A theory of homogeneous, isotropic turbulence of incompressible fluids

ANNALS

OF PHYSICS:

73, 326-371 (1972)

A Theory

Department

of Homogeneous, of Incompressible

of Physics,

Isotropic Fluids*

TOHRU

NAKANO

University

of Illinois,

Urbana,

Turbulence

Illinois

61801

Received August 30, 1971 An eddy description of the Navier-Stokes equation is used to study the homogeneous, isotropic turbulence of incompressible fluids. Eddies can be split into two groups, corresponding to eddies appreciably smaller and appreciably larger than a given eddy under study. It is shown that the influence of the smaller eddies may be described by an effective turbulent viscosity whose coefficient is calculated as the functional of the energy spectrum of small eddies with the aid of a diagram technique; the viscosity coefficient is similar to that introduced by Heisenberg and may be interpreted by Prandtl’s mixing-length theory. On the other hand, larger eddies act to diffuse the eddy in k space, exerting a diffusive force, and feeding energy to it; the diffusion coefficient and diffusive force are calculated diagrammatically as functionals of the spectrum of large eddies. An equation for the energy spectrum which holds in both the inertial and dissipative regions is then derived by combining the contributions from small and large eddies. In the inertial region the contributions from small eddies, which appear as the effective turbulent viscosity, are in equilibrium with the contributions from large eddies, which appear as the diffusion coefficient and diffusive force. It is shown that the interaction in the inertial region is local in wavenumber space, as assumed by Kolmogorov, and the scaling laws which lead to the Kolmogorov universal spectrum are derived. The equation for the energy spectrum in this region may be related to that of Heisenberg. The numerical predictions of the theory, such as the coefficient of the Kolmogorov spectrum, are shown to be in good accord with experiment. In the dissipative region the energy spectrum at wavenumber k obtained from our basic equation is shown to be proportional to exp (- 2/[15] k2/k$), where kd determines the boundary between the inertial and dissipative regions. The relationship of these results to those obtained by Townsend is established. To the extent that vertex corrections to the operator which describes the interaction between different eddies may be neglected, the theory is shown to be self-consistent.

I. INTRODUCTION

Fully developed turbulent flow at Reynolds numbers much larger than the critical Reynolds number is characterized by the presence of an extremely irregular * Work supported by Army Research Office (Durham) under contract DA-HCO4-69-0007. Based on part of author’s Ph.D. thesis, University of Illinois. Copyright All rights

0 1972 by Academic Press, Inc. of reproduction in any form reserved.

326

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variation of the velocity with time at each point, superposed on the regular mean flow. This irregular motion may be regarded as the superposition of turbulent eddies of different sizes; by the size of an eddy is meant the order of magnitude of the distance over which the velocity varies appreciably. Let us consider the physical features associated with the mathematical description of an eddy. When a fluctuating velocity component u,(x, t) is Fourier-analyzed in space as u,(x, t) = C u,(k, t) eik.x

(1.1)

k

each Fourier component, u&, t), may be regarded as representing the intensity of an eddy of size k-l. A single component u,(k, t) eik’x of (1.1) corresponds, then, to a sequence of eddies of size k-l. In this eddy description the eddies are individual entities even though they interact with each other strongly. This aspect may be contrasted to the wave case in which u,(k, t) eik.x represents a single wave with wavenumber k which spreads in space over many wavelengths. A further difference between the eddy and wave descriptions appears when the interaction of two components u,(k - q, t) and u&q, t) is considered. In the wave description a new wave u,,(k, t) is created as a result of the interaction. On the other hand, in the eddy description it is not correct to conclude that an eddy of the size q-l interacts with an. eddy of the size 1k - q j-l, to provide a new eddy of the size k-l. To see this, consider, for example, the interaction between a very small eddy and a large eddy. The small eddy looks, in general, like a particle to the large eddy; the consequence of the interaction is an energy transfer from the large eddy to the small eddy, not another tiny eddy being created by the interaction. We shall work with the eddy description throughout this paper. [In order to emphasize the importance of the eddy description it would be helpful to use an expansion which is more evocative of the physical process taking place than the plane-wave expansion employed in (1.1). Such an expansion would likely involve a mixed representation of space and wavenumber; we have not, however, as yet been able to develop a suitable expansion.] In this paper we treat the homogeneous and isotropic turbulence of an incompressible fluid. For this purpose it must be discussed whether such ideal turbulence exists in practice. Although the energy of the small-scale motion in the inertial and dissipative regions is supplied by the inhomogeneous and anisotropic largescale motion in the energy-containing region [l], the influence of the energycontaining region is gradually weakened during the energy-transfer process in wavenumber space and disappears for those high wavenumbers which are widely separated from the energy-containing region-e.g., for the dissipative region. Moreover, if the inertial region is big enough as is the case at sufficiently large Reynolds numbers, its structure may likewise be regarded as homogeneous and

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isotropic. It must be added that we ignore the inhomogeneity which comes from the molecular structure of a fluid because we restrict ourselves to wavenumbers much smaller than the inverse of a mean-free path of molecules of a fluid. Another simplifying condition, the incompressibility of a fluid, obtains under circumstances that density variations may be neglected. A measure of the importance of these density variations is provided by the ratio of a typical fluid velocity to the average velocity of sound in the fluid. As far as turbulence in the laboratory, atmosphere and oceans is concerned, the ratio is very small compared to unity; fluids may be treated as incompressible in these cases. On the other hand, in some astrophysical problems one must consider the turbulence of a compressible fluid. The difficulties encountered in the study of turbulence may be ascribed to the fact that the Navier-Stokes equation is strongly nonlinear. If we set up the equation for pair correlation of velocity components from Navier-Stokes equation, it involves triple correlations in an essential way. The equation for the triple correlation involves quadruple correlations in an essential way and so on. In view of the difficulties associated with the nonlinearity of the Navier-Stokes equation it has proved desirable to develop a phenomenological model of turbulence. Kolmogorov [2] suggested that the small-scale motion belonging to the inertial region might be homogeneous and isotropic because it is widely separated from the energy-containing region. Based on the postulate that the transfer of energy from small to large wavenumbers is a cascade process and that the structure of the inertial region depends only on the energy flow rate E per unit mass, wavenumber k and not on the viscosity v, he predicted the energy spectrum E(k) as E(k) - &Wk--5/3 (1.2) with the help of dimensional analysis. The Kolmogorov spectrum (1.2) has been verified experimentally [3,4]; a detailed account of Kolmogorov’s theory can be found in the standard monographs on turbulence [5-71. Dimensional analysis was a feature of the qualitative theories by Obukhov [8], Heisenberg [9], von Karman [lo] and Kovasznay [ll]. These authors truncated the hierarchy by approximating the triple correlation in terms of pair correlation with the aid of somewhat different intuitive and dimensional considerations [12]. Of these perhaps the most interesting and far-reaching theory is that due to Heisenberg [9]. Heisenberg’s approach was based on the physical picture that small eddies act on large eddies as an effective viscosity; with it he succeeded in deriving the c2/3k-5/3 spectrum in the inertial region and estimating the wavenumber at which the energy spectrum begins to deviate from e2/3k-5/3. He also derived an energy spectrum which is proportional to k-’ in the dissipative region, a prediction which is in considerable doubt [13]. Quantitative theories were formulated by Proudman and Reid [14], Chandrasekhar [15], and Tatsumi [ 161 who approximated the quadruple corre-

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lation by the products of pair correlations based on the hypothesis of a normal distribution of the velocity field. These theories, however, failed to take account of higher-order correlations which may be seen to be as important as the lowerorder correlations which have been included. It is essential for understanding turbulence to take into consideration the collective character of the eddy interaction process. Kraichnan [17] was the first to emphasize this with the use of a “direct interaction approximation.” However, Kraichnan could not recover the k-5/3 spectrum because he overestimated the effects of large eddies on small eddies [I 81. As we shall see, this overestimation may be avoided by considering the physical significance of the eddy description. Stimulated by Kraichnan’s pioneer work, Wyld [I93 presented a systematic method of obtaining equations for the energy spectrum, velocity propagator, and vertex function by applying the diagrammatic techniques of quantum field theory to hydrodynamic turbulence. He showed that the simplest approximation which ignores the renormalization effect on the eddy propagator yields Chandrasekhar’s equation [15] and that if the renormalization is included it leads to Kraichnan’s equation [ 171. Recently Balescu and Senatorski [20] developed a similar diagrammatic technique smrting with the Liouville equation; they failed to obtain the k-5/3 spectrum for the same reason as Kraichnan. We turn now to a physical description of the interaction between eddies. As we shall demonstrate, the approximations required to achieve a quantitative theory of turbulence based on the Navier-Stokes equation must be physically motivated at every stage in the development of the theory. Consider, for example, the interaction of a large eddy with small eddies. The small eddies are carried together by the large eddy as they move individually with their own relatively small velocities. Parts of the large eddy are pushed back and forth at random by individual motions of small eddies, and it is this which may lead to the damping of the intensity of the large eddy. It is, however, improbable that the size of the large eddy is changed as a result (of the interaction. The situation is quite different in the case in which a small eddy interacts with large eddies. While the small eddy is carried by large eddies, the size of the small eddies stretches or shrinks because each part of the small eddy is carried with the slightly different velocity. The distortion is caused by a velocity variation over the small eddy due to the large eddies. Our theory begins with the Navier-Stokes equation without an external force. With the: aid of a diagrammatic technique we calculate two fundamental quantities as functions of the energy spectrum. One is a turbulent effective viscosity which arises from the coupling of small eddies to the eddy under study; the expression for the viscosity has nearly the same form as that due to Heisenberg [9]. The second fundamental quantity is an energy-transfer function which arises from coupling

