Penetrative turbulence, which occurs in a convectively unstable fluid layer and penetrates into an adjacent, originally stably stratified layer, is numerically and theoretically analyzed. As example we pick the canonical Rayleigh-Bénard geometry, but now with the bottom plate temperature $T_b>4{}^{\circ }\mathrm{C}$, the top plate temperature $T_t\le 4{}^{\circ }\mathrm{C}$, and the density maximum around $T_m\approx 4{}^{\circ }\mathrm{C}$ in between, resulting in penetrative turbulence. Next to the Rayleigh number Ra, the crucial new control parameter as compared to standard Rayleigh-Bénard convection is the density inversion parameter ${\theta }_m\equiv (T_m-T_t)/(T_b-T_t)$. The crucial response parameters are the relative mean midheight temperature ${\theta }_c$ and the overall heat transfer (i.e., the Nusselt number Nu). We numerically show (for Ra up to ${10}^{10}$) and theoretically derive that ${\theta }_c\left({\theta }_m\right)$ and $\text{Nu}\left({\theta }_m\right)/\text{Nu}\left(0\right)$ are universally(i.e., independently of $\text{Ra}$) determined only by the density inversion parameter ${\theta }_m$ and succeed to derive these universal dependences. In particular, ${\theta }_c\left({\theta }_m\right)=(1+{\theta }_m^2)/2$, which holds for ${\theta }_m$ below a $\text{Ra}$-dependent critical value, beyond which ${\theta }_c\left({\theta }_m\right)$ sharply decreases and drops down to ${\theta }_c=1/2$ at ${\theta }_m={\theta }_{m,c}$. This critical density inversion parameter ${\theta }_{m,c}$ can be precisely predicted by a linear stability analysis. Finally, we numerically identify and discuss rare transitions between different turbulent flow states for large ${\theta }_m$.

• Received 15 December 2020
• Accepted 7 June 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.063502