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# Prandtl-Meyer function

Prandtl-Meyer function describes the angle through which a flow can turn isentropically for the given initial and final Mach number. It is the maximum angle through which a sonic (M = 1) flow can be turned around a convex corner. For an ideal gas, it is expressed as follows,

begin\left\{align\right\} nu\left(M\right)
& = 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{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 $nu\left(1\right) = 0. ,$

As Mach number varies from 1 to $infty$, $nu ,$ takes values from 0 to $nu_\left\{max\right\} ,$, where

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

For isentropic expansion, $nu\left(M_2\right) = nu\left(M_1\right) + theta ,$
For isentropic compression, $nu\left(M_2\right) = nu\left(M_1\right) - theta ,$

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.