Physical model for turbulent friction on rough surfaces
  We present a physical model for turbulent friction on rough surfaces with regularly distributed roughness elements. Wall shear stresses are expressed as functions of physical quantities. Surfaces with varying roughness densities and roughness elements with different aspect ratios are considered. We propose a straight forward method based on the conservation of momentum to deduce the drag on elements by expressing it as functions of the maximum drag and drag reductions ratios, as the drag on individual elements decreases as packing density increases. A drag reduction effect of momentum redistribution is proposed and the mutual sheltering effect is studied. These two drag reduction mechanisms for individual elements are significant for sparse and dense surfaces, respectively. Reduction ratios for redistribution effect and mutual sheltering effect are deduced, for the two different types of rough surfaces. The shear stress on elements and the total wall shear stress are obtained as the result of the drag analysis. The estimated wall shear stresses of the proposed model are consistent with classical experimental measurements. 
Zhuoqun Li and Xiaojing Zheng
physics.flu-dyn