Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T07:16:13.375Z Has data issue: false hasContentIssue false
  • English
  • Français

The axisymmetric convective regime for a rigidly bounded rotating annulus

Published online by Cambridge University Press:  28 March 2006

Michael E. Mcintyre
Michael E. Mcintyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The axisymmetric flow of liquid in a rigidly bounded annular container of height H, rotating with angular velocity Ω and subjected to a temperature difference ΔT between its vertical cylindrical perfectly conducting side walls, whose distance apart is L, is analysed in the boundary-layer approximation for small Ekman number v/2ΩL2, with gαΔTHv/4Ω2L2K ∼ 1. The heat transfer across the annulus is then convection-dominated, as is characteristic of the experimentally observed ‘upper symmetric regime’. The Prandtl number v/k is assumed large, and H is restricted to be less than about 2L. The side wall boundary-layer equations are the same as in (non-rotating) convection in a rectangular cavity. The horizontal boundary layers are Ekman layers and the four boundary layers, together with certain spatial averages in the interior, are determined independently of the interior flow details. The determination of the latter comprises a ‘secondary’ problem in which viscosity and heat conduction are important throughout the interior; the meridional streamlines are not necessarily parallel to the isotherms. The secondary problem is discussed qualitatively but not solved. The theory agrees fairly well with an available numerical experiment in the upper symmetric regime, for v/k [bumpe ] 7, after finite-Ekmannumber effects such as finite boundary-layer thickness are allowed for heuris-tically.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 32 , Issue 4 , 18 June 1968 , pp. 625 - 655
Copyright
© 1968 Cambridge University Press

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

Barcilon, V. 1964 Role of the Ekman layers in the stability of the symmetric regime obtained in a rotating annulus J. Atmos. Sci. 21, 291.Google Scholar
Barcilon, V. & Pedlosky, J. 1967 Linear theory of rotating stratified fluid motions J. Fluid Mech. 29, 1.Google Scholar
Bowden, M. & Eden, H. F. 1965 Thermal convection in a rotating fluid annulus: temperature, heat flow and flow field observations in the upper symmetric regime. J. Atmos. Sci. 22, 185.Google Scholar
Courant, R. & Hilbert, D. 1962 Methods of Mathematical Physics, vol. II. New York: Interscience.
Davies, T. V. 1953 The forced flow of a rotating viscous liquid which is heated from below. Phil. Trans. A 246, 81.Google Scholar
Eckert, E. R. G. & Carlson, W. O. 1961 Natural convection in an air layer enclosed between two vertical plates with different temperatures Int. J. Heat Mass Trans. 2, 106.Google Scholar
Elder, J. W. 1966 Numerical experiments with free convection in a vertical slot J. Fluid Mech. 24, 823.Google Scholar
Fowlis, W. W. & Hide, R. 1965 Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J. Atmos. Sci. 22, 541.Google Scholar
Fultz, D., Long, R. R., Owens, G. V., Bohan, W., Kaylor, R. & Weil, J. 1959 Studies of thermal convection in a rotating cylinder with some implications for large-scale atmospheric motions Meteorol. Monographs, 4, no. 21.Google Scholar
Gill, A. E. 1966 The boundary-layer regime for convection in a rectangular cavity J. Fluid Mech. 26, 515.Google Scholar
Goldish, L. H., Koutsky, J. A. & Adler, R. J. 1965 Tracer introduction by flash photolysis Chem. Eng. Sci. 20, 1011.Google Scholar
Hide, R. 1958 An experimental study of thermal convection in a rotating liquid. Phil. Trans. A 250, 441.Google Scholar
Hide, R. 1967 Theory of axisymmetric thermal convection in a rotating fluid annulus Phys. Fluids, 10, 56.Google Scholar
Hunter, C. 1967 The axisymmetric flow in a rotating annulus due to a horizontally applied temperature gradient J. Fluid Mech. 27, 753.Google Scholar
Kuo, H.-L. 1954 Symmetrical disturbances in a thin layer of fluid subject to a horizontal temperature gradient and rotation J. Meteorol. 11, 399. (Also in Saltzman, B. 1962 Theory of Thermal Convection. New York: Dover.)Google Scholar
Lambert, R. B. & Snyder, H. A. 1966 Experiments on the effect of horizontal shear and change of aspect ratio on convective flow in a rotating annulus J. Geophys. Res. 71, 5225.Google Scholar
Lorenz, E. N. 1953 A proposed explanation for the existence of two regimes of flow in a rotating symmetrically-heated cylindrical vessel. Proc. First Symp. on the Use of Models in Geophys. Fluid Dyn., p. 73 (ed. R. R. Long).
Mcintyre, M. E. 1967 Convection and baroclinic instability in rotating fluids. Ph.D. Thesis, University of Cambridge.
Ooyama, K. 1966 On the stability of the baroclinic circular vortex: a sufficient criterion for instability. J. Atmos. Sci. 23, 43.Google Scholar
Piacsek, S. 1966 Thermal convection in a rotating annulus of liquid: numerical studies of the axisymmetric regime of flow. Ph.D. Thesis, Mass. Inst. of Technology.
Quon, C. 1967 Numerical study of thermal convection in a rotating fluid system. Ph.D. Thesis, University of Cambridge.
Riehl, H. & Fultz, D. 1957 Jet stream and long waves in a steady rotating-dishpan experiment: structure of the circulation. Quart. J. Roy. Meteorol. Soc. 83, 215.Google Scholar
Robinson, A. R. 1959 The symmetric state of a rotating fluid differentially heated in the horizontal J. Fluid Mech. 6, 599.Google Scholar
Williams, G. P. 1967 Thermal convection in a rotating fluid annulus, Parts I and II. J. Atmos. Sci. 24, 144, 162.Google Scholar