Elena Meneguz * M. W. Reeks

Elena Meneguz* M. W. Reeks

*School of Mechanical & Systems Engineering, Newcastle University, U.K.

Statistical properties of particle segregation in homogeneous isotropic

turbulenceSingularities , intermittency and random uncorrelated motion

DUST STORM/TORNADOS VOLCANIC ERUPTIONS RAIN CLOUDS

Unmixing by turbulent flows

‘Particle segregation ’ Open Stats Phys 6-3-12

Stokes no St=0.1 St=1.0 St=10.0particles vorticity

Wang & Maxey JFM 1993Caustics (Wilkinson & Mehlig 2007) -4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

Random uncorrelated motion (RUM)

Février et. al JFM, 2005

Mesoscopic Eulerian particle velocity field


MEPVF+RUM

222ppp qqq

Compressibility of a particle flow

• zero for particles which follow an incompressible flow

• non zero for particles with inertia• measures the change in particle concentration

Divergence of the particle velocity field along a particle trajectory

particlestreamlines

),(),(

stxYyp sy

Compressibility (rate of compression of elemental particle volume along particle trajectory)

‘Particle segregation’ Mathematics, Strathclyde, 14/2/12

Measurement of the compressibility

ijj

ipij JJ

xtxx

J det ;),(

,0

0,

Deformation of elemental volume

can be obtained directly from solving the particle eqns. of motion

Avoids calculating the compressibility via the particle velocity field

Can determine the statistics of ln J(t) easily.

Jdtd

dtdJJtxp ln,v 1

0

- xp(t),vp(t),Jij(t),J(t)) - Fully Lagrangian Method (FLM) Osiptsov(200)

‘Particle segregation’ Mathematics, Strathclyde, 14/2/12

ijij JJdtd Evolution equations:

ijk

ikj

pij J

xu

JJdtd

1

Compression - fractional change in elemental volume of particlesalong a particle trajectory

Particle trajectories in a periodic array of vortices


Deformation Tensor J


Simple 2-D flow field of counter rotating vortices

Compressibility of the pvf: v||ln J

dtd

Particle average compressibility

‘Particle segregation ’ Maths, Strathclyde, 13/2/12

3D Carrier flow field:• Fully described by 200 random Fourier modes (Spelt & Biesheuvel 1997)• Incompressible, periodic in space• Smoothly varying in time and space• Not a solution to Navier-Stokes equations

10-2

100

102

10-4

10-2

100

k/k0

E(k

)

Energy spectrum from Kraichnan (1970)

• Relatively small separation of scales• Valid for low Reynolds number turbulence

Model of synthetic turbulent flow


Particle average compressibility

divergence

compression

For a given flow field, there is a threshold St below which the segregation goes on indefinitely with time, and above which the dilation prevails over segregation.

Compressibility of the pvf: v||ln Jdtd

KS


Moments of particle number density

)(|)(| 1 tntJ || Jn || Jn 11 || Jnn


St=0.1

• Particle number density is spatially strongly intermittent• The segregation goes on with time!• The peaks reveal the presence of singularities!

)(|)(| 1 tntJ || Jn || Jn 11 || Jnn

St=0.5

0 5 10 15 20 2510

-5

100

105

1010

1015

1020

t/

=3=2=0

DNS


Singularities in the ptcl concentration fieldSingularities correspond to |J|=0 events

10 20 30 40 500

0.05

0.1

0.15

Ns

PDF(

Ns)

10-2 10-1 100 101 10210-10

10-8

10-6

10-4

10-2

St

s

p

Simulation data

A*exp(-B/St)*StC

Distribution of singularities Frequency of singularities

max at

St=0.5

St=1

St=5

:The distribution of singularities follows a Poisson curveThe maximum frequency of singularity events occurs for

1St

1St


-4 -2 0 2 410-8

10-6

10-4

10-2

100

ln|J|

Gaussian PDFPDF(logJ)

-30 -25 -20 -15 -10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ln|J|

PDF(

ln|J

|)

time=25, mean=-2.97, skewst=-0.15, kurtst=3.10



Statistics of the compression C=ln|J|

• The PDF of the compression looks Gaussian but..it is not!• Singularities correspond to ln|J|-> -inf..what is the cause for the

deviation from Gaussianity visible on the left tail of the curve?

St=0.5 St=0.5



Février et. al JFM, 2005

Mesoscopic Eulerian particle velocity field


MEPVF+RUM

222ppp qqq

Decomposition of the compression (C) • We want to separate the mesoscopic and RUM component of the compression

C=ln|J|

• The domain is subdivided in 80x80x80 cells

• mesoscopic contribution is evaluated as:

• RUM contribution as:

where:

i ji

ij N

CC,

|

jjiji CC |

CC jj |

jiC

j-th cell

Compression experienced by the i-th particle in the j-th cell

Average of compression for the j-th cell

C Average of compression calculated all over the domain

-6 -4 -2 0 2 4 610-4

10-3

10-2

10-1

100

ln|J|

PDF(ln|J|)mesRUM

The effect of RUM on C=ln|J|

• It is the RUM component of the compression which causes the deviation from Gaussianity! =>Singularities and RUM are intrinsically related

mesRUMJJJ lnlnln


Trajectories

Velocity (RUM)

|J| (singularities)

Singularities- ”sling shot”- RUM


Conclusions

• Segregation of particles: the accumulation of particles in preferred zones and shapes

1. The clustering of inertial particles can be quantified for any given St

2. It is not a stationary process (as so far assumed) but goes on until particles collide with one another

• Spatially random contribution (RUM)

The RUM component of the velocity of nearby particle pairs

2. The deviation from Gaussianity of the clustering process has shown to be due to the presence of this component

The study is relevant to particle dispersion and de-mixing processes, first of all droplet coalescence and the onset of rain:

1. Easily detected in contrast to traditional box-counting methods (due to spatial resolution limits)

• Singularitiesinstantaneous events which correspond to very large concentration

2. For the first time, their distribution and has been studied and found to be Poisson

The FLM is a very powerful technique;Extension to in-homogeneous case is possible


THANKS FOR YOUR ATTENTION

Any questions?

Elena Meneguz , 13th European Turbulence Conference, 12-15 th September 2011 20

Particle trajectories in a periodic array of vortices


Deformation Tensor J


Singularities in particle concentration


Model of synthetic turbulent flow

3D Carrier flow field:• Fully described by 200 random Fourier modes (Spelt & Biesheuvel 1997)• Incompressible, periodic in space• Smoothly varying in time and space• Not a solution to Navier-Stokes equations

10-2

100

102

10-4

10-2

100

k/k0

E(k

)

Energy spectrum from Kraichnan (1970)

• Relatively small separation of scales• Valid for low Reynolds number turbulence


Compressibility

Simple 2-D flow field of counter rotating vortices KS random Fourier modes: distribution of scales, turbulence energy spectrum


• Along particle trajectory: particle number density n related to J by:

)(|)(| 1 tntJ || Jn || Jn

• Particle averaged value of is related to spatially averaged value:n

11 || Jnn

n

Trivial limits: ,10 n 11 n (equivalent to counting particles)

• Any space-averaged moment is readily determined, if J is known for all particles in the sub-domain


St=0.05 St=0.5

• Particle number density is spatially strongly intermittent• Sudden peaks indicate singularities in particle velocity field


Random uncorrelated motion • Quasi Brownian Motion - Simonin et al• Decorrelated velocities - Collins • Crossing trajectories - Wilkinson • RUM - Ijzermans et al.• Free flight to the wall - Friedlander (1958)• Sling shot effect - Falkovich

Falkovich and Pumir (2006)

12

2L1L ),2(v),1(v)(

rrr

rrrRL


Radial distribution function (RDF) g(r)

rg(r)

)()( Strrg


Legitimate questions …

?

