Vortical flow outside a sphere and sound generation

Formulas are presented for an incompressible inviscid velocity field V with a vorticity field Ω outside of a rigid sphere and for the far-field sound generation. The velocity V is expressed as the sum of an image velocity v* and a known velocity v in ℜ³, which is induced by the same vorticity field Ω outside the sphere and the extension Ω = 0 inside. We derive formulas for the image velocity v* and the corresponding image potential Φ*, which in turn yields the far-field sound. These formulas are applied to define the velocity of a slender vortex filament in the presence of a rigid sphere and the associated far-field sound.