Equilibrium and traveling - wave solutions of plane Couette flow

We survey equilibria and traveling waves of plane Couette flow in small periodic cells at moderate Reynolds number Re, adding in the process eight new equilibrium and two new traveling wave solutions to the four solutions previously known. Bifurcations under changes of Re and spanwise period are examined. These non-trivial solutions can only be computed numerically, but they are 'exact' in the sense that they converge to solutions of the Navier-Stokes equations as the numerical resolution increases. We find two complementary visualizations of these flow-invariant solutions particularly insightful. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in moderate Re turbulence. Projections of these solutions and their unstable manifolds from ∞-dimensional state space onto suitably chosen 2-or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulent boundary shear flows.