Circulation (fluid dynamics)

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In fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted Γ. If \mathbf{V} is the fluid velocity and the closed curve is denoted C:

\Gamma=\oint_{C}\mathbf{V}\cdot\mathbf{ds}

The units of circulation are length squared over time.

For a body in an inviscid flow field, lift is equal to the product of the circulation about the body, the air density, and the velocity. That is:-

l = \rho V \times \Gamma

where ρ is the air density, V is the free-stream airspeed, and Γ is the circulation. This is known as the Kutta-Joukowski Theorem.

This equation applies both around aerofoils, where the circulation is generated by aerofoil 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. The circulation around an airfoil can be finite, but the vorticity of the fluid outside of the airfoil can be zero.

Circulation is highly related to vorticity. By Stokes' theorem:

\Gamma=\oint_{C}\mathbf{V}\cdot\mathbf{ds}=\int\!\!\!\int_S(\nabla\times\mathbf{V})\cdot\mathbf{dS}

but only if the integration path is a boundary, not just a closed cycle. Thus vorticity is the circulation per unit area, taken around an infinitesimal loop.

Circulation was first used independently by Frederick Lanchester, Wilhelm Kutta, and Nikolai Zhukovsky.

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