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 = ∞ {\displaystyle \infty } . For an ideal gas, it is expressed as follows,
ν ( M ) = ∫ M 2 − 1 1 + γ − 1 2 M 2 d M M = γ + 1 γ − 1 ⋅ arctan γ − 1 γ + 1 ( M 2 − 1 ) − arctan M 2 − 1 {\displaystyle {\begin{aligned}\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{aligned}}}
where ν {\displaystyle \nu \,} is the Prandtl–Meyer function, M {\displaystyle M} is the Mach number of the flow and γ {\displaystyle \gamma } is the ratio of the specific heat capacities. By convention, the constant of integration is selected such that ν ( 1 ) = 0. {\displaystyle \nu (1)=0.\,}
As Mach number varies from 1 to ∞ {\displaystyle \infty } , ν {\displaystyle \nu \,} takes values from 0 to ν max {\displaystyle \nu _{\text{max}}\,} , where
ν max = π 2 ( γ + 1 γ − 1 − 1 ) {\displaystyle \nu _{\text{max}}={\frac {\pi }{2}}{\bigg (}{\sqrt {\frac {\gamma +1}{\gamma -1}}}-1{\bigg )}}
where, θ {\displaystyle \theta } is the absolute value of the angle through which the flow turns, M {\displaystyle 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 Liepmann, Hans W.; Roshko, A. (2001) [1957]. Elements of Gasdynamics. Dover Publications. ISBN 978-0-486-41963-3.
