Direct numerical simulations of the supersonic Taylor--Green vortex via the Boltzmann equation

Abstract

We explore the dynamics of the three-dimensional compressible Taylor--Green vortex from the perspective of kinetic theory by directly solving the six-dimensional Boltzmann equation. This work studies the connections between molecular-scale information encoded in the high-dimensional distribution function (e.g., molecular entropy measures) and macroscopic turbulent flow characteristics. We present high-order direct numerical simulations at Mach numbers of $0.5$ to $1.25$ and Reynolds numbers of $400$ to $1600$ performed using up to $137{\times}10^{9}$ degrees of freedom. The results indicate that the Kullback--Leibler divergence of the distribution function $f$ and its local equilibrium state $M[f]$ (i.e., the relative entropy functional $\langle f \log (f/M[f]) \rangle$) is strongly related to the macroscopic viscous dissipation rate, with the relative entropy value matching the sum of the solenoidal and dilatational dissipation rates very closely across a range of Mach and Reynolds numbers. Furthermore, we present the behavior of subgrid-scale quantities under spatial averaging/filtering operations performed directly on the particle distribution function, which shows notable similarities between the subgrid-scale dissipation and subgrid-scale relative entropy. These observations imply that information encoded in the distribution function, particularly certain measures of its deviation from equilibrium, may be useful for closure-modeling for compressible turbulence.