Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-21T17:24:41.658Z Has data issue: false hasContentIssue false
  • English
  • Français

A model of confined thermal convection driven by non-uniform heating from below

Published online by Cambridge University Press:  19 April 2006

Peter D. Killworth  and
Peter C. Manins
Peter D. Killworth
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England
Peter C. Manins
Affiliation:
Division of Atmospheric Physics, C.S.I.R.O., P.O. Box 77, Mordialloc, Victoria 3195, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

A model is presented of the two-dimensional boundary-layer and interior flow in a rectangular box resulting from the application of a quadratic temperature variation on its lower surface. The other walls are insulating. It is shown that a similarity form exists for the narrow, thermocline-like layer near the lower surface, and that this can satisfy all known consistency conditions with the interior, together with either laminar or turbulent side-wall regions. The interior temperature and Nusselt number are shown to be insensitive to Prandtl number, and to be primarily functions of the horizontal Rayleigh number RaL.

Specifically, the interior temperature, relative to the coldest applied value, is 60% of the total applied temperature range. The Nusselt number is predicted to vary as $0.26 Ra^{\frac{1}{5}}_L$ for a box with unit aspect ratio. The dynamics of the side-wall region, and the details of the imposed temperature variation appear to be unimportant in determining the overall buoyancy exchange. The solutions are compared with numerical and observational results, and generally good agreement is found.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 3 , 12 June 1980 , pp. 587 - 607
Copyright
© 1980 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Beardsley, R. C. & Festa, J. F. 1972 A numerical model of convection driven by a surface stress and non-uniform horizontal heating. J. Phys. Oceanog. 2, 444–455.Google Scholar
Duncan, I. B. 1966 Axisymmetric convection between two rotating disks. J. Fluid Mech. 24, 417–449.Google Scholar
Goody, R. M. & Robinson, A. R. 1966 A discussion of the deep circulation of the atmosphere of Venus. Astrophys. J. 146, 339–355.Google Scholar
Goldstein, S. 1965 Modern Developments in Fluid Dynamics, vol. 1. Dover.
Hignett, P. 1979 Experiments on thermal convection in a rotating fluid annulus, driven by non-uniform heating from below. Ph.D. dissertation, University of Reading.
McKenzie, D. P., Roberts, J. M. & Weiss, N. O. 1973 Convection in the earth's mantle: towards a numerical simulation. J. Fluid Mech. 62, 465–538.Google Scholar
Manins, P. C. 1973 Confined convective flows. Ph.D. dissertation. University of Cambridge.
Manins, P. C. 1979 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 91, 773–800.Google Scholar
Norman, A. C. 1972 A system for the solution of initial and two-point boundary value problems. Proc. Ass. Comp. Mach. (25th Anniv. Conf., Boston), pp. 826–834.Google Scholar
Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 9–16.Google Scholar
Somerville, R. C. J. 1967 A non-linear spectral model of convection in a fluid unevenly heated from below. J. Atmos. Sci. 24, 665–676.Google Scholar
Stern, M. E. 1975 Ocean Circulation Physics. Academic.
Stommel, H. 1950 An example of thermal convection. Trans. Am. Geophys. Union 31, 553–554.Google Scholar
Stommel, H. 1962 On the smallness of sinking regions in the ocean. Proc. Natn. Acad. Sci. 48, 766–772.Google Scholar
Stone, P. H. 1968 Some properties of Hadley regimes on rotating and non-rotating planets. J. Atmos. Sci. 25, 644–657.Google Scholar
Walin, G. 1971 Contained non-homogeneous flow under gravity or how to stratify a fluid in the laboratory. J. Fluid Mech. 48, 647–672.Google Scholar
Wesseling, P. 1969 Laminar convection cells at high Rayleigh number. J. Fluid Mech. 36, 625–637.Google Scholar