modelandexperimentalvisualizationsoftheinteractionofabubbl ...physics of fluids 13(1), 45–57.] and...

Chemical Engineering Science 63 (2008) 1914–1928www.elsevier.com/locate/ces

Model and experimental visualizations of the interaction of a bubble with aninclined wall

B. Podvina,∗, S. Khojab, F. Moragac,1, D. Attingerb

aLIMSI-CNRS UPR 3251, Université Paris-Sud, FrancebDepartment of Mechanical Engineering, Columbia University, USA

cRensselaer Polytechnic Institute, Troy, USA

Received 31 May 2007; received in revised form 10 December 2007; accepted 14 December 2007Available 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 anextension of the model derived by Klaseboer, Chevailier, Mate, Masbernat, Gourdon [2001. Model and experiments of a drop impinging onan immersed wall. Physics of Fluids 13(1), 45–57.] and Moraga, Drew, Larreteguy, Lahey [2005. Modeling wall-induced forces on bubblesfor inclined walls. Multiphase Science and Technology 17(4), 483–505.] in the case of a horizontal wall. We consider bubbles of diameter1–2 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 dynamicalbehavior of bubbles is observed to depend on the wall inclination. We find that the model reproduces the bubble trajectories for wall inclinationssmaller than 55◦–60◦. This critical value for the wall inclination corresponds to an experimentally observed transition in the bubble bouncingbehavior, which agrees with the observations of Tsao and Koch [1997. Observations of high Reynolds number bubbles interacting with a rigidwall. Physics of Fluids 468, 271.]. We show that the main features of our lubrication-based model for rebound with an inclined wall can beexpressed with a simple force model proposed by Moraga et al., suitable for use in direct numerical simulations of multiphase flow.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Bubbles; Lubrication theory; Wall interaction

1. Introduction

The numerical simulation of multiphase flows in complex,realistic geometries involves a wide variety of scales and there-fore requires adequate modelling to describe some aspects ofthe fluid–particle, particle–particle, particle–wall interaction. Inindustrial processes such as cooling of nuclear reactors or phar-maceutical processes (Mudde, 2005), the phase ratio of bubblesto the liquid phase tends to be high, so that collisions of bubbleswith each other or the wall happen frequently. Bubble–bubble

∗ Corresponding author.E-mail address: [email protected] (B. Podvin).

1 Currently at General Electric Research and Development Center,Niskayuna, NY, USA.

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.12.023

and bubble–wall interactions have been found to be of crucialinterest for medical applications such as echography (Becherand Burns, 2000). Drag reduction using injected microbubbleshas also been the focus of intensive study (Xu et al., 2002),with promises to reduce the costs and increase the performanceof sea transport by as much as 30%. Despite ever-increasingcomputer power, direct numerical simulation of large numbersof bubbles is not yet possible. Moreover, even the dynamicsof a single bubble are not completely understood (Prosperetti,2004). Generally speaking, research studies involving bubblestend to focus only on a few selective aspects of the physics,while using crude modelling for those aspects considered lessimportant for the application at hand (Theofanous, 2004). Theinteraction of a bubble with a wall constitutes only a relativelysmall piece of the puzzle in many applications of multiphaseflows, and as such has often been neglected in the past. The

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B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914–1928 1915

simplest and most commonly used model consists in defininga restitution coefficient for the bubble velocity (Canot et al.,2003). We also observe that in most theoretical and numeri-cal developments (Shopov et al., 1990) the wall is chosen tobe horizontal or vertical, whereas in real-life applications, thewall may well be slanted. Yet wall inclination strongly con-ditions the behavior of the bubbles such as their velocity ortheir shape, as shown by the observations of Tsao and Koch(1997) or Perron et al. (2006). A ship hull for instance haswalls of various inclinations, each corresponding to a differentforce balance, which has consequences for the effectiveness ofa drag reduction strategy.

