Model and experimental visualizations of the interaction of a bubble with an inclined wall
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Chemical Engineering Science 63 (2008) 1914 – 1928
www.elsevier.com/locate/ces
Model and experimental visualizations of the interaction of a bubble with an
inclined wall
B. Podvin a,∗ , S. Khoja b , F. Moraga c,1 , D. Attinger b
a LIMSI-CNRS UPR 3251, Université Paris-Sud, France
b Department of Mechanical Engineering, Columbia University, USA
c Rensselaer Polytechnic Institute, Troy, USA
Received 31 May 2007; received in revised form 10 December 2007; accepted 14 December 2007
Available online 23 December 2007
Abstract
In this paper we derive a model based on lubrication theory to describe the interaction of a bubble with an inclined wall. The model is an
extension of the model derived by Klaseboer, Chevailier, Mate, Masbernat, Gourdon [2001. Model and experiments of a drop impinging on
an immersed wall. Physics of Fluids 13(1), 45–57.] and Moraga, Drew, Larreteguy, Lahey [2005. Modeling wall-induced forces on bubbles
for inclined walls. Multiphase Science and Technology 17(4), 483–505.] in the case of a horizontal wall. We consider bubbles of diameter
1–2 mm, which corresponds to high Reynolds numbers Re ∼ O(1 0 0), and moderate deformation effects (with a Weber number of O(1)).
Predictions of the model are compared with experimental visualizations of air bubbles rising in water toward an inclined wall. The dynamical
behavior of bubbles is observed to depend on the wall inclination. We find that the model reproduces the bubble trajectories for wall inclinations
smaller than 55◦ –60◦ . This critical value for the wall inclination corresponds to an experimentally observed transition in the bubble bouncing
behavior, which agrees with the observations of Tsao and Koch [1997. Observations of high Reynolds number bubbles interacting with a rigid
wall. Physics of Fluids 468, 271.]. We show that the main features of our lubrication-based model for rebound with an inclined wall can be
expressed with a simple force model proposed by Moraga et al., suitable for use in direct numerical simulations of multiphase flow.
䉷 2007 Elsevier Ltd. All rights reserved.
Keywords: Bubbles; Lubrication theory; Wall interaction
1. Introduction and bubble–wall interactions have been found to be of crucial
interest for medical applications such as echography (Becher
The numerical simulation of multiphase flows in complex, and Burns, 2000). Drag reduction using injected microbubbles
realistic geometries involves a wide variety of scales and there- has also been the focus of intensive study (Xu et al., 2002),
fore requires adequate modelling to describe some aspects of with promises to reduce the costs and increase the performance
the fluid–particle, particle–particle, particle–wall interaction. In of sea transport by as much as 30%. Despite ever-increasing
industrial processes such as cooling of nuclear reactors or phar- computer power, direct numerical simulation of large numbers
maceutical processes (Mudde, 2005), the phase ratio of bubbles of bubbles is not yet possible. Moreover, even the dynamics
to the liquid phase tends to be high, so that collisions of bubbles of a single bubble are not completely understood (Prosperetti,
with each other or the wall happen frequently. Bubble–bubble 2004). Generally speaking, research studies involving bubbles
tend to focus only on a few selective aspects of the physics,
while using crude modelling for those aspects considered less
∗ Corresponding author. important for the application at hand (Theofanous, 2004). The
E-mail address: podvin@limsi.fr (B. Podvin). interaction of a bubble with a wall constitutes only a relatively
1 Currently at General Electric Research and Development Center, small piece of the puzzle in many applications of multiphase
Niskayuna, NY, USA. flows, and as such has often been neglected in the past. The
0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2007.12.023B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1915
simplest and most commonly used model consists in defining 2. Experimental configuration
a restitution coefficient for the bubble velocity (Canot et al.,
2003). We also observe that in most theoretical and numeri- In this study an apparatus is built to generate air bubbles in a
cal developments (Shopov et al., 1990) the wall is chosen to liquid and observe their rebound on a wall with controllable in-
be horizontal or vertical, whereas in real-life applications, the clination. The apparatus is a rectangular water tank with 0.5 in
wall may well be slanted. Yet wall inclination strongly con- thick Plexiglas walls. It is 30 cm long, 30 cm high and 3 cm
ditions the behavior of the bubbles such as their velocity or wide. A rotating solid wall is placed 15 cm above the bubble
their shape, as shown by the observations of Tsao and Koch injection point. Air bubbles were first generated using a syringe
(1997) or Perron et al. (2006). A ship hull for instance has and a Hamilton needle with 50 m diameter. The bubble diam-
walls of various inclinations, each corresponding to a different eter ranged from 0.4 to 2 mm. In order to control the bubble size
force balance, which has consequences for the effectiveness of and therefore ensure the reproducibility of the experiments, we
a drag reduction strategy. used an solenoid valve through which pressurized air entered a
The goal of this paper is to evaluate the relevance of an ex- nozzle. The inner diameter of the nozzle was 127 m, which al-
tension of Moraga et al.’s (2005) force model to describe the lowed us to produce bubbles of diameter between 1 and 2 mm.
rebound of a bubble on an inclined wall. An additional objec- Typical opening times of the valve ranged from 0.5 to 5 ms.
