Two-dimensional Rayleigh-Benard convection

Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R, for linear instability. This range is extended to 1000 R c. Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number N 2 (R/Rc) ^1{3}, independently of p. The heat flux is a maximum for cells whose width is between 1·2 and 1·4 times the layer depth. This regime is found when 5 R/Rc p^3{2}. At higher Rayleigh numbers advection of vorticity becomes important and N ∞ R 0·365. When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p < 1 it is also found that N ∞ R 0·365, with a constant of proportionality equal to 1·90 when p Lt 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.