Circulation around a Vortex

Let us now calculate the circulation (see Circulation) around a free vortex. We have

$$\Gamma = \oint_C d \phi = \int_0 ^{2 \pi} {K \over r} ds = \int_0 ^ {2 \pi} {K \over r} (r d\theta) = 2 \pi K$$ (4.78)

which is non-zero. Where is the flaw in our integration then? It is a simple matter to find out. We have a singularity in our region, namely, r = 0! If we exclude the singularity by making a small cut around the origin, we will in fact get the result that circulation around the vortex is zero.

It is usual to write the equation for velocity potential and stream function in terms of circulation $\Gamma$, thus

$$\phi = {\Gamma \over {2 \pi}} \theta,   \psi = - {\Gamma \over {2 \pi}} \ln r$$ (4.79)

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney