Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: P1.74 - Lagrangian Stochastic modeling of gas-solid flows with two-way coupling
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 Material Information
Title: P1.74 - Lagrangian Stochastic modeling of gas-solid flows with two-way coupling Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Zeren, Z.
Bédat, B.
Simonin, O.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Lagrangian stochastic model
Langevin equation
two-way coupling
 Notes
Abstract: Two Langevin equations are proposed to model the particle-laden flows where the carrier phase’s structure is modified by the solid uniform property particles. The first equation is the one taking into account the changes in the fluid velocity along the fluid element trajectories and the second equation takes into account the fluid velocity along the particle trajectories. The equations are the same as the ones proposed by Simonin et al. Simonin, O., Deutsch, E. and Minier, J.P., Applied Scientific Research, vol 51, p275-283, 1993 except an additional term to take into account the two-way coupling effect. The additional term is modeled in terms of mean drag from the mean transport equations of both phases. The model is compared to the DNS simulations corresponding to homogeneous isotropic turbulence kept stationary with stochastic forcing. Results showed that the mean drag term provides results in good agreement with the DNS results implying that the mean drag model takes into account the particle dispersion using the fluid statistics seen by the particles. However, the fluid statistics seen by the fluid elements are sensibly different than the fluid statistics seen by the particles which requires further consideration for adequate physical modeling.
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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Resource Identifier: P174-Zeren-ICMF2010.pdf

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


Lagrangian Stochastic modeling of gas-solid flows with two-way coupling


Zafer Zeren~t, Beno~it B~dat~t and Olivier Simonin~t

University de Toulouse; INPT, UPS; IMFT; 31400 Toulouse, France

t CNRS; Institut de Mecanique des Fluides de Toulouse; 31400 Toulouse, France
zeren~imft.fr, bedat~imft.fr and simonin~imft.fr

Keywords: Lagrangian stochastic model, Langevin equation, Two-way coupling




Abstract

Two Langevin equations are proposed to model the particle-laden flows where the carrier phase's structure is modified
by the solid uniform property particles. The first equation is the one taking into account the changes in the fluid
velocity along the fluid element trajectories and the second equation takes into account the fluid velocity along the
particle trajectories. The equations are the same as the ones proposed by Simonin et al. [Simonin, O., Deutsch,
E. and Minier, J.P., Applied Scientific Research, vol 51, p275-283, 1993] except an additional term to take into
account the two-way coupling effect. The additional term is modeled in terms of mean drag from the mean transport
equations of both phases. The model is compared to the DNS simulations corresponding to homogeneous isotropic
turbulence kept stationary with stochastic forcing. Results showed that the mean drag term provides results in good
agreement with the DNS results implying that the mean drag model takes into account the panticle dispersion using
the fluid statistics seen by the particles. However, the fluid statistics seen by the fluid elements are sensibly dif-
ferent than the fluid statistics seen by the particles which requires further consideration for adequate physical modeling.


Nomenclatu re


Introduction


Two-phase flows are important in terms of many indus-
trial and natural processes. Among many different pos-
sibilities of two different flowing matters moving cor-
related to each other, solid particles in a turbulent gas
flow receive much attention from the academic and in-
dustrial partners. Different physical phenomena occur
during these complex motions related to a particular flow
considered. Interactions between the particles, interac-
tions between the fluid and the particles form diversity
of problems that can be encountered.
In comparison to the other physical mechanisms, the
two-way phase interactions are relatively less under-
stood. Interactions between the fluctuating velocities
of the phases, especially in presence of a mean motion,
are very complex in terms of "what scale of particles'
collective motion effects what scale (or scales) of the
fluid turbulence", that the momentum transfer between
the phases, probably, does not depend on specific parti-
cle parameters but depends on other parameters such as
initial conditions of turbulent field (Simonin and Squires
2003).
This scale way of thinking in turn calls for spectral
analysis that has been a widespread method of explain-


Roman symbols
fi Force acting on fluid due to particles
Fi Turbulence forcing
q2 Fluid kinetic energy
ql Fluid kinetic energy seen
q" Particle kinetic energy
qf Fluid-particle covariance
uf< Instantaneous local fluid velocity
SStochastic particle velocity
Uf~ Mean fluid velocity
Instantaneous fluid velocity seen
SInstantaneous particle velocity
UMean particle velocity
Vd1 Fluid-particle turbulent drift velocity


Greek



symbols
Particle volumetric loading
Particle mass loading
Turbulent dissipation rate
Two-way coupling term of Langevin model







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


source approximation to take into account the effect of
the particles on the fluid turbulence. Turbulent flow
is homogeneous and isotropic kept stationary with the
stochastic algorithm of Eswaran and Pope (1988). A pri-
ori tests showed that the modeling the two-way coupling
term using the mean drag force produces accounts accu-
rately for the particle dispersion modeled using the fluid
statistics seen by the particle phase.


