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, which prevails in realistic high energy density systems. The generalized solutions assume a material model with power law temperature dependent opacities and a material energy density which is proportional to the radiation energy density, as well as a surface radiation temperature drive which 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.

Figures (13)

• Figure 1: Various solutions for the radiation $f^{1/4}\left(\xi\right)$ (in red) and matter $g^{1/4}\left(\xi\right)$ (in blue) temperature similarity profiles. The dimensionless defining parameters, namely the total and absorption opacity temperature powers $\alpha$ and $\alpha'$, the heat capacity ratio $\epsilon$ and the coupling parameter $\mathcal{A}$, are listed in the figure titles together with the resulting heat front coordinate $\xi_{0}$ and temperatures ratio at the origin, $g_{0}^{1/4}$. As discussed in the text, it is evident that the radiation and matter temperature become close only if $\epsilon\mathcal{A}$ is large.
• Figure 2: The matter-radiation temperature ratio at the origin, as a function of $\epsilon$ and $\mathcal{A}$ for the absorption opacity temperature powers $\alpha'=1.5$ (upper pane) and $\alpha'=3$ (lower pane).
• Figure 3: The matter-radiation temperature ratio at the origin, as a function of the absorption opacity temperature power $\alpha'$ and $\epsilon\mathcal{A}$.
• Figure 4: Radiation and material temperature profiles for Test 1. Results are shown at times $t=0.2,\ 0.6$ and 1ns, as obtained from a gray diffusion simulation and from Implicit-Monte-Carlo (IMC) and discrete ordinates ($S_{N}$) transport simulations, and are compared to the analytic solution of the gray diffusion equation (given in equations \ref{['eq:xh_1']},\ref{['eq:Tr_anal_I']}-\ref{['eq:Tm_anal_I']}).
• Figure 5: A comparison between the surface (red line) and bath (dashed line) driving temperatures for Test 1 (given in equation \ref{['eq:Tabth_1']}).