A common approach says an incompressible flow has velocity divergence of 0; use this to solve for pressure in the Navier Stokes momentum equation. Or, using the Helmholtz Decomposition "project" the zero divergence component of the velocity field.
Instead, can we treat the positive/negative divergence as expansion/compression, and obtain pressure from the gas equation? This seems straightforward, copying how nature does it. (Similar to the shallow water equations, where the depth of water gives hydrostatic pressure directly).
Is it simply naive? And worse than solving in terms of efficiency/accuracy/stability? But I haven't seen it discussed at all - and it might work well for some cases (though not the engineering/heliophysics applications where much of cfd seems to originate). I'm curious.
My understanding so far: an incompressible flow doesn't require that the fluid is incompressible. e.g. air is compressible, yet can be treated as an incompressible flow. This requires velocities in the fluid to have a magnitude much less than the speed of sound (pressure waves) in the fluid. The speed of sound doesn't directly cause the incompressibility, but indicates how stiff the fluid is; the strength of the elastic reaction to pressure. The ratio of fluid velocity to speed of sound is the Mach number $M$; the Cauchy number is $M^2$, and is the "ratio between inertial and the compressibility force (elastic force) in a flow". So I think that: if the pressure is transmitted more quickly than the fluid itself, we get an "incompressible flow".
BTW: I've mainly looked at cfd for computer animation, so I'm probably missing a lot.