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Lagrangian approach for modal analysis of fluid flows

Published online by Cambridge University Press:  15 October 2021

Vilas J. Shinde  and
Datta V. Gaitonde Open the ORCID record for Datta V. Gaitonde [Opens in a new window]
Vilas J. Shinde*
Affiliation:
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
Datta V. Gaitonde
Affiliation:
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
*
†Email address for correspondence: [email protected]
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Abstract

Common modal decomposition techniques for flow-field analysis, data-driven modelling and flow control, such as proper orthogonal decomposition and dynamic mode decomposition, are usually performed in an Eulerian (fixed) frame of reference with snapshots from measurements or evolution equations. The Eulerian description poses some difficulties, however, when the domain or the mesh deforms with time as, for example, in fluid–structure interactions. For such cases, we first formulate a Lagrangian modal analysis (LMA) ansatz by a posteriori transforming the Eulerian flow fields into Lagrangian flow maps through an orientation and measure-preserving domain diffeomorphism. The development is then verified for Lagrangian variants of proper orthogonal decomposition and dynamic mode decomposition using direct numerical simulations of two canonical flow configurations at Mach 0.5, i.e. the lid-driven cavity and flow past a cylinder, representing internal and external flows, respectively, at pre- and post-bifurcation Reynolds numbers. The LMA is demonstrated for several situations encompassing unsteady flow without and with boundary and mesh deformation as well as non-uniform base flows that are steady in Eulerian but not in Lagrangian frames. We show that application of LMA to steady non-uniform base flow yields insights into flow stability and post-bifurcation dynamics. LMA naturally leads to Lagrangian coherent flow structures and connections with finite-time Lyapunov exponents. We examine the mathematical link between finite-time Lyapunov exponents and LMA by considering a double-gyre flow pattern. Dynamically important flow features in the Lagrangian sense are recovered by performing LMA with forward and backward (adjoint) time procedures.

JFM classification

Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 928 , 10 December 2021 , A35
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Albensoeder, S. & Kuhlmann, H.C. 2005 Accurate three-dimensional lid-driven cavity flow. J. Comput. Phys. 206 (2), 536–558.CrossRefGoogle Scholar
Allshouse, M.R. & Peacock, T. 2015 Lagrangian based methods for coherent structure detection. Chaos 25 (9), 097617.CrossRefGoogle ScholarPubMed
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. Theor. Comput. Fluid Dyn. 2 (5), 339–352.CrossRefGoogle Scholar
Aubry, N., Guyonnet, R. & Lima, R. 1991 Spatiotemporal analysis of complex signals: theory and applications. J. Stat. Phys. 64 (3-4), 683–739.CrossRefGoogle Scholar
Balasuriya, S., Kalampattel, R. & Ouellette, N.T. 2016 Hyperbolic neighbourhoods as organizers of finite-time exponential stretching. J. Fluid Mech. 807, 509–545.CrossRefGoogle Scholar
Balasuriya, S., Ouellette, N.T. & Rypina, I.I. 2018 Generalized Lagrangian coherent structures. Phys. D: Nonlinear Phenom. 372, 31–51.CrossRefGoogle Scholar
Batchelor, G.K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Beam, R.M. & Warming, R. 1978 An implicit factored scheme for the compressible Navier–Stokes equations. AIAA J. 16 (4), 393–402.CrossRefGoogle Scholar
