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

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.

Time-series image denoising of pressure-sensitive paint data by projected multivariate singular spectrum analysis

Time-series data, such as unsteady pressure-sensitive paint (PSP) measurement data, may contain a significant amount of random noise. Thus, in this study, we investigated a noise-reduction method that combines multivariate singular spectrum analysis (MSSA) with low-dimensional data representation. MSSA is a state-space reconstruction technique that utilizes time-delay embedding, and the low-dimensional representation is achieved by projecting data onto the singular value decomposition (SVD) basis. The noise-reduction performance of the proposed method for unsteady PSP data, i.e., the projected MSSA, is compared with that of the truncated SVD method, one of the most employed noise-reduction methods. The result shows that the projected MSSA exhibits better performance in reducing random noise than the truncated SVD method. Additionally, in contrast to that of the truncated SVD method, the performance of the projected MSSA is less sensitive to the truncation rank. Furthermore, the projected MSSA achieves denoising effectively by extracting smooth trajectories in a state space from noisy input data. Expectedly, the projected MSSA will be effective for reducing random noise in not only PSP measurement data, but also various high-dimensional time-series data.

Stable Dynamic Mode Decomposition Algorithm for Noisy Pressure-Sensitive Paint Measurement Data

In this study, we proposed the truncated total least squares dynamic mode decomposition (T-TLS DMD) algorithm, which can perform DMD analysis of noisy data. By adding truncation regularization to the conventional TLS DMD algorithm, T-TLS DMD improves the stability of the computation while maintaining the accuracy of TLS DMD. The effectiveness of the proposed method was evaluated by the analysis of the wake behind a cylinder and pressure-sensitive paint (PSP) data for the buffet cell phenomenon. The results showed the importance of regularization in the DMD algorithm. With respect to the eigenvalues, T-TLS DMD was less affected by noise, and accurate eigenvalues could be obtained stably, whereas the eigenvalues of TLS and subspace DMD varied greatly due to noise. It was also observed that the eigenvalues of the standard and exact DMD had the problem of shifting to the damping side, as reported in previous studies. With respect to eigenvectors, T-TLS and exact DMD captured the characteristic flow patterns clearly even in the presence of noise, whereas TLS and subspace DMD were not able to capture them clearly due to noise.