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Hyperbolic neighbourhoods as organizers of finite-time exponential stretching

Published online by Cambridge University Press:  20 October 2016

Sanjeeva Balasuriya ,
Rahul Kalampattel  and
Nicholas T. Ouellette
Sanjeeva Balasuriya*
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
Rahul Kalampattel
Affiliation:
School of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005, Australia
Nicholas T. Ouellette*
Affiliation:
Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Hyperbolic points and their unsteady generalization – hyperbolic trajectories – drive the exponential stretching that is the hallmark of nonlinear and chaotic flow. In infinite-time steady or periodic flows, the stable and unstable manifolds attached to each hyperbolic trajectory mark fluid elements that asymptote either towards or away from the hyperbolic trajectory, and which will therefore eventually experience exponential stretching. But typical experimental and observational velocity data are unsteady and available only over a finite time interval, and in such situations hyperbolic trajectories will move around in the flow, and may lose their hyperbolicity at times. Here we introduce a way to determine their region of influence, which we term a hyperbolic neighbourhood, that marks the portion of the domain that is instantaneously dominated by the hyperbolic trajectory. We establish, using both theoretical arguments and empirical verification from model and experimental data, that the hyperbolic neighbourhoods profoundly impact the Lagrangian stretching experienced by fluid elements. In particular, we show that fluid elements traversing a flow experience exponential boosts in stretching while within these time-varying regions, that greater residence time within hyperbolic neighbourhoods is directly correlated to larger finite-time Lyapunov exponent (FTLE) values, and that FTLE diagnostics are reliable only when the hyperbolic neighbourhoods have a geometrical structure that is ‘regular’ in a specific sense.

JFM classification

Mixing: Chaotic advection Mixing: Mixing Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 807 , 25 November 2016 , pp. 509 - 545
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© 2016 Cambridge University Press 

