Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
TwoPhase SubgridScale (SGS) Stress Models for
TwoFluid LargeEddy Simulation of GasParticle Flows
Lixing Zhou1, Yang Liu2
1 Department of Engineering Mechanics, Tsinghua University
Beijing, 10084 China, Zhoulximail.tsinghua.edu.cn
2 Marine Engineering College, Dalian Maritime University,
Daliani 16026, China, liuvaA@mail.tsinghua.edu.cn
Keywords: subgrid scale model; largeeddy simulation; gasparticle flows
Abstract
In present EulerianLagrangian and EulerianEulerian (twofluid) largeeddy simulations (LES) of gasparticle flows most
investigators use singlephase subgrid scale (SGS) stress models. The Interaction between the twophase SGS stresses is
totally not or at least not fully taken into account. In this paper, a unified secondorder moment (USM) twophase SGS stress
model and a twophase SGS energy equation (kkp) model for the twofluid LES of gasparticle flows are proposed, in which
the interaction between the twophase SGS stresses is fully taken into account. The proposed model is used in the LES of
swirling and suddenexpansion gasparticle flows, together with RANS modeling using the USM twophase turbulence model.
The twophase timeaveraged velocities predicted by LESUSM/LESkkp and RANSUSM models are almost the same and
are in good agreement with the experimental results. However, for twophase RMS fluctuation velocities, the LESUSM and
LESkkp results are better than the RANSUSM results.
Introduction
In recent years, largeeddy simulation (LES) of
gasparticle flows attracts more and more attention (e,g.
Boivin et al., 2000; Riber et al., 2009). It is by and by
becoming an advanced CFD simulation tool. For LES of
gasparticle flows, one of the key problems is the selection
or development of the twophase subgrid scale (SGS)
stress models. For LES of singlephase gas flows, the
widely used SGS stress models are Smagorinsky (1963)
eddy viscosity model, Germano (1991) dynamic eddy
viscosity model and Kim (1995) SGS energy equation
model. For EulerianLagrangian LES of gasparticle flows,
many investigators use the singlephase SGS stress models,
for example, Apte et al i2 '" ). However, Zhou, H. et al.
(2007) proposed a gas SGS energy equation model,
accounting for the effect of particles on gas SGS stresses.
In the framework of twofluid or EulerianEulerian LES,
Xiang & Guo ("'ii4) proposed a particle Smagorinsky
SGS eddy viscosity model, which is a simple imitation of
the single gas phase SGS model. Similarly, Boileau et al.
(2008) adopt the Smagorinsky SGS eddy viscosity model
for both gas and droplet phases in their twofluidLES
modeling. In all these approaches the interaction between
the twophase SGS stresses is not or at least not fully taken
into account.
In this paper, extending the idea of the twophase
turbulence models in RANS modeling, a unified
secondorder moment (USM) twophase SGS stress model
and a twophase SGS energyequation stress model are
proposed for the twofluid LES of gasparticle flows. The
proposed model may fully account for the interaction
between the gas and particle SGS stresses. The USMSGS
and kkpSGS twophase stress models are used in the
twofluid LES of swirling and suddenexpansion
gasparticle flows respectively, and the statistical results
are validated by the measurement results reported in
references (Sommerfeld & Qiu, 1991; Xu & Zhou, 1999 ),
and also are compared with the RANSUSM modeling
results.
