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Effective conditions for the reflection of an acoustic wave by low-porosity perforated plates

Published online by Cambridge University Press:  05 March 2014

S. Laurens ,
E. Piot ,
A. Bendali ,
M’B. Fares  and
S. Tordeux
S. Laurens*
Affiliation:
IMT, University of Toulouse, INSA, 135 avenue de Rangueil, F-31077, Toulouse, France CERFACS, 42 avenue Gaspard Coriolis, F-31100, Toulouse, France
E. Piot
Affiliation:
ONERA – The French Aerospace Lab, F-31055, Toulouse, France
A. Bendali
Affiliation:
IMT, University of Toulouse, INSA, 135 avenue de Rangueil, F-31077, Toulouse, France CERFACS, 42 avenue Gaspard Coriolis, F-31100, Toulouse, France
M’B. Fares
Affiliation:
CERFACS, 42 avenue Gaspard Coriolis, F-31100, Toulouse, France
S. Tordeux
Affiliation:
INRIA and University of Pau, LMA, avenue de l’Université, F-64000, Pau, France
*
†Email address for correspondence: [email protected]
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Abstract

This paper describes an investigation of the acoustic properties of a rigid plate with a periodic pattern of holes, in a compressible, ideal, inviscid fluid in the absence of mean flow. Leppington & Levine (J. Fluid Mech., vol. 61, 1973, pp. 109–127) obtained an approximation of the reflection and transmission coefficients of a plane wave incident on an infinitely thin plate with a rectangular array of perforations, assuming that a characteristic size of the perforations is negligible relative to that of the unit cell of the grating, itself assumed to be negligible relative to the wavelength. One part of the present study is of methodological interest. It establishes that it is possible to extend their approach to thick plates with a skew grating of perforations, thus confirming recent results of Bendali et al. (SIAM J. Appl. Math., vol. 73 (1), 2013, pp. 438–459), but in a much simpler way without using complex matched asymptotic expansions of the full wave or to a grating of multipoles. As is well-known, effective compliances for the plate can then be derived from the corresponding approximations of the reflection and transmission coefficients. These compliances are expressed in terms of the Rayleigh conductivity of an isolated perforation. Consequently, in one other part of the present study, the methodology recently introduced by Laurens et al. (ESAIM, Math. Model. Numer. Anal., vol. 47 (6), 2013, pp. 1691–1712) to obtain sharp bounds for the Rayleigh conductivity has been extended to include the case for which the openings of the perforations on the upper and lower sides of the plate are elliptical in shape. This not only enables the determination of these bounds and of the associated reflection and transmission coefficients for actual plates with tilted perforations but also yields single expressions covering all usual cases of perforations: straight or tilted with a circular or an elliptical cross-section.

JFM classification

Acoustics: Acoustics Mathematical Foundations: Mathematical Foundations Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 743 , 25 March 2014 , pp. 448 - 480
Copyright
© 2014 Cambridge University Press 

