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On the evolution of thermally driven shallow cavity flows

Published online by Cambridge University Press:  26 April 2006

P. G. Daniels  and
P. Wang
P. G. Daniels
Affiliation:
Department of Mathematics, City University, Northampton Square, London EC1V 0HB, UK
P. Wang
Affiliation:
Department of Mathematics, City University, Northampton Square, London EC1V 0HB, UK
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Abstract

The temporal evolution of thermally driven flow in a shallow laterally heated cavity is studied for the nonlinear regime where the Rayleigh number R based on cavity height is of the same order of magnitude as the aspect ratio L (length/height). The horizontal surfaces of the cavity are assumed to be thermally insulating. For a certain class of initial conditions the evolution is found to occur over two non-dimensional timescales, of order one and of order L2. Analytical solutions for the motion throughout most of the cavity are found for each of these timescales and numerical solutions are obtained for the nonlinear time-dependent motion in end regions near each lateral wall. This provides a complete picture of the evolution of the steady-state flow in the cavity for cases where instability in the form of multicellular convection does not occur. The final steady state evolves on a dimensional timescale proportional to l2/κ, where l is the length of the cavity, κ is the thermal diffusivity of the fluid and the constant of proportionality depends on the ratio R/L.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 259 , 25 January 1994 , pp. 107 - 124
Copyright
© 1994 Cambridge University Press

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