Chaotic mixing of viscous fluids Topological entanglement and - - PowerPoint PPT Presentation

Chaotic mixing of viscous fluids

Topological entanglement and transport barriers

Emmanuelle Gouillart

Joint Unit CNRS/Saint-Gobain, Aubervilliers

Olivier Dauchot, CEA Saclay Jean-Luc Thiffeault, University of Wisconsin

Outline

1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows

Plan

1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows

Fluid mixing is ubiquituous in the industry...

Mixing step in many processes Low Reynolds number : no turbulence Closed (batch) or open flows

... or in the environment

What we would like to know about fluid mixing

What we would like to know about fluid mixing

Can we avoid unmixed regions ? Homogenization mechanisms : black/white → gray ? Mixing speed ? Quality of mixing ? How to define it and how to measure it ?

Plan

1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows

Mixing = transport, stretching and diffusion

Arratia et al. 2004

Large-scale mixing : particles of a fluid patch should disperse (fast) in the whole domain. This imposes to stretch the initial patch Diffusion is more efficient for thin filaments

Chaotic advection : chaos in the fluid space !

How can one disperse and stretch fluid patches ? Slow (linear with time) stretching. No mixing across streamlines.

Meunier and Villermaux 2003

Chaotic advection : chaos in the fluid space !

How can one disperse and stretch fluid patches ? Slow (linear with time) stretching. No mixing across streamlines.

Meunier and Villermaux 2003

Successive shears in different directions → rapid stretching and fluid dispersion.

Chaotic advection : chaos in the fluid space ! t l0 l(t) = l0eλt

Successive shears in different directions → rapid stretching and fluid dispersion. Neighbouring particles separate exponentially with time. Enough flow complexity ⇒ chaotic trajectories.

Chaotic advection : chaos in the fluid space !

Lagrangian trajectories are determined by a dynamical system, that can be chaotic if it has enough degrees of freedom (≥ 3). 2-D incompressible flows : Hamiltonian dynamics.

• ˙

x = vx(t) = −ψ,y ˙ y = vy(t) = ψ,x Need for time dependency. Phase space = real fluid space Aref 1984, Ottino 1989 chaotic advection 3-D flows

    

˙ x = vx ˙ y = vy ˙ z = vz Fountain et al., 2000

Poincaré sections reveal mixed and unmixed regions

Chaiken et al., 1986

Lyapunov exponents measure stretching

t l0 l(t) = l0eλt Muzzio et al., PoF 1991 Lyapunov exponent : mean stretching rate A measure of mixing efficiency

Topological mixing

A new description of chaotic advection

Principle : characterize chaotic advection by the braiding/entanglement of fluid particles (as opposed to the stretching of fluid particles). Topological description = metric description

Topological chaos imposed by stirrers

Boyland, Aref, Stremler JFM 2000 Clever motion of rods ⇒ entangled trajectories of the rods, on which material lines are stretched exponentially with time. The topological complexity is characterized by the topological entropy (the entanglement) of the braid traced by the rods.

+

smart protocol ⇒ entangled braid →

t

• The motion of rods (its braid) gives a lower bound on the stretching

rate of filaments.

Topological chaos

Topological entropy hbraid of a braid : index of entanglement

t

• Topological

entropy

• f

the flow : rate

• f

exponential stretching

• f

material lines L(t) = L0 exp(hflowt)

hbraid ≤ hflow

Topological chaos with ghost rods

Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Elliptical islands are ghost rods

Topological chaos with ghost rods

Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Elliptical islands are ghost rods Poincaré section displays many islands Rod + islands ⇒ entangled braid. Material line stretched

• n

the braid.

Topological chaos with ghost rods

Poincaré section displays many islands Rod + islands ⇒ entangled braid. Material line stretched

• n

the braid. Elliptical islands account for topological chaos and behave like ghost rods. A simple glimpse at the Poincaré section can give a lower bound on the stretching rate of material lines.

Topological chaos with ghost rods

Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Unstable (point-like) periodic orbits are ghost rods as well

0.5 1 1.5 2 2.5 5 10 15 20 25 30 hbraid number of strands in braid

A small number of orbits can account for the topological entropy

• f the flow.

