Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations

Abstract

Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible Navier-Stokes equations, based on variations of an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches for handling the incompressibility condition are considered. The variational methods are applied to the specific case of periodic, two-dimensional Kolmogorov flow and compared against existing Newton iteration-based shooting methods. While computationally slow, our methods converge from very inaccurate initial guesses.

References in corpus (13)

• The Significance of Simple Invariant Solutions in Turbulent Flows
• Recurrent motions within plane Couette turbulence
• Snakes and ladders: localized solutions of plane Couette flow
• Variational method for finding periodic orbits in a general flow
• An adjoint-based approach for finding invariant solutions of Navier-Stokes equations
• Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics
• Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow
• Coarse graining the state space of a turbulent flow using periodic orbits
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits
• Stabilisation of exact coherent structures in two-dimensional turbulence using time-delayed feedback
• Periodic orbits in Large Eddy Simulation of Box Turbulence
• Adjoint-based variational method for constructing periodic orbits of high-dimensional chaotic systems

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• Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow
• Ghost states underlying spatial and temporal patterns: how non-existing invariant solutions control nonlinear dynamics
• Data-driven guessing and gluing of unstable periodic orbits
• Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback
• Three-dimensional gravity-capillary standing waves: computation, resonance and instability
• On nonlinear transitions, minimal seeds and exact solutions for the geodynamo