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Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of measures

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Abstract

The analytical formulas for spectrum of turbulence on the basis of the new theory of stochastic hydrodynamics are 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 solutions based on these stochastic equations for the formation of the turbulence spectrum in the form of the spectral function \( E(k)_j\) depending on wave numbers k in form \(E(k)_{{j}}\sim k^{n}\). At the beginning of the article two formulas for the viscous interval were obtained. The first analytical formula gives the law \(E(k)_{j}\sim k^{-3}\) and agrees with the experimental data for initial period of the dissipation of turbulence. The second analytical formula gives the law which is in a satisfactory agreement with the classical Heisenberg’s dependence in the form of \(E(k)_{j}\sim k^{-7}\). The final part of the paper presents four analytical solutions for a spectral function on the form \(E(k)_{j}\sim k^{n}\), \(\hbox {n}=(-1,4;-5/3;-3;-7)\) which are derived on the basis of stochastic equations and equivalence of measures. The statistical deviation of the calculated dependences for the spectral function from the experimental data is above 20%. It should be emphasized that statistical theory allowed to determine only two theoretical formulas that were determined by Kolmogorov \(E(k)_{j}\sim k^{{-5/3}}\) and Heisenberg \(E(k)_{j}\sim k^{{-7}}\).

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Acknowledgements

This work was supported by the program of increasing the competitive ability of National Research Nuclear University MEPhI (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013, Project No. 02.a03.21.0005).

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Authors and Affiliations

  1. Department of Thermal Physics, National Research Nuclear University “MEPhI”, Kashirskoye shosse 31, Moscow, Russian Federation, 115409

    A. V. Dmitrenko

  2. Department of Thermal Engineering, Russian University of Transport “MIIT”, Obraztsova Street 9, Moscow, Russian Federation, 127994

    A. V. Dmitrenko

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Correspondence to A. V. Dmitrenko.

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Communicated by Andreas Öchsner.

This article is dedicated to the memory of Academician N.A. Anfimov.

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Dmitrenko, A.V. Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of measures. Continuum Mech. Thermodyn. 33, 603–610 (2021). https://doi.org/10.1007/s00161-020-00890-4

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