Exploring Invariant Symmetry Subspaces of Channel Flow*

Exact coherent structures (ECS) such as equilibria, traveling waves, periodic and relative periodic orbits, are known to be important in arranging the dynamics of a turbulent attractor. Several ECS have been found for canonical shear flows such as the plane Couette flow (PCF) and the pipe flow. However, less progress has been made for the case of plane Poiseuille flow (PPF). Many studies so far have used homotopy continuation, numerically continuing ECS from PCF conditions to PPF conditions. In the present study, some important invariant subspaces of PPF are explored. In particular, several new nonlinear traveling waves are identified in the <σ_y>,  <σ_y, σ_z> and <σ_y, σ_zτ_x> subspaces, where σ represents reflection symmetry about the specified axis and τ_x denotes half-box shift in the streamwise x-direction, making σ_zτ_x a shift-reflect symmetry. Instead of homotopy continuation, guesses for the Newton-Krylov solver are taken directly from turbulent time-series, with appropriately imposed symmetries. Numerical continuation is used to continue the converged traveling wave solutions in Reynolds number. Subsequent bifurcations, including several symmetry-breaking bifurcations, are analyzed.