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 \(\nu_\text{max} \,\), where
\[\nu_\text{max} = \frac{\pi}{2} \bigg( \sqrt{\frac{\gamma+1}{\gamma-1}} -1 \bigg)\]
| 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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