The goal of this paper is to evaluate the relevance of an ex-tension of Moraga et al.’s (2005) force model to describe therebound of a bubble on an inclined wall. An additional objec-tive of the paper is to verify and validate the 2D lubricationequation as a first step before coupling it with a 3D solver.Coupling the lubrication solver to a simple bubble trajectoryequation as Klaseboer et al. (2001) and Moraga et al. (2006)did, instead of to a 3D solver, has the advantages of produc-ing a much simpler algorithm that can still solve the bubblerebound problem if the bubble and the wall inclination angleare kept small enough. However the coupling to the trajec-tory equation forces the introduction of simplifying assump-tions for the boundary conditions of the 2D lubrication equationsolver. These assumptions include a simplification of the bub-ble shape and the extent of the 2D lubrication solver: it isassumed that (a) the deformation of the portion of the bub-ble interphase further away from the wall is negligible and (b)that the 1D flow assumptions needed to derive the lubricationequations hold in a volume of fluid of small extent separat-ing the bubble from the wall, i.e. with a characteristic radiusof r < rmax. These assumptions can be removed by couplingthe lubrication equation solver to a 3D solver. The feasibilityof this coupled multiscale solvers has been already proven byShopov et al. (1990), which show the development of a filmbetween the bubble and the wall. Canot et al. (2003) have cou-pled an analytical model based on lubrication theory with aboundary element method to simulate a 2D (cylindrical) bub-ble approaching a horizontal wall. They compare the bubbletrajectory to the experimental observations of Tsao and Koch(1997), which were made for horizontal as well as inclinedwalls. Klaseboer et al. (2001) have developed a model for therebound of a drop impacting a horizontal plane wall. The modelis based on a force balance for the drop and the use of lubrica-tion approximation to compute the force exerted by the wall.The model prediction was satisfactorily compared with exper-imental results. Moraga et al. (2005) solved the model for ahorizontal wall to derive a simple law which could be imple-mented in a two-fluid simulation of bubbly flows. To betterunderstand the physics of the interaction, we use experimentalvizualizations of air bubbles rising in water through buoyancyand bouncing against an inclined wall. We describe the experi-mental configuration in Section 2. The model equations are pro-vided in Section 3. Comparison of experimental observationswith the model is given in Section 4. Conclusions are given inSection 5.

2. Experimental configuration

In this study an apparatus is built to generate air bubbles in aliquid and observe their rebound on a wall with controllable in-clination. The apparatus is a rectangular water tank with 0.5 inthick Plexiglas walls. It is 30 cm long, 30 cm high and 3 cmwide. A rotating solid wall is placed 15 cm above the bubbleinjection point. Air bubbles were first generated using a syringeand a Hamilton needle with 50 �m diameter. The bubble diam-eter ranged from 0.4 to 2 mm. In order to control the bubble sizeand therefore ensure the reproducibility of the experiments, weused an solenoid valve through which pressurized air entered anozzle. The inner diameter of the nozzle was 127 �m, which al-lowed us to produce bubbles of diameter between 1 and 2 mm.Typical opening times of the valve ranged from 0.5 to 5 ms.

The main difference between the apparatus and that of Tsaoand Koch (1997) is that we use a plate instead of a channel asa solid impact boundary. This modification reduces the amountof interfaces between the illumination and the camera objectivefrom 8 to 4, which allows us to improve the quality of ourpictures. The setup allowed varying the angle between the walland the bubble vertical trajectory from 90◦ to 5◦. The liquidphase consisted of distilled water (Type 2). The system wasbacklit from one side with a halogen lamp. A schematic ofthe apparatus is shown in Fig. 1. The total height of the tankwas chosen so that the bubbles could reach their correspondingterminal velocity before they interact with the solid surface.

Let � and � be the respective density and viscosity of thefluid. Let � be the surface tension at the air–water interface�=0.07. The air bubble is characterized by its equivalent diame-ter d=2R and terminal velocity VT . Three adimensional param-eters govern the wall–bubble interaction. One is the Reynoldsnumber defined as

Re = �VT d

�

which relates the importance of inertial effects to that of viscouseffects. The other one is the Weber, which characterizes thebubble capacity for deformation:

We = �V 2T d

2�.

The third one is the wall inclination

0 < � < 90◦.

