Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Analysis of Drag and Lift Coefficient Models of Bubbly Flow System for Low to Median
Reynolds Number
M.J. Pang, J.J. Wei*
Xi'an Jiaotong University, State Key Laboratory of Multiphase Flow in Power Engineering
Xi'an 710049, China
Email: jjwei@mail.xjtu.edu.cn
Keywords: Bubble; Drag fore; Lift force; coefficient model; Multiphase flow
Abstract
It is very significant to calculate interphase forces correctly in multiphase flow systems. Among numerous interphase forces,
drag and lift forces are of special importance, which have direct influences on terminal velocities and lateral distribution of
particles. A variety of drag and lift coefficient models were developed by different researchers. In the face of divers and
numerous expressions of the drag and lift coefficient models, it is quite hard to choose a pertinent model to correctly compute
the drag and lift forces for one who is not specially employed to this job. To solve this problem, some typical drag and lift
coefficients, to the best of the authors' knowledge, were collected and summarized in this paper. Combining the forming
mechanisms of the drag and lift forces into their influencing factors, we discussed each model of the drag and lift coefficients in
detail, and qualitatively compared them with each other. Analysis showed that most of the existing drag coefficients are simply
functions of bubble Reynolds number (Reb), and they display similar profiles. Differences among them are that the critical Reb
corresponding to the inertial region are different, varying amplitudes of the drag coefficients with increasing Reb are different,
and even some profiles show discontinuation. Comparing with the drag coefficient, profiles of the lift coefficient (CL) are much
more complicated. The major parameters influencing the lift coefficient are the bubble Reynolds number (Reb) and
dimensionless shear rate (Srb). The profiles of different lift coefficient models display quite different trends with the change of
Reb or Srb. Even though some expressions of CL are independent of Srb, they violate physical laws. We recommended several
typical drag and lift coefficient models for application in numerical computation of multiphase flows.
Introduction
Bubbly flows can frequently be seen in industrial processes,
such as smelting of metals, refine of fine chemical products,
cloud cavitation in hydraulic systems, coal liquefaction,
fermentation reactions, waste water treatment, phasechange
heat and mass transfer, bubble column reactors and so on. In
order to understand the flow pattern of bubbly flows well and
improve efficiency of bubble column reactors, it is very
necessary to conduct deep study on the bubbly flow. Among
numerous contents of study on gasliquid twophase flows,
the most important and intractable problem considered by
Sankaranarayanan and Sundaresan (2002) is the closure
relations for the effective stresses and interphase force term
appearing in control equations of gas and liquid phases which
are not fully understood. And the correct model of interphase
forces and turbulence is of prime importance for capturing the
physics correctly. As long as interphase forces and turbulence
models are correctly developed, the physics behind
phenomena occurring in bubble columns can be fully captured
by developing accurate CFD code. If the turbulence model is
given, accurate prediction of local hydrodynamics strongly
depends on properly selecting simulation parameters of
interphse forces like lift fore, drag force, and virtual mass
force etc. One sees easily from the above analysis that it is
extremely significant to correctly and accurately calculate
interphase forces.
Presently, a number of works on computational models of
interphase forces based on a variety of assumptions have been
published by using theoretical analysis, numerical simulations
or experimental investigations. Among numerous interphase
forces, the drag and lift forces have great influences on
hydrodynamic characteristics of bubbly flows such as average
gas hold up and average liquid velocity in the streamwise
direction (Zhang et al., 2006; Tabib et al. 2008), so they
should not be neglected under any conditions for study of the
bubbly flows. And thus, the published literatures on them are
quite many compared to other interphase forces. Presently, the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
computational models for the drag and lift forces have come
to an agreement and can be expressed respectively as
FD = CDPfU (1)
S2
FD = C fU 2. (2)
Where U, is the mean velocity of fluid on the streamline
through the centre of bubbles; and CD, CL, pf, rb are drag
coefficient, lift coefficient, fluid density and radius of
bubbles respectively. And the computational models can be
further specified as
F, (3)
4
FL= 4CLpfVf. (4)
3
Where V, is the slip velocity between bubbles and liquid
(V.=uiUb), af the angular velocity of liquid at the bubble
place. Currently the focus on the drag and lift forces is not
the force model but the coefficient model. Different authors
presented different models of the drag and lift coefficients.
