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UDC 532.5.045
HYDROMECHANICS
F. R. ULINICH
STATISTICAL DYNAMICS OF A TURBULENT INCOMPRESSIBLE FLUID
(Presented by Academician M. A. Leontovich on 11 IV 1968)
I. Basic equations. Elimination of pressure. The Navier—Stokes equations for describing the motion of a viscous incompressible fluid have the form:
\[ \partial v_\alpha/\partial t+v_\beta \partial v_\alpha/\partial x_\beta+\partial p/\partial x_\alpha=\nu \Delta v_\alpha, \qquad \partial v_\beta/\partial x_\beta=0. \tag{1} \]
From these two equations the pressure can be eliminated in the usual way:
\[ p(\mathbf{x})=-\frac{1}{4\pi}\int \frac{\partial v'_\alpha}{\partial x'_\beta} \frac{\partial v'_\beta}{\partial x'_\alpha} \frac{d\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|} +\psi(\mathbf{x}). \tag{2} \]
The function \(\psi(\mathbf{x})\) is harmonic,
\[ \Delta \psi(\mathbf{x})=0. \tag{3} \]
Although in what follows we shall regard the velocity field as random, we shall show that the function \(\psi\) is not a random variable. We shall denote averaging over the ensemble by an overbar and introduce a new random function
\[ \psi_1(\mathbf{x})=\psi(\mathbf{x})-\overline{\psi(\mathbf{x})} \]
and its two-point second moment
\[ \overline{\psi_1(\mathbf{x})\psi_1(\mathbf{x}+\mathbf{r})}=f(\mathbf{x},\mathbf{r}). \tag{4} \]
For fixed \(\mathbf{x}\), the function \(f(\mathbf{x},\mathbf{r})\) is harmonic. In addition, it satisfies the following physically evident requirements: it must have no singularities and must tend to zero for large \(\mathbf{r}\). A harmonic function possessing such properties is equal to zero. Passing in formula (4) to the limit \(\mathbf{r}\to 0\), we obtain
\[ \overline{\psi_1^2(\mathbf{x})}=0, \]
which means that the random function \(\psi\) coincides exactly with its mean value:
\[ \psi(\mathbf{x})\equiv \overline{\psi(\mathbf{x})}. \]
Thus, in our equations we may regard \(\psi(\mathbf{x})\) as not being a random variable.
After integration by parts in formula (2), the equations acquire the form
\[ \frac{\partial v_\alpha}{\partial t} +v_\beta \frac{\partial v_\alpha}{\partial x_\beta} -\frac{1}{4\pi}\int v_\beta(\mathbf{x}')v_\gamma(\mathbf{x}') T_{\alpha\beta\gamma}(\mathbf{x}-\mathbf{x}')\,d\mathbf{x}' = -\frac{\partial \overline{\psi}}{\partial x_\alpha} +\nu \Delta v_\alpha, \]
\[ \frac{\partial v_\beta}{\partial x_\beta}=0, \tag{5} \]
where
\[ T_{\alpha\beta\gamma}(\mathbf{x}-\mathbf{x}') = \frac{\partial^3}{\partial x_\alpha \partial x_\beta \partial x_\gamma} \frac{1}{|\mathbf{x}-\mathbf{x}'|}. \]
II. Chain of equations for distribution functions. Introduce distribution functions statistically describing the behavior
fluids such that the probability that at the points \(\mathbf{x}_1,\ldots,\mathbf{x}_n\) at the times \(t_1,\ldots,t_n\), respectively, the fluid velocities will lie in the intervals \(d\mathbf{v}_1,\ldots,d\mathbf{v}_n\) is
\[ F_n(\mathbf{v}_1,\ldots,\mathbf{v}_n;\mathbf{x}_1,\ldots,\mathbf{x}_n;\quad t_1,\ldots,t_n)d\mathbf{v}_1\ldots d\mathbf{v}_n . \tag{6} \]
It is clear that the functions \(F_n\) give a complete statistical description of the turbulent motion of a fluid. Knowing \(F_n\), one can find any mean characteristics, for example the mean dissipated energy
\[ \varepsilon(\mathbf{x}_1,t_1)=\nu\, \frac{\partial v_\alpha(\mathbf{x}_1,t_1)}{\partial x_{1\beta}}\, \frac{\partial v_\alpha(\mathbf{x}_1,t_1)}{\partial x_{1\beta}} = \]
\[ =\nu\int \frac{\partial^2 F_2(\mathbf{v}_1,\mathbf{v}_2;\mathbf{x}_1,\mathbf{x}_2;t_1,t_2)} {\partial x_{1\beta}\,\partial x_{2\beta}}\, v_{1\alpha}v_{2\alpha}\, \delta(\mathbf{x}_1-\mathbf{x}_2)\delta(t_1-t_2)\, d\mathbf{v}_1d\mathbf{v}_2d\mathbf{x}_2dt_2 . \]
Let us derive the chain of equations satisfied by the functions \(F_n\). To this end consider an arbitrary function
\(\varphi(\mathbf{v}_1(\mathbf{x}_1,t'),\ldots,\mathbf{v}_n(\mathbf{x}_n,t_n))\) of the velocities at the points \(\mathbf{x}_1,t_1,\ldots,\mathbf{x}_n,t_n\) of space-time. It is clear that
\[ \frac{\partial\varphi}{\partial t_s} = \frac{\partial\varphi}{\partial v_{s\alpha}}\, \frac{\partial v_{s\alpha}(\mathbf{x}_s,t_s)}{\partial t_s}, \]
or, using equation (5), we can write
\[ \frac{\partial\varphi}{\partial t_s} = \frac{\partial\varphi}{\partial x_{s\alpha}} \left[ -\,v_{s\beta}(\mathbf{x}_s,t_s) \frac{\partial v_{s\alpha}(\mathbf{x}_s,t_s)}{\partial x_{s\beta}} +\right. \]
\[ \left. +\frac{1}{4\pi}\int v'_\beta(\mathbf{x}',t_s)v'_\gamma(\mathbf{x}',t_s) T_{\alpha\beta\gamma}(\mathbf{x}_s-\mathbf{x}')\,d\mathbf{x}' +\nu\Delta v_{s\alpha}(\mathbf{x}_s,t_s) \right]. \tag{7} \]
Take the mean of relation (7). We have
\[ \overline{\frac{\partial\varphi}{\partial t_s}} = \int \frac{\partial F_n}{\partial t_s}\, \varphi\,d\mathbf{v}_1\ldots d\mathbf{v}_n, \]
\[ \overline{ \frac{\partial\varphi}{\partial v_{s\alpha}} \,v_{s\beta} \frac{\partial v_{s\alpha}}{\partial x_{s\beta}} } = \int \frac{\partial F_n}{\partial x_{s\beta}}\, v_{s\beta}\varphi(\mathbf{v}_1,\ldots,\mathbf{v}_n)\, d\mathbf{v}_1\ldots d\mathbf{v}_n . \]
The other terms in relation (7) are transformed analogously. Using the fact that \(\varphi\) is an arbitrary function, we obtain the chain of equations for the functions \(F_n\)
\[ \frac{\partial F_n}{\partial t_s} = -\,v_{s\beta}\frac{\partial F_n}{\partial x_{s\beta}} -\frac{1}{4\pi}\int \frac{\partial F_{n+1}}{\partial v_{s\alpha}}\, v_{n+1,\beta}v_{n+1,\gamma} T_{\alpha\beta\gamma}(\mathbf{x}_s-\mathbf{x}_{n+1})\times \]
\[ \times\delta(t_{n+1}-t_s)\, d\mathbf{v}_{n+1}dt_{n+1}d\mathbf{x}_{n+1} -\frac{\partial F_n}{\partial v_{s\alpha}}\, \frac{\partial \overline{\psi}(\mathbf{x}_s,t_s)}{\partial x_{s\alpha}} - \]
