M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Inertial
particles in turbulence
Massimo Cencini
CNR

ISC
Roma
INFM

SMC
Università “La Sapienza” Roma
Massimo.Cencini@roma1.infn.it
In collaboration with:
J. Bec, L. Biferale, G. Boffetta, A. Celani,
A. Lanotte, S. Musacchio & F. Toschi
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Problem:
P
articles differ from fluid
tracers
their
dynamics is dissipative due to inertia one has
preferential concentration
Goals
:
understanding physical mechanisms at work,
characterization of dynamical & statistical properties
Main assumptions
:
collisionless heavy & passive particles in
the absence of gravity
In many situations it is important to consider finite

size
(
inertial
) particles transported by incompressible flows.
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Rain drops in clouds
(
G. Falkovich et al.
Nature
141
, 151 (2002))
clustering
enhanced collision rate
Formation of planetesimals in the
solar system
(
J. Cuzzi
et al.
Astroph. J.
546
, 496
(2001);
A. Bracco
et al.
Phys. Fluids
11
, 2280 (2002)
)
Optimization of combustion processes
in
diesel engines
(
T.Elperin
et al.
nlin.CD/0305017)
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Equations of motion & assumptions
Dissipative range physics
Heavy particles
Particle Re <<1
Dilute suspensions: no collisions
a
f
p
1
v
Re
a
a
a
Stokes number
Response time
Stokes Time
(Maxey & Riley Phys. Fluids
26
, 883 (1983))
Kolmogorov ett
u(x,t) (incompressible) fluid velocity field
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Phenomenology
Preferential concentration:
particle trajectories detach from those of
tracers due to their inertia inducing
preferential concentration
in peculiar flow regions.
Used in flow visualizations in experiments
Dissipative dynamics:
The dynamics is uniformly contracting in phase

space with rate
As St increases spreading in velocity direction

> caustics
This is the only effect present in Kraichnan models
Note that as an effect of dissipation the fluid velocity is
low

pass filtered
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Direct numerical simulations
After the fluid is stabilized
simulation box seeded with millions of
particles
and
tracers
injected randomly
& homogeneously with
For a subset the initial positions of
different Stokes particles coincide at t=0
~2000 particles at each St tangent
dynamics integrated for measuring LE
Statistics is divided in transient(1

2ett) +
Bulk (3ett)
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
DNS summary
Resolution
128
3
,
256
3
,
512
3
Pseudo spectral code
Normal viscosity
Code parallelized MPI+FFTW
Platforms: SGI Altix 3700, IBM

SP4
Runs over
7

30 days
N
3
512
3
256
3
128
3
Tot #particles
120Millions
32Millions
4Millions
Fast 0.1
500.000
250.000
32.000
Slow 10
7.5Millions
2Millions
250.000
Stokes/Lyap
(15+1)/(32+1)
(15+1)/(32+1)
15+1
Traject. Length
900 +2100
756 +1744
600+1200
Disk usage
1TB
400GB
70GB
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Particle Clustering
Important in optimization of reactions,
rain drops formation….
Characterization of fractal aggregates
Re and St dependence in turbulence?
Some studies on clustering:
Squires & Eaton Phys. Fluids
3
, 1169 (1991)
Balkovsky, Falkovich & Fouxon Phys. Rev. Lett.
86,
2790 (2001)
Sigurgeirsson & Stuart Phys. Fluids
14
, 1011 (2002)
Bec. Phys. Fluids
15
, L81 (2003)
Keswani & Collins New J. Phys.
6
, 119 (2004)
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Two kinds of clustering
Particle preferential concentration is observed
both
in the
dissipative
and in
inertial
range
Instantaneous p. distribution in a slice
of width
≈ 2.5
.
St
= 0.58
R
= 185
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Small scales clustering
•
Velocity is smooth we expect fractal distribution
•
Probability that
2
particles are at a distance
•
correlation dimension D
2
Use of a tree algorithm to
measure dimensions at
scales
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Correlation dimension
D
2
weakly depending on Re
Maximum of clustering for
Particles preferentially concentrate in
the hyperbolic regions of the flow.
Maximum of clustering seems to be
connected to preferential concentration
but
Counterexample: inertial p. in Kraichnan flow
(Bec talk)
Hyperbolic
non

hyperbolic
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Multifractal distribution
Intermittency
in the mass
distribution
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Lyapunov dimension
d
D
1
provides information similar to D
2
can be seen as a sort of “effective” compressibility
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Inertial

