Penetrative Convection at High Rayleigh Numbers

We study penetrative convection of a fluid confined between two horizontal plates, the temperatures of which are such that a temperature of maximum density lies between them. The range of Rayleigh numbers studied is $Ra = \left[10^6, 10^8 \right]$ and the Prandtl numbers are $Pr = 1$ and $11.6$. An evolution equation for the growth of the convecting region is obtained through an integral energy balance. We identify a new non-dimensional parameter, $\Lambda$, which is the ratio of temperature difference between the stable and unstable regions of the flow; larger values of $\Lambda$ denote increased stability of the upper stable layer. We study the effects of $\Lambda$ on the flow field using well-resolved lattice Boltzmann simulations, and show that the characteristics of the flow depend sensitively upon it. For the range $\Lambda = \left[0.01, 4\right]$, we find that for a fixed $Ra$ the Nusselt number, $Nu$, increases with decreasing $\Lambda$. We also investigate the effects of $\Lambda$ on the vertical variation of convective heat flux and the Brunt-V\"{a}is\"{a}l\"{a} frequency. Our results clearly indicate that in the limit $\Lambda \rightarrow 0$ the problem reduces to that of the classical Rayleigh-B\'enard convection.