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Eigenmodes of lined flow ducts with rigid splices

Published online by Cambridge University Press:  28 November 2011

E. J. Brambley ,
A. M. J. Davis  and
N. Peake
E. J. Brambley*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
A. M. J. Davis
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
N. Peake
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: E.J.Brambley@damtp.cam.ac.uk
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Abstract

This paper presents an analytic expression for the acoustic eigenmodes of a cylindrical lined duct with rigid axially running splices in the presence of flow. The cylindrical duct is considered to be uniformly lined except for two symmetrically positioned axially running rigid liner splices. An exact analytic expression for the acoustic pressure eigenmodes is given in terms of an azimuthal Fourier sum, with the Fourier coefficients given by a recurrence relation. Since this expression is derived using a Green’s function method, the completeness of the expansion is guaranteed. A numerical procedure is described for solving this recurrence relation, which is found to converge exponentially with respect to number of Fourier terms used and is in practice quick to compute; this is then used to give several numerical examples for both uniform and sheared mean flow. An asymptotic expression is derived to directly calculate the pressure eigenmodes for thin splices. This asymptotic expression is shown to be quantitatively accurate for ducts with very thin splices of less than 1 % unlined area and qualitatively helpful for thicker splices of the order of 6 % unlined area. A thin splice is in some cases shown to increase the damping of certain acoustic modes. The influences of thin splices and thin boundary layers are compared and found to be of comparable magnitude for the parameters considered. Trapped modes at the splices are also identified and investigated.

JFM classification

Acoustics: Aeroacoustics Wakes/Jets: Shear layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 690 , 10 January 2012 , pp. 399 - 425
Copyright
Copyright © Cambridge University Press 2011

