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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
L
z
sU : bulk velocity
D
Driving force : fixed mass flux
The laminar flow is stable to infinitesimal disturbances
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Incompressible N.S. equations
Additional boundary conditions for numerics :
Numerical DNS code developed by A.P. Willis
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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
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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 Brmin =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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Eu
Es
Eu
A solution with only at least two unstable eigenvectors remains a saddle point on the laminar-turbulent boundary
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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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L ~ 2.5D, Re=2400, m0=2
Imposing symmetries can simplify the dynamics and show new solutions
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Local attractors on the edge
2b_1.25 (Kerswell & Tutty, 2007) C3 (Duguet et. al., 2008, JFM 2008)
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LAMINAR FLOW
TURBULENCE
A
B
C
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Longer periodic domains
2.5D model of Willis : L = 50D, (35, 256, 2, m0=3) generate edge trajectory
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Edge trajectory for Re=10,000
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Edge trajectory for Re=10,000
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A localised Travelling Wave Solution ?
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Dynamical interpretation of slugs ?
« Slug » trajectory?
relaminarising trajectory
Extended turbulence
localised TW
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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 Pringle’s talk)
• Method seems applicable to tackle transition in real flows (implying localised structures)