Periodic orbits exhibit oblique stripe patterns in plane Couette flow

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.