论文标题
在无限prandtl数字上进行内部加热对流的严格缩放定律
Rigorous scaling laws for internally heated convection at infinite Prandtl number
论文作者
论文摘要
在平均垂直对流热传输中，新的边界已被证明，$ \ overline {\ langle wt \ rangle} $，用于内部加热（ih）对流，在无限prandtl数字的限制中。 For fluid in a horizontally-periodic layer between isothermal boundaries, we show that $\overline{\langle wT \rangle} \leq \frac12 - c R^{-2}$, where $R$ is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and $c = 216$.然后，$ \叠加{\ langle wt \ rangle} = 0 $单独对应于传导垂直的热传输，而$ \ overline {\ langle wt \ rangle}> 0 $代表对对流运动引起的垂直热传输的增强。相反，下边界是热绝缘子，则我们获得$ \ edline {\ langle wt \ rangle} \ leq \ frac12 -c r^{ - 4} $，$ c \ of $ c \ of 0.0107 $。该结果意味着Nusselt编号$ NU $定义为总导电热传输的比率，满足$ NU \ lyssim r^{4} $。通过将背景方法与流体温度的最低原理和硬性 - 雷利希不等式相结合，以利用垂直速度和温度之间的联系来获得这两个边界。在这两种情况下，对$ r $的权力依赖都改善了以前最著名的界限，尽管在无限和有限的prandtl数字上均有效，但它以$ r $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ R $的限制。
New bounds are proven on the mean vertical convective heat transport, $\overline{\langle wT \rangle}$, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that $\overline{\langle wT \rangle} \leq \frac12 - c R^{-2}$, where $R$ is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and $c = 216$. Then, $\overline{\langle wT \rangle} = 0$ corresponds to vertical heat transport by conduction alone, while $\overline{\langle wT \rangle} > 0$ represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain $\overline{\langle wT \rangle} \leq \frac12 - c R^{-4}$, with $c\approx 0.0107$. This result implies that the Nusselt number $Nu$, defined as the ratio of the total-to-conductive heat transport, satisfies $Nu \lesssim R^{4}$. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on $R$ improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with $R$.
