Scaling in two-dimensional Rayleigh-Bénard convection
Erik Lindborg
TL;DR
This work analyzes scaling in two-dimensional Rayleigh-Bénard convection by deriving a mean-energy evolution equation and exploiting enstrophy conservation in 2D. Under the classical Nu ~ Ra^{1/3} scaling and assuming mean enstrophy scales as $E/L^2$, it shows 2D Re scales as $Re \sim Pr^{-1} Ra^{2/3}$—a stronger Ra-dependence than in 3D—and establishes a slow convergence time $\tilde{\tau} > c Pr^{-1/2} Ra^{1/2}$, challenging the applicability of the ultimate-state theory in 2D. Despite the stark difference in interior dynamics between 2D and 3D, the Nu scaling remains similar, supporting boundary-layer control of heat transfer as proposed by Malkus. DNS comparisons for stress-free and no-slip boundaries largely corroborate the classical scalings, while convergence data indicate slow attainment of stationarity at high $Ra$, underscoring the need for long simulations to distinguish between competing theories. The findings emphasize that 2D results, while informative, should be applied to 3D contexts with care due to fundamental dynamical differences.
Abstract
An equation for the evolution of mean kinetic energy, $ E $, in a 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, which is a much stronger scaling than in the 3-D system. Using the evolution equation and the Reynolds number scaling, it is shown that $ \tildeτ > c Pr^{-1/2}Ra^{1/2} $, where $ \tildeτ $ is the non-dimensional convergence time scale and $ c $ is a non-dimensional constant. For the 3-D system, we make the estimate $ \tildeτ \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 2-D and 3-D. The results of the analysis are compared to DNS data and it is concluded that the theory of the `ultimate state' is not valid in 2-D. Despite the big difference between the 2-D and 3-D systems in the scaling of $ Re $ and $ \tildeτ $, the Nusselt number scaling is similar. This observation supports the hypothesis of Malkus (1954) that the heat transfer is not regulated by the dynamics in the interior of the convection cell, but by the dynamics in the boundary layers.