In this paper we estimate the relative strengths of various terms of the Rayleigh-Bénard equations. Based on these estimates and scaling analysis, we derive a general formula for the large-scale velocity U or the Péclet number that is applicable for arbitrary Rayleigh number Ra and Prandtl number Pr. Our formula fits reasonably well with the earlier simulation and experimental results. Our analysis also shows that the wall-bounded convection has enhanced viscous force compared to free turbulence. We also demonstrate how correlations deviate the Nusselt number scaling from the theoretical prediction of Ra1/2 to the experimentally observed scaling of nearly Ra0.3.
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Pandey, A., Kumar, A., Chatterjee, A. G., & Verma, M. K. (2016). Dynamics of large-scale quantities in Rayleigh-Bénard convection. Physical Review E, 94, . https://doi.org/10.1103/PhysRevE.94.053106