Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 2.5.3 - A Simple Model for Particle Turbulence Interaction Effect in the PDF Kinetic Equation
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 Material Information
Title: 2.5.3 - A Simple Model for Particle Turbulence Interaction Effect in the PDF Kinetic Equation Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Aguinaga, S.
Simonin, O.
Borée, J.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: gas particle flow
probability density function
kinetic equation
BGK model
boundary condition
 Notes
Abstract: The direct resolution of the particle velocity Probability Density Function (PDF) equation is found to be a very powerful approach to predict the particle statistical properties in turbulent flow when the particle velocity distribution is far from equilibrium, as measured in dilute suspension of very inertial particles in non-homogeneous flow and in the near wall region with particle deposition or inelastic frictional bouncing on smooth and rough wall. But a difficult part in such an approach is the modeling and the numerical treatment of the term representing the interaction of the particle with the carrier fluid turbulence. To overcome this difficulty, the particle-turbulence effect in the PDF equation is approximated by using a BGK type model, called the SAB model, representing a relaxation effect of the particle velocity distribution towards an equilibrium velocity distribution (Gaussian or Grad’s expansion function) written in the frame of the local particle turbulence equilibrium assumption (Tchen’s assumption). The efficiency of the relaxation term is assumed to be proportional to the inverse of the particle relaxation time. In a similar manner than the BGK model in kinetic theory of dilute gases, the SAB model is consistent with the number density, mean velocity and kinetic stress transport equations and the discrepancies arise when deriving the triple velocity correlation equations. The paper presents first the model and then its application to particles suspended in a wall bounded homogenous turbulence as described by Swailes and Reeks (1994). The simulation results are in good accordance with the theory and the data from literature. In case of turbulent channel flow, the SAB model seems be able to model the beginning of the transition between the inertial particle deposition regime to the diffusion regime.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


A Simple Model for Particle Turbulence Interaction Effect in the PDF Kinetic
Equation


S. Aguinaga and 0. Simonint I and J. Bor6e&

Centre Scientifique et Technique du Batiment (C.S.T.B.), Universit6 Paris-Est, 11 rue Picherit 44300 Nantes, France
t University de Toulouse; INPT, UPS; IMFT; All6e Camille Soula, 31400 Toulouse, France
SCNRS; Institut de Mecanique des Fluides de Toulouse; 31400 Toulouse, France
Laboratoire d'Etudes A6rodynamiques (L.E.A.); ENSMA; Teleport 2, 86960 Chasseneuil Futuroscope, France
sylvain.aguinaga@cstb.fr and simonin@imft.fr and jacques.boree@lea.ensma.fr
Keywords: Gas particle flow, Probability Density Function, Kinetic equation, BGK model, Boundary condition




Abstract

The direct resolution of the particle velocity Probability Density Function (PDF) equation is found to be a very
powerful approach to predict the particle statistical properties in turbulent flow when the particle velocity distribution
is far from equilibrium, as measured in dilute suspension of very inertial particles in non-homogeneous flow and
in the near wall region with particle deposition or inelastic frictional bouncing on smooth and rough wall. But a
difficult part in such an approach is the modeling and the numerical treatment of the term representing the interaction
of the particle with the carrier fluid turbulence. To overcome this difficulty, the particle-turbulence effect in the PDF
equation is approximated by using a BGK type model, called the SAB model, representing a relaxation effect of the
particle velocity distribution towards an equilibrium velocity distribution (Gaussian or Grad's expansion function)
written in the frame of the local particle turbulence equilibrium assumption (Tchen's assumption). The efficiency of
the relaxation term is assumed to be proportional to the inverse of the particle relaxation time. In a similar manner
than the BGK model in kinetic theory of dilute gases, the SAB model is consistent with the number density, mean
velocity and kinetic stress transport equations and the discrepancies arise when deriving the triple velocity correlation
equations. The paper presents first the model and then its application to particles suspended in a wall bounded
homogenous turbulence as described by Swailes and Reeks (1994). The simulation results are in good accordance
with the theory and the data from literature. In case of turbulent channel flow, the SAB model seems be able to model
the beginning of the transition between the inertial particle deposition regime to the diffusion regime.


