Why does density decrease more above Mach 0.3, and not as much below Mach 0.3? From what I've heard, it isn't a linear relationship but why is that? Asked differently, what makes it a non linear relationship?
Thanks!
Link to chat room of this question.
Also, the last paragraph of this question has my guess of why the density doesn't change much below Mach ~0.3. It might be useful to see where I'm coming from.
When you combine the equations for isentropic flow you can plot "density compared to density at zero speed, M=0", i.e. $\rho / \rho_0$, as a function of M.
Anderson's Fundamentals of Aerodynamics presents that as $\rho_0 / \rho = (1+ \frac {\gamma -1}{2} M^2)^{\frac{1} {\gamma-1}}$ with $\gamma = 1.4$ for air (page 567, eq. 8.43).
Inverting, $\rho / \rho_0 = (1+ \frac {\gamma -1}{2} M^2)^{\frac{1} {1-\gamma}}$.
I've copied a plot of this below, but you can calculate it yourself if you like. The red curve shows that M = 0.3 isn't magical. (At M=0.3, the density ratio is 0.956.) It's just a convenient, practically useful threshold for everyday calculations.
Why isn't the red curve a straight line? Because these equations have proven to be a useful, accurate, predictive model of fluid flow in many fields, and these equations are nonlinear. You could replace them with linear ones if you liked, but the calculations you'd make with them would be less accurate.
Anderson's own plot, p. 573 fig. 8.6, agrees:
