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

$\backslash Gamma=\backslash oint\_\{C\}\backslash mathbf\{V\}\backslash cdot\backslash mathbf\{ds\}$

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:-

$l\; =\; \backslash rho\; V\; \backslash times\; \backslash Gamma$

where $\backslash rho$ is the air density, $V$ is the free-stream airspeed, and $\backslash Gamma$ is the circulation.

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:

$\backslash Gamma=\backslash oint\_\{C\}\backslash mathbf\{V\}\backslash cdot\backslash mathbf\{ds\}=\backslash int\backslash !\backslash !\backslash !\backslash int\_S(\backslash nabla\backslash times\backslash mathbf\{V\})\backslash cdot\backslash mathbf\{dS\}$

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. Fluid dynamics

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