Lissajous curve

2014-02-01T15:09:37Z

120.59.169.130: /* Application for the case of a = b */

{{Continuum mechanics| cTopic=Fluid mechanics}}

In [[fluid dynamics]], '''circulation''' is the [[line integral]] around a closed curve of the [[velocity field]]. Circulation is normally denoted Γ ([[Greek alphabet|Greek]] [[uppercase]] [[gamma]]). Circulation was first used independently by [[Frederick Lanchester]], [[Martin Wilhelm Kutta|Wilhelm Kutta]], and [[Nikolai Yegorovich Zhukovsky|Nikolai Zhukovsky]].

==Definition==

If '''V''' is the fluid [[velocity]] on a small element of a defined curve, and ''d'''''l''' is a vector representing the [[Differential (infinitesimal)|differential]] length of that small element, the contribution of that differential length to circulation is ''d''Γ:

:<math>d\Gamma=\mathbf{V}\cdot \mathbf{dl}=|\mathbf{V}||d\mathbf{l}|\cos \theta</math>

where ''θ'' is the angle between the vectors '''V''' and ''d'''''l'''.

The circulation around a [[closed curve]] ''C'' is the [[line integral]]:<ref>{{cite book
| title = Introduction to Fluid Mechanics
| author1 = Robert W. Fox
| author2 = Alan T. McDonald
| author3 = Philip J. Pritchard
| edition = 6
| publisher = [[John Wiley & Sons|Wiley]]
| year = 2003
| isbn = 0-471-20231-2
}}</ref>

:<math>\Gamma=\oint_{C}\mathbf{V}\cdot d\mathbf{l}</math>

The [[dimensional analysis|dimension]]s of circulation are [[length]] [[square number|square]]d, divided by [[time]]; L<sup>2</sup>⋅T<sup>−1</sup>, which is equivalent to velocity times length.

== Kutta–Joukowski theorem ==
{{main|Kutta–Joukowski theorem}}

The [[lift (force)|lift]] force acting per unit span on a body in a two-dimensional inviscid flow field can be expressed as the product of the circulation Γ about the body, the fluid density ''ρ'', and the speed of the body relative to the free-stream '''V'''. Thus,

:<math>l = \rho V \Gamma\!</math>

This is known as the Kutta–Joukowski theorem.<ref>{{cite book
| title = Foundations of Aerodynamics
| author1 = A. M. Kuethe
| author2 = J. D. Schetzer
| edition = 2
| publisher = [[John Wiley & Sons]]
| year = 1959
| isbn = 0-471-50952-3
}}</ref>

This equation applies around airfoils, where the circulation is generated by airfoil action, and around spinning objects, experiencing the [[Magnus effect]], where the circulation is induced mechanically.

Circulation is often used in [[computational fluid dynamics]] as an intermediate variable to calculate forces on an [[airfoil]] or other body. When an airfoil is generating lift the circulation around the airfoil is finite, and is related to the vorticity of the [[boundary layer]]. Outside the boundary layer the vorticity is zero everywhere and therefore the circulation is the same around every circuit, regardless of the length of the circumference of the circuit.

== Relation to vorticity ==

Circulation can be related to [[vorticity]]:

:<math>\mathbf{\omega} = \nabla\times\mathbf{V}</math>

by [[Stokes' theorem]]:

:<math>\Gamma=\oint_{\partial S}\mathbf{V}\cdot d\mathbf{l}=\int\!\!\!\int_S \mathbf{\omega} \cdot d\mathbf{S}</math>

only if the integration path is a [[boundary (topology)|boundary]] (indicated by "∂") of a [[closed surface]] ''S'', not just a closed curve. Thus vorticity is the circulation per unit area, taken around an infinitesimal loop. Correspondingly, the [[flux]] of vorticity is the circulation.

==See also==

* [[Biot-Savart law#Aerodynamics applications|Biot-Savart law]]
* [[Kutta condition]]
* [[Kutta–Joukowski theorem]]
* [[Kelvin circulation theorem]]

==References==
{{reflist}}

[[Category:Fluid dynamics]]

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Lissajous curve