Introduction


Particles dispersion in turbulent flows has been studied
since a long time as it has many industrial and environ-
mental applications. Therefore such two-phase flows are
difficult to model as particles partially response to tur-
bulence. The macroscale Stokes number St -= may
be used to evaluate the particle response to turbulence,
where Tp pdP is the particle relaxation time and Tf
Pj 1ivy
a fluid turbulence characteristic time. If St > > 1 parti-
cles are very inertial with respect to the local fluid flow
unsteadyness and do not react to the energetic turbulent
eddies. On the contrary, if St << 1 particles are ex-
pected to follow the smallest turbulent structures and act
like tracers.


In the near wall region, turbulent properties are evolv-
ing rapidly with the distance to the wall. So does the
Stokes number and particles partially respond to the tur-
bulence of the boundary layer. The particle-turbulence
interaction within the near wall region is found to have
a major impact on the deposition process and remains
difficult to model accurately with practical approaches.
This problem has been extensively studied, especially
in the academic case of the 2D channel flow ( Kallio
and Reeks (1989), Marchioli et al. (2007)). This case
is particularly well described in the literature with a lot
of experimental databases, the most famous is the Liu &
Agarwal's one (Liu and Agarwal (1974)).
In a channel flow, three deposition regimes are gen-
erally observed according to the dimensionless particle
relaxation time T* = .2 with u* is the dimensionless
Vr











fluid characteristic velocity of the boundary layer:

1. T* << 1: diffusion regime, corresponding to low
inertia particles which respond to the whole bound-
ary layer turbulence and cross the laminar sub layer
thanks to a Brownian diffusion process.

2. >> 40: inertial regime, corresponding to high
inertia particles which are ejected from the bulk
flow and do not interact with the boundary layer
turbulence. This mechanism is similar to the free
flight process described by Friedlander and John-
stone (1957)

3. 1 < p* < 40: diffusion-impaction regime, corre-
sponding to particles which partially respond to the
boundary layer turbulence. This regime is the most
difficult to model accurately.

Particle laden channel flows have been particularly
studied using Lagrangian approach for the solid phase
coupled with Direct Numerical Simulation (DNS) ap-
proach for the fluid phase (Kallio and Reeks (1989),
Botto et al. (2005),Narayanan et al. (2003),Marchioli
et al. (2007)). This simulations allowed to improve
the understanding of the particle behavior in the bound-
ary layer and to evaluate the influence of other forces
than linear drag on the deposition process. This ap-
proach is very accurate, but calculations time are pro-
hibitive. Lagrangian simulations using Reynolds Av-
erage Navier Stokes (RANS) approach have also been
considered (Matida et al. (2000)). Those models require
stochastic approach in order to recreate the turbulent dis-
persion from the mean value of the fluid turbulent kinetic
energy and dissipation rate. Unfortunately, RANS mod-
els generally offer a poor description of the near wall
region and the prediction of the deposition rate is gener-
ally inaccurate( Aguinaga et al. (2009)).
Other authors have chosen the Euler/Euler approach
to model particles dispersion and deposition in the near
wall region. Those models suit particularly the diffusion
regime. Authors generally refer to Young and Leeming
(1997) model which is also supposed to deal with inertial
particles. But Reeks (2005) has shown that this kind of
model, referred as ADE (Advection Diffusion Equation)
models, suffer from a lack of consistency and may not
be suitable for inhomogeneous turbulence. Reeks espe-
cially highlights that the equations for mass, momentum
and kinetic stresses are uncoupled, thanks to some ques-
tionable simplifications. More, those models are gen-
erally based on ensemble average rather than particles
weighted average. Swailes and Reeks (1994), and more
recently, Van Dijk and Swailes (2007), proposed an in-
teresting alternative with a model based on the Particles
Density Function (PDF) approach.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Let us introduce fp(cp,x,t), the particle velocity
PDF, where fp(cp, x, t)dxdcp is the mean number of
particles, whose centre of mass at time t is located in
the volume [x, x + dx[ with a translation velocity up E
[cp, Cp + dcp[.
The mean number density of particles np (x, t), at po-
sition x and time t, can be obtained by integration of
fp over all the possible particle velocities c, (integration
over the velocity space):


fp(cp,,x,t) [ dcpi (1)
i=1,3


More, the dispersed phase average of any function y
may be obtained by integration over the particle velocity
space following:

np()p =/ (cp) f (cp)dcp (2)

With the appropriate function y, one can compute di-
rectly from fp the mean velocity (first order moment)
and kinetic stresses (second order moments) as:


Up, ([p, = )p
Rp,ij =[p,i I1.. - Up,j])p


The evolution of fp is governed by the kinetic trans-
port equation which can be written as follow in the case
of inertial particles without collision (Reeks (1991)):


at i [0] 0 [gt f

a) 1 aP, ,icp)fp ]
0 1 ^'^


0 (4)