The Reynolds numbers in the experiment range from 40 to 560and the Weber number varies between 0.02 and 1.80. As thebubbles rose at their terminal velocities, their trajectory wasrecorded using a high-speed camera (Pixelink PLA-741), whichwas able to capture up to 1000 frames per second (FPS). Acompromise had to be found in order to have both a sufficientlyhigh-time resolution and a sufficiently large field. In practicethe frame rate varied around 300 fps, while the total extent ofthe field of view was 40 mm×30 mm. The centroid of the bub-ble was found by processing individual frames in ScionImage,and coupling it with a MATLAB routine, which determined

1916 B. Podvin et al. / Chemical Engineering Science 63 (2008) 1914–1928

Camera

Nozzle

Valve

Opens at 1.2 Volts for DC

Opens at 3.2 Volts for AC

Pressurized air inlet

Function

generator

Light Source

Fig. 1. Experimental apparatus.

the interface of the bubble. The centroid was taken to be themidpoint between the measured major and minor axes of thebubble. The bubble speed was found by measuring the bub-ble centroid position in successive frames and multiplying thetraveled distance by the frame rate. Since the bubble shape wasoblate, we also calculated the bubble aspect ratio, defined asthe ratio of the horizontal and vertical axes. The pixel incerti-tude yielded a maximum error of 6% in the calculation of thebubble diameter for the range of bubbles studied. The fluctu-ation in frame rate caused an error of 2% in frame rate mea-surement. Since the velocity was computed as the product ofdistance traveled in one frame and the frame rate, the total errorin the velocity measurements was 8%.

3. The numerical model

3.1. Equations

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 fluidtoward a wall inclined at an angle, as illustrated in Fig. 2.

Let � and � be, respectively, the fluid density and viscosity.We choose to describe the motion in a reference frame mov-

ing with the bubble tangential velocity relative to the wall. Weassume that the bubble tangential acceleration is small enoughto be neglected, so that the reference frame can be assumed tobe Galilean. Let (i, j, k) resp. (x, y, z) be the coordinate systemresp. coordinate variables associated with the reference frame.

h(x,z)

0

θ

x

yVT

Fig. 2. Physical configuration.

The origin O of the reference frame coincides with the normalprojection of the bubble centroid onto the wall.

Let us write the bubble centroid velocity U as U =Ui +V j .The equations of motion for the bubble centroid are obtainedby estimating the different forces acting on the bubble:

�bVdU

dt= Fbuoyancy + Fdrag + Fadded mass

+ Fhistory + Fwall. (1)


Specifically

• V is the bubble volume.• Fbuoyancy is the buoyancy force

Fbuoyancy = ((� − �b)43�R3g sin �)i

− ((� − �b)43�R3g cos �)j , (2)

where g is the gravity.• Fdrag is the drag force (excluding wall effects)

Fdrag = −�CDRev4�RUi − �CDRev4�RV j , (3)

where Rev is the Reynolds number based on the actual bubblevelocity

Rev = 2R�√

U2 + V 2

�

and CD is the drag coefficient based on the actual bubble ve-locity, which is typically determined in an empirical fashion.

• Fadded mass is the added mass force

Fadded mass = −CAM�V

(dU

dti + dV

dtj

), (4)

where CAM is the added mass coefficient• Fhistory is the history force.

Fhistory = −6√

��c�cR2∫ t

0

1√t − �

dV

d�d�. (5)

Note that unlike Moraga et al. (2005), we have chosen to in-clude the history force in the equations. We found that addingthe history force term resulted in a small shift in the ampli-tude and characteristic time scales of the rebound, leading toa slightly better agreement with experimental observations.We use the Basset formulation to represent the force. A fulldiscussion of the effect of the history force can be found inKlaseboer et al. (2001).The effect of the wall is represented by a force Fwall. Theidea is that the wall makes itself felt through an excess pres-sure exerted on the top of the bubble, which corresponds toa deformation of the interface. The flow between the bubbleand the wall constitutes a film which can be described us-ing lubrication theory. To be able to use lubrication theorywe have to assume that the pressure and velocity field areuniform across the film, an hypothesis that limits the rangeof validity of the model. The deformation corresponds to theheight h(x, z) of the film between the bubble and the wall,since the effects of deformation at the interface farther fromthe wall are neglected by the model.

