DERIVING THE SEMI-EMPIRICAL FORMULA TO COMPUTE THE FRICTION FACTOR λ FOR TURBULENT FLOW IN PIPE

Abstract

Based on law of shear stress in turbulent flow, Prandtl’s mixing length theory, and Bakhmeteff’s 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 V are obtained, after the integration, Q=  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 R_e or relative roughness e/D. Finally, the value of friction factor λ can be derived as a function of R_e or e/D for the two above-mentioned cases. These formations of λ 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.