Transition to the Ultimate Regime in Two-Dimensional Rayleigh-Bénard Convection

The possible transition to the so-called ultimate regime, wherein both the bulk and the boundary layers are turbulent, has been an outstanding issue in thermal convection, since the seminal work by Kraichnan [Phys. Fluids 5, 1374 (1962)]. Yet, when this transition takes place and how the local flow induces it is not fully understood. Here, by performing two-dimensional simulations of Rayleigh-Bénard turbulence covering six decades in Rayleigh number Ra up to $10^{14}$ for Prandtl number $\mathrm{Pr}=1$, for the first time in numerical simulations we find the transition to the ultimate regime, namely, at ${\mathrm{Ra}}^{*}={10}^{13}$. We reveal how the emission of thermal plumes enhances the global heat transport, leading to a steeper increase of the Nusselt number than the classical Malkus scaling $\mathrm{Nu}\sim {\mathrm{Ra}}^{1/3}$ [Proc. R. Soc. A 225, 196 (1954)]. Beyond the transition, the mean velocity profiles are logarithmic throughout, indicating turbulent boundary layers. In contrast, the temperature profiles are only locally logarithmic, namely, within the regions where plumes are emitted, and where the local Nusselt number has an effective scaling $\mathrm{Nu}\sim {\mathrm{Ra}}^{0.38}$, corresponding to the effective scaling in the ultimate regime.

Figure 1

Nu(Ra) plot compensated by ${\mathrm{Ra}}^{0.35}$. The data agree well with the previous results in the low-Ra regime [39]. The flow structures of the three colored data points (blue for $\mathrm{Ra}={10}^{11}$, green for $\mathrm{Ra}={10}^{13}$, and gray for $\mathrm{Ra}={10}^{14}$) are displayed in Fig. 2.

Figure 2

The instantaneous temperature fields for (a) $\mathrm{Ra}={10}^{11}$ and (b) $\mathrm{Ra}={10}^{14}$. The corresponding movies are shown in the Supplemental Material [40]. (c) The mean temperature and velocity field for $\mathrm{Ra}={10}^{13}$. The contours represent the mean temperature field, while the vectors show the direction of the velocity, scaled by its magnitude. The plate surfaces have been divided into equal-sized plume-ejecting and -impacting regions.

Figure 3

Mean velocity (a) and temperature (b) profiles in wall units ($u^{+}$ for velocity, $T^{+}$ for temperature, and $y^{+}$ for wall distance) at four Ra. The dashed lines show the viscous sublayer behavior and the log-layer behavior. A log layer is seen for the velocity (with inverse slope ${\kappa }_v=0.4$) but not for the temperature. (c) Local temperature profiles averaged in plume-ejecting and -impacting regions [see Fig. 2 for definitions]. The dashed lines again show the viscous sublayer behavior and the log-layer behavior. A log layer is seen for the temperature in the plume-ejecting regions (with inverse slope ${\kappa }_{\theta }=4.0$) but not in impacting regions.

Figure 4

Local wall-heat flux as a function of Ra, separately for the plume-ejecting region (${\mathrm{Nu}}_e$) and the plume-impacting region (${\mathrm{Nu}}_i$). At ${\mathrm{Ra}}^{*}={10}^{13}$, ${\mathrm{Nu}}_e$ starts to undergo a transition to the ultimate regime with an effective scaling exponent of 0.38, while ${\mathrm{Nu}}_i(\mathrm{Ra})$ has a much smaller effective scaling exponent of 0.28.