Exact coherent states for grooved Couette flows.

Recent progress indicates that highly symmetric recurring solutions of the Navier-Stokes equations, such as equilibria and periodic orbits, provide a skeleton for turbulence dynamics in state-space. Many of these solutions have been found for flat-walled plane Couette, channel, and pipe flows. Rough-walled flows are of great practical significance, yet no recurring solutions are known for these flows. We present a numerical homotopy method to continue solutions from flat-walled plane Couette flow (PCF) to grooved PCF, demonstrated here at a Reynolds number of 400, to act as a starting point for similar continuation to rough-walled flows. Loss of spanwise homogeneity in grooved PCF reduces continuous families of solutions (identical up to translational shifts) in flat-walled Couette flow to multiple, discrete families in grooved Couette flow; this can manifest in turbulence as spatially anchored exact coherent structures near the wall, so that turbulent statistics reflect symmetry-restricted structure of exact recurring solutions. Furthermore, the vortex-streak structures characteristic of these equilibria are squeezed out of the grooves when the groove-wavelength is smaller than the characteristic spanwise size of the structures. This produces reduced shear stress at the wall even at the low Reynolds numbers considered, and the mechanism is consistent with the drag reduction observed in some riblet-mounted turbulent flows.