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The Rayleigh number $Ra$ dependence of the Nusselt number $Nu$ in turbulentRayleigh--B\'enard convection is numerically investigated for a moderate andlow Prandtl number, $Pr=0.7$ and $0.021$, respectively. Here we specificallyaddress the case of a Boussinesq fluid in a planar configuration with smoothhorizontal walls and notionally infinite aspect ratio. Numerical simulations upto $Ra=10^{16}$ for $Pr=0.7$ and up to $Ra=8\times10^{13}$ for $Pr=0.021$ aremade feasible on state-of-the-art workstations by utilising the stochasticone-dimensional turbulence (ODT) model. The ODT model parameters were estimatedonce for two combinations $(Pr,Ra)$ in the classical regime and kept fixedafterwards in order to address the predictive capabilities of the model. TheODT results presented exhibit various effective Nusselt number scalings$Nu\propto Ra^b$. The exponent changes from $b\approx1/3$ to $b\approx1/2$ whenthe $Ra$ number increases beyond the critical value $Ra_*\simeq6\times10^{14}$($Pr=0.7$) and $Ra_*\simeq6\times10^{11}$ ($Pr=0.021$), respectively. This isconsistent with the literature. Furthermore, our results suggest that thetransition to the ultimate regime is correlated with a relative enhancement ofthe temperature-velocity cross-correlations in the bulk of the fluid ashypothesised by Kraichnan, R. H., Phys. Fluids, 5, 1374 (1962).

The transition to the ultimate regime of thermal convection from a stochastic one-dimensional turbulence perspective