Vortical Flow Outside a Sphere and Sound Generation

Abstract

Abstract. 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.

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

“…This approach also leads analytical expressions of the far-field sound [1,2,7]. The latter show the interaction between the filament and the rigid sphere results in the generation of O(M ) dipoles and O(M 2 ) quadrupoles, where M is the Mach number.…”

“…The latter has been studied by Knio & Ting [2] who extend the classical results of Weiss [9] and Lighthill [10]. In particular, analytical formulas are provided in [2] for the "image" potential and the associated velocity field. These formulas express "image" velocity as a line integral along the image of the filament centerline with regular weight functions.…”

“…The slender filament model used in the present work is adapted from previous efforts discussed in [1,2,3,4,7]. Thus, only a brief outline is provided.…”

“…This setup is motivated by some of our own previous computations [1] which featured a simplified analysis of the far-field sound generated during the interaction of a 3D filament with a rigid sphere. In [1] the effect of the sphere was approximated with a potential flow [2], and viscous effects were restricted to the core of the vortex filament. The evolution of the core Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1999034 under the combined effects of stretching and viscosity were described using an asymptotic model [3,4,5].…”

Hamiltonian Structure and Dynamics of a Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings

“…These equations and Hamiltonian structure will be derived as a special case of the model described in [1]. The simple geometry of the sphere allows an explicit representation of the image velocity field of the rings and we will follow the work of [2] for this. Second, with a view to studying dynamic orbits of such a system, we focus on the case of an axisymmetric configuration in which the rings are all circles (in parallel planes) with centers along a common line passing through the center of the sphere, as in Figure 2.…”