Nonlinear Dynamics of Fluid Mixing and Flow Separation

该提案讨论了使用非线性动力系统理论来解决流体混合和流动分离的几个长期存在的问题。在数学方面，它将推进非双曲不变流形，惯性流形，对流扩散和发电机方程，有限时不变集，非自治系统的稳定性和非周期平均理论。在流体力学方面，本文的工作将拓宽对三维非定常流体分离、示踪剂扩散混合、三维涡的拓扑结构以及诱导方程中发电机作用的起源的理解，从更广泛的意义上讲，本文描述了研究海洋和空气动力学流动中非线性动力学现象的新方法。这些现象在传统上理解得很少，因为先进的数学理论，特别是动力系统理论，在他们的研究中没有得到充分利用。研究人员提出了新的理论工具来描述不同物质如何混合，以及高速流体如何从边界分离。该提案讨论了这些结果如何改善海洋特征检测和预测，推进空气动力学设计和航空技术，并影响高效化学混合器的设计。

This proposal discusses the use of nonlinear dynamical systems theory in solving several long-standing problems of fluid mixing and flow separation. On the mathematical side, it will advance the theory of nonhyperbolic invariant manifolds, inertial manifolds, advection-diffusion and dynamo equations, finite-time invariant sets, stability of nonautonomous systems, and aperiodic averaging. On the fluid mechanical side, the proposed work will broaden the understanding of three-dimensional unsteady fluid separation, diffusive tracer mixing, the topology of three-dimensional vortices, and the origin of dynamo action in the induction equations.In a broader sense, the proposal describes new approaches to nonlinear dynamical phenomena in oceanic and aerodynamic flows. Such phenomena have traditionally been poorly understood, because advanced mathematical theories, notably dynamical systems theory, have been underutilized in their study. The investigator proposes new theoretical tools for describing how different substances mix, and how high-speed fluid separates from boundaries. The proposal discusses how the results can improve ocean feature detection and prediction, advance aerodynamic design and aviation technology, and impact the design of efficient chemical mixers.