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References

1. Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions, 9th edn. Dover.Google Scholar
2. 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.Google Scholar
3. Amos, D. E. 1986 Algorithm 644: a portable package for Bessel functions of a complex argument and non-negative order. ACM Trans. Math. Softw. 12, 265–273.CrossRefGoogle Scholar
4. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. & Sorensen, D. 1999 LAPACK Users’ Guide. Society for Industrial and Applied Mathematics.Google Scholar
5. Aurégan, Y., Starobinski, R. & Pagneux, V. 2001 Influence of grazing flow and dissipation effects on the acoustic boundary conditions at a lined wall. J. Acoust. Soc. Am. 109, 59–64.CrossRefGoogle Scholar
6. Bi, W. 2008 Calculations of modes in circumferentially non-uniform lined ducts. J. Acoust. Soc. Am. 123, 2603–2612.Google Scholar
7. Bi, W. P., Pagneux, V., Lafarge, D. & Aurégan, Y. 2006 Modelling of sound propagationin a non-uniform lined duct using a multi-modal propagation method. J. Sound Vib. 289, 1091–1111.Google Scholar
8. Bi, W. P., Pagneux, V., Lafarge, D. & Aurégan, Y. 2007 Characteristics of penalty mode scattering by rigid splices in lined ducts. J. Acoust. Soc. Am. 121 (3), 1303–1312.CrossRefGoogle ScholarPubMed
9. Bi, W. P., Pagneux, V., Lafarge, D. & Aurégan, Y. 2009 Trapped modes at acoustically rigid splices. AIAA paper 2009-3105.Google Scholar
10. Brambley, E. J. 2009 Fundamental problems with the model of uniform flow over acoustic linings. J. Sound Vib. 322, 1026–1037.CrossRefGoogle Scholar
11. Brambley, E. J. 2011a Acoustic implications of a thin viscous boundary layer over a compliant surface or permeable liner. J. Fluid Mech. 678, 348–378.CrossRefGoogle Scholar
12. Brambley, E. J. 2011b A well-posed boundary condition for acoustic liners in straight ducts with flow. AIAA J. 49 (6), 1272–1282.CrossRefGoogle Scholar
13. Brambley, E. J. & Peake, N. 2006 Classification of aeroacoustically relevant surface modes in cylindrical lined ducts. Wave Motion 43, 301–310.Google Scholar
14. Campos, L. M. B. C. & Oliveira, J. M. G. S. 2004 On the acoustic modes in a cylindrical duct with an arbitrary wall impedance distribution. J. Acoust. Soc. Am. 116 (6), 3336–3347.Google Scholar
15. Cargill, A. M. 1993 Scattering from joins between liners in intake ducts with application to BR710 buzz-saw noise. Technical Report TSG0688. Rolls–Royce.Google Scholar
16. Davis, A. M. J. & Llewellyn Smith, S. G. 2007 Perturbation of eigenvalues due to gaps in 2d boundaries. Proc. R. Soc. Lond. A 463, 759–786.Google Scholar
17. Duta, M. C. & Giles, M. B. 2006 A three-dimensional hybrid finite element/spectral analyis of noise radiation from turbofan inlets. J. Sound Vib. 296, 623–642.Google Scholar
18. Eversman, W. & Beckemeyer, R. J. 1972 Transmission of sound in ducts with thin shear layers – Convergence to the uniform flow case. J. Acoust. Soc. Am. 52, 216–220.CrossRefGoogle Scholar
19. Fuller, C. R. 1984 Propagation and radiation of sound from flanged circular ducts with circumferentially varying wall admittances, I: Semi-infinite ducts. J. Sound Vib. 93, 321–340.Google Scholar
20. Gabard, G. & Astley, R. J. 2008 A computational mode-matching approach for sound propagation in three-dimensional ducts with flow. J. Sound Vib. 315, 1103–1124.Google Scholar
21. Joubert, L. 2010 Asymptotic approach for the mathematical and numerical analysis of the acoustic propagation in a strong shear flow. PhD thesis, École Polytechnique (in French).Google Scholar
22. Koch, W. & Möhring, W. 1983 Eigensolutions for liners in uniform mean flow ducts. AIAA J. 21 (2), 200–213.CrossRefGoogle Scholar
23. LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.Google Scholar
24. Lighthill, M. J. 1978 Waves in Fluids. Cambridge.Google Scholar
25. McAlpine, A. & Wright, M. C. M. 2006 Acoustic scattering by a spliced turbofan inlet duct liner at supersonic fan speeds. J. Sound Vib. 292, 911–934.Google Scholar
26. Munt, R. M. 1977 The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe. J. Fluid Mech. 83, 609–640.Google Scholar
27. Myers, M. K. 1980 On the acoustic boundary condition in the presence of flow. J. Sound Vib. 71, 429–434.CrossRefGoogle Scholar
28. Osipov, A. V. & Norris, A. N. 1999 The malyuzhinets thoery for scattering from wedge boundaries: a review. Wave Motion 29, 313–340.CrossRefGoogle Scholar
29. Pagneux, V., Amir, N. & Kergomard, J. 1996 A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation. J. Acoust. Soc. Am. 100, 2034–2048.CrossRefGoogle Scholar
30. Pridmore-Brown, D. C. 1958 Sound propagation in a fluid flowing through an attenuating duct. J. Fluid Mech. 4, 393–406.Google Scholar
31. Rademaker, E. R., Sarin, S. L. & Parente, C. A. 1996 Experimental investigation on the influence of liner non-uniformities on prevailing modes. AIAA paper 96-1682.Google Scholar
32. Regan, B. & Eaton, J. 1999 Modelling the influence of acoustic liner non-uniformities on duct modes. J. Sound Vib. 219, 859–879.Google Scholar
33. Rienstra, S. W. 2003 A classification of duct modes based on surface waves. Wave Motion 37, 119–135.Google Scholar
34. Rienstra, S. W. 2007 Acoustic scattering at a hard–soft lining transition in a flow duct. J. Engng Maths 59, 451–475.Google Scholar
35. Rienstra, S. W. & Darau, M. 2011 Boundary-layer thickness effects of the hydrodynamic instability along an impednace wall. J. Fluid Mech. 671, 559–573.CrossRefGoogle Scholar
36. Rienstra, S. W. & Vilenski, G. G. 2008 Spatial instability of boundary layer along impedance wall. AIAA paper 2008-2932.Google Scholar
37. Sarin, S. L. & Rademaker, E. R. 1993 In-flight acoustic mode measurements in the turbofan engine inlet of fokker 100 aircraft. AIAA paper 93-4414.Google Scholar
38. Swinbanks, M. A. 1975 The sound field generated by a source distribution in a long duct carrying sheared flow. J. Sound Vib. 40, 51–76.Google Scholar
39. Tam, C. K. W. & Ju, H. 2009 Finite difference computation of acoustic scattering by small surface inhomogeneities and discontinuities. J. Comput. Phys. 228, 5917–5932.CrossRefGoogle Scholar
40. Tam, C. K. W., Ju, H. & Chien, E. W. 2008 Scattering of acoustic duct modes by axial liner splices. J. Sound Vib. 310, 1014–1035.Google Scholar
41. Tester, B. J. 1973 Some aspects of sound attenuation in lined ducts containing inviscid mean flows with boundary layers. J. Sound Vib. 28, 217–245.Google Scholar
42. Tester, B. J. & de Mercato, L. 2006 Far-field directivity of rotor-alone tones radiated from fan intakes with spliced liners for different intake shapes, with flow. AIAA paper 2006-2456.Google Scholar
43. Tester, B. J., Powles, C. J., Baker, N. J. & Kempton, A. J. 2006 Scattering of sound by liner splices: a Kirchhoff model with numerical validation. AIAA J. 44 (9), 2009–2017.Google Scholar
44. Veitch, B. & Peake, N. 2008 Acoustic propagation and scattering in the exhaust flow from coaxial cylinders. J. Fluid Mech. 613, 275–307.Google Scholar
45. Vilenski, G. G. & Rienstra, S. W. 2007 On hydrodynamic and acoustic modes in a ducted shear flow with wall lining. J. Fluid Mech. 583, 45–70.Google Scholar
46. Watson, W. R. 1981 Noise suppression characteristics of peripherally segmented duct liners. Technical Report TP-1904. NASA.Google Scholar
47. Watson, W. R., Nark, D. M. & Jones, M. G. 2008 Assessment of 3D codes for predicting liner attenuation in flow ducts. AIAA paper 2008-2828.Google Scholar
48. Wright, M. C. M. 2006 Hybrid analytical/numerical method for mode scattering in azimuthally non-uniform ducts. J. Sound Vib. 2006, 583–594.Google Scholar
49. Yang, B. & Wang, T. Q. 2008 Investigation of the influence of liner hard-splices on duct radiation/propagation and mode scattering. J. Sound Vib. 315, 1016–1034.Google Scholar