Energy transfer in compressible channel flows with two-dimensional sinusoidal rough walls

We perform direct numerical simulations to investigate the effect of two-dimensional sinusoidal roughness in the compressible channel flows with varying roughness height at Mach numbers M0=0.8 and 1.5. We observed the strong oblique shock waves and alternating compression/expansion regions are generated due to the roughness at higher Mach number, which also results in higher temperature in the channel center. The effects of roughness height on the transfer between the kinetic and internal energies are analyzed in detail. We found that the roughness significantly enhances the production of the turbulent kinetic energy while the Mach number has little influence on this term. The transfer terms between kinetic and internal energies are pressure- and viscosity-related, and is dominated by the viscous terms. The roughness-induced shock waves strongly affect the local distributions of the pressure-related terms, but its spatial average is only slightly modified. The energy transfer from the mean kinetic energy to both the internal energy and the turbulent kinetic energy is amplified by the roughness through the viscous terms. The average effect of roughness is intensified as the roughness height increases, but is insensitive to the Mach number variation.