Prandtl–Meyer function

Variation in the Prandtl–Meyer function ($\nu$) with Mach number ($M$) and ratio of specific heat capacity ($\gamma$). The dashed lines show the limiting value ${\nu }_{\text{max}}$ as Mach number tends to infinity.

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

$\begin{array}{cc}\nu (M)& =\int \frac{\sqrt{M^2-1}}{1+\frac{\gamma -1}{2}{M}^2}\frac{dM}{M}\\ & =\sqrt{\frac{\gamma +1}{\gamma -1}}\cdot \arctan \sqrt{\frac{\gamma -1}{\gamma +1}(M^2-1)}-\arctan \sqrt{M^2-1}\end{array}$

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

${\nu }_{\text{max}}=\frac{\pi }{2}(\sqrt{\frac{\gamma +1}{\gamma -1}}-1)$

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

• Template:Cite book

Template:Fluiddynamics-stub