tive of the paper is to verify and validate the 2D lubrication The main difference between the apparatus and that of Tsao
equation as a first step before coupling it with a 3D solver. and Koch (1997) is that we use a plate instead of a channel as
Coupling the lubrication solver to a simple bubble trajectory a solid impact boundary. This modification reduces the amount
equation as Klaseboer et al. (2001) and Moraga et al. (2006) of interfaces between the illumination and the camera objective
did, instead of to a 3D solver, has the advantages of produc- from 8 to 4, which allows us to improve the quality of our
ing a much simpler algorithm that can still solve the bubble pictures. The setup allowed varying the angle between the wall
rebound problem if the bubble and the wall inclination angle and the bubble vertical trajectory from 90◦ to 5◦ . The liquid
are kept small enough. However the coupling to the trajec- phase consisted of distilled water (Type 2). The system was
tory equation forces the introduction of simplifying assump- backlit from one side with a halogen lamp. A schematic of
tions for the boundary conditions of the 2D lubrication equation the apparatus is shown in Fig. 1. The total height of the tank
solver. These assumptions include a simplification of the bub- was chosen so that the bubbles could reach their corresponding
ble shape and the extent of the 2D lubrication solver: it is terminal velocity before they interact with the solid surface.
assumed that (a) the deformation of the portion of the bub- Let and be the respective density and viscosity of the
ble interphase further away from the wall is negligible and (b) fluid. Let be the surface tension at the air–water interface
that the 1D flow assumptions needed to derive the lubrication =0.07. The air bubble is characterized by its equivalent diame-
equations hold in a volume of fluid of small extent separat- ter d=2R and terminal velocity VT . Three adimensional param-
ing the bubble from the wall, i.e. with a characteristic radius eters govern the wall–bubble interaction. One is the Reynolds
of r < rmax . These assumptions can be removed by coupling number defined as
the lubrication equation solver to a 3D solver. The feasibility VT d
of this coupled multiscale solvers has been already proven by Re =
Shopov et al. (1990), which show the development of a film
between the bubble and the wall. Canot et al. (2003) have cou- which relates the importance of inertial effects to that of viscous
pled an analytical model based on lubrication theory with a effects. The other one is the Weber, which characterizes the
boundary element method to simulate a 2D (cylindrical) bub- bubble capacity for deformation:
ble approaching a horizontal wall. They compare the bubble
trajectory to the experimental observations of Tsao and Koch VT2 d
We = .
(1997), which were made for horizontal as well as inclined 2
walls. Klaseboer et al. (2001) have developed a model for the
The third one is the wall inclination
rebound of a drop impacting a horizontal plane wall. The model
is based on a force balance for the drop and the use of lubrica- 0 < < 90◦ .
tion approximation to compute the force exerted by the wall.
The model prediction was satisfactorily compared with exper- The Reynolds numbers in the experiment range from 40 to 560
imental results. Moraga et al. (2005) solved the model for a and the Weber number varies between 0.02 and 1.80. As the
horizontal wall to derive a simple law which could be imple- bubbles rose at their terminal velocities, their trajectory was
mented in a two-fluid simulation of bubbly flows. To better recorded using a high-speed camera (Pixelink PLA-741), which
understand the physics of the interaction, we use experimental was able to capture up to 1000 frames per second (FPS). A
vizualizations of air bubbles rising in water through buoyancy compromise had to be found in order to have both a sufficiently
and bouncing against an inclined wall. We describe the experi- high-time resolution and a sufficiently large field. In practice
mental configuration in Section 2. The model equations are pro- the frame rate varied around 300 fps, while the total extent of
vided in Section 3. Comparison of experimental observations the field of view was 40 mm × 30 mm. The centroid of the bub-
with the model is given in Section 4. Conclusions are given in ble was found by processing individual frames in ScionImage,
Section 5. and coupling it with a MATLAB routine, which determined1916 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
Light Source
Camera
Nozzle
Valve Function
Opens at 1.2 Volts for DC generator
Opens at 3.2 Volts for AC
Pressurized air inlet
Fig. 1. Experimental apparatus.
the interface of the bubble. The centroid was taken to be the
midpoint between the measured major and minor axes of the
bubble. The bubble speed was found by measuring the bub-
θ
ble centroid position in successive frames and multiplying the
traveled distance by the frame rate. Since the bubble shape was
oblate, we also calculated the bubble aspect ratio, defined as x
the ratio of the horizontal and vertical axes. The pixel incerti- 0
tude yielded a maximum error of 6% in the calculation of the VT y
bubble diameter for the range of bubbles studied. The fluctu- h(x,z)
ation in frame rate caused an error of 2% in frame rate mea-
surement. Since the velocity was computed as the product of
distance traveled in one frame and the frame rate, the total error
in the velocity measurements was 8%.
3. The numerical model
3.1. Equations
Fig. 2. Physical configuration.
Our purpose is to describe the motion of a bubble of density
B , of volume V and equivalent radius R (or diameter d =
2R), rising initially at terminal velocity VT in a quiescent fluid The origin O of the reference frame coincides with the normal
toward a wall inclined at an angle, as illustrated in Fig. 2. projection of the bubble centroid onto the wall.
Let and be, respectively, the fluid density and viscosity. Let us write the bubble centroid velocity U as U = U i + V j .
We choose to describe the motion in a reference frame mov- The equations of motion for the bubble centroid are obtained
ing with the bubble tangential velocity relative to the wall. We by estimating the different forces acting on the bubble:
assume that the bubble tangential acceleration is small enough
to be neglected, so that the reference frame can be assumed to dU
be Galilean. Let (i, j, k) resp. (x, y, z) be the coordinate system b V = Fbuoyancy + Fdrag + Fadded mass
dt
resp. coordinate variables associated with the reference frame. + Fhistory + Fwall . (1)B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1917
Specifically As the bubble deforms, the pressure jump becomes
• V is the bubble volume. 1 1
PL − PB = − + , (7)
• Fbuoyancy is the buoyancy force Rx Rz
Fbuoyancy = (( − b ) 43 R 3 g sin )i where Rx and Rz are the principal curvature radii of the de-
− (( − b ) 43 R 3 g cos )j , (2) formed interface. The effect of the wall can therefore be com-
puted as the spatial integral of the pressure difference between
where g is the gravity. the spherical and the deformed interface. The deformation is
• Fdrag is the drag force (excluding wall effects) assumed to take place only on the top surface of the bubble
within a region of size |x|, |z| < rmax with rmax >R. In prac-
Fdrag = −CD Rev 4RU i − CD Rev 4RV j , (3) tice, we take rmax ∼ R and we check a posteriori that the ex-
cess pressure is indeed negligible on most of this domain. This
where Rev is the Reynolds number based on the actual bubble excess pressure p can be written as
velocity
√ 2 1 1
2R U 2 + V 2 p = − + . (8)
Rev = R Rx Rz
The curvature radii can be expressed as a simple function of
and CD is the drag coefficient based on the actual bubble ve-
the film height h(x, z) if its deformation is small enough:
locity, which is typically determined in an empirical fashion.
