turbulent dynamics of pipe flows captured in a ‘2+ ɛ ’-dimensional model ashley p. willis...
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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
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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
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Peixinho & Mullin (2006), PRL
t
Energy
suddenrelaminarisation
Temporal characteristics
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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)
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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.
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• → 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’
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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
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Reduced model preserves temporal characteristics...
long-term transients, sudden decay, memoryless
(Lz = 50 diameters, Re=2800)
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(W&K 2007)
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Probability of surviving to time T Lz = 32 π diameters
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Short pipes
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τ ~ 1/(Rec-Re)5
Rec = 3450
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Disturbance amplitude
Re
exact travelling waves
Lower branch associated with `edge’
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Laminar
‘Edge’
Turbulent
t
Amplitude
TW
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The laminar-turbulent ‘edge’ (short pipe Lz = π diameters; 10,000 d.f.)
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Model has exact TW solutions (Newton-Krylov code by Yohann Duguet)
fast axial flow slow axial flow
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The laminar-turbulent ‘edge’ (short pipe Lz = π diameters; 10,000 d.f.)
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Roll + wave energy
Amplitude ~ Re-3/2
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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
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Edge at high flow rate, Re = 4000
Re = 10000
see Yohann’s talk
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• 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
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Time-averaged profiles within the puff
2+ɛ-dim model, Re = 3000Full 3-dim, Re = 2000