Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Abstract

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 P1 approximation, which is a combination of the asymptotic P1 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 SiO2 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, onedimensional 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 P1 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

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

“…Although these benchmarks are characterized in highly-anisotropic flux profile, these modifications that are based on the exact asymptotic solutions, yield good results using P 1 -form equations. In addition, a gray comparison in one Marshak wave experiment (using SiO 2 foam) showed a good agreement between the discontinuous asymptotic P 1 on the one hand, and the IMC and the experimental results on the other [16]. The experimental result is of course non-gray and thus different, due to the specific measure in a specific energy band.…”

“…This approximation was tested via wellknown benchmarks, showing better results than other common approximations [17]. In addition, it was also compared to the results in one of the main Marshak wave experiments (using SiO 2 foam [6]) and IMC gray (mono-energetic) simulations of the same problem, showing good agreements [16].…”

“…The full P 1 equations, that give rise to the Telegrapher's equation, has a hyperbolic form, but with an incorrect finite velocity, c/ √ 3 [13,14]. Possible solutions, such as flux-limiters (FL) solutions (in the form of a non-linear diffusion notation), or Variable Eddington Factor (VEF) approximations (in the form of full P 1 equations), yield a gradient-dependent nonlinear diffusion coefficients (or a gradient-dependent Eddington factor) [12,15], and are still not exact enough to be trusted to make a full reconstruction of experimental results [16].…”

Key to understanding supersonic radiative Marshak waves using simple models and advanced simulations

“…This model is based on the 1D heat-wave propagation analytic model of Hammer and Rosen (HR) [27], while a primary version, that includes only some of the physical phenomena was introduced in [19]. In the present work, we expand the model, and derive a full version that take into account all the main physical phenomena that affects the general propagation of a radiative heat wave inside a finite tube.…”

“…In [19] this assumption was checked against a more exact Henyey-like temperature profile, but the effect was small.…”