Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Colliding Particlepair Velocity Correlation Function in Turbulent Flows
Santiago Laina, Martin Ernstb and Martin Sommerfelda'b
a Universidad Autonoma de Occidente (UAO), Energetics and Mechanics Department, FMRG, Cali, Columbia
Email: slain@uao.edu.co
b MartinLutherUniversitit HalleWittenberg, Mechanische Verfahrenstechnik, Halle (Saale), 06099, Germany
Email: martin.ernst@iw.unihalle.de and martin.sommerfeld@iw.unihalle.de
Keywords: Lagrangian simulations, preferential concentration, particle pair velocity correlation function
Abstract
As an alternative to the much more expensive CPU time deterministic models, Sommerfeld (2001) developed an efficient
stochastic interparticle collision model. A key parameter in such model was the velocity correlation between colliding particles,
depending on particle Stokes number, which at that stage was estimated by trial and error comparing the results with LES
computations. This work is focused on the determination of such particlepair correlation function between colliding particles in
homogeneous isotropic turbulence as a function of the particle Stokes number. In order to overcome the extremely high CPU
time demands of Direct Numerical Simulations (DNS), the model presented in Zaichik & Alipchenkov (2i 1"'), which describes
the particle pair dispersion and preferential concentration of identical particles on homogeneous isotropic turbulence, has been
used to estimate such particlepair correlation function between colliding particles, R, depending on particle Stokes number.
Such functional form has been validated also versus DNS performed by the Lattice Boltzmann Method. Additionally, the
morphology of heavy particle clusters in homogeneous isotropic turbulence depending on Stokes number has been investigated
with the help of the code developed by Kerscher et al. (1997).
Introduction
Suspensions of dust, impurities, droplets, bubbles, and other
dispersed particles advected by incompressible turbulent
flows are commonly encountered in many natural
phenomena and industrial processes. These inertial particles,
whose density generally differs from that of the underlying
fluid, cannot follow the fluid motion. Often, they are
characterized by the presence of strong inhomogeneities in
their spatial distribution. Such a phenomenon is known as
preferential concentration (Eaton & Fessler, 1994). The
inhomogeneities then appearing in the particle spatial
distribution are important because they affect the probability
to find close particle pairs and thus influence their possibility
to collide, or to have biological and chemical interactions.
Examples showing the importance of this phenomenon are
the formation of rain drops by coalescence in warm clouds,
or the coexistence between several species of plankton in the
hydrosphere. Engineering applications encompass
optimization of spray combustion in diesel engines and in
rocket propellers. Inertial particles are also important for the
problem of dispersion of dust, chemicals or aerosols in the
atmosphere.
Quite a number of theoretical studies on the collision rate of
particles or droplets in turbulent flows have been published
in the past. A detailed review was given for example by
Williams & Crane (1983) and Pearson et al. (1984). Also the
method of direct numerical simulation (DNS) is being
extensively used for the analysis of interparticle collisions
mainly in isotropic turbulent flows applying a particle
tracking approach with pointparticles (Sundaram & Collins,
1997; Wang et al., 1998; Zhou et al., 1998; Mei & Hu, 1999).
Modelling of interparticle collisions in the frame of the
Euler/Lagrange method for the numerical calculation of
twophase flows has been based so far mainly on two
approaches: a direct simulation deterministicc model) and a
stochastic model based on concepts of the kinetic theory of
gases. The most straightforward approach to account for
interparticle collisions is the direct simulation approach.
This requires that all the particles have to be tracked
simultaneously through the flow field. Thereby, the
occurrence of collisions between any pair of particles can be
judged based on their positions and relative motion during
one time step. Once a collision occurs the change in
translational and angular particle velocities can be
determined by solving the equations for the conservation of
linear and angular momentum in connection with Coulomb's
law of friction. When the duration of the collision process is
negligibly small compared to the time of collisionless motion,
the size of the colliding particles is not too different, and the
ratio of solid particle density to the fluid density is much
larger than unity, fluid dynamic effects during the collision
process can be neglected and the collision efficiency may be
assumed as 100%.
