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HHeavy particles in eavy particles in turbulent turbulent flowsflows
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
with: J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, S. Musacchio, F. Toschi
Alessandra Lanotte
CNR ISAC Lecce (Italy)
a.lanotte@isac.cnr.it
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
OutlineOutline
• Introduction Physical systems Observations Model Details of numerical simulations What we measured
Short summary of some results
• Core of the talk Small scales clustering Inertial scales clustering
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Where do we find heavy Where do we find heavy particles?particles?
Formation of planetesimals in Formation of planetesimals in the solar systemthe solar system
(A. Bracco et al. Phys. Fluids 2002)
Control of combustion processes in diesel engines
(see T.Elperin et al. nlin.CD/0305017)
In clouds, dust storms, firesvolcano eruption..
(see e.g. K. Sassen, Nature 2005)
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
What can we observe/measure in a What can we observe/measure in a lab?lab?
Lagrangian turbulence has always suffered the lack of accurate space & time measurements
now particles can be accurately tracked !
QuickTime™ and aVideo decompressor
are needed to see this picture.
From Cornell group: frame rate : 1000fps; 4x4 cm area.
State-of-the-art Lagrangian experiments (tracers) Ott & Mann exp. at Risø, 3D PTV - Re 300 Pinton exp. at ENS, Doppler track. Re =740 Bodenshatz exp. at Cornell, fast CCD Re =1000
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Heavy particles in wind tunnel Heavy particles in wind tunnel turbulenceturbulence
Z. Warhaft experiment at Cornell
Re 250 water droplets <d> = 20 micronHigh-speed camera: 2D frames
Sampling time 1/100
then also other experiments in complex geometries: e.g. channel flows,..
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
The ModelThe Model finite size impurities of size much smaller than the flow dissipative scale
much heavier than the fluid
particle Reynolds number low
very dilute suspension : no role of collisions
no back reaction on the flow
a
fp
Rea a | V - u | 1
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Simplified equationsSimplified equations
Under previous assumption we can simplify original eqs:(M. Maxey & J. Riley, Phys Fluids 1983)
Parameters:
Stokes time --> Stokes number
Density ratio
Xonly Stokes drag(water in air =0.001)
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Something we know about inertia…Something we know about inertia…
Try to understand physical mechanisms
and identify relevant parameters for statistical description…
1. Ejection of heavy particles from vortices --> experience smaller
acceleration
3. Very strong concentration fluctuation --> particle distribute on
clusters
2. Particle have finite response time to fluid fluctuations
--> smoothing and filtering of fast time scales
(since Maxey, Eaton, Fessler, Squires, …)
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
A large numerical A large numerical “experiment”“experiment”
To start with the simplestsimplest situation
To have good statistics
To build up a database for common use
The lab particles in the flow box
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Details about the DNSDetails about the DNS
N3 5123 2563 1283
Tot #particles 120Millions 32Millions 4Millions
Fast 0.1 500.000 250.000 32.000
Slow 10 7.5Millions 2Millions 250.000
Stoke/Lyap (15+1)/(32+1) (15+1)/(32+1) 15+1
Traject. Length
900 +2100 756 +1744 600+
1200
Disk usage 1TB 400GB 70GB
Lagrangian Particles with 15
Lagrangian Tracers
Initial conditionsInitial conditions
particles and tracers particles and tracers injected injected randomly & homogeneously randomly & homogeneously with initial veloc. = with initial veloc. = fluid veloc.fluid veloc.
STATISTICSTRANSIENT (1-2 T)+BULK ( 3-4 T)
ReRe= 65, 105, 185= 65, 105, 185
Pseudo Spectral Code, Pseudo Spectral Code, MPIMPI
Normal viscosity Normal viscosity
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
How long do we wait for the stationary How long do we wait for the stationary mass distribution?mass distribution?
Coarse-grained mass in the j-th cell of side l=2x
St=0
St=0.48
St=0.27
St=0.9
St=1.6
St=3.3
St=0.16
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Just a quick overview about Just a quick overview about few things:few things:
Acceleration
Conditioned analysis
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Why study acceleration ?Why study acceleration ?Urban reshape, Old Shangai
Steel factory, Taranto
Acceleration is relevant for Lagrangian Stochastic Models for relative dispersion
(see e.g. Sawford, Ann. Rev. Fluid Mech. 2001)
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Acceleration for Acceleration for tracerstracers
a P(a)
K41prediction
Multifractal
(Biferale, Boffetta , Celani, Devenish, AL, Toschi 2004)
Tracers acceleration can be very well described in terms of the multifractal model
Phenomenological modelfor small scale fluctuations
1024^3 DNS
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Acceleration for heavy particlesAcceleration for heavy particles
Increase St
Increase Re
Two coexisting effects preferential concentration at low St filtered dynamics at higher St
No simple phenomenological model for particles at varying
Stokes and Reynolds numbers !
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Comparison with experimentsComparison with experiments
A. Gylfason, S. Ayyalasomayajula, E. Bodenschatz, Z. Warhaft, PRL submitted 2006
St=0.09
St=0.15
Water drops in air:clearly polydisperse flow !
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Particles and 3d flow Particles and 3d flow structuresstructures
White: non-hyper regionsBlack: hyperbolic regions
St =0.16
St = 0.8
St = 3.3
Particles preferentially concentrate in hyperbolic regions
Such effect is clearly evident by Such effect is clearly evident by looking at the fluid acceleration looking at the fluid acceleration
conditioned on particle conditioned on particle positions positions aa((XX,t),t)
(Bec, Biferale, Boffetta, Celani, Cencini, AL, (Bec, Biferale, Boffetta, Celani, Cencini, AL,
Musacchio & Toschi 2006)Musacchio & Toschi 2006)
hyperbolic non-hyperbolic
Pnonhyper
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
SummarisingSummarisingAcceleration statistics depends on two mechanisms:
1. Preferential concentrat. of particles effective at small St 2. Filtering due to particles response time effective at large St
A very small amount of inertia expels particles out of intense structures: strong correlation with flow at small St;
at larger St, because of filtering, particles can not follow the flow: no correlation with flow at larger St
Can we better understand clustering ?
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
One motivationOne motivationStrong particle concentration fluctuations have an impact on climate in different ways
Reflective power of the atmosphere
due to aerosols scattering and absorption
is crucial for climatological models
Desert dusts are particularly active ice-forming agents.
They can affect clouds formation.
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Rain droplets formation due to Rain droplets formation due to clusteringclustering
Enhanced collision rates may explain rapid rain formation
Rain drop size2mm
coalescence
Droplet size0.02mm
condensation
CCN size 0.2-2micronnucleation
preferentialconcentration + gravity
(warm) cloud large scale L=100m; dissipative scale = 1mm; Re=107
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Only a small scale Only a small scale feature?feature?
Particle clusters & voids are observed both
in the dissipativedissipative and in inertialinertial range
Slice of width ≈ 2.5. Particles with St = 0.58; R = 185
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Observables at small scales r Observables at small scales r < <
Space density of particles pairs (useful for collisions, pair dynamics)Probability to find 2 particles at a distance smaller
than r
Another common observable is the radial distribution function g(r)
It is O(1) for tracers, it diverges as r--> 0 for inertial particles (or in compressible flows).
is the correlation dimension (Grassberger 1983 ; Hentschel Procaccia, 1983)
r
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Probability and DProbability and D22• Velocity is smooth: we expect fractal distribution (with power law tails)
• At these scales, the only relevant time scale is thus
everything should depend on St & Re only
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Shape of correlation dimension Shape of correlation dimension DD22
• Optimal Stokes number for maximal clusterization
• No Reynolds dependence (as in Collins & Keswani 2004)
• Similar behaviour at higher order Dq
• Particles positions correlate with low values of acceleration (for 2d flows Chen, Goto, Vassilicos 2006)
Maximum of clustering seems to be connected Maximum of clustering seems to be connected toto preferential concentration, confirming preferential concentration, confirming
classical scenarioclassical scenario
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
What happens at larger What happens at larger scalesscales
< r < L? < r < L?
Can particles of Stokes time feel effects
of time scales tr>> ?
How do particles distribute out
of vortical regions?
What are the proper parameters to describe
the mass distribution?
