Direct numerical simulation of decaying compressible turbulence and shocklet statistics

We present results from $1283$ and $2563$ direct numerical simulations (DNS) of decaying compressible, isotropic turbulence at fluctuation Mach numbers of $Mt∼0.1–0.5$ and at Taylor Reynolds numbers $Reλ=O(50–100).$ The presence or absence of fluctuations of thermodynamic quantities as well as velocity divergence in the initial conditions are found to have a negligible effect on the decay of turbulent kinetic energy. The decay of the turbulent kinetic energy shows no significant effect of $Mt$ and power laws fitted to the timewise decay exhibit exponents $n=1.3–1.7$ that are similar to those found for decaying incompressible turbulence. The main new phenomenon produced by compressibility is the appearance of random shocklets which form during the main part of the decay. An algorithm is developed to extract and quantify the shocklet statistics from the DNS fields. A model for the probability density function (PDF) of the shocklet strength $Mn−1$ $(Mn$ is the normal shock Mach number) is derived based on combining weak-shock theory with a model of the PDF of longitudinal velocity differences in the turbulence. This shows reasonable agreement with PDFs obtained from the shocklet extraction algorithm. The model predicts that at moderate $Mt$ the most probable shocklet strength is proportional to $Mt/Reλ1/2$ and also that the PDF for the shock thicknesses has an inverse cubic tail. The shock thickness statistics are found to scale on the Kolmogorov length rather than the mean free path in the gas.