When the air can be regard as incompressible and how to get this conclusion? The air density changes negligibly when the air velocity $<0.3$ Sound velocity, so in this case it can be regarded as incompressible, How to get to this conclusion?
 A: From the Navier-Stockes equations for a perfect gas, it can be derived that the variation of density is proportional to the term:
$1+\frac{(\gamma−1)V²}{2a²}$
benign $V$ airflow's speed, $a$ speed of sound and $\gamma$ the heat capacity ratio. $V/a$ is defined as the Mach number $M$.
This term can be expanded as:
$1+\frac{(\gamma−1)V²}{2a²}=1+ \frac{1}{2}M²+…$
from where it can be seen that if $M$ is smaller than some 0.3 (rule of thumb), density is basically constant. This condition simplifies the Navier-Stockes equations in that the energy equation becomes decoupled from the equations of conservation of mass and conservation of momentum.
A: In theory, any fluid is compressible to some degree. Whether a particular flow of a compressible fluid can be treated as incompressible depends on several conditions. You can find derivations of these conditions in
G. Bachelor (1967), An Introduction to Fluid Dynamics, Cambridge University Press, Section 3.6
P. Thompson (1972), Compressible-Fluid Dynamics, McGraw-Hill, p. 137ff.
In the simplest case, the conditions reduce to requiring that the Mach number $M=u/c$, where $u$ is flow speed and $c$ is the sound speed, is "small enough". For practical purposes, $M<0.3$ is often taken to be "small enough" based on the following argument. The ratio of static density $\rho$ to stagnation density $\rho_0$ is given by
$\frac{\rho}{\rho_0} = \left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{1}{\gamma-1}}$.
where $\gamma$ is the ratio of specific heats. (For air, we can take $\gamma=1.4$ as a good approximation.) If you plot this equation as a function of the Mach number, you get the following (from J. Anderson (2001), Fundamentals of Aerodynamics, Third edition, McGraw-Hill, p. 483):

The figure shows that for $M<0.3$, the error in assuming the density to be constant is less than 5%, which is adequate for most purposes.