What happens if we take into consideration a more complex flow field model e.g. a model with a broader range of scales and which takes into account the sweeping of the small scales by the large ones?

• e.g. Do we find the same threshold value?

Direct Numerical Simulations will give us the answer!!!

DNS: details of the code

• Statistically stationary HIT• Pseudo-spectral code• Grid 128x128x128• Re =65• Forcing is applied at the lowest wavenumbers

• NSE for an incompressible viscous turbulent flow:

• In a DNS of HIT, the solution domain is in a cube of size L, and: k

xik tet )(k,u)u(x,

• 100.000 inertial particles are random distributed at t=0 in a box of L=2• • Interpolation of the velocity fluid @ the particle position with a 6th order Lagrangian

polynomial• Trajectories and equations calculated by RK4 method• Initial conditions so that volume is initially a cube

)0,u()0v( txt p

Averaged value of compressibility vs time

Elena Meneguz 32

• Qualitatively the same trend with respect to KS• We expect a different threshold value

0 0.1 0.2 0.3 0.4 0.5 0.6-2

-1

0

1

2

3

4

time

d<ln

|J|>

dt

St=0.7St=1St=0.1St=10

WHAT CAUSES THE POSITIVE VALUES???

Moments of particle number density (KS/RUM)

St=0.05

• Particle number density is spatially strongly intermittent• The segregation goes on with time!• The peaks reveal the presence of singularities!

)(|)(| 1 tntJ || Jn || Jn 11 || Jnn

St=0.5

0 5 10 15 20 2510

-5

100

105

1010

1015

1020

t/

=3=2=0

St=0.5

DNS


0

50

100

150

200

250

PDF Poisson distribution (scaled)

PDF of SINGULARITIES: Poisson distr.!

St=0.5 St=1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

50

100

150

200

250


1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 580

50

100

150

200

250


St=5

Elena Meneguz 34

Compressibility in the MEPVF• MEPVF = PVF + RUM (According to Février et al. 2005)

PVF =

jN

jj

jNt

1

1),( vxv

i-th cell

Spatially uncorrelated component (for large inertia)

Elena Meneguz 35

“sling effect” (Falkovich et al. 2002) and “crossing of trajectories” (Wilkinson & Mehlig 2005) Smoothly varying


Random Uncorrelated Motion (RUM)

Février et. al (2005)

It is the manifestion of the decorrelation of two nearby particles.

Elena Meneguz 36

-30 -25 -20 -15 -10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

log|J|

PDF(

log|

J|)




Here is where singularities take place!

SINGULARITIES and RUM

The distribution of log|J| approaches a Gaussian for(St=0.5)

t

For other Stokes numbers…


SINGULARITIES

Elena Meneguz 38

-5 -4 -3 -2 -1 0 1 2 3 410

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

log|J|

Gaussian trendPDF(logJ)

Deviation from the Gaussian trend!!!

What is the cause for this behaviour which manifests itself in the limit of log|J|->-inf, in other words |J|=0 (singularities)?

St=1


Decomposition of the compression (C) • We want to separate the mesoscopic and RUM component of the compression

C=ln|J|

• The domain is subdivided in 80x80x80 cells

• Mesoscopic contribution is evaluated as:

• RUM contribution as:

where:

i ji

ij N

CC,

|

jij CCi |)( ,

CCi j |)(

ijC ,

C

j-th cell

Compression experienced by the i-th particle in the j-th cell

Average of compression for the j-th cell

Average of compression calculated all over the domain

-6 -4 -2 0 2 4 610

-4

10-3

10-2

10-1

100

PDF(logJ)mesRUMmes+RUM

C=Log|J|

Singularities and RUM(C)

It’s the RUMComponent that is causing the deviation!

Singularities are related to RUM, but how?

