Revealing the state space of turbulent wall-bounded shear flows

低维混沌动力学理论与无限维非线性湍流动力学理论之间存在着巨大的概念鸿沟。实验成像、计算方法和动力系统理论的进步为弥合这一差距提供了一条途径，并在我们对湍流的理解上取得了根本性的突破。最近发现，在有壁的剪切流（如管道和边界层）中观察到的循环相干结构是由接近Navier-Stokes方程的弱不稳定不变解引起的。这些三维的、完全非线性的解决方案（平衡、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。研究者计算了一个典型的有壁剪切流的不变解的层次结构，并使用这些解来开发流动的湍流动力学的几何和定量描述。他结合了新颖和成熟的数值和分析技术，如周期轨道理论、群表示理论、非线性搜索方法、变分求解器和计算流体动力学。该项目由日本、德国、英国和美国的合作者进行，所有结果和数值软件都通过研究者的合作电子书www.ChaosBook.org传播。湍流不仅是经典物理学中尚未解决的重大基本问题，而且在技术和工程中具有重要的实际意义，其中心脏系统、电磁等离子体、海洋和大气都表现出湍流。湍流现象的数学描述必然是高维的，而准确的预测需要解决数百万个方程。现代计算和实验数据分析的进步使这样的计算成为可能。更近的发现是“循环相干结构”，人们在云等结构中反复观察到的螺旋。这种结构使我们能够以数学上精确的方式对湍流状态进行分类，并预测它们的演变，而无需诉诸进一步的大规模计算。该项目旨在利用这些“连贯结构”作为一个平台，从中绘制所有可能的湍流状态的极高维度世界，创建重要状态的地图集，并给出湍流运动的详细预测（而不是统计）描述。任何对湍流基础理解的进步都会影响到从抑制磁约束聚变反应堆中的等离子体不稳定性到天气预报，再到减少湍流阻力的关键挑战等一系列应用。即使是增加的减少，例如，通过这样的见解，湍流阻力可能会产生重大的经济影响，因为阻力在运输中消耗的燃料中占很大一部分。

CvitanovicDMS-1211827 A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. 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. The investigator calculates a hierarchy of invariant solutions for a canonical wall-bounded shear flow and uses these solutions to develop a geometrical and quantitative description of the flow's turbulent dynamics. He uses a combination of novel and proven numerical and analytical techniques, such as periodic orbit theory, group representation theory, nonlinear search methods, variational solvers, and computational fluid dynamics. The project is conducted with collaborators in Japan, Germany, the UK, and the US, and all results and numerical software are disseminated through the investigator's collaborative e-book www.ChaosBook.org. Turbulence is not only the great unsolved fundamental problem of classical physics, but a problem of great practical importance for technology and engineering, where turbulence is exhibited by cardiac systems, electromagnetic plasmas, oceans, and the atmosphere. A mathematical description of turbulent phenomena is of necessity high-dimensional, and accurate prediction requires solving millions of equations. Modern advances in computation and experimental data analysis have brought such computations within reach. Even more recent is the discovery of "recurrent coherent structures," whorls that one observes over and over in structures such as clouds. Such structures enable us to classify types of turbulent states in a mathematically precise manner and to predict their evolution without resorting to further large-scale computations. The project aims to deploy these "coherent structures" as a platform from which to chart the extremely high-dimensional world of all possible turbulent states, create an atlas over the important ones, and give a detailed predictive (as opposed to statistical) description of turbulent motions. Any progress in fundamental understanding of turbulence affects applications that range from the suppression of plasma instabilities in magnetic confinement fusion reactors, to weather prediction, to the key challenge of reducing turbulent drag. Even an incremental reduction of, for instance, turbulent drag by insights so achieved might have a significant economic impact, because drag is responsible for a significant part of the fuel consumed in transportation.