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Prandtl–Meyer function

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Variation in the Prandtl–Meyer function ( ν {\displaystyle \nu } ) with Mach number ( M {\displaystyle M} ) and ratio of specific heat capacity ( γ {\displaystyle \gamma } ). The dashed lines show the limiting value ν max {\displaystyle \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 = ∞ {\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}}\\&={\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 )}}
For isentropic expansion, ν ( M 2 ) = ν ( M 1 ) + θ {\displaystyle \nu (M_{2})=\nu (M_{1})+\theta \,}
For isentropic compression, ν ( M 2 ) = ν ( M 1 ) − θ {\displaystyle \nu (M_{2})=\nu (M_{1})-\theta \,}

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

References


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