Generalized analytical solution of the plane Poiseuille problem for the turbulent regime of incompressible fluid flow

DOI: 10.34759/trd-2022-123-08

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

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 (NSE) in a phase space expanded by the introduction of an additional stochastic variable. As a result, an additional term appears in the expression for the total time derivative, characterized by the production of entropy due to the excitation of stochastic perturbations.

For laminar flow modes, entropy production takes zero value, the additional term disappears, and the transition to the NSE in their standard form is carried out, while the corresponding solutions describe only laminar flow modes.

The inclusion of an additional term, characterized by entropy production (which is always non-negative), into the equations allows us to take into account the time irreversibility of physical processes in cases where this production is non-zero.

It is shown that the occurrence and maintenance of nondeterministic – stochastic processes – in a liquid is possible in those systems, where incompatible boundary condition occurs. 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 an uncertainty domain that generates a stochastic process.

As generalized solutions to problems with two asymptotes of the solution, functions were considered that are the sum of two terms, each of which is the product of two functions: one of which determines one of the asymptotes of the solution, and the second determines the degree of influence of this asymptote on the overall solution at each point of the studied domain.

The “laminar” and generalized “turbulent” solutions of the plane Poiseuille problem are found in this formulation. The comparison of the found solution with experimental data of the wall-mounted universal velocity profile is given for different values of the Reynolds number.

Keywords:

turbulent Poiseuille flow, universal velocity profile, laminar-turbulent transition

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