Snakes and ladders: localized solutions of plane Couette flow

TL;DR

This paper discovers numerous localized exact solutions in plane Couette flow, revealing a complex structure akin to pattern-forming systems, advancing the understanding of turbulence in extended flows.

Contribution

It introduces a large set of localized solutions in plane Couette flow, organized in a snakes-and-ladders pattern, extending the dynamical systems perspective to spatially extended turbulence.

Findings

Localized solutions share topology with known periodic solutions.

Solutions are organized in a snakes-and-ladders structure.

Advances understanding of transitional turbulence in extended flows.

Abstract

We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming PDE systems. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.