Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The Subgrid Hydrodynamic Behavior of Accelerating Gassolid Flows
Christian Costa Milioli and Fernando Eduardo Milioli
University of S~io Paulo, School of Engineering of Sio Carlos, Department of Mechanical Engineering,
Av. Trabalhador S~iocarlense, 400, 13566590, S~io Carlos, SP, Brasil
ccosta asc.usp.br and milioli~sc.usp.br
Keywords: twofluid model, subgrid simulation, gassolid flow, riser
Abstract
Literature presents subgrid twofluid simulations (SGS) that are intended to support largescale simulations (LSS) of gassolid
riser flows. Small periodic domains are considered and an additional gas phase pressure gradient is introduced in the
gravitational direction to account for the flow driving force. Such additional term is chosen to exactly match the gravity acting
on the gassolid mixture, so that the simulations give rise to low velocity gassolid suspensions. The present work aims to
verify whether or not the lwdrodynamics that prevails in such suspensions is relevant to rapid gassolid flows. In order to do
this, we perform SGS simulations applying an additional gas phase pressure gradient in excess to that required to match the
gravity acting on the gassolid mixture. As a consequence, instead of reaching a suspension statistical steady state regime, the
flow keeps accelerating throughout a suitable range of gas phase velocities. We considered a high Stokes number particulate
typical of low density risers (520 Cpm diameter, 2620 kg/m3 density), and run simulations for solid phase average volume
fractions between 0.015 and 0.09. We found that the solid phase subgrid effective stress components increase by up to two
orders of magnitude as the gas velocity increases from about 3 to about 9 m/s. This result suggests that suspension
hydrodynamics does not suit well rapid gassolid flows, at least from the present twofluid modelling point of view.
Introduction
It is expected that twofluid modelling under large cluster
simulations (LCS) can accurately predict gassolid flows in
real risers. This expectation is far from realized mainly due
to modelling shortcomings like the absence of validation for
granular thermodynamic related theories such as the kinetic
theory of granular flows, and for the continuum Newtonian
fluid NavierStokes equations applied to particulate phases.
Other very significant unresolved difficulties relate to the
absence of appropriate treatments for solid phase's
turbulence, related to the inexistence of proper closure for
drag at a microscale level, and related to the lack of
separation of scales that prevails in gassolid flows. Of
course, the previous issues must be dealt with and resolved
before any prediction at all can be called realistic. Grace &
Taghipour (21 r 4,i do provide us with a clear insight into the
present capabilities for verification and validation of CFD
based models for granular fluidized bed flows. The authors
stress that verification is virtually impossible, while no
properly validated CFD models are still available. They
observe, in addition, that no current CFD based model can
perform better than simpler mechanistic models. In view of
the above, a warning must be placed regarding the validity,
or the lack of validity of the twofluid model to be described
in the next section. In spite of the many problematic and
open questions, the model seems to produce a qualitatively
correct topology of a riser flow. Accordingly, only
qualitative behaviors shall be analyzed here, and even so
with much care.
Notwithstanding all the previously mentioned difficulties,
there is a continuing research effort to provide subgrid data
for LCS obtained from subgrid scale simulations (SGS)
with twofluid modelling (Sundaresan, 2000; Agrawal et al.,
2001, Andrews IV et al., 2005, Igci et al., 2008). Regarding
the solid phase, SGS can also be numerical direct simulation
(NDS). For a typical LCS grid cell (order of centimeters)
taken to be an SGS domain, NDS is computationally
feasible for the solid phase. In that case SGS requires the
microscale of the solid flow to be properly described. Here
we call microscale the scale free of any solid phase
coherent structure. The current state of the art does not
include microscale experimental information of gassolid
flows, and the so called kinetic theory of granular flows
(KTGF) is widely applied. This theory is an analogy with
the kinetic theory of dense gases, which is modified to
account for the inelastic particle collisions characteristic of
gassolid flows (Bagnold, 1954, Jenkins & Sarage, 1983,
Lun et al., 1984, Gidaspow, 1994).
