Flow control-oriented coherent mode prediction via Grassmann-kNN manifold learning

Abstract

A data-driven method using Grassmann manifold learning is proposed to identify a low-dimensional actuation manifold for flow-controlled fluid flows. The snapshot flow field are twice compressed using Proper Orthogonal Decomposition (POD) and a diffusion model. Key steps of the actuation manifold are Grassmann manifold-based Polynomial Chaos Expansion (PCE) as the encoder and K-nearest neighbor regression (kNN) as the decoder. This methodology is first tested on a simple dielectric cylinder in a homogeneous electric field to predict the out-of-sample electric field, demonstrating fast and accurate performance. Next, the present model is evaluated by predicting dynamic coherence modes of an oscillating-rotation cylinder. The cylinder's oscillating rotation amplitude and frequency are regarded as independent control parameters. The mean mode and the first dynamic mode are selected as the representative cases to test present model. For the mean mode, the Grassman manifold describes all parameterized modes with 8 latent variables. All the modes can be divided into four clusters, and they share similar features but with different wake length. For the dynamic mode, the Grassman manifold describes all modes with 12 latent variables. All the modes can be divided into three clusters. Intriguingly, each cluster is aligned with clear physical meanings. One describes the near-wake periodic vortex shedding resembling Karman vortices, one describes the far wake periodic vortex shedding, and one shows high-frequency K-H vortices shedding. Moreover, Grassmann-kNN manifold learning can accurately predict the modes. It is possible to estimate the full flow state with small reconstruction errors just by knowing the actuation parameters. This manifold learning model is demonstrated to be crucial for flow control-oriented flow estimation.