscid case can be generalized to the solution of the system (3.33)-(3.34), as an analytic solution is known for the tube sections with constant wall temperature, i.e., between 0 I 1 I x and 1 Ix IL, respectively. At x = 1, p and $ must be continuous, and this leads to a transcendental equation for w. The analytical and numerical solution of this equation has been discussed extensively by the author in previous work (1969, 1973). Here, no further details are given; it is only noted that E as given by (3.37) is also a constant along the isothermal sections; all the change of this quantity is “concentrated in the “point” x = 1 of the temperature jump. In summary, then, an alternative approach, not using a step-bystep integration, exists for the temperature distribution (3.7). In the following discussion of the results, cases based on the discontinuous model are presented almost exclusively. This is not so much a consequence of the preference for a method of solution. Calculations have shown that for a fixed ct, the temperature distribution with the steepest gradient (i.e., with a jump, in the extreme limit) leads to the most unstable situation. (This result can also be supported by analytical investigations.) Stability curves based on the jump distribution lead to a conservative approach to the most dangerous situations.

Thermoacoustics

155

To illustrate this point, the following example is presented: a heliumfilled straight tube, open at x = 0 and closed at x = L, is investigated for a series of temperature distributions that all have the maximum temperature gradient in the middle of the tube, at 1 = L / 2 . The family of curves is characterized by the width W of the temperature transition region, defined as (3.41) The transition is assumed to be given for all W by a sine curve. For W < 2/71, constant temperature plateaus are reached at both ends of the tube with a continuous tangent; for W 2 2/71, sine curve sections are fitted to give distributions as indicated in Fig. 2, with a straight line in the limit W = 1. The case of a temperature jump is given by W = 0. For the temperature distributions thus defined, the minimum value of OL for which the stability limit is reached was calculated. The search involved the determination of the appropriate value of the dynamic similarity parameter Ycc,.It was assumed that the thermal properties of the tube wall permitted to set N = 1; this value is used for all examples that follow. The constants of the helium-gas are given on page 147.

50

20

10

5

1

Fig. 2. The influence of the “spread” W , (3.41), on the minimum attainable temperature ratio r for sustained undamped oscillations in a helium-filled tube. The T, distributions are shown in the insert; the arrow indicates the biggest value of W (=2/n)for which dT,/dx = 0 at both ends. Also shown is the dynamic similarity parameter (3.40)for which the minimal a values occur; however, the reference frequency is always the one found at W = 0, and the parameter thus defined is called Y,,,,,.

156

Nikolaus Rott

The results are shown in Fig. 2. For W = 0, the critical value of a is 5.75. The growth of a with W is first very slow: for a width W = 0.2 by definition (3.41), one finds a = 6.45. The growth increases rapidly and a reaches for W = 1, i.e., for a linear distribution of T , between the two ends, a value of almost 50. It must be emphasized again that only these results, for which Fig. 2 shows a typical plot, justify the extensive use of the discontinuous temperature distribution. It has been noted before that the present theory in the limit W = 0 does not cover the effect in which a real jump (over a fraction of a radius, say) of the tube wall temperature is maintained. However, the results for W = 0 in this theory differ only very little from those obtained for W << 1, i.e., for the cases with small transition regions where (2.4) is fulfilled. A breakdown of the theory is expected in case the amplitude of the gas motion reaches the width of the transition region, but such problems lie beyond the scope of the linear theory of the stability. Also plotted in Fig. 2 is the value of the dynamical similarity parameter, for which the minimum critical value of a was found. Its definition is changed against (3.40) inasmuch as the reference frequency that is used for all W is that for the inviscid mode for W = 0. The change of wo with W is not included in the quantity shown, which is called Y,,,, because only for W = 0 does an elementary formula for wo exist; its discussion is deferred to the next section. A typical value for KO,is about 10, as long as W is not too large. 2. Straight Tubes The next problem in the treatment of the stability limit for straight tubes is the question of the most dangerous position of the temperature jump along the tube. The parameter (3.42) 5 = ( L - U/k that is, the ratio between the lengths of the hot and the cold parts, is used. The lowest critical temperature ratio a for a half-open tube filled with helium is plotted in Fig. 3 as a function of 5, and it is seen that the minimum of tl lies almost exactly at 5 = 1, the value used in the example of the previous section. The minimum around 5 = 1 is flat; thus 5 = 1 is a suitable choice for representative examples. The quantities KO,which belong to the critical values of a, are also plotted in Fig. 3. Here, definition (3.40) is used, and the inviscid frequency oois calculated for each 5. This frequency was determined in Section III,A and is given by (3.9). With the notations introduced in Sections III,D,l and III,D,2, Eq. (3.9) is rewritten in the form (3.43) 52, tan A, = A, cot I , ,

Thermoacoustics

157

0 Fig. 3. The influence of the position of the temperature jump (discontinuous T , distribution) on the minimum attainable temperature ratio a for sustained oscillations in straight halfopen tubes filled with helium. The ordinate (, (3.42),is the length ratio of the hot part to the cold part of the tube. Also shown are the correspondingvalues of Yco,

where

1, (with a,

=

=

wol/a,

(3.44)

speed of sound in the cold section of the length 1) and

I,

= ta-

1/22,.

(3.45)

In case c1 >> 1, the factor c1-l’’ in (3.45)(expressing the speed-of-sound ratio) leads in (3.43) to the limiting formula

t i , tan I ,

= 1,

a >> 1,

(3.46)

which is sufficiently accurate in m’any cases. It shows that for high enough a, I , (and thus wo) only depends on t. It has been stipulated before that wo is defined for Y, -, 00, and for a temperature distribution corresponding to marginal stability for --* co. For helium, the stability curve for r, -+ co was found by the author (1969) to grow as a YClic1‘ 8 ) (see the discussion in Section II1,B). Thus w o is defined with both r, and a tending to infinity, for helium; it is determined therefore, by (3.46).The solution of this equation for 5 = 1 is, for instance, A, = 0.860. In the case of diatomic gases, a finite “Kramers limit” exists for the critical value of a for stability (see again Section III,B), and 1,is determined from (3.43) with (3.45). As the critical asymptotic values of a are still high, N

158

Nikolaus Ro tt

only small corrections of (3.46)are obtained, which i’s discussed in connection with the results for nitrogen. Discussion of cases with t differing strongly from 1 were published by the author (1973); here, a full stability diagram is given only for t = 1 in Fig. 4, again for helium in a half-open tube. The figure shows curves of the temperature ratio c( against Y., with the imaginary part of w as the parameter. With the definition co = 0,

+ iw, = w,(l

- it),

(3.47)

[ is a measure of the excitation per period; [ = 0 is what is called the stability

curve. In Fig. 4, this curve displays for the minimum of c( the value given in Fig. 3 for 5 = 1, and shows two growing “branches.” The right branch leads to the characteristic asymptote for helium, given by the author in 1969, and discussed here in Section II1,B. The left branch rises sharply and ultimately turns to the right; this has the following interpretation: (1) no unstable modes exist when yofalls under a limiting value ( z8 in Fig. 4), and (2) there exist values of c( that are too high to lead to excited oscillations. The reason for this phenomenon is that with the kinematic viscosity growing strongly with T,,, [see (2.3)], temperatures that are too high cause the hot part to become “clogged” by the viscous effects. Further discussions and the determination of the asymptote for the left branch were given by the

Fig. 4. The temperature ratio c1 producing given constant values of the excitation [,(3.47), as a function of KO,for a helium-filled straight tube with the temperature jump in the middle (( = 1). The curve i = 0 is the Stability limit; its asymptote is dashed. An arrow indicates the asymptotic direction of the curves [ > 0, and a dashed-dotted line connects the minimum values of a for a given [.

Thermoacoustics

159

author in 1973. A striking confirmation of this character of the stability curve was given by von Hoffmann et al. (1973). A half-open tube with Y,, 2 10 was stuck into a dewar, containing helium vapor and liquid in equilibrium, at a temperature of 4.2 K and a pressure of 1 atm. With the hot closed end of the tube at room temperature, a value of a = 70 was obtained; there were no oscillations. Next the hot part was cooled with liquid N 2 to 77 K, leading to c1 = 18, i.e., to a point within the instability region. After this strong reduction of the temperature ratio, spontaneous oscillations were indeed observed. Such effects can practically only occur with the cold end at a very low absolute temperature. The influence on this effect caused by a change in cross section along the tube is discussed later. A very satisfactory agreement with the whole theoretical stability curve for helium was obtained in experiments carried out by Yazaki, et a/. (1979). A U tube closed at both ends with symmetrical temperature distribution with respect to the midpoint was used, an arrangement that is naturally (in linear theory) fully equivalent to the half-open tube, but permits a better way of regulating and measuring the temperature of the cold midpart. The lower temperature was varied up to 45 K, which is significant because when the lowest temperatures (4.2 K) are approached, the thermal properties of the tube materials undergo significant changes; these in turn could influence the experimental results in a way that is not yet fully covered by the existing theories. Figure 4 also shows a series of curves for constant values of the excitation parameter i;these are taken from an unpublished dissertation of U. Muller (1981). The right branch of a curve for a constant > 0 has ultimately an asymptote that becomes parallel to a line on which lrwl in the hot part is a constant. In Fig. 5 the frequencies that belong to the curves shown in Fig. 4 are plotted, completing the results of the stability calculation. Curves of the real part of the frequency w,./wo are shown for constant values of against yo.The value of w,/wo remains near 1 except for a region in the vicinity of the left branch of the stability curve for high temperatures, where viscous effects are strong. The upper asymptote of the stability curve 5 = 0 corresponds to the left branch of Fig. 4. The frequency in this (highly unrealistic) limit is obtained by assuming a closed end at the point of the temperature jump and neglecting viscosity in the cold part (see Rott, 1973). When the tube is filled with nitrogen instead of helium, the most important change is found in the asymptotic behavior of the right branch of the stability curve. The values of the material constants chosen for N, are y = 1.4, (r = 0.74, and = 0.85; these are appropriate for the lower temperature ranges (77-300K) and are not strongly different from those for air. The influence of 5 on the stability limit runs practically parallel to the results for helium

c,

160

Nikolaus Rott

FIG.5. The real part of the oscillation frequency correspondingto the curves shown in Fig. 4. The reference frequency oois found on the stability curve (‘ = 0 for x,, = 03. Examples of lines a = const are cross-plotted.

and is not shown; again, t = 1 is (almost exactly) the “most dangerous” configuration, for which a stability diagram is given in Fig. 6 . The minimum value of a on the stability curve [ = 0 is now 5.08, somewhat less than for helium (a = 5.75). The right branch of the curve [ = 0 leads asymptotically to a constant value of tl = 14.55 for yo+ 03. As explained in Section III,B, this result can be deduced from Kramers’ theory. However, it is important to realize that the curves for a constant finite excitation, e.g., for [ = 0.05, show very little difference in character for helium and for nitrogen; indeed, in both cases, the asymptotes of the right branches are characterized by lqwl at the hot end having a certain constant value. Thus, even when the Kramers limit exists, it does not provide all the information needed for the discussion of the stability behavior for KO-+ 03. The frequency curves belonging to the stability diagram for nitrogen are shown in Fig. 7. They have the same character as those for helium. However, the value of wo has to be determined now from (3.43)and (3.45)with the finite value of a given by the Kramers limit: a = 14.55. It follows in this

Thermoacoustics

161

1.8

1.7

1.6

Wr -

1.5

WO

1.L 1.3

1.2 1. I

1.0

0.9

5

10

20

50

100

Yco 200

500

FIG.7. Same as Fig. 5 , for a nitrogen-filled tube.

case that ,Ic = 0.855 for = 1 instead of 0.860 for ct -,00, a small correction but necessary to have the lower branch of the curve 5 = 0 Fig. 7 reach the asymptotic value 1 exactly. Final remarks on straight tubes: when both ends are closed, and there is only one temperature jump, the danger of instability is strongly reduced.

162

Nikolaus Rott

For a helium-filled tube, the most unstable situation occurs when the warm section has 40% of the length of the cold section, and the lowest value of a for which an instability can occur is 11.7. With two open ends and only one temperature jump, no instability exists. 3.3 Tubes with Variable Cross Section

Of the many shapes with the cross section changing along the axis of the tube, those investigated here lead to a low critical temperature ratio for stability. The design principle for such devices is clear, as it has been found that the steepest temperature gradients are the most effective. Thus, optimal conditions should be provided (by the proper choice of 1 ~ ~ over 1) a short section of the tube where the temperature changes strongly, i.e., where the “driving” takes place. The tube portions with constant temperature should have cross sections as big as practically feasible, to reduce the damping. This leads to a design of a tube with a contraction; there is no true optimum from the point of view of linear theory, which predicts a continuous improvement with growing cross section of the isothermal portions, albeit with diminishing return. Actually, an abrupt change in cross section is always connected with a “loss” that manifests itself in a pressure drop growing essentially with the square of the flow velocity. Such an effect would not influence the stability limit obtained from the linear theory, but the amplitudes of the excited oscillations would be restricted to rather low values. A short contraction with two abrupt changes leads to particularly strong losses. Therefore, the following discussion is restricted to tubes with only one change in cross section. As the kinematic viscosity is higher in the hot part, the increase in cross section is most effective when it is provided at the hot closed end. The effectiveness of such a measure has been described by Rott and Zouzoulas (1976; see also Zouzoulas, 1975). A typical example is that of a nitrogen-filled half-open tube, with its radius doubled in the hot part; the change of the cross section is in the middle of the tube, and the temperature jump is adjacent to this spot on the small-cross-section side. The minimum critical temperature ratio for stability is a = 3.1 for this device, in contrast to a value somewhat over 5 for a straight tube (see Fig. 6). This particular tube configuration with a 2:l change in the radius was chosen for a precision check of the theoretically calculated stability curve. For all experiments, the cold temperature was maintained at 77 K, the boiling point of liquid N, for 1 atm, and the hot part was at room temperature, 293 K; this gives a temperature ratio of 3.8, only slightly above the minimum value 3.1, so that excited oscillations occur only in near-optimal devices. All tubes investigated had radii of 1 resp. 2 mm; the parameter varied was the length and thus the frequency ooand thereby ultimately the quantity

Thermoacoustics

163

Yco. The agreement between experiment and theory was excellent; for further details, see Rott and Zouzoulas (1976). (This particular experiment was chosen for a precision measurement because cryogenic nitrogen could be used, which is easier to handle than liquid helium.) When a further increase of the radius of the hot section is considered, the theory shows that finally only the volume of the hot part has an effect; the shape of this part becomes immaterial. Such a device is identical with the classical Sondhauss tube, described in the beginning of Section III,B. It consists typically of a cold straight tube section of length 1, and radius rw, with a volume V attached at the closed end, in most cases a sphere with a radius much bigger than r w . A formula for the natural frequency of such devices, based on observations, was given by Sondhauss (1850) and derived by Rayleigh (1896) in the inviscid case. Rayleigh considered the system as an oscillator with the gas in I/ acting as the spring and the gas in the tube providing the mass. The frequency derived from this model is immediately given by the formula

(A, =)

o()lc/uc= (7cr:1,/V)~’z,

(3.48)

independently of the temperature of the hot gas in V , a result already known to Sondhauss and to Rayleigh. This effect is related to the independence of A, from c( in (3.46). [Formula (3.46) becomes identical to (3.48) for tan Ac E 2, and I/ = n r i ( L --[).I The fact that CI does not have to be specified for the determination of wo is helpful in avoiding the small complications in the definition of KOencountered previously for diatomic gases. For best effects, the temperature jump has to be located just adjacent to the hot volume I/ but on the tube with the radius r w ,an observation already made by Sondhauss. The stability theory for the configuration with a big bulb and the discontinuities of temperature and cross section coinciding in this particular way becomes very simple (Rott and Zouzoulas, 1976). The frequency equation is uz/dl= [1 -

f(vlw,)l~(vlw,)/~(vlw*)

(3.49)

with coo given by (3.48) and

where v, and vh are the kinematic viscosities in the gas on the cold and the hot side, respective1y;f and E are given by (3.32) and (3.37). For lyw,l + 00, (3.49) leads to an asymptotic formula for the critical temperature ratio for stability: (3.51)

164

Nikolaus Rott

independently of y ; low values of cr are (understandably) advantageous. For air, a, is 2.66, considerably lower than values attainable in straight tubes even under optimal conditions. Figure 8 shows the stability diagram evaluated for air (y = 1,4,0 = 0.70, fi = 0.73,values appropriate for the temperature range 300-500 K) according to formula (3.49). The stability limit C = 0 shows for optimal conditions (KO= 5.86) only a small dip below the asymptotic value: the minimum is a = 2.27, The left branch now rises without “bending over” to the right, as was found for straight tubes. This is the consequence of the assumption of a very big volume of the hot part, which cannot be “clogged.” Finite values of V could lead to clogging, but only at unrealistically high temperature ratios. It is significant that again the curves C > 0 show a remarkable growth of a on the “right” branch, leading to asymptotes where Iqwt) is a constant. Thus, for an appreciable excitation of the oscillations, it is important to be near to the optimal values of KO,a fact that cannot be deduced from considering the stability curve C = 0 only. The stability diagram is completed by the frequency plot w,/oo (Fig. 9). Only for high viscosity (low Yc0)and for high temperatures does or/oo differ appreciably from 1. To design an “optimal” Sondhauss tube, one proceeds as follows: a gas is chosen (e.g., air) and v, is fixed; a frequency of operation is chosen, e.g.,

FIG.8. The temperature ratio a producing given constant values of the excitation for an air-filled ideal Sondhauss tube (with the temperature jump at the “neck”). Notation is the same as in Fig. 4.

