Revealing the state space of turbulent wall-bounded shear flows

CvitanovicDMS-1211827 低维混沌动力学理论与湍流的无限维非线性动力学理论之间存在着很大的概念鸿沟。 在实验成像，计算方法和动力系统理论的进步建议一种方法来弥合这一差距，使我们的理解湍流的根本突破。 最近发现，在壁面剪切流（如管道和边界层）中观察到的周期性相干结构是由Navier-Stokes方程的弱不稳定不变解的近距离通过引起的。 这些3D完全非线性解（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。 调查人员计算出一个规范的壁有界剪切流的不变的解决方案的层次结构，并使用这些解决方案来开发一个几何和定量描述的流动的湍流动力学。 他使用了新颖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，变分求解器和计算流体动力学。 该项目是与日本，德国，英国和美国的合作者进行的，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未解决的重大基本问题，而且对技术和工程具有重要的实际意义，其中湍流表现在心脏系统，电磁等离子体，海洋和大气中。 湍流现象的数学描述必然是高维的，而准确的预测需要解数百万个方程。 计算和实验数据分析的现代进步使这种计算触手可及。 更近的是发现了“循环相干结构”，人们在云等结构中一遍又一遍地观察到螺旋。 这样的结构使我们能够以数学上精确的方式对湍流状态的类型进行分类，并预测它们的演变，而不需要进一步的大规模计算。 该项目旨在将这些“相干结构”作为一个平台，从中绘制所有可能的湍流状态的极高维世界，创建重要状态的地图集，并对湍流运动进行详细的预测（而不是统计）描述。 对湍流的基本理解的任何进展都会影响到从磁约束聚变反应堆中等离子体不稳定性的抑制到天气预报，再到减少湍流阻力的关键挑战的应用。 即使是通过这样获得的见解来逐步减少湍流阻力，也可能产生重大的经济影响，因为阻力是运输中消耗的很大一部分燃料的原因。

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.