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.

In compressible flow, information propagates at the speeds: u, u + c, and u - c, where c is the sound speed. Mathematically the timestep restriction is expressed as

Δ t < min { Δ x / (|u| + c) }

For very low Mach number flows, this is

Δ t ~ Δ x / c

This means that for an interface moving at a Mach number M << 1, it takes 1/M timesteps for that interface to move just one zone!

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| }

Therefore, our algorithm will require far fewer timesteps (~1/M fewer) to simulate low Mach number flows.

Instantaneous Equilibration -- An Example

As 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.

On the left, the top panel shows the fully compressible solution, while the bottom solution is calculated using a low Mach number method.

We initialize the calculation by placing three small temperature perturbations at varying heights in the atmosphere. In the compressible solution, a signal travels from each bubble at a finite speed. At this point in time (~0.04s after initialization), the bubbles are just starting to interact.

In the low Mach number solution, the acoustic signal travels infinitely fast, so the velocity field is already non-zero at points further than the physical sound travel time from each bubble. Because this flow is very subsonic, sound waves aren't important to the overall evolution, and the shapes and locations of the bubbles themselves are the same in the two calculations.

Different Low Mach Number Approximations

Both figures and much of the text are courtesy of Michael Zingale.