Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien-Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance 2H apart, and in a domain of width 2πH and length 2πH, the subcritical instability to TS waves sets in at Rec 5815 and extends down to ReTE ≈ 4884. The bypass route becomes available above Ree = 459 with the appearance of three-dimensional, finite-amplitude travelling waves. Below Rec, TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above Rec, the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.
- nonlinear instability
- transition to turbulence