Compressible Turbulence: Cascade, Locality, and Scaling

While Kolmogorov's 1941 phenomenology forms the cornerstone for our understanding of incompressible turbulence, no analogous results exist for compressible flows. We present a rigorous framework to analyzing highly compressible turbulence. We show how the sole requirement that viscous effects on the dynamics of large-scale flow be negligible naturally leads to a density weighted coarse-graining of the velocity field, also known as Favre averaging. We prove that there exists a range of scales over which viscous and large-scale forcing contributions are negligible in the kinetic energy budget. An important part of our work proves that the non-linear transfer of kinetic energy to small scales is in the form of a local cascade process. Using scale-locality, we show that the \emph{average} pressure-dilatation only acts at large-scales and that the mean kinetic and internal energy budgets statistically decouple beyond a ``conversion'' scale-range. We rigorously prove that over the ensuing inertial range, scaling exponents of velocity structure functions $\langle|\delta{\bf u}|^{p}\rangle^{1/p}\sim \ell^{\sigma_p}$ are constrained by $1/3 \ge\sigma_p$ for all $p\ge3$. By assuming self-similarity, we show semi-rigorously that $\sigma_p=1/3$ for $p>0$ which implies a Kolmogorov spectrum $E^{u}(k)\sim k^{-5/3}$ for the velocity field.