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

Yuya Ohmichi

TL;DR

The paper introduces VMD-NCS, a data-driven framework that blends POD-based dimensionality reduction with multivariate variational mode decomposition to extract intrinsic coherent structures (ICSs) that can evolve in time in both their spatial distribution and amplitude. By applying MVMD to POD coefficients, ICSs capture nonstationary phenomena in high-dimensional flows, such as transient wake growth behind a square cylinder and vortex shedding modulated by pitching in an airfoil, providing a single, time-evolving representation of complex dynamics. The results demonstrate that ICSs can separate multiple nonstationary phenomena more effectively than conventional POD or DMD approaches, with the parameter $K$ (number of ICSs) having a pronounced impact on the balance between nonstationary and quasi-periodic content, while $\alpha$ controls temporal coherence. This approach enhances interpretability and offers a potential path for improved prediction and control of nonstationary fluid flows.

Abstract

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 ($α$), 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.

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

TL;DR

The paper introduces VMD-NCS, a data-driven framework that blends POD-based dimensionality reduction with multivariate variational mode decomposition to extract intrinsic coherent structures (ICSs) that can evolve in time in both their spatial distribution and amplitude. By applying MVMD to POD coefficients, ICSs capture nonstationary phenomena in high-dimensional flows, such as transient wake growth behind a square cylinder and vortex shedding modulated by pitching in an airfoil, providing a single, time-evolving representation of complex dynamics. The results demonstrate that ICSs can separate multiple nonstationary phenomena more effectively than conventional POD or DMD approaches, with the parameter (number of ICSs) having a pronounced impact on the balance between nonstationary and quasi-periodic content, while controls temporal coherence. This approach enhances interpretability and offers a potential path for improved prediction and control of nonstationary fluid flows.

Abstract

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 () and the penalty factor related to the temporal coherence (), was investigated. The results revealed that has a significant impact on the VMD-NCS analysis results. In the case of a relatively high , the VMD-NCS analysis tends to extract more periodic spatiotemporal patterns resembling the results of dynamic mode decomposition. In the case of a small , it tends to extract more nonstationary spatiotemporal patterns.
Paper Structure (24 sections, 22 equations, 15 figures, 1 algorithm)

This paper contains 24 sections, 22 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Variational Mode Decomposition
  4. Multivariate Variational Mode Decomposition
  5. Alternate Direction Method of Multipliers for Multivariate Variational Mode Decomposition
  6. Variational Mode Decomposition--Based Nonstationary Coherent Structure Analysis
  7. Input data
  8. Dimensionality reduction
  9. Intrinsic coherent structure
  10. Amplitude of the intrinsic coherent structure
  11. Setting parameters
  12. Transient wake behind a cylinder
  13. Problem Setting and Input data
  14. Results and Discussion
  15. Extracted coherent structures ($K = 5,~\alpha = 100$)
  16. ...and 9 more sections

Figures (15)

  • Figure 1: Time history of the velocity in the $x$-direction at $(x,~y)=(5,~0.5)$ and spatial distributions of the velocity in the $x$-direction at time $t = 100$, 150, and 200.
  • Figure 2: (a) Time history of the ICS amplitudes and (b) time-frequency distribution of the $x$-direction velocity component at position $(x,~y)=(5,~0.5)$ for each ICS. The shade of the points in (b) represents the amplitude. $K = 5,~\alpha = 100$.
  • Figure 3: Instantaneous spatial distributions of the input flow field (with mean components) and the ICSs during the transient growth period. Input flow field (left), ICS of $St_c=0.112$ (middle), and ICS of $St_c=0.147$ (right). The $x$-direction velocity component is shown.
  • Figure 4: Instantaneous spatial distributions of the ICSs during the transient growth period. ICS of $St_c=0.003$ (left), $St_c=0.289$ (middle), and $St_c=0.443$ (right). The $x$-direction velocity component is shown.
  • Figure 5: The spatial distribution of the $x$-direction velocity component for the (a) first, (b) third, (c) fifth, (d) seventh, and (e) ninth POD modes.
  • ...and 10 more figures