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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: [email protected]
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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

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