Rayleigh-Bénard convection

Sammanfattning: This report considers Rayleigh-Bénard convection, i.e. the 　ow between two large parallel plates where the lower one is heated. The change in density due to temperature variations gives rise to a 　ow generated by buoyancy. This motion is opposed by the viscous forces in the 　uid. The balance between these forces determines whether the 　ow is stable or not and the goal of this report is to nd a condition giving this limit as well as analyzing other aspects of the 　ow. The starting point of the analysis is the incompressible Navier- Stokes equations and the thermal energy equation upon which the Boussinesq approximation is applied. Using linear stability analysis a condition for the stability is obtained depending solely on a nondimensional parameter, called the Rayleigh number, for a given wavenumber k . This result is conrmed to be accurate after comparison with numerical simulations using a spectral technique. Further non-linear two- and three-dimensional simulations are also performed to analyze dierent aspects of the 　ow for various values of the Rayleigh number.