Geometry of state space in plane Couette flow

一个巨大的概念鸿沟将低维混沌动力学理论与无限维湍流非线性动力学理论分开。在实验成像、计算方法和动力系统理论方面的最新进展表明，一种方法可以弥合这一差距，并在我们对湍流的理解上取得根本性突破。最近发现，在壁面剪切流(如管道和边界层)中观察到的循环相干结构是由Navier-Stokes方程的闭合不变解到弱不稳定不变解的结果。这些完全非线性的3D解(平衡点、行波和周期轨道)构成了湍流的状态空间，并为分析其动力学提供了框架。我们建议计算正则壁边界剪切流的不变解的层次，并使用这些解(1)发展流动的湍流动力学的定量描述，(2)直接从基本方程预测物理量，如体积流量和平均壁面阻力。我们将使用新的和经过验证的数值和分析技术的组合，如周期轨道理论、群表示理论、非线性搜索方法和变分求解器，以及计算流体力学。拟议的研究将与日本、德国、英国和美国的合作者一起进行，所有结果和数值软件将通过我们小组的合作电子书www.ChaosBook.org和开源CFD软件和不变解数据库www.Channelflow.org发布。浑浊不仅是经典物理中尚未解决的重大基本问题，也是一个具有重大现实意义的问题。我们对湍流的基本理解上的任何进步都会影响到技术和工程。应用范围从抑制磁约束聚变反应堆中的等离子体不稳定性到减少湍流阻力的关键挑战。在汽车、航空和船舶工业以及流体运输中，阻力占燃料消耗的很大一部分。即使是通过改进流量控制方法来逐步减少阻力，也会对广泛的行业产生重大的经济影响。

A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. Recent advances in experimental imaging, computational methods, and dynamical systems theory suggest a way to bridge this gap and make a fundamental breakthrough in our understanding of turbulence. It has recently been discovered that recurrent coherent structures observed in wall-bounded shear flows (such as pipes and boundary layers) result from close passes to weakly unstable invariant solutions of the Navier-Stokes equations. These 3D, fully nonlinear solutions (equilibria, traveling waves, and periodic orbits) structure the state space of turbulent flows and provide a skeleton for analyzing their dynamics. We propose to calculate a hierarchy of invariant solutions for a canonical wall-bounded shear flow and to use these solutions (1) to develop a quantitative description of the flow's turbulent dynamics, and (2) to predict, directly from the fundamental equations, physical quantities such as bulk flow rate and mean wall drag. We will use a combination of novel and proven numerical and analytical techniques, such as periodic orbit theory, group representation theory, nonlinear search methods and variational solvers, and computational fluid dynamics. The proposed research will be conducted with collaborators in Japan, Germany, the UK, and the US, and all results and numerical software will be disseminated through through our group's collaborative e-book www.ChaosBook.org and open-source CFD software and invariant solution database www.Channelflow.org.Turbulence is not only the great unsolved fundamental problem of classical physics, but a problem of great practical importance. Any progress in our fundamental understanding of turbulence impacts technology and engineering. Applications range from the suppression of plasma instabilities in magnetic confinement fusion reactors to the key challenge of reducing turbulent drag. Drag is responsible for a significant part of the fuel consumed in automotive, aviation and shipping industry as well as in transport of fluids. Even an incremental reduction of drag by improving flow control methodology would have a significant economic impact across a broad range of industries.