The conventional modal analysis techniques face difficulties in handling nonstationary phenomena, such as transient, nonperiodic, or intermittent 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 the 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 spatial and temporal directions. Unlike many conventional modal analysis techniques, the proposed method accounts for the temporal changes in the spatial distribution with time. Tthe 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. Furthermore, 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. The impact of two parameters that control 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. In the case of a small $K$, it tends to extract more nonstationary spatiotemporal patterns.

翻译：传统模态分析技术在处理瞬态、非周期或间歇性等非平稳现象时面临困难。本文提出一种基于变分模态分解的非平稳相干结构分析方法（VMD-NCS），能够从高维时空数据中提取并分析非平稳现象中的相干结构。VMD-NCS分析将输入时空数据分解为本征相干结构（ICS），这些结构表征非平稳时空模式，并在空间和时间维度上均展现相干性。与多种传统模态分析技术不同，本方法考虑了空间分布随时间的变化。基于圆柱绕流瞬态增长现象的验证表明，描述涡脱落瞬态增长（远尾区波动逐渐逼近近尾区）的空间分布时间变化可被表征为单一本征相干结构。此外，在俯仰机翼准周期流场分析中，受机翼俯仰运动影响而变化的机翼后方涡脱落空间分布与振幅的时间演化，也被捕捉为单一本征相干结构。本文进一步研究了控制本征相干结构数量的参数$K$和与时间相干性相关的惩罚因子$\alpha$的影响，结果表明$K$对VMD-NCS分析结果具有显著作用：当$K$值较高时，方法倾向于提取类似动态模态分解结果的周期性时空模式；当$K$值较低时，则更倾向于提取非平稳时空模式。