General proper orthogonal decomposition, modal correction, inter-modal interactions, and sequencing
University of New Brunswick
This study revisits key points, and develops new tools for a better understanding of the Proper Orthogonal Decomposition method (POD), a popular approach to educate researchers about the underlying mechanisms in fluid flow. First, the implementation of a Jacobian matrix is revised to develop a general form that is implementable on arbitrary grids, coordinate systems, and variations of POD with ease. Using the general approach developed herein, the scalar and vectorial kernel are analytically compared to show the superiority of the latter. The commonly accepted concept of POD equivalency with the Fourier transform in homogeneous directions is challenged using an analytical test function to show it does not always hold true. Moreover, a summary of the practical difficulties of the Fourier transform are presented and a discussion of how POD overcomes these difficulties is presented. Next, tools for selecting the number of modes for low-dimensional reconstruction and choosing the best instances to represent such reconstructions are developed. Then, the eigenvalue degeneracy and its effect on the globality of POD modes are investigated for the first time. A correction method has been developed to increase the locality of the modes through the use of a blind source separation algorithm and eigenvalue uncertainty. Furthermore, two new criteria are developed to discover the modes associated with traveling waves with the least possible assumptions. Finally, a novel method is developed to provide sequences of instances from a non-time-resolved dataset.