Scale-by-scale energy transfers and fluxes in compressible turbulence

Understanding compressible turbulence is critical for modeling atmospheric, astrophysical, and engineering flows. However, compressible turbulence poses a more significant challenge than incompressible turbulence. We construct a novel mathematical framework to compute mode-to-mode energy transfer rates and energy fluxes for compressible flows. We simulate forced compressible turbulence for Mach numbers 0.15, 0.30, and 0.45 and compute the energy spectra and fluxes. We show that the compressive component follows Burgers $k^{-2}$ spectrum, whereas the rotational component follows Kolmogorov $k^{-5/3}$ spectrum. The energy injected into the solenoidal (rotational) velocity component primarily cascades to small scales where it is viscously dissipated. The solenoidal component also transfers a tiny fraction of energy to the internal energy via pressure dilatation. In contrast, a significant part of the compressive component is transferred to internal energy via pressure dilatation. The compressive component also exhibits a forward energy cascade. In addition, there is a small amount of energy transfer from the solenoidal component to the compressive component. The root-mean-square velocity and energy flux of the compressive component follow Burgers scaling. Thus, our work details the energy transfers in a compressible flow. We are in the process of extending this study to compressible convection, compressible magnetohydrodynamics, and quantum turbulence.