7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Numerical simulation of turbulent gasparticle flow in a riser using a
quadraturebased moment method
A. Passalacqua and R. O. Fox
Department of Chemical and Biological Engineering, Iowa State University, Ames, IA 500112230, USA
Salbertop~iastate.edu and rofox~iastate.edu
Keywords: Quadraturebased moment method, gasparticle flow, riser flow
Abstract
Gaspanticle flows are used in many industrial applications in the energy, oil and gas fields, such as coal gasification,
production of light hydrocarbons by fluid catalytic cracking, catalytic combustion and different treatments aiming to
reduce or eliminate pollutants. The particle phase of a gasparticle flow is described by analogy to a granular gas, by
finding an approximate solution of the kinetic equation in the velocitybased number density function. In the recent
past, many studies have been published on the mathematical modeling of gasparticle flows using hydrodynamic
models (e.g. Enwald et al. 1996), where NavierStokestype equations are solved to describe the panticle phase as a
continuum, computing its stress tensor using moment closures from kinetic theory (Gidaspow 1994). These closures,
however, are obtained assuming that the flow is dominated by collisions and near equilibrium, which corresponds to
considering a very small particlephase Knudsen number. This assumption leads to inconsistencies and erroneous
predictions of physical phenomena when these models are applied to dilute fluidparticle flows, where rarefaction
effects are not negligible. In these flows, the wall Knudsen layers extend inside the bulk of the fluid, and cannot be
accounted for with the simple addition of partialslip boundary conditions. Recently Desjardin et al. (2008) showed
that twofluid models are unable to correctly capture particle trajectory crossing, seriously compromising their
ability to correctly describe any velocity moment for finite Stokes numbers. These authors clarified that the particle
segregation captured by twofluid models for finite Knudsen numbers is artificially high due to their mathematical
formulation, which leads to the formation of deltashocks. In order to overcome these shortcomings, Fox (2008)
developed a thirdorder quadraturebased moment method for dilute gaspanticle flows, which has been successfully
coupled to a fluid solver to compute dilute and moderately dilute gaspanticle flows by Passalacqua et al. (2010) in
two dimensions. These authors validated their model against EulerLagrange and twofluid simulations. In this work,
the fully coupled quadraturebased fluidparticle code described in Passalacqua et al. (2010) is applied to simulate
turbulent gaspanticle flow in the riser described by He et al. (2009), using a threedimensional configuration. This
application shows the predictive capabilities and the robustness of the quadraturebased moment method to predict
the behavior of gasparticle flows in accordance with experiments (He et al. 2009).
Nomenclature K11 Knudsen number
Ma Mach number
Roman symbols
CD drag coefficient in, particle mass (kg)
rl, particle diameter (m1) n, quadrature weight
t: particleparticle restitution coefficient
e, particlewall restitution coefficient Mf, momentum exchange due to drag (kg/(112 s2))
f distribution function Mi k velocity moment ((11/s)')
F force acting on each particle (N)
p pressure (Pa)
g gravitational acceleration (m1/s )
Pr Prandtl number
go radial distribution function
Re Reynolds number
I identity matrix
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
able to properly predict the behavior of a gassolid flow
in a riser if all the interactions between the phases are
accounted for, and correlated these interactions to the
averaged and fluctuating components of the phase ve
locity fields. Sinclair and Jackson (1989) also devel
oped the first stationary hydrodynamic model able to
predict the particulate phase stresses through the kinetic
theory of the granular flow, as a function of the parti
cle fluctuating energy (granular energy). Pita and Sun
daresan (1991) modified the Sinclair and Jackson (1989)
model and validated it against Bader et al. (1988) exper
imental data, showing its high sensitivity to the value
of the restitution coefficient, whose reduction may lead
to a wrong prediction of the particles segregation pat
tems inside the duct. Lounge et al. (1991) introduced
a oneequation turbulence model to describe the gas
phase turbulence, and adopted standard wallfunctions
for the zone near the wall. Gidaspow and Therdthi
anwong (1993) adopted a zeroequation closure for the
gas phase turbulence. Ocone et al. (1993) extended the
work of Sinclair and Jackson (1989) to arbitrarily in
clined ducts, accounting for the effects of particles slid
ing and rotation. Bolio et al. (1995) adopted a low
Reynolds twoequation /ce model for the gas phase, to
eliminate the need for wall functions. Hrenya and Sin
clair (1997) studied the influence of turbulence both on
the transport equations and on the kinetic theory clo
sure equations, leading to a reformulation of the dissi
pation term of the granular energy, which resulted in
a reduced sensitivity of the model to the value of the
particle restitution coefficient. Benyahia et al. (2000);
Arastoopour (2001) simulated a FCC riser using the ki
netic theory of granular flow, neglecting the gasphase
turbulence. Peirano and Leckner (1998); Peirano et al.