A direct simulation method of interparticle collisions was for
example applied by Tanaka & Tsuj i (1991) for the calculation
of vertical gasparticle flow. This calculation was feasible
only by considering a relatively short element of a pipe and
applying periodic boundary conditions. Furthermore, rather
coarse particles were considered (i.e. particle diameters of
Paper No
0.4 and 1.0 mm), whereby the number of particles for a given
mass loading was relatively small and hence only about 1000
particles had to be tracked simultaneously through the flow
field. When more complex flow configurations and smaller
particles are considered, a direct simulation of interparticle
collisions is not feasible due to the high computational effort
and the large storage requirements.
Following Sommerfeld (2001), stochastic interparticle
collision models rely on the generation of fictitious collision
partners and the calculation of the collision probability
according to kinetic theory. The advantage of this model is
that it does not require information on the location of the
surrounding particles and hence it is also applicable if a
sequential tracking of the particles is adopted, as usually
done when applying the Euler/Lagrange approach to
stationary flows. During each time step of the trajectory
calculation of the considered particle a fictitious second
particle is generated. The size and velocity of this fictitious
particle are randomly sampled from local distribution
functions.
Nomenclature
A union set of a body/morphology ()
B Eulerian twopoint velocity correlation (m2/s2)
D diameter (m)
f frequency (1/s)
f longitudinal Eulerian spatial correlation ()
g transverse Eulerian spatial correlation ()
k kinetic energy (m2/s2)
L length of one period [m]
n number per unit volume (1/m3)
P probability ()
R correlation function ()
R particle location (m)
Re Reynolds number ()
RN random number ()
r separation distance (m)
S structure function (m2/s2)
St Stokes number ()
T integral time scale (s)
t time (s)
u velocity (m/s)
V Minkowski functionals
v normalized Minkowski functionals
W mean relative velocity (m/s)
w relative velocity (m/s)
x location (m)
Greek letters
a correlation parameter ()
a volume fraction ()
, correlation parameter ()
r particle pair density
y correlation parameter ()
8 Kronecker delta ()
E dissipation rate (m2/s3)
r7 Kolmogorov length scale (m)
v kinematic viscosity (m2/s)
a local rms value of velocity (m/s)
 time scale (s)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Subsripts
coll collision
fict fictitious
i,j tensor subscripts
K Kolmogorov
L Lagrangian quantity
N number
nn transverse
11 longitudinal
p particle
r radius
real real
T Taylor
t turbulent
g morphological characterization
0 volume Ii iL.:.. ,Li functional)
1 surface area iJ l!im.:.v Li functional)
2 mean curvature .I JI!,.:.. Li functional)
3 Euler characteristic .I ImL!,.:.. .L functional)
S fluctuating components
Stochastic interparticle collision model
The sampling of the fictitious particle size requires
information on the local particle size distribution. Since in
practical situations the particle size distribution may change
throughout the flow field due to the different responses of
different sized particles, the particle size distribution (i.e., the
number frequency distribution) has to be sampled and stored
for each control volume of the entire computational domain.
This requires the size distribution to be resolved by a number
of size classes. Hence the size of the fictitious particle is
randomly sampled from such a distribution function. The
velocity components of the fictitious particle are composed
of the local mean velocities and fluctuating components
sampled from Gaussian velocity distributions with the local
rmsvalue. In case, a particle size distribution is considered,
also the particle sizevelocity correlation has to be obtained
for each control volume, i.e., the particle mean and rms
velocities are sampled and stored for each size class.
In generating the fictitious particle fluctuating velocities the
correlation with the velocity of the considered particle due to
turbulence has to be respected (see section 2). The degree to
which the particle fluctuating velocities are correlated
depends on their response to the turbulent fluctuations. The
response of particles to turbulent fluctuations is characterized
in terms of the Stokes number, St, defined as the ratio of the
particle response time, p, to some proper time scale of
turbulence, z. In Sommerfeld (2001) the correlation of the
fluctuating velocity components of the fictitious particle with
those of the real particle u'real,, is accounted for in the
following way by using the turbulent Stokes number:
Sfict,= R(St)u'real,1i+ p,1 1R(St)
Here op,, is the local rms value of the particle velocity
component i and 4 is a Gaussian random number with zero
mean and a standard deviation of one. Hence, the sampled
fluctuating velocity components are composed of a
correlated and a random part. With increasing Stokes number
the correlated term (first term in Eq. (1)) decreases and the
Paper No
random term increases accordingly. Comparing model
calculations with large eddy simulations the following
dependence of the correlation function R(St) on the Stokes
number was found in Sommerfeld (2001) under a trial and
error basis:
R(St)= exp(0.55x St4) (2)
where ; was taken as the Lagrangian integral time scale TL.
