Решение проблемы турбулентности, отсутствие аналитического решения уравнений Навье-Стокса / The solution to the pboblem of turbulence, lack of analyt - страница 4

Write down the Navier-Stokes equation for a compressible fluid:

For a compressible fluid in the equation

Notation in the equation and its conclusion-see the work of L. N. Landau [1].

A. N. Kolmogorov in [2,p. 294] showed a physical model of turbulence (in accordance with Taylor and Richardson), which consists in superimposing various-scale turbulent pulsations on the averaged flow. The largest scale is the mastshab L of the "mixing path", the smallest scale is the value λ, on which the viscosity exerts an influence. Of ripples from the big scale transfer energy pulsations of smaller scales. As a result, there is a flow of energy, whose dissipation is due to the forces of the viscous torus on the scale λ. Kolmogorov proposed the following equations of turbulent motion based on local properties of turbulence [2, p. 295]:

In equations-notation according to the cited work of A. N. Kolmogorov.

L. D. Landau noted [2, p. 296] that these equations are true for the local turbulence structure, but in a turbulent flow, the presence of a velocity rotor is limited to a finite space and the equations should show exactly this distribution of turbulent vortices.

Henri Navier in [3] when formulating equations of fluid motion, he proceeded from the notation for a single point in the space of a continuous medium.

Landau calls Henri Navier's representations model [2, p. 73] (in a footnote).

We show a model of the physical flow pattern on which the Navier equations are based:

What is described by this model, a solution of the Navier-Stokes equations can be found for this. This model can perfectly describe special cases.

Let's compare the motion model from the work of A. Navier with the turbulence model proposed by A. N. Kolmogorov.

Model Of Kolmogorov:

Some authors point out that the Navier-Stokes equations contain turbulence. Apparently, this is not the case. The Navier-Stokes equations can be applied at the lowest level of the Kolmogorov model. In General, the Henri Navier model is physically incorrect compared to the correct Ko

Let's compare the motion model from the work of A. Navier with the turbulence model proposed by A. N. Kolmogorov.

The volume for which the Navier-Stokes equations are compiled is chosen with minimal dimensions that ensure the continuity of the medium. However, this is not essential. Obviously, the cube is incomparably smaller than the space R3.

For a cube, the description of the physical process consists of describing the flow of liquid into and out of it, as well as the effect of viscosity.

For a space R3 with a complex structure of turbulent flow, the physical process is much more complex and the descriptions used for the cube in the derivation of the Navier-Stokes equations are not enough to describe it!

In existing attempts to solve the Navier-Stokes equations, the space R3 is conditionally divided (discretized) by a grid with cubic elements.

Attempts at analytical solutions, for example, in [4], are reduced to assigning boundary conditions for equations and searching for solutions.

The boundary conditions for a cube with sides x, y, x and step Q are written as:

It is obvious that the movement of a liquid in the space R3 and in any space, the Navier model and its representations are not described. The area around the point does not exceed the Kolmogorov scale.

The Navier-Stokes equations are compounded for a physical model of a small Kolmogorov scale and do not correspond to the physical processes of turbulent movement of large volumes of liquid.

In the case of analytically accurate solutions of the Navier-Stokes Equations for the case of the Poiseuille flow, the solution is performed for the physical process described by the process for the cube.

There are methods for direct numerical solution of Navier-Stokes equations[5], [6], [7].

In these (finite-difference) methods, the space is discretized by a grid. The derivative is replaced by an algebraic relation.

It is obvious that numerical methods for the space R3 solve Navier-Stokes equations that do not describe a physical process on the space R3. However, the results of solutions for each grid cube are carried over for an integral solution for the entire grid, i.e. for the space R3.