Following the above, computational experiments have been
performed by some researchers to provide subgrid scale
correlations to be used in LCS of gassolid riser flows
(Agrawal et al., 2001, and then the following works of
Andrews IV et al., 2005, and Igci et al., 2008). In general,
the simulations are performed in small periodic domains
which are thought to repeat themselves throughout the
whole volume of a riser. As periodic boundaries are applied
an additional gas phase pressure gradient is introduced in
the gravitational direction to account for the flow driving
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
representative. We concentrate the current analysis on the
behavior of the solid phase subgrid effective stresses in
view of their determining effect over the gassolid flow
topology.
Nomenclature
Paper No
force. Such additional term is chosen to exactly match the
gravity acting on the gassolid mixture, so that the
simulations give rise to low velocity gassolid statistical
steady state suspensions. This assumption is brought from
previous studies on the instabilities that develop in unforced
granular materials when the inelasticity of the collisions
among particles is accounted for, which ultimately leads to
the formation of clusters (Goldhirsch et al., 1993: Tan &
Goldhirsch, 1997). Those studies stand for regimes where
particulates arrange themselves in low velocity suspensions.
In spite of that, the clustering mechanism that prevails is
believed to be also relevant to rapid gassolid flows (Tan &
Goldhirsch, 1997). We will call this statement as Tan &
Goldhirsh s assumption throughout the article.
In addition to the above, Agrawal et al. (2001) imposed
macroscale shear rates through opposing parallel vertical
boundaries in their small scale periodic domains. By volume
averaging their subgrid scale predictions the authors
determined mesoscale parameters of the flow that were
analyzed as a function of the imposed macroscale shear
rates and volume average solid phase volume fractions.
Andrews IV et al. (2005) further extended the analysis of
Agrawal et al. (2001) by actually deriving expressions for
the mesoscale parameters. The imposition of macroscale
shear rates on those works appears as a first attempt to deal
with the lack of scale separation issue. Extending the
previous works of Agrawal et al. (2001) and Andrews IV et
al. (2005), Igci et al. (2008) evaluated the effects on the
mesoscale results of different LCS filter sizes applied over
the subgrid scale predictions. They found mesoscale
results quite dependent on filter size.
Following the community effort, we have also performed
some twofluid subgrid simulations of gassolid flows in
periodic domains (Milioli & Milioli, 2007a,b). All the
previous literature works considered a particulate typical of
high density catalytic cracking fluidized bed reactors (75
pLm size, with density of 1500 kg/m ), while our previous
simulations included also a particulate typical of low
density circulating fluidized bed combustors (520 pLm size,
with density of 2620 kg/m ). We have also imposed a flow
driving force through an additional gas phase pressure
gradient chosen to balance the gravity over the mixture and,
therefore, like in the previous literature works, our
predictions provided low velocity gassolid suspensions.
Notwithstanding the efforts, how representative such low
velocity fields are regarding the subgrid scale of high
velocity real risers, it remains to be shown. In the present
article we address this particular issue. We propose.
differently from all the previous works, to apply an
additional gas phase pressure gradient in excess to that
required to match the gravity acting on the gassolid mixture.
