Heavy particles in turbulent flows
Alessandra Lanotte
CNR ISAC
Lecce (Italy)
[email protected]
with:
J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, S. Musacchio, F. Toschi
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Outline
• 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 particles?
Formation of planetesimals in
the solar system
(A. Bracco et al. Phys. Fluids 2002)
In clouds, dust storms, fires
volcano eruption..
(see e.g. K. Sassen, Nature 2005)
Control of combustion processes
in diesel engines
(see T.Elperin et al. nlin.CD/0305017)
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
What can we observe/measure in a lab?
Lagrangian turbulence has always suffered
the lack of accurate space & time measurements
now particles can be accurately tracked !
QuickTime™ and a
Video decompressor
are needed to see this picture.
State-of-the-art Lagrangian experiments
(tracers)
From Cornell group:
frame rate : 1000fps; 4x4 cm area.
Istituto di Scienze dell’Atmosfera e del Clima
 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
Alessandra Lanotte
Heavy particles in wind tunnel turbulence
Z. Warhaft experiment
at Cornell
Re  250
water droplets <d> = 20 micron
High-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 Model
 finite size impurities of size much
a  
 much heavier than the fluid
 p   f
smaller than the flow dissipative scale
 particle Reynolds number low

Re a  a | V - u |  1
 very dilute suspension : no role of collisions
 no back reaction on the flow

Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Simplified equations
Under previous assumption we can simplify original eqs:
(M. Maxey & J. Riley, Phys Fluids 1983)
X
Parameters:
dragnumber
Stokesonly
time Stokes
-->
Stokes
(water in air b=0.001)
Density ratio
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Something we know about inertia…
(since Maxey, Eaton, Fessler, Squires, …)
1. Ejection of heavy particles from vortices
--> experience smaller acceleration
2. Particle have finite response time to fluid fluctuations
--> smoothing and filtering of fast time scales
3. Very strong concentration fluctuation
--> particle distribute on clusters
Try to understand physical mechanisms
and identify relevant parameters for
statistical description…
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
A large numerical “experiment”
To start with the simplest situation
To have good statistics
To build up a database for common use
The lab
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particles in the flow box
Alessandra Lanotte
Details about the DNS
Lagrangian Particles with 15
Lagrangian Tracers
Initial conditions
particles and tracers injected
randomly & homogeneously
with initial veloc. = fluid veloc.
Re= 65, 105, 185
Pseudo Spectral Code, MPI
Normal viscosity
Istituto di Scienze dell’Atmosfera e del Clima
STATISTICS
TRANSIENT (1-2 T)+BULK ( 3-4 T)
3
3
3
3
N
512
256
128
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
Alessandra Lanotte
How long do we wait for the stationary mass
distribution?
St=0.9
St=1.6
St=0.48
St=3.3
St=0.27
St=0.16
St=0
Coarse-grained mass in the j-th cell of side l=2x
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Just a quick overview about
few things:
 Acceleration
 Conditioned analysis
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
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 tracers
Tracers acceleration can be very well
described in terms of the multifractal model
Phenomenological model
for small scale fluctuations
a P(a)
1024^3 DNS
Multifractal
K41
prediction
(Biferale, Boffetta , Celani,
Devenish, AL, Toschi 2004)
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Acceleration for heavy particles
Increase Re
Increase St
Two coexisting
effects
No simple phenomenological
model
for particles at varying
preferential concentration
low Reynolds
St
filtered dynamics
at higher St
Stokesatand
numbers
!
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Comparison with experiments
St=0.15
St=0.09
Water drops in air:
clearly polydisperse flow !
A. Gylfason, S. Ayyalasomayajula, E. Bodenschatz, Z. Warhaft, PRL submitted 2006
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Particles and 3d flow structures
hyperbolic
non-hyperbolic
Such effect is clearly evident by
looking
St =0.16 at the fluid acceleration
St = 0.8
conditioned on particle
positions a(X,t)
(Bec, Biferale, Boffetta, Celani, Cencini, AL,
Pnonhyper
Musacchio & Toschi 2006)
St = 3.3
White: non-hyper regions
Black: hyperbolic regions
Particles preferentially concentrate in hyperbolic regions
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Summarising
Acceleration 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 motivation
Strong particle concentration fluctuations
have an impact on climate in different ways
Desert dusts are particularly active
ice-forming agents.
They can affect clouds formation.
Reflective power of the atmosphere
due to aerosols scattering and absorption
is crucial for climatological models
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Rain droplets formation due to clustering
(warm) cloud large scale L=100m;
Rain drop size
2mm
coalescence
dissipative scale = 1mm; Re=107
Droplet size
0.02mm
condensation
preferential
concentration
+
gravity
CCN size
0.2-2micron
nucleation
Enhanced collision rates may explain rapid rain formation
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Only a small scale feature?
Particle clusters & voids are observed both
in the dissipative and in inertial 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 < 
Space density of particles pairs (useful for collisions, pair dynamics)
Probability to find 2 particles at a distance smaller than r
r
is the correlation dimension
(Grassberger 1983 ; Hentschel Procaccia, 1983)
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).
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Probability and D2
• 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 D2
• 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 to
preferential concentration, confirming classical
scenario
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
What happens at larger scales
 < 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 observables
Probability Distribution Function
of the coarse-grained particle density:
r
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
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Quasi-Lagrangian mass density
r=L/16

Tracers behave according to uniform Poisson distribution
Particle show deviations, already there for very small 
such deviations become stronger with 
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Algebraic tails at low density  <<1
we have
(tracers limit, uniform)
St
(non zero prob. to have empty areas)
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 PDFs?
Particles should not distribute self-similarly
i.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 PDF
these two limits are equivalent:
• fixed  and r
• fixed r and 
∞ (large observation scale)
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, rescaling
the 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 relevant
particle clustering in
the inertial range?
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
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)
But it does not work in real turbulence where
all these features are present…
X
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
A different observation
Effective compressibility
good for r<< for St<<1
[Maxey (1987)]
particle
flow
[Balkovsky, Falkovich & Fouxon (2001)]
Suppose the argument remains valid also for finite r & 
This is the contraction- rate of a particle volume of of size r and
Stokes time 
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Numerical Results
Collapse of the
coarse-grained mass PDF
for different values of 
Uniformity is recovered
going to the large scales
But very slowly
Non-dimensional contraction rate
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
Conclusions
We gave a description of particle clustering
for moderate St  and moderate Re200 numbers
1.
scales
r <scales

2. clustering
clustering at
at small
inertial
range
< r < L
 The only relevant number for particle dynamics is St=/
 concentration fluctuations are relevant also for the
inertial range scales
 Particles concentrate onto a multi-fractal set, whose
dimension depends on the Stokes number only
(or
uniformity
mass distribution
is recovered very slowly
just veryofweakly
depends on Reynolds)
at large scale
 Optimal finite Stokes number for clusterization: St ~ 0.6
 if the(unpredictable..)
contraction rate , and not Str, is the proper number to
rescale mass statistics ----> sweeping is important
This global picture is the same as in smooth random flow
(Bec, Biferale, Cencini, AL, Musacchio, Toschi PRL submitted 2006)
(see Bec 2005; Bec, Celani, Cencini,Musacchio 2005)
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Perspectives
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
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
A common database : http://cfd.cineca.it/
END
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte
Particles with different inertia
inside a vortex
St=3.31
St=0
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Alessandra Lanotte
Clusters & voids
2d slice (512x512x4) at
Stokes 0.16 (blue) 0.8 (red) 1.33 (green)
Istituto di Scienze dell’Atmosfera e del Clima
Alessandra Lanotte