a kinetic theory for the transport of small particles in turbulent flows

Open Statistical Physics Open University,10/03/2010

A kinetic theory for the transport of small particles in turbulent flows

Michael W ReeksSchool of Mechanical & Systems Engineering,

Newcastle University


Environmental /industrial processes

• Mixing & combustion

• pollutant dispersion

• fouling / deposition

• clean up

• radioactive releases

• slurry /pneumatic conveying

• aerosol formation


Modelling Particle Flows

• Particle Tracking (Lagrangian)– track particles through a random flow by solving

particle equation of motion

• Two-Fluid Model (Eulerian)– Continuum equations for continuous (carrier flow) and

dispersed phase (particles)

– constitutive relations /closure approximations

– boundary conditions


Objectives

• application of formal closure methods in dilute flows to– derive continuum equations /constitutive relations for

the particle phase• compare with traditional heuristic approach• criteria for their validity

– incorporate the influence of turbulent structures on particle motion into continuum equations

• One particle dispersion• Two particle (pair) dispersion• Drift in inhomogeneous turbulence


particle motion in plain vortex and straining flow


Caustics – Mehlig & Wilkinson


Settling in homogeneous turbulence Maxey 1988, Maxey & Wang 1992

fieldy velocitparticle ousinstantane theis ),(

),(

),(),(

),(

1

),(

0),(

)(

sy

y

u

y

usy

sytxu

p

stxYyk

m

m

k

stxYyp

t

stxYypo

gg


– Begin with particle equation of motion e.g. for gas-solid flows

– Separate particle velocities & aerodynamic forces into mean & fluctuating components

– Average over all realisations of the flow

iiijij

i uuxDt

D

Reynoldsstresses Inter-phase momentum

transferMass X

accel

,;

mean ighteddensity we particle /average ensemble

iiiiiii

ii

uuuu

Two-Fluid Model

timerelaxation particle drag Stokes;)( 1 ii

i udt

d

mean and fluctuating carrier flow velocities


Kinetic/ PDF Approach

– analogous to the kinetic theory of gases– uses an equation analogous to the Maxwell-

Boltzmann Equation to derive the two-fluid equations for a dispersed flow

• mass-momentum and energy equations (c.f. RANS for continuous phase)

• constitutive relations

• boundary conditions


Modelling of Particle Flows 11

Two PDF approaches

Apply closure to transport equations for •<W(,x,t)> particle phase space density•<P(,up,x,t)> up is the fluid velocity seen by the particle at x (Simonin approach)

)(diffusive

force drivingturbulent convective

WuWux

W

t

W


Closure approximations for Reeks, Zaichek, Swailes, Minier

)(diffusive

force drivingturbulent convective

WuWux

W

t

W

WWtxu

,

flowcarrier theof gradients local upon the

depend and Lagrangian are ,

space-in xdiffusion yin velocitdiffusion

,0,, ,0,, txutxxtxutx

0s

0s

x

tx,,

Wu Wu

Wu


Modelling of Particle Flows 13

iijiiji u

xDt

D

Body forceMass x acceleration

Turbulent stress

ijjiijp

Momentum Equation - PDF Approach

Equation of state


Momentum equation as a diffusion equation

t

d

dC

C

x

u

Dt

D

x

1

sisturbophore

1

1


15

Particle Reynolds stress transport eqns

forces by volume work of Rate

stressesshear by work of Rate

flux stress Reynolds

2 nmnmnm

i

mni

i

nmi

nmii

nm

uu

xx

xDt

D

- Reynolds stresses depend on shearing of both phases- Requires closure for Reynolds stress flux


16

Chapman Enskog Approximation

...

phase continuous for the Harlow &Daly with Compare

31

kjl

liD

skji

jil

lkkil

ljkjl

likji

x

kC

xxx


Application- PDF solutions • Transport and deposition in turbulent boundary layers

Deposition velocity

k

particle relaxation time

Velocity distribution at a wall for particles settling under gravity in a turbulent flow with particla absorption at the wall


Divergence of the particle velocity field

along a particle trajectory

•measures the change in particle concentration•zero for particles which follow an incompressible flow •non zero for particles with inertia

particlestreamlines

),(

),(stxYyp sy


19

Application to kinetic approach

Gaussian are )( )(

,,

citydrift velo

0

tuandtif

WdsstxYtxuWx

WWu

p

t

p


Pair dispersion and segregationTwo colliding spheres radii r1 , r2

r1

r2

n

Collision sphere

rg(r)

)()( Strrg


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

wr

Pw

t

P

),,(

convection β = St-1 , St=Stokes number

Probability density(Pdf)

0

wxt i

mass


Kinetic Equation predictionsZaichik and Alipchenkov, Phys Fluids 2003


Summary / Conclusions• Particle transport and segregation in a turbulent flow

– Kinetic / pdf approach (single particle transport)• Treatment of the dispersed particle phase as a fluid

– Continuum equations

– Constitutive relations

– Boundary conditions (perfectly / partially absorbing)

– Kinetic approach for particle pair transport• radial distribution function

• Role of compressibility in the formulation of a kinetic equation

– Net relative drift velocity between particle pairs

– enhancement local concentration of neighbouring particles


Moments of particle number density

St=0.05 St=0.5

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


Influence of turbulent structures

tx

sstxX

dsstxXtxXtx

,t)x(x

,t)x( tdt

d

,t)x( t

p

t

ppp

pp

p

at time at wasparticle

thegiven that at timeposition particle theis , where

,vexp0,0,,

:issoln The

v

0

x,t

Xp,s

Consider the instantaneous concentration (x,t)derived froman initial concentration (x,0) and a particle velocity field vp(x,t).The conservation of mass equation is


