Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 14.4.1 - A Priori Test of Effective Drag Modeling for Filtered Two-Fluid Model Simulation of Circulating and Dense Gas-Solid Fluidized beds
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 Material Information
Title: 14.4.1 - A Priori Test of Effective Drag Modeling for Filtered Two-Fluid Model Simulation of Circulating and Dense Gas-Solid Fluidized beds Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Özel, A.
Parmentier, J.F.
Simonin, O.
Fede, P.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: two-fluid model, LES, fluidized bed, drag force
LES
fluidized bed
drag force
 Notes
Abstract: Eulerian two-fluid approach is generally used to simulate gas-solid flows in industrial fluidized beds. Because of limitation of computational resources, simulations of large vessels are usually performed using too coarse mesh to capture the influence of the fine flow scales which can play an important role in the dynamic behaviour of the beds. In particular, neglecting the particle segregation effect at small scale leads to an inadequate modelling of the mean interfacial momentum transfer between phases. Then, an appropriate modelling approach which accounts for influences of unresolved structures has to be proposed for "coarse simulations". For this purpose, computational grids are refined to get mesh-independent results for a dense and a periodic circulating fluidized beds in which statistical quantities do not change with further mesh refinement. These mesh-independent results are filtered by volume averaging and then used to perform a priori analysis on the filtered drag term. Results show that filtered momentum equation can be computed on "coarse simulations" but must take into account the particle to fluid drift velocity due to the subgrid correlation between the local fluid velocity and the local particle volume fraction. In the present paper we propose a model, for subgrid the drift velocity, written in terms of the difference between the averaged of gas velocity weighted by solid volume fraction and the averaged of gas velocity weighted by gas volume fraction. We use a systematic procedure to provide constitutive closures for the subgrid drift velocity and the closure depends of both the filtered solid volume fraction and a characteristic filter size.
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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Bibliographic ID: UF00102023
Volume ID: VID00352
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 1441-Ozel-ICMF2010.pdf

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


A Priori Test of Effective Drag Mlodeling for Filtered Two-Fluid Mlodel Simulation of
Circulating and Dense Gas-Solid Fluidized Beds


A. iizel", J.F. Parmentier ', O. Simonind and P. Fededi

University de Toulouse: INPT, UPS: IMFT: 31400 Toulouse, France
SCNRS: Institut de M~canique des Fluides de Toulouse: 31400 Toulouse, France
ozeld imft.fr, parmend imft.fr, simonin a uft.fr and feded imft.fr

Keywords: Two-fluid model, LES, fluidized bed, drag force




Abstract

Eulerian two-fluid approach is generally used to simulate gas-solid flows in industrial fluidized beds. Because of
limitation of computational resources, simulations of large vessels are usually performed using too coarse mesh
to capture the influence of the fine flow scales which can play an important role in the dynamic behaviour of the
beds. In particular, neglecting the particle segregation effect at small scale leads to an inadequate modelling of the
mean interfacial momentum transfer between phases. Then, an appropriate modelling approach which accounts for
influences of unresolved structures has to be proposed for "coarse simulations". For this purpose, computational grids
are refined to get mesh-independent results for a dense and a periodic circulating fluidized beds in which statistical
quantities do not change with further mesh refinement. These mesh-independent results are filtered by volume
averaging and then used to perform a priori analysis on the filtered drag term. Results show that filtered momentum
equation can be computed on "coarse simulations" but must take into account the particle to fluid drift velocity due
to the sub grid correlation between the local fluid velocity and the local particle volume fraction. In the present paper
we propose a model, for subgrid the drift velocity, written in terms of the difference between the averaged of gas
velocity weighted by solid volume fraction and the averaged of gas velocity weighted by gas volume fraction. We use
a systematic procedure to provide constitutive closures for the subgrid drift velocity and the closure depends of both
the filtered solid volume fraction and a characteristic filter size.


