Prediction of Sound Generated by Complex Flows at Low Mach Numbers Yaser Khalighi, * Ali Mani, * Frank Ham, † and Parviz Moin ‡ Stanford University, Stanford, California 94305 DOI: 10.2514/1.42583 We present a computational aeroacoustics method to evaluate sound generated by low Mach number flows in complex configurations in which turbulence interacts with arbitrarily shaped solid objects. This hybrid approach is based on Lighthill’s acoustic analogy in conjunction with sound source information from an incompressible calculation. In this method, Lighthill’s equation is solved using a boundary element method that allows the effect of scattered sound from arbitrarily shaped solid objects to be incorporated. We present validation studies for sound generated by laminar and turbulent flows over a circular cylinder at Re  100 and 10,000, respectively. Our hybrid approach is validated against directly computed sound using a high-order compressible flow solver as well as the solution of the Ffowcs Williams and Hawkings equation in conjunction with compressible sound sources. We demonstrate that the sound predicted by a second-order hybrid approach is as accurate as sound directly computed by a sixth-order compressible flow solver in the frequency range in which low-order numerics can accurately resolve the flow structures. As an example of an engineering problem, we calculated the sound generated by flow over an automobile side-view mirror and compared it to experimental measurements. Nomenclature C D , C L = drag and lift coefficients, respectively c = speed of sound d = dimension of the problem e ij = viscous stress tensor f = frequency G = Green’ s function of the Helmholtz operator k = wave number L, L c = width of the mirror, recirculation length M, M i = freestream Mach number, freestream vector Mach number n i = unit outward to the boundary @ p, p a = pressure, acoustic pressure defined as c 2 0  0 Re = Reynolds number r = distance from the sound source region St = Strouhal number T ij = Lighthill’ s stress tensor t = time u, v = compressible and incompressible velocities, respectively u = streamwise velocity x, y = locations of the observer and the source, respectively x i = Cartesian coordinate  = domain-dependent geometrical factor  ij = Kronecker delta  = fluid density  = power spectral density , @ = acoustic medium, boundary of the acoustic medium, that is, solid boundary nfxg = domain  excluding the point x ! = angular frequency k = modulus of a complex quantity  x = quantity evaluated at observer location x Subscript 0 = reference quantity Superscripts 0 = difference from the reference quantity  = quantity in the frequency domain  = time mean of the quantity rms = rms. of the quantity I. Introduction I N MANY practical applications, sound is generated by the interaction of turbulent flow with solid objects. Here, sound waves experience multiple reflections from solid objects before they propagate to an observer. To predict the acoustic field in such situ- ations requires a general aeroacoustic framework to operate in com- plex environments. Furthermore, the method employed must avoid making simplifying assumptions about the geometry, compactness, or frequency content of sound sources. The present work aims to develop, validate, and demonstrate the functionality of such a com- putational framework. The prediction of flow-generated sound must account for the physics of both unsteady flow and sound simultaneously. Because these two phenomena exhibit very different energy and length scales, prediction of flow-generated sound is challenging, especially from a numerical perspective. Sound waves carry only a minuscule fraction of flow energy, and accurate numerical schemes with low dissipation and low dispersion are required in a direct computation to keep the sound waves intact. Additionally, in the low Mach number regime, the acoustic Courant-Friedrichs–Lewy (CFL) number imposes ex- tremely small time steps on numerics for resolving both acoustics and hydrodynamics. To avoid such computational difficulties, vari- ous hybrid approaches have employed Lighthill’ s acoustic analogy [1], which allows for the use of separate numerics suited to each physical phenomenon. In particular, at low Mach numbers, the un- steady hydrodynamic field is computed by an incompressible flow solver in which the time step is not restricted by the acoustic CFL number. This solution is then used to represent the sound sources in a separate acoustic solver. Received 16 December 2008; revision received 21 September 2009; accepted for publication 18 October 2009. Copyright © 2009 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. * Graduate Student, Department of Mechanical Engineering, Center for Turbulence Research. † Research Associate, Center for Turbulence Research. ‡ Franklin and Caroline Johnson Professor on Engineering, Department of Mechanical Engineering. Fellow AIAA. AIAA JOURNAL Vol. 48, No. 2, February 2010 306