A numerical study of the Benard cell

When a layer of liquid is heated from below at a rate whichexceeds a certain critical value, a two- or three-dimensional motionis generated. This motion arises from the combined actions of buoyancyand surface tension forces, the latter being due to variationsin the temperature of the liquid surface.The two-dimensional form of the flow has been studied bya numerical method. The equations and boundary conditions describingthe problem have been approximated using finite differences andsolved at the nodes of a rectangular grid superimposed on the domain.Variable time steps have been used to accelerate the approach tosteady state conditions and an analysis has been made of the stabilityconsistency and convergence of the technique.The resulting flow consists of a series of rolls, rotatingalternately clockwise and anticlockwise, and which are shown to besymmetrical about the dividing streamlines. As well as a detaileddescription of the motion and the temperature of the liquid, and ofthe effects on these characteristics of variations in the Rayleigh,Marangoni, Prandtl and Biot numbers, a study has been made of theconditions under which the motion first starts, the wavelength ofthe rolls and the rate of heat transfer through the liquid layer.Two quantities have also been defined which characterise the strengthof convective motion