UDC 532.517.45

HYDROMECHANICS

B. Ya. LYUBIMOV, F. R. ULINICH

ON THE PROBLEM OF CLOSURE IN THE THEORY OF TURBULENCE

(Presented by Academician M. D. Millionshchikov, 4 VIII 1969)

One of the schemes of approximate closure of the system of coupled equations for moments is based on the assumption that the Eulerian velocity field of a fluid is close to normally distributed. The cumulant approximation of the \(n\)-th order consists in neglecting the cumulants of moments of higher order \((^{1})\).

For the theory of turbulence, formulated as a system of coupled equations for the joint probability distributions of the values of the velocity field at a collection of fixed points, or of other random quantities characterizing the flow \((^{2})\), one can construct a closure method similar to the quasinormal approximation for the Friedmann–Keller system of moments.

The equation for the distribution function of \(n\) random variables usually contains, in addition to the \(n\)-th function, the distribution function of the \((n+1)\)-st variable. The closure scheme for the chain of equations consists in constructing a closed system of equations for a finite number of distribution functions.

A closure variant different from \((^{3})\) can be proposed on the basis of an exact expansion for the joint probability distribution of related variables. Namely, for the characteristic function of the distribution \(\varphi_{12}\) of two random variables \(u_1\) and \(u_2\) one may write

\[ \varphi_{12}(\theta_1,\theta_2)=\varphi_1(\theta_1)\varphi_2(\theta_2)\exp\left[\sum_{k,l=1}^{\infty}\frac{S_{k,l}}{k!\,l!}(i\theta_1)^k(i\theta_2)^l\right]. \tag{1} \]

Here \(S_{kl}\) is the cumulant of the moment \(\langle u_1^k u_2^l\rangle\). Expression (1) is the expansion of the characteristic function

\[ \varphi_{12}=\exp\left[\sum_{k,l=0}^{\infty}\frac{S_{kl}}{k!\,l!}(i\theta_1)^k(i\theta_2)^l\right], \]

partially summed with respect to the cumulants \(S_{0m}\) and \(S_{q0}\); here \(\varphi_1(\theta_1)=\varphi_{12}(\theta_1,\theta_2=0)\) and \(\varphi_2(\theta_2)=\varphi_{12}(\theta_1=0,\theta_2)\). The corresponding inversion of expansion (1) has the form

\[ P_{12}(u_1,u_2)=\exp\left[\sum_{k,l=1}^{\infty}\frac{S_{kl}}{k!\,l!}\frac{\partial^{k+l}}{\partial u_1^k\partial u_2^l}\right]P_1(u_1)P_2(u_2). \tag{2} \]

The quantities

\[ P_1=\int P_{12}(u_1,u_2)\,du_2,\qquad P_2(u_2)=\int P_{12}(u_1,u_2)\,du_1 \]

are the exact distribution functions of the quantities \(u_1\) and \(u_2\), respectively. In essence, the operator

\[ \exp\left[\sum_{k,l=1}^{\infty}\frac{S_{kl}}{k!\,l!}\frac{\partial^{k+l}}{\partial u_1^k\partial u_2^l}\right] \]

is a difference integral operator. The generalization to the case of a larger number of variables is obvious.

Possible closure methods may consist in limiting the number of cumulants in (2). The system of equations for the distribution functions of velocities \(F_n\) at \(n\) fixed points can, for example, be closed if one restricts oneself to the simplest cumulants in the expansion of the distribution function \(F_3\) of the velocities \(V_1, V_2, V_3\) at three points \(x_1, x_2, x_3\)

\[ F_3=\exp\left[ S_{11}^{\alpha\beta}(x_1,x_3)\frac{\partial^2}{\partial V_1^\alpha \partial V_3^\beta} + S_{11}^{\alpha\beta}(x_2,x_3)\frac{\partial^2}{\partial V_2^\alpha \partial V_3^\beta} \right]F_2(V_1,x_1,V_2,x_2)F_1(V_3,x_3); \tag{3} \]

here
\[ S_{11}^{\alpha\beta}(x_i,x_3)=\langle V_i^\alpha V_3^\beta\rangle-\langle V_i^\alpha V_3^\beta\rangle,\quad i=1,2. \]
In this case, since the cumulants are completely expressed in terms of \(F_2\) and \(F_1\), we obtain closed equations for the latter.

