Publication 26-CNA-013

Anisotropic Thermalization in Far-from-Equilibrium Flows

Arnab Debnath
Program in Computational Mechanics
Department of Civil and Environmental Engineering
Carnegie Mellon University
Pittsburgh, PA 15213

Timothy Breitzman
Materials and Manufacturing Directorate
Air Force Research Laboratory

Kaushik Dayal
Department of Civil and Environmental Engineering
Center for Nonlinear Analysis
Department of Mechanical Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
Kaushik.Dayal@cmu.edu

Abstract: We present a deterministic discontinuous Galerkin (DG) finite-element solution of the Boltzmann equation, without moment-closure approximations, under a class of far-from-equilibrium deformations. Specifically, we consider affine flows which reduce the Boltzmann equation to a purely velocity-space problem for the reduced distribution function in a reduced velocity field. We solve the reduced equation using a tensor-product Lagrange DG discretization for four representative flows: simple shear, pressure shear, bi-directional shear, and a vortex flow. Our principal finding is that the velocity distribution is well-approximated by an anisotropic Gaussian throughout the evolution, despite the non-equilibrium conditions. Further, we show the evolution of the covariance tensor of the Gaussian distribution is equal to the inverse of the right Cauchy-Green tensor in the free-streaming limit without collisions. This prediction compares very well with the numerical solution at short times; at longer times, they grow apart, reflecting the influence of particle collisions.

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