Drag due to regular arrays of roughness elements of varying geometry

Abstract

Comparisons are made of experimental studies on the drag, at high Reynolds number, due to regular arrays of roughness elements of various shapes immersed in a turbulent boundary layer. Using a variant of Millikan's dimensional analysis, the form of the velocity profile is deduced in terms of the dimensions and concentration of the roughness elements. A drag formula results which is shown to be in good agreement with data. Available measurements of the partition of drag between the elements and the intervening surface indicates that equipartition occurs at quite low concentrations. The interaction between elements is then small, so that the drag coefficient of a typical roughness element is nearly constant. A re-examination of some of O'Loughlin's velocity-profile data, obtained below the tops of the roughness elements, suggests the existence of a nearly constant-stress layer scaled to the shear stress of the intervening surface. Above the roughness elements, the mean-velocity profile undergoes a transition to the form appropriate to the total shear stress exerted by the roughened surface. A formula is given which describes the one-dimensional velocity profile over the entire range, excluding the viscous sublayer on the intervening surface. The viscous sublayer appears to correspond quite closely to that on a smooth plate.