  Direct numerical simulations were performed to characterize fully developed
supersonic turbulent channel flows over isothermal rough walls. The effect of
roughness was incorporated using a level-set/volume-of-fluid immersed boundary
method. Turbulence statistics of five channel flows are compared, including one
reference case with both walls smooth and four cases with smooth top walls and
bottom walls with two-dimensional (2D) and three-dimensional (3D) sinusoidal
roughnesses. Results reveal a strong dependence of the turbulence on the
roughness topography and the associated shock patterns. Specifically, the 2D
geometries generate strong oblique shock waves that propagate across the
channel and are reflected back to the rough-wall side. These strong shocks are
absent in the smooth-wall channel and are significantly weaker in cases with 3D
roughness geometries, replaced by weak shocklets. At the impingement locations
of the shocks on the top wall in the 2D roughness cases, localized
augmentations of turbulence shear production are observed. Such regions of
augmented production also exist for the 3D cases, at a much weaker level. The
oblique shock waves are thought to be responsible for a more significant
entropy generation for cases with 2D surfaces than those with 3D ones, leading
to a higher irreversible heat generation and consequently higher temperature
values in 2D roughness cases. In the present supersonic channels, the effects
of roughness extend beyond the near-wall layer due to the shocks. This suggests
that outer layer similarity may not fully apply to a rough-wall supersonic
turbulent flow.
