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By means of the similarity laws of the turbulent flow in boundary layers, a general relation is developed for the frictional resistance of flat plates with arbitrary roughness from which logarithmic formulas are obtained for the specal cases of the fully rough regime, the quasi-smooth regime, and engineering roughness. Among other uses, this general relation permits for the first time a simple rational extrapolation to full-scale conditions of the frictional resistance of model plates covered uniformly with any irregular full-scale roughness encountered in practice. Furthermore, for calculations of turbulent boundary layers in pressure gradients on rough surfaces, relations are derived for the local skin friction and velocity-profile shape parameter in terms of momentum-thickness Reynolds number. Simplified relations also are derived for the velocity profile of the transitional sublayer in the vicinity of a rough surface.
Paul S. Granville (Mon,) studied this question.