## Low Mach Number Approximations## Why Use A Low Mach Number Approach?A large number of interesting fluid dynamical phenomena occur at low Mach numbers. For low speed flows in which the energy carried by the soundwaves is unimportant to the overall solution, numerical simulation based on the fully compressible form of the governing equations is inefficient, because of the need to follow the sound waves. For an explicit time-discretization (i.e., the new state is expressed solely in terms of the present state), stability considerations constrain the size of the allowable timesteps -- the CFL condition. A timestep is restricted such that information may only propagate across one zone in the computational grid per timestep.
This means that Our desire is to reformulate the equations of hydrodynamics to filter out sound waves, while retaining the compressibility effects important to the problem at hand. This will result in a timestep constraint of the form Δ t < min{ Δ x / |u| } ## Instantaneous Equilibration -- An ExampleAs an example, we consider a set of reacting bubbles in a stratified stellar atmosphere. This plot of the Mach number demonstrates the effect of "instantaneous equilibration" in the low Mach number method.
## Different Low Mach Number Approximations### Incompressible hydrodynamics### Anelastic hydrodynamics### Low Mach Number Combustion### Low Mach Number Stellar 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. 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 ∇ ⋅ (ρ
_{0}U) = 0This 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. In the low Mach number combustion model, the pressure is decomposed
into a dynamic, π, and thermodynamic component, p ∇ ⋅ 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, and is used in our work on laboratory-scale flames, small-scale flames and small-scale thermonuclear flames in Type I supernovae. To model a full star we need to include compressibility effects from both heating and stratification. To derive this model, we begin with the low Mach number combustion model described above, but no longer assume that the background pressure is constant; instead, the background is hydrostatically stratified as in the anelastic approximation. This results in a constraint on the velocity field of: ∇ ⋅ ( β
_{0}U) = &beta_{0}[
S - 1/(Γ_{1}p_{0})
∂p_{0}/∂t]Here, β Both figures and much of the text are courtesy of Michael Zingale. |