Ultimate Regime of High Rayleigh Number Convection in a Porous Medium

Well-resolved direct numerical simulations of 2D Rayleigh-Bénard convection in a porous medium are presented for Rayleigh numbers $\mathrm{Ra}\le 4\times {10}^4$ which reveal that, contrary to previous indications, the linear classical scaling for the Nusselt number, $\mathrm{Nu}\sim \mathrm{Ra}$, is attained asymptotically. The flow dynamics are analyzed, and the interior of the vigorously convecting system is shown to be increasingly well described as $\mathrm{Ra}\to \infty$ by a simple columnar “heat-exchanger” model with a single horizontal wave number $k$ and a linear background temperature field. The numerical results are approximately fitted by $k\sim {\mathrm{Ra}}^{0.4}$.

Figure 1

A snapshot of the temperature field at $\mathrm{Ra}=2\times {10}^4$, illustrating the different regions described in the text.

Figure 2

The time-averaged Nusselt number Nu, showing the onset of convection at $\mathrm{Ra}=4{\pi }^2$, and the transition to the high-Ra regime at $\mathrm{Ra}\approx 1300$. Inset: Nu scaled by Ra in the high-Ra regime, for different aspect ratios $L$, together with the highest data from [10] for comparison. This plot shows a clear trend in $\mathrm{Nu}/\mathrm{Ra}$ towards a constant as Ra increases.

Figure 3

Measurements of the interior region: (a) temporally and horizontally averaged temperature profile $⟨\overline{T}⟩$ for $\mathrm{Ra}=1,2,4\times {10}^4$, and (b) time-averaged root-mean-square values, $T_{\mathrm{rms}}$, $u_{\mathrm{rms}}$, and $w_{\mathrm{rms}}$, of the temperature perturbations and velocity components at $z=0.5$.

Figure 4

Time-averaged horizontal wave number $k$ scaled by ${\mathrm{Ra}}^{0.4}$, measured from the Fourier spectrum at $z=0.5$. We attribute the variability in these measurements to very long-time-scale fluctuations in the dominant wave number that are not fully time averaged.

Figure 5

Space-time plots of the temperature in a slice just above the boundary layer, for (a) $\mathrm{Ra}=5\times {10}^3$ at $z=2\times {10}^{-2}$, and (b) $\mathrm{Ra}=2\times {10}^4$ at $z=5\times {10}^{-3}$. These plots show both the directly measured temperature ($t<120$) and the results of a plume-tracking algorithm ($t>120$), which gives a way to analyze the dynamics of plumes in more detail. Megaplume roots are highlighted, and the “ribs” of the fish-bone structures (see the text) mark the formation and entrainment of protoplumes.