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[[Image:Prandtl meyer function.png|thumb|300px|Varition in the Prandtl–Meyer function (<math>\nu</math>) with Mach number (<math>M</math>) and ratio of specific heat capacity (<math>\gamma</math>). The dashed lines show the limiting value <math> \nu_\text{max} </math> as Mach number tends to infinity.]]
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'''Prandtl–Meyer function''' describes the angle through which a flow can turn [[Isentropic process#Isentropic flow|isentropically]] for the given initial and final [[Mach number]]. It is the maximum angle through which a sonic ([[Mach number|M]] = 1) flow can be turned around a convex corner. For an [[ideal gas]], it is expressed as follows,
 
: <math>\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} </math>
 
where, <math>\nu \,</math> is the Prandtl–Meyer function, <math>M</math> is the Mach number of the flow and <math>\gamma</math> is the [[heat capacity ratio|ratio of the specific heat capacities]].
 
By convention, the constant of integration is selected such that <math>\nu(1) = 0. \,</math>
 
As Mach number varies from 1 to <math>\infty</math>, <math>\nu \,</math> takes values from 0 to <math>\nu_\text{max} \,</math>, where
 
: <math>\nu_\text{max} = \frac{\pi}{2} \bigg( \sqrt{\frac{\gamma+1}{\gamma-1}} -1 \bigg)</math>
 
{|
|-
|For isentropic expansion,
|<math>\nu(M_2) = \nu(M_1) + \theta \,</math>
|-
|For isentropic compression,
|<math>\nu(M_2) = \nu(M_1) - \theta \,</math>
|-
|}
 
where, <math>\theta </math> is the absolute value of the angle through which the flow turns, <math>M</math> 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 ==
* {{cite book
  | last = Liepmann | first = Hans W. | coauthors = Roshko, A.
  | title = Elements of Gasdynamics    | origyear = 1957
  | publisher = [[Dover Publications]] | year = 2001
  | isbn = 0-486-41963-0 }}
 
{{DEFAULTSORT:Prandtl-Meyer function}}
[[Category:Aerodynamics]]
[[Category:Fluid dynamics]]
 
 
{{fluiddynamics-stub}}

Revision as of 03:09, 17 February 2014

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