Let PB be the pressure inside the bubble and PL be thepressure just outside the bubble interface. Initially, in the caseof a spherical bubble, one has

PL − PB = −�2

R. (6)

As the bubble deforms, the pressure jump becomes

PL − PB = −�

(1

Rx

+ 1

Rz

), (7)

where Rx and Rz are the principal curvature radii of the de-formed interface. The effect of the wall can therefore be com-puted as the spatial integral of the pressure difference betweenthe spherical and the deformed interface. The deformation isassumed to take place only on the top surface of the bubblewithin a region of size |x|, |z| < rmax with rmax>R. In prac-tice, we take rmax ∼ R and we check a posteriori that the ex-cess pressure is indeed negligible on most of this domain. Thisexcess pressure �p can be written as

�p = �

(2

R−(

1

Rx

+ 1

Rz

)). (8)

The curvature radii can be expressed as a simple function ofthe film height h(x, z) if its deformation is small enough:

1

Rx

+ 1

Rz

= �2h

�x2+ �2h

�z2. (9)

Finally, the pressure force can be expressed as

Fwall = −∫ ∫

(�p)nx dx dzi −∫ ∫

(�p)ny dx dzj , (10)

where nx and ny are the components of the bubble normal, andthe normal can be approximated by

n = �h

�xi + �h

�zk − j . (11)

Combining Eqs. (2)–(10) together, one can then rewriteEq. (1) as

4

3��R3(CAM + �B

�)

dU

dt

= (� − �b)4

3�R3g sin � − �CDRe4�RU

− 6√

��c�cR2∫ t

0

1√t − �

dU

d�d�

−∫ ∫

�

(2

R−(

1

Rx

+ 1

Rz

))�h

�xdx dz (12)

and

4

3��R3(CAM + �B

�)dV

dt

= −(� − �b)4

3�R3g cos � − �CDRe4�RV

− 6√

��c�cR2∫ t

0

1√t − �

dV

d�d�

+∫ ∫

�

(2

R−(

1

Rx

+ 1

Rz

))dx dz. (13)


In the reference frame, at t = 0

U = VT sin � (14)

and

V = −VT cos �. (15)

To solve for the bubble centroid velocity in Eqs. (12)–(13) withinitial conditions (14) and (15), we need an evolution equationfor the liquid film height as it is drained between the bubbleand the wall. This is readily obtained from lubrication theory,in which we assume that the flow in the film is in the planeof the wall and dominated by viscous effects. Given that evena very low concentration of impurities in the flow is sufficientto make the interface immobile, we assume a no-slip boundarycondition for the liquid at the bubble interface, rather than ano-shear condition (see for instance Klaseboer et al., 2001 for afull discussion). However, this assumption may no longer holdwhen the wall is slanted. Moreover, we do not have enoughcontrol on water quality so we have to see the immobile andmobile descriptions as limit cases, with the real case fallingsomewhere in between. Thus the interface will be consideredimmobile. Mass conservation for a slab of fluid within the filmyields the modified lubrication equation:

�h

�t+ 1

2

�(Uh)

�x= 1

12�

�

�x

(h3 �PL

�x

)+ 1

12�

�

�z

(h3 �PL

�z

),

(16)

where PL is the pressure in the liquid film. Assuming that thepressure variations within the bubble are negligible comparedto those in the film, one can replace the film pressure in this lastequation using Eq. (7). This leads to a fourth-order elliptic equa-tion which requires two boundary conditions on the boundaryat all times. These are provided by the assumption that outsidethe film domain the pressure difference is zero and the bubble isno longer deformed by the wall i.e., it moves with the centroidvelocity. This means that on the boundary (x2 + z2)1/2 = rmax,

�p = 0 (17)

and

dh

dt= −V . (18)

Note that if we assume that the interface is mobile, we obtaina modified equation for the film height

�h

�t+ 1

2

�(Uh)

�x= 1

3�

�

�xh3 �PL

�x+ 1

3�

�

�zh3 �PL

�z. (19)

Since the bubble is initially spherical, the initial conditionfor the film is

h(x, z) = h0 − (x2 + z2)

2R(20)

Δ p=0 Δ p=0

Fig. 3. Numerical domain.

and we also have

�p = 0. (21)

Eqs. (12)–(21) constitute a coupled system (S) in which thebubble centroid position (U, V ) and its top interface h(x, z) aredetermined. The numerical domain along with the boundaryconditions is shown in Fig. 3. Numerical solving of the system(S) is discussed in the next paragraph.

3.2. Numerical implementation

Here we solve a 2D version of the lubrication and bubbletrajectory equation due to the fact that the inclined wall breaksthe axial symmetry of the horizontal wall case. The equationsconstituting (S) are discretized using second-order finite differ-ences and linearized about the current solution. The system isno longer pentadiagonal as in 1-D, but has 13 non-zero diago-nals over a bandwidth of 2 N + 1, where N is the number ofgrid points in either direction x or z. NAG Fortran routines wereused to solve for the band-diagonal system within a toleranceof 10−8. The bubble trajectory was integrated in time using asecond-order finite difference scheme, with implicit treatmentof the drag and the added mass force, and explicit treatment ofthe wall force. The time step was constant and equal to 0.002in most of the simulations. We checked that smaller time stepsdid not change the results, while larger time steps occasion-ally led to the blow-up of the solution. The code was tested fordifferent spatial resolutions. An increase of 30% in the spatialresolution did not produce any noticeable changes. The effectof larger increases in the spatial resolution were not tested dueto the large increase in computational time associated with im-proved resolution. The lubrication equations are discretized incartesian coordinates, in which defining a circular or ellipticboundary was not straightforward. We chose instead to definea square boundary interface

|x| = rmax, |z| = rmax.

We expect the pressure and interface variations to be significantonly near the very top part of the bubble, so that the exactcontour of the interface boundary should be irrelevant. Thishypothesis was checked a posteriori. A variety of models wereused to represent the drag and the added mass coefficients, aswell as the correction due to the influence of the wall. Forspherical quasi-rigid bubbles, we used Schiller and Nauman’s


law (Clift et al., 1978):

CD = 24

Re(1 + 0.15Re0.687) (22)

which has been validated for Reynolds numbers up to 1000. Inorder to account for the effect of the wall on the drag coefficient,following Van der Geld (2002), this law was replaced by

CD = 24

ReC(y)(1 + 0.15Re0.687), (23)

where

C(y) =(

1 −(

R

2y

)3)−2

. (24)

For a spherical rigid bubble, the added mass coefficient isequal 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:

C(y) = 3

2

(1 −

(R

2y

)3)−1

− 1. (25)

For non-spherical bubbles of aspect ratio �, where� = rmax/rmin with rmin and rmax, respectively, characterizingthe smallest and largest dimensions of the bubble, we usedMoore’s (1990) functions:

CDRe = 55G(�)(1 + H(�)Re−0.5), (26)

where

G(�) = �4/3

3(�2 − 1)1/2

√�2 − 1 − (−�2 + 2) cos−1 �−1

(−√�2 − 1 + �2 cos−1 �−1)2

(27)

and

H(�) = 0.00195�4 − 0.213�3 + 1.703�2 − 2.146� − 1.573.

(28)

For a bubble of aspect ratio smaller than 2.5, the added masscoefficient is (see Klaseboer et al., 2001)

CAM = 0.62� − 0.12. (29)

The aspect ratio of the bubbles was kept constant through-out the simulation, identical to the initial aspect ratio observedin similar cases experimentally. However, this is different fromwhat usually happens in experiments where the aspect ratio ofthe bubble changes sharply after the rebounds (see next sec-tion). The reason to keep the aspect ratio constant is that thecorrelations above account for the effect of drag for a bubbleaway from walls, where the aspect ratio is due to the com-petition of surface tension and buoyancy. But when a wall ispresent, the wall also contributes to deform the bubble in waysnot accounted for in the expansions derived by Moore. In thepresent study we focus in air bubbles rising in water. Mea-suring the bubble radius should be sufficient to determine theterminal velocity from balancing the drag force with the buoy-ancy force. However, since the terminal velocity is sensitive to

the presence of impurities (Clift et al., 1978), we restricted ourstudy to experimentally determined terminal velocities.

All quantities appearing in Eqs. (13) and (14) can be adi-mensionalized with the following scales:

v = v

VT

,

t = tVT

R,

x = x

R, z = z

R.

4. Results

4.1. Interaction with a flat wall

The interaction of bubbles with a flat wall has been welldocumented in Tsao and Koch (1997) and in Klaseboer et al.(2001). A first validation of the 2-D model was obtained bycomparing the prediction with observations of air bubbles ris-ing through water toward a horizontal wall. The bubble radiuswas 0.85 mm. The Reynolds number ReT was 522 and the We-ber number was 0.866. These characteristics are close to oneof Tsao and Koch’s experiments investigated in Moraga et al.(2005), which constitutes an additional check on the validityof our experimental data. Following Tsao and Koch (1997), itwas assumed that the interface was mobile. As explained inKlaseboer et al. (2001), the motion of the bubble induces a dis-placement of the surfactants from the front of the bubble towardthe rear. The bubble front may therefore be considered mobile,while the accumulation of surfactants on its rear surface gen-erates a drag equivalent to that obtained on a rigid bubble. Ittherefore makes sense to assume mobility at the bubble surfacein the film close to a horizontal wall. However, this assump-tion no longer holds when the wall is slanted (the bubble front

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

time (s)

y (

mm

)

modelexperiment

Fig. 4. Trajectory of the centroid of a bubble of radius 0.85 mm with Re=522and We = 0.866 rising toward a horizontal wall—solid lines: 2D lubricationmodel (the 1D model gives identical results); circles: experiments.


