Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, p. 1529-1555
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of a discontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuous one leads to still open modeling questions, as in particular the singularity of the solution at the abrupt transition and the choice of the right unknown to formulate the scattering problem. To address these questions we propose a mathematical approach based on variational formulations set in weighted Sobolev spaces. Considering the discontinuous impedance as the limit of a continuous boundary condition, we prove that only the problem formulated in terms of the velocity potential converges to a well-posed problem. Moreover we identify the limit problem and determine some Kutta-like condition satisfied by the velocity: its convective derivative must vanish at the ends of the impedance area. Finally we justify why it is not possible to define limit problems for the pressure and the displacement. Numerical examples illustrate the convergence process.

DOI : https://doi.org/10.1051/m2an/2014008
Classification:  35J20,  35J05
Keywords: aeroacoustics, scattering of sound in flows, treated boundary, Myers condition, finite elements, variational formulations
@article{M2AN_2014__48_5_1529_0,
     author = {Luneville, Eric and Mercier, Jean-Francois},
     title = {Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     pages = {1529-1555},
     doi = {10.1051/m2an/2014008},
     zbl = {1301.35079},
     mrnumber = {3264364},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_5_1529_0}
}
Luneville, Eric; Mercier, Jean-Francois. Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, pp. 1529-1555. doi : 10.1051/m2an/2014008. http://www.numdam.org/item/M2AN_2014__48_5_1529_0/

[1] K. Ingard, Influence of Fluid Motion Past a Plane Boundary on Sound Reflection, Absorption, and Transmission. J. Acoust. Soc. Am. 31 (1959) 1035-1036.

[2] M. Myers, On the acoustic boundary condition in the presence of flow. J. Acoust. Soc. Am. 71 (1980) 429-434. | Zbl 0448.76065

[3] W. Eversman and R.J. Beckemeyer, Transmission of Sound in Ducts with Thin Shear layers-Convergence to the Uniform Flow Case. J. Acoust. Soc. Am. 52 (1972) 216-220. | Zbl 0238.76020

[4] B.J. tester, Some Aspects of “Sound” Attenuation in Lined Ducts containing Inviscid Mean Flows with Boundary Layers. J. Sound Vib. 28 (1973) 217-245 | Zbl 0255.76094

[5] G. Gabard and R.J. Astley, A computational mode-matching approach for sound propagation in three-dimensional ducts with flow. J. Acoust. Soc. Am. 315 (2008) 1103-1124.

[6] G. Gabard, Mode-Matching Techniques for Sound Propagation in Lined Ducts with Flow. Proc. of the 16th AIAA/CEAS Aeroacoustics Conference.

[7] R. Kirby, A comparison between analytic and numerical methods for modeling automotive dissipative silencers with mean flow. J. Acoust. Soc. Am. 325 (2009) 565-582

[8] R. Kirby and F.D. Denia, Analytic mode matching for a circular dissipative silencer containing mean flow and a perforated pipe. J. Acoust. Soc. Am. 122 (2007) 71-82.

[9] Y. Aurégan and M. Leroux, Failures in the discrete models for flow duct with perforations: an experimental investigation. J. Acoust. Soc. Am. 265 (2003) 109-121

[10] E.J. Brambley, Low-frequency acoustic reflection at a hardsoft lining transition in a cylindrical duct with uniform flow. J. Engng. Math. 65 (2009) 345-354. | MR 2557210 | Zbl 1180.76053

[11] S. Rienstra and N. Peake, Modal Scattering at an Impedance Transition in a Lined Flow Duct. Proc. of 11th AIAA/CEAS Aeroacoustics Conference, Monterey, CA, USA (2005).

[12] S.W. Rienstra, Acoustic Scattering at a Hard-Soft Lining Transition in a Flow Duct. J. Engrg. Math. 59 (2007) 451-475. | Zbl 1198.76136

[13] S. Rienstra, A classification of duct modes based on surface waves. Wave Motion 37 (2003) 119-135. | MR 1949360 | Zbl 1163.74431

[14] E.J. Brambley and N. Peake, Surface-waves, stability, and scattering for a lined duct with flow. Proc. of AIAA Paper (2006) 2006-2688.

