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Direct and adjoint problems for sound propagation in non-uniform flows with lined and vibrating surfaces

Published online by Cambridge University Press:  06 December 2022

Sophie Le Bras Open the ORCID record for Sophie Le Bras [Opens in a new window] ,
Gwénaël Gabard Open the ORCID record for Gwénaël Gabard [Opens in a new window]  and
Hadrien Bériot Open the ORCID record for Hadrien Bériot [Opens in a new window]
Sophie Le Bras*
Affiliation:
Siemens Industry Software SAS, 107 Avenue de la République, 92320 Châtillon, France
Gwénaël Gabard
Affiliation:
LAUM (UMR CNRS 6613), Institut d'Acoustique - Graduate School (IA-GS), Le Mans Université, 72085 Le Mans CEDEX 9, France
Hadrien Bériot
Affiliation:
Siemens Industry Software N.V., Interleuvenlaan 68, 3001 Leuven, Belgium
*
†Email address for correspondence: [email protected]
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Abstract

This paper presents a systematic analysis of direct and adjoint problems for sound propagation with flow. Two scalar propagation operators are considered: the linearised potential equation from Goldstein, and Pierce's equation based on a high-frequency approximation. For both models, the analysis involves compressible base flows, volume sources and surfaces that can be vibrating and/or acoustically lined (using the Myers impedance condition), as well as far-field radiation boundaries. For both models, the direct problems are fully described and adjoint problems are formulated to define tailored Green's functions. These Green's functions are devised to provide an explicit link between the direct problem solutions and the source terms. These adjoint problems and tailored Green's functions are particularly useful and efficient for source localisation problems, or when stochastic distributed sources are involved. The present analysis yields a number of new results, including the adjoint Myers condition for the linearised potential equation, as well as the formulation of the direct and adjoint Myers condition for Pierce's equation. It is also shown how the adjoint problems can be recast in forms that are readily solved using existing simulation tools for the direct problems. Results presented in this paper are obtained using a high-order finite element method. Several test cases serve as validation for the approach using tailored Green's functions. They also illustrate the relative benefits of the two propagation operators.

