Circulation (fluid dynamics)

In fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted $\Gamma\;$.

If $\mathbf{V}$ is the fluid velocity on a small element of a defined curve, and $\mathbf{dl}$ is a vector representing the differential length of that small element, the contribution of that differential length to circulation is $d\Gamma\;$:

$d\Gamma=\mathbf{V}\cdot \mathbf{dl}=|\mathbf{V}||\mathbf{dl}|\cos \theta$ where $\theta\;$ is the angle between the vectors $\mathbf{V}$ and $\mathbf{dl}$.

The circulation around a closed curve C is the line integral:[1]

$\Gamma=\oint_{C}\mathbf{V}\cdot\mathbf{dl}$

The dimensions of circulation are length squared, divided by time.

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

Kutta–Joukowski theorem

The 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,

$l = \rho V \Gamma\!$

This is known as the Kutta–Joukowski theorem.[2]

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 by Stokes' theorem:

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

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

References

1. ^ Robert W. Fox; Alan T. McDonald; Philip J. Pritchard (2003). Introduction to Fluid Mechanics (6 ed.). Wiley. ISBN 0471202312.
2. ^ A. M. Kuethe; J. D. Schetzer (1959). Foundations of Aerodynamics (2 ed.). John Wiley & Sons. ISBN 0471509523.

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