Concrete proposals for truncating the cumulant series must satisfy additional conditions: preservation of normalization, symmetry (if the functions considered are symmetric with respect to permutation of some groups of arguments), incompressibility, and positive definiteness of the approximate functions introduced; for example, expression (3) for \(F_3\) preserves the normalization of the distributions, but is not symmetric. Naturally, the closure scheme must correspond to plausible ideas about the structure of the velocity field.

An example of a physically and experimentally better substantiated method of decoupling may be the closure method based on the phenomenon of approximate statistical independence of the substantial acceleration, determined mainly by small-scale motions, from the velocity at the given point. The first equation of the chain for the joint distributions of the quantities \(V,\ \dot V=A_1,\ \ddot V=A_2,\ldots\), where the dot denotes the total time derivative, has the following form:

\[ \frac{\partial F_1}{\partial t} + V^\alpha\frac{\partial F_1}{\partial x^\alpha} + \frac{\partial}{\partial V^\alpha} \int F_2(V,A_1,x)A_1^\alpha\,dA_1 =0. \tag{4} \]

The joint distribution of velocity and acceleration \(F_2\) can, as was already said, be represented in the form

\[ F_2=\exp\left[ \sum_{k,l=1}^{\infty} \frac{S_{kl}^{\alpha\beta}}{k!\,l!} \frac{\partial^{k+l}}{(\partial V^\alpha)^k(\partial A_1^\beta)^l} \right]F_1(V)\Phi_1(A_1), \]

where \(F_1(V)\) and \(\Phi_1(A_1)\) are the exact distribution functions of velocity and acceleration at the point \(x\), and \(S_{kl}^{\alpha\beta}\) are the cumulants of the corresponding correlation moments. The assumption of approximate statistical independence of velocity and acceleration can be realized by regarding the cumulants \(S_{kl}^{\alpha\beta}\) as small and decreasing. Restricting ourselves to the first terms of the expansion, we find the approximate expression for \(F_2\)

\[ F_2(V,A_1)=F_1(V)\Phi_1(A_1)+S_{11}^{\alpha\beta}(x)\frac{\partial F_1}{\partial V^\alpha}\frac{\partial\Phi_1}{\partial A_1^\beta}. \]

Substituting this expression into (4), we obtain the following equation for \(F_1(V)\):

\[ \frac{\partial F_1}{\partial t} + V^\alpha\frac{\partial F_1}{\partial x^\alpha} + S_{11}^{\alpha\beta}\frac{\partial^2 F_1}{\partial V^\alpha\partial V^\beta} + S_{01}^{\alpha}\frac{\partial F_1}{\partial V^\alpha} =0, \tag{5} \]

\[ S_{11}^{\alpha\beta} = \langle V^\alpha A_1^\beta\rangle - \langle V^\alpha\rangle\langle A_1^\beta\rangle, \qquad S_{01}^{\alpha}=\langle A_1^\alpha\rangle. \]

Equation (5) contains the unknown moment \(S_{11}^{\alpha\beta}\), which is not expressed through \(F_1(V)\); therefore the equation obtained is, generally speaking, not closed, and for the cumulant \(S_{11}^{\alpha\beta}\), defined in the case of locally homogeneous turbulence by the dissipation \(\varepsilon(x,t)\) as \(S_{11}^{\alpha\beta}=\frac{1}{3}\varepsilon(x,t)\delta_{\alpha\beta}\), an additional equation is necessary.

Received
11 VI 1969

References Cited

1. M. D. Millionshchikov, DAN 32, 611 (1941).
2. F. R. Ulinich, DAN, 183, 535 (1968); A. S. Monin, PMM, 31, 1057 (1968); B. Ya. Lyubimov, DAN, 184, 1069 (1969).
3. F. R. Ulinich, B. Ya. Lyubimov, ZhETF, 55, 951 (1968).