JFM classification

Acoustics: Aeroacoustics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 953 , 25 December 2022 , A16
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Amestoy, P.R., Duff, I.S., L'Excellent, J.-Y. & Koster, J. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Applics. 23 (1), 15–41.CrossRefGoogle Scholar
Astley, R.J. 2009 Numerical methods for noise propagation in moving flows, with application to turbofan engines. Acoust. Sci. Technol. 30 (4), 227–239.CrossRefGoogle Scholar
Bayliss, A. & Turkel, E. 1982 Far field boundary conditions for compressible flows. J. Comput. Phys. 48 (2), 182–199.CrossRefGoogle Scholar
Bériot, H. & Gabard, G. 2019 Anisotropic adaptivity of the $p$-FEM for time-harmonic acoustic wave propagation. J. Comput. Phys. 378, 234–256.CrossRefGoogle Scholar
Bériot, H., Gabard, G. & Perrey-Debain, E. 2013 Analysis of high-order finite elements for convected wave propagation. Intl J. Numer. Meth. Engng 96 (11), 665–688.CrossRefGoogle Scholar
Bériot, H. & Modave, A. 2021 An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape. Intl J. Numer. Meth. Engng 122 (5), 1239–1261.Google Scholar
Bériot, H., Prinn, A. & Gabard, G. 2016 Efficient implementation of high-order finite elements for Helmholtz problems. Intl J. Numer. Meth. Engng 106 (3), 213–240.CrossRefGoogle Scholar
Bertoluzza, S., Decoene, A., Lacouture, L. & Martin, S. 2018 Local error estimates of the finite element method for an elliptic problem with a Dirac source term. Numer. Meth. P. D. E. 34 (1), 97–120.CrossRefGoogle Scholar
Blokhintzev, D. 1946 The propagation of sound in an inhomogeneous and moving medium I. J. Acoust. Soc. Am. 18 (2), 322–328.Google Scholar
Bogey, C. & Bailly, C. 2002 Three-dimensional non-reflective boundary conditions for acoustic simulations: far field formulation and validation test cases. Acta Acust. United AC 88 (4), 463–471.Google Scholar
Chaillat, S., Cotté, B., Mercier, J.-F., Serre, G. & Trafny, N. 2022 Efficient evaluation of three-dimensional Helmholtz Green's functions tailored to arbitrary rigid geometries for flow noise simulations. J. Comput. Phys. 452, 110915.CrossRefGoogle Scholar
Dahl, M.D. (Ed.) 2004 Fourth Computational Aeroacoustics (CAA) Workshop on Benchmark Problems.Google Scholar
Dennery, P. & Krzywicki, A. 2012 Mathematics for Physicists. Dover.Google Scholar
DLMF 2022 NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15, (ed. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl & M.A. McClain).Google Scholar
Eversman, W. 2001 a The boundary condition at an impedance wall in a non-uniform duct with potential mean flow. J. Sound Vib. 246 (1), 63–69.CrossRefGoogle Scholar
Eversman, W. 2001 b A reverse flow theorem and acoustic reciprocity in compressible potential flows in ducts. J. Sound Vib. 246 (1), 71–95.CrossRefGoogle Scholar
Ewert, R. & Schröder, W. 2003 Acoustic perturbation equations based on flow decomposition via source filtering. J. Comput. Phys. 188 (2), 365–398.CrossRefGoogle Scholar
Gabard, G. 2010 Mode-matching techniques for sound propagation in lined ducts with flow. 16th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2010-3940.Google Scholar
Gabard, G. 2014 Noise sources for duct acoustics simulations: broadband noise and tones. AIAA J. 52 (9), 1994–2006.CrossRefGoogle Scholar
Gabard, G., Bériot, H., Prinn, A. & Kucukcoskun, K. 2018 Adaptive, high-order finite-element method for convected acoustics. AIAA J. 56 (8), 3179–3191.CrossRefGoogle Scholar
Givoli, D. 2004 High-order local non-reflecting boundary conditions: a review. Wave Motion 39 (4), 319–326.Google Scholar
Godin, O.A. 1997 Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid. Wave Motion 25 (2), 143–167.Google Scholar
Goldstein, M. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (3), 433–468.Google Scholar
Hamiche, K., Le Bras, S., Gabard, G. & Beriot, H. 2019 Hybrid numerical model for acoustic propagation through sheared flows. J. Sound Vib. 463, 114951.CrossRefGoogle Scholar
Howe, M.S. 2003 Theory of Vortex Sound. Cambridge University Press.Google Scholar
Hu, F., Guo, Y. & Jones, A. 2005 On the computation and application of exact Green's function in acoustic analogy. 11th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2005-2986.CrossRefGoogle Scholar
Koppl, T. & Wohlmuth, B. 2014 Optimal a priori error estimates for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 52 (4), 1753–1769.CrossRefGoogle Scholar
Luneville, E. & Mercier, J.-F. 2014 Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition. ESAIM: Math. Model. Numer. Anal. 48 (5), 1529–1555.CrossRefGoogle Scholar
Marchner, P., Beriot, H., Antoine, X. & Geuzaine, C. 2021 Stable perfectly matched layers with Lorentz transformation for the convected Helmholtz equation. J. Comput. Phys. 433, 110180.Google Scholar
Möhring, W. 1978 Acoustic energy flux in nonhomogeneous ducts. J. Acoust. Soc. Am. 64 (4), 1186–1189.Google Scholar
Möhring, W. 1999 A well posed acoustic analogy based on a moving acoustic medium. In Aeroacoustic Workshop SWING, pp. 1–14.Google Scholar
Möhring, W. 2001 Energy conservation, time-reversal invariance and reciprocity in ducts with flow. J. Fluid Mech. 431, 223–237.Google Scholar
Myers, M. 1980 On the acoustic boundary condition in the presence of flow. J. Sound Vib. 71 (3), 429–434.CrossRefGoogle Scholar
Pierce, A.D. 1990 Wave equation for sound in fluids with unsteady inhomogeneous flow. J. Acoust. Soc. Am. 87 (6), 2292–2299.CrossRefGoogle Scholar
Pierce, A.D. 2019 Acoustics: An Introduction to its Physical Principles and Applications. Springer.CrossRefGoogle Scholar
Rienstra, S.W. 2007 Acoustic scattering at a hard–soft lining transition in a flow duct. J. Engng Maths 59 (4), 451–475.Google Scholar
Schram, C. 2009 A boundary element extension of Curle's analogy for non-compact geometries at low-Mach numbers. J. Sound Vib. 322 (1–2), 264–281.CrossRefGoogle Scholar
Spieser, É. 2020 Modélisation de la propagation du bruit de jet par une méthode adjointe formulée pour l'acoustique potentielle. PhD thesis, Ecole Centrale de Lyon.Google Scholar
Spieser, É. & Bailly, C. 2020 Sound propagation using an adjoint-based method. J. Fluid Mech. 900, A5.CrossRefGoogle Scholar
Tam, C.K. & Auriault, L. 1998 Mean flow refraction effects on sound radiated from localized sources in a jet. J. Fluid Mech. 370, 149–174.CrossRefGoogle Scholar
Tam, C.K. & Auriault, L. 1999 Jet mixing noise from fine-scale turbulence. AIAA J. 37 (2), 145–153.Google Scholar
Tam, C.K. & Webb, J.C. 1993 Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107 (2), 262–281.Google Scholar
Tournour, M., Cremers, L., Guisset, P., Augusztinovicz, F. & MArki, F. 2000 Inverse numerical acoustics based on acoustic transfer vectors. In 7th International Congress on Sound and Vibration, pp. 1069–1076.Google Scholar