Symmetry groups and invariant solutions of plane Poiseuille flow

Equilibrium, travelling-wave and periodic-orbit solutions of the Navier–Stokes equations provide a promising avenue for investigating the structure, dynamics and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille and pipe flow. However, the organisation of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organisation and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new travelling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one travelling-wave solution whose symmetry involves shifts by one third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behaviour of shear flows.