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

Abstract

Fully 3-dimensional computations of flow through a long pipe demand a huge number of degrees of freedom, making it very expensive to explore parameter space and difficult to isolate the structure of the underlying dynamics. We therefore introduce a `2+epsilon' dimensional model of pipe flow which is a minimal 3-dimensionalisation of the axisymmetric case: only sinusoidal variation in azimuth plus azimuthal shifts are retained, yet the same dynamics familiar from experiments are found. In particular the model retains the subcritical dynamics of fully resolved pipe flow, capturing realistic localised `puff'-like structures which can decay abruptly after long times, as well as global `slug' turbulence. Relaminarisation statistics of puffs reproduce the memoryless feature of pipe flow and indicate the existence of a Reynolds number about which lifetimes diverge rapidly, provided that the pipe is sufficiently long. Exponential divergence of the lifetime is prevalent in shorter periodic domains. In a short pipe, exact travelling-wave solutions are found nearby to flow trajectories on the boundary between laminar and turbulent flow. In a long pipe, the attracting state on the laminar-turbulent boundary is a localised structure which resembles a smoothened puff. This `edge' state remains localised even for Reynolds numbers where the turbulent state is global.

21 pages, 9 figures; as accepted, J. Fluid Mech

References in corpus (7)

• Scaling of the turbulence transition threshold in a pipe
• Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
• Laminar-turbulent boundary in plane Couette flow
• Reply to Comment on 'Critical behaviour in the relaminarization of localized turbulence in pipe flow'
• Recurrence of Travelling Waves in Transitional Pipe Flow
• Statistical analysis of coherent structures in transitional pipe flow
• Critical Behaviour in the Relaminarisation of Localised Turbulence in Pipe Flow

Cited by in corpus (6)

• The rise of fully turbulent flow
• Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos
• Speed and structure of turbulent fronts in pipe flow
• Transition to turbulence in pulsating pipe flow
• Heteroclinic path to spatially localized chaos in pipe flow
• Propagation speed of turbulent fronts in pipe flow at high Reynolds numbers