Analytical method for velocity profile determining of the turbulent fluid flow in the flat Couette problem

Fluid, gas and plasma mechanics


Khatuntseva O. N.



Despite the great progress related to the solution of hydrodynamical problems both for the laminar and for turbulent flow modes based on the numerical solution of Navier-Stokes equations (NSE), the main question is not resolved at the mathematical level of rigorousity: whether the NSE describe both these modes.

The Navier-Stokes equations represent the Newton second law for the selected small enough but finite volume of the isothermal liquid, and describe this volume acceleration under the action of the force caused, on one hand, by the pressure gradient and external forces, and, on the other hand, by viscous force action on the surface of this volume.

In case of the deterministic, i.e. laminar, liquid flow mode the NSE correctness for such process describing is undoubtful. However, while transition to the turbulent liquid flow mode a great number of additional stochastic degrees of freedom occur. In this regard, the issue on the possibility of such system description by the deterministic Navier-Stokes equations remains open.

It is obvious that the issue on the possibility or impossibility of the turbulent flow mode description based on the NSE can be considered in the simplest way on the example of solving those hydrodynamic problems, which allow analytical solutions. Regrettably, due to their complexity, the NSE have such solutions only for the restricted set of problems for very simple geometries.

The flat Couette problem is one of such problems. The transition to the analysis of Navier-Stokes equations in the space expanded via the additional variable, which specifies the entropy production due to the excitation of the stochastic pulsation in the fluid flow enables to find two solution of this problem. One of them corresponds the laminar flow mode, and the second corresponds the turbulent flow mode. is characterized by the linear velocity profile over the complete liquid flow domain. The second one is realized for the high enough Reynolds numbers, and is specified by the velocity profile proportional to the hyperbolic sine with parameters dependent on the Reynolds number.

The critical Reynolds number, at wich the laminar to turbulent flow mode transition is possible, is determined. Comparison with the available experimental data is presented.


stochastic systems, probability density, turbulence, plane Couette flow, critical Reynolds number


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