231

数理解析研究所講究録
第 661 巻 1988 年 231-241

A COMMENT ON THE DISTANT INTERACTION ALGORITHM

Toshiyuki Gotoh andNagoya Institute of TechnologyYukio Kaneda
Nagoya University

1. INTRODUCTION
Recently $Kraichnan^{1.2}$ has presented the ‘Distant Interaction Algorithm’(DSTA) in order to examine a turbulence theory of Yakhot and Orszag3 which

is based on the dynamic renormarization group and an extrapolation of theinteractions among widely separated wavenumbers to the local ones. He has
shown that the essence of the theory of Yakoht and Orszag can also be interpreted by DSTA of the perturbative closure and that the DSTA yields goodinertial range constants such as the Kolmogorov constant $K_{o}=1.56$ and theObukhov Corrsin constant $C_{o}=0.958^{1,2}$ as well as $K_{o}=1.62$ and $C_{o}=1.16$ ofYakhot and Orszag.3 If both the DSTA and the theory of Yakhot and Orszag

are reliable, they would provide the great advantages to the computation of theturbulence because they are very simple approximations and the eddy viscosity like expressions can be supported. On the other hand it has been believedthat the local interactions among the wavenumbers are very important for theenergy transfer in the inertial range.
In this paper we shall consider why the DSTA can provide good inertial
range constants and discuss its reliability on the basis of a systematic La--1-

232

grangian closure theory.

II. THE DISTANT-INTERACTION ALGORITHM
We consider the evolution of the energy specrum $E(k)$ of the turbulence

which obeys the incompressible Navier Stokes equation. The equation for theenergy spectrum $E(k)$ can be written as$( \frac{\partial}{\partial t}+2\nu_{m}k^{2})E(k,t)=T(k,t)$,(1)

where $T(k,t)$ is the energy transfer function due to the nonlinear term in theNavier Stokes equation. The requirement of energy conservation by the nonlin-ear interaction implies the following relation$\Pi(P)\equiv-\int_{O}^{p}dqT(q)=2\int_{O}^{p}\nu(k|p)k^{2}E(k)dk$,(2)$-2\nu(k|p)k^{2}E(k)\equiv T(k|P)$ ,(3)where $T(k|p)$ is the contributions to the energy transfer function arising fromthe all triad wavenumbers interactions among $k<pand/orq$ . The relations (1)to (3) are exact.
In the DSTA we assume,(4)$\nu(k|p,t)=\nu(0|p,t)$ ,(5)$\nu(k,t)=\nu(k|\beta k,t)$

where$\beta\geq 1$

as DIA$\tau$

is a cut off parameter. In any perturbative two point closures such

ALHDIA8, SBALHDIA9, $TFM^{10}$ , EDQNM11 and LRA,5,6 ($\nu$$k$I p) is

given by the form$\nu(k|p)=\frac{1}{15}\int_{p}^{\infty}dq\theta(q,t)(5E(q)+q\frac{\partial}{\partial q}E(q))$$-2-$,for$k\ll p$,(6)

233

and
$\theta(q,t)=\int_{0}^{\infty}ds\{G(q,t,s)\}^{2}$

where$\theta(G)$,(7)is the so-called triple relaxation time (response function).

In order to specify$G$in (7), Kraichnan

assumed that the dynamics ofthe Eulerian $ve$ locity amplitude of the incompressible Navier-Stokes equation

can be modeled by the Langevin equation$( \frac{\partial}{\partial t}+\nu_{m}k^{2}+k^{2}\nu(k,t))u_{i}(k,t$where$\nu_{m}$} $=f_{i}(k, t)$ ,(8)is molecular viscosity, $\nu(k,t)$ is a dynamical viscosity, and $f:(k,t)$ thesolenoidal forcing which has isortopy and fluctuates in time like white noise. As
regards the interpretation of these quantities readers may refer Refs. 1 and 2.
It follows from (8) that the response function $G(k,t,s)$ is given by$G(k,t,s)=\exp(-l^{\ell_{ds’}}(\nu_{m}k^{2}+k^{2}\nu(k,s’)))$,(9)and that the fluctuation dissipation relation holds as$Q(k,t,s)=G(k,t,s)Q(k,t,t)$,$t\geq s$,(10)where$P_{1j}(k)Q(k,t;s)=<u_{i}(k,t)u_{j}(-k,s)>$

.

