Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow

The stochastic equations of continuum are used for determining the hydraulic drag coefficients. As a result, the formulas for the hydraulic drag coefficients dependent on the turbulence intensity and scale instead of only on the Reynolds number are proposed for the classic flows of an incompressible fluid along a smooth flat plate and a round smooth tube. It is shown that the new expressions for the classical drag coefficients, which depend only on the Reynolds number, should be obtained from these new general formulas if to use the well-known experimental data for the initial turbulence. It is found that the limitations of classical empirical and semiempirical formulas for the hydraulic drag coefficients and their deviation from the experimental data depend on different parameters of initial fluctuations in the flow for different experiments in a wide range of Reynolds numbers. On the basis of these new dependencies, it is possible to explain that the differences between the experimental results for the fixed Reynolds number are caused by the difference in the values of flow fluctuations for each experiment instead of only due to the systematic error in the processing of experiments. Accordingly, the obtained general dependencies for the smooth flat plate and the smooth round tube can serve as the basis for clarifying the results of experiments and the experimental formulas, which used for continuum flows in different devices.