A consistent approach to solving the radiation diffusion equation

Abstract

Diffusive x-ray-driven heat waves are found in a variety of astrophysical and laboratory settings, e.g. in the heating of a hohlraum used for ICF, and hence are of intrinsic interest. However, accurate analytic diffusion wave (also called Marshak wave) solutions are difficult to obtain due to the strong non-linearity of the radiation diffusion equation. The typical approach is to solve near the heat front, and by ansatz apply the solution globally. This works fairly well due to "steepness" of the heat front, but energy is not conserved and it does not lead to a consistent way of correcting the solution or estimating accuracy. We employ the steepness of the front through a perturbation expansion in E = b/(4+a), where the internal energy varies as TB and the opacity varies as T". We solve using an iterative approach, equivalent to asymptotic methods that match outer (away from the front) and inner (near the front) solutions. Typically E < 0.3. Calculations are through first order in E and are accurate to-lo%, which is comparable to the inaccuracy from assuming power laws for material properties. We solve for supersonic waves with arbitrary drive time history, including the case of a rapidly cooling surface, and generalize the method to arbitrary temperature dependence of opacity and internal energy. We also solve for subsonic waves with drive temperature varying as a

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

“…The numerical values for the internal energy and the opacity in the heat wave region, which are best fit for a temperature range of 100 − 300eV and are taken from [10], are presented in Table I.…”

“…1) [13], a single EOS was used for both regions (r 1 = r 2 = 0.25). We remind that this value is a fit for the region T = 100 − 300eV are the results of a full EOS SESAME table [14] and an ideal gas EOS with numerical value r = γ − 1 = 0.25 (proposed by Hammer & Rosen [10]). …”

“…a ≡ 4σ /c is the radiation density constant (σ is Stefan-Boltzmann constant) and c is the speed of light. In order to find a self-similar solution, we assume the internal energy of the matter e and the opacity κ R can be expressed in the form of power laws of the density and the temperature T (we adopt the notation first presented by [10]):…”

“…Subsonic case solutions were investigated by many authors [7][8][9][10][11]. Most of the solutions focused on the ablation region alone, since the full problem (including the shock region) is not self-similar for the general case.…”

“…Using a single ideal-gas for the EOS (which is fitted for the ablation regime as in [10]) may cause large errors when describing the shock region, since the temperatures and densities vary widely between the ablation region and the shock region. This can be shown by comparing the semi-analytic solution to a numerical simulation with an EOS tables.…”