Nomenclature


117,; sub grid drift velocity
Subscripty
<; Gas
p Particle


Symbols
Si

6;y
As


gravity
mean volume fraction of phase k
Kronecker delta
bulk viscosity of phase k
viscosity of phase k


Introduction

Numerical analysis of fluidized beds by Eulerian two-
fluid approach appear to be a powerful tool to improve
design and performance of industrial facilities. How-
ever, in most industrial applications involving large de-
vices, because of limited computational resources, two-
fluid model equations for unsteady gas-particle flows
in dense and circulating fluidized bed risers are rou-
tinely simulated over too coarse spatial grids which can-
not resolve all the fine-scale structures. Comparisons
with fine-grid simulation results showed that unresolved
structures of gas-particle fluidized bed can have a dras-


<1, particle diameter
e, elasticity coefficient during inter-particle collision
Isinterphase momentum exchange
P, gas pressure
q~i disperse phase fluctuations
Re particle Reynolds number
7,, characteristic time scale of
gas-particle momentum transfer
us,; velocity ofphase k
Ux, = ux,) Amean velocity of phase k
11; mean relative velocity







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


tic influence on the flow dynamic due to the inadequate
modeling of the resolved drag term coupling the gas and
particle resolved momentum equations (Andrews et al.
2005). To account for the effect of unresolved struc-
tures on the macroscopic behavior in coarse grid simula-
tions, suitable sub grid model can be proposed to model
accurately the effective drag term. In this study, a pri-
ori analysis of the effective drag modeling is performed
by spatial filtering of highly resolved 2D and 3D nu-
merical simulations of both dense and periodic circu-
lating fluidized beds. The numerical simulations have
been carried out using an unsteady Eulerian multi-fluid
approach implemented in the unstructured parallelized
code NEPTUNE CFD V1.ll' &Thei NEPTUNE CFD
is a multiphase flow software developed in the frame-
work of the NEPTUNE project, financially supported by
CEA (Commissariat g l'Energie Atomique), EDF (Elec-
tricit6 De France), IRSN (Institut de Radioprotection et
de Stiret6 Nucl~aire) and AREVA-NP.


as follows:


i3 i3

( i)P ?)~i
-at -- p ;+ A;+ R


with P, the mean pressure, 9; acceleration due to
gravity, Ex,;y is the effective stress tensor, IA~i is the
mean transfer momentum transfer rate after substraction
of the mean gas pressure effect.

Interfacial Momentum Transfer
The term It s accounts for momentum transfer rate
between the gas phase (carrier or continuous phase) and
particle (disperse) phase. This term can be modeled by
taking into account only the drag force between phases:


Qppp
7
Qp


Mathematical Modeling Approach


The modelling approach is based on the two-fluid
model formalism that involves mean separate transport
equations of mass, momentum and energy for each
phases. Interactions between phases are coupled
through interphase transfers. The transport equation for
disperse phase fluctuations, q~ developed in the frame
of kinetic theory of granular media supplemented by
the interstitial fluid effect and the interaction with the
turbulence (Balzer et al. (1995), Gobin et al. (2003)),
are resolved with taking into account inter-particle
collisions on the dispersed phase hydrodynamic. The
effect of the fluctuations of the gas velocity at small
scales is neglected. Concerning the transfers between
the phases with non-reactive isothermal flow, drag
force was only taken into account for the transfer of
momentum. Neither terms of added mass nor lift force
were considered.


Transport Equation
The two-fluid mass balance equation for the phase k is
written:


(asPk)+- (asPaLT<,;)= 0 (1)
at dir

with asi, p e, ET the volume fraction, the density and the
mean velocity of phase k (when subscript k = g, we
refer to the gas and k = p to the particle phase). The
momentum balance equation for the phase k is defined


with the particle relaxation time scale, 7,'
aMI, di V | and (.) the ensemnble aver-
ag~e orator over the particle (Simonin 1996).
The mean drag coefficient of a single particle,
(C D) can be written as function of a particle
Reynolds number and defined by Wen & Yu (1966),
Dp)i -24/(Rr p) 1+ 0.15(Reg~6i Qr) a with
the definition of the mean particle Reynolds number,
(Re p) = ad,(|. |\p/v where d, is the particle
diameter and v, is the molecular viscosity of gas,
respectively. In this study, only the Wen&Yu correlation
is used while the combination of Ergun's and Wen&Yu
correlations is generally employed in fluidized beds
(Gobin et al. 2003). The term, (|vr )p, in (3) represents
the local instantaneous relative velocity, v,,;, and is
equal to difference between the local particle velocity,
undisturbed by presence of the particle at the particle
position. V,,; is the averaged of the local relative
velocity and equal to (il,,; up,;) p. It can be expressed
in terms of the averaged velocity between phase,


Effective Solid Stress Tensor
The effective stress tensor for particle phase has two
contributions. The first contribution is the kinetic stress
tensor which represents the transport of the momentum
by the particle velocity fluctuations. The second one
is the collisional stress tensor which accounts for
destruction and exchange of the momentum during
inter-particle collisions. The constitutive equation of the







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


Free outlet


effective stress tensor is:


1F: 17, iif 6
i~"' i


OWall
g Periodicity


H=0 22 m


1


~sij~] (4)


Walls


.oslmii'~


The constitutive relations for viscosity and diffusivity
are derived in frame of the extension of kinetic theory
of dry granular flow and given by Balzer et al. (1995).
The collisional pressure and the bulk viscosity are writ-
ten according to Balzer et al. (1995) as follow:

PP = 3apPPq ~[1+299,,(1+ec)]

Xp n pylppun(1 + ec) 2q (5)
3 V 3 x

with the random kinetic energy of particles, q, the resti-
tution coefficient, e 0, that determines energy loss dur-
ing inter-particle collisions and go is the pair correla-
tion function. The shear viscosity is the linear combi-
nation of the collisional and the kinetic stress: ps =
a~ppp [va~n, + y<>i] where



21~p
c 2


g it Aag (1+ec v'" + cld q (6)
5 ; 3 x

with ac = (1+ e ) (3 e ) and Ae =
(1 c)(30c -1).Such closure laws were also
used by Agrawal et al. (2001) in the case of 2D-periodic
flows.

Flow Configurations

Gas-particle flows were simulated for two configura-
tions: a two-dimensional dense and a three-dimensional
periodic circulating fluidized bed. Typical FCC parti-
cles, <1, 75 pm, pp 1500 ky/m3, are interact-
ing with the ambient gas (p, 1.186 ky/m3, p,
1.8 x 10-") for both cases. The computational domains
of cases are shown in Figure 1. The dense fluidized bed
is initialized by the homogenous distribution with the
particle volume fraction equal to 0.55 and the fluidiza-
tion velocity, Uf, is set to 0.2 at/s. The periodic cir-
culating fluidized bed is initialized by the homogenous
distribution of the particle volume fraction equal to 0.05
and very small fluctuations applied on the particle vol-
ume fraction to destabilize the bed. The flow in the pe-
riodic fluidized bed is driven by the pressure gradient


Uniform gaz velocity


Figure 1: The computational domain of two-
dimensional dense and three-dimensional periodic cir-
culating fluidized bed.



due to the total mass which is defined as the opposite di-
rection to the gravity. For both cases, no-slip condition
were imposed at the walls.



Mesh Independent Results


Agrawal et al. (2001) stated that statistics quantities over
the whole domain is strongly dependent on the mesh size
and they became mesh-independent when mesh sizes
are the order of few particle diameters. In this study,
the mesh refinement studies were carried out to be com-
pletely sure that the mesh resolution is sufficient and all
spatial and temporal scales of particle and gas phases
are captured. Figure 2 shows different instantaneous
particle volume fraction fields in the dense and periodic
circulating fluidized bed obtained by numerical simula-
tions using different mesh sizes. As the resolution of
mesh increases, inhomogenous structures are better re-
solved. These structures have a drastic influence on the
height of dense bed and the solid holdup of periodic bed.
The influence of mesh size on the time averaged bed
height of dense fluidized bed and the vertical solid flux,
# ff a, Up clS where the integral is performed on a
horizontal surface, in the periodic circulating fluidized
bed is shown in Fig. 3. Dramatic changes of macro-
scopic behaviors of beds can be attributed to poor pre-
diction of drag force. These converged results can be
called "DNS" results by analogy with single phase flows.
They are statistically sufficient and not needed to have
further mesh refinement, then used to perform a priori
analysis on the drag term filtered by volume averaging
to propose subgrid model for the modeling of the effec-
tive drag term.


'7=0 0275 m
L=0 0275 m








































.
0 0. .


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


Figure 2: Instantaneous particle volume fraction field in the dense (le f t) and periodic circulating fluidized bed (rig ht)
for different grid mesh sizes. >From left to right, the mesh resolution increases for both cases. White color corresponds
to ap = 0. Black color corresponds to ap = am = 0.64.


0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0


0.14
0.12
0.1
0.08
0.06
0.04
0.02

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Figure 3: The influence of grid size on the height of dense fluidized bed (le ft) and the vertical solid flux in the
periodic circulating fluidized bed brightt), 0: h~omogenlous case, conv~:: converged case, A/(ist)ag: non-dimensional
mesh size with characteristic mesh size, (AxAyAz)l/ and Stokes' relaxation time, 7,st


Filtered Two-Fluid Model Equations and


The two-fluid model equations are spatially averaged
over some chosen filter length scales. Let a,(x, t) de-
notes the particle volume fraction at location, x, and
time, t, obtained by solving the two-fluid equations. We
can define the filtered particle volume fraction 0, (x, t)



ap,(x, t) Glx Ciu) a(ut) du (7)

where G is a weight function which satisfies
fff G(u) du 1 Filtered phase velocities
are defined according to


U,(xi) =X G- u) a,(u, t) U (u, t) du
as . .(9)
Applying such a filter to the continuity equations of
phases, one can obtain


8agpa pkcasUlc4


0 (10)


Upx t) = GJ )a(,t pu )d Repeating this filtering process to momentum balances,
(8) the filtered momentum balance for particle and gas







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


as filtered particle volume fraction, gas and particle ve-
locities at several time instants. We carry out correlation
analysis between the filtered drag term and the filtered
variables. Finally, we perform conditional average of the
filtered drag term for given values of the filtered particle
volume fraction. We propose to decompose the filtered
in two contributions: the difference between filtered gas
and particle velocity Up,4 Up,s and a sub grid drift ve-
locity Vd,i defined by


phase are

i3i3

dP,
-agc- (P + arPpagg + Irc g +


8xy 8xyprcc~rar~ij (11)

Additional terms arise in (11) due to the filtering pro-
cess and require closure models. The term, cpi, repre-
sents the contribution between fluctuations of the vol-
ume fractions of phases and gas pressure and defined as:

dP, dP,
(Pi =r an 8 ag 8xy (12)

The modeling of this term was proposed by De Wilde
(2005) with the introduction of the global added mass
coefficient. A Reynolds stress-like contribution com-
ing from the gas or particle phase velocity fluctuations,
arc~ij, is defined by the following equation:

Ercarc,ij = ag Uc,iUlc,j ag Elc,i1p,j (13)

This term can be modeled by using Boussinesq approx-
imation with introducing the subgrid viscosity, psy,,~,
for the gas phase:


appp pppy ~ p.


~ ~
U + Va
gi -


This decomposition allows us to identify physically the
meaning of sub grid drift velocity, and then to understand
the origin of the difference between the filtered and re-
solved drag force. A correlative analysis has shown that
the filtered drag force can be approximated by the fol-
lowing epxression.

Qpp' V apV17
7p ,i --pr,i (7

Both sides of (17) are correlated with more than 99 %,
even for large filter size. It shows that the subgrid mo-
mentum transfer is occurred by the filtered relative ve-
locity averaged by particle volume fraction, a,V,,s. Us-
ing (16) and (17), the subgrid drift velocity is approxi-
mated by:


3 8xm


V4,4 g~p~i- g(i8)
where Ugep~i =pUg,~i / op iS the filtered gas velocity
seen by the particle phase.


where 6ij is the Kronecker delta. The subgrid viscosity
can be modelled by sub-grid scale model proposed by
Deardoff (1971):

Psas,9 = a~~Pp(ctA)ay22/SjSi (15)

where a = (AxAyAz)l/ and Sy,j,
4 [+i~dzj1'. This term for disperse phase
was investigated by Moreau (2005) with mesoscopic
Eulerian approach. The term, I,, is the filtered drag
term and we focus on this term in this study. To define
variables contributed to the subgrid drag term, correla-
tion analysis can be performed between the filtered drag
term and computed variables. To fill up the main goal,
suitable model accounted for main contributions of the
subgrid drag term will be proposed by the definition
of the effective drag term. We present a priori test on
the filtered drag term and briefly show primary results
obtained by the under-developing model.