St=1

41

The answer is… CAUSTICS!Wilkinson & Mehlig 2005

• The particle velocity field is multivalued as a consequence of folding;• The frequency of singularities is ultimately the frequency of activation of the caustics events • This is related to RUM (crossing of trajectory) as this takes place between caustics (1D

example)• This has an impact on the rate of collision of particles: preferential concentration is not the

only effect (dependency on the St number)

Caustics, singularities and RUM

Conclusions…

We have exploited a FLM

• We have calculated quantities such as particle averaged compressibility and moments of the particle number density

INITIAL PROBLEM:

To investigate and quantify the clustering of inertial particles in turbulent flow from a theoretical and numerical point of view.

• We have compared results in different models of turbulent flows – from simple to more complex ones

• We have investigated some detailed features such as the presence of singularities – not detected with box counting methods

• We were trying to conclude our investigation by looking at a way to establish a link between the singularities and the occurrence of Random Uncorrelated Motion.

• Caustics are the answer!

Thank you for you attention

Particle kinetic stress transport equation

nmDtD

i

nmi

i

mni x

px

p Work done by total stresses

Smnnm 2

nmiix

Kinetic stress flux

niin

kinetic stresses

stresses from turbulent forces

Particle-fluid velocity covariances

nmnm uu vv21

...

phase continuous for the Harlow &Daly with Compare

kj

lli

Dskji x

kC

jil

lkkikl

ljkjl

likji xxxvvvvvvvvv 3

1

Swailes, Sergeev at al

Chapman Enskog Approximation

trackingparticle


Predictions versus experimental measurementsSimonin et al.

Work done by shear stresses

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5 Turbulent round jet: radial rms velocit-ies x/DI =20 , Hishida et al.

Radial rms veloc-ity-kinetic stress transport modelRadial rms veloc-ity- local homo-geneous modelAxial rms velocity -kinetic stress transport model

radial distance r/DIpa

rticle

rms

velo

city

m/s

0 0.01 0.02 0.03 0.04-0.2

-1.66533453693773E-16

0.2

0.4

0.6

0.8

1

1.2

-5.55111512312578E-17

0.02

0.04

0.06

0.08

0.1Vertical channel flow: kinetic stress transport predic-tions versus experimental measurements

Rogers and Eaton

mean streamwise velocityrms normal velocity rms streamwise ve-locity

normalised distance from wall

mea

n st

ream

wise

veloc

ity

rms

velocit

y


Near-wall behaviour

PEQp 2/exp

• p, the resuspension rate constant• ω , the typical frequency of

vibration, • Q height of adhesive potential

well, • <PE> average potential energy of

particle in the well.

particle resuspension

Q

Particle escape from potential well

boundary conditions

ugivenofondistributiu

udutxuPutxpnu

v )v(

)v(),,(),,(v0.

influence of particle wall interactions scattering & absorption/deposition/ bounce

/resuspension


Particle wall impacts with absorption

turbulence

particleVg=5

Velocity v

P(v)

Critical impact velocity vc=5, settling velocity vg=5(normalized on particle rms velocity for perfect reflection)

Swailes & Reeks (1997)


Deposition in turbulent boundary layer

rapid decay of turbulence near the wallparticle not in local equilibrium with flowbreak down of gradient transport

comparison of PDF results with experimental results

y+

τ+van Dijk JCP (2010)

van Dijk JCP (2010)τ+

y+

1 20-1-2

-1

-3

-2

4

5

Zaichik & Alipchenkov, (2009)

deposition velocity

Pair dispersion and segregationTwo colliding spheres radii r1 , r2

r1

r2

n

Collision sphere

rg(r)~rα(St)

)()( Strrg

wtrudtwdw

dtrd

),(; Eqns of Motion

PDF Equation for relative dispersion(Zaichik and Alipchenkov 2003,2009 )

1 / /~)(,)( 222 KKr rrruru

w = relative velocity between identical particle pairs, distance r apartΔu(r,t) = relative velocity between 2 fluid pts, distance r apart at time t

