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)