Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 2.4.4 - A Fully Coupled Fluid-Particle Flow Solver Using a Quadrature-Based Moment Method with High-Order Realizable Schemes on Unstructured Grids
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00055
 Material Information
Title: 2.4.4 - A Fully Coupled Fluid-Particle Flow Solver Using a Quadrature-Based Moment Method with High-Order Realizable Schemes on Unstructured Grids Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Vikas, V.
Wang, Z.J.
Passalacqua, A.
Fox, R.O.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: kinetic theory of granular flow
QMOM
multiphase flow
unstructured grid
high-order realizable
 Notes
Abstract: Kinetic Equations containing terms for spatial transport, gravity, fluid drag and particle-particle collisions can be used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the problem in terms of moments of the velocity distribution function. A quadrature method of moments (QMOM) was derived by Desjardins et al. 1 for approximating solutions to the kinetic equation for arbitrary Knudsen number. Fox 2, 13 derived a third-order QMOMfor dilute particle flows, including the effect of the fluid drag on the particles. Passalacqua et al. 4 and Garg et al. 3 coupled an incompressible finite-volume solver for the fluid-phase and a third order QMOM solver for particle-phase on Cartesian grids. In the current work a compressible finite-volume fluid solver is coupled with a particle-phase solver based on third-order QMOM on unstructured grids. The fluid and particle-phase are fully coupled by accounting for the volume displacement effects induced by the presence of the particles and the momentum exchange between the phases. The success of QMOM is based on the moment inversion algorithm that allows quadrature weights and abscissas to be computed from the moments of the distribution function. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. Desjardins et al. 1 showed that realizability is guaranteed only with the 1st-order finite-volume scheme that has excessive numerical diffusion. The authors 5, 6 have derived high-order finite-volume schemes that guarantee realizability for QMOM. These high-order realizable schemes are used in this work for the particle-phase solver. Results are presented for a dilute gas-particle flow in a lid-driven cavity with both Stokes and Knudsen numbers equal to 1. For this choice of Knudsen and Stokes numbers, particle trajectory crossing occurs which is captured by QMOM particle-phase solver.
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: VID00055
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 244-Vikas-ICMF2010.pdf

Full Text



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


A fully coupled fluid-particle flow solver using quadrature-based moment method
with high-order realizable schemes on unstructured grids


V. Vikas*, Z.J. Wang*, A. Passalacquat and R.O. Foxt

Department of Aerospace Engineering, 2271 Howe Hall, Iowa State University, Ames, IA 50011, USA
t Department of Chemical and Biological Engineering, 2114 Sweeney Hall, Iowa State University, Ames,
IA 50011, USA
wikas@iastate.edu, zjw@iastate.edu, albertop@iastate.edu and rofox@iastate.edu
Keywords: Kinetic theory of granular flow, QMOM, multiphase flow, unstructured grid, high-order realizable




Abstract

Kinetic Equations containing terms for spatial transport, gravity, fluid drag and particle-particle collisions can be
used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical
simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the
problem in terms of moments of the velocity distribution function. A quadrature method of moments (QMOM) was
derived by Desjardins et al. [1] for approximating solutions to the kinetic equation for arbitrary Knudsen number.
Fox [2, 13] derived a third-order QMOM for dilute particle flows, including the effect of the fluid drag on the particles.
Passalacqua et al. [4] and Garg et al. [3] coupled an incompressible finite-volume solver for the fluid-phase and a
third order QMOM solver for particle-phase on Cartesian grids. In the current work a compressible finite-volume
fluid solver is coupled with a particle-phase solver based on third-order QMOM on unstructured grids. The fluid and
particle-phase are fully coupled by accounting for the volume displacement effects induced by the presence of the
particles and the momentum exchange between the phases. The success of QMOM is based on the moment inversion
algorithm that allows quadrature weights and abscissas to be computed from the moments of the distribution function.
The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative
weights. Desjardins et al. [1] showed that realizability is guaranteed only with the 1st-order finite-volume scheme that
has excessive numerical diffusion. The authors [5, 6] have derived high-order finite-volume schemes that guarantee
realizability for QMOM. These high-order realizable schemes are used in this work for the particle-phase solver.
Results are presented for a dilute gas-particle flow in a lid-driven cavity with both Stokes and Knudsen numbers equal
to 1. For this choice of Knudsen and Stokes numbers, particle trajectory crossing occurs which is captured by QMOM
particle-phase solver.


