Brief Overview of Low Speed Approximations

The IAMR code can be used as a basis for more general low Mach number flow algorithms (e.g., low Mach number combustion, low Mach number astrophysics, porous media flow) There are many low speed formulations of the equations of hydrodynamics in use, each with their own applications. All of these methods share in common a constraint equation on the velocity field that augments the equations of motion.

The simplest low Mach number approximation is incompressible hydrodynamics. This approximation is formally the \(M \rightarrow 0\) limit of the Navier-Stokes equations. In incompressible hydrodynamics, the velocity satisfies a constraint equation:

\[\nabla \cdot {{\bf U}}= 0\]

which acts to instantaneously equilibrate the flow, thereby filtering out soundwaves. The constraint equation implies that

\[D\rho/Dt = 0\]

(through the continuity equation) which says that the density is constant along particle paths. This means that there are no compressibility effects modeled in this approximation.

IAMR uses a constraint of the form

\[\nabla \cdot {{\bf U}} = S\]

which filters sound waves while capturing compressibilty effects due to thermal diffusion. Projection methodology is used to enforce the constraint. To achieve second order accuracy, IAMR includes two projections per timestep. The first (the ‘MAC’ projection [2]) operates on the half-time, edge-centered advective velocities, making sure that they satisfy the divergence constraint. These advective velocities are used to construct the fluxes through the interfaces to advance the solution to the new time. The second/final projection operates on the cell-centered velocities at the new time, again enforcing the divergence constraint. Additional information on the projections is in AMReX-Hydro’s documentation: Projection Methods.