(uf |cp) is the ensemble averaged fluid velocity "seen"
by particles with a given velocity up cp. The (a) term
of equation (4) represents the fluid-particle coupling via
the drag force.
The main closure problem of the PDF approach is the
modeling of the conditional average of the fluid turbu-
lent velocity "seen" by the particles, see for example
Reeks (1991) or Derevich and Zaichik (1988). Then the
transport equations for any velocity moments (number
density, mean velocity or kinetic stresses) may be de-
rived by integration on the velocity space of the PDF
kinetic equation (4). Transport properties, such as vis-
cosity or diffusivity, can also be derived from the same
equations by using classical approaches of the kinetic
theory of dilute gases such as Chapmann Enskog or
Grad's approximations.


00 00 0
n P(Xft) f f











Swailes and Reeks (1994) shown that the PDF ap-
proach based on the full computation of the kinetic trans-
port equation may be very effective for the prediction
of the deposition of high inertia particles. More re-
cently, Van Dijk and Swailes (2007) have also proposed
a PDF approach coupling the moment method, for the
bulk flow, with the PDF kinetic equation, in the near-
wall region. This hybrid approach allows an improved
accuracy and a faster computational time to solve the
dispersion and deposition within the boundary layer.
The models presented by Swailes and Reeks (1994)
and Van Dijk and Swailes (2007) were very interesting
and innovating as they directly solve the evolution of the
PDF rather than the moment equations obtained by in-
tegration of the PDF kinetic equation. However the nu-
merical approach proposed by Swailes and Reeks (1994)
is quite difficult to implement in inhomogeneous flows
and complex geometries.
The main goal of this paper is to propose and evaluate
an approximate PDF closure model for the fluid-particle
turbulence interaction term, called the "SAB model".

The SAB Model

Main closure assumptions
Let us write the fluid-particle interaction term of equa-
tion (4) in two terms :


9- [1(cp,i (uf,i|Cp))fP =
80p,i -p
Up,i Uf p,i Df p A (5)

(a) (b)

where (a) is a first contribution which accounts for
the particle interaction with the mean fluid flow and (b)
is a second contribution which represents the particle in-
teraction with the fluid turbulence.
Ufup is the mean fluid velocity "seen" by the particles
given by

npUfop j (cf, cp)fp(cp)dCp (6)

fiq is a given PDF corresponding to particles "in-
equilibrium" with the local fluid turbulence according to
the Tchen's theory. So (b) represents the turbulence ef-
fect on the particles as a relaxation of the velocity distri-
bution towards a Tchen's equilibrium PDF with a char-
acteristic time Tp. This modeling approach is inspired
from the so-called "BGK model" developed in the frame
of kinetic theory of dilute gases to model the relax-
ation towards Maxwell equilibrium distribution by inter-
particle collisions. The coefficient A is taken equal to 2


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


in order to insure that the second order moment (Rp,ij)
transport equations derived from equation (4) with the
fluid-particle interaction model given by (5) are identi-
cal to the one derived with the exact term.
fq represents the velocity PDF of particles which
are in equilibrium with the fluid turbulence in the frame
of the Tchen's theory, so the corresponding moments
should obey the following equations,


[ np
.. Ueq
I p,i
\ R'"1
I p,a 3p~


- np(Rfp,ij + Rfp,ji)/2


where Rfp,ij is the fluid-particle velocity correlation
given by,


npRfp,ij i Up,](uy,lcp) fp(cp)dcp (8)

So, following Pialat et al. (2007a) feq may be writ-
ten as an elliptic Gaussian distribution centered on the
particle mean velocity (c* cp Up),

f np ---exp[1(9)
feq 8ex3det(R -- c (R ') .cio] (9)
8w3det (R ')

Finally, the SAB modeling approach is using alge-
braic equations to compute the fluid-particle velocity
correlations Uf p and Rfp, j. Following, Simonin et al.
(1993), the mean fluid velocity "seen" by the particles
may be written


Ufp = Uf + Vd


where Vd is the fluid-particle turbulent drift velocity
which represents the correlation between the turbulent
velocity and the particle distribution. In the case of par-
ticles suspended in homogeneous isotropic turbulence,
the fluid-particle turbulent drift velocity may be given
in terms of the particle number density using a gradient
approximation and is written,