Introduction

Gas-particle flows are relevant in many engineering ap-
plications. A detailed understanding of such flows is es-
sential to the improvement of these applications. Cur-
rently, there exist several different ways for numerical
simulation of gas-particle flows. All of them use the
same fluid solver. They differ in the way in which parti-
cle phase is treated:

1. Direct solver that discretizes velocity phase space
of particle number density function [7, 8].

2. Lagrangian solver that tracks all the particles indi-
vidually [9].


3. Hydrodynamic models with kinetic theory moment
closures [10].

4. Quadrature Method Of Moment (QMOM) solver
that solves for moments of particle number den-
sity function with quadraure-based closures [1, 2,
4, 11].

A direct solution of the kinetic equation is pro-
hibitively expensive due to the high dimensionality of
the space of independent variables, while Lagrangian
solvers are computationally very expensive for many en-
gineering and industrial applications, since the number
of particles to be tracked is very large. Hydrodynamic
models are developed assuming that the Knudsen num-











ber of the flow is nearly zero, which is equivalent to as-
suming a Maxwellian (or nearly Maxwellian) equilib-
rium velocity distribution. This, however, is not correct
in relatively dilute gas-particle flows, where the Knud-
sen number is high, the collision frequency is small and
phenomena like particle trajectory crossing can happen.
In particular, Desjardins et al. [1] showed that the as-
sumption that a gas-particle flow can be described by
accounting for only the mean momentum of the parti-
cle phase leads to incorrect prediction of all the velocity
moments, including the particle number density, show-
ing the need of using a multi-velocity method, in order
to correctly capture the physics of the flow.
QMOM for gas particle flow [2, 12, 13] is based on
the idea of tracking a set of velocity moments of arbi-
trarily high order, providing closures to the source terms
and the moment spatial fluxes in the moment transport
equations by means of a quadrature approximation of
the number density function. Fox [2, 13] derived a third
order QMOM for dilute particle flows, including the ef-
fect of the fluid drag on the particles. Passalacqua et
al. [4] and Garg et al. [3] coupled a third order QMOM
solver with an incompressible finite volume solver for
the fluid-phase on Cartesian grids In the current work,
a compressible finite-volume fluid-phase solver is cou-
pled with a particle phase solver based on third-order
QMOM on unstructured grids. The fluid and particle
phases are fully coupled by accounting for the volume
displacement effects induced by the presence of the par-
ticles, and accounting for the momentum exchange be-
tween the phases.
The key to the success of QMOM is an inversion al-
gorithm which allows to uniquely determine a set of
weights and abscissas from the set of transported mo-
ments. Condition for the inversion algorithm to be ap-
plied is that the set of moments is realizable, meaning it
actually corresponds to a velocity distribution. This con-
dition is not generally ensured by the traditional finite-
volume methods used in computational fluid dynamics.
Desjardins et al. [1] showed that realizability is guaran-
teed only with the lst-order finite-volume scheme. But
the lst-order finite-volume scheme has excessive numer-
ical diffusion. The authors [5, 6] have recently derived
high-order finite-volume schemes that guarantee realiz-
ability for QMOM. These high-order realizable schemes
are used in this work for the particle phase solver.
The remainder of the paper is organized as follows.
First the governing equations for the fluid and particle
phases are described. Then the details of the two solvers
and the coupling algorithm are briefly explained. Fi-
nally, numerical results are presented for a dilute gas-
particle flow in a lid-driven cavity. For simplicity quan-
tities with subscript will be associated with fluid-phase.
For the particle-phase subscript p may or may not appear


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


explicitly. Also, any repetition of variable indices will
denote summation as per Einstein notation.


