Skip to main content Accessibility help
×
Home

Non-resonant viscous theory for the stability of a fluid-filled gyroscope

  • JEAN-PIERRE LAMBELIN (a1), FRANÇOIS NADAL (a1), ROMAIN LAGRANGE (a2) and ARTHUR SARTHOU (a3)

Abstract

In the case of a gyroscope including a cylindrical fluid-filled cavity, the classic Poinsot's coning motion can become unstable. For certain values of the solid inertia ratio, the coning angle opens under the effect of the hydrodynamic torque. The coupled dynamics of such a non-solid system is ruled by four dimensionless numbers: the small viscous parameter ε = Re−1/2 (where Re denotes the Reynolds number), the fluid–solid inertia ratio κ which quantifies the proportion of liquid relative to the total mass of the gyroscope, the solid inertia ratio σ and the aspect ratio h of the cylindrical cavity. The calculation of the hydrodynamic torque on the solid part of the gyroscope requires the preliminary evaluation of the possibly resonant flow inside the cavity. The hydrodynamic scaling used to derive such a flow essentially depends on the relative values of κ and ε. For small values of the ratio /ε (compared to 1), Gans derived an expression of the growth rate of the coning angle. The principles of Gans' approach (Gans, AIAA J., vol. 22, 1984, pp. 1465–1471) are briefly recalled but the details of the whole calculation are not given. At the opposite limit, that is for large values of /ε, the dominating flow is given by a linear inviscid theory. In order to take account of viscous effects, we propose a direct method involving an exhaustive calculation of the flow at order ε. We show that the deviations from Stewartson's inviscid theory (Stewartson, J. Fluid Mech., vol. 5, 1958, p. 577) do not originate from the viscous shear at the walls but rather from the bulk pressure at order ε related to the Ekman suction. Physical contents of Wedemeyer's heuristic theory (Wedemeyer, BRL Report N 1325, 1966) are analysed in the view of our analytical results. The latter are tested numerically in a large range of parameters. Complete Navier–Stokes (NS) equations are solved in the cavity. The hydrodynamic torque obtained by numerical integration of the stress is used as a forcing term in the coupled fluid–solid equations. Numerical results and analytical predictions show a fairly good quantitative agreement.

Copyright

Corresponding author

Email address for correspondence: Francois.Nadal@cea.fr

References

Hide All
D'Amico, W. P. 1977 Inertial mode corrections for the large amplitude motion of a liquid-filled gyroscope. PhD Thesis, University of Delaware, Newark.
D'Amico, W. P. & Rogers, T. H. 1981 Yaw instabilities produced by rapidly rotating, highly viscous liquids. AIAA paper 81-0224.
Fortin, M. & Glowinski, R. 1982 Méthodes de Lagrangien augmenté. Application à la résolution numérique de problèmes aux limites. Paris, Dunod.
Fultz, D. 1959 A note on overstability and elastoid-inertia oscillations of Kelvin, Solberg and Bjerknes. J. Meteorol. 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 measurements. J. Guid. 9 (3), 357361.
Greenhill, A. G. 1880 On the general motion of a liquid ellipsoid. Proc. Camb. Phil. Soc. 4 (4).
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Harlow, F. H & Welsh, J. E. 1965 Numerical calculation of time dependant viscous incompressible flows. Phys. Fluids 8, 21822189.
Karpov, B. G. 1962 Experimental observations of the dynamic behaviour of liquid-filled shell. Tech. Rep. BRL Report 1171. Aberdeen Proving Ground, MD.
Karpov, B. G. 1965 The effect of Reynolds number on resonance. Tech. Rep. BRL Report 1302. Aberdeen Proving Ground, MD.
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283306.
Kerswell, R. R. & Barenghi, C. F. 1995 On the viscous decay rates of inertial waves in a rotating circular cylinder J. Fluid Mech. 285, 203214.
Kobine, J. J. 1995 Inertial wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233352.
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, Cambridge, MA.
Lagrange, R., Meunier, P., Eloy, C. & Nadal, F. 2008 Instability of a fluid inside a precessing cylinder. Phys. Fluids 20, 081701.
Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external Coriolis force. Phys. Fluids A 5 (4), 891900.
Manasseh, R. 1992 Breakdown regimes of inertia waves in precessing cylinder. J. Fluid Mech. 243, 261296.
Manasseh, R. 1994 Distortions of inertia waves in a rotating fluid cylinder forced near its fundamental mode resonance. J. Fluid Mech. 265, 345370.
Manasseh, R. 1996 Nonlinear behaviour of contained inertia waves. J. Fluid Mech. 315, 151173.
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603640.
Meunier, P., Eloy, C., Lagrange, R. & Nadal, F. 2008 Rotating fluid cylinder subject to weak precession. J. Fluid Mech. 599, 405440.
Milne, E. A. 1940 On the stability of a liquid-filled shell (U). EBD report No. 6.
Murphy, C. H. 1982 Angular motion of a spinning projectile with a viscous payload. Tech. Rep. ARBRL-TR-02422. US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD.
Scott, W. E. & D'Amico, W. P. 1973 Amplitude-dependent behavior of a liquid-filled gyroscope. J. Fluid Mech. 60, 751758.
Stewartson, K. 1958 On the stability of a spinning top containing liquid. J. Fluid Mech. 5, 577592.
Thomson, W. Sir 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.
Thomson, 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.
Vincent, S. & Caltagirone, J.-P. 2000 A one-cell local multigrid method for solving unsteady incompressible multiphase flow. J. Comput. Phys. 163, 172215.
van der Vorst, H. A. 1992 Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear system. SIAM J. Sci. Stat. Comput. 13, 631644.
Waleffe, F. 1989 The three-dimensional instability of a strained vortex and its relation to turbulence. PhD thesis, Massachussets Institute of Technology, Cambridge, MA.
Ward, G. N. 1959 Appendix to K. Stewartson “On the stability of a spinning top containing liquid”. J. Fluid Mech. 5, 577592.
Watson, G. N. 1952 Theory of Bessel Functions. Cambridge University Press.
Wedemeyer, E. H. 1965 Dynamics of a liquid-filled shell: theory of viscous corrections to Stewartson's stability problem. BRL Report N 1287. US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD.
Wedemeyer, E. H. 1966 Viscous corrections to Stewartson's stability criterion. BRL Report N 1325. US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD.
Whiting, R. D. & Gerber, N. 1981 Dynamics of a liquid-filled gyroscope: update of theory and experiment. Tech. Rep. ARBRL-TR-02221. US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Non-resonant viscous theory for the stability of a fluid-filled gyroscope

  • JEAN-PIERRE LAMBELIN (a1), FRANÇOIS NADAL (a1), ROMAIN LAGRANGE (a2) and ARTHUR SARTHOU (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed