Vortex Lattice Method with eliptical panels

Question

I implemented vortex lattice method (or deeper here) into my aircraft simulation program. The problem is that sometimes trailing vortex filaments from Horse Shoe Vortexes of main wing hits panels of rudder/elevator which leads to singularities (infinite coupling between the two panels). This is because rectangular panels generate sharp singular vortex filament at the end of wing. (see https://scicomp.stackexchange.com/posts/30062/edit)

I'm thinking to use eliptical panels as a basis set for VLM to avoid those singularities - then I hope my simulator can work with smaller number of panels (I wan't to make it real-time).

This is normally ploted in texbook:

My questions:

• I search for formula / derivation how the down-wash from eliptical wing looks like in the whole 3D space. Usually what is plotted in texbook (constant rectangular downwash) is true just immediately behind the wing.
• Is it fine to express wing of complex shape as linear combination of eliptical wings and it's downwash as linear combination of downwashes from each of them? I mean if potential flow is linear problem, than I see no problem - ut maybe I forgot about some catch.

Answer 1

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:

1. Rankine $$V_{\theta}=\begin{cases}\overline{r} & \text{if }\overline{r} \leq 1 \\ 1 / \overline{r} & \text{if }\overline{r}>1\,.\end{cases}$$

2. Lamb–Oseen $$V_\theta=\frac{1}{\overline{r}}\left(1-\mathrm{e}^{-1.2526 \overline{r}^{2}}\right).$$

3. Bumham–Hallock $$V_{\theta}=\frac{\overline{r}}{\overline{r}^{2}+1}\,.$$

4. Proctor

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

5. Proctor–Winkelman

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

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

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