Lipps, F. B., 1976: Numerical simulation of three-dimensional Bénard convection in air. Journal of Fluid Mechanics, 75 (Part 1), 113-148.

Abstract: A numerical model is developed to simulate three-dimensional Bénard convection. This model is used to investigate thermal convection in air for Rayleigh numbers between 4000 and 25000. According to experiments, this range of Rayleigh numbers in air covers three regimes of thermal convection: (i) steady two-dimensional convection, (ii) time-periodic convection and (iii) aperiodic convection. Numerical solutions are obtained for each of these regimes and the results are compared with the available experimental data and theoretical predictions.
At the Rayleigh number Ra = 4000 the present model is able to produce experimentally realistic wavelengths for the two-dimensional convection. The small amplitude wave disturbances at Ra = 6500 have period t = 0.24. When they become finite amplutude travelling waves, the period is t = 0.27. These values are in good agreement with theoretical and experimental results. A detailed study of the form of these waves and of their energetics is given in appendix A. As the Rayleigh number is increased to Ra = 9000 and 25000, the convection manifests progressively more complex spatial and temporal variations.
The vertical heat transport and other mean properties of the convection are calculated for the range of Ra considered and compared with experimental and theoretical data. A detailed comparison is also made between the mean properties of two- and three-dimensional convection at the larger values of Ra. It is found that the heat flux Nu is nearly independent of the dimensionality of the convection.