Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of
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O R I G I NA L A RT I C L E
A. V. Dmitrenko
Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of measures This article is dedicated to the memory of Academician N.A. Anfimov.
Received: 24 October 2019 / Accepted: 17 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 ∼ 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 ∼ 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 ∼ k −7 . The final part of the paper presents four analytical solutions for a spectral function on the form E(k) j ∼ k n , 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 ∼ k −5/3 and Heisenberg E(k) j ∼ k −7 .
1 Introduction Ideas about the nature of turbulence were formulated in [1–9]. Certain mathematical methods for solving the Navier–Stokes equations and methods of the theory of strange attractors for studying turbulence are defined in articles [10–18]. Statistical and stochastic equations and numerical methods for studying turbulent processes are presented in [19–28]. The main attention was focused to the theoretical description of the different range of the spectral density. The turbulence spectrum in the viscous interval in the form of spectral function E(k) depending on wave numbers k in form E(k) j ∼ k n was presented [1–3]. The main solution was presented as the classical Heisenberg’s dependence in the form of E(k) j ∼ k −7 [29]. However, when using Heisenberg’s dependence works, it was impossible to derive the dependence in form E(k) j ∼ k n for initial viscous interval, which has spectral function E(k) j depending on wave numbers as E(k) j ∼ k −3 [30,31]. So, the theory and practice demand a single mathematical apparatus that would allow us to derive Communicated by Andreas Öchsner. A. V. Dmitrenko (B) Department of Thermal Physics, National Research Nuclear University “MEPhI”, Kashirskoye shosse 31, Moscow, Russian Federation 115409 E-mail: AVDmitrenko@m
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