Lower-branch travelling waves and transition to turbulence in pipe flow Dr Yohann Duguet, Linné Flow Centre, KTH, Stockholm, Sweden, formerly : School.

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Presentation transcript:

Lower-branch travelling waves and transition to turbulence in pipe flow Dr Yohann Duguet, Linné Flow Centre, KTH, Stockholm, Sweden, formerly : School of Mathematics, University of Bristol, UK

Overview Laminar/turbulent boundary in pipe flow Identification of finite-amplitude solutions along edge trajectories Generalisation to longer computational domains Implications on the transition scenario

Colleagues, University of Bristol, UK Rich Kerswell Ashley Willis Chris Pringle

Cylindrical pipe flow L z s U : bulk velocity D Driving force : fixed mass flux The laminar flow is stable to infinitesimal disturbances

Incompressible N.S. equations Additional boundary conditions for numerics : Numerical DNS code developed by A.P. Willis

Parameters Re = 2875, L ~ 5D, m 0 =1 (Schneider et. Al., 2007) Numerical resolution(30,15,15)  O(10 5 ) d. o. f. Initial conditions for the bisection method Axial average

‘Edge’ trajectories

Local Velocity field

Measure of recurrences?

Function r i (t)

r min (t)

r min along the edge trajectory

Starting guesses A B r min =O(10 -1 )

Convergence using a Newton-Krylov algorithm r min = O( )

The skeleton of the dynamics on the edge Recurrent visits to a Travelling Wave solution …

EuEu EsEs EuEu A solution with only at least two unstable eigenvectors remains a saddle point on the laminar-turbulent boundary

A solution with only one unstable eigenvector should be a local attractor on the laminar-turbulent boundary EuEu EsEs EsEs

L ~ 2.5D, Re=2400, m 0 =2 Imposing symmetries can simplify the dynamics and show new solutions

Local attractors on the edge 2b_1.25 (Kerswell & Tutty, 2007) C3 (Duguet et. al., 2008, JFM 2008)

LAMINAR FLOW TURBULENCE A B C

Longer periodic domains 2.5D model of Willis : L = 50D, (35, 256, 2, m 0 =3)  generate edge trajectory

Edge trajectory for Re=10,000

A localised Travelling Wave Solution ?

Dynamical interpretation of slugs ? « Slug » trajectory? relaminarising trajectory Extended turbulence localised TW

Conclusions The laminar-turbulent boundary seems to be structured around a network of exact solutions Method to identify the most relevant exact coherent states in subcritical systems : the TWs visited near criticality Symmetry subspaces help to identify more new solutions (see Chris Pringle’s talk) Method seems applicable to tackle transition in real flows (implying localised structures)