Large-scale flow in a cubic Rayleigh-B ́enard cell: Long-term turbulence statistics and Markovianity of macrostate transitions

We investigate the large-scale circulation (LSC) in a turbulent Rayleigh-B ́enard convection flow in a cubic closed convection cell by means of direct numerical simulations at a Rayleigh number Ra = 106. The numerical studies are conducted for a single flow trajectory up to 105 convective free-fall times to obtain a sufficient sampling of the four discrete LSC states and the two crossover configurations which are taken in between for short periods. It is found that the statistics and time history depends strongly on the Prandtl number Pr of the working fluid which takes values of 0.1, 0.7, and 10. It changes from very rapid switches for the lowest Prandtl number to the spontaneous lock in one of the four states for the whole period for the largest one. Alternatively, we run ensembles of up to 1800 short-term simulations to study the transition probabilities between the discrete LSC states. This second approach is also used to probe the Markov property of the dynamics. The ensemble analysis revealed that the sample size might still be too small to conclude firmly the Markovianity of the transition process from one LSC state to another even though it is indicated. PACS numbers: 47.20.Bp, 47.27-i., 02.50.G