Exact recurrent states in the transition to turbulence in pipe

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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
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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

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Colleagues, University of Bristol, UK

• Rich Kerswell
• Ashley Willis
• Chris Pringle

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Cylindrical pipe flow
U bulk velocity
s
D
z
L
Driving force fixed mass flux The laminar
flow is stable to infinitesimal disturbances
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Incompressible N.S. equations
Numerical DNS code developed by A.P. Willis
Additional boundary conditions for numerics
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Parameters
Re 2875, L 5D, m01 (Schneider et. Al.,
2007)
Numerical resolution
(30,15,15) ? O(105) d. o. f.
Initial conditions for the bisection method
Axial average
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Edge trajectories
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Local Velocity field
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Measure of recurrences?
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Function ri(t)
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Function ri(t)
rmin(t)
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rmin along the edge trajectory
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Starting guesses
A
B
rmin O(10-1)
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Convergence using a Newton-Krylov algorithm
rmin O(10-11)
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The skeleton of the dynamics on the edge
Recurrent visits to a Travelling Wave solution

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A solution with only at least two unstable
eigenvectors remains a saddle point on the
laminar-turbulent boundary
Eu
Es
Eu
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A solution with only one unstable eigenvector
should be a local attractor on the
laminar-turbulent boundary
Eu
Es
Es
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Imposing symmetries can simplify the dynamics
and show new solutions
L 2.5D, Re2400, m02
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Local attractors on the edge
C3 (Duguet et. al., 2008, JFM 2008)
2b_1.25 (Kerswell Tutty, 2007)
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TURBULENCE
B
A
C
LAMINAR FLOW
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Longer periodic domains
2.5D model of Willis L 50D, (35, 256, 2,
m03) ? generate edge trajectory
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Edge trajectory for Re10,000
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Edge trajectory for Re10,000
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A localised Travelling Wave Solution ?
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(No Transcript)
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Dynamical interpretation of slugs ?
Extended turbulence
 Slug  trajectory?
localised TW
relaminarising trajectory
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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 Pringles talk)
• Method seems applicable to tackle transition in
  real flows (implying localised structures)