So it is very difficult to choose the pertinent coefficient
model from the existing literatures for researches engaging
in study of bubbly flows. Thus, some analyses and
discussions on the drag and lift coefficient models for dilute
bubbly systems are presented in this paper.
Analysis of drag coefficient models for low to
median bubble Reynolds number
The origin of drag fore is due to resistance experienced by a
bubble moving in liquid. The drag force consists of two
parts of form drag and viscous drag. The form drag is
caused by uneven pressure distribution around the moving
bubble, and it is associated with bubble size and shape. The
viscous drag is caused by viscous stress due to viscosity of
fluid, and it is mainly produced within the boundary layer.
The magnitude of two parts varies with the bubble
Reynolds number. With the increase of the bubble Reynolds
number, the flow begins to separate and form vortices
behind bubbles, which results in an increase of the form
drag. When the bubble Reynolds number is higher than a
critical value (100), the viscous drag can be neglected, and
the drag almost exactly comes from the form drag due to
the start of the flow separatation behind bubbles. Two parts
can be described by the uniform drag coefficient CD.
Factors influencing the drag coefficient are quite many. It
was reported that the drag coefficient strongly depends on
the bubble shape and orientation with respect to the flow as
well as on flow parameters such as bubble Reynolds
number, Mach number, Edtvos number, turbulent level and
so on (Crowe et al., 1998). If the bubble deformation is
neglected, CD only depends on flow regime and liquid
properties (Pan et al., 1999). It was reported that the effect
of shear on the drag force is very weak for low to moderate
values of the shear rate (Sridhar and Katz, 1995; Legendre
and Magnaudet, 1998). The drag coefficient models given
in existing literature were overwhelmingly related to the
bubble Reynolds number, and small part of models
presented the drag coefficient as a function of the E6tvds
number as shown in Table 1. As a matter of fact, factors
influencing the drag coefficient are numerous. Hameed
(2"'111 discussed influences of void fraction, the bubble
Reynolds number, Weber number, the Edtvos number and
Froude number on the bubble drag coefficient in detail.
Table 1 lists some typical models of the drag coefficient in
our reading scopes. In Table 1, Reb, Eo and of denote the
bubble Reynolds number ( Reb = IVd/v ), the E6tvds
number (Eo = g(p pg )d/ ) and the liquid volume
fraction respectively. Here pf, Pb, db, g, o, Vf are liquid
density, bubble density, bubble diameter, gravity
acceleration, surface intension and kinematic viscosity of
liquid. According to the bubble Reynolds number, the flow
region influencing the drag coefficient can be divided into
the following four pars: Stokes drag region, viscous drag
region, transitional and turbulent region. It can be seen from
Table 1 that the drag coefficient models corresponding to
different authors are very different from one another. Thus,
it is very necessary to analyze each of them in depth. In this
paper, we analyze and compare the influences of Reb or Eo
on the drag coefficient in the region of the bubble Reynolds
number at Reb <100.
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Table 1. Drae coefficient model.
