Periodic orbits exhibit oblique stripe patterns in plane Couette flow

TL;DR

This paper identifies unstable periodic orbits in plane Couette flow that exhibit oblique stripe patterns, providing insight into the transition dynamics between laminar and turbulent states.

Contribution

It discovers and characterizes unstable periodic orbits with oblique pattern features, extending understanding of flow structures beyond steady solutions.

Findings

Periodic orbits show oblique large-scale amplitude modulation.

Orbits propagate across velocity streaks on viscous time scales.

Embedded in the edge of chaos, they may mediate flow transition.

Abstract

Spatio-temporally chaotic dynamics of transitional plane Couette flow may give rise to regular turbulent-laminar stripe patterns with a large-scale pattern wavelength and an oblique orientation relative to the laminar flow direction. A recent dynamical systems analysis of the oblique stripe pattern demonstrated that the Navier-Stokes equations have unstable equilibrium solutions that capture the three-dimensional spatial structure of the oblique stripe patterns. Here we identify unstable periodic orbits that not only show oblique large-scale amplitude modulation but also have a characteristic time-evolution, unlike steady equilibrium solutions. The periodic orbits represent standing waves that slowly propagate across wavy velocity streaks in the flow on viscous diffusion time scales. The unstable periodic orbits are embedded in the edge of chaos in a symmetry subspace of plane Couette flow and thereby may mediate transition to and from turbulent flows with oblique patterns.