A local and hierarchical Koopman spectral analysis of fluid dynamics

IF 1.7 4区 工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Wei Zhang, Mingjun Wei
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Abstract

A local and hierarchical Koopman spectral analysis is proposed to extend Koopman spectral analysis typically used in a linear system or an ergodic process to its application in general nonlinear dynamics. The continuous and analytic Koopman eigenfunctions and eigenvalues, derived from operator perturbation theory, are capable of dealing with a nonlinear transition process with mathematical rigorousness. A proliferation rule is identified to derive high-order eigenvalues and eigenfunctions from lower-order ones, thus various spectral patterns may be generated through recursive proliferations. The locally linear map around each state constructs base local Koopman eigenvalues and eigenfunctions from an algebraic eigenvalue problem, and high-order ones are generated via the proliferation rule to express the systematic nonlinearity. The aforementioned hierarchy simplifies the Koopman spectral analysis and is verified by studying the development of Kármán vortex streets. The triangular chain and the lattice distribution of Koopman eigenvalues confirm the critical role of the proliferation rule and the hierarchy structure of Koopman eigenvalues. The local spectral analysis on the transition process shows that the periodic flow forms as the growth rates of the critical Koopman modes reduce to zero, and meanwhile, the Koopman modes at the same frequency superpose on each other to form the well-known Fourier or Floquet modes, where the latter are the enhanced nonlinear motions due to the alignment of Koopman eigenvalues with the critical ones.

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流体动力学的局部和分层库普曼谱分析
本文提出了一种局部和分层库普曼谱分析方法,将通常用于线性系统或遍历过程的库普曼谱分析方法扩展到一般非线性动力学中。从算子扰动理论推导出的连续和解析库普曼特征函数和特征值,能够以严谨的数学方法处理非线性过渡过程。从低阶特征值和特征函数导出高阶特征值和特征函数的扩散规则被确定下来,因此可以通过递归扩散产生各种频谱模式。每个状态周围的局部线性图从代数特征值问题中构造出基本的局部库普曼特征值和特征函数,并通过扩散规则生成高阶特征值和特征函数,以表达系统非线性。上述层次结构简化了库普曼谱分析,并通过研究卡尔曼涡街的发展得到了验证。库普曼特征值的三角链和晶格分布证实了扩散规则和库普曼特征值层次结构的关键作用。对过渡过程的局部频谱分析表明,当临界库普曼模式的增长率减小到零时,周期性流动就形成了,与此同时,相同频率的库普曼模式相互叠加,形成了众所周知的傅里叶模式或 Floquet 模式,后者是由于库普曼特征值与临界特征值一致而产生的增强非线性运动。
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来源期刊
International Journal for Numerical Methods in Fluids
International Journal for Numerical Methods in Fluids 物理-计算机:跨学科应用
CiteScore
3.70
自引率
5.60%
发文量
111
审稿时长
8 months
期刊介绍: The International Journal for Numerical Methods in Fluids publishes refereed papers describing significant developments in computational methods that are applicable to scientific and engineering problems in fluid mechanics, fluid dynamics, micro and bio fluidics, and fluid-structure interaction. Numerical methods for solving ancillary equations, such as transport and advection and diffusion, are also relevant. The Editors encourage contributions in the areas of multi-physics, multi-disciplinary and multi-scale problems involving fluid subsystems, verification and validation, uncertainty quantification, and model reduction. Numerical examples that illustrate the described methods or their accuracy are in general expected. Discussions of papers already in print are also considered. However, papers dealing strictly with applications of existing methods or dealing with areas of research that are not deemed to be cutting edge by the Editors will not be considered for review. The journal publishes full-length papers, which should normally be less than 25 journal pages in length. Two-part papers are discouraged unless considered necessary by the Editors.
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