Self-similar Solution of the Subsonic Radiative Heat Equations Using a Binary Equation of State

Abstract

Radiative subsonic heat waves, and their radiation driven shock waves, are important hydroradiative phenomena. The high pressure, causes hot matter in the rear part of the heat wave to ablate backwards. At the front of the heat wave, this ablation pressure generates a shock wave which propagates ahead of the heat front. Although no self-similar solution of both the ablation and shock regions exists, a solution for the full problem was found in a previous work. Here, we use this model in order to investigate the effect of the equation of state (EOS) on the propagation of radiation driven shocks. We find that using a single ideal gas EOS for both regions, as used in previous works, yields large errors in describing the shock wave. We use the fact that the solution is composed of two different self-similar solutions, one for the ablation region and one for the shock, and apply two ideal gas EOS (binary-EOS), one for each region, by fitting a detailed tabulated EOS to power laws at different regimes. By comparing the semi-analytic solution with a numerical simulation using a full EOS, we find that the semi-analytic solution describes both the heat and the shock regions well.

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

“…In this section we introduce a simple model that is derived from the analytical solutions of one-dimensional radiative heat waves, both supersonic and subsonic [20][21][22][23][24][25][26][27]. First, we introduce a one-dimensional (1D) model, which is based on the analytic solution for the supersonic Marshak waves of Hammer and Rosen (HR) [25], pointing out the importance of the different radiation temperatures that are involved on the problem [5,28,29].…”

“…Then, we expand the model to include the two-dimensional (2D) effect of the energy loss to the gold walls. This extension is based on the self-similar one-dimensional subsonic Marshak waves solutions for gold [26,27].…”

“…There is a vast literature covering self-similar solutions of both supersonic and subsonic radiative (Marshak) heat-waves, assuming LTE conditions, i.e. E( r, t) ≈ aT 4 m ( r, t) [20][21][22][23][24][25][26][27]. In these solutions, one assumes that the Rossland mean opacity κ (which is connected to the absorption cross-section σ a (T m ( r, t)) = κρ, when ρ is the material's density) and the internal energy e(T, ρ) can be approximated in a power-law form (using [25] notations):…”

Self-similar solutions for high-energy density radiative transfer with separate ion and electron temperatures

“…The classic Marshak wave is a self-similar solution in a single similarity variable, and producing those solutions has been the study of several previous studies [11–13]. There have been extensions to the original problem to include two-dimensional effects [14], non-constant drives [15], non-stationary material [16], binary equations of state [17] and magnetic effects [18]. To date there have been no self-similar solutions presented for a multi-temperature model.…”

The Discontinuous Asymptotic Telegrapher’s Equation (P1) Approximation

“…This problem has long been a subject of theoretical astrophysics research [5,6], and of experimental studies testing radiative-hydrodynamics macroscopic modeling [7,8]. Specifically, the propagating radiative Marshak waves in optically thick media are well described by a simple local thermodynamic equilibrium (LTE) diffusion model, yielding self-similar solutions of both supersonic and subsonic regimes [9][10][11][12]. However, in optically thin media, the diffusion limit fails to describe the exact physical behavior of the problem.…”