• Fadded mass is the added mass force
1 1 j2 h j2 h
+ = 2 + 2. (9)
dU dV Rx Rz jx jz
Fadded mass = −CAM V i+ j , (4)
dt dt
Finally, the pressure force can be expressed as
where CAM is the added mass coefficient
• Fhistory is the history force.
Fwall = − (p)nx dx dzi − (p)ny dx dzj , (10)
t
√ 1 dV
Fhistory = −6 c c R 2 √ d. (5)
0 t − d where nx and ny are the components of the bubble normal, and
the normal can be approximated by
Note that unlike Moraga et al. (2005), we have chosen to in-
clude the history force in the equations. We found that adding jh jh
the history force term resulted in a small shift in the ampli- n= i+ k − j. (11)
jx jz
tude and characteristic time scales of the rebound, leading to
a slightly better agreement with experimental observations. Combining Eqs. (2)–(10) together, one can then rewrite
We use the Basset formulation to represent the force. A full Eq. (1) as
discussion of the effect of the history force can be found in
Klaseboer et al. (2001). 4 dU
R 3 (CAM + B )
The effect of the wall is represented by a force Fwall . The 3 dt
idea is that the wall makes itself felt through an excess pres- 4 3
sure exerted on the top of the bubble, which corresponds to = ( − b ) R g sin − CD Re4RU
3 t
a deformation of the interface. The flow between the bubble √ 1 dU
and the wall constitutes a film which can be described us- − 6 c c R 2 √ d
t − d
ing lubrication theory. To be able to use lubrication theory 0
2 1 1 jh
we have to assume that the pressure and velocity field are − − + dx dz (12)
R Rx Rz jx
uniform across the film, an hypothesis that limits the range
of validity of the model. The deformation corresponds to the and
height h(x, z) of the film between the bubble and the wall,
since the effects of deformation at the interface farther from 4 dV
R 3 (CAM + B )
the wall are neglected by the model. 3 dt
4 3
Let PB be the pressure inside the bubble and PL be the = −( − b ) R g cos − CD Re4RV
3
pressure just outside the bubble interface. Initially, in the case √ t 1 dV
of a spherical bubble, one has − 6 c c R 2 √ d
t − d
0
2 2 1 1
PL − PB = − . (6) + − + dx dz. (13)
R R Rx Rz1918 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
In the reference frame, at t = 0
U = VT sin (14)
Δ p=0 Δ p=0
and
V = −VT cos . (15)
To solve for the bubble centroid velocity in Eqs. (12)–(13) with
initial conditions (14) and (15), we need an evolution equation
for the liquid film height as it is drained between the bubble Fig. 3. Numerical domain.
and the wall. This is readily obtained from lubrication theory,
in which we assume that the flow in the film is in the plane
of the wall and dominated by viscous effects. Given that even and we also have
a very low concentration of impurities in the flow is sufficient p = 0. (21)
to make the interface immobile, we assume a no-slip boundary
condition for the liquid at the bubble interface, rather than a Eqs. (12)–(21) constitute a coupled system (S) in which the
no-shear condition (see for instance Klaseboer et al., 2001 for a bubble centroid position (U, V ) and its top interface h(x, z) are
full discussion). However, this assumption may no longer hold determined. The numerical domain along with the boundary
when the wall is slanted. Moreover, we do not have enough conditions is shown in Fig. 3. Numerical solving of the system
control on water quality so we have to see the immobile and (S) is discussed in the next paragraph.
mobile descriptions as limit cases, with the real case falling
somewhere in between. Thus the interface will be considered 3.2. Numerical implementation
immobile. Mass conservation for a slab of fluid within the film
yields the modified lubrication equation: Here we solve a 2D version of the lubrication and bubble
trajectory equation due to the fact that the inclined wall breaks
jh 1 j(U h) 1 j jPL 1 j 3 jPL the axial symmetry of the horizontal wall case. The equations
+ = 3
h + h ,
jt 2 jx 12 jx jx 12 jz jz constituting (S) are discretized using second-order finite differ-
(16) ences and linearized about the current solution. The system is
no longer pentadiagonal as in 1-D, but has 13 non-zero diago-
where PL is the pressure in the liquid film. Assuming that the nals over a bandwidth of 2 N + 1, where N is the number of
pressure variations within the bubble are negligible compared grid points in either direction x or z. NAG Fortran routines were
to those in the film, one can replace the film pressure in this last used to solve for the band-diagonal system within a tolerance
equation using Eq. (7). This leads to a fourth-order elliptic equa- of 10−8 . The bubble trajectory was integrated in time using a
tion which requires two boundary conditions on the boundary second-order finite difference scheme, with implicit treatment
at all times. These are provided by the assumption that outside of the drag and the added mass force, and explicit treatment of
the film domain the pressure difference is zero and the bubble is the wall force. The time step was constant and equal to 0.002
no longer deformed by the wall i.e., it moves with the centroid in most of the simulations. We checked that smaller time steps
velocity. This means that on the boundary (x 2 + z2 )1/2 = rmax , did not change the results, while larger time steps occasion-
ally led to the blow-up of the solution. The code was tested for
p = 0 (17) different spatial resolutions. An increase of 30% in the spatial
resolution did not produce any noticeable changes. The effect
and of larger increases in the spatial resolution were not tested due
to the large increase in computational time associated with im-
dh
= −V . (18) proved resolution. The lubrication equations are discretized in
dt cartesian coordinates, in which defining a circular or elliptic
Note that if we assume that the interface is mobile, we obtain boundary was not straightforward. We chose instead to define
a modified equation for the film height a square boundary interface
|x| = rmax , |z| = rmax .