The next step in the collision model is the determination of
the probability for the occurrence of a collision between the
considered and the fictitious particle within the time step.
This probability is essentially the number of collisions within
the time step which should be smaller than unity, if a proper
time step constraint is applied. The collision probability is
calculated as the product of the time step size At and the
collision frequency given by kinetic theory:
Pol = fIAt = (Dp1 + Dp, ) p, Up, nAt (3)
4
where Dp, and Dp, are the particle diameters, p, Up, is
the instantaneous relative velocity between the considered
and the fictitious particle and np is the number of particles per
unit volume in the respective control volume. In order to
decide whether a collision takes place, a random number RN
from a uniform distribution in the interval [0,1] is generated.
A collision is calculated when the random number becomes
smaller than the collision probability, i.e. if RN < Pooll
(Sommerfeld, 2001).
This article is focused on the determination of the
particlepair fluctuating velocity correlation function
between colliding particles in homogeneous isotropic
turbulence as a function of the particle Stokes number. In
order to overcome the extremely high CPU time demands of
DNS, the model presented in Zaichik & Alipchenkov (2 I"' ),
which describes the particle pair dispersion and preferential
concentration of identical particles in homogeneous isotropic
turbulence, has been used to estimate such particlepair
correlation function between colliding particles, R(St). The
proposed functional form for the correlation has been
validated also versus DNS performed by the Lattice
Boltzmann Method. Finally, the morphology of heavy
particle clusters in homogeneous isotropic turbulence
depending on Stokes number has been investigated with the
help of the code developed by Kerscher et al. (1997).
Particle dispersion in isotropic, homogeneous
turbulence
When modelling preferential concentration of particles
immersed in a dilute disperse mixture, that is, when the
volume fraction of the disperse phase is small, the attention
should be focused on the interaction of particles with
turbulent eddies of the carrier flow, since the role of
interparticle interactions is negligible. However, the
contribution of interparticle interactions to momentum and
energy transport in the disperse phase grows with volume
fraction and size of particles.
The simplest and most extensively studied type of turbulence
is statistically homogeneous isotropic turbulence of an
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
incompressible fluid. In such configuration, it is well known
that smallscale fields of velocity and temperature at large
Reynolds numbers can be thought of being more or less
homogeneous and isotropic. Small scale vortex structures are
responsible for dissipation of turbulent energy and play a key
role in the accumulation (clustering) of particles in a
turbulent flow. Study of isotropic turbulence is thus of
fundamental importance for both singlephase and twophase
flows.
Formation of clusters, that is, regions with significantly
higher concentration of the disperse phase surrounded by
areas of low concentration, represents one of the most
interesting and complex phenomena caused by the
interaction of particles with turbulent eddies (Squires &
Eaton, 1991). We should distinguish between two types of
flows in which clusters may be formed: inhomogeneous and
homogeneous turbulent flows. The phenomenon of
clustering of heavy particles in inhomogeneous turbulent
flows is explained by the effect of turbulent migration
(turbophoresis) from regions of high intensity of turbulent
velocity fluctuations to regions of low turbulence
(Caporaloni et al., 1975; Reeks, 1983). Clustering of inertial
particles also takes place in homogeneous turbulence, where
the gradients of the mean velocity fluctuations of the carrier
flow are zero and consequently, particle transport via
turbophoresis does not take place in the conventional sense.
Fractal dimension of the resulting cluster structures may be
less than the dimensionality of the physical space (Bec, 2003,
2005). However, despite the stochastic character of
turbulence, the distribution of heavy inertial particles in
turbulent flows is not random, and their interaction with the
coherent vortex structures of the turbulent flow may give rise
to some significant clustering.
