7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Influence of collisions on particle behavior in a turbulent wallbounded flow
V. Lavezzof and A. Soldati *
Dipartimento di Energetica e Macchine and Centro Interdipartimentale di Fluidodinamica ed Idraulica
UniversitO degli Studi di Udine, via delle Scienze, 208 33100 Udine, Italy
SApplied Physics Department, Eindhoven University of Technology
Den Dolech, 2 5612AZ Eindhoven, The Netherlands
v.lavezzo@tue.nl and soldati@uniud.it
Keywords: DNS, Lagrangian tracking, Turbulent scales, Particleparticle collisions
Abstract
In this work, a Direct Numerical Simulation coupled with Lagrangian particle tracking is used to study the influence of
binary collisions occurring in a turbulent boundary layer. Large samples of particles characterized by different inertia
have been released in an incompressible and Newtonian air flow at Re, = 150. The reference geometry consists of
two flat, infinite, parallel walls separated by a distance of 2h = 0.04 m. The domain is periodic in the streamwise
(x) and spanwise (y) direction and noslip boundary is enforced at the walls. A hard sphere collision model has been
implemented to account for particleparticle and particlewall interactions using a neighbour list approach with linked
lists method to optimize the collision detection in the domain (Sundaram and Collins (1996)). Particle kinetic energy
calculated in presence of collisions changes depending on both particle inertia and collision rate, and it is found to be
higher than in the case without collisions. Reasons for this behavior are associated with the multiple collision events
close to the wall which are able to resuspend particles to the channel core and so, to a region characterized by higher
velocity and energy. The relative velocity between two colliding particles is also evaluated to characterize different
types of collisions responsible for particle energy modification and accumulation/resuspension.
Nomenclature
Roman symbols
Re, Reynolds number ()
h channel half height (m)
g gravitational constant (rn/s1)
p pressure (N/mr2)
dp particle diameter (/tm)
Re, particle Reynolds number ()
St Stokes number ()
Greek symbols
p fluid density (Kg/m13)
pp particle density (Kg/mr3)
v fluid kinematic viscosity (rn2/s)
Introduction
Twophase flows (gasparticle and gasdroplets) are
present in a wide range of applications in both natural
and industrial fields, e.g. aerosol processing, pneumatic
conveying in pipes or channels, sand transport, dust sep
aration in cyclones, pollutant formation and control, etc.
Thus, accurate prediction of the interactions between
the fluid flow and the particles is important to under
stand the underlying physics, to characterize them from
a quantitative viewpoint and to optimize process and
product quality. Twophase flows are extremely complex
since they are governed by simultaneous and interre
lated processes: (i) particleturbulence interaction at dif
ferent length scales, (ii) particleparticle collisions and
(iii) surface or external forces e.g. Van der Waals forces,
particle wetness, electrostatic charges, lubrication force
etc.. In past years, many works attempted at the de
scription of the complex interplay between particles and
turbulent structures in wallbounded flows (Rouson and
Eaton (2001), Kaftori et al. (1995) among many others),
but only few of them focused on the effects of particle
particle rebounds on the dispersion process. Reasons for
this reside on both the difficulties in performing well
controlled experiments in which is possible to separate
the effect of one mechanism from another and the condi
tion commonly used in numerical works, of a dilute flow
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
for which collisions become of secondary importance.
Previous studies on a channel flow laden with micro
particles (see Picciotto et al. (2005), Narayanan et
al. (2003) among others) under dilute conditions have
demonstrated that the motion of particles is strongly af
fected by the fluid coherent structures (sweps and ejec
tions), which are responsible for particle transport and
dispersion in the vicinity of the walls. When ejections
are not strong enough to promote particle resuspension
or when particles are highly inertial, they remain trapped
at the walls and segregate in regions characterized by
high strain and low vorticity. When this occurs, the spe
cific assumption for dilute system is no longer valid and
the flow, in the viscous sublayer, becomes locally dense.
The motion of the particles at the wall is dominated by
collisions and by the interplay between large scale ed
dies and particles. Previous studies (see e.g. Wang et
al. (2000)) have shown that large scale energetic eddies
are the dominant factor contributing to the relative ve
locity between two colliding particles (turbulent trans
port effect), whereas smallscale dissipative eddies can
enhance the collision rate significantly by inducing local
nonuniform particle distribution (accumulation effect).
The turbulent transport effect is most noticeable when
the particle inertial response time Tp is of the order of
the flow integral timescale and the accumulation effect
is most pronounced when Tp is comparable to the flow
Kolmogorov time.
From these considerations two types of collisions can
be identified: highly energetic and low energy. The for
mer refers to a collision between a particle traveling with
high velocity from the channel core toward the wall and
particles that are already accumulated at the wall. Such
collision finds its origin in the turbulent transport and
is responsible for a possible particle resuspension to
wards the channel core. Low energy collisions, on the
other hand, are characterized by low relative velocities
and occur within swarms of particles already accumu
lated at the wall or trapped on the edges of the same
vortex. This collision is mainly responsible for particle
dispersion in a direction parallel to the wall. It becomes
clear that particleparticle collisions play an important
role in particle dispersion since they are able to decor
relate the motion of the particles from the coherent ed
dies responsible for their accumulation. This results in
an overall increase in particle turbulent kinetic energy
since particles are spread in regions of higher velocity,
thus, experiencing an acceleration.
To single out the effects of particleparticle collisions
in a wallbounded flow, a Direct Numerical Simulation
(DNS) coupled with Lagrangian particle tracking has
been employed in this work. The Navier Stokes equa
tions are solved using a pseudospectral method in a
horizontal channel flow of air seeded with 0(106) par
/a o 8 ? / = 30o wv
./
j ,
I'.o : C
"440
'^
Figure 1: Schematics of the channel geometry.
tides. Since the collision type and in turn the collision
rate are affected by the sampling and filtering action ex
erted by the particles on the turbulent flow depending
on their inertia, six swarms of particles characterized
by different Stokes numbers have been released in the
domain (St 0.8, 1.6, 10.7, 20, 40 and 100). Re
sults confirm the work by Sundaram and Collins (1997)
where it was found that finitesize particles with negli
gible Stokes number have either multiple low energetic
collisions or they do not collide at all. On the con
trary, large particles show both a high number of high
energetic collision due to their stronger acceleration and
multiple low energetic collisions due to their preferential
segregation at the wall. This result has been confirmed
also by calculating the Probability Density Function of
the relative velocity between two colliding particles.
Methodology
Fluid flow and Lagrangian tracking A pseudo
spectral Direct Numerical Simulation has been used to
compute the turbulent Poiseuille flow of air (assumed to
be incompressible and Newtonian) flowing in a horizon
tal channel. The domain (shown schematically in Figure
1), is bounded by two infinite walls of length 47h in the
x (streamwise), 27 h in the y (spanwise) and separated by
a distance of 2h in the z (wallnormal) direction, where
h is equal to 0.04 m. The fluid flowing between the two
walls is air (p 1.2 kg/m, v 1.5 10 6 m2/s).
The computational domain has been discretized with
128 x 128 x 129 nodes in the x, y and z direction, re
spectively. Periodic boundary conditions are imposed in
the stramwise (x) and spanwise (y) direction and noslip
boundary condition is enforced at the upper and lower
wall.
The equations for the fluid are the continuity and
Table 1: Summary of gas and particle properties used in
computations.
Variable Value Units
Fluid p 1.2 Kg/m3
v 1.5 105 rn2/s
U, 0.112 m/s
Tf 1.25. 103 s
Particles pp 1000 kg/m3
St 0.80, 1.60, 10.7
St 20, 40, 100
Navier Stokes equations, respectively shown below:
OVu
t + u V
V+ V2u
The shear velocity is defined as u, (Tr,/p)12
where T, is the shear stress at the wall. Velocity, length,
and time are normalized in wall units using the wall
shear stress T,, the fluid density, and the kinematic vis
cosity as u+ u/UT U/(T,/pf)1/2, h+ = hut/v,
t+ tu2/v. The shear Reynolds number is based on
the channel half height h and the shear velocity u, giv
ing Re, = uh/v 150. Summary of fluid param
eters is given in Table 1. The parameter T7 represents
the fluid characteristic time scale and it is equal to the
ratio of the fluid viscosity to the shear velocity squared
as: Tf v/I.
