Multi-Scale Proper Orthogonal Decomposition of Complex Fluid Flows

Abstract

Data-driven decompositions are becoming essential tools in fluid dynamics, allowing for tracking the evolution of coherent patterns in large datasets, and for constructing low order models of complex phenomena. In this work, we analyze the main limits of two popular decompositions, namely the Proper Orthogonal Decomposition (POD) and the Dynamic Mode Decomposition (DMD), and we propose a novel decomposition which allows for enhanced feature detection capabilities. This novel decomposition is referred to as Multiscale Proper Orthogonal Decomposition (mPOD) and combines Multiresolution Analysis (MRA) with a standard POD. Using MRA, the mPOD splits the correlation matrix into the contribution of different scales, retaining non-overlapping portions of the correlation spectra; using the standard POD, the mPOD extracts the optimal basis from each scale. After introducing a matrix factorization framework for data-driven decompositions, the MRA is formulated via 1D and 2D filter banks for the dataset and the correlation matrix respectively. The validation of the mPOD, and a comparison with the Discrete Fourier Transform (DFT), DMD and POD are provided in three test cases. These include a synthetic test case, a numerical simulation of a nonlinear advection-diffusion problem, and an experimental dataset obtained by the Time-Resolved Particle Image Velocimetry (TR-PIV) of an impinging gas jet. For each of these examples, the decompositions are compared in terms of convergence, feature detection capabilities, and time-frequency localization.