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
. 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.
2011. , 29 p.