Coarse graining the state space of a turbulent flow using periodic orbits

Abstract

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than $10^5$ degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits.

Supplementary video and data will be made available following publication in a journal

References in corpus (2)

• The Openpipeflow Navier--Stokes Solver
• Reduction of SO(2) symmetry for spatially extended dynamical systems

Cited by in corpus (18)

• Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations
• Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence
• Exact coherent structures and shadowing in turbulent Taylor-Couette flow
• Exact coherent structures and phase space geometry of pre-turbulent 2D active nematic channel flow
• Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders
• Data-Driven Stabilization of Periodic Orbits
• Deep Learning of Conjugate Mappings
• Decomposing the Dynamics of the Lorenz 1963 model using Unstable Periodic Orbits: Averages, Transitions, and Quasi-Invariant Sets
• Direct path from turbulence to time-periodic solutions
• Observing a Dynamical Skeleton of Turbulence in Taylor-Couette Flow Experiments
• Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach
• Marginally-Stable Thermal Equilibria of Rayleigh-Bénard Convection
• Scale-dependent Error Growth in Navier--Stokes Simulations
• Minimal model of quasi-cyclic behaviour in turbulence driven by Taylor--Green forcing
• Computing Chaotic Time-Averages from Few Periodic or Non-Periodic Orbits
• Crises and chaotic scattering in hydrodynamic pilot-wave experiments
• Far-from-equilibrium complex landscapes
• Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics