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.

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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

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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

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Advances in Applied Mechanics Volume 20

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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

(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 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,

(4.31)

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 > 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)- 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 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, 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 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.

(6.51)

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 > 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.

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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.

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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

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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

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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

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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 > 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)

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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