We started our calculations from a fully developed
fluid flow field. The time step used for both fluid and
particle advancement has been chosen equal to 1/10 the
smaller particle response time equal to At 1 10 4 in
order to satisfy the Nyquist sampling criterion to accu
rately follow curved trajectories.
A Lagrangian particle tracking coupled with the DNS
code is used to calculate particles trajectories in the
flow field. Since particles are small and their density is
much larger than the fluid, the full equation presented by
Maxey and Riley to describe the motion of a particle can
be reduced only to a balance of the Stokes drag, gravi
tational and buoyancy forces. The equation of particle
motion becomes:
3 'Dp p
4 dp pp
v(u v) (1
) g (3)
Pp
where v and u are the particle and fluid velocity vectors,
dp is particle diameter and g the gravitational accelera
tion. CD represents the corrected drag coefficient and
depends on the particle Reynolds number as proposed
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
by Schiller and Naumann (1935). It can be written as:
CD 24 (1 + 0.15Re 7) (4)
Rep
where the particle Reynolds number is equal to Rep
dclu v/v, being v the fluid kinematic viscosity. It
is worth noting that in the present work the correction
was necessary only for the three larger Stokes number
considered, for which the particle Reynolds number was
greater than unity. A 4th order RungeKutta scheme
is used to integrate the equation of motion and 6th or
der Lagrangian polynomials are employed to interpolate
fluid velocities at particle position. Periodic boundary
conditions are imposed on particles in both streamwise
and spanwise directions and elastic reaction is applied
when the particle centre is a distance less than dp/2
from the wall. Elastic reaction was chosen as it is the
most conservative assumption when studying particle
behaviour in a turbulent boundary layer. At the begin
ning of the simulation, particles are distributed homoge
neously over the computational domain and their initial
velocity is set equal to that of the fluid at particle posi
tion. Moreover, particles are assumed to be pointwise,
rigid and spherical.
Swarms of 320.000 particles characterized by differ
ent relaxation times were considered in the simulation
to evaluate the effect of inertia on particle behavior.
Particle inertia is expressed in terms of Stokes number
which is the ratio between particle relaxation time to a
characteristic time scale of the fluid (Tr). In this work
the Stokes number is equal to St T Tp/T7f
dpp,/18vTf. Summary of all the values used in the
computations can be found in Tablel. For each Stokes
number two simulations have been considered: with and
without collisions. Details of the collision algorithm can
be found in the following paragraph.
Collision algorithm A hard sphere collision model
has been implemented in the particle tracking code. This
implies that: (i) collisions are instantaneous, elastic and
frictionless, (ii) particle deformation is neglected, as that
the distance between the particle center of mass is equal
to the sum of the radii throughout the collision process
and (iii) particle momentum is perfectly conserved i.e.
the restitution coefficient is equal to 1.
Particles are allowed for multiple collisions within a
single time step (proactive approach proposed by Sun
daram and Collins (1997)) and their time advancement
can be broadly divided into five steps: (i) update particle
velocity by solving the equation of motion described in
the previous section; (ii) use this velocity to identify the
particles that will collide within the current time step by
calculating the time to collision for each particle pair;
(iii) advance particles to the minimum time to collision;
(iv) enact the elastic collision between the two identi
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
fled particles; (v) search for any other possible collision
within the same time step or go to the next time step.