Nomenclature
D
G
k
P
p
R
t
V,v
a
A
V
p
T
i,j,k,l
g
1
Diffusion term
Source term
Subgrid scale kinetic energy
Production term
Pressure, Pa
Correlation term
time, s
velocity, m s1
Volume fraction
KronicDelta unit tensor
Filtered space scale, m
Dissipation term
Dynamic viscosity, kg m3 *s1
Kinematic viscosity, m2Zs'
Pressurestrain term
Density, kg m3
Stress, kg m' s2
Coordinates directions
Gas
Laminar
Paper No
p
r Particle
Relaxation
Subgrid scale
LES Governing Equations
For a twofluid LES, neglecting the forces other than
the drag force acting on the particles, the filtered continuity
and momentum equations for gas and particle phases can
be obtained as (k=g,p):
a a
(akPk)+ (kPk uki) = 0
(aXgpgu g)+ (agpgugiUg,)
t ax J
S gij
+ +
apg
Oxj
_gs ij+ agPg( pij)
C _X T Up
S pij
axj
+ + p (ugi upi)
Bxj T r
where the filtered gas and particle viscous forces are:
GgiB 2 ugi9j
Zg,ij [tgl( x + xi 1gl  ij
pUPi + pj)2. Upj
"p,ij = p xj i 3 P xj 3ij
The gas and particle subgrid scale (SGS) stresses are
defined as:
gsij = pgRgs,ij = Pg(ugigj Ugiugj)
ps,ij = PRps.i = Pp (Upipj UpiUpj)
The USM SGS Stress Model
Using the idea that used in modeling of twophase
Reynolds stresses in RANS modeling (Zhou, L., 2002), the
SGS stresses of gas and particle phases can be given by the
following transport equations:
(aggpgRgs,ij) + (agPg gkRgsij) =
ct cgk (4)
Dsgs + psgs + Gsg+ 1sgs sgs
g g pg g g
(apppR ps ij) + (appppkRps,ij)
Ct Cxk (5)
Dsgs + sgs +sgs
p p p
where Dsgs, sgs ,1sgs sgs and Dsgs Psgs sgs
g g g hg d sn, pd ps n
are the diffusion, production, pressurestrain and
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
dissipation rate terms. Except the dissipation terms, all
other terms have the similar forms as those for the
corresponding terms in the twophase Reynolds stress
equations in RANS modeling. For example, the closed
twophase diffusion terms are:
a ksgs aR
Dsgs a g s Pg Rgskl gs
g axk c g gsk
a ksgs SS r
Dsgs (apCppy Rps,k )xl
p xk k e
The phase interaction term in the gas SGS stress equation
pg = Y (Rpgs,ij Rpgsji
P 'rp
2Rgs,ij)
where Rpgs,ij = (Upiug UpiUgj) is the twophase
SGS stress correlation.
The SGS kinetic energies for gas and particle phases kgs
s
and kp are defined as:
1 3 ksgs
ksgs= IRgs,ii ksgs Rps,
g 2 j~ gs,n lp Y Rj ps,u
2 Jj=l
The dissipation rates of SGS kinetic energies for gas and
particle phases e and Sp are given as:
=(ksgs 3/2/; p ( 3sgs 3/2
(=kl)g j /A; 8p = kp /A
where A is the subgrid scale size. The transport equation
of the twophase SGS stress correlation is:
(Rpgsij) + (Upk + Uk) (Rpgs,ij)
at kk
a IRpgs,ij
x [(vg + vp) ]
8xk axk
+ P [ ppRps,ij + agPgRgs,ij
Pg'crp
(apPp + agpg)Rpgs,ij] (Rpgs,kj k
cxk
+ Rpgs,ik )
Oxk
g
sgs pgs,ij ij
ksg
(6)
where Vg = Cg(k gs)2/g, Vp = Cp(ksgs)2/p
So, the USM SGS stress model can fully account for
interaction between the twophase SGS stresses and the
anisotropy of twophase SGS stresses.
