Spectral measures with residual DMD for identifying intended-observable-relevant flow structures

Dynamic Mode Decomposition (DMD) is a data-driven analysis technique used to extract structures and dynamic information from data. DMD identifies point spectra of the system, enabling the extraction of modes associated with specific frequencies. However, applying DMD to complex flows with broadband continuous spectra, such as turbulent flows, remains challenging due to the difficulty in handling continuous spectra. To address this limitation, Residual DMD (ResDMD) has been recently proposed as an extended framework capable of incorporating continuous spectra.

In this study, we develop a novel data analysis method that enables the identification of flow structures relevant to specific observables of interest (e.g., lift and drag), leveraging the spectral measures with ResDMD that include continuous spectral information. The proposed method is demonstrated on a three-dimensional flow past a circular cylinder at a Reynolds number of 200, where Mode A emerges and the system exhibits continuous spectra. In the proposed method, we first apply Hankel DMD to the three-dimensional flow data, and the residuals of the resulting eigenvalues are evaluated to remove spectral pollution. Then, spectral measures are estimated using the lift and drag as the intended observables. By reconstructing the flow fields from the spectral sets based on the obtained spectral measures, we successfully identify the flow structures that contribute to the generation of pure lift and drag separately.