论文标题
平衡源和水槽之间不可压缩流的传热界限
Bounds on heat transfer by incompressible flows between balanced sources and sinks
论文作者
论文摘要
内部加热的对流涉及通过流体运动在源和水槽的分布之间传递热量。关注源添加的总热量与水槽吸收的热量相匹配的平衡案例，我们获得了\ emph {a先验}在最小平均热量耗散$ \ langle | \ nabla t |^2 \ rangle $上，以衡量运输效率低的运输效率。在对流极限中，我们的边界尺度与流的平均动能相反。这种缩放定律中的常数取决于源分布，正如我们在涉及振荡或浓缩加热和冷却的一对示例中所解释的那样，以及通过一般渐近变异原理来优化运输。我们分析的关键是纯对流方程的解决方案，我们为通过细胞和“捏”流的极端传热示例来解决。当流量遵守动量方程时，我们的界限会根据基于通量的瑞利号$ ra $产生$ \ langle | \ nabla | \ nabla t |^2 \ rangle \ geq cra^cra^{ - α} $。功率$α$为$ 0、2/3 $或$ 1 $，具体取决于来源和沉没相对于重力的排列。
Internally heated convection involves the transfer of heat by fluid motion between a distribution of sources and sinks. Focusing on the balanced case where the total heat added by the sources matches the heat taken away by the sinks, we obtain \emph{a priori} bounds on the minimum mean thermal dissipation $\langle |\nabla T|^2\rangle$ as a measure of the inefficiency of transport. In the advective limit, our bounds scale with the inverse mean kinetic energy of the flow. The constant in this scaling law depends on the source--sink distribution, as we explain both in a pair of examples involving oscillatory or concentrated heating and cooling, and via a general asymptotic variational principle for optimizing transport. Key to our analysis is the solution of a pure advection equation, which we do to find examples of extreme heat transfer by cellular and `pinching' flows. When the flow obeys a momentum equation, our bound is re-expressed in terms of a flux-based Rayleigh number $Ra$ yielding $\langle |\nabla T|^2\rangle\geq CRa^{-α}$. The power $α$ is $0, 2/3$ or $1$ depending on the arrangement of the sources and sinks relative to gravity.
