Dissipation closure of fluid turbulence from energy spectrum

Direct method of the turbulence closure consists of obtaining expression for small-scale velocity and inserting this expression in the Reynolds tensor and in the transport equations for turbulent energy, dissipation, eddy and so on, implementing some averaging. Previously such approaches were used for Burgers-Lundgren vortex. Here we used ‘naïve’ perturbation method for the transport equation for small-scale velocity incompressible fluid - the equation suggested by one of the authors earlier. It is assumed that the energy spectrum of turbulence is given.


Introduction
Turbulence in fluids and gases is the most difficult problem of mathematics and physics as considered by many researchers today.Scientists believe that turbulence can be described by Navier-Stokes equations.But new evidence was appeared that turbulence in gases cannot be described by these equations due to thermal noise [1].Thus we consider only turbulence of incompressible fluid here.
It can be assumed that description of fully developed turbulence can be divided into two steps.The first step is a derivation of energy spectrum, then, the second step is a derivation of transport equations of a turbulence model.Central role in option of energy spectrum is its agreement with Kolmogorov's law for inertional range and fast decrement in dissipation range.Since these spectra contain dissipation rate of turbulence, it can be suggested that the simplest closure of turbulence is consists of constructing transport equation for dissipation rate, and expression for Reynolds stresses through this important physical quantity.

Transport equation for small-scale velocity and perturbation method of its solution
Transport equation for small-scale velocity, following [2], has the form: We have identities: Reynolds stresses can be defined as the following expression for given spectral tensor , and turbulent energy T E is defined as We can rewrite these expressions via polarization components of velocity: Inserting small parameter  in this equation before the sigma sign and nonlinear terms, we have: We seek solution equation (2) as ...
Equating terms with the same power, we obtain the next equations: Let's confine to considering the simple shear flow: only Where range of the angular spherical variables are : Let us consider two components polarization velocity, evolving from isotropic state.Then expressions (4) is system of two linear homogeneous equations, and an expressions (5) is system of two linear inhomogeneous differential equations.Their solutions can be found with the help of the Cauchy matrix is the fundamental matrix [3].Changing sum by integral, according to the rule: where − L is the integral scale of turbulence.
Neglecting viscosity, linear part of equation ( 6) became: We seek eigenvalues    6) in zero-th and first orders 'naïve' perturbation will be where The integral scale L can be defined from the dimension consideration as We can write down equation (12) analogically to equation ( 6), changing the sum by the integral: Equation (13) should be averaged over time and space.Transport equation for dissipation rate should be accomplished by inverse Fourier transformation equation (13).

Conclusion
Here we have put forward a strategy for closure of fully developed turbulence.The further steps to simplify equation (13) will include the derivation of the infrared asymptotes of this equation.Phenomenological one-equation model, based on transport equation of dissipation rate, was put forward in [7].Unfortunately, we could not find a way of derivation that model from Navier -Stokes equations.
We think that Navier -Stokes equations with random forces can be applied only for calculation Reynolds stresses but not for derivation of the dissipation transport equation as in [8,9].Review of the most popular empirical models of turbulence was given in [10].
It should be noted that the bounds on dissipation rate  [11]   Thus, dissipation rate  can be considered as small parameter for asymptotic expansion for model [7] and for using the naïve expansion approach here in our paper.

U 3 .
-i-th component of large-scale velocity U .We have for the simple shear Transfer of dissipation rate in anisotropic turbulence Dissipation rate is usually determined via polarization components as: Differentiating this formula with respect time t using equation (1), we shall have APITECH-V-2023 Journal of Physics: Conference Series 2697 (2024) 012009 : 