Self-similar solutions for the non-equilibrium nonlinear supersonic Marshak wave problem

Abstract

Similarity solutions to the nonlinear non-equilibrium Marshak wave problem with a time dependent radiation driving source are presented. The radiation transfer model used is the gray, non-equilibrium diffusion approximation in the supersonic regime. These solutions constitute an extension of existing non-equilibrium supersonic Marshak wave solutions, which are linear to the nonlinear regime that prevails in realistic high energy density systems. The generalized solutions assume a material model with power law temperature dependent opacities, a material energy density that is proportional to the radiation energy density, and a surface radiation temperature drive that obeys a temporal power law. The solutions are analyzed in detail, and it is shown that they take various qualitatively different forms according to the values of the opacity exponents. The solutions are used to construct a set of standardized benchmarks for supersonic non-equilibrium radiative heat transfer, which are nontrivial but straightforward to implement. These solutions are compared in detail to implicit Monte Carlo and discrete-ordinate transport simulations as well gray diffusion simulations, showing a good agreement, which demonstrates the usefulness of these solutions as a code verification test problem.