Thermoacoustics

165

FIG.9. The real part of the oscillation frequency correspondingto the curves shown in Fig. 8. The reference frequency wois that of an ideal inviscid Sondhauss tube, given by (3.48).

oo= 2.rr . 100 sec-’ (audible). The radius rw follows from rwopt

= yc,p,(vc/~o)”2.

With KO,, 6, one obtains an optimal radius of 0.9 mm. It remains to fix I/ and 1, so that the oscillator has the desired frequency. In the recent literature, only a few papers have been published on the Sondhauss tube (see, e.g., Feldman and Carter, 1970). The device is simple but it is not an effective noisemaker, mostly because the optimal devices have small radii. Big optimal radii are obtained for low frequencies; these in turn are best produced when in the oscillator the gas mass is replaced by a liquid. A few remarks on these devices complete this section.

4. Gas-Liquid Oscillations When in the oscillator described in the previous section the mass of the gas is replaced by a liquid column of length I and density pfl, then the frequency becomes (neglecting gravity)

(3.52)

166

Nikolaus Rott

where V is the volume of the gas with mean pressure p m .This formula reduces to (3.48) for pfl = pm.In case that the liquid is contained in a vertical U tube with constant radius rw, the frequency changes to wg = (w;

+ 29/1)”2.

(3.53)

Zouzoulas and Rott (1976) have given a detailed account of the theory and of certain experiments with gas-liquid oscillations. Here only the same idealized limit, leading to the lowest values of a, is discussed, which corresponds closely to the ideal Sondhauss tube of the previous section. Again, the temperature jump is assumed at the “neck” adjacent to the hot bulb, over a very short gas column in the tube; its length is negligible compared to that of the liquid. Then, a formula analogous to (3.49) is obtained: w2/w; = [1

- f(vn)IE(Ylw,)/E(Yl,,).

(3.54)

The only changes compared to (3.49) are the replacement of wo by tog, and a different argument for the function f , namely,

(3.55) m = (vfl/vc)1’2,

(3.56)

where vfl is the kinematic viscosity of the liquid, which enters only the quantity f . A counterpart for the asymptotic formula (3.51) can also be found:

(3.57) differing from (3.51) only by the appearance of the factor m (3.56). As an example, consider the case when the hot bulb is filled with air and drives a column of water. The value of m for these substances, both at room temperature, is 0.258, and (3.57) gives u,, = 1.406, a very low value. The physical reason for this result is that the kinematic viscosity of the liquid is much lower than that of the gas. In practice the value given by (3.57) cannot be attained as the length of the gas column in the tube-which does all the “driving”-has been assumed to be negligibly short; the theory including the effects of a gas column of finite length is discussed by Zouzoulas and Rott (1976). The evaluation of (3.54) for air and water gives a stability diagram, Fig. 10, which has the same character as Fig. 8 for the Sondhauss tube. The minimum of the stability curve = 0 gives a = 1.274, not too far below the asymptotic value ass, but the minimal values of ct for a given excitation are defined much sharper, for values of yC, r 4. Figure 11 is very similar to Fig. 9.

Thermoacoustics

FIG. 10. The temperature ratio a producing given constant values of the excitation for an ideal gas-liquid oscillator, with air in the hot volume and a water column providing the inertia.

3

Wr

2

1

Nikolaus Rott

168

A device was built by the author in which hot air of about 600 K was driving a column of water at room temperature, with V = 0.25 liters, r, = 6 mm, and 1 (adjustable) of about 2 m. With a value of ct z 2, amplitudes of almost 20 cm were reached. This device, dubbed a “rudimentary Stirling motor,” is also described by the author elsewhere in more detail (1976). It raises the hope that thermoacoustic oscillations could become useful.

IV. Thermoacoustic Streaming A. GENERAL THEORY

As stated in Section I, thermoacoustic streaming has its perfect analog in the acoustic streaming of mass, discovered by Rayleigh (1883); nevertheless, not before the work of Merkli and Thomann (1975) was an analytic treatment of the thermoacoustic effect published. Previous treatments of the energy problem connected with acoustic oscillations in a tube were restricted to overall energy balances (see, e.g., Temkin, 1968). Observations of the energy distribution (Sprenger, 1954; Merkli and Thomann, 1975) have led to the investigation of the detailed streaming problem. Actually, the theory of thermoacoustic streaming is much simpler than the corresponding massstreaming problem, if one restricts the calculations to the distribution of the energy exchange between gas and wall. In particular, the restrictive assumption of a thin boundary layer is not needed, in contrast to the analogous Rayleigh theory, which becomes analytically unmanageable without this restriction. The deriviation is based on the full nonlinear energy equation; however, as stated in the first paragraph of Section III,C, neglected are the radial pressure gradient and all dissipative terms involving differentiation in axial direction. No restriction is applied to the extent of the dissipative region relative to the tube cross section. Not surprisingly, the nonlinear energy equation under these assumptions has essentially the same form as the wellknown boundary layer energy equation, except that now the flow in the tube is described by cylindricalcoordinates x and r. A further modification through the continuity equation leads to the following form, which is also analogous to a well-known form of the boundary layer equation: a(PH) a(PuH) at ax +

~

+

_a (_P_r_w_ rdr

aP at

Thermoacoustics

169

where H

= c,T

+.'.3

(4.2) Multiplication of (4.1) by r dr and integration from r to rw gives, with u = u = 0 at r = rw, after taking the time-average, the following result: rw

(G)wa

= -~

dx

w p uHr dr.

0

(4.3)

Now the result is specialized to the order required for the description of thermoacoustic streaming, i.e., to an equation of second order in an amplitude expansion. In this case it suffices to replace H by cpT, and to expand T = T , + T , + T 2 . . and all other functions of state likewise, whereby the subscript 1refers to the linear acoustic solution considered in the previous sections. The velocities u = u1 + u2 + . . . are expanded in the same manner. Equation (4.3),when restricted to second-order terms, takes the form

+

Q2 = 2n

so''" pmcpTFlrdr,

(4.5)

whereby use has been made of the second-order continuity equation

Ji'"p,&rdr + Jiw p,u,rdr

= 0.

(4.6)

The quantity ij2 is the mean second-order heat flux to the tube wall per unit area, and Q2 is the total mean second-order axial heat flux in the gas, over the whole tube cross section, provided that Q2 = 0 at x = 0 (a permissible assumption when u1 = 0 at x = 0). It remains to introduce u , from (3.26), and T , from (3.17) and (3.28), in (4.5). The result is, after a lengthy but straightforward calculation :

where i dp u,, - __-

o p , dx '

and

170

Nikolaus Rott

A tilde again denotes a complex conjugate quantity; the symbols on the right-hand side of (4.9) are defined by (3.27) and (3.22). All applications considered thus far in the literature are obtainable as special cases of (4.7).

B. ISOTHERMAL WALLS In case T,,, = const, Eqs. (4.4) and (4.5) are identical with results obtained by Merkli and Thomann (1975). Equation (4.7) for Q2 then reduces to the first term, which introduced in (4.4) gives, upon differentiation with respect to x for 0 = 0, the numerical method used by Merkli and Thomann to evaluate if2. They have obtained excellent agreement between their theory and the experiments they carried out with a “resonance tube,” i.e., a closed tube in which standing acoustic waves were produced by a piston at one end. The experiments of Merkli and Thomann were carried out for conditions that resulted in a rather thin boundary layer compared to the tube radius. It is definitely of interest to find the results of their theory in this limit. It is, however, just as easy to obtain the thin-boundary-layer results directly, by making a fresh start beginning from the fundamental equations, as it is to specialize the results of Merkli and Thomann. Thus, the derivation starts again from (4.1). Let a boundary layer coordinate y = r - rw be defined and introduced into (4.1), and ub = - v by positive in the y direction. The full nonlinear boundary layer equation corresponding to (4.1) is

The time-averaged equation integrated across the boundary layer (formally, from y = 0 to co)is

a

-

ax

J-

m -

0

puHdy

+ pv,HI,

=

-q.

(4.11)

Now, the second-order equation is extracted by the same method as before, and again simplified by the second-order continuity equation, which is (4.12)

The result is

171

Thermoacoustics

Finally, as the entropy of the order 1 solution at y -+ GO vanishes, one has P m ~ p T 1 ( = ~ ) Pl(.o)

(4.14)

= P1,

so that one finds (4.15) (This equation was obtained by the author, 1974b.) The first term in (4.15) has been already considered in Section III,A (in the more general case when dT,/dx # 0; unfortunately, the generalization of (4.15) to this case is very cumbersome, and the discussion is restricted here to isothermal walls). Introducing ubl(m) from (2.20) and p1 again from (3.2),one finds (allowing for the slight changes in notation necessitated by the presentation in different sections and with n = N) pmpmo

( + ;$)(

p,v,,(GO) = - 1

__

~

8

)

a2 d u l c d C l c

zdxdx‘

(4.16)

Similarly, with u1 from (2.2) and TI from (2.10), and taking account again of the modified boundary condition (2.14) by use of the constant (2.15) [an effect that has been ignored in the derivation of (4.7)], one obtains

Now the results are applied to the case in which the “core” solution is an acoustic standing wave with the amplitude distribution between x = 0 and x = L given by (4.18) Introduced in (4.15)and (4.16), one obtains, after some simple manipulations, the result L

(4.19)

where c=

[Y - ( N - 1)&](1 [Y + ( N - l)&](l

+ a) + (1 - a)(l + &) + a) - (1 + a)(l - 6)’

(4.20)

a result previously given by the author (1974b) in the special case N = 1. Through c, the appearance of the result is a strong function of the Prandtl

Nikolaus Rott

172

number o, an effect that has been already noted and discussed by Merkli and Thomann (1975). In the following, special values of o are considered. First, for the rather singular case o -+ co, heat conduction is negligible altogether, and it is consistent to put N = 1. One obtains in this limit c = - 1 and qz vanishes at both ends of the tube, with the maximum in the middle. The distribution of qz is identical with that of the local dissipation due to viscosity. With vanishing heat conduction in the gas, this result is physically understandable. However, the profound change in ?j2 caused by nonzero heat conduction is hard to interpret. This is strikingly illustrated by considering the special case o = 1; c then has the value C=

7 - ( N - 1) y ( N - 1)’

+

and in the important particular case N = 1, we have c = 1. Then q2 is maximal at both ends (nodes) of the tube, and vanishes in the middle! It is also of interest to note that the term (4.17) vanishes for o = N = 1, so that ljz is restricted to the expression given by (4.16). Thus, the energy exchanged between the acoustic core and the boundary layer is directly transmitted to the wall in this special case. Finally, for o < 1, as is found for practically all gases, and for N sufficiently near to 1, one has c > 1 and a cooling effect (negativeijz)is produced around x = L/2. Merkli and Thomann (1975) have experimentally found this effect for air as well as for an air-helium mixture with a particularly low o = 0, 5. The extreme value o 0 is obtained for vanishing viscosity ,u + 0; the value of c is --f

c = (Y

+ 1 ) h J- 11,

and the maximum cooling at x = L/2 is l/y times the maximum heating at x = 0. In this case, viscous dissipation does not exist and the main dissipative effectis the energy exchange between the core and the boundary layer, whose thickness fluctuates due to heat conduction. However, this effect, which is biggest at the nodes, is already present for o = 1, and the physical interpretation of the “redistribution” for o < 1, caused by the term (4.17), is very difficult. An interesting discussion, based on (4.5)rather then (4.17), is given by Merkli and Thomann (1975),leading to the interpretation of the direction of the thermoacoustic streaming (toward the nodes) for o --t 0. In a later paper, Thomann (1976)investigated the case in which the viscous region completely fills the tube, and the “shear wave number” a = r , ( o / ~ ) ‘ ’ ~ is small compared to 1. In this case, the interaction of acoustic streaming of mass and heat becomes very important. The effect is that only a small fraction of the heat energy corresponding to viscous dissipation is transferred to the wall. Almost all of this energy is expended as mechanical work done

173

Thermoacoustics

by the acoustic streaming of mass. The situation is similar to the case of the steady flow of a gas in a tube, where the heat energy of dissipation is expended in the work needed to overcome the pressure drop due to friction. For completeness it is noted that the author has investigated the effect of heat transfer on the Rayleigh acoustic streaming (1974a). As predicted (but not yet proven) by previous investigators, the effect turned out to be quite small. C . NONISOTHERMAL WALLS

In the case of nonisothermal walls, the full equation (4.7) is needed. Numerical evaluations pose no problem when a continuous basic temperature stratification is assumed; however, the analytical solutions based on a discontinuous model (temperature jump) lead to serious problem of divergence, as pointed out by the author in 1975. The most obvious (but not the unique) difficulty is the explicit appearance of 6 in (4.7), which makes Q2 infinite for the discontinuous model. One can conclude, however, that for sharp gradients the second term in (4.7) is strongly dominant. It is possible to calculate this term in the approximation valid for thin boundary layers, whether by expanding the function g or by use of the temperature distribution Q 2 of (2.11) in the evaluation of the enthalpy flux. The result is

Q,

=

pmpma -*nrw(g)

1+,/“G+Na . w2 c p dx N(1 + o)(& a)

ulCiilc dT, ~

+

(4.21)

It is convenient to introduce an “effective” coefficient of heat conduction by the relation (4.22) One finds k

rw

N(1

+ &)(1 + a)

where s1 = ulE/w is the amplitude of the particle displacement in the core. The value of the last bracket in (4.23) is 0.67 for CT = 2/3 and N = 1. It is significant that in (4.23) the boundary layer thickness (or the square root of the kinematic viscosity) appears in the denominator. Thus it is possible that the effective coefficient of heat conductivity can be orders of magnitude bigger than the proper (molecular) conductivity k of the gas. This is in agreement with experience.

174

Nikolaus Rott

To judge the efficiencyof thermally driven acoustic oscillations as a means of obtaining mechanical work from heat, one can adapt the following idealized procedure. The heat flux inevitably associated with a certain configuration is Q 2 given by (4.7);its value grows as long as heat is absorbed from the hot surroundings, and it decreases when heat is deposited at the cold spot. Thus the maximum of Q2 is the heat to be provided to the engine. This quantity is quadratic in the amplitude, and for its calculation (as mentioned before) the discontinuous model is not suitable. A numerical method for the solution of the oscillation problem is needed, which works for continuous wall temperature distributions. An optimization problem can already be recognized, as it was found that the excitation-which is connected with the mechanical work obtainable-has its maximum with steep gradients, but so has Q2,,,. The latter becomes singular with the discontinuous model, while the former remains finite. The ideally obtainable mechanical energy is found by considering a state in which, with the boundary conditions “one end closed-one end open” the oscillations are excited. Now the boundary condition is changed such that at a certain point of the tube, a piston is driven and energy is extracted until the steady state is reached. This energy again is a quadratic form in the amplitude. Thus, an expression results for the efficiency independently of the amplitude (valid in the limit of small amplitude), which depends only on the parameters of the configuration. A final question is concerned with the choice of the point where power is extracted. The open end of the original configuration offers itself as a first choice; however, one has to realize that the whole cold part only acts as inert mass, which can be replaced, e.g., by a column of liquid, as was already discussed before; or, alternatively, by the mass of whatever piston is provided to extract the mechanical energy. Thus, in a program worked out for the search of best efficiencies (Muller, 1981) the piston is always situated at the point where the gas reaches its lowest temperature. The difficulty encountered in this search is that the requirement of high efficiency leads to configurations with insufficient power density. Thus a compromise has to be found, a problem which is not yet resolved at the time of the writing of this report. REFERENCES CARRIER, G.F. (1954). The mechanics of the Rijke tube. Q. Appl. Math. 12, 383-395. W. (1804) Letter to the Editor. Gilbert’s Ann. der Physik (Halle)17, 482. CASTBERG, FELDMAN, K.T.,and CARTER, R. L. (1970). A study of heat driven pressure oscillations in a gas. J . Heat Transfer 92, 536-540. IBERALL,A. S. (1950). Attenuation of oscillatory pressures in instrument lines. J. Res. Natl. Bur. Stand. 45,85-108. KEMPTON, A. J. (1976). Heat diffusion as a source of aerodynamic sound. J.FluidMech. 78,l-31.