Fluid-phase governing equations

The fluid-phase is described by Navier-Stokes equations
modified for multi-fluid models. The fluid-phase conti-
nuity, momentum and energy equations are given as:


Wf dHfj (Wf)
at +
-- + i~z
O Oxj


OHj (Wf)
S + Sf.
dx


In (1), Wf, H (Wf), HU (Wf) and Sf denote the set of
conserved variables, inviscid fluxes, viscous fluxes and
source terms respectively. These terms are given by


Wfe




Hfj(Wf) [-
a


afpfU

afpfEfI


QfPfUfj
f (pfUf UfU + pf)
f (pfEf + pf) UfJ


0
Hj (Wf) f j ,
a(fij Ufi


Sf [


0
Mfpi + afgi
Qfp


In (2)-(5), af, pf, Ufi and pf are fluid-phase volume-
fraction, density, velocity components and pressure re-
spectively. The total energy Ef can be written as:


E Pf 1
(b 1)pf 2


where 7 is the ratio of specific heats. In (4), the compo-
nents of the viscous stress tensor afij are given by


fij [ {f (OUf + %Uf


2 8Uf
3 8xk


where pf is the fluid dynamic viscosity and ,ij denotes
Kronecker delta. The body force due to gravity is ac-
counted for by afgi. For the current work, gravity is not
considered. The other two source terms, Mfpi and Qfp
account for momentum and energy exchange between
the fluid and particle phases. Details about these two
source terms will be discussed in a later section.











Particle-phase governing equations

Kinetic equation. Dilute gas-particle flows canbe mod-
eled by a kinetic equation [14, 15, 16] of the form:

Otf + v axf + av (fF) = C, (8)

where f(v, x, t) is the velocity based number density
function, v is the particle velocity, F is the force act-
ing on individual particle, and C is the collision term
representing the rate of change in the number density
function due to collisions. The collision term can be de-
scribed using Bhatnagar-Gross-Krook (BGK) collision
operator [17]:
C (feq f), (9)
TC
where T, is the characteristic collision time, and feq is the
Maxwellian equilibrium number density function given
by:

Jl" IV Up12)
feq(V) exp (10)
7TTCeq)"3 \ 1eq)
in which Up is the mean particle velocity, ,eq is the
equilibrium variance and I'" f fdv is the particle
number density. In fluid-particle flows, the force term is
given by the sum of the gravitational contribution and
the drag term exerted from the fluid on the particles.

Moment transport equations. In the quadrature-
based moment method of Fox, a set of moments of
number density function f are transported and their
evolution in space and time is tracked. Each element
of the moment set is defined through integrals of the
velocity distribution function. For the first few moments
the defining integrals are:

/"' I fdv,

M = vi fdv,

= f dv,

M34 ,= fdv.

In these equations, the superscript of M represents the
order of corresponding moment. The particle-phase vol-
ume fraction a, and mean particle velocity Up are re-
lated to these moments by:

ap V .1 (12)

and
ptpcUppi mpMA (13)
where m, ppVp is the mass of a particle with density
p, and volume Vp. For 2D cases, Vp 7rd2/4 and for


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


3D cases, Vp 7rd /6. Likewise, the particle temper-
ature is defined in terms of the trace of the particle ve-
locity covariance matrix, which is found from 1 [-, and
lower-order moments. By definition, ap + af = 1 and
this relation must be accounted for when solving a fully
coupled system for the fluid and particle phases.
Moment transport equations are obtained by applying
the definition of moments to (8). The transport equations
for moments in (11) can be written as:


aW, aHp,(Wp)
t + OHx Dp + G + Cp.
8Ot 0x1


In (14), Wp and Hp (Wp) are the conserved moments
and spatial fluxes respectively and are given as:


M1
Wp, = I (15)



Mill
M2
H p(W,) = M I (16)
M4

The source terms on right hand side of (14), Dp, Gp and
Cp respectively denote drag, gravity and collision terms
and can be written as:


0
D1

Di


0

g~I? + gjM +
.1 + j+ gM .gkMij


0
Cp, 2j (19)

Cijk

Gravity is not considered in the current work. Hence,
Gp 0. The details of drag and collision terms will be
discussed later.
According to the third order QMOM derived by
Fox [2, 13], following set of moments are transported
in 2D and 3D respectively:

W2D 1/", M 2,M1 M M2 M M3
p 2M 1, 12 122, (ll 0
M3 M3 3
112, 122, 222]


Gp















W3D mI, M M, M31, M2, 2 23,

Mpt2, Mj%3 A1\ 11, Aj 12, 1s(3, 2
M3 M3 M3 M3 M3
122 123 133 222 223
^233, 333]'



For simplicity, hereinafter we will assume that all of the
moments have been multiplied by Vp, so that the zero
order moment corresponds to the particle-phase volume
fraction i.e. /" a= p. This simplification helps in
handling of coupling terms.

Quadrature-based closures. Using the BGK
model [17], the collision terms in (19) can be closed.
Details of closure of collision terms can be found
in [4]. However, the set of transport equations in (14)
is still unclosed because of the spatial flux and drag
terms. Each equation contains the spatial fluxes of the
moments of order immediately higher. In quadrature-
based moment methods, quadrature formula are used to
provide closures to these terms in the moment transport
equations, by introducing a set of weights and abscissas.
The number density function f is written in terms of the
quadrature weights (n) and abscissas (Ua) using Dirac
delta representation:


3
f(v) -L n(v
a=l


U,).


The method based on (22) is called 3-node quadrature
method. The moments can be computed as a function
of quadrature weights and abscissas by using the above
definition of f in (7):


_ll" = ^"a
a=l
f3

M12 E nagiUaj,
asl
3
1[e. 1:naUatUaj,
asl
3
Mt -L anUatUajUak-
a-l


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


The source terms in (17) due to drag are computed as:
/3
a=: n"F p,


Dij L n (F aUja + FjaUta), (24)

3
D 3 n -(F.FiUjUk. + FjiUkaUa+

FkaUt Usa),
where the drag force term F,, is given by

Fia = (Uf, Ut ). (25)
Td
In (25), the drag time Td is given by

T 4dppp (26)
3acfpfCdlUf Ual
The drag coefficient Cd is given by Schiller and Nauman
correlation [18]:


24
fRep =
afRepo


[1 + 0.15(af Repa)0"687] f2"65, (27)


in which Rep pfdplUf Ual/pf. The coupling
source terms for the fluid-phase in (5) are given by:
n3
Mfpi =- Q ,Fi) (28)



Qfp E Fa, ,) (29)

The next few sections discuss the details of the fluid
and particle phase solvers and the coupling between
them. Although, for the numerical simulations a two-
stage Runge-Kutta scheme is used, for simplicity, all the
discussion on solver details and coupling algorithm will
be based on a single-stage time-integration.

Fluid-phase solver

Let I and DI denote any cell in the domain and its
boundary respectively. Also let e E DI be a face of cell
I, A, be its area and 1,b, be the neighboring cell cor-
responding to this face. The finite-volume scheme using
single-stage explicit time-integration for (1) can be writ-
ten as:
1 Wf& V0 olEt Gf ( 1fw We ) A}


=1 5 {AGC (Wn1 WfnW ) A6}


AtSf,







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


where W' and WV1 are the cell averaged values
while W and W are the values reconstructed
on different sides of the face e. Also, voli denotes the
volume of cell I. In (30), Gf and G' denote numerical
inviscid and viscous fluxes respectively. Roe flux [19]
is used to calculate Gf. For calculation of viscous flux,
gradient of velocity field is required which is obtained
using a least-squares linear reconstruction [20, 21]. In
the current work, results are presented using 1st-order
and 2nd-order finite-volume schemes. For the 1st-order
finite-volume scheme, a piecewise-constant reconstruc-
tion is used i.e. W I W'. For the 2nd-order finite-
volume, a least-squares linear reconstruction is obtained
using cell averaged values of neighboring cells. No slip
boundary conditions are applied at walls using a ghost-
cell approach.