A Priori Analysis on the Filtered Drag term

We apply top hat volume filters with various sizes on lo-
cal instantaneous drag term and computed variables such


Modeling of the subgrid drift velocity
Heynderick (2l1***4, Andrews et al. (2005) and Igci
(2007) introduce an effective drag coefficient Pe to
express the filtered drag force term as


appp ri=# 7~ ~


Heynderick (21 *4) and Andrews et al. (2005) write the
effective drag coefficient as a function of the filtered par-
ticle volume fraction while Igci (2007) suggest that this
coefficient is a function of the filter size. We propose
to write the filtered drag force by modeling the subgrid
drift velocity as:


Vas = g(A, a,) ET,,s


rET,,4)


with a function g that will be determined by the condi-
tional averaging procedure applied on the high resolu-
tion simulations. Then, the effective drag term can be
written as follows:


appy pp


ET,, (21)


agge,ij = psys,,g 8 Us,jdz 8Ug,idz













0


~-0.25
0 0.2 0.
Fitrdsld ouofato


Figure 4: The function "y" for dense fluidized bed
casefor different filter sizes: 0 : 11A/ADNS, O :
15A/ADNS, o2 2 DDN.



The function "9"
The function "g" can be calculated by the ratio between
the sub grid drift velocity and the difference between the
filtered gas and particle velocities which are condition-
ally averaged by the filtered volume fraction of the dis-
persed phase. This allows us to derive the following re-
lation:


(Mid.


9 a, Gr)


The computed values of the function g are shown for
dense and periodic circulating fluidized bed in Figure
4 and Figure 5. The function "g" can be written as a
product of two functions: h (o,) (for the filtered vol-
ume fraction dependance) and f (A) (for the filter size
effect).

U(a, Oap) = f (a) h(op) (23)

>From our database, we propose the following form of
the function h:

h(K4) = [iK,-" '(ap,,, a (24)

with contents m and I which are equal to 1.2 and 1.6 ,
respectively. The function h is shown for different filter
sizes in Figure 6. The function f is modeled as the
following equation:


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


-0.1
-0.2
-0.3
-0.4
-0.5
S-0.6. x
-0.7 x
-0.8 x


0 0.05 0.1 0.15 0.2 0.25



Figure 5: The function "g" for periodic circulating flu-
idized bed case for different filter sizes; : 3A/ADNS,
x :5A/ADNS,* /DNS 09/DNS.


1.8
1.6 *
1.4 -
1.2

0.8
0.6
0.4
0.2 f(A)
0 1, Proposed f(A)
0 0.2 0.4 0.6 0.8 1 1.2 1.4



Figure 6: The function f". Symbols stand for the
measured values in highly resolved simulation and the
dashed line is (25).


A Posteriori Test

To validate the model, a posteriori test has been done
with two-dimensional dense fluidized bed. The non-
dimensional bed heights with and without model for
coarses meshes are shown in Figure 7 and results are
promising. Herein, we present results briefly because the
non-dimensional mesh size, A/(TS') 9, is questionable
parameter and more detailed analysis have to be done by
different type of particles and different bed widths to be
completely sure that this parameter represents physical
interpretations.


IP
( S) 9


Conclusions


f (a)= 1.75


The global dynamics of fludized beds are strongly de-
pendent on the mesh size and simulations conducted by
coarse meshes can not predict accurately the statistics


















n a
.


