Surface roughness plays a substantial role in many flows for which Reynolds averaged prediction is needed. The transformation used in the Formula: see text model is extended to rough surfaces by adding an effective origin. The log-layer offset is computed as a function of this effective origin, thereby creating a correspondence between effective origin and equivalent sandgrain roughness. A formula is derived for the virtual origin of the fully rough log law. It is shown how the present model is consistent with the fully rough limit.
Durbin et al. (Thu,) studied this question.