Periodic orbits exhibit oblique stripe patterns in plane Couette flow

Spatiotemporally 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. While equilibrium solutions are embedded in the turbulence supporting set and capture spatial features of the flow, a description of the dynamics requires evolving time-periodic solutions. These periodic orbits are unstable, expected to lie dense in the invariant set supporting turbulence, are shadowed by the chaotic trajectory and may allow for a quantitative description of turbulent statistics via periodic orbit expansions. Here we identify unstable periodic orbits that not only show oblique large-scale amplitude modulation in space but also have a characteristic time evolution. The periodic orbits represent standing waves that slowly propagate across wavy velocity streaks in the flow on viscous diffusion timescales. 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.