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Sound propagation using an adjoint-based method

Published online by Cambridge University Press:  31 July 2020

Étienne Spieser Open the ORCID record for Étienne Spieser [Opens in a new window]  and
Christophe Bailly Open the ORCID record for Christophe Bailly [Opens in a new window]
Étienne Spieser*
Affiliation:
Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, F-69134Écully, France Safran Aircraft Engines, 77500 Moissy-Cramayel, France
Christophe Bailly
Affiliation:
Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, F-69134Écully, France
*
†Email address for correspondence: etienne.spieser@ec-lyon.fr
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Abstract

In this study, a comprehensive description of the adjoint formulation based on a systematic use of Lagrange's identity is proposed to compute acoustic propagation effects induced by the presence of a mean flow. The adjoint method is a clever approach introduced by Tam & Auriault (J. Fluid Mech., vol. 370, 1998, pp. 149–174) in aeroacoustics to predict noise of distributed stochastic sources in a complex environment. A clear statement is also provided about the application of the flow reversal theorem, and its restriction to self-adjoint wave equations. As an illustration, sound propagation is computed numerically over a sheared and stratified mean flow for Lilley's and Pierce's wave equations. Acoustic solutions obtained with the adjoint approach are then compared with predictions obtained with the flow reversal theorem. Additionally Pierce's wave equation for potential acoustics is identified as an outstanding candidate to compute accurately acoustic propagation while removing possible instability waves.

JFM classification

Acoustics: Aeroacoustics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A5
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Afsar, M. Z. 2009 Solution of the parallel shear layer Green's function using conservation equations. Intl J. Aeroacoust. 8 (6), 585–602.CrossRefGoogle Scholar
Afsar, M. Z. 2010 Asymptotic properties of the overall sound pressure level of subsonic jet flows using isotropy as a paradigm. J. Fluid Mech. 664, 510–539.CrossRefGoogle Scholar
Afsar, M. Z., Dowling, A. P. & Karabasov, S. A. 2006 Comparison of jet noise models. In 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, AIAA Paper 2006–2593.Google Scholar
Afsar, M. Z., Dowling, A. P. & Karabasov, S. A. 2007 Jet noise in the ‘zone of silence’. In 13th AIAA/CEAS Aeroacoustics Conference, Rome, AIAA Paper 2007–3606.Google Scholar
Afsar, M. Z., Sescu, A. & Leib, S. J. 2016 a Predictive capability of low frequency jet noise using an asymptotic theory for the adjoint vector Green's function in non-parallel flow. In 22nd AIAA/CEAS Aeroacoustics Conference, Lyon, AIAA Paper 2016–2804.Google Scholar
Afsar, M. Z., Sescu, A., Sassanis, V. & Lele, S. K. 2017 Supersonic jet noise predictions using a unified asymptotic approximation for the adjoint vector Green's function and LES data. In 23rd AIAA/CEAS Aeroacoustics Conference, Denver, CO, AIAA Paper 2017–3030.Google Scholar
Afsar, M. Z., Sescu, A., Sassanis, V., Towne, A., Bres, G. A. & Lele, S. K. 2016 b Prediction of supersonic jet noise using non-parallel flow asymptotics and LES data within Goldstein's acoustic analogy. In Proceedings of the Summer Program, Center for Turbulence Research, pp. 253–262.Google Scholar
Alonso, J. S. & Burdisso, R. A. 2007 Green's functions for the acoustic field in lined ducts with uniform flow. AIAA J. 45 (11), 2677–2687.CrossRefGoogle Scholar
Bailly, C. & Bogey, C. 2003 Radiation and refraction of sound waves through a two-dimensional shear layer. In Fourth Computational Aeroacoustics (CAA) Workshop on Benchmark Problems. NASA/CP2004-212954. Citeseer.Google Scholar
