Interaction of a slender vortex with a rigid sphere: Dynamics and far- field sound

Interactions between a slender vortex filament and a stationary rigid sphere are analyzed using a vortex element scheme which tracks the motion of the filament centerline. The filament velocity is expressed as the sum of a self-induced velocity and potential velocity due to the presence of the sphere. The self-induced velocity is estimated numerically using a line Biot- Savart integral which is carefully desingularized so as to reflect the correct asymptotic behavior of the core vorticity distribution under the influence of stretching and viscous diffusion. Meanwhile, the potential velocity is evaluated from a recently derived formula, which expresses it as a line integral along the image of the filament centerline in the sphere with regular weight functions. From the far-field behavior of an unsteady vortical flow outside a stationary sphere, formulas for the acoustic far field are obtained. It is shown that the interaction between the slender vortex filament and the sphere generates dipoles and quadrupoles in addition to the quadrupoles generated by the filament alone in space. The strengths and orientations of the dipoles and quadrupoles are completely determined by the time evolution of the weighted first and second moments of vorticity. The formulas are applied to compute the far-field sound generated by the passage of a slender vortex ring over the sphere. Both coaxial and noncoaxial passage events are analyzed in the computations, as well as the effects of initial core size and asymmetric perturbations.