Mathematical formulation of mode-to-mode energy transfers and energy fluxes in compressible turbulence

We present a novel mathematical framework to compute mode-to-mode energy transfers and fluxes for compressible flows. This formalism captures detailed energy conservation within triads and allows decomposition of transfers into rotational, compressive, and mixed components, providing a clear picture of energy exchange among velocity and internal energy modes. The key image shows the decomposed energy fluxes.