Thermoacoustics

175

KIRCHHOFF, G. ( 1 868).Ueberden EinflussderWarmeleitungineinemGasaufdieSchallbewegung. Ann. Phys. (Leipzig) [2] 134,177-193. KRAMERS, H. A. (1949).Vibrations of a gas column. Physica (Utrecht) 15, 971 -984. MmX, C. (1841). Ueber das Tonen erhitzter glaserner Rohren. J . prakt. Chem. 22, 129-135. MAWARDI, 0. K. (1956). Aero-thermoacoustics. Rep. Prog. Phys. 19, 156-187. MERKLI,P.. and THOMANN, H. (1975). Thermoacoustics effects in a resonance tube. J . Fluid Mech. 70, 161-177. MORFEY, C. L. (1976). Sound radiation due to unsteady dissipation in turbulent flows. J . Sound Vib. 48, 95-1 1I . MULLER,U. (1981). T o be published, Diss., ETH Zurich. RAYLEIGH, J. W. S. LORD(1883). On the circulation of air observed in Kundt’s tubes, and on some allied acoustical problems. Philos. Trans. R . SOC.London 75, 1-21. RAYLEIGH, J. W. S. LORD(1896).“Theory of Sound,” 2nd ed., Vol. 11 (reprinted at Macmillan, London, 1944). RIJKE,P. L. (1859). Notiz uber eine neue Art, die in einer an beiden Enden offenen Rohre enthaltene Luft in Schwingungen zu versetzen. Ann. Phys. (Leipzig) [2] 107, 339-345. ROTT,N. (1969). Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z . Angew. Math. Phys. 20,230-243. ROTT,N. (1973). Thermally driven acoustic oscillations. Part 11. Stability limit for helium. Z . Angew. Math. Pliys. 24, 54-72. ROTT,N. (1974a). The influence of heat conduction on acoustic streaming. Z . Angew. Math. Phys. 2 5 4 1 7-421. ROTT,N. (1974b). The heating effect connected with non-linear oscillations in a resonance tube. Z . Angew. Math Phys. 25,619-634. ROTT, N. (1975). Thermally driven acoustic oscillations. Part 111. Second-order heat flux. Z . Angew. Math. Phys. 26, 43-49. ROTT,N. (1976). Ein “rudimentarer” Stirlingmotor. Neue Zuercher Ztg. 197, No. 210. ROTT,N., and ZOUZOULAS. G. (1976). Thermally driven acoustic oscillations. Part 1V.Tubes with variable cross-section. Z . Angew. Math. Phys. 27, 197-224. SONDHAUSS, C. (1850). Ueber die Schallschwingungen der Luft in erhitzten Glasrohren und in gedeckten Pfeifen von ungleicher Weite. Ann. Phys. (Leipzig) [2] 79, 1-34. H. (1954). Ueber thermische Effekte in Resonanzrohren. Mitt. Inst. f: A e . , Eidg. SPRENGER, Tech. Hochschule Zurich 21, 18-35. TACONIS, K. W. (1949). Physica IS, footnote on p. 738. TEMKIN, S. (1968). Nonlinear gas oscillations in a resonant tube. Phys. Fluids 11, 960-963. THOMANN. H. (1976).Acoustical streaming and thermal effects in pipe flow with high viscosity. Z . Angew. Math. Phys. 27, 709-715. VON HOFFMANN, T., LIENERT. U., and QUACK,H. (1973). Experiments on thermally driven gas oscillations. Cryogenics 13, 490--492. YAZAKI,T., TOMINAGA, A,, and NARAHARA, Y . (1979). Stability limit for thermally driven acoustic oscillations. Cryogenics 19. 393-396. ZOUZOULAS, G. (19751. Thermisch getriebene Gasschwingungen in Rohren veriinderlichen Querschnitts. Diss., No. 5520. ETH Zurich. ZOUZOULAS, G., and ROTT,N. (1976). Thermally driven acoustic oscillations. Part V. Gasliquid oscillations. Z . Angew. Math. Phys. 27. 325 -334.

This Page Intentionally Left Blank

ADVANCES IN APPLIED

MECHANICS, VOLUME 20

Simple Non-Newtonian Fluid Flows ERNST BECKER Institut fur Mechanik Technische Hochschule Barmstadt Darmstadr, Federal Republic of Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . II. Non-Newtonian Flow Behavior. . . . . . . , . . . , . . A. Nonlinear Flow Behavior. . . . . . . . . . . . . . . . . B. Normal Stress Effects . . . . . . . . . . . . . . . . . . . C. Memory Effects; Elasticity . . . . . . . . . . , . , . . . 111. The Constitutive Equation of Simple Fluids . . . . . . . IV. Fully Developed Pipe Flow . . . . . . . . . . , . . . . . V. Peristaltic Pumping . . . , . . . . . . . . . . , . . . . . VI. Viscosity Pumps . . . . . . . . . . . . . . . . . . . . . . VII. VIII. IX. X. XI.

... ......, .. . , .......... ............. ............. . . ........... . .... .. ..... .... . ...... . .. ....... . .. . .... ....... Effective Viscosities . . . . . . . . . . . , . . , . . . . . . . . . . . . . . . . . Extruder Flow . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . Nearly Viscometric Flow , . . . . . . . . , . . . , . . . . . . . . . . . . . . , Plane Boundary Layer Flow of a Fluid with Short Memory . . . . . . . . . . JournalBearing . . . . . . . . . . . . , . . . , . . , . . . , . . . , , . . . . . References . . . . , . , . , . . . . , . . . . , . . , . . . , . , , . . . . . , , .

. . . . . . . . . . . . . . ,

177 179 179 182 183 184 187 192 197 204 210 212 216 219 225

I. Introduction In this chapter simple rheological problems of engineering importance are discussed. Rheology is here understood as a branch of continuum mechanics; molecular theories are not considered. Continuum mechanics is based on a number of balance equations, in particular, on the balance equations for mass, momentum, and energy. Thermodynamic effects are often negligible and therefore the energy equation need not be explicitly taken into account. In such cases a purely “mechanical” theory, with the balance equations for mass and momentum as basis, is sufficient for describing the motion of 177 Copyright @ 1980 by Academic Ress, Inc. All rights of reproduction in any form reserved. ISBN 0-12-002020-3

178

Ernst Becker

continuous media. This chapter concerns the mechanical theory only. However, it should be mentioned in passing that thermal effects, like heating by dissipation, may play a role in many of the technologically important flows studied in the following sections. The balance equations have to be supplemented by constitutive equations, which characterize special classes of materials. Briefly stated, a constitutive equation connects stress with the motion, or deformation, of the material. The different branches of continuum mechanics are distinguished by different constitutive equations. For example, classical fluid dynamics is the mechanics of materials that are described by the constitutive equation

T = -PI

+ qD + qv(trD)I,

(1.1)

where T(x,t ) denotes the symmetric Cauchy stress tensor at position x and time t, p(x, t ) is the pressure in the fluid (I being the unit tensor), and D is the symmetric part of the gradient tensor of the velocity field v(x, t):

(Cartesian coordinates, vectors, and tensors are used throughout.) The coefficients q and qv are called the shear viscosity and the volume viscosity, respectively. They may depend on the local thermodynamic state of the moving fluid, particularly on the temperature, but by assumption they do not explicitly depend on the motion. A material whose stress during motion is appropriately described by (1.1) is called a Newtonian fluid. If the fluid is density preserving, or incompressible, the mass balance equation, also called the equation of continuity, reduces to t r D = dui/dxi= 0.

(1.3)

The last term on the right-hand side of (1.1)drops out in that case, and hence volume viscosity plays no role. It should be mentioned here that for flow of an incompressible fluid the absolute value of the pressure p is irrelevant, and only differences of pressure matter: Changing the pressure everywhere in such a flow field by the same amount does not at all affect the flow. In conformity herewith the constitutive equation (1.1) connects only the “extra stress tensor” T + pI uniquely with the local motion of the fluid, but leaves the pressure p indeterminate. This applies also to the constitutive equations of incompressible non-Newtonian fluids. Only incompressible fluids are treated in what follows. Water and air, the most important fluids for life, are Newtonian fluids. Our everyday familiarity with water and air makes us consider the flow of Newtonian fluids as normal. However, there are many fluid materials of

Simple Non-Newtonian Fluid Flows

179

importance in nature and technology that do not obey (1.1).Certain aspects of the flow behavior of such non-Newtonian fluids are markedly different from the flow of Newtonian fluids and, at first sight, seem abnormal or even paradoxical.* Polymer solutions and polymer melts, paints, and suspensions are non-Newtonian, as are many household fluids like cake dough, mayonnaise, and whipped cream. This chapter is concerned with simple flows of such non-Newtonian fluids that are of technological import.

I I

///////////////////////

x3

X1

FIG.1.

Simple shear Row.

11. Non-Newtonian Flow Behavior

The difference in behavior of Newtonian and non-Newtonian fluids can be roughly characterized by three types of phenomena:

A. NONLINEAR FLOWBEHAVIOR Figure 1 shows simple shear flow: the lower wall is fixed, the upper wall moves with constant velocity U . The space between the two walls is filled with fluid, which under nearly all circumstances sticks to the walls, so that no slip occurs between the motion of the wall and of the fluid. The velocity field is given by 01 = K X 2 ,

u2

=ug =0

(2.1)

where K

5

av,/ax2 = U / h

* Walker (1978) describes many experiments, easy to perform by a layman, that clearly show the striking features of non-Newtonian fluid Row.

180

Ernst Becker

is called the shear velocity. For a Newtonian fluid (1.1) yields the following stress components in simple shear flow: (2.3) All other stress components vanish. According to (2.3) the three normal stress components are equal and the only nonvanishing shear stress 7 is proportional to the shear velocity K. For non-Newtonian fluids in simple shear flow, the only nonvanishing shear stress is again 7 = z I 2 = 7 2 1 . However, for most non-Newtonian fluids 7 depends nonlinearly on K : 7

= 7[u].

(2.4)

For reasons of isotropy 7 is an odd function K : Reversing the shear direction (K -+ - K), by reversing the direction in which the upper wall moves, reverses the stress direction (7 -+ - 7). It is very useful to write (2.4) in the nondimensionalized form Here z* is a reference stress, q* a reference value with the dimension of a viscosity, and f the dimensionless“flow function” of the dimensionless argument q * ~ / 7 * . Although in principle q* and 7* can be chosen arbitrarily, a choice that takes account of the properties of the particular fluid studied is advisable. Therefore, in this chapter v.+ is always chosen as the lower Newtonian viscosity, defined as q* = lim (z/K). K - 0

This choice is subject to the assumption that the limit in (2.6) exists and is finite. Experience with many non-Newtonian fluids shows this to be the case. This experience is corroborated by a theorem due to Coleman and No11 (Truesdelland Noll, 1965; Truesdell, 1974; Becker and Burger, 1975),which states that under weak conditions the class of “simple fluids” (see Section 111) behaves asymptotically for slow motion like Newtonian fluids. However, it must be mentioned that the so-called power law fluids, which enjoy widespread but somewhat undeserved popularity among engineers, do not possess a finite lower Newtonian viscosity. Having chosen the reference viscosity q*, one chooses the reference stress 7* in such a way that dimensionlessnumbers entering the analytic expression of the flow function f attain simple values. By that choice z* will usually have the physical meaning of shear stress for which the deviation from linear flow behavior first becomes appreciable (see Fig. 2). Depending on whether

Simple Non-Newtonian Fluid Flows

181

shear - thickening

FIG.2. Flow functions.

I

1

c

I. x I T.

shear stress T grows more or less rapidly than linearly with shear velocity rc, non-Newtonian fluids are divided into shear-thickening and shear-thinning fluids (Fig. 2). Polymer solutions and polymer melts, but also many biological fluids, like blood, are shear thinning; many suspensions are shear thickening. It should be noted that temperature 8, pressure p, and other physical parameters, like the molecular weight M in the case of a polymer, enter the flow function f as parameters. Remarkably, many measurements with polymeric liquids have shown (Hellwege et al., 1967; Schonewald, 1970) that these quantities enter only through q,[O,p,M]; z* is independent of 8, p, M ! Therefore, in order to determine the complete dependency of shear stress on IC, 8, p, M , it suffices to determine experimentally how the lower Newtonian viscosity q* depends on t?, p , M, and then to measure the dependency of q on IC for one set of values of 8, p, M . Recent investigations have demonstrated that z* for polymeric substances is influenced by the molecular weight distribution; details are given by Christmann (1977). For some purposes it is useful to invert (2.5),

Two widely used simple analytic expressions for the inverse flow function g, which can be realistically fitted to real fluids, are (sinh w,

Prandtl-Eyring,

(2.8)

Reiner-Philippoff.

(2.9)

The Prandtl-Eyring fluid is always shear thinning. The Reiner-Philippoff fluid is shear thinning if qm/v* < 1 and shear thickening if qm/q* > 1. The parameter qm is the upper Newtonian viscosity: For w + co,(2.9) shows that

182

Ernst Becker

g[w] -+ q,o/q,, or z -+ q,~. Therefore, for large values of o,and hence of K, the fluid behaves again in a Newtonian way, with viscosity qm (see, however, the footnote on p. 207). This agrees with observations for many non-Newtonian fluids. A large part of the engineering literature on non-Newtonian fluid flow is concerned only with nonlinear flow behavior as explained above. The rather elementary investigations reported in that literature are based on various special forms of the flow function, and often the advantages and the insights to be gained by appropriate nondimensionalization are not exploited. A profusion of special results is thereby generated, which could be incorporated into a few basic and widely applicable results by consistent nondimensionalization. Use of q* and z*, as advocated here, is a proper means of achieving such consistent nondimensionalization, and it is hoped that Sections IV-VIII show this. If, instead of nonlinear flow behavior, memory effects of the fluid are of dominant importance or interest (see Sections 111, IX-XI) it might be advisable to use the quantities q* and I instead of q* and z* for nondimensionalization, where I is a characteristic relaxation or memory time (see Sections II,C and 111).In that case a suitable value z*, if needed, can always be defined as z, = q*/I.

B. NORMAL S T ~ EFFECTS S

As mentioned, for a Newtonian fluid in simple shear flow the three normal stresses are equal [cf. (2.3)]. In simple shear flow of a non-Newtonian fluid, the three normal stresses usually have different values. Since in incompressible fiow the absolute normal stress values are irrelevant (as has already been noted with respect to pressure in Section I), only the normal stress differences (2.10)

are of import. The normal stress functions o1and o2 are even functions of K , i.e., they are functions of x 2 ; reversal of the shear direction does not affect the normal stresses. The three functions ~ [ I c ]( ,T ~ [ I C ~ J] ~, [ K ’ ]are usually called viscometric functions. The finite normal stress differences are the cause of so-called normal stress effects in nowNewtonian fluid flow. For example, in many cases nonNewtonian fluids climb at a rotating rod immersed in a mass of fluid with free surface (Weissenbergeffect), whereas Newtonian fluids climb at the wall of the surrounding vessel. Die-swell, the drastic widening of a jet of liquid after leaving a pipe with circular cross section, is another manifestation of normal stress effects. Many secondary flow phenomena, for example, the

Simple Non-Newtonian Fluid Flows

183

flow within the noncircular cross section of a straight pipe with main flow in axial direction, are normal stress effects that Newtonian fluids do not exhibit. Because T , ol,6, vanish in a fluid at rest, it is customary to write (2.4), and (2.10) in the form

4.1 ni[tc]

(2.11)

= “l[KZ1, = I C ~ N ~ [ K ’ ] ,i =

(2.12)

1,2,

where 1 is the shear-dependent viscosity, and the N iare the shear-dependent normal stress coefficients. These quantities can be nondimensionalized : 1CK’I

= ?*“rl*“/7*)21,

Ni[Kz] = (r]:/T*)ni[(r*IC/T*)’],

(2.13) i = 192.

(2.14)

The nondimensionalization suggests that the dependency of Ni(0)= limK+o( ~ J I c ’on ) the physical parameters 8, p , M should be determined by the dependence of q* and z* on these parameters. This surmise is borne out by experimental evidence for high polymeric liquids at least for the dependence on molecular weight M ; the measurements show that r]* M3.’ and Ni(0) M 7 . 0 . Incidentally, for many fluids o1 > 0 and o, < 0, hence also N1 > 0 and N , < 0. Finally, the ratio ol/t2is often nearly independent of IC and therefore about equal to its value for IC = 0: OJT’ = n,(O)/z, (Truesdell, 1974).

