heavy particles in turbulent flows istituto di scienze dell’atmosfera e del clima alessandra...

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)

[email protected]


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


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)


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


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,..


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


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)


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, …)


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


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


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


Just a quick overview about Just a quick overview about few things:few things:

Acceleration

Conditioned analysis


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)


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


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 !


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 !


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


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 ?


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.


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


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


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


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


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


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?


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


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


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


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…


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?


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)]


[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


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


Deviation from uniformity: 2nd moment

can give some better information


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)


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/


END


St=0St=0

St=3.31

Particles with different inertia Particles with different inertia inside a vortexinside a vortex


Clusters & voids

2d slice (512x512x4) at Stokes 0.16 (blue) 0.8 (red) 1.33 (green)

heavy particles in turbulent flows istituto di scienze dell’atmosfera e del clima alessandra...

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