(2002); DeWilde et al. (2002) adopted a model that cou
ples a twoequation turbulence model with a set of two
equations for the particulatephase turbulent kinetic en
ergy and for the gasparticle velocity correlation Gi
daspow and Therdthianwong (1993); Samuelsberg and
Hjertager (1996); Mathiesen et al. (2000); Huilin and
Gidaspow (2003); Huilin et al. (2006) introduced the
Smagorinsky (1963) subgrid stress model to describe
the turbulence of the gas phase following the principles
of largeeddy simulation. Recently Ibsen et al. (2004)
compared discrete methods and multifluid models in
circulating fluidised beds. Zeng and Zhou (2006) devel
oped a twoscale secondorder moment particle turbu
lence model for dense gasparticle flows. Moreau et al.
(2009) proposed a new largeeddy simulation approach
for particleladen turbulent flows in the framework of the
Eulerian formalism for inertial particles. He et al. (2009)
performed experiments in a riser with monodispersed
particles and compared the measurements with the re
sults of numerical simulations performed using the La
St Stokes number
t time (s)
U velocity (m/s)
Uai quadrature abscissa (m/s)
v particle velocity (m/s)
x position vector
Greek symbols
volume fraction
agje set of equilibrium moments
8 granular temperature (m /s )
Dynamic viscosity (Pa s)
p density (kg/m )
ogy particlephase stress tensor components
Ty fluidphase tensor
7o collision time (s)
Subscripts
f Fluid
p Particle
Superscipts
f Fluid
p Particle
Introduction
Gasparticle flows in risers have been the topic of ex
tensive research in order to develop reliable computa
tional models capable of describing their peculiarities,
Generally speaking, two kind of approaches are possi
ble to describe the particle phase: the Lagrangian ap
proach, where each particle trajectory is resolved inde
pendently, applying fundamental laws of mechanics, and
the Eulerian approach, in which the particle phase is de
scribed by transport equations of moments of the par
ticle velocity distribution function. The computational
convenience and the absence of statistical noise char
acteristic of Eulerian models made them very attractive
both for research and applications, and \ignilk~.llll effort
to improve their formulation has been spent in the last
two decades. Syamlal and Gidaspow (1985); Gidaspow
(1986) developed hydrodynamic models for CFB reac
tors, accounting for heat transfer and introducing a nor
mal stress modulus for the particulate phase. Gidaspow
et al. (1989); Tsuo and Gidaspow (1990) adopted the
Wen and Yu (1966) drag correlation in their model, and
properly predicted flow regimes typical of the circulat
ing fluidised bed risers. Sinclair and Jackson (1989);
Sinclair (1997) showed that a hydrodynamic model is
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The collision term C is described using the Bhatnagar
GrossKrook (BGK) collision operator (Bhatnagar et al.
1954):
C 1 (es ) ,(5)
where 7, is the collision time and fes is the equilibrium
distribution function, extended to account for inelastic
collisions:
grangian discrete particle model (DPM) approach.
In this work we present the results of the simulations
of the same system considered by He et al. (2009) ob
tained with a fully threedimensional implementation of
the quadraturebased moment method developed by Fox
(2008) and coupled with a fluid solver by Passalacqua
et al. (2010). Results are compared to experimental data,
statistics of the particle phase are computed and dimen
sionless parameters such as the particlephase Mach,
Knudsen and Stokes numbers of the flow are examined.
Model description
Fluidphase governing equations. The fluid phase,
assumed to be incompressible and isothermal, is de
scribed by a continuity and a momentum equation as
in traditional twofluid models (Drew 1971; Gidaspow
1994; Enwald et al. 1996). The fluid continuity equation
has the form
fes =
[det (2i7r)]1/
1 ((
ct~x 1~?1 tl) 
where A1 is the inverse of the the matrix A, defined by
A = yW 8,I + (yW2 2yw + 1) o (7)
with y = 1/ Pr, w = (1 + e)/2, \Il" the number den
sity of particles (zeroorder moment), U, the mean par
ticle velocity, e the restitution coefficient, 8, the gran
ular temperature, and 0 the velocity covariance matrix.
In this work y 1 so that Pr 1 in the standard BGK
model (Struchtrup 2005).
In this work, following Fox (2008); Passalacqua et al.
(2010), a set of twenty moments of f up to third order
is considered. Each moment is defined through integrals
of the distribution function as
i)
and the fluid momentum equation is
(QfPpUf) + V (QfPpUfUf) =
Of
0, (1)
V (agef) ar~lp + afppg + Mfp, (2)
ii" = fdy,
Afi' = c; lfdy,
Aff = Sv
where at, pc, Uf are the fluidphase volume fraction,
density and mean velocity, Mf, represent the momen
tum exchange term due to the drag between the fluid
and particle phases, and g is the gravitational acceler
ation vector.
The stress tensor 7f is
Transport equations for the moments are obtained by ap
plying the definition (8) to both sides of the kinetic equa
tion, leading to (Fox 2008)
Pr (V Ur) I, (3)
.i = pr VOUr + (VUr)'
a~t itrj
where pr is the dynamic viscosity of the fluid and I the
unit tensor.
Particlephase governing equations. The particle
phase is described in analogy to a gas made of smooth,
monodisperse, noncohesive spheres. Its governing
equation is represented by a kinetic equation for the par
ticle number density function f (t, x, v) (Chapman and
Cowling 1961; Cercignani et al. 1994; Struchtrup 2005)
VYi)Af i)Af"k
dt itrg
"3 3
Cs A3
where ,4,1, A24@ and 4 p are the source terms due to the
acceleration acting on each particle, and C,' and CL're
are those due to the collision operator.
The set of transport equations (9) is not closed, be
cause each equation contains the spatial flux of the mo
ments of order immediately higher, and the source terms
due to the force and to collisions. Gaussian quadrature
O~f O~f i) F
+ v + f
DLt i~x i"vin,
where F is the force acting on each particle, including
gravity and drag, and C represents the rate of change
in the number density function due to binary collisions
between the particles.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
fonnulae are used to provide closures for these source
tenns as a function of a set of weights n, and abscissas
U,. Weights and abscissas are computed from the set
of transported moments by means of an inversion algo
rithm. Following Fox (2008), a set of eight weights and
abscissas per each velocity component is used.
The moments can be computed as a function of the
quadrature weights and abscissas using the definition
(8):
1
(10)
M = n ni~a,
The source tenns due to drag and gravity are computed
rnFD ; mP, (1
+ ye~ + gy Use ,
(Fp
m'P (11 UloU
( FD," Se,
Thedrg tme copue gyr Ukach veoiyasis,
(F 1(3
Ther h dragfocle trmsae givesn byr isdfndfre
The drsag tie optdfrec vlct bcsas
and the drag coefficient CD is modeled according to Wen
and Yu (1966):
24 26
CD(Rep, ar) = [1 + 0.15(at Re,)0.687] 2.6
at Rep
(15)
The source terms in the moment transport equations
due to collisions are given by
in which the collision time is defined by
To = ,(17)
with 8, defined in tenns of the moments by
1 1 1 \
933
The radial distribution function of Carnahan and Starling
(1969) is adopted:
go = 1 30 + (20)
1 p 2 (1 ap)2 2(1 ap)3
The moment spatial fluxes are represented by the sec
ond tenn on the lefthand side of (9), and are computed
according to their kinetic definition (Perthame 1990;
Desjardin et al. 2008; Fox 2008) in order to ensure the
realizability of the set of moments.
Boundary conditions. Boundary conditions for the
moment transport equations are specified in terms of the
quadrature approximation, as shown in Fox (2008); Pas
salacqua et al. (2010). The specular reflective condition
at walls is defined as
Uon2 ewUn2 (1
where ew is the restitution coefficient for collisions be
tween a particle and the wall, and i = 0 represents the
wall, assumed to be along the second direction in the
considered reference frame. Periodic boundary condi
tions are imposed by enforcing the periodicity on the set
of weights and abscissas.
Repo
pfd,  Uf U, 
W
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Table 1: Dimensions of the riser and flow properties
Variable Value Units
if 0.05 11
H 1.5 11
D 0.015 11
L 0.30 11
d, 335.0 p
p, 2500.0 kg/m13
e, 0.97
Gs 10.0 kg/(In' s)
Pc 1.2 kg/m13
py 1.8 105 Pa s
Uf 2.3 11/s
Ut 2.7 li/s
The fluidphase mass flux has been imposed to match
the mean particle mass flux Gs, by adapting the pressure
gradient along the height of the computational domain.
The grid density used in the simulations, taken from
the work of He et al. (2009), is 25 grid points along 10,
60 along L and 10 along D. Simulations were performed
with an adaptive time stepping, based on the conver
gence of the fluidphase residuals (Syamlal et al. 1993),
on the QMOM CFL condition, on the particle collision
time and the drag time. Simulations are performed for
a total of 10 s of simulation time, while time averages
are computed on the last 5 s of simulation time. Since
the domain is periodic, averages are computed assuming
the z direction to be homogeneous, and considering a
symmetry plane, located at 1C/2, normal to the Jr direc
tion. The RMS values are found by subtracting the time
averaged values from the instantaneous fields, and time
averaging the square of the differences. The final RMS
value are then the square root of timeaveraged square
differences.
The fluidphase equations are solved with an itera
tive procedure based on the SIMPLE (Patankar 1980;
Ferziger and Peric 2002), and coupled with the QMOM
equations using the partial elimination algorithm (Spald
ing 1980), as shown in Passalacqua et al. (2010). The
secondorder scheme with superbee limiter was used to
discretize the fluidphase equations. As convergence cri
terion for all the variable of the fluidphase, a reduction
of the residuals below 104 was required to consider the
solution converged.
Results and discussion
An example of the instantaneous solids volume fraction
field at t = 8 s obtained in the simulation performed
using the quadraturebased moment method is reported
Figure 1: Schematic representation of the riser consid
ered in He et al. (2009). The hashed area rep
resents the actual portion of the riser consid
ered in the simulations.
Test case description
The model presented above has been implemented in the
CFD code MFIX (Syamlal et al. 1993; Syamlal 1998)
as detailed in Passalacqua et al. (2010), and is validated
here against the experimental measurements in a gas
particle flow riser, with monodisperse particles, realized
by He et al. (2009). The experimental setup used in this
reference is constituted by a riser column of rectangular
section, as schematically represented in Fig. 1. The di
mensions of the riser and the properties of the fluid and
particle phases are reported in Table 1
Simulation setup
The computational domain considered in this work rep
resents only a portion of the whole riser examined in the
experiments of He et al. (2009), in the center of the col
umn, with a height L, as represented by the dashed vol
ume in Fig. 1. Periodic boundary conditions were im
posed at the top and at the bottom of the computational
domain, while noslip conditions were used for the fluid
phase at the wall, and reflective conditions with resti
tution coefficient e w were used for the particle phase.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.025
0.020
$ 0.015
$ 0.010
s
P~0.005
~0.04
.0.03
0.01
0.006
0.000
0.000
0.005 0.010
0.015 0.020
0.025
x [m]
Figure 3: Timeaveraged particle volume fraction.
in acceptable agreement with the experimental measure
ments of He et al. (2009), and similar to those obtained
by the same authors performing Lagrangian simulations.
The convergence of the averages is considered satis
factory, in spite of the relatively short averaging time,
since the values of the timeaveraged velocity along the
x direction (Fig. 5), are approximately zero, as expected
when the flow reaches the steadystate condition. The
rootmeansquare (RMS) of the vertical component of
the velocity of the particle phase is reported in Fig. 6,
and is in qualitative agreement with the experiments of
He et al. (2009), but not in quantitative agreement. Both
the experimental and the computational profiles show a
minimum at the riser centerline, and the maximum value
in proximity of the wall, but at a certain distance from
it. The RMS velocity in the x: direction, reported in
Fig. 7 is in quantitative agreement with the experimental
data. Differences in the numerical predictions compared
to the experimental data might depend on the simplified
numerical configuration used in the simulation, where
only one portion of the system is considered, and peri
odicity is assumed in the axial direction. Further reasons
that could explain the differences are the choice of the
drag law, as well as other submodels used in the numer
ical model, and the systematic errors in the experimental
measurements (He et al. 2009).
