Abstract
The Boussinesq equations are reduced to a two-dimensional form governing convection between two free surfaces maintained at a constant temperature difference. These equations are then transformed to a set of ordinary differential equations governing the time variations of the double-Fourier coefficients for the motion and temperature fields. The system is then highly truncated by taking into account only a very limited number of terms of the Fourier expansions. Numerical integrations of the system have been performed at supercritical conditions, initiating the motion by the introduction of various sets of initial conditions. In all cases investigated, the system achieves a steady state and only one mode is present at the steady state. The results show however that slightly different initial conditions give rise the motion of different modes for the same Rayleigh number. Once established, the motion of that selected mode is found to be very stable against the superposition of perturbations of other modes.
