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 $\text{Nu}$ and the Reynolds number $\text{Re}$ on the Rayleigh number $\text{Ra}$ and the Prandtl number $Pr$. We focus on theoretical arguments as well as on numerical simulations for the case of large-$\text{Pr}$ natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-$\text{Pr}$ boundary-layer dominated regime is ${\mathrm{I}}_{\infty }^{<}$, introduced and defined by Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], with the scaling relations $\text{Nu}\sim {\text{Pr}}^0{\text{Ra}}^{1/3}$ and $\text{Re}\sim {\text{Pr}}^{-1}{\text{Ra}}^{2/3}$. Our direct numerical simulations for $\text{Ra}$ from ${10}^4$ to ${10}^9$ and $Pr$ from 0.1 to 200 show that the regime ${\mathrm{I}}_{\infty }^{<}$ is almost indistinguishable from the regime ${\mathrm{III}}_{\infty }$, where the kinetic dissipation is bulk-dominated. With increasing $\text{Ra}$, the scaling relations undergo a transition to those in ${\mathrm{IV}}_u$ of Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], where the thermal dissipation is determined by its bulk contribution.

Figure 1

Sketch of the conducted DNS (symbols) in an $(\text{Ra},Pr)$ plane. White region corresponds to the scaling regimes ${\mathrm{I}}_{\infty }^{<}$ and ${\mathrm{III}}_{\infty }$. Shadowed region corresponds to transition to other regimes.

Figure 2

Nusselt number, Reynolds number and their combinations, as functions of the Prandtl number (a), (c), (e) and of the Rayleigh number (b), (d), (f), as obtained in the DNS of RBC in a cylindrical container of the aspect ratio 1, for (a), (c), (e) $\text{Ra}={10}^4$ (red), $\text{Ra}={10}^5$ (green), $\text{Ra}={10}^6$ (blue), $\text{Ra}={10}^7$ (magenta), $\text{Ra}={10}^8$ (cyan), and $\text{Ra}={10}^9$ (gray), and for (b), (d), (f) $Pr=0.1$ (red circles), $Pr=0.2$ (green circles), $Pr=0.3$ (blue circles), $Pr=0.5$ (magenta circles), $Pr=0.7$ (cyan circles), $Pr=1$ (red squares), $Pr=2$ (green squares), $Pr=3$ (blue squares), $Pr=5$ (magenta squares), $Pr=10$ (cyan squares), $Pr=20$ (red triangles), $Pr=30$ (green triangles), $Pr=50$ (blue triangles), $Pr=100$ (magenta triangles), $Pr=200$ (cyan triangles).