槽道流中亚临界转捩的研究

Subcritical transition to turbulence in fluid motions remains one of the unsolved problems of physics, despite its importance in various fields of science and engineering ranging from aerodynamics to astrophysics. However, the past quarter century has seen numerous new developments for the understanding of chaotic behavior of turbulence and a promising dynamical framework to study turbulence has been proposed. .Most prominent contribution, among the new developments, is the discovery of invariant solutions to the Navier-Stokes equations, called the exact coherent states, which constitute, in a figurative way of speech, the skeleton of the turbulence. In other words, turbulence can be viewed as a walk through a forest of the exact solutions in the infinite-dimensional state space of the governing equations..The first discovery of the exact coherent state was made by the applicant in the system of plane Couette flow, the simplest shear flow with the linear velocity profile (Nagata, 1990)..Since then, numerous exact coherent states have been identified in both plane Couette flow and pipe flow and progress has been made towards the understanding of the subcritical transition, one of the most active research fields in fluid dynamics. It is now widely agreed that the spatial structures of the exact coherent states carrying a substantial share of the Reynolds stresses are important ingredients of shear turbulent flows, where they have led to applications such as improved control and drag reduction. .The same author that made the historical discovery of the exact coherent state in plane Couette flow proposes here a project on the study of subcritical transition in channel flow. In contrast to a large number of studies of plane Couette flow and pipe flow, known exact coherent states in channel flow remains relatively few. .Therefore, firstly we intend to gather a sufficient number of exact coherent states in channel flow. For this purpose, we use our other expertise on the method of Homotopy, whereby solutions in rotating channel flow (Wall & Darren, 2013) shall be continued to non-rotating case. We note that another Homotopy approach has been successfully conducted from plane Couette flow to channel flow by the applicant (Nagata & Deguchi, 2013)..Secondly, we address a classification of the exact coherent states by exploiting our expertise on symmetry identification procedure. We shall re-visit the nonlinear analysis on secondary flow in channel flow by Ehrenstein & Koch (1991), possibly adding the number of new solutions to theirs. The classification is a valuable tool to understand turbulent transition. The structures of the exact coherent states play a key role in the dynamics of fluid motions. The important components of structures that can organize the state-space dynamics of the disturbed flow can be described and thereby we can ultimately explain the sustenance of turbulence in channel flow, which is the goal of our current project.

亚临界转捩是航空航天等学科中重要的研究课题，在工程中也应用广泛，但其物理机制尚未解决。近25年来，对湍流混沌系统的理解给我们提供了一个湍流研究的新动力学框架。而寻找N-S方程精确相干态意义重大。申请人(Nagata, 1990)在线性稳定的平面库塔流中首次找到了其精确相干态。此后，在平面库塔流和管流中也相继找到了很多精确相干态。这加深了人们对亚临界转捩的理解。自由剪切湍流中，相干结构承载着相当大份额的雷诺应力，故被减阻等流动控制关注。但目前在对槽道流中的精确相干态研究很少，因此本项目主要研究槽道流的亚临界转捩。首先，我们拟寻找到大量的槽道流精确相干态，并用同伦方法把解从旋转槽道流(Wall & Darren, 2013)继续推广至无旋槽道流；其次，通过对称性分析，对这些相干态分类，将对Ehrenstein & Koch (1991)槽道流中的二次流问题重新进行非线性分析，以补充可能的新解。

流动转捩在科学和工程学的各个领域中都很重要，但其仍然是物理学尚未解决的问题之一。在过去的十年中，对高维系统中混沌动力学的理解为研究湍流带来了新的发展，包括发现了Navier-Stokes方程的不变解，即精确相干态，它构成了湍流的“骨架”。.本项目的研究路径为获得无法被普通的DNS方法获得的不稳定非线性状态。我们通过沿流向/展向的傅里叶级数和沿壁面法线方向的切比雪夫多项式来表达有限振幅扰动。将它们代入Navier-Stokes方程组后，通过牛顿法对所得的扰动幅值二次方程进行数值求解。此外，我们利用扰动空间形式的对称性分析，基本流或二次流的线性稳定性分析所产生的本征模态被视为要获得的非线性解的对称性的主要组成部分。我们还使用同伦方法，从其他系统中的已知解决方案继续引入新的非线性解决方案。我们的分析还辅以DNS方法。.本项目在理解亚临界/超临界到湍流过渡方面已经取得了如下进展：.1.该项目通过总结和汇集了由同伦分析获得的平面库埃特流和管道流（Nagata，2016）中的精确相干结构， 指出了湍流转捩可以在不存在线性不稳定的情况下发生，因此基本层流状态到“突然“非线性亚临界转捩是可能的。.2.该项目开始于旋转平面库埃特流，阐明了小旋转速率下各种三维流的分叉结构。特别是，从理论上证明了倾斜涡旋结构的存在。.3. 我们的工作就Ehrenstein＆Koch（1991）在槽道流转捩的开创性论文进行了重大的修正。与他们的结论相反，我们证明在二维最小雷诺数以下，没有任何三次流存在。

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