Direct numerical simulation (DNS) and
Lagrangian particle tracking (DPS)

Turbulence simulation and characterization of the
flow. Precise computation of the particle trajectories de-
pends on the accurate prediction of the fluid statistics at
the particle position. In this work, the focus is on the
dynamical evolution of a particle cloud formed by N,
number of inertial particles in a turbulent flow where the
particles interact with the turbulent eddies and modifies
their cascade of energy. To this purpose, Direct Numer-
ical Simulation (DNS) method will be used which pro-
vides the information about all the scales of turbulent
flow. Thus comprehension and true modeling of the two-
way coupling mechanism is more feasible with DNS due
to the fact that the range of fluid scales influenced by the
particles is not known a priori.
DNS performed in this study solves the non-
dimensional compressible unsteady Navier-Stokes equa-
tions along with the continuity equation written as:


ing the phase interactions. Many works were reported so
far focusing on spectral analysis in order to understand
the mechanism of two-way coupling as a function of par-
ticle inertia, mass loading etc... Among these works, we
can cite the works of Squires and Eaton (1990), Elgob-
ashi and Truesdell (1993), Boivin (1998). The main re-
sult according to these studies is that highly inertial par-
ticles have more uniform modification on the turbulent
energy spectrum; on the other hand, low inertia particles
have more non-uniform modification of the spectrum.
One efficient method to investigate the phase interac-
tions is to use Lagrangian stochastic approach, which
is a powerful tool and largely used in statistical physics
to simulate random processes. Langevin equation, the
simplest stochastic model equation, has received much
attention not only in gas-solid flows but also in other do-
mains such as combustion. Its one of the advantages is
the easy adaptation to the inherently Lagrangian motion
of discrete particles and fluid elements.
In this work, the focus is on the modeling of gas-solid
flows using stochastic Langevin equation in presence of
particles smaller than the Kolmogorov scales. Particles
are of uniform property, spherical in shape and inde-
formable. Volumetric loading of particles <1, is very low
so that the particle-particle interactions are not taken into
account. However, mass-loading of particles 97, is con-
siderably high that the fluid turbulence structure is mod-
ified significantly.
Lagrangian stochastic method was developed initially
for monophase flows in which the trajectories of fluid el-
ements are modeled using stochastic Langevin equation
(Haworth and Pope 1986). These fluid element trajecto-
ries are perturbed by the effect of particles in two-way
coupled gas-solid flows which should be taken into ac-
count properly. Langevin equation was also shown to
provide the unperturbed fluid velocity at particle posi-
tion which is required to calculate the drag force (Si-
monin et al. 1993). This fluid velocity is unperturbed by
the presence of particle in question, however, it is mod-
ified by all the other particles of the system. Therefore
two equations of Langevin type are proposed where the
originality is an additional term, which accounts for the
modification of the fluid velocity by the effect of sur-
rounding particles. First equation is to simulate the tra-
jectories of fluid elements and the second one is to sup-
ply the fluid velocities experienced by particles during
their trajectories. Because of the lack of the spectral un-
derstanding of two-way coupling mechanism, the form
of the two-way coupling term should be imposed prop-
erly.
Additional two-way coupling term is modeled using
the mean drag force. Model results are compared to
the DNS simulations corresponding to the solution of
Navier-Stokes equations written in the frame of the point


8~ 8.r; ~= 0 (1)

8p? 1 8 8.r; Re, dze.32
+ fe, + F,,, (2)


Sdt fi S z /it/
+


where p is the density of the gas, p is pressure and fe,
is the force acting on the fluid due to the presence of
particles which will be modeled later on. F,,, is the
turbulence forcing term to obtain statistically stationary
flow. Re ac is the acoustic Reynolds number based on the
speed of sound. The turbulent Reynolds number Re L Of
the flow defined as:


U 3


Re L
Vf


where vf is the kinematic viscosity, Lf is the length of
the large scales, u is the characteristic velocity of tur-
bulence where qy is the fluid turbulence kinetic energy
defined as:


97 = d' ,i; "7,







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


takes into account the effect of a particle on the short and
long range turbulent fluid flow. Koch (1990) showed
that in Stokes' regime only the long range interactions
are important reducing the Point Source approach to
only the monopole force. This is written as:


where (-) is the ensemble average operator. These defini-
tions are proper to the homogeneous isotropic turbulence
which is the turbulence conducted in this study.
The balance equations are solved using the in-house
code called NTMIX3D which is a finite difference base
code. It solves time dependent, three-dimensional mo-
tion of fluid turbulence in a Cartesian geometry using
the 6t" order compact scheme developed by Lele (1992)
for the space derivatives. Time advancement is obtained
using 3rd order Runge-Kutta scheme. Flow is solved on
1283 grid in a cubical domain whose one side is Lb = 2x7.
Periodic boundary conditions are used at the boundaries
of the cube.


