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

We believe that the system of Navier-Stokes equations is consistent. It follows that it is impossible to answer the possibility or not of a solution for the R3 space within this system. To respond, you need to go outside the system – to the extended system. And in the framework of the extended system, it is possible to solve the problem of the presence or absence of a solution to the Navier-Stokes equation for the space R3. Perform this task. And the result will be the proof and solution of the Millennium problem (the Millennium problem in the formulation of the Glue Institute).

Considering the turbulence model proposed by Kolmogorov as a single system, we will build a hierarchy of this system. The lower level is designated by the base system, and the upper level is designated by the extended system.

The Navier-Stokes equations were derived for the base system, for the cubic element. All attempts made earlier by different authors to solve the Navier-Stokes equations were an attempt to obtain a solution for the extended system by means of the base system.

In the solution by numerical method on the calculated grid in the computer program, the solution is made for the basic systems, which are the cells of the calculated grid. Then, by the solution for the cells, there is a solution for all the grids, that is, for the extended system. The solution for the entire grid is the integral level.

The solution in the calculation grid ensures that the solution is correctly observed in the system hierarchy.

But the solution based on the calculated grid does not ensure the correctness of the physical side of the turbulent flow process. Since the Navier-Stokes equations are valid for an elementary volume, as indicated earlier. And for the space R3, you need to solve a system of equations that takes into account physical phenomena that were not taken into account when deriving the Navier-Stokes equations.

To make the numerical calculation correct, a system of equations describing the hierarchical energy transition from large-scale vortices to smaller ones and the subsequent energy dissipation due to viscous friction forces must be introduced into the computer program's calculation apparatus.

The claims of a number of authors that the Navier-Stokes equations apparently contain a complete description of turbulence are fundamentally incorrect.

The results obtained for the space R3 can be extended to the surface of the torus.

Conclusion

1. The physical principles on the basis of which the Navier-Stokes equations (energy balance taking into account viscous friction) are derived are given. The views of Henri Navier, using which he derived his equations, are given.

2. A diagram of Kolmogorov turbulence is given describing the physical principles of energy transfer from upper-level to shallow vortices and the transfer of energy to heat due to viscous friction forces.

3. The discrepancy between the physical principles described by Navier-Stokes equations and the principles of turbulent flow is shown using the Kolmogorov turbulence model as an example. In other words, the Navier-Stokes equations do not correspond to the physical picture of the flow at the level of space R3.

2. To apply Goedel's theorem, the Kolmogorov turbulence model is considered systemically, and a hierarchy of levels is compiled.

The Navier-Stokes equations are assigned to the smallest level of the system, called the base system.

The upper level is called the extended system in relation to the base system.

3. It is shown that the solution by numerical methods corresponds to the hierarchy of the model, but does not take into account the fact that the Navier-Stokes equations cannot describe the flow of a liquid at the upper level, that is, the numerical methods are based on a non-compact theoretical basis.

5. It is shown that based on Kurt Goedel's incompleteness theorem, it is impossible to obtain a solution for the extended system by means of the base system.

6. The results obtained for the space R3 can be extended to the surface of the torus.

The fomulation of the proof

The existence and smoothness of the solution of the Navier – Stokes equation on the space R3 is absent on the basis of:

–the Navier-Stokes equations do not describe turbulence in contrast to the Kolmogorov model;

– the Navier-Stokes equations are derived for the base system ( the lower level in the hierarchy model of Kolmogorov), which means you can not get the solution to solve the extended system (upper level according to Kolmogorov).

Bibliography

1 Landau L. D., Lifshits E. M. Hydrodynamics. – ed. 3-E. M.: Nauka, 1986. – 736 p. – Theoretical physics, vol. VI.

2 Kolmogorov A. N. Equations of turbulent motion of incompressible liquid / / Selected works. Mechanics and mathematics. M. Nauka. 1985. – 470 p.

3. Navier. Mémoire sur les rois du mouvement des fluides // Mémoires de l'académie des sciences de l'institut de France. 1822. Vol. 6.