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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

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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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