Bailly, C., Bogey, C. & Candel, S. 2010 Modelling of sound generation by turbulent reacting flows. Intl J. Aeroacoust. 9 (4–5), 461–489.CrossRefGoogle Scholar
Barone, M. F. & Lele, S. K. 2005 Receptivity of the compressible mixing layer. J. Fluid Mech. 540, 301–335.CrossRefGoogle Scholar
Berland, J., Bogey, C., Marsden, O. & Bailly, C. 2007 High-order, low dispersive and low dissipative explicit schemes for multiple-scale and boundary problems. J. Comput. Phys. 224 (2), 637–662.CrossRefGoogle Scholar
Blokhintzev, D. I. 1946 The propagation of sound in an inhomogeneous and moving medium I. J. Acoust. Soc. Am. 18 (2), 322–328.CrossRefGoogle Scholar
Bogey, C. & Bailly, C. 2002 Three-dimensional non-reflective boundary conditions for acoustic simulations: far field formulation and validation test cases. Acta Acoust. United with Acoust. 88, 463–471.Google Scholar
Bogey, C. & Bailly, C. 2004 A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J. Comput. Phys. 194 (1), 194–214.CrossRefGoogle Scholar
Bojarski, N. N. 1983 Generalized reaction principles and reciprocity theorems for the wave equations, and the relationship between the time-advanced and time-retarded fields. J. Acoust. Soc. Am. 74 (1), 281–285.CrossRefGoogle Scholar
Cheung, L. C., Pastouchenko, N. N., Mani, R. & Paliath, U. 2015 Fine-scale turbulent noise predictions from non-axisymmetric jets. Intl J. Aeroacoust. 14 (3-4), 457–487.CrossRefGoogle Scholar
Cho, Y. C. 1980 Reciprocity principle in duct acoustics. J. Acoust. Soc. Am. 67 (5), 1421–1426.CrossRefGoogle Scholar
Crighton, D. G. & Leppington, F. G. 1970 Scattering of aerodynamic noise by a semi-infinite compliant plate. J. Fluid Mech. 43 (4), 721–736.CrossRefGoogle Scholar
Crighton, D. G. & Leppington, F. G. 1971 On the scattering of aerodynamic noise. J. Fluid Mech. 46 (3), 577–597.CrossRefGoogle Scholar
Crighton, D. G. & Leppington, F. G. 1973 Singular perturbation methods in acoustics: diffraction by a plate of finite thickness. Proc. R. Soc. Lond. A 335 (1602), 313–339.Google Scholar
Dahl, M. D. 2004 Fourth CAA workshop on benchmark problems. Tech. Rep. 2004-212954. NASA.Google Scholar
Davis, T. A. 2004 Algorithm 832: UMFPACK v4.3 an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. (TOMS) 30 (2), 196–199.CrossRefGoogle Scholar
Depuru Mohan, N. K., Dowling, A. P., Karabasov, S. A., Xia, H., Graham, O., Hynes, T. P. & Tucker, P. G. 2015 Acoustic sources and far-field noise of chevron and round jets. AIAA J. 53 (9), 2421–2436.CrossRefGoogle Scholar
Dowling, A. P. 1983 Flow-acoustic interaction near a flexible wall. J. Fluid Mech. 128, 181–198.CrossRefGoogle Scholar
Dowling, A. P., Ffowcs Williams, J. E. & Goldstein, M. E. 1978 Sound production in a moving stream. Phil. Trans. R. Soc. Lond. A 288 (1353), 321–349.Google Scholar
Eisler, T. J. 1969 An introduction to Green's functions. Tech. Rep. Catholic University of America Washington DC Institute of Ocean Science and Engineering.Google Scholar
Eversman, W. 1976 A reciprocity relationship for transmission in non-uniform hard walled ducts without flow. J. Sound Vib. 47 (4), 515–521.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
Galbrun, H. 1931 Propagation d'une onde sonore dans l'atmosphère et théorie des zones de silence. Gauthier-Villars et Cie., Éditeurs, Libraires du Bureau des longitudes, de l’École polytechnique.Google Scholar
Giles, M. B. & Pierce, N. A. 1997 Adjoint equations in CFD: duality, boundary conditions and solution behaviour. In 13th Computational Fluid Dynamics Conference, Snowmass Village, CO, AIAA Paper 1997–1850.Google Scholar
Godin, O. A. 1997 Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid. Wave Motion 25, 143–167.CrossRefGoogle Scholar
Godin, O. A. & Voronovich, A. G. 2004 Fermat's principle for non-dispersive waves in non-stationary media. Proc. R. Soc. Lond. A 460 (2046), 1631–1647.CrossRefGoogle Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (3), 433–468.CrossRefGoogle Scholar