,t fp 1 9n
Vd,i -7f@p p 1 (11)
3 np, Dxi

where 7 p is the eddy-particle interaction time and
qfp Rfp,i is the fluid-particle velocity covariance. In
addition, the fluid-particle velocity correlation is written
in terms of the fluid turbulent kinetic energy q ,


-
f 2 Tf p
ffp 2q
TiOp TP


Wall boundary conditions











An interesting advantage of the full PDF approach,
compared to the moment approach, is its ability to pre-
scribe complex boundary conditions. Indeed, using only
the three first order moments n,, Up and Rp, the bound-
ary condition derivation is significantly restricted to sim-
ple cases, generally using an hi\p.lIhsis, of an incident
Gaussian distribution of particle velocities (Sakiz and
Simonin (1999)). While, the full PDF approach allows
to prescribe very easily more complex boundary con-
ditions as deposition or rebound on rough wall (Konan
et al. (2009))
In order to prescribe boundary conditions, fp will be
divided in two PDF f+ and fp following:


{fP+ (Cs)
fP+ (cs)


fp (Cs) 0 fp(cp)


0 for cp e] oc, 0[
fp(cp) for cp E]0, oc[
and


for cp e] oo, 0[
for cp E]0, oo[


and

Vcp fp(cp) fp (cp)+ fp (cp) (15)
This decomposition illustrated figure 1 will also be
used for the numerical resolution of fp.


Figure 1: fp decomposition in two PDF f+ et fp

The model is monodimensionnal but will be ap-
plied in the two-dimensional finite phase-space cp E
[cpmin, cpmax] and y E [y = YH]. The model re-
quires the following boundary conditions:

1. at the upper bound of the domain (y = YH),
f (cp, YH) is entirely imposed (by a prescribed
law or by an external model as the coupling method
presented by Van Dijk and Swailes (2007) or Pialat
et al. (2007b))

2. at the wall (y -= ), fp+(c, j) is entirely im-
posed

f+ (cp, YH) and fp (cp, dP) are obtained with the cal-
culation of the PDF in the phase space domain. The pre-
scription of f+ (cp, L~) depends of the type of boundary
condition modeled:


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


1. for a perfect absorption condition, no particles are
reemitted:

f(,y ) 0

2. for an elastic rebound without energy loss

fp+(C, Y = f(-CpY )

3. for an elastic rebound with energy loss (where e,
is the coefficient of restitution, Sakiz and Simonin
(1999)):


fp(cp, Y=-


1
~ fp (-ewCp,Y


More complex boundary conditions can also be ap-
plied as partially absorbing wall, only the corresponding
law for f+ (cp, y ) has to be specified.
Numerical implementation
This model was implemented in Scilab which allows
a graphical representation of the PDF during the com-
putation. The decomposition of fp using fp(cp)
f (cp) + fp (c) is used. The PDF is calculated in two
times. First fp is calculated with an upwind scheme
starting from the upper bound at y YH till the wall at
y j. In the case of particles rebounding to the wall
fp (y c p) is calculated thanks to f (y p Cp)
evaluated at the current time step and a appropriate re-
bound model. Then fp is resolved from the wall to YH
with an upwind scheme. The resolution method is im-
plicit using a relaxation loop.
This model has many applications. The present paper
will focus on two of them:

1. particles deposition in homogeneous turbulent flow
for high inertia particles. Results obtained by
Swailes & Reeks will be used to asses the SAB
model in that particular case.
2. particles deposition in a boundary layer turbulence.
The experimental database of Liu & Agarwal will
be used to assess the model.

Application to particle deposition in
homogeneous turbulence

The PDF approach is applied to the monodimensional
problem of particles dispersion in the near wall region.
y is the wall normal coordinate with y 0 at the wall. cp
is the wall normal particles velocity, uf the wall normal
fluid velocity. Although the physical space is monodi-
mensionnal, the PFD is defined on the two-dimensional
phase space (y, Cp). The evolution of fp is ruled by the







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Boltzmann equation and can be written in this simplified
case (SimoninSimonin (2000)):
Swailes & Reeks have performed a lot of calcula-
tions in this case using particularly high inertial parti-
cles. Those particles are expected to have a low interac-
tion with the fluid turbulence. They injected the particles
at various distances from the wall

YH NL L

calculated using L the mean free path of particles, with:

L = T) (16)

TL is the Lagrangian integral time scale calculated
according to the Tchen theory:

U/ 2f
TL = f (17)
P -P(2)

Finally in the frame of the Tchen theory, the mean free
path can be written as:

L = Tfp Ku2) f Tf1 p) (18)

Swailes & Reeks prescribed Tf p = 1 and Tp 101.
The particles Stokes number St - is then equal
TfSp
to 101. So, the particles are expected to be very inertial.
The other fluid and particles properties are deduced from
Tf p and Tp.

