The compact Green’s function for multiple bodies

Abstract

The acoustic emissions of compact bodies may be analysed using the compact Green's function. Whilst it is straightforward to construct the compact Green's function for single bodies, the construction for multiple bodies is far more challenging. In this paper, we present analytic solutions for the compact Green's function for multiple bodies. By writing the solution in terms of the transcendental Schottky-Klein prime function, the compact Green's function has exactly the same functional form for any number of bodies in the system. Moreover, the formulation permits the inclusion of a low Mach number background flow. The compact Green's function is illustrated for some simple examples for flat plates and cylinders and the sound generation of a Kasper wing is considered in detail.

Periodic Schwarz–Christoffel mappings with multiple boundaries per period

“…For example, the mapping can be used to calculate the trajectories of point vortices through periodic domains by making use of known analytical formulae (in terms of the same prime functions used here) for the so-called Kirchhoff-Routh path function [28], as illustrated in figure 7. Such a geometry is an example of case 2, and could be used to model the aeroacoustic interaction of a point vortex with bioinspired noise reduction mechanisms [29] using the compact Green's function for multiple bodies [30]. In a similar vein, we may use a calculus for two-dimensional vortex dynamics [31] to calculate the uniform potential flow through a 'finned channel' as illustrated in figure 8.…”