In this case the flow becomes accelerated, and never reaches
a statistical steady state regime. We present instantaneous
domain averaged predictions at suitable gas phase velocities
inside a range typical of circulating fluidized beds, and then
analyze the results as for the effect of the gas phase velocity
on the flow hydrodynamics. While the predictions at any
particular mesh point should not be regarded as significant
in view of the instability of the flow, the subgrid scale
domain averaged results, on the other side, are quite
CD
dp
D
go
I
NI
P
Re
t
u v~
UVsrw
xshp
drag coefficient (nd)
particle diameter (m)
strain rate tensor (s )
restitution coefficient (nd)
gravity acceleration (ms )
radial distribution function (nd)
unit tensor (nd)
interface drag force (Nm )
pressure (Nm 
additional gas pressure gradient (Nm )
Reynolds number (nd)
time (s)
velocity vector (ms ')
Cartesian components of velocity (ms ')
8Xi81 Slipvelocity (Fv,) (ms ')
Cartesian coordinates (m)
Greek letters
volume fraction (nd)
P gassolid friction coefficient (kgm s ')
o granular temperature (m~s )
n bulk viscosity (Nsm )
i" dynamic viscosity (Nsm )
P density (kgm )
r viscous stress tensor (Nm )
T, effective stress tensor (Nm 2)
u9 particle sphericity (nd)
w driving force factor (nd)
Subscripts
e mesoscale or effective
a gas phase
I interface
n~r maximum
P particle
solid phase
x. L = Cartesian directions
Others
_LCS resolved
(.) volume average
Mathematical Modelling
Multiphase flow twofluid models stand on the major
hypothesis of continuum for all of the phases, no matter
fluid or particulate. The phases are treated as
interpenetrating dispersed continue in thermodynamic
equilibrium. The theory of twofluid models has been
developed by many researchers. Some classical reference
developments on this matter can be found in the works of
Anderson & Jackson (1967), Ishii (1975), Drew (1983),
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
Gidaspow (1994), Enwald et al. (1996), among many others.
The lwdrodynamic twofluid models comprise a basic set of
mass and momentum averaged conservative equations plus
closure laws for stress tensors, viscosities, pressures and
drag.
We must formulate our twofluid model to perform SGS,
which remains large eddy simulation alike regarding the gas
phase, but is required to become NDS regarding the solid
phase if all the scales of clusters are to be captured. In this
way, the gas phase would require closures at both the micro
and the mesoscales. Agrawal et al. (2001) found the
turbulence of the gas phase not to significantly affect the
volume averaged solid phase parameters, so that a constant
molecular viscosity approach seems to be good enough for
the gas phase. For our present purposes, turned mamnly to
solid phase lwdrodynamics, tlus assumption shall be good
enough. The microscale closure for the solid phase is
established by applying the kinetic theory of granular flows
(KTGF), where solid phase microscale properties are
derived as a function of a granular temperature determined
from a pseudo thermal energy balance.
In addition to the conventional formulation, as periodic
boundaries are to be applied, an additional gas phase
pressure gradient must be enforced in the gravitational
direction to account for the flow driving force. We present
next a subgrid scale lwdrodynamic formulation of the
twofluid model including closure laws based on the KTGF
(Gidaspow, 1994: Syamlal et al., 1993: Agrawal et al., 2001),
and including gravity compensation for applying periodic
boundary conditions. The formalism involved in the
derivation of the large eddy simulation alike formulation for
the gas phase can be found, for instance, in Sagaut (2001)
and Lesieur et al. (2005).
The continuity and momentum conservative equations for
the gas and the solid phases are, respectively:
Eg +a, = 1
The additional gas phase pressure gradient in the
gravitational direction for exactly matching the gravity
acting on the gassolid mixture is given by:
V7~= (pa, +p,a,)g
The filtered viscous stress tensor for the gas phase is
disregarded as of lower order in relation to the effective
stress tensor, which is assumed to be given by:
_ pg, ug Yu n (Ae Vui
It isalso assumed that u, = constant and n,, = o.
The viscous stress tensor for the solid phase is given by:
The dynamic and bulk viscosities of the solid phase are
obtained by analogy with the kinetic theory of dense gases,
and usually include terms due to kinetic and collisional
effects. Here we exclude the kinetic term since it brings the
solid phase stresses to unrealistically high values at very
low solid fractions: the collisional term, on the other side,
correctly vanishes as expected at lower solid fractions (Lun
et al, 1984). The dynamic and bulk viscosities of the solid
phase are, respectively (as presented in Gidaspow, 1994):
Y = afpdg(+ (9)
(10)
(1) '' ,= a~pXd~o\1+e8
"6 ) #s lly) o
er(apsp) + V (asPsus) = o
The radial distribution function is taken form Ding &
(2) Gidaspow (1990), and is given by:
np(O, VIVPe~)
The solid phase pressure also comes from the analogy with
the kinetic theory of dense gases, and is given by (Gidaspow,
1994):
at (ap,u,) + V (asppusu) = a/VP, + grP,+ Vp s
+V*(a,r,)+ asp,g + M,, (4)
As the formulation for the gas phase is for large eddy
simulation, the conserved variables are the filtered
parameters to be resolved. For the solid phase, on the other
hand, the flow field variables are fully resolved as NDS is
assumed.
