Finding a generalized analytical solution to the plane Couetta problem for a turbulent flow regime

DOI: 10.34759/trd-2022-122-07

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

Despite the fact that random processes are ubiquitous, doubts remain as to how random they are, and whether this randomness is seeming, based only on the limitations of our knowledge.

If the uncertainty of the processes occurring at the quantum level is already largely beyond doubt, the stochasticity of macroprocesses, and turbulence in particular, is still a subject of scientific discussion, if only because they are described by deterministic equations.

The article shows that the Navier-Stokes equations used to describe hydrodynamic flows lose their deterministic properties when they are integrated by computational methods. This can explain the rather successful application of these equations in solving many practical hydrodynamic problems in the implementation of turbulent flow regimes. However, this approach to describing turbulent flows can be correlated rather with analog modeling of turbulence.

In order to describe both laminar and turbulent flow regimes in a controlled manner, the author proposes to consider the Navier-Stokes equations in the phase space expanded by introducing an additional variable characterizing the entropy production. For laminar flow regimes, the entropy production takes on a zero value, the additional term disappears, and the transition to the Navier-Stokes equations in their standard form is being realized.

The article considers the issues of the possibility of emergence and maintenance of non-deterministic, i.e. stochastic processes in a liquid due to the existence of incompatible between each other boundary conditions, as well as the ways of describing them relating to the turbulent flow. It is shown that the velocity profile of a turbulent flow can be described as a generalized solution of the problem, which is the sum of the two terms, each of which is the product of two functions: one of which determines one of the asymptotes of the solution, while the second one determines the degree of influence of this asymptote on the general solution in each point of the area of interest.

The plane Couette flow problem was solved by dint of this approach. Good agreement of the results with experimental data is demonstrated.

Keywords:

stochastic systems, probability density, turbulent flow, flat Couette flow

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