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Computing heteroclinic orbits using adjoint-based methods. (English) Zbl 1415.76007
Summary: 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.

76A02 Foundations of fluid mechanics
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76M30 Variational methods applied to problems in fluid mechanics
Full Text: DOI
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