Closures are required for various parameters that appear in
the conservative equations. The volumetric continuity
imposes that:
P,= asp,O[1+2(1+ e)goas]
The granular temperature required in the previous equations
is determined here from the algebraic equation of Syamlal
et al. (1993), which is derived by assuming that the
pseudothermal energy is locally generated by viscous stress
and dissipated by inelastic collisions. It is given by:
r,=p, V.cu,+@usn +(,yi)/uji
go
5 a
+ V. a(r,rge)] pgagg + NQ,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
application of either free slip, partial slip, or periodic
COnditions to vertical boundaries gives rise to the same flow
(13) topology. In the present work we apply the simpler free slip
condition.
Finally, if averaged subgrid or mesoscale data are to be used
for solid phase's closures in LCS, then a bridge needs to be
built between the SGS solution and the filtered parameters
(14) required in LCS. Following the large eddy simulation theory,
the components of the solid phase subgrid effective stress
tensor result (Sagaut, 2001; Lesieur et al., 2005):
Klatr(D,)+ Ki7tr ID
)a, +4K arK tr ID )+2K tr(D 2
where
Ds 1 Yus+ Yusn
K, = 2(1+e)pso
r,~se Ps tsitsTisis)
Z~ =Z
uzse u~se s 1ss 1ss
rz,se, rese= Ps("sts\s's)
4dpps(1+e)asgo 2
K, = K3
d~p~js 8asgo(1+e)
K3 [1+0.4(1+e)(3e 1)asEOl
2 33e) 5 ~
_12 1 e PsEO
K,
A stationary drag force is applied to account for
interface momentum transfer between the gas and the
phases, which is given by:
Ms;= MgI s gu)
The gassolid friction coefficient is determined by app]
the procedure frequently referred to as Gidasp
procedure, where Ergun's equation (Ergun, 1952) is us
high solid fractions and Wen & Yu's correlation (Wen &
1966) is used at low solid fractions. The procedure
follows (Gidaspow, 1994):
Pgasl ug us
P =150 +1.75 for as >0.2
ag dpa~s) (ps ,
(25)
(26)
the
solid
(19)
lying
Iol's
ed at
iYu,
is as
If a subgrid filter is assumed that exactly fits the SGS
domain, then the filtered parameters become equal to their
volume averaged subgrid values, so that:
r ,,= P (NUz) (Ils)(rts) (31)
r = p, (v:v, (~,)(v~,) (32)
(20)
se.~ se Ps 11s s 11 s)~) (34)
(21) r = =p r e)( )(s (35)
r , = c ,, = p (i, e) (v,)( ty)) (36)
Equations (31) to (36) are the components of the subgrid
(22) effective stress tensor that are required to be introduced in the
Solid phase's momentum equation, in LCS.
S im ulations
4 (dpa~s
for a, 50.2
The drag coefficient is given by (Rolve, 1961):
S1 +0.15Re 0.08i for Re, <1000
Ds 
0.44 for Rep 21000
were
The complex set of partial differential nonlinear coupled
equations of the twofluid models can only be solved
through numerical procedures. In this work, the numerical
model available in the software CFX (CFX, 2005) was used.
The simulations were performed for a high Stokes number
particulate typical of low density risers (520 Cpm diameter,
2620 kg/m' density), for solid phase average volume
fractions of 0.015, 0.03, 0.05, 0.07 and 0.09. Accelerating
flows were generated and the results were analysed for
l i, us d, pp
Re, =(23)
Periodic conditions are applied at entrance and exit, i.e. in the
horizontal boundaries normal to the vertical gravitational
direction. This means that the flow at the inlet is exactly the
same that flows at the outlet. Free slip is applied in all of the
vertical boundaries. Agralval et al. (2001) showed that the
S 6
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
time (s)
Figure 2: Averaged axial velocities and axial slip velocity
as a function of time, for y = 1 and 1.5, for an average
solid volume fraction of 0.05.