The TwoPhase SGS Energy Equation Model
For isotropic twophase SGS stresses we have
ata Xpppp p)+ 0r~ p p ~i ,
Paper No
'Cgsij PgK gs,ij = 2Pg kgs,ij
2pg ( vgiVg1
V gi gj)
'ps,ij PpKpsij = 2pp kpsij
2pp VpiVpj
2
VpiVpj)
The twophase SGS energies are closed by
( gKgs,ij )+ (gvgjKgs,ij)
a OXj
a 8e Ogsij + sgs sgs
xj,, j xk gk + pg,
SjpKpsij)+ pvpjKpsij )
xa j p psiji + Gsgs + p
Ox j \Y x p P pk P
J y P J
pg8g
The terms on the righthand side of Eqs.(7) and (8) have
the similar meanings as those in the twophase turbulent
kinetic energy equations in RANS modeling. For
example,
sgs [np pi ksgs
pg,
1
p =[(VgiVpi
Tr
ugs p (v p)anp
p np Oxi
2k~gSX
CYP Pifi
The SGS twophase velocity correlation v vp, i
determined by :
_(VgiVpi) + (Vk + Vpk) (Vgipi)
[(Ue + p) (VgiVpi)] +
Oxk Cxk
1 _
P [PgVpiVpi + PgVgigi (Pg +Pp)VgiVpi]
Pg'rp
, pi Vgk
pkgi k gkVpi X
C_ k SK k
s; 
g pig
kgs
(9)
LargeEddy Simulation of Swirling GasParticle
Flows
Swirling gasparticle flows with s=0.47, measured by
Sommerfeld and Qiu (1991) was simulated using LES with
the proposed USM SGS stress model. The geometrical
configuration and sizes of the swirl chamber are given in
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Fig.1.The inlet flow parameters are: central flow rate 9.9
g/s, annular flow rate 38.3 g/s, inlet Reynolds number
53256 and particle loading 0.034. The particles are glass
beads with a size of 60gum, the material density is 2500
kg/m3 and the mass flow rate is 0.34 g/s. The grid size is
taken as 1 mm and the grid number is about 100000 for a
half of the 2D computation domain..
The numerical method is an extension of the
singlephase SIMPLEC algorithm (Patankar, 1980) to
twophase flows. The LESUSM code is written in
Fortan77 language, consisting of 12000 statements. The
time step is taken as 106s. The convergence criteria for gas
and particle phases are 105. For boundary conditions, the
inlet twophase velocities, normal components of Reynolds
stresses and particle number density or volume fraction
are given by experiments, the shear stresses are given by
the eddyviscosity assumption ; The fullydeveloped flow
condition of two phases are taken at the exit ; At the walls
noslip condition is used for gas velocity, and the gas
Reynolds stress components as well as gas velocities are
determined via production term including the effect of wall
functions for nearwall grid nodes. Zero normal velocity,
zero gradients of longitudinal and tangential velocities and
normal components of Reynolds stresses, and zero mass
flux at the walls are used for the particle phase. At the axis
the symmetrical condition is adopted for both phases.
Periodic inlet fluctuations are superposed to the inlet
velocities
Figures 2 and 3 give the predicted gas and particle
timeaveraged axial and tangential velocities by both
LESUSM and RANSUSM modeling and their
comparison with the experimental data respectively. It can
be seen that both modeling results are in good agreement
with the measured results and there is only a slight
difference between these two modeling results for axial
velocities. For the twophase tangential velocities the
LESUSM results are somewhat better than the
RANSUSM results. Predictions can well give wshaped
axial velocity profiles with an annular reverseflow zone
and a typical Rankinevortex structure with solid body
rotation plus free vortex in tangential velocity profiles.
Since only the timeaveraged velocity field is of interest
for engineering design, one can conclude that the
RANSUSM modeling is good enough for engineering
applications.
Figures 4 and 5 give the gas and particle axial and
tangential RMS fluctuation velocities predicted by both
LESUSM and RANSUSM modeling and their
comparison with the experimental data respectively. It can
be seen that the LESUSM results are obviously better than
the RANSUSM results, in particular for the particle
fluctuation velocities.
The gasparticle fluctuation velocity correlation is an
important term in the RANS USM twophase turbulence
model. It represents the turbulence interaction between the
gas and particle Reynolds stresses. Figure 6 gives the
correlations (U'u') and (w;w') predicted by both
LESUSM and RANSUSM modeling. In general, the LES
and RANS modeling give the same trend, however, the
LES predicted values are higher than the RANS modeling
values. Comparing with Figs 4 and 5, it is seen that both
Paper No
LES and RANS modeled gasparticle correlation
distribution is similar in shape to that of gas and particle
RMS fluctuation velocities, but its values are smaller than
the gas and particle RMS fluctuation velocities.
Largeeddy Simulation of SuddenExpansion
GasParticle Flows
The kkp SGS energy equation model is used for
largeeddy simulation of suddenexpansion gasparticle
flows measured by Xu and Zhou (1999). The geometrical
configuration of the suddenexpansion chamber is shown
in Fig.7. The heavy particles are glass beads with an
average size of 30nm and material density of 2500kg/m3.
The length of the chamber is 1000mm. The inlet gas
volumetric flow rate is 212.4m3/h, and the particle mass
loading is 0.005. For LES the grid size is taken as 1mm,
the grid number in a half 2D computation domain is
60000. The time step is taken as 106s.
Figure 8 gives the LESkkp predicted twophase
axial timeaveraged velocities and their comparison with
the measurement results. It is seen that predictions are in
good agreement with the experimental results. Figure 9
shows the predicted twophase axial RMS fluctuation
velocities using both LESkkp and RANSUSM and their
comparison with the measurement results. Both modeling
results are in good agreement with the experimental results,
and the LES results are somewhat better than the RANS
modeling results. Figure 10 gives the predicted axial and
radial components of gasparticle velocity correlation
using LESkkp and RANSUSM and their comparison
with the measurement results. Both modeling results give
the same tendency in agreement with the experimental
results, and the LES predicted values are greater than the
RANS predicted values. In most regions the LES results
are closer to the experimental results than the RANS
modeling results. Comparing with Fig.9 it is seen that the
gasparticle correlation distribution is similar in shape to
that of gas and particle RMS fluctuation velocities, but its
values are smaller than the gas and particle RMS
fluctuation velocities.
Conclusions
(1) The proposed USM twophase SGS stress model and
twophase SGS energy equation model can fully
account for interaction between the gas and particle
SGS stresses.
(2) These twophase SGS models work well for LES of
gasparticle flows
(3) The LESUSM, LESkkp and RANSUSM modeling
can well predict timeaveraged gas and particle
velocity fields with only a slight difference among
them.
(4) For gas and particle RMS fluctuation velocities the
LESUSM and LESkkp results are better than the
RANSUSM results.
(5) For the gasparticle fluctuation velocity correlation
distribution, the LESUSM, LESkkp and
RANSUSM modeling give the same trend: its shape is
similar to that of gas and particle RMS fluctuation
velocities, but the former is smaller in values than the
latter. The LES predicted results are closer to the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
experimental results than the RANSUSM predicted
results.
(6) There is still some discrepancy between the LES
prediction results and the measurement results for gas
and particle RMS fluctuation velocities. This is due to
the shortcomings of 2D LES and the errors of the
secondorder difference scheme.
Acknowledgement
This study was sponsored by the Projects of National
Natural Science Foundation of China under the Grants
50606026 and 50736006.
References
Apte S V, Mahesh K, Moin P, Oefelein J C., Largeeddy
simulation of swirling particleladen flows in a
coaxialjet combustor, International Journal of
Multiphase Flow, 29:13111331 (2003)
Boileau, M., Pascaud, S., Riber, E., Cuenot, B., Gicquel, L.
Y. M., Poinsot, T. J., Cazalens, M., Investigation of
twofluid methods for large eddy simulation of spray
combustion in gas turbines, Flow, Turbulence and
Combustion, 80:291321 (2008)
Boivin M, Simonin O, Squires K., On the prediction of
gassolid flows with twoway coupling using large
eddy simulation, Physics of Fluids, 12:20802090,
(2000)
Germano M, Piomelli U, Moin P Cabot W. H., A dynamic
subgridscale eddy viscosity model, Physics of Fluids.