The timeaveraged granular temperature is reported in
Fig. 8. (Note that the granular temperature is not re
lated to the RMS velocity statistics!) The values of 8,
Figure 2: Instantaneous solids volume fraction field at
t 8 s across the centerline (D/2).
in Fig. 2. It can be seen that particles segregated at the
walls of the riser, forming ensembles at higher volume
fraction, which tends to move downward, since their
weight wins the resistance exerted by the fluid. As ex
plained in Passalacqua et al. (2010), the flow evolves
from the uniform initial conditions through an interme
diate state where particles, due to the reflective condi
tions at the walls, tend to originate two vertical stripes
parallel to the walls, with lower particle concentration.
In these stripes the fluid accelerates, and the difference in
shear causes the flow to become unstable. The instabil
ity quickly propagates to the whole flow, leading to the
segregation of particles typically observed in riser flows.
The timeaveraged volume fraction, shown in Fig. 3 (ex
perimental data is not available for this quantity), is low
est in the center of the flow and highest at the walls,
which is typical of coreannular flow.
Results of the timeaveraged vertical component of
the velocity obtained in the simulation performed us
ing the quadraturebased moment method are reported
in Fig. 4 and compared with the experiments of He et al.
(2009). In all the reported plots, x: 0 represents the
channel centerline, and x: = 0.025 m indicates the chan
nel wall. The predicted values of the axial velocity show
a coreannular behavior, with negative values at the wall,
indicating particles fall in that region of the system, and
positive values in the center of the riser. Results are
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
m
E o.
x
QMOM
a Exp, Ug= 2
m/s
0.03
SExp, Ug =2.3
m/s
0.02
0.00
0.01
0.000 0.005 0.010 0.015 0.020 0.025
x (m)
Figure 5: Timeaveraged x: component of the particle
velocity.
cording to the region where it is computed, following
Kogan (1969), so that
0.000 0.005 0.010 0.015 0.020 0.025
x (m)
Figure 4: Timeaveraged z component of the particle
velocity.
present a minimum at the wall of the channel, where
the collision frequency is higher due to higher particle
concentration, due to particle segregation, and a maxi
mum in the core of the riser, where the flow is dilute
and collisions are not predominant, as shown in Fig. 10.
Before proceeding to consider the behavior of the Knud
sen number in detail, it is worth noticing that the flow
is in transonic conditions, meaning that there are parts
of the computational domain where the particle veloc
ity is above or equal to the local particlephase 'speed of
sound' (Ma > 1), and other parts of the system where
the particle velocity is below the the local value of the
speed of sound (Ma < 1). This is evident from the
values of the timeaveraged local particle Mach num
ber Ma  Up/0,1/ in Fig. 9, which shows the Ma
is approximately between 6 x 103 and 1.8. Under
these conditions, two regimes are present in the system.
Where Ma < 1, the flow is dominated by diffusive pro
cesses, regulated by the local value of the granular tem
perature, which has to be used to compute the character
istic velocity in this regime. When Ma > 1, the flow is
dominated by convective phenomena, meaning that the
transport of properties is mainly due to the convective
transport of particles more than to diffusive phenomena,
and the granular temperature has to be replaced by the
local mean velocity magnitude in the definition of the
characteristic velocity of the flow. The two zones are
separated by a dashed line in Fig. 9. Since the flow un
dergoes a transition between two different regimes, the
definition of the Knudsen number has to be modified ac
K~n =~V L Ol/
Ma< 1
Ma> 1
where the collision time is given by
To = 12goap 8/ (23)
The values of the Knudsen number, computed as
suming L = 2WD/(W + D), according to the def
inition of hydraulic diameter of the riser, are reported
in Fig. 10. The diagram shows the flow transitions
from the slip regime, where 0.01 < K~n < 0.1 (Bird
1994), in the region adjacent to the wall, to a more rar
efied regime (transitional regime) in the center of the
riser, where nonequilibrium phenomena are expected
to happen. The two regions are separated by the dot
ted line. The lower value of K~n at the wall is justi
fied by the higher particle concentration in that region
of the system, which leads to higher values of the col
lision frequency, making the flow locally dominated by
collisions. It is worth noticing that if an hydrodynamic
model (Gidaspow 1994; Enwald et al. 1996), derived in
the by! polhesc'i\ of nearly zero Knudsen number (Contin
uum regime, K~n < 0.01) were used to perform the sim
ulation, the adoption of partial slip boundary conditions
(Johnson and Jackson 1987) would have been necessary
to describe the behavior of the flow in the zone adjacent
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
S0.04
a Exp.
