Development of supersonic turbulent boundary layers over prism-shaped rough surfaces

Surface roughness is often present in flight systems travelling at high speeds, but its interaction with compressible turbulence is not well understood. Using direct numerical simulations, we study how prism-shaped roughness influences supersonic turbulent boundary layers at a free-stream Mach number $M_\infty =2$ . The dataset includes four simulations featuring cubic- and diamond-shaped elements in aligned and staggered configurations. All cases have an initial smooth region where a fully turbulent boundary layer transitions to a rough wall with positively skewed roughness elements relative to the smooth-wall zero plane. This causes a sudden boundary layer growth at the smooth-to-rough transition, generating an oblique shock wave. Individual roughness elements downstream do not generate shock or expansion waves, as they do not protrude into the supersonic region. For cubical elements, the staggered arrangement increases drag and produces more pronounced boundary layer growth than the aligned case. Rotating the cubes along their vertical axis further enhances these effects, yielding the highest drag. Interestingly, diamond-shaped elements in a staggered arrangement exhibit a dynamics similar to aligned cubes, producing lower drag than other cases. We explain the relative drag induced by each roughness shape by examining viscous and pressure drag components separately. The analysis reveals that, for staggered diamonds, the flow skims more easily over roughness, drastically reducing recirculation in troughs and gaps. In other cases, wake interactions are more prominent, causing spikes of highly positive and negative skin friction, a feature often neglected in reduced-order model formulations.