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References

Abrahams, I. D. 1999 Sound radiation from a line forced perforated elastic sandwich panel. J. Acoust. Soc. Am. 105 (6), 3009–3020.CrossRefGoogle Scholar
Allard, J. F. 1993 Propagation of Sound in Porous Media. Elsevier Applied Sciences.CrossRefGoogle Scholar
Andreini, A., Bianchini, C., Facchini, B. & Simonetti, F. 2011 Assessment of numerical tools for the evaluation of the acoustic impedance of multi-perforated plates. In Proceedings of ASME Turbo Expo 2011. Vancouver, British Columbia, Canada.CrossRefGoogle Scholar
Andreini, A., Facchini, B., Ferrari, L., Lenzi, G. & Simonetti, F. 2012 Experimental investigation on effusion liner geometries for aero-engine combustors: evaluation of global acoustic parameters. In Proceedings of ASME Turbo Expo 2012. Copenhagen, Denmark.CrossRefGoogle Scholar
Atalla, N. & Sgard, F. 2007 Modeling of perforated plates and screens using rigid frame porous models. J. Sound Vib. 303, 195–208.CrossRefGoogle Scholar
Bellucci, V., Flohr, P. & Paschereit, C. O. 2004 Numerical and experimental study of acoustic damping generated by perforated screens. AIAA J. 42 (8), 1543–1549.CrossRefGoogle Scholar
Bendali, A., Fares, M. B., Piot, E. & Tordeux, S. 2013 Mathematical justification of the Rayleigh conductivity model for perforated plates in acoustics. SIAM J. Appl. Math. 73 (1), 438–459.CrossRefGoogle Scholar
Bennetts, L. G. & Squire, V. A. 2012 On the calculation of an attenuation coefficient for transects of ice-covered ocean. Proc. R. Soc. Lond. A 468 (2137), 136–162.Google Scholar
Beranek, L. L. 1992 Noise and Vibration Control Engineering. Wiley.Google Scholar
Copson, E. T. 1947 On the problem of the electrified disc. Proc. Edinburgh Math. Soc. 2 8 (01), 14–19.CrossRefGoogle Scholar
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics. vol. I. Interscience Publishers, Inc..Google Scholar
Cummings, A. 1984 Acoustic nonlinearities and power losses at orifices. AIAA J. 22, 786–792.CrossRefGoogle Scholar
Cummings, A. 1986 The effects of grazing turbulent pipe-flow on the impedance of an orifice. Acoustica 61, 233–242.Google Scholar
Cummings, A. 1987 The response of a resonator under a turbulent boundary layer to a high amplitude non-harmonic sound field. J. Sound Vib. 115 (2), 321–328.CrossRefGoogle Scholar
Dowling, A. P. & Hughes, I. J. 1992 Sound absorption by a screen with a regular array of slits. J. Sound Vib. 156 (3), 387–405.CrossRefGoogle Scholar
Eldredge, J. D., Bodony, D. J. & Schoeybi, M. 2007 Numerical investigation of the acoustic behavior of a multi-perforated liner. In 13th AIAA/CEAS Aeroacoustics Conference, AIAA paper 2007-3683.Google Scholar
Eldredge, J. D. & Dowling, A. P. 2003 The absorption of axial acoustic waves by a perforated liner with bias flow. J. Fluid Mech. 485, 307–335.CrossRefGoogle Scholar
Fok, V. A. 1941 Teoreticheskoe issledovanie provodimosti kruglogo otverstiya v peregorodke, postavlennoi poperek truby. Doklady Akademii nauk SSSR 31, 875–878.Google Scholar
Guess, A. W. 1975 Calculation of perforated plate liner parameters from specified acoustic resistance and reactance. J. Sound Vib. 40 (1), 119–137.CrossRefGoogle Scholar
Howe, M. S. 1998 Acoustics of Fluid–Structure Interactions. Cambridge University Press.CrossRefGoogle Scholar
Howe, M. S. 1979 On the theory of unsteady high Reynolds number flow through a circular aperture. Proc. R. Soc. Lond. 366, 205–223.Google Scholar
Howe, M. S. 1980 The influence of vortex shedding on the diffraction of sound by a perforated screen. J. Fluid Mech. 97, 641–653.CrossRefGoogle Scholar
Howe, M. S. 1996 The influence of tangential mean flow on the Rayleigh conductivity of an aperture. Proc. R. Soc. Lond. 452, 2303–2317.Google Scholar
Hughes, I. J. & Dowling, A. P. 1990 The absorption of sound by perforated linings. J. Fluid Mech. 218, 299–335.CrossRefGoogle Scholar
Ingard, K. U. 1994 Notes on Sound Absorption Technology. Noise Control Foundation.Google Scholar
Ingard, U. 1953 On the theory and design of acoustic resonators. J. Acoust. Soc. Am. 25, 1037–1061.CrossRefGoogle Scholar
