Critical Behaviour in the Relaminarisation of Localised Turbulence in Pipe Flow

The statistics of the relaminarisation of localised turbulence in a pipe are examined by direct numerical simulation. As in recent experimental data (Peixinho & Mullin, Phys. Rev. Lett. 96, 094501, 2006), the half life for the decaying turbulence is consistent with the scaling (Re_c-Re)^{-1}, indicating a boundary crisis of the localised turbulent state familiar in low-dimensional dynamical systems. The crisis Reynolds number, is estimated as Re_c=1870, a value within 7% of the experimental value 1750. We argue that the frequently-asked question of which Re and initial disturbance are needed to trigger sustained turbulence in a pipe, is really two separate questions: the `local phase space' question (local to the laminar state) of what threshold disturbance at a given Re is needed to initially trigger turbulence, followed by the `global phase space' question of whether Re exceeds Re_c at which point the turbulent state becomes an attractor.