Vortex pair (equal circulation)

Figure 6.9: A pair of PVs with equal circulation.

In Figure (6.9), the point at $(x,y) = (\ell/2, 0)$ has ${\bf v}= 0$. The two PVs turn about this point following the circumference of a circle of radius $\ell/2$ with speed $\Gamma_{0}/2 \pi \ell$. The time for one rotation is

$$T = \frac{2 \pi (\ell/2)}{\Gamma_{0}/2 \pi \ell} = 2 \pi^{2}\ell^{2}/\Gamma_{0}$$

The angular velocity of the PVs about the centre is

$$\Omega = \frac{2 \pi}{T} = \frac{\Gamma_{0}}{\pi \ell^{2}}$$

.