Fast Field Solutions in Acoustic Analogy Formulations

Abstract

The computation of acoustic eld solutions due to aeroacoustic sources is performed for a large number of observer locations. Sound generation by vortex shedding is computed by a hybrid method and direct calculation and the results are compared. The hybrid approach uses direct calculation for neareld source computations and the Ffowcs Williams-Hawkings (FW-H) equation as the acoustic analogy formulation. The integration of surface dipole and volume quadrupole source terms appearing in the FW-H formulation are accelerated by a multi-level adaptive Fast Multipole Method (MLFMM). Convection eects are included in both the direct FW-H and the MLFMM formulations. The method developed in this work is applied to 2-D calculations. However, it can be extended to 3-D calculations of surface monopole and dipole source terms and volume quadrupole source terms. Results for acoustic eld solutions obtained by the accelerated FW-H formulation are 100 times faster compared to the direct computation of FW-H equation. Noise regulations have become more stringent and to achieve the required noise reductions it is important to develop more sophisticated physics-based noise prediction tools. The design of 3-D realistic congurations requires the use of time consuming numerical simulations for the study and mitigation of jet and airframe noise sources. Direct simulation of noise remains prohibitively expensive for engineering problems because of resolution requirements. Therefore, hybrid approaches that consist of predicting neareld o w quantities by a suitable CFD simulation and fareld sound radiation by an acoustic analogy formulation are more attractive. The o w physics associated with sound generation must be accurately captured in the CFD calculation in order to be used in this context. The Ffowcs Williams-Hawkings 1 (FW-H) acoustic analogy formulation is used when moving bodies are present. In this formulation, acoustic pressure uctuations are predicted by solving an inhomogeneous wave equation with surface monopole and dipole and volume quadrupole source terms. These volume sources are often neglected in sound calculations from low Mach number o w simulations since monopole and dipole sound contributions are dominant. However, for jet o ws, quadrupole terms have to be computed since they are the dominant noise sources and for wake and shear layer o ws, quadrupole terms have an important contribution to noise generation. For design purposes, it is important to compute acoustic eld solutions in order to understand the interaction between noise sources and complex geometries, including scattering and diraction of sound waves by sharp corners and cavities, for example. Moreover, complete directivity mappings may be required for the study of fareld noise radiated from realistic 3-D congurations. However, the solution of acoustic analogy formulations for eld plots and spherically resolved directivity mappings requiring many observer locations presents a high computational cost. In Ref. 2, Lockard discusses a parallel implementation of the FW-H formulation in order to overcome this drawback. In this paper, a hybrid method is used to study the sound generation by low Mach number o ws past aerodynamic geometries. In order to reduce the computational cost of eld solutions at many observer locations, the integration of surface dipole and volume quadrupole source terms are accelerated by a multi-level adaptive Fast Multipole Method 3, 4 (MLFMM). Although the method developed in this paper is applied for 2-D dipole and quadrupole integrations, it can accelerate 3-D surface integrations of monopole and dipole source terms as well as 3-D volume integrations of quadrupole