-

-

C . MEMORYEFFECTS; ELASTICITY When the upper wall velocity in the simple shear flow depicted in Fig. 1 depends on time, then the shear velocity also depends on time ~ [ t=] U [ t ] / h . Here, we tacitly assume that, despite the unsteadiness, the velocity field is still given by (2.1).This is true if the fluid inertia can be neglected. A criterion for the validity of this neglect is ph2/r]to<< 1; here p is the density of the fluid, to a characteristic time for the change of U [ t ] ,and q a characteristic value of the viscosity. For a Newtonian fluid the instantaneous shear stress ~ [ r in ] the unsteady flow field is still determined by the instantaneous shear velocity ~ [ t through ] the relation T = ~ I Cwith , constant r]. In particular, if the upper wall is suddenly started at time t = 0 from rest and then moves with constant velocity U for t > 0, so that IC = 0 for t < 0 and K = U / h = const for t > 0, the shear stress in a Newtonian fluid is represented by the dashed line in Fig. 3. Non-Newtonian fluids usually exhibit a memory for past deformations and hence for past shear. The following elementary model equation describes

Ernst Becker

184

‘t

elastic

/

such memory effects in simple shear flow: z[t] =

som

e - ’ h c [ t - s] ds

(2.15)

where y is a constant viscosity and 1a constant relaxation time. The “Maxwell fluid” described by (2.15) has “fading memory” because the exponential function weighs values of the shear velocity in the far past less than those in the recent past. For the wall suddenly set into motion, (2.15) gives z[t] = r/lc(l - exp[-t/1]) for t > 0; the graph of this function is the full line in Fig. 3. For t >> I the shear stress is given by z = ylc, as for a Newtonian fluid. For t << 1the shear stress is given by z = (y/1)lct = Glct (dash-dot line in Fig. 3). Therefore, the shear stress immediately after the start of the motion is proportional to shear ~ tThis . reflects elastic properties of the liquid; G = r//1 has the meaning of the elastic shear modulus. It should be mentioned that nonlinear flow behavior in steady shear flow, normal stress effects, and memory or elastic effects are intimately coupled and, in nearly all cases, occur simultaneously.This feature is, of course, not adequately taken account of by (2.15), because this equation yields the Newtonian relation z = ylc for steady shear flow. 111. The Constitutive Equation of Simple Fluids

In order to describe theoretically the non-Newtonian phenomena mentioned in Section I1 one has to formulate a suitable constitutive equation that generalizes (1.1). Many such equations have been proposed (Bird, 1976; Bird et al., 1977; Huilgol, 1975; Truesdell, 1974; Truesdell and Noll, 1965). A rather general formulation of a constitutive equation is due to No11 and Coleman (Becker and Burger, 1975; Coleman et a/., 1966; Leigh, 1968; Truesdell, 1974;Truesdell and Noll, 1965).Before presenting this formulation a few preliminaries concerning the kinematics of continuous media have to be mentioned.

Simple Non-Newtonian Fluid Flows

185

Y FIG.4.

Motion and deformation of fluid particle.

Consider a material point* that at the present time t is at position x [ t ] . At the past time t - s, with 0 5 s < co,the same point was at position x [ t - s ] . Another material point in the infinitesimal neighborhood of the first one has position x [ t ] d x [ t ] at time t and x [ t - s] + d x [ t - s ] at the past time t - s (see Fig. 4). By the relation

+

dxi[t - s]

= Fik[S; t , x ]

dXk[t],

(3.1)

a matrix F[s; t , ~ with ] elements Fi, is defined, where F is the relative deformation gradient. The square of the distance between the two material points at time t - s is related to the squared distance at time t by dXi[t - s ] d x i [ t - s ]

FikFildXk[t] dx,[t] = C k l [ s ;t , x] dXk[t] d X l [ t ] . =

(3.2)

The matrix C = FTFrepresents a symmetric tensor, which is called relative right Cauchy-Green tensor (RRCG tensor). This tensor is a local measure of the past deformation of the medium relative to its present configuration. In other words, C describes the (relative) deformation history of the infinitesimal neighborhood of a material point that at time t is at position x . For s = 0 (present time) C reduces to C[O; t , x ] = I. It is easy to show (Becker and Burger, 1975; Leigh, 1968) that the ratio between a r infinitesimal material volume at time t - s and at time t (Fig. 4) is given by dV[t - s] = (det F ) d V [ t ] . Hence, in a density-preserving fluid det F = 1, and therefore det C = 1. Finally, if the motion of the medium is a rigid-body motion, C = I, because distances between any two points remain unchanged in rigid body motion. For the simple, steady shear flow depicted in Fig. 1, the relative deformation gradient and the RRCG tensor * In a fluid a “material point” can be approximated and made visible by marking a small blob of fluid, e.g., with color.

186

Ernst Becker

are given by

;

1 -ICs

b(;

;I, 0

c=

[-is

-ICS

1 + y

;I. 0

(3.3)

Because of the steadiness of the flow, F and C do not depend on time t ; because of the spatial homogeneity of the simple shear flow field, neither F nor C depends on x. Both F and C depend only on s, the time variable extending from the present time (s = 0) into the past (s > 0), and on the constant shear velocity IC. An incompressible (density-preserving)simple fluid in the sense of No11 and Coleman is defined by the constitutive equation for the extra stress

TCx, t l

+ PI =

10

[C[s; t, X I ]

(3.4)

s=o

where 8 denotes a general, tensor-valued functional of C[s], with x and t as parameters. According to (3.4) the extra stress at time t and position x is a functional of the deformation history of the infinitesimal neighborhood of the material point that occupies position x at time t. Because of the indeterminacy of pressure p , a corresponding indeterminacy of 8 prevails. The indeterminacy can be removed by an auxiliary condition, for example, by t r g = 0. This condition is of no import for the subsequent discussion because the absolute values of the normal stress components are of no import but only the normal stress differences. These, however, are unaffected by the auxiliary condition. Finally, the functional 8 is usually required to be continuous with respect to a norm of fading memory. This requirement ensures that deformations have progressively less influence on the present stress the farther in the past they have been experienced. For details the reader is referred to the pertinent literature (e.g., Becker and Burger, 1975; Leigh, 1968; Truesdell and Noll, 1965). The concept of a fading memory introduces a characteristic memory time A of the fluid. Deformations that have been undergone by a particle at the past time t - s, with s >> A, have virtually no influence on the stress at the present time t. A simple example of a constitutive equation that satisfies all requirements characteristic of the class of simple fluids is given by T f pI =

so"

K[s](C[s] - 1)ds.

(3.5)

(Note that the dependence of C on x and t will no longer be explicitly indicated.) A closely related equation, which is usually named after Lodge, is T

+ pI =

JOm

K[s](I - C-'[s])d~.

Simple Non-Newtonian Fluid Flows

187

In both cases K [ s ] denotes a kernel function that, for reasons of fading memory, has to vanish for s + co. To make matters as simple as possible, let us assume

K[s] =

- (r]/12)e-”/”,

(3.7)

with two positive constants r] and 1. Using (3.7) in (3.5) and inserting the expression (3.3) for C, one obtains the viscometric functions z = ylc,

a, = - a 2

= 2r]11c2.

(3.8)

Equation (3.6) yields with the same assumptions z = qlc,

a, = 2qllc2,

a, = 0.

(3.9)

Obviously, (3.5) and (3.6), together with (3.7), describe fluids for which shear stress in simple steady shear flow is exactly the same as for a Newtonian fluid with viscosity r]. However, contrary to a Newtonian fluid the present fluids exhibit normal stress effects, with constant normal stress coefficients N , = - N , = 2ql, and N , = 2 ~ 1 N, 2 = 0, respectively. For unsteady shear flow one must substitute & l c [ t - (1 d( for s in the expression (3.3) for C. Insertion of the resulting expression for C into (3.5) or (3.6) and partial integration with respect to s yields the result (2.15) for the shear stress z, which has already been discussed. An important consequence of the theory of simple fluids, already mentioned in Section II,A, is repeated here. For sufficiently slow flow the general constitutive equation (3.4) reduces to the constitutive equation of a Newtonian fluid provided the functional 8 satisfies rather nonrestrictive regularity conditions. Roughly stated, this means that in sufficiently slow motion “nearly all” fluids behave like Newtonian fluids. This explains the importance as well as the tremendous success of the constitutive equation (1.1). It goes without saying that physical quantities not discussed in this section, particularly temperature, enter the fclnctional as parameters, as mentioned in Section 11. In what follows the influence of such parameters is not discussed and hence no further mention is made of them.

IV. Fully Developed Pipe Flow Steady simple shear flow (Fig. 1) is an example of a class of flows called viscometric flows because they are realized in many devices used as viscometers for measuring the viscometric functions z[K], al[lc2], G,[K’] (or, equivalently, q [K’], N , [ K ’ ] ) . The kinematics of steady viscometric flows is

188

Ernst Becker

X

FIG.5.

Viscometric pipe flow.

simple. In such flows one can identify material surfaces, i.e., surfaces consisting of the same material particles at all times, that, while sliding past each other, remain rigid. In simple shear flow these are the planes x2 = const parallel to the bounding walls (Fig. 1). Another example of viscometric flow is the fully developed flow through a straight pipe with circular cross section, which is studied in this section. Here the sliding rigid surfaces are the circular cylinders that are coaxial to the wall of the pipe (Fig. 5). Of course, infinitely many flows with the special kinematic properties of viscometric flows are imaginable. However, most of these flows could be realized only by providing complicated fields of body forces, which is extremely difficult. Steady simple shear flow and flow through a pipe are viscometric flows that are easy to realize: shear flow is induced by the motion of the bounding walls, and pipe flow by applying a pressure difference between inlet and exit of the pipe. All viscometric flows are equivalent to steady simple shear flow in the sense that the deformation history of each fluid particle in such flows is essentially determined by a constant shear velocity K , such that the stress in the fluid is determined by the three viscometric functions T [ K ] , C J ~ [ K ’ ] ,o ~ [ K ’ ] The . shear stress t is much more important here than the normal stress functions c l , cr2. In fully developed flow through a pipe with circular cross section the velocity has axial direction and it depends only on radial distance r from the axis: u = u[r].* The shear velocity is therefore given by ~ [ r =] du/dr.

(4.1)

The relevant shear stress ,z, is given by ,z, = z [ ~ [ r ] ]Because . the fluid particles move with acceleration zero, the stress tensor T must satisfy the equilibrium condition V * T = 0, and this leads to t=

-Ar/2,

(4.2)

where - A is the constant pressure gradient along the axis of the pipe: dpldx = -A.

* It might be mentioned that the flow in the entrance section of the pipe, before it is fully developed, is rather tricky for nowNewtonian fluids and not yet satisfactorily understood.

Simple Non-Newtonian Fluid Flows

189

Inserting (4.1) and (4.2) into (2.7), one obtains for the velocity u(r) the differential equation

dr

(4.3) yI*

(Note that g is an odd function!) Of paramount interest is the relation between the flow volume per unit time $ and the pressure gradient A. The flow volume is given by

soR

sf

r2g [F]d r , (4.4) 22, where R denotes the radius of the pipe. The second expression follows from the first by a single partial integration, with the no-slip condition u(R)= 0 taken into account. If one substitutes w = Ar/27* in the last integral and uses the abbreviation

I;= 2n

ru[r] dr =

- n JoR

rz @ dr dr

5 =w

= 71t*

r*

(4.5)

z * ,

one can write

P =(n~4w3~,)~[51,

(4.6)

with

d 5 1 =4 64 Jo '"w 2 g [ w ]dw.

(4.7)

For a Newtonian fluid with viscosity y ~ * one has g [ w ] = w ; then q[5] = 1, and (4.6) reduces to the well-known Hagen-Poiseuille law. Obviously, the function q characterizes the deviation from the Hagen-Poiseuille law due to the nonlinear flow behavior (cf. Section 11,A). An elementary calculation yields for the Prandtl-Eyring fluid (2.8) (4.8)

2

and for the Reiner-Philippoff fluid (2.9)

Both functions are displayed in Fig. 6. The function qRPhas the limits lim qRP= 1, 6'0

lim qRP= y c+

~ * / y ~ ~ .

(4.10)

Ernst Becker

190 50

t P 10

5

2

1

2

0-5

5

02

10

0

10

20

-

30

b

FIG.6. Dimensionless flow volume for Prandtl-Eyring fluid (qpE)and Reiner-Philippoff fluid ((PRP).

This means that for very small values of 5, and hence of A, the HagenPoiseuille law is satisfied, with q* as the relevant viscosity; for large values of A the Hagen-Poiseuille-law again applies, albeit with the upper Newtonian viscosity qa, as relevant. Several complementary remarks are in order:

(a) By setting r = R in (4.2) we obtain the shear stress at the wall of the pipe: 5, = AR/2. Therefore, the parameter 5 in (4.5) can also be written as

5 = 22,/2*. (b) The function cp[c] can be determined experimentally by measuring the relation between flow volume and pressure gradient A in a capillary viscometer. From q[5] the inverse flow function g [ o ] can be determined by solving (4.7) for g : g [ w ] = o'p[2o]

+ 7o2 p'[2W].

The symbol cp' denotes the first derivative of cp.

(4.11)

Simple Non-Newtonian Fluid Flows

191

TABLE I Model fluid

9 [wl

Rabinowitsch

w

+[51

+ w3

1+-

t2 6

Ellis Reiner (I-")/"

Ostwald-de Waele

(01' "

Bingham

0 for

signw

(Iwl

10

<1

0 forO<[
- 1) signw

for

10

>1

16

3t4

for 5 > l

(c) In many engineering applications the friction factor cf for pipe flow is used. It is defined by the equation A

P-

1 d

= cf - u2 -,

2

(4.12)

where d = 2R denotes the diameter of the pipe, p the density of the fluid, and ii the mean velocity: ii = v/nR2. From (4.12) one obtains (4.13) Defining the Reynolds number Re = pii2R/q,

(4.14)

and using (4.6)for rearranging the last factor on the right-hand side of (4.13), one is led to the result (4.15) The first factor, 64/Re, represents the well-known result for Newtonian fluids (with viscosity v*). The deviation of non-Newtonian pipe flow from Newtonian pipe flow is obviously governed by the function cp [t]. Therefore, this section concludes with Table I, which gives cp for an additional number of fluids, which are characterized by different inverse flow functions q[o].

Ernst Becker

192

control volume

t

I L+I - - .

FIG.7. Simple model for peristaltic pumping.

V. Peristaltic Pumping The result of Section IV is now used to study a simple model for peristaltic pumping of non-Newtonian fluids (Fig. 7). Along a pipe of total length L + 1 a construction of length 1 travels with constant velocity U. The constriction reduces the pipe radius from R , to R 2 < R , . Inlet and exit pressure of the pipe are pi and p e , respectively; inlet and exit pressure of the constricted part of the pipe are denoted by p , and p 2 . Due to the motion of the constriction a fluid volume, will flow per unit time from left to right even if pe > pi. The device therefore acts as a pump. This peristaltic pumping effect plays an important role in many physiological processes of fluid transport, and is also exploited in technology, for example, in “roller pumps.” In the following simple theory of peristaltic pumping, fully developed, and hence viscometric, flow in the wide as well as in the narrow part of the pipe is assumed. Hydrodynamic entrance and exit effects and memory effects in the fluid are thereby neglected. The flow volume in both of the wider parts of the pipe has, of course, the same value and hence the pressure gradient in both these parts is also the same. In order to find a relation among U , pi,perthe geometric parameters R , , R 2 , L, 1, and the fluid properties we start from the identity

vl

v,,

p 1. - p e = p .I - p I + P z - P ~ + P I - P ~ *

vl,

(5.1)

With the definitions

-P1 2 - - -

-P2R2 1 ’

z*

(5.3)

Simple ~Qn-New~Qnian Fluid Flows

I93

(5.1) can be written as (1 + Y)t

=

+ (Y/V)t2?

51

where the abbreviations =

1/L,

v = R2/Rl

have been used. Next, the mass balance for the fixed control volume indicated in Fig. 7 yields

vl v2+ n(R: - R:)U, =

(5.7)

where r', denotes the flow volume in the narraw part of the pipe. Equation (4.6) is now used in a slightIy changed form that is more convenient in the present context:

Using (5.8) one can write (5.7) in the form

$[tll - v3$Ct2I = (1 - v2)x. The newly introduced abbreviation defined as

(5.9)

x is the nondimensionalized velocity U ,

x = r*U/z,R,.

(5.10)

In the problem of peristaltic pumping, the parameters 7, v, 5, and x can be assumed as given. Equations (5.5) and (5.9) then yield tl, t2. From t1the flow volume r', can be calculated according to (5.8). The result can be written in dimensionless form as q=

wl.wJ q [ y , v, < , X I . =

For a Newtonian fluid with viscosity for q assumes the simple form

y~,,

(5.11)

i.e., for I) = t/S, the expression

(5.12)

I ) denoting the global pressure gradient between inlet and exit (Fig. 7). It is easy to understand that 4 and x enter the Newtonian result only through the combination t/x: The flow behavior of an incompressible Newtonian fluid is completely characterized by the viscosity yl*. Therefore, the reference quantity z* must drop out of the result for q. This is the case if the result contains the ratio
FIG.8. Dimensionless flow volume for peristaltic pumping of Prandtl-Eyring fluid.

2,10-2

0

0.5

10

15

-

2.5

t; FIG.9. Dimensionless flow volume for peristaltic pumping of Prandtl-Eyring fluid.

Simple Non-Newtonian Fluid Flows

195

1.o

I 0.6

FIG. 10. Maximum sustainable pressure gradient for peristaltic pumping (Prandtl-Eyring fluid); 7 = 0.1, y = 0.9.

0.1

0.2 0

5

10

__c

x

20

fluid; note that qx = q*vl/(nR:~*). Figure 10 shows the maximum possible value toof the adverse pressure gradient that can be sustained without reversal of the flow direction, i.e., the pressure gradient for q = 0. Of some interest is the pumping efficiency, denoted by p. The work to be done per unit time in moving the constriction against the pressure difference p 2 - p1 is given by n(R: - R$)(p2- pl)U. The useful pumping work per unit time is vl(pe - pi). Hence,

Figure 11 contains results for p(y, v, t, x) for the Prandtl-Eyring fluid. Figures 8-11 are taken from an unpublished paper by W. Ochs (1978). If v = 1, (5.9) yields $[tl] = $[t2] and hence, granted that $[t] is a monotonic function, one has t1= t2. Equation (5.5) then shows that t = t1= t2. This suggests the derivation of simple approximate analytical formula for peristaltic pumping by putting*

t1=t+6

v=l-E,

and expanding with respect to yields

E

(5.14)

and 6. Equation (5.9), together with ( 5 3 ,

$[t + 61 - (1 - ~ ) ~ $ [-( El) ( <

- 6/y)]

=

(1 - (1 - E ) ’ } x .

