Prandtl–Meyer function



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 = $$\infty$$. For an ideal gas, it is expressed as follows,


 * $$\begin{align} \nu(M)

& = \int \frac{\sqrt{M^2-1}}{1+\frac{\gamma -1}{2}M^2}\frac{\,dM}{M} \\[4pt] & = \sqrt{\frac{\gamma + 1}{\gamma -1}} \cdot \arctan \sqrt{\frac{\gamma -1}{\gamma +1} (M^2 -1)} - \arctan \sqrt{M^2 -1} \end{align} $$

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 $$\infty$$, $$\nu \,$$ takes values from 0 to $$\nu_\text{max} \,$$, where


 * $$\nu_\text{max} = \frac{\pi}{2} \bigg( \sqrt{\frac{\gamma+1}{\gamma-1}} -1 \bigg)$$

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.