0 10.1 20.2 30.3 40.4 50.5 60.6 70.7 80.8 90.9 101 222

0 mm

2 mm

4 mm

6 mm

8 mm

10 mm

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 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

r

Δ p

t=24 ms

t=25 ms

t=26 ms

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r/rmax

h (

r)

t=24 ms

t=25 ms

t=26 ms

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 towarda horizontal wall. Here rmax = R.

no longer faces the wall), in which case the interface will beconsidered immobile.

Fig. 4 compares the bubble trajectory observed in the exper-iment with its prediction from the 2D model. We found thatthe 2D code yielded nearly undistinguishable results from the1D version. Both models capture the correct amplitude reboundand characteristic time scale. This is all the more remarkableas during the rebound the aspect ratio of the bubble varies sub-stantially, as is evident in Fig. 5. Magnaudet et al. (2003) havedeveloped a theory that relates the trajectory of the bubble cen-troid with the bubble interface deformation. Changes in thebubble aspect ratio should affect the intensity of the drag forceand the added mass force. Moreover, a portion of the kineticenergy is converted into surface oscillations.

Fig. 6 shows the evolution of the excess pressure and of thefilm height as a function of the radial position, during the re-bounds. All substantial variations occur at locations r>rmax =R, which therefore justifies a posteriori the original assumptionmade on rmax when deriving the model. We observe that the use

of square boundary conditions in the 2-D code did not generatesome discrepancy with the 1D code, where radial boundary con-ditions were used. The pressure variations are concentrated in aregion close to the top of the bubble so that boundary conditionsat the exact boundary are not significant. This also validatesthe assumption that pressure variations occur over a character-istic scale which is small compared to the bubble radius.

Furthermore, the much coarser resolution (by a factor of 5)in the 2D case does not appear to perturb significantly the pres-sure profiles. Owing to the complexity of the problem in 2D,we have not yet attempted a resolution comparable to the 1Dcase.

4.2. Observations of bubbles interacting with an inclined wall

Several bubbles in the range diameter of 1.2–1.8 mm weresent toward the wall for various inclinations and the behaviorof the bubbles was captured by the cameras. The trajectory ofabout 20 bubbles was examined in detail. Three different types


0 10 20 30 40 50 60 70 80

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

wall inclination θ

d (

mm

)

bouncing

sliding

Fig. 7. Classification of the bubble trajectories as a function of the bubblediameter d = 2r and the wall inclination—the bubble at � = 0 represents alimit case of sliding as it sticks in fact on the wall.

of behaviors could be distinguished. The wall angle of incli-nation � appeared to be the determining factor for the bubbletrajectory.

• When the wall inclination was less than 10◦, the bubblebounced a few times (two or three) against the wall andstopped moving, as in Fig. 5.

• For moderate wall inclinations up to a critical angle �c,10 < � < �c = 55◦–60◦ the bubble slided against the wallwith a constant speed. In some of these cases, some transientbouncing—a few rebounds of decreasing amplitude—wereobserved 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 bubblediameter or its velocity, since the bubble velocity dependson the bubble diameter, which results in an implicit couplingbetween the Weber number and the Reynolds number.

• Finally, for large wall inclinations � > �c, the bubble ex-perienced steady bouncing. The amplitude of the reboundsappeared to be constant over time. We note that due to thelimited scope of the fixed camera, we were not able to recordmore than a couple of steady rebounds at a time. However, wechecked by moving the camera along the vertical axis, thatrebounds of the same amplitude were still observed severalrebound lengths away from the point where the bubble firsthit 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 samerange of bubbles. Fig. 7 shows that the critical angle seemsto be around �c ∼ 55 + / − 5◦ for bubbles in the diameterrange of 1–2 mm. This is in agreement with the observationsof Tsao and Koch (1997) made for slightly smaller bubbles inthe diameter range of 1.0 – 1.4 mm.