fui = F6x


where


F )du ()
7,s m(" di


. n this work, parti- and 6(x) is the Dirac delta function.
in shape, all with the The application of the point source approach in a DNS
.Particle diameter is code considers first, the calculation of the drag force at
Scales of the flow: the particle position by interpolation the fluid velocity
to this position and then opposing this force in sign and
(4) projecting back onto the 8 Eulerian grids surrounding
nterctios de tothe the particle. These projections could cause the fluid ve-
locity at the position of the particle n to be perturbed by
,rces acting on a parti-
the particle resulting in a bias of drag force which
Each particle is time
is based on the unperturbed fluid velocity by the exis-
ns:
tence of particle. This can be validated by a simple
P,i(t) test using two populations of particles (Vermorel et al.
di Up,s (t) (5) 2003). One of the populations is two-way coupled and
the other group is one-way coupled (so-called ghost par-
.* ,, p~i) (6) ticles). Then fluid-particle covariance of the two groups
is compared to each other. This test controls simply that
Article, xp,4 is the posi- whether the fluid velocity at the position of the parti-
,, is the density of the cle is perturbed by all the particles of the system includ-
It. For Re, < 1000, it ing particle.
As shown in figure 1, for the most critical simulation
performed in this paper (~ = 1.32 and 7, = 3.0), the
dp IPVr 7 statistics between the two groups is different to a %b 1
Vf error margmn validating the point source approach.


Particle trajectory computation
cles are accepted solid, spherical
same diameter d, and density p,
very small before the Kolmogorov

d, << 4

which avoids the hydrodynamic i
wakes of particles. The external fo
cle then reduces to the drag force.
advanced according to the equatio


dx:


dup,4(t) 3 p CD
|ufep Up
di 4 p, d'

where up,4 is the velocity of the pa
tion of the center of the particle, p
particle. CD is the drag coefficien
can be written as:

C'D = 2 [1 + 0.15Re0.687
Rep


v, is the local relative velocity between the particle and
the surrounding fluid written as vr = u, ufep. ufop
is the locally undisturbed fluid velocity by the particle
in question. This velocity is obtained by interpolating
the fluid velocities at the grid nodes, where the fluid ve-
locities are stored, at the particle position, using third
order Lagrangian interpolation scheme which is precise
enough to accurately calculate the fluid velocities.

Handling of two-way coupling. The back effect of
the particles on the fluid, the term fe, in Navier-Stokes
equations, is taken into account using the Point Source
In Cell approach developed by Crowe et al. (1977).
Point Source approach accepts particles which are very
smaller than the Kolmogorov scales and it depends on
the multipolar formulation of the Stokes' equations for
a rigid sphere subjected to a turbulent flow. Each pole


Reference simulation and parameterization of two-
Way coupled simulations. The simulations performed
in this study consider the same initial fluid turbulence
field. The statistics of the reference field is tabulated
in Table 1. Two-way coupled simulations are character-
ized by two dimensional numbers; mass-loading and
the Stokes number St which are defined as:


pJ Tf


where up is the volumetric loading of particles and 7,
is the particle relaxation time. Tf is the Lagrangian
timescale of the fluid elements. They are given by:


N,nd
6L


pp 4 dp 1
p 3 CD IUp Ufpg}







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



Table 2: Parametrization of simulations. Values inside the boxes show the number of particles NI, (all numbers are
non-dimensional).

-T, = 1.5 y, = 3.O Ty, = 6.O Ty, =12.O
0.16 1 x 1283 0.5 x 1283 0.25 x 1283 0.125 x 1283
0.32 2 x 1283 1 x 1283 0.5 x 1283 0.25 x 1283
0.65 4 x 1283 2 x 1283 1 x 1283 0.5 x 1283
1.32 8 x1283 4 x1283 2 x1283 1 x1283


Table 1: Statistics of the reference monophase simula-
tion (all numbers are non-dimensional).
Variable Symbol Value

Characteristic velocity a 0.088
Fluid kinetic energy q~ 0.0118
Dissipation 7.7E-4
Turbulent Reynolds Number Re L 96
Taylor Reynolds Number Rex 48
Length of the box Lb 2i
Longitudinal large scale Ly /Lb 0.0864
Transversal large scale Ly /L, 1.9797
Taylor length scale A,/lLb 0.0438
Kolmogorov length scale ty/Lb 0.0032
Maximum wavenumber ky,,,,. 1 1.2608
Eulerian integral timescale
/ eddy turnover time TE/lT, 1.02
Lagrangian integral
timescale Tr//TE 0.8275
Kolmogorov timescale r,,/Te 0.131


