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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,
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| : <math>\begin{align} \nu(M)
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| & = \int \frac{\sqrt{M^2-1}}{1+\frac{\gamma -1}{2}M^2}\frac{\,dM}{M} \\
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| & = \sqrt{\frac{\gamma + 1}{\gamma -1}} \cdot \arctan \sqrt{\frac{\gamma -1}{\gamma +1} (M^2 -1)} - \arctan \sqrt{M^2 -1} \\
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| \end{align} </math>
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| 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]].
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| By convention, the constant of integration is selected such that <math>\nu(1) = 0. \,</math>
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| As Mach number varies from 1 to <math>\infty</math>, <math>\nu \,</math> takes values from 0 to <math>\nu_\text{max} \,</math>, where
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| : <math>\nu_\text{max} = \frac{\pi}{2} \bigg( \sqrt{\frac{\gamma+1}{\gamma-1}} -1 \bigg)</math>
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| {|
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| |-
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| |For isentropic expansion,
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| |<math>\nu(M_2) = \nu(M_1) + \theta \,</math>
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| |-
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| |For isentropic compression,
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| |<math>\nu(M_2) = \nu(M_1) - \theta \,</math>
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| |-
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| |}
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| 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.
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| == See also == | |
| * [[Gas dynamics]]
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| * [[Prandtl–Meyer expansion fan]]
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| == References ==
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| * {{cite book
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| | last = Liepmann | first = Hans W. | coauthors = Roshko, A.
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| | title = Elements of Gasdynamics | origyear = 1957
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| | publisher = [[Dover Publications]] | year = 2001
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| | isbn = 0-486-41963-0 }}
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| {{DEFAULTSORT:Prandtl-Meyer function}}
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| [[Category:Aerodynamics]]
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| [[Category:Fluid dynamics]]
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| {{fluiddynamics-stub}}
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Friends contact her Claude Gulledge. His day occupation is a cashier and his salary has been really fulfilling. For years she's been living in Kansas. The preferred hobby for him and his children is to perform badminton but he is having difficulties to find time for it.
My web page; Demonknights.madrealms.net