Direct numerical simulation of supersonic turbulent flows over rough walls

We carry out direct numerical simulation of supersonic turbulent channel flow over distributed roughness to quantify the effect of compressibility on added drag and heat transfer. We fix the ratio between the roughness height and the channel half height to k/h= 0.08, and carry out geometrically increasing simulations at friction Reynolds number Re_τ ≈ 500, 1000 spanning equivalent sand-grain roughness Reynolds numbers from transitional to fully rough (k_s⁺ ≈ 80, 160). We study compressibility effects by systematically changing the bulk Mach number M_b = u_b/c_b = 0.3, 2, 4, where u_b and c_b are the bulk velocity and bulk speed of sound, respectively.

We show that the outer layer similarity hypothesis is valid for the mean velocity and Reynolds stresses, which allow us to use the Hama roughness function to quantify the added drag. Furthermore, we apply different compressibility transformations on the Hama roughness function and show that most of them account for compressibility effects when the roughness height is scaled by the wall-to-crest viscosity ratio.