0.00
0.000 0.005 0.010 0.015 0.020 0.025
x (m)
Figure 7: RMS of the Jr component of the particle ve
locity.
mulation of particles (deltashocks), as shown in Des
jardin et al. (2008), who simulated the behavior of par
ticles in TaylorGreen flow at different Stokes numbers,
and compared the solutions obtained with Lagrangian
simulations, Eulerian quadraturebased simulations, and
hydrodynamic models.
Conclusions
A portion of a riser of a circulating fluidized bed have
been simulated using a thirdorder quadraturebased mo
ment method in a fully threedimensional numerical
setup, showing the robustness and the capability of the
quadraturebased moment method for this kind of ap
plications. Results for the timeaveraged mean particle
velocity were found in satisfactory agreement with the
experimental results in spite of the simplifying assump
tions made in the simulation about the periodicity of the
computational domain. RMS velocities were found in
qualitative agreement with experiments, but not in quan
titative agreement, for what concerns the vertical com
ponent of the velocity.
Characteristic dimensionless parameters of the parti
cle flow were examined. The Mach number showed that
the panticle flow is in transonic conditions, with the sub
sonic region adjacent to the wall. The values of the local
Knudsen number pointed out the flow is in a condition
across two regimes: the slip regime (zone adjacent to the
wall) and the transition regime (center of the channel),
indicating that the adoption of hydrodynamic models in
QMOM
1.3 a Exp.
1.1
0.9
S0.7
S0.5
0.3 a"
0.1.
0.1
0.000 0.005 0.010 0.015 0.020 0.025
x (m)
Figure 6: RMS of the z component of the particle ve
locity.
to the walls, where K~n < 0.1, however the model would
have deviated from the correct behavior in the center of
the riser, where higher values of the Knudsen number are
present, and higherorder approximations of the kinetic
equation than the hydrodynamic are necessary.
The Stokes number profile is reported in Fig. 11, and
reports values between 66.1 and 123.5, which indicate
particles reaction to changes in the local conditions of
the fluid flow are not instantaneous, but delayed and,
since the flow is dilute, might lead to panticle trajec
tory crossing. As pointed out in Desjardin et al. (2008),
such phenomenon cannot be predicted by hydrodynamic
models, since they only consider moments up to the sec
ond order, and define only one local velocity in each
computational cell, whereas multiple local velocities are
necessary to be able to capture the discontinuous veloc
ity field that originates when panticle trajectory cross
ing occurs. This becomes clear considering the panti
cle velocity distribution function. The hydrodynamic
models are obtained assuming equilibrium or nearly
equilibrium conditions, imposing that the velocity distri
bution is Maxwellian. When particle trajectory crossing
occurs faster particles pass slower particles, locally the
velocity becomes multivalued and the distribution func
tion strongly deviates from the equilibrium condition,
originating discontinuities in the velocity field, which
cannot be described under the hydrodynamic hy! po lhes\is
since singlevalued velocity is assumed in these models.
Hydrodynamic models hypothesize the flow is always
dominated by collisions, and predict unphysical accu
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.060
E~0.055
a 0.050
S0.045
0.040
S0.035
0.030
0.000 0.005 0.010 0.015 0.020 0.025
x [m]
Figure 8: Timeaveraged granular temperature profile.
0.000 0.005 0.010 0.015 0.020 0.025
x (m)
Figure 10: Timeaveraged Knudsen number profile.
2.03
1.83
1.63
1.43
1.23
cj 1.03
0.83
0.63
0.43
0.23
0.03
0.000
120
110
100
90
80
70
60
0.000
0.005 0.010 0.015 0.020 0.025
x [m]
0.005 0.010 0.015
x (m)
0.020 0.025
Figure 9: Timeaveraged Mach number profile.
Figure 11: Timeaveraged Stokes number profile.
such a regime would be inappropriate. This was further
confirmed by considering the local values of the Stokes
number, which clarified that particles do not immedi
ately react to the fluid flow, and can originate particle
trajectory crossing wherever the flow is not dominated
by collisions (high K~n).
Future work will involve the simulation of systems
with more realistic boundary conditions and geometries
than those considered in the present work and the exami
nation of denser flow conditions. Additionally, idealized
simulations in domains with periodic conditions in all
the spatial directions will be performed to further inves
tigate the formation of clusters and the conditions that
lead to flow instabilities at the base of the particle segre
gation.
Acknowledgements
This work has been supported by the National Energy
Technology Laboratory of the U.S. Department of En
ergy under the award number DEFC2607NT43098.
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