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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{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 γ 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, θ 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

  • Liepmann, Hans W.; Roshko, A. [1957] (2001). Elements of Gasdynamics. Dover Publications. ISBN 0-486-41963-0. 


 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Prandtl-Meyer_function". A list of authors is available in Wikipedia.
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