Tp 100
Tf p 1
Uf 0


(I)


(u'U'),f = 101

LU) 101
L= 101


fp is prescribed at y yH assuming a Gaussian dis-
tribution of particles in equilibrium with the turbulence:


for c E] oc, 0[


the wall (YH = 10L). The gravity is defined using the
Froude number Fr:

Fr= P == 5


The PDF prescribed at y YH is centered on the
settling velocity Tpg. With such inertial particles, all the
PDF fp(yH, Cp) is defined on the negative part c, < 0.
Two calculations are done with two different bound-
ary conditions:

1. perfect absorption: f+ (y d, C) 0

2. elastic rebound without energy loss: fp(y =

d2 ,cp) fp (y 2, -cp)
fp is set to 0 at t 0 Figure 2 represents the evolu-
tion of fp at four different times.


ABSORPTION


REBOUND


t = T 011


- y
L


02r


-10
6ieL


02,


t =2rp 0.1- 0




02 0

t = 3r .

y I ,., ,- y :
0 0
3 r p

0.2

t = 4- "


(! (p'


1(u ')(H)
j27rat'pU'f (yH')


exp -(cp -Tg)2
2(u'U'f)p(yH)


for c e]0O, oc[
fp (YH, p) 0 (20)
(21)

Injection with gravity
Figure 2 presents two calculations made with gravity.
The injection is set at NL 10 mean free paths from


Figure 2: Time evolution of the PDF with two different
boundary conditions


In the perfect absorption case the PDF shape is not
influenced by the turbulence. Particles are absorbed at
the wall and fp = 0 in all the domain, i.e. no particles
are going away from the wall. In the elastic rebound
case, the shape of fp is becoming more complex. Parti-
cles reach the wall at t 2p. One can see that f+ is no
longer equal to 0. The negative velocities are reflected in


f, (yH, Cp)







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


the positive part. At t 3p the reflected positive PDF is
even more visible. Close to the wall, the PDF as a com-
plex shape with an important population of particle with
low normal velocities. This PDF describes particles do-
ing successive rebounds at the wall, losing velocity due
to drag force and gravity as they go away from the wall.
Finally, at t 4T7, one can see the important accumu-
lation of particles with a low velocity. This behavior is
typically Lagrangian, it is quite interesting that it is so
well described using the PDF approach which remains
an Eulerian approach.
Comparison to Swailes & Reeks results
Swailes & Reeks evaluated their model in homoge-
neous turbulence, with perfect absorption at the wall
and without gravity. They made simulations with dif-
ferent injection distances YH NL L from the wall.
In order to compare their results to those obtained by
SAB, the following distances were considered: NL =
0.1, 0.5,1, 2, 5, 10, 20, 30 and 100. For each case, the
calculation is run until the convergence is reached.
Figure 3 represents the PDF obtained for NL 0.1, 1
and 30. For NL 0.1 the PDF shape is merely af-
fected by the turbulence and remains constant till the
wall. For NL 1 the shape of the PDF is close to a
Gaussian at the top of the domain. It is reminded that
f+(y = YH) is obtained via the calculation and is not
prescribed. f, (y YH) corresponds to a population of
particle which has been driven by the turbulence away
from the wall. For NL 30 the shape of the PDF at the
upper bound of the domain is Gaussian. One can see that
the particles concentration is rapidly decreasing. Parti-
cles are driven by turbulence away from the wall, and
finally few particles are then able to reach the near wall
region to deposit.
Swailes & Reeks calculated the concentration mean
velocity and kinetic stresses profiles for all their simu-
lations. Those moments are obtained by an integration
over the property space of particles (equation 3). Graph-
ically, for a given distance y, the mean concentration
corresponds to the area under the PDF, the mean veloc-
ity to the center of the PDF, and the kinetic stresses to
its width. Swailes & Reeks compared the results to ran-
dom walk Lagrangian simulations made with the same
parameters. They also reported the deposit flux J:

f0
J / cpjfpdcp (22)

It is made dimensionless using the bulk concentration:
It is made dimensionless using the bulk concentration:


Ic Cp fp dcp
kd = =
S0 fdc d .00

Y y=0 1 cp-oo


N=0.1



A c 10

C P 3 6U

N=1 7--~


0.04
0.01
0.02
0.0

C 6


L100
: y


N=30

0.04
0.02
0.02 :',00
0.0 1 -',:,:,I

u 3 6

Figure 3: PDF obtained by SAB with NL = 0.1, 1 and
30.


