Prandtl-Meyer function

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{align} 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{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_{max} ,, where

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

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

References

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Category : Aerodynamics Category : Fluid dynamics

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