Drag coefficient model
Schiller and Naumaan(1935)
Dalle Ville (1948)
Clift et al.(1978)
Ishii and Zuber (1979)
Ma and Ahmadi (1990)
Grevskott et al. (1996)
Tsuchiya et al. (1997)
S l24 (1+0.15Re687)
0.44
CD =(0.63+ 4.1
29.1667 3.8889
Reb Re2
CD
S(1+0.15Re68 )
K Reb)
CD jo
3
Reb <1000
Reb 1000
1 < Reb <10
10 < Reb < 200
CD 24 (l+0.1Re)
Reb
5.645
D Eo1 +2.835
C, = Fx min16 ( 0.15Re 68 ) 48 8 Eo 1+17.6'7.' "
m Reb Reb )'3Eo+47 18.67.,
Kurose et al. (2001)
Lain et al I ,II2'
Zhang and Vanderheyde (2002)
Tomiyama (21" 114
16
CD Reb
S16 (1+0.15Reb)
16
Reb
14.9
RRe078
CD bRe
48 2(21 756
48 (1 2.21 )+1.86 x10 15Reb4756
Reb VReb
2.61
24 6
CD = 0.44 ++4
Reb 1+IReb
8 EoE2
CD 8 E Eo+(1E2) F(E)2
3E2/3Eo+161E24/3
Reb <1
1 Reb
Reb 1.5
1.5 Reb < 80
80 < Reb < 1500
1500 Reb
1 (E) sin l1E2 EV1E2
E +0.163 7 2
1+0.163Eo05 1E2
24
Reb
S24 R 3.6 Re l1
Murray et al. (2007) CD Rb 313 1
Reb Re 19
24 (1+0.15Re 68 )
Reb
In order to conveniently perform analysis and discussion, Reynolds number
we classify the drag coefficient models into two groups: one Fig. 1 (a) that the
is only related to the bubble Reynolds number, and the other on the drag coeff
is only associated with the E6tvos number. Fig. 1 shows the the influence of t
drag coefficient profiles with respect to the bubble coefficient become
Reb <1
1< Reb <20
20 < Reb
or the E6tvos number. It can be seen from
influence of the bubble Reynolds number
icients is very great at Reb<10, however,
he bubble Reynolds number on the drag
es smaller and smaller with continuously
Paper No
Investigators
V
Paper No
increasing bubble Reynolds number. The drag coefficient
profiles have similar profiles, whereas their changing
amplitudes are different with increases of the bubble
Reynolds number. At low bubble Reynolds numbers
(Reb
drag coefficient quickly decreases with increasing bubble
Reynolds number. With further increasing Reynolds number,
the drag coefficient nearly approaches a constant value, and
the flow gradually transits to the inertial range. In the
Stokes flow region, the flow is thought as a creeping flow in
which the inertial terms in the NavierStokes equations are
unimportant. With increasing the bubble Reynolds number,
10 10 10
Reb
(a) Drag coefficient varying with Reb, at 1
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
the inertial terms become more and more significant. Fig. 1
(d) shows drag coefficient profiles of Sshiii and zuber
(1979), Grevskott et al. (1996) and Tomiyama (2li14) with
respect to the E6tvos number. One can see from Fig. 1 (d)
that variation of three drag coefficient models with the
Edtvds number exhibits great differences. The drag
coefficient model of Grevskott et al. (1996) is greatly
affected by the E6tvos number (at Eo<10), whereas the drag
coefficient keeps a constant, CD=1.9 when Eo are higher
than 10. The drag coefficients of Sshiii and zuber (1979)
and Tomiyama (21 4) continue to increase with increasing
the E6tvos number in the present region of Eo.
10 10
Reb
(b) Drag coefficient varying with Reb, atl
10 Re 10 0 10 20 30 40 50 60
Eo
(c) Drag coefficient varying with Reb, at 10
Figure 1: Drag coefficient profiles.
The slip velocity can be considered as the signature of the The slip velocity can be solved from Eq. (5) and is
multiphase system under given flow conditions (Tabib et al., expressed as:
2008). In order to understand the influences of different 4db Pf P Pb
drag coefficient models on the bubbly system, variations of 3CD f (6)
the slip velocity with bubble diameter for different drag It is clearly seen from Eq. (6) that the slip velocity changes
models are presented in Fig. 2. For a single bubble, rising at with bubble diameter for a given value of the drag
steady state, the force balance is given by coefficient. To understand interrelations between drag
FD p Vf VK. =l,73(p Pb)g. (Joshi, 2001) (5) coefficient, bubble diameter and slip velocity, the slip
velocity has been plotted as a function of bubble size for
0 Schiller and Naumaan (1935)
Dalle Ville (1948)
 Cilft et al. (1978)
1Ma and Ahmadi (1990)
 Tsuchiya et al. (1997)
VKurose et al. (2001)
 Lain et al. (2002)
Zhang and Vanderheyde (2002)
 Murray et al. (2007)
Paper No
different coefficient models. From Fig. 2, one can see that
values of the slip velocity, which correspond to different
combinations of the bubble diameter and drag coefficient
model, are different. Similarly, for a particular bubble
diameter, several different values of slip velocity can be
obtained due to selection of different drag coefficient model.