Figure 4 compares the results extracted from Swailes
& Reeks to those obtained by SAB. Swailes & Reeks
results are compared to Lagrangian random-walk simu-
lations results. Results obtained by SAB are very similar
for the mean velocity and kinetic stresses. Profiles con-
verge to a single curve as NL augments. Close to wall,
the mean velocity and the kinetic stresses obtained by
SAB are smaller than those obtained by the Swailes &
Reeks model. However they seemed to be in a better
accordance to their random walk simulations.
Swailes & Reeks showed that the concentration pro-
file converge to a linear profile as NL increases. More,
for each profile, np 0.5 at y NL L. Concen-
tration profiles obtained by SAB are in accordance for
NL < 10, but do not converge toward a linear profile for
NL > 10.
Swailes & Reeks compared the deposition rates ob-
tained with their PDF model to random walk simula-
tions. Performances of the PDF model (referred as the
'kinetic' model in the figure) are better than free-flight
or gradient diffusion based models. The results obtained
by SAB are in good agreement with Swailes & Reeks
results, even for NL > 10 although the concentration
profiles are different.
To conclude, the predictions obtained by SAB are
very close to those obtained by Swailes & Reeks in the








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


simple case of homogeneous turbulence for high inertia
particles.


Swailes & Reeks



"I I] ) ......

/U-

Swo-
,- 4




00 lo-- - ---r- lO
r a-i









-- .------




:10 10~ 16 20 25 30


For y < 5

T p (7.122 + 0.5731 (5) 0.001290 (5)2) V'
For ) > 5
For y+ > 5


TfUp (7.122 + 0.5731 (y*)


-0.001290. (y*)2) Vf
(U*)2


Those profiles are plotted figure 5.

8-4


.0 1 . . .. .. . . .
Y/Y,





o .. --.... . ...


Figure 4: Comparisons of results obtained by SAB and
Swailes & Reeks' model. Top: particles mean
velocity and kinetic stresses profiles. Contin-
uous lines in Swailes & Reeks results corre-
sponds to random walk simulation. Middle:
particles concentration profiles. Bottom: Di-
mensionless deposition rate.


Application to particle deposition in boundary
layer turbulence


The model is now applied to the boundary layer turbu-
lence case. The only difference with the previous ap-
plication comes from the fluid profile prescribed. Cor-
relations used by Kallio and Reeks (1989) in a two di-
mensional channel flow Lagrangian simulation are pre-
scribed:



( 0.005. (5)2 2
(u, )f 1 +0.002923 (5)2218)
f f* 1 0.005(y*)2 2 or

O 1+0.002923(y*)2.21 for


a,,
08 / *ul I ulI
/
04 /
,,
0 1 _ -ii . .
o0 -' --7 ---------
D 10 20 30 40 50 .6 0 0 90 100
y


Figure 5: FLuid correlations extracted from Kallio and
Reeks (1989). Left (u'u')f and (u''),p.
Right TfJp.


(u' )p is calculated using the Tchen theory follow-
ing (u'u')p (u'u')f- T-P and is also reported
figure 5.
In order to assess the model in this configuration,
the experimental database of Liu and Agarwal (1974)
is used. The model is evaluated for various 7" obtained
using different couples of dp and u*. Thus, a wide range
of p* is investigated. The different configurations are
reported table 1:


Table 1: List of configuration used in the boundary layer
simulations
+<5

Particles are injected at y YH with y* -Y
+> 00. fP is prescribed assuming a Gaussian distribution
in equilibrium with the fluid.


