Abstract Generated abstract
The paper addresses closure in turbulence theory for chains of equations governing joint probability distributions of velocity and related random variables. It proposes a closure approach based on an exact cumulant expansion of joint characteristic and probability distributions, in which approximate closures are obtained by truncating the cumulants while preserving constraints such as normalization, symmetry, incompressibility, and positive definiteness. The authors illustrate the scheme for velocity distributions at several fixed points and discuss a physically motivated variant using the approximate statistical independence of velocity and substantial acceleration. This leads to an approximate one-point distribution equation, but the resulting equation still contains an undetermined velocity-acceleration cumulant, indicating the need for an additional relation, such as one involving turbulent dissipation in locally homogeneous turbulence.
Full Text
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
- M. D. Millionshchikov, DAN 32, 611 (1941).
- F. R. Ulinich, DAN, 183, 535 (1968); A. S. Monin, PMM, 31, 1057 (1968); B. Ya. Lyubimov, DAN, 184, 1069 (1969).
- F. R. Ulinich, B. Ya. Lyubimov, ZhETF, 55, 951 (1968).