Highly-symmetric travelling waves in pipe flow

Chris Pringle*, Yohann Duguet*† & Rich Kerswell**University of Bristol

†Linné Flow Centre, KTH Mechanics

Highly-symmetric travelling waves in pipe flow

Pipe Flow

• Linearly stable for all Reynolds numbers• Sustained turbulence possible after Re ≈ 1700-2000 Re based upon mean velocity and pipe diameter

Poiseuille 1840Hagen 1839

Travelling Waves

S2

S3

Asymmetric(S1)

Mirror Symmetric

Faisst & Eckhardt (2003), Wedin & Kerswell (2004), Pringle & Kerswell (2007)

Travelling Waves in Phase spaceS2

S3Faisst & Eckhardt (2003), Wedin & Kerswell (2004), Pringle & Kerswell (2007)

Travelling Waves within the Edge

• Strikingly different cross-sections• Alternative axial evolution• Additional mirror symmetry

A3 C3 S2

Duguet, Willis & Kerswell (2008)

Symmetries

• All of the TWs originally discovered only possess shift-&-reflect symmetry

M-class Travelling Waves

• Double layer of streaks• Rolls bisect layers• Relatively quiescent center

M2 M3 M4

N-class Travelling Waves

•Stronger, more active rolls•Larger streaks

N2 N3 N4

• M1 is known from Pringle & Kerswell (2007)

• N1 is entirely new

M1 and N1

Travelling Waves in Phase space

Travelling Waves in Phase space

S3 and N3

Stability of N2

Stability of N2

Summary

• Two new classes of TW have been explored

• They occur earlier than previously seen TWs

• They exhibit much higher friction factors – they are ‘more nonlinear’

• Appear to be more fundamental – the original TWs bifurcate off them