Acoustic Analogy Formulations Accelerated by Fast Multipole Method for Two-Dimensional Aeroacoustic Problems

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.

[1]  C. Bailly,et al.  Flow-induced cylinder noise formulated as a diffraction problem for low Mach numbers , 2005 .

[2]  R. Coifman,et al.  The fast multipole method for the wave equation: a pedestrian prescription , 1993, IEEE Antennas and Propagation Magazine.

[3]  S. Lele,et al.  Sound generation due to transient motion of bluff body: direct computation and acoustic analogy predictions , 2008 .

[4]  Sanjiva K. Lele,et al.  Development of Large Eddy Simulation for Aero-Thermal Prediction in High Pressure Turbine Cascade , 2008 .

[5]  N. Nishimura.  Fast multipole accelerated boundary integral equation methods , 2002 .

[6]  David P. Lockard,et al.  AN EFFICIENT, TWO-DIMENSIONAL IMPLEMENTATION OF THE FFOWCS WILLIAMS AND HAWKINGS EQUATION , 2000 .

[7]  B. Dembart,et al.  The accuracy of fast multipole methods for Maxwell's equations , 1998 .

[8]  David P. Lockard,et al.  A COMPARISON OF FFOWCS WILLIAMS-HAWKINGS SOLVERS FOR AIRFRAME NOISE APPLICATIONS , 2002 .

[9]  V. Rokhlin.  Rapid Solution of Integral Equations of Scattering Theory , 1990 .

[10]  Stéphane Caro,et al.  Analysis of the noise source interpolation within a hybrid CAA method: application to the case of the Helmholtz resonator , 2009 .

[11]  L. Greengard,et al.  A Fast Adaptive Multipole Algorithm for Particle Simulations , 1988 .

[12]  Sanjiva K. Lele,et al.  Aeroacoustic Integrals Accelerated by Fast Multipole Method , 2011 .

[13]  Kenneth S. Brentner,et al.  SIMULATION OF ACOUSTIC SCATTERING FROM A TRAILING EDGE , 2000 .

[14]  Joel H. Ferziger,et al.  A robust high-order compact method for large eddy simulation , 2003 .

[15]  Stéphane Moreau,et al.  Trailing Edge Noise of a Controlled-Diffusion Airfoil at Moderate and High Angle of Attack , 2009 .

[16]  Romesh C. Batra,et al.  EXACT CORRESPONDENCE BETWEEN EIGENVALUES OF MEMBRANES AND FUNCTIONALLY GRADED SIMPLY SUPPORTED POLYGONAL PLATES , 2000 .

[17]  Eric Darve,et al.  The Fast Multipole Method I: Error Analysis and Asymptotic Complexity , 2000, SIAM J. Numer. Anal..

[18]  Zydrunas Gimbutas,et al.  A wideband fast multipole method for the Helmholtz equation in three dimensions , 2006, J. Comput. Phys..

[19]  Osamu Inoue,et al.  Sound generation by a two-dimensional circular cylinder in a uniform flow , 2002, Journal of Fluid Mechanics.

[20]  A BOUNDARY INTEGRAL FORMULATION FOR TWO-DIMENSIONAL ACOUSTIC RADIATION IN A SUBSONIC UNIFORM FLOW , 1996 .

[21]  S. Lele,et al.  Sound Generation by Unsteady Airfoil Motions: A Study Using Direct Computation and Acoustic Analogy , 2005 .

[22]  Joel H. Ferziger,et al.  Leading-edge effects in bypass transition , 2007, Journal of Fluid Mechanics.

[23]  R. F. Warming,et al.  An Implicit Factored Scheme for the Compressible Navier-Stokes Equations , 1977 .

[24]  D. L. Hawkings,et al.  Sound generation by turbulence and surfaces in arbitrary motion , 1969, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[25]  Parviz Moin,et al.  Computation of quadrupole noise using acoustic analogy , 1996 .