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Topological colouring of fluid particles unravels finite-time coherent sets

Published online by Cambridge University Press:  27 July 2021

Gisela D. Charó Open the ORCID record for Gisela D. Charó [Opens in a new window] ,
Guillermo Artana Open the ORCID record for Guillermo Artana [Opens in a new window]  and
Denisse Sciamarella Open the ORCID record for Denisse Sciamarella [Opens in a new window]
Gisela D. Charó*
Affiliation:
CONICET – Universidad de Buenos Aires, Centro de Investigaciones del Mar y la Atmósfera (CIMA), C1428EGA CABA, Argentina CNRS – IRD – CONICET – UBA, Institut Franco-Argentin d’Ètudes sur le Climat et ses Impacts (IRL 3351 IFAECI), C1428EGA CABA, Argentina CONICET – Consejo Nacional de Investigaciones Científicas y Técnicas, C1425FQD CABA, Argentina
Guillermo Artana
Affiliation:
CONICET – Consejo Nacional de Investigaciones Científicas y Técnicas, C1425FQD CABA, Argentina Laboratorio de Fluidodinámica, Facultad de Ingeniería, Universidad de Buenos Aires, C1063ACV CABA, Argentina
Denisse Sciamarella
Affiliation:
CNRS – IRD – CONICET – UBA, Institut Franco-Argentin d’Ètudes sur le Climat et ses Impacts (IRL 3351 IFAECI), C1428EGA CABA, Argentina CNRS – Centre National de la Recherche Scientifique, 75795 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

