Prandtl-Meyer Expansion
Prandtl-Meyer theory describes smooth, isentropic turning of supersonic flow through an expansion fan.
Prandtl-Meyer Function
For Mach number \(M\) and ratio of specific heats \(\gamma\):
\[
\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\,\tan^{-1}\!\left(\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}\right) - \tan^{-1}\!\left(\sqrt{M^2-1}\right)
\]
Across an isentropic expansion:
\[
\nu_2 = \nu_1 + \theta
\]
where \(\theta\) is the flow turning angle.
References
Anderson, J. D. (2003). Modern Compressible Flow: With Historical Perspective, 3rd ed. McGraw-Hill, Chapter 4.