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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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References

REFERENCES

Arbabi, H. & Mezić, I. 2017 Study of dynamics in post-transient flows using Koopman mode decomposition. Phys. Rev. Fluids 2 (12), 124402.CrossRefGoogle Scholar
Auteri, F., Parolini, N. & Quartapelle, L. 2002 Numerical investigation on the stability of singular driven cavity flow. J. Comput. Phys. 183 (1), 1–25.CrossRefGoogle Scholar
Balajewicz, M., Dowell, E. & Noack, B. 2013 Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285–308.CrossRefGoogle Scholar
Blum, A. L. & Langley, P. 1997 Selection of relevant features and examples in machine learning. Artif. Intell. 97 (1–2), 245–271.CrossRefGoogle Scholar
Boppana, V. B. L. & Gajjar, J. S. B. 2010 Global flow instability in a lid-driven cavity. Intl J. Numer. Meth. Fluids 62 (8), 827–853.Google Scholar
Botella, O. & Peyret, R. 1998 Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27 (4), 421–433.CrossRefGoogle Scholar
Brasseur, J. & Wei, C. 1994 Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interactions. Phys. Fluids 6, 842–870.CrossRefGoogle Scholar
Brunton, S. L., Noack, B. R. & Koumoutsakos, P. 2019 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508.CrossRefGoogle Scholar
Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113 (15), 3932–3937.CrossRefGoogle ScholarPubMed
Cazemier, W., Verstappen, R. W. C. P. & Veldman, A. E. P. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10 (7), 1685–1699.CrossRefGoogle Scholar
Chen, K. K., Tu, J. H. & Rowley, C. W. 2012 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887–915.CrossRefGoogle Scholar
Cordier, L., El Majd, B. A. & Favier, J. 2010 Calibration of POD reduced-order models using Tikhonov regularization. Intl J. Numer. Meth. Fluids 63 (2), 269–296.Google Scholar
Couplet, M., Basdevant, C. & Sagaut, P. 2005 Calibrated reduced-order POD-Galerkin system for fluid flow modelling. J. Comput. Phys. 207 (1), 192–220.CrossRefGoogle Scholar
Couplet, M., Sagaut, P. & Basdevant, C. 2003 Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow. J. Fluid Mech. 491, 275–284.CrossRefGoogle Scholar
Fick, L., Maday, Y., Patera, A. T. & Taddei, T. 2018 A stabilized POD model for turbulent flows over a range of Reynolds numbers: optimal parameter sampling and constrained projection. J. Comput. Phys. 371, 214–243.CrossRefGoogle Scholar
Fletcher, C. A. J. 1984 Computational Galerkin Methods, 1st edn. Springer.CrossRefGoogle Scholar
Friedman, J., Hastie, T. & Tibshirani, R. J. 2008 The Elements of Statistical Learning. Springer.Google Scholar
Hastie, T., Tibshirani, R. & Wainwright, M. 2015 Statistical Learning with Sparsity: The LASSO and Generalizations. Chapman and Hall/CRC.CrossRefGoogle Scholar
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.CrossRefGoogle Scholar
Jovanović, M. R., Schmid, P. J. & Nichols, J. W. 2014 Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26 (2), 024103.CrossRefGoogle Scholar
Kaiser, E., Kutz, J. N. & Brunton, S. L. 2018 Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proc. Math. Phys. Engng Sci. 474 (2219), 20180335.Google ScholarPubMed
Kraichnan, R. H. 1971 Inertial-range transfer in two-and three-dimensional turbulence. J. Fluid Mech. 47 (3), 525–535.CrossRefGoogle Scholar
Laval, J. P., Dubrulle, B. & Nazarenko, S. 1999 Nonlocality of interaction of scales in the dynamics of 2D incompressible fluids. Phys. Rev. Lett. 83 (20), 4061.CrossRefGoogle Scholar
Loiseau, J. C. & Brunton, S. L. 2018 Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42–67.CrossRefGoogle Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Lumley, J. L. 1979 Computational modeling of turbulent flows. In Advances in Applied Mechanics, vol. 18, pp. 123–176. Elsevier.Google Scholar
Mendez, M. A., Balabane, M. & Buchlin, J.-M. 2019 Multi-scale proper orthogonal decomposition of complex fluid flows. J. Fluid Mech. 870, 988–1036.CrossRefGoogle Scholar
Mezić, I. 2013 Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378.CrossRefGoogle Scholar
Nair, A. G., Brunton, S. L. & Taira, K. 2018 Networked-oscillator-based modeling and control of unsteady wake flows. Phys. Rev. E 97, 063107.CrossRefGoogle ScholarPubMed
Nair, A. G. & Taira, K. 2015 Network-theoretic approach to sparsified discrete vortex dynamics. J. Fluid Mech. 768, 549–571.CrossRefGoogle Scholar