Figure 2 also shows that, while both the phase axial velocities
grow in time, the slip velocity is kept approximately constant.
This result suggests that the clustering topology of the flow is
not significantly affected by the increasing gas axial velocity.
This is an evidence in favor of Tan & Goldhirsch's
assumption. Plots similar to those of Figure 2, for solid
volume fractions from 0.015 to 0.09, show the same behavior.
Figures 3 to 7 present subgrid effective shear stresses of the
solid phase as a function of the averaged gas phase axial
velocity, for various average solid volume fractions. The
effective shear stresses resulted very oscillating, and seem to
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
becomes accelerated. Notice that the volume averaged solid
phase axial velocity results negative in the statistical steady
state regime, in the stage of the simulation where the gravity
acting on the suspension is exactly matched. This seems to be
a clear consequence of the clustering that develops in the flow,
since the gravity does not simply act on a bunch of individual
particles, but on a relatively heavier bunch of clusters of
particles.
Paper No
increasing gas velocities from about 3 to about 9 m/s.
A 2x2 cm wide and 8 cm tall vertical hexahedral domain was
considered, applying a 1xlx1 mm uniform hexahedral
numerical mesh. The flow enters the domain through the
bottom and exits at the top. The density and viscosity of the
gas phase were, respectively, 1.1614 kg/m3 and 1.82 x 10 5
N.s/m A solid phase volume fraction at maximum packing of
0.38 was applied following Gidaspow and Ettehadieh (1983),
and a restitution coefficient of 0.9 was taken following
Agrawal et al. (2001).
The driving force factor (W) multiplying the additional gas
phase pressure gradient required to match the gravity over the
mixture was set at 1.5. This value allowed the simulations to
go along a range of gas axial velocities typical of circulating
fluidized beds in a reasonable computing time. Initial
conditions for the accelerating runs were obtained by running
previous simulations applying = 1 until statistical steady
state regimes were found. This is done because, if the
accelerating runs are to be realistic, then they must depart
from realistic gassolid flow conditions. Therefore, the
simulations developed in two stages, the first for 7y = 1, and
the second for y = 1.5 The initial conditions for the first
stage were of uniform quiescent suspensions with fixed
uniform solid volume fractions. It should be noted that the
axial periodic boundary condition that was applied and the
continuity constraint cause the volume average solid volume
fraction to be sustained throughout the whole simulation.
A time step of 5x10' s was applied which is suitable for solid
phase NDS. The lower characteristic time scale of clusters of
the order of 102 s (Sharma et al., 2000) is expected to be fully
captured. Also, for the present 520 pLm particulate size the
smaller clusters on the flow are expected not to be larger than
5.2 mm (following Agrawal et al., 2001). Therefore, regarding
the solid phase, both the spatial and temporal meshes which
were applied are suitable for NDS. The convergence criterion
for the numerical procedure was a rms of 1 x 10 5
Results
We will now use the results of the present simulations to
collect arguments both in favor and against the assumption of
Tan & Goldhirsch (1997), that the clustering mechanism in
low velocity suspensions also prevails in rapid gassolid flows.
Figure 1 shows grayscale plots of the solid volume fraction in
an axial section of the domain for increasing volume averaged
gas phase velocities, for the average solid volume fraction of
0.05. It is clearly seen that the flow topology progressively
becomes more and more homogeneous as the gas velocity
increases towards pneumatic transport. This is an evidence
against Tan & Goldhirsch's assumption. Plots similar to those
of Figure 1, for solid volume fractions from 0.015 to 0.09,
show the same behavior.
Figure 2 shows the time development of the phase's volume
averaged axial velocities and the axial slip velocity for the
simulation with an average solid volume fraction of 0.05.
