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On the boundary-layer structure of high-Prandtl-number horizontal convection

Published online by Cambridge University Press:  19 May 2010

P. G. DANIELS
P. G. DANIELS*
Affiliation:
Centre for Mathematical Science, City University, Northampton Square, London EC1V 0HB, UK
*
†Email address for correspondence: p.g.daniels@city.ac.uk
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Abstract

This paper describes the boundary-layer structure of the steady flow of an infinite Prandtl number fluid in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated and the upper surface is assumed to be either shear-free or rigid. In the limit of large Rayleigh number (R → ∞), the solution involves a horizontal boundary layer at the upper surface of depth of order R−1/5 where the main variation in the temperature field occurs. For a monotonic temperature distribution at the upper surface, fluid is driven to the colder end of the cavity where it descends within a narrow convection-dominated vertical layer before returning to the horizontal layer. A numerical solution of the horizontal boundary-layer problem is found for the case of a linear temperature distribution at the upper surface. At greater depths, of order R−2/15 for a shear-free surface and order R−9/65 for a rigid upper surface, a descending plume near the cold sidewall, together with a vertically stratified interior flow, allow the temperature to attain an approximately constant value throughout the remainder of the cavity. For a shear-free upper surface, this constant temperature is predicted to be of order R−1/15 higher than the minimum temperature of the upper surface, whereas for a rigid upper surface it is predicted to be of order R−2/65 higher.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 299 - 331
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Batchelor, G. K. 1954 Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12, 209–233.CrossRefGoogle Scholar
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.CrossRefGoogle Scholar
Chiu-Webster, S., Hinch, E. J. & Lister, J. R. 2008 Very viscous horizontal convection. J. Fluid Mech. 611, 395–426.Google Scholar
Daniels, P. G. & Punpocha, M. 2005 On the boundary-layer structure of cavity flow in a porous medium driven by differential heating. J. Fluid Mech. 532, 321–344.Google Scholar
Gramberg, H. J. J., Howell, P. D. & Ockenden, J. R. 2007 Convection by a horizontal thermal gradient. J. Fluid Mech. 586, 41–57.Google Scholar
Hughes, G. O., Griffiths, R. W., Mullarney, J. C. & Peterson, W. H. 2007 A theoretical model for horizontal convection at high Rayleigh number. J. Fluid Mech. 581, 251–276.Google Scholar
Killworth, P. D. & Manins, P. C. 1980 A model of confined thermal convection driven by non-uniform heating from below. J. Fluid Mech. 98, 587–607.Google Scholar
Krause, D. & Loch, H. (Eds) 2002 Mathematical Simulation in Glass Technology. Springer.Google Scholar
Mullarney, J. C., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181–209.Google Scholar
Rossby, T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep Sea Res. 12, 9–16.Google Scholar
Rossby, T. 1998 Numerical experiments with a fluid heated non-uniformly from below. Tellus 50A, 242–257.CrossRefGoogle Scholar
Siggers, J. H., Kerswell, R. R. & Balmforth, N. J. 2004 Bounds on horizontal convection. J. Fluid Mech. 517, 55–70.CrossRefGoogle Scholar
Stern, M. E. 1975 Ocean Circulation Physics. Academic Press.Google Scholar
Stommel, H. 1962 On the smallness of the sinking regions in the ocean. Proc. Natl Acad. Sci. 48, 766–772.Google Scholar
Wang, W. & Huang, R. X. 2005 An experimental study on thermal circulation driven by horizontal differential heating. J. Fluid Mech. 540, 49–73.CrossRefGoogle Scholar