Equilibrium and traveling-wave solutions of plane Couette flow

TL;DR

This paper discovers new equilibrium and traveling-wave solutions in plane Couette flow, providing insights into turbulence organization and solution classifications at low Reynolds numbers.

Contribution

It introduces ten new equilibrium solutions and two traveling-wave solutions, along with a classification of isotropy groups and visualization methods for understanding turbulence.

Findings

New equilibrium solutions at low Re

Traveling-wave solutions identified

Visualization techniques reveal turbulence structures

Abstract

We present ten new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number (Re) and two new traveling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low Re turbulence. Projections of these solutions and their unstable manifolds from their infinite-dimensional state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.