Tf@p.-
3e4
2.-4
0 10 20 30 40 50 6 7010 8 90 100 110
y


u*(m/s)
1 1.5 2
d,(/mum) *T
5. 6. 12. 22.
10. 22. 48. 85.
12. 31. 69. 122
15. 48. 107. 190.
16. 55. 122. 217.
20. 85. 190. 338.
30. 190. 428. 760.
40. 338. 760. 1351.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


For c E] oc, 0[


exp -( J ))
exp -(A
2(u'U'f),(yH)


For cp e]0, oo[
fp (YH,Cp) 0 (26)
The gravity is not taken in account at the moment. The
simulations are conducted until the PDF has converged
with the following criterion:
Y=YH Y=YH

o0 f)dcpdy < Econ ,= (f)2 dcpdy
(27)
with fp the value of the PDF at the previous time
step. This criterion is severe as conv 10 8. Fig-
ure 6 presents four PDF calculated for = 2, 30, 47
and 190 (corresponding to u* I1m/s and respectively
dp = 3pm, 12pm, 15pm and 30pm).
Figure 7 to 9 represents the evolution of the three first
moments np, Up and (u') against and T. Those
moments are extracted integrating the PDF fp over cp.
Four profiles, corresponding to the PDF plotted figure 6
are also presented.
The concentration profile linearly decreases toward
the wall for high inertia particles. An important particles
accumulation appears close to the wall for < 50 It
has already been spotted in the literature and is referred
as the "build-up". It corresponds to particles which don't
have enough inertia to cross the viscous layer turbulence,
but which are too inertial to be driven by the smaller
structures of the viscous layer toward the wall. This
build-up is diminishing for 7* < 5, this phenomenon is
also reported in the literature, it corresponds to particles
with very small inertia which can be now transported to
the wall by a diffusion mechanism.
The kinetic stresses of high inertia particles are rel-
atively constant as their interaction with the fluid turbu-
lence is low. On the contrary, (u'p) profile for low inertia
particles is very close to fluid one.
The particles mean velocity is maximum at the tran-
sition between the inertial regime and the diffusion-
impaction regime, actually where the deposition is ex-
pected to be the most important. Up is increasing from
the top of domain till y* z 12. There Up is decreasing
rapidly to the wall. This behavior is characteristic of par-
ticles which have just sufficient inertia to cross the vis-
cous layer turbulence. High inertia particles are merely
affected by the viscous layer turbulence and their mean
velocity is still increasing as they cross the viscous layer.
Finally, figure 10 represents the PDF at the wall (y
2-) and at the top of the domain (y YH). It is re-


=5
Jp '


2.5




190


2.5
1.5
Cp


Figure 6: PDF calculated by SAB. u* = m/s.


f, (YH, p)


1
27r(u'4U'1p(yHH)








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


0


10 20 30 40 50 0 70 80 90 100 110
y


Figure 7: Top: np against T' and y*. Bottom: Profiles
of np calculated using the PDF plotted figure
6.


Figure 8: Top: Up against of Up calculated using the PDF plotted figure
6.


2.2
1.8
1.4
n
P 1.0
0.6
0.2
0



2.0


1.5


p 1.0


-0.05

-0.10

U -0.15-

-0 20
U
-0 25







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


minded that fp it is not prescribed at y = yH and is
obtained via the calculation. One can see that f+ / 0
at the wall although the boundary condition is a perfect
absorption. Actually, the PDF is calculated in a cell cor-
responding to y E [t, + Ay[,Ay is the width of the
cell. So there may be some particles in that cell which
are driven away for the wall by the turbulence before
they deposit. A PDF in the positive part can be then ex-
pected, especially for low inertia particles. For low iner-
tia particles, the PDF is close to a Gaussian distribution.
The more inertial the particles are, the less Gaussian the
PDF shape is. At y = yH all the PDF are close to a
Gaussian distribution, except for high inertia particles.
This lack of symmetry is induced by the mass conserva-
tion flux of particles which deposit at the wall.


0.9
0.12
U 0.6 tau_p'=5

oo o03.o- tau_p'=30
0. 0
u 00.0



tau_p'=5a -
0.9 taup=30 3 -2 -1 1 2
S0.8 taup047 au-p'=47



0.0 00 0.20*

0.3 .15
-2 --- -0 .02




0.2 0.10
0.1 0.05

0 10 20 30 40 50 60 70 80 90 100 110 0"00 . -. .
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Y c

Figure 9: Top: (4up against T7 and y*. Bottom: Figure 10: Left: PDF at the wall, right PDF at the top of
Profiles of (upu), calculated using the PDF the domain.
plotted figure 6.

