Proper orthogonal decomposition-based estimations of the flow field from particle image velocimetry wall-gradient measurements in the backward-facing step flow

In this paper, particle image velocimetry (PIV) results from the recirculation zone of a backward-facing step flow, of which the Reynolds number is 2800 based on bulk velocity upstream of the step and step height (h = 16.5 mm), are used to demonstrate the capability of proper orthogonal decomposition (POD)-based measurement models. Three-component PIV velocity fields are decomposed by POD into a set of spatial basis functions and a set of temporal coefficients. The measurement models are built to relate the low-order POD coefficients, determined from an ensemble of 1050 PIV fields by the 'snapshot' method, to the time-resolved wall gradients, measured by a near-wall measurement technique called stereo interfacial PIV. These models are evaluated in terms of reconstruction and prediction of the low-order temporal POD coefficients of the velocity fields. In order to determine the estimation coefficients of the measurement models, linear stochastic estimation (LSE), quadratic stochastic estimation (QSE), principal component regression (PCR) and kernel ridge regression (KRR) are applied. We denote such approaches as LSE-POD, QSE-POD, PCR-POD and KRR-POD. In addition to comparing the accuracy of measurement models, we introduce multi-time POD-based estimations in which past and future information of the wall-gradient events is used separately or combined. The results show that the multi-time estimation approaches can improve the prediction process. Among these approaches, the proposed multi-time KRR-POD estimation with an optimized window of past wall-gradient information yields the best prediction. Such a multi-time KRR-POD approach offers a useful tool for real-time flow estimation of the velocity field based on wall-gradient data.