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between the given eddy and large eddies. By combining the two quantities we formulate an equation for the energy spectrum which holds both in the inertial region and in the dissipative region. The equation is self-consistent in the sense that it involves only the energy spectrum. From it we are able to justify Kolmogorov’s assumption about an energy flow in wavenumber space, and to obtain Kolmogorov’s spectrum in the inertial region. The connection of the equation with that of Heisenberg may likewise be established. The calculated numerical coefficient of the spectrum agrees with experiment [3,4]. In good accordance with experiment the theory also estimates the wavenumber at which the spectrum begins to deviate from a k-5/3 power law. In the dissipative region the spectrum takes the asymptotic form exp(-d[ 15]k2/kd2), where kd = (c/v3)lj4. The theory is closely related with that of Townsend [21] who argued that the eddies at wavenumbers k > kd form concentrated, but weak, vortex sheets which are stretched by large eddies and weakened by dissipative effects at the same time. One of the open questions in turbulence is why a universal region (such as the inertial region) exists. A necessary condition for its existence is that the rate of energy transfer depends only on local (in k space) value of the flow [22]. It is shown that the present theory has that desirable feature. Section II is devoted to a diagrammatic analysis of the Navier-Stokes equation. In Section III we consider the coupling to small eddies and develop the approximations which lead to a turbulent viscosity coefficient. In Section IV we consider the coupling to large eddies and derive the energy transfer and diffusion contribution to the eddy transport equation. In Section V we consider the energy spectrum and derive the Kolmogorov scaling law. In Section VI we compare our theory with those of Kolmogorov, Heisenberg, and Townsend, while in Section VII it is compared with experiment. The results of the present investigations are summarized in Section VIII. Some of the computational details are given in Appendices. II. DIAGRAMMATIC

ANALYSIS

OF THE NAVIER-STOKES

EQUATION

In this section we consider the interaction of a given eddy with other eddies with the aid of a diagrammatic analysis of the Navier-Stokes equation and determine the self-energy of the eddy in terms of the correlation function by taking account of the collective interaction between eddies in the framework of the RPA. As first shown by Taylor [23], because of the irregular variation of fluid velocities in turbulent flow, the velocity-velocity correlation function provides a useful starting point for a description of turbulent motion. The correlation function &(x + X, t + T; X, T) between velocities at two space-time points is defined as &3(X + x, t + T; x 0 = <%b + x, t + T) %3(X, T)),

(2.1)

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where the bracket denotes a suitable average. This average may be an ensemble average,, a time average over r, or a space average over X. Since the system is homogeneous in space as well as in time, R,, is a function of only the differences between the space-time variables. On substituting (1.1) into (2.1), we may write R&x,

t) = 1 (u,(k, k.q

t + T) z+(q, T)) eik.xei(k+Q).X,

(2.2)

where k and q are discrete wavenumbers which satisfy periodic boundary conditions for a cube of volume V of the system. Since the right-hand side of (2.2) must be independent of X the correlation function vanishes unless q = -k. R,&x, t) thus may be written as R&x, t) = 1 (u,(k,

t + T) u,(-k,

T))

eik.x

k

(2.3) = v

T G&k,

t) eik.x,

where Ga,(k 0 = ( WW3)
t + T> q4--h,

7’)).

(2.4)

Since we are interested in isotropic and incompressible turbulence, we may further express G in the form G&,

t> = 4,(k)

g(k 0

(2.5)

where g(k, t) is a scalar function which depends only on the magnitude of k and is the projection operator which picks out a perpendicular component to k;

A,,(k)

&@)

= b

- W#2~.

(2.6)

The energy per unit mass associated with the velocity fluctuation, which is usually referred to as the energy spectrum, E(k), is simply related to g(k, 0) according to E(k) = 4n-k2g(k,0).

(2.7)

Thus, E(k) takes the form WV

= 4~WW)
T) d-k,

~))[v/(243],

which tells us that Jr E(k) dk is the fluctuation energy per unit mass. Our basic analysis is based on diagrammatic representation of the Navier-Stokes equation. which is similar to the representation introduced first by Wyld [ 191.

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NAKANO

The starting equation is the Fourier-transformed an external force ( aat

+ vk’) @,

t) = V’&(k)

Navier-Stokes

equation without

1 u,(k - q, t) dq, t>,

(2.8)

9

which is derived in Appendix A. v is the kinematical interaction operator defined by %dk)

= -@L40

viscosity, while 9&,(k),

k + 4x,(k) k,),

the (2.9)

is symmetric with respect to the exchange of /3 and y. The reason why we have dropped the external force term in (2.8) is obvious; the inertial and dissipative regions which concern us are not acted on by an external force and energy is transferred to the both regions only through the nonlinear term in (2.8). Integrating (2.8) over time we find

u,(k, t) = @&(k) c sl, ds e-“RB(t-s)~g(k- q, s)u,,(q,s) + e-“lca(t-io’u,(k, to), P

(2.10)

where u,(k, to) is the initial value of the velocity. The initial term is not important because the nonlinear term destroys the memory of the initial condition quickly; indeed, it vanishes if we choose the initial time to as minus infinity. Equation (2.10) then becomes dk

t) = +%dk> c f,

ds e-vka(t-s)ug(k- q, s) uJq, s),

a

(2.11)

a strongly nonlinear equation. We note that since there are two identical terms which are obtained by exchanging k - q and q on the right-hand side of (2.1 l), if we do not distinguish between two u’s in (2.11) it may be written as u,(k, t) = K,,(k)

c j”,

ds e--vk8(t-s)~,g(k - q, s) uy(q, s).

(I

(2.12)

Y, a ax t

=

----_ ah t -<

S P.G

FIG. 1. The diagrammatic expressionfor (2.12). Associateudq,S) with the solid line with y, q which ends at s, e~‘~‘t~-~) with the dotted line with k which begins at s and ends at t, and V&k) with the vertex part denoted by a dot. We do not count twice the same diagram which is obtained by the interchange of two solid lines U&C - q, S) and u,(q,s). Integrate over dummy wavenumber and time as &Jf-ao ds.

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333

A diagrammatic expression of (2.12) is given in Fig. 1; an eddy with wavenumber k - q interacts with an eddy with wavenumber q at time S; as a result a new eddy with wavenumber k propagates from s to t. The calculational recipe is the following; associate u,,(q, S) with a solid line, y, q, which ends at S, the propagator e-vk*(t-s) with a dotted line, k, which begins at s and ends at t, YEaBY(k)with a vertex part denoted by a dot, and integrate over dummy wavenumber and time as 2, fToo ds; the two solid lines which stand for us(k - q, s) and u,(q, S) are indistinguishable. To compute the response function, we assume that an eddy with wavenumber k is present in the system. The eddy couples to an eddy with wavenumber q1 , to provide a new eddy with wavenumber k + q, , which couples to an eddy with wavenumber qz , to provide k + q1 + qz and so on. In deciding what diagrams to include, we will first of all make the random phase approximation, hereafter referred to as RPA. In this approximation one assumes that there is no correlation between the phases of different eddies. There are two important consequences of the RPA.: first, if one applies an external impulse with wavenumber k to the system one observes only the excitation with wavenumber k (this is also a consequence of the spatial homogeneity of the system); second, any higher-order correlation function may be decomposed in terms of the appropriate products of pair correlation functions. Kraichnan’s “maximal randomness” approximation and “direct interaction” approximation as well as that of Wyld are equivalent to the above approximation. The appropriateness of the RPA will be discussed later. Regarding u,(k, s) as an external impulse let us calculate the induced disturbance, u,(k, t) at time t by starting from (2.12). Applying an iteration method to (2.12) we find rl,(k, t) as

where the bracket stands for the average process introduced

in (2.1). In (2.13)

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the subscript p signifies that one must take only proper terms. By a proper term or diagram is meant the diagram which has no internal propagator line with wavenumber k. The first term of the right-hand side of (2.13) implies that u,(k, S) couples to u?(q) s), to provide u,(k, t). The second term shows that u,(k, SJ interacts with uYz(qz, s2) and then with u,,(q, , sl), to provide u,(k, t) and so on. Figure 2 is a pictorial representation of (2.13); in it the cross implies that the intersecting eddies are correlated.

FIG.

2.

The

diagrammatic

expression for (2.13).

The cross implies that the intersecting eddies

are correlated.

The first term in (2.13) has no contribution if we carry out the calculation in a system moving with the velocity (u,(q, t)) S,,, . In the RPA the correlations of odd number of velocity components vanish and the correlations of even number of velocity components are decomposed into the products of pair correlation functions; for instance a proper quadruple correlation function which appears in Fig. 2 is decomposed as

which is interpreted in Fig. 3. It should be added that the above decomposition does not include such a term as

which is equivalent to Fig. 4, because it is not a proper term as seen from its internal propagator line with k. This improper diagram is contained in the second diagram of the right-hand side in Fig. 2.

335

HOMOGENEOUS, ISOTROPIC TURBULENCE

FIG. 3. The decomposition correlations.

of a proper quadruple

correlation

into the products of pair

FIG. 4. The improper diagram which is not included inI(2.14).

We see in Fig. 3 that two basic types of diagrams enter in the perturbationtheoretic expansion of u,(k, t), crossing and noncrossing diagrams. As Wyld showed, Kraichnan’s direct-interaction approximation may be obtained if one neglects all crossing diagrams; Fig. 2 then reduces to the simple series of diagrams shown in Fig. 5. Here a double dotted line stands for an exact propagator which takes account of the collective interaction between eddies. The integral equation

___=--i ----xYL +li!L

+-itxAL.lLLL +--fmLh- +.. FIG. 5. The diagrammatic expression for u,(k, t) which arises from the interaction with other eddies. A double dotted line stands for an exact propagator of an eddy which takes account of the renormalization effect due to the collective interaction of eddies. 59517312-3

336

NAKANO

for the exact propagator is derived with the aid of the diagrammatic Fig. 6. To this end we associate

(2.15)

L?(k) hk@ - to)

= ak I------t

67

+ t

to

FIG. 6. The diagrammatic

expression

S

ir+T

s’

1,

equation for the exact propagator.

with the exact propagator which starts with /I, k at t, and ends with 01, k at t. A,,(k) appears in (2.15) because disturbances propagate in perpendicular to k. The resulting equation is A,,(k) h,(t - to) = A,,(k) e-“k2(t-to) + ;

x 4&) x

hk(s’

%,v,(k) -

to)

d&O

jl

ds jl ds’ c”~~(~-~

4tc+&

- ~‘1 &,a,@ + 4 %g,s,,Jk + d

<%1(-(1’

S) %,@I~

(2.16)

S’)).

