Free convection in fluid layers with two rigid sidewalls

The influence of sidewalls on free convection behaviour of a fluid layer heated from below is investigated. This layer of infinite extent in x-direction is limited by two vertical and two horizontal sidewalls. The problem is characterized by the height to width ratio A. Because of the sidewalls the convection depends on all three spatial coordinates. However for a large variety of aspect ratios A and Rayleigh numbers, only two velocity components are of importance. To describe the temperature and the velocity fields the Boussinesq approximation is applied on the Navier-Stokes equations and on the energy equation. The resulting system of differential equations is solved by means of a Galerkin method. For a wide range of the aspect ratio A between A = 0 (Benard convection) to A = 100 (Hele-Shaw convection) solutions are presented ranging from the onset of free convection up to Rayleigh numbers being about 30 times above the critical linear stability value. The influence of sidewalls on the critical Rayleigh number and wave number and on the heat transport is investigated for several combinations of boundary conditions (horizontal boundaries: isotherm and rigid or free; vertical boundaries: rigid and insulated or perfectly conducting). Transient convection is discussed in the case of Hele-Shaw convection. Where comparison with experimental results is possible, good agreement is found. (orig.)