Symmetry-Constrained Exact Coherent Structures in Plane Poiseuille Flow

TL;DR

This paper reports the discovery and analysis of five new exact coherent structures in plane Poiseuille flow, revealing their bifurcation structures, stability properties, and role in turbulence organization.

Contribution

The study introduces five new ECS in plane Poiseuille flow, computed in symmetry-invariant subspaces, and analyzes their bifurcation and stability properties using continuation methods.

Findings

Two RPOs are linearly stable within their symmetry subspace.

Three TWs are saddle-type solutions with distinct instability modes.

Continuation diagrams show complex bifurcation structures, including saddle-node and multi-branch formations.

Abstract

Turbulence in wall-bounded shear flows is increasingly understood through exact coherent structures (ECS) -- invariant solutions of the Navier-Stokes equations that act as organising centres in the high-dimensional state space. Here we report five new ECS of plane Poiseuille flow: two relative periodic orbits (RPOs) and three travelling waves (TWs), that are computed in four distinct symmetry-invariant subspaces using a Newton-Krylov-hookstep solver initialised from direct numerical simulations. We trace each state through one-parameter continuations in both the Reynolds number $Re$ and the spanwise period $L_z$. All five states are organised around counter-rotating rolls sustaining streamwise velocity streaks, yet they exhibit qualitatively different stability properties: the two RPOs are linearly stable within their symmetry subspace, while the three TWs are saddle-type solutions whose instabilities are, respectively, mixed oscillatory-and-monotone, purely monotone and purely oscillatory. The continuation diagrams reveal their bifurcation geometry, from simple saddle-node folds for the RPOs to a pronounced S-shaped multi-branch structure with three coexisting dissipation levels for one of the travelling waves. Along each branch, the roll-streak topology is preserved across the folds, with the different branches distinguished by their amplitude and gradient intensity rather than by a change in spatial organisation. Floquet spectra evaluated at multiple points along each branch shows that folds generically act as sites of reduced instability for the travelling waves, while upper branches develop stronger and sometimes qualitatively different unstable modes. These include the emergence of linearly stable segments on the intermediate and upper branches of the multi-branch travelling wave, despite the solution being unstable at its reference parameters.