Behara, S. & Mittal, S. 2010 Wake transition in flow past a circular cylinder. Phys. Fluids 22 (11), 114104.CrossRefGoogle Scholar
Bergamo, L.F., Gennaro, E.M., Theofilis, V. & Medeiros, M.A.F. 2015 Compressible modes in a square lid-driven cavity. Aerosp. Sci. Technol. 44, 125–134.CrossRefGoogle Scholar
Berkooz, G., Holmes, P. & Lumley, J.L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1), 539–575.CrossRefGoogle Scholar
BozorgMagham, A.E. & Ross, S.D. 2015 Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty. Commun. Nonlinear Sci. Numer. Simul. 22 (1-3), 964–979.CrossRefGoogle Scholar
Branicki, M. & Wiggins, S. 2010 Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents. Nonlinear Process. Geophys. 17 (1), 1–36.CrossRefGoogle Scholar
Browne, O.M.F., Rubio, G., Ferrer, E. & Valero, E. 2014 Sensitivity analysis to unsteady perturbations of complex flows: a discrete approach. Intl J. Numer. Meth. Fluids 76 (12), 1088–1110.CrossRefGoogle Scholar
Bruce, P. & Colliss, S. 2015 Review of research into shock control bumps. Shock Waves 25 (5), 451–471.CrossRefGoogle Scholar
Bruneau, C.-H. & Saad, M. 2006 The 2D lid-driven cavity problem revisited. Comput. Fluids 35 (3), 326–348.CrossRefGoogle Scholar
Canuto, D. & Taira, K. 2015 Two-dimensional compressible viscous flow around a circular cylinder. J. Fluid Mech. 785, 349–371.CrossRefGoogle Scholar
Chandler, G.J., Juniper, M.P., Nichols, J.W. & Schmid, P.J. 2012 Adjoint algorithms for the Navier–Stokes equations in the low mach number limit. J. Comput. Phys. 231 (4), 1900–1916.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
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Citriniti, J.H. & George, W.K. 2000 Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166.CrossRefGoogle Scholar
Dauch, T.F., Ates, C., Rapp, T., Keller, M.C., Chaussonnet, G., Kaden, J., Okraschevski, M., Koch, R., Dachsbacher, C. & Bauer, H.-J. 2019 Analyzing the interaction of vortex and gas–liquid interface dynamics in fuel spray nozzles by means of Lagrangian-coherent structures (2D). Energies 12 (13), 2552.CrossRefGoogle Scholar
d'Ovidio, F., Fernández, V., Hernández-García, E. & López, C. 2004 Mixing structures in the mediterranean sea from finite-size Lyapunov exponents. Geophys. Res. Lett. 31 (17), L17203.CrossRefGoogle Scholar
Falkovich, G., Gawedzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73 (4), 913.CrossRefGoogle Scholar
Finn, J. & Apte, S.V. 2013 Integrated computation of finite-time Lyapunov exponent fields during direct numerical simulation of unsteady flows. Chaos 23 (1), 013145.CrossRefGoogle ScholarPubMed
Gaitonde, D.V. & Visbal, M.R. 2000 Padé-type higher-order boundary filters for the Navier–Stokes equations. AIAA J. 38 (11), 2103–2112.CrossRefGoogle Scholar
Garaboa-Paz, D., Eiras-Barca, J., Huhn, F. & Pérez-Muñuzuri, V. 2015 Lagrangian coherent structures along atmospheric rivers. Chaos 25 (6), 063105.CrossRefGoogle ScholarPubMed
Ghia, U., Ghia, K.N. & Shin, C.T. 1982 High-re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48 (3), 387–411.CrossRefGoogle Scholar
Goldhirsch, I., Sulem, P.-L. & Orszag, S.A. 1987 Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. Phys. D: Nonlinear Phenom. 27 (3), 311–337.CrossRefGoogle Scholar
González, D.R., Speth, R.L., Gaitonde, D.V. & Lewis, M.J. 2016 Finite-time Lyapunov exponent-based analysis for compressible flows. Chaos 26 (8), 083112.CrossRefGoogle ScholarPubMed
Gordnier, R.E. & Visbal, M.R. 2002 Development of a three-dimensional viscous aeroelastic solver for nonlinear panel flutter. J. Fluids Struct. 16 (4), 497–527.CrossRefGoogle Scholar
Goza, A. & Colonius, T. 2018 Modal decomposition of fluid–structure interaction with application to flag flapping. J. Fluids Struct. 81, 728–737.CrossRefGoogle Scholar
