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A rotating fluid cylinder subject to weak precession

  • PATRICE MEUNIER (a1), CHRISTOPHE ELOY (a1), ROMAIN LAGRANGE (a1) and FRANÇOIS NADAL (a2)
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.

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Agrawal, B. N. 1993 Dynamics characteristics of liquid motion in partially filled tanks of a spinning spacecraft. J. Guid. Control Dyn. 16, 636640.
Bao, G. W. & Pascal, M. 1997 Stability of a spinning liquid filled spacecraft. Arch. Appl. Mech. 67, 407421.
Eloy, C., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85, 34003403.
Eloy, C., Le Gal, P. & Le Dizès, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357388.
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235274.
Fultz, D. 1959 A note on overstability and elastoid-inertia oscillations of Kelvin, Solberg and Bjerknes. J. Met. 16, 199208.
Gans, R. F. 1970 On the precession of a resonant cylinder. J. Fluid Mech. 476, 865872.
Gans, R. F. 1984 Dynamics of a near-resonant fluid-filled gyroscope. AIAA J. 22, 14651471.
Garg, S. C., Furunoto, N. & Vanyo, J. P. 1986 Spacecraft nutational instability prediction by energy dissipation measurments. J. Guid. Control Dyn. 9, 357361.
Goto, S., Ishii, N., Kida, S. & Nishioka, M. 2007 Turbulence generator using a precessing sphere. Phys. Fluids 19, 061705.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Greenspan, H. P. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36, 257264.
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283306.
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83113.
Kerswell, R. R. & Barenghi, C. F. 1995 On the viscous decay rates of inertial waves in a rotating cylinder. J. Fluid Mech. 285, 203214.
Kobine, J. J. 1995 Inertial wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233252.
Kobine, J. J. 1996 Azimuthal flow associated with inertial wave resonance in a precessing cylinder. J. Fluid Mech. 319, 387406.
Kudlick, M. 1966 On the transient motions in a contained rotating fluid. PhD thesis, Massachussets Institute of Technology.
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111128.
Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external coriolis-force. Phys. Fluids A 5, 891900.
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, 123134.
Manasseh, R. 1992 Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 243, 261296.
Mason, D. M. & Kerswell, R. R. 1999 Nonlinear evolution of the elliptical instability: an example of inertial breakdown. J. Fluid Mech. 396, 73108.
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603640.
Meunier, P. & Leweke, T. 2003 Analysis and minimization of errors due to high gradients in particule image velocimetry. Exps. Fluids 35, 408421.
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.
Poincaré, H. 1910 Sur la précession des corps déformables. Bull. Astron. 27, 257264.
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, 291321.
Sipp, D. 2000 Weakly nonlinear saturation of short-wave instabilities in a strained Lamb-Oseen vortex. Phys. Fluids 12, 17151729.
Stewartson, K. 1958 On the stability of a spinning top containing liquid. J. Fluid Mech. 5, 577592.
Thompson, R. 1970 Diurnal tides and shear instabilities in a rotating cylinder. J. Fluid Mech. 40, 737751.
Vanyo, J. P. 1993 Rotating Fluids in Engineering and Science. Dover.
Waleffe, F. 1989 The 3d instability of a strained vortex and its relation to turbulence. PhD thesis, Massachusetts Institute of Technology.
Wood, W. W. 1965 Properties of inviscid, recirculating flows. J. Fluid Mech. 22, 337346.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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