Studies of the Exact Coherent States that control turbulence and transition to turbulence in shear flows

该奖项支持的研究的主要目标是发展“低分支相干态”的渐近理论，这是最近发现的一类用于剪切流（平面Couette，通道和管道流）的Navier-Stokes方程的三维行波解。数值计算表明，这些解是不稳定的，但具有非常低维的不稳定流形（一维或二维），并且它们控制着向湍流的过渡。数值计算还表明，解具有显著的涉及临界层的渐近结构。需要一个渐近理论来澄清、巩固和推广数值结果到大雷诺数。由于奇异临界层结构的非线性自相互作用，这种渐近理论将对Navier-Stokes方程解的高雷诺数极限具有有趣的数学特征和意义。这个研究项目将继续发展这样一个理论。本研究计划的第二个方面是继续对各种相干态及其与湍流的关系进行数值研究。部分研究涉及美国和国际合作，特别是P. Cvitanovic， J. Gibson， D. Viswanath和M. Graham（美国），R. Kerswell（英国），G. Kawahara， S. Toh， S. Kida和M. Nagata（日本）以及B. Eckhardt（德国）。我们已经在月球上行走了，但我们仍然不理解水管里的水流。在低速时，水以有序的方式流动（“层流”），但在高速时，水流变得非常无序（“湍流”）。湍流是一种无处不在的流体现象，它也发生在管道中的石油流动、汽车、飞机和建筑物周围的空气中，以及托卡马克内部的等离子体、太阳和恒星中。湍流的能量和环境影响是主要的。紊流比层流消耗更多的能量，混合效果也更好。湍流已被积极研究了120多年，被广泛认为是经典物理学中未解决的主要问题。该研究的主要重点是开发用于工程计算的湍流半经验模型。这些模型总是基于对湍流性质的各种假设。这个研究项目是世界上为数不多的开创性项目之一，这些项目已经发现了一系列以前未被怀疑的其他可能的流体流动状态，介于层流和湍流状态之间。这些新的流动状态包括不稳定的平衡状态和周期状态，它们似乎控制着湍流的开始以及完全发展的湍流。这些状态的发现迫使人们从根本上重新思考湍流的本质。这些发现是由实验可视化方法以及计算机和计算机计算方法的进步提出并使之成为可能的。

A primary goal of the research that is supported with this award is to develop an asymptotic theory for `lower branch coherent states', a recently discovered class of 3D traveling wave solutions of the Navier-Stokes equations for shear flows (plane Couette, channel and pipe flows). Numerical calculations reveal that these solutions are unstable but with very low dimensional unstable manifolds (1 or 2 dimensional) and that they control transition to turbulence. Numerical calculations also reveal that the solutions have a striking asymptotic structure involving a critical layer. An asymptotic theory is needed to clarify, solidify and extend the numerical results to large Reynolds number. Such an asymptotic theory would have interesting mathematical features and implications about the high Reynolds number limit of solutions to the Navier-Stokes equations because of the nonlinear self-interaction of a singular critical layer structure. This research project will continue to develop such a theory.A second aspect of this research program is to continue the numerical study of the various coherent states and their connection with turbulent flows. Parts of this research involve US and international collaborations, in particular with P. Cvitanovic, J. Gibson, D. Viswanath and M. Graham (USA), R. Kerswell (UK), G. Kawahara, S. Toh, S. Kida and M. Nagata (Japan) and B. Eckhardt (Germany).We have walked on the moon but we still do not understand flow of water down a pipe. For low speeds, the water flows in an orderly manner (`laminar flow'), but at higher speeds the flow becomes very disordered (`turbulent flow'). Turbulence is an ubiquitous fluid phenomenon that also occurs for flow of oil in pipelines, air around cars, airplanes and buildings, as well as in plasmas inside tokamaks, the sun and stars. The energetic and environmental effects of turbulence are major. Turbulent flows consume a lot more energy and mix things much better than laminar flows. Turbulence has been actively studied for over 120 years and is widely considered as the major unsolved problem of classical physics. A main emphasis of that research has been to developed semi-empirical models of turbulence for engineering calculations. These models are invariably based upon various presumptions about the nature of turbulence. This research program is one of an handful of pioneering programs around the world that have uncovered a series of previously unsuspected other possible states of fluid flow, intermediate between laminar and turbulent states. These new flow states consists of equilibria and periodic states that are unstable but appear to control onset of turbulence as well as fully developed turbulent flows. The discovery of these states forces a fundamental rethinking of the nature of turbulence. These discoveries were suggested and made possible by advances in experimental visualization methods as well as computers and computer calculation methods.