Sub critical transition to turbulence in three-dimensional Kolmogorov flow

TL;DR

This paper investigates the transition to turbulence in three-dimensional Kolmogorov flow, identifying stable and unstable invariant solutions and their bifurcations, which elucidates the flow's stability and transition mechanisms.

Contribution

It provides a concise proof of linear stability and identifies invariant solutions and bifurcations that characterize the transition to turbulence in 3D Kolmogorov flow.

Findings

Stable laminar flow at low Reynolds numbers

Existence of an edge state with spatial structure similar to plane Couette flow

Bifurcation of solution branches at finite Reynolds number

Abstract

We study Kolmogorov flow on a three dimensional, periodic domain with aspect ratios fixed to unity. Using an energy method, we give a concise proof of the linear stability of the laminar flow profile. Since turbulent motion is observed for high enough Reynolds numbers, we expect the domain of attraction of the laminar flow to be bounded by the stable manifolds of simple invariant solutions. We show one such edge state to be an equilibrium with a spatial structure reminiscent of that found in plane Couette flow, with stream wise rolls on the largest spatial scales. When tracking the edge state, we find an upper and a lower branch solution that join in a saddle node bifurcation at finite Reynolds number.