To illustrate the effect of clustering, Figure 1 shows the
results of kinematic simulation of instantaneous fields of
particle distribution in homogeneous isotropic turbulence for
different values of Stokes number StK= p/K, which
characterizes particle inertia and is equal to the ratio between
the response time of particles, p, and the Kolmogorov time
microscale, TK. A local rise of concentration of heavy
particles is observed in the regions of low vorticity due to the
action of the centrifugal force and is caused primarily by the
interaction of particles with smallscale vortex structures.
Therefore, the effect of clustering is most pronounced when
particle response time approaches the Kolmogorov time
microscale of turbulence.
From DNS visualisations (Chen et al., 2006) it is known that
inertial particle position fields starting from an initial
uniform distribution develop welldefined nearempty spaces
as time progresses (see Figure 1). It is striking that the
locations of these nearempty spaces at given integration
times are the same for all different Stokes numbers. What
differs from one Stokes number to another is the size of these
regions, the average size of which increases with increasing
Stokes number. All these observations are valid for any
integration time that is long enough in comparison to p.
Paper No
,
*.
if '..
, . 
Figure 1: Illustration of preferential concentration obtained
during this work combining Kinematic Simulation and
particle Lagrangian tracking. Particle inertia increases from
top to bottom.
Moreover, following Fevrier et al. (2005), interactions
between the dispersed and continuous phases not only
structure the particle number density, but also lead to
correlation of the velocities of neighboring particles. As
discussed in Abrahamson (1975), it seems legitimate to
assume that the twoparticle velocity distribution will have
the following asymptotic behaviours. For smallinertia
particles, neighboring velocities will be spatially correlated
through the interactions with the same local fluid flow. In
contrast, for largeinertia particles with response times that
are long compared to the fluid turbulence macroscales,
neighboring particle velocities are uncorrelated since these
particles maintain a stronger connection (memory) to their
interactions with very distant, and independent, turbulent
eddies. In the large inertia limit, statistics of the particle
velocity distribution will satisfy the assumption of molecular
chaos and can be described using kinetic theory (e.g. see
Reeks, 1977). An important consequence is that in the
largeinertia limit, the particle velocity distribution cannot be
assumed to correspond to a spatially continuous velocity
field. The random nature of the particle motion will lead to a
crossing of individual trajectories (obviously inducing a
collision in physical systems).
The above small and largeinertia limit cases illustrate that
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
substantially different spatial features in the particle velocity
are possible, with these differences depending on particle
inertia. In the scalar limit, the spatial correlation function
between any two particle velocities should be accurately
modelled by a decaying exponential, analogous to the
correlation describing the fluid turbulence (Hinze, 1975). On
the other hand, in the limit of very inertial particles, motion
becomes stochastically equivalent to a Brownian motion
with independent random velocities, even for those particles
passing close to each other (Abrahamson 1975; Reeks 1977).
For intermediate values of the particle response time, the
particle velocity distribution should be expected to exhibit
spatial features with characteristics representative of these
two limiting regimes.
In Fevrier et al (2005) a theoretical approach based on
application of an average conditioned on a realization of the
carrier fluid flow was used to develop relations governing
two contributions to the velocity of a finiteinertia particle.
The first contribution the mesoscopic Eulerian particle
velocity field, MEPVF is a continuous velocity field shared
by all the particles of the system. The remaining contribution
to the particle velocity represents a random component the
quasiBrownian velocity distribution, QBVD that accounts
for the fact that a portion of the particle velocity corresponds
to a distribution that is not spatially correlated, which also
implies that the quasiBrownian velocities of two
neighboring particles are not correlated. In gassolid
systems, partitioning of the particle velocity into the MEPVF
and QBVD is dictated by particle inertia. The MEPVF
accounts for all fluidparticle and particleparticle velocity
correlations. For lowinertia particles that follow the flow,
the contribution of the QBVD is not large and the particle
velocities possess spatial correlations similar to that of the
underlying carrier flow. With increase in particle inertia, the
spatial velocity correlations are increasingly affected by the
QBVD, especially for small separations.
The Zaichik and Alipchenkov (Z&A) model
Zaichik & Alipchenkov (t2 I"' ), hereafter called Z&A model,
presented a model for describing the particle pair dispersion
and preferential concentration of particles in isotropic
turbulence. In the case of a steadystate suspension of
particles immersed in isotropic homogeneous turbulence, the
pair relative velocity statistics and the particle accumulation
effect are described with the following model:
2(Spii Sp ) dS1 dlnF
2p n +(Spi+grSn) =0
r dr dr
St dr F(spiu + gr S1t + 2F(fr Sii Sp = 0
dr I dr I
St2
 (a + b)+ 2(fSnn Spnn) = 0
3r
d 2 dS7pnn
a r F(Spil + grS1) d
dr dr
"''
Paper No
b= 2d [rr(Spnn + gr Snn)(Spi Spnn)]
dr
In Eqs. (4), the overbar stands for normalisation by the
Kolmogorov length scale r7 or the Kolmogorov velocity scale
uK = ('i" ', StK is the Stokes number based on the
Kolmogorov time scale rK, and F is the particle pair density
normalised with the particle pair density expected when r 
co. S11 and S.n are the fluid longitudinal and transverse
structure functions of the velocity increments at two points:
Sj(r)=((u,(x+r,t)uj(x,t))(uj(x+r,t)u,(x,t))) (5)
In order to introduce the structure functions for the particle
relative velocity, the following definitions must be
introduced:
where rp and Wp are the separation distance and the relative
velocity between two particles, located at Rpi and Rp2 with
velocities upl and up2. The relative velocity of particles pairs
have a mean value W= , which allows to define a
fluctuation w' = w W. Therefore the particle structure
functions of the relative velocity are defined as:
SplJ =(w:w>) (7)
The relevant boundary conditions for Eqs. (4) are given by:
dSpli dSpn 
dS= 0 r=0 (8)
dr dr
Spll =frSll; Spnn= fSn; F =1 ro (9)
Conditions (8) express the balance between the radial relative
inward velocity and outward fluxes at the origin, and they are
valid if the particle size is less than the Kolmogorov length
scale. Relations (9) point to the fact that the particle
velocities become nearhomogeneous at large separations,
whereas the particles are randomly distributed.
The fluid structure functions as well as thef, gr functions are
assumed to be known for isotropic homogeneous turbulence
and the specific forms can be found in the original paper of
Zaichik and Alipchenkov (2' i;).
This model can be solved numerically and the results relevant
to the estimation of the correlation function of particle pair
fluctuating velocities can be extracted and they will
presented in the following. To this end, Zaichik &
Alipchenkov (2 11') define the Eulerian twopoint velocity
correlation as:
2 2
Bpl(r)=(uIpl(x, t)up(x+r,t) =2 kpl2Spl(r) (10)
where kp is the kinetic energy of particles. Figure 2 shows the
longitudinal space correlation, 3Bp1/2kp, as a function of the
separation distance related to the integral length scale. It is
apparent that, in accordance with DNS by Fevrier et al.
(2005) for dispersed twophase flows and in contrast to the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
wellknown result for fluid turbulence, this correlation for
inertial particles is not tending towards unity in the limit of r
 0. This specific feature of the spatial correlation indicates
that as consequence of the inertial bias in their trajectories,
the velocities of particles are not completely correlated even
at r = 0. Overall, however, the agreement with the DNS
results is reasonable, so the model captures the main
characteristics of particle pair correlation velocities in
isotropic, homogeneous turbulence.
3 Bp,
2 kp
0.8
So  St=0.32
0 + St= 1.9
+o0 St=9.4
0.6 a0 A0  St=21.8
0.4
+* St =30.9
0.2
S0.01 0.02 0.03 0.04
r/L
Figure 2: Influence of particle inertia on the longitudinal
space correlation R. Results obtained solving the Z&A model
versus DNS results of Fevrier et al. (2005).
In the case of colliding particles, the distance between them
tends to zero (obviously it can not be less than the sum of the
radii of both particles), so the particle pair velocity
correlation is given by the behaviour near the origin of the
longitudinal space correlation, R = 3BpI(r 0)/2k. This fact
allows us build Figure 3 where R'. i,.) is plotted versus the
Stokes number based on the Kolmogorov time scale, StK.
Based in the results obtained solving the Z&A model (black
dots), the following functional form for the particle velocity
correlation of colliding particles is proposed:
R(StK) =exp 1+ 
Eq. (11) uses three parameters instead of the two originally
proposed by Sommerfeld (2001), R(St) = exp(oSt). With
the values a= 0.019, 8= 1.725 and y= 0.044 the black dots
fit is fairly good. In the Figure 3 also the optimum fit with the
two parameter expression to the DNS results of Fevrier et al.
(2005) is presented. From that figure it is obvious that the
three parameter correlation Eq. (11) is more appropriated
than that using only two parameters in the configuration of
isotropic homogeneous turbulence. Moreover, the values of
a= 0.55 and 8= 0.4 employed by Sommerfeld (2001) were
obtained under a trial and error approximation whereas the
actual values for a, 8 and y were "measured" by comparing
the results of the Z&A model with DNS which is a much
more rigorous approach.
rp =Rp2 Rpl,; Wp =Up2 Upl
Paper No
S04 Z&A model (Lain)
0,4 3 DNS Fevrier et al. (2005)
 Fit two parameters DNS results
Sommerfeld correlation
0,2  Fit three parameters Lain
10 101 100 101 102 io3
StK
Figure 3: Correlation function of particle pair fluctuating
velocities. Proposed correlations versus the DNS results of
Fevrier et al. (2005) and results obtained in this work (black
dots) with the Z&A model.
Comparison with Direct Numerical Simulations
based on the Lattice Boltzmann Method
Direct numerical simulations of colliding particles suspended
in a cube with homogeneous isotropic turbulence were
carried out to further validate the proposed particlepair
correlation function, Eq. (11). The computation of the fluid
flow is performed using a threedimensional Lattice
Boltzmann Method (LBM) which is originated from
molecular dynamics. Whereas conventional models are
based on the conservation laws formulated on the
macroscopic level, the Boltzmann equation describes the
behaviour of fluids on the mesoscopic level. The Lattice
Boltzmann equation characterizes the temporal and spatial
development of a discrete probability distribution function
and is solved with the help of the single relaxation time
collision operator (Bhatnagar et al., 1954).
The spectral forcing scheme of Eswaran & Pope (1988) is
used to generate isotropic turbulence. The turbulence is
realized by generating a force in spectral space and
introducing it as a change of velocity in the flow field. As a
result, motion is created at large length scales. This is the
basis for the development of motion at small length scales in
form of an energy cascade which dissipates over time.
Simulations were carried out for a mesh size of 643 yielding a
Taylor Reynolds number of 82.1. Figure 4 shows three
typical snapshots of the temporal and spatial varying
turbulent velocity field.
Under the assumption of pointparticles, the transport of
spherical particles within the fluid is modelled by the
Lagrangian approach. Modification of fluid turbulence by the
particles is not considered. In the present simulations,
interparticle collisions are taken into account. In the
framework of the presented study the detection and
modelling of inelastic particleparticle collisions is computed
using a deterministic collision model proposed by Sundaram
& Collins (1997). A detailed validation of the implemented
collision model was presented in Ernst & Sommerfeld
According to Figure 4 (top), the motion of particles with
small Stokes numbers is completely correlated with the fluid
motion. They are transported in the same turbulent eddy until
a collision occurs. Due to centrifugal effects, particles with
intermediate Stokes numbers tend to concentrate in regions
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
of low vorticity or on the edge of eddy structures (Figure 4,
centre). This locally enhances the collision rate whereby the
particles are redispersed. Due to the random process of
interparticle collisions, the relative motion between fluid
structures and particles becomes more and more uncorrelated
with increasing particle inertia (Figure 4, bottom).
t, '' .' r" '
,., .,.;, , . .
I
,, '. ", .d "'  "
c...
'c*
t
N~' F
4
1,,
4,
'Si..
I
;g j
Figure 4: Cross section of fluid turbulence (vector plot, DNS
643 cells, ReT = 82.1, rl = 0.3 mm) and field distribution of
pointparticles (particle volume fraction, o = 0.02)
considering interparticle collisions. Particles with different
Stokes number: StK = 0.5 (top), StK = 3.7 (centre) and StK =
13.4 (bottom).
Figure 5 shows the comparison between the proposed
correlation function and the results obtained from the Direct
Numerical Simulations for different solid phase volume
fractions, %a. In the range of small particle Stokes numbers
the correlations show a good agreement. Furthermore,
different volume fractions of the particle phase have no
considerable influence on the evolution of the correlation.
However, there is certain influence of the volume fraction for
higher Stokes numbers. In this study, the proposed fit with
the three parameters expression (Eq. (11)) matches the DNS
Paper No
results for a volume fraction of 0.005. At higher particle
Stokes numbers a clear difference is observed for different
volume fractions. Further simulations are currently being
done for higher particle Stokes numbers.
1.0
1 . .....i . . ..... i . .....
0.8
0.6
o DNS: a =0.001
f 0.4 DNS: ap =0.005 3
A DNS: aP =0.010
0.2 < DNS: a =0.020
S Fit three parameters Lain
0.0
102 101 100 101 102 103
StK
Figure 5: Particle pair fluctuating velocities correlation
function: proposed correlation compared to results of
performed Lattice Boltzmann simulations as a function of
particle Stokes number StK and volume fraction of the
particle phase %.
Morphology of particle clusters in isotropic,
homogeneous turbulence
From the results shown in previous sections, it is clear that
particles cluster in turbulent flows, and such clustering
depends on particle inertia. Moreover, maximal clustering
happens for an optimal Stokes number.
Calzavarini et al. (2" 1") demonstrated that tracers, heavy
particles and bubbles cluster in a different way in isotropic,
homogeneous turbulence: tracers (fluid particles) keep
uniformly distributed, particles tend to cluster in 2Dlike
structures whereas bubbles accumulate in IDlike filaments.
Following these authors, the idea for distinguishing the
clustering on spatially extended, eventually interconnected,
sheets from clouds or filamentary clustering can be expressed
as follows: consider the union set of balls of radius r around
N particles at positions x,, i = 1, 2, ..., N, thereby creating
connections between neighboring balls.
Ar =UI Br (xi) (12)
The global morphology of the union set of these balls
changes with radius r, which is employed as a diagnostic
parameter. Global geometrical and topological measures of
e.g. A, are additive, invariant under rotations and translations,
and satisfy a certain continuity requirement. With these
prerequisites Hadwiger (1957) proved that in three
dimensions the four Minkowski functionals V,(r), u = 0, 1, 2,
3, give a complete morphological characterization of the
body A,. The Minkowski functional V,(r) simply is the
volume of A,, Vl(r) is a sixth of its surface area, V2(r) is its
mean curvature divided by 37E, and V3(r) is its Euler
characteristic. Volume and surface area are well known
quantities. The integral mean curvature and the Euler
characteristic are defined as surface integrals over the mean
and the Gaussian curvature respectively. This definition is
only applicable for bodies with smooth boundaries. In our
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
case we have additional contributions from the intersection
lines and intersection points of the spheres. Mecke et al.
(1994) discuss the definitions of the integral mean curvature
and the Euler characteristic for unions of convex bodies.
The Euler characteristic as a topological invariant allows for
several other definitions, like X = #(isolated bodies) 
#(tunnels) + #(completely enclosed cavities). Minkowski
functionals have been developed for the morphological
characterization of the large scale distribution of galaxies by
Mecke et al. (1994) and have successfully been used in
cosmology by Kerscher et al. (1997).
The code used for the calculations of the Minkowski
functionals is an updated version of the code developed by
Kerscher et al. (1997), based on the methods outlined in
Mecke et al. (1994). The code is public via the link
www.mathematik.unimuenchen.de/kerscher/software/
Calzavarini et al. ('" I") discuss the behaviour of the four
Minkowski functionals v,(r) = V,(r) /L3, u = 0, 1, 2, 3,
determined from the particle distributions with St = 0.6 for
three cases: bubbles, heavy particles, and neutral tracers. For
the neutral tracers, the functionals coincide with the
analytically known values (Mecke & Wagner (1991)) for
randomly distributed objects, i.e. following a Poisson
distribution. In such case, as the balls' radii increase, the
volume is filled until reaching complete coverage where the
volume density vo(r), i.e. the filling factor, reaches unity.
The density of the surface area, measured by vi(r), increases
with the radius r. As more and more balls overlap the growth
of vl(r) slows down and the surface area reaches a maximum.
For large radii the balls fill up the volume and no free surface
area is left.
