Scaling of rough-wall turbulence by the roughness height and steepness

A roughness scaling behavior is tested by performing the direct numerical simulation (DNS) of a turbulent channel flow over three-dimensional sinusoidal rough walls. The effects of roughness height (k$^{\mathrm{+}})$ and roughness steepness (S) on the turbulent instantaneous field and statistics are examined. The results show that the mean velocity and Reynolds stresses are highly dependent on both k$^{\mathrm{+\thinspace }}$and S, and a good scaling behavior is obtained on the roughness function and the peak of the streamwise turbulent intensity by using a coupling scale k$^{\mathrm{+}}$S. Then we define a rough-wall drag increasing ratio based on the roughness function. Accordingly, the wall resistance can be estimated directly from the coupling scale k$^{\mathrm{+}}$S for a given rough surface.