Topological chaos with ghost rods

Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Unstable (point-like) periodic orbits are ghost rods as well

0.5 1 1.5 2 2.5 5 10 15 20 25 30 hbraid number of strands in braid

A small number of orbits can account for the topological entropy

• f the flow.

The complexity of chaotic advection arises from the entanglement of ghost rods’ trajectories. Different characterization of fluid mixing in terms of braiding and topological entanglement, as opposed to metric stretching (Lyapunov exponents).

[E. Gouillart, J.-L. Thiffeault, M. Finn, Phys. Rev. E, 73, 036311, 2006] [M. Finn, J.-L. Thiffeault, E. Gouillart, Physica D, 221, 92, 2006]

Designing mixers for efficient topological chaos

Finn and Thiffeault, 2010

√

Maximize topological entropy per rods exchange. Best configurations reach the silver ratio for the entropy. Easy implementation with planetary gears. Same idea in some industrial planetary mixers !

Applications of ghost rods

Extension to non-periodic orbits Oceanography Floats in the Labrador sea Thiffeault, Chaos 2010 Dynamics of granular media Puckett et al. 2009

Statistics of concentration fluctuations

Evolution according to the advection-diffusion equation ∂C ∂t + v · ∇C = D∆C One fluid particle is stretched and thinned up to the Batchelor scale wB =

• D/λ

Statistics of concentration fluctuations

One fluid particle is stretched and thinned up to the Batchelor scale wB =

• D/λ

The concentration levels of several fluid particles are averaged inside boxes of size wB → fluctuations of C decay.

Statistics of concentration fluctuations

The concentration levels of several fluid particles are averaged inside boxes of size wB → fluctuations of C decay. A particle (concentration fluctuation) is mixed once it has reached wB and is averaged with other particles. Finite-size Lyapunov exponents [Boffetta et al. 2000, 2001]

Plan

1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows

What is the speed of homogenization ?

Eulerian description

Pierrehumbert 1994, Rothstein et al. 1999 Jullien et al. 2000

self-similar concentration field : strange eigenmode The variance σ2(C) decays exponentially (at long times)

What is the speed of homogenization ?

Eulerian description

Pierrehumbert 1994, Rothstein et al. 1999 Jullien et al. 2000

self-similar concentration field : strange eigenmode The variance σ2(C) decays exponentially (at long times) Lagrangian description Antonsen et al. 1996 Exponential stretching should yield an exponential decay. Villermaux and Duplat, 2003 The concentration PDF evolves by auto-convolution due to (random) self-averaging by diffusion [Turbulence].

(Non-asymptotic) homogenization mechanisms

Stretching of filaments

(Non-asymptotic) homogenization mechanisms

Stretching of filaments Self-averaging by diffusion

(Non-asymptotic) homogenization mechanisms

Stretching of filaments Self-averaging by diffusion Persistent contrast in low-stretching regions

(Non-asymptotic) homogenization mechanisms

Stretching of filaments Self-averaging by diffusion Persistent contrast in low-stretching regions Chaotic region ⇒ propagation

• f poorly stretched patches

(Non-asymptotic) homogenization mechanisms

Stretching of filaments Self-averaging by diffusion Persistent contrast in low-stretching regions Chaotic region ⇒ propagation

• f poorly stretched patches

Mixing is controlled by the worst-stretched elements !

Wall-controlled slow mixing

1 rod, figure-eight periodic protocol (∞). Chaotic advection : stretching and folding.

Wall-controlled slow mixing

Wall-controlled slow mixing

Slow (algebraic) mixing

Wall-controlled slow mixing

No-slip condition ⇒ unmixed fluid at the wall leaks into the mixed region (d(t) ∼ 1/t) ⇒ contamination of the mixing pattern

Wall-controlled slow mixing

Model : modified baker’s map Accounts for algebraic dynamics. No-slip condition ⇒ unmixed fluid at the wall leaks into the mixed region (d(t) ∼ 1/t) ⇒ contamination of the mixing pattern

Insulate the mixing region from the wall... ... to recover exponential mixing dynamics

fixed wall

Insulate the mixing region from the wall... ... to recover exponential mixing dynamics

fixed wall rotating wall

10 20 30 40

t

10-1 100 101

(d)

• =0.2

=0.13

=0.21

(C)

10 20 30 40 10-1 100 101

=0.3

Insulate the mixing region from the wall... ... to recover exponential mixing dynamics

fixed wall rotating wall global rotation

10 20 30 40

t

10-1 100 101

(d)

• =0.2

=0.13

=0.21

(C)

10 20 30 40 10-1 100 101

=0.3

Insulate the mixing region from the wall... ... to recover exponential mixing dynamics

fixed wall rotating wall

[Chertkov and Lebedev 2003, Boffetta et al. 2009] [Gouillart et al. 2007, 2008, 2010, Thiffeault 2008]

Another example of slow transport and mixing

Yield-stress fluid Eggbeater + rotating wall

Another example of slow transport and mixing

Yield-stress fluid Eggbeater + rotating wall

Another example of slow transport and mixing

Yield-stress fluid Eggbeater + rotating wall

Plan

1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows

In open flows as well...

mixing is controlled by least-stretched elements

Open-flow mixing : fluid flows through the stirrers

In open flows as well...

mixing is controlled by least-stretched elements

Open-flow mixing : fluid flows through the stirrers

short residence times : poor mixing

In open flows as well...

mixing is controlled by least-stretched elements

Open-flow mixing : fluid flows through the stirrers

short residence times : poor mixing

In open flows as well...

mixing is controlled by least-stretched elements

Open-flow mixing : fluid flows through the stirrers

long residence times : mixing by diffusion

Mixing is efficient for particles that diffuse into other particles initially far from them.

In open flows as well...

mixing is controlled by least-stretched elements

Open-flow mixing : fluid flows through the stirrers

long residence times : mixing by diffusion invariant pattern at long times : open-flow strange eigenmode

[Gouillart et al., POF 21, 023603 (2009)]

New measures of transient mixing...

... derived from the strange eigenmode

From the invariant pattern (strange eigenmode), one can derive the fraction and the location of unmixed particles. New index of mixing : normalized standard deviation of the eigenmode : σSE = σ(C)/C [Gouillart et al. 2010]

Conclusions

The speed of mixing is controlled by poorly-stretched regions ⇒ Need for measures of chaos and mixing that are relevant for homogenization. The speed of mixing can often be "read" from mixing patterns (phase portrait, ghost rods, etc.)

Perspectives and open questions

Can we extend topological-mixing arguments to retrieve the rate of dye homogenization ? Identify a simple mixing criterion that accounts for stretching heterogeneities and homogenization rate. Spatial characterization of mixing : typical size of heterogeneities. Other systems : 3-D flows, complex fluids, ...

Collaborations

CEA Saclay : Olivier Dauchot, Bérengère Dubrulle, François Daviaud, Arnaud Chiffaudel Imperial College : Jean-Luc Thiffeault, Matthew Finn LMT, ENS Cachan : Stéphane Roux Saint-Gobain Recherche : Franck Pigeonneau, Jacques-Olivier Moussafir, Jean-Marc Flesselles, Laurent Pierrot Students : Natalia Kuncio (CEA), Benoît Roche, Ianis Lallemand (SGR)

References

• T. M. Antonsen, Z. Fan, E. Ott, and E. Garcia-Lopez.

The role of chaotic orbits in the determination of power spectra of passive scalars. 8 :3094, 1996.

• H. Aref.

Stirring by chaotic advection.

• J. Fluid Mech., 143 :1–21, 1984.
• P. E. Arratia, J. P.Lacombe, T. Shinbrot, and F. J. Muzzio.

Segregated regions in continuous laminar stirred tank reactors.

• Chem. Eng. Sci., 59 :1481–1490, 2004.
• G. Boffetta, A. Celani, M. Cencini, and G. Lacorata.

Nonasymptotic properties of transport and mixing. Chaos, 10 :50, 2000.

• G. Boffetta, F. De Lillo, and A. Mazzino.

Peripheral mixing of passive scalar at small Reynolds number. Journal of Fluid Mechanics, 624 :151–158, 2009.

• G. Boffetta, G. Lacorata, G. Redaelli, and A. Vulpiani.

Detecting barriers to transport : a review of different techniques. Physica D Nonlinear Phenomena, 159 :58–70, November 2001.

• P. L. Boyland, H. Aref, and M. A. Stremler.

Topological fluid mechanics of stirring.

• J. Fluid Mech., 403 :277–304, 2000.
• J. Chaiken, R. Chevray, M. Tabor, and Q. M. Tan.

Experimental study of Lagrangian turbulence in a Stokes flow.

• Proc. Roy. Soc. Lond., 408 :165–174, 1986.

References

• M. Chertkov and V. Lebedev.

Boundary effects on chaotic advection-diffusion chemical reactions.

• Phys. Rev. Lett., 90 :134501, 2003.
• M. Chertkov and V. Lebedev.

Decay of scalar turbulence revisited.

• Phys. Rev. Lett., 90 :034501, 2003.
• M. D. Finn, J.-L. Thiffeault, and E. Gouillart.

Topological chaos in spatially periodic mixers. Physica D, 221 :92–100, 2006.

• G. O. Fountain, D. V. Khakhar, I. Mezic, and J. M. Ottino.

Chaotic mixing in a bounded three-dimensional flow.

• J. Fluid Mech., 417 :265, 2000.
• E. Gouillart, O. Dauchot, B. Dubrulle, S. Roux, and J.-L. Thiffeault.

Slow decay of concentration variance due to no-slip walls in chaotic mixing.

• Phys. Rev. E, 78(2) :026211–+, August 2008.
• E. Gouillart, O. Dauchot, J.-L. Thiffeault, and S. Roux.

Open-flow mixing : Experimental evidence for strange eigenmodes.

• Phys. Fluids, 2009.
• E. Gouillart, N. Kuncio, O. Dauchot, B. Dubrulle, S. Roux, and J.-L. Thiffeault.

Walls inhibit chaotic mixing.

• Phys. Rev. Lett., 99 :114501, September 2007.
• E. Gouillart, J.-L. Thiffeault, and M. D. Finn.

Topological mixing with ghosts rods.

• Phys. Rev. E, 73 :036311, 2006.

References

• E. Gouillart, J.L. Thiffeault, and O. Dauchot.

Rotation Shields Chaotic Mixing Regions from No-Slip Walls. Physical Review Letters, 104(20) :204502, 2010. M.-C. Jullien, P. Castiglione, and P. Tabeling. Experimental observation of batchelor dispersion of passive tracers.

• Phys. Rev. Lett., 85 :3636, 2000.
• P. Meunier and E. Villermaux.

How vortices mix.

• J. Fluid Mech., 476 :213, 2003.
• F. J. Muzzio, P. D. Swanson, and J. M. Ottino.

The statistics of stretching and stirring in chaotic flows.

• Phys. Fluids A, 3 :5350–5360, 1991.
• J. M. Ottino.

The Kinematics of Mixing : Stretching, Chaos, and Transport. Cambridge University Press, Cambridge, U.K., 1989.

• R. T. Pierrehumbert.

Tracer microstructure in the large-eddy dominated regime. Chaos Solitons Fractals, 4 :1091–1110, 1994. J.G. Puckett, F. Lechenault, and K.E. Daniels. Generating ensembles of two-dimensional granular configurations. Chaos : An Interdisciplinary Journal of Nonlinear Science, 19 :041108, 2009.

• D. Rothstein, E. Henry, and J. P. Gollub.

Persistent patterns in transient chaotic fluid mixing. Nature (London), 401 :770, 1999.

References

J.L. Thiffeault. Braids of entangled particle trajectories. Chaos : An Interdisciplinary Journal of Nonlinear Science, 20 :017516, 2010. J.L. Thiffeault, M.D. Finn, E. Gouillart, and T. Hall. Topology of chaotic mixing patterns. Chaos : An Interdisciplinary Journal of Nonlinear Science, 18 :033123, 2008.

• E. Villermaux and J. Duplat.

Mixing as an aggregation process.

• Phys. Rev. Lett., 91 :184501, 2003.