(5.15)

Performing the expansion mentioned and retaining only terms linear in and 6 leads, after some elementary calculations, to

wcti = E

&

( 2~

ti- 3 m ) .

* This derivation is due to H. Buggisch, who communicated it orally in a seminar.

E

(5.16)

196

1

P

2,102

1

0.1

(13

(17

05 P

b FIG.11. Efficiency of peristaltic pumping (Prandtl-Eyring fluid).

Now using (5.8) for $ and (4.7) for q one verifies easily the relation

3 m i + t $ ~ r= i g~ri21.

(5.17)

Therefore, (5.16) can be written as (5.18) The dimensionless flow volume is given by (5.19) This yields, together with (5.18), q= %

I)];[

($[t1 + +$)(zx - 9

(5.20)

Equation (5.20) is valid for all fluids provided the constriction is small, so that E = 1 - v << 1. The equation can be checked by putting $[
Simple Non-Newtonian Fluid Flows

197

g[t/2] = 5/2 (Newtonian fluid); thereby (5.20) reduces to a formula that is also obtained from (5.12) by linearization in E = 1 - v.

VI. Viscosity Pumps Because no slip occurs between a wall and an adjacent fluid, a moving wall drags fluid along with itself. This effect can be used to pump fluid against an adverse pressure gradient, as shown in Fig. 12, where it is assumed that the pressure gradient is positive, dp/dx > 0, and that the velocity U of the upper wall is large enough to effect a net transport of fluid from the low to the high-pressure region. A technical realization of a viscosity pump, based on this effect, is sketched in Fig. 13. Essentially the same pumping effect is used in screw extruders, which play an important role in the technology of high polymers and plastics; some remarks on extruder flow are found in Section VII.

FIG.12. Simple model for viscosity pump.

inlet

exit

FIG. 13. Viscosity pump.

Ernst Becker

198

The shear velocity IC of the flow shown in Fig. 12 is given by IC = du/dz. Hence the velocity distribution u(z) is determined by the equation

Because each fluid particle moves with constant velocity, the momentum balance reduces to the equilibrium condition V T = 0, which by a single integration leads to a linear distribution of shear stress over the breadth of the flow channel: T = z0 + Az, (6.2) where zo is the stress at the lower wall, and A = dp/dx denotes the pressure gradient. By inserting (6.2) into (6.1) and integrating with the boundary condition u(z = 0) = 0, one obtains

The shear stress T,, at the upper wall is given by The no-slip condition at the upper wall, u(z = h) = U , yields

If the dimensionless quantities

t =W T , ,

x = rl*U/T*h,

T

= TOIT,,

(6.6)

are introduced (6.5) assumes the form

t x = JTT+‘ g [ w ] dw.

(6.7)

This relation yields T, the dimensionless shear stress at the lower wall, as a function of 5 (dimensionless pressure gradient) and x (dimensionless wall velocity): T = T ( ~ , x ) . ‘The flow volume (per unit breadth perpendicular to the flow plane) is given by =

u [ z ] dz.

Using (6.3), performing a partial integration, and taking account of (6.7) finally leads to the nondimensionalized flow volume

Simple Non-Newtonian Fluid Flows

199

If the pressure gradient vanishes, 5 + 0, the integral on the right-hand side vanishes like C3 and (6.9) reduces to q = +,which is the obvious result for pure “drag flow.” The pumping efficiency p is defined as the ratio of useful work VA, performed by transporting per unit time the volume against the pressure gradient A, to the work z,U, expended by moving the upper wall with velocity U against the shear stress q,.Hence (6.10) Specialization of these formulas to a Newtonian fluid with viscosity q*, for which g[o]= w, yields

5

(6.11a)

T=X-Z1

q = - -1- , 5 2 12x

(6.11b) (6.11~)

Inserting the definitions (6.6) for 5 and 2, one obtains from (6.11b) the wellknown formula for the flow volume (6.12) According to (6.11~)the maximum possible value of p is pmax= f . This value is attained for (/x = 2, and hence if T = 0 [cf. (6.11a)l. Therefore, the maximum efficiency is attained for that flow for which the shear stress at the lower wall vanishes. This feature pertains not only to Newtonian fluids but is valid generally, as was shown by Bohme and Nonn (1978); see also the discussion of Fig. 18 below. Another fluid for which the formulas derived above take a simple explicit form is the Prandtl-Eyring fluid. Here,

T

= arcsinh[

“

]

2sinh
q=f{l-[l+( 2

-

(6.13)

2’

)]

5x 512 2 sinh

‘j2

(cothi-:)).

(6.14)

Figures 14 and 15 show q and p for the Prandtl-Eyring fluid in the range of parameters that corresponds to pumping (Becker, 1977). For small values of x the fluid behaves like a Newtonian one, and the maximum possible

Ernst Becker

200 0.5

4f

0.3

0.2 0.I

FIG.14. Dimensionless flow volume for viscosity pump ; Prandtl-Eyring fluid.

0.3

It

0.2

03

0

2

4

6 -

f FIG. IS. Efficiency of viscosity pump; Prandtl-Eyring fluid.

pumping efficiency approaches the value f. For larger values of x the nonNewtonian shear-thinning behavior makes itself felt, and the maximum possible value of u , for fixed x falls below 3. Figures 16 to 18 show analogous results for the Reiner-Philippoff fluid in a wide range of the relevant parameters. These results are taken from the paper by Bohme and Nonn (1978), where T , the dimensionless shear stress at the lower wall, was chosen as the parameter distinguishing the individual curves in the several diagrams. Therefore, Fig. 16 is added here as an auxiliary diagram that shows the

Simple Non-Newtonian Fluid Flows

/’ 2

2

/’

/’

/

I

0

’

-3’

1

I-1;

20 1

-2;

-3;

i

10‘

I

x loo

5

2

16’

2

5

10’

loo

(b)

c

z

l2

FIG. 16. (a]Auxiliary diagram for shear-thinning Reiner-Philippoff fluid. (b)Auxiliary diagram for shear-thickening Reiner-Philippoff fluid.

(b) FIG. 17. (a) Dimensionless flow volume for viscosity pump; shear-thinning Reiner-Philippoff fluid. (b) Dimensionless flow volume for voscosity pump ; shear-thickening Reiner-Philippoff fluid.

203

Simple Non-Ne wtonian Fluid Flows

10-l

2

5

loo

10’

(b)

lo2

-

lo3

5

FIG. 18. (a) Efficiency of viscosity pump; shear-thinning Reiner-Philippoff fluid. (b) Efficiency of viscosity pump; shear-thickening Reiner-Philippoff fluid.

relation between T and the parameters 5, x used in this section. Figure 17 displays qx = Vq,/(r,hz), i.e., a nondimensional measure of the flow volume. Figure 18 shows that the curves for T = 0 are the upper envelopes of the efficiency curves. If the fluid is shear thinning (v],/q, < l),the highest possible value of the efficiency fi is f ; this value is attained for very low and very high values of shear, for which the Reiner-Philippoff fluid behaves Newtonian.

204

Ernst Becker

=-

The shear-thickening fluid (qm/q* 1) has a maximum efficiency above 5. The inset of Fig. 18b shows how ,urn,, increases with increasing ratio rm/q*. As is shown by Becker (1978a), this behavior can be understood qualitatively with the aid of the concept of effective viscosities; the explanation of that concept is the subject of Section VII. Bohme and Nonn (1978) present many more results and diagrams for viscosity pumps. Particularly, their paper also contains curves for negative values of T. These curves correspond to situations with reverse flow (in negative x direction) near the lower wall. Incidentally, if the upper wall (Fig. 12) is at rest, i.e., if U = 0, x = 0, the flow studied in this section is plane channel flow, induced by a pressure gradient 5. Equation (6.7) then yields T = - 5/2 (note that g [ o ] is an odd function!). Multiplying (6.9) by Uh, performing the limit U -,0, and rearranging the result leads to

.

Ah3

- V = -4[5], 12Y*

with #[6] =

24

so

r12

w2g[o]dw.

(6.15)

This formula corresponds to (4.6) for pipe flow. The first factor is the result for the Newtonian fluid; the function 4[5] describes the deviation therefrom due to nonlinear flow behavior. For the Prandtl-Eyring fluid and the ReinerPhilippoff fluid the explicit expressions for Cp are (6.16)

4 , , [ r ] =q*- + , rcc

2 4 (~ * y (2: ~- 1

5

Ym

){:

- - (:m)1-5 12

arc tan[(^)''^;]). (6.17)

VII. Effective Viscosities In many engineering-type theoretical investigations non-Newtonian fluids are treated like Newtonian fluids with a value of viscosity that is thought to approximate the flow behavior in the respective situation. This procedure can be justified, within certain limits, as shown now (Becker, 1978a). For that purpose we study flow between two parallel plates, z = 0 and z = h, with the velocity vector everywhere parallel to the plates and depending on z only (Fig. 19a):

+I,

VCZ3 = (u[z],

0).

(7.1)

Simple Non-Newtonian Fluid Flows

205

2

r

7 X

FIG.19. Flow field between two parallel walls.

Such a viscometric flow can be generated by moving one of the bounding walls with constant velocity within its plane and/or applying a pressure gradient in the direction parallel to the walls. Without loss of generality we can assume that the upper wall ( z = h) moves with velocity U in the x direction. The components of the pressure gradient in the x and y directions are denoted by A, and A,, respectively. The flow studied in Section VI is the special case of the flow discussed now, for which Ay = 0, A, = A. We define a shear vector as K = dvldz.

(7.2)

The shear velocity in the sense used earlier is given by

The shear stress vector in the planes z

= const

is then

(7.41 Here, Z(K) is the shear stress, which is determined by the shear velocity through the flow function (2.5). Equation (7.4) generalizes the ideas discussed in Section II,A to flow fields of the form (7.1); the generalization is obvious and therefore taken as granted without further discussion. Let us change the shear vector by a small increment: z = Z(K)K/K.

K + K + d K ,

K+K+dK.

(7.5)

206

Ernst Becker

Such a change can be effected by a slight change of the pressure gradient and/or the velocity of the upper wall. By differentiating (7.4) we obtain

(7.6) Writing BK = 6rcll + B K ~ where , 6~~~and 6xc,denote the increments of BIC in directions parallel and perpendicular to IC, we can put (7.6) into the form

] the differential Here, q [ ~ is] the viscosity in the usual sense and q ’ [ ~ denotes viscosity; they are defined as

q’[K] = ~

z [ K ] / ~ K .

(7.8)

For Newtonian fluids both viscosities are equal and (7.7)reduces to 6z = q 6 ~ . Equation (7.7) shows that for a slight change of the shear vector in its own direction the change of shear stress is the same as in a Newtonian fluid with viscosity q’, whereas for a change perpendicular to the direction of the shear vector the relevant viscosity is q. If the pressure gradient A = (A$ + A;)1’2 is sufficiently small, its effect can be assumed to be a small disturbance only of the basic shear flow with shear velocity K = U/h. The two viscosities r] and r]’, defined by (7.8), are then, to a first approximation, independent of z because the shear velocity is approximately given by U / h for all values of z. In that case the fluid flow can be said to be effectively Newtonian, although it reacts with two different viscosities to slight changes parallel and perpendicular to the basic shear flow direction. An early mention of the two viscosities is due to Pipkin (1968); however, no further use was made of them. It is noteworthy that, to the best of this author’s knowledge, the separate role of these two viscosities has never been recognized in the engineering literature on non-Newtonian fluid flow, although the values of r] and q’ can differ markedly and although this difference is the cause of a number of technologically relevant effects. A first indication of the aptness of that statement is provided in Section VIII. As an aside it may be remarked that the differential viscosity is important for the Orr-Sommerfeld instability of non-Newtonian fluid flow. Because q‘ can be much smaller than q for shear-thinning fluids, a destabilization is to be expected for such fluids, which together with the influence of normal stress and memory effects may lead to instability even at extremely low Reynolds numbers. This important effect provides a probable explanation of “viscoelastic turbulence,” or “melt fracture,” and is borne out by as yet tentative and unpublished investigations. However, communication of detailed results and a pertinent discussion go beyond the scope of the present chapter.

Simple Non-Newtonian Fluid Flows

207

FIG. 20. Viscosity ratio for Reiner-Philippoff fluids.

The dimensionless viscosities q/q* and are functions of dimensionless shear velocity ~ * K / z or, , equivalently, of T/T*. Hence, for small pressure gradients the dimensionless viscosities depend on x = q*U/z,h. For a Prandtl-Eyring fluid the two viscosities are given by ?I/?*

(7.10) The ratio q'/q has value 1 for IC = 0 and tends to zero for IC + 00. For a fluid that behaves again Newtonian for high shear velocities the limit of q'/q has value 1 for both K = 0 and K -+ co. An example is the ReinerPhilippoff fluid, for which numerical data are shown in Fig. 20.* * For qm/q* > 9 the function g [ w ] for a Reiner-Philippoff fluid [cf. (2.9)] is no longer monotonic. Therefore q' becomes infinite and even negative for certain values of r/z*. Whether this reflects possible behavior of real substances or whether the simple expression (2.9) is not usable for q,,/q, > 9 is a question left open here. Nevertheless, Figs. 17b and 18b are thought to reflect correctly the behavior of shear-thickening fluids in viscosity pumps although the value qrn/q* = 10 is already slightly above the critical value 9. It might be remarked that there are some indications that for certain shear-thinning fluids q' might fall below zero with corresponding instability and hysteresis of the flow behavior (Uhland, 1978). The concept of differential viscosity is, of course, only useful as long as q' remains finite and positive.

Ernst Becker

208

After these preliminaries we return to the flow sketched in Fig. 19a. By transforming to a frame of reference that moves with velocity U/2 in the x direction the situation depicted in Fig. 19b is obtained. The flow volumes (per unit breadth) in the x and y directions are denoted by and in the original frame of reference and by Q, and Q, in the new frame of reference. The definition of these quantities is

v,

<

v, = fo” U ( Z ) ~ Z= Q, + Uh/2,

~(z)d= z Q,.

=

<

(7.11)

The dimensionless parameters

5,

= A,hJT*,

5, = A,hIT*?

x = 9*U/Q

(7.12)

are now introduced. The nondimensionalized flow volume components Q, and Q, are qx = Q,P, qy = Q,IUh.

The quantities q,, qy are functions oft,, properties: qs~-5x,ty,xI

(7.13)

t,, x; they have obvious symmetry (7.14)

= -qxCtx,5y,xI,

-5,Jl = 4x[5x,5y,xI, 4v[ - 5x9 t y l XI = q y [ L ry,XI, qy[<x, -5,,Xl = -qy[5x,ty,xl.

(7.15)

4X[5X?

(7.16) (7.17)

Equation (7.14) states that reversal of the pressure gradient in the x direction reverses the flow volume in the x direction. This is obvious because reversal of the pressure gradient in the x direction is equivalent to rotating the whole flow field by 180” about the y axis, which reverses the flow in the x direction. Equations (7.15)-(7.17) can be explained by analogous reasoning. As a consequence of these symmetry properties the expansions of qx and 4, with respect to t, and 5, start with the terms

+ a2535, + a352 + . . q y = ( b , + b2535, + b35; + . .

4,

= (a1

*

. 9

(7.18)

*

(7.19)

The coefficients a,, bi are odd functions of x. By neglecting terms that are of third or higher order in t,, tY,one obtains linear relations between flow volume and pressure gradient in the respective direction. Because of the absence of quadratic terms in (7.18) and (7.19) it is to be expected that these linear relations are a good approximation even for moderately large values of Cx, 5,. This guess is corroborated, for example, by Fig. 14, which shows

Simple Non-Newtonian Fluid Flows

209

that the graph of q as a function of t does not differ very much from a straight line for most values of a!. The usefulness of the concept of effective viscosities q and v’ depends to a large degree on that fact! For a Newtonian fluid (with viscosity q = q’ = q*) the relations between q x , qy and t,, 5, are exactly linear and are given by q x = -tx/12a!,

q y = -ty/12x.

(7.20)

[Note that only the combinations 5,/x and t,,/x enter; this has already been explained in connection with (5.12).] The previous discussion of the role played by the viscosities q and q’ makes clear that these relations remain valid for non-Newtonian fluids under neglect of third- and higher order terms in t,, t,, provided that v]‘ is substituted for the viscosity y ~ * in the expression for q x and q in the expression for q y . In other words, instead of a! = q*U/z,h we have to substitute (?’/?*)a! = q‘U/z,h in qx and (q/q*)x = v]U/t,h in q y .This yields (7.21) Incidentally, it is easy to verify the expression for qx by expanding the result (6.9) with respect to and keeping the linear term only [note that in (6.9) q = q x + 1/2, t = t,]. Transforming back to the original frame of reference (Fig. 19a) and using dimensional quantities again we obtain from (7.21)

<

(7.22) A simple application of these formulas is discussed in Section VIII. Before turning to these applications, a helpful and plausible remark is in order: The velocity components u, v of the flow sketched in Fig. 19a are, to first order, given by the Newtonian expressions with the appropriate viscosity r] resp. q ’ : (7.23) v[.]

