  Motivated by recent experimental studies of Bodenschatz et al. [E.
Bodenschatz, J.R. de Bruyn, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. {\bf
67}, 3078 (1991) ], we present a numerical study of a generalized two
dimensional Swift-Hohenberg equation to model pattern formation in
Rayleigh-B\'enard convection in a non-Boussinesq fluid. It is shown that many
of the features observed in these experiments can be reproduced by this
generalized model that explicitly includes non-Boussinesq and mean flow
effects. The spontaneous formation of hexagons, rolls, and a rotating spiral
pattern is studied, as well as the transitions and competition among them. Mean
flow, non-Boussinesq effects, the geometric shape of the lateral wall, and
sidewall forcing are all shown to be crucial in the formation of the rotating
spirals. We also study nucleation and growth of hexagonal patterns and find
that the front velocity in this two dimensional model is consistent with the
prediction of marginal stability theory for one dimensional fronts.