range clustering
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Characterization of clustering in the inertial range
(Preliminary & Naive)
From Kraichnan model ===> we do not expect fractal distribution
(Bec talk and Balkovsky, Falkovich, Fouxon 2001)
Range too short to use local correlation dimension or similar characterization
Coarse grained mass:
St=0 ==>
Poissonian
St
0 ==>
deviations from Poissonian. How do behave
moments and PDF of the coarse grained mass?
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
PDF of the coarse

grained mass
r
s
Deviations from Poissonian are strong & depends on
s
, r
Is inertial range scaling inducing a scaling for
Kraichnan results suggest invariance for
(bec talk)
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Collapse of CG

mass
moments
Inertial range
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Sketchy argument for
s
/r
5/3
True for St<<1
(Maxey (1987) & Balkovsky, Falkovich & Fouxon (2001))
Reasonable also for St(r)<<1
(i.e. in the inertial range)
<

Rate of volume contraction
<

from the equation of motion
The relevant time scale for the distribution of particles
is that which distinguishes their dynamics from that of tracers
can be estimated as
The argument can be made more rigorous in terms of the
dynamics
of the quasilagrangian mass distribution
and using the rate of
volume contraction.
But the crucial assumption is
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Scaling of acceleration
Controversial result about pressure and pressure gradients
(see e.g. Gotoh & Fukayama Phys. Rev. Lett.
86
, 3775 (2001) and references therein)
Our data are compatible with the latter
Note that this scaling comes from assuming
that the sweeping by the large scales is the
leading term
We cannot exclude that the other spectra may be observed at higher Re
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Single point acceleration properties
Some recent studies on fluid acceleration:
Vedula & Yeung Phys. Fluids
11
, 1208 (1999)
La Porta et al. Nature
409
, 1011 (2001) ; J. Fluid Mech
469
, 121 (2002)
Biferale et al. Phys. Rev. Lett.
93
, 064502 (2004)
Mordant et al. New J. Phys.
6
, 116 (2004)
Probe of small scale intermittency
Develop Lagrangian stochastic models
What are the effect of inertia?
Bec, Biferale, Boffetta, Celani, MC, Lanotte, Musacchio & Toschi
J. Fluid. Mech.
550
, 349 (2006); J. Turb.
7
, 36 (2006).
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Acceleration statistics
At increasing St: strong depletion of both
rms acc. and pdf tails.
Residual dependence on Re very similar
to that observed for tracers.
(
Sawford et al. Phys. Fluids 15, 3478 (2003);
Borgas Phyl. Trans. R. Soc. Lond A
342
, 379 (1993))
DNS data are in agreement with
experiments by Cornell group
(Ayyalasomayajula et al. Phys. Rev. Lett. Submitted)
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Two mechanisms
Preferential concentration
Filtering
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Preferential concentration & filtering
Heavy particles acceleration
Fluid acc. conditioned on p. positions good at St<<1
Filtered fluid acc. along fluid traj. good at St>1
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Preferential concentration
Fluid acceleration
Fluid acc. conditioned on particle positions
Heavy particle acceleration
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Filtering
Fluid acceleration
Filtered fluid acc. along fluid trajectories
Heavy particle acceleration
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Dynamical features
From passive tracers studies we know
that
wild
acceleration events come
from
trapping in strong vortices
.
(La Porta et al 2001)
(Biferale et al 2004)
Inertia expels particles from strong
vortexes ==> acceleration depletion
(a different way to see the effect of
preferential concentration)
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Conclusions
Two kinds of preferential concentrations in turbulent
flows
:
Dissipative range:
intrinsic clustering (dynamical attractor)
tools borrowed from
dynamical system
concentration
in hyperbolic region
Inertial range:
voids
due to ejection from eddies
Mass distribution recovers uniformity in a self

similar
manner (
DNS at higher resolution required, experiments?
)
open
characterization of clusters (
minimum spanning tree….??
)
Preferential concentration together with the dissipative nature of
the dynamics affects small scales as evidenced by the behavior
of acceleration
New experiments are now available for a comparative study with
DNS, preliminary comparison very promising!
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Thanks
M.Cencini
Inertial particles in turbulent flows
Warwick, July
2006
Then assuming
With the choice
Mass conservation
One sees that p
r,
(t) can be
Related to
p
r,
(t

T(r,
)) hence all the statistical
Properties depend on T(r,
).
From which
Hence if a
=a
0
Where we assumed that a p.vel.
Field can be defined
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