Multi-Group Discontinuous Asymptotic $P_1$ Approximation in Radiative Marshak Waves Experiments

TL;DR

This paper introduces a modified discontinuous asymptotic $P_1$ approximation for modeling radiative Marshak wave propagation, demonstrating improved accuracy over traditional methods through experimental and Monte Carlo comparisons.

Contribution

It develops and tests a new discontinuous asymptotic $P_1$ approximation that better predicts heat wave velocities in optically-thin media compared to classic models.

Findings

The new approximation aligns well with experimental Marshak wave data.

It outperforms classic diffusion and $P_1$ approximations in accuracy.

Monte Carlo simulations confirm its superiority across various conditions.

Abstract

We study the propagation of radiative heat (Marshak) waves, using modified $P_1$-approximation equations. In relatively optically-thin media the heat propagation is supersonic,~i.e. hydrodynamic motion is negligible, and thus can be described by the radiative transfer Boltzmann equation, coupled with the material energy equation. However, the exact thermal radiative transfer problem is still difficult to solve and requires massive simulation capabilities. Hence, there still exists a need for adequate approximations that are comparatively easy to carry out. Classic approximations, such as the classic diffusion and classic $P_1$, fail to describe the correct heat wave velocity, when the optical depth is not sufficiently high. Therefore, we use the recently developed discontinuous asymptotic $P_1$ approximation, which is a time-dependent analogy for the adjustment of the discontinuous asymptotic diffusion for two different zones. This approximation was tested via several benchmarks, showing better results than other common approximations, and has also demonstrated a good agreement with a main Marshak wave experiment and its Monte-Carlo gray simulation. Here we derive energy expansion of the discontinuous asymptotic $P_1$ approximation in slab geometry, and test it with numerous experimental results for propagating Marshak waves inside low density foams. The new approximation describes the heat wave propagation with good agreement. Furthermore, a comparison of the simulations to exact implicit Monte-Carlo slab-geometry multi-group simulations, in this wide range of experimental conditions, demonstrates the superiority of this approximation to others.