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I've read that Mach 0.3 is pretty much the upper limit for treating air as an incompressible fluid. The sources I've read seem to treat this as a given, without proof or justification.
Why is this the limit? Is there a mathematical justification for this? Also, does this limit only apply to air? If not, then what does the limit depend on?
Wikipedia gives the reason for Mach 0.3 as due to the fact that this achieves ~5% change in density.
I found a NASA page that describes (analytically!) the relationship. I cited the source, but I'll reproduce the work here for posterity, in the event their links change.
Start with conservation of momentum:
$$ (\rho V) dV = -dp \\ $$
where $\rho$ is the fluid density, $V$ is the velocity, and $p$ is the pressure. for isentropic flow:
$$ \frac{dp}{p} = \gamma \frac{d\rho}{\rho} \\ dp = \left( \frac{\gamma p}{\rho} \right) d\rho \\ $$
where $\gamma$ is the specific heat ratio. The ideal gas law gives:
$$ p = \rho R T \\ $$
where $R$ is the specific gas constant and $T$ is the absolute temperature. So, substituting:
$$ dp = \gamma R T d\rho $$
The speed of sound can be calculated by:
$$ \gamma R T = a^2 \\ $$
where $a$ is the speed of sound, so:
$$ dp = a^2 d\rho \\ $$
Substituting the expression above into the conservation of momentum equation gives:
$$ (\rho V)dV = -a^2 d\rho \\ -\left(\frac{V^2}{a^2}\right)dV/V = d\rho/\rho \\ -M^2 dV/V = d\rho/\rho \\ $$
where $M$ is the Mach number. This gives a Mach number of 0.3 to be approximately a 5% change in density.
As a note, this is based on the Mach number, which in turn is dependent on the speed of sound in the gas, so it's automatically adjusted on a per-gas basis.