0 5 10 15 20 25

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

y/R

t VT/R

Fig. 8. Normal position of a bubble of radius 0.51 mm, Re = 103, We = 0.12as it interacts with a wall inclined at � = 45◦—the axis orientation here ischosen to match that of the experiment.

Fig. 9. Visualization of a bubble of radius 0.51 mm, Re = 103, We = 0.12rising toward a wall inclined at � = 45◦—the frame rate is 100 s−1.

4.3. Comparison of model–experiments

We now examine to which extent the model is able to re-produce the experimental observations. Accordingly we showresults for three typical cases corresponding to the three differ-ent types of bubble dynamics observed.


−1 −0.5 0 0.5 1

0

0.05

0.1

0.15

0.2

0.25

x/rmax

h (

x,0

)

t=5.20t=5.60t=6.0t=8.0

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x/rmax

Δ p (

x,0

)

t=5.20t=5.60t=6.0t=8.0

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 5 10 15 20 25

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

t

tangential fo

rces

wall forcedragadded massbuoyancy

0 5 10 15 20 25

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t

norm

al fo

rces

wall forcedragadded massbuoyancy

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 wallinclined at � = 45◦—all units are adimensional as defined in Section 3.2.

4.3.1. Sliding case without transient bouncingThe first case involves the case of a bubble of diameter

1.03 mm, with corresponding Reynolds 103 and Weber num-ber 0.12. Fig. 8 shows that the bubble trajectory predicted bythe model is a pure sliding motion, which indeed correspondsto the experimental observations. Images of the experimentalbubble are gathered in Fig. 9. Since the Weber number is verysmall, deformation effects are expected to be very small and thebubble interface shown in Fig. 10 remains relatively spherical,in agreement with experimental observations. It only flattensclose to the top of the bubble. Note that the deformation is notentirely symmetrical in x, as the minimum film height is located

at x ∼ 0.1 rmax = 0.1R. The lack of symmetry corresponds tothe jump in the pressure profile which is shown in Fig. 10b.

Fig. 11 shows the force balance in each direction as thebubble approaches the wall. Since the bubble density is smallcompared to that of the fluid, the rate of change of momentumfor the bubble (not plotted) is small compared to the forcesacting on the bubble so that the forces appear to cancel eachother.

In the tangential direction, the balance between the dragand gravity is only slightly disrupted as the bubble approachesthe wall, which gives rise to a small added mass component.The magnitude of the tangential wall force remains negligible


0 0.01 0.02 0.03 0.04 0.05 0.06

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

t (s)

y (

mm

)

experiment

model

Fig. 12. Normal position of a bubble of radius 0.85 mm, Re=522, We=0.866interacting with the wall inclined at � = 28◦.

throughout the evolution. In the normal direction, the wall forcebalances the added mass component as the bubble comes nearthe wall. Then it balances buoyancy as the bubble slides alongthe wall. As it can be seen from Eq. (24) we did not introduce acorrection to account for increased drag in the tangential direc-tion due to the proximity of the wall. Thus we are likely under-estimating the drag in the tangential direction when the bubbleis close to the wall. This is consistent with the results shownin Figs. 13 and 18 that show good agreement between modeland experiment before the first rebound, but not after it, as thedrag correction would grow abruptly as the bubble approachesthe wall. These corrections can be taken from Van der Geld(2002) for example and they would probably help improve theagreement with the experimental data.

4.3.2. Sliding case with transient bouncingSpecial attention was given to this type of motion. We

used the model to predict the trajectory of a bubble of radius0.81 mm, for an angle of 28◦. The Reynolds number of thebubble is 522 and the Weber number is 0.89. Experimental ob-servations show that the bubble experiences transient bouncingbefore sliding along the wall. As evidenced in Fig. 12, the evo-lution of the distance wall–bubble matches the experiments intime scale as well as in amplitude. However, Fig. 13 shows thetangential velocity does decrease slightly during the rebound,which the model fails to predict. This decrease in the velocitycan be related to the surface oscillations taking place at thewall as the velocity changes sign. The variation in aspect ratiois not taken into account in the model in its present form. Weassume that the bubble retains its equilibrium shape (far fromthe wall) throughout the entire wall interaction. Fig. 14 showsthe normal force balance. As the bubble approaches the wall,the dominant forces in the normal direction are the wall forceand the added mass force, which almost compensate eachother, since their magnitudes are both large compared to that ofbuoyancy. Then, as the bubble leaves the wall and prepares to

0 0.02 0.04 0.06 0.08 0.1

0

2

4

6

8

10

12

t (s)

x (

mm

)

experiment

model

Fig. 13. Tangential position of a bubble of radius 0.85 mm, Re = 522,We = 0.866 interacting with the wall inclined at � = 28◦.

