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
Random uncorrelated motion (RUM)
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Deformation Tensor J
‘Particle segregation ’ Open Stats Phys 6-3-12
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 segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Moments of particle number density
)(|)(| 1 tntJ || Jn || Jn 11 || Jnn
Moments of particle number density
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
-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
time=10, mean=-1.19, skewst=-0.28, kurtst=3.40
time=1, mean=-0.18, skewst=-2.22, kurtst=13.27
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Random uncorrelated motion (RUM)
Février et. al JFM, 2005
Mesoscopic Eulerian particle velocity field
Random uncorrelated motion (RUM)
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Trajectories
Velocity (RUM)
|J| (singularities)
Singularities- ”sling shot”- RUM
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Deformation Tensor J
‘Particle segregation ’ Open Stats Phys 6-3-12
Singularities in particle concentration
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Compressibility
Simple 2-D flow field of counter rotating vortices KS random Fourier modes: distribution of scales, turbulence energy spectrum
Moments of particle number density
• 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
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Radial distribution function (RDF) g(r)
rg(r)
)()( Strrg
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
PDF Poisson distribution (scaled)
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
PDF Poisson distribution (scaled)
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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|)
time=25, mean=-2.97, skewst=-0.15, kurtst=3.10
time=10, mean=-1.19, skewst=-0.28, kurtst=3.40
time=1, mean=-0.18, skewst=-2.22, kurtst=13.27
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…
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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 segregation ’ Open Stats Phys 6-3-12
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)
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
Diffusion coefficient versus drift
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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)
‘Particle segregation ’ Open Stats Phys 6-3-12
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
‘Particle segregation ’ Open Stats Phys 6-3-12
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)
‘Particle segregation ’ Open Stats Phys 6-3-12