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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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