Combining Fig. 1 with Fig. 2, one can see that under the
same condition a big value of the drag coefficient is
035. I I I I I I I I I
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
associated with a small slip velocity as reported by zhang et
al. (2006). Tabib et al. (2008) reported that, under the same
study condition, the higher slip velocity value can cause the
greater streamwise liquid velocity, which also leads to the
steeper void fraction profile. Therefore, it is of prime
importance to choose the correct combination of drag model
and bubble diameter to model the gasliquid flow.
015 / 0 Schiller and Naumaan (1935) 025
0 Dalle Ville (1948)
 Cilft et al. (1978)
o01 / L Maand Ahmadi (1990) 02
0 Tsuchiya et al. (1997)
S Kurose et al. (2001)
005 A Lain et al. (2002) o 15
Zhang and Vanderheyde (2002)
Murray et al. (2007)
0002 0004 0006 0008 001 0012 0014 0016 0018 002 010 300
,0 10 20 30 40 50 60
db (m) db (m)
(a) Variation of Vs with db for drag coefficient as a function ofReb (b) Variation of Vs with db for drag coefficient as a function ofEo
Figure 2: Variation of slip velocity with bubble size for different drag laws
Although all drag coefficient models had successfully been
applied to simulate bubbly flows and were able to predict
some important parameters of bubbly flows, deviations of
simulating results for different drag coefficient models from
experimental values are different. Tabibi et al. (2008)
simulated the gasliquid flow using drag coefficient models
developed by Schiller and Naumaan (1935), Dalle Ville,
(1948), Ishii and Zuber (1979), Ma and Ahmadi (1990),
Grevskott et al. (1996) and Zhang and Vanderheyde (2002),
respectively, and discovered that the drag coefficient model
of Zhang and Vanderheyde (2002) is in better agreement
with experimental results than other models. In face of
numerous drag coefficient models, therefore, it is very
difficult to choose a proper model to predict hydrodynamics
of bubbly flows due to lack of a selecting standard on the
drag coefficient model. Additionally, accuracy of different
drag coefficient model predicting hydrodynamics of bubbly
flows is related to specific operating parameters, which
increases difficulty of selecting the drag coefficient model.
If an experiment is in advance used to validate which drag
coefficient model is more appropriate than others before the
simulation of bubbly flow using CFD each time, we think
that CFD lose the essential meaning. Consequently, in our
view, some principles of selecting drag coefficient model
should be developed from the theoretical point of view and
existing results of literature. For an excellent drag
coefficient model, in our opinion, it should have the
following features: firstly, it must comply with actual
physics; secondly, it should be in good agreement with
generally accepted knowledge of fluid mechanics; thirdly,
the profile of drag coefficient model should be continuous
and should not show abnormal changes; fourthly, the drag
coefficient model should be a formula as simple and general
as possible; fifthly, influencing factors on the drag
coefficient model are taken into account as
comprehensively as possible.
It can be seen from Fig. 1 that the profile of the drag
coefficient model reported by Murray et al. (2007) becomes
very flat at Reb<100, which means that the inertial terms are
predominant at the small bubble Reynolds number; and this
profile shows discontinuity. This seems to be in
disagreement with physical laws. The profiles of the drag
coefficient models of Schiller and Naumaan (1935) and
Clift et al. (1978) almost overlap in the range, 1
but the latter is applied to a limited range 1
can not describe the Stokes flow region. The profile of the
drag coefficient model of Tsuchiya et al. (1997) is not
smooth, and there is a mutation phenomenon. As far as the
drag coefficient models of Ishii and Zuber (1979),
Grevskott et al. (1996) and Tomiyama (2k1 '4), the influence
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
of slip velocity on the drag coefficient is neglected, and thus
they seems to violate physical laws too. As is well known,
the classic Stokes drag coefficient model (Reb
expressed as
CD 24 (6)
Reb
It can be seen from Fig. 3 that profiles of the drag
coefficient models of Kurose et al. (2001) and Lain et al.
