A unified theory of the self-similar supersonic Marshak wave problem

TL;DR

This paper develops a comprehensive set of similarity solutions for the supersonic Marshak wave problem under LTE diffusion approximation, providing benchmarks and insights into heat front behavior across different regimes and parameters.

Contribution

It introduces a unified theoretical framework for self-similar solutions in supersonic LTE radiative heat transfer, including analytical and highly accurate approximate solutions, and benchmarks for validation.

Findings

Existence of self-similar solutions for specific power law drives.

Conditions for linear or flat heat fronts in nonlinear conduction.

Good agreement with Monte Carlo and discrete-ordinate simulations.

Abstract

We present a systematic study of the similarity solutions for the Marshak wave problem, in the local thermodynamic equilibrium (LTE) diffusion approximation and in the supersonic regime. Self-similar solutions exist for a temporal power law surface temperature drive and a material model with power law temperature dependent opacity and energy density. The properties of the solutions in both linear and nonlinear conduction regimes are studied as a function of the temporal drive, opacity and energy density exponents. We show that there exists a range of the temporal exponent for which the total energy in the system decreases, and the solution has a local maxima. For nonlinear conduction, we specify the conditions on the opacity and energy density exponents under which the heat front is linear or even flat, and does posses its common sharp character; this character is independent of the drive exponent. We specify the values of the temporal exponents for which analytical solutions exist and employ the Hammer-Rosen perturbation theory to obtain highly accurate approximate solutions, which are parameterized using only two numerically fitted quantities. The solutions are used to construct a set of benchmarks for supersonic LTE radiative heat transfer, including some with unusual and interesting properties such as local maxima and non sharp fronts. The solutions are compared in detail to implicit Monte-Carlo and discrete-ordinate transport simulations as well gray diffusion simulations, showing a good agreement, which highlights their usefulness as a verification test problem for radiative transfer simulations.