Structure functions

iijij

i uwwrDt

D

wwmomentum

0

wxt i

mass

1)(2

1 Strrg St

),,w(w

trPr

Net turbulent Force (diffusive)mass Pu

wtrwPw

wrPw

tP

),,(

convectionβ = St -1

Kinetic Equation predictions

mean of magnitude of relative velocity rwRDF radial distribution functionKrrStrg / ,),(

Zaichik and Alipchenkov, NJP 2009

mean of magnitude of relative velocity rw


Current Actvities• Inter particle collisions Zaichik / Simonin/Fede

– elastic- non elastic– binary collisions

• Random uncorrelated motion (RUM)– transport Equations for RUM in shear flows– influence on particle collisions (Masi/Simonin, IMFT)

• PDF/two fluid models for ρp/ρf ~ 1– (added mass, pressure gradients) Swailes / Skartlian IFE

• Numerical methods – QMOM (Fox et al.)– Hybrid numerical schemes (van Dijk /Swailes

PDF eqns

particle surface interactions

Matching interface

Continuum RST eqns

221122 vv),,,v( dkdktkdxxpd

tP

ppcoll


Current activities

• Two particle dispersion– Segregation / demixing processes

• Application to cloud physics – Growth and coalescence of water droplets– Role of singularities and intermittency

» Statistics and relation to caustics

St=0.5


Summary / Conclusions– Kinetic / pdf approach (one particle approach)

• Treatment of the dispersed particle phase as a fluid– PDF eqn Master eqn

» Eqns for P(v,x,t) and P(v,x,up,t)– Continuum equations– Constitutive relations – Reproduces exact results for generic flows– Influence of turbulent structures – body force– Boundary conditions (particle surface interactions)

– Kinetic approach for particle pair transport• radial distribution function• collision rate


Traditional two-fluid modelling

• Density weighted averages of the instantaneous conservation equations for both phases

k=1 continuous k=2 particle0,

ikkki

kk uxt

mass balance

ikikki

kikjkkjk

ikkk Ig

xpuu

xtDuD

,,,,

momentum balance

Closure of equations requires closure models for Reynolds/kinetic stresses Interfacial drag term

Vol. frac.

Density weighted velocity


Continuous phase Reynolds stresses

jin

nmm

sij uu

xuuq

xCD ,1,1,1,1

1

21

111

,1

Diffusion

m

imj

m

jmiij x

uuuxuuuP

,1

,1,1,1

,1,1,1Production

ijmmijijjiij PPCquuq

C ,13

1,12

213

2,1,12

1

11,1 redistribution

2,12,1

11

22,1 jjjiij ufuf

Fluid-particle interaction

ijijijijijjj PD

tDuuD

.1,1,1,11

,1,111


The viscous dissipation rate f

iij

jii

ii

qC

xuuqC

x

CP

CqDt

D

,121

13

1

1

21

21111

11,1

121

1

1

1

1

2

Particle influence


Why use a PDF Approach?• It’s a rational theory• a linear equation (more success with simple

closure)• exact closure for Gaussian forces• exact solutions for simple flows (bench marking)• continuum equations/constitutive relations (CRs)

developed in formal way • natural length scale for validity of simple CRs• use of natural boundary conditions


Diffusion coefficient versus drift


Closure of momentum/energy equations– GLM approach

jpijj

ijpippj

jipj

j

ip uGxu

uuux

uxDt

uD

12v

jmpimjpipjipF

m

ip

mpjm

jmip

m

ipjmipjm

m

jip

uGuuu

xu

ux

uxu

uxDt

uD

vv1

vv

vvvvvv

12

12

Mean carrier flow encountered by particle

Particle-carrier flow velocity covariances

Hybrid numerical schemes• Complex wall bounded flows with partially

absorbing walls Ladd Issa, vanDijk, Swailes

PDF eqns

particle surface interactions

Matching interface

Continuum RST eqns


Kinetic Equation for P(w,r,t)and moment equations

1 / /~)(,)( 222 KKr rrruru

citydrift velo

0

,,

ww

t

p

ijij

i

dsstxYtxux

u

uwwxDt

D

momentum

w = relative velocity between identical particle pairs, distance r apartΔu(r) = relative velocity between 2 fluid pts, distance r apart

