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Numerical evaluation of the compact acoustic Green’s function for scattering problems. (English) Zbl 1446.76029

Summary: The reduction of noise generated by new and existing engineering products is becoming of increasing commercial importance. Noise prediction schemes are important tools available to help us understand and develop a means of controlling noise. Hybrid noise prediction schemes alleviate many issues associated with exclusively numerical or analytical approaches. These schemes often make use of a Green’s function to compute the sound field – the Green’s function representing geometrical scattering effects. Current hybrid schemes are limited to propagating noise in simple geometries for which the Green’s function is known. In order to extend hybrid schemes to more general geometries, we develop here a robust, semi-analytical computational method to compute Green’s functions for more general geometries in both 2D and 3D. The class of Green’s functions considered here can be constructed through conformal mapping of the geometry to a canonical domain. Traditionally, this would only be possible if the mapping could be expressed analytically. Here we combine the traditional algorithm with a numerical mapping procedure to allow the Green’s function to be computed for more general geometries. The accuracy is assessed through application to 2D benchmark problems for which analytical solutions are known. Although we assess the accuracy and speed of the method on 2D problems only, the extension to 3D only requires an additional execution of the same computational procedure for the extra dimension with a predictable effect on these two properties. We compute a Green’s function for a baffle in a 2D channel, an important geometry in vortex sound problems, and a 3D projection from the half-plane. The semi- analytical method presented here demonstrates calculation of the Green’s function accurately and robustly by avoiding particular conformal transformations and the evaluation of potential models containing singularities.

MSC:

