For the Kolmogorov turbulence model, this approach would mean calculating the scattered energy at all small scales and summing the values obtained for the upper scale. Kolmogorov's model describes the real picture of the fluid flow.
The error in numerical methods of solution is found in their very theoretical basis. We solve a model with a physical process that does not describe the flow on the space R3 for each grid cell and then obtain the calculation result for the entire grid, and therefore for the space R3. In other words, the numerical method applies an incorrect physical model to the solution on the space R3, which does not describe the processes on the space R3.
Proof of the absence of a solution to the Navier-Stokes equations on the space R3 based on the application of Kurt Goedel's incompleteness theorem
To solve the problem, we use Kurt Goedel's incompleteness theorem. Goedel's theorem is given in [7].
The system of equations, including the Navier-Stokes system of equations, is a formal system. As you know, there are questions within the formal system that cannot be answered within these systems.
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.
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.