2025/06/16 by Erik Lindborg, Lindborg, Erik
Engineering · #Convection #Fluid Dynamics and Turbulent Flows #Limit (mathematics) #Nanofluid Flow and Heat Transfer #Nusselt number #Prandtl number #Rayleigh number #Reynolds number #Scaling
paper · pdf · doi:10.48550/arxiv.2506.13213
openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/08/05
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 ∼ Ra1/3 , and that mean enstrophy, in the absence of a downscale energy cascade, scales as ∼ E/L2 , we find that the Reynolds number scales as Re ∼ Pr-1Ra2/3 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 τ > c Pr-1/2Ra1/2 , where τ is the non-dimensional convergence time scale and c is a non-dimensional constant. For the 3-D system, we make the estimate τ \gtrsim Ra1/6 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 τ , 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.