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A numerical bifurcation study of natural convection in a tilted two-dimensional porous cavity

Published online by Cambridge University Press:  26 April 2006

D. S. Riley  and
K. H. Winters
D. S. Riley
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
K. H. Winters
Affiliation:
Theoretical Physics Division, Harwell Laboratory, Didcot, Oxon OX11 0RA, UK
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Abstract

Techniques derived from bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh–Bénard problem, Lapwood convection in a two-dimensional saturated porous cavity heated from below. The objective of the study is the explanation of how the multiplicity of solutions observed for lower boundary heating evolves to an apparently unique solution for sidewall heating. The change in boundary conditions from floor to sidewall heating can be effected by smoothly tilting the cavity through 90°. The present study aims to demonstrate the mechanisms that reduce the multiplicity for increasing tilt angle.

The many solutions in the untilted cavity arise from a complex bifurcation structure. The effect of tilting the cavity is to unfold all bifurcations, except those that break the centro-symmetry, and so to create branches disconnected from the primary flow. As the angle of tilt, ϕ, increases most of the limit points at which these branches arise move to higher Rayleigh number Ra. Unexpectedly, for a square cavity, the critical Rayleigh number of the most important limit point (that gives rise to an anomalous stable unicellular flow) is found to be almost independent of the angle of tilt. Moreover the two branches arising at this limit point merge again at higher Rayleigh number to form a continuous closed loop, or isola. As the tilt increases, the upper limit point approaches the lower one until they coalesce at an isola formation point at a critical angle ϕc of 10.72°, the maximum angle at which this anomalous mode can exist. Symmetry-breaking bifurcations destabilize part of the branch and determine a smaller critical angle of 10.23°, the maximum angle for which the anomalous mode is stable. At very small angles of tilt, the path of limit points forms the expected cusp catastrophe in the (ϕ, Ra)-plane and at larger angles the path itself turns back at the isola formation point.

The results reveal as too simplistic the conjecture that the reduction of multiplicity for increasing tilt derives from the movement of disconnected branches to increasingly higher Rayleigh number. The predicted collapse and disappearance of branches at an isola formation point is a further novel mechanism which ensures that only the unicellular primary branch remains at a tilt of 90°, in accord with the expected uniqueness of the flow in a square cavity with sidewall heating.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 215 , June 1990 , pp. 309 - 329
Copyright
© 1990 Cambridge University Press

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