Soledad Le Clainche

Higher Order Dynamic Mode Decomposition

This paper deals with an extension of dynamic mode decomposition (DMD), which is appropriate to treat general periodic and quasi-periodic dynamics, and transients decaying to periodic and quasi-periodic attractors, including cases (not accessible to standard DMD) that show limited spatial complexity but a very large number of involved frequencies. The extension, labeled as higher order dynamic mode decomposition, uses time-lagged snapshots and can be seen as superimposed DMD in a sliding window. The new method is illustrated and clarified using some toy model dynamics, the Stuart--Landau equation, and the Lorenz system. In addition, the new method is applied to (and its robustness is tested in) some permanent and transient dynamics resulting from the complex Ginzburg--Landau equation (a paradigm of pattern forming systems), for which standard DMD is seen to only uncover trivial dynamics, and the thermal convection in a rotating spherical shell subject to a radial gravity field.

Higher order dynamic mode decomposition to identify and extrapolate flow patterns

This article shows the capability of using a higher order dynamic mode decomposition (HODMD) algorithm both to identify flow patterns and to extrapolate a transient solution to the attractor region. Numerical simulations are carried out for the three-dimensional flow around a circular cylinder, and both standard dynamic mode decomposition (DMD) and higher order DMD are applied to the non-converged solution. The good performance of HODMD is proved, showing that this method guesses the converged flow patterns from numerical simulations in the transitional region. The solution obtained can be extrapolated to the attractor region. This fact sheds light on the capability of finding real flow patterns in complex flows and, simultaneously, reducing the computational cost of the numerical simulations or the required quantity of data collected in experiments.

A reduced-order model for compressible flows with buffeting condition using higher order dynamic mode decomposition with a mode selection criterion

This study proposes an improvement in the performance of reduced-order models (ROMs) based on dynamic mode decomposition to model the flow dynamics of the attractor from a transient solution. By combining higher order dynamic mode decomposition (HODMD) with an efficient mode selection criterion, the HODMD with criterion (HODMDc) ROM is able to identify dominant flow patterns with high accuracy. This helps us to develop a more parsimonious ROM structure, allowing better predictions of the attractor dynamics. The method is tested in the solution of a NACA0012 airfoil buffeting in a transonic flow, and its good performance in both the reconstruction of the original solution and the prediction of the permanent dynamics is shown. In addition, the robustness of the method has been successfully tested using different types of parameters, indicating that the proposed ROM approach is a tool promising for using in both numerical simulations and experimental data.

Structural analysis on a hemisphere-cylinder at moderate Reynolds number and high angle of attack

Three-dimensional DNS has been performed on a hemisphere-cylinder at Reynolds number Re= 1000 and angle of attack AoA= 20◦ in order to analyze flow structures and wake frequencies. PIV experiments have also been carried out over the same geometry and flow conditions to validate the numerical results. Critical point theory has been applied in order to determine the topology patterns over the surface of the body. Critical points and separation lines on the body surface show the presence of three different flow patterns: separation bubble, ”horn vortices” and ”leeward vortices”. Both, ”horn vortices” and ”leeward vortices” are found to be asymmetric and unsteady. The frequency related to ”leeward vortices” oscillations has been identified both experimentally and numerically. Two more dominant frequencies, related to two different wake shedding modes have been found. On the other hand, POD has been performed and the four most energetic POD modes is found to be composed of a mixture of these three frequencies. They are modes on the wake shedding. Finally, DMD modes associated with these three frequencies are found on the symmetry plane close to the nose area. These modes represent different shear layer instabilities. Flow separation was found to be intrinsically linked with the observed shear-layer instability.

Flow Scales in Cross-Flow Turbines

This work presents analytical estimates for various flow scales encountered in cross-flow turbines (i.e. Darrieus type or vertical axis) for renewable energy generation (both wind and tidal). These estimates enable the exploration of spatial or temporal interactions between flow phenomena and provide quantitative and qualitative bounds of the three main flow phenomena: the foil scale, the vortex scale and wake scale. Finally using the scale analysis, we show using an illustrative example how high order computational methods prove beneficial when solving the flow physics involved in cross-flow turbines.

Spatio-Temporal Koopman Decomposition

This paper deals with a new purely data-driven method, called the spatio-temporal Koopman decomposition, to approximate spatio-temporal data as a linear combination of (possibly growing or decaying exponentially) standing or traveling waves. The method combines (i) either standard singular value decomposition (SVD) or higher-order SVD and (ii) either standard dynamic mode decomposition (DMD) or an extension of this method by the authors, called higher-order DMD. In particular, for periodic or quasiperiodic attractors, the method gives the spatio-temporal pattern as a superposition of standing and/or traveling waves, which are identified in an efficient and robust way. Such superposition may give the whole pattern as a modulated, periodic or quasiperiodic, standing or traveling wave. The method is illustrated in some simple toy-model dynamics, and its performance is tested in the identification of standing and traveling waves in the Ginzburg–Landau equation and of azimuthal waves in a rotating spherical shell with thermal convection.

MATHEMATICS APPLIED TO ENGINEERING PROCESSES: A PRACTICAL GUIDE TO INCREASE STUDENTS' MOTIVATION

The main goal of this work is to motivate undergraduate students, following mechanical and aerospace engineering subjects, to increase their interest and deepen their knowledge in fundamental sciences such as mathematics. To this end, we present a practical course of ten hours that will show the students the link between mathematics and engineering and will also show them how important is to study fundamental mathematics and how to solve real practical problems. This course will not only increase the motivation of the students in maths, but also will help them to develop some additional important skills such as practical thinking or the ability of linking and applying theory to practice.