Mathematical formulation of mode-to-mode energy transfers and energy 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 present a novel mathematical framework to compute $\mathit{\text{mode-to-mode}}\mathit{\text{energy}}\mathit{\text{transfer}}\mathit{\text{rates}}$ and energy fluxes for compressible flows. The 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. We also establish analogies with incompressible hydrodynamic and magnetohydrodynamic flows, highlighting the framework’s universality in studying energy transfers.