Dynamics on the pipe flow critical threshold

For high enough values of the Reynolds number (Re ≥ 1750), two attracting states coexist in pipe flow: a steady laminar solution (the Hagen-Poiseuille flow) and a complex, unsteady, turbulent state. While the former is believed to be stable for all values of Re, the sustained nature of the latter is currently at stake and under thourough scrutiny. A combined shooting-refining technique is used to track trajectories on the boundary separating the basins of attraction of these states. These trajectories are essentially chaotic but their main features do not seem to depend on the initial condition from which they are approached. Furthermore, they all seem to periodically visit a slowly evolving coherent state consisting of a slow streak excentrically placed towards the wall of the pipe and sandwiched between a pair of fast streaks. One of the visits to such a state is set as the starting point in the search for a trevelling wave solution that seems to underlie the chaotic trajectory on the boundary.