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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:

enter image description here

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.
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  • $\begingroup$ Interesting problem. I however don't really get the idea of elliptical panels. Your textbook figure approximates infinitesimally small panels, yielding an elliptical lift distribution. (There is probably a previous figure in which the discrete vortices / panels are still visible.) Have you already taken a look at the (open source) codes of Drela's AVL and Tornado (by Tomas Melin)? Especially the latter takes a somewhat different approach in wake modelling, based on Eppler (1997), that might help/inspire you. $\endgroup$ – Bram Aug 29 '18 at 7:18
  • $\begingroup$ maybe "panels" is not propper word. I basically want to express down-wash as a linear combination of some nice smooth basis functions (rather than singular horse-shoe vortexes produced by rectangular panels). And one of the possibilities which comes to my mind is eliptical lift distribution, which produce constant rectangular down-wash functions. $\endgroup$ – Prokop Hapala Aug 29 '18 at 11:14
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    $\begingroup$ The elliptical lift distribution is the result, but if you discretize that - you'll end up with some form of panels. Of course, the more you have, the more continuous the distribution's going to get, but if you want to use it for real-time applications, that's probably too computationally expensive. Again I point you to Tornado, which uses vortex slings instead of horseshoe vortices. These consist of 7 vortex filaments, and might provide some more flexibility. Also, I'm wondering if you can't prevent the singularity in numerically - e.g. adding some noise to prevent the vortices lining up. $\endgroup$ – Bram Aug 29 '18 at 12:03
  • $\begingroup$ I prevent the singularity now by using 1/(r^2+w^2) instead of 1/(r^2) in expression of vortex strenght. But then the vorticity is not stricly conserved, so I don't want to use too large w. Than the velocity field is not very smooth and I need rather fine panels to sample it properly. $\endgroup$ – Prokop Hapala Aug 30 '18 at 6:39
  • $\begingroup$ ad The elliptical lift distribution is the result, but if you discretize that - you'll end up with some form of panels ... No, I don't want to discretize eliptical distribution using some other basis-functions (e.g. by panesl and correspedning horse shoe vortexes). I want single analytical basis-function describing analytically velocity field generated by the whole eliptical lift distribution. Than I want to express other lift-distrubutions by linear combination by such basis - with 1 expansion coef. assigned to each eliptical basis function. $\endgroup$ – Prokop Hapala Aug 30 '18 at 6:46
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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

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

  1. Proctor–Winkelman

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

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

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

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