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

Abstract

The Rayleigh number $Ra$ dependence of the Nusselt number $Nu$ in turbulent Rayleigh--B\'enard convection is numerically investigated for a moderate and low Prandtl number, $Pr=0.7$ and $0.021$, respectively. Here we specifically address the case of a Boussinesq fluid in a planar configuration with smooth horizontal walls and notionally infinite aspect ratio. Numerical simulations up to $Ra=10^{16}$ for $Pr=0.7$ and up to $Ra=8\times10^{13}$ for $Pr=0.021$ are made feasible on state-of-the-art workstations by utilising the stochastic one-dimensional turbulence (ODT) model. The ODT model parameters were estimated once for two combinations $(Pr,Ra)$ in the classical regime and kept fixed afterwards in order to address the predictive capabilities of the model. The ODT results presented exhibit various effective Nusselt number scalings $Nu\propto Ra^b$. The exponent changes from $b\approx1/3$ to $b\approx1/2$ when the $Ra$ number increases beyond the critical value $Ra_*\simeq6\times10^{14}$ ($Pr=0.7$) and $Ra_*\simeq6\times10^{11}$ ($Pr=0.021$), respectively. This is consistent with the literature. Furthermore, our results suggest that the transition to the ultimate regime is correlated with a relative enhancement of the temperature-velocity cross-correlations in the bulk of the fluid as hypothesised by Kraichnan, R. H., Phys. Fluids, 5, 1374 (1962).