(2002) are very different from profile of the classic Stokes
drag coefficient in the Stoke flow region (Reb
one can see from the above analyses that the drag
coefficient models reported by Schiller and Naumaan
(1935), Dalle Ville (1948), Ma and Ahmadi (1990), and
Zhang and Vanderheyde (2002) seem to be more reasonable
and better applied to calculate the drag force in bubbly
flows than other models. Combining the above analyses
with the results of Tabibi et al. (2008), one can conclude
that the drag coefficient model of Zhang and Vanderheyde
(2002) seems to be more properly applied to describe the
drag coefficient in bubbly flows than other models.
2500 D  a
Sch d Nau an(1935)
BD&J V&1 (1948)
Ma and Ahmad, (1990)
K ose etd (201)
Lametal (22) M
10 10 10
Reb
Figure 3: Drag coefficient profiles in the Stoke region.
Analysis of lift coefficient models for low to
median bubble Reynolds number
Compared with the drag force, the lift force appears much
more complicated. Just like the drag force, the lift force also
plays an important role in predicting the bubble motion in
complex flows. The common definition of the lift force is
that, owing to the asymmetrical flow, the bubble
experiences a lift force perpendicular to its relative motion.
The lift force is not only associated with the flow feature
but also is related to bubble parameters. Factors affecting
the lift force are numerous, and they include the relative
velocity between bubbles and ambient liquid, the shear rate
of a flow field, bubble parameters (such as size,
deformation, velocity, rotational speed and so on) and the
boundary condition etc (Hibiki and Ishii, 2007).
Consequently, it is very difficult to study the lift force,
whatever the methodology employed for that purpose
because it is governed by the instantaneous and local
structure of the flow and not only by the average velocity
distribution (Legendre and Magnaudet, 1998). Despite
extensive studies on the lift coefficient model over past six
decades, there is not a universally accepted lift coefficient
model. This is the reason why many ad hoc models of the
lift coefficient are used in computational processes. It was
investigated by Hibiki and Ishii (2007) that the
computational models on the lift coefficient are more than
twenty. They summarized that the lift force models and
some remarks on them in detail, but they did not perform a
deep contrast analysis. In order to qualitatively understand
each computational model, we choose some typical models
(listed in Table 2) and carry out qualitative analysis.
According to the forming mechanism, the lift force can be
divided into three kinds such as Saffman lift force, Magnus
force and the wall lift force. The Saffman lift force results
from the inhomogeneous pressure distribution developed
around a bubble due to the fluid rotation induced by the
velocity gradient; the Magnus lift force is the lift developed
due to rotation of bubble; and the wall lift force is caused by
the fact that the drainage rate of liquid is restrained by the
noslip condition at walls which leads to pressure near the
wall higher than that in the certain distance far away from
the wall, so bubbles near the wall receive the lift force
towards the center of channel. For the existing models of
the lift force, most models only take into account the
Saffman lift force and Magnus lift force, whereas only does
individual model consider the Saffman and wall lift force.
Some typical models of the lift force are shown in Table 2.
We can see from Table 2 that each model almost holds
water on the basis of special assumptions. In order to
exhibit complete comparison, we ignore all assumptions in
the following analysis.
Paper No
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Table 2. Lift force models.
Investigators Lift force equations Major assumptions
Bagchiand FL =C 0.57p rr2v2 C =12 (1+0.5971:c' )Srb 10Reb, <100, 0.04 5Srb 0.1
Balachandar (2002) Reb
)db d2
Reb = <<1, Rebn = b < 1
Saffman (1965) FL, = 6.46pf 5 )vr 05 of U
f d2 Reo 5d
Reb b <>
Of RebR
c .4 3pf X(VVf) Srb <
Auton (1987) 3F = CL P V v
CL, = 0.5 uniform density in uniform weak shear
F 4 3 Lx f Magnus effect was not included in the
Zun (1980) 3 case of bubbly flow, because a spinning
CL 0 = 0.0.3 bubble interface was not expected
FL= 4 C rb pfaf V x ( f)
Wang et al. 3
(1987) (001 0.49 ,1(ln+9.3168
C =ab0.01+cot 0.1963
L= 2/ 0.1963
2 05
Mei and rFL 1.72J(, ) 3Srb05
F r Xrrb2Srb05x
Klausner (1994) 2 Re O 4
4
Legendre and FL = CL. 3 b PfvAc Clear bubble surface,
Magnaudet (1998) CL wR C= (RebSrb )} + L Reb)}2 0.1
CLF owRe(Reb,Srb LhRe(Reb
4 3 5.51logo Mo< 2.8,
FL =CL Pf 7rb,
Tomiyama et al. 3 1.39
(2002) CL min[0.288tanh(0.121Reb),f(Eod)] Eod < 4 g(f , pg)4
{f(Eod) 4 Eod <10.7 p2 3
In order to conveniently compare and analyze, the models
of the lift force are rewritten by using Equation (4). The lift
coefficient models rewritten by Eq. (4) are listed in Table 3.