Green, M.A., Rowley, C.W. & Smits, A.J. 2010 Using hyperbolic lagrangian coherent structures to investigate vortices in bioinspired fluid flows. Chaos 20 (1), 017510.CrossRefGoogle ScholarPubMed
Guckenheimer, J. & Holmes, P. 2013 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer Science & Business Media.Google Scholar
Haller, G. 2002 Lagrangian coherent structures from approximate velocity data. Phys. Fluids 14 (6), 1851–1861.CrossRefGoogle Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162.CrossRefGoogle Scholar
Haller, G. & Yuan, G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D: Nonlinear Phenom. 147 (3-4), 352–370.CrossRefGoogle Scholar
Hardy, G.H., Littlewood, J.E. & Pólya, G. 1952 Inequalities, 2nd edn. Cambridge Mathematical Library. Cambridge University Press.Google Scholar
Hill, D. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183–204.CrossRefGoogle Scholar
Iorio, M.C., González, L.M. & Ferrer, E. 2014 Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile. Intl J. Numer. Meth. Fluids 76 (3), 147–168.CrossRefGoogle Scholar
Karhunen, K. 1946 Zur spektraltheorie stochastischer prozesse. Ann. Acad. Sci. Fennicae, AI 34.Google Scholar
Kasten, J., Petz, C., Hotz, I., Noack, B.R. & Hege, H.-C. 2009 Localized finite-time Lyapunov exponent for unsteady flow analysis. In VMV, pp. 265–276.Google Scholar
Koopman, B.O. 1931 Hamiltonian systems and transformation in Hilbert space. Proc. Natl Acad. Sci. USA 17 (5), 315.CrossRefGoogle ScholarPubMed
Kosambi, D. 1943 Statistics in function space. J. Ind. Math. Soc. 7, 76–88.Google Scholar
Koseff, J.R., Street, R.L., Gresho, P.M., Upson, C.D., Humphrey, J.A.C. & To, W.M. 1983 Three-dimensional lid-driven cavity flow: experiment and simulation. Tech. Rep.. Stanford University, CA. Department of Civil Engineering; Lawrence Livermore.Google Scholar
Lekien, F., Shadden, S.C. & Marsden, J.E. 2007 Lagrangian coherent structures in n-dimensional systems. J. Math. Phys. 48 (6), 065404.CrossRefGoogle Scholar
Loeve, M. 1948 Functions aleatoires du second ordre. In Processus stochastique et mouvement Brownien, pp. 366–420.Google Scholar
Lopez, J.M., Welfert, B.D., Wu, K. & Yalim, J. 2017 Transition to complex dynamics in the cubic lid-driven cavity. Phys. Rev. Fluids 2 (7), 074401.CrossRefGoogle Scholar
Lu, H. & Tartakovsky, D.M. 2020 Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena. J. Comput. Phys. 407, 109229.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.CrossRefGoogle Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation (ed. A.M. Yaglom & V.I. Tatarski), pp. 166–178. Nauka.Google Scholar
Lumley, J.L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Mancho, A.M., Wiggins, S., Curbelo, J. & Mendoza, C. 2013 Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18 (12), 3530–3557.CrossRefGoogle Scholar
Marquet, O., Lombardi, M., Chomaz, J.-M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21.CrossRefGoogle Scholar
Meneveau, C., Lund, T.S. & Cabot, W.H. 1996 A lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech. 319, 353–385.CrossRefGoogle Scholar
Menon, K. & Mittal, R. 2020 Dynamic mode decomposition based analysis of flow over a sinusoidally pitching airfoil. J. Fluids Struct. 94, 102886.CrossRefGoogle Scholar
Mezić, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41 (1), 309–325.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
Mohan, A.T., Gaitonde, D.V. & Visbal, M.R. 2016 Model reduction and analysis of deep dynamic stall on a plunging airfoil. Comput. Fluids 129, 1–19.CrossRefGoogle Scholar
Moin, P. & Moser, R.D. 1989 Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–509.CrossRefGoogle Scholar
