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Particles in turbulence
Federico Toschi
NCAR, Boulder CO, 14 August 2012
ICTR International Collaboration for Turbulence Research
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TOC
• Introduction
• Motivation to study particles in turbulence
• Short review of results on point particles
• A “point particle” model for finite size particles
• Fully deformable droplets...
• Conclusions
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Dictionary...
• Tracer: a tiny particle moving following the streamlines of the flow (e.g. a fluid molecule)
• Heavy particle: • density >> fluid density • size << dissipative scale
• Light particle: • density << fluid density• size << dissipative scale
• Large particle:• D <= 10 η
• Deformable “particle”• deforming droplets with surface tension
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Where is Lagrangian turbulence ?
Plankton
Pollu)onDust,stoms
Combus)on
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Eulerian vs. Lagrangian turbulence
Leonardo da Vinci; circa 1500 (translated by Ugo Piomelli): “Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to the random and reverse motion.”
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Eulerian turbulence
Inertial range
Richardson energy cascade
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The “standard model”
Multi-"actal modelParisi-Frisch 1995
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Tracers acceleration
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Acceleration multi-fractal view
with probability
and for the large scale:
The small scale fluctuates !!!
The large scale fluctuates !!!
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Acceleration multi-fractal view
From standard multifractal arguments:
Supposing without intermittency (K41 case)
Also able to make other predictions, i.e. acceleration variance conditioned to velocity value:
…with h=1/3
L. Biferale, G. Boffetta, A. Celani, B. Devenish, A. Lanotte, and F. Toschi. Multi#actal statistics of lagrangian velocity and acceleration in turbulence. Physical Review Letters, 93:064502, 2004.
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Acceleration p.d.f. result for tracers
Multifractal predictionK41prediction
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Lagrangian velocity statistics
Does it exist and how to estimate ?
We do not have much fantasy! So we “copy” from Eulerian turbulence: structure functions
In Eulerian turbulence we have
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Magnifying glass: Local Scaling Exponents
The local exponents ζp(τ) act as magnifying glass, probing locally the value of the scaling exponents. By comparing with S2 we also take advantage of the Extended Self Similarity (ESS)
Power law scaling plateaux for local scaling exponents
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Biferale, L., Bodenschatz, E., Cencini, M., Lanotte, A. S., Ouellette, N. T., Toschi, F., & Xu, H. (2008). Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation. Physics Of Fluids, 20(6), 065103. doi:10.1063/1.2930672
DNS & Experiments: no discrepancies !!
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S2 vs. τ : resolution far from satisfactory!!
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Local scaling exponents
DNS and experiment show a very good agreement
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MF: Lagrangian velocity statistics
Bridge between eulerian and lagrangian description:
We assume that and are linked by the typicaleddy turn over time at the given spatial scale
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MF: Lagrangian structure functions
Multifractal prediction for the Lagrangian structure functions
whereSame D(h) that forthe Eulerian field !!
This allow us to actually predict the following value:
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Velocity structure functions (tracers)
Eulerian (space) -> Lagrangian (time)
Start from Eulerian
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Lagrangian turbulence IS universal
ICTR A. Arneodo, R. Benzi, J. Berg, L. Biferale, E. Bodenschatz, A. Busse, E. Calzavarini, B. Castaing, M. Cencini, L. Chevillard, R. Fisher, R. Grauer, H. Homann, D. Lamb, A. S. Lanotte, E. Leveque, B. Luthi, J. Mann, N. Mordant, W.-C. Muller, S. Ott, N.T. Oullette, J.-F. Pinton, S.B. Pope, S.G. Roux, F. Toschi, X. Xu, P.K. Yeung
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How to model (point) particles in turbulence ?
