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Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

Published online by Cambridge University Press:  11 February 2011

SJOERD W. RIENSTRA  and
MIRELA DARAU
SJOERD W. RIENSTRA*
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
MIRELA DARAU
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
*
†Email address for correspondence: [email protected]
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Abstract

The Ingard–Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a two-dimensional incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wavelength or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contour plot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard–Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem.

JFM classification

Instability: Absolute/convective instability Acoustics: Aeroacoustics Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 559 - 573
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Aurégan, Y. & Leroux, M. 2008 Experimental evidence of an instability over an impedance wall in a duct flow. J. Sound Vib. 317, 432–439.CrossRefGoogle Scholar
Aurégan, Y., Leroux, M. & Pagneux, V. 2005 Abnormal behavior of an acoustical liner with flow. In Forum Acusticum, Budapest.Google Scholar
Bauer, A. B. & Chapkis, R. L. 1977 Noise generated by boundary-layer interaction with perforated acoustic liners. J. Aircraft 14 (2), 157–160.CrossRefGoogle Scholar
Bers, A. 1983 Space–time evolution of plasma instabilities: absolute and convective. In Handbook of Plasma Physics, vol. 1: Basic Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), chap. 3.2, pp. 451–517. North Holland.Google Scholar
Brambley, E. J. 2008 Models for acoustically-lined turbofan ducts. In 14th AIAA/CEAS Aeroacoustics Conference, 5–7 May 2008, The Westin Bayshore Vancouver, Vancouver, Canada. AIAA 2008–2879.Google Scholar
Brambley, E. J. 2009 Fundamental problems with the model of uniform flow over acoustic linings. J. Sound Vib. 322, 1026–1037.CrossRefGoogle Scholar
Brambley, E. J. 2010 A well-posed modified Myers boundary condition. In 16th AIAA/CEAS Aeroacoustics Conference, 7–9 June 2010, Stockholm, Sweden. AIAA 2010–3942.Google Scholar
Brambley, E. J. & Peake, N. 2006 Surface-waves, stability, and scattering for a lined duct with flow. In 12th AIAA/CEAS Aeroacoustics Conference, 8–10 May 2006, Cambridge, MA. AIAA-2006–2688.Google Scholar
Brambley, E. J. & Peake, N. 2008 Stability and acoustic scattering in a cylindrical thin shell containing compressible mean flow. J. Fluid Mech. 602, 403–426.CrossRefGoogle Scholar
Brandes, M. & Ronneberger, D. 1995 Sound amplification in flow ducts lined with a periodic sequence of resonators. In 1st AIAA/CEAS Aeroacoustics Conference, 12–15 June 1995, Munich, Germany. AIAA-95–126.Google Scholar
Briggs, R. J. 1964 Electron–stream interaction with plasmas. Monograph no. 29. MIT Press.Google Scholar
Campos, L. M. B. C., Oliveira, J. M. G. S. & Kobayashi, M. H. 1999 On sound propagation in a linear shear flow. J. Sound Vib. 219 (5), 739–770.CrossRefGoogle Scholar
Chevaugeon, N., Remacle, J.-F. & Gallez, X. 2006 Discontinuous Galerkin implementation of the extended Helmholtz resonator impedance model in time domain. In 12th AIAA/CEAS Aeroacoustics Conference, 8–10 May 2006, Cambridge, MA. AIAA-2006–2569.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
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 (1), 216–220.CrossRefGoogle Scholar
Hubbard, H. H. 1995 Aeroacoustics of Flight Vehicles: Theory and Practice, vol. 2: Noise Control. Acoustical Society of America.Google Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
Ingard, K. U. 1959 Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmission. J. Acoust. Soc. Am. 31 (7), 1035–1036.CrossRefGoogle Scholar
Lingwood, R. J. & Peake, N. 1999 On the causal behaviour of flow over an elastic wall. J. Fluid Mech. 396, 319–344.CrossRefGoogle Scholar
Marx, D., Aurégan, Y., Baillet, H. & Valière, J. 2010 Evidence of hydrodynamic instability over a liner in a duct with flow. J. Sound Vib. 329, 3798–3812.CrossRefGoogle Scholar
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23 (3), 521–544.CrossRefGoogle Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199.CrossRefGoogle Scholar
Myers, M. K. 1980 On the acoustic boundary condition in the presence of flow. J. Sound Vib. 71 (3), 429–434.CrossRefGoogle Scholar
Peake, N. 1997 On the behaviour of a fluid-loaded cylindrical shell with mean flow. J. Fluid Mech. 338, 387–410.CrossRefGoogle Scholar
Peake, N. 2002 Structural acoustics with mean flow. In Sound-Flow Interactions (ed. Aurégan, Y., Maurel, A., Pagneux, V. & Pinton, J.-F.), pp. 248–264. Springer.CrossRefGoogle Scholar
Pridmore-Brown, D. C. 1958 Sound propagation in a fluid flowing through an attenuating duct. J. Fluid Mech. 4, 393–406.CrossRefGoogle Scholar
Richter, C. 2009 Liner impedance modeling in the time domain with flow. PhD dissertation, Technische Universität Berlin.Google Scholar
Rienstra, S. W. 2003 A classification of duct modes based on surface waves. Wave Motion 37 (2), 119–135.CrossRefGoogle Scholar
Rienstra, S. W. 2006 Impedance models in time domain, including the extended Helmholtz resonator model. In 12th AIAA/CEAS Aeroacoustics Conference, 8–10 May 2006, Cambridge, MA. AIAA-2006–2686.Google Scholar
Rienstra, S. W. 2007 Acoustic scattering at a hard–soft lining transition in a flow duct. J. Engng Maths 59 (4), 451–475.CrossRefGoogle Scholar
Rienstra, S. W. & Darau, M. 2010 Mean flow boundary layer effects of hydrodynamic instability of impedance wall. In Proc. IUTAM Symp. on Computational Aero-Acoustics for Aircraft Noise Prediction, Southampton.Google Scholar
Rienstra, S. W. & Tester, B. T. 2008 An analytic Green's function for a lined circular duct containing uniform mean flow. J. Sound Vib. 317, 994–1016.CrossRefGoogle Scholar
Rienstra, S. W. & Vilenski, G. G. 2008 Spatial instability of boundary layer along impedance wall. In 14th AIAA/CEAS Aeroacoustics Conference, 5–7 May 2008, The Westin Bayshore Vancouver, Vancouver, Canada. AIAA-2008–2932.Google Scholar
Tester, B. J. 1973 a The propagation and attenuation of sound in ducts containing uniform or ‘plug’ flow. J. Sound Vib. 28 (2), 151–203.CrossRefGoogle Scholar
Tester, B. J. 1973 b Some aspects of ‘sound’ attenuation in lined ducts containing inviscid mean flows with boundary layers. J. Sound Vib. 28 (2), 217–245.CrossRefGoogle Scholar