Analytical method for velocity profile determining of the turbulent liquid flow in the flat Poiseuille problem

Fluid, gas and plasma mechanics


Khatuntseva O. N.

Korolev Rocket and Space Corporation «Energia», 4а, Lenin str., Korolev, Moscow region, 141070, Russia



The present paper continues the circle of publications devoted to solving the hydrodynamic problems allowing an analytical approach to their consideration. Searching for analytical solutions in rather simple «model» problems is not an end in itself. Developing an approach, enabling accounting for the laminar and turbulent flow modes specifics in Navier-Stokes equations (NSE), for further correct NSE integrating by both analytical and numerical methods is the main motivation of this work. However, studying specifics of various flow modes is possible namely while solving the problems allowing exact (analytical) solutions. Unfortunately, due to their complexity the Navier-Stokes equation have such solutions for a very restricted set of problems, namely for very simple geometries. The most well-known of them all are:

– The Hagen-Poiseuille problem, describing the flow of the incompressible thermally non-conductive liquid in a tube of a circular cross-section;

– The flat Couette problem describing the flow of incompressible thermally non-conductive fluid between two infinite parallel plates, moving under a constant velocities in opposite to each other directions in their own planes;

– The flat Poiseuille problem describing the flow of incompressible thermally non-conductive fluid under the pressure drop between two infinite parallel plates.

There is unique analytic solution describing the laminar profile of velocity for any Reynolds numbers in all these problems while Navier-Stokes equations integration under the absence of entropy production caused by excitation of velocity stochastic pulsations.

Two solutions were determined analytically due to the account of the entropy production in NSE for both the flat Poiseuille problem and other above-listed problems (considered in the previous articles). One solution corresponds to the laminar flow mode, and the second to the turbulent one. The first one is realized for any Reynolds numbers and is specified by the parabolic velocity profile in the total fluid flow domain, and the second one is realized for relatively high Reynolds numbers and is specified by the logarithmic velocity profile in the center of the flat channel. The multiplier prior to the logarithm function is the von Karman constant. The article presents comparison of the results with the available experimental data.


turbulence, plane Poiseuille flow, laminar-turbulent transition, critical Reynolds number


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