Dispersion in a random compressible flow

t

ppp

t

ppp

p

t

ppp

t

pp

p

t

ppppp

ststttdsdsdssstxXtxD

tandtif

txdssstxXtx

txdssstxXtx

txtx

tandtif

dsstxXtxXtxtxtx

0

2121

0

p

0

0

p

p

p

0

vvv,,v,v

Gaussian-non are )(v )(

,,,v,v

,,,,v

,,v

Gaussian are )(v )(

,exp0,0,,v,,v

Drift velocity(Maxey)

Diffusion tensor D (Taylor’s theory)

Net particle flux


Segregation of inertial particles in turbulent flows

M. W. Reeks, R. IJzermans, E. Meneguz, Y.AmmarNewcastle University, UKM. Picciotto, A. SoldatiUniversity of Udine, It

‘Fractals, singularities, intermittency, and random uncorrelated motion’


De-mixing of particles• Particles suspended in a turbulent flow do not mix but

segregate – depends upon the particles inertial response to:

• structure and persistence of the turbulence

• Important in mixing and particle collision processes – growth of PM10 and cloud droplets in the atmosphere

• the onset of rain.

• Presentation is about quantifying segregation– analysing statistics and morphology of the segregation

• using a Full Lagrangian Method (FLM)– use of compressibility to reveal

» fractal nature » intermittency» random uncorrelated motion


particle motion in a vortex and straining flow

Stokes number St = τp/τf~1


Segregation in isotropic turbulence


Segregation - dependence on Stokes number St=τp/τf


-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

Segregation in counter-rotating vortices

Flow pattern translated randomly in space with finite life- time


Caustics – Mehlig & Wilkinson


Compressibility of a particle flowFalkovich, Elperin,Wilkinson, Reeks

•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)


Measurement of the compressibility

ijj

ipij JJ

x

txxJ det ;

),(

,0

0,

Deformation of elemental volume

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

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.

The process is strongly non-Gaussian – highly intermittent

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

Jdt

d

dt

dJJtxp ln,v 1

0


Particle trajectories in a periodic array of vortices


Deformation Tensor J


Singularities in particle concentration


Compressibility

Simple 2-D flow field of counter rotating vorticesKS 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


Comparison with analytical estimate

If St is sufficiently small: )exp( tn

),( St

•For first time, numerical support for theory of Balkovsky et al (2001, PRL): “ is convex function of ”.


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)

r

g(r)

)()( Strrg


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 46

• 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=0.7

St=1St=0.1

St=10

WHAT CAUSES THE POSITIVE VALUES???


Moments of particle concentration

0 0.2 0.4 0.6 0.8 1 1.2 1.410

0

102

104

106

108

1010

time

St=1

alpha=0

alpha=2alpha=3

n

intermittency due to the presence of singularities in the pvf


Turbulent Agglomeration

rr wnj 21

nrjK

rrj

rrr

cr

ccr

c

/)( areacollision

kernelCollison

at particles colliding ofcurrent )(

spherecollision of radius 21

Two colliding spheres radii r1 , r2

r1

r2

test particle

Saffman & Turner model

21121211

32/1

22/12

222

)(),()(

15

8

15 ;

22

rnrnrrKdt

rdn

rK

x

ur

x

u)σ(r

)σ(rrπwπrK

S

cS

cc

cπcrcS

Agglomeration in DNS turbulence L-P Wang et al. - examined S&T model•Frozen field versus time evolving flow field•Absorbing versus reflection Brunk et al. - used linear shear model to assess influence of persistence of strain rate, boundary conditions, rotation

n

Collision sphere


Agglomeration of inertial particlesSundarim & Collins(1997) , Reade & Collins (2000): measurement of rdfs and impact velocities as a function of Stokes number St

)(),(4),( 221 StwStrgrrrK rcc Net relative velocity between colliding

spheres along their line of centresRDF at rc

)( crg


Inertial collisions (RUM)

Ratio of the RMS of the relative velocity of colliding particles over the corresponding RMS of the relative fluid velocity; collision radius rc/ηk =0.1

r1

r2

rc=r1+r2

particle Stokes number St

2/12

2/12

w

w

rf

rc


Probabalistic Methods


Kinetic Equation and its Moment equationsZaichik, Reeks,Swailes, Minier)

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

iijij

i uwwrDt

D

wwmomentum

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 (diffusive)mass Pu

wtrwPw

wr

Pw

t

P

),,(

convection β = St-1

Probability density(Pdf)

0

wxt i

mass

1)(2

1 Strrg St


Kinetic Equation predictionsZaichik and Alipchenkov, Phys Fluids 2003


Dispersion and Drift in compressible flows (Elperin & Kleorin, Reeks, Koch & Collins, Reeks)

)(flux Drift )(flux Diffusive dD qqw

tdttrwtrtrwDt

D t

0

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

•w(r,t) the relative velocity between particle pairs a distance r apart at time t•Particles transported by their own velocity field w(r,t) •Conservation of mass (continuity)

1/

1Stfor Only works

,(,

,(,)(,)(

22

0

0

KSt

t

idi

t

jiijj

ijDi

rg(r) ~r

tdttrwtrwq

tdttrwtrwrDr

rDq

Random variable


Summary / Conclusions• Overview

– Transport, segregation, agglomeration dependence on Stokes number

– Use of particle compressibility d/dt(lnJ)

– Singularities, intermittency, fractals, random uncorrelated motion

– Measurement) and modeling of agglomeration• (RDF and de-correlated velocities

• PDF (kinetic) approach, diffusion / drift in a random compressible flow field

– New PDF approaches – statistics of acceleration points (sweep/stick mechanisms)(Coleman & Vassilicos)

a kinetic theory for the transport of small particles in turbulent flows

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