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Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. Part 1. Linear theory

Published online by Cambridge University Press:  08 January 2008

J.-P. RACZ  and
J. F. SCOTT
J.-P. RACZ
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, ECL, UCBL, INSA, CNRS, 36 avenue Guy de Collongue, 69134 Ecully, France
J. F. SCOTT
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, ECL, UCBL, INSA, CNRS, 36 avenue Guy de Collongue, 69134 Ecully, France
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Abstract

An analysis is presented of parametric instability in a finite-length rotating cylinder subject to periodic axial compression by small sinusoidal oscillations of one of its ends (the ‘piston’). The instability is due to resonant interactions between inertial-wave (Kelvin) modes of the cylinder and the oscillatory compression and is resisted by viscosity, acting both through thin boundary layers and throughout the volume, the two mechanisms proving crucial for a satisfactory description. Instability is found to take the form of either a single axisymmetric mode with frequency near to half that of compression, or a pair of non-axisymmetric modes of the same azimuthal and axial orders and oppositely signed frequencies, whose difference is close to the compression frequency. Thus, in the axisymmetric case, instability leads to spontaneous growth of a system of one or more oscillating toroidal vortices encircling the cylinder axis, while the differing frequencies of the two modes of non-axisymmetric instability implies an oscillatory aperiodic flow. The neutral curves giving the threshold for instability are determined for all modes/mode pairs. For a given mode or mode pair, the neutral curve shows a critical piston amplitude dependent on rotational Reynolds number and cylinder aspect ratio, below which instability does not occur, and above which there is instability provided the compression frequency is chosen to lie in a band centred on the exact resonance condition.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 265 - 290
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37, 307–323.CrossRefGoogle Scholar
Baines, P. G. 1967 Forced oscillations of an enclosed rotating fluid. J. Fluid Mech. 30, 533–546.CrossRefGoogle Scholar
Duguet, Y., Scott, J. & Le Penven, L. 2005 Instability inside a rotating gas cylinder subject to axial periodic strain. Phys. Fluids 17, 114103.CrossRefGoogle Scholar
Eloy, C. & Le Dizés, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13, 660–676.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizés, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.CrossRefGoogle Scholar
Fultz, D. 1959 A note on overstability, and the elastoid-inertia oscillations of Kelvin, Solberg and Bjerknes. J. Met. 16, 199–208.2.0.CO;2>CrossRefGoogle Scholar
Gans, R. F. 1970 On the precession of a resonant cylinder. J. Fluid Mech. 41, 865–872.CrossRefGoogle Scholar
Gledzer, E. B. & Ponomarev, V. M. 1992 Instability of bounded flows with elliptical streamlines. J. Fluid Mech. 240, 1–30.CrossRefGoogle Scholar
Graftieaux, L., Le Penven, L., Scott, J. F. & Grosjean, N. 2002 A new parametric instability in rotating cylinder flow. In Advances in Turbulence IX (ed. Castro, I. P. & Hancock, P. E.). CIMNE, Barcelona, Spain.Google Scholar
Greenspan, H. P. 1969 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72, 107–144.CrossRefGoogle Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–114.CrossRefGoogle Scholar
Kerswell, R. R. & Barenghi, C. F. 1994 On the viscous decay of inertial waves in a rotating circular cylinder. J. Fluid Mech. 285, 203–214.CrossRefGoogle Scholar
Kobine, J. J. 1995 Inertial wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233–252.CrossRefGoogle Scholar
Kudlick, M. 1966 On the transient motions in a contained rotating fluid. PhD thesis, MIT.Google Scholar
Le Noble, J. 1995 Etude de l'écoulement d'un fluide dans un cylindre en rotation uniforme autours de son axe. DEA dissertation, Laboratoire de Mécanique des Fluides et d'Acoustique, Ecole Centrale de Lyon, Ecully, France.Google Scholar
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.CrossRefGoogle Scholar
Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external Coriolis force. Phys. Fluids A 5 (4), 891–900.CrossRefGoogle Scholar
Mannaseh, R. 1994 Distortions of inertia waves in a rotating fluid cylinder forced near its fundamental mode resonance. J. Fluid Mech. 265, 345–370.CrossRefGoogle Scholar
Mansour, N. N. & Lundgren, T. S. 1990 Three-dimensional instability of rotating flows with oscillating axial strain. Phys. Fluids A 2, 2089–2091.CrossRefGoogle Scholar
Racz, J.-P. & Scott, J. F. 2008 Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. Part 2. Weakly nonlinear theory. J. Fluid Mech. 595, 291–321.CrossRefGoogle Scholar
Rieutord, M. 1991 Linear theory of rotating fluids using spherical harmonics. Part II, time-periodic flows. Geophys. Astrophys. Fluid Dyn. 59, 185–208.CrossRefGoogle Scholar
Stergiopoulos, S. & Aldridge, K. D. 1982 Inertial waves in a fluid partially filled cylindrical cavity during spin-up from rest. Geophys. Astrophys. Fluid Dyn. 21, 89–112.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar