论文标题
三维分支管流以最佳标量传输在墙之间
Three dimensional branching pipe flows for optimal scalar transport between walls
论文作者
论文摘要
我们考虑了“壁到壁最佳运输”的问题，在这种问题中，我们试图最大程度地在热板和冷板之间传输被动温度场。具体而言，我们优化了受胚胎约束的对流扩散方程中无差速器场的选择（可以将其理解为对生成流量所需的功率的约束）。先前的工作在运输上建立了一个先验的上限，将其缩放为流动肠的1/3强力。最近，烟草和二氧化碳（Phys。Phys。Thy.Lett。Vol.118，2017，p.264502}）和Doering＆Tobasco（Comm。PureAppl。Math。Math。72，2019年，P.2385--2448}）构建的构建的自动化二维稳定的稳定的分支稳定的稳定的支支稳定的分支流动饱和，以此为止。这种对数校正似乎是由于二维稳定分支流固有的拓扑阻塞而产生的。我们提出了三维“分支管流”的构造，该结构消除了对数校正的可能性，因此将最佳缩放标识为清洁的1/3-power定律。我们的流量类似于Motoki，Kawahara和Shimizu的三维壁到壁问题的数值研究（J. Fluid Mech。Vol.851，2018，P.R4}）。我们还讨论了结果对雷利 - 贝纳德对流中传热问题的含义以及被动标量中异常耗散的问题。
We consider the problem of "wall-to-wall optimal transport" in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field in the advection-diffusion equation subject to an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established an a priori upper bound on the transport, scaling as the 1/3-power of the flow's enstrophy. Recently, Tobasco & Doering (Phys. Rev. Lett. vol.118, 2017, p.264502}) and Doering & Tobasco (Comm. Pure Appl. Math. vol.72, 2019, p.2385--2448}) constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction. This logarithmic correction appears to arise due to a topological obstruction inherent to two-dimensional steady branching flows. We present a construction of three-dimensional "branching pipe flows" that eliminates the possibility of this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara & Shimizu (J. Fluid Mech. vol.851, 2018, p.R4}). We also discuss the implications of our result to the heat transfer problem in Rayleigh--Bénard convection and the problem of anomalous dissipation in a passive scalar.
