Proper orthogonal decomposition-based spatial refinement of TR-PIV realizations using high-resolution non-TR-PIV measurements

A proper orthogonal decomposition (POD)-based spatial refinement approach is proposed to increase the resolution of time-resolved particle image velocimetry (TR-PIV) realizations. POD analysis of the high-resolution non-TR-PIV measurements is used to construct the high-resolution POD modes, and the TR-PIV instantaneous realization with relatively low spatial resolution is cross-projected onto the POD modes to obtain the time-varying mode coefficients. Subsequently, the high-resolution time-resolved flow fields, which inherit the information of the superimposed multi-scale structures from the non-TR-PIV measurements, are reconstructed using the linear combination of the cross-projected mode coefficients and the corresponding high-resolution POD modes. Two main sources of error in the novel strategy are discussed: the cross-projected reconstruction of the instantaneous fields, which are excluded in the determination of the high-resolution POD modes, and the interpolation that maps the high-resolution POD modes onto the coarse grid in the cross-projection process when estimating the mode coefficients. To evaluate the method, the flow fields of a free round jet at Reynolds number Re\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Re$$\end{document} = 3000 are separately measured in the same field-of-view region using non-TR-PIV and TR-PIV setups with a spatial resolution ratio of 42:1. The high- and low-resolution realizations of the instantaneous flow fields measured by the non-TR-PIV setup are determined by applying fine (32 × 32 pixels) and coarse (128 × 128 pixels) interrogation windows, respectively, to ensure accurate fine-to-coarse mapping of the POD modes and for use as a reference to assess the refinement accuracy. This evaluation shows that the combination of these two error sources can be minimized when the POD-sample size (the number of snapshots employed in the decomposition) and the number of POD modes for refinement are optimized. Finally, application of the present method to TR-PIV data shows that the high-resolution time-resolved fields are successfully achieved with the temporal pattern similar to the original TR-PIV realizations, while much smaller structures are captured by the spatially refined flow.