1. 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

2. 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

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

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

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

6. 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

7. ‘Edge’ trajectories

8. Local Velocity field

9. Measure of recurrences?

10. Function r i (t)

11. r min (t)

12. r min along the edge trajectory

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

14. Convergence using a Newton-Krylov algorithm r min = O(10 -11 )

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

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

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

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

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

20. LAMINAR FLOW TURBULENCE A B C

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

22. Edge trajectory for Re=10,000

24. A localised Travelling Wave Solution ?

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

27. 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)