Prediction of laminar-turbulent transition on flat plate on the basis of stochastic theory of turbulence and equivalence of measures

In this article, on the basis of the stochastic theory of turbulence and the regularity of equivalence of measures, the calculation of the friction coefficient is presented for the laminar-turbulent transition on the flat plate. As a result, the formula for the friction coefficient depending on the turbulence intensity, the scale of turbulence, the velocity-profile index, and the Reynolds number for the laminar-turbulent regime of flow of an incompressible fluid along a smooth flat plate is proposed. For each of the listed parameters included in the equation for the drag coefficient, the relations determining these parameters for each Reynolds number in the region of the laminar-turbulent transition are obtained. It is also determined that the equation for the friction coefficient obtained previously on the basis of stochastic equations for a fully developed turbulent flow can be obtained on the basis of a new dependence for the laminar-turbulent transition with taking into account the initial perturbations in the deterministic motion. The parameters of these perturbations may be determined from the well-known experimental data for the initial turbulence in the flow on the flat plate. Using new dependence of the friction coefficient for the laminar-turbulent transition, it is possible to understand that the differences between the experimental results both for the laminar-turbulent transition and for a fully developed turbulent flow with the same Reynolds number are caused by the difference in the magnitudes of flow fluctuations for concrete experiment instead of only due to the systematic error in the processing of experimental data. The friction coefficient for a laminar-turbulent transition on a smooth flat plate is calculated in the of Reynolds number range of 5x10(5)divided by 2x10(7) up to the region of developed turbulent flow. The results of calculations of the friction coefficient show both qualitative and quantitative agreement with the experimental data. So, the law of equivalence of measures and stochastic equations presents both the physical and mathematical essence between interacting deterministic and random states. Thus, the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales does not reflect the law of mechanism of interaction of a deterministic state with the fluctuation.

Automated summary

This article presents a method for calculating the friction coefficient in the laminar-turbulent transition based on the stochastic theory of turbulence and the regularity of equivalence of measures. The formula for the friction coefficient is proposed, taking into account the turbulence intensity, turbulence scale, velocity-profile index, and Reynolds number. By considering initial perturbations, the authors show that the equation for the friction coefficient obtained from a fully developed turbulent flow can also be derived for the laminar-turbulent transition. The calculated results of the friction coefficient agree well with experimental data.