7.1.1 Governing equations

The flow studied here is that due to a diffusing point vortex of strength $\bar{\Gamma}$. We will normalize this problem by means of $\bar{\Gamma}/2\pi$ and an arbitrary characteristic length $\bar \ell$. We take the Reynolds number of the flow correspondingly as $Re={\bar \Gamma}/2\pi{\bar \nu}$. Thus normalized, the problem is governed by the nondimensional vorticity equation

$$\omega_\tau = \omega_{xx} + \omega_{yy} - Re (u \omega_x + v \omega_y) \quad \ ,$$ (7.1)

where $\tau$ is a diffusion time defined as $\tau=t/Re$.

The initial condition is

$$\omega(x,y,0) = 2 \pi \delta(x,y) \quad \ ,$$ (7.2)

where $\delta(x,y)$ is the two-dimensional delta function.

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

$$\nabla^2 \psi = - \omega \qquad u= \psi_y \qquad v = - \psi_x \quad \ .$$ (7.3)

Note that the problem is independent of the length scale $\bar \ell$, since the Reynolds number is. The only relevant length scale is the computational vortex spacing as compared to the diffusion distance $\sqrt{4\,\nu\,t}$. Our computation will therefore proceed from initial stages with no numerical resolution at all to later stages of increasing resolution. Despite the lack of resolution in the early stages, our computation turns out to be accurate (see section 9.1 for related theoretical results).