Neutral HeavyLight
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Faxen pointwise particle model
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Numerical study of particles in turbulence
Bubbles β=3, St=0.6 and Reλ=78
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Year N Rλ Reference
1972 32 35 Orszag, Patterson, PRL 28 76
1981 128 84 Rogallo, NASA rep. 1981PRL 58 547 (1987)
1991 256 150 Vincent, Meneguzzi, JFM 25 1Sanada, PRA 44 6480
1993 512 200 She et al., PRL 70 3251
2001 1024 460 Gotoh, Fukuyama, PRL 86 3775
2003 2048 730 Kaneda et al. Phys.Fluids 15 L21
2006 4096 1200 Kaneda et al. J. Turb. 7 20
Floating Point Operation for eddy turnover time: FLOP ∝ N3 log(N) x N (at high N flop=660 N3 log(N) for time step)Memory: ram ∝ N3 (at high N, RAM=182 N3 byte)
best fit: Reλ=2.26 N0.75
K41: Reλ≈N2/3
Moore law for DNS of HI turbulence
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k-5/3Spectral flux
N Reλ η L TL τη T δx Np
512 183 0.01 3.14 2.1 0.048 5 0.012 0.96 106
1024 284 0.005 3.14 1.8 0.033 4.4 0.006 1.92 106
Lagrangian database(x(t),v(t),a(t)=-∇p+ν∆u)with high temporal resolution
Energy spectrum
CINECA keyproject 10243 DNS+tracers
Pseudo spectral code - dealiased 2/3 rule - normal viscosity - 2 millions of passive tracers- code fully parallelized with MPI+FFTW - Platform IBM SP4 (sust. Performance 150Mflops/proc) - 50000 cpu hours - duration of the run: 40 days
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N Reλ η L TL τη T δx Np
2048 400 0.0025 3.14 1.8 0.02 5.9 0.003 2 109
Lagrangian database (x(t),v(t),u(t),∂iuj (t)) at high resolution
Energy spectrum
DEISA 20483 DNS with tracers & heavy
Pseudo spectral code - dealiased 2/3 rule - normal viscosity - 2 billions of passive tracers & heavy particles- code fully parallelized with MPI+FFTW - Platform SGI Altix 4700 - 400000 cpu hours - duration of the run: 40 days over 3 months.
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N Reλ η L TL τη T δx Np
128
512 183 0.01 3.14 2.1 0.048 5 0.012 1.0 108
Lagrangian database (x(t),v(t),u(t),∂iuj (t)) at high resolution
CASPUR 5123 DNS tracers & heavy & light
Pseudo spectral code - dealiased 2/3 rule - normal viscosity - 100 millions of passive tracers & heavy/light particles- code fully parallelized with MPI+FFTW - Platform IBM SP5 1.9 GHz - 30000 cpu hours - duration of the run: 30 days.
tracerbubble heavy
64 different particles classes (ß,St)
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Heavy particles - parametersL3 2563 5123 20483
Total particles 32 Mparticles 120 Mparticles 2,1 Gparticles
Stokes/ LyapStokes 16/32 16/32 21
Slow dumps 10 τη 2.000.000 7.500.000 101.888.000
Fast dumps 0.1 τη 250.000 500.000 203.776
dt 8 10-4 4 10-4 1.1 10-4
Time step ch0+ch1 756 + 1744 900 + 2100 11000+39000
τη 0.0746 0.0466 0.02
τ 0.0, 0.0120, 0.0200, 0.0280, 0.0360, 0.0440, 0.0520, 0.0600, 0.0680, 0.0760, 0.0840, 0.1000, 0.1200, 0.152, 0.200, 0.248
0.0, 0.00753454, 0.0125576, 0.0175806, 0.0226036, 0.0276266, 0.0326497, 0.0376727, 0.0426957, 0.0477187, 0.0527418, 0.0627878, 0.0753454, 0.0954375, 0.125576, 0.155714
0, 0, 0, 0.0032, 0.0032, 0.0032, 0.012, 0.012, 0.012, 0.02, 0.02, 0.02, 0.04, 0.06, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.4
Disk space used 400 GByte 1 TByte 6.3 Tbytes
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http://mp0806.cineca.it/icfd.php
iCFDdatabase2
Heavyparticle
Filtered tracer
Tracers at particle position
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All effects of inertia, in a slide...
