Question 10.21: What percent density change would be expected for a flow of a...
What percent density change would be expected for a flow of air with a characteristic velocity equivalent to M = 0.3? What is the corresponding percent change in pressure and temperature?
This exercise is solved by using Eqs. 10.71 to estimate the various changes. Noting that γ = 1.4 for air and M² = (0.3)² = 0.09, we find:
\frac{dp }{p}=\frac{\gamma }{2}M^2, \ \ \frac{d\rho }{\rho }=\frac{1}{2}M^2, \text{and} \ \ \frac{dT }{T}=\frac{\gamma -1}{2}M^2 (10.71a–c)
\frac{d\rho }{\rho }=\frac{1}{2}M^2=\frac{1}{2}(0.09)=0.045
which is a 4.5% density change. The changes in pressure and temperature are calculated as
\frac{dp }{p}=\frac{\gamma }{2}M^2=\frac{1.4}{2}(0.09)=0.063=6.3\%
\frac{dT }{T}=\frac{\gamma -1}{2}M^2=\frac{0.4}{2}(0.09)=0.018=1.8\%
A Mach number of 0.3 is generally accepted as the upper limit for assuming that air may be modeled as an incompressible fluid. Nevertheless, since the sound speed in air is about 340 m/s (1130 ft/s), a constant density model for air is appropriate in many practical applications.