7.2.1 Governing equations

The flow studied here starts from two counter-rotating point vortices. We use the same nondimensionalization as for a single point vortex in subsection 7.1.1, but in addition we now chose the characteristic length $\bar \ell$ as the distance between the initial point vortices. This leads again to the vorticity diffusion equation

$$\omega_\tau = \omega_{xx} + \omega_{yy} - Re (u \omega_x + v \omega_y) \quad \ .$$ (7.4)

However, the initial condition is now:

$$\omega(x,y,0) = 2 \pi \delta(x-{\textstyle\frac{1}{2}},y) - 2 \pi \delta(x+{\textstyle\frac{1}{2}},y) \quad \ .$$ (7.5)

This vortex system will drift in the direction normal to the line connecting the vortices. Using our normalizations, the initial drift velocity will be unity (which explains our choice of normalizations). The Reynolds number $Re={\bar \Gamma}/2\pi{\bar \nu}$ can therefore also be considered to be based on the initial drift velocity and the vortex spacing.

The velocity again follows from the Biot-Savart law [14], or equivalently from the streamfunction $\psi$:

$$\nabla^2 \psi = - \omega \qquad u= \psi_y \qquad v = - \psi_x \quad \ .$$ (7.6)