Turbulent dynamics of pipe flows captured in a â€2+ ›...

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  • Turbulent dynamics of pipe flows captured in a 2+-dimensional modelAshley P. WillisLadHyX, cole Polytechnique, Palaiseau, France.,

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

    + Yohann Duguet, Chris Pringle

  • Parabolic laminar flowD, diameterU, mean axial flow (constant)

    Re = UD / Pipe flowNo linear instability

  • Peixinho & Mullin (2006), PRLt EnergysuddenrelaminarisationTemporal characteristics

  • Disturbance amplitudeReexact travelling waves(all unstable)Issues : Does turbulence become sustained?How are TWs related to turbulence?How does localised turbulence remain localised!?F&E (2003)W&K (2004)

  • Parabolic laminar flowD, diameterU, mean axial flow (constant)Re = UD / Localised puff plot of axial vorticityExpanding slugQ: Can we reduce the system but capture localised structures?

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

    i.e. only 3 degrees of freedom in :a mean mode,a sinusoidal variation in ,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+-dimensional model.11

  • 3-dim. 3,600,000 d.f.2+-dim. 160,000 d.f.(Lz = 50 diameters, 25 shown)Do we capture the spatial characteristics?puff slow delocalisation slugRe = 2600, 3200, 4000Re = 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 TLz = 32 diameters

  • Short pipes

  • ~ 1/(Rec-Re)5

    Rec = 3450

  • Disturbance amplitudeReexact travelling wavesLower branch associated with `edge

  • LaminarEdgeTurbulenttAmplitudeTW

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

  • Model has exact TW solutions (Newton-Krylov code by Yohann Duguet) fast axial flowslow axial flow

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

  • Roll + wave energyAmplitude ~ Re-3/2

  • The edge in a long pipe (work in progress)Edge at high flow rate, Re = 4000Re = 2600, 3200, 4000exact soln. (Chris Pringle)exact soln. (Yohann Duguet)Turbulence at increasing Re

  • Edge at high flow rate, Re = 4000Re = 10000 see Yohanns 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 puff2+-dim model, Re = 3000Full 3-dim, Re = 2000