Comparison between two and three dimensional Rayleigh-B\'enard convection

Abstract

Two dimensional (2D) and three dimensional (3D) Rayleigh-B\'enard convection is compared using results from direct numerical simulations and prior experiments. The explored phase diagrams for both cases are reviewed. The differences and similarities between 2D and 3D are studied using Nu(Ra) for Pr = 4.38 and Pr = 0.7 and Nu(Pr) for Ra up to $10^8$. In the Nu(Ra) scaling at higher Pr, 2D and 3D are very similar; differing only by a constant factor up to $\text{Ra} = 10^{10}$. In contrast, the difference is large at lower Pr, due to the strong roll state dependence of Nu in 2D. The behaviour of Nu(Pr) is similar in 2D and 3D at large Pr. However, it differs significantly around Pr = 1. The Reynolds number values are consistently higher in 2D and additionally converge at large Pr. Finally, the thermal boundary layer profiles are compared in 2D and 3D.