Non-Boussinesq simulations of Rayleigh–Bénard convection in a perfect gas

We present direct numerical simulations of Boussinesq and non-Boussinesq Rayleigh–Bénard convection in a rigid box containing a perfect gas. For small stratifications, which includes Boussinesq fluids, the first instability after steady rolls was an oscillatory instability ~a Hopf bifurcation!. The resulting convection was characterized by two hot and two cold blobs circulating each convective roll. The same sign thermal perturbations ~blobs! are at diametrically opposite points on the circular rolls, i.e., they are symmetric about the roll center. The time for a hot ~or cold! blob to circulate a roll was between two and three roll turnover times. When the stratification was of sufficient strength, there was a dramatic change in the nature of the bifurcation. The sign of the thermal perturbations became antisymmetric with respect to the roll center, i.e., a hot blob was diametrically opposite a cold blob. In this case, a hot or cold blob circulated around each roll in about one turnover time. In a stratified layer, the Rayleigh number varies with height. We found that at the Hopf bifurcation, the Rayleigh number at the base was closest to the Boussinesq value. The change in instability appeared to be related to an increase in the speed ~or Mach number! of the circulating rolls. It did not seem to be affected by the transport property variation with temperature. If the along roll aspect ratio was less than 2 or the walls perpendicular to the roll axis periodic, then only the symmetric instability could be found. We describe how our results might be reproduced in a laboratory experiment of convection in cryogenic helium gas. © 2004 American Institute of Physics. @DOI: 10.1063/1.1689350#