Sound generation by the interaction of a vortex ring with a rigid sphere

¹ Laboratoire de Mécanique, Université du Havre, BP 540, 76058 Le Havre, France
² Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218-2686, USA

Abstract

High-Reynolds-number interactions between a vortex ring and a stationary rigid sphere are computed using two Lagrangian particle models. The first is a 3D slender vortex filament scheme which tracks the motion of the filament centerline. The centerline velocity is expressed as the sum of a self-induced velocity and potential velocity added to satisfy potential boundary conditions on the surface of the sphere. The self-induced velocity is computed numerically from the line Biot-Savart integral, which is carefully desingularized so as to provide the correct behavior of the vorticity distribution in the asymptotic thin-core limit. The second model is a particle scheme for the simulation of axisymmetric viscous flow. The scheme is based on discretization of the vorticity field into desingularized vortex elements that are advected in a Lagrangian frame. The velocity of the particles is expressed as the sum of a vortex interaction velocity expressed in terms of a desingularized Biot-Savart law, a potential velocity expressed in terms of the image of vortex elements, and a diffusion velocity. The filament and the particle schemes are applied to compute the passage of axisymmetric vortex rings over a stationary rigid sphere in the high-Reynolds-number limit, and to analyze the far-field sound generated by this interaction. Both models show that as the ring passes over the sphere, its radius increases while its core shrinks due stretching. In the parameter regime considered, the two models yield very close predictions of vortex trajectories and speeds. The filament model indicates that the passage leads to the generation of a pressure spike in the acoustic far-field. Meanwhile, the particle computations reveal that in addition to a pressure spike the far-field sound can also exhibit a substantial high-frequency quadrupole emission. Analysis of the computations reveals that this high-frequency noise emission is due to filamentation within the vortex core. The results also show that the high-frequency quadrupole noise may be dominant, especially when the vortex core is thin and the Mach number is not very small.