It can be seen from Table 3 that the lift coefficient models
are very different from one another, and factors influencing
the lift coefficient models are numerous compared with the
drag coefficient models. Through rough comparison, it can
be discovered that key factors influencing the lift coefficient
model are the bubble Reynolds number (Reb) and
dimensionless shear rate (Srb). In order to qualitatively
understand every lift coefficient model, threedimensional
profiles of the lift coefficient models are plotted in Fig. 4
with respect to the bubble Reynolds number (Reb) and
dimensionless shear rate (Srb). Fig. 4 shows the profiles of
the lift coefficient models based on following conditions:
bubble diameter, db=22x 106m; bubble density, Pb
=1.3kg/m3; average void fraction, ao=1.338x104; liquid
density, pf =1000 kg/m3; surface intension, a =0.0728 N/m2;
gravity acceleration, g = 9.8 N/kg; E6tv6s number, Eo =
5.97x 105; pipe diameter, D =lm; bubble velocity,
Vb=20m/s; the bubble Reynolds number range, Reb=0.0150;
the dimensionless shear rate, Srb=0.0014.
Table 3. Lift coefficient models
Investigators Lift coefficient expressions
Saffman (1965) CL = 9.69 (Reb. Srb,)
Auton (1987) CL = 0.5
Zun (1980) CL = 0.3
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Wang et al.
(1987)
Mei & Klausner
(1994)
Legendre &
Magnaudet
(1998)
Tomiyama et al.
(2002)
Bagchi and
Balachandar
(2002)
One can see from Table
models of Tomiyama
Balachandar (2002) are
( 2
0.49 ,Inl+9.3168^ fb 1 v,
C, =a, 0.01+cot L=e Srb
/L K 0.1963 ) D pm Reb 1.18(og/pf)14
3 (S. 2 1.72J( ) 4 2 21/2
8 Srb 2
J(s )= 0.6765[1+tanh(2.51gs +0.191)x [0.667+tanh(6(s 1 .
CLF = L/&lowRe(RebSrb) + (ghRRe L (leSr)=lowR(Re) Srb 05 + .255 5'
R 1(R )= I1+16/Reb *221
2 1+29/Re 2 exp(.d
C Iminm[.288tanh(O.121Reb.),f(Eod)] Eod < 4 g(pf, Pg
CL f (Eod) 4 Eod 10.7 od
f(Eod) = 0.00105Eo O 0.0159Eo
d~ d
CL 3 24 (1+0.597Re0593)
8 Re b
3 and Fig. 4 that the lift coefficient
et al. (2002) and Bagchi and
independent of the dimensionless
shear rate and only associated with the bubble Reynolds
number. As far as the lift coefficient model of Bagchi and
Balachandar (2002) is concerned, its profile is a little
similar to that of the drag coefficient. With increasing
bubble Reynolds number, the lift coefficient quickly
decrease at Reb<10 and then continue to slowly diminish.
