The scattering of sound waves by a vortex: numerical simulations and analytical solutions

The scattering of plane sound waves by a vortex is investigated by solving the compressible Navier-Stokes equations numerically, and analytically with asymptotic expansions. Numerical errors associated with discretization and boundary conditions are made small by using high-order-accurate spatial differentiation and time marching schemes along with accurate non-reflecting boundary conditions. The accuracy of computations of flow fields with acoustic waves of amplitude five orders of magnitude smaller than the hydrodynamic fluctuations is directly verified. The properties of the scattered field are examined in detail. The results reveal inadequacies in previous vortex scattering theories when the circulation of the vortex is non-zero and refraction by the slowly decaying vortex flow field is important. Approximate analytical solutions that account for the refraction effect are developed and found to be in good agreement with the computations and experiments. The prediction of the sound produced by turbulent flow requires a detailed knowledge of acoustic source terms. Direct computation of both the acoustic sources and far-field sound using the unsteady Navier-Stokes equations allows direct validation of aeroacoustic theories. In a recent review by Crighton (1988), the difficulties involved in direct computations of aeroacoustic fields are discussed. These include: the large extent of the acoustic field compared with the flow field; the small energy of the acoustic field compared to the flow field; and the possibility that numerical discretization may introduce a significant sound source due to the acoustic inefficiency of low-Mach-number flows. In order to address these difficulties, Crighton proposed that direct computations be performed on elementary model aeroacoustic problems whose physics are well understood. For this reason, and to validate our numerical scheme for direct computation of aeroacoustic problems, we investigate the scattering of sound waves by a compressible viscous vortex. This problem has received significant attention, and thus provides a large database of theory, numerics and experiment with which detailed comparisons may be made. Yet there is significant disagreement amongst the various theories, which has not yet been fully rectified. Therefore, the purpose of the current work is twofold: to validate our numerical scheme for direct computation of aeroacoustic problems using the unsteady Navier-Stokes equations, and to investigate the scattering of sound waves by a compressible viscous vortex.