Certificate of Registration Media number Эл #ФС77-53688 of 17 April 2013. ISSN 2308-6033. DOI 10.18698/2308-6033
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Landau — Lifshits vibrator in the equations of gas dynamics

Published: 23.03.2018

Authors: Ovsyannikov V.M.

Published in issue: #4(76)/2018

DOI: 10.18698/2308-6033-2018-4-1739

Category: Mechanics | Chapter: Mechanics of Liquid, Gas, and Plasma

The problem of explaining the causes of turbulence excited L.D. Landau back in the 1940s. Much later, after the presentation of the famous method of the acoustic analogy of M.J. Lighthill, in the textbook “Hydrodynamics” by L.D. Landau and E.M. Lifshits (1988) members responsible for the formation of periodic waves against a background of steady laminar flow appeared in a complete nonstationary system of equations of gas dynamics.They were the convective terms of the equation of motion. Convective terms, as the source of the oscillations appearance, attracted attention of A.S. Predvoditelev, who introduced the coefficient 1-β to the equation of motion to describe turbulence. In the article it became possible to represent the inhomogeneous terms of the wave equation arising from the convective terms, through second-order Jacobians of the velocity vector, previously given by Euler in the equation of continuity. The inhomogeneous part of the wave equation for the velocity vector having derived by L.D. Landau and E.M. Lifshits, transformed to the sum of second-order Jacobians, makes it possible to specify the velocity fields of the hydrodynamic flows calculated earlier, to obtain an estimate of the generation intensity of periodic waves produced by a stationary flow. L.D. Landau andE.M. Lifshitz established that when using the M.J. Lighthill method of acoustic analogy the convective terms of the gas dynamics equation of motion penetrate into the inhomogeneous part of the wave equation, which corresponds to the generation of sound and self-oscillations not associated with external influences on the flow. The wave equation which they derived from the system of gas dynamics equations without involving extraneous relations can have the same solution as the indicated system of equations.This inhomogeneous wave equation can be regarded as a vibrator building-up both its analytical solution and the solution of problems of gas flow around bodies by numerical methods.

[1] Landau L.D. Zhurnal tekhnicheskoy fiziki AN SSSR — Journal of Technical Physics USSR AS, 1944, vol. XIV, p. 240.
[2] Landau L.D., Lifshits E.M. Gidrodinamika [Hydrodynamics]. Moscow, Nauka Publ., 1988, 736 p.
[3] Lighthill M.J. On sound generated aerodynamically. Part I. General theory. Part II. Turbulence a source sound. Proc. Royal Society, A211, 1952, A222, 1954.
[4] Ovsyannikov V.M. Volnoobrazovanie i konechno-raznostnoe uravnenie nerazryvnosti Leonarda Eylera [Wave formation and the finite-difference continuity equation of Leonhard Euler]. Moscow, Sputnik+ Publ., 2017, 487 p.
[5] Ovsyannikov V.M. Istoriya vyvoda uravneniya nerazryvnosti [The history of the equation of continuity derivation]. Sbornik dokladov XI vserossiyskogo sezda po fundamentalnym problemam teoreticheskoy i prikladnoy mekhaniki. Kazan,20–24 avgusta 2015 g. [Proceedings of the XI All-Russian congress on the fundamental problems of theoretical and applied mechanics. Kazan, August 20–24, 2015]. Kazan, 2015, pp. 2823–2824.
[6] Godunov S.K. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1971, 16 p.
[7] Ovsyannikov V.M.. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Ser. Estestvennye nauki — Bulletin of the Moscow City Pedagogical University. Ser. Natural Sciences, 2016, no. 4 (24), pp. 51–59.
[8] Chetverushkin B.N. Kineticheskie skhemy i kvazigazodinamicheskaya sistema uravneniy [Kinetic schemes and quasi-gas-dynamic system of equations]. Moscow, MAKS Press Publ., 2004, 332 p.
[9] Elizarova T.G. Kvazigazodinamicheskie uravneniya i metody rascheta vyazkikh techeniy [Quasi-gasdynamic equations and methods for analysis of viscous flows]. Moscow, Nauchnyy mir Publ., 2007, 352 p.
[10] Sheretov Yu.V. Regulyarizovannye uravneniya gidrodinamiki [Regularized equations of hydrodynamics]. Tver, Tver State University Publ., 2016, 222 p.
[11] Ovsyannikov V.M. Comparison of Additional Second-Order Terms in Finite-Difference Euler Equations and Regularized Fluid Dynamics Equations. Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 5, pp. 876–880. Pleiades Publishing, Ltd., 2017. Original Russian Text published in Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, 2017, vol. 57, no. 5, pp. 876–880. DOI: 10.1134/S0965542517050098