  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.