A3:17601765 (1991)
Kim W W, Menon S S., A new dynamic oneequation
subgridscale model for large eddy simulation, AIAA
950356 (1995)
Patankar, S. V, Numerical Heat Transfer and Fluid Flow,
Hemisphere, New York (1980)
Riber E, Moureau V, Poinsot T, Simonin O., Evaluation of
numerical strategies for large eddy simulation of
particulate twophase recirculating flows, Journal of
Computational Physics, 228:539564 (2009)
Smagorinsky J. General circulation experiments with the
primitive equation (I) : The basic experiment. Monthly
Weather Review, 91 (3) : 99164 (1963)
Sommerfeld M, Qiu H H., Detailed measurements in a
swirling particulate twophase flow by a phase Doppler
anemometer, International Journal of Heat and Fluid
Flow, 12: 2028 (1991)
Xiang P, Guo Y C., Modeling the hydrodynamics of dense
gasparticle flow in a riser, Journal of Engineering
Thermophysics (in Chinese), 25: 7579 (2" 14)
Xu, Y, Zhou, L X, Experimental studies of twophase
fluctuation velocity correlation in suddenexpansion
flows, 8th International Symposium on GasParticle
Flows, ASMEFED Summer Meeting, San Francisco,
CDROM, Paper FEDSM997909 (1999)
Zhou H S, Flamant G Gauthier D., Modelling of the
turbulent gasparticle flow structure in a
twodimensional circulating fluidized bed riser,
Chemical Engineering Science, 62:269280 (2007)
Zhou, L X, Dynamics of Multiphase Turbulent Reacting
Fluid Flows (in Chinese), Beijing, Defense Industry
Press, 105110 (2002)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Figure 1 The Swirl Chamber (Sommerfeld & Qiu)
(1 primary gasparticle flow; 2 annular swirling flow,
Dl=32mm, D2=38mm, D3=64mm, D4=70mm,
D5=194mm)
X=3mm X=52mm X=155mm X=195mm
o Exp LES RANSUSM
(a) gas
X=315mm
0  \
,, =! 1
5 0 5 10 15 0 5 10 0 5 0 5 0 5 10
X=3mm X=52mm X=155mm X=195mm X=315mm
a Exp LES RANSUSM
(b) particle
Figure 2 Twophase timeaveraged axial velocities
/ ,
\. /
I),
.I)"
ID,
I) i
X=3mm X=52mm X=155mm X=195mm X=315mm
o Exp LES RANSUSM
(b) particle
Figure 3 Twophase tangential timeaveraged
velocities
X=3mm X=52mm X=155mm X=195mm X=315mm
[ Exp LES RANSUSM
(a) gas
X=3mm X=52mm X=155mm X=195mm X=315mm
o Exp LES RANSUSM
(b) particle
Figure 4 Twophase axial RMS fluctuation velocities
Paper No
2 12
1 1 1
9i c
I \I j 1 I I
0 5 10 0 4 80 2 4 0 2 4 0 2 4
X=3mm X=52mm X=155mm X=195mm X=315mm
Exp LES RANSUSM
(a) gas
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
X=3mm X=52mm X=155mm X=195mm X=315mm
o Exp LES RANSUSM
(a) gas
X=3mm X=52mm X=155mm X=195mm X=315mm
n Exp LES RANSUSM
(b) particle
Figure 5 Twophase tangential RMS fluctuation
velocities
0 05 10 00 05 10 00 05 10 00 05 10 00 05 10
X=3mm X=52mm X=155mm X=195mm X=315mm
LES RANSUSM
(a) Axial
R (mm)
39 119 248 495
Di=120mm
Q=50mm 1
X(mm)
Figure 7 The SuddenExpansion Chamber (Zhou & Xu)
1 [
I \\
rv\ '
X=38.5mm X=119mm X=245mm X=494mm
Exp. LES
D 1I 2A 36 A 2 3D 0 1A 20 A 5
X=38.5mm X=119mm X=245mm X=494mm
Exp. LES
Figure 8 Twophase timeaveraged axial velocities
( Uppergas; Lowerparticle )
0 u
I ,,
0 I I I I I I
00 05 1000 05 10 00 05 10 00 05 1 0 00 05 10
X=3mm X=52mm X=155mm X=195mm X=315mm
LES RANSUSM
(b)Tangential
Figure 6 Gasparticle fluctuation velocity correlations
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
X=38.6mm X=119mm X=246mm X=494mm
Exp. LES RANKS
X=38.5mm X=119mm X=245mm X=494mm
Exp. LES RANS
Figure 9 Twophase RMS axial fluctuation velocities
(Uppergas; lowerparticle)
SExp. LES .RANS
I 2 0 2002
X=38.5mm X=119mm X=245mm X=494mm
Exp. LES RANS
Figure 10 Gasparticle velocity correlation
(Upperaxial; Lowerradial)
Paper No
1.i
DB
D2
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