On the validity of two-dimensional numerical approaches to time-dependent thermal convection

High-resolution computer simulations of two-dimensional (2D) convection are often used to investigate turbulent flows. In this paper we compare numerical simulations in 2D with three-dimensional (3D) simulations. We investigate flows at a fixed Rayleigh number of Ra = 106. The velocity boundary conditions are rigid for the upper and lower boundary and stress-free for the side walls. For high values of the Prandtl number the flow structure and global quantities such as the Nusselt number (Nu) and the Reynolds number (Re) show similar behaviour in 3D and 2D simulations. For values of the Prandtl number smaller than unity, however, the 2D results diverge strongly form the 3D findings. This is true not only for global properties (Nu, Re) but also for local properties such as the structure of the boundary layer and the shapes of the up- and down-wellings. We relate this behaviour to the different large-scale structure of the flow and the toroidal component of the kinetic energy.