Equilibrium and traveling-wave solutions of plane Couette flow

Authors:
  • JHJonathan Halcrow
    Jonathan Halcrow
  • PCPredrag Cvitanović
    Predrag Cvitanović
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Abstract
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 flow-invariant unstable 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 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 infinite-dimensional state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

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