Prandtl–Meyer function explained

In aerodynamics, the Prandtl–Meyer function describes the angle through which a flow turns isentropically from sonic velocity (M=1) to a Mach (M) number greater than 1. The maximum angle through which a sonic (M = 1) flow can be turned around a convex corner is calculated for M =

inf ty . For an ideal gas, it is expressed as follows,\begin{align} \nu(M) & = \int (\sqrt{M ^{2-1})/() }\frac \\[4pt]& = \sqrt \cdot \arctan \sqrt - \arctan \sqrt\end

where

\nu is the Prandtl–Meyer function, M is the Mach number of the flow and \gamma is the ratio of the specific heat capacities.

By convention, the constant of integration is selected such that

\nu( 1 ) = 0 .

As Mach number varies from 1 to

inf ty , \nu takes values from 0 to \nu _{max} , where\nu _{max} = (\pi)/(2) ( \sqrt{ (\gamma +1)/(\gamma -1) } -1 \bigg)

For isentropic expansion,	\nu(M _{2)} = \nu(M _{1)} + \theta
For isentropic compression,	\nu(M _{2)} = \nu(M _{1)} - \theta

where,

\theta is the absolute value of the angle through which the flow turns, M is the flow Mach number and the suffixes "1" and "2" denote the initial and final conditions respectively.

See also

• Gas dynamics
• Prandtl–Meyer expansion fan

References

• Book: Liepmann, Hans W. . Roshko, A. . Elements of Gasdynamics . 1957 . . 2001 . 978-0-486-41963-3 .