Goldstein, M. E. 2006 Hybrid Reynolds-averaged Navier-Stokes/large eddy simulation approach for predicting jet noise. AIAA J. 44 (12), 3136–3142.CrossRefGoogle Scholar
Goldstein, M. E. & Leib, S. J. 2008 The aeroacoustics of slowly diverging supersonic jets. J. Fluid Mech. 600, 291–337.CrossRefGoogle Scholar
Goldstein, M. E., Sescu, A. & Afsar, M. Z. 2012 Effect of non-parallel mean flow on the Green's function for predicting the low-frequency sound from turbulent air jets. J. Fluid Mech. 695, 199–234.CrossRefGoogle Scholar
Gryazev, V., Markesteijn, A. P. & Karabasov, S. A. 2018 Temperature effect on the apparent position of effective noise sources in a hot jet. In 24th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2018–2827.Google Scholar
Helmholtz, H. 1870 Theorie der Luftschwingungen in Röhren mit offenen Enden. J. Reine Angew. Maths 57, 1–72.Google Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183–204.CrossRefGoogle Scholar
Howe, M. S. 1975 The generation of sound by aerodynamic sources in an inhomogeneous steady flow. J. Fluid Mech. 67 (3), 597–610.CrossRefGoogle Scholar
Hu, F. Q. 2008 Development of PML absorbing boundary conditions for computational aeroacoustics: a progress review. Comput. Fluids 37 (4), 336–348.CrossRefGoogle Scholar
Jameson, A. 1988 Aerodynamic design via control theory. J. Sci. Comput. 3 (3), 233–260.CrossRefGoogle Scholar
Karabasov, S. A., Bogey, C. & Hynes, T. P. 2013 An investigation of the mechanisms of sound generation in initially laminar subsonic jets using the Goldstein acoustic analogy. J. Fluid Mech. 714, 24–57.CrossRefGoogle Scholar
Karabasov, S. A. & Hynes, T. P. 2005 An efficient frequency-domain algorithm for wave scattering problems in application to jet noise. In 11th AIAA/CEAS Aeroacoustics Conference, Monterey, CA, AIAA Paper 2005–2827.Google Scholar
Karabasov, S. A. & Sandberg, R. D. 2015 Influence of free stream effects on jet noise generation and propagation within the Goldstein acoustic analogy approach for fully turbulent jet inflow boundary conditions. Intl J. Aeroacoust. 14 (3–4), 413–429.CrossRefGoogle Scholar
Khavaran, A. & Bridges, J. 2005 Modelling of fine-scale turbulence mixing noise. J. Sound Vib. 279 (3–5), 1131–1154.CrossRefGoogle Scholar
Lamb, G. L. Jr. 1995 Introductory Applications of Partial Differential Equations: With Emphasis on Wave Propagation and Diffusion. John Wiley & Sons.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Course of Theoretical Physics, vol. 6. Fluid mechanics. Pergamon.Google Scholar
Levine, H. & Schwinger, J. 1948 On the radiation of sound from an unflanged circular pipe. Phys. Rev. 73 (4), 383–406.CrossRefGoogle Scholar
Lilley, G. M., Plumblee, H. E., Strahle, W. C., Ruo, S. Y. & Doak, P. E. 1972 The generation and radiation of supersonic jet noise. Volume IV. theory of turbulence generated jet noise, noise radiation from upstream sources, and combustion noise. Tech. Rep. Lockheed-Georgia Co. Marietta.Google Scholar
Luchini, P. & Bottaro, A. 1998 Görtler vortices: a backward-in-time approach to the receptivity problem. J. Fluid Mech. 363, 1–23.CrossRefGoogle Scholar
Lyamshev, L. M. 1961 On certain integral relations in the acoustics of a moving medium. In Doklady Akademii Nauk, vol. 138, pp. 575–578. Russian Academy of Sciences.Google Scholar
Maestrello, L., Bayliss, A. & Turkel, E. 1981 On the interaction of a sound pulse with the shear layer of an axisymmetric jet. J. Sound Vib. 74 (2), 281–301.CrossRefGoogle Scholar
Miller, S. A. E. 2014 a The prediction of jet noise ground effects using an acoustic analogy and a tailored Green's function. J. Sound Vib. 333 (4), 1193–1207.CrossRefGoogle Scholar
Miller, S. A. E. 2014 b Toward a comprehensive model of jet noise using an acoustic analogy. AIAA J. 52 (10), 2143–2164.CrossRefGoogle Scholar
Morris, P. J. & Farassat, F. 2002 Acoustic analogy and alternative theories for jet noise prediction. AIAA J. 40 (4), 671–680.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics – Part I. McGraw-Hill.Google Scholar
