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 $4^\circ\rm{C}$ 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 $θ_m$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thickness ($F_λ=λ_t^θ/λ_b^θ$) increases with increasing $θ_m$, and the relationship between $F_λ$ and $θ_m$ seems to be independent of $Ra$. The centre temperature $θ_c$ is enhanced compared to that of Oberbeck-Boussinesq (OB) cases, as $θ_c$ is related to $F_λ$ with $1/θ_c=1/F_λ+1$, $θ_c$ is also found to have a universal relationship with $θ_m$ which is independent of $Ra$. Both the Nusselt number $Nu$ and the Reynolds number $Re$ decrease with increasing $θ_m$, the normalized Nusselt number $Nu(θ_m)/Nu(0)$ and Reynolds number $Re(θ_m)/Re(0)$ also have universal relationships with $θ_m$ which seem to be independent of both $Ra$ and the aspect ratio $Γ$. The scaling exponents of $Nu\sim Ra^α$ and $Re\sim Ra^β$ are found to be insensitive to $θ_m$ despite of the remarkable change of the flow organizations.