Jing, X. & Sun, X. 1999 Experimental investigations of perforated liners with bias flow. J. Acoust. Soc. Am. 106 (5), 2436–2441.CrossRefGoogle Scholar
Jing, X. & Sun, X. 2000 Effect of plate thickness on impedance of perforated plates with bias flow. AIAA J. 38 (9), 1573–1578.CrossRefGoogle Scholar
Kirby, R. & Cummings, A. 1998 The impedance of perforated plates subjected to grazing gas flow and backed by porous media. J. Sound Vib. 217 (4), 619–636.CrossRefGoogle Scholar
Laurens, S. & Tordeux, S. 2013 Explicit computation of the electrostatic energy for an elliptical charged disc. Appl. Maths Lett. 26 (2), 301–305.CrossRefGoogle Scholar
Laurens, S., Tordeux, S., Bendali, A., Fares, M. & Kotiuga, R. 2013 Lower and upper bounds for the Rayleigh conductivity of a perforated plate. ESAIM, Math. Model. Numer. Anal. 47 (6), 1691–1712.CrossRefGoogle Scholar
Lee, S. H., Ih, J. G. & Peat, K. S. 2007 A model of the acoustic impedance of perforated plates with bias flow considering the interaction effect. J. Sound Vib. 303, 741–752.CrossRefGoogle Scholar
Lefebvre, A. H. 1999 Gas Turbines Combustion. Taylor and Francis.Google Scholar
Leppington, F. G. 1990 The effective boundary conditions for a perforated elastic sandwich panel in a compressible fluid. Proc. R. Soc. Lond. A 427, 385–399.Google Scholar
Leppington, F. G. & Levine, H. 1973 Reflexion and transmission at a plane screen with periodically arranged circular or elliptical apertures. J. Fluid Mech. 61, 109–127.CrossRefGoogle Scholar
Linton, C. M. 2010 Lattice sums for the Helmholtz equation. SIAM Rev. 52 (4), 630–674.CrossRefGoogle Scholar
Linton, C. M. & Thompson, I. 2009 One-and two-dimensional lattice sums for the three-dimensional helmholtz equation. J. Comput. Phys. 228 (6), 1815–1829.CrossRefGoogle Scholar
Luenberger, D. G. 1997 Optimization by Vector Space Methods. Wiley-Interscience.Google Scholar
Luong, T., Howe, M. S. & McGowan, R. S. 2005 On the Rayleigh conductivity of a bias-flow aperture. J. Fluids Struct. 21, 769–778.CrossRefGoogle Scholar
Maa, D. Y. 1998 Potential of microperforated panel absorber. J. Acoust. Soc. Am. 104 (5), 2861–2866.CrossRefGoogle Scholar
Melling, T. H. 1973 The acoustic impedance of perforates at medium and high sound pressure levels. J. Sound Vib. 29 (1), 1–65.CrossRefGoogle Scholar
Mendez, S. & Eldredge, J. D. 2009 Acoustic modeling of perforated plates with bias flow for large-eddy simulations. J. Comput. Phys. 228, 4757–4772.CrossRefGoogle Scholar
Morfey, C. L. 1969 Acoustic properties of openings at low frequencies. J. Sound Vib. 9, 357–366.CrossRefGoogle Scholar
Nedelec, J. C. & Starling, F. 1991 Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations. SIAM J. Math. Anal. 6, 1679–1701.CrossRefGoogle Scholar
Rayleigh, J. W. S. 1945 The Theory of Sound. vol. 2. Dover publications.Google Scholar
Roberts, J. E. & Thomas, J.-M. 1991 Mixed and hybrid methods. In Handbook of Numerical Analysis vol. 2, Elsevier Science Publishers.Google Scholar
Rschevkin, S. N. 1963 A Course of Lectures on the Theory of Sound. vol. 31. Pergamon.Google Scholar
Sauter, S. & Schwab, C. 2010 In Boundary Element Methods Computational Mathematics, vol. 39, Springer.Google Scholar
Scarpato, A., Tran, N., Ducruix, S. & Schuller, T. 2012 Modeling the damping properties of perforated screens traversed by a bias flow and backed by a cavity at low Strouhal number. J. Sound Vib. 331 (2), 276–290.CrossRefGoogle Scholar
Sun, X., Jing, X., Zhang, H. & Shi, Y. 2002 Effect of grazing-bias flow interaction on acoustic impedance of perforated plates. J. Sound Vib. 254 (3), 557–573.CrossRefGoogle Scholar
Tam, C. K. W., Ju, H. & Walker, B. E. 2008 Numerical simulation of a slit resonator in a grazing flow under acoustic excitation. J. Sound Vib. 313, 449–471.CrossRefGoogle Scholar
Tuck, E. O. 1975 Matching problems involving flow through small holes. Adv. Appl. Mech. 15, 89–158.CrossRefGoogle Scholar
Wang, R. & Shen, H. H. 2006 Gravity waves propagating into an ice-covered ocean: A viscoelastic model. J. Geophys. Res.: Oceans 115, C06024.Google Scholar
Yu, J., Kwan, H. W., Echternach, D. K., Kraft, R. E. & Syed, A. A. 1999 Acoustic treatment design scaling method. Tech. Rep.. NASA/CR-1999-209120/Vol3.Google Scholar