Unified theory on power laws for flow resistance

Two general power formulas, one for hydraulically smooth flows and the other for fully rough flows, are derived in a rational way from the widely accepted logarithmic formulas for the velocity profile and the Darcy‐Weisbach friction factor. A regression analysis based on the method of least squares is used to determine the valid range of the local velocity (or normal distance from the wall) in the power formula. Some older empirical formulas, such as Lacey's, Manning's, Blasius', and Hazen‐Williams', and their valid ranges, are actually explained analytically by the results. Incomplete self‐similarity of the power law, in which the exponent and the associated coefficient vary with the similarity parameters, such as the Reynolds number and the relative roughness, is elucidated through the parametric representations of the power formulas and their counterparts based on the logarithmic law. This paper examines the concept and rationale behind the power formulation of uniform turbulent shear flows, thereby addressing some critical issues in the modeling of flow resistance based on the power law.