(11)

The DSTA by Kraichnan is defined by (2), (4), (5), (6), (7) and (9). It is tobe noted that the damping of$G$(and also ) is essentially determined by the$\theta$”viscosity” $\nu(k, t)$ or $\nu(k|p)$ appearing in (4) to (6) as implied by (9).

However this is not true in any systematic perturbative closures such as

DIA and Lagrangian closures including the ALHDIA, SBALHDIA and LRA.
We therefore reconsider the validity of the DSTA in the framework of such-3-234

a systematic perturbative closure. Here we use the Lagrangian renormalized
approximation(LRA), because the LRA is the simplest among the systematicclosures compatible with the the Kolmogorov inertial range law $k^{-6/3}$ and has
been found in good agreement with the numerical and the field experiments.6
In the LRA we have$( \frac{\partial}{\partial t}+\nu_{m}k^{2}+k^{2}\eta(k,t,s))v;(k,t;s)=0$,$( \frac{\partial}{\partial t}+\nu_{m}k^{2}+k^{2}\eta(k,t,s))G_{\dot{\iota}j}(k,t;s)=0$$G(k,t;t)=\delta_{:j}$(12)$t\geq s$,$(13a)$$t\geq s$,$(13b)$instead of (8) and (9), where $v:(k,t;s)$ is the Fourier transform of the generalized
velocity field $v_{i}(x,t;s)$ whose values are measured at time

whose trajectory passes the$x$$t$of the fluid particle

at time $s\leq t$ , and the $G(k,t,s)$ is the responseto the infinitesimal disturbance. The $\eta(k,t,s)$ is a eddy damping factor whichcomes from the lagrangian acceleration of the $pressure^{5,6}$ ;$\eta(k,t,s)=k\int_{O}^{\infty t}dqq^{3}J(\frac{q}{k})l^{d_{S}G(q,t,s)}$

,

(14)

where
$J(x)= \pi((a^{2}-1)^{2}\log\frac{1+a}{|1-a|}-2a+\frac{10}{3}a^{3})(2a^{4})^{-1}$,(15)$16\pi$for$\sim\overline{15}$$a=2x/(1+x^{2})$$x\ll 1$,.

It follows from eqs.(12) and (13) that the fluctuation dissipation relation holds$Q(k,t;s)=G(k,t,s)Q(k,s,s)$ ,$-4-$$t\geq s$,(16)2 ?, 5

where
$P:j(k)Q(k,t;s)=P_{:l}(k)<v_{l}(k,t;s)v_{j}(-k,s;s)>$ ,(17)(cf. (11)) and from the white noise assumption we can write as$Q(k,t,s)=G(k,t,s)Q(k,t,t)$ ,$t\geq s$.

(18)

The DSTA by the LRA is defined by (2), (4), (5), (6), (7) and (13).
Ill. ENERGY TRANSFER RATE IN THE INERTIAL RANGE
In the inertial range of steady turbulence the energy transfer rate $\Pi(p)$ is aconstant$\Pi(p)=\epsilon$

where$\epsilon$,(16)is a total rate of energy transfer. The DSTA and (16) yield thependent Kolmogorov constant$K(\beta)$

de-in the steady state inertial range spectrum$E(k)=K(\beta)\epsilon^{2/3}k^{-6/3}$

Kraichnan$\beta$.

assumed that there is a gap between

(17)
$k$

and

$\beta k$in the energy

spectrum and that the built up time of the triple correlation can be expressedby the dynamic viscosity as(18)$\theta(k)=\frac{1}{2k^{2}\nu(k|\beta k)}$and then obtained the approximate energy transfer rate$\tilde{\Pi}(p)=(\frac{21}{80})^{1/2}\{K(\beta)\}^{3/2}\beta^{-2/3}(1+\frac{4}{3}\ln\beta)\epsilon$$\nu(k|p)=A(\beta)\epsilon^{1/3}p_{\beta}^{-4/3}$,$(p \rho=\max(p,\beta k),$

$-5-$

,

$p\geq k$(19)),

(20)

236

and
$\theta(k)=\frac{1}{2A(\beta)}\beta^{4/3}\epsilon^{-1/3}k^{-2/3}$

,

(21)

where
$A( \beta)=(\frac{7}{60}K(\beta))^{1/2}\beta^{2/3}$.

