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Stability and acoustic scattering in a cylindrical thin shell containing compressible mean flow

Published online by Cambridge University Press:  25 April 2008

E. J. BRAMBLEY  and
N. PEAKE
E. J. BRAMBLEY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
N. PEAKE
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
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Abstract

We consider the stability of small perturbations to a uniform inviscid compressible flow within a cylindrical linear-elastic thin shell. The thin shell is modelled using Flügge's equations, and is forced from the inside by the fluid, and from the outside by damping and spring forces. In addition to acoustic waves within the fluid, the system supports surface waves, which are strongly coupled to the thin shell. Stability is analysed using the Briggs–Bers criterion, and the system is found to be either stable or absolutely unstable, with absolute instability occurring for sufficiently small shell thicknesses. This is significantly different from the stability of a thin shell containing incompressible fluid, even for parameters for which the fluid would otherwise be expected to behave incompressibly (for example, water within a steel thin shell). Asymptotic expressions are derived for the shell thickness separating stable and unstable behaviour.

We then consider the scattering of waves by a sudden change in the duct boundary from rigid to thin shell, using the Wiener–Hopf technique. For the scattering of an inbound acoustic wave in the rigid-wall section, the surface waves are found to play an important role close to the sudden boundary change. The solution is given analytically as a sum of duct modes.

The results in this paper add to the understanding of the stability of surface waves in models of acoustic linings in aeroengine ducts. The oft-used mass–spring–damper model is regularized by the shell bending terms, and even when these terms are very small, the stability and scattering results are quite different from what has been claimed for the mass–spring–damper model. The scattering results derived here are exact, unique and causal, without the need to apply a Kutta-like condition or to include an instability wave. A movie is available with the online version of the paper.

JFM classification

Acoustics: Acoustics Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 602 , 10 May 2008 , pp. 403 - 426
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions, 9th edn. Dover.Google Scholar
Bers, A. 1983 Space–time evolution of plasma instabilities — absolute and convective. In Basic Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), Handbook of Plasma Physics, vol. 1, pp. 451–517. North-Holland.Google Scholar
Brambley, E. J. & Peake, N. 2006 Classification of aeroacoustically relevant surface modes in cylindrical lined ducts. Wave Motion 43, 301–310.Google Scholar
Brandes, M. & Ronneberger, D. 1995 Sound amplification in flow ducts lined with a periodic sequence of resonators. AIAA Paper 95–126.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas, chap. 2. MIT Press.Google Scholar
Chevaugeon, N., Remacle, J.-F. & Gallez, X. 2006 Discontinuous Galerkin implementation of the extended Helmholtz resonator model in time domain. AIAA Paper 2006-2569.CrossRefGoogle Scholar
Crighton, D. G. & Leppington, F. G. 1974 Radiation properties of the semi-infinite vortex sheet: the initial-value problem. J. Fluid Mech. 64, 393–414.CrossRefGoogle Scholar
Crighton, D. G. & Oswell, J. E. 1991 Fluid loading with mean flow. I. Response of an elastic plate to localized excitation. Phil. Trans. R. Soc. Lond. A 335, 557–592.Google Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89, 433–468.CrossRefGoogle Scholar
Jones, D. S. 1977 The scattering of sound by a simple shear layer. Phil. Trans. R. Soc. Lond. A 284, 287–328.Google Scholar
Jones, D. S. & Morgan, J. D. 1972 The instability of a vortex sheet on a subsonic stream under acoustic radiation. Proc. Camb. Phil. Soc. 72, 465–488.CrossRefGoogle Scholar
Jones, D. S. & Morgan, J. D. 1974 A linear model of a finite amplitude Helmholtz instability. Proc. R. Soc. Lond. A 338, 17–41.Google Scholar
Leissa, A. W. 1973 Vibration of Shells, chap. 2. NASA SP-288.Google Scholar
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.CrossRefGoogle Scholar
Morgan, J. D. 1975 The interaction of sound with a subsonic cylindrical vortex layer. Proc. R. Soc. Lond. A 344, 341–362.Google Scholar
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.CrossRefGoogle Scholar
Myers, M. K. 1980 On the acoustic boundary condition in the presence of flow. J. Sound Vib. 71, 429–434.CrossRefGoogle Scholar
Noble, B. 1958 Methods based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. Pergamon.Google Scholar
Païdoussis, M. P. 2004 Fluid-Structure Interactions Vol. 2. Elsevier.Google Scholar
Peake, N. 1997 On the behaviour of a fluid-loaded cylindrical shell with mean flow. J. Fluid Mech. 338, 387–410.CrossRefGoogle Scholar
Richter, C. & Thiele, F. H. 2007 The stability of time explicit impedance models. AIAA Paper 2007-3538.CrossRefGoogle Scholar
Rienstra, S. W. 2003 A classification of duct modes based on surface waves. Wave Motion 37, 119–135.CrossRefGoogle Scholar
Rienstra, S. W. 2006 Impedance models in time domain, including the extended Helmholtz resonator model. AIAA Paper 2006-2686.Google Scholar
Rienstra, S. W. 2007 Acoustic scattering at a hard–soft lining transition in a flow duct. J. Engng. Maths. 59, 451–475.CrossRefGoogle Scholar
Rudin, W. 1991 Functional Analysis, p. 380. McGraw-Hill.Google Scholar
Tam, C. K. W. & Auriault, L. 1996 Time-domain impedance boundary conditions for computational aeroacoustics. AIAA J. 34, 917–923.Google Scholar
Tam, C. K. W. & Ju, H. 2006 Scattering of acoustic duct modes by axial liner splices. AIAA Paper 2006-2459.CrossRefGoogle Scholar

Brambley and Peake supplementary movie

Movie 1 (best played on repeat). A cylindrical duct contains uniform mean flow from left to right. The movie shows the amplitude of pressure oscillations (Re(p(x, r, t))) for a wave inbound from the left (the first-radial-order downstream-propagating mode) scattering off a sudden change from a rigid-wall to a thin-shell boundary at x=0. x is the horizontal axis, which varies from -10 to 10, and r (the radius from the duct centreline) is the vertical axis, with r=0 at the bottom and r=1 at the top. Red indicates a positive pressure perturbation and blue a negative perturbation. The fluid is air at standard temperature and pressure and the boundary is aluminium. The mean flow Mach number is U=0.5. The pressure oscillations have azimuthal order m=1 and Helmholtz number ω=16. The thin shell has external resistivity R=0.5, external spring forcing b=0, thickness h=10-4, sound speed cl=15.8, density ρs=2200, and Poisson's ratio ν=0.33. The movie is an animation of figure 11 of the paper, and is generated in the same way, by numerically summing equation (4.13) of the paper for x<0 and equation (4.14) of the paper for x>0.

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