With the aid of (2.4)-(2.6) and (2.9) Eq. (2.16) is simplified becomes

considerably;

it

&(t) = e-vke”+ j dq b(k, q) j: ds j; ds’ e-vk2(t-s)h,k+q,(s - s’) g(q, s - s’) h&‘), (2.17) where W, Q) = U - /-4/l k + q I”](-k”

- 2Wp + Wp)

(2.18)

is a geometric factor and p is cosine of the angle between k and q. The relationship

T -&

f4

(2.19)

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HOMOGENEOUS, ISOTROPIC TURBULENCE

has been used in (2.17). On differentiating

(;;

(2.17) with respect to t we find

+ yk2) h(t) = s 4 W, q) f ds dq, t - s) hlr+ql(t - s) h&),

(2.20)

0

which was derived by Kraichnan [ 171 and Wyld [ 191. Kraichnan attempted to solve (2.20) for the turbulent spectrum; however, he was not able to reproduce the results of Kolmogorov or make connection with the theories of Heisenberg and others. The reason for his lack of success is that he included in (2.20) the contributions from large eddies. In Section IV, where the role of large eddies is studied, we will show in detail that we have to ignore the contributions from large eddies to the right-hand side of (2.20). In other words, the Kraichnan equation (2.20) is only valid for the coupling between an eddy of wavevector k and those eddies which are appreciably smaller than it. III.

COUPLING

TO SMALL

EDDIES

In this section we focus our attention on the interaction of a given eddy with those eddies which are distinctly smaller in size, i.e., have shorter wavelengths. We show that under the Kraichnan approximation (which we have seen is equivalent to the RPA plus the neglect of all crossing diagrams) the consequences of that interaction may be expressed through a coefficient of turbulent viscosity. We make connection with the Heisenberg theory. Restricting our attention to the coupling of our eddy of wavenumber k is equivalent to putting a lower limit, say mk, on the wavenumbers of the eddies which couple to it. (m may be 22, but the results do not depend on a choice of m critically.) We therefore arrive at the modified equation which expresses the coupling to small eddies,

(+ +vk2) M)= s,,,,4 b(k, d j-” ds dq, 0

t - s) h+sdt - s) MS).

(3.1)

Let us consider hk(t) for times larger compared to a relaxation time of h,(t). Since q is larger than k, hlkfql(t) will die away more rapidly than hk(t) (as will be seen in :more detail later). Therefore the right-hand side of (3.1) may be approximated as

U

e>mk

4 W, 4 f dsdq,t - s>h&t - d] hdt) 0

- [s

4

0,

4

j-”

zzz

4

Wk

d

j-m

n>mk

[S o>mk

-cc

0

ds

da

ds dq,

s> h,+,,(s)]

$1 hc+,~(s)]

MO

h(t),

(3.2)

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NAKANO

because g(q, s) and hl,+,l(s) are even functions of s. Therefore (3.1) takes the form

[$ + vk”- j,,,, 4 W, 9)j” dsg(q,s>hi,+,l(s)] Mf) = 0, 0

(3.3)

whose solution is

The above expression reduces to a more transparent form if b(k, q) and ham+,&) are expanded in series of k/q. Keeping the first nonvanishing terms we find

h,(t) = q-d-(v

+ veff(k)) k2tl,

(3.4)

where

vedk) = $

s,:, dq q2 j,” ds g(q, s) (h,(s)

- ; y).

(3.5)

We consider now the simplification of our expression for v&k), postponing until later a discussion of the validity of the above results. We note first that in (3.5) we may replace g(q, s) by g(q, 0) since, as will be shown in Section V, for the states in which we are interested g(q, s) cannot damp so rapidly as h,(s). Next we note that the idea of an effective viscosity is meaningful only in the inertial region where v is assumed to be much smaller than v&k); we thus approximate h,(s) by exp(-v&q) q2s). Under these circumstances (3.5) becomes w(k)

16n

= 15

j

m

dq

v,ff;,

mk

q2 4 $ (veffk) q2)).

(3.6)

It is interesting to note that (3.6) may be derived in part through application of the mixing-length theory of Taylor [24] and Prandtl [25] to the present case; Taylor and Prandtl assumed that the coefficient of eddy viscosity was equal to product of a “mixing” length and some suitable velocity. Here the mixing length is of the order of u(q)/v&q) q2 while the velocity is of order u(-q); it follows that w(k)

-

‘(-d) - mdqg(% ‘) f mdq(u(q> mk

veff(q)

Cl2

s mk

vern(q>

’

(3.7)

which is the same expression as (3.6) if one neglects the q dependence of v&q) q2. To calculate verr , we assume that v,ff(k) CCk-a (a will be shown later to be 4/3); then veff(k) = m%ft(mk).

(3.8)

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HOMOGENEOUS, ISOTROPIC TURBULENCE

On substituting (3.8) into (3.6) and differentiating a differential equation for v&k);

mW&P~l

= -

tW4

with respect to mk we derive

- 4/WIdp, Ww(dl,

(3.9)

where p = mk. (3.9) may be written in integral form as

with the boundary condition that limy+m v&p) = 0. With the aid of (2.7) and (3.8) we then find the following form for the turbulent viscosity: ve&)

4(4 - a) ---m”

=

(3.10)

To compare (3.10) with the result of Heisenberg we note that in this theory the effective viscosity, v,H(k), is defined by &(k)

= A j:,

&$$)

dq,

where A is a numerical factor. Equations (3.10) and (3.11) do not make any difference for E(q) which obeys the power of q. Then the rate of the change of u,(k, t) in time as a result of interaction with small eddies is governed by [h(k,

~)/a~lsmalleddies

=

-Veff(k>

k%Ak,

t>.

(3.12)

The origin of this viscosity may be understood by referring to Fig. 7 in which is depicted the coupling of our given eddy E, to a number of smaller eddies, denoted by El, E, ,... . These small eddies are carried by the eddy E,, as they move individually with their own relatively small velocities. Parts of the eddy E,

FK;. 7. Interaction

of an eddy E. with many small eddies denoted by El, E2 ,....

340

NAKANO

are pushed back and forth at random by the individual small eddies, and it is this coupling which leads to the damping of the intensity of the eddy E, . One may expect, however, that the size of the eddy EO is not appreciably changed by its interaction with small eddies, a situation which certainly exists to the extent that small eddies may be regarded as particles which move at random. Now we are in a position to discuss the consistency of the approximation we did. Since v&k) - k-4/3, as will be shown later, v&k) k2 - k2J3, which implies that the relaxation time of hk(t) decreases as k increases. Therefore the approximation which was made on passing from (3.1) to (3.2) is consistent with the result. The limitations on the use of (3.4) are clear. In deriving it, we made use of a power-series expansion in k/q in b(k, q) hltiql(s). The next nonvanishing terms are of order of k4/q4: the l/m2 corrections from these are comparatively small if m 2 3; in any event, they give rise to fluctuating terms in the equation of motion for the eddy which cannot be represented by an effective viscosity.

IV. COUPLING TO LARGE EDDIES We begin by considering the nature of large-eddy coupling, and show that the most important terms come from gradients in the velocity field of the large eddies at the position of the given eddy. With the aid of diagrammatic analysis, we find that the large eddy coupling gives rise to both an effective diffusion in k-space and a k-dependent force which acts to transfer energy from large to small eddies. A. The Role of the Average Velocity Field The interaction of an eddy E, with a large eddy denoted by E is depicted in Fig. 8. While the eddy E,, is carried by the large eddy E, the size of the eddy EO stretches or shrinks because each part of the eddy E,, is not carried at the same velocity. The distortion is caused by a velocity variation over the eddy E,, due to a large eddy E. This intuitive picture may be understood more explicitly by applying (2.8).

RG. 8. Interaction of an eddy E. with a largeeddyE. Eachpart of the eddyE. hasa velocity determinedby the largeeddyE.

HOMOGENEOUS,

ISOTROPIC

TURBULENCE

Let u,(q, t) represent eddies of wavenumber 4 < nk where n, say, is 51/2; may then expand u,(k - q, t) as u,(k, t) - q * V,U&, t) + -.. and so obtain

341 we

The numberical factor l/2 does not appear in (4.1) because there are two identical terms in (2.8). The first term of the right-hand side of (4.1) describes the interaction of an eddy of the size k-l with the locally average velocity zq&k z&(-q, t) due to large eddies, the second with the average gradient of velocity CaGnk q&-q, t) and so on. Let us keep only the first term in (4.1) for the time being. In this case u,(k, t) has the solution

u,(k, t) = u,(k, 0) Pkat exp (--i ,: A k, ,c;cnk4q> d),

(4.2)

where u,(k, 0) is an initial value of u,(k, t). Since u,(q, s) fluctuates in time the longtime behavior of u,(k, t) is obtained by taking the ensemble average of (4.2);

u,(k, t) = u,(k, 0) e--vkst(exp (-i 1: ds k ,gk dq, s))).

(4.3)

Let us consider the relaxation function (=P

(-i

j: ds k ,,cnk u&, d)),

(4.4)

which decays with time as long as uY(q, s) behaves at random. (This kind of relaxation is familiar in magnetic resonance problems [26].) The relaxation process may be interpreted in the following way. An eddy with wavenumber k is carried with the average velocity &&k uy(q, s) due to large eddies. The presence of this random velocity causes u,(k, t) to lose it 5 memory in some time I, , determined by the relaxation of the random component itself, that is, by the relaxation time for large eddies; then the relaxation function begins to damp after a time of the order of the relaxation time for large eddies. However, this large-eddy relaxation process is relatively inefficient compared to the coupling of u,Q, t) to the small eddies which was discussed in the preceding section; put another way, because the large

342

NAKANO

eddies relax more slowly than the eddy under study, they cannot play a role in relaxing it. It should be noted that the large-eddy contribution to (2.20) corresponds to the above term. Kraichnan attempted to calculate explicitly the influence of that and was led to results which are not consistent with the Kolmogorov spectrum. We shall now show if one neglects the above artificial dissipative effects entirely and considers instead the next-order term-that corresponding to gradient coupling-one arrives at an equation which is consistent with the results of Kolmogorov. B. Gradient Coupling If we drop the first term of the right-hand side of (4.1) and keep only the nextlargest term in the expansion, (4.1) becomes

(--;t +

vkz) u,(k,

t)=

%asy(k)

c

cw,(-q,

t> &

Mk,

t).