A neighbour list approach with the linked list method
proposed by Sundaram and Collins (1996) has been used
to reduce the computational cost required to seek for
possible collisions in the domain. To this aim, the chan
nel has been divided into a special exploring lattice, so
that the search is limited to the 26 neighboring cells of
the one containing the particle. The time to collision is
calculated as the time required by a particle pair to reach
a relative distance equal to the sum of the radii using the
following equation:
1000
Particle concentration in the wall normal direction
St=0 80
St=l 60
St=10 76
St=20
St 40
St= 100
01 1 10 100
rij(t+tij)\ = rij + I dp
where rij and . are the relative distance and velocity
between particle i and j, respectively. Defining bij as
rij . and substituting in Eq. 5 we obtain:
 2 2 
+ l2. + r d = 0
3 p
bij Vj i
The postcollisional particle velocity is calculated in a
cartesian reference frame centered in the center of mass
of one of the particles and it is then translated back into
the global reference frame. Based on the previous as
sumptions the postcollisional velocity of the two par
ticles is derived from a simple momentum and energy
conservation law as:
miVil + mjVjl miVi2 + mjVj2
1 o 1 2
. L+m vj 2
2 2 i
S 1
2 2
where i and j subscripts refer to the two particles and 1
and 2 to the pre and postcollisional velocities, respec
tively. It is necessary to recall here that in the present
study particles are not allowed to change their mass
(mi mj) so Eq. 8 and 9 can be simplified by eliminat
ing the masses. The approaching angle between two col
liding particles, calculated from the dot product between
the particle relative velocity and their relative distance,
as proposed by Chen et al. (1998), is used to have a
better comprehension on particle collision dynamics. If
the value of the angle is low the two particles are collid
ing almost horizontally and multiple collisions are ex
pected, whereas if its value is close to 90 degrees the
particles are almost perpendicular with one of the two
particles coming from the buffer layer or being thrown
away from an intense vortex so a high relative velocity
and a highly energetic collision is expected.
Figure 2: Particle concentration profile varying with
Z+. The profile is obtained as time average of
the number of particles contained in a specific
slab of the domain normalized by the initial
particle concentration in the same slab.
Results
In this section results on particleparticle collisions are
presented. To have a better insight on the collision dy
namics and on the influence of inertia it is first useful
to understand in which region of the flow particles are
mainly distributed and where the majority of collisions
occur. To this aim particle concentration profile with re
spect to the wall normal direction has been calculated
and shown in Figure 2. The profile has been obtained as
time average of the number of particles residing in a spe
cific horizontal slab of the domain, nondimensionalised
with respect to the initial concentration of the same slab.
It is well known from the literature that particles tend
to collect at the wall depending on their inertia, due to
both a turbophoretic and a gravitational mechanism: the
more inertial the particle the faster the segregation. As
visible in Figure 2, particle concentration slightly devi
ates from the expected behaviour having a higher con
centration of small particles (St < 20) at the wall with
respect to larger ones. This is the result of two possible
mechanisms which prevent large particles from actually
reaching the wall: (i) high inertial particles are undergo
ing a high number of collisions at large distances from
the wall (see Figure 3) due to sling effects (Falkovic
and Pumir (2007)) and (ii) the majority of collisions oc
curring at the walls are highly energetic, so associated
to particle resuspension to the channel core. Medium
sized particles (St = 10.7) are experiencing multiple
collisions close to the channel walls and almost no col
lisions in the core of the channel. Their motion towards
the wall is, thus, almost undisturbed as well as for small
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
15000
10000
5000
Number of colhding particles varying with Z
St=0 80
St 1 60
St10 76
St=20
St=40
St=100
Figure 3: Number of collisions varying with respect to
the wall normal direction.
particles which are not colliding throughout the simu
lated time. This result is in agreement with the work
by Sundaram and Collins (1997) where it was suggested
that finitesize particles with negligible Stokes number
have either multiple low energetic collisions or they do
not collide at all.
A sketch of the two collision types is visible in Figure
4, where particles are coloured with their acceleration.