Newman, M. 2018 Networks. Oxford University Press.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Noack, B., Morzynski, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control, vol. 528. Springer Science & Business Media.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
Noack, B. R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P. & Tadmor, G. 2008 A finite-time thermodynamics of unsteady fluid flows. J. Non-Equil. Thermodyn. 33 (2), 103–148.Google Scholar
Noack, B. R., Stankiewicz, W., Morzyński, M. & Schmid, P. J. 2016 Recursive dynamic mode decomposition of transient and post-transient wake flows. J. Fluid Mech. 809, 843–872.CrossRefGoogle Scholar
Ohkitani, K. 1990 Nonlocality in a forced two-dimensional turbulence. Phys. Fluids A 2 (9), 1529–1531.CrossRefGoogle Scholar
Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al. . 2011 Scikit-learn: machine learning in Python. J. Machine Learning Res. 12, 2826–2830.Google Scholar
Peng, Y. H., Shiau, Y. H. & Hwang, R. R. 2003 Transition in a 2-D lid-driven cavity flow. Comput. Fluids 32 (3), 337–352.CrossRefGoogle Scholar
Perret, L., Collin, E. & Delville, J. 2006 Polynomial identification of POD based low-order dynamical system. J. Turbul. 7, N17.CrossRefGoogle Scholar
Poliashenko, M. & Aidun, C. K. 1995 A direct method for computation of simple bifurcations. J. Comput. Phys. 121 (2), 246–260.CrossRefGoogle Scholar
Pope, S. B. 2001 Turbulent Flows. Cambridge University Press.Google Scholar
Ramirez, C., Kreinovich, V. & Argaez, M. 2013 Why $l_1$ is a good approximation to $l_0$: a geometric explanation. J. Uncertain Syst. 7 (3), 203–207.Google Scholar
Rempfer, D. & Fasel, H. F. 1994 a Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 275, 257–283.CrossRefGoogle Scholar
Rempfer, D. & Fasel, H. F. 1994 b Evolution of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 260, 351–375.CrossRefGoogle Scholar
Rowley, C. W. & Dawson, S. T. M. 2017 Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49, 387–417.CrossRefGoogle Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127.CrossRefGoogle Scholar
Schlegel, M. & Noack, B. R. 2015 On long-term boundedness of Galerkin models. J. Fluid Mech. 765, 325–352.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schmidt, O. T. 2020 Bispectral mode decomposition of nonlinear flows. arXiv:2002.04146.Google Scholar
Schmidt, M. & Lipson, H. 2009 Distilling free-form natural laws from experimental data. Science 324 (5923), 81–85.CrossRefGoogle ScholarPubMed
Sieber, M., Paschereit, C. O. & Oberleithner, K. 2016 Spectral proper orthogonal decomposition. J. Fluid Mech. 792, 798–828.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structure. Part I, II, III. Q. Appl. Maths 3, 583.CrossRefGoogle Scholar
Taira, K., Brunton, S. L., Dawson, S. T. M., Rowley, C. W., Colonius, T., McKeon, B. J., Schmidt, O. T., Gordeyev, S., Theofilis, V. & Ukeiley, L. S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55, 4013–4041.CrossRefGoogle Scholar
Taira, K., Nair, A. G. & Brunton, S. L. 2016 Network structure of two-dimensional decaying isotropic turbulence. J. Fluid Mech. 795, R2.CrossRefGoogle Scholar
Terragni, F., Valero, E. & Vega, J. M. 2011 Local POD plus Galerkin projection in the unsteady lid-driven cavity problem. SIAM J. Sci. Comput. 33 (6), 3538–3561.CrossRefGoogle Scholar
Thomas, V. L., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27 (10), 105104.CrossRefGoogle Scholar
Thomas, V. L., Lieu, B. K., Jovanović, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26 (10), 105112.CrossRefGoogle Scholar
Tibshirani, R. 1996 Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. B 58 (1), 267–288.Google Scholar
Tibshirani, R. J. 2013 The LASSO problem and uniqueness. Electron. J. Stat. 7, 1456–1490.CrossRefGoogle Scholar
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.CrossRefGoogle Scholar
Xie, X., Mohebujjaman, M., Rebholz, L. G. & Iliescu, T. 2018 Data-driven filtered reduced order modeling of fluid flows. SIAM J. Sci. Comput. 40 (3), B834–B857.CrossRefGoogle Scholar
Yeung, P. K., Brasseur, J. G. & Wang, Q. 1995 Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and fourier-space views. J. Fluid Mech. 283, 43–95.CrossRefGoogle Scholar
Zhang, L. & Schaeffer, H. 2019 On the convergence of the SINDy algorithm. Multiscale Model. Simul. 17 (3), 948–972.CrossRefGoogle Scholar

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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