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
Workshop on Turbulence in Clouds
Definition of particle inertia
Turbulent gas/solid flows
Dilute mixture/ one way coupling
d
1
u     g ;

dt  p
pd 2
 p  Stokes relaxation time 
( sphere )
18
Scaling Parameters in Shear Flows
turbulent Stokes no.   p TL or  p TE ; mean shear Stokes no.   p u L
settling velocity  g
u ,  g u
For particles to follow the turbulenc e  p TL or  p TE  1
For particles to follow the mean flow  p u L  1
Workshop on Turbulence in Clouds
Overview of scales in turbulent clouds
Turbulence:
Large scales:
Small scales:
L0 ~ 100 m,
Lk ~ 1 mm,
0 ~ 103 s,
k ~ 0.04 s,
u0 ~ 1 m/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
CONDENSATION
Microscales:
rd ~ 10-5 m,
St = d/k ~ 0.02,
vT/uk ~ 0.3
COLLISIONS / COALESCENCE
Rain drops:
rd ~ 10-3 m,
St = d/k ~ 200,
vT/uk ~ 3000
Collisions / coalescence process vastly enhanced
if droplet size distribution at microscales is broad
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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
Long time Diffusion coefficien t


Dij()   i (0) j ( s ) ds  
ui (0)u j ( s ) ds

0
0




fluid velocityalong particletrajectory
( p)
( p)
Taylor's formula

  ui (0,0)u j (Y p ( s ), s ) ds ; Y p ( s ) is the position of a particle at time s
( p)
D ( p )   TLf
 (f)
(f)
D   TLf
0
; TLf( p )  Fluid Lagrangian integral time - scale along a particle trajector y
TLf( f )  Fluid point Lagrangian integral time - scale
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
g  g
 
t
(o)

  u ( x, t )    p ( y , s )

0
p
( y, s)

y Y ( x ,t s )
  St S 2
p

y Y ( x ,t s )
 R2
p

 p ( y, s ) is the instantane ous particle
velocity field; S , R are the strain rate and
rotation t ensors of the fluid along a particle tr
g
v (g0)  settling velocity in still gas (without t urbulence)
v g  settling velocity in turbule nt gas
Maxey & Wang vg>vg(0)
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Davila & Hunt: settling
around free vortices vg>,<vg(0)
Compressibility of a particle flow
Falkovich, Elperin,Wilkinson, Reeks
Compressibility (rate of compression of elemental
particle volume along particle trajectory)
particle
streamlines
 
p

( y, s) y Y ( x ,t s ) Divergence of the particle velocity
field along a particle trajectory
•zero for particles which follow an incompressible flow
•non zero for particles with inertia
•measures the change in particle concentration
Workshop on Turbulence in Clouds

The statistics of the process v p t , X p (t ), ln J (t )
J ij 
xi (t )
; J  det J ij
x j (0)
  v p X p x,0 t , t  
d
ln J
dt

Compression - fractional change
in elemental volume of particles
along a particle trajectory
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 field
Can determine the statistics of ln J(t) easily.
The process is strongly non-Gaussian
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Particle trajectories in a periodic array of vortices
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Deformation Tensor J
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Singularities in a particle concentration
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Compressibility
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Intermittency –
Balkovsky, Falkovich (2001), Ijzermans et al (2008)
Moments of the spatially averaged number density, St=.5
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Caustics - Wilkinson
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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
R L (r )  v L (1,r1 )v L (2,r 2 )
r  r 2  r1
Falkovich and Pumir (2006)
Radial distribution function g(r)
r
g (r )  r  ( St )
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g(r)
Compressibility in DNS isotropic turbulence
Piccioto and Soldati (2005)
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Turbulent Agglomeration
Two colliding spheres volume v1, v2
r2
n
r1
Saffman & Turner model
radius of collision sphere rc  r1  r2
jr (rc )  current of colliding particles at rc
Collison kernel
K  collision area  / n
jr  12 n wr
test particle
Collision sphere
Agglomeration in DNS turbulence
L-P Wang et al. critically examined S&T model
•Frozen field versus time evolving flow field
•Absorbing versus reflection
Brunk et al. – used linear shear model to asess
influence of persistence of strain rate, boundary
conditions, rotation
Workshop on Turbulence in Clouds
K S  2πrc
2
wr  2π 2 rc
 u 
σ(rc )   
 x 
2
1/ 2
2
π
σ(rc )
 u 
rc ;  
 x 
2
 8 
3
KS  
 rc
 15 
dn(r1 )
  12 K S (r1 , r2 )n1 (r1 )nr2 
dt
1/ 2


15
Agglomeration of inertial particles
Sundarim & Collins(1997) , Reade & Collins (2000): measurement of
rdfs and impact velocities as a function of Stokes number St
K (r1 , r2 )  4rc g (rc , St ) wr (St )
2
RDF at rc
Ghost = interpenetrating
Finite particles = elastic particles
DNS -5%, 25% agglomeration
Workshop on Turbulence in Clouds
Net relative velocity between colliding
spheres along their line of centres
Probabalistic Methods
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Kinetic Equation and its Moment equations
Zaichik, Reeks,Swailes, Minier)
w = relative velocity between identical particle pairs, distance r apart
Δu(r) = relative velocity between 2 fluid pts, distance r apart
Structure functions
ur (r ) , u (r ) ~ r /  K 
n
n
2
r /  K   1
Probability density(Pdf)
P
P


 w  mass 
 wP( w, r , t )  
  u P
t
r
w
w
convection
mass
momentum
 
t


β = St-1

 w0
xi
D

wi  
 wiwj   w i     ui
Dt
rj
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Net turbulent
Force (diffusive)
Kinetic Equation predictions
Zaichik and Alipchenkov, Phys Fluids 2003
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Dispersion and Drift in compressible flows
(Elperin & Kleorin, Reeks, Koch & Collins, Reeks)
•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)
Random variable
•Conservation of mass (continuity)
D
   w(r , t )     (r (t ),0) exp     w(r , t t )dt 
Dt
0
t
 w  Diffusive flux (q D )  Drift flux (qd )
q   Dij (r )
D
i
 
rj
t
, Dij (r )   wi r , t w j (r , t t  dt 
t
q   wi r , t   w(r , t t  dt  
d
i
0
Only works for St<<1
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0
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)
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