Theoretical calculation of the laminar-turbulent transition in the round tube on the basis of stochastic theory of turbulence and equivalence of measures

On the basis of stochastic differential equations of conservation laws and the theory of equivalence of measures, the dependence is obtained for the coefficient of hydraulic friction in a smooth round pipe for a laminar-turbulent flow of an incompressible fluid depending on the intensity of turbulence, the scale of turbulence, the index of the velocity profile, and the Reynolds number. All the listed parameters included in the final formula of hydraulic friction are also determined by the corresponding formulas based on stochastic equations for the Reynolds number in a laminar-turbulent flow regime. In carrying out the calculations, the parameters of the initial disturbances are determined from known experimental data on the initial turbulence in a flow through a smooth round pipe. It is shown that, by applying a new dependence for the laminar-turbulent regime, which takes into account the parameters of the initial disturbance in the flow of deterministic motion, the dependence for the friction coefficient can be determined, which is obtained for a fully developed turbulent flow on the basis of stochastic equations. The friction coefficient for flow through a smooth pipe in a laminar-turbulent regime is calculated in the Reynolds number range from 2300 to 4500 and includes the fully developed turbulent flow regime in the Reynolds number range of 4500-7000. The obtained results of the values of the coefficient of friction show both quantitative and qualitative agreement with the known experimental data of Nikuradze. It is important to emphasize that, applying the dependence of the friction coefficient derived in the study for the laminar-turbulent flow through a smooth circular tube, it is possible to explain the differences between the values of the experimental results both for the laminar-turbulent regime and for the fully developed turbulent flow in a tube with the same Reynolds number, which can be caused by different values of the initially existing flow pulsations in each specific experiment. Results which are presented in article probably open up prospects both for the development of new experimental measuring instruments and for the development of a new calculation method-direct theoretical-numerical simulation (DTNS).

Automated summary

Based on stochastic differential equations and the theory of equivalence of measures, this study investigates the dependence of hydraulic friction coefficient in a smooth round pipe on the intensity, scale, velocity profile index, and Reynolds number of laminar-turbulent flow of an incompressible fluid. The results show agreement with known experimental data and have implications for the development of new experimental instruments and theoretical-numerical simulation methods.