Modeling of Supersonic Radiative Marshak waves using Simple Models and Advanced Simulations

We study the problem of radiative heat (Marshak) waves using advanced approximate approaches. Supersonic radiative Marshak waves that are propagating into a material are radiation dominated (i.e. hydrodynamic motion is negligible), and can be described by the Boltzmann equation. However, the exact thermal radiative transfer problem is a nontrivial one, and there still exists a need for approximations that are simple to solve. The discontinuous asymptotic $P_1$ approximation, which is a combination of the asymptotic $P_1$ and the discontinuous asymptotic diffusion approximations, was tested in previous work via theoretical benchmarks. Here we analyze a fundamental and typical experiment of a supersonic Marshak wave propagation in a low-density $\mathrm{SiO_2}$ foam cylinder, embedded in gold walls. First, we offer a simple analytic model, that grasps the main effects dominating the physical system. We find the physics governing the system to be dominated by a simple, one-dimensional effect, based on the careful observation of the different radiation temperatures that are involved in the problem. The model is completed with the main two-dimensional effect which is caused by the loss of energy to the gold walls. Second, we examine the validity of the discontinuous asymptotic $P_1$ approximation, comparing to exact simulations with good accuracy. Specifically, the heat front position as a function of the time is reproduced perfectly in compare to exact Boltzmann solutions.