Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T22:29:54.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. Part 2. Weakly nonlinear 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
Get access Rights & Permissions [Opens in a new window]

Abstract

A weakly nonlinear analysis is presented of parametric instability in a rotating cylinder subject to periodic axial compression by small sinusoidal oscillations of one of its ends (‘the piston’). Amplitude equations are derived for the pair of parametrically resonant (primary) inertial modes which were found to arise from linear instability in Part 1. These equations introduce an infinity of geostrophic mode amplitudes, representing a nonlinear modification of the mean flow, for which evolution equations are also derived. Consequences of the total system of equations are investigated for axisymmetric modes. Different possible outcomes are found at large times: (a) a fixed point, representing a saturated state in which the oscillatory toroidal vortices of the primary mode are phase-locked to the piston motion with half its frequency; (b) a limit cycle or chaotic attractor, corresponding to slow-time oscillations of the primary mode; or (c) exponential divergence of the amplitudes to infinity. The latter outcome, a necessary condition for which is derived in the form of a threshold piston amplitude for divergence, invalidates the theory, inducing a gross change in the character of the flow and providing a route out of the weakly nonlinear regime. Non-zero fixed-point branches arise via bifurcations from both sides of the linear neutral curve, where the basic flow changes local stability. The lower-amplitude branch is shown to be unstable, while the upper one may lose local stability, resulting in a Hopf bifurcation to a limit cycle, which can subsequently become aperiodic via a series of further bifurcations. Typically, during the resulting oscillations, whether periodic or not, the perturbation first grows from small amplitude owing to basic-flow instability, then nonlinear detuning of the parametric resonance causes decay back to small amplitude in the second half of the cycle, which then restarts.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 291 - 321
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33, 739–751.CrossRefGoogle Scholar
Douady, S. 1990 Experimental study of Faraday instability. J. Fluid Mech. 221, 383–409.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 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
Graftieaux, L. 2003 Etude expérimentale de l'instabilité d'un cylindre de gaz tournant soumis à une compression périodique. PhD thesis, Ecole Centrale de Lyon.Google 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. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36, 257–264.CrossRefGoogle Scholar
Gunn, J. S. & Aldridge, K. D. 1990 Inertial wave eigenfrequencies for a non-uniformly rotating fluid. Phys. Fluids A 2 (11), 2055–2060.CrossRefGoogle Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating flows: inertial wave breakdown. J. Fluid Mech. 382, 283–306.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–114.CrossRefGoogle Scholar
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.Google Scholar
Malkus, W. V. R. 1989 An experimental study of the global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.Google Scholar
Manasseh, R. 1992 Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 243, 261–296.CrossRefGoogle Scholar
Mason, D. M. & Kerswell, R. R. 1999 Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown. J. Fluid Mech., 396 73–108.CrossRefGoogle Scholar
Miles, J. 1993 On Faraday waves. J. Fluid Mech. 248, 671–683.Google Scholar
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81–100.Google Scholar
Racz, J.-P. 2006 Instabilité paramétrique dans un cylindre de gaz en rotation soumis à la compression périodique d'un piston. PhD thesis, Ecole Centrale de Lyon.Google Scholar
Racz, J.-P. & Scott, J. F. 2008 Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. Part 1. Linear theory. J. Fluid Mech. 595, 265–290.CrossRefGoogle Scholar
Waleffe, F. 1989 The 3D instability of a strained vortex and its relation to turbulence. PhD thesis, MIT.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar
Supplementary material: PDF

Racz supplementary material

Racz supplementary material

Download Racz supplementary material(PDF)
PDF 176.2 KB