Figure 11 plots the particle deposition flux at the wall.
It also presents results obtained by Swailes & Reeks
model. Both are compared to measurement done by Liu
and Agarwal (1974). Swailes & Reeks demonstrated
that their model was accurate in the inertial regime, but
failed to reproduce the diffusion-impaction regime. Fig-
ure 11 shows that the SAB model reproduces well the
inertial regime. One can see the so called "roll-off', i.e.
the deposition rate decreases with increasing inertia of
particles (for <7 > 100). Those particles disengage from
the large structures of the buffer layer and cannot be cen-
trifuged toward the wall. The interesting point is that
the SAB model also reproduces the transition toward the












diffusion-impaction regime for 7* 40. Results are in
accordance with Liu & Agarwal database for 7* > 8.
For T* < 8 the deposition rate predicted by SAB is
too important. This problem highlights a first limita-
tion of the model. Additional researches as to be con-
ducted in order to improve the performances for low in-
ertia particles. However it is encouraging that the model
is able to reproduce the transition toward the diffusion
impaction regime, which wasn't possible for Swailes &
Reeks model. It is believed that the bad performances of
the SAB model for low inertia particles may come from
the numerical implementation of the model, rather than
the model itself.

Swailes & Reeks


oop'PB aoo-
0
00
0


o paint
hiutk ood


10' 10 10' 10' 10 10'
rS
SAB


1 0 i 2
10 10 10


Figure 11: Deposition rate predictions. Left: Swailes
& Reeks model, Right: SAB model. Each
one is compared to Liu & Agarwal's mea-
surements (dots).

Even if the model presents a lack of accuracy for low
inertia particle regarding the deposition rate, it allows
to reproduce interesting features of the particles dynam-
ics in the boundary layer, especially at the transition be-
tween the inertial and diffusion-impaction regime. The
analysis of the PDF at different wall distances is very
interesting and allows to formulate additional conclu-
sions. More, the PDF model is computational cost ef-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


fective compared to Lagrangian simulations.


Conclusions & Perspectives

A new PDF model has been presented. It allows the
particles wall normal velocity PDF fp calculation in a
monodimensional problem. The model is based on the
Boltzmann equation. The fluid-particles interaction term
has been simplified using a closure inspired by the so-
called 'BGK' models. The model has been assessed
in homogeneous turbulence, and its results compared to
Swailes & Reeks PDF calculations. Results are in good
agreement excepted for the concentration profile in some
limited configurations. The model has also been used to
reproduce particles dispersion in a turbulent boundary
layer. It reproduces interesting features of particles dy-
namics which has been spotted in the literature. The re-
production of the deposition rate was good in the inertial
regime. The interesting point is that the model is able
to reproduce the transition between the inertial regime
and the diffusion-impaction regime. The evolution of
the PDF close to that transition, at r* 40, is very
interesting, and is well illustrated by the model. Unfor-
tunately the model presents limitations for low inertia
particles with "* < 8. The numerical implementation
of the model may be improved to increase the perfor-
mances with such low inertia particles.
The model can also been improved including a bet-
ter model for the turbulent fluid-particles drift velocity
Udrift accounting for inhomogeneous turbulence (Si-
monin et al. (1993)). The Brownian effect for the dif-
fusion regime could also been introduced, but it will in-
troduce second order derivatives on c, which will con-
siderably increase the complexity of the numerical im-
plantation.
Additional simulations have been carried out and are
not presented in this paper. The purpose of those simu-
lations was to analyze the evolution of the PDF and the
deposition rate when the PDF fp prescribed at the top
of the domain was out of equilibrium. Interesting results
have been found. Some are close to observations made
by Mito and Hanratty (2006). This study may be pre-
sented in an another paper. The effect of the gravity on
the deposition has also been studied, and coupled to the
prescription of a PDF out of equilibrium at the top of the
domain.
This model as two main objectives. First, it is a
didactic tool, computational time efficient, in order to
understand complex behavior for particles dispersion
thanks to the PDF tool. Second, it may be useful to de-
velop additional boundary conditions for Eulerian and
Lagrangian simulations. Indeed, the SAB model can be
used to calculate transfer function between the PDF at
the top of the domain and the PDF at the wall, especially


oo
o

A, A Llu & Agarwal
o SAB











when the PDF at the top of the domain in out equilib-
rium, which is generally observed in complex flow.
Finally, considering the very interesting work pre-
sented by Van Dijk and Swailes (2007), coupling of the
PDF model with other models can also be envisaged.
The PDF models constitute a very interesting alternative
to classical Lagrangian or Eulerian method for academic
research problems.

Acknowledgements

This research was supported by PSA Peugeot Citroen
in the context of PSA-CNRS research project "Droplets
dispersion an deposition: experiments and numerical
modeling".

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