Scale-Locality: Insights into the energy cascade across scales in a shock*

Inter-scale energy transfer (or the cascade) is of relevance to both LES modeling and turbulence theory. In incompressible homogeneous isotropic turbulence, there has been compelling theoretical and empirical support that the scale-transfer of kinetic energy (KE) is local. Here, we analyze the locality of KE scale-transfer in compressible turbulence. There is a common notion that shocks and discontinuities that pervade compressible turbulence necessarily imply a non-local scale-transfer. We show this not to be the case by demonstrating rigorous proofs of scale-locality using two examples: (i) solution to the Burgers equation and (ii) the 1D normal shock solution. Proofs of scale-locality in compressible turbulence hold in broad generality, at any Mach number, for any equation of state, and without the requirement of homogeneity or isotropy. Rather, locality rests on assumptions about the scaling of velocity, pressure, and density structure functions, which are weak and enjoy broad empirical support.