Modal exchange mechanisms in Lapwood convection

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Abstract
Techniques of 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. Two particular aspects of the problem are focused upon: (i) the existence of multiple steady solutions and (ii) the influence of aspect ratio.Convection begins only when the applied temperature difference (say) exceeds a critical value defined by linear stability theory. The resulting convective flow pattern depends both on the magnitude of the temperature difference and on the aspect ratio of the cavity. A weakly nonlinear analysis reveals the roles played by so-called secondary bifurcations in determining the formation of further, anomalous patterns at fixed aspect ratio. In addition to giving rise to alternative stable flows for identical operating conditions, the secondary bifurcations are required for the modal exchanges which take place as the aspect ratio varies, a process which causes an abrupt change in preferred flow pattern at certain critical values of the aspect ratio.As a complement to and an extension of the weakly nonlinear analysis, numerical methods are used to determine the bifurcation processes and to elucidate the modal exchange mechanisms in both weakly and strongly convective flows. The effect of container size is studied by continuation methods to predict the variation of the critical Rayleigh number of the bifurcation points for aspect ratios in the range 0.5 to 2.0. In this way a stability map is obtained which shows the alternative patterns expected for particular operating conditions. The Nusselt number is computed and it is found that the alternative stable modes transfer significantly different amounts of heat through the medium.The study has provided new information on the existence and characteristics of, and interactions between, alternative steady modes of two-dimensional Lapwood convection. The results have important ramifications for the modelling and design of physical systems in which convective flow in a saturated porous medium is stimulated by an imposed unstable temperature gradient.