jh 1 j(U h) 1 j 3 jPL 1 j 3 jPL
+ = h + h . (19)
jt 2 jx 3 jx jx 3 jz jz We expect the pressure and interface variations to be significant
only near the very top part of the bubble, so that the exact
Since the bubble is initially spherical, the initial condition contour of the interface boundary should be irrelevant. This
for the film is hypothesis was checked a posteriori. A variety of models were
used to represent the drag and the added mass coefficients, as
(x 2 + z2 ) well as the correction due to the influence of the wall. For
h(x, z) = h0 − (20)
2R spherical quasi-rigid bubbles, we used Schiller and Nauman’sB. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1919
law (Clift et al., 1978): the presence of impurities (Clift et al., 1978), we restricted our
study to experimentally determined terminal velocities.
24
CD = (1 + 0.15Re0.687 ) (22) All quantities appearing in Eqs. (13) and (14) can be adi-
Re mensionalized with the following scales:
which has been validated for Reynolds numbers up to 1000. In
v
order to account for the effect of the wall on the drag coefficient, v̂ = ,
following Van der Geld (2002), this law was replaced by VT
VT
CD =
24
C(y)(1 + 0.15Re0.687 ), (23) tˆ = t ,
Re R
x z
where x̂ = , ẑ = .
R R
3 −2
R
C(y) = 1 − . (24) 4. Results
2y
For a spherical rigid bubble, the added mass coefficient is 4.1. Interaction with a flat wall
equal to 0.5. If the effect of the wall is taken into account, an-
other expression for C(y) (Van der Geld, 2002) can be derived: The interaction of bubbles with a flat wall has been well
documented in Tsao and Koch (1997) and in Klaseboer et al.
3 −1 (2001). A first validation of the 2-D model was obtained by
3 R
C(y) = 1− − 1. (25) comparing the prediction with observations of air bubbles ris-
2 2y
ing through water toward a horizontal wall. The bubble radius
For non-spherical bubbles of aspect ratio , where was 0.85 mm. The Reynolds number ReT was 522 and the We-
= rmax /rmin with rmin and rmax , respectively, characterizing ber number was 0.866. These characteristics are close to one
the smallest and largest dimensions of the bubble, we used of Tsao and Koch’s experiments investigated in Moraga et al.
Moore’s (1990) functions: (2005), which constitutes an additional check on the validity
of our experimental data. Following Tsao and Koch (1997), it
CD Re = 55G()(1 + H ()Re−0.5 ), (26) was assumed that the interface was mobile. As explained in
Klaseboer et al. (2001), the motion of the bubble induces a dis-
where
placement of the surfactants from the front of the bubble toward
4/3 2 2 − 1 − (−2 + 2) cos−1 −1 the rear. The bubble front may therefore be considered mobile,
G() = ( − 1) 1/2
while the accumulation of surfactants on its rear surface gen-
3 (− 2 − 1 + 2 cos−1 −1 )2
erates a drag equivalent to that obtained on a rigid bubble. It
(27)
therefore makes sense to assume mobility at the bubble surface
and in the film close to a horizontal wall. However, this assump-
tion no longer holds when the wall is slanted (the bubble front
H () = 0.001954 − 0.2133 + 1.7032 − 2.146 − 1.573.
(28)
0
For a bubble of aspect ratio smaller than 2.5, the added mass
coefficient is (see Klaseboer et al., 2001) −1
−2
CAM = 0.62 − 0.12. (29)
−3
The aspect ratio of the bubbles was kept constant through-
−4
out the simulation, identical to the initial aspect ratio observed
y (mm)
in similar cases experimentally. However, this is different from −5
what usually happens in experiments where the aspect ratio of model
−6 experiment
the bubble changes sharply after the rebounds (see next sec-
tion). The reason to keep the aspect ratio constant is that the −7
correlations above account for the effect of drag for a bubble −8
away from walls, where the aspect ratio is due to the com-
−9
petition of surface tension and buoyancy. But when a wall is
present, the wall also contributes to deform the bubble in ways −10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
not accounted for in the expansions derived by Moore. In the
time (s)
present study we focus in air bubbles rising in water. Mea-
suring the bubble radius should be sufficient to determine the Fig. 4. Trajectory of the centroid of a bubble of radius 0.85 mm with Re=522
terminal velocity from balancing the drag force with the buoy- and We = 0.866 rising toward a horizontal wall—solid lines: 2D lubrication
ancy force. However, since the terminal velocity is sensitive to model (the 1D model gives identical results); circles: experiments.1920 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
10 mm
8 mm
6 mm
4 mm
2 mm
0 mm
0 10.1 20.2 30.3 40.4 50.5 60.6 70.7 80.8 90.9 101 222
Time (ms)
Fig. 5. Visualization of a bubble of radius 0.85 mm with Re = 522 and We = 0.866 rising toward a horizontal wall—each frame is taken each 1/99 s.
0.6 0.35
t=24 ms t=24 ms
t=25 ms t=25 ms
0.4 t=26 ms 0.3 t=26 ms
0.2 0.25
0 0.2
h (r)
Δp
−0.2 0.15
−0.4 0.1
−0.6 0.05
−0.8 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
r r/rmax
Fig. 6. Evolution of (a) the adimensional excess pressure; (b) the adimensional film height for a bubble of radius 0.85 mm, Re = 522, We = 0.866 rising toward
a horizontal wall. Here rmax = R.
no longer faces the wall), in which case the interface will be of square boundary conditions in the 2-D code did not generate
considered immobile. some discrepancy with the 1D code, where radial boundary con-
Fig. 4 compares the bubble trajectory observed in the exper- ditions were used. The pressure variations are concentrated in a
iment with its prediction from the 2D model. We found that region close to the top of the bubble so that boundary conditions
the 2D code yielded nearly undistinguishable results from the at the exact boundary are not significant. This also validates
1D version. Both models capture the correct amplitude rebound the assumption that pressure variations occur over a character-
and characteristic time scale. This is all the more remarkable istic scale which is small compared to the bubble radius.
as during the rebound the aspect ratio of the bubble varies sub- Furthermore, the much coarser resolution (by a factor of 5)
stantially, as is evident in Fig. 5. Magnaudet et al. (2003) have in the 2D case does not appear to perturb significantly the pres-
developed a theory that relates the trajectory of the bubble cen- sure profiles. Owing to the complexity of the problem in 2D,
troid with the bubble interface deformation. Changes in the we have not yet attempted a resolution comparable to the 1D
bubble aspect ratio should affect the intensity of the drag force case.
and the added mass force. Moreover, a portion of the kinetic
energy is converted into surface oscillations. 4.2. Observations of bubbles interacting with an inclined wall
Fig. 6 shows the evolution of the excess pressure and of the
film height as a function of the radial position, during the re- Several bubbles in the range diameter of 1.2–1.8 mm were
bounds. All substantial variations occur at locations r>rmax = sent toward the wall for various inclinations and the behavior
R, which therefore justifies a posteriori the original assumption of the bubbles was captured by the cameras. The trajectory of
made on rmax when deriving the model. We observe that the use about 20 bubbles was examined in detail. Three different typesB. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1921
1.7 −0.5
bouncing
sliding
1.65 −1
−1.5
1.6
−2
1.55
d (mm)
y/R
−2.5
1.5
−3
1.45
−3.5
1.4 −4
1.35 −4.5
0 10 20 30 40 50 60 70 80 0 5 10 15 20 25
wall inclination θ t VT/R
Fig. 7. Classification of the bubble trajectories as a function of the bubble Fig. 8. Normal position of a bubble of radius 0.51 mm, Re = 103, We = 0.12
diameter d = 2r and the wall inclination—the bubble at = 0 represents a as it interacts with a wall inclined at = 45◦ —the axis orientation here is
limit case of sliding as it sticks in fact on the wall. chosen to match that of the experiment.
of behaviors could be distinguished. The wall angle of incli-
nation appeared to be the determining factor for the bubble
trajectory.
• When the wall inclination was less than 10◦ , the bubble
bounced a few times (two or three) against the wall and
stopped moving, as in Fig. 5.
• For moderate wall inclinations up to a critical angle c ,
10 < < c = 55◦ –60◦ the bubble slided against the wall
with a constant speed. In some of these cases, some transient
bouncing—a few rebounds of decreasing amplitude—were
observed before the sliding motion. Transient bouncing (resp.
pure sliding) seemed to occur for bubbles with a diame-
ter larger (resp. smaller) than 1.2 mm. It is, however, diffi-
cult to isolate a single critical parameter such as the bubble
diameter or its velocity, since the bubble velocity depends
on the bubble diameter, which results in an implicit coupling
between the Weber number and the Reynolds number.
• Finally, for large wall inclinations > c , the bubble ex-
perienced steady bouncing. The amplitude of the rebounds
appeared to be constant over time. We note that due to the
limited scope of the fixed camera, we were not able to record
more than a couple of steady rebounds at a time. However, we
checked by moving the camera along the vertical axis, that
rebounds of the same amplitude were still observed several
Fig. 9. Visualization of a bubble of radius 0.51 mm, Re = 103, We = 0.12
rebound lengths away from the point where the bubble first
rising toward a wall inclined at = 45◦ —the frame rate is 100 s−1 .
hit the wall. Persistent bouncing was observed at an angle of
= 60◦ for the whole range of bubbles of diameter 1–2 mm.
Sliding was shown to occur at angles of 50◦ over the same 4.3. Comparison of model–experiments
range of bubbles. Fig. 7 shows that the critical angle seems
to be around c ∼ 55 + / − 5◦ for bubbles in the diameter We now examine to which extent the model is able to re-
range of 1–2 mm. This is in agreement with the observations produce the experimental observations. Accordingly we show
of Tsao and Koch (1997) made for slightly smaller bubbles in results for three typical cases corresponding to the three differ-
the diameter range of 1.0 – 1.4 mm. ent types of bubble dynamics observed.1922 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
0.25 0.8
t=5.20 t=5.20
t=5.60 t=5.60
t=6.0 0.6 t=6.0
t=8.0 t=8.0
0.2 0.4
0.2
0.15
Δ p (x,0)
0
h (x,0)
−0.2
0.1
−0.4
0.05 −0.6
−0.8
0 −1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
x/rmax x/rmax
Fig. 10. Evolution of (a) the film height; (b) the excess pressure for a bubble of radius 0.51 mm, Re = 103, We = 0.12 rising toward a wall inclined at
= 45◦ —all units are adimensional (see definition in Section 3.2).
