particle transport in turbulence and the role of inertia

31
Workshop on Turbulence in Clouds Particle transport in turbulence and the role of inertia Michael Reeks School of Mechanical & Systems Engineering University of Newcastle-upon-Tyne, UK ingularities, fractals,and random uncorrelated moti

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Particle transport in turbulence and the role of inertia. Singularities, fractals,and random uncorrelated motion. Michael Reeks School of Mechanical & Systems Engineering University of Newcastle-upon-Tyne, UK. Definition of particle inertia. Turbulent gas/solid flows. - PowerPoint PPT Presentation

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Page 1: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Particle transport in turbulence and the role of inertia

Michael ReeksSchool of Mechanical & Systems EngineeringUniversity of Newcastle-upon-Tyne, UK

Singularities, fractals,and random uncorrelated motion

Page 2: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Definition of particle inertia

Turbulent gas/solid flows

)( 18

timerelaxation Stokes ; 1

2p sphered

gudt

dp

p

1 flowmean thefollow toparticlesFor

1 or e turbulenc thefollow toparticlesFor

, velocity settling

no. Stokesshear mean ; or no. Stokesturbulent

Lu

TT

uu

LuTT

p

EpLp

gg

pEpLp

Dilute mixture/ one way coupling

Scaling Parameters in Shear Flows

Page 3: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Overview of scales in turbulent clouds

Turbulence:Large scales: L0 ~ 100 m, 0 ~ 103 s, u0 ~ 1 m/s,Small scales: Lk ~ 1 mm, k ~ 0.04 s, uk ~ 0.025 m/s.

Droplets: Radius: Inertia: Settling velocity:Formation: rd ~ 10-7 m, St = d/k ~ 2 × 10-6, vT/uk ~ 3 × 10-5

Microscales: rd ~ 10-5 m, St = d/k ~ 0.02, vT/uk ~ 0.3

Rain drops: rd ~ 10-3 m, St = d/k ~ 200, vT/uk ~ 3000

COLLISIONS / COALESCENCE

CONDENSATION

Collisions / coalescence process vastly enhancedif droplet size distribution at microscales is broad

Page 4: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Purpose and Objectives

• Overview / Historical Development• Relevance to Cloud Physics

– Segregation / demixing /collisions/ agglomeration

• Analogies and similarities to related processes – Deposition in a turbulent boundary layer

• Role of KS and DNS• Awareness and appreciation

Page 5: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Outline

• Turbulent diffusion– Homogeneous turbulence

• Particle diffusion coefficients• Crossing trajectories

– Simple shear– Inhomogeneous turbulence

• Turbulent boundary layer

• Segregation– Characteristics– Agglomeration

Page 6: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Particle dispersion in homogeneous stationary turbulence

Fundamental result

ssYdsssYuu

dssuudss)(D

ppji

pj

pijiij

at time particle a ofposition theis )( ; )),(()0,0(

)()0()()0(

tcoefficienDiffusion timeLong

0

0y trajectorparticle along velocity fluid

)()(

formula sTaylor'

0

scale- timeintegral Lagrangianpoint Fluid

y trajectorparticle a along scale- timeintegral Lagrangian Fluid ;

)(

)()(

)(

)(

)(

fLf

pLff

Lf

pLf

f

p

T

TT

T

D

D

Page 7: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Diffusion coefficient versus inertia

K. Squires PhD thesis

Page 8: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Diffusion coefficient versus drift

Crossing trajectories Yudine (1959) Csanady (1970?)Wells & Stock (1983)Wang-Stock (1988)

Page 9: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Segregation• Quantifying segregation

– Historical development – Compressibility – Singularities – Random uncorrelated motion– Radial distribution function

• Agglomeration– Simulation– Probabalistic methods

Page 10: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

particle motion in vortex and straining flow

Stokes number St ~1

Page 11: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Segregation in isotropic turbulence

Page 12: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Segregation simple random flow field

Page 13: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Settling in homogeneous turbulence , Maxey 1988, Maxey & Wang 1992, Davila & Hunt

y trajectorparticle a along fluid theof ensorsrotation t

and ratestrain theare field; velocity

particle ousinstantane theis ),(

),(

),(),(

22

),(

0),(

)(

R , S

sy

RSStsy

sytxu

p

ppstxYyp

t

stxYypo

gg

Maxey & Wang vg>vg(0)

Davila & Hunt: settling around free vortices vg>,<vg

(0)

g

gasnt in turbule velocity settling v

urbulence)(without t gas stillin velocity settling v

g

)0(

g

Page 14: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

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)

Page 15: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

)(ln),(,v process theof statistics The tJtXt pp

Jdt

dttxX

JJx

txJ

pp

ijj

iij

ln,0,v

det ;)0(

)(

can be obtained directly from solving the eqns. of motion x(t),v(t),Jij(t),J(t))

Avoids calculating the compressibility via the particle velocity fieldCan determine the statistics of ln J(t) easily.The process is strongly non-Gaussian

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

Page 16: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Particle trajectories in a periodic array of vortices

Page 17: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Deformation Tensor J

Page 18: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Singularities in a particle concentration

Page 19: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Compressibility

Page 20: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Intermittency – Balkovsky, Falkovich (2001), Ijzermans et al (2008)

Moments of the spatially averaged number density, St=.5

Page 21: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Caustics - Wilkinson

Page 22: Particle transport  in turbulence and the role of inertia

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

Page 23: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Radial distribution function g(r)r

g(r)

)()( Strrg

Page 24: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Compressibility in DNS isotropic turbulence

Piccioto and Soldati (2005)

Page 25: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Turbulent Agglomeration

rr wnj 21

nK

rrj

rrr

ccr

c

/ areacollision

kernelCollison

at particles colliding ofcurrent )(

spherecollision of radius 21

Two colliding spheres volume v1, v2

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. critically examined S&T model•Frozen field versus time evolving flow field•Absorbing versus reflectionBrunk et al. – used linear shear model to asess influence of persistence of strain rate, boundary conditions, rotation

n

Collision sphere

Page 26: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

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

Ghost = interpenetratingFinite particles = elastic particles

DNS -5%, 25% agglomeration

Page 27: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Probabalistic Methods

Page 28: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

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

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

0

w

xt i

1 / /~)(,)( 2 KKnn

r rrruru Structure functions

Net turbulent Force (diffusive)mass Pu

wtrwPw

wr

Pw

t

P

),,(

convection β = St-1

Probability density(Pdf)

mass

Page 29: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Kinetic Equation predictionsZaichik and Alipchenkov, Phys Fluids 2003

Page 30: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

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)

tdttrwtrwq

tdttrwtrwrDr

rDq

t

idi

t

jiijj

ijDi

0

0

,(,

,(,)(,)(

Random variable

Only works for St<<1

Page 31: Particle transport  in turbulence and the role of inertia

Workshop on Turbulence in Clouds

Summary Conclusions• Overview

– Transport, segregation, agglomeration dependence on Stokes number

– Use of particle compressibility d/dt(lnJ)

– Singularities, caustics, 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)