A Consistent Approach to Solving the Radiation Diffusion Equation
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 {var_epsilon} = {beta}/(4+{alpha}), where the internal energy varies as T{sup {beta}} and the opacity varies as T{sup -{alpha}}. We solve using an iterative approach, equivalent to asymptotic methods that match outer (away from the front) and inner (near the front) solutions. Typically {var_epsilon} < 0.3. Calculations are through first order in {var_epsilon} and are accurate to {approx} 10%, 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 power of time. In the subsonic case, the specific heat, (pressure/density) and opacity are each assumed to vary as density to a small power, of order {var_epsilon}. We find the theory compares well with radiation hydrodynamics code calculations of the heat front position, absorbed flux and ablation pressure.
- Research Organization:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- US Department of Energy (US)
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 15002150
- Report Number(s):
- UCRL-JC-149170; TRN: US200408%%145
- Resource Relation:
- Journal Volume: 10; Journal Issue: 5; Conference: 44th Annual Meeting of the Division of Plasma Physics, Orlando, FL (US), 11/11/2002--11/15/2002; Other Information: PBD: 6 Nov 2002
- Country of Publication:
- United States
- Language:
- English
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