O.N. Khatuntseva | Generalized analytical solution of the problem of determining the universal...
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2022-08-01

O.A. Ladyzhenskaya centennial conference on PDE’s, St. Petersburg, July 16 – July 22, 2022
Olga N. Khatuntseva (S.P. Korolev Rocket and Space Corporation ”Energia”, 
 Moscow Institute of Physics and Technology) 
Generalized analytical solution of the problem of determining the universal profile of the turbulent flow of an incompressible fluid

The possibility of describing both laminar and turbulent modes of fluid flow based on the same equations has been investigated. It is proposed to consider the Navier-Stokes equations in a phase space expanded by the introduction of an additional stochastic variable and to supplement the full time derivative with a term characterized by entropy production due to the excitation of stochastic disturbances. In this approach, the problems of Hagen-Poiseuille, Poiseuille and the plane Couette flow are solved. It is shown that the occurrence and maintenance of stochastic processes in a liquid is possible in those systems where there are incompatible boundary conditions. In this case, the existence of one smooth solution becomes impossible, and we can only talk about the presence of two or more non-intersecting or non-smoothly intersecting asymptotes of the solution. The region located between these asymptotes (or in the vicinity of the point of "discontinuity" of derivatives) is a domain of uncertainty that generates a stochastic process. In this approach, the problems of Hagen-Poiseuille,
Poiseuille and the plane Couette flow are solved. In all these problems, both
"laminar" and generalized "turbulent" solutions are found. It is shown that,
despite the difference between the solved equations for the Hagen-Poiseuille problems, the plane Poiseuille flow, and also for the flow along a flat plate (as the limiting case of the Poiseuille problem), the velocity profiles for the turbulent flow regime are the same, that may indicate the existence of a universal wall velocity profile for these problems.
References
[1] O.N. Khatuntseva, On finding a generalized analytical solution to the
Hagen–Poiseuille problem for a turbulent flow mode, ”Trudy‘ MAI” 118
(2021), paper ID 158211.