Bubbles Tracers Heavy
Preferential concentration Filtering of turbulent fluctuations
Toschi and Bodenschatz. Lagrangian properties of particles in Turbulence. Ann. Rev. Fluid Mech. (2009) vol. 41 pp. 375-404
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Heavy particle’s acceleration
Filteredacceleration
Fluid acceleration at particle’s position -20%
Acceleration at varying Stokes no model
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More general picture of forces
• Behavior of acceleration vs. (St, β)
arms
Toschi and Bodenschatz. Lagrangian properties of particles in Turbulence. Ann. Rev. Fluid Mech. (2009) vol. 41 pp. 375-404
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Preferential concentration
β = 0, heavyβ = 1, tracerβ = 3, bubble
Poisson distribution
St=0.6
tracerbubble heavy
105 particles
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Kaplan-Yorke dimension: DKY
As in Bec Phys. Fluids (2003), Bec JFM (2005), Bec et al. Phys. Fluids (2006)
Particle equations of motion defines a dissipative dynamical systemAttractor’s dimension in the (x,v) space: Kaplan-Yorke dimension DKY
DKY
6 Lyapunov exponents computed by tracking
stretchingrates
Standard orthonormalization Gram-Schmidt procedure adopted
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Federico Toschi - ETC12 Marburg 2009
Kaplan-Yorke dimension Balance between contraction and expansion
DKY
DKY
Heavy min. at St≈0.5, DKY ≈2.6 Light min at
St≈1, DKY≈1.4
Close to fractal dimension of vortex filaments in turbulence? Dω≈1.1(Moisy & Jimenez JFM04)
DKY =3 ± 0.01
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P2(r) Probability to find a couple of particle whose distance is below r.
At r << η P2(r) = A rD2
Correlation dimension D2
r
Similar features as DKY
fractal dimension hierarchy: D2 ≤ D1 = DKY
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Finite size particles
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Motivation: Marine snow
Particle aggregation processes and how they affect particles in the marine environment. Biological aggregation (e.g., fecal pellet production) and physical aggregation by (a) shear and (b) differential sedimentation form large, heterogeneous, rapidly settling particles in the surface waters. In deeper waters, fragmentation and repackaging of this material by zooplankton are the dominant processes that affect aggregate sizes and properties. Microbes decompose material throughout the water column.
Burd, A. B., & Jackson, G. A. (2011). Particle Aggregation. Annu. Rev. Marine. Sci., Annual Review of Marine Science, 1(1), 65-90. Annual Reviews.
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Bacteria: small scale view
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Bacteria: large scale view
http://earthobservatory.nasa.gov/IOTD/view.php?id=41385
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Plankton: small scale view
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Plankton: large scale view
"Van Gogh" AlgaeImage courtesy EROS/USGS/NASA
In the style of Van Gogh's "Starry Night," massive congregations of greenish phytoplankton swirl in dark water around Sweden's Gotland (see map) island in a satellite picture released this week by the U.S. Geological Survey (USGS).
The image of the Baltic Sea island is 1 of 40 in the new Earth as Art 3 collection, the latest compilation of Landsat pictures chosen for their artistic quality.
"The collected images are authentic and original in the truest sense," Matt Larsen, the USGS's associate director for Climate and Land Use Change, said in a statement. "These magnificently engaging portraits of Earth encourage us all to learn more about our complex world."
Population explosions, or blooms, of phytoplankton, like the one shown here, occur when deep currents bring nutrients up to sunlit surface waters, fueling the growth and reproduction of these tiny plants, according to the USGS.
(Related: "The Best Pictures of Earth: Reader Picks of NASA Shots.")
Published November 19, 2010
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Experiments
Recent experiments on acceleration statistics with D ≥ η particles:
- G. Voth et al. J.Fluid Mech. (2002)
- N. Qureshi et al. Phys. Rev. Lett. (2007)
- R. Volk et al. Physica D (2008)
- H. Xu et al. Physica D (2008)
- N.Qureshi et. al EPJ B (2008)
- R. D. Brown et. al PRL (2009)
- R. Volk et al. JFM (2011)
- Experimentally most studied case: neutrally buoyant particles.
Von Karman Flow, Corne! USA, Goettingen, Lyon
Wind tunnelgrid turbulenceGrenoble
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Instrumented particle
• Instrumented tracer for Lagrangian measurementsin Rayleigh-Bénard convectionInstrumented tracer for Lagrangian measurementsin Rayleigh-Bénard convection
Shew, W. L., Gasteuil, Y., Gibert, M., Metz, P., & Pinton, J. (2007). Instrumented tracer for Lagrangian measurements in Rayleigh-Benard convection. Review of Scientific Instruments, 78(6), 065105. doi:10.1063/1.2745717
Instrumented tracer for Lagrangian measurements in Rayleigh-Bénard convection
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Tracking the dynamics of translation and absolute orientation of a sphere in a turbulent flowRobert Zimmermann, Yoann Gasteuil, Mickael Bourgoin,Romain Volk, Alain Pumir, and Jean-Francois Pinton
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Finite size particles
These clear, colorless spheres made of sodium polyacrylate - the superabsorbent polymer found in diapers - grow from 4 to 20 mm in diameter. Hydrated spheres have the refractive index of water, so they are invisible in water. Students can monitor growth rate or calculate before and after volumes as spheres hydrate over 12 hr.
Each pack contains 50 g (about 1,100 spheres). Re-use objects multiple times. Allow 12 hr to air dry.
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Finite size particles experiments: GoettingenAbstract: CA.00007 : Flow around finite-size neutrally buoyant Lagrangian particles in fully developed turbulence DFD 2010
Mathieu Gibert (Max Planck Institute for Dynamics and Self-Organization)
Simon Klein (Max Planck Institute for Dynamics and Self-Organization)
Antoine B\'erut (Max Planck Institute for Dynamics and Self-Organization)
Eberhard Bodenschatz (Max Planck Institute for Dynamics and Self-Organization)
By using an innovating technique based on Lagrangian Particle Tracking (LPT), we have been able to follow the motion of finite-size neutrally buoyant particles together with the trajectories of tracer particles in the surrounding fluid. The particles we study have diameters of about 200 times the dissipative scale of the flow, and their density is almost that of the fluid. The experiments are conduced in a von Karman swirling water flow at Taylor microscale Reynolds numbers up to 500. By measuring the full motion of the big particles (translation and rotation), we are able to ``sit'' in their frame of reference and measure the flow properties around them. We will report experimental results on the flow properties and its correlations with the big particle trajectories in this Lagrangian frame.
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Finite size particle: Lyon
Figure 1. (Colour online) Experimental set-up. (a) Geometry of the turbulence generator. (b)Schematics of the von Karman flow in water. (c) Principle of the LDV using wide beams
(eLDV) – top view of the experiment. PM denotes location of the photomultiplier which
detects scattering light modulation as a particle crosses the interference pattern created at
theintersection of the laser beams. The eLDV measures, for one particle at a time, the
evolution of its velocity component ux (t ) along the particle trajectory.
Volk, R., Calzavarini, E., Leveque, E., & Pinton, J. F. (2011). Dynamics of inertial particles in a turbulent von Kármán flow. Journal Of Fluid Mechanics, 668, 223-235. doi:10.1017/S0022112010005690
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Phenomenology (neutrally buoyant particle)
D-2/3
D-2/3
Voth et al. (2002) Qureshi et al. (2007)
For D>η, acceleration variance decreases when D is increased
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Phenomenology (neutrally buoyant particle)
Qureshi et al. (2007)
D/η
Early experiments were showing acceleration PDF very weakly dependent on D and non-Gaussian, recently things changed.
Volk et al. (2011)
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From the Point Particle model (PP)...
dilute suspension (no collisions), no particle feedback on the flow
2a
Point-like heavy/neutral/light inertial particles: Stokes drag, fluid acceleration & added mass
0 ≤ β ≤3
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A minimal model for finite-sized particles: the Faxén Corrected model (FC)
2rp
Acceleration statistics of finite-sized particles in turbulent flow: the role of Faxen forcesE. Calzavarini , R. Volk, M. Bourgoin, E. Leveque, J-F. Pinton and F. Toschi, J. Fluid Mech. (2009)
Accounts for the nonuniformity of the flow at the particle scale
volume average
surface average
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Sphere Average
Approximation:
Gaussian Average
with first order Faxén corrections
Efficient numerical implementation
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PP model (neutrally buoyant particles)
• Acceleration variance is D independent• Acceleration PDF is D independent• Correlation time of Acceleration does not
grow with D, but reduces.
arms / <Dtu>rms
=1same acc.as the fluid
St=4 D/η ≈ 7
See:Volk et al. Physica D (2008)
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Acceleration Variance from:Point-Particle and Faxén corrected models
DNS (1283 ) Reλ =75
PP
FCheavy
light
neutral
<a2 >
/<a f2 >
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Acceleration Flatness
DNS (1283 ) Reλ =75
3
F( af ) PP
FCheavy
light
neutral
F(a) → F(af)
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PDF of acceleration: exp vs. FC simulations
DNS (5123 ) Reλ =180EXP (Qureshi et al. 2007) Reλ =160 EXP (Volk et al. 2011)
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Acceleration Variance: comparison
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Acceleration Variance: comparison
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More refined FC model for finite size particles
Van Dyke, M. (1982). An Album of Fluid Motion. (M. Van Dyke, Ed.) (12nd ed. p. 176). Parabolic Press, Inc.
Impact of trailing wake drag on the statistical properties of finite-sized particles in turbulence
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More refined particle model
Non-Stokesian drag force(model for the wake drag)
History force(unsteady Stokes drag)
Faxen corrected modelwith Stokes drag
Schiller-Naumann (1933)
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Particle Reynolds number
Focus on the case Reλ ~ 31 and neutrally buoyant particles to compare with fully resoled DNS by
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Velocity variance
particle
fluid
=
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Acceleration variance
a = [Du/Dt]v (sphere)
a = [Du/Dt]v (Gauss)
(with improved volume averages)
Acceleration variance
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Large Eddy Simulation
Filtered velocity field Filtered energy spectra
Resolved scales
Unresolved scales
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Effects of droplet deformability
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Droplets deformability: key issues
• Physics: finite size particles plus surface tension• Transfer of energy from fluid to elastic modes (and
viceversa)• How is turbulence affected by the presence of droplets?• How do properties of (deformable) droplets differ from
rigid droplets ?
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Experimental investigations
Gopalan and Katz. Turbulent Shearing of Crude Oil Mixed with Dispersants Generates Long Microthreads and Microdroplets. PRL, 104, 054501 (2010)
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Prakash, Tagawa, Martinez-Mercado, Sun, LohseExperiment in the Twente water tunnel.
Bubbles deform in presence of flows.