3

q
(4.5)

The integral form of (4.5), u&,

t) = jl,

ds e+‘ke(t-s) K,,OQ

q;mk wd-q,

a

s> akj u,(k

~1,

(4.6)

possesses the diagrammatic representation in Fig. 9; here a helical line stands for the larger eddies and the vertex, given by Y&(k) q * (a/ak), is represented by a circle.

r,-7 t

0.7

a,7 = -----L 1

L?T S

FIG. 9. The diagrammatic expression for (4.6). A helical line stands for a large eddy and V&,,(k) q * (ajak) is associated with a vertex represented by a circle.

We now consider the iteration of (4.6); the diagrammatic representation of the resulting expression for u,(k, t) is shown in Fig. 10 (compare to Fig. 2). If we now apply the same RPA once more, Fig. 10 reduces to Fig. 11. However, of these diagrams only the first one is retained for the following reason. As shown later the first diagram has the interpretation that large eddies act to diffuse the stationary eddy distribution in k space. The dissipation loss due to small eddies is balanced by the diffusion gain by large eddies. In the second diagram the eddy k interacts

HOMOGENEOUS,

ISOTROPIC

TURBULENCE

343

with a large eddy, shifting to the intermediate state which is not steady. The nonsteady intermediate state is subject to diffusion in k space and therefore may be expected to damp very rapidly. Consequently the contributions from the second diagram may be expected to be small compared to those from the first. A similar argument may be applied to the higher-otder diagrams.

FIG. 10. The diagrammatic large eddies.

expression for u,(k, t) which takes account of interaction with

~kT”-T!L+... +---Kx.) FIG. 11. The reduction of Fig. 10 according to the RPA.

Nevertheless the first diagram does not suffice because we have not yet taken into account the possibility that the eddy in question may collide with many small eddies during its interaction with the superposed large eddy. This process is included if we permit the internal propagator line to be renormalized by small eddies as shown in Fig. 12. (The vertex correction by small eddies is neglected

FIG. 12. A diagrammatic expression for u,(k, t) which takes account of interaction both large eddies and small eddies.

with

344

NAKANO

here as in the preceding section.) From Fig. 12, we obtain the following analytical expression for u,(k, t): u,(k, t) = ,;,

ds j-1, ds’ e--vk2(t--s) c

V&(k)

q
After a lengthy calculation (given in Appendix B), (4.7) may be cast in the following more transparent form: u,@, t) = A,,(k) j;, x

ds j-I &(k,

ds’ e-vk2(t-8)

s - s’) &

3

-

a CA s - s') -ak, I

q¶(k, s’),

(4.8)

where Dij(k, s - s’) = h,(s - s’) j c
4 qiq&,,,(q,

s - s’> k&z

(4.9)

and C,(k, s - s’) = hk(S - s’) s,<.,

dq n 6(nk - q) 2

qiGylyg(q, s - s’) kylk,,* . (4.10)

On differentiating

(4.8) we find

( ata + yk2) ua(k, t)

We now make the assumption that the turbulent velocity distribution is quasistationary (we shall show that it is self-consistent); that is, u,(k, t) is assumed to be nearly constant in time. uB(k, s) may be taken out from under the integral sign on the right-hand side of (4.11). The equation thus simplifies to ($

+ vk2) uci(k, t) = 444

[&

Wk)

-&

-

C,(k) &]

us@, t),

(4.12)

HOMOGENEOUS,

ISOTROPIC

345

TURBULENCE

where &(k)

= j-I

ds

--m

&(k, t - s) (4.13)

Gy&q, 0) =s q
= St ds C,(k, t - s) --m

(4.14)

and G&q, t - s) has been approximated by G,,,&q, 0). To summarize, we find from (4.12) that the rate of the change of u,(k, t) in time as a result of the interaction with large eddies is

largeeddies

= 4,(k) [& &,(k) &

-

C,(k) -&]

3

K&, t>. (4.15)

The two terms on the right-hand side possess a simple physical interpretation. The first corresponds to an effective diffusion in wavenumber space, as we shall show below. The second corresponds to the influence of the wavenumber-dependent force C,(k), and will likewise be discussed later. To consider diffusion in k space, we note that a diffusion coefficient, D(k), may be defined according to

W) - KW2/4,

(4.16)

where Ak is the change of the wavenumber of an eddy with wavenumber k during time At. Since the size h of the eddy is roughly related to k according to h - (l/k), we may likewise write D(k)

(4.17)

- k4[(Ah)2/At],

where AA - -Ak/k2 is the variation of the size of the eddy during At. In order to know how much the size of the eddy changes during At let us look at Fig. 8. Since A and B move with different velocities AB stretches by the amount Ah - At@(B)

- u(A)) N Ath[du(x)/dx],

(4.18)

where u(A) and u(B) are velocities of a large eddy at A and B, respectively. Substituting (4.18) into (4.17) we find D(k)

- k2 A t( 1du(x)/dx

j2),

(4.19)

346

NAKANO

where the bracket stands for some average (for instance, a time average). Since

D(k) becomes D(k) -kZdt

s q
4 q2gk 0).

(4.12)

What should dt be ? If the eddy of the size k-l is sufficiently stable, d t will correspond to the life time of large eddies. If, however, this life time is long compared to that of the eddy of wavenumber k, then it is the latter which enters into (4.21). Such is the present case, since as a result of its interaction with small eddies, our eddy of wavenumber k will lose its memory in a time interval of the order of (v&k) k2)-l, a time much shorter than a typical large-eddy life time. Therefore At - (wf(k) k2)-l, and D(k) takes the form

4q2&, 0)/~&L D(k) - s,,,,

(4.22)

which is the same as (4.13). Although Cj(k) seems to arise as a result of the sharp cutoff we have imposed on q(q < nk), it is quite generally of importance, and thus is present even for a slowly-varying cutoff. To see this we note that C,(k) is related to Dii(k) according to (4.23) with the aid of (4.13) and (4.14). In this expression, there is no reference to a cutoff in q. Furthermore, on making use of (4.19) and (4.20) v&k) k2Dij(k) may be regarded as the square of the effective velocity which carries the eddy of the size k-l in wavenumber space. If it depends on k there may be a preferable direction for this transport in wavenumber space. This process may be characterized by the diffusion force C(k). We shall see in the following section that in the inertial region v&k) k2 varies as k2/3(Dij(k) - k8/3); (4.23) may be then approximated as

G(k) - (Wi) D&L

(4.24)

which shows directly that C,(k) arises from the k dependence of the diffusion coefficient.

HOMOGENEOUS, ISOTROPIC TURBULENCE

341

In this and the previous sections we have introduced the cutoffs mk and nk to denote the fact that we have limited ourselves to the interaction of a given eddy of wavenumber k with eddies which are either appreciably smaller or appreciably larger. It is to be expected from the principle of action and reaction that just as an eddy with wavenumber q acts on the eddy with wavenumber k as an effective viscosity, the eddy with wavenumber k distorts the eddy with wavenumber q with an equal reaction. It follows that the conditions q 3 mk and nq >, k should be equivalent, or m N l/n. (4.25)

V. THE ENERGY SPECTRUM In this section we determine the turbulent energy spectrum by combining the principal results of the preceding two sections, that is, by taking into account the coupling of both small and large eddies to our given eddy. A basic equation for the energy spectrum is derived and put in a form in which the consequencesof energy conversation are rendered transparent. From this equation the Kolmogorov spectrum in the inertial region and a Townsend-type spectrum in the dissipative region are obtained. A. The Equation of Motion for an Eddy Our basic eddy equation of motion may be written symbolically as

(& + @) udkt>= [Al,,,,

eddfes + [-Ilarge eddies . (5.1)

Substituting (3.12) and (4.15) into the above equation one finds

where the first term of the right-hand side takes into account through v dissipation produced by random thermal-particle motion and through verf the coupling to small eddies. The second describesdiffusion in wavenumber space, while the third corresponds to an internal force on the eddy. As a result of the balance between the three contributions a steady state may be established. It should be added that (5.2) is correct for times comparable to or larger than (v&k) k2)-l. Let us now consider the extent to which a stationary turbulent state may be

348

NAKANO

established. The complete equation of motion the aid of (3.12) and (4.11) as

($

+ vk2) u,(k, t) = - w(k)

k%(k,

for the eddy may be written with

t) + A&)

jt--m ds

+ c’ T&m %4l, t) 430(- q, t>, a

(5.3)

where in the second term of the right-hand side it is not assumed that u,(k, s) is nearly constant in time. The last term in (5.3) corresponds to those interactions which have not been included in the first two terms and may be thought of as giving rise to a fluctuating force on u,(k, t). The fluctuating force may feed energy to u,(k, t), and it may affect the relaxation of u,(k, t); in such a case u,(k, t) is not necessarily nearly constant in time even in the stationary state. The fluctuating force, however, may be neglected as long as our physical picture about eddy interactions is correct in describing the turbulence. Therefore, in the framework of our physical picture about turbulence the last term in (5.3) may be ignored and we obtain (5.2) instead of (5.3). B. The Energy-Flow

Equation

Let us derive the equation for the pair correlation in (2.4). Multiplying (5.2) by us(-k, t) one obtains

$

G&c,

0) = -2(v

+ w&W

k2G&

0)

