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An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

Published online by Cambridge University Press:  14 April 2016

M. Farazmand
M. Farazmand*
Affiliation:
Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
†Email address for correspondence: mfarazmand6@gatech.edu
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Abstract

We consider the incompressible Navier–Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and travelling-wave solutions of the Navier–Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of $100\,\%$ is observed, leading to the discovery of $21$ new steady-state and travelling-wave solutions at Reynolds number $Re=40$ . Some of the new invariant solutions have spatially localized structures that were previously believed to exist only on domains with large aspect ratios. We show that one of the newly found steady-state solutions underpins the temporal intermittencies, i.e. high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.

JFM classification

Nonlinear Dynamical Systems: Chaos Turbulent Flows: Intermittency Instability: Transition to turbulence

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Papers
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Journal of Fluid Mechanics , Volume 795 , 25 May 2016 , pp. 278 - 312
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© 2016 Cambridge University Press 

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