A h2 z

z2 h2)*

= -Y _ - _

1 2 ~( h

(7.24)

The neglected terms are of second order in the pressure gradient. However, these second-order terms have the property that their integrals over z from 0 to h vanish because second-order terms do not contribute to the flow volume, as we have seen above.

Ernst Becker

210

R I _

FIG.21. Screw extruder.

VIII. Extruder Flow Extruders, as they are widely used in the technology of plastics, are essentially viscosity pumps (Fenner, 1970; Han, 1976; Tadmor and Klein, 1970). Molten plastic is transported against a pressure gradient through the helical channel of a screw rotating with angular velocity R in a fixed casing (Fig. 21). The cross section of the channel is usually rectangular. If h/R << 1 the flow in the curved channel can be approximated by flow in a straight channel (Fig. 21, lower part); a is the helix angle. If, furthermore, h/l<< 1 the flow in the straight channel can be further approximated by the viscometric flow studied in Section VI. The deviation from that flow due to the sidewalls, or flanges, is restricted to a neighborhood of the flanges with spatial extension of magnitude h and therefore can be neglected. Yet, the influence of the flanges makes itself felt insofar as they are impervious to the flow. Therefore, the relation

V; = t a n a t

(8.1) must be satisfied between the flow volumes in the x and y directions. Insert% and rearranging yields ing the expressions (7.22) for

vx,

A,, = tan 01 (Ax

-6

$).

The viscosities q and q' are determined by the basic shear velocity U / h (with U = RR). Equation (8.2) is a relation between the pressure gradients in the x and y directions. The pressure gradients parallel and perpendicular to the

Simple Non-Newtonian Fluid Flows

21 1

flanges are given by

+ A ysin a, A, = -AI\,sina + Aycosu. A,

= A, cos u

With the aid of (8.2) one derives the relation

between A, and A,, (Becker, 1978a). The dimensionless quantities

t s = Ash/r*,

5,

= AnhIr,,

x = ~*Ulz*h

(8.6)

have been used here. For a Newtonian fluid, q = q' = q,, and (8.5) reduces to 5, = - 6 ~ s i n u . In that case the pressure gradient perpendicular to the flanges is independent of the gradient parallel to the flanges. Knowledge of these gradients is a prerequisite for calculation of the leak flow in the narrow slit between the flanges and the casing (Winter, 1979). The flow volume parallel to the flanges (again per unit breadth) is given by

<

= V, cos a +

V; sin a.

Using these results and defining the dimensionless flow volume as 4, = l$-Jh,

we obtain by elementary algebra

These simple and useful results can be applied to extruders only if the values of ltsland Itn\are sufficiently small, because (7.22) are valid for small values of the dimensionless pressure gradient only. Equation (8.5) shows that, even if I(,) is small, can attain appreciable values if the angle CI is not small and if xq/q.+ is appreciable. Therefore, application of (8.9), for example, will generally have to be restricted to small values of a. In Fig, 22 the flow volume according to (8.9) is compared, for a Rabinowitsch fluid (see Table I), with exact numerical values (Ludwigs, 1978). The increasing deviation of (8.9) from the exact result for increasing u is clearly visible. For a more detailed theoretical investigation of extruder flow the reader is referred to a recent doctoral thesis (Schweinfurth, 1979).

\<, I

Ernst Becker

212 036

t U

C ._ m

cn U

0.08

//I

I

ff-50

0.OL

zoo

5

10

' 25

15

f;,/sinu FIG.22. Dimension1 s flow volume for screw extruder; (-) wigs (1978), (---) approximate result according to (8.9).

numerical result of Lud-

IX.Nearly Viscometric Flow The kinematics of viscometric flows is rather restricted; in many applications viscometric flow does not prevail. However, frequently the flow is at least approximately viscometric. This applies, for example, to the plane, steady flow in the gap between a plane wall, x2 = 0, moving with constant velocity U,and a wavy wall, x2 = h ( x , ) = b + e cos 271xl/l, provided that e/b << 1 (Fig. 23). If e = 0, the flow is simple shear flow with shear velocity

I

-

I

L

FIG.23. Flow between moving straight wall and fixed wavy wall.

Simple Non-Newtonian Fluid Flows

213

U/b. For e > 0 the flow is periodic in the x1 direction, with wavelength 1, and it is close to simple shear flow, and hence nearly viscometric, if e/b << 1. Even if r/b is not small, but if b/l<<1, the flow can still be considered nearly viscometric, provided that the characteristic time 1 of the fading memory of the fluid (see Section 111) is small compared with the time l/U, which is of the order of the time needed by a fluid particle to travel through one wavelength in the x1 direction: XJ/l<<1. If b/l<< 1 (narpow gap) the deformation of a fluid particle is predominantly a shear deformation and if, in addition, AU/l<<1 (short memory), the shear velocity of a particle will not change appreciably during a time span of the order of its memory time. Therefore, the particle will respond to its deformation with nearly the same stress as in simple shear flow, where the shear velocity is exactly constant. The flow in a narrow gap of changing width is of boundary layer type because changes of the flow variables in the x2 direction are much more pronounced than changes in the main flow (xJ direction. The concepts of nearly viscometric flow are applicable to other boundary layer-type flows if the memory time of the fluid is sufficiently short (Section X). The idea of nearly viscometric flow was first expounded by Pipkin and Owen (1967). Their fundamental paper is rather complicated, and no applications of importance seem to have been made until recently. A first attempt to apply the theory of nearly viscometric flow to plane boundary layers was made by Diehn (1974, 1975). Recently, flow in a slightly eccentric journal bearing has been studied as an example of nearly viscometric flow (Akbay, 1978; Akbay and Becker, 1979); this corresponds to the flow situation in Fig. 23. In this section a brief exposition of the ideas of Pipkin and Owen is given; the restriction to steady plane flow allows considerable simplifications in comparison to the treatment by Pipkin and Owen (1967). Section X shows how these ideas can be further simplified and thereby be brought into an easily applicable form, for boundary layer flows. For steady, nearly viscometric, plane flow the RRCG-tensor C (see Section 111) is split into its viscometric part C', pertinent to the basic viscometric flow, and a deviation E therefrom: K =

C = C'

+ E.

(9.1)

In all the applications to be discussed in the rest of this chapter the basic viscometric flow is simple shear flow, as in Fig. 23, with shear velocity K ; therefore,*

* Because of the restriction to plane flow, only the 2 x 2 matrix representations of the relevant tensors need be considered.

Ernst Becker

214

FIG. 24. Simple shear flow in two different coordinate systems.

The tensor E, with elements E l l , El, = E 2 1 , E , , , is assumed to be small compared with the viscometric part C’ of C.The concept of smallness can be quantified by using the norm of fading memory mentioned in Section 111. To avoid a lengthy discussion of that point (see, e.g., Becker and Burger, 1975; Truesdell and Noll, 1965), we must be satisfied here with the statement that in the example of Fig. 23 the tensor E is proportional to the geometric parameter e/b and therefore becomes small for small values of e/b. The stress tensor of the fluid is also split into its viscometric part and a deviation. The only relevant stress quantities in plane incompressible flow are the normal stress difference T,l

- T,2

= 01[IC2]

+ o’,

(9.3)

and the shear stress 21,

= T [ K ] f z’

(9.4)

where O~[IC’] and ~ [ I c ]are the values of the viscometric functions pertaining to the basic viscometric flow with shear velocity IC. The theory of nearly viscometric flow is linearized with respect to E. The linearization has two immediate consequences:

1. The three elements E l l , El, = E , , , E,, of E are linearly dependent on each other. This is simply a consequence of the density preservation of an incompressible fluid, which yields det C = 1 (cf. Section 111). Using (9.1) for C and (9.2) for C’, expanding det C and retaining only terms linear in E , one obtains, for example, E l , in terms of El, and E 2 , as E,, Ell = -

+~ICSE,,

1 + IC2s2

’

(9.5)

2. The additional stress terms o’,z’ [(9.3), and (9.4)] are linear functionals of E. Under weak and scarcely restrictive conditions these functionals may

Simple Non-Newtonian Fluid Flows

215

be written as integrals

t' =

som

(2B12E12 + B,,E,,)ds.

Here, we have indicated that the four kernel functions A , , , A , , , B , , , B,, depend on the basic viscometric flow, and hence on the shear velocity K of that flow. The elements Ei, depend on x. Of couse, the kernel functions and the elements Eik are furthermore functions of the time variable s extending from the present instant (s = 0) into the past (s > 0). Because only two elements of E are independent of each other, only these two enter the integrals; the factor 2 in the integrands has been introduced for convenience only. It is evident that the kernel functions have to vanish for s -+ 00 in order that the integrals have appropriate convergence properties. Vanishing of the kernel functions is a manifestation of the fading memory of the fluid. The four kernel functions are not completely independent of the viscometric functions T [ K ] and a , [ ~ ~or] v,[ K ' ] , and N , [ K ' ] ; they must statisfy four compatibility conditions (Akbay, 1978; Akbay and Becker, 1979; Diehn, 1975; Pipkin and Owen, 1965). Because only two of these conditions are needed in the following sections, only these two are derived here. For that purpose we take simple shear flow in the x1 direction with constant shear velocity (Fig. 24). The same shear flow can be assessed from an ,:x x z system that is rotated by the small angle a against the x I , x, system. In the x:, x t system the flow can be considered a simple shear flow in the x: direction, with shear velocity K, upon which a disturbance proportional to a is superposed. Therefore, we write the RCCG tensor in this system in the form C* = C' + E, (9.8) with C' given by (9.2). If we denote by Q the matrix that transforms from the x , , x, system to the x:, x z system, we can write C* as

C* = QTCVQ,

(9.9)

because C' is the RCCG tensor in the xl,x , system. In the approximation, which is linear in a, Q is given by Q = ( -a

'). 1

(9.10)

Inserting (9.10) into (9.9) and neglecting higher order terms in a leads to

E l , = -uK's',

(9.11)

E,, = - 2aKS.

(9.12)

Ernst Becker

216

The stress tensor in the x l , x 2 system has components z l l , z12 = ~ 2 1 7, 2 2 , with z12 = Z [ K ] and z l l - zZ2 = nl[~’]. The stress components in the x:, xf systems, i.e., zY1, zf, , are given by the transformation formula

T* = QTTQ.

(9.13)

Calculation of z$ according to (9.13) yields in the a linear approximation zT1 - z:2 = t l l - ,z, 772

= 212

- 4az12,

+ a b l l - 722).

(9.14) (9.15)

This is equivalent to g‘ =

-4az = - 4 a 1 4

z‘ = an1 = aNlrc2.

(9.16) (9.17)

Using (9.6) and (9.7) for 6‘ and z’, together with (9.11) and (9.12) for El, and E z z , finally leads from (9.16) and (9.17) to the compatibility conditions Jom

+ A,,s)ds = 21,

(A,,KSZ

(9.18) (9.19)

X. Plane Boundary Layer Flow of a Fluid with Short Memory The theory of nearly viscometricflow can be further simplified by assuming that the fluid has short memory, such that in a steady plane flow field the motion of a fluid particle during a time span of the order of the memory time A may be approximated with sufficient accuracy by x i [ t - s]

= xi[t] - ui[x[t]]s

+ +ai[x[t]]s2.

(10.1)

Here, x i [ t ] denotes the position of the particle at time t, x i [ t - s] its position at the previous time t - s; ui and ai in (10.1)are the velocity and the acceleration components of the particle at time t. In steady flow the acceleration is the “convective” acceleration ai = v k u i , k .

(10.2)

For simplification of the notation the comma notation for partial derivatives has been introduced here: u i , k = dvi/dxk.The elements of the relative deformation gradient F, defined by (3.1), are easily calculated from (10.1) as Fik

= dik

- Vi,$

+ fai,ks2.

(10.3)

Simple Non-Newtonian Fluid Flows

217

FIG.25. Boundary layer flow.

Hence, the elements of the RCCG tensor, Cik= F l i F l k , are, up to order s2, given by cik

= 6 , - (0i.k

+

uk,i)s

+

i(ai,k

+

ak,i

+ 20e,iue,k)s2.

(1 0.4)

is much larger in magniIn boundary layer flow (Fig. 25) the derivative tude than the other first-order derivatives of the velocity components. Therefore, we define at every point xl, x2 an “osculating” simple shear flow with shear velocity =

aul/ax2.

(10.5)

This shear flow is the basic viscometric flow, from which the real flow deviates only slightly in the vicinity of the reference point xl, x 2 . Of course, the osculating shear flow is usually different for different reference points because the derivative u1,2 usually depends on xl, x2. Subtracting the RCCG tensor C‘ of the basic shear flow (9.2)from C given by (10.4)one obtains El2 = -02,1s E22 =

-&s

+ (u1.1v1.2 + u2,1%,2 + &+ + ( 6 . 2 + a2,2)s2.

h2,1)s2,

(10.6) (10.7)

At this stage the boundary layer approximation is invoked: Denoting the characteristic velocity and characteristic length in the x1 direction by U and I, respectively, and the characteristic length in the x2 direction by b (see Figs. 23 and 25), one verifies that only the underlined terms in (10.6) and (10.7) contain the geometric ratio b/l linearly, all the other terms being of higher order in b/l. Because for boundary layer-type flow b/l << 1, we

218

Ernst Becker

keep only the underlined terms.* Hence the following theory can be considered to be an expansion with respect to powers of b/!, which in view of the boundary layer assumption is restricted to the linear terms. As explained presently, the short-memory assumption is invoked to justify further neglect of terms that are of order higher than first in UA/l. Substitution of the underlined terms for El, and E z 2 into (9.6) and (9.7) yields

+ = 1om(2JCs2A12+ 2SA2,)dS

+ B12s2ds UI,, + SomA1,s2ds

z’ = JOm (2m2B12 2sB2,)ds * u l , l (T’

*

JOm

*

a,,,,

(10.8)

u1,2.

(10.9)

(Note that IC = u ~ , ~ In . ) view of the compatibility relations (9.18)and (9.19), this reduces to z’ =

-JCN1u1,1

= 4qul,,

(T’

+

(10.10)

h , 2 ,

+ aa,,,.

(10.11)

The abbreviations a[.]

=

1; A12s2ds,

B[K] =

1; BI2s2ds

(1 0.12)

have been used here. Further simplification is achieved by the following three-step argument: (a) We note that the orders of magnitude of the quantities and appearing in (10.10)and (lO.ll),are JC

= O(U/b),

u ~ =,O(U/I), ~

JC

=

= O(U2/bI).

u,,,,

(10.13)

(b) The ratios N1/q, a/q, p/q have the dimension of time and therefore should be of the order of the memory time of the fluid; hence N,,M,B = Wrl).

(10.14)

(c) In the equations for plane boundary layer flow only the stress derivatives az,,/dx, and a(zl1 - Z , ~ ) / ~ play X ~ a role (see Section XI). Using (10.13) and (10.14)together with (10.10)and (10.11)one confirms the following statements: The leading term in dz12/ax2,i.e., the viscometric term az/ax2, is of order qU/b2. The other terms, i.e., az‘/dx2, are of order (qU/b2)AU/I.The x , ,the viscometric term aol/ax,, is of leading term in d(zll - z ~ ~ ) / ~ i.e.,

* For the sake of completeness it is mentioned that the terms of third or higher order in s, which are neglected in the short-memory approximation,contain b/l only in order higher than linear.

Simple Non-Newtonian Fluid Flows

219

order ( q U / b 2 ) X J / l .The other two terms, i.e., dd/axl, are of order (qU/b2)(b/l)AU/l.Therefore, if one retains only terms linear in b/l (boundary layer, or narrow-gap assumption) and terms linear in UA/l (short-memory assumption), 0’has to be dropped completely. The constitutive equations for plane, steady boundary layer flow of a fluid with short memory are therefore reduced to Z , =T[K] -

z,

au p--,aa ax aZ

(10.15)

with

(10.16)

N I K - -k

- z,, = G ~ [ I c ~ ] ,

K = aqaz.

Here we have reverted to a conventional notation (xl -,x, x2 -+ z, o1 -, u, u2 + w, a , -+ a ; see also Fig. 25), which is more convenient for the applications to be made of (10.15) and (10.16).Z[K] and o , [ K ’ ] = K ’ N , [ K ~are ] the viscometric functions. It is noteworthy that apart from the viscometric functions only one additional material function of K , namely, / ? [ K ] , enters the constitutive relations (10.15) and (10.16)! These constitutive relations lend themselves easily to diverse applications; one such application is the subject of Section XI. Concluding this section we remark that (10.15) and (10.16) are valid also for unsteady boundary layer flow, provided the flow changes only slowly with time on a time scale determined by the memory time of the fluid: Alto << 1.

