Abstract
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.
Highlights
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
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 neural networks (NNs)
The number of modes k used in the reconstructions is equal to the number of probes p. This was chosen based on the underlying principles of the calculation for the optimal placement using the Q-discrete EIMs (DEIM); revealing optimal probe locations for reconstruction Method 1 (Section 3.3.1)
Summary
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. Sparse reconstruction is a technique used to obtain accurate details about the full scale features of a system using a sparse subset of information (e.g., a few pixels or measurements within the system) and has been the subject of interest for some decades (Candès, 2006; Donoho, 2006) Applications for such state estimation problems range from reconstructing faces from limited or corrupted data (Wright et al, 2008) to deblurring and improving image resolution (Dong et al, 2011) to estimating global sea surface temperatures (Manohar et al, 2018; Callaham et al, 2019). If the entire flow field in the neighborhood of a gas turbine blade is simulated and a global basis tabulated, the flow field in the neighborhood of a real turbine blade fitted with flow sensors can be estimated Such a reconstruction technique is widely applicable to any system exhibiting many degrees of freedom. For example: how do variations of nondimensional parameters that characterize a system impact the reconstruction? How many known full-state snapshots are required to generate a global basis? How many sparse sensors or probes are needed to achieve a desired reconstruction accuracy? Where should the probes be placed? These are some of the underlying questions that motivate the current study seeking to expand the application of sparse sensing to engineering problems
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