Symmetry-Constrained Exact Coherent Structures in Plane Poiseuille Flow

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.

Figures (21)

• Figure 1: Raw $(D,I)$ (dissipation--input) projections of DNS trajectories used to initialise the Newton--Krylov searches for the ECS reported in this paper. Near approaches to the identity line $D=I$ (small $|D-I|$) and near-closure of loops in the $(D,I)$ plane provide a compact signature of candidate recurrence windows (RPOs) or near-steady episodes in a comoving frame (TWs).
• Figure 2: Phase-plane diagnostics for RPO1 at $Re=1000$. Repeated overlap of $T$-shifted loops indicates a robust relative recurrence consistent with a drift $(a_x,a_z)$ over one period.
• Figure 3: Structure of RPO1 at $Re=1000$ shown via streak--roll projections. Background colour: streamwise velocity $u$ (or $\langle u\rangle_x$ in the middle panel). Arrows: in-plane velocity components in the plotted plane.
• Figure 4: Floquet spectrum of RPO1 at $Re=1000$. (a) Multipliers $\varLambda$ in the complex plane; the dashed circle marks $|\varLambda|=1$. (b) Corresponding exponents $\lambda=(1/T)\log\varLambda$; the dashed vertical line marks $\Re(\lambda)=0$. In both panels, blue markers show all computed eigenvalues and red markers highlight the 15 eigenvalues with largest $|\varLambda|$ (left) or largest $\Re(\lambda)$ (right).
• Figure 5: Phase-plane diagnostics for RPO2 at $Re=1500$. Near-closure of the loop in the $(D,I)$ plane indicates a robust relative recurrence consistent with a drift $(a_x,a_z)$ over one period.