Turbulent kinetic energy spectra transport in particle-laden compressible turbulent boundary layers

In the present study, we perform direct numerical simulations to investigate the kinetic energy transfer in the physical and spectral spaces in compressible turbulent boundary layers at the free-stream Mach number of 6.0 laden with particles at different mass loadings. We found that the non-monotonic variation of the streamwise velocity fluctuation intensities with the increasing mass loading should be attributed to the two counteracting factors, i.e. the reduced velocity streaks that participate in the near-wall self-sustaining cycles, and the strengthened larger-scale velocity fluctuations induced by the comparatively high Stokes number particles. The analysis of the turbulent kinetic energy spectra transport equation shows that the production term is reduced in the near-wall region but enhanced in the outer region, which is balanced by the inter-scale energy transfer and the work of the particle force on the fluid. The cross-stream velocity fluctuations, on the other hand, manifest a monotonic reduction in magnitude. It is found that the abatement of the cross-stream velocity fluctuations near the wall is ascribed to the lower pressure-strain term that transfers the energy from the streamwise component, while the suppression in the outer region to the weaker spatial diffusion, the stronger inter-scale energy transfer and the negative work of the particle feedback force. Compressibility effects, reflected by the work of pressure on flow dilatation, are alleviated at higher particle mass loadings.