Two-dimensional effects in the edge sound of vortices and dipoles

Abstract

Edge sound consists of the pressure waves generated by the fast modification of the local velocity field when a flow inhomogeneity passes near an edge. The modifications take place in the near field; they locally allow a description in terms of incompressible flow. The incompressible disturbances are referred to as pseudosound. The disturbances progress into the far field as diffraction waves that, in 3D space, dominate the pseudosound. In 3D space the diffraction waves have a 2D character in their time history; they behave as the halfth time derivative of the pseudosound. This latter effect is not manifest in the analogous, purely 2D diffraction problem where the diffracted wave in the far field has the form of retarded pseudosound. In this paper, an answer is presented to the intriguing question: what is the rationale behind this difference in behavior? The analysis is based on time integral representations having as a kernel the closed form of Green’s functions for half-plane problems in 2D and in 3D space. The Green’s functions are presented in this paper. The integral representations inherently represent the relations between wave-sound and incompressible pseudosound. After performing an integration by parts, appropriate approximations reveal the incompressible near-field behavior or the far-field wave behavior. Examples are given of the edge sound of moving vortices and dipoles in 2D and in 3D space. In free 2D space the potential field of a vortex is proportional to the logarithm of the distance; the field of a dipole is proportional to the inverse of the distance. However, the edge sound pressure of vortices and of dipoles does not show a difference in far field behavior. It falls off with the square root of distance in 2D space and linear with distance in 3D space.