Setting j3 equal to (Yand summing over 01,(5.4) becomes

- $+,@)

a~& t> a%%(--k, t> (F ak, >’

function

Gas(k, 0) defined

HOMOGENEOUS,

ISOTROPIC

349

TURBULENCE

where g(!c, 0) is related to G&k, 0) according to (2.5). Because g(k, 0) depends only on the magnitude of k the second term of the right-hand side of (5.5) may be simplified:

Since Dij(k) kj/k is a vector with the index i, it may be written as D&)(kj/k)

= (Wk)(kknlk2)

D,m(k)*

On substituting this expression into (5.6) we find [ & z Ddk) & 3 - C,(k) $-I =

$2 D(k) & I z

= 37 = -&

&

-

dk C(k) $1

g(k, 0) + ki 2 $

D(k) Y& dk,

0) + 2 q

0) g(k, 0) q

&

&

g(k, 0) -

g(k, 0) - ‘3)

C(k) 4

g(k, 0)

(5.7)

Y& g(k 01,

where D(k) and C(k) are defined as D(k) = +

&n(k)

= 15ve;f(k) 1”” &I q2%d

(5.8)

0

and ‘3)

2 kl = k cd&) = 15veff(k) I om 4 n W

with the help of (4.13) and (4.14). Substituting ; g(k, 0) = - 2(v

+

wt(k))

k2g(k,

- 4) q2E(q)

(5.9)

(5.7) into (5.5) one finds 0)

+

&

D(k)

-$

dk

0)

+ 2 D(k) a g(k0)- C(k)4 dk 0) k ak - &

D&) (%@ aua(--;’“).

(5. IO)

350

NAKANO

We now pass to the energy-flow equation with the aid of (2.7). On substituting (2.7) into (5.10) we obtain

y& I

= & (D(k)k2& F) - &

4,i-k2 l&(k)

- 2(v + veff(k)) k2E(k) (%$$

““^‘j$

-

C(k) k2 &

“‘)

F

(5.11)

as our basic equation describing turbulent motion. Each term of the right-hand side of (5.11) possesses a simple physical interpretation. The first term describes a diffusive energy flow in wavenumber space arising through coupling to large eddies. The second represents energy loss through the dissipative processes; v describes the effect of thermal-particle motion, verr that of the random motion of small eddies. The third describes energy transfer from large eddies through the wavenumber-dependent force C(k). The last is nothing but an energy loss due to diffusion in wavenumber space. If the energy flux in wavenumber space is defined by E(k), (5.11) becomes g E(k) + div c(k) = -C(k) + $+

k2 -$ F 4n-k2&(k)

-

[2(v i- v&k)) (w

k2E(k)

“‘=$$

“‘)I,

(5.12)

where (5.13)

c(k) = --D(k) k2(8/8k)[E(k)/k2].

In (5.12) div e(k) means &(k)/ak. One may easily interpret (5.12) as the equation of energy conservation. There are two distinct regions corresponding to possible solutions of (5.12). In the first, which we identify as the inertial region, there exists a balance between energy transfer and energy loss such that the net flux of energy vanishes according to (a/i%) E(k) + div e(k) = 0 (5.14) and the explicit energy-balance equation - k2 - 2(v + w(k)) k2E(k) -C(k) k2 za E(k) [ + &

4,&s D,,(k) (v

aum’&;’ “)I

= 0.

(5.15)

HOMOGENEOUS,

ISOTROPIC

351

TURBULENCE

In the second region, which we identify as the dissipative region, the above energy balance between energy transfer and energy loss no longer holds. In this case we have to solve the entire equation - 2(v + v&k)) - -&

4&2 &@)

(w

k2E(k) - C(k) k2 &

aum’;;y

“)

p

= 0,

(5.16)

which is obtained from (5.11) by setting aE(k)/at equal to zero. On the basis of the above criterion, we may expect that the energy spectrum takes different forms in the inertial and dissipative regions, so we proceed to consider these separately. C. The Energy Spectrum in the Inertial Region The inertial region is characterized by the fact that the diffusive energy flow in wavenumber space is independent of k; E(k) = const E E,

(5.17)

where E is the wavenumber-independent energy flow in wavenumber space. Only with (5.17) one can separately satisfy (5.14) and aI?(k)/& = 0. With (5.13) and (5.17) one finds E = --D(k) k2(+?k)[E(k)/k2].

(5.18)

Suppose that E(k) takes the form (5.19)

f (4 k-3 wheref(e) is some function of E. Substituting D(k)

-

(5.19) into (5.8) one finds

f(E)Vk--B/2+7/2

(5.20)

with the help of (3.10). On the other hand one also finds D(k) N cf(+’

ke+l

(5.21)

from (5.18). The two expressions, (5.20) and (5.21), must be the same; hence

f (W2 - lf(c)-1 -f(e) - 9,

k-S/2+7/2 - kBf’ -+ ,B = 513, 595/7312-4

352

NAKANO

so that in the inertial region E(k) takes the form E(k) - E2/3k-5/3

(5.22)

which is the Kolmogorov spectrum [2]. In order to calculate the numerical coefficient of the spectrum we assume E(k)

In this case v&k),

= Fc2/3k-5/3.

(5.23)

D(k) and C(k) are computed as v,n(k)

= (4/15)li2 m-2/3F1/2&3k-4J3,

(5.24)

D(k)

= (3/@))1/2 m--2/3Fl/2&/3k~/3,

(5.25)

c(k) = (l/15)112 m--2l31;1/2&/3k~/3

(5.26)

with the aid of (3.10), (5.8), (5.9) and (4.25). Substituting (5.18) the numerical coefficient F reduces to

(5.23) and (5.25) into

and

F = (240/121)li3 m4j9.

(5.27)

Kolmogorov also argued that the boundary between the inertial and dissipation regions is specified by the wavenumber of the order of kd = (c/v3)lj4,

(5.28)

the only quantity which can be set up dimensionally from v and E. If the wavenumber k, at the boundary is determined in such a way that u&kJ = v, k, is related to k, according to kl = Gkd,

(5.29)

G = (24O/121)1/8 (4/15)3/8 rn-li3

(5.30)

where on making use of (5.24) and (5.27). Comparison of our results with experiment will be made in the following section. Let us check the energy-balance equation (5.15) which shows that the energydissipation term represented by the second term is compensated by the energy supply represented by the first. We do not know how to estimate the last term in the bracket in (5.15); we, therefore, simply ask that it be determined by (5.15) itself. Substituting (5.23), (5.24), (5.26) and (5.27) into (5.15) we obtain ?(W3

4nk2 D&)

(*

au (k, t) ih,(-k, akj

t)

=

4 E ---9 33 k

(5.31)

HOMOGENEOUS,

which is numerically

ISOTROPIC

353

TURBULENCE

small compared to the dissipation due to small eddies 21&k)

k2E(k) = (16/l l)(~/k).

(5.32)

[In (5.31) the kinematical viscosity term has been neglected because it is much smaller than u&k) in the inertial region.] We conclude that to something like 10 % accuracy we might neglect (5.31). D. The Energy Spectrum in the Dissipative Region In the dissipative region each term of the energy-conservation equation (5.12) has the same meaning as in the inertial region. However, there are two significant differences in the calculation of the various contributions in the dissipative region. One, which we have already mentioned, is the fact that the diffusive energy flow e(k) depends on k because of energy loss due to dissipative processes. The second, because k in the dissipative region is much larger than kd , is that V, the usual viscosity coefficient, is much larger than v&k). In order to calculate D(k) and C(k) let us go back to the original expressions (4.13) and (4.14), where (v&k) k2)-l corresponds to the life time of h,(t). Since hk(t) takes the form (3.4) the life time of hK(t) is (vk2 + v&k) k2)-l - (vk2)-l in the dissipative region. Therefore, in (4.13) and (4.14) v,ff(k) k2 must be replaced by uk2. Consequently, from (5.8) and (5.9) we may calculate D(k) and C(k) as

D(k)= &, s,” 4 q2E(q), (5.34) in the dissipative region. As will be shown later, E(k) decreases exponentially as k increases, so that the upper limit of the integration in D(k) can be safely extended to infinity (with exponentially small error). Thus

D(k)= & s,”4 q2EGd I

(5.35)

where E’ is the total dissipation energy per unit mass per unit time defined by E’ = 2v m 4 q2E(ds0

354

NAKANO

Since the energy which flows in the inertial region without any appreciable dissipation is eventually dissipated in the dissipative region, the energy-flow rate E in the inertial region must be the same as the dissipation rate E’ [27]. Therefore D(k) becomes simply D(k)

= E/15V2

(5.36)

= D,.

On the other hand, since E(k) N exp(--k2) C(k) = (2/l 5V) nWE(nk) is exponentially small. For large wavenumbers v&k) is also exponentially small [compare (3.10)]. Consequently, the equation (5.16) which determines the spectrum in the dissipative region becomes D, &

(k2 &

- -&

F)

- 2vk2E(k)

4&2 D,,(k) (w

““8-,f’ “) = 0.

