Penetrative turbulent Rayleigh-Bénard convection in two and three dimensions

Abstract

Penetrative turbulent Rayleigh-Bénard convection which depends on the density maximum of water near $4^{\circ }\mathrm{C}$ is studied using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations (DNS). The working fluid is water near $4^{\circ }\mathrm{C}$ with Prandtl number $Pr=11.57$. The considered Rayleigh numbers $Ra$ range from $10^7$ to $10^{10}$. The density inversion parameter ${\theta }_m$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thickness ($F_{\lambda }={\lambda }_t^{\theta }/{\lambda }_b^{\theta }$) increases with increasing ${\theta }_m$, and the relationship between $F_{\lambda }$ and ${\theta }_m$ seems to be independent of $Ra$. The centre temperature ${\theta }_c$ is enhanced compared to that of Oberbeck-Boussinesq (OB) cases, as ${\theta }_c$ is related to $F_{\lambda }$ with $1/{\theta }_c=1/F_{\lambda }+1$, ${\theta }_c$ is also found to have a universal relationship with ${\theta }_m$ which is independent of $Ra$. Both the Nusselt number $Nu$ and the Reynolds number $Re$ decrease with increasing ${\theta }_m$, the normalized Nusselt number $Nu({\theta }_m)/Nu(0)$ and Reynolds number $Re({\theta }_m)/Re(0)$ also have universal relationships with ${\theta }_m$ which seem to be independent of both $Ra$ and the aspect ratio $\Gamma$. The scaling exponents of $Nu\sim R{a}^{\alpha }$ and $Re\sim R{a}^{\beta }$ are found to be insensitive to ${\theta }_m$ despite of the remarkable change of the flow organizations.

17 pages,10 figures