Time-Dependent Instabilities of Rayleigh-Bénard Convection in Boxes

Abstract

The sequence of instabilities in Rayleigh-Benard convection is studied for the case of rectangular boxes with low aspect ratio (length:width:height) which are heated from below and cooled from above. The different steady, transient and time-dependent solutions are calculated by means of an explicit finite difference method for selected values of Rayleigh number, Prandtl number and aspect ratio. The numerical results are compared with experimental observations using optical measuring techniques like interferometry and streakline-photography. Physical Model, Basic Equations, Experimental MethodsRayleigh-Benard convection is a suitable object to study the laminar-turbulent transition in low order systems. Because turbulence in its essential properties is threedimensional, those problems should be selected, which show a high three-dimensionality also in the primary steady flow field. By this reason we have investigated the sequence of instabilities in thermal convection flows using closed rectangular boxes of low aspect ratio -especially cubical boxes -which are heated from below and cooled from above. This problem is characterized by the following dimensionless parameters: the Rayleigh number, the Prandtl number and the two aspect ratios Hz and H II , as defined below:Two types of geometries are selected: flat boxes of low aspect ratio, where flow regimes of cellular convection can be observed and cubical boxes with pronounced threedimensional flow pattern. All sidewalls are perfectly conducting.The Boussinesq-approximation is applied which leads to the following set of basic equations for the conservation of mass, momentum and energy: