Turbulent dynamics of pipe flows captured in a

‘2+ɛ’-dimensional model

Ashley P. WillisLadHyX, École Polytechnique, Palaiseau, France.,

Rich KerswellDept. of Maths., University of Bristol, U.K.

+ Yohann Duguet, Chris Pringle

Parabolic laminar flow

r

θ z

D, diameterU, mean axial flow (constant)

Re = UD / ν

Pipe flow

No linear instability

Localised “puff” plot of axial vorticity

Expanding “slug”

Distinct spatial structures:

Re >~ 1650

Re >~ 2250

Peixinho & Mullin (2006), PRL

t

Energy

suddenrelaminarisation

Temporal characteristics

Disturbance amplitude

Re

exact travelling waves

(all unstable)Issues :

Does turbulence become sustained?

How are TWs related to turbulence?

How does localised turbulence remain localised!?

Repuff-slug~2250Re ~ 750

F&E (2003)W&K (2004)

Parabolic laminar flow

r

θ z

D, diameterU, mean axial flow (constant)

Re = UD / ν

Localised “puff” plot of axial vorticity

Expanding “slug”

Q: Can we reduce the system but capture localised structures?

Problems: System has a vast number of degrees of freedom!Short periodic pipes cannot capture the spatio-temporal characteristics.

Full 3-dim calculations are expensive.

Aim: Want to study pipe flow as a subcritical dynamical system.

• → leaves only θ . Minimal 3-dimensionalisation: Fourier modes, m = −m0, 0, m0 only.

i.e. only 3 degrees of freedom in θ :a mean mode,a sinusoidal variation in θ,an azimuthal shift of the sinusoid.

• localised in z → keep axial resolution

• near-wall structures important, detachment from wall during relaminarisation; r → keep radial resolution

‘2+ɛ’-dimensional model.

1

1

‘ɛ’

3-dim. 3,600,000 d.f.

‘2+ɛ’-dim. 160,000 d.f.(Lz = 50 diameters, 25 shown)

Do we capture the spatial characteristics?

puff → slow delocalisation → slug

Re = 2600, 3200, 4000

Re = 2000, 2300, 2700

Reduced model preserves temporal characteristics...

long-term transients, sudden decay, memoryless

(Lz = 50 diameters, Re=2800)

(W&K 2007)

Probability of surviving to time T Lz = 32 π diameters

Short pipes

τ ~ 1/(Rec-Re)5

Rec = 3450

Disturbance amplitude

Re

exact travelling waves

Lower branch associated with `edge’

Laminar

‘Edge’

Turbulent

t

Amplitude

TW

The laminar-turbulent ‘edge’ (short pipe Lz = π diameters; 10,000 d.f.)

Model has exact TW solutions (Newton-Krylov code by Yohann Duguet)

fast axial flow slow axial flow

The laminar-turbulent ‘edge’ (short pipe Lz = π diameters; 10,000 d.f.)

Roll + wave energy

Amplitude ~ Re-3/2

The ‘edge’ in a long pipe (work in progress)

Edge at high flow rate, Re = 4000

Re = 2600, 3200, 4000

exact soln. (Chris Pringle)

exact soln. (Yohann Duguet)

Turbulence at increasing Re

Edge at high flow rate, Re = 4000

Re = 10000

see Yohann’s talk

• Model has TWs, long-term transients, memoryless, spatially localised puffs, slugs, chaotic edge state.

• Amplitude of rolls+waves ~ Re-3/2 .

• Can we use the model to explain why puffs stay localised?

• Is there a measure that predicts the puff-slug transition?

• ..or a sensitive measure that pre-empts relaminarisation?

• What is the effect of ‘noise’ on transitions, relam., puff-slug...?

• What is the ‘minimal flow unit’ that produces transients?

• Are there exact localised solutions on the edge? periodic orbits?

Ref.: arXiv.org:0712.2739

Time-averaged profiles within the puff

2+ɛ-dim model, Re = 3000Full 3-dim, Re = 2000