∗ Low Mach number variations
There are several popular methods for filtering soundwaves, which we summarize below.
The simplest low Mach number approximation is incompressible hydrodynamics. This approximation is formally the M → 0 limit of the Navier-Stokes equations. In incompressible hydrodynamics, the velocity satisfies a constraint equation
The constraint equation implies that
In the anelastic approximation small amplitude thermodynamic perturbations are carried with respect to a hydrostatic background. The density perturbation is ignored in the continuity equation, resulting in a constraint equation
The low Mach number combustion model
In the low Mach number combustion model, the pressure is decomposed into a dynamic, π, and thermodynamic component, p0, the ratio of which is O(M2). The total pressure is replaced everywhere by the thermodynamic pressure, except in the momentum equation. This decouples the pressure and density and filters out the sound waves. Large amplitude density and temperature fluctuations are allowed. The only requirement is that the total pressure stay close to the background pressure, which is assumed constant. This requirement can be expressed as
Since the background pressure is taken to be constant, we cannot model flows that cover a large fraction of a pressure scale height. However, this method is ideal for exploring the physics of flames. We formulated this algorithm for astrophysical flows and used it to explore the dynamics of Rayleigh-Taylor unstable flame fronts in Type Ia supernovae in two- and three-dimensions. This was the predecessor to Maestro.
The Maestro algorithm for low Mach number stellar flows
To model the full star, we begin with the low Mach number combustion model described above, but no longer assume that the background pressure is constant, but instead, is hydrostatically stratified. This results in a constraint on the velocity field of: