Panels vs Vortex Rings
Taking a panel method with a constant source doublet distribution, would it be worthwile to replace the panels with their doublets by a lattice structure of vortex rings?
I have seen that replacement in the wake panels of some methods, why is this done? And why is this not done on the geometry itself?
If so, could one then solve on unstructured triangulated surfaces instead?
As far as I remember, using a vortex ring or a doublet should lead to the same results. I m very sure you can solve for doublets using triangular meshes also. I did it during a short project during my studies. Regarding your question: I have no idea why they used a mixed combination, using the vortex rings in the wake...
In theory, you can use vortex rings for every panel, in the wake as well as on the body. However, because a vortex ring is equal to a constant doublet panel you can't use it for closed surfaces (i.e., the body) because the neumann boundary condition for all the panels will give you a singular matrix. Therefore, the difference between the wake panels and the body panels stems from the fact that any wake is, by definition, an open surface and, in addition, you don't even need to apply the neumann boundary condition on it (indeed the wake panels just need to move with the local flow velocity).
The current practice in 3D panel methods is to use a constant source distribution coupled with a dirichlet boundary condition for the flow potential function and to solve for the doublet distribution (tipically constant).
However, you can find some papers by Srivastava from the Bangalore tech institute, which actually uses vortex rings for closed bodies and wakes. He shows that, despite the matrix singularity, the solution given by the difference of the circulation between two adjacent panels (i.e., the actual local circulation) is unique and does not change whatever of the infinite results of the singular problem you choose. As a consequence he suggests to just set the value of the circulation on one panel for each closed surface and the method should work (actually he shows that it works!).
It is worth mentioning that in the computation of the local velocity induced by all the panels on a point on the surface of the body, this method needs the addition of the principal value part of the integral of the singularity distribution and this equals half the gradient of the circulation along the body surface.
With these two modifications any vortex lattice method will equally work for closed surfaces too. Of course it has no requirement on the number of sides of the panel so thay can be triangular or polygonal as well without any coordinate transformation, but just the addition of the velocities induced by the sides of the panel.
Wow, I posted this question in 2006 and it's purely a coincidence that I hang around at CFD-online now.
Thank you for your answer it's still helpful! Grazie! :-)
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