Rayleigh-Bénard convection at high Rayleigh number and infinite Prandtl number: asymptotics and numerics

The problem of fast viscous steady Rayleigh-Bénard convection in a rectangular enclosure is revisited using asymptotic and numerical methods. There are two generic cases: in the first, there is zero shear stress at all boundaries; in the second, there is zero shear stress at the vertical boundaries, but no slip at the horizontal ones. For the first case, we reconcile our new numerical solutions to the full equations with earlier asymptotic results for large Rayleigh number and effectively infinite Prandtl number. For the second case, we first derive the corresponding asymptotic theory and then reconcile it also with the relevant full numerical solutions. However, the latter also indicate behavior which the asymptotic theory does not predict, for Rayleigh numbers in excess of just over 106 and aspect ratios in excess of around 1.1. C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4829450