Data-driven sparse reconstruction of flow over a stalled aerofoil using experimental data

Abstract Recent work has demonstrated the use of sparse sensors in combination with the proper orthogonal decomposition (POD) to produce data-driven reconstructions of the full velocity fields in a variety of flows. The present work investigates the fidelity of such techniques applied to a stalled NACA 0012 aerofoil at $ {Re}_c=75,000 $ at an angle of attack $ \alpha ={12}^{\circ } $ as measured experimentally using planar time-resolved particle image velocimetry. In contrast to many previous studies, the flow is absent of any dominant shedding frequency and exhibits a broad range of singular values due to the turbulence in the separated region. Several reconstruction methodologies for linear state estimation based on classical compressed sensing and extended POD methodologies are presented as well as nonlinear refinement through the use of a shallow neural network (SNN). It is found that the linear reconstructions inspired by the extended POD are inferior to the compressed sensing approach provided that the sparse sensors avoid regions of the flow with small variance across the global POD basis. Regardless of the linear method used, the nonlinear SNN gives strikingly similar performance in its refinement of the reconstructions. The capability of sparse sensors to reconstruct separated turbulent flow measurements is further discussed and directions for future work suggested. Impact Statement Sparse reconstruction of full-field information using a limited subset of data is widely relevant to data-centric engineering applications; from reconstructing human faces with limited pixels to predicting laminar and turbulent flow fields from limited sensors. The focus of the present study is of the latter example with high relevance to active flow control in aerospace and related industry. There are multiple data-driven methodologies for obtaining flow field reconstructions from sparse measurements ranging from the linear unsupervised proper orthogonal decomposition to the use of nonlinear supervised NNs. The feasibility of such methods to flow fields that are highly turbulent as well as obtained via experiment remains an open area of research. The present study reveals the capability of these techniques to create a time-invariant library that can predict instantaneous states of the flow from sparse measurements alone (provided that these states are within the bounds of the applied training data). The proposed linear methods, as well as the NN architecture, provide well-characterized frameworks for future efforts in sparse sensing and state estimation applications: particularly for highly nonlinear underlying systems such as turbulent flow.