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

Abstract

An equation for the evolution of mean kinetic energy, $E$ , in a two-dimensional (2-D) or 3-D Rayleigh–Bénard system with domain height $L$ is derived. Assuming classical Nusselt-number scaling, $Nu \sim Ra^{1/3}$ , and that mean enstrophy, in the absence of a downscale energy cascade, scales as $\sim E/L^2$ , we find that the Reynolds number scales as $Re \sim Pr^{-1}Ra^{2/3}$ in the 2-D system, where $Ra$ is the Rayleigh number and $Pr$ the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that $\tilde {\tau } \gtrsim Pr^{-1/2}Ra^{1/2}$ , where $\tilde {\tau }$ is the non-dimensional convergence time scale. For the 3-D system, we make the estimate $\tilde {\tau } \gtrsim Ra^{1/6}$ for $Pr = 1$ . It is estimated that the total computational cost of reaching the high $Ra$ limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.