76-10 Mathematical modeling or simulation for problems pertaining to fluid mechanics
76Q05 Hydro- and aero-acoustics
76M99 Basic methods in fluid mechanics
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[1] Tam, C. K.W., Computational aeroacoustics: an overview of computational challenges and applications, Int. J. Comput. Fluid Dyn., 18, 6, 547-567 (2004) · Zbl 1065.76180
[2] Wang, M.; Freund, J. B.; Lele, S. K., Computational prediction of flow-generated sound, Annu. Rev. Fluid Mech., 38, 1, 483-512 (2006) · Zbl 1100.76058
[3] Howe, M. S., Theory of Vortex Sound (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1098.76003
[4] McGowan, R. S.; Howe, M. S., Compact Green’s functions extend the acoustic theory of speech production, J. Phon., 35, 2, 259-270 (2007)
[5] Harwood, A. R.G.; Dupère, I. D.J., Numerical evaluation of the compact Green’s function for the solution of acoustic flows, Noise Control and Acoustics Division Conference at InterNoise 2012 (2012), American Society of Mechanical Engineers
[6] Lighthill, M. J., On sound generated aerodynamically. I. general theory, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci., 211, 1107, 564-587 (1952) · Zbl 0049.25905
[7] Tang, S. K.; Lau, C. K., Two-dimensional model of low Mach number vortex sound generation in a lined duct, J. Acoust. Soc. Am., 126, 3, 1005-1014 (2009)
[8] Curle, N., The influence of solid boundaries upon aerodynamic sound, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci., 231, 1187, 505-514 (1955) · Zbl 0067.43104
[9] Howe, M. S., The generation of sound by aerodynamic sources in an inhomogeneous steady flow, J. Fluid Mech., 67, 03, 597-610 (1975) · Zbl 0297.76068
[10] Ffowcs-Williams, J. E.; Howe, M. S., The generation of sound by density inhomogeneities in low Mach number nozzle flows, J. Fluid Mech., 70, 3, 605-622 (1975) · Zbl 0311.76039
[11] Howe, M. S., Flow-surface interaction noise, J. Sound and Vib., 314, 113-146 (2008)
[12] Schinzinger, R.; Laura, P., Conformal Mapping: Methods and Applications (2003), Dover Publications: Dover Publications Mineola, New York · Zbl 1063.30007
[13] Driscoll, T.; Trefethen, L., Schwarz-Christoffel Mapping (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1003.30005
[14] Sridhar, K. P.; Davis, R. T., A Schwarz-Christoffel method for generating two-dimensional flow grids, J. Fluids Eng., 107, 3, 330-337 (1985)
[15] Papamichael, N., Numerical conformal mapping onto a rectangle with applications to the solution of Laplacian problems, J. Comput. Appl. Math., 28, 63-83 (1989) · Zbl 0683.30010
[16] Peters, M. C.A. M.; Hoeijmakers, H. W.M., A vortex sheet method applied to unsteady flow separation from sharp edges, J. Comput. Phys., 120, 1, 88-104 (1995) · Zbl 0843.76063
[17] Driscoll, T. A., Algorithm 843: improvements to the Schwarz-Christoffel toolbox for MATLAB, ACM Trans. Math. Softw., 31, 2, 239-251 (2005) · Zbl 1070.30500
[18] Gysen, B. L.J.; Lomonova, E. A.; Paulides, J. J.H.; Vandenput, A. J.A., Analytical and numerical techniques for solving Laplace and Poisson equations in a tubular permanent-magnet actuator: part I. semi-analytical framework, IEEE Trans. Magn., 44, 7, 1751-1760 (2008)
[19] Murray, P. R.; Howe, M. S., Compact Green’s function for a generic Rijke burner, Int. J. Spray and Combust. Dyn., 3, 3, 191-208 (2011)
[20] Dupère, I. D.J., Sound Vortex Interaction in Pipes (1997), Department of Engineering, University of Cambridge, (Ph.D. thesis)
[21] Takaishi, T.; Ikeda, M.; Kato, C., Method of evaluating dipole sound source in a finite computational domain, J. Acoust. Soc. Am., 116, 3, 1427-1435 (2004)
[22] Ffowcs-Williams, J. E.; Hall, L. H., Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half plane, J. Fluid Mech., 40, 04, 657-670 (1970) · Zbl 0201.29001
[23] Crighton, D. G.; Leppington, F. G., On the scattering of aerodynamic noise, J. Fluid Mech., 46, 03, 577-597 (1971) · Zbl 0224.76083
[24] Crighton, D. G., Radiation from vortex filament motion near a half plane, J. Fluid Mech., 51, 02, 357-362 (1972) · Zbl 0227.76118
[25] Bjorstad, P.; Grosse, E., Conformal mapping of circular arc polygons, SIAM J. Sci. Stat. Comput., 8, 1, 19-32 (1987) · Zbl 0615.30007
[26] Tietjens, O., Fundamentals of Hydro- and Aero-Mechanics (1957), Dover: Dover New York · Zbl 0078.39603
[27] Milne-Thomson, L., Theoretical Hydrodynamics (1960), Macmillan: Macmillan New York · Zbl 0089.42601
[28] Harwood, A. R.G., Numerical Evaluation of Acoustic Green’s Functions (2014), School of Mechanical, Aerospace & Civil Engineering, The University of Manchester, UK, (Ph.D. thesis)
[29] Trefethen, L., Numerical computation of the Schwarz-Christoffel transformation, SIAM J. Sci. Stat. Comput., 1, 82-102 (1980) · Zbl 0451.30004
[30] Howell, L., Computation of Conformal Maps by Modified Schwarz-Christoffel Transformations (1990), Massachusetts Institute of Technology, (Ph.D. thesis)
[31] Trefethen, L.; Driscoll, T., Schwarz-Christoffel mapping in the computer era, Proc. Int. Congr. Math. (1998) · Zbl 0896.30005
[32] Howe, M. S., Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute, J. Fluid Mech., 71, 04, 625-673 (1975) · Zbl 0325.76117
[33] Borth, W., Vibration and resonance phenomena in pipes of piston fans, Zeitschrift des VDI, 60, 28, 565-569 (1916)
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