Determining critical Reynolds number of laminar-turbulent transition in a flat Poiseuille problem based on “discontinuous functions” method


DOI: 10.34759/trd-2019-108-3

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

The existing methods for determining critical Reynolds numbers of laminar-turbulent transition are related mainly to determining the loss of linear stability of laminar flow mode [1-3]. However, the attempts for resolving the best-known “classical” problems of the hydrodynamics are far from the success. Thus, the Hagen-Poiseuille problem and the flat Couette problem possess linear stability for any Reynolds numbers. There is a limitation on stability for laminar flow mode of the flat Poiseuille flow. However, the critical Reynolds number for stability loss computed by Orszag is 5772, which is exceeds experimental data almost six times.

The presented article continues the author’s circle of publication devoted to the solution of the hydro-dynamical problems, assuming an analytical approach existence for their consideration. The possibilities of “discontinuous functions” method application for theoretical estimation of the critical Reynolds number value of laminar-turbulent transition for the Hagen-Poiseuille problem and the flat Couette problem are studied in [17-18]. The method may be applied in the cases, when different functions specifying various physical processes exist, such as laminar and turbulent flow modes, as well as the jump-like transition from one physical process to another.

The author succeeded to analytically define functions, describing quasi-stationary turbulent and stationary laminar flow modes for the Hagen-Poiseuille problem and the flat Couette problem. These problems solutions were obtained at the cost of accounting for the entropy production in the Navier-Stokes equations, stipulated by stochastic pulsations excitation in the fluid flow. The similar approach to the velocity profiles determining both in laminar and turbulent flows applied in [19] for the flat Poiseuille problem solution. The presented work is devoted to the Reynolds number critical value determining by the “discontinuous functions” method in the for the flat Poiseuille problem. The critical value for Reynolds number at which laminar to turbulent flow mode transition was possible, was found. The article presents the results comparison with the available experimental data.

Keywords:

stochastic systems, probability density, turbulence, flat Poiseuille flow, critical Reynolds number

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