Symmetry reduction in high dimensions, illustrated in a turbulent pipe.
Symmetry reduction in high dimensions, illustrated in a turbulent pipe.
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高维对称性降低，如湍流管道所示。
DOI：
10.1103/physreve.93.022204
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Willis AP
中科院分区：
文献类型：
--
作者：
Willis AP
平衡解被认为构成了动力系统中遍历轨迹的路径。然而，对于具有连续对称性的系统，即具有均匀空间维度的系统，平衡是非典型的，而相对平衡(行波)是通用的。为了可视化这类解的不稳定流形，需要一种实用的对称性约化方法，将相对平衡转化为平衡，将相对周期轨道转化为周期轨道。在这篇文章中，我们将以前应用的一维偏微分方程组的固定傅立叶模切片方法扩展到空间三维流体流动，并表明它比我们以前的方法更有效。将这种方法应用于最小流动单元管道，发现了许多相对周期轨道，这些轨道似乎填充了状态空间的湍流区。我们通过揭示这些相对周期轨道之间的相互关系和它们塑造湍流吸引子几何形状的方式的投影(只有在对称缩减的空间中才可能的投影)进一步证明了这种方法对于对称性约化的价值。
Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereasrelativeequilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.