Particle-phase solver



The particle-phase equations evolve the moments due
to three kinds of terms spatial fluxes, collisions and
drag. These three terms are treated sequentially using
an operator-splitting technique. First the moments
are updated using spatial flux terms, then using drag
terms and finally using collision terms. A detailed
solution algorithm involving all the terms can be found
in [1, 4, 13].

Spatial flux terms. Consider a 3D domain. Again,
let I and dI denote any cell in the domain and its
boundary respectively. Also, let e E dI be a face of
cell I, A, be its area and lnb be the neighboring cell
corresponding to this face. The finite-volume scheme
using single-stage explicit time-integration for the
spatial flux terms in (14) can be written as:




W* W V{ W" p Ae
P PI vol pel' pelb ) '
(31)
where Wp and W* are the cell averaged values while
W" and W." are the values reconstructed on dif-
pel pelnbe
ferent sides of the face e. In (31), voli denotes the vol-
ume of cell I. Let n = [n,' n n"'] denote the outward
unit normal for cell I at face e. The numerical flux Gp


is computed as:


G W", Wn" -
pe pe

13 n,
n U.a

naUtaUjaUka

nal I a

naU aUjaUka


Un




) } nbe


where UZ, =max(UlanI + U_ ,.e + U .., 0) and
U, min(Ulan + U_ + UU ,.. ,O). In the
current work, results are presented using 1st-order and
quasi-2nd-order [5, 6] finite-volume schemes. For the
1st-order finite-volume scheme, a piecewise-constant
reconstruction is used for both weights and abscissas.
For the quasi-2nd-order finite-volume, a least-squares
linear reconstruction [20, 21] is used for weights while
for abscissas, a piecewise-constant reconstruction is
used. Moreover, a limiter [20, 22] is applied to the
least-squares reconstruction of weights to avoid spuri-
ous oscillations. Wall boundary conditions as described
in [4] are applied using a ghost-cell approach.

Collision terms. Collisions only affect the second
and third order moments. These moments are updated
using BGK model as:

Wp; = Ap + (Wp, Ap,)exp(-At/T/), (33)

where T, is the collision time and Apr denotes the set
of equilibrium moments. Details about the calculation
of Tc and Apr can be found in [4].

Drag terms. Drag terms do not affect the weights
because they do not change the number of particles.
The weights obtained after accounting for collisions in
(33) are updated using:

U u+ +1 ** At (34)
7Tip


Coupling algorithm


The coupling between fluid and particle phase solvers is
obtained by following the underlying steps:

1. Initialize parameters and flow variables for both
fluid-phase and particle-phase solvers.

2. For the fluid-phase solver calculate Atf using a pre-
specified value of CFL.







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


3. Pass Atf, pf, /tf, Uf from fluid-phase solver to
particle-phase solver.

4. For the particle-phase solver calculate At,. Details
of calculation of At, can be found in [6].

5. Calculate global time step, At = min(Atf, Atp).

6. Advance particle-phase solver by At.
a) Advance moments by At due to spatial flux
terms using a finite-volume approach.
b) Advance moments by At due to collision
terms.
c) Advance weights by At due to drag force
terms and compute the coupling source terms
Mfpi and Qfp for fluid-phase solver.

7. Pass At, Mfpi, Qfp and a4f( 1 ap) from particle-
phase solver to fluid-phase solver.

8. Advance fluid-phase solver by At.

9. Repeat steps 2 through 8 at each timestep.

Numerical Results

Numerical results are presented for a dilute gas-particle
flow in a lid-driven cavity. The lid has a length L and
moves with a constant velocity Ulid, as schematized in
Figure 1. The cavity is filled with the gas phase and with
initially uniformly distributed particles. Both the phases
have zero initial velocity as initial condition. The evo-
lution of the flow fields are tracked for a time sufficient
to the lid to go through twenty lid lenghts. The param-
eters that characterize the system are the Knudsen num-
ber (Kn), the Reynolds number (Re), the Stokes number
(St) and the mass loading (A). The Knudsen number is
defined as:
Kn dp (35)
6apLv2"
The Reynolds number is defined on the base of the lid
length and the lid velocity as:

PflUlidlL|
Re p L (36)

The mass loading is given by the ratio

A ppX (37)
afpf
while Stokes number is defined as:

St -18 (d L) Re. (38)
18 jf Lf

Results are presented for the case with Kn = 1, St = 1,
Re = 100, A = 2.5. This case is of particular interest as


U


Figure 1: Schematic representation of lid-driven cavity.


it involves particle trajectory crossing which cannot be
captured by two-fluid models [3]. Particles are driven by
the fluid velocity field. At the top-right comer particles
hit the wall and are reflected back. Because of particles
with opposing velocities, trajectroy crossing occurs near
the top-right comer. Figure 2 shows the grid with rectan-
gular cells near the boundary and triangular cells in the
core region. Total number of cells is 6904. A two stage
Runge Kutta scheme is used for time-integration. Fig-
ure 3 and Figure 4 show particle volume-fraction fields
at the final time. Figure 3 shows results when 1st-order
finite-volume scheme is used for both fluid and particle
phase solvers while Figure 4 shows results when 2nd-
order finite-volume scheme is used for fluid-phase solver
and quasi-2nd-order [5, 6] finite-volume scheme is used
for particle-phase solver. Both Figure 3 and Figure 4,
show the trajectory crossing near the top-right comer.
The results are in agreement with the ones presented
in [3]. Second-order finite-volume solver for the fluid
phase gives better resolution of the fluid velocity field.
As the particles are driven by the fluid velocity field, a
second-order finite-volume solver for fluid-phase leads
to better prediction of particle volume-fraction. The use
of quasi-2nd-order finite-volume scheme for particle-
phase further improves the solution.


Conclusions


In the current work, a compressible finite-volume fluid
solver is coupled with a particle-phase solver based on
third-order QMOM on unstructured grids. The fluid and
particle-phase are fully coupled by accounting for the
volume displacement effects induced by the presence of
the particles and the momentum exchange between the
phases. High-order realizable finite volume schemes are


__








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


Figure 2: Grid (6904 cells).


0006
000
00055
0005
0,00o5
0004
0.00355
0003
0.0025
0002
00015
0001
00005
0.0001


Figure 3: Particle-phase volume-fraction using 1st-order
finite-volume solver for both fluid and particle phases.



used for particle-phase QMOM solver. Numerical re-
sults are presented for a dilute gas-particle flow in a lid-
driven cavity. Complex features like particle-trajectory
crossing are captured easily. The coupling can be ex-
tended to practical problems as it is relatively inexpen-
sive compared to Lagrangian and direct kinetic solvers.




Acknowledgements


The study was funded by NSF grant CISE-0830214. The
views and conclusions herein are those of the authors
and should not be interpreted as necessarily representing
the official policies or endorsements, either expressed or
implied, of NSF or the U.S. Government.


- - - -


------------------------------- .............
--------------------------------------------
..........


Q 1 ..............
W W51-
ma ""
MMMMN
..............

MMMME
P-Q . .. ..............
";rAV, ..........
"_5'e
IIIIINIIIIIII

MNENNE
BIMMMM:
HER.

----------------------


0006
000
00055
0005
0.00+5
0004

0003
00025
0002-
00015
00010

0.0005




Figure 4: Particle-phase volume-fraction using 2nd-
order finite-volume solver for fluid-phase and quasi-
2nd-order finite-volume solver for particle phase.



References

[1] O. Desjardins, R.O. Fox, P. Villedieu, A
quadrature-based moment method for dilute fluid-
particle flows, Journal of Computational Physics
227 (2008) 2514-2539.

[2] R.O. Fox, A quadrature-based third-order moment
method for dilute gas-particle flows, Journal of
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