(5.37)

For the moment let us ignore the last term in (5.37). Then the differential equation for E(k) is -$

&

(k2 $

F)

- k2E(k)

= 0,

(5.38)

which possesses the asymptotic solution, E(k) -

exp - - ~‘(30) - k2 i

2

kz2 1

for k > k, . Equation (5.39) implies that large wavenumber disturbances are exponentially damped by thermal viscous process, as one might expect. The above result suggests that u,(k, t) may possess the same asymptotic behavior; we proceed to assume that u,(k) N exp(-(q/2) k2) and exploit the consequence of this assumption. [The phase factor of uoL(k)will not be important compared to the exponential.] Then aUrn& t)/aki - -@iudk

(5.40)

t)

and the last term in (5.37) becomes - &

4rk2q2kikj

D&k)

(u,Q, t) u,(-k,

t)> -

-2+k2

DOE(k)

(5.41)

HOMOGENEOUS,

ISOTROPIC

TURBULENCE

with the aid of (2.4), (2.5), (2.7), (5.8) and (5.36). Substituting and (5.41) into (5.37) and keeping the largest term one finds 2q2k2D,E(k)

- 2vk2E(k)

355

E(k) - exp(-nk2)

= 0,

which yields

The asymptotic form of E(k) is thus seen to be

E(k) - q-

dW)tk2/kd2))

rather than (5.39). The coefficient of (5.42) may be calculated once the last term in (5.39:) is treated more accurately. The exponential form, (5.42), however, must be correct. Our result, (5.42), involves the same type of spectrum as was proposed by Townsend [21]. The connection of our theory with that of Townsend will be made in the following section. E. Consistency and Validity of the Present Theory The theory of turbulence we have developed may be characterized as a selfconsistent rather than a deduced theory, in the sense that nowhere does there appear a small parameter which guarantees its validity. Throughout our treatment we have made a series of approximations (RPA, smaller time scales for decay of energy spectrum, etc.) of which we have only demanded that they be self-consistent, i.e., that the results we obtain be consistent, a posteriori, with the assumptions made in deriving them. We review here the extent to which our solutions are self-consistent, and then consider the larger question of the physical circumstances under which such a self-consistent theory may be expected to be valid. For example, let us consider the RPA which forms the basis for our diagrammatic treatment. The eddy equation of motion (5.2) shows that couplings to other eddies enter into wf(k), D&) and C,(k) only. Inspection of (3.10), (4.13) and (4.14) makes one notice that wf(k), Dij(k) and C,(k) depend on the intensities, but not the phases of these other eddies. Therefore the solutions we have obtained to the Navier-Stokes equation are such that the phase of u,(k, t) does not couple to the phases of other eddies. The solutions are therefore consistent with the assumption which led to them, namely, that the phases of different eddies may be random.. Next we consider assumptions made about time scales. In our theory we have made extensive use of the assumption that the relaxation time of large eddies is large compared to that of small eddies. The relaxation time of the eddy with wavenumber k may be calculated by h,(t) as is done in (2.20). We make use of

356

NAKANO

this result, and the above assumption to show in Section IV that larger eddies cannot play an important role in relaxing a given eddy. Its relaxation is determined only by its interaction with smaller eddies and is characterized by he(t) = (-(v + v,rr(k)) k2t). In the inertial region where K&) is much larger than v, v&k) k2 is proportional to k2/3 which implies that a larger eddy has a longer life time in the inertial region. In the dissipative region where v > wf(k) the statement likewise is true. By inspection, therefore, the results we have obtained are consistent with the assumption concerning time scale we have made in deriving them. We made use of the assumption that the turbulent velocity distribution is stationary. As a result of this stationary state the net flux of the energy in the inertial region vanishes and consequently the scaling argument becomes possible. With the aid of the scaling law we were able to calculate the energy dissipation due to smaller eddies and the energy transfer due to larger eddies, which almost satisfy the energy-balance equation. Therefore the result is consistent with the above assumption. Assuming that we have complete self-consistency it is natural to inquire as to the physical circumstances under which our description is valid. There is only one global parameter which describes the extent to which fully developed turbulence may exist, the Reynolds number R. More specifically, we may inquire, for what value of R does the inertial region exist ? To answer this let us assume the inertial region is located between k, and k, , where k, is the wavenumber of a typical eddy in the energy-containing region. The turbulent viscosity v&kJ at wavenumber k, is related to the Reynolds number and the kinematical viscosity according to [28] w&)/v

- RI&,

(5.43)

where R is the Reynolds number for the flow configuration of interest and R,, is the corresponding critical Reynolds number. On the other hand, at the boundary k, between the inertial and issipative regions v&J Combining

= v.

(5.44)

(5.43) and (5.44) one obtains

w&,)/w&)

- R/R,, .

(5.45)

With the aid of the fact that v&k) cc k-4/3 in the inertial region shown in (5.24), (5.45) reduces to (5.46) Wk, - W&33’4 which estimates the size of the inertial region roughly. Detailed connection with experiment will be developed in Section VII.

discussion in

HOMOGENEOUS,

VI.

COMPARISON

ISOTROPIC WITH

357

TURBULENCE OTHER

THEORIES

In this section we compare our results with the theories of Kolmogorov, Heisenberg, and Townsend. We show that the theory we have presented supports the physical assumptions of Kolmogorov, and that it provides both a natural justification for and an extension of Heisenberg’s theory in the inertial region. In the dissipative region it is closely related to Townsend’s theory. A. Kolmogorov’s

Theory [2]

Kolmogorov’s basic assumption was that the eddy interaction in the inertial region was nearly local in wavenumber space. This led him to argue that the equation for the energy spectrum of an eddy with wavenumber k in the inertial region might be determined by the characteristics of eddies with wavenumbers near k. Under these circumstances, the inertial region would be relatively independent of the energy-containing region, and its properties could be calculated. -- ----

0

FIG.

13.

Profiles

of E&)/V,

S,(p) representJ$ dk E(k)k2

ko

P

E(k) and E(k)/?. {E(k) and ~-1~ dk E(k)/kz.

E(k)/k’ E(k) E(k)k*

k,

is taken

k

from

Ref.

[6,

p.

189.11 S,(p)

and

Our theory supports that assumption. In Eq. (5.12) for the energy spectrum there appear two fundamental functions, D(k) and v&k) [C(k) behaves similarly to D(k)]. Let us consider what parts of the spectrum make the major contributions to D(k) and v&k). To this end it is helpful to draw profiles of E(k)/k2, E(k), and E(k) k2 [29] in Fig. 13. It is convenient to rewrite down the expressions for D(k) and we(k) here: (6.1)

w@)-

(6.2)

358

NAKANO

with the aid of (3.10) and (5.8). Let us consider an eddy with wavenumber p in the inertial region. If we define

J‘tp4 q2w=UP), smdqE!$L S(P),

rnP

then D(P) - ~dP)ld/[&(P)l~ W(P)

-

2/L(P)l*

It may be seen in Fig. 13, where the shaded areas denote S,(p) and S,(p), that these quantities are mainly determined by the spectrum with wavenumbers near p, so that D(p) and v&p) are, too. It follows that the eddy interaction in the inertial region is nearly local in wavenumber space. This locality guarantees the existence of the inertial region. In order that the p-5/3 spectrum may exist v&p) and D(p) must be vern(p)

-

p-4’3,

D(P) - P”~,

which correspond to S,(P) - P4’3, S,(p) - p-E/3.

Therefore, the minimum wavenumber, k, , of the k-5J3 spectrum region is the minimum wavenumber for which S,(p) - p 4/3. The upper limit k, of the k-5/3 spectrum will be the maximum wavenumber for which S,(p) ~p-*/~. If one chooses p in the dissipative region, D(p) must be defined as D(P) -

; s”* 4 qWq) 0

(6.3)

as shown in (5.33). Then D(P) - &(P)/v

and is mainly determined by eddies around k - k, . Therefore, the interaction in the dissipative region is not local.

HOMOGENEOUS,

ISOTROPIC

359

TURBULENCE

B. Heisenberg’s Theory [9] We consider now the relationship of our theory to that of Heisenberg [9]. Heisenberg combined dimensional analysis with physical insight to arrive at the following equation for the integrated energy spectrum

a k at j 0 dq Qd + 2,~ j” 0 dq q2&)

= -2d%)

jk dq q2Wd, 0

(6.4)

where &k) is an effective viscosity introduced to describe the influence of small eddies on large eddies, defined by

vzf(k) = A j:, 4 d(y). Differentiating

(3.11)

(6.4) with respect to k, we obtain 4 E(k) + &

(2&n(k)

j” dq q2&q)) = --2vE(k) k2, 0

which implies that the energy flux G(k) in wavenumber space in Heisenberg’s theory is &t)

= 2&k)

j: dq q2W).

(6.6)

If now, in place of the Heisenberg expression for the effective viscosity we use that derived in Section 111, v&k)

= A [j:,

4 T’]1’2~

(3. IO)

the energy-flux term (6.6) becomes

&V = 2(veft(kN2( j: dq q’E(q)/w(k)) (6.7) - D(k) jIk 4 y, where D(k) is defined in (5.8). On the other hand, our expression for the energy flux is c(k) = --D(k) k2(i3/ak)[E(k)/K2]. (5.13) The differences between (6.7) and (5.13) are the following: according to (6.7) the eddy diffusion D(k) acts on all eddies smaller than the eddy with wavenumber k;

360

NAKANO

on the other hand, (5.13) describes an eddy diffusion process in which D(k) acts essentially only on eddy with wavenumber k. E(k)/k2, however, is a very rapidly varying function of k (see Fig. 13) so that the major contributions to Jzlc dqE(q)/q2 in (6.7) come from the eddies very near k. It follows that in the inertial region (6.7) has the same structure as (5.13), and one can in fact write

K

-k2

a

ak

E(p)

k2

so that in this region our theory and that of Heisenberg are in close agreement. C. Townsend’s Theory [21] Townsend argued that the eddies at wavenumbers k > kd form concentrated, but weak vortex sheets which are stretched by large eddies and weakened by dissipation effects at the same time. The balance between these processes determines the configuration of the vortex sheets. Suppose that a vortex sheet lies on the X-Z plane and that it has only an x component, [, which is a function only of y. This vortex sheet is supported by a large shear motion expressed by (LXX,--o~y, 0). Then t(y) obeys the equation (6.8)

which has been derived from the equation for vorticity o

(a/a+ = -curl(vxo) where v is a shear motion.

+ vv20,

According to (6.8) the energy spectrum for large k is E(k) N (l/k2) exp(-(a/v)

k2).