This work describes the application of a technique that extracts branched manifolds from time series to study numerically generated fluid particle behaviour in the wake past a cylinder performing a rotary oscillation at low Reynolds numbers, and compares it with the results obtained for a paradigmatic analytical model of Lagrangian motion: the driven double gyre. The approach does not require prior knowledge of the underlying equations defining the dataset. The time series taken as input corresponds to the evolution of a position coordinate of an individual fluid particle. A delay embedding is used to reconstruct the dynamics in phase space, and a cell complex is built to characterize the topology of the embedding. Fluid particles are said to belong to the same topological class when the Betti numbers, orientability chains and weak boundaries of the associated cell complexes coincide. Topological colouring consists of labelling or ‘colouring’ advected particles with the topological class obtained in their finite-time analyses. The results suggest that topological colouring can be used to distinguish between regions of the flow where trajectories exhibit different finite-time dynamics.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Mathematical Foundations: General fluid mechanics Mixing: Chaotic advection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A17
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abarbanel, H.D.I. & Gollub, J.P. 1996 Analysis of observed chaotic data. Phys. Today 49, 86.CrossRefGoogle Scholar
Allshouse, M.R. & Peacock, T. 2015 a Lagrangian based methods for coherent structure detection. Chaos 25 (9), 097617.CrossRefGoogle ScholarPubMed
Allshouse, M.R. & Peacock, T. 2015 b Refining finite-time Lyapunov exponent ridges and the challenges of classifying them. Chaos 25 (8), 087410.CrossRefGoogle Scholar
Balasuriya, S. 2017 Transport between two fluids across their mutual flow interface: the streakline approach. SIAM J. Appl. Dyn. Syst. 16 (2), 1015–1044.CrossRefGoogle Scholar
Balasuriya, S., Ouellette, N.T. & Rypina, I.I. 2018 Generalized Lagrangian coherent structures. Physica D 372, 31–51.CrossRefGoogle Scholar
Birman, J.S. & Williams, R.F. 1983 Knotted periodic orbits in dynamical systems I: Lorenz's equations. Topology 22 (1), 47–82.CrossRefGoogle Scholar
Boltcheva, D., Canino, D., Aceituno, S.M., Léon, J.-C., De Floriani, L. & Hétroy, F. 2011 An iterative algorithm for homology computation on simplicial shapes. Comput.-Aid. Des. 43 (11), 1457–1467.CrossRefGoogle Scholar
Broomhead, D.S., Jones, R. & King, G.P. 1987 Topological dimension and local coordinates from time series data. J. Phys. A: Math. Gen. 20 (9), L563.CrossRefGoogle Scholar
Charó, G.D., Artana, G. & Sciamarella, D. 2020 Topology of dynamical reconstructions from Lagrangian data. Physica D 405, 132371.CrossRefGoogle Scholar
Charó, G.D., Sciamarella, D., Mangiarotti, S., Artana, G. & Letellier, C. 2019 Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems. Chaos 29 (12), 123126.CrossRefGoogle ScholarPubMed
Choi, S., Choi, H. & Kang, S. 2002 Characteristics of flow over a rotationally oscillating cylinder at low Reynolds number. Phys. Fluids 14 (8), 2767–2777.CrossRefGoogle Scholar
Coutanceau, M. & Bouard, R. 1977 Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part I. Steady flow. J. Fluid Mech. 79 (2), 231–256.CrossRefGoogle Scholar
Cremer, J., Segota, I., Yang, C.-Y., Arnoldini, M., Sauls, J.T., Zhang, Z., Gutierrez, E., Groisman, A. & Hwa, T. 2016 Effect of flow and peristaltic mixing on bacterial growth in a gut-like channel. Proc. Natl Acad. Sci. 113 (41), 11414–11419.CrossRefGoogle Scholar
D'Adamo, J., Godoy-Diana, R. & Wesfreid, J.E. 2015 Centrifugal instability of Stokes layers in crossflow: the case of a forced cylinder wake. Proc. R. Soc. A 471 (2178), 20150011.CrossRefGoogle Scholar
Dennis, S.C.R., Nguyen, P. & Kocabiyik, S. 2000 The flow induced by a rotationally oscillating and translating circular cylinder. J. Fluid Mech. 407, 123–144.CrossRefGoogle Scholar
Farazmand, M. & Haller, G. 2012 Computing Lagrangian coherent structures from their variational theory. Chaos 22 (1), 013128.CrossRefGoogle ScholarPubMed
Filippi, M., Rypina, I.I., Hadjighasem, A. & Peacock, T. 2021 An optimized-parameter spectral clustering approach to coherent structure detection in geophysical flows. Fluids 6 (1), 39.CrossRefGoogle Scholar