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References

Allshouse, M. & Peacock, T. 2015 Refining finite-time Lyapunov exponent ridges and the challenges of classifying them. Chaos 25, 087410.Google Scholar
Allshouse, M. R. & Thiffeault, J.-L. 2012 Detecting coherent structures using braids. Physica D 241, 95–105.Google Scholar
Babiano, A., Cartwright, J. H. E., Piro, O. & Provenzale, A. 2000 Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems. Phys. Rev. Lett. 84, 5764–5767.Google Scholar
Balasuriya, S. 2010 Optimal frequency for microfluidic mixing across a fluid interface. Phys. Rev. Lett. 105, 064501.CrossRefGoogle ScholarPubMed
Balasuriya, S. 2011 A tangential displacement theory for locating perturbed saddles and their manifolds. SIAM J. Appl. Dyn. Syst. 10, 1100–1126.Google Scholar
Balasuriya, S. 2012 Explicit invariant manifolds and specialised trajectories in a class of unsteady flows. Phys. Fluids 24, 12710.Google Scholar
Balasuriya, S. 2015a Dynamical systems techniques for enhancing microfluidic mixing. J. Micromech. Microengng 25, 094005.Google Scholar
Balasuriya, S. 2015b Quantifying transport within a two-cell microdroplet induced by circular and sharp channel bends. Phys. Fluids 27, 052005.Google Scholar
Balasuriya, S. 2016a Flow Barriers and Transport in Unsteady Flows: A Melnikov Approach. SIAM (in press).Google Scholar
Balasuriya, S. 2016b Local stable and unstable manifolds and their control in nonautonomous finite-time flows. J. Nonlinear Sci. 26, 895–927.Google Scholar
Balasuriya, S. & Finn, M. D. 2012 Energy constrained transport maximization across a fluid interface. Phys. Rev. Lett. 108, 244503.Google Scholar
Balasuriya, S. & Padberg-Gehle, K. 2014 Accurate control of hyperbolic trajectories in any dimension. Phys. Rev. E 90, 032903.Google Scholar
Basdevant, C. & Philipovitch, T. 1994 On the validity of the ‘Weiss criterion’ in two-dimensional turbulence. Physica D 73, 17–30.Google Scholar
Blazevski, D. & Haller, G. 2014 Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows. Physica D 273–274, 46–64.Google Scholar
Bourgoin, M., Ouellette, N. T., Xu, H., Berg, J. & Bodenschatz, E. 2006 The role of pair dispersion in turbulent flow. Science 311, 835–838.Google Scholar
Budišić, M. & Mezić, I. 2012 Geometry of the ergodic quotient reveals coherent structures in flows. Physica D 241, 1255–1269.Google Scholar
Budišić, M. & Thiffeault, J.-L. 2015 Finite-time braiding exponents. Chaos 25, 087407.CrossRefGoogle ScholarPubMed
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777.Google Scholar
Coppel, W. A. 1978 Dichotomies in Stability Theory, Lecture Notes Math., vol. 629. Springer.Google Scholar
Dellnitz, M. & Junge, O. 2002 Set oriented numerical methods for dynamical systems. In Handbook of Dynamical Systems, vol. 2, pp. 221–264. Elsevier.Google Scholar
Doan, T. S., Karrasch, D., Nguyen, T. & Siegmund, S. 2012 A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents. J. Differ. Equ. 252, 5355–5554.Google Scholar
Froyland, G., Lloyd, S. & Santitissadeekorn, N. 2010 Coherent sets for nonautonomous dynamical systems. Physica D 239, 1527–1541.Google Scholar
Froyland, G. & Padberg, K. 2009 Almost-invariant sets and invariant manifolds – connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D 238, 1507–1523.Google Scholar
Froyland, G. & Padberg-Gehle, K. 2012 Finite-time entropy: a probabilistic approach for measuring nonlinear stretching. Physica D 241, 1612–1628.Google 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, 087406.Google Scholar
Grigoriev, R. O. 2005 Chaotic mixing in thermocapillary-driven microdroplets. Phys. Fluids 17, 033601.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.Google Scholar
Haller, G. 2011 A variational theory of hyperbolic Lagrangian coherent structures. Physica D 240, 574–598.Google Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–161.Google Scholar
Haller, G., Hadjighasem, A., Farazmand, M. & Huhn, F. 2016 Defining coherent vortices objectively from the vorticity. J. Fluid Mech. 795, 136–173.CrossRefGoogle Scholar
He, G.-S., Pan, C., Feng, L.-H., Gao, Q. & Wang, J.-J. 2016 Evolution of Lagrangian coherent structures in a cylinder-wake disturbed flat boundary layer. J. Fluid Mech. 792, 274–306.Google Scholar
Hua, B. L. & Klein, P. 1998 An exact criterion for the stirring properties of nearly two-dimensional turbulence. Physica D 113, 98–110.Google Scholar
Hunt, J. C. R., Wray, A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Tech. Rep. CTR-S88. Center for Turbulence Research.Google Scholar
Huntley, H., Lipphardt, B., Jacobs, G. & Kirwan, A. 2015 Clusters, deformation, and dilation: Diagnostics for material accumulation regions. J. Geophys. Res. 120, 6622–6636.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Johnson, P. & Meneveau, C. 2015 Large deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence. Phys. Fluids 27, 085110.Google Scholar
Karrasch, D. 2013 Linearization of hyperbolic finite-time processes. J. Differ. Equ. 254, 256–282.Google Scholar
Karrasch, D. 2015 Attracting Lagrangian coherent structures on Riemannian manifolds. Chaos 25, 087411.Google Scholar
