Accurate calculation of friction in tubes, channels, and oscillatory flow: A unified formulation

The dynamics of estuaries and coastal waters is strongly influenced by friction, which must be modeled with great accuracy in order to construct reliable hydrodynamic and morphodynamic models of these regions. We show that using appropriate global scaling (which does not depend on the detailed distribution of mean velocity and momentum) together with a single numerical coefficient, which is analytically calculated, the entire friction laws in tubes, channels, and oscillatory flow can be made to collapse into one single curve for both smooth and rough (granular type) walls. This suggests that wall friction has a global nature, which allows the derivation of a new unified expression for turbulent friction that is valid for all these flows and is noticeably more accurate than existing formulae (including Prandtl's universal law of friction for smooth pipes and subsequent refinements based on the Princeton superpipe experiment). In the light of previous work by the authors we consider finally the case of high roughness, for which simple turbulent friction laws cease to be valid, showing that observation is consistent with an upper bound for the flux of longitudinal momentum toward the wall that turbulence can generate. This leads to a new expression for the law of friction appropriate to high roughness cases which is shown to be in excellent agreement with observation.