Based on law of shear stress in turbulent flow. Prandd's mixing length theory, and Bakhmeteflfs point of view on "wall velocity", turbulent velocity distribution u on wetted area can be derived for smooth pipe and complete turbulence, rough pipe. Discharge Q and average velocity y are obtained, after the integration, Q= ∫∫wu.dw is done. Relying on the properties of uniform flow, relationship between V, friction factor λ, and shear velocity u is set up. After eliminating u*. velocity V is obtained as a function of Reynolds number Re or relative roughness e/D. Finally, the value of friction factor z can be derived as a function of Re or e/D for the two above-mentioned cases. These formations of z formulas are almost same as the experimental ones introduced by Nikuradse with minor deviations in the factors and their relative errors do not exceed 1% for smooth pipe, and 2% for complete turbulence, rough pipe. Through this research result, the rightness of Prandtl's mixing length theory is almost asserted.
Discussion: “Turbulent Flow in the Inlet Region of a Smooth Pipe” (Barbin, A. R., and Jones, J. B., 1963, ASME J. Basic Eng., 85, pp. 29–33)
