Circulation and Vorticity

Circulation is the line integral around a closed curve of the velocity field.It is normally denoted by \(\tau \).\[\tau = - \oint_c {V.ds} \]

Circulation should not be confused by its general dictionary meaning, in aerodynamics it is the mathematical line integral around a closed curve of a velocity field - a technical term. Circulation is related to vorticity through stoke's theorem\[\tau = - \oint_c {V.ds} = - \iint\limits_s {\left( {\nabla \times V} \right)}.ds\]Example: Consider a velocity field where x and y components of velocity are \(u = 5y/\left( {{x^2} + {y^2}} \right)\) and \(v = - 5x/\left( {{x^2} + {y^2}} \right)\). Calculate circulation around a circular path of radius \(3m\) . Let the units of \(u\) and \(v\) be in \(m/s\).

Solution: Let \(x = r\cos \theta ,\,y = r\sin \theta \), therefore \({x^2} + {y^2} = {r^2}\). In polar- coordinates \({V_r} = u\cos \theta + v\sin \theta \) and \({V_\theta } = - u\sin \theta + v\cos \theta \)

Hence,
\(u = \frac{{5y}}{{{x^2} + {y^2}}} = \frac{{5r\sin \theta }}{{{r^2}}} = \frac{{5\sin \theta }}{r}\) , \(v = - \frac{{5x}}{{{x^2} + {y^2}}} = - \frac{{5r\cos \theta }}{{{r^2}}} = - \frac{{5\cos \theta }}{r}\)

\({V_r} = \frac{{5\sin \theta }}{r}\left( {\cos \theta } \right) + \left( { - \frac{{5\cos \theta }}{r}} \right)\sin \theta = 0\) , \({V_\theta } = - \frac{{5\sin \theta }}{r}\sin \theta + \left( { - \frac{{5\cos \theta }}{r}} \right)\cos \theta = - \frac{5}{r}\)

\(V.ds = \left( {{V_r}{e_r} + {V_\theta }{e_\theta }} \right).\left( {dr{e_r} + rd\theta {e_\theta }} \right)\)\( = \left( {{V_r}dr + r{V_\theta }d\theta } \right)\)\( = 0 + r\left( { - \frac{5}{r}} \right)d\theta \)\( = - 5d\theta \)

Therefore, \(\tau = - \oint_c {V.ds} = - \int\limits_0^{2\pi } { - 5d\theta } = 5\int\limits_0^{2\pi } {d\theta } \)\( = 5 \times 2\pi \ = \,10\,{m^2}/s\).
Here, value of circulation, is independent of diameter of circular path.

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Aerodynamics

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