Adaptive low Mach number simulations of nuclear flame microphysics

Abstract

We introduce a numerical model for the simulation of nuclear flames in Type Ia supernovae. This model is based on a low Mach number formulation that analytically removes acoustic wave propagation while retaining the compressibility effects resulting from nuclear burning. The formulation presented here generalizes low Mach number models used in combustion that are based on an ideal gas approximation to the arbitrary equations of state such as those describing the degenerate matter found in stellar material. The low Mach number formulation permits time steps that are controlled by the advective time scales resulting in a substantial improvement in computational efficiency compared to a compressible formulation. We briefly discuss the basic discretization methodology for the low Mach number equations and their implementation in an adaptive projection framework. We present validation computations in which the computational results from the low Mach number model are compared to a compressible code and present an application of the methodology to the Landau-Darrieus instability of a carbon flame.

Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics

“…which is consistent with the low Mach number, nuclear burning model used in Bell et al (2004). The assumption that…”

“…We begin with the fully compressible equations governing motion in the stellar environment as described, for example, in Bell et al (2004) …”

“…[5]), namely, p( ; X k ; T ) ¼ p 0 (z). To complete the low Mach number model we re-pose the equation of state as a constraint on the velocity field, following closely the derivation in Bell et al (2004) but retaining stratification effects.…”

“…We note that for domains sufficiently smaller than a pressure scale height in which :p 0 can be neglected, equation (12) reduces exactly to the divergence constraint, equation (5) in Bell et al (2004). For the larger domains that are the target of this paper, we use the thermodynamic identities as outlined in Appendix A, to write…”

“…Generalizations that incorporate variations in density include the Boussinesq approximation (Boussinesq 1903), which allows heating-induced buoyancy in a constant-density background, and the anelastic atmospheric (Batchelor 1953;Ogura & Phillips 1962;Dutton & Fichtl 1969;Gough 1969;Lipps & Hemler 1982, 1985Lipps 1990;Wilhelmson & Ogura 1972) and stellar ( Latour et al 1976;Gilman & Glatzmaier 1981;Glatzmaier 1984) approximations that include the effect of large-scale background stratification in the fluid density and pressure but assume small thermodynamic perturbations from the background. Low Mach number models for chemical combustion (Rehm & Baum 1978;Majda & Sethian 1985;Day & Bell 2000) and nuclear burning (Bell et al 2004) incorporate large compressibility effects due to chemical nuclear reactions and thermal processes with a spatially constant background pressure.…”