Mosson, A., Binet, D. & Caprile, J. 2014 Simulation of the installation effects of the aircraft engine rear fan noise with ACTRAN/DGM. In 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, AIAA Paper 2014–3188.Google Scholar
Möhring, W. 1978 Acoustic energy flux in nonhomogeneous ducts. J. Acoust. Soc. Am. 64 (4), 1186–1189.CrossRefGoogle Scholar
Möhring, W. 1979 Modelling low mach number noise. In IUTAM Symposium on Mechanics of Sound Generation in Flows, Göttingen, pp. 85–96. Springer.CrossRefGoogle Scholar
Möhring, W. 1999 A well posed acoustic analogy based on a moving acoustic medium. In Aeroacoustic workshop SWING, Dresden, pp. 1–11.Google Scholar
Möhring, W. 2001 Energy conservation, time-reversal invariance and reciprocity in ducts with flow. J. Fluid Mech. 431, 223–237.CrossRefGoogle Scholar
Pastouchenko, N. N. & Tam, C. K. W. 2007 Installation effects on the flow and noise of wing mounted jets. AIAA J. 45 (12), 2851–2860.CrossRefGoogle Scholar
Phillips, O. M. 1960 On the generation of sound by supersonic turbulent shear layers. J. Fluid Mech. 9 (1), 1–28.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
Raizada, N. & Morris, P. J. 2006 Prediction of noise from high speed subsonic jets using an acoustic analogy. In 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, AIAA Paper 2006–2596.Google Scholar
Roberts, P. H. 1960 Characteristic value problems posed by differential equations arising in hydrodynamics and hydromagnetics. J. Math. Anal. Appl. 1 (2), 195–214.CrossRefGoogle Scholar
Semiletov, V. A. & Karabasov, S. A. 2013 A 3D frequency-domain linearised Euler solver based on the Goldstein acoustic analogy equations for the study of nonuniform meanflow propagation effects. In 19th AIAA/CEAS Aeroacoustics Conference, Berlin, AIAA Paper 2013–2019.Google Scholar
Stone, M. & Goldbart, P. 2009 Mathematics for Physics. Cambridge University Press. http://goldbart.gatech.edu/PG_MS_MfP.htmCrossRefGoogle Scholar
Strutt, J. W. 1877 The Theory of Sound, 2nd ed. (1945), vol. 1. Dover.Google Scholar
Tam, C. K. W. & Auriault, L. 1998 Mean flow refraction effects on sound radiated from localized source in a jet. J. Fluid Mech. 370, 149–174.CrossRefGoogle Scholar
Tam, C. K. W. & Auriault, L. 1999 Jet mixing noise from fine-scale turbulence. AIAA J. 37 (2), 145–153.CrossRefGoogle Scholar
Tam, C. K. W. & Pastouchenko, N. N. 2002 Noise from fine-scale turbulence of nonaxisymmetric jets. AIAA J. 40 (3), 456–464.CrossRefGoogle Scholar
Tam, C. K. W., Pastouchenko, N. N. & Viswanathan, K. 2005 Fine-scale turbulence noise from hot jets. AIAA J. 43 (8), 1675–1683.CrossRefGoogle Scholar
Tam, C. K. W., Pastouchenko, N. N. & Viswanathan, K. 2010 Continuation of near-acoustic fields of jets to the far field. Part I: Theory. In 16th AIAA/CEAS Aeroacoustics Conference, Stockholm, AIAA Paper 2010–3728.Google Scholar
Tam, C. K. W. & Webb, J. C. 1993 Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 262–281.CrossRefGoogle Scholar
Vasconcelos, I., Snieder, R. & Douma, H. 2009 Representation theorems and Green's function retrieval for scattering in acoustic media. Phys. Rev. E 80 (3), 036605.CrossRefGoogle ScholarPubMed
Wapenaar, C. P. A. 1996 Reciprocity theorems for two-way and one-way wave vectors: a comparison. J. Acoust. Soc. Am. 100 (6), 3508–3518.CrossRefGoogle Scholar
Wei, M. & Freund, J. B. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123–152.CrossRefGoogle Scholar
Xu, X., He, J., Li, X. & Hu, F. Q. 2015 3-D jet noise prediction for separate flow nozzles with pylon interaction. In 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, AIAA Paper 2015–512.Google Scholar
Yates, J. E. 1978 Application of the Bernoulli enthalpy concept to the study of vortex noise and jet impingement noise. Tech. Rep. NASA Contractor Report 2987.Google Scholar