On accounting for stochastic disturbances effect on Navier-Stokes equations solution in Hagen-Poiseuille problem

Fluid, gas and plasma mechanics


Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

The Navier-Stokes equations (NSE) are equations of the law of conservation of momentum (or the second Newton Law) for the selected fluid volume. They describe the acceleration of this volume under the action of forces caused by a pressure gradient and external forces. Notwithstanding that the NSE numerical solutions are widely used in many scientific and practice applications, the issue of proving of the possibility (or impossibility) to describe turbulent flow modes by NSE is still open. It is associated, in particular, with the fact, that the porlems allowing analytical solution (such as Hagen-Poiseulle or Couette problem) do not have solutions corresponding to the turbulent mode of the flow.

If ask a question what aspects are not accounted for while turbulence modelling by NSE, on assumption of the first principles, it can be noted that the turbulence mode as well as other stochastic processes, possesses an important statistical feature, namely, excitation of a large number of the independent degrees of freedom (pulsations) at various scales of the system consideration. The law of conservation of momentum herewith for the selected fluid volume in the NSE form, written with no accounting for this process, is violated since not all of the resultant impact, directed at the selected volume, goes into its acceleration. A part of this impact should go into excitation of the extra internal degrees of freedom.

The stochastic system entropy is the parameter, characterizing the relation of micro and macro processes. Thus, accounting for the entropy generation in the selected liquid volume in such process is obligatory. On this basis, the Navier-Stokes equations may be rewritten by introducing the total time derivative to its left part as an extra member responsible for the velocity changing while changing the differential entropy of the selected volume.

Navier-Stokes equations modification by accounting for the extra degrees of freedom, associated with stochastic pulsations excitation in a fluid flow, allowed obtain two solutions of a problem of a fluid flow in a tube with circular cross-section (the Hagen-Poiseulle problem).

One of these solutions exists at any Reynolds numbers and corresponds to the laminar flow mode. The second one is realized only at relatively high Reynolds numbers, and corresponds to a turbulent flow. The Von Karman constant was determined analytically in the expression, describing the logarithmic velocity profile in the central part of the tube.

Keywords:

stochastic systems, probability density, turbulence, Hagen-Poiseulle problem

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