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The l1-based sparsification of energy interactions in unsteady lid-driven cavity flow

Published online by Cambridge University Press:  26 October 2020

Riccardo Rubini Open the ORCID record for Riccardo Rubini [Opens in a new window] ,
Davide Lasagna Open the ORCID record for Davide Lasagna [Opens in a new window]  and
Andrea Da Ronch Open the ORCID record for Andrea Da Ronch [Opens in a new window]
Riccardo Rubini
Affiliation:
Faculty of Engineering and Physical Sciences, University of Southampton, SO17 1BJSouthampton, UK
Davide Lasagna*
Affiliation:
Faculty of Engineering and Physical Sciences, University of Southampton, SO17 1BJSouthampton, UK
Andrea Da Ronch
Affiliation:
Faculty of Engineering and Physical Sciences, University of Southampton, SO17 1BJSouthampton, UK
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, sparsity-promoting regression techniques are employed to automatically identify from data relevant triadic interactions between modal structures in large Galerkin-based models of two-dimensional unsteady flows. The approach produces interpretable, sparsely connected models that reproduce the original dynamical behaviour at a much lower computational cost, as fewer triadic interactions need to be evaluated. The key feature of the approach is that dominant interactions are selected systematically from the solution of a convex optimisation problem, with a unique solution, and no a priori assumptions on the structure of scale interactions are required. We demonstrate this approach on models of two-dimensional lid-driven cavity flow at Reynolds number $Re = 2 \times 10^{4}$, where fluid motion is chaotic. To understand the role of the subspace utilised for the Galerkin projection in the sparsity characteristics, we consider two families of models obtained from two different modal decomposition techniques. The first uses energy-optimal proper orthogonal decomposition modes, while the second uses modes oscillating at a single frequency obtained from discrete Fourier transform of the flow snapshots. We show that, in both cases, and despite no a priori physical knowledge being incorporated into the approach, relevant interactions across the hierarchy of modes are identified in agreement with the expected picture of scale interactions in two-dimensional turbulence. Yet, substantial structural changes in the interaction pattern and a quantitatively different sparsity are observed. Finally, although not directly enforced in the procedure, the sparsified models have excellent long-term stability properties and correctly reproduce the spatio-temporal evolution of dominant flow structures in the cavity.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A15
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Rubini et al. Supplementary Material

Time evolution of the reconstructed vorticity field obtained from the l-1 sparsified model (center) and from the dense model obtained by Galerkin projection (right), compared with DNS evolution (left).

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