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 ${\theta }_m$.

中文翻译：

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穿透湍流瑞利-贝纳对流的普遍性质

渗透湍流发生在对流不稳定的流体层中，并渗透到相邻的、最初稳定分层的层中，进行了数值和理论分析。作为示例，我们选择规范的 Rayleigh-Bénard 几何，但现在使用底板温度${吨}_{乙}>4{}^{\circ }\mathrm{C}$, 顶板温度 ${吨}_{吨}\le 4{}^{\circ }\mathrm{C}$, 以及周围的密度最大值 ${吨}_{米}\approx 4{}^{\circ }\mathrm{C}$在两者之间，导致穿透湍流。除了瑞利数 Ra，与标准瑞利-贝纳德对流相比，关键的新控制参数是密度反演参数${\theta }_{米}\equiv ({吨}_{米}-{吨}_{吨})/({吨}_{乙}-{吨}_{吨})$. 关键的响应参数是相对平均中高温度${\theta }_C$和总传热（即努塞尔数 Nu）。我们用数字表示（对于 Ra 高达${10}^{10}$) 并从理论上推导出 ${\theta }_C\left({\theta }_{米}\right)$ 和 $\text{怒}\left({\theta }_{米}\right)/\text{怒}\left(0\right)$是普遍的（即，独立于$\text{镭}$) 仅由密度反演参数决定 ${\theta }_{米}$并成功推导出这些普遍依赖。特别是，${\theta }_C\left({\theta }_{米}\right)=(1+{\theta }_{米}^2)/2$，这适用于 ${\theta }_{米}$ 低于 $\text{镭}$-依赖临界值，超过该临界值 ${\theta }_C\left({\theta }_{米}\right)$ 急剧下降并下降到 ${\theta }_C=1/2$ 在 ${\theta }_{米}={\theta }_{米,C}$. 这个临界密度反演参数${\theta }_{米,C}$可以通过线性稳定性分析精确预测。最后，我们数值识别和讨论了不同湍流状态之间的罕见转变${\theta }_{米}$.