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Dynamics of a fluid inside a precessing cylinder

Published online by Cambridge University Press:  05 August 2009

Romain Lagrange ,
Patrice Meunier ,
Christophe Eloy  and
François Nadal
Romain Lagrange
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre / Université de Provence, IRPHE – UMR 6594, Technopôle de Château-Gombert, 49, rue Joliot Curie, B.P. 146, 13384 Marseille Cedex 13, France
Patrice Meunier
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre / Université de Provence, IRPHE – UMR 6594, Technopôle de Château-Gombert, 49, rue Joliot Curie, B.P. 146, 13384 Marseille Cedex 13, France
Christophe Eloy
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre / Université de Provence, IRPHE – UMR 6594, Technopôle de Château-Gombert, 49, rue Joliot Curie, B.P. 146, 13384 Marseille Cedex 13, France
François Nadal
Affiliation:
Commissariat à l'Énergie Atomique (CEA/CESTA), France
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Abstract

The instability of a fluid inside a precessing cylinder is studiedtheoretically and experimentally. This study is motivated byaeronautics and geophysics applications. Precessional motion forceshydrodynamics waves called Kelvin modes whose structure andamplitude are predicted by a linear inviscid theory. When a forcedKelvin mode is resonant, a viscous and weakly nonlinear theory hasbeen developed to predict its saturated amplitude. We show that thisamplitude scales as Re1/2 for low Reynolds numbers and asθ1/3 (where θ is the precessing angle) for highReynolds numbers. These scalings are confirmed by PIV measurements.For Reynolds numbers sufficiently large, this forced flow becomesunstable. A linear stability analysis based on a triadic resonancebetween a forced Kelvin mode and two free modes has been carriedout. The precessing angle for which the flow becomes unstable ispredicted and compared successfully to experimental measurements. Aweakly nonlinear theory was developed and allowed to show that thebifurcation of the instability of precession is subcritical. It alsoshowed that, depending on the Reynolds number, the unstable flow canbe steady or intermittent. Finally, this weakly nonlinear theoryallowed to predict, with a good agreement with experiments, the meanflow in the cylinder; even if it is turbulent.

Keywords

PrecessionKelvin modesinstabilitytriadic resonancePrécessionmodes de Kelvininstabilitérésonance triadique
Type
Research Article
Information
Mechanics & Industry , Volume 10 , Issue 3-4: 19e Congrès français de mécanique (CFM 2009) , May 2009 , pp. 187 - 194
Copyright
© AFM, EDP Sciences, 2009

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References

Malkus, W.V.R., An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder, Geophys. Astrophys. Fluid Dynamics 48 (1989) 123–134 CrossRef
McEwan, A., Inertial oscillations in a rotating fluid cylinder, J. Fluid Mech. 40 (1970) 603–640 CrossRef
Manasseh, R., Breakdown regimes of inertia waves in a precessing cylinder, J. Fluid Mech. 243 (1992) 261–296 CrossRef
Manasseh, R., Distortions of inertia waves in a rotating fluid cylinder forced near its fundamental mode resonance, J. Fluid Mech. 265 (1994) 345–370 CrossRef
Manasseh, R., Nonlinear behaviour of contained inertia waves, J. Fluid Mech. 315 (1996) 151–173 CrossRef
Kobine, J., Azimuthal flow assiociated with inertial wave resonance in a precessing cylinder, J. Fluid Mech. 319 (1996) 387–406 CrossRef
Kerswell, R., Secondary instabilities in rapidly rotating fluids: inertial wave breakdown, J. Fluid Mech. 382 (1999) 283–306 CrossRef
Kerswell, R., Elliptical instability, Ann. Rev. Fluid Mech. 34 (2002) 83–113 CrossRef
Eloy, C., Le Gal, P., Le Dizès, S., Elliptic and triangular instabilities in rotating cylinders, J. Fluid Mech. 476 (2003) 357–388 CrossRef
Meunier, P., Leweke, T., Analysis and minimization of errors due to high gradients in particle image velocimetry, Exp. Fluids 35 (2003) 408–421 CrossRef
Meunier, P., Eloy, C., Lagrange, R., Nadal, F., A rotating fluid cylinder subject to weak precession, J. Fluid Mech. 599 (2008) 405–440 CrossRef
H. Greenspan, The theory of rotating fluids, Cambridge University Press, 1968
Gans, R., On the precession of a resonant cylinder, J. Fluid Mech. 41 (1970) 865–872 CrossRef
Lagrange, R., Instability of a fluid inside a precessing cylinder, Physics of Fluids. 20 (2008) 081701 CrossRef
M. Kudlick, On the transient motions in a contained rotating fluid, P.h.D. Thesis, MIT, 1966
Fukumoto, Y., The three-dimensional instability of a strained vortex tube revisited, J. Fluid Mech. 493 (2003) 287318 CrossRef