\[ -\nu\int \delta(\mathbf{x}_s-\mathbf{x}_{n+1})\delta(t_s-t_{n+1})\,d\mathbf{x}_{n+1}\Delta_{n+1} \int \frac{\partial F_{n+1}}{\partial v_{s\alpha}}\, v_{n+1,\alpha}\,d\mathbf{v}_{n+1}dt_{n+1}\,* . \tag{8} \]
If one restricts oneself only to simultaneous functions, then in the right-hand sides of equations (8) one must pass to the limit \(t_1=t_2=\ldots=t\), take into account that
\[ \lim_{t_1\to t,\;t_2\to t,\ldots,\;t_n\to t} \left[ \sum_{s=1}^{n}\frac{\partial F_n}{\partial t_s} \right] = \frac{\partial F_n(\mathbf{v}_1,\ldots,\mathbf{v}_n;\mathbf{x}_1,\ldots,\mathbf{x}_n;t)} {\partial t} \]
and obtain for the simultaneous correlation functions the chain of equa-
\[ \text{* Equations (8), as we learned while preparing the article for press, were obtained by another method for the case of homogeneous turbulence in work }(^{1}). \]
\[
\frac{\partial F_n}{\partial t}
=
-\sum_{k=1}^{n} v_{k\beta}\frac{\partial F_n}{\partial x_{k\beta}}
-\frac{1}{4\pi}\sum_{k=1}^{n}\int
\frac{\partial F_{n+1}}{\partial v_{k\alpha}}\,
v_{n+1,\beta}v_{n+1,\gamma}T_{\alpha\beta\gamma}(\mathbf{x}_k-\mathbf{x}_{n+1})
\,d\mathbf{v}_{n+1}d\mathbf{x}_{n+1}
-
\]
\[
-\nu\sum_{k=1}^{n}\int
\delta(\mathbf{x}_k-\mathbf{x}_{n+1})\,d\mathbf{x}_{n+1}\Delta_{n+1}
\int \frac{\partial F_{n+1}}{\partial v_{k\alpha}}\,v_{n+1,\alpha}\,d\mathbf{v}_{n+1}.
\tag{9}
\]
III. Additional conditions
We present the conditions that the functions \(F_n\) must satisfy.
- Normalization conditions:
\[ \int F_{n+1}\,d\mathbf{v}_{n+1}=F_n,\qquad \int F_n\,d\mathbf{v}_1\ldots d\mathbf{v}_n=1. \]
- Continuity conditions:
\[ \lim_{\mathbf{x}_{n+1}\to \mathbf{x}_n,\; t_{n+1}\to t_n} F_{n+1} = F_n\delta(\mathbf{v}_{n+1}-\mathbf{v}_n). \]
- Symmetry conditions:
\[
F_n(\ldots,\mathbf{v}_s,\ldots,\mathbf{v}_k,\ldots;\ldots,\mathbf{x}_s,\ldots,\mathbf{x}_k,\ldots;\ldots,t_s,\ldots,t_k,\ldots)
=
\]
\[
=
F_n(\ldots,\mathbf{v}_k,\ldots,\mathbf{v}_s,\ldots;\ldots,\mathbf{x}_k,\ldots,\mathbf{x}_s,\ldots;\ldots,t_k,\ldots,t_s,\ldots).
\]
- Incompressibility condition:
\[ \int \frac{\partial F_n}{\partial x_{k\alpha}}\,v_{k\alpha}\,d\mathbf{v}_k=0. \]
- Compatibility conditions:
\[ \frac{\partial F_n}{\partial x_{k\alpha}} = -\frac{\partial}{\partial v_{k\beta}} \int \delta(\mathbf{x}_{n+1}-\mathbf{x}_k) \frac{\partial F_{n+1}}{\partial x_{n+1,\alpha}}\, v_{n+1,\beta}\,d\mathbf{v}_{n+1}d\mathbf{x}_{n+1} \delta(t_{n+1}-t_k)\,dt_{n+1}, \]
\[ \frac{\partial F_n}{\partial t_k} = -\frac{\partial}{\partial v_{k\beta}} \int \delta(\mathbf{x}_{n+1}-\mathbf{x}_k)\delta(t_{n+1}-t_k) \frac{\partial F_{n+1}}{\partial t_{n+1}}\, v_{n+1,\beta}\,d\mathbf{v}_{n+1}d\mathbf{x}_{n+1}dt_{n+1}. \]
Only when all the conditions are fulfilled can the functions \(F_n\) be regarded as functions with independent variables.
Received
28 I 1968
CITED LITERATURE
- A. S. Monin, PMM, 31, no. 6 (1967).