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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.
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.
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.
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
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