Reynolds-number scaling and convergence time scale in two-dimensional Rayleigh-Bénard convection

Abstract

An equation for the evolution of mean kinetic energy, in a two-dimensional (2-D) or 3-D Rayleigh-Bénard system with domain height is derived. Assuming classical Nusselt-number scaling, and that mean enstrophy, in the absence of a downscale energy cascade, scales as, we find that the Reynolds number scales as in the 2-D system, where is the Rayleigh number and the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that, where is the non-dimensional convergence time scale. For the 3-D system, we make the estimate for. It is estimated that the total computational cost of reaching the high limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.