# Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition

• Volume: 48, Issue: 5, page 1529-1555
• ISSN: 0764-583X

top

## Abstract

top
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.

## How to cite

top

Luneville, Eric, and 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 48.5 (2014): 1529-1555. <http://eudml.org/doc/273229>.

@article{Luneville2014,
abstract = {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.},
author = {Luneville, Eric, Mercier, Jean-Francois},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {aeroacoustics; scattering of sound in flows; treated boundary; Myers condition; finite elements; variational formulations; scattering},
language = {eng},
number = {5},
pages = {1529-1555},
publisher = {EDP-Sciences},
title = {Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition},
url = {http://eudml.org/doc/273229},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Luneville, Eric
AU - Mercier, Jean-Francois
TI - Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1529
EP - 1555
AB - 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.
LA - eng
KW - aeroacoustics; scattering of sound in flows; treated boundary; Myers condition; finite elements; variational formulations; scattering
UR - http://eudml.org/doc/273229
ER -

## References

top
1. [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. [2] M. Myers, On the acoustic boundary condition in the presence of flow. J. Acoust. Soc. Am.71 (1980) 429–434. Zbl0448.76065
3. [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. Zbl0238.76020
4. [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 Zbl0255.76094
5. [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. [6] G. Gabard, Mode-Matching Techniques for Sound Propagation in Lined Ducts with Flow. Proc. of the 16th AIAA/CEAS Aeroacoustics Conference.
7. [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. [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. [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. [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. Zbl1180.76053MR2557210
11. [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. [12] S.W. Rienstra, Acoustic Scattering at a Hard-Soft Lining Transition in a Flow Duct. J. Engrg. Math.59 (2007) 451–475. Zbl1198.76136
13. [13] S. Rienstra, A classification of duct modes based on surface waves. Wave Motion37 (2003) 119–135. Zbl1163.74431MR1949360
14. [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. [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 Zbl0258.76057
16. [16] P.G. Daniels, On the Unsteady Kutta Condition. Quarterly J. Mech. Appl. Math.31 (1985) 49-75. Zbl0389.76018MR489358
17. [17] D.G. Crighton, The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech.17 (1985) 411–445. Zbl0596.76037
18. [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. [19] Y. Aurégan, M. Leroux and V. Pagneux, Abnormal behaviour of an acoustical liner with flow. Forum Acusticum, Budapest (2005).
20. [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. [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. Zbl0982.76529
22. [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. [23] S.N. Chandler-Wilde and J. Elschner, Variational Approach in Weighted Sobolev Spaces to Scattering by Unbounded Rough Surfaces. SIAM J. Math. Anal. SIMA42 (2010) 2554–2580. Zbl1241.35044MR2733260
24. [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. Zbl1121.35098MR2209521
25. [25] M. Dambrine and G. Vial, A multiscale correction method for local singular perturbations of the boundary. ESAIM: M2AN 41 (2007) 111–127. Zbl1129.65084MR2323693
26. [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. Zbl1126.78004MR2334544
27. [27] S. Tordeux, G. Vial and M. Dauge, Matching and multiscale expansions for a model singular perturbation problem. C. R. Acad. Sci. Paris Ser. I343 (2006) 637–642. Zbl1109.35013MR2271738
28. [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. Zbl0955.74048MR1634297
29. [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. Zbl1023.76044
30. [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. [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. Zbl1198.76134MR2211495

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.