Visualizing the geometry of state space in plane Couette flow

TL;DR

This paper explores the high-dimensional state space of plane Couette flow at Re=400, visualizing invariant solutions and manifolds to better understand turbulence dynamics.

Contribution

It introduces a new visualization method for invariant manifolds in high-dimensional state space and computes new equilibrium solutions at moderate Reynolds numbers.

Findings

Invariant manifolds form a web of heteroclinic connections.

Visualization reveals the structure of turbulence in state space.

New equilibrium solutions expand understanding of flow dynamics.

Abstract

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.