Penetrative turbulent Rayleigh-B\'enard convection in two and three dimensions

Penetrative turbulent Rayleigh-B\'enard convection which depends on the density maximum of water near $4^\circ\rm{C}$ is studied using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations (DNS). The working fluid is water near $4^\circ\rm{C}$ 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 $Nu\sim Ra^\alpha$ and $Re\sim Ra^\beta$ are found to be insensitive to $\theta_m$ despite of the remarkable change of the flow organizations.