I read several introductory texts about potential flow and vortex lattice method. Basically, it is fitting of some velocity field described by conditions on velocity at some control points using basis functions placed in other points.
In order to fullfill Helmholdz theorem that vortex filaments $\Gamma$ are closed loops (does not end in the air) horse-shoe vortex is used as basis function. But it is not very nice because:
- It is quite computationally costly to evaluate resulting velocity field using Biot–Savart law
- It imposes that vortex fillament is stright line behind the wing while in reality vortex fillaments curve and merge.
Delocalized trailing vortex:
Since we don't know how exactly filament looks like far behind the wing it would be better not to impose particular shape, but rather assume its position is uncertain (=delocalized) => vorticity $\Gamma$ smoothly blurred behind the wing. Do we really need strictly irotational vortex which concentrate vorticity in singular line?
So I'm thinking if it would be possible to use somethink like rotational dipole which is velocity field generated by one infinitisimal element of vortex filament $d\Gamma$. I'm not exaclty sure about formula but something like:
$\vec \phi(\vec r)=(v_x,v_y,v_z) = (0, z/|\vec r|^3, y/|\vec r|^3)$
This function is both faster to evaluate than horse-shoe and smoothly decays behind the wing. But I'm not sure how well it fulfills condition for proper potential flow basis function. Maybe with little modifications the errors (residual divergence and rotation) can be minimized ?
Maybe residual of such simple basisfunctions can be fitted to capture viscosity (like Lamb–Oseen vortex )? Possibly somebody already have done this?