Comparison between two- and three-dimensional Rayleigh–Bénard convection

Two-dimensional and three-dimensional Rayleigh–Bénard convection is compared using results from direct numerical simulations and previous experiments. The phase diagrams for both cases are reviewed. The differences and similarities between two- and three-dimensional convection are studied using $Nu(Ra)$ for $\mathit{Pr}= 4. 38$ and $\mathit{Pr}= 0. 7$ and $Nu(Pr)$ for $Ra$ up to $1{0}^{8} $ . In the $Nu(Ra)$ scaling at higher $Pr$ , two- and three-dimensional convection is very similar, differing only by a constant factor up to $\mathit{Ra}= 1{0}^{10} $ . In contrast, the difference is large at lower $Pr$ , due to the strong roll state dependence of $Nu$ in two dimensions. The behaviour of $Nu(Pr)$ is similar in two and three dimensions at large $Pr$ . However, it differs significantly around $\mathit{Pr}= 1$ . The Reynolds number values are consistently higher in two dimensions and additionally converge at large $Pr$ . Finally, the thermal boundary layer profiles are compared in two and three dimensions.