Unstable Equilibria and Invariant Manifolds in Quasi-Two-Dimensional Kolmogorov-like Flow

Abstract

Recent studies suggest that unstable, non-chaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role of unstable equilibrium solutions and their invariant manifolds in a weakly turbulent, electromagnetically driven, shallow fluid layer. Identifying instants when turbulent evolution slows down, we compute 31 unstable equilibria of a realistic two-dimensional model of the flow. We establish the dynamical relevance of these unstable equilibria by showing that they are closely visited by the turbulent flow. We also establish the dynamical relevance of unstable manifolds by verifying that they are shadowed by turbulent trajectories departing from the neighborhoods of unstable equilibria over large distances in state space.

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• Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence
• Exact coherent structures and shadowing in turbulent Taylor-Couette flow
• Heteroclinic and Homoclinic Connections in a Kolmogorov-Like Flow
• Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence
• Observing a Dynamical Skeleton of Turbulence in Taylor-Couette Flow Experiments
• Predicting chaotic statistics with unstable invariant tori
• Stably stratified exact coherent structures in shear flow: the effect of Prandtl number
• Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics
• Data-driven detection of drifting system parameters