XI. Journal Bearing Figure 26 is a sketch of a journal bearing. An inner cylinder of radius R rotates with circumferential speed U within a stationary outer cylinder of slightly larger radius R + b. The axes of the two cylinders are a distance

FIG.26. Journal bearing

220

Ernst Becker

e < b apart. The narrow gap between the two cylinders is filled by a nonNewtonian liquid lubricant. An important aim of the theory of lubrication is the calculation of the force acting on the rotating cylinder as a function of the geometric parameters, the speed of rotation, and the rheological properties of the lubricant (Akbay, 1978; Akbay and Becker, 1979; Becker 1978a,b). Assuming b / R << 1 (narrow gap), one can neglect effects of curvature of the flow channel on the motion of the liquid. The flow can therefore be assumed the same as in the plane channel sketched in Fig. 23. The length I is identified with the circumference 271R of the rotating cylinder. Furthermore, for b / R + 0, effects of fluid inertia vanish. Therefore, the motion of the fluid in the narrow gap is governed by the equilibrium conditions aP --

ax

+ -= 0,

(11.1)

aZ

For convenience the pressure p has been normalized in such a way that it coincides with the normal stress in the x direction. Elimination of p by cross differentiation yields

(11.3) The third term is of order b2/RZsmaller than the first term and therefore is neglected. The remaining equation can be integrated twice, with the result (11.4) where zo(x) is the shear stress at the lower (moving) wall ( z = 0) and A ( x ) is given by (11.5) Because p condition

+ zxx - ,z,

is periodic in x with period 2nR, A must satisfy the JOZnR

A [ x ] dx = R

SoznA[9] d9

= 0,

(11.6)

with 9 = x/R. The force on the rotating cylinder (per unit length in axial direction) is split into a component FIIparallel to the displacement e of the

Simple Non-Newtonian Fluid Flows

22 1

two centers and a component F I perpendicular to the displacement (Fig. 26). The two components are given by F I I= R

Jo2'

4 = R Jo2n

+ ( p +,,z ( p ,,z

- zzz) cos 9 d9 = - R2

so2'

A[$] sin 9 d9, (11.7)

- t z zsin ) 9 d9 = R 2 Jo2n A [ $ ] cos 9 d9.

(11.8)

Obviously, the determination of the force components necessitates the calculation of A[9]. For calculating A[$] we use as starting point the constitutive relations (10.15)and (10.16);with the aid of these relation (11.4)can be transformed into (11.9) where z' denotes the deviation from the viscometric shear stress z[K], which according to (10.15) or (10.10) is given by au aa z'= - N , K - + p - . ax az

We now use (2.7) for the relation between (11.9) we obtain

K = au/az

(11.10) and z; together with

In order to obtain simple analytical results the calculations are from now on restricted to small values of the eccentricity by the assumption that (11.12) e/b K 1. The value E = 0 corresponds to simple shear flow, with A = 0, z' = 0, Ba,/ax = 0. Therefore, (11.11)can be expanded for small values of E as E =

where g' denotes the first derivative of g. Retaining only the first term on the right-hand side is equivalent to neglecting memory effects, because z' and crl are proportional to the memory time [cf. (10.14)]. The solution of this first-order problem for A[$] is now sought. Then this solution is inserted into the second term on the right-hand side of (11.13) and a correction of order UA/R, due to memory effects, is thereby calculated. The determination of A[$] for the case of neglected memory is easy when one uses the concept of effective viscosities (Section VII): The constant flow

222

Ernst Becker

volume at every position x in the gap is given by (7.23),with A, = A. Hence, (11.14)

Thereby a constant ho has been defined; q’ is the differential viscosity belonging to the shear velocity - U/b of the basic shear flow. Solving (11.14) for A, using h

= b(l

+ ECOS~),

and taking (11.6) into account yields, to first order in in that approximation, A is given by A=-

61’U & cos 9. bZ

(11.15) E,

h,

=

b. Therefore, (11.16)

From (11.8) one obtains then F , = 6nq‘U(R2/b2)&,

(11.17)

whereas (11.7) yields FII= 0. These results are completely analogous to the results of the classical Reynolds-Sommerfeld theory for Newtonian lubricants, the only difference being the appearance of the differential viscosity q’, instead of viscosity q, in (11.17). It is customary to introduce the nondimensionalizedforce, the Sommerfeld number, which is defined as SO= F,b2/R2Uq, = 671~ri/q,.

(11.18)

For a Newtonian fluid (11.18) reduces to the well-known result So = 6 n ~ . For a Prandtl-Eyring fluid (11.18) yields (Becker, 1978a,b)

SO = 6 n ~ / ( + l x2)l”.

(11.19)

Here x = q*U/T,b; see (6.6) or (7.13). In the limit x + co, So + 6m/x, or, F,

+

6net,R2/b.

(11.20)

In that limit the force acting on the rotating cylinder becomes independent of the rotational speed U , in marked contrast to the situation for a Newtonian lubricant where F , always increases with U . For further discussion the reader is referred to Becker (1978a). The calculation of the correction due to the finite memory of the fluid is somewhat lengthy and is not communicated in detail. The main steps of the procedure are as follows:

Simple Non-Newtonian Fluid Flows

223

At the outset one notes that according to definition (7.8) of differential viscosity q’ and in view of (2.7) I/?’ = d K / d Z = g’/V,.

(11.21)

Therefore, the first factor of the second term on the right-hand side of (11.13) is equal to l/$. In the &-linearapproximation it is sufficient to use for q-’ the value belonging to the basic shear velocity - U/b. Furthermore,

Here, again within the bounds of the linearization in E, the dependency of do,/dK on z could be neglected; do1/dK can be approximated by its value for K = - U/b. With (11.21), (11.22), and (11.10) one can write (11.13) as au

z*

(11.23)

?*

The coefficients M and B have the meaning M = -?,( 1 - N l ~ + ~ ) = ~ ( N l + B~= ?~B. ) (11.24) ,

They are determined by their values for K = - U/b. Integrating (11.23) once and taking account of au/dx = -aw/dz (equation of continuity) and of the evident boundary conditions a(z = 0) = 0, u(z = 0) = U,w(z = 0) = 0, we obtain (11.25) At the upper wall, z = h, the no-slip condition u Therefore, from (11.25) one concludes that

= w = 0 has

to be satisfied.

(11.26) This condition is exactly the same as the condition one obtains by neglecting the terms Mw and Ba a priori, i.e., by neglecting the memory of the fluid. Therefore, the relation among U , h [ x ] , zo[x], given by (11.26), is the same as in the first-order theory, which neglects memory. Consequently, the velocity u is given by

224

Ernst Becker

The terms within braces are derived from the terms within braces in (11.25); they are the result for a fluid with neglected memory and are therefore given by (7.23),with a difference in the first term because the lower instead of the upper wall is moving now. Integration of (11.23) yields the flow volume

v.

(11.28) [For the terms within braces compare with (7.22).] For evaluating the two integrals in (11.24) we must first calculate w [ x , z ] and a [ x , z ] . Here, we note that the first-order result for u, i.e., the underlined expression in (11.23), can be written, in view of (11.16), as u = u ( 1 -;)+3uEcos9(;-;).

(11.29)

This expression for u is used for calculating w from the equation of continuity, au/dx = - aw/az, and u from the identity u = u au/ax + w au/az [cf. (10.2)]. Inserting the results into the two integrals in (11.24), solving for A, and satisfying condition (11.6) finally yields

A = 6q’U - cos 9 + b2

UEsin 9

E

~

b2

The second term takes the memory effects into account. The result (11.30) can now be used to calculate the force components from (11.7) and (11.8). The force component F I is unchanged and again given by (11.17). The component FIIis now different from zero and has the value (11.31) We note again that N1 and 8 are determined by K = - U/b.Taking account of (10.14) one easily verifies that F , , / F , = O(UA/R). The result (11.31) agrees with the first term of an expansion with respect to eccentricity E of (8.4-17) given by Bird et al. (1977). Though (8.4-17) by Bird et aI. holds for arbitrary eccentricities, it is restricted to seconcjorder Rivlin-Ericksen fluids (see, e.g., Becker and Burger, 1975; Bird et al., 1977). For such fluids one has N 1 = -28 = const (=pl in the notation and y ~ ’= q = const. The present results (11.17) and (11.31) of Bird et d.), hold for quite arbitrary fluids, provided the eccentricity of the bearing is small and the memory of the fluid is short. The restriction to small eccentricity was made here only for the purpose of simplifying the analysis and for obtaining simple analytical results; the restriction can be lifted without principal changes in the calculation procedure explained above. In view

Simple Non-Newtonian Fluid Flows

225

of the agreement between (11.31) and (8.4-17) of Bird et al. (1977) in the corresponding range of overlap, it may be expected that (11.31) can be extrapolated with some confidence beyond the validity of the short-memory assumption, since this assumption has not been explicitly used in deriving (8.4-1 7). In the literature on the lubrication of bearings, the periodicity condition (11.6) is usually called the Sommerfeld boundary condition. The solution found with this condition is relevant for weakly loaded bearings or for pressurized bearings. In many situations of technological importance the bearings are neither weakly loaded nor pressurized. For such cases the so-called Reynolds boundary conditions are thought to provide a better approximation to the real behavior of the lubricant than the Sommerfeld condition (see, e.g., Cameron, 1966). The theory described in this section can also be adapted to the Reynolds conditions. Thereby only details of the calculation are changed, the principal ideas explained above remain unchanged. REFERENCES AKBAY,U. (1978). Fastviskometrische periodische Stromungen im Ringspalt zwischen zwei Kreiszylindern. Dissertation D 17, Darmstadt. AKBAY,U., and BECKER,E. (1979). Fastviskometrische Gleitlagerstromungen. Rheol. Acta 18, 217-228. BECKER, E. (1977). Schleppstromungspumpe fur nicht-newtonsche Fluide. Mech. Rex. Comrnun. 4,235-240. BECKER, E. (1978a). Die effektive Viskositat bei viskometrischer Druck-Schleppstromung mit Anwendung auf Extruder und Gleitlager. 2. Werkstofltech. 9,452-459. BECKER,E. (1978b). Nicht-newtonsche Effekte bei der Gleitlagerstromung. Z . Angew Math. Mech. 58, T245 -T246. BECKER, E., and BURGER, W. (1975). “Kontinuumsmechanik.” Teubner, Stuttgart. BIRD,R. B. (1976). Useful non-Newtonian models. Annu. Rev. Fluid Mech. 8, 13-32. BIRD,R. B., ARMSTRONG, R. C., and HASSAGER, 0. (1977). “Dynamics of Polymeric Liquids,” Vol. I. Wiley, New York. B~HME G., , and NONN,G. (1978). Kennfelder fur Schleppstromungspumpen zur Forderung nicht-newtonscher Fliissigkeiten. Rheol. Actu 17, 81-97. CAMERON, A. (1966). “The Principles of Lubrication.” Longmans, Green, New York. CHRISTMANN, L. (1977). Untersuchungen zur molekulargewichtsinvarianten Darstellung des rheologischen Verhaltens von hochpolymeren Schmelzen. Dissertation D 17, Darmstadt. H., and NOLL,W. (1966). Viscometric Flow of Non-Newtonian COLEMAN, D. B., MARKOWITZ, Fluids. “Berlin and Springer-Verlag. New York. DIEHN,T. (1974). Grenzschichten einfacher Fluide mit kurzem Gedachtnis. Dissertation D 17, Darmstadt. DIEHN,T. (1975). Grenzschichten von einfachen Fliissigkeiten mit kurzem Gedachtnis. 2. Angew. Math. Mech. 55, T119-Tl22. FENNER, R..T. (1970). “Extruder Screw Design.” Iliffe, London. HAN,C. D. (1976). “Rheology in Polymer Processing.” Academic Press, New York.

Ernst Becker HELLWEGE, K. H., KNAPPE,W., PAUL,F., and SEMJONOV, V. (1967). Druckabhangigkeit der Viskositat einiger Polystyrolschmelzen. Rheol. Acta 6 , 165-170. HUILGOL,R. R. (1975). “Continuum Mechanics of Viscoelastic Liquids.” Hindustan Pubi. Corp., New Delhi. LEIGH,D. C. (1968). “Nonlinear Continuum Mechanics.” McGraw-Hill, New York. LUDWIGS,J. (1978). “Berechnung von Kennlinien fur Schraubenextruder zur Forderung viskoelastischer Fliissigkeiten,” Internal Report. Institut fur Stromungslehre und Stromungsmaschinen, Hochschule der Bundes-wehr Hamburg. OCHS, W. (1978). Peristaltisches Pumpen nicht-newtonscher Fliissigkeiten. Diplomarbeit, Darmstadt. PIPKIN,A. C. (1968). Small displacements superposed on viscometric flow. Trans. Soc. Rheoi. 12, No. 3, 397-408. PIPKIN,A. C., and OWEN,D. R. (1967). Nearly viscometric flows. Phys. Fluids 10, 836-843. SCHONEWALD, H. J . (1970).Anwendung der temperatur- und druckinvarianten Auftragung der Viskositat von Kunststoffschmelzen bei der Auslegung von Dusen. Dissertation D 17, Darmstadt. H. (1979). Die isotherme Stromung eines Second-Order-Fluids im ForderSCHWEINFURTH, kana1 einer Schneckenpumpe. Dissertation D 17, Darmstadt. TADMOR, Z., and KLEIN,I. (1970). “Engineering Principles of Plasticating Extrusion.” Van Nostrand-Reinhold, Princeton, New Jersey. TRUESDELL, C. (1974). The meaning of viscometry in fluid dynamics. Annu. Rev. Fluid Mech. 6 , 1 1 1-146. TRUESDELL, C., and NOLL,W. (1965).The nonlinear field theories of mechanics. In “Handbach der of Physik” (S. Fliigge, ed.), Vol. 3, Part 111. Springer-Verlag, Berlin and New York. UHLAYD,E. (1978). Theoretische und experimentelle Untersuchungen zum anormalen Stromungsverhalten von extrahochmolekularem Polyathylen hoher Dichte (HDPG). Dissertation, Stuttgart. WALKER, J. (1978). Serious fun with Polyox, Silly Putty, Slime, and other nowNewtonian fluids. Sci. Am. 239, 142-149. WINTER,H. H. (1979). Schlepp- und Druckstromung im engen Spalt uber dem Schneckensteg. 6 Stuttgarter Kunststoffkolloquium, Art. 2.2.

Author Index Numbers in italics refer to the pages on which the complete references are listed.

A Ablowitz, M . S., 23, 36 Abramowitz, M., 34,36, 69, 130 Airy, G. B., 3, 36 Akbay, U., 213, 215, 220,225 Andre, J. C., 108, 132 Anker, D. A,, 7, 20, 36 Armstrong, R. C., 184, 224, 225,225

B Basdevant, C., 108, 132 Batchelor, G. K., 40,50, 80, 83, 102, 103, 107, 111, 130 Becker, E., 180, 184, 185, 186, 199, 204, 211, 213,214, 215, 220, 222, 224,225 Bird, R. B., 184.224, 225,225 Birkhoff, G., 83, 130 Bohme, G., 199. 200, 204,225 Bordas, L. A., 7, 23.36 Boussinesq, J., 1. 3, 7,36 Burger, W., 180, 184, 185, 186, 214, 225 Burgers, J. M., 116, 130

C Calogero, F., 30, 36 Cameron, A., 225,225 Canavan, G . H., 40, 125, 130 Carrier, G. F., 137. 174 Carter, R. L., 165, 174 Champagne. F. H.. 63, 130 Chen, W. Y . , 50, 51, 52, 53, 55, 58, 59, 133 Chou, P. Y . , 66, 130 Christmann, L., 181, 225 Chu, F. Y. F., 2, 37

Cole, J . D., 116, 130 Coleman, D. B., 184,225 Comte-Bellot, G., 102, 130 Corrsin, S., 102, 130 Crow, S. C., 40, 125, 130

D Deen, G . S., 112, 130 Desgasparis, A,, 30, 36 De Vries, G., 3, 8, 36 Diehn, T., 213, 215, 225

E Earnshaw, S., 3,36 Edwards, S. F., 73, 130

F Feldman, K. T., 165, 174 Fenner, R. T., 210,225 Fenton, J., 12,36 Fornberg, B., 112, 130 Fox, P. G., 112, 115, 131 Freeman, N. C., 7, 20, 36 Frenkiel, F. N., 51, 53, 54, 55, 56, 57, 59, 60, 131 Frisch, U., 65. 131

G Gad-eh-Hak, M., 102, 131 Gardner, C. S., 2, 6, 15, 36

227

Author Index Gel'fand, I., 16.36 Gibson, C. H., 63, 131, 132 Gibson, M. M., 63, 131 Grant, H. L., 63,131 Greene, J. M., 2, 6, 15,36

Kramers, H. A., 147, 154,175 Kruskal, M. D., 1, 2, 6, 15, 36,37

L

H Han, C. D., 210,225 Hassager, O., 184,224,225,225 Hellwege, K. H., 181,226 Herring, J. R., 73, 112, 115, 131 Hirota, R., 6,36 Hopf, E., 45,46, 116, 131 Hosokawa, I., 125,133 Huilgol, R. R., 184,226

I Iberall, A. S., 148, 174 Its, A. R., 7,23,36

J Johnson, R. S.,8, 29, 34, 35,36

K Kadomtsev, B. B., 2, 6, 8, 22, 23, 36 Kamotani, Y.,102, 133 Kaup, D. J., 35,36 Kawahara, T., 72, 131 Kelland, P., 3, 36 Kempton, A. J., 136, 137, 174 Kida, S.,75, 78, 79, 81, 82, 84, 85, 87, 88, 89, 90,95,96, 100, 103, 104, 105, 116, 122 124, 125,129, 131,133 Kirchhoff, G., 136, 148, 174 Klebanoff, P. S.,52, 53, 54, 55, 56, 57, 59, 60,131 Klein, I., 210, 226 Kolmogorov, A. N., 40,46,61,63,64,86, 131 Korteweg, D. J., 1, 3, 8,36 Knappe, W., 181,226 Kraichnan, R. H., 73, 98, 107, 112, 115, 131

Laitone, E. V., 12,36 Lamb, H., 76,131 Landau, L. D., 64,131 Lax, P., 15,36 Leigh, D. C., 184, 185, 186,226 Leith, C. E., 73, 107, 131, 132 Lesieur, M., 108, 132 Levitan, B. M., 16,36 Lienert, U., 159, 175 Lifschitz, E. M., 64, I31 Lighthill, M. J., 115, 132 Lilly, D. K., 107, 112, 132 Ling, S. C., 102, 132 Loitsiansky, L. G., 83, 132 Ludwigs, J., 211, 212, 226 Lumley, J. L., 52, 132

M McLaughlin, D. W., 2.37 Manakov, S.V., 7, 23,36 Markowitz, H., 184,225 Marx, C., 146, 175 Matveev, V. B., 7, 23,36 Mawardi, 0. K., 137, 175 Maxon, S., 30, 36 Merkli, P., 137, 168, 170, 175 Miles, J. W., 2, 7, 8, 10, 12, 36.37 Millionshtchikov, M., 66, 132 Mizushima, J., 75, 78, 74, 81, 82, 84, 85, 89,95, 104, 118, 119,132,133 Moilliet, A., 63, 131 Monin, A. S.,41. 64, 132 Morfey, C. L., 136, 137, 175 Muira, R. M., 2, 6, 15,36,37 Miiller, U., 159, 174, 175

N Narahara, Y., 159, 175 Nelkin, M., 65, 131

Author Index Newell, A. C., 35, 36 Noll, W., 180, 184, 186, 214,225,226 Nonn, G., 199, ,200, 204, 225

0 Obukhov. A. M., 64,132 Ochs, W., 195,226 Ogura, Y., 40, 71, 132 Orszag, S. A., 41, 65, 73, 76, 77, 98, 104, 112, 115, 131, 132 Owen, D. R., 213, 215,226

P Pao, Y.H., 104, 133 Patterson, G . S., 104, 132 Paul, F., 181,226 Petviashvili, V. I., 2, 6, 8, 22, 23,36 Pipkin, A. C., 206, 213, 215,226 Pouquet, A., 108,132 Proudman, I., 66, 69, 70, 83, 92, 130, 132

Q Quack, H., 159, 175

Scott, A. C., 2,37 Scott-Russell, J., 1, 2, 3, 6, 7, 35, 37 Semjonov, V., 181,226 Shabat, A. B., 2, 13, 15, 31, 37 Sondhauss, C., 146, 163, 175 Sprenger, H., 168, 175 Stegen, G. R., 63, 132 Stegun, I. A., 34, 36, 69, 130 Stewart, R. W., 51, 63, 103, 131, 132 Sulem, P. L., 65, 131

T Taconis, K. W., 147, 175 Tadmor, Z., 210,226 Tanaka, H., 72, 132 Tassa, Y.,102, 133 Tatsumi, T., 66, 71, 75, 78, 79, 81, 82, 84, 85, 87, 88, 89, 90, 95, 96, 101, 103, 104, 105, 108, 110, 115, 116, 118, 122, 125, 129, 133 Taylor, G. I., 40, 62, 133 Temkin, S., 168, 175 Thomann, H., 137, 168, 170, 172, 175 Thompson, S., 8, 29, 34,36 Titt, E. W., 45,46, 131 Tokunaga, H., 115,133 Tominaga, A., 159, 175 Townsend, A. A., 103, 130 Truesdell, C., 180, 183, 184, 186,214, 226

R Rayleigh, J. W. S . Lord, I. 37, 136, 146, 163, 168, 175 Reid, W. H., 66, 69, 70, 92, 132 Rijke, P. L., 137, 175 Rosenblatt, M., 52, 132 Rott, N., 146, 147, 148, 149, 150, 151, 154, 157, 158, 159, 162, 163, 166, 168, 171, 173, 175

S Saffman, P. G., 41, 112,132 Satsuma, J., 2, 7, 13, 23,36,37 Schedvin, J., 63, 132 Schonewald, H. J., 181,226 Schwarz, W. H.. 63, 131 Schweinfurth, H.. 21 I , 226

U Uberoi, M. S., 104, 133 Uhland, E., 207,226

V Van Atta. C. W... 50.. 51, 52, 53, 55, 58, 59, 60. 133 Viecelli, J ., 30. 36 Von Hoffmann, T., 159, 175

w Walker, J., 179,226 Walton, J. J., 125, 133

230

Author Index

Wan, C. A., 102,132 Whitharn, G. B., 2, 6, 37 Winter, H. H., 21 1,226 Wyngaard, J. C., 104, 133

Y Yaglorn, A. M., 41,64,132 Yamamoto, K., 125, 133

Yanase, S., 108, 110, 133 Yazaki, T., 159, 175 Yeh, T. T., 60, 133

Z Zabusky, N. J., 1,6,37, 112, 130 Zhakarov, V. E., 2, 7, 13, 15, 23, 31,36, 37 Zouzoulas, G., 148, 149, 162, 163, 166, 175

Subject Index D

A Acoustic oscillations, 137 damping and excitation, 143- 148 in gas column, 143-148 temperature stratification and, 143-148 in tube, 143-168 Acoustic streaming, I37 of heat, 172 of mass, 168, 173 Rayleigh, heat transfer effect, 173 Air-filled gas oscillations, Sondhauss tube, 164-1 65

Damping and excitation, gas column oscillations, 143-168 Differential viscosity, defined, 206 Dispersion, positive, Kadomtsev Petviashvili, 22-35

E Effective viscosities, 204-210 Elasticity, non-Newtonian fluid flow, 183- 184 Energy decay, isotropic turbulence, 101 -102 dissipation range Kolomogrov’s similarity law, 94-96 similarity law, 92-99 wavenumber integration domain, 97 spectrum backward energy transfer, 109 cumulant approximation, 71 -74 equation, isotropic turbulence, 77-78 energy transfer function, 80-83 enstrophy, 114-1 15 isotropic turbulence, 80-88 quasi-equilibrium range, 93, 113-1 14 Taylor’s turbulence, 79 two-dimensional turbulence, 108-1 14 wavenumber range energy transfer range, 100 intermediate range, 99-101 zero-cumulant approximation, 71 -74 80 spectrum, negative zero-cumulant approximation, 72-74, 78 transfer function, 80-83

B Balance equation, non-Newtonian fluid, constitutive equation and, 178 Bearing, journal, non-Newtonian lubricant fluid, 219-225 Boundary layer plane, memory fluid flow and, 216-219 thin, gas column acoustics, 143-146

C Characteristic functional, isotropic turbulence, 44-46 Constitutive equation non-Newtonian fluid, 184-187 defined, 178 balance equation and, 178 Cumulant expansion approximation, 71 -74, 80 Markovianization, 76 turbulence, 65-78, 127 Cylindrical Korteweg-de Vries equation, 30-35

231

232

Subject Index

Energy-containing range similarity exponent, 90 similarity law, 90-92, 109-1 13 Enstrophy growth, zero-cumulant approximation, 69-70

energy spectrum, 114-1 15 Exponential soliton, rational soliton interaction, 29-30 Extruder flow, 210-212

F Fluids compressible, see Newtonian fluid incompressible, see Non-Newtonian fluid

G Gas column acoustics, 143- 148 damping, 143- 148 excitation, 143- 148 oscillations in helium-filled tube, 156-165 Kirchhoff theory, 148 stability limits, 156-165 stability theory, 148-153 in straight tube, 156-165 Gas-liquid oscillations, 138- 140 stability theory, 165-168 Gas oscillations in column, 143-148 heat flux effect, 140-141 over nonisothermal wall, 140-141 Sondhauss tube, 164-165 thermoacoustics, 138-1 74 Gel'fand-Levitan equation, soliton, 33 Glowing glass-harmonica, 146 Grid turbulence. small-scale motions, 61

H Homogeneous turbulence, see Isotropic turbulence

Heat flux effect temperature oscillations, 140-141 oscillations along nonisothermal wall, 140-141

Heat transfer effect, Rayleigh acoustic streaming, 173 Helium-filled tube, gas column oscillations, 155- 157

I Incompressible isotropic turbulence, see Isotropic turbulence Inverse scattering theory Korteweg-de Vries equation and, 15 soliton, 14-27, 35 Inviscid enstrophy dissipation, twodimensional turbulence, 106-108 Isotropic turbulence characteristic functional, 44-46 cumulant expansion, 65-78 moments and, 44-49 dynamical equation, 42-44 energy spectrum, 80-101 equation, 77-78 grid turbulence, 61 homogeneous, theory, 39-130 hypo thesis inviscid similarity, 63-64 local isotropy, 62-64 local similarity, 62-64 incompressible, 78, 80-105 energy decay, 101-102 energy spectrum, 80-88 fluid, 39-127 microscale derivation, 104-105 skewness, 102-104 statistical quantities, derivation, 101-105

velocity derivation, 102-104 large-scale motions frequency distribution of velocity components, 50-53 grid-generated, 50-53 quasi-equilibrium state, 50-61 quasi-normal approximation, 60 quasi-normality, 50-61 statistical state, 49-65 time-correlation. 55-60

233

Subject Index time-kurtosis, 53-55 time-skewness, 53-55 mathematical formulation, 42-49 moments and cumulants, 44-49 one-dimensional, 1 15- 127 energy-containing range, 11 8-122 energy, 118-127 inviscid energy dissipation, 117 triangular shock waves, 116 turbulence of Burgers, 115- 127 quasi-equilibrium state, 50-65 quasi-normal approximation, 60 quasi-normality, 50-6 I similarity law, 88-101 small-scale motions grid turbulence, 61 hypothesis, 61 -64 inertial subrange spectrum, 63 Kolmogorov’s equilibrium theory, 61 -65 quasi-equilibrium state, 61-65 similarity law, 63 statistical state, 61 -65 wavenumber range, 63-64 spectrum, energy, 80-101 statistical state, 49-65 Taylors expansion, 47-49 theory, 39-127 turbulence of Burgers, 115-127 turbulent velocity field, 44-49 two-dimensional energy spectrum, 108-1 14 incompressible fluid, 106-1 15 inviscid enstrophy dissipation, 106-108 wavenumber range, 63-64 zero-cumulant approximation, 65-78

J Journal bearing, non-Newtonian lubricant fluid, 219-225

K K-de V, see Korteweg-de Vries Kadomtsev-Petviashvili equation, positive dispersion and, 22-35

Kirchhoff theory, gas column oscillations, 148 Kolmogrov equilibrium theory, small-scale motions, 61 -65 hypothesis, 62-63 inertial subrange spectrum, 62-63 similarity law, 62-63 energy dissipation range, 92-99 energy spectrum and Reynold’s number, 88-90 Korteweg-de Vries cylindrical equation, 30-35 single soliton solution, 32-33 equation two soliton interaction, 8-14, 35 inverse scattering theory, 15 Kramers stability theory, gas column oscillations, 147-148

L Large-scale motions, see Isotropic turbulence Local isotropy, hypothesis, 62-63 Lubrication, journal bearing, 219-225 Ludwigs approximation, screw extruder, 212

M Markovianization, cumulant expansion, 76 Mass balance equation, defined, 178 Memory effect, non-N-wtonian fluid flow, 183-184,216-2 19 Microscale, isotropic turbulence, 104-105 Multisoliton, see also Soliton, Two-soliton interaction interaction, 7, 13, 16-22 resonant triad formation, 19 solution, 6, 17-22

N Newtonian fluid, see also Non-Newtonian fluid defined, 178 viscosity, 204-206

234

Subject Index

Non-Newtonian fluid constitutive equation, 178, 184-187 defined, 177 flow, 177-226 behaviour, 179-184 dimensionless, 189-194 elasticity, 183- 184 fluid particle, motion and deformation, 185 melt fracture, 206 memory effect, 183-184, 216-219 nonlinear, 179- 182,206 normal stress effect, 182-183 peristaltic pumping, 192-197 in pipe, viscometric, 187-192 plane boundary layer, 2 16-21 9 Prandtl-Eyring fluid, 189-190, 194-196 Reiner -Philippoff fluid, 190 shear velocity, slip change response to, 184 simple shear, 179-226 viscoelastic turbulence, 206 viscometric, between parallel walls, 204-205 viscosity, 204-210 Nonisothermal surface, oscillating flow and, 138-143 Nonlinear flow, non-Newtonian fluid, 179-182,206 Nonstationary turbulence, 79

0 One-dimensional turbulence, see Isotropic turbulence Oscillating flow, nonisothermal surface and, 138-143

P Peristaltic pumping, 192- 197 efficiency, 196 pressure gradient, 195 Pipe flow non-Newtonian fluid, 187-192, 204 Newtonian fluid, 191 Plane boundary layer flow, short memory fluid and, 216-219

Positive dispersion, Kadomtsev-Petviashvili equation and, 22-35 Prandtl-Eyring fluid, 189-190, 194-196

Q Quasi-equilibrium range, energy spectrum, 93, 113-1 14 state, isotropic turbulence, 50 zero-cumulant approximation, 76 Quasi-normal approximation, isotropic turbulence, 60

R Rayleigh acoustic streaming, heat transfer effect, 173 Rational soliton, exponential soliton interaction, 29-30 Reiner-Philippoff fluid, 190 Resonant soliton, 29 Rheology, see also Non-Newtonian fluid flow; Newtonian fluid engineering applications, 177-226 theory mechanical, 177-226 thermal effects, 178-226 Russell’s water channel experiment, wave formation, 3-4

S Scott-Russell solitary waves, 3 Screw extruder, Ludwigs approximation, 212 Shear velocity, slip change response to, 184 Short memory fluid, plane boundary layer flow, 216-219 Similarity law, energy spectrum, 88-101 Similar evolution in turbulence, defined, 50 Sine-Gordon equation, soliton, 2 Skewness, velocity derivative, isotropic turbulence, 102-104 Small-scale motiois, see Isotropic turbulence Solitary waves, 3-8 defined, 1-2 exponential, 29-30 Scott-Russell, 3

Subject Index Soliton formation, cosine wave, 6 interactions Gel’fand-Levitan equation. 33 two-dimensional, 1-37 Korteweg-de Vries, 19 multidimensional, 1-37 one-dimensional, I resonant, 29 two-dimensional, 1-37 Zakharov-Manatov rational, 24-30 Sondhauss tube, air-filled gas oscillations, 164-165 Sound producing tube, 143-168 Stability limits, oscillations in straight tube, 156-165 in tubes, variable cross section, 162-165 problems, history, 146-148 theory gas-column oscillations, 148-153 gas-liquid oscillations, 165-168 Straight tube, gas column oscillations, 165- 168

235

Triangular shockwaves, one-dimensional turbulence, 116 Turbulence, see Isotropic turbulence Two-dimensional turbulence, see Isotropic turbulence Two-soliton interactions, Korteweg-de Vries equation and, 8 - 14

V Velocity oscillations perpendicular to wall, 141-143 thermoacoustic effect, 138-139 Viscometric flow, 212-216 between parallel walls, 212-213 simple shear, 214 Viscosities, effective, 204-210 Viscosity pumps, 197-204 dimensionless flow, 202-203 Prandtl-Eyring fluid, 199-200 Reiner-Philippoff, 202 efficiency, 199-200 extruders, 210-212 model, 197 shear thinning behaviour, 200-203

T Taylors expansion, 47-49, 127 turbulence, 79 Temperature oscillations, thermoacoustic effect, 138- 139 stratification in gas column, 143-148 Thermoacoustics, 135-1 75 defined, 135 effect, heated surfaces, 138-143 gas oscillations, 138-174 streaming, 137, 168-174 in isothermal walls, 170-173 in nonisothermal walls, 173-174 theory, 168-170 Three-dimensional turbulence, 46, 98 -99, 114-130, see also Isotropic turbulence Transfer function, zero-cumulant approximation, 67-68,74

W Wavenumber range energy transfer range, 100 intermediate range, 99-101 Wave reflection, Russell experiment, 3-5

Z Zakharov-Manatov rational interaction, 25-30 time-increasing, 25-27 rational soliton, 23-30 ;era-cumulant approximation energy spectrum, 71 -74, 80 isotropic turbulence, 65-78

This Page Intentionally Left Blank