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Lower-branch travelling waves and transition to turbulence in pipe flow

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

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Lower-branch travelling waves and transition to turbulence in pipe flow

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

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

  6. Parameters Re = 2875, L ~ 5D, m0=1 (Schneider et. Al., 2007) Numerical resolution (30,15,15)  O(105) d. o. f. Initial conditions for the bisection method Axial average

  7. ‘Edge’ trajectories

  8. Local Velocity field

  9. Measure of recurrences?

  10. Function ri(t)

  11. Function ri(t) rmin(t)

  12. rmin along the edge trajectory

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

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

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

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

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

  18. Imposing symmetries can simplify the dynamics and show new solutions L ~ 2.5D, Re=2400, m0=2

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

  20. TURBULENCE B A C LAMINAR FLOW

  21. Longer periodic domains 2.5D model of Willis : L = 50D, (35, 256, 2, m0=3)  generate edge trajectory

  22. Edge trajectory for Re=10,000

  23. Edge trajectory for Re=10,000

  24. A localised Travelling Wave Solution ?

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

  26. 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)

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