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Very viscous horizontal convection

Published online by Cambridge University Press:  25 September 2008

S. CHIU-WEBSTER ,
E. J. HINCH  and
J. R. LISTER
S. CHIU-WEBSTER
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
E. J. HINCH
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
J. R. LISTER
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
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Abstract

‘Horizontal convection’ arises when a temperature variation is imposed along a horizontal boundary of a finite fluid volume. Here we study the infinite-Prandtl-number limit relevant to very viscous fluids, motivated by the study of convection in glass furnaces. We consider a rectangular domain with insulating conditions on the sides and bottom, and a linear temperature gradient on the top. We describe steady states for a large range of aspect ratio A and Rayleigh number Ra, and find universal scalings for the transition from small to large Rayleigh numbers. At large Rayleigh number, the top boundary-layer thickness scales as Ra−1/5, with the circulation and heat flux scaling as Ra1/5. These scalings hold for both rigid and shear-free boundary conditions on the top or on the other boundaries, which is initially surprising, but is because the return flow is dominated by a horizontal intrusion immediately beneath the top boundary layer. A downwelling plume also forms on one side, but because of strong stratification in the interior, the volume flux it carries is much smaller than that of the horizontal intrusion, decaying as the inverse of the depth below the top boundary. The fluid in the plume detrains into the interior and then returns to the top boundary, thus forming a ‘filling box’. We find analytic solutions for the interior temperature and streamfunction and match them to a similarity solution for the plume. At depths comparable to the length of the top boundary the streamfunction has O(1) values and the temperature variations scale as 1/Ra. Transient calculations with a large, but finite, Prandtl number, show how the steady state is reached from hot and cold initial conditions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 611 , 25 September 2008 , pp. 395 - 426
Copyright
Copyright © Cambridge University Press 2008

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