(6.9)

Townsend set 01equal to (c/~v)‘/“; then E(k) N (l/k3 exp(-2k2/k,“),

(6.10)

which Townsend found was in fair agreement with experiment [30]. Our result (5.42) resembles (6.10) closely, in that it contains a similar exponential term. Townsend’s treatment, however, is open to question because 01is a fluctuating quantity. In order to see what happens for a fluctuating IX, let us Fourier-analyze (6.8) as

361

HOMOGENEOUS, ISOTROPIC TURBULENCE

Substituting fk(t) = -,“, into the right-hand

ds e-uk2ft-s)ak $

&(s)

side of (6.11) and taking an average over a2, (6.11) becomes

where the last term has been obtained by approximating case in the steady state. Finally in the steady state vk2&(t) = k &

$

k$

&&s) by &(t) as is the

&c(f)

from which one finds the equation for (I tic 1”) as k --$ $

k 4

(I & I”> - 2vk2<1 &x 1”) - 2 q

(I 2

1’) = 0.

(6.12)

If we approximate (a2> by a quantity of the order of E/V instead of setting 01equal to (~/4v)l/~ as Townsend did, we may write (~2>1v - D,

(6.13)

except the unknown numerical factor. Substituting (6.13) into (6.12) we see that since (I & 12) - E(k), (6.12) takes almost the same form as (5.37).

VII. COMPARISON WITH EXPERIMENTS A. Energy Spectrum in the Inertial Region As we have discussed in Section V, a high Reynolds number is required to establish a substantial inertial region. It is, however, difficult to achieve highReynolds-number turbulence in the laboratory because the characteristic size L is limited geometrically by the containers with which one works. Indeed, in ordinary laboratory flow typical Reynolds numbers are only lo4 - 105, so that the flow is likely not “turbulent” enough to guarantee the existence of a large inertial region. In nature, on the other hand, where L may be as large as one likes, one might hope to observe large-scale motion and high-Reynolds-number turbulence-for example, in the atmosphere [31], oceans, rivers, and interstellar space [32].

362

NAKANO

Grant, Stewart and Moilliet [3] have carried out experiments in a tidal channel where the Reynolds number is up to 2.8 x IO*. In this experiment the inertial region extends roughly from 1O-3 cm-l to 10 cm-’ in wavenumber space, and one would hope to observe the Kolmogorov spectrum within these limits. In practice they were able to verify the c2/3k-5/3 spectrum for two or three decades. In the laboratory, Gibson [4] has produced turbulent flow with a Reynolds number of the order of lo’, for which the inertial region extends roughly from 0.4 cm-l to 70 cm-l; he was able to observe the Kolmogorov spectrum for one or two decades of k. The above experiments thus support the existence of a universal Kolmogorov spectrum in the inertial region of the fully developed turbulence. We now consider the comparison of our theoretical results with experiment. As shown in Section V our theory derives the Kolmogorov spectrum in the inertial region; this means that the scaling of D(k) is correct. In order to check the correctness of the numerical factors which enter our expression for D(k), we compare the numerical coefficient F of the Kolmogorov spectrum, defined by E(k) = Fc2/3k--5/3 (5.23) with experiment.

Since the energy spectrum is related to D(k) through -(a/ak)[E(k)/k2]

determination of F from the F’s calculated from m = 2, 3 and 4. Given E(k), the agreement may

(5.18)

= e/D(k) k2,

(5.23) suffices to determine D(k). In Table I we compare (5.27) with experiments using three different cutoffs, the difficulties of making quantitative measurement of be seen to be comparatively good.

TABLE I The Comparison of F with Experiment a Grant et al. [3] 1.4

Gibson [4] Kistler et al. [33] 1.6

2.7

m=2 1.7

m=3 2.1

m=4

2.3

a F’s are calculated from (5.27) in the cases of m = 2, 3 and 4.

Next let us consider the turbulent viscosity v&k). In the inertial region wf(k) is proportional to d/3k-4/3 as calculated in Section V based on the scaling argument; again, how accurate is our estimate of the numerical factor of proportionality ? The best way of checking it is to compare the calculated energy spectrum in the whole wavenumber region to that observed in experiment, especially around the wevenumber at which a deviation from the Kolmogorov spectrum begins. The energy spectrum is, however, calculated in the inertial region and in the

HOMOGENEOUS,

ISOTROPIC

363

TURBULENCE

dissipative region at large k. One way to overcome this difficulty is to develop a suitable extrapolated spectrum [such as E2/3k-5/3 exp( - d( 15) k2/kd2) 1; however, we have not as yet been able to find a good extrapolated spectrum which agrees with experiment [3]. We therefore use a less sophisticated method to check the numerical factor of v&k). The Kolmogorov spectrum exists as long as v&k) > v. It is a reasonable assumption that the spectrum begins to deviate from the Kolmogorov spectrum when w(k) - 5v, that is, v&k,) - 5v, where k, is the wavenumber at which a deviation from the Kolmogorov spectrum begins. Making use of the relation, w(k) -d k-4/3, k, may be related to k, k, - k1/5314 = kJ3.34,

(6.1)

where k, is determined in such a way that v&k,) = v. On making use of our result for k, , (5.30) we obtain the values k, listed in Table II. Grant et al. [3] observed that k, - 0.1 kd , and Gibson [4] observed that k, ,- (0.1 - 0.2) k, . The agreement of our admittedly crude estimate with experiment is again good. TABLE II The Estimated Wavenumber at Which a Deviation from the Kolmogorov Spectrum Begins m=2

m=3

m=4

0.17 kd

0.14 kd

0.13

kd

B. Energy Spectrum in the Dissipative Region Townsend’s spectrum at large k takes the form (l/k3 exp(-2k2/kd2) which he argued was reasonable fit to the experiment of Stewart and Townsend [30]. However, their experiment measured the spectrum of wavenumber only up to 1.6 kd . In such a low wavenumber region the difference between Townsend’s spectrum and ours, exp(- d(15) k2/kd2), is not sufficiently large to permit one to distinguish between the two forms for the spectrum. It is, in principle, possible to verify our spectrum in the region of large wavenumbers where only the exponential form is important. To our knowledge such experiments have not yet been carried out, primarily because of the difficulty in making measuring instruments (e.g., a hot wire) small enough to observe such fine scale motion.

364

NAKANO

VIII.

CONCLUSION

In conclusion we summarize our results and comment briefly on possible future developments. We have seen that the irregular variation of motion in turbulence may be regarded as the superposition of eddies of different sizes. We have found it convenient to split these eddies into two groups; those which are appreciably smaller than the eddy under study and those appreciably larger. Small eddies resemble particles which move thermally at random; they push parts of the eddy back and forth, and produce a viscous drag. Their effect is thus described by effective turbulent viscosity which is a functional of the energy spectrum of small eddies and is interpreted as the product of a mixing length and a suitable velocity as proposed by Prandtl. On the other hand, large eddies carry the eddy while they distort it, that is, the size of the eddy is changed by large eddies. Large eddies thus act to diffuse the eddy in k space by exerting a diffusion force and feeding energy to it; the diffusion coefficient and the diffusion force are determined as functionals of the energy spectrum of large eddies. In the inertial region the contributions from small and large eddies are in equilibrium. The effective viscosity, the diffusion coefficient in k space and the diffusion force are mainly determined by the energy spectrum near the wavenumber of the eddy. In other words, the interaction is local in wavenumber space, which is a version of Kolmogorov’s assumption. One thereby justifies the scaling hypothesis and finds the effective viscosity -k-+13, the diffusion coefficient -kai3 and the diffusive force -k5/3. The resulting energy spectrum is the universal Kolmogorov spectrum, proportional to k-5/3. In the dissipative region the scaling argument is useless because the kinematical viscosity dominates the turbulent effective viscosity. Here the distortion of the eddy due to large eddies is balanced by the viscous effect. The energy spectrum takes an exponentially decaying form similar to that found by Townsend. The discussion is developed self-consistently throughout this paper. We first assumed the random phase approximation, the existence of hierarchy of relaxation times for eddies of different sizes and so on. These assumptions are then confirmed by the results which are derived from them. In our self-consistent approximations the Reynolds number does not come in explicitly so that our results would seem to be true for turbulence of any intensity. This is, however, not the case, since we have limited ourselves to homogeneous and isotropic turbulence which may only be realized in the small-scale motions characteristic of the inertial and dissipation regions. In order that there may exist an appreciable inertial region which may be described by homogeneous and isotropic turbulence we show that a comparatively high Reynolds number is needed. We estimated that this Reynolds number must be of the order of ten times the corresponding critical Reynolds number.

HOMOGENEOUS, ISOTROPIC TURBULENCE

365

The results we have obtained are not applicable to the energy-containing region where inhomogeneity and anisotropy are important. However, we may expect that the interaction mechanism there is the same as developed in the present theory; smaller eddies give rise to an effective viscosity and larger eddies distort a given eddy. In this region, however, the effective viscosity, diffusion coefficient and diffusion force may depend on position and direction. The study of the energy-containing region is necessary to understand how the instability, which leads to turbulence, is suppressed and how transport of energy and momentum is modified by the presence of turbulence. It is hoped that the approach developed in this paper may be extended to a study of the energy-containing region. In the present paper all vertex corrections have been neglected. Although the equation for a vertex operator can be written down with the aid of diagrammatic analysis introduced by Wyld, it is difficult to solve. One can take account of the first few vertex corrections, but one has no guarantee that in so doing one has achieved. a better result. This expectation is not unreasonable when we reflect on the fact that we would have been led to an incorrect result in Section II if we had not included a certain infinite series of diagrams in the calculation of the self-energy of an ed.dy. It is hoped to examine questions of vertex renormalization in future work.

APPENDIX

A:

EQUATION

The motion of incompressible

FOR INCOMPRESSIBLE FLUID

fluid is governed by the Navier-Stokes

(au/at) + (u * V)u = -(l/p)

vp + vv2u

equation (A4

where II, p and v are the fluid velocity, mass density and viscosity, andp the pressure. The condition for incompressibility is v*u=o.