[15] B.J. tester, The Propagation and Attenuation of sound in Lined Ducts containing Uniform or “Plug” Flow. J. Acoust. Soc. Am. 28 (1973) 151-203 | Zbl 0258.76057

[16] P.G. Daniels, On the Unsteady Kutta Condition. Quarterly J. Mech. Appl. Math. 31 (1985) 49-75. | MR 489358 | Zbl 0389.76018

[17] D.G. Crighton, The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech. 17 (1985) 411-445. | Zbl 0596.76037

[18] M. Brandes and D. Ronneberger, Sound amplification in flow ducts lined with a periodic sequence of resonators. Proc. of AIAA paper, 1st AIAA/CEAS Aeroacoustics Conference, Munich, Germany (1995) 95-126.

[19] Y. Aurégan, M. Leroux and V. Pagneux, Abnormal behaviour of an acoustical liner with flow. Forum Acusticum, Budapest (2005).

[20] B. Regan and J. Eaton, Modeling the influence of acoustic liner non-uniformities on duct modes. J. Acoust. Soc. Am. 219 (1999) 859-879.

[21] K.S. Peat and K.L. Rathi, A Finite Element Analysis of the Convected Acoustic Wave Motion in Dissipative Silencers. J. Acoust. Soc. Am. 184 (1995) 529-545. | Zbl 0982.76529

[22] W. Eversman, The Boundary condition at an Impedance Wall in a Non-Uniform Duct with Potential Mean Flow. J. Acoust. Soc. Am. 246 (2001) 63-69.

[23] S.N. Chandler-Wilde and J. Elschner, Variational Approach in Weighted Sobolev Spaces to Scattering by Unbounded Rough Surfaces. SIAM J. Math. Anal. SIMA 42 (2010) 2554-2580. | MR 2733260 | Zbl 1241.35044

[24] B. Guo and C. Schwab, Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces. J. Comput. Appl. Math. 190 (2006) 487-519. | MR 2209521 | Zbl 1121.35098

[25] M. Dambrine and G. Vial, A multiscale correction method for local singular perturbations of the boundary. ESAIM: M2AN 41 (2007) 111-127. | Numdam | MR 2323693 | Zbl 1129.65084

[26] P. Ciarlet and S. Kaddouri, Multiscaled asymptotic expansions for the electric potential: surface charge densities and electric fields at rounded corners. Math. Models Methods Appl. Sci. 17 (2007) 845-876. | MR 2334544 | Zbl 1126.78004

[27] S. Tordeux, G. Vial and M. Dauge, Matching and multiscale expansions for a model singular perturbation problem. C. R. Acad. Sci. Paris Ser. I 343 (2006) 637-642. | MR 2271738 | Zbl 1109.35013

[28] M. Costabel, M. Dauge and M. Surib, Numerical Approximation of a Singularly Perturbed Contact Problem. Computer Methods Appl. Mech. Engrg. 157 (1998) 349-363. | MR 1634297 | Zbl 0955.74048

[29] A.-S. Bonnet-Ben Dhia, L. Dahi, E. Lunéville and V. Pagneux, Acoustic diffraction by a plate in a uniform flow. Math. Models Methods Appl. Sci. 12 (2002) 625-647. | Zbl 1023.76044

[30] D. Martin, Code éléments finis MELINA. Available at http://anum-maths.univ-rennes1.fr/melina/danielmartin/melina/www/somm˙html/fr-main.html

[31] S. Job, E. Lunéville and J.-F. Mercier, Diffraction of an acoustic wave in a uniform flow: a numerical approach. J. Comput. Acoust. 13 (2005) 689-709. | MR 2211495 | Zbl 1198.76134