壁间标量最佳输运的三维分支管流
Three dimensional branching pipe flows for optimal scalar transport between walls
摘要：
我们考虑“墙到墙最优输运“的问题，我们试图最大限度地提高运输热板和冷板之间的被动温度场。具体来说，我们优化的选择的无发散的速度场的平流-扩散方程受到拟能约束（这可以理解为一个约束的功率产生的流量）。以前的工作建立了一个先验的上界的运输，缩放为1/3的功率流的拟能。最近，Tobasco & Doering（Phys. Rev. Lett. vol.118，2017，p.264502}）和Doering & Tobasco（Comm. Pure Appl.Math.vol.72，2019，p.2385--2448}）构建了自相似二维稳态分支流，使该约束饱和到对数校正。这种对数校正似乎是由于二维定常分支流固有的拓扑障碍而产生的。我们提出了一个三维的“分支管流”，消除了这种对数校正的可能性，因此确定了一个干净的1/3幂律的最佳缩放的建设。我们的流动类似于Motoki，Kawahara & Shimizu之前对三维壁到壁问题的数值研究（J. Fluid Mech. vol.851，2018，p.R4}）。我们还讨论了该结果对Rayleigh-Bénard对流中的热传导问题和被动标量中的反常耗散问题的意义。
英文摘要：
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.
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