Universal properties of penetrative turbulent Rayleigh--Bénard convection in cold water near $4^\circ\rm{C}$

Abstract

Penetrative turbulence, which occurs in a convectively unstable fluid layer and penetrates into an adjacent, originally stably stratified layer, is numerically and theoretically analyzed. We chose the most relevant example, namely thermally driven flow of water with a temperature around $T_m\approx 4^\circ\rm{C}$, where it has its density maximum. We pick the Rayleigh-Bénard geometry with the bottom plate temperature $T_b > 4^\circ\rm{C}$ and the top plate temperature $T_t \le 4^\circ\rm{C}$. Next to the overall thermal driving strength set by the temperature difference $Δ= T_b - T_t$ (the Rayleigh number $Ra$ in dimensionless form), the crucial new control parameter as compared to standard Rayleigh-Bénard convection is the density inversion parameter $θ_m \equiv (T_m - T_t ) / Δ$. The crucial response parameters are the relative mean mid-height temperature $θ_c$ and the overall heat transfer (i.e., the Nusselt number $Nu$). We theoretically derive the universal (i.e., $Ra$-independent) dependence $θ_c (θ_m) =(1+θ_m^2)/2$, which holds for $θ_m$ below a $Ra$-dependent critical value, beyond which $θ_c (θ_m)$ sharply decreases and drops down to $θ_c=1/2$ at $θ_m=θ_{m,c}$. Our direct numerical simulations with $Ra$ up to $10^{10}$ are consistent with these results. The critical density inversion parameter $θ_{m,c}$ can be precisely predicted by a linear stability analysis. The heat flux $Nu(θ_m)$ monotonically decreases with increasing $θ_m$ and we can theoretically derive a universal relation for the relative heat flux $Nu(θ_m)/Nu(0)$. Finally, we numerically identify and discuss rare transitions between different turbulent flow states for large $θ_m$.