Structure functions

Net turbulent Force mass Pu

wtrwPw

wrPw

tP

),,(

convection β = St-1 , St=Stokes number

Probability density(Pdf)

0

wxt i

mass

PDF Probabalistic Approach

N

nnpnNN tXXtXXXP

tX, XtXXPrTxX

tTrxtrTxP

1,21

21212

1

1

))((),;...;(

at time coordintes space phase have particles 2 that PDF);;( vectorspace phase ,,..,,,v

at time re temperatu, size ,position ,vvelocity a has particle a that (PDF)density y probabilit ),,,,v(

Pai, M. G & Subramaniam, S. JFM. 628, 181–228, 2009

Euler-Euler ; same flow field for both phases as solution of N-S equations (with bcs imposed at interphase boundaries) ≡ flow with moving deformable boundaries (see Drew, Prosperetti)

Lagrange-Euler : dispersed phase composed of distribution of discrete elements (particles) whose (Lagrangian) individual transport is obtained by solving their eqm’s in a (Eulerian) random flow field (particle tracking)


dtttxuudttuu pppp

00

Lagrangian

1 )),(()0,0()()0(

Homogeneous turbulence

Momentum equation as a diffusion equation

turbophoresis Drift due spatial inhomogeneity

s

p xdtxGtxuuhypothesi Corrsins

),(),()0,0(Diffusion coefficient

txutxxxx

u ddC ,0,υ 1

Momentum eqn.DtD

xC

1

Generalised Langevin Model approach Pope/ Simonin

P(v,x,up,t) , up = velocity of carrier flow seen by particle

jipp uu v ,for equationsnsport Obtain tra ,

Simonin Deustch & Minier

j

ijpj x

uu

,viCwwhite noise

force due to mean flow generalised Langevin model

1 122

jjijmm

i

if

i uuGxx

uxp

dtdu

uPC

utuxPF

uu

xt pp ,,,vvv

v

jjpij uuG ,12

dtdu ip ,

Kinetic versus GLM Approaches

GLM is a realizable model for up(t) Exact closures Compatible with GLM for the carrier flow More physics but more demanding computationally

Kinetic approach simpler to apply Closures are approximate

Both approaches the same for Gaussian differ only in detail e.g. Simonin-Pope model gives dependence

of fluid co-variances and timescale on the local shearing which must be inputs to Kinetic equations


Dispersion and drift in compressible flows ( Chun, Koch & Collins, JFM 536, 219--251, 2005)

tdttrwtrtrwDtD t

0

),(exp)0),((),(

• w(r,t) the relative velocity between particle pairs a distance r apart at time t

• Particle pairs are separated by their own relative veslity field w(r,t) • Conservation of mass (continuity)

)(flux Drift )(flux Diffusive dD qqw

1/

1Stfor Only works

,(,

,(,)(,)(

22

0

0

KSt

t

idi

t

jiijj

ijDi

rg(r) ~r

tdttrwtrwq

tdttrwtrwrDr

rDq

Random variable

tdttrwtrwwwx

qt

id

i0

1 ,(,

RDF radial distribution functionKrrStrg / ,),(

Satisfying the well-mixed condition

dsstxtxux

Wdsstxtxu

Wx

W

Wu

t

d

t

0

1

0

,,vvυ

,,

txWxX ,,v,,v,

particlestreamlines

compressibility along a particle trajectory

),(

),(stxYyp sy

Eleprin, Kleorin, Reeks, Maxey

ilitycompressibnet

Particle drift velocity

txutxxxxd ,0,υ 1

dsstxux

txud ,,0

Limiting case β-1 >>1 (fluid point motion)

No guarantee that this is zero (spurious drift)


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Elena Meneguz * M. W. Reeks