0.4 0.8
wall force
drag
0.3 0.6 added mass
buoyancy
0.2 wall force
drag 0.4
added mass
tangential forces
0.1
normal forces
buoyancy
0.2
0
0
−0.1
−0.2
−0.2
−0.3 −0.4
−0.4 −0.6
0 5 10 15 20 25 0 5 10 15 20 25
t t
Fig. 11. Force balance in (a) the tangential direction (b) the normal direction for a bubble of radius 0.51 mm, Re = 103, We = 0.12 rising towards a wall
inclined at = 45◦ —all units are adimensional as defined in Section 3.2.
4.3.1. Sliding case without transient bouncing at x ∼ 0.1 rmax = 0.1R. The lack of symmetry corresponds to
The first case involves the case of a bubble of diameter the jump in the pressure profile which is shown in Fig. 10b.
1.03 mm, with corresponding Reynolds 103 and Weber num- Fig. 11 shows the force balance in each direction as the
ber 0.12. Fig. 8 shows that the bubble trajectory predicted by bubble approaches the wall. Since the bubble density is small
the model is a pure sliding motion, which indeed corresponds compared to that of the fluid, the rate of change of momentum
to the experimental observations. Images of the experimental for the bubble (not plotted) is small compared to the forces
bubble are gathered in Fig. 9. Since the Weber number is very acting on the bubble so that the forces appear to cancel each
small, deformation effects are expected to be very small and the other.
bubble interface shown in Fig. 10 remains relatively spherical, In the tangential direction, the balance between the drag
in agreement with experimental observations. It only flattens and gravity is only slightly disrupted as the bubble approaches
close to the top of the bubble. Note that the deformation is not the wall, which gives rise to a small added mass component.
entirely symmetrical in x, as the minimum film height is located The magnitude of the tangential wall force remains negligibleB. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1923
0 12
−0.5
10
−1
−1.5 8
−2
y (mm)
x (mm)
6
−2.5
−3
4
−3.5
−4 experiment 2
model experiment
−4.5 model
0 0.01 0.02 0.03 0.04 0.05 0.06 0
t (s) 0 0.02 0.04 0.06 0.08 0.1
t (s)
Fig. 12. Normal position of a bubble of radius 0.85 mm, Re =522, We=0.866
interacting with the wall inclined at = 28◦ . Fig. 13. Tangential position of a bubble of radius 0.85 mm, Re = 522,
We = 0.866 interacting with the wall inclined at = 28◦ .
throughout the evolution. In the normal direction, the wall force
balances the added mass component as the bubble comes near 0.8
wall force
the wall. Then it balances buoyancy as the bubble slides along drag
0.6 added mass force
the wall. As it can be seen from Eq. (24) we did not introduce a
buoyancy
correction to account for increased drag in the tangential direc-
0.4
tion due to the proximity of the wall. Thus we are likely under-
estimating the drag in the tangential direction when the bubble
0.2
2
Fw R/VT
is close to the wall. This is consistent with the results shown
in Figs. 13 and 18 that show good agreement between model
0
and experiment before the first rebound, but not after it, as the
drag correction would grow abruptly as the bubble approaches −0.2
the wall. These corrections can be taken from Van der Geld
(2002) for example and they would probably help improve the −0.4
agreement with the experimental data.
−0.6
4.3.2. Sliding case with transient bouncing 0 5 10 15 20
Special attention was given to this type of motion. We t VT/R
used the model to predict the trajectory of a bubble of radius
0.81 mm, for an angle of 28◦ . The Reynolds number of the Fig. 14. Time-varying normal forces exerted on a bubble of radius 0.85 mm,
Re = 522, We = 0.866 interacting with the wall inclined at = 28◦ . Here we
bubble is 522 and the Weber number is 0.89. Experimental ob- use adimensional time units: 10 adimensional time units correspond to about
servations show that the bubble experiences transient bouncing 0.028 s.
before sliding along the wall. As evidenced in Fig. 12, the evo-
lution of the distance wall–bubble matches the experiments in
time scale as well as in amplitude. However, Fig. 13 shows the bounce again, the wall force falls down to zero and the added
tangential velocity does decrease slightly during the rebound, mass force is now balanced by the drag force. Typically after
which the model fails to predict. This decrease in the velocity at most three rebounds the bubble settles down to a sliding
can be related to the surface oscillations taking place at the motion in which the buoyancy force is now compensated by
wall as the velocity changes sign. The variation in aspect ratio the wall force in the normal direction and the drag force in the
is not taken into account in the model in its present form. We tangential direction.
assume that the bubble retains its equilibrium shape (far from Fig. 15 shows the evolution of the film height as the bubble
the wall) throughout the entire wall interaction. Fig. 14 shows approaches the wall for the first two rebounds. The bubble
the normal force balance. As the bubble approaches the wall, interface flattens as the bubble moves toward the wall then
the dominant forces in the normal direction are the wall force becomes spherical again as it leaves the wall. The interface
and the added mass force, which almost compensate each deformation is weaker for the second rebound. At all times
other, since their magnitudes are both large compared to that of the interface remains generally symmetrical with respect to the
buoyancy. Then, as the bubble leaves the wall and prepares to bubble normal axis. Fig. 16 show the evolution of the pressure1924 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
0.35 0.35
t=4.2
t=4.8
0.3 t=5.4 0.3
t=5.4
t=6.0
t=6.6
0.25 0.25 t=7.2
0.2 0.2
h (x,0)
h (x,0)
0.15 0.15
0.1 0.1
0.05 0.05
0 0
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
x/rmax x/rmax
0.35 0.35
0.3 0.3
t=16.2
t=14.4 t=16.8
0.25 t=15.0 0.25 t=17.4
t=15.6 t=18.0
t=16.2
0.2 0.2
h (x,0)
h (x,0)
0.15 0.15
0.1 0.1
0.05 0.05
0 0
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
x/rmax x/rmax
Fig. 15. Film height at the bubble spanwise centerline during the first two rebounds—the bubble characteristics are given by R = 0.85 mm, Re = 522, We = 0.866.
