Inertial particles in turbulence

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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