(22)

If the result of the perturbative closure (here LRA, but its choice is not im-portant)$K_{o}=1.72^{5}$is used instead of$K(\beta)$, then we can see the performanceof the approximation of the DSTA to the exact result $(\Pi(p)=\epsilon)$ of the closure
theory with respect to the energy transfer rate in the inertial range as$\tilde{\Pi}(p)=1.16\epsilon$,

for$\beta=1$,(23)$\tilde{\Pi}(p)=1.09\epsilon$,

for$\beta=8$,(24)

and the approximate Kolmogorov constant based on

can be estimated as$K_{o}$

follows$K=1.56$ ,for$\beta=1$,(25)$K=1.62$ ,for$\beta=8$.

(26)

It seems, however, to be natural to consider that there is not the spectral
gap in the inertial range and there is no reason for$\theta(k,t)$

to be expressed by thedynamic viscosity. Within the perturbative closure theory$\theta$must be the time

integral of the response function. If we use the exact results of the $LRA^{5}(no$extrapolation in the response $eq.(3))$ , then we obtain$\theta_{LRA}(k)=\epsilon^{-1/3}k^{-2/3}\gamma I_{2}’$,$I_{2}’= \int_{0}^{\infty}dr\overline{G}\{\tau$$=1.49$.

-6-$\gamma=\sqrt{\frac{2\pi}{K}}$

)

,

(27)

(28)

237

Thus the dynamic viscosity and the approximate energy transfer rate become,$\nu(k|p)=\frac{\sqrt{2\pi}}{6}\{K(\beta)\}^{1/2}\epsilon^{1/3}I_{2}’p_{\beta}^{-4/3}$(29)$\tilde{\Pi}(P)=\frac{\sqrt{2\pi}}{4}\{K(\beta)\}^{3/2}I_{2}’\beta^{-4/3}(1+\frac{4}{3}\ln\beta)\epsilon$

If we use the value$K_{o}$

instead of$K(\beta)$

.

(30)

, we obtain the following estimates$\tilde{\Pi}(p)=0.58\epsilon$, $K=2.47$ , for$\beta=1$,(31)$\tilde{\Pi}(p)=0.14\epsilon$, $K=6.38$ , for$\beta=8$.

(32)

More consistent treatment of$\theta$in the context of the DSTA nay be an

extrapolation of the response function $G(k,t,s)$ . Under the same assumption

which lead to (27), we estimate $\eta(k,t,s)$ . By using (4) and the asymptotic
expression of (5), we obtain$\eta(k|p)=B(\beta)\epsilon^{1/3}p_{\beta}^{-4/3}k^{2}$

,

(33)

$\theta_{LRA,DSTA}(k)=\frac{1}{2B(\beta)}\epsilon^{-1/3}k^{-2/3}\beta^{4/3}$$B( \beta)=(\frac{2}{5}K(\beta))^{1/2}\beta^{2/3}$

,

(34)

,

(35)

and,

(36)

$\nu(k|p)=\frac{1}{12B(\beta)}K(\beta)\epsilon^{1/3}\beta^{4/3}p_{\beta}^{-4/3}$$\tilde{\Pi}(P)=(\frac{5}{128})^{1/2}\{K(\beta)\}^{3/2}\beta^{-2/3}(1+\frac{4}{3}\ln\beta)\epsilon$

If we use again the value$K_{o}$

instead of$K(\beta)$

.

(37)

, we obtain the following estimates$\tilde{\Pi}(p)=0.45\epsilon$, $K=2.95$ , for$\beta=1$,(38)$\tilde{\Pi}(p)=0.42\epsilon$, $K=3.(n$ , for$\beta=8$

.

(39)

$-7-$

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For other conbinations of the assunptions with respect to the energy gap andthe choice of the , the results are shown in the Table. From these observations$\theta$

it is found that the use ofto$\theta$

due to the response function instead of the one dueleads to poor approximation of the energy transfer rate$\nu$$\tilde{\Pi}(p)$. This reason

will be discussed in the next section.

