The modal analysis techniques face difficulties in handling nonstationary
phenomena. This paper presents a variational mode decomposition-based
nonstationary coherent structure (VMD-NCS) analysis that enables the extraction
and analysis of coherent structures in case of nonstationary phenomena from
high-dimensional spatiotemporal data. The VMD-NCS analysis decomposes the input
spatiotemporal data into intrinsic coherent structures (ICSs) that represent
nonstationary spatiotemporal patterns and exhibit coherence in both the spatial
and temporal directions. Furthermore, unlike many conventional modal analysis
techniques, the proposed method accounts for the temporal changes in the
spatial distribution with time. The performance of the VMD-NCS analysis was
validated based on the transient growth phenomena in the flow around a
cylinder. It was confirmed that the temporal changes in the spatial
distribution, depicting the transient growth of vortex shedding where
fluctuations arising in the far-wake region gradually approach the near-wake
region, were represented as a single ICS. Further, in the analysis of the
quasi-periodic flow field around a pitching airfoil, the temporal changes in
the spatial distribution and the amplitude of vortex shedding behind the
airfoil, influenced by the pitching motion of the airfoil, were captured as a
single ICS. Additionally, the impact of two parameters, adjusting the number of
ICSs ($K$) and the penalty factor related to the temporal coherence ($\alpha$),
was investigated. The results revealed that $K$ has a significant impact on the
VMD-NCS analysis results. In the case of a relatively high $K$, the VMD-NCS
analysis tends to extract more periodic spatiotemporal patterns resembling the
results of dynamic mode decomposition, whereas in the case of a small $K$, the
analysis tends to extract more nonstationary spatiotemporal patterns.

该文章介绍了一种基于变分模态分解的非平稳相干结构分析方法，能够从高维时空数据中提取和分析非平稳现象中的相干结构，该方法对时空分布的时变性进行了建模，并验证了其在流体力学中的应用。

基于变分模式分解的非平稳相干结构分析方法对于时空数据

Variational Mode Decomposition-Based Nonstationary Coherent Structure  Analysis for Spatiotemporal Data

We present applications of modal analysis techniques to study, model, and
control canonical aerodynamic flows. To illustrate how modal analysis
techniques can provide physical insights in a complementary manner, we selected
four fundamental examples of cylinder wakes, wall-bounded flows, airfoil wakes,
and cavity flows. We also offer brief discussions on the outlook for modal
analysis techniques, in light of rapid developments in data science.

本文介绍了模态分析技术在研究、建模和控制典型空气动力学流体中的应用。通过选取四个基本实例，展示了模态分析技术如何以互补方式提供物理洞见。另外，针对数据科学的快速发展，对模态分析技术的展望也作了简要讨论。