Sparse sensor-based flow estimation with spectral proper orthogonal decomposition

The application of Artificial Neural Networks (ANNs) in developing sensor-based estimators for unsteady flows has become an active area of research over the last decade. One of the challenges in this area is the selection of a low-dimensional subspace that enables the ANN to reconstruct relevant spatiotemporal dynamics in the flow, as both sparsity and interpretability are simultaneously desired. The present study demonstrates the use of a flow-estimation framework based on Long Short-Term Memory (LSTM) neural networks and the time-domain Spectral Proper Orthogonal Decomposition (SPOD) [Sieber et al., “Spectral proper orthogonal decomposition,” J. Fluid Mech. 792, 798–828 (2016)], which was proposed as an extension to the traditional POD. The two-cylinder flow selected for analysis in this study is referred to as the “flip-flop” regime for exhibiting intermittent changes in the phase alignment of the vortices shed by the two cylinders. This flow has dynamics occurring over a wide range of frequencies, and it was selected to demonstrate the usefulness of SPOD in characterizing the wake dynamics of periodic wake flows and to determine if it can provide a better subspace for training and estimation when compared to POD. It was found that a particular SPOD basis obtained with an empirically determined filter length completely separated the frequency centered phenomena present in the spectrum of the most energetic POD modes into different modes. These new SPOD modes were observed to have a direct relationship with the vortex dynamics in the flow, providing direct access to the antiphase and in-phase flow states. The LSTM neural networks estimation capacities were very similar across all the modal spaces investigated, performing well regardless of whether the frequency content of the modal space used for training and estimation was found superimposed in the spectrum of the most energetic modes (POD) or separated into different modes (SPOD). Further investigation is required to determine if this result holds for turbulent flows.