IV. MECHANISM OF DSTA
We consider here why the DSTA yields good inertial range constants. Thebuilt up time of the triple correlations$\theta$

is symbolically written as$\theta(k)=\frac{1}{2\eta(k|p)}$

.

(40)

As seen in Sec.III, the various approximations for$\eta$

can be ordered as follows$\eta_{LRA}>\eta_{tRA.DS’PA}>\nu_{DSTA}k^{2}$

where$\nu_{DS24}$

,

(41)

means (20). On the other hand the factor $\{5E(q)+q\partial/\partial qE(q)\}$has less contribution to$\tilde{\Pi}(P)$than the exact one, then the two elements of theapproximations in the DSTA work in a way to compensate each error. Moreoverwhen$\nu_{DST4}k^{2}$is used in$\theta$

the residual error (say$R(\beta)$) of the Taylor expansionof the energy transfer function $T(k|p)$ can be reduced in the expression of$\nu_{DST4}$

is,, because the equation for$\nu_{DST4}$$\nu_{DSTA}$

must be self-consistently solved. Thatmay be symbolically expressed as$\nu_{DST\lambda}=(\int dq\frac{1}{2k^{2}}\{5E(q)+q\frac{\partial}{\partial q}E(q)\})^{1/2}(1+\frac{1}{2}R(\beta))$.

When the Kolmogorov constant is computed, the reduction of the error$R(\beta)$

becomes larger than that in the situation for$-8-$$\nu_{DST4}$. The equation for$\tilde{\Pi}$239

is symbolically written as$\epsilon=\tilde{\Pi}=K^{3/2}C(\beta)\epsilon(1+\frac{1}{2}R(\beta))$,thus we have
$K=C( \beta)^{-2/3}(1-\frac{1}{3}R(\beta))$where$C$,(42)is a constant. That is the Kolmogorov constant is given as the rootof the cubic equation. From (42) it is found that the error to the Kolmogorov

constant is only 33% even if$R$is unity.

In the LRA, the essence of the reason why the DSTA gives good numer-ical constants is found to be in the compensation of errors in both the triple
relaxation time and the Taylor expansion of the energy transfer function, andto lie in solving the self-consistent equation with identifying the dynamic eddyviscosity with the eddy damping$\eta$. At present we do not know whether suchcompensation of errors occurs or not for quantities other than the $\Pi(p)$ in the$k^{-5/3}$

inertial range.

-9-240

REFERENCES

1 R.H.Kraichnan, Phys.Fluids

30 2400 (1987).

2 R.H.Kraichnan, Phys.Fluids

30 1583 (1987).

3 V.Yakhot and S.A.Orszag, J.Sci.Comput.
4 Y.Kaneda, J.Fluid Mech.

5 Y.Kaneda, Phys.Fluids

1 3 (1986).

107 131 (1981).

29 701 (1986).

6 T.Gotoh, Y.Kaneda and N.Bekki, J.Phys.Soc.Jpn.
7 R.H.Kraichnan J.Fluid Mech.
8 R.H.Kraichnan

57 866 (1988).

5 497 (1959).

Phys.Fluids 8 575 (1966).

9 R.H.Kraichnan and J.R.Herring J.Fluid Mech.
io R.H.Kraichnan J.Fluid Mech.

11 S.A.Orszag J.Fluid Mech.

47 513 (1971).

41 363 (1970).

-10-88 355 (1978).

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TABLE

$\nu_{Dsrs}k^{2}$$\mu_{LRA,DSPA}$$\mu_{LR4}$$\beta=1$$\beta=8$$\beta=1$$\beta=8$$\beta=1$$\beta=8$

0.58

0.14

0.45

0.42

0.98

0.92

0.81

0.19

0.62

0.59

1.16’

1.09’2.47

6.38

2.95

3.06

1.74

1.82

2.00

5.20

2.36

2.44

1.56’

1.62’

$\tilde{\Pi}/\epsilon$

$K$

Table
Comparison of the energy transfer rate and the Kolmogorov constant by thevarious Distant Interaction Algorithms. The upper lines of each comparison

term.show the values for no energy spectral gap and the lower for the energygap. The value of the Kolmogor $ov$ constant by the exact closure theory is forexample $K_{o}=1.72(LRA)$ . The DSTA of $Kraichnan^{1,2}$ corresponds to the values

with$*$-11-