Abstract Generated abstract
This paper formulates a statistical description of turbulent motion in a viscous incompressible fluid starting from the Navier-Stokes equations. After eliminating pressure, it argues that the harmonic part of the pressure is nonrandom under regularity and decay assumptions, and derives a hierarchy of equations for multipoint velocity distribution functions, including the simultaneous-time case. The work also states normalization, continuity, symmetry, incompressibility, and compatibility conditions required for these distribution functions to represent independent statistical variables of the turbulent flow.
Full Text
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).