For the lift coefficient model of Tomiyama et al. (2002), its
profile shows the contrary trend to that of Bagchi and
Balachandar (2002) at Reb<10. The lift coefficient of
Tomiyama et al. (2002) gradually approaches a constant
value of 0.3 with increasing Reb. For the lift coefficient
model of Saffman (1965) corresponding to Fig. 4 (a), the
lift coefficient depends on Reb and Srb, and its value
becomes great when the product of Reb and Srb is very
small. At the same Srb (or Reb), the smaller Reb (or Srb)
exhibits the greater influence on the lift coefficient. The
influence of Reb or Srb on the lift coefficient is of equal
importance. For the model of Wang et al. (1987)
0.0204Eod +0.474 ,, dH =db(+0.163Eo0757)
corresponding to Fig. 4 (b), the lift coefficient is quite small
compared with other models when Reb is higher than 1 and
Srb is less than 1. When Reb is higher than 1, the lift
coefficient is independent of Srb; but the influence of Srb on
the lift coefficient seems quite complicated at Reb less
than 1. The models of Mei and Klausner, (1994) and
Legendre and Magnaudet (1998) have a similarly
distributing trend, and they have nearly nothing to do with
Srb when Reb is relatively high. Similarly they are
independent of Reb for high Srb. When both Srb and Reb
have small values at the same time, the lift coefficient can
reach large values. But they approach a constant value of
0.5 with increasing bubble Reynolds number. Additionally,
the models of Auton (1987) and Zun (1980) are independent
of Srb and Reb, and the lift coefficients keep constant values.
Joshi (2001) concluded through analysis of literature that
both the direction and magnitude of force depend on the
local flow condition such that using mean values of the lift
coefficient is misleading. It shows that the models of Auton
(1987) and Zun (1980) are not accurate.
10
10
101
106
4
(b) Lift coefficient model of Wang et al. (1987)
Paper No
3 50
2 00
Sb 01 20Reb
00
(a) Lift coefficient model of Saffman (1965)
Paper No
1 0
a s
(c) Lift coefficient model of Mei and Klausner, (1994)
10,
10
1 04
3 50
Srb Reb
00
10'
10,
4
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
3 0' 50
Srb I bbo
(d) Lift coefficient model of Legendre and Magnaudet,(1998)
(e) Lift coefficient model of Bagchi and Balachandar (2002) (f) Lift coefficient model of Tomiyama et al. (2002)
Figure 4: Lift coefficient profiles as Srb and Reb function.
In order to clearly show differences among all lift
coefficient models, profiles of all models as functions of
Reb or Srb are plotted in Figure 5 against fixed Reb or Srb.
Fig. 5 (a) shows the lift coefficient profiles as the function
of Reb at Srb = 0.2. For models of Saffman (1965) and
Bagchi and Balachandar (2002), the lift coefficients linearly
decrease with increases of the bubble Reynolds number in
the loglog grid, whereas profiles of other models have a
common feature that the lift coefficients approach constant
values with increasing the bubble Reynolds number. The
model of Wang et al. (1987) is almost independent of Reb
and keeps a finite value CL=0.0085 at Srb= 0.2. For the
model of Tomiyama et al. (2002), the lift coefficient
increases to a constant value CL=0.3 with increases of Reb.
Compared with the model of Tomiyama et al. (2002), the
lift coefficient of Mei and Klausner (1994) gradually
decreases to 0.5 with increasing Reb. However, the lift
coefficient of Legendre and Magnaudet (1998) firstly
quickly decreases and then slowly increases to 0.5 with
increasing Reb. Fig. 5 (b) and (c) show the lift coefficient
profiles as the function of Srb at fixed Reb = 0.1, 10. One
can see from Fig. 5 (b) and (c) that Srb has the definitive
influence on the lift coefficient models of Saffman (1965),
Wang et al. (1987), Mei and Klausner (1994) and Legendre
and Magnaudet (1998) at Reb=0.1 and 10. Compared with
the models of Wang et al. (1987), Mei and Klausner (1994)
and Legendre and Magnaudet (1998), the dependence of
Saffman model (1965) on Srb is more obvious and its
value rapidly decreases and then almost keeps a constant
value 0.5 with increasing Srb for fixed Reb=0.1 and 10. For
the fixed Reb=0.1, the lift coefficients of Mei and Klausner
(1994) and Legendre and Magnaudet (1998) show
complicated changes with increasing Srb. However, for the
fixed Reb=10, the lift coefficients of Mei and Klausner
(1994) and Legendre and Magnaudet (1998) keep
unchanged with increasing Srb. It shows that the lift
coefficients of Mei and Klausner (1994) and Legendre and
Magnaudet (1998) are not related to Srb when Reb is higher
than a certain value. The lift coefficient of Wang et al.