Fountain, G.O., Khakhar, D.V., Mezić, I. & Ottino, J.M. 2000 Chaotic mixing in a bounded three-dimensional flow. J. Fluid Mech. 417, 265–301.CrossRefGoogle Scholar
Fraser, A.M. & Swinney, H.L. 1986 Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33 (2), 1134.CrossRefGoogle ScholarPubMed
Froyland, G. 2013 An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems. Physica D 250, 1–19.CrossRefGoogle Scholar
Froyland, G. & Padberg-Gehle, K. 2015 A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data. Chaos 25 (8), 087406.CrossRefGoogle ScholarPubMed
Gilmore, R. 1998 Topological analysis of chaotic dynamical systems. Rev. Mod. Phys. 70 (4), 1455.CrossRefGoogle Scholar
Gonzalez, C.E., Lainscsek, C., Sejnowski, T.J. & Letellier, C. 2020 Assessing observability of chaotic systems using delay differential analysis. Chaos 30 (10), 103113.CrossRefGoogle ScholarPubMed
Hadjighasem, A., Karrasch, D., Teramoto, H. & Haller, G. 2016 Spectral-clustering approach to Lagrangian vortex detection. Phys. Rev. E 93 (6), 063107.CrossRefGoogle ScholarPubMed
Haller, G., Hadjighasem, A., Farazmand, M. & Huhn, F. 2016 Defining coherent vortices objectively from the vorticity. J. Fluid Mech. 795, 136–173.CrossRefGoogle Scholar
Haller, G., Karrasch, D. & Kogelbauer, F. 2018 Material barriers to diffusive and stochastic transport. Proc. Natl Acad. Sci. 115 (37), 9074–9079.CrossRefGoogle ScholarPubMed
Hathcock, J.J. 2006 Flow effects on coagulation and thrombosis. Arterioscler. Thromb. Vasc. Biol. 26 (8), 1729–1737.CrossRefGoogle ScholarPubMed
Kantz, H. & Schreiber, T. 2004 Nonlinear Time Series Analysis. Cambridge University Press.Google Scholar
Karrasch, D. & Haller, G. 2013 Do finite-size Lyapunov exponents detect coherent structures?. Chaos 23 (4), 043126.CrossRefGoogle ScholarPubMed
Kelley, D.H., Allshouse, M.R. & Ouellette, N.T. 2013 Lagrangian coherent structures separate dynamically distinct regions in fluid flows. Phys. Rev. E 88 (1), 013017.CrossRefGoogle ScholarPubMed
Kennel, M.B., Brown, R. & Abarbanel, H.D.I. 1992 Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45 (6), 3403.CrossRefGoogle ScholarPubMed
Kinsey, L.C. 2012 Topology of Surfaces. Springer Science & Business Media.Google Scholar
Lathrop, D.P. & Kostelich, E.J. 1989 Characterization of an experimental strange attractor by periodic orbits. Phys. Rev. A 40 (7), 4028.CrossRefGoogle ScholarPubMed
Maletić, S., Zhao, Y. & Rajković, M. 2016 Persistent topological features of dynamical systems. Chaos 26 (5), 053105.CrossRefGoogle ScholarPubMed
Mangiarotti, S., Coudret, R., Drapeau, L. & Jarlan, L. 2012 Polynomial search and global modeling: two algorithms for modeling chaos. Phys. Rev. E 86 (4), 046205.CrossRefGoogle ScholarPubMed
Mangiarotti, S. & Letellier, C. 2021 Topological characterization of toroidal chaos: A branched manifold for the Deng toroidal attractor. Chaos 31 (1), 013129.CrossRefGoogle ScholarPubMed
Meiss, J.D. 1992 Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (3), 795.CrossRefGoogle Scholar
Mendoza, C., Mancho, A.M. & Wiggins, S. 2014 Lagrangian descriptors and the assessment of the predictive capacity of oceanic data sets. Nonlinear Process Geophys. 21 (3), 677–689.CrossRefGoogle Scholar
Mindlin, G.B., Hou, X.-J., Solari, H.G., Gilmore, R. & Tufillaro, N.B. 1990 Classification of strange attractors by integers. Phys. Rev. Lett. 64 (20), 2350.CrossRefGoogle ScholarPubMed
Mindlin, G.B. & Solari, H.G. 1997 Tori and Klein bottles in four-dimensional chaotic flows. Physica D 102 (3–4), 177–186.CrossRefGoogle Scholar
Mindlin, G.B., Solari, H.G., Natiello, M.A., Gilmore, R. & Hou, X.J. 1991 Topological analysis of chaotic time series data from the Belousov–Zhabotinskii reaction. J. Nonlinear Sci. 1 (2), 147–173.CrossRefGoogle Scholar
Muldoon, M.R., MacKay, R.S., Huke, J.P. & Broomhead, D.S. 1993 Topology from time series. Physica D 65 (1–2), 1–16.CrossRefGoogle Scholar
Natiello, M.A., et al. 2007 The User'S Approach to Topological Methods in 3D Dynamical Systems. World Scientific.CrossRefGoogle Scholar
Packard, N.H, Crutchfield, J.P., Farmer, J.D. & Shaw, R.S. 1980 Geometry from a time series. Phys. Rev. Lett. 45 (9), 712.CrossRefGoogle Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190 (2), 572–600.CrossRefGoogle Scholar
