Regime transitions in high-Rayleigh number vertical convection

Vertical convection (VC) is investigated using direct numerical simulations over wide range of Rayleigh numbers $10^5\le Ra\le10^{14}$, in a two-dimensionl convection cell with unit aspect ratio. The Prandtl numbers $Pr$ vary from 0.71 to 30. The dependence of the mean vertical center temperature gradient [$S=\left<\partial T/\partial z\right>_{c,t}$] on $Ra$ shows three different regimes. In Regime I where $Ra \le 5\times10^{10}$, $S$ hardly depends on $Ra$. In the newly-identified Regime II ($5\times10^{10} \le Ra \le 5\times10^{12}$), $S$ first increases with increasing $Ra$ and then reaches its maximum as function of $Ra$, before decreasing again. In Regime III where $Ra\ge5\times10^{12}$, $S$ again becomes weakly dependent on $Ra$, with a smaller value than that of Regime I. It is further found that the change of $S$ is closely related to the change of global flow organization: The flow in Regime III is characterized by well-mixed bulk region due to continuous ejection of plumes over large fraction of the plate, thus $S$ is smaller than that of the first regime. The scaling exponent $\beta$ in the effective scaling $Nu\sim Ra^\beta$ reaches a value very close to 1/3 in Regime II and III, which is larger than the value around 1/4 in Regime I.