Multiscale Proper Orthogonal Decomposition Research Articles

Multi-scale proper orthogonal decomposition of complex fluid flows

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 analyse 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 multi-scale proper orthogonal decomposition (mPOD) and combines multi-resolution 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 one- and two-dimensional 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.

Experimental analysis of the stability of the jet wiping process, part II: Multiscale modal analysis of the gas jet-liquid film interaction

This second part of a two-part article on the stability of the jet wiping process focuses on the interaction between the impinging gas jet and the coating liquid film. Using a simplified laboratory model, the first part of the study (Gosset et al., 2019) has shown that this coating process is unstable and leads to wavy coating films. The scaling laws of the problem suggest that this coating nonuniformity does not originate from the liquid coating stability itself, but from a hydrodynamic feedback between the liquid and the impinging jet. This hydrodynamic feedback is herein investigated by combining several optical techniques and data processing methods. Time-Resolved Light Absorption (LAbs) technique is used for the 3D characterization of the coating layer thickness downstream the wiping region, while Time-Resolved Particle Image Velocimetry (TR-PIV) is used to analyze the dynamics of the impinging gas jet, in combination with liquid interface tracking via image processing and Laser Induced Fluorescence (LIF). A link between the frequency content in the liquid film and the jet flow is established. Specifically, the dynamics of the gas jet is analyzed via Multiscale Proper Orthogonal Decomposition (mPOD) of the TR-PIV data and reveals large-scale oscillations which are coupled with waves in the liquid film upstream the wiping, referred to as runback flow. The interaction between the flow structures responsible for the oscillation and the runback liquid film is discussed, showing how this self-sustained mechanism is potentially responsible for the nonuniformity of the final coating.

Multiscale modal analysis of an oscillating impinging gas jet

Oscillating impinging gas jets are encountered in many flow configurations, including fluidic devices, jet flow discharges into cavities and jet impinging onto deformable interfaces. For an incompressible flow, the self-sustained mechanism is referred to as hydrodynamic feedback and it is linked to the interaction between the flow field and its geometric confinement. This work presents an experimental characterization of the flow structures responsible for the hydrodynamic feedback, considering both a steady confinement and a pulsing confinement. The selected configuration consists of a planar gas jet impinging on an asymmetric interface consisting of a flat wall on one side of the stagnation region and bidimensional bump on the other. This bump is fixed in the steady condition and pulses in the unsteady conditions. Jet oscillation and interface tracking are carried out by means of high-speed flow visualization and image processing, combined with a time-frequency analysis in the Fourier and Wavelet spaces. The flow field is characterized via Time-Resolved Particle Image Velocimetry (TR-PIV) and analyzed with a novel modal decomposition, referred to as multi-scale Proper Orthogonal Decomposition (mPOD). This decomposition uses multiresolution analysis (MRA) on the correlation matrix to produce a set of PODs at multiple scales.Two physical mechanisms of the hydrodynamic feedback have been captured at largely different scales and the dynamic of their characteristic flow structures is presented. The first, referred to as fluidic mode, consists of a self-sustained oscillation of the impinging jet; the second, referred to as coupled mode, consists of a periodic downward deflection of the impinging jet, at the frequency of the interface pulsation.

Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014

Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations

Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations