Lower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for tilted perforations. The main techniques are a proper use of the Dirichlet and Kelvin variational principlesin the context of Beppo-Levi spaces. The derivations are validated by numerical experiments in 2D for the axisymmetric case as well as for the full three-dimensional problem.
Keywords: Rayleigh conductivity, perforated plate, Kelvin principle, Dirichlet principle
@article{M2AN_2013__47_6_1691_0,
author = {Laurens, S. and Tordeux, S. and Bendali, A. and Fares, M. and Kotiuga, P. R.},
title = {Lower and upper bounds for the {Rayleigh} conductivity of a perforated plate},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1691--1712},
year = {2013},
publisher = {EDP Sciences},
volume = {47},
number = {6},
doi = {10.1051/m2an/2013082},
mrnumber = {3123372},
zbl = {1283.35088},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2013082/}
}
TY - JOUR AU - Laurens, S. AU - Tordeux, S. AU - Bendali, A. AU - Fares, M. AU - Kotiuga, P. R. TI - Lower and upper bounds for the Rayleigh conductivity of a perforated plate JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1691 EP - 1712 VL - 47 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2013082/ DO - 10.1051/m2an/2013082 LA - en ID - M2AN_2013__47_6_1691_0 ER -
%0 Journal Article %A Laurens, S. %A Tordeux, S. %A Bendali, A. %A Fares, M. %A Kotiuga, P. R. %T Lower and upper bounds for the Rayleigh conductivity of a perforated plate %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1691-1712 %V 47 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2013082/ %R 10.1051/m2an/2013082 %G en %F M2AN_2013__47_6_1691_0
Laurens, S.; Tordeux, S.; Bendali, A.; Fares, M.; Kotiuga, P. R. Lower and upper bounds for the Rayleigh conductivity of a perforated plate. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1691-1712. doi: 10.1051/m2an/2013082
[1] , Improved calculation of resonant frequencies of Helmholtz resonators. J. Sound Vibr. 24 (1972) 63-85.
[2] , and , Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator an approach in weighted sobolev spaces. J. Math. Pures Appl. 76 (1997) 55-81. | Zbl | MR
[3] and , Boundary integral equations methods in acoustics. Computer Methods for Acoustics Problems, Chapter 1. Edited by F. Magoules. Saxe-Coburg Publications, Kippen, Stirlingshire, Scotland (2008) 1-36.
[4] , Effects of geometry on the resonance frequency of Helmholtz resonators, part II. J. Sound Vibr. 204 (1997) 829-834.
[5] , On the problem of the electrified disc. Proc. Edinburgh Math. Soc. (Ser. 2) 8 (1947) 14-19. | Zbl | MR
[6] and , Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc., New York, N.Y. (1953). | Zbl | MR
[7] and . Les espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1953) 54. | Zbl | MR | Numdam
[8] and , Absorbing boundary conditions for the numerical simulation of wave. Math. Comput. 31 (1977) 629-651. | Zbl | MR
[9] , Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Ph.D. Thesis. Paris VI (1987).
[10] , On the theory of unsteady high Reynolds number flow through a circular aperture. Proc. Roy. Soc. London A. Math. Phys. Sci. 366 (1979) 205. | Zbl | MR
[11] , Influence of wall thickness on Rayleigh conductivity and flow-induced aperture tones. J. Fluids Struct. 11 (1997) 351-366.
[12] , Acoustics of fluid-structure interactions. Cambridge University Press (1998). | Zbl | MR
[13] and , Boundary Integral Equations. Springer, Berlin-Heidelberg (2008). | Zbl | MR
[14] , On the theory and design of acoustic resonators. J. Acoust. Soc. America 25 (1953) 1037-1061.
[15] and , Exact non-reflecting boundary conditions. J. Comput. Phys. 82 (1989) 172-192. | Zbl | MR
[16] , and , Analytical Modeling of Helmholtz Resonator Based Powered Resonance Tubes, in 2nd AIAA Flow Control Conference, AIAA Paper 2004-2691 (2004).
[17] , Optimization by vector space methods. Wiley-Interscience (1997). | Zbl | MR
[18] and , Evaluation of the acoustic impedance of a screen. J. Australian Math. Soc. Ser. B Appl. Math. 20 (1977) 46-61. | Zbl | MR
[19] , Étude théorique et expérimentale de l'impédance acoustique de matériaux en présence d'un écoulement d'air tangentiel. Ph.D. thesis. University du Maine (2000).
[20] , Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, UK, and New York, USA (2000). | Zbl | MR
[21] , The acoustic impendance of perforates at medium and high sound pressure levels. J. Sound Vibr. 29 (1973) 1-65.
[22] and , Acoustic modeling of perforated plates with bias flow for Large-Eddy Simulations. J. Comput. Phys. 228 (2009) 4757-4772. | Zbl
[23] , Helmholtz resonators with large aperture. Acta Acoust. United Acoust. 85 (1999) 751-763.
[24] , Acoustic properties of openings at low frequencies. J. Sound Vibr. 9 (1969) 357-366.
[25] and , Resonant frequencies of cylindrical Helmholtz resonators. J. Acoust. Soc. America 57 (1975) 1533-1535.
[26] , The Theory of Sound, vols. 1 and 2. Dover Publications, New York (1945). | Zbl | MR
[27] and , Mixed and hybrid methods, in Handbook of numerical analysis. Vol. 2. Elsevier Science Publishers (1991). | Zbl | MR
[28] and , Boundary Element Methods, vol. 39 of Springer Series in Computational Mathematics. Springer, Heidelberg (2010). | Zbl | MR
[29] , Mixed boundary value problems in potential theory. North-Holland Pub. Co. (1966). | Zbl | MR
[30] , Matching problems involving flow through small holes. Adv. Appl. Mech. 15 (1975) 89-158. | MR
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