Roca, P., Cammilleri, A., Duriez, T., Mathelin, L. & Artana, G. 2014 Streakline-based closed-loop control of a bluff body flow. Phys. Fluids 26 (4), 047102.CrossRefGoogle Scholar
Ruskeepää, H. 2017 Chaotic data, delay time and embedding dimension. Wolfram Demonstrations Project.Google Scholar
Rypina, I.I., Llewellyn Smith, S.G. & Pratt, L.J. 2018 Connection between encounter volume and diffusivity in geophysical flows. Nonlinear Process Geophys. 25 (2), 267–278.CrossRefGoogle Scholar
Rypina, I.I. & Pratt, L.J. 2017 Trajectory encounter volume as a diagnostic of mixing potential in fluid flows. Nonlinear Process Geophys. 24 (2), 189–202.CrossRefGoogle Scholar
Rypina, I.I., Pratt, L.J., Wang, P., Özgökmen, T.M. & Mezic, I. 2015 Resonance phenomena in a time-dependent, three-dimensional model of an idealized eddy. Chaos 25 (8), 087401.CrossRefGoogle Scholar
Rypina, I.I., Scott, S.E., Pratt, L.J. & Brown, M.G. 2011 Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures. Nonlinear Process Geophys. 18 (6), 977–987.CrossRefGoogle Scholar
Sauer, T., Yorke, J.A. & Casdagli, M. 1991 Embedology. J. Stat. Phys. 65 (3), 579–616.CrossRefGoogle Scholar
Schlueter-Kuck, K.L. & Dabiri, J.O. 2017 Coherent structure colouring: identification of coherent structures from sparse data using graph theory. J. Fluid Mech. 811, 468–486.CrossRefGoogle Scholar
Sciamarella, D. & Mindlin, G.B. 1999 Topological structure of chaotic flows from human speech data. Phys. Rev. Lett. 82 (7), 1450.CrossRefGoogle Scholar
Sciamarella, D. & Mindlin, G.B. 2001 Unveiling the topological structure of chaotic flows from data. Phys. Rev. E 64 (3), 036209.CrossRefGoogle ScholarPubMed
Shadden, S.C., Lekien, F. & Marsden, J.E. 2005 Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212 (3–4), 271–304.CrossRefGoogle Scholar
Soulvaiotis, A., Jana, S.C. & Ottino, J.M. 1995 Potentialities and limitations of mixing simulations. AIChE J. 41 (7), 1605–1621.CrossRefGoogle Scholar
Sulalitha Priyankara, K.G.D., Balasuriya, S. & Bollt, E. 2017 Quantifying the role of folding in nonautonomous flows: the unsteady Double-Gyre. Intl J. Bifurcation Chaos 27 (10), 1750156.CrossRefGoogle Scholar
Taneda, S. 1978 Visual observations of the flow past a circular cylinder performing a rotatory oscillation. J. Phys. Soc. Japan 45 (3), 1038–1043.CrossRefGoogle Scholar
Thiria, B., Goujon-Durand, S. & Wesfreid, J.E. 2006 The wake of a cylinder performing rotary oscillations. J. Fluid Mech. 560, 123–147.CrossRefGoogle Scholar
Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O. & Goldstein, R.E. 2005 Bacterial swimming and oxygen transport near contact lines. Proc. Natl Acad. Sci. 102 (7), 2277–2282.CrossRefGoogle ScholarPubMed
Vieira, G.S., Rypina, I.I. & Allshouse, M.R. 2020 Uncertainty quantification of trajectory clustering applied to ocean ensemble forecasts. Fluids 5 (4), 184.CrossRefGoogle Scholar
Villermaux, E. 2019 Mixing versus stirring. Annu. Rev. Fluid Mech. 51, 245–273.CrossRefGoogle Scholar
Williams, M.O., Rypina, I.I. & Rowley, C.W. 2015 Identifying finite-time coherent sets from limited quantities of Lagrangian data. Chaos 25 (8), 087408.CrossRefGoogle ScholarPubMed
Williamson, C.H.K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477–539.CrossRefGoogle Scholar
You, G. & Leung, S. 2014 An Eulerian method for computing the coherent ergodic partition of continuous dynamical systems. J. Comput. Phys. 264, 112–132.CrossRefGoogle Scholar
Zdravkovich, M.M. 1997 Flow around circular cylinders; Vol. I. Fundamental. J. Fluid Mech. 350 (1), 377–378.Google Scholar
Zomorodian, A. 2010 Fast construction of the Vietoris-Rips complex. Comput. Graph. 34 (3), 263–271.CrossRefGoogle Scholar

Charó et al. Supplementary Movie 1

Advection of particles in the rotary oscillatory cylinder system. Particles are coloured following their topological classification: class I in green, class II in magenta and class III in blue (table 1).

Download Charó et al. Supplementary Movie 1(Video)
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Charó et al. Supplementary Movie 2

Advection of particles in the driven double gyre system. Particles are coloured following their topological classification: class I in green, class II in magenta, class III in blue, class IV in red, and class V in orange (table 2).

Download Charó et al. Supplementary Movie 2(Video)
Video 1.7 MB