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∗ Low Mach number variations

There are several popular methods for filtering soundwaves, which we summarize below.

Incompressible hydrodynamics

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

∇ ⋅ U = 0
which acts to instantaneously equilibrate the flow, thereby filtering out soundwaves.

The constraint equation implies that

Dρ/Dt = 0
which says that the density is constant along particle paths. This means that there are no compressibility effects modeled in this approximation.

Anelastic hydrodynamics

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

∇ ⋅ (ρ0U) = 0
This properly captures the compressibility effects due to the stratification of the background. Because there is no evolution equation for the perturbational density, approximations are made to the buoyancy term in the momentum 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

p = p0,
and differentiating this along particle paths leads to a constraint on the velocity field:
∇ ⋅ U = S
This looks like the constraint for incompressible hydrodynamics, but now we have a source term, S, representing the compressibility effects due to the energy generation and thermal diffusion.

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:

∇ ⋅ ( β0U) = β0[ S - 1/(Γ1p0) ∂p0/∂t]
Here, β0 is a density-like variable. Again, large amplitude density and temperature variations are allowed. The only restriction is that the pressure remain close to the background pressure. In the presence of heating, the star will expand, and therefore we need to evolve the background state in response to the local heating. This is an extension of the pseudo-incompressible method introduced in the atmospheric science community. As with anelastic hydrodynamics, this constraint incorporates the effects of the background stratification. However, in contrast to anelastic, the right-hand side is non-zero, and represents the compressibility effects due to heating and the change in the background state.