Circulation and point vortices

Consider a contour $(C)$ that encloses a Rankine Vortex tube (Figure (6.5)).

Figure 6.5: Calculation of the circulation $\Gamma$ for a Rankine Vortex tube.

The circulation $(\Gamma)$ is defined as

$$\Gamma = \oint_{C}{\bf v}\cdot d{\bf l} = \int_{S}(\nabla \times {\bf v})\cdot d{\bf S}$$ (6.18)

by Stoke's Theorem. ($S$ is bounded by $C$).

$$\Gamma = \int_{S}{\bf\omega}\cdot d{\bf S} = \pi a^{2}\omega_{0} = \textrm{const.}$$ (6.19)

Thus the circulation $\Gamma$ does not depend upon the details of $C$.
(We have a similar result in electromagnetism: $\oint{\bf B}\cdot d{\bf l} = \int_{S}(\nabla\times{\bf B})\cdot d{\bf S} = \mu\int{\bf j}\cdot d{\bf S}.)$

The flow velocity $(R>a)$ depends only upon the circulation, $\Gamma$. If $a\rightarrow a/2$ and $\omega_{0}\rightarrow 4\omega_{0}$ then $\Gamma$ (by (6.19)) = const., and the flow velocity will be unchanged.

The limit of $a\rightarrow 0$ and $\omega_{0}\rightarrow\infty$, such that $\Gamma = \pi a^{2}\omega_{0} =$ const. results in a point vortex (PV), and is illustrated in Figure (6.6).

Figure 6.6: A point vortex (PV) is an ordinary vortex tube in the limit of vanishing cross section.

What is the flow around a PV? Take $C$ to be a circle of radius $R$, then (6.18) becomes

$$\Gamma = 2 \pi Rv_{\phi}(R),{\rm i.e.},~~~~ v_{\phi}(R) = \frac{\Gamma}{2\pi R}$$ (6.20)

The flow associated with a PV of circulation $\Gamma$ consists of circular streamlines (see Figure (6.7)).

Figure 6.7: The flow velocity surrounding a point vortex (PV).

Thus one PV just sits there with flow circulating around it (if there is no uniform flow at $R\rightarrow \infty$.) If there is a uniform flow ${\bf v}_{0}$ as $R\rightarrow \infty$ we can find the solution by adding on ${\bf v}_{0}$ to everything - this is equivalent to a frame transformation. The result is that the PV appears to move with a steady velocity ${\bf v}_{0}$.