Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T03:40:44.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower and upper bounds for the Rayleigh conductivityof a perforated plate

Published online by Cambridge University Press:  11 October 2013

S. Laurens ,
S. Tordeux ,
A. Bendali ,
M. Fares  and
P.R. Kotiuga
S. Laurens
Affiliation:
CERFACS – EMA, 42 avenue Gaspar Coriolis,31100 Toulouse, France. sophie.laurens@insa-toulouse.fr
S. Tordeux
Affiliation:
INRIA Bordeaux Sud-Ouest – Magique 3D  Université de Pau, LMA (UMR-CNRS 5142), avenue de l’Université,64013 Pau, France; sebastien.tordeux@inria.fr
A. Bendali
Affiliation:
INSA-Mathematical Institute of Toulouse (UMR-CNRS 5219),135 avenue de Rangueil, 31077 Toulouse, France; abderrahmane.bendali@insa-toulouse.fr
M. Fares
Affiliation:
CERFACS - EMA, 42 avenue Gaspar Coriolis, 31100 Toulouse, France; fares@cerfacs.fr
P.R. Kotiuga
Affiliation:
Boston University, Department of Electrical and Computer Engineering, 8 Saint Mary’s Street, Boston MA, 02215, USA; prk@bu.edu
Get access

Abstract

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 conductivityperforated plateKelvin principleDirichlet principle
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1691 - 1712
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alster, M., Improved calculation of resonant frequencies of Helmholtz resonators. J. Sound Vibr. 24 (1972) 63–85. Google Scholar
Amrouche, C., Girault, V. and Giroire, J., 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. Google Scholar
A. Bendali and M. Fares, 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.
Chanaud, R.C., Effects of geometry on the resonance frequency of Helmholtz resonators, part II. J. Sound Vibr. 204 (1997) 829–834. Google Scholar
Copson, E.T., On the problem of the electrified disc. Proc. Edinburgh Math. Soc. (Ser. 2) 8 (1947) 14–19. Google Scholar
R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc., New York, N.Y. (1953).
Deny, J. and Lions, J.L.. Les espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1953) 54. Google Scholar
Enquist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of wave. Math. Comput. 31 (1977) 629–651. Google Scholar
J. Giroire, Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Ph.D. Thesis. Paris VI (1987).
Howe, M.S., On the theory of unsteady high Reynolds number flow through a circular aperture. Proc. Roy. Soc. London A. Math. Phys. Sci. 366 (1979) 205. Google Scholar
Howe, M.S., Influence of wall thickness on Rayleigh conductivity and flow-induced aperture tones. J. Fluids Struct. 11 (1997) 351–366. Google Scholar
M.S. Howe, Acoustics of fluid-structure interactions. Cambridge University Press (1998).
G. C. Hsiao and W.L. Wendland, Boundary Integral Equations. Springer, Berlin-Heidelberg (2008).
Ingard, U., On the theory and design of acoustic resonators. J. Acoust. Soc. America 25 (1953) 1037–1061. Google Scholar
Keller, J.B. and Givoli, D., Exact non-reflecting boundary conditions. J. Comput. Phys. 82 (1989) 172–192. Google Scholar
E. Kerschen, A. Cain and G. Raman, Analytical Modeling of Helmholtz Resonator Based Powered Resonance Tubes, in 2nd AIAA Flow Control Conference, AIAA Paper 2004-2691 (2004).
D.G. Luenberger, Optimization by vector space methods. Wiley-Interscience (1997).
Macaskill, C. and Tuck, E.O., Evaluation of the acoustic impedance of a screen. J. Australian Math. Soc. Ser. B Appl. Math. 20 (1977) 46–61. Google Scholar
C. Malmary, 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).
William McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, UK, and New York, USA (2000).
Melling, T.H., The acoustic impendance of perforates at medium and high sound pressure levels. J. Sound Vibr. 29 (1973) 1–65. Google Scholar
Mendez, S. and Eldredge, J.D., Acoustic modeling of perforated plates with bias flow for Large-Eddy Simulations. J. Comput. Phys. 228 (2009) 4757–4772. Google Scholar
Mohring, J., Helmholtz resonators with large aperture. Acta Acoust. United Acoust. 85 (1999) 751–763. Google Scholar
Morfey, C.L., Acoustic properties of openings at low frequencies. J. Sound Vibr. 9 (1969) 357–366. Google Scholar
Panton, R.L. and Miller, J.M., Resonant frequencies of cylindrical Helmholtz resonators. J. Acoust. Soc. America 57 (1975) 1533–1535. Google Scholar
J.W.S. Rayleigh, The Theory of Sound, vols. 1 and 2. Dover Publications, New York (1945).
J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of numerical analysis. Vol. 2. Elsevier Science Publishers (1991).
S. Sauter and C. Schwab, Boundary Element Methods, vol. 39 of Springer Series in Computational Mathematics. Springer, Heidelberg (2010).
I.N. Sneddon, Mixed boundary value problems in potential theory. North-Holland Pub. Co. (1966).
Tuck, E.O., Matching problems involving flow through small holes. Adv. Appl. Mech. 15 (1975) 89–158. Google Scholar