profile during the rebounds. A key feature seems to be that the tion is neglected by our model. Second, the strong deformation
pressure profile remains symmetrical before the rebounds, but of the film and its lack of symmetry may make the flow highly
is characterized by strong asymmetry in the tangential direction multidimensional at r ≈ rmax ≈ R. Indeed as Tsao and Koch
as the bubble moves away from the wall. argue (see their Fig. 5) we expect an unsteady vortical structure
to form during the rebound. Although this vortical structure is
4.3.3. Steady bouncing present even in the case of a horizontal wall, the model can
At large inclination angles, the bubble experiences constant, perform well as long as most of the pressure contribution to
bouncing motion. A typical case can be examined in Figs. 17 the wall force comes from the center of the bubble. However,
and 18. We point out that in that case the lubrication model as the inclination angle increases and the flow becomes asym-
fails to capture the steady bouncing behavior. Several simpli- metrical the vortical structure is expected to make a bigger
fications in our model may play a role in the model inability impact. The multidimensionality of the flow introduced is not
to predict the steady bouncing. First, as our experiments and consistent with the assumptions used to derive the lubrication
those of Tsao and Koch put in evidence, the angle between the equation, which requires 1D flow. In addition the boundary con-
bubble largest axis and the wall changes considerably during a ditions assumed in Eq. (17) and to a lesser extent in Eq. (18)
rebound. That is not only the top portion of the bubble deforms are no longer appropriate. For all this reasons we believe that
but also the bottom one. This deformation of the bottom por- in order to predict the steady bouncing regime accurately, theB. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1925
0.7 0.8
t=4.2 t=5.4
t=4.8 0.6 t=6.0
0.6 t=5.4 t=6.6
0.4 t=7.2
0.5
0.2
0.4
0
Δ p (x,0)
Δp
0.3 −0.2
−0.4
0.2
−0.6
0.1
−0.8
0
−1
−0.1 −1.2
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
x/rmax x/rmax
0.7 0.8
t=14.4 t=16.2
0.6 t=15.0 t=16.8
t=15.6 0.6 t=17.4
t=16.2 t=18.0
0.5
0.4
0.4
0.3 0.2
Δp
Δp
0.2 0
0.1
−0.2
0
−0.4
−0.1
−0.2 −0.6
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
x/rmax x/rmax
Fig. 16. Excess pressure at the bubble spanwise centerline during the first two rebounds—the bubble characteristics are given by R = 0.85 mm, Re = 522,
We = 0.866.
lubrication model has to be coupled with a 3D solver that ac- where
counts for the bubble deformation and the flow dynamics in the
region where a lubrication equation model and its associated • H ( ) is a smoothed Heaviside function over a region of extent
boundary conditions are not appropriate. Shopov et al. (1990) 2 around a height of 0 .
have demonstrated that this kind of multiscale model is indeed • Fw is an empirical constant that depends on the Weber
feasible. number.
Eq. (30) is based on the observation that when the bubble is
4.4. Validation of the force model extension close to the wall the bubble normal velocity is small but its
acceleration is large. Thus the only force that can balance the
Moraga et al. (2006) have proposed in the case of a horizontal wall force is the added mass. The analytical solution of Tsao
wall that the wall force be modeled as a function of the bubble and Koch (1994) is consistent with this observation. When the
acceleration: bubble reaches equilibrium, the wall force exactly compensates
buoyancy. These characteristic trends are adequately captured
dV by the force model. Moraga et al. have proposed an extension
Fwall = −H ( )Fbuoyancy 1 + Fw , (30) in the case of a inclined wall by pointing out that the driving
dt1926 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
35 other two adimensional parameters Re and We are coupled, so
that they cannot be modified independently (Brennen, 1995).
30 Our first test case is a bubble of radius R = 0.83 mm, rising
toward the wall with a velocity of 30 cm/s. The corresponding
25 Reynolds and Weber numbers Re and We are, respectively, 449
and 1.89. We solved the model for different inclinations: 0◦ ,
20 15◦ , 30◦ , 45◦ . Results in Fig. 19 show that the normal wall force
x (mm)
acts on a time scale that is almost independent of the wall angle.
15 The intensity of the wall force was found to decrease as the wall
inclination increases. Moraga et al. suggested that the depen-
10 dance was proportional to cos , in other words that the force is
proportional to the normal velocity of the bubble. However the
5 wall force is expected to scale with the acceleration of the bub-
experiment
model
ble rather than the velocity. So an appropriate scaling would be
0 V /, where V is the normal velocity component and a char-
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 acteristic time scale of the rebound. We have ∼ h/V where h
t (s) is the film height. As shown by Tsao and Koch (1997), the am-
plitude of the pressure necessary to create surface deformations
Fig. 17. Experimental determination of the centroid tangential coordinate for
a bubble of radius 0.80 mm, Re = 603, We = 1.60 interacting with a wall must be on the order of p = O(/R) so that the characteristic
inclined at = 72◦ degrees and comparison with the model prediction. film height h scales as h = O((W e/Re)2/3 ). The dependence
of h with the wall inclination is therefore h ∝ (cos )2/3 . The
characteristic time scale varies like (cos )−1/3 and the wall
0 force should therefore scale with (cos )4/3 . A linear regres-
sion performed on the wall forces computed for different wall
−1 inclinations yielded a scaling in (cos ) with = 1.366.
