Stability and acoustic scattering in a cylindrical thin shell containing compressible mean flow. (English) Zbl 1170.76016
Summary: We consider the stability of small perturbations to a uniform inviscid compressible flow within a cylindrical linearly elastic thin shell. The thin shell is modelled using Flügge 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.
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.
MSC:
| 76E17 | Interfacial stability and instability in hydrodynamic stability |
| 76Q05 | Hydro- and aero-acoustics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74K25 | Shells |
References:
| [1] | Pa?doussis, Fluid-Structure Interactions Vol. 2 (2004) |
| [2] | DOI: 10.1017/S0022112097005144 · Zbl 0920.76034 · doi:10.1017/S0022112097005144 |
| [3] | DOI: 10.1016/0022-460X(80)90424-1 · Zbl 0448.76065 · doi:10.1016/0022-460X(80)90424-1 |
| [4] | DOI: 10.1017/S0022112077001384 · doi:10.1017/S0022112077001384 |
| [5] | Morgan, Proc. R. Soc. Lond. 344 pp 341– (1975) |
| [6] | DOI: 10.1017/S0022112078002682 · Zbl 0401.76018 · doi:10.1017/S0022112078002682 |
| [7] | DOI: 10.1016/j.jsv.2005.09.010 · doi:10.1016/j.jsv.2005.09.010 |
| [8] | DOI: 10.1098/rsta.1991.0060 · Zbl 0735.76030 · doi:10.1098/rsta.1991.0060 |
| [9] | DOI: 10.1017/S0022112074002461 · Zbl 0298.76045 · doi:10.1017/S0022112074002461 |
| [10] | Jones, Proc. R. Soc. Lond. 338 pp 17– (1974) |
| [11] | Jones, Proc. Camb. Phil. Soc. 72 pp 465– (1972) |
| [12] | Briggs, Electron-Stream Interaction with Plasmas (1964) |
| [13] | DOI: 10.1098/rsta.1977.0011 · Zbl 0353.76055 · doi:10.1098/rsta.1977.0011 |
| [14] | DOI: 10.1016/j.wavemoti.2006.01.001 · Zbl 1231.76261 · doi:10.1016/j.wavemoti.2006.01.001 |
| [15] | Bers, Basic Plasma Physics pp 451– (1983) |
| [16] | Abramowitz, Handbook of Mathematical Functions (1964) |
| [17] | Tam, AIAA J. 34 pp 917– (1996) |
| [18] | Rudin, Functional Analysis pp 380– (1991) |
| [19] | DOI: 10.1007/s10665-007-9193-z · Zbl 1198.76136 · doi:10.1007/s10665-007-9193-z |
| [20] | DOI: 10.1016/S0165-2125(02)00052-5 · Zbl 1163.74431 · doi:10.1016/S0165-2125(02)00052-5 |
| [21] | Noble, Methods based on the Wiener?Hopf Technique for the Solution of Partial Differential Equations (1958) · Zbl 0082.32101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
