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 <3, 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.