Stabilizing CFD - Compressibility, Avoiding incompressible
 

Does adding the complication of compressibility help stabilize a CFD?

The incompressible assumption seems numerically harder to solve because of the infinite stiffness of the fluid. For instance, if you are simulating a large room of fluid and you displace a small amount of fluid, then the incompressible assumption requires that the fluid on the other side of the room must move. However, with compressibility, the coupling of fluid particles far away from the displaced fluid is weak meaning the that far aware particles aren't necessary displaced. Adding compressibility seems to add energy dissipation that the numerical effect of damping on the Navier Stokes equations.

I am thinking more of weak compressible flow at Ma<0.3. Pressure differentials can be large such that fluid compresses a lot.

Thoughts?

No, actually the compressible form is more critical at low Mach number and requires special care.

Whether to use compressible model or not should depend on whether the fluid particles experience significant density changes. It is not a question of stabilizing the model or numerics by adding compressibility effects.

The Mach number is not the only parameter that determines the question of compressible vs incompressible flows. I recommend reading [1] for a nice discussion on this question.

Incompressible flow does seem somewhat unnatural since it has instantaneous action at a distance type behaviour.

In your example of gas in a room, you are right that incompressible flow predicts fluid everywhere will start moving if you displace any small portion of the wall by a small amount. But the velocity induced by this far away will be small, so the incompressible results may be still a good approximation. But if you displace the walls by a large amount, then gases being more compressible, the density will change, and the incompressible results will not give a good approximation.

[1] Mohamed Gad-el-Hak, Flow Control, Section 2.3