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首页> 外文期刊>Journal of Mathematical Physics >Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows
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Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows

机译：针对固定的能量，固定的功率或固定的古营养流进行最佳混合和最佳搅拌

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We consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget, or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin et al. ["Optimal stirring strategies for passive scalar mixing," J. Fluid Mech.675, 465 (2011)]10.1017/S0022112011000292 that determines the flow field that instantaneously maximizes the depletion of the H-1 mix-norm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. ["Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations," SIAM J. Numer. Anal.50(1), 126-150 (2012)]10.1137/110834901 we establish that the H~(-1) mix-norm decays at most exponentially in time if the two-dimensional incompressible flow is constrained to have constant palenstrophy. Finite-time perfect mixing is thus ruled out when too much cost is incurred by small scale structures in the stirring. Direct numerical simulations in two dimensions suggest the impossibility of finite-time perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H~(-1) mix-norm in this case. We also discuss some related problems from other areas of analysis that are similarly suggestive of an exponential lower bound for the H-1 mix-norm.
机译：我们考虑在周期框中通过规定的无散度速度矢量场进行无源标量混合，并解决以下问题：从给定的无源示踪剂初始初始不均匀分布开始，并给出一定的能量预算，功率预算或有限的古营养预算，什么不可压缩流场最能混合标量？我们关注Lin等人最近提出的最佳搅拌策略。 [“用于被动标量混合的最佳搅拌策略，” J。Fluid Mech.675，465（2011）] 10.1017 / S0022112011000292，其确定瞬时使H-1混合模耗量最大化的流场。在这项工作中，我们弥合了最好的可用先验分析和模拟结果之间的一些鸿沟。在回顾一些先前的分析之后，我们给出一个明确的示例，说明在搅拌流上具有有限能量约束的有限时间完美混合。另一方面，使用Wirosoetisno等人的最新结果。 [“二维Navier-Stokes方程的经典有效方案的长时间稳定性，” SIAM J. Numer。 Anal.50（1），126-150（2012）] 10.1137 / 110834901我们确定，如果二维不可压缩流被约束为具有恒定的古营养，则H〜（-1）混合范数随时间最多以指数方式衰减。因此，当在搅拌中由于小规模结构而导致太多成本时，可以排除有限时间的完美混合。二维直接数值模拟表明，对于具有固定功率约束的流，不可能进行有限时间完美混合，并且在这种情况下，我们推测H〜（-1）混合范数上的指数下界。我们还将讨论来自其他分析领域的一些相关问题，这些问题同样暗示了H-1混合范数的指数下限。

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