Nelson, D.A. & Jacobs, G.B. 2015 Dg-ftle: Lagrangian coherent structures with high-order discontinuous-Galerkin methods. J. Comput. Phys. 295, 65–86.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
Nolan, P.J., Serra, M. & Ross, S.D. 2020 Finite-time Lyapunov exponents in the instantaneous limit and material transport. Nonlinear Dyn. 100 (4), 3825–3852.CrossRefGoogle Scholar
Ohmichi, Y. & Suzuki, K. 2017 Compressibility effects on the first global instability mode of the vortex formed in a regularized lid-driven cavity flow. Comput. Fluids 145, 1–7.CrossRefGoogle Scholar
Oseledets, V. 1968 A multiplicative ergodic theorem. Lyapunov characteristics numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231.Google Scholar
Ottino, J.M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport, vol. 3. Cambridge University Press.Google Scholar
Peacock, T. & Dabiri, J. 2010 Introduction to focus issue: Lagrangian coherent structures. Chaos 20 (1), 017501–017501.CrossRefGoogle ScholarPubMed
Pope, S. 1994 Lagrangian pdf methods for turbulent flows. Annu. Rev. Fluid Mech. 26 (1), 23–63.CrossRefGoogle Scholar
Price, J.F. 2006 Lagrangian and Eulerian Representations of Fluid Flow: Kinematics and the Equations of Motion. MIT OpenCourseWare.Google Scholar
Ramanan, N. & Homsy, G.M. 1994 Linear stability of lid-driven cavity flow. Phys. Fluids 6 (8), 2690–2701.CrossRefGoogle Scholar
Ranjan, R., Unnikrishnan, S. & Gaitonde, D. 2020 A robust approach for stability analysis of complex flows using high-order Navier–Stokes solvers. J. Comput. Phys. 403, 109076.CrossRefGoogle Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.CrossRefGoogle Scholar
Rowley, C.W. & Dawson, S.T. 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. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641 (1), 115–127.CrossRefGoogle Scholar
Samelson, R.M. 2013 Lagrangian motion, coherent structures, and lines of persistent material strain. Annu. Rev. Mar. Sci. 5, 137–163.CrossRefGoogle ScholarPubMed
Schmid, P.J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P.J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schmid, P.J. & Brandt, L. 2014 Analysis of fluid systems: stability, receptivity, sensitivity: lecture notes from the FLOW-NORDITA summer school on advanced instability methods for complex flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66 (2), 021003.CrossRefGoogle Scholar
Schmid, P.J., Meyer, K.E. & Pust, O. 2009 Dynamic mode decomposition and proper orthogonal decomposition of flow in a lid-driven cylindrical cavity. In 8th International Symposium on Particle Image Velocimetry, pp. 25–28.Google Scholar
Shadden, S.C. & Arzani, A. 2015 Lagrangian postprocessing of computational hemodynamics. Ann. Biomed. Engng 43 (1), 41–58.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. Phys. D: Nonlinear Phenom. 212 (3-4), 271–304.CrossRefGoogle Scholar
Sharma, A. & McKeon, B. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Shen, J. 1991 Hopf bifurcation of the unsteady regularized driven cavity flow. J. Comput. Phys. 95 (1), 228–245.CrossRefGoogle Scholar
Sheu, T.W.H. & Tsai, S.F. 2002 Flow topology in a steady three-dimensional lid-driven cavity. Comput. Fluids 31 (8), 911–934.CrossRefGoogle Scholar
Shinde, V. 2020 Proper orthogonal decomposition assisted subfilter-scale model of turbulence for large Eddy simulation. Phys. Rev. Fluids 5 (1), 014605.CrossRefGoogle Scholar
Shinde, V.J., Gaitonde, D.V. & McNamara, J.J. 2021 Supersonic turbulent boundary-layer separation control using a morphing surface. AIAA J. 59 (3), 912–926.CrossRefGoogle Scholar