Kelley, D. H. & Ouellette, N. T. 2011a Onset of three-dimensionality in electromagnetically forced thin-layer flows. Phys. Fluids 23, 045103.Google Scholar
Kelley, D. H. & Ouellette, N. T. 2011b Spatiotemporal persistence of spectral fluxes in two-dimensional weak turbulence. Phys. Fluids 23, 115101.CrossRefGoogle Scholar
Khurana, N., Blawzdziewicz, J. & Ouellette, N. T. 2011 Reduced transport of swimming particles in chaotic flow due to hydrodynamic trapping. Phys. Rev. Lett. 106, 198104.Google Scholar
Khurana, N. & Ouellette, N. T. 2012 Interactions between active particles and dynamical structures in chaotic flow. Phys. Fluids 24, 091902.Google Scholar
Lee, Y.-K., Shih, C., Tabeling, P. & Ho, C.-M. 2007 Experimental study and nonlinear dynamics analysis of time-periodic micro chaotic mixers. J. Fluid Mech. 575, 425–448.Google Scholar
Liao, Y. & Ouellette, N. T. 2013 Spatial structure of spectral transport in two-dimensional flow. J. Fluid Mech. 725, 281–298.Google Scholar
Ma, T. & Bollt, E. 2014 Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting. SIAM J. Appl. Dyn. Syst. 13, 1106–1136.Google Scholar
Mathur, M., Haller, G., Peacock, T., Ruppert-Felsot, J. E. & Swinney, H. L. 2007 Uncovering the Lagrangian skeleton of turbulence. Phys. Rev. Lett. 98, 144502.Google Scholar
Mezić, I. 2013 Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378.Google Scholar
Mezić, I., Loire, S., Fonoberov, V. A. & Hogan, P. 2010 A new mixing diagnostic and Gulf oil spill movement. Science 330, 486–489.Google Scholar
Mordant, N., Crawford, A. M. & Bodenschatz, E. 2004 Experimental Lagrangian probability density function measurement. Physica D 193, 245–251.Google Scholar
Nishikawa, T., Toroczkai, Z. & Grebogi, C. 2001 Advective coalescence in chaotic flows. Phys. Rev. Lett. 87, 038301.Google Scholar
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445–454.Google Scholar
Olascoaga, M. J. & Haller, G. 2012 Forecasting sudden changes in environmental pollution patterns. Proc. Natl Acad. Sci. USA 109, 4738–4743.Google Scholar
Ouellette, N. T. & Gollub, J. P. 2007 Curvature fields, topology, and the dynamics of spatiotemporal chaos. Phys. Rev. Lett. 99, 194502.Google Scholar
Ouellette, N. T. & Gollub, J. P. 2008 Dynamic topology in spatiotemporal chaos. Phys. Fluids 20, 064104.Google Scholar
Ouellette, N. T., Hogg, C. A. R. & Liao, Y. 2016 Correlating Lagrangian structures with forcing in two-dimensional flow. Phys. Fluids 28, 015105.Google Scholar
Ouellette, N. T., O’Malley, P. J. J. & Gollub, J. P. 2008 Transport of finite-sized particles in chaotic flow. Phys. Rev. Lett. 101, 174504.Google Scholar
Ouellette, N. T., Xu, H. & Bodenschatz, E. 2006 A quantitative study of three-dimensional Lagrangian particle tracking algorithms. Exp. Fluids 40, 301–313.Google Scholar
Peacock, T., Froyland, G. & Haller, G. 2015 Introduction to focus issue: objective detection of coherent structures. Chaos 25, 087201.Google Scholar
Peacock, T. & Haller, G. 2013 Lagrangian coherent structures: the hidden skeleton of fluid flows. Phys. Today 66, 41.Google Scholar
Perry, A. E. & Chong, M. S. 1987 A description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19, 125–155.Google Scholar
Pierrehumbert, R. 1991 Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves. Geophys. Astrophys. Fluid Dyn. 58, 285–319.Google Scholar
Rivera, M. K., Aluie, H. & Ecke, R. E. 2014 The direct enstrophy cascade of two-dimensional soap film flows. Phys. Fluids 26, 055105.Google Scholar
Rivera, M. K. & Ecke, R. E. 2005 Pair dispersion and doubling time statistics in two-dimensional turbulence. Phys. Rev. Lett. 95, 194503.Google 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.Google Scholar
Samelson, R. 1992 Fluid exchange across a meandering jet. J. Phys. Oceanogr. 22, 431–440.Google Scholar
Samelson, R. 2013 Lagrangian motion, coherent structures, and lines of persistent material strain. Annu. Rev. Marine Sci. 5, 137–163.Google Scholar
Sawford, B. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33, 289–317.CrossRefGoogle Scholar
Senatore, C. & Ross, S. D. 2011 Detection and characterization of transport barriers in complex flows via ridge extraction of the finite time Lyapunov exponent field. Intl J. Numer. Meth. Engng 86, 1163–1174.Google Scholar
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, 271–304.Google Scholar
Solomon, T. H. & Gollub, J. P. 1988 Chaotic particle transport in time-dependent Rayleigh–Bénard convection. Phys. Rev. A 38, 6280–6286.Google Scholar
Stroock, A., Dertinger, S., Adjari, A., Mezić, I., Stone, H. & Whitesides, G. 2002 Chaotic mixer for microchannels. Science 295, 647–651.Google Scholar
Torney, C. & Neufeld, Z. 2007 Transport and aggregation of self-propelled particles in fluid flows. Phys. Rev. Lett. 99, 078101.Google Scholar
Truesdell, C. & Noll, W. 2004 The Non-Linear Field Theories of Mechanics. Springer.Google Scholar
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88, 254501.Google Scholar
Waugh, D., Abraham, E. & Bowen, M. 2007 Spatial variations of stirring in the surface ocean: a case study of the Tasman Sea. J. Phys. Oceanogr. 36, 526–542.Google Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.Google Scholar
Wiggins, S. 1992 Chaotic Transport in Dynamical Systems. Springer.Google Scholar