Optimal cooling of an internally heated disc
Optimal cooling of an internally heated disc
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DOI：
10.1098/rsta.2021.0040
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发表时间：
2022-06-13
期刊：
影响因子：
5
通讯作者：
Tobasco, Ian
中科院分区：
文献类型：
--
作者：
Tobasco, Ian
在寻找湍流换热的尖锐界限和设计最佳换热器的激励下，我们考虑了最有效地冷却内部加热的阀瓣的不可压缩流动。热量通过分布式源进入，被被动地平流和扩散，并在固定温度下通过边界离开。我们寻求一种平流流来优化这种交换。以前关于恒定源的能量约束冷却的工作已经推测，全局优化器应该类似于对流卷；我们证明了能量约束冷却的单边界对应于这个猜想，但不是解决这个猜想。在Enstrophy约束的情况下，我们的结果更完整：我们构造了一族自相似的树状‘分支流’，我们证明了其冷却在全局最优的对数范围内。这些结果适用于一般与空间和时间相关的源汇分布，这些分布增加的热量比带走的热量更多。我们的主要技术工具是求解具有无散度速度的非齐次对流扩散方程的边界解的非局部Dirichlet类变分原理。本文是《物理流体动力学数学问题(上)》的一部分。
Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like 'branching flows' whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source-sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection-diffusion equation with a divergence-free velocity.This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.