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Computing heteroclinic orbits using adjoint-based methods

Published online by Cambridge University Press:  12 November 2018

M. Farano Open the ORCID record for M. Farano [Opens in a new window] ,
S. Cherubini Open the ORCID record for S. Cherubini [Opens in a new window] ,
J.-C. Robinet ,
P. De Palma  and
T. M. Schneider
M. Farano*
Affiliation:
DMMM, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy DynFluid Laboratory, Arts et Metiers ParisTech, 151 Bd de l’Hopital, 75013 Paris, France ECPS, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
S. Cherubini*
Affiliation:
DMMM, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy
J.-C. Robinet
Affiliation:
DynFluid Laboratory, Arts et Metiers ParisTech, 151 Bd de l’Hopital, 75013 Paris, France
P. De Palma
Affiliation:
DMMM, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy
T. M. Schneider*
Affiliation:
ECPS, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
*
†Email addresses for correspondence: mirko.farano@epfl.ch, stefania.cherubini@poliba.it, tobias.schneider@epfl.ch
†Email addresses for correspondence: mirko.farano@epfl.ch, stefania.cherubini@poliba.it, tobias.schneider@epfl.ch
†Email addresses for correspondence: mirko.farano@epfl.ch, stefania.cherubini@poliba.it, tobias.schneider@epfl.ch
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Abstract

Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.

JFM classification

Mathematical Foundations: Mathematical Foundations Mathematical Foundations: Variational methods Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , R3
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Copyright
© 2018 Cambridge University Press 

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Farano et al. supplementary movie

All 9 heteroclinic connections. (Left) State-space projection onto the 3D orthonormal basis. Symbols represent equilibria. (Right) Streamwise averaged velocity field, color represent streamwise velocity (blue negative, red positive) and arrows represent in-plane velocity.

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