Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T09:00:20.514Z Has data issue: false hasContentIssue false
  • English
  • Français

A rotating fluid cylinder subject to weak precession

Published online by Cambridge University Press:  06 March 2008

PATRICE MEUNIER ,
CHRISTOPHE ELOY ,
ROMAIN LAGRANGE  and
FRANÇOIS NADAL
PATRICE MEUNIER
Affiliation:
IRPHE, CNRS, Universités Aix Marseille I & II, 49 rue Joliot–Curie, 13013 Marseille, France
CHRISTOPHE ELOY
Affiliation:
IRPHE, CNRS, Universités Aix Marseille I & II, 49 rue Joliot–Curie, 13013 Marseille, France
ROMAIN LAGRANGE
Affiliation:
IRPHE, CNRS, Universités Aix Marseille I & II, 49 rue Joliot–Curie, 13013 Marseille, France
FRANÇOIS NADAL
Affiliation:
Commissariat à l'Energie Atomique, CESTA, 33114 le Barp, France
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we report experimental and theoretical results on the flow inside a precessing and rotating cylinder. Particle image velocimetry measurements have revealed the instantaneous structure of the flow and confirmed that it is the sum of forced inertial (Kelvin) modes, as predicted by the classical linear inviscid theory. But this theory predicts also that the amplitude of a mode diverges when its natural frequency equals the precession frequency. A viscous and weakly nonlinear theory has therefore been developed at the resonance. This theory has been compared to experimental results and shows a good quantitative agreement. For low Reynolds numbers, the mode amplitude scales as the square root of the Reynolds number owing to the presence of Ekman layers on the cylinder walls. When the Reynolds number is increased, the amplitude saturates at a value which scales as the precession angle to the power one-third for a given resonance. The nonlinear theory also predicts the forcing of a geostrophic (axisymmetric) mode which has been observed and measured in the experiments. These results allow the flow inside a precessing cylinder to be fully characterized in all regimes as long as there is no instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 405 - 440
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

Agrawal, B. N. 1993 Dynamics characteristics of liquid motion in partially filled tanks of a spinning spacecraft. J. Guid. Control Dyn. 16, 636–640.CrossRefGoogle Scholar
Bao, G. W. & Pascal, M. 1997 Stability of a spinning liquid filled spacecraft. Arch. Appl. Mech. 67, 407–421.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85, 3400–3403.CrossRefGoogle ScholarPubMed
Eloy, C., Le Gal, P. & Le Dizès, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.CrossRefGoogle Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235–274.CrossRefGoogle Scholar
Fultz, D. 1959 A note on overstability and 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. 476, 865–872.CrossRefGoogle Scholar
Gans, R. F. 1984 Dynamics of a near-resonant fluid-filled gyroscope. AIAA J. 22, 1465–1471.CrossRefGoogle Scholar
Garg, S. C., Furunoto, N. & Vanyo, J. P. 1986 Spacecraft nutational instability prediction by energy dissipation measurments. J. Guid. Control Dyn. 9, 357–361.CrossRefGoogle Scholar
Goto, S., Ishii, N., Kida, S. & Nishioka, M. 2007 Turbulence generator using a precessing sphere. Phys. Fluids 19, 061705.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36, 257–264.CrossRefGoogle Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google 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, 83–113.CrossRefGoogle Scholar
Kerswell, R. R. & Barenghi, C. F. 1995 On the viscous decay rates of inertial waves in a rotating 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
Kobine, J. J. 1996 Azimuthal flow associated with inertial wave resonance in a precessing cylinder. J. Fluid Mech. 319, 387–406.CrossRefGoogle Scholar
Kudlick, M. 1966 On the transient motions in a contained rotating fluid. PhD thesis, Massachussets Institute of Technology.Google Scholar
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128.CrossRefGoogle Scholar
Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external coriolis-force. Phys. Fluids A 5, 891–900.CrossRefGoogle Scholar
Malkus, W. V. R. 1989 An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.CrossRefGoogle 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 breakdown. J. Fluid Mech. 396, 73–108.CrossRefGoogle Scholar
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2003 Analysis and minimization of errors due to high gradients in particule image velocimetry. Exps. Fluids 35, 408–421.CrossRefGoogle Scholar
Noir, J., Jault, D. & Cardin, P. 2001 Numerical study of the motions within a slowly precessing sphere at low Ekman number. J. Fluid Mech. 437, 283–29.CrossRefGoogle Scholar
Poincaré, H. 1910 Sur la précession des corps déformables. Bull. Astron. 27, 257–264.CrossRefGoogle Scholar
Racz, J.-P. & Scott, J. F. 2007 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
Sipp, D. 2000 Weakly nonlinear saturation of short-wave instabilities in a strained Lamb-Oseen vortex. Phys. Fluids 12, 1715–1729.CrossRefGoogle Scholar
Stewartson, K. 1958 On the stability of a spinning top containing liquid. J. Fluid Mech. 5, 577–592.CrossRefGoogle Scholar
Thompson, R. 1970 Diurnal tides and shear instabilities in a rotating cylinder. J. Fluid Mech. 40, 737–751.CrossRefGoogle Scholar
Vanyo, J. P. 1993 Rotating Fluids in Engineering and Science. Dover.Google Scholar
Waleffe, F. 1989 The 3d instability of a strained vortex and its relation to turbulence. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Wood, W. W. 1965 Properties of inviscid, recirculating flows. J. Fluid Mech. 22, 337–346.CrossRefGoogle Scholar