Regime transitions in thermally driven high-Rayleigh number vertical convection

Vertical convection is investigated using direct numerical simulations over a wide range of Rayleigh numbers $10^7\le Ra\le10^{14}$ with fixed Prandtl number $Pr=10$, in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient $S$ on $Ra$ shows three different regimes: In regime I ($Ra \lesssim 5\times10^{10}$), $S$ is almost independent of $Ra$; In the newly identified regime II ($5\times10^{10} \lesssim Ra \lesssim 10^{13}$), $S$ first increases with increasing $Ra$ (regime ${\rm{II}}_a$), reaches its maximum and then decreases again (regime ${\rm{II}}_b$); In regime III ($Ra\gtrsim10^{13}$), $S$ again becomes only weakly dependent on $Ra$, being slightly smaller than in regime I. The transitions between diffeereent regimes are discussd. In the three different regimes, significantly different flow organizations are identified: In regime I and regime ${\rm{II}}_a$, the location of the maximal horizontal velocity is close to the top and bottom walls; However, in regime ${\rm{II}}_b$ and regime III, banded zonal flow structures develop and the maximal horizontal velocity now is in the bulk region. The different flow organizations in the three regimes are also reflected in the scaling exponents in the effective power law scalings $Nu\sim Ra^\beta$ and $Re\sim Ra^\gamma$. In regime I, the fitted scaling exponents ($\beta\approx0.26$ and $\gamma\approx0.51$) are in excellent agreement with the theoretical predication of $\beta=1/4$ and $\gamma=1/2$ for laminar VC (Shishkina, {\it{Phys. Rev. E.}} 2016, 93, 051102). However, in regimes II and III, $\beta$ increases to a value close to 1/3 and $\gamma$ decreases to a value close to 4/9. The stronger $Ra$ dependence of $Nu$ is related to the ejection of plumes and larger local heat flux at the walls.