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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
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
sU : 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, m0=1
(Schneider et. Al., 2007)
Numerical resolution (30,15,15) O(105) d. o. f.
Initial conditions for the bisection method
Axial average
‘Edge’ trajectories
Local Velocity field
Measure of recurrences?
Function ri(t)
Function ri(t)
rmin(t)
rmin along the edge trajectory
Starting guesses
A Brmin =O(10-1)
Convergence using a Newton-Krylov algorithm
rmin = O(10-11)
The skeleton of the dynamics on the edge Recurrent visits to a Travelling Wave solution
…
Eu
Es
Eu
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
Eu
Es
Es
L ~ 2.5D, Re=2400, m0=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, m0=3) generate edge trajectory
Edge trajectory for Re=10,000
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)