Experimental investigations
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Some recent numerical work
Qian et al., Simulation of bubble breakup dynamics in homogeneous turbulence. Chem. En#. Commun. 193, 1038 (2006)
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Some recent numerical work
J.J. Dersksen and H.E.A. Van Den Akker, Chem. En#. Res. (2006)
• Turbulence
• Inertial force
• Surface tension force
• Weber number
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Dimensionless numbers
J.O. Hinze, A.I.Ch.E, (1955)
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Elongation in a stationary flow
H. Stone, “Dynamics of drop deformation and breakup in viscous fluids” Annu Rev Fluid Mech (1994) vol. 26 pp. 65-102
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d > dmax: Droplet breaksd < dmax: Droplet does not break
Hinze 1955
J.O. Hinze, A.I.Ch.E, (1955)
K41
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Numerical approach
Biferale, L., Perlekar, P., Sbragaglia, M., & Toschi, F. (2012). Convection in Multiphase Fluid Flows Using Lattice Boltzmann Methods.Physical Review Letters, 108(10).
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D3Q19 BGK LB model
with multicomponent Shan-Chen
Technique inspired to the continuum Boltzmann equation
Lattice Boltzmann Method (LBM)
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Shan and Chen. Lattice Boltzmann Model for Simulating Flows with Multiple Phases and Components. Phys. Rev. E 47, 1815 (1993).
LBM: multicomponent SC
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Random phases generated from Ornstein-Uhlenbeck process
Convincing LBM to go turbulent
Forcing: Large scale forcing in first two Fourier modes
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LBM: Energy and enstrophy
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Acceleration of a fluid parcel
SC: acceleration for single component flow
LBM: Energy and acceleration
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• JUGENE (FZJ-JSC IBM Blue Gene/P)• 23.5RM (about 15Mhours)• 32-64 kprocs• I/O HDF5• Fully parallel code
Simulations
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Droplet breakup in turbulence
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Towards a stationary state…
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Droplet radius vs. time
0
30
Nd
60
90
0 5 t/!eddy 15 20 25
"=0.5%"=5%
"=10%
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Re N density ratio
(liquid /saturated
vapor)
viscosity
G dissipative
scale(LU)
D Hinze
D(LBE)
We Volume fraction
%
Eddy turnove
r
RUN0 80 512 0.5/0.5 0.005 - 6 - - -
RUN1 80 512 2.038/0.362
0.005 0.03 6 24.2 24+/-1 0.075 0.3%RUN2 80 512 1.757/0.08
80.005 0.08 6 39.5 36+/-1 0.033 0.3%
RUN0 40 128 0.5/0.5 0.005 - 3
RUN1RUN2RUN3RUN4
40 128 2.038/0.362
0.005 0.03 3 8.65 11,13,15,18
0.12 0.07,0.5,5,10
50,50,50,560
Simulation parameters
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pdf of radius and volumes
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d32 (LBM)
40 14.5 3.5 10-8
80 24 2.85 10-9
Droplet PDF: Re dependence
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Droplet size distribution
0.1
0.2
0.4
0.6
0.8
1.0
1.2
0.6 1 5 10 20
P(R
)/[P
(R)]
max
R
!=0.07%
!=0.5%
!=5%
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Verification of Hinze criterion
1!10-2
"#<D>
1!10-1
3!10-1
1!101 1!102$/#"
4 1!104 5!104
#=1.6e-3,%=0.3%, N512A#=1.7e-3,%=0.3%, N512B
#=1.6e-3,%=0.07%, N128A #=1.6e-3,%=0.5%, N128B
#=1.6e-3,%=5%, N128C #=1.6e-3,%=10%, N128D
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Trajectories look smooth but acceleration is very noisy
Droplet trajectory vs. acceleration
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PDF of acceleration
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Droplet deformation
Volume VSurface S
Deformation parameter is the dimensionless surface i.e. normalized wrt the surface of the equivalent sphere with same volume V
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Deformation, dissipation and breakups
0
2
4
Wes
6
8
10
7 14 t/!eddy 21 28 0
0.2
S*
0.4
0.6S*
Wes
! ! !
