DynSyst_Special_Topics: Dynamics Of Turbulent Flow Via Unstable Exact Navier-Stokes Solutions: Connecting Theory & Numerics With Experiments

该研究计划的目标是开发并通过实验测试一种革命性的新方法来建模和预测二维湍流。将识别一组弱不稳定不变纳维-斯托克斯解，并对不变解之间的转换进行表征，以提供湍流非线性动力学的粗略全局描述。由洛伦兹力持续驱动的浅电解质层中的准二维流动为该方法的理论、分析和实验开发提供了基础。新颖且经过验证的技术，如周期轨道理论、群表示理论、克雷洛夫子空间数值方法、牛顿和变分求解器将用于发展这一观点，并将在实验中进行测试，在整个流域中以完整的空间和时间分辨率测量流动。如果成功，这项研究的结果将影响科学、工程和医学的多个领域。尽管这项研究的重点是二维流体湍流，但所提出的方法有可能提供新的方法来建模并最终控制各种时空混沌系统，例如磁约束聚变反应堆和异常心脏动力学（从轻度心律失常到潜在致命的颤动）。然而，最直接的实际应用是减少湍流阻力，湍流阻力是汽车、航空和航运业消耗的大部分燃料的原因。即使通过所提出的流量控制方法逐步减少阻力也会产生巨大的经济影响。该研究项目产生的所有软件和数据都将公开，其中心目标是通过为最先进的数值算法提供记录齐全、易于使用的界面来降低动力系统研究的进入门槛。

The objective of this research program is to develop and to test experimentally a revolutionary new approach to modeling and predicting two-dimensional turbulent flows. A set of weakly unstable invariant Navier-Stokes solutions will be identified and transitions between invariant solutions will be characterized to provide a coarse global description of the nonlinear dynamics of turbulent flow. Quasi-2D flow in a shallow electrolyte layer continually driven by Lorentz forces provides the setting for theoretical, analytic and experimental development of this approach. Novel and proven techniques, such as periodic orbit theory, group representation theory, Krylov-subspace numerical methods, Newton and variational solvers will be used to develop this viewpoint, which will be tested in experiments where the flow can be measured with full spatial and temporal resolution throughout the entire flow domain.If successful, the results of this research will impact several areas of science, engineering, and medicine. Although the focus of this investigation is on fluid turbulence in two dimensions, the proposed approach has the potential to provide new ways to model, and ultimately control, a wide range of spatiotemporally chaotic systems, such as magnetic confinement fusion reactors and abnormal cardiac dynamics (from mild arrhythmias to potentially lethal fibrillation). The most immediate practical application, however, is the reduction of turbulent drag responsible for a significant part of the fuel consumed in the automotive, aviation, and shipping industries. Even an incremental reduction of drag by the proposed flow control methodology would have a tremendous economic impact. All software and data produced by the research program will be made publicly available with a central aim of lowering the barrier of entry to dynamical systems research by providing well-documented, easy-to-use interfaces to state-of-the-art numerical algorithms.