The oscillating and vibrating cylinder: a benchmark to study low Mach number aeroacoustics

Abstract

Hydrodynamic and acoustic scales separate as the Mach number decreases, making the modelling of aeroacoustic phenomena singular in this flow regime. The benchmark of the flow developing around an oscillating and vibrating cylinder is one of the scarce configuration that is fully analytically tractable, and is thus precious in the validation of new theories or solvers. This work carefully derives the complete incompressible flow solution for this case, extending the axisymmetric results of the literature to more complex cylinder motion. High-order finite element method solution of the incompressible Navier-Stokes equations provide a reference to validate the analytical formulae derived here. Both analytical and numerical investigations agree on an independence of the configuration to the Reynolds number. Up to the critical Reynolds number, found to lie above $10^4$, the hydrodynamic solution of this configuration is solely governed by the Stokes number. The exact expression of the sound radiated in the far field is computed by accounting for the cylinder scattering by means of tailored Green's functions.