Acoustic Analogy Formulations Accelerated by Fast Multipole Method for Two-Dimensional Aeroacoustic Problems William R. Wolf * and Sanjiva K. Lele † Stanford University, Stanford, California 94305-4035 DOI: 10.2514/1.J050338 The calculation of acoustic field 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 an accurate two-dimensional direct calculation, and the results are compared. The hybrid approach uses direct calculation for near-field source computations and the Ffowcs-Williams–Hawkings equation as the acoustic analogy formulation. The integrations of surface dipole and volume quadrupole source terms appearing in the Ffowcs-Williams–Hawkings formulation are accelerated by a wideband multilevel adaptive fast multipole method. The wideband multilevel adaptive fast multipole method presented here applies a plane-wave expansion formulation for calculations in the high-frequency regime and a partial-wave expansion formulation in the low-frequency regime. The method is described in detail for the solution of a two-dimensional Green’s function that incorporates convective effects. The method presented in this work is applied to two-dimensional calculations. However, it can be easily extended to three-dimensional calculations of surface monopole and dipole source terms and volume quadrupole source terms. Results for acoustic field solutions obtained by the accelerated Ffowcs-Williams–Hawkings formulation are 2 orders of magnitude faster when compared with the direct computation of the Ffowcs-Williams–Hawkings equation. Nomenclature c = speed of sound D = diagonal operator in plane-wave expansion formulation F i = dipole source f = Ffowcs-Williams–Hawkings surface G = Green’ s function H = Heaviside function H 2 n = Hankel function of the second kind and order n I = shifting operator in plane-wave expansion formulation i = imaginary unit J n = Bessel function of the first kind and order n K = modified wave number k = wave number L HF = local expansion for high-frequency regime L LF = local expansion for low-frequency regime M = Mach number M HF = multipole expansion for high-frequency regime M LF = multipole expansion for low-frequency regime p = pressure Q = monopole source R = regular function in partial-wave expansion formulation S = singular function in partial-wave expansion formulation St = Strouhal number T ij = quadrupole source (Lighthill stress tensor) t = time U i = uniform velocity vector u i = fluid velocity vector X i = observer location in transformed coordinates x i = observer location Y i = source location in transformed coordinates y i = source location  = Dirac delta function  ij = Kronecker delta  = polar angle of arbitrary vector  = density  ij = viscous stress tensor  = polar angle of plane wave ! = angular frequency Subscript 0 = freestream property Superscript 0 = acoustic property I. Introduction N OISE 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 three-dimensional (3-D) realistic configurations requires the use of time-consuming numerical simulations for the study and mitigation of jet, fan, 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 near-field flow quantities by a suitable computational fluid dynamics (CFD) simulation and far-field sound radiation by an acoustic analogy formulation are more attractive. The flow 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 (FW–H) [1] acoustic analogy formulation is used when moving rigid or flexible bodies are present. In this formulation, acoustic pressure fluctuations are predicted by solving an inhomogeneous wave equation with surface monopole and dipole and volume quadrupole source terms. Quadrupole sources are often neglected in sound calculations from low-Mach- number flow simulations, since monopole and dipole sound contri- butions are dominant. However, for jet flows, they have to be computed, since they are the dominant noise sources and, for wake and shear layer flows, quadrupole sources have an important contri- bution to noise generation. Presented as Paper 2009-3231 at the 15th AIAA/CEAS Aeroacoustics Conference, Miami, FL, 11–13 May 2009; received 10 November 2009; revision received 11 February 2010; accepted for publication 09 April 2010. Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/10 and $10.00 in correspondence with the CCC. * Ph.D. Candidate, Department of Aeronautics and Astronautics. † Professor, Department of Aeronautics and Astronautics, Department of Mechanical Engineering. AIAA JOURNAL Vol. 48, No. 10, October 2010 2274