Table 3: Particle physical properties (all numbers are
non-dimensional).
7,, 1.5 3.0 6.0 12.0
p> 2000 4000 8000 16000
cl 0.00266 0.00266 0.00266 0.00266


troduce it from the corresponding side of the domain. In
this way, particles have an infinite space of free motion,
Two-way coupling is turned on from the initial condi-
tions and particles are time-stepped for 4 or 5 relaxation
time so that all the statistics concerning the fluid and the
particle phases become stationary in time. All the statis-
tics, reported at the results and discussion section, are
calculated after this time for at least 10 eddy turnover
times to obtain robustness.
Additional to the solid particles used in the simula-
tions, a population of fluid elements is also introduced
into the reference flow field to calculate the Lagrangian
statistics of the fluid in the stationary level after the tur-


~$0.4



-0.2

0 2 4 6 8
Tf/71



Figure 1: Ghost particle test. qj, is the covariance of
one-way coupled (ghost) particles, qp is the
covariance of physical particles. T+ is the
non-dimensional time of the simulation,


It is obligatory to observe in the DNS simulations the dy-
namic evolution of the system for a fixed 9 changing 7ia
and vice versa. However, the Lagrangian timescale af-
ter the coupling is turned on is not known a priori and it
is difficult to obtain the same Stokes number for a given
particle relaxation time. For simulations conducted here,
particle relaxation time is constant, it does not change
whether there is two-way coupling or not. Therefore,
the reasoning of the simulations is based on the mass-
loading and the particle relaxation time. The simulation
parameters are presented in Table 2. Particle parame-
ters to impose different particle relaxation times are pre-
sented in Table 3.

Performing simulations. For each simulation, NI,par-
ticles are released into the reference turbulence field with
a spatially homogeneous distribution. Initially, particle
velocities are put equal to the fluid velocity at parti-
cle positions. Periodic boundary conditions ensure the
homogeneous distribution of particles during their dy-
namic motion. These boundary conditions take a parti-
cle exiting domain from a boundary of the cube and in-











bulence is modulated by solid particles.


Statistical approach

Ensemble averages are used to calculate the statistical
quantities in this study. Ensemble averaging operator
considers the arithmetic average calculation over infinite
number of realizations of the phenomenon in question.
Applied to gas-solid flows, we can define the statistical
averaging operator as:


where Hf&, is the number of fluid-particle flow real-
izations. Using this averaging we can define mean and
fluctuating velocities of both phases as:

Uf,i = (uf,i) a7,i = uf,i Uf,i (13)
Up,, = (Up,i) Up,i = Up,i Up,i (14)

We can also quantify the statistics calculated at the par-
ticle positions using the simplified writing (), which
is related to the ensemble averaging operator by (-) =
up (-)p. n, here is the particle density number in a unit
volume. As will be seen in the course of this paper, this
statistics concern the properties of the fluid seen by the
particles .


Lagrangian stochastic modeling of gas-solid
flows

Stochastic modeling turbulent flows began with the fa-
mous paper of Taylor (1921) and continue to receive
much attention from the researchers from different do-
mains. The application is simple. A turbulent flow is
represented by a number of fluid particles which are time
advanced according to the stochastic trajectory equa-
tions (Pope 1994).
In gas-solid flows, Lagrangian stochastic modeling is
used to provide the fluid statistics encountered by the
particles during their trajectories (Simonin et al. 1993).
This approach not only allows the writing of the trans-
port equations for the turbulent drift velocity and the
fluid-particle correlations, but also it provides closure
to the Boltzmann type joint probability density function
transport equation (Simonin 2000).

Langevin equation for fluid velocity along fluid el-
ement trajectories. According to Haworth and Pope
(1986), the increase in the fluid velocity of a fluid ele-
ment can be modeled as a stochastic process. Applica-
tion to turbulence comes up as decomposing the velocity


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


and the pressure in Navier-Stokes equations into mean
and fluctuating parts:

1 BP
du,, (t + bt) = + v VUy, d



+ f, di (15)

and then modeling the fluctuating part of the equation
with the stochastic Langevin equation as:

1 BP
,iPf d8 i 2 f~'i dL"
+Afgi (u; (t) Uf,j dt + Bf .'TU
-II, di (16)

where the terms containing the tensors Af, y and Bf,ij
are the drift and diffusion terms, respectively. 6W, is the
isotropic Wiener process with zero mean and variance
dt. The last term is the two-way coupling term modeling
of which closes the Langevin equation completely. The
two-way coupling term in the Navier-Stokes equation is
known in the context of point sources (see the section
'Handling of two-way coupling' above), however, the
corresponding term in Langevin equation is not known
a priori because one-to-one correspondence between the
drift and diffusion terms of the Langevin equation and
the fluctuating part of the Navier-Stokes equation can
not be made.
Langevin equation is qualitatively incorrect for in-
finitesimal time separations, it is consistent with the in-
ertial zone where the timestep is in the range 7, << di <<
Tf. The diffusion coefficient is derived in this context
and usually it is justified by the Kolmogorov theory for
the turbulent structures of the inertial zone. However, we
are going to stay general enough and using the isotropy,
the writing will be simplified as:


