Vortex Lattice Method with eliptical panels
Typically in numerical simulations, instead of an ideal(inviscid) vortex, a viscous vortex with a finite core is used to avoid the singularity at the centre. In these viscous vortices, the swirl velocity inside the core is assumed to be linearly increasing from zero at the centre. While outside the core, the induced velocity remains the same as that of the ideal vortex. 

There are multiple models that model these finite core vortices; some of which are:


Rankine
$$V_{\theta}=\begin{cases}\overline{r} &amp; \text{if }\overline{r} \leq 1 \\ 1 / \overline{r} &amp; \text{if }\overline{r}&gt;1\,.\end{cases}$$
Lamb&ndash;Oseen
$$V_\theta=\frac{1}{\overline{r}}\left(1-\mathrm{e}^{-1.2526 \overline{r}^{2}}\right).$$
Bumham&ndash;Hallock
$$V_{\theta}=\frac{\overline{r}}{\overline{r}^{2}+1}\,.$$
Proctor


$$V_{\theta}=\begin{cases}
  \frac{1.4}{\overline{r}}\left(1-\mathrm{e}^{-\beta(1 / \overline{B})^{0.75}}\right) \times\left(1-\mathrm{e}^{-1.2526 \overline{r}^{2}}\right) &amp; \text{if }\overline{r} \leq 1 \\
  \frac{1}{\overline{r}}\left(1-\mathrm{e}^{-\beta(\overline{r} / \overline{B})^{0.75}}\right) &amp; \text{if }\overline{r}&gt;1\,.
\end{cases}$$


Proctor&ndash;Winkelman


$$V_{\theta}=\frac{1}{\overline{r}}\left\{1-\exp \left[-\beta_{i}(\overline{r} / \overline{B})^{2} /\left\{1+\left[\left(\beta_{i / \beta_{o}}\right)\left(\overline{r} /_{\overline{B}}\right)^{1.2}\right]^{p}\right\}^{1 / p}\right]\right\}\,.$$


Vatistas
$$V_{\theta}=\frac{\overline{r}}{\left(\overline{r}^{2 n}+1\right)^{1 / n}}\,.$$


In these equations, $\overline{r}=r/r_\mathrm{core}$.