Departing from a unitary driving force factor, the flow
develops and reaches a statistical steady state regime. Then
the driving force coefficient is turned to 1.5, and the flow
Figure 1: Grayscale plots of solid volume fraction in an
axial section of the domain for increasing volume averaged
gas phase velocities (3, 4, 6, 8 and 9 m/s from the left), for an
average solid volume fraction of 0.05.
rre
VV I I I I r
3 4 56 789
=V, (nu s)
Figure 5: Subgrid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.05.
Figure 6: Subgrid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.07.
100
0.01
3456789
Figure 7: Subgrid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.09.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
vary randomly at the same range, no mater the averaged gas
phase axial velocity. This pattern is observed for all the
average solid phase volume fractions, except for 0.015. In this
case there is a rising tendency of up to two orders of
magnitude as the gas phase axial velocity increases. Here we
have arguments both in favor and against Tan & Goldhirsch's
assumption.
Figures 8 to 12 show subgrid effective normal stresses of the
solid phase as a function of the averaged gas phase axial
velocity, for various average solid volume fractions. The
effective normal stresses resulted much less oscillating than
the effective shear stresses. In most of the cases there is a
decreasing tendency as the averaged gas phase axial velocity
increases. Variations of up to one order of magnitude are
observed. Again, the above pattern is observed for all of the
average solid phase volume fractions, except for 0.015. In this
case a clear rising tendency is not seen. Here we also have
arguments both in favor and against Tan & Goldhirsch's
assumption.
 o I I I I I
vs
"A h
'y 8 A
~ ~vse
use
yrse
iv
i
1
i', (r; s)
3456789
~v~ pn s)
Figure 3: Subgrid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity
for an average solid volume fraction of 0.015.
l i e l l i e l
3456789
Figure 4: Subgrid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.03.
1456789
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
10 
3456789
nms
Figure 11: Subgrid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.07.
10 
::456789
Figure 12: Subgrid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.09.
Conclusion
We developed a twofluid subgrid scale simulation of an
accelerated gassolid flow covering a range of gas velocities
typical of risers. Our main objective was to verify the
conunonly applied Tan and Goldhirsch's assumption that
the clustering mechanism in low velocity suspensions also
prevails in rapid gassolid flows. We considered a high
Stokes number particulate (520 Cpm diameter, 2620 kg/m3
density). The analysis was concentrated on the behavior of
the subgrid effective stresses of the solid phase. From the
predictions, we have collected a number of evidences both
in favor and against the concerning assumption.
For increasing averaged gas phase velocities, we found that:
(i). slip velocities are kept approximately constant; (ii) the
subgrid effective shear stresses of the solid phase seem to
be mostly unaffected, except for a solid volume fraction of
0.015, where a variation of up to two orders of magnitude is
I e I I I I e I
Paper No
lo
At ~se
Figure 8: Subgrid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.015.
345678 im
Figure 9: Subgrid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.03.
~100
lo
3456789
Figure 10: Subgrid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.05.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Gidaspow, D. Multiphase flow and fluidization. San Diego.
Academic Press (1994).
Paper No
observed; and (iii) the subgrid effective normal stresses of
the solid phase decrease by up to one order of magnitude for
most of the cases. Here we have arguments both in favour
and against Tan and Goldhirsch's assumption.
Notwithstanding the above conclusions, all we can say, for
the moment, is that Tan & Goldhirsh's assumption seems
not to hold, at least for the particulate that was considered,
and from the present twofluid modelling point of view. Of
course, the current predictions need much improvement
before better conclusions may be drawn. It is wise to keep
in mind that even though twofluid modelling can provide
apparently correct qualitative predictions of riser flows, the
same can not be said regarding quantitative predictions.
Acknowledgements
This work was supported by The State of Sio Paulo
Research Foundation (FAPESP), The National Council for
Scientific and Technological Development (CNPq), and The
Coordination for the Improvement of Higher Level
Personnel (CAPES).
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