(A.4

Taking the divergence of (A.l) with the help of (A.2) we find v2p = -pv

* ((u . V) u),

which can be solved as PW

= &

$

V’ . ‘;(;”

J’,’ 4x’>)

with VT(x) = 0.

dx’ + I;(x)

(A.3)

366

NAKANO

F(x) must be chosen so as to satisfy the boundary condition. of motion takes the form

--a

( at

16'~)u,(x, t) = fPt(x,

t)

+ 4j

dx’ ?‘&(x,

Then the equation

x’) ua(x’, t) u,,(x’, t),

(A.4)

where

+

&-

[$

L

-

ax, ax, a2

]

&

j

x

A

XI

1

(A.5)

and c11,j3 and y denote the components of the rectangular coordinates. fext(x, t) is the external field defined by = -(l/p)

f-t(X)

with the condition

W(x)

that v . fext(x) = 0.

64.6)

The Fourier transformation u(x, t) = C u(k, t) eik.x, k

u(k, t) = (l/V)

1 dx u(x, t) e-ik.x,

casts (A.4) into the following form in k space: i&

+ vk2j urn@, t) = ftxt(k

t) + V&(k)

c u,@ - q, t> dq, t>, cl

(A.7)

where

W-3 and -‘l&k>

= L - bW,/k2).

(A.9)

367

HOMOGENEOUS, ISOTROPIC TURBULENCE

APPENDIX

B:

DERIVATION

OF (4.8) FROM (4.7)

We begin with (4.7) which we rewrite here for the sake of convenience: u,(k, t) = -s”

-co

ds 1’ ds’ e-vka(t-sW’&(k) -cc

x *
s) q . &

MS - s’> %,&)

u,,(q, s’>q * &)

Y.& 0. (4.7)

z pCnk and alaki do not commute.

With the help of the identity

where 0(x) is such that e(x) = 1 for x > 0 and 0 for x < 0, (4.7) becomes dk

t> = - jl,

ds jl,

ds’ e-vk2(t-s)~s,,(k)

[&

a x MS - s? Go&) - akj - s dq ‘2 x G,,,(-q,

s - 8’) MS - s’> %Jk)

2

j,,,, \

4 qaGl,2(-q,

s - s’)

n SW - d qj &]

3

Mk, 0,

(B-1)

where (2.4) and (2.19) have been used. Since UC&Or,s - s’) = -‘f’&(k) -

s

[&

jgGn, 4 wiG,&-q,

dq n S(nk - q) k-q k GCy,(-q,

s - s’> h@ - ~‘1 %dW s - s? US - s’>G&)

$1

& 3

3 WV

is a tensor with the indices 01and /3 which is characterized only by a vector k, it may be expressed as G&

s - ~‘1 = &(k)

A@, s - s’) + (k,k,/k2) B(k, s - s’)

in the isotropic case. A and B include derivatives with respect to k. 595/73/%

(B.3)

368

NAKANO

Let us calculate A(k, s - s’) first: A(k, s -

s’) = ; A,,(k)

U&k,

s -

s’)

= - '2Cd4 [+ j,,,, 4 qiq&,,,(-q,s - s')W - $7 a __akj - I dq n S(nk - q) kk*q q&,,(-q, s - 8’) x %Y*@) a (B-4) x Ms - s’)%3,,or) akj 1) with the help of (2.6) and (2.9), &Or)

+‘isr,@> = %dk)-

Exchanging the order of V&(k)

and a/i3ki in (B.4) we find

46

j,,,,

4 qiqjG,,,(--4,

a --akj

s 4 n Gk

s - ~‘1 = - ; [s x ?%#>

x MS - ~‘1 %,,(k) x G&-q,

s - ~‘1 Ms - 8’) 5&,(k) k-q - d qj k

%;ss,Or) $-I

s - ~‘1 Ms - ~‘1

G,,(-q,

+ ; j,,,,

s - ~‘1

4 qiqi

a

W&(k)

ski %,,(k>xi -

(B.5)

With the aid of (2.6) and (2.9) we obtain the identities +‘&Q

w-9

%ssv#) = -2JcyA

and

P-tr,,,,(WW %&‘W = --3k&,, + k,L, . Substituting &k

(B.7)

(B.6) and (B.7) into (B.5) A(k, s - s’) becomes

s - ~‘1 = & -

j,,,, s

dq w&t&

s - ~‘1 k&&As

dq n 6(nk - q) k-q k G&&,

- ~‘1 & 3

s - s? k$,,Ms

a

- ~‘1 akj 03.8)

HOMOGENEOUS,

ISOTROPIC

because of the transversality of G,&-q, that G,,,(-q,

369

TURBULENCE

s - 3’). In (B.8) we have used the fact

s - ~3 = Gv,,(q, s - 0.

In the same way B(k, s - s’) is calculated as B(k, s - s’) = (k,k,/k”“) U&k,

s - s’).

However,

so that B(k, s - s’) = 0. Substituting u&k,

(B.9)

(B.8) and (B.9) into (B.5) we obtain

s - s’) = A,,(k)

[&

D,&

3 - S’) +$

-

Cdk, s - s’) +]>

(B.1o)

where DJk,

s - s’) = j.

PGnk

dq qiq,Gy,y,(qy s -

S’)

k,,ky,h,(s - s’)

(B.11)

and C&

s -- s’) = 1 dq II 6(nk - q) +$!

qjGp&q,

s - s’) k,,k,h,(s

- 0

U3.12)

s - s’) L]akj

u,(k, s’).

(B.13)

Finally (B.l) takes the form

x

& D,(k, 1 I

S

- S’) &

I

-

Cj(k

370

NAKANO ACKNOWLEDGMENTS

The author wishes to express his appreciation to Professor David Pines for his continual advice, guidance and encouragement as well as for many enlightening discussions and a critical reading of the paper. It is pleasure to thank Professor Setsuo Ichimaru for constant guidance, encouragement and helpful discussions. The author also wishes to thank Professors Atlee Jackson, Christopher Pethick, Manfred Raether, Michael Wortis, and Dr. Jacob Shaham for stimulating discussions.

1. The classification of the wavenumber region into three parts is well described by E. M. MONTROLL, Some remarks on turbulence, in “Contemporary Physics, Trieste Symposium 1968,” Vol. 1, International Atomic Energy Agency, Vienna, 1969. 2. A. N. KOLMOGOROV, Compt. Rend. Acad. Sci. USSR 30 (1941), 301 lEnglish transl. Sou. Phys.-Usp. 10 (1968), 7341. 3. H. L. GRANT, R. W. STEWART, AND A. MOILLIET, J. Fluid Mech. 12 (1962), 241. 4. M. M. GIBSON, Nature 195 (1962), 1281; J. Fluid Mech. 15 (1963), 161. 5. G. K. BATCHELOR, “The Theory of Homogeneous Turbulence,” Cambridge University Press, London, 1953. 6. J. 0. HINZE, “Turbulence,” McGraw-Hill, New York, 1959. 7. L. D. LANDAU AND E. M. LIPSHITZ, “Fluid Mechanics,” Addison-Wesley, MA, 1959. 8. A. M. OBUKHOV, Compt. Rend. Acad. Sci. USSR 32 (1941), 19. 9. W. I-IEISENBERG,Z. Physik 124 (1948), 628. 10. T. VON -MAN, Compt. Rend. Acad. Sci. Paris 226 (1948), 2108. 11. L. S. G. KOVASZNAY, J. Aeron. Sci. 15 (1948), 745. 12. The above theories are summarized in Ref. [6, p. 1921. 13. The breakdown of Heisenberg’s theory at large wavenumbers is discussed in Ref. 15, p. 1611. 14. P. I. PROUDMAN AND W. H. REID, Phil. Trans. A 247 (1954), 163. 15. S. CHANDRASEKHAR, Proc. Roy. Sot. A 229 (1955), 1; Phys. Rev. 102 (1956), 94. 16. T. TATSUMI, Proc. Roy. Sot. A 239 (1957), 16. 17. R. H. KRAICHNAN, J. Fluid Mech. 5 (1959), 497. 18. Kraichnan overestimated the role of large eddies but such an overestimate is inevitable if the Fourier-transformed NavierStokes equation is treated in a mathematically straightforward manner. In order to suppress this overestimate he modified his method and developed the so-called “Lagrangian history direct interaction approximation” in his more recent papers [Phys. Fluids 7 (1964), 1724; 8 (1965), 575; 9 (1966), 17281, and thereby obtained the k--5/3 spectrum in the inertial region. However, we do not believe that those papers have revealed fully the relevant physical features of turbulence. 19. H. W. WYLD, JR., Ann. Phys. (N.Y.) 14 (1961), 143. 20. R. BALESCU AND A, SENATORSKI,Ann. Phys. (N.Y.) 58 (1970), 587. 21. A. A. TOWNSEND, Proc. Roy. Sot. A 208 (1951), 534. 22. S. A. ORSZAG AND M. D. KRUSKAL, Phys. Fluids 11 (1968), 43. 23. G. I. TAYLOR, Proc. Roy. Sot. A 151 (1935), 421. 24. G. I. TAYLOR, Proc. London Math. Sot. 20 (1921), 196. 25. L. PRANDTL, Z. Angew, Math. Mech. 5 (1925), 136; see S. GOLDSTEIN, “Modern Developments in Fluid Dynamics,” Vol. 1, p. 205, Oxford University Press, New York, 1938.

HOMOGENEOUS,

ISOTROPIC

TURBULENCE

371

26. R. Kuao, A stochastic theory of line-shape and relaxation, in “Fluctuation, Relaxation and Resonance in Magnetic Systems” (D. Ter Haar, Ed.), Plenum Press, New York, 1962. 27. The connection of the total energy dissipation rate to the energy flow flux in k space is discussed in Ref. [7, p. 1211. 28. See p. 120 of Ref. [7]. 29. E(k) is illustrated in Ref. [6, p. 1891. 30. R. W. STEWART AND A. A. TOWNSEND, Phil. Trans. A 243 (1951), 359. 31. J. L. LUMLEY AND H. A. PANOFSKY,“The Structure of Atmospheric Turbulence,” Interscience, New York, 1964. 32. S. A. KAPLAN, “Interstellar Gas Dynamics,” Pergamon Press, New York, 1966. 33. A. K. KLWLER AND T. VREBALQVITCH, Bull. Amer. Phys. Sot. 6 (1961), 207; their experimental data is available in Ref. [4].