Local Mach number for an airfoil

Question

I would like to know how to calculate the local Mach number on the upper surface of an airfoil given the ambient temperature, the local velocity on the airfoil surface, the freestream velocity, and the ambient pressure.

I am assuming from my knowledge that the Mach number depends only the temperature, so I would need to calculate the local temperature on the upper airfoil surface, which is pretty trivial using the isentropic gas equations.

So $$a ~=~ \sqrt{\gamma * R * T}.$$

And then just divide the local velocity by the local speed of sound. Is this the right way to do it?

Answer 1

The local Mach number is $$ M = V/a $$ where, $a$ is the local speed of sound which depends on the fluid speed $V$ via the total enthalpy (1st Law) relation, and on the ideal gas relations. All together these give $$ a^2 = a_\infty^2 + 0.5 (\gamma-1) V_\infty^2 - 0.5 (\gamma-1) V^2 $$

where

$V_\infty$ = freestream fluid speed

$a_\infty$ = freestream speed of sound

$\gamma$ = ratio of specific heats ($\gamma=1.4$ for air, and diatomic gases in general)

Therefore, to calculate the local Mach $M$ it's not enough to know only the local speed $V$. The freestream quantities and the fluid properties are also required.