Shinde, V., Longatte, E., Baj, F., Hoarau, Y. & Braza, M. 2016 A galerkin-free model reduction approach for the Navier–Stokes equations. J. Comput. Phys. 309, 148–163.CrossRefGoogle Scholar
Shinde, V., Longatte, E., Baj, F., Hoarau, Y. & Braza, M. 2019 a Galerkin-free model reduction for fluid–structure interaction using proper orthogonal decomposition. J. Comput. Phys. 396, 579–595.CrossRefGoogle Scholar
Shinde, V., McNamara, J. & Gaitonde, D. 2020 Control of transitional shock wave boundary layer interaction using structurally constrained surface morphing. Aerosp. Sci. Technol. 96, 105545.CrossRefGoogle Scholar
Shinde, V., McNamara, J., Gaitonde, D., Barnes, C. & Visbal, M. 2019 b Transitional shock wave boundary layer interaction over a flexible panel. J. Fluids Struct. 90, 263–285.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Maths 45 (3), 561–571.CrossRefGoogle Scholar
Sreenivasan, K.R., Strykowski, P.J. & Olinger, D.J. 1987 Hopf bifurcation, Landau equation, and vortex shedding behind circular cylinders. In Forum on unsteady flow separation, vol. 1, pp. 1–13. ASME.Google Scholar
Symon, S., Rosenberg, K., Dawson, S.T. & McKeon, B.J. 2018 Non-normality and classification of amplification mechanisms in stability and resolvent analysis. Phys. Rev. Fluids 3 (5), 053902.CrossRefGoogle Scholar
Taira, K., Brunton, S.L., Dawson, S.T., 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 (12), 4013–4041.CrossRefGoogle Scholar
Taira, K., Hemati, M.S., Brunton, S.L., Sun, Y., Duraisamy, K., Bagheri, S., Dawson, S.T.M. & Yeh, C.-A. 2020 Modal analysis of fluid flows: applications and outlook. AIAA J. 58 (3), 998–1022.CrossRefGoogle Scholar
Talpaert, Y.R. 2002 Lagrangian and Eulerian Descriptions, pp. 147–169. Springer.CrossRefGoogle Scholar
Thiffeault, J.-L. & Boozer, A.H. 2001 Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions. Chaos 11 (1), 16–28.CrossRefGoogle ScholarPubMed
Thomas, P. & Lombard, C. 1979 Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17 (10), 1030–1037.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
Truesdell, C. & Noll, W. 2004 The non-linear field theories of mechanics. In The non-linear field theories of mechanics, pp. 1–579. Springer.CrossRefGoogle Scholar
Vastano, J.A. & Moser, R.D. 1991 Short-time Lyapunov exponent analysis and the transition to chaos in Taylor–Couette flow. J. Fluid Mech. 233, 83–118.CrossRefGoogle Scholar
Visbal, M.R. & Gaitonde, D.V. 1999 High-order-accurate methods for complex unsteady subsonic flows. AIAA J. 37 (10), 1231–1239.CrossRefGoogle Scholar
Visbal, M.R. & Gaitonde, D.V. 2002 On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181 (1), 155–185.CrossRefGoogle Scholar
Visbal, M.R. & Gordnier, R.E. 2004 Numerical simulation of the interaction of a transitional boundary layer with a 2-D flexible panel in the subsonic regime. J. Fluids Struct. 19 (7), 881–903.CrossRefGoogle Scholar
Wiggins, S. 2005 The dynamical systems approach to Lagrangian transport in oceanic flows. Annu. Rev. Fluid Mech. 37, 295–328.CrossRefGoogle Scholar
Williamson, C.H.K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477–539.CrossRefGoogle Scholar
Xie, X., Nolan, P.J., Ross, S.D., Mou, C. & Iliescu, T. 2020 Lagrangian reduced order modeling using finite time Lyapunov exponents. Fluids 5 (4), 189.CrossRefGoogle Scholar
Yeung, P. 2002 Lagrangian investigations of turbulence. Annu. Rev. Fluid Mech. 34 (1), 115–142.CrossRefGoogle Scholar
Yeung, P.-K. & Pope, S.B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.CrossRefGoogle Scholar
Yoden, S. & Nomura, M. 1993 Finite-time Lyapunov stability analysis and its application to atmospheric predictability. J. Atmos. Sci. 50 (11), 1531–1543.2.0.CO;2>CrossRefGoogle Scholar