Is Incompressible Good Enough?

       In fluids, aerodynamics, and thermodynamics, sometimes it is important to distinguish whether a gas stream needs to be treated as compressible flow or incompressible flow. In this context, these terms have a little different implication than they might have in common use. Gases are pretty much always compressible in the sense that their density changes significantly with changes in pressure. However, in the specialized context of “compressible flow” we are talking about situations where the gas velocity is high enough that the kinetic energy of the flow plays a significant part in determining the properties and changes in properties of the gas. A common rule of thumb is that a flow should be treated as compressible if the velocities involved exceed about 1/3 the speed of sound in the fluid. Of course, that is a general guideline, not a sharp limit. In this post we’ll explore that guideline a little bit.
      This figure shows the percentage error in calculating the static temperature and static pressure for air while neglecting compressibility effects, as a function of Mach number.

     The Mach number is just the dimensionless velocity—the velocity divided by the local speed of sound. The static temperature and pressure differ from the total temperature and pressure due to the kinetic energy in the flow. Both temperature and pressure error lines are thickened in order to represent a range of air temperatures between 100 deg F (37.8 deg C), and 2000 deg F (1093.3 deg C). Since the speed of sound is very different for air at the two temperatures (1160 ft/s for 100 deg F vs 2353 ft/s for 2000 deg F), a range of air velocities corresponding to a few labeled Mach numbers are shown for a few points below the X-axis.  
      As you would expect, the error increases quickly with increasing Mach number, getting close to 90% for the error in calculating static pressure at Mach=1. Obviously, by the time you get that close to sonic velocity, you would never want to neglect compressibility effects. However, we are interested in looking more closely at the rule of thumb boundary of Mach=1/3.

     This figure is the same as the one above, but zoomed in to look more closely at the low-speed region.

At Mach=1/3, the error range in temperature occurs at about 2% and the error range in pressure occurs at about 7.5%. The error varies continuously, so if that is too much error for your application, you should invoke compressible flow equations at a lower limit than Mach=1/3, and if you can stand a little more error, you might use incompressible relations up to Mach=0.5, or higher.
      Like all good rules of thumb, this one is succinct, clear, and easy to remember. As long as we understand the limitations of it, it will be endlessly useful.