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Fully 3-dimensional computations of flow through a long pipe demand a hugenumber of degrees of freedom, making it very expensive to explore parameterspace and difficult to isolate the structure of the underlying dynamics. Wetherefore introduce a `2+epsilon' dimensional model of pipe flow which is aminimal 3-dimensionalisation of the axisymmetric case: only sinusoidalvariation in azimuth plus azimuthal shifts are retained, yet the same dynamicsfamiliar from experiments are found. In particular the model retains thesubcritical dynamics of fully resolved pipe flow, capturing realistic localised`puff'-like structures which can decay abruptly after long times, as well asglobal `slug' turbulence. Relaminarisation statistics of puffs reproduce thememoryless feature of pipe flow and indicate the existence of a Reynolds numberabout which lifetimes diverge rapidly, provided that the pipe is sufficientlylong. Exponential divergence of the lifetime is prevalent in shorter periodicdomains. In a short pipe, exact travelling-wave solutions are found nearby toflow trajectories on the boundary between laminar and turbulent flow. In a longpipe, the attracting state on the laminar-turbulent boundary is a localisedstructure which resembles a smoothened puff. This `edge' state remainslocalised even for Reynolds numbers where the turbulent state is global.

Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised `edge' states