Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-26T02:10:44.389Z Has data issue: false hasContentIssue false
  • English
  • Français

On fluid flows in precessing narrow annular channels: asymptotic analysis and numerical simulation

Published online by Cambridge University Press:  20 May 2010

KEKE ZHANG ,
DALI KONG  and
XINHAO LIAO
KEKE ZHANG*
Affiliation:
Department of Mathematical Sciences, University of Exeter, EX4 4QF, UK
DALI KONG
Affiliation:
Department of Mathematical Sciences, University of Exeter, EX4 4QF, UK
XINHAO LIAO
Affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
*
†Email address for correspondence: kzhang@ex.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a viscous, incompressible fluid confined in a narrow annular channel rotating rapidly about its axis of symmetry with angular velocity Ω that itself precesses slowly about an axis fixed in an inertial frame. The precessional problem is characterized by three parameters: the Ekman number E, the Poincaré number ε and the aspect ratio of the channel Γ. Dependent upon the size of Γ, precessionally driven flows can be either resonant or non-resonant with the Poincaré forcing. By assuming that it is the viscous effect, rather than the nonlinear effect, that plays an essential role at exact resonance, two asymptotic expressions for ε ≪ 1 and E ≪ 1 describing the single and double inertial-mode resonance are derived under the non-slip boundary condition. An asymptotic expression describing non-resonant precessing flows is also derived. Further studies based on numerical integrations, including two-dimensional linear analysis and direct three-dimensional nonlinear simulation, show a satisfactory quantitative agreement between the three asymptotic expressions and the fuller numerics for small and moderate Reynolds numbers at an asymptotically small E. The transition from two-dimensional precessing flow to three-dimensional small-scale turbulence for large Reynolds numbers is also investigated.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 116 - 146
Copyright
Copyright © Cambridge University Press 2010

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Bullard, E. C. 1949 The magnetic flux within the Earth. Proc. R. Soc. Lond. A 197, 433–453.Google Scholar
Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 136, 739–751.CrossRefGoogle Scholar
Busse, F. H. 1994 Convection driven zonal flows and vortices in the major planets. CHAOS 4, 123–134.CrossRefGoogle ScholarPubMed
Busse, F. H. 2005 Convection in a narrow annular channel rotating about its axis of symmetry. J. Fluid Mech. 537, 145–154.CrossRefGoogle Scholar
Chorin, A. J. 1968 Numerical solutions of Navier–Stokes equations. Math. Comp. 22, 745–762.CrossRefGoogle Scholar
Davies-Jones, R. P. & Gilman, P. A. 1971 Convection in a rotating annulus uniformly heated from below. J. Fluid Mech. 46, 65–81.CrossRefGoogle Scholar
Eccles, F. J. R., Read, P. L., Castrejn-Pita, A. A. & Haine, T. W. N. 2009 Synchronization of modulated travelling baroclinic waves in a periodically forced, rotating fluid annulus. Phys. Rev. E 79, 015202.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizs, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.CrossRefGoogle Scholar
Gans, R. F. 1970 On the precession of a resonant cylinder. J. Fluid Mech. 476, 865–872.CrossRefGoogle Scholar
Gilman, P. A. 1973 Convection in a rotating annulus uniformly heated from below. Part 2. Nonlinear results. J. Fluid Mech. 57, 381–400.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hollerbach, R. & Kerswell, R. R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327–339.CrossRefGoogle Scholar
Jones, C. A., Rotvig, J. & Abdulrahman, A. 2003 Multiple jests and zonal flow on Jupiter. Geophys. Res. Lett. 30, 1–4.CrossRefGoogle Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72, 107–144.CrossRefGoogle Scholar
Kerswell, R. R. 1996 Upper bounds on the energy dissipation in turbulent precession. J. Fluid Mech. 321, 335–370.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Kobine, J. J. 1995 Inerital wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233–252.CrossRefGoogle Scholar
Lagrange, R., Eloy, C., Nadal, F. & Meunier, P. 2008 Instability of a fluid inside a precessing cylinder. Phys. Fluids 20, 081701CrossRefGoogle Scholar
Li, L., Liao, X., Chan, K. H. & Zhang, K. 2008 Linear and nonlinear instabilities in rotating cylindrical Rayleigh–Bénard convection. Phys. Rev. E 78, 056303-1–056303-12.CrossRefGoogle ScholarPubMed
Liao, X. & Zhang, K. 2009 Inertial oscillation, inertial wave and initial value problem in rotating annular channels. Geophys. Astrophys. Fluid Dyn. 103, 199–222.CrossRefGoogle Scholar
Liao, X., Zhang, K. & Chang, Y. 2006 On boundary-layer convection in a rotating fluid layer. J. Fluid Mech. 549, 375–384.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. 1968 Precession of the Earth as the cause of geomagnetism. Science 136, 259–264.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, R. M. & Kerswell, R. R. 2002 Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer. J. Fluid Mech. 471, 71–106.CrossRefGoogle Scholar
Meunier, P., Eloy, C., Lagrange, R. & Nadal, F. 2008 A rotating fluid cylinder subject to weak precession. J. Fluid Mech. 599, 405–440.CrossRefGoogle Scholar
Noir, J., Cardin, P., Jault, D. & Masson, J. P. 2003 Experimental evidence of nonlinear resonance effects between retrograde precession and the tilt-over mode within a spheroid. Geophys. J. Intl 154, 407–416.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, 283–299.CrossRefGoogle Scholar
Roberts, P. H. & Stewartson, K. 1965 On the motion of a liquid in a spheroidal cavity of a precessing rigid body. II. Math. Proc. Camb. Phil. Soc. 61, 279–288.CrossRefGoogle Scholar
Tilgner, A. 1999 Magnetohydrodynamic flow in precessing spherical shells. J. Fluid Mech. 136, 303–318.CrossRefGoogle Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104–034106.CrossRefGoogle Scholar
Tilgner, A. 2007 Kinematic dynamos with precession driven flow in a sphere. Geophys. Astrophys. Fluid Dyn. 100, 1–9.CrossRefGoogle Scholar
Tilgner, A. & Busse, F. H. 2001 Fluid flows in precessing spherical shells. J. Fluid Mech. 426, 387–396CrossRefGoogle Scholar
Vanyo, J. P. 1993 Rotating Fluids in Engineering and Science. General Publishing.Google Scholar
Vanyo, J. P., Wilde, P., Cardin, P. & Olson, P. 1995 Experiments on precessing flows in the Earth's liquid core. Geophys. J. Intl 121, 136–142.CrossRefGoogle Scholar
Wu, C. C. & Roberts, P. H. 2008 A precesionally-driven dynamo in a plane layer. Geophys. Astrophys. Fluid Dyn. 102, 1–19.CrossRefGoogle Scholar
Wu, C. C. & Roberts, P. H. 2009 On a dynamo driven by topographic precession. Geophys. Astrophys. Fluid Dyn. 103, 467–501.CrossRefGoogle Scholar
Zhang, K., Earnshaw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103–119.CrossRefGoogle Scholar
Zhang, K. & Liao, X. 2004 A new asymptotic method for the analysis of convection in a rotating sphere. J. Fluid Mech. 518, 319–346.CrossRefGoogle Scholar
Zhang, K. & Liao, X. 2008 On the initial value problem in a rotating circular cylinder. J. Fluid Mech. 610, 425–443.CrossRefGoogle Scholar