Radiation from line vortex filaments exhausting from a two-dimensional semi-infinite duct

Abstract

The low Mach number sound field induced by the motion of line vortex filaments coupled to a two-dimensional semi-infinite duct is determined by means of a singular perturbation technique. Using the method of matched asymptotic expansions, solutions for the sound field are obtained by matching with an ‘inner’ region of incompressible flow. The radiation field induced by the emergence of a single vortex from the channel exhibits the edge scattering effects typical of half-plane problems. The sound field intensity is found to have angular dependence on sin2½ θ, where θ = 0 defines the exterior axis of symmetry. When a vortex pair propagates out of the duct however, the special symmetry of the fluid motion causes cancellation of the scattered field from the duct edges. In that case the sound field is driven from sources located at the duct exit. We show that this result is consistent with the general theories of both Curle and Powell. The sound field is essentially induced by a dipole at the exit plane of the duct, part of which drives a coupled weak monopole, while the remainder corresponds to an axial ‘edge force’ originating in the r−½ velocity singularities at the duct edges.