
Full Citation 
Material Information 

Title: 
P1.74  Lagrangian Stochastic modeling of gassolid flows with twoway coupling ParticleLaden 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 twoway coupling 
Notes 

Abstract: 
Two Langevin equations are proposed to model the particleladen 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, p275283, 1993 except an additional term to take into
account the twoway 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. ICMF2010 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: BioFluid 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 NanoScale Multiphase Flows; Microgravity in TwoPhase Flow; Multiphase Flows with Heat and Mass Transfer; NonNewtonian Multiphase Flows; ParticleLaden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows 
Record Information 

Bibliographic ID: 
UF00102023 

Volume ID: 
VID00465 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved by the source institution and holding location. 

Resource Identifier: 
P174ZerenICMF2010.pdf 

Full Text 
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Lagrangian Stochastic modeling of gassolid flows with twoway 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, Twoway coupling
Abstract
Two Langevin equations are proposed to model the particleladen 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, p275283, 1993] except an additional term to take into
account the twoway 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
Twophase 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
twoway 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 Fluidparticle covariance
uf< Instantaneous local fluid velocity
SStochastic particle velocity
Uf~ Mean fluid velocity
Instantaneous fluid velocity seen
SInstantaneous particle velocity
UMean particle velocity
Vd1 Fluidparticle turbulent drift velocity
Greek
symbols
Particle volumetric loading
Particle mass loading
Turbulent dissipation rate
Twoway 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 twoway 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 NavierStokes 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 twoway 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 nonuniform 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 gassolid 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 gassolid
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 particleparticle interactions are not taken into
account. However, massloading 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 twoway
coupled gassolid 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 twoway coupling mechanism, the form
of the twoway coupling term should be imposed prop
erly.
Additional twoway coupling term is modeled using
the mean drag force. Model results are compared to
the DNS simulations corresponding to the solution of
NavierStokes 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 inhouse
code called NTMIX3D which is a finite difference base
code. It solves time dependent, threedimensional 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 RungeKutta 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 twoway coupled and
the other group is oneway coupled (socalled ghost par
.* ,, p~i) (6) ticles). Then fluidparticle 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 twoway coupling. The back effect of
the particles on the fluid, the term fe, in NavierStokes
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. Twoway coupled simulations are character
ized by two dimensional numbers; massloading 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
nondimensional).
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 nondimensional).
Variable Symbol Value
Characteristic velocity a 0.088
Fluid kinetic energy q~ 0.0118
Dissipation 7.7E4
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
nondimensional).
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,
Twoway coupling is turned on from the initial condi
tions and particles are timestepped 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
oneway coupled (ghost) particles, qp is the
covariance of physical particles. T+ is the
nondimensional 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 twoway 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 gassolid flows, we can define the statistical
averaging operator as:
where Hf&, is the number of fluidparticle 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 gassolid
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 gassolid 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
fluidparticle 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 NavierStokes 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 twoway coupling term modeling
of which closes the Langevin equation completely. The
twoway coupling term in the NavierStokes equation is
known in the context of point sources (see the section
'Handling of twoway coupling' above), however, the
corresponding term in Langevin equation is not known
a priori because onetoone correspondence between the
drift and diffusion terms of the Langevin equation and
the fluctuating part of the NavierStokes 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 twoway
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
NavierStokes 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 pressurevelocity 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
pressurevelocity 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 twoway 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 = p8 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
NavierStokes 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 twoway 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 fluidparticle 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 twoway 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.
Fluidparticle covariance transport equation. Trans
port equation for the fluidparticle 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
twoway 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 twoway
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)
Twoway 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 pressuredeformation, 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 fluidparticle 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 TehenHinze 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 twoway coupling term
If we follow the reasoning of Haworth and Pope (1986),
twoway coupling term in the NavierStokes 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:
Fluidparticle 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 TehenHinze given by
(50). It is to be noted that this relation is the same as
the relation in oneway 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
TehenHinze.
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)
Twoway 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
twoway coupling has no direct effect on these statistics,
thus we obtain the same set of equations as for the one
way coupled flows.
TehenHinze 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 fluidparticle 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 twoway
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 massloading 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 fluidparticle 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 TehenHinze. The line
and the symbols represent the same condi
tions as in figure 3
Effect of twoway 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 oneway
coupled simulations to compare our results with the re
sults of Deutsch and Simonin (1991). Twoway 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 oneway
coupled simulations. The results are in coherence with
the results of Deutsch and Simonin (1991).
Twoway 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. Oneway
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 oneway
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, twoway 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 massloadings 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 TehenHinze 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 TehenHinze (56) have ten
dency to overestimate the fluid statistics for increasing
massloadings. 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 twoway coupled
gassolid flows.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
B* *
results for twoway coupled flows shows that decreas
ing inertia makes particle behave like fluid elements. In
oneway 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 twoway 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 oneway 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 twoway 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 twoway 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 oneway
coupled DNS simulations
Conclusions
The statistics of solid small particles suspended in a tur
bulent flow were investigated in presence of twoway
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 oneway cou
pled and twoway coupled flows. Blank sym
bols represent the same conditions as in fig
ure 2, *: Numerical results of Deutsch and Si
monin (1991), *: Results of oneway coupled
DNS simulations
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