In the tangential direction, we found that both wall force and
added mass maxima were at most 6% of the buoyancy force,
−2
so that as a first approximation, the influence of the wall force
can be neglected. These findings are in agreement with Moraga
y (mm)
−3 et al.’s conjecture.
We therefore propose the following form for the force model:
−4
dV
Fwall = −H ( )Fbuoyancy 1 + Fw (32)
dt
−5
experiment with
model
−6
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Fbuoyancy = Fbuoyancy cos ,
t (s)
Fw = Fw (cos )1/3 .
Fig. 18. Experimental determination of the centroid normal coordinate for
a bubble of radius 0.80 mm, Re = 603, We = 1.60 interacting with a wall An approach similar to that used by Moraga et al. (2005)
inclined at = 72◦ and comparison with the model prediction. can be used to determine the dependence of Fw on the Weber
number. To test this hypothesis, we build a modified version of
the 1D model corresponding to the assumptions made above
force for the bubble motion—-buoyancy—-needs to be rescaled
and compare it with the equivalent 2D model. The buoyancy
so that on a wall inclined at angle , one has
force is accordingly replaced with its wall-normal component,
dV and the added mass force was rescaled with a factor (cos )1/3
Fwall = −H ( )Fbuoyancy 1 + Fw (31) corresponding to the modification of the characteristic rebound
dt
time scale. We simulate a bubble of R = 0.85 mm rising against
with a wall inclined at an angle of 28◦ . Fig. 20 shows that the ve-
locity and position of the bubble is adequately predicted. The
Fbuoyancy = Fbuoyancy cos .
wall force is correctly estimated to within less than 5%. We
To determine whether this expression for the force model found that when the added mass coefficient is not modified, the
holds for inclined walls, we examine the dependence of the second rebound predicted by the 1D model occurs slightly ear-
model on the wall inclination. We point out that it is difficult lier, which is no longer the case when a modified time scale is
to investigate the effect of the other parameters, as the bubble introduced for the added mass force. The 1D modified model
average radius and bubble terminal velocity, and therefore the is clearly a good approximation for the 2D model, so thatB. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928 1927
0.08 0.7
θ=0 θ=0
θ=15 θ=15
0.06 0.6 θ=30
θ=30
θ=45 θ=45
0.5
0.04
0.4
Fwall/cos(θ)4/3
Fdrag/cos(θ)
0.02
0.3
0
0.2
−0.02
0.1
−0.04 0
−0.06 −0.1
0 5 10 15 20 0 5 10 15 20
t VT/R t VT/R
Fig. 19. (a) Normal drag force exerted on a bubble of radius 0.64 mm, Re = 449, We = 1.89 interacting with a wall with different inclinations: 0◦ , 15◦ , 30◦ ,
45◦ ; (b) Normal wall force exerted on the same bubble.
0.8 0
2−D model
0.7 1−D modified model −0.5
0.6 −1
−1.5
0.5
−2
0.4
Fwall
y/R
−2.5
0.3
−3
2−D model
0.2 1−D modified model
−3.5
0.1 −4
0 −4.5
−0.1 −5
0 5 10 15 20 0 5 10 15 20
t VT/R t VT/R
Fig. 20. (a) Normal wall force exerted on a bubble of radius 0.85 mm, Re = 522, We = 0.866 interacting with the wall inclined at = 28◦ ; (b) Normal position
of the bubble centroid.
expression (32) represents a quick, adequate estimate for the wall inclination: (a) sticking motion, for very low wall incli-
wall force. nations (smaller than 10◦ ); (b) sliding motion, for moderate
inclinations 10 < < 55◦ ; (c) repeated bouncing, for larger in-
5. Conclusion clinations. Transient bouncing could be observed in cases (a)
and (b).
The goal of this paper was to investigate the validity of We have shown that when the wall inclination is less than
a model to describe the bubble–wall interaction in the case 55◦ –60◦ , the model provides the correct time scale and rebound
of an inclined wall. We used a model based on lubrication amplitude for the bubble. However, it does not seem able to re-
theory, which leads to the derivation of a wall force model produce the slight variations in the tangential velocity that are
that could be plugged into complex multiphase simulations. associated with the rebound. This is likely to be associated with
High-speed vizualizations of the bubble trajectories made it the large interface deformations associated with our relatively
possible to identify three types of motion depending on the large bubbles (W e ∼ O(1)). When the wall inclination is larger1928 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914 – 1928
than 55◦ –60◦ , the model is not able to capture the steady bounc- Moore, D.W., 1990. The boundary layer on a spherical gas bubble of a
ing motion observed in experiments. This is likely to be due to deformable bubble with a rigid wall at moderate Reynolds numbers. Journal
the absence of lift effects in the current lubrication model. of Fluid Mechanics 219, 242–271.
Moraga, F.J., Cancelos, S., Lahey Jr., R.T., 2005. Modeling wall-induced
In order to make this model useful for large-scale multiphase forces on bubbles for inclined walls. Multiphase Science and Technology
flow simulations, the extension of the force model proposed 17 (4), 483–505.
by Moraga et al. is tested against experimental data. We find Moraga, F.J., Drew, D.A., Larreteguy, A., Lahey Jr., R.T., 2006. A center-
that a modified 1D model is quite appropriate to describe the averaged two-fluid model for wall-bounded bubbly flows. Computers and
wall–bubble interaction for moderate wall inclinations, even for Fluids 3 (4), 429–461.
Mudde, R.F., 2005. Gravity-driven bubbly flows. Annual Review of Fluid
moderate Weber numbers W e ∼ O(1). Mechanics 37, 393–423.
Perron, A., Kiss, L.I., Poncsak, S., 2006. An experimental investigation of the
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