Formulations for both are found in many books on fluid mechanics and turbulent flows. However, I am looking for a way to understand the two approaches from a common point of view and would like to find a derivation which goes from one to the other, if possible.
It is unclear for me from the question what kind of problem you try to solve? If you are looking for the way to calculate pressures on the surfaces when the is a vorticity in domain nearby then the approach is presented in paperworks of Dynnikova G.Ya. and the numerical method is described in works of Setukha A.V..
I don't know the reference, but on one of the conferences I saw presentation from Dynnikova where she was presenting numerical simulation examples very similar to those you wrote above but only 2D case. I think you can write her a letter with your question to get more information.
I haven't looked into this in detail yet, but if vorticity is involved in the pressure equation, yes, then it is related to the Biot-Savart law. The Biot-Savart law, when stripped down to the basics is simply angular momentum, which in turn is closely related to vorticity.
Jeffrey,
I'm not answering your direction question about Biot-Savart pressure Poisson equations, but rather the one that appears later: " the relation between the pressure fluctuations at a body surface and their relation to the generation of vorticity during self motion."
I think that the (second) relation can be seen in our recent work, Stalnov et al. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0134582
what we tried to get is to link the vorticity in the wake of a free flying bird to the forces developed on the wings (drag appears in another, preceding work) and cite the central works in this direction (von Karman and Sears, Theodorsen, Panda, Wu). The caveat is that it's all under inviscid approximation that Panda and Wu extended to viscous cases. I think that in liquids the added mass will be one of the dominant forces (acting through pressure on a body) but would not appear specifically in the vorticity equation.
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0134582
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