Scaling relations in large-Prandtl-number natural thermal convection

In this study, we follow Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], who derived various scalings regimes for the dependence of the Nusselt number Nu and the Reynolds number Re on the Rayleigh number Ra and the Prandtl number Pr. We focus on theoretical arguments as well as on numerical simulations for the case of large-Pr natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-Pr boundary-layer dominated regime is I-infinity(<), introduced and defined by Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], with the scaling relations Nu similar to Pr-0 Ra-1/3 and Re similar to Pr-1 Ra-2/3. Our direct numerical simulations for Ra from 10(4) to 10(9) and Pr from 0.1 to 200 showthat the regime I-infinity(<) is almost indistinguishable from the regime III infinity, where the kinetic dissipation is bulk-dominated. With increasing Ra, the scaling relations undergo a transition to those in IVu of Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], where the thermal dissipation is determined by its bulk contribution.