Turbulence kinetic energy transfers in direct numerical simulation of shock-wave-turbulence interaction in a compression/expansion ramp

A direct numerical simulation is performed for a supersonic turbulent boundary layer interacting with a compression/expansion ramp at an angle alpha = 24 degrees, matching the same operating conditions of the direct numerical simulation by Priebe & Martin (J. Fluid Mech., vol. 699, 2012, pp. 1-49). The adopted numerical method relies on the high-order spectral difference scheme coupled with a bulk-based, low-dissipative, artificial viscosity for shock-capturing purposes (Tonicello et al., Comput. Fluids, vol. 197, 2020, 104357). Filtered and averaged fields are evaluated to study total kinetic energy transfers in the presence of non-negligible compressibility effects. The compression motions are shown to promote forward transfer of kinetic energy down the energy cascade, whereas expansion regions are more likely to experience backscatter of kinetic energy. A standard decomposition of the subgrid scale tensor in deviatoric and spherical parts is proposed to study the compressible and incompressible contributions in the total kinetic energy transfers across scales. On average, the correlation between subgrid scale dissipation and large-scale dilatation is shown to be caused entirely by the spherical part of the Reynolds stresses (i.e. the turbulent kinetic energy). On the other hand, subtracting the spherical contribution, a mild correlation is still noticeable in the filtered fields. For compressible flows, it seems reasonable to assume that the eddy-viscosity approximation can be a suitable model for the deviatoric part of the subgrid scale tensor, which is exclusively causing forward kinetic energy cascade on average. Instead, more complex models are likely to be needed for the spherical part, which, even in statistical average, provides an important mechanism for backscatter.