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Inertial range observablesInertial range observables
Probability Distribution Function of the coarse-grained particle density:
Given N particles, we compute number density of particles within a cell of scale r,
weighting each cell with the mass it contains:
Quasi-Lagrangian measure
a natural measure to reduce finite N effects at <<1 due to voids
r
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Quasi-Lagrangian mass densityQuasi-Lagrangian mass density
Tracers behave according to uniform Poisson distribution
Particle show deviations, already there for very small
such deviations become stronger with
r=L/16r=L/16
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Algebraic tails at low density Algebraic tails at low density <<1 <<1
we have(tracers limit, uniform)
(non zero prob. to have empty areas)
St
These empty regions can play a relevant role in many physical issues
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
How do we understand this How do we understand this PDFs?PDFs?
Particles should not distribute self-similarlyi.e. Deviations from a uniform distribution are not scale-invariant(Balkovsky, Falkovich & Fouxon 2001)
No simple rescaling of the mass distributions
We note however that for the mass PDFthese two limits are equivalent:
• fixed and r ∞ (large observation scale)• fixed r and small inertia)
Both limits give a uniform particle distribution. So…
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
So there could be a parameter, rescalingthe mass distribution , which relates
Stokes times and observations scales r
At scale r, the eddy-turn-over time scale is r=-1/3r2/3, in analogy with dissipative scales, we could define:
Is this time scale Is this time scale relevantrelevant
particle clustering particle clustering in in
the inertial range?the inertial range?
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Unfortunately not so simple! Unfortunately not so simple!
This simple analogy works in synthetic flows: e.g. Kraichnan flows
• no time correlation• no spatial structures• no large scale-sweeping
(Bec, Cencini & Hillerbrand 2006)(Bec, Cencini & Hillerbrand 2006)
But it does not work in real turbulence where
all these features are present…
X
A different observation
[Maxey (1987)][Maxey (1987)]
Effective compressibilityEffective compressibilitygood for r<<good for r<< for St for St<<1<<1
[Balkovsky, Falkovich & Fouxon (2001)][Balkovsky, Falkovich & Fouxon (2001)]
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
[Maxey (1987)][Maxey (1987)]
Suppose the argument remains valid also for Suppose the argument remains valid also for finite finite r r & &
This is the contraction- rate of a particle volume
of of size r and Stokes time
particle flow
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Numerical ResultsNumerical Results
Non-dimensional contraction rateNon-dimensional contraction rate
Collapse of the coarse-grained mass PDFfor different values of
Uniformity is Uniformity is recoveredrecoveredgoing to the large going to the large scalesscalesBut very slowlyBut very slowly
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Deviation from uniformity: 2nd moment
can give some better information
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
ConclusionsConclusions
1. clustering at small scales r <1. clustering at small scales r <
The only relevant number for particle dynamics is St=/
Particles concentrate onto a multi-fractal set, whose dimension depends on the Stokes number only (or just very weakly depends on Reynolds)
Optimal finite Stokes number for clusterization: St ~ 0.6 (unpredictable..)
This global picture is the same as in smooth random flow
(see Bec 2005; Bec, Celani, Cencini,Musacchio 2005)
We gave a description of particle clustering for moderate St and moderate Re200
numbers2. clustering at inertial range scales 2. clustering at inertial range scales < r < r < L< L
concentration fluctuations are relevant also for the inertial range scales
uniformity of mass distribution is recovered very slowly at large scale
if the contraction rate , and not Str, is the proper number to rescale mass statistics ----> sweeping is important
(Bec, Biferale, Cencini, AL, Musacchio, Toschi PRL submitted 2006)
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
PerspectivesPerspectives
A better understanding of the statistics of fluid acceleration (rather than vorticity) seems crucial to understand clustering
Conversely inertial particles can be used as probes for acceleration properties
Larger Re studies are necessary to confirm the picture
Currently performing DNS to study rain drops growth
A common database : http://cfd.cineca.it/
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
END
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
St=0St=0
St=3.31
Particles with different inertia Particles with different inertia inside a vortexinside a vortex
Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte
Clusters & voids
2d slice (512x512x4) at Stokes 0.16 (blue) 0.8 (red) 1.33 (green)
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