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


References

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Coarse Grid Simulation of Gas-Particle Flows in Ver-
tical Risers, Ind. Eng. Chem. Res., Vol. 44, pp. 6022-
6037, 2005

Balzer G., Boelle A., Simonin O., Eulerian Gas-Solid
Flow Modelling of Dense Fluidized Bed, FLUIDIZA-
TION VIII, Proc. International Symposium of the Engi-
neering Foundation, pp 409-418, 1995.

K. Agrawal, P. N. Loezos, M. Syamlal, and S. Sundare-
san, The Role of Mesoscales Structures in Rapid Gas-
solid Flows, J. Fluid Mech. 445, 151, 2001.

J. De Wilde, Reformulating and Quantifying the Gener-
alized Added Mass in Filtered Gas-solid Flow Models,
Plws. Fluids 17, 113304, 2005.

O. Simonin, Continuum Modelling of Dispersed Two-
Phase Flows, in Combustion and Turbulence in Two-
Phase Flows, Lecture Series 1996-02, von Karman In-
stitute for Fluid Dynamics, Rhode Saint Gendse (Bel-
gium), 1996.

W. Wang, J. Li, Simulation of Gas-solid Two Phase Flow
a Multi-scale CFD Approach Extension of the EMMS
model to the sub-grid level, Chem. Eng. Sci. 62, 208,
2007.

G. J. Heynderickx, A. K. Das, J. De Wilde, and G. B.
Marin, Effect of Clustering on Gas-solid Drag in Dilute
Two-phase Flow, Ind. Eng. Chem. Res. 43, 4635, 2004.

Y. Igci, A. T. Andrews IV, S. Sundaresan, S. Pannala,
T. O'Brien, Filtered Two-fluid Models for Fluidized Gas
Particle Suspensions, AIChE J. 54, 1431, 2008.

C. Y. Wen, Y. H.Yu, Mechanics of Fluidization, Chem.
Eng. Prog. Sym. Ser., 62, 100-113, 1966.

J. W.Deardoff, On the Magnitude of the Subgrid Scale
Eddy Coefficient, J. Comp. Phy. 7, 120-133, 1971.

Moreau, M., Simonin, O., Bedat, B., Development
of Gas-Particle Euler-Euler LES Approach: A Priori
Analysis of Particle Subgrid Models in Homogeneous
Isotropic Turbulence, Flow Turbulence Combust. 84,
295-324, 2010.

Gobin A., Neau H., Simonin O., Llinas J.R., Reiling
V, Selo J.L., Fluid Dynamic Numerical Simulation of
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cal Methods in Fluids 43, 1199-1220, 2003.


0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02


0.5 0.6 0.7 0.8


1 1.1


Figure 7: The influence of grid size on the height of
dense fluidized bed: with model a, without model O.



quantities in the beds. This poor predictions of gas-solid
flows in beds can be attributed to lack of an appropri-
ate model of the drag term for coarse mesh simulations.
For this propose, the mesh refinement studies were car-
ried out for two-dimensional dense fludized and three-
dimensional circulating fludized bed to get mesh inde-
pendent results. Then, these results are used to perform
a priori analysis on drag term and the subgrid contribu-
tion of drag is presented with plwsical explanation. It is
stated that the filtered momentum equation can be com-
puted on "coarse simulations" but must take into account
the particle to fluid drift velocity due to the subgrid cor-
relation between the local fluid velocity and the local
particle volume fraction. We propose a model for the
subgrid drift velocity which can be written the differ-
ence between the averaged of gas velocity weighted by
solid volume fraction and the averaged of gas velocity
weighted by gas volume fraction. This model is a func-
tion of the filtered particle volume fraction and the non-
dimensionless mesh size. A posteriori test is performed
on two-dimensional dense fluidized bed and results have
good agreement. However, the non-dimensionless mesh
size is the questionable part of the approach and it has to
be investigated by different flow configurations. Addi-
tion that, closures are still missing for terms in (12) and
(13) and it will be the object of further work.




Acknowledgements


This work was granted access to the HPC resources
of CINES under the allocation 2010-026012 made by
GENCI (Grand Equipement National de Calcul Inten-
sif) and of CALMIP under the allocation P0 111 (Calcul
en Midi-Pyr~ndes ).




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