The Spectrum of the turbulence based on theory of stochastic equations and equivalenceof measures

The formation of the spectrum of turbulence in the inertial interval on the basis of the new theory of stochastic hydrodynamics is presented. This theory is based on the theory of stochastic equations of continuum laws and equivalence of measures between random and deterministic movements. The purpose of the article is to present a solution based on these stochastic equations for the formation of the turbulence spectrum in the inertial interval in the form of the spectral function E(k)j depending on wave numbers k in form E(k)j∼kn. The results of analytical solutions showed a satisfactory correspondence of the obtained dependence with the classical Kolmogorov’s dependence in the form of E(k)j∼k5/3.


Introduction
The development of the theory of turbulence with the use of different ideas is presented in . Special attention was focused on the theoretical description of the spectral density. It should be noted that the most well-known ratio based on of the theory of dimension was, as is known, determined by Kolmogorov [37][38][39][40][41], who wrote the formula for the spectral density in the wave number function as -5/3. But on the basis of these works, it was not possible to obtain a single mathematical apparatus that would allow to determine analytically all the important characteristics of turbulence. At the same time, stochastic turbulence theory based on stochastic equations and the theory of equivalent measures makes it possible to derive analytical dependences for the first and second critical Reynolds numbers, profiles of averaged velocity and temperature fields, friction and heat transfer coefficients, and second-order correlations [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61].
In [46,47,53] analytical formula of the spectral density were presented, which are consistent with the classical dependencies of the statistical theory. Vortexes has a formula of spectrum E(k) j depending on wave numbers k for interval of generation of turbulence in form E(k) j~k n , n=-1.2÷-1.5, -1.66<n<-1. This ratio was named as the ratio of uncertainty in turbulence generation.

Stochastic equations of conservation
Equations were derived in [42][43][44][45][46][47][48][49][50][51][52][53][54] the momentum equation and the energy equation Here, are the energy, the density; the velocity vector; the velocity components in directions x i , x j , x l (i, j, l = 1, 2, 3); the dynamic viscosity; the time; and stress tensor τ i,j Р is the pressure of liquid or gas;  is the thermal conductivity; p c and v c are the specific heat at constant pressure and volume, respectively; F is the external force , and 2 3 . Further, L = L U, P = L U is the scale of turbulence. Indexes ( U, P ) and ( U ) refer to the velocity field and index ( T ) refers to the temperature field. L y on x 2 =y, or L x , x 1 = x. Here, x 1 and x 2 are coordinates along and normal to the wall. Index "col st" refers to components, which are actually the deterministic. Index "st" refers to component, which are actually the stochastic. Then for the non-isothermal motion of the medium, using the definition of equivalency measures between deterministic and random process in the critical point, the sets of stochastic equations of energy, momentum, and mass are defined for the next space-time areas: 1) the beginning of the generation (index 1,0 or 1 ); 2) generation (index 1,1); 3) diffusion (1,1,1) and 4) the dissipation of the turbulent fields.

Sets of stochastic equations
The set of equations of mass, momentum, and energy (1) Using the sets (4),(5) formulas for the velocity and temperature fields may be obtained [45,54].
The area 3) is the diffusion of the turbulence. So, for the area of 3) diffusion, in accordance with [41][42][43][44][45][46][47][48][49][50][51][52] we have two fractal equations. The first equation is written as Here (E st ) is the field energy component, which is actually the stochastic one (index "st"), index j=1 refers to the space-time area of the diffusion of turbulence 3). The the solution may be written for ( The value k is the wave number and index cor1 determines the characteristics of the single or main perturbation generating turbulence. The value k cr -is the wave number of the beginning of the interaction between the deterministic field and random field that leads to turbulence in critical point. (E st /ρ) cr -is turbulence energy, which corresponds to the wave number k cr . Note that the written expression determines the turbulence energy in the case of a continuous spectrum for the fully turbulence flow.

The spectral function in the inertial interval
The specificity of the importance of the inertial interval as shown by the experience of experimental study of turbulence is the independence of this interval from the molecular viscosity. In this regard, to find a solution, equation (8) can be transformed to a stochastic equation of the following form The solution is

Conclusions
The results of analytical solutions showed a satisfactory correspondence of the obtained dependence with the classical Kolmogorov's dependence in the form of E(k) j~k 5/3 or the law of 2/3.