The density of the integral mean curvature v2(r) allows us to
differentiate convex from concave situations. For small radii
the balls are growing outward. The main contributions to the
integral mean curvature is positive, coming from the convex
parts. Increasing the radius further a maximum is attained.
For uniformly distributed objects the empty holes start to fill
up and the structure is growing into the cavities. Now, the
main contribution to the integral mean curvature is negative,
stemming from the holes and tunnels through the structure.
For large radii the balls fill up the volume, no free surface and
hence no curvature is left.
The topology undergoes a number of changes which is
measured with the Euler characteristic. For small radii r z 0
the balls remain separated and the volume density of the
Euler characteristic v3(r 0) equals the number density of the
particles. As the radius increases, balls merge and the Euler
characteristic decreases. When further increasing the radius r,
more and more tunnels start to form resulting in a negative
v3(r). For randomly distributed particles this behavior
reaches a turning point when these tunnels are blocked to
form closed cavities and a second positive maximum of v3(r)
can be seen.
In this paper, this analysis based on Minkowski functionals
has been applied to dispersion of heavy particles in isotropic
homogeneous turbulence depending of the Stokes number.
This situation was not explored in Calzavarini et al. (2" I").
Therefore, the positions of particles for different Stokes
number (based on Kolmogorov time scale) evolving in the
same underlying turbulent field, are examined with the
software of Kerscher mentioned before. The results are
shown in the Figure 6.
Vo shows clearly the effect of preferential concentration,
Paper No
where the volume filling is delayed regarding the tracers,
represented here by St = 0. The existence of a Stokes number
where the clustering is maximal can be idenfied. Also can be
seen that the larger inertia particles (i.e., high St) tend to be
more uniformly distributed because its curve is closer to the
Poisson distributions than the others.
r [ill
Figure 6: Minkowski functionals for heavy particles with
different Stokes number in isotropic homogeneous
turbulence.
Also the clustering shows smaller maximal surface for
inertial particles, reflected in the smaller maxima of the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
curves for v1. This fact corresponds to situations close to
optimal clustering, i.e, with St around 1. Also, for
intermediate radii (around two Kolmogorov length scales) a
slightly skewed shape of vl(r) is observed, meaning an
excess of surface area on intermediate scales compared to
Poisson. Particles cluster in sheet like structures and the
surface area of the balls grows into the empty space between
them.
The lower maxima in v2, regarding the uniform distribution,
for small separation values is interpreted as the loss of
convexity of the cluster structure for inertial particles. Also
the concavity of the structures as r increases is lower than in
the case of tracers, meaning a reduction of holes and tunnels.
Finally, the Euler characteristic, v3, reduces for small r faster
than for tracers, indicating clustering. Also, the tunneling for
heavy particles (negative part in v3) is reduced as clustering
increases and that still there is a small positive maximum for
large r, indicating the presence of a small number of cavities
due to the tunnel blocking. This effect is more pronounced
for the largest Stokes number particles where the distribution
tends to be uniform.
In summary, the topological method based on the Minkowski
functionals allows to describe the main characteristics of the
heavy particle clusters in isotropic homogeneous turbulence.
In particular also provide a tool to determine the optimal
Stokes number for maximum clustering.
Conclusion
In this work, the Zaichik & Alipchenkov (2 "11') model for
dispersion of particle pairs in isotropic, homogeneous
turbulence has been numerically solved. The results have
been employed to build a correlation expression for the
particle pair correlation function versus the particle Stokes
number, necessary for the development of stochastic
interparticle collision models. This correlation is based on
three parameters instead of the two proposed by Sommerfeld
(2001), and it has been validated versus Direct Numerical
Simulations based on the Lattice Boltzmann Method.
Additionally, the morphology of heavy particle clusters in
isotropic homogeneous turbulence depending on Stokes
number has been investigated with the help of the code
developed by Kerscher et al. (1997) which is public. As a
result, it has been shown that for St 0 and oo particles tend
to follow a uniform Poisson distribution filling the full 3D
space, but for intermediate St particles tend to cluster in 2D
sheetlike structures.
Acknowledgements
The financial support for a part of the present studies by the
Deutsche Forschungsgemeinschaft under contract number
SO 204/332 is gratefully acknowledged.
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