! breakup
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Deformation vs. local Weber
-1.0 -0.5 log10 Wes 0.5 1.0
-2.2
-1.8
-1.4
-1.0
-0.6
-0.2lo
g10(S
* )
S*=We0.066
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The boring (?) life of non-breaking drop
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Minimal bibliography
Droplets in turbulenceDroplet size distribution in homogeneous, isotropic turbulence, P. Perlekar, L. Biferale, M. Sbragaglia, and F. Toschi, arXiv:1112.6041(submitted to Phys. Fluids).Convection in Multiphase Fluid Flows Using Lattice Boltzmann Methods, L. Biferale, P. Perlekar, M. Sbragaglia and F. Toschi, Physical Review Letters, 108(10) (2012).
Numerical TechniqueA Lattice Boltzmann method for turbulent emulsions, L. Biferale, M. Sbragaglia, S. Srivastava, and F. Toschi, J. Phys.: Conf. Ser., 318, 052017 (2011) .Unified %amework for a side-by-side comparison of different multicomponent algorithms: lattice Boltzmann vs phase field model, L. Scarbolo, D. Molin, P. Perlekar, A. Soldati, and F. Toschi, arXiv:1112.5547 (submitted to J. Comp. Phys.)
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The end.
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PublicationsL. Biferale and G. Boffetta and A. Celani and A. Lanotte and F. Toschi,Lagrangian statistics in fu!y developed turbulenceJournal of Turbulence, 7 (2006) 1-12 10.1080/14685240500460832.
T. H. Van Den Berg and S. Luther and I. M. Mazzite!i and J. M. Rensen and F. Toschi and D. Lohse,Turbulent bubbly flowJournal of Turbulence, 7 (2006) 1-12 10.1080/14685240500460782,
L. Biferale and F. Toschi,Joint statistics of acceleration and vorticity in fu!y developed turbulenceJournal of Turbulence, 6 (2006) 1-8 10.1080/14685240500209874.
J. Bec and L. Biferale and G. Boffetta and A. Celani and M. Cencini and A. Lanotte and S. Musacchio and F. Toschi,Acceleration statistics of heavy particles in turbulenceJournal of Fluid Mechanics, 550 (2006) 349-358 10.1017/S002211200500844X M. Cencini and J. Bec and L. Biferale and G. Boffetta and A. Celani and A. Lanotte and S. Musacchio and F. Toschi,Dynamics and statistics of heavy particles in turbulent flows
L. Biferale and J. Bec and G. Boffetta and A. Celani and M. Cencini and A. Lanotte and S. Musacchio and F. Toschi,Inertial Particles in Turbulence
Luca Biferale and Massimo Cencini and Alessandra S. Lanotte and Federico Toschi,Velocity Statistics of Tracers and Inertial Particle in Turbulent Flows: the effects of vortex trapping
J. Bec and L. Biferale and G. Boffetta and M. Cencini and A. Lanotte and S. Musacchio and F. Toschi,Lyapunov exponents
/ TN & WI Federico Toschi
Publications
L. Biferale and G. Boffetta and A. Celani and A. Lanotte and F. Toschi,Particle trapping in three-dimensional fu!y developed turbulencePhysics of Fluids, 17 (2005) 021701 10.1063/1.1846771 L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte and F. Toschi,Lagrangian statistics of particle pairs in homogeneous isotropic turbulencePhysics of Fluids, 17 (2005) 115101 10.1063/1.2130742
F. Toschi and L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte,Acceleration and vortex filaments in turbulenceJournal of Turbulence, 6 (2005) 1-10 10.1080/14685240500103150
L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte and F. Toschi,Multiparticle dispersion in fu!y developed turbulencePhysics of Fluids, 17 (2005) 111701 10.1063/1.2130751 L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte and F. Toschi,Multi&actal statistics of Lagrangian velocity and acceleration in turbulencePhysical Review Letters, 93 (2004) 064502 10.1103/PhysRevLett.93.064502
Irene M. Mazzite!i and Detlef Lohse and Federico Toschi,Effect of microbubbles on developed turbulencePhysics of Fluids, 15 (2003) L5-L8, 10.1063/1.1528619
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