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{{Continuum mechanics| cTopic=Fluid mechanics}}
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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]]

Latest revision as of 19:16, 18 December 2014

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