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

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.