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

这项研究计划的目标是开发和实验测试一种革命性的新方法来模拟和预测二维湍流。将识别一组弱不稳定的不变Navier-Stokes解，并对不变解之间的转变进行表征，以提供对湍流非线性动力学的粗略全局描述。在Lorentz力的驱动下，浅电解层中的准2D流动为该方法的理论、分析和实验发展提供了环境。新的成熟的技术，如周期轨道理论、群表示理论、Krylov子空间数值方法、牛顿和变分求解器将被用来发展这一观点，它将在实验中得到验证，可以在整个流域内以全时空分辨率测量流动。如果成功，这项研究的结果将影响到科学、工程和医学的多个领域。虽然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模拟并最终控制广泛的时空混沌系统，例如磁约束聚变反应堆和异常的心脏动力学(从轻微的心律失常到潜在的致命的纤颤)。然而，最直接的实际应用是减少湍流阻力，这是汽车、航空和航运行业消耗的很大一部分燃料。即使通过建议的流动控制方法逐渐减少阻力，也会产生巨大的经济影响。该研究计划产生的所有软件和数据都将向公众开放，其核心目标是通过为最先进的数值算法提供文件齐全、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.