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 mot ion of the filament centerline. The filament velocity is expressed as the sum of a self-induced velocity and potential velocity due to the p resence of the sphere. The self-induced velocity is estimated numerica lly using a line Biot-Savart integral which is carefully desingularize d so as to reflect the correct asymptotic behavior of the core vortici ty distribution under the influence of stretching and viscous diffusio n. Meanwhile, the potential velocity is evaluated from a recently deri ved formula, which expresses it as a line integral along the image of the filament centerline in the sphere with regular weight functions. F rom the far-field behavior of an unsteady vortical flow outside a stat ionary sphere, formulas for the acoustic far field are obtained. It is shown that the interaction between the slender vortex filament and th e sphere generates dipoles and quadrupoles in addition to the quadrupo les generated by the filament alone in space. The strengths and orient ations 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 a s the effects of initial core size and asymmetric perturbations. (C) 1 998 Acoustical Society of America.