On the left a low energy collision occurring at the chan
nel wall between two St = 10.7 particles is presented. It
is possible to notice that particle acceleration is almost
unaffected by the collision event and the two particles
remain segregated at the wall. An opposite behavior is
presented in Figure 4b where a high energy collision
occurring for St = 100 particles is depicted. A particle
coming from the buffer layer is hitting one already seg
regated at the wall causing particle resuspension and a
consequent change in the particle accelerations as well
as in the particle turbulent kinetic energy. Confirmation
of this is given by the PDF of the relative velocity be
tween two colliding particles. St 10.7 and St 100
are taken as examples. As visible in Figure 5top, the
peak of the pdf can be found in correspondence to low
values of the relative velocity (about 1 in wall units),
suggesting collisions between particles already segre
gated at the wall. In Figure 5bottom the peak in the pdf
is shifted toward higher values of the relative velocity,
corresponding to more energetic collisions.
To quantify the effects of collisions on the overall par
ticle behaviour, the average profile of the particle tur
bulent kinetic energy varying with respect to the wall
normal direction has been computed. Results are pre
sented in Figure 6. Values are averaged in both space
(over the streamwise and spanwise directions) and time
(over the entire time span available for each simulation).
4
(a) (b)
Figure 4: Schematics of the two collision types. (a)
Low energy collision occurring at the chan
nel wall for St = 10.7 particles and (b) high
energetic collision associated with particle re
suspension for St = 10.7 particles.
For each Stokes number considered, the total turbulent
kinetic energy is found to be higher if collisions are
taken into account. Results for St=0.80 and 1.60 are not
shown since the two simulations with and without colli
sions are almost equal. For St=100 particles the profile
obtained in absence of collisions is higher at the channel
wall with respect to the one with collisions. This again
can be related to either the particle bouncing at the wall
or the energetic collisions occurring to these particles as
previously explained.
Conclusions
This paper addresses the issue of particle energy modi
fication in a developed turbulent boundary layer due to
particleparticle interactions under oneway assumption.
It is well known from the literature that particles tend to
segregate at the channel walls depending on their iner
tia. In this region the effect of the interaction between
particles is no longer negligible and the dilute flow as
sumption is locally no more valid.
To better understand the role of collision events on
particle behaviour, a Direct Numerical Simulation of a
horizontal channel laden with microparticles has been
performed. Particles with different inertia (St number
ranging from 0.8 to 100) have been injected into the flow
and simulations with and without collisions for each
Stokes number have been carried out. Particle concen
tration profile has shown that particles tend to collect
at the channel walls. However a deviation is found for
large particles which are still suspended into the flow at
the end of the simulation. Reasons for this can be ex
plained by two possible mechanisms: (i) particles are
undergoing several collisions at large distances from the
wall which slow down their motion toward the boundary
and (ii) the majority of particles is experiencing highly
energetic collisions which are responsible for particle
resuspension. Confirmation of this result has been ob
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Turbulent particle lnetic energy
St=10 76nocoll
St=10 76withcoll
PDF of the relative velocity between two colliding particles for St=10.
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5 10 15 20
Relative velocity between two particles [wu]
PDF of the relative velocity between two colliding particles for St=100
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 5 10 15 20
Relative velocity between two particles [wu]
Figure 5: Probability Density Function of the relative
velocity between two colliding particles for
St 10.7 (top) and St 100 (bottom)
trained by calcutating the Probability Density Function
of the relative velocity between two colliding particles.
Large particles exhibit larger relative velocities with re
spect to smaller ones which are undergoing several low
energy collisions. For all Stokes number considered in
the simulation, an overall increase in the particle turbu
lent kinetic energy was found. This was related to the
ability of a collision event to decorrelate particles from
the coherent eddies responsible for their accumulation.
After a collision, particles are spread in regions of higher
fluid velocity, thus experiencing an acceleration. Exep
tion is found for St = 100 particles in the vicinity of
the wall. This has been explained using the same two
mechanisms previously described.
1 1 10 100
z+
Turbulent particle lnetlc energy
St=20 no coll
.. St=20 with coll
,
Turbulent particle lnetic energy
St=40 no coll
St=40 with coil
50
30
20
10
01
Turbulent particle hnetc energy
St=100 no coll
St=100 with coil
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Figure 6: Particle turbulent kinetic energy. Lines refer
to the simulation without collisions, symbols
to the computations in presence of collisions.
'
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
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