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
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
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
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
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
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
Workshop on Turbulence in Clouds
Diffusion coefficient versus inertia
K. Squires PhD thesis
Workshop on Turbulence in Clouds
Diffusion coefficient versus drift
Crossing trajectories Yudine (1959) Csanady (1970?)Wells & Stock (1983)Wang-Stock (1988)
Workshop on Turbulence in Clouds
Segregation• Quantifying segregation
– Historical development – Compressibility – Singularities – Random uncorrelated motion– Radial distribution function
• Agglomeration– Simulation– Probabalistic methods
Workshop on Turbulence in Clouds
particle motion in vortex and straining flow
Stokes number St ~1
Workshop on Turbulence in Clouds
Segregation in isotropic turbulence
Workshop on Turbulence in Clouds
Segregation simple random flow field
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
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)
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
Workshop on Turbulence in Clouds
Particle trajectories in a periodic array of vortices
Workshop on Turbulence in Clouds
Deformation Tensor J
Workshop on Turbulence in Clouds
Singularities in a particle concentration
Workshop on Turbulence in Clouds
Compressibility
Workshop on Turbulence in Clouds
Intermittency – Balkovsky, Falkovich (2001), Ijzermans et al (2008)
Moments of the spatially averaged number density, St=.5
Workshop on Turbulence in Clouds
Caustics - Wilkinson
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
Workshop on Turbulence in Clouds
Radial distribution function g(r)r
g(r)
)()( Strrg
Workshop on Turbulence in Clouds
Compressibility in DNS isotropic turbulence
Piccioto and Soldati (2005)
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
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
Workshop on Turbulence in Clouds
Probabalistic Methods
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
Workshop on Turbulence in Clouds
Kinetic Equation predictionsZaichik and Alipchenkov, Phys Fluids 2003
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
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)