Af7ij = Af64,
Bf,ij = By da


In this paper, the purpose is to model the two-way
coupling term in order to obtain the diffusion and drift
coefficients. Closed Langevin equation for fluid el-
ements then should cover the following macroscopic
properties of fluid turbulence:
Same Tf as the one measured from DNS of homo-
geneous isotropic turbulence

Same mean fluid velocity Uf,i transport equation

Same q~ equation as the one derived from the
Navier-Stokes equations


(-) = lim 1 E (.)
Hyg, oo LH/&P Hyg







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


where the turbulent transport term is written as:


This methodology includes following steps consistent
with the work of Pope (2002):
To determine Af using the Lagrangian autocorrela-
tion function

To determine Bf in terms of Af using the second
order stress tensor transport equation

Fluid autocorrelation function. For isotropic turbu-
lence, autocorrelation function is defined as:

R(i(t) = (u;,4(to)u;,4(to +t)) (19)
where statistical operator is defined in (12). Differen-
tial equation governing the Lagrangian autocorrelations
is derived from (16) as:


Df,i x j,9 a,


and production term is written as:


Pf ,is = / of ,m U,,y

i \ 8Uys
a ofmi 8xm


%f,ij is the pressure deformation term which will not
be written explicitly here. Comparing the Reynolds
stress equation above to the one derived from the Navier-
Stokes equation gives:

7f,ij tf,ij + nRs,i + Has =


+B 6 4 + SSII (28)

where #7,ij is the fluctuating pressure-velocity correla-
tion tensor, e7,ij is the dissipation tensor, IF~g is derived
from the forcing term in (1).
For decaying homogeneous isotropic turbulence, we
can simplify the relation. Using the isotropy, the
pressure-velocity correlation term can be taken as zero,
/f,ij = 0 and we do not have the term IFgy. Once Af
is known using the autocorrelation function, Bf can be
obtained from (28) in terms of Af and the two-way cou-
pling terms as:

3BZf = -2Aiq- 2ef + I,,d II (29)

where the contraction of indices i and j is used.
For the forced turbulence, contracting the indices of
(28) results in:




where q~ is the fluid kinetic energy as defined in 3. This
relation forms the 3rd condition to be verified by the
Langevin equation for the fluid elements.

Langevin equation for fluid velocity along solid
particle trajectories


dRf (t)
di = AfRf(t)


u},4~i(to)Inul(to +t))


Lagrangian timescale can be defined as:

T/f = T R (t)di (21)
SR((0) Jo

Fluid mean momentum equation. Volumetric loading
of particles is negligible in this study, ap << 1. This
leads to the mean momentum equation derived from (16)
as:


8~pUf ~ B'U
+ pUf,y
8 8xy


BP
-- + p V2 Uf,i4
8xi
+De (II,i) (22)


where the turbulent transport term is written as:

De = p-8 a(23)

where the symbol is omitted in the fluid velocities be-
cause the high order moments from the Langevin equa-
tion are assumed equal to the statistics obtained from
Navier-Stokes equations. The necessary condition for
the Langevin equation to represent the mean motion of
the fluid is written as:


(nu, )=(fi)


Fluid Reynolds stress equation. Reynolds stress
equation derived from (16) is written as:


In presence of inertial particles, the trajectories of the
fluid elements and solid particles are not the same. As
the calculation of the drag force acting on a particle is
based on the information of the fluid velocity at the par-
ticle's position, Simonin et al. (1993) proposes using a
Langevin type equation to model the fluid velocity en-
(25) countered by a particle during its trajectory where the


a a + U 7au 3

D y,ij + Pf,ij + 7f,

+Alb,64 (u;,ju;~m + A764ml *f;,iu;,m )
+B 64+ I







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


Fluid Reynolds stress seen by the particles equa-
tion. The methodology using the Langevin equation for
the fluid velocity seen by the particles allows the writing
of Reynolds stress seen by the particles equation written
as:


deviation between the trajectory of the particle and a
fluid element, which are assumed to be at the same initial
position, are accounted by an additional term.
We use the same equation as proposed by Simonin
et al. (1993) to calculate the fluid velocity at the par-
ticle position, however, as this velocity is modified by
the surrounding particles in case of two-way coupling
mechanism, we also add an additional term to take into
account this effect. The equation is written as:

'"~('"';[ f ~z1 BP

du}Op~ / 2 ,i s


+(. ; -. ;) t (31)

where the last term accounts for the deviation of the
particle trajectories from the fluid element trajectories.
There are different possibilities for this term (Minier and
Peirano 2001). The form written above is proposed by
Simonin et al. (1993) and it takes into account the correct
behaviour of the fluid-particle covariance. Therefore it
will be used in this study. Using the isotropy, the writing
of the drift and diffusion coefficients will be simplified
as:

Afep,ij = AfepGif (32)
Bfep,ij = fepGsy (33)
It is to be noted that this equation is not an acceler-
ation of a fluid element, it is simply a time derivative
of the fluid velocity computed along the trajectory of a
solid particle.

Fluid autocorrelation function seen by the particles.
The differential equation governing the fluid autocorre-
lation function viewed by the particles can be written as:


ddtp = Afep ~ept

f O,i 0 us 0 (t+ t)) (34)
The integration of this autocorrelation function gives the
fluid Lagrangian timescale viewed by the particles as:

T Op =R e (0) Opt~ (35)

This time scale is the most critical for the two-way cou-
pling studies because it sees the highest modification due
to the particles, as will be seen in results section,
The definition of the Stokes number will then be mod-
ified as:

St (36)
T Op


d(UfOp~i f Op,@
'"+ Up,m


D fep,ij Pfep,ij


+nAf~pim, ( i~il, fpm



+B Op +nRfos~, (37)


where the turbulent transport term is written as:


ftpd s3 ftp,i" .' i u
Df dxmi = 7


and production term is written as:


I'~"=~;i,17; p7i BU
Pfpi f~ ftp.


Contracting the indices for the homogeneous isotropic
stationary turbulence, it is obtained that:


2Afep9 op op op


0 (40)


where q Op iS the fluid kinetic energy seen by the parti-
ClCS defined as:


9 :Op f O(; p,i f Op~ip


It is worth noting that the transport equation (37) for
the Reynolds stresses seen by the particles is not nec-
essarily the same as the one derived from the Langevin
equation for the fluid element trajectories due to the av-
eraging operator weighted by the particle number den-
sity up in the case of fluid velocities along the particle
trajectories.

Fluid-particle covariance transport equation. Trans-
port equation for the fluid-particle covariance tensor can
be derived using the Langevin equation for fluid velocity







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


Then we can think of a mean drag contribution for the
two-way coupling term in the Langevin equation and
leaving the fluctuating part of the drag force to the drift
and diffusion terms. Thus we can formulate the two-way
coupling term as:


along the particle trajectories. It can be written as:


u2p + npUp,m =

Dfp,ij + Pfp,

+2 f,; p F,

+Ipiad nyUfp,. i i


+np~~l ./ ;IIy (42)

The terms on right hand side are, respectively:
Turbulent dispersion of the covariance by velocity
fluctuations written as:

Dfp,ij = 8m fpi, 4)

- Production terms by the mean particle and fluid veloc-
ity gradients, written as:



FI~s=-np i f: Op,i pi~m





-n, ;upm Va(44)



Two-way coupling term takes into account the inter-
phase turbulent momentum transfer, such as dragging of
the particles along the turbulent eddies
Last two terms are derived from the drift and two-
way coupling terms of the Langevin equation and takes
into account the pressure-deformation, viscous dissipa-
tion and the crossing trajectories effect (Boivin 1996)
By contracting the indices of the tensor for the
isotropic turbulence, we can define the fluid-particle co-
vaniance as:


IL, = '(F,,4)
pp


u2pm' Up y~ 44
PfTp


Results and discussion


Turbulent particle dispersion. The modeling steps
mentioned in section Lagrangian stochastic modeling
are to determine the drift and diffusion coefficients.
Model is tested against the DNS simulation results
according to the particle dispersion theory of Tehen
(1947)-Hinze (1975) where the particle agitation is de-
fined in terms of the agitation of the fluid. Hinze (1975)
obtained, using spectral analysis, the results of Tehen
(1947) and showed that the agitation of the particles is
in equilibrium with the agitation of the fluid after suf-
ficiently long time. According to the theory of Tehen-
Hinze, particle dispersion is controlled by the inertia of
the particle when there is no relative motion between the
phases. Deutsch and Simonin (1991) showed that for
certain Stokes' numbers, the particle dispersion coeffi-
cient D~ is higher than the fluid element dispersion coef-
ficient D) and this increase is related to the Lagrangian
timescale seen by the particles T/Oep. This timescale
shows the interaction time of particles with the largest
turbulent eddies and according to Deutsch and Simonin
(1991), this increase in the dispersion is induced by en-
hanced correlation of the particles with eddies of largest
scale. Thus they derive the particle dispersion coefficient
in terms of fluid statistics seen by the particles as:

D 2"
D = q gT Op (49)

Assuming an exponential form for the Lagrangian
timescale, they derive the relation for the extended the-
ory of Tehen-Hinze as:
T/Op
4f p = 2q OF f (50)
L7P
Developing the differential equations governing the
fluid autocorrelation functions seen by the fluid elements
and by the solid particles given by (20) and (34), using
the mean drag model given by (48), the coefficients Af
and Af Op iS obtained as:

A, = A fep p (51)

Deutsch and Simonin (1991) showed that there is a
negligible bias between the Lagrangian timescale seen


'/v = (1 f Op;i p~i p


Modeling of two-way coupling term

If we follow the reasoning of Haworth and Pope (1986),
two-way coupling term in the Navier-Stokes equation
can also be decomposed into mean and fluctuating parts
as:

fui = ( fui)+ fli (47)







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


by the fluid elements and the solid particles. Then we
can formulate a first 1 1 ghs'l'is

Tfe = Tf (52)

which leads to the writing:


is proportional to the Eulerian characteristic timescale
by a constant.
Using (30) and (40) written for homogeneous
isotropic stationary turbulence, coefficients Bf and
By up can be written as:


Bf=


Bfyp= dfort op, (55)


4;op =A4f


In turbulence modeling context, Lagrangian timescale
can, be assumed proportional to a characteristic Eu-
lerian timescale of turbulence which is usually given in
terms of the fluid kinetic energy q~ and dissipation rate
t ratio as:


Fluid-particle covariance tensor equation (42) can be
simplified using homogeneity and isotropy along with
stationarity which leads to the writing of:


qy,- = 29ep ,, (56

where using the A4; o given by (51), we obtain the re-
lation for the extended theory of Tehen-Hinze given by
(50). It is to be noted that this relation is the same as
the relation in one-way coupled flows which brings us
to the conclusion that with mean drag model, the two-
way coupling has no direct effect on the equilibrium of
Tehen-Hinze.
Deutsch and Simonin (1991), Fevrier (2000) showed
that the difference between the fluid kinetic energy seen
by the fluid elements and the solid particles is negligible
which brings us to the second Il)1s'lllc'\i\

q,4,, = q, (57)
Two-way coupling terms in the second order stress
tensor transport equations (25) and (37) and the fluid-
particle covariance transport equation (42) is zero with
mean drag modeling. This modeling dictates that the
two-way coupling has no direct effect on these statistics,
thus we obtain the same set of equations as for the one-
way coupled flows.
Tehen-Hinze relation (56) is tested against DNS ex-
periments and the results are shown in figure 3. As
seen, the evolution of the numerical results as function
of the particle inertia follows the theoretical line for the
high inertia particles (the left side of the graph, small

It is remarked that the increasing mass loading re-
duces the particles inertia and therefore measure points
move towards the low inertia part of the graph (right
hand side). In fact, in the study conducted here, par-
ticles can not produce or enhance turbulence, they can
only attenuate. Attenuated turbulence has higher inte-
gral timescales which reduces the inertia (Stokes num-
ber) of the particles (see the definition of the Stokes'
number).
Increasing mass loading also corresponds to the in-
crease in the fluid-particle covariance. G.In~islc'llll with
the reduction in particle Stokes number, particles are
more and more controlled by the turbulence dynamics.


TL =C


where ('7 is a constant. This relation is accounted ac-
curately by the drift and diffusion terms of the Langevin
equation in homogeneous isotropic turbulent monophase
flows (Pope 1994).
To test the behaviour of the constant (', in two-way
coupled turbulent flows, the DNS measures of the La-
grangian timescales and the fluid kinetic energy, dissi-
pation rate ratios are plotted in figure 2. As seen on the


0.


'T 0.4


0.2


Oa oao


100 101
T/ /4 7


Figure 2: Ratio of the Lagrangian timescale to the Eu-
lerian characteristic timescale of turbulence as
function of the particle inertia; 0:p = 0.16,
o: = 0.32, n: = 0.65, 0:p = 1.32.

figure, the ratio of q /t to the Lagrangian timescale T/
depends on 9 and the inertia of the particles. For the
high mass-loading simulations, it varies sensibly with
the modifications in the particle inertia. This variation
of the Lagrangian timescale should be properly taken
into account in the Langevin equations proposed which
is difficult and requires further work. A priori, we ignore
this variation and assume that the Lagrangian timescale







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


The deviation observed for small inertia particles
(right hand side of the graph) indicates an underestima-
tion of the fluid-particle covariance and it is in fact due
to the assumption of exponential form for the autocorre-
lations viewed by the particles. Indeed, due to the low
Reynolds number of 96 in the study conducted here, the
autocorrelation functions deviate from the exponential
form for small time increments (Boivin 1996). The low
Reynolds number effects can be taken into account using
the model proposed by Sawford (1991) for the correla-
tion curves. This will not be presented in this study.


0.8

N9 0.6

S0.4

0.