(1987) slowly decreases with increasing Srb for fixed
Reb=0.1, whereas it slowly decreases to a fixed value of
0.008 with increasing Srb for fixed Reb=0.1. In short, one
can see from the above analysis that, except the model of
Saffman (1965), other models of the lift coefficient have
nearly nothing to do with Srb when Reb is higher than a
certain value.
It can be seen from the above analysis that the models of
Bagchi and Balachandar (2002) and Tomiyama et al. (2002)
are independent of Srb, whereas other models depend on Reb
and Srb simultaneously. It was reported that the lift
coefficient strongly depends on both Srb and Reb at low Reb,
whereas for moderate to high Reb this dependence are
found to be very weak (Legendre and Magnaudet, 1998;
Sankaranarayanan and Sundaresan, 2002).
Paper No
(a) Lift coefficient profiles as Reb function, at Srb =0.2.
102 Bagchi and Balachandar(2002)
0 egendre and Magnaudet,(1998)
10965)
SMel and Klausner (1994)
10' CL 03
10
~Wa etal (1987)
Tomivama et al (2002)
0 05 1 15 2 25 3 35 4
Sr
(b) Lift coefficient profiles as Srb function, at Reb=0.1
101
Bagchi and Balachandar (2002)
Saffman (1965)
10
MemandKlausner(1994) C,05
Tomlmaeal(200)  ^
Tomlyama et al (2002)
101 C03 Legendre andMagnaudet,(1998)
102
Wang etal (1987)
10A
0 05 1 15 2 25 3 35 4
Srb
(c) Lift coefficient profiles as Srb function, at Reb= 10.
Figure 5: Lift coefficient profiles as Srb or Reb function.
Dijkhuizen et al (2010) experimentally showed that there is
a very strong effect of the shear rate on the lift force in a
contaminated liquid. Additionally, some authors reported
that the higher shear rate corresponds to the larger CL when
the fixed Reb is less than 10 (Legendre and Magnaudet,
1998; Cherukat et al., 1999). It is thus clear that the
influence of Srb on CL can not be neglected. A number of
studies showed that the lift coefficient for a stationary
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
sphere in a linear shear flow rapidly decreases with
increasing particle Reynolds number. In short, one can
conclude that the lift coefficient should depend on the Reb
and Srb at the same time for the shear flows, and this
dependence of CL on them varies with changes of flow
regions. And thus, in our view, the models of Mei and
Klausner (1994) and Legendre and Magnaudet (1998) seem
to be good agreement with the flow rule and have a wide
range of application. Compared with the model of Mei and
Klausner (1994), the model developed by Legendre and
Magnaudet (1998) in the singlebubble system, has been
extended to the multibubble system and seems to be
applied well to describe the lift coefficient in bubbly flows.
In short, whether the drag coefficient model or the lift
coefficient model must conform to the physical law, and
they should have a broad scope of application. Developing
perfectly computational models of the drag and lift
coefficient is a great challenge for every one.
Conclusions
Through a large number of literature reviews, we
summarized some typical expressions for the drag and lift
coefficient models, qualitatively analyzed each model, and
performed comparison among all models. The following
results were obtained:
(1) The drag coefficients have similarly distributing trends
although their models are not same, whereas varying
amplitudes for different models are unequal with increasing
Reb. The critical bubble Reynolds number corresponding to
the inertial region is different for different drag coefficient
models.
(2) The key factors influencing the lift coefficients are Reb
and Srb, but dependence of different lift coefficients
models on them are different. At the fixed Reb or Srb, the
profiles of the lift coefficient models as functions of Reb or
Srb show great differences.
(3) According to our analysis and some conclusions from
the literature, in contrast, the drag coefficient model
developed by Zhang and Vanderheyde (2002) seems to be
more reasonably applied to describe the drag coefficient,
and the lift coefficient models of Legendre and Magnaudet
(1998) may be good agreement with flow rules.
Acknowledgement
We gratefully acknowledge the financial support from the
NSFC Fund (No. 10602043, 50821064, 50876114), the
Specialized Research Fund for the Doctoral Program of
Higher Education of China (No. 20090201110002) and
Paper No
Jiangsu Provincial Natural Science Foundation
(BK2009145).
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