7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
A fully coupled fluidparticle flow solver using quadraturebased moment method
with highorder 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, highorder realizable
Abstract
Kinetic Equations containing terms for spatial transport, gravity, fluid drag and particleparticle collisions can be
used to model dilute gasparticle 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 thirdorder 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 finitevolume solver for the fluidphase and a
third order QMOM solver for particlephase on Cartesian grids. In the current work a compressible finitevolume
fluid solver is coupled with a particlephase solver based on thirdorder QMOM on unstructured grids. The fluid and
particlephase 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 momentinversion algorithm does not work if the moments are nonrealizable, which might lead to negative
weights. Desjardins et al. [1] showed that realizability is guaranteed only with the 1storder finitevolume scheme that
has excessive numerical diffusion. The authors [5, 6] have derived highorder finitevolume schemes that guarantee
realizability for QMOM. These highorder realizable schemes are used in this work for the particlephase solver.
Results are presented for a dilute gasparticle flow in a liddriven 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
particlephase solver.
Introduction
Gasparticle 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 gasparticle 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 quadraurebased 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 gasparticle 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 gasparticle 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 multivelocity 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 fluidphase on Cartesian grids In the current work,
a compressible finitevolume fluidphase solver is cou
pled with a particle phase solver based on thirdorder
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 lstorder finitevolume scheme. But
the lstorder finitevolume scheme has excessive numer
ical diffusion. The authors [5, 6] have recently derived
highorder finitevolume schemes that guarantee realiz
ability for QMOM. These highorder 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 liddriven cavity. For simplicity quan
tities with subscript will be associated with fluidphase.
For the particlephase 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.
Fluidphase governing equations
The fluidphase is described by NavierStokes equations
modified for multifluid models. The fluidphase 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 fluidphase 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.
Particlephase governing equations
Kinetic equation. Dilute gasparticle 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 BhatnagarGrossKrook (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 fluidparticle 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 particlephase 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
lowerorder 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 particlephase volume
fraction i.e. /" a= p. This simplification helps in
handling of coupling terms.
Quadraturebased 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 3node 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
al
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 fluidphase 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 RungeKutta scheme is used, for simplicity, all the
discussion on solver details and coupling algorithm will
be based on a singlestage timeintegration.
Fluidphase 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 finitevolume scheme using
singlestage explicit timeintegration 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 leastsquares linear reconstruction [20, 21]. In
the current work, results are presented using 1storder
and 2ndorder finitevolume schemes. For the 1storder
finitevolume scheme, a piecewiseconstant reconstruc
tion is used i.e. W I W'. For the 2ndorder finite
volume, a leastsquares linear reconstruction is obtained
using cell averaged values of neighboring cells. No slip
boundary conditions are applied at walls using a ghost
cell approach.
Particlephase solver
The particlephase equations evolve the moments due
to three kinds of terms spatial fluxes, collisions and
drag. These three terms are treated sequentially using
an operatorsplitting 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 finitevolume scheme
using singlestage explicit timeintegration 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 1storder and
quasi2ndorder [5, 6] finitevolume schemes. For the
1storder finitevolume scheme, a piecewiseconstant
reconstruction is used for both weights and abscissas.
For the quasi2ndorder finitevolume, a leastsquares
linear reconstruction [20, 21] is used for weights while
for abscissas, a piecewiseconstant reconstruction is
used. Moreover, a limiter [20, 22] is applied to the
leastsquares reconstruction of weights to avoid spuri
ous oscillations. Wall boundary conditions as described
in [4] are applied using a ghostcell 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
fluidphase and particlephase solvers.
2. For the fluidphase 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 fluidphase solver to
particlephase solver.
4. For the particlephase solver calculate At,. Details
of calculation of At, can be found in [6].
5. Calculate global time step, At = min(Atf, Atp).
6. Advance particlephase solver by At.
a) Advance moments by At due to spatial flux
terms using a finitevolume 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 fluidphase solver.
7. Pass At, Mfpi, Qfp and a4f( 1 ap) from particle
phase solver to fluidphase solver.
8. Advance fluidphase solver by At.
9. Repeat steps 2 through 8 at each timestep.
Numerical Results
Numerical results are presented for a dilute gasparticle
flow in a liddriven 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 liddriven cavity.
it involves particle trajectory crossing which cannot be
captured by twofluid models [3]. Particles are driven by
the fluid velocity field. At the topright comer particles
hit the wall and are reflected back. Because of particles
with opposing velocities, trajectroy crossing occurs near
the topright 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 timeintegration. Fig
ure 3 and Figure 4 show particle volumefraction fields
at the final time. Figure 3 shows results when 1storder
finitevolume scheme is used for both fluid and particle
phase solvers while Figure 4 shows results when 2nd
order finitevolume scheme is used for fluidphase solver
and quasi2ndorder [5, 6] finitevolume scheme is used
for particlephase solver. Both Figure 3 and Figure 4,
show the trajectory crossing near the topright comer.
The results are in agreement with the ones presented
in [3]. Secondorder finitevolume solver for the fluid
phase gives better resolution of the fluid velocity field.
As the particles are driven by the fluid velocity field, a
secondorder finitevolume solver for fluidphase leads
to better prediction of particle volumefraction. The use
of quasi2ndorder finitevolume scheme for particle
phase further improves the solution.
Conclusions
In the current work, a compressible finitevolume fluid
solver is coupled with a particlephase solver based on
thirdorder QMOM on unstructured grids. The fluid and
particlephase are fully coupled by accounting for the
volume displacement effects induced by the presence of
the particles and the momentum exchange between the
phases. Highorder 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: Particlephase volumefraction using 1storder
finitevolume solver for both fluid and particle phases.
used for particlephase QMOM solver. Numerical re
sults are presented for a dilute gasparticle flow in a lid
driven cavity. Complex features like particletrajectory
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 CISE0830214. 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
PQ . .. ..............
";rAV, ..........
"_5'e
IIIIINIIIIIII
MNENNE
BIMMMM:
HER.

0006
000
00055
0005
0.00+5
0004
0003
00025
0002
00015
00010
0.0005
Figure 4: Particlephase volumefraction using 2nd
order finitevolume solver for fluidphase and quasi
2ndorder finitevolume solver for particle phase.
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