10 100 101 102



Figure 4: Classical theory of Tehen-Hinze. The line
and the symbols represent the same condi-
tions as in figure 3


Effect of two-way coupling on the fluid statistics
seen by the particles. The results presented in the pre-
vious section put in question the validity of the hypothe-
ses made (52) and (57). We performed several one-way
coupled simulations to compare our results with the re-
sults of Deutsch and Simonin (1991). Two-way coupled
simulations are also compared. As seen in figure 5, the
fluid kinetic energy seen by the particles does not change
with inertia of the particles and it is sensibly equal to the
kinetic energy seen by the fluid elements for one-way
coupled simulations. The results are in coherence with
the results of Deutsch and Simonin (1991).
Two-way coupled simulations, on the other hand,
have the fluid statistics viewed by the particles signifi-
cantly different than the ones viewed by fluid elements.
This figure suggests to look also the fluid Lagrangian
timescale seen by the particles because these two quan-
tities determine the particle dispersion, see (49).
Evolution of the fluid Lagrangian timescale viewed
by particles with inertia is shown in figure 6. One-way
coupled simulations are also plotted in the same graph
with comparison to the numerical results of Deutsch and
Simonin (1991). For very low inertia particles (the right
hand side of the graph), it can be noted that in one-way
coupled flows, the Lagrangian timescale seen by the par-
ticles and by the fluid elements is fairly equal indicating
the the particles behave like fluid elements (see also the
figure 5). For, two-way coupled flows, we do not have
any simulation in the very low inertia limit due to the
computing source restrictions. Indeed, for decreasing in-
ertia, we need larger number of particles in order to ob-
tain the mass-loadings considered in this paper (see the
tables 2 and 3). In any case, the tendency of the DNS


10 100o


Figure 3: Extended theory of Tehen-Hinze using the
fluid characteristics seen by the particles.
Symbols are DNS results and represent the
same conditions as in the figure 2. -: Model
given in (56) obtained using mean drag force
given in (48).

From the other part, this deviation observed indicates
that the fluid statistics seen by the fluid elements and
the solid particles are sensibly different for small Stokes
numbers. This in turn puts in question the by plkses c' c
made in the development of the model. Thus the same
graph is plotted using the fluid statistics viewed by fluid
elements, e.g., replacing the q ,, by q) and T/Oep by
T .
In contrast to the fluid statistics seen, the true fluid
quantities shows that the Tehen-Hinze (56) have ten-
dency to overestimate the fluid statistics for increasing
mass-loadings. This behavior of the experimental mea-
sures ensures that, as noted above, the statistics are dif-
ferent for fluid elements and solid particles.
We can conclude that the fluid statistics seen produces
good dispersion results for the theoretical line of Tehen-
Hinze which justifies the use of the fluid statistics seen
in modeling the particle dispersion for two-way coupled
gas-solid flows.







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












B-* *


results for two-way coupled flows shows that decreas-
ing inertia makes particle behave like fluid elements. In
one-way coupled simulations, it can be concluded that
in coherence with the results of Deutsch and Simonin
(1991), particle dispersion is controlled by the changes
in Tf "I with varying inertia.

Our simulations show that in two-way coupled flows,
the dispersion can be controlled by the variations in
both fluid kinetic energy viewed and the Lagrangian
timescale viewed by the particles. As suggested by
Squires and Eaton (1991) and Deutsch and Simonin
(1991), in one-way coupled flows, the augmentation in
the Lagrangian timescale seen by the particles corre-
sponds to an improved correlation of the particles with
the large eddies and this phenomenon is related to the
preferential concentration of the particles into special re-
gions in the flow which is also verified in the simulations
of Fevrier (2000). We remark more pronounced increase
in Tf "I in two-way coupled flows which can be related
to the same phenomenon. The changes in the fluid ki-
netic energy seen is also \ignilk~.llll and there is contin-
uous exchange of energy between the phases. However,
we do not have a good explanation for these changes,
yet. More study is required for the preferential concen-
tration mechanism in two-way coupled flows.


-1
I


0.8

101


100 101
T f/7p


Figure 6: Effect of particle inertia on the fluid La-
grangian timescale seen by the particles.
Blank symbols represent the same conditions
as in figure 2. *: Numerical results of Deutsch
and Simonin (1991), *: Results of one-way
coupled DNS simulations


Conclusions


The statistics of solid small particles suspended in a tur-
bulent flow were investigated in presence of two-way
coupling. The modulation of turbulence by a particle
cloud is a difficult phenomenon because the interactions
occur in particle positions which is difficult to take into
account in Eulerian models. From this point of view,
Lagrangian stochastic models provide powerful tools to
simulate the fluid element trajectories and the fluid ve-
locities encountered by particles along their trajectories.
To the Langevin equation, we added an additional
term to take into account the effect of the particles on
the undisturbed flow. Modeling of this term with mean
drag force produces results in agreement with DNS sim-
ulations' results in terms of particle dispersion and ad-
... .. .... vantageous due to the simplifications in transport equa-
lo2 103 tions of second order. However, the by~l polkssc~ are very
open to be questioned and the mechanism of preferential
concentration should be investigated in more detail.


+*

\ a- ;


10


100 10
T/f/7


Figure 5: Effect of panticle inertia on the fluid kinetic
energy seen by the particles for one-way cou-
pled and two-way coupled flows. Blank sym-
bols represent the same conditions as in fig-
ure 2, *: Numerical results of Deutsch and Si-
monin (1991), *: Results of one-way coupled
DNS simulations


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7th International Conference on Multiphase Flow,
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