0 5 10 15 20

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8wall force

drag

added mass forcebuoyancy

t VT/R

Fw R

/VT2

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 weuse adimensional time units: 10 adimensional time units correspond to about0.028 s.

bounce again, the wall force falls down to zero and the addedmass force is now balanced by the drag force. Typically afterat most three rebounds the bubble settles down to a slidingmotion in which the buoyancy force is now compensated bythe wall force in the normal direction and the drag force in thetangential direction.

Fig. 15 shows the evolution of the film height as the bubbleapproaches the wall for the first two rebounds. The bubbleinterface flattens as the bubble moves toward the wall thenbecomes spherical again as it leaves the wall. The interfacedeformation is weaker for the second rebound. At all timesthe interface remains generally symmetrical with respect to thebubble normal axis. Fig. 16 show the evolution of the pressure


−1 −0.5 0 0.5 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/rmax

h (

x,0

)

t=4.2t=4.8t=5.4

−1 −0.5 0 0.5 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/rmax

h (

x,0

)

t=5.4t=6.0t=6.6t=7.2

−1 −0.5 0 0.5 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/rmax

h (

x,0

)

t=14.4t=15.0t=15.6t=16.2

−1 −0.5 0 0.5 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/rmax

h (

x,0

)

t=16.2t=16.8t=17.4t=18.0

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 thepressure profile remains symmetrical before the rebounds, butis characterized by strong asymmetry in the tangential directionas the bubble moves away from the wall.

4.3.3. Steady bouncingAt large inclination angles, the bubble experiences constant,

bouncing motion. A typical case can be examined in Figs. 17and 18. We point out that in that case the lubrication modelfails to capture the steady bouncing behavior. Several simpli-fications in our model may play a role in the model inabilityto predict the steady bouncing. First, as our experiments andthose of Tsao and Koch put in evidence, the angle between thebubble largest axis and the wall changes considerably during arebound. That is not only the top portion of the bubble deformsbut also the bottom one. This deformation of the bottom por-

tion is neglected by our model. Second, the strong deformationof the film and its lack of symmetry may make the flow highlymultidimensional at r ≈ rmax ≈ R. Indeed as Tsao and Kochargue (see their Fig. 5) we expect an unsteady vortical structureto form during the rebound. Although this vortical structure ispresent even in the case of a horizontal wall, the model canperform well as long as most of the pressure contribution tothe wall force comes from the center of the bubble. However,as the inclination angle increases and the flow becomes asym-metrical the vortical structure is expected to make a biggerimpact. The multidimensionality of the flow introduced is notconsistent with the assumptions used to derive the lubricationequation, which requires 1D flow. In addition the boundary con-ditions assumed in Eq. (17) and to a lesser extent in Eq. (18)are no longer appropriate. For all this reasons we believe thatin order to predict the steady bouncing regime accurately, the


−1 −0.5 0 0.5 1

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x/rmax

Δ p (

x,0

)

t=4.2t=4.8t=5.4

−1 −0.5 0 0.5 1

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x/rmax

Δ p

t=5.4t=6.0t=6.6t=7.2

−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x/rmax

Δ p

t=14.4t=15.0t=15.6t=16.2

−1 −0.5 0 0.5 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x/rmax

Δ p

t=16.2t=16.8t=17.4t=18.0

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-counts for the bubble deformation and the flow dynamics in theregion where a lubrication equation model and its associatedboundary conditions are not appropriate. Shopov et al. (1990)have demonstrated that this kind of multiscale model is indeedfeasible.

4.4. Validation of the force model extension

Moraga et al. (2006) have proposed in the case of a horizontalwall that the wall force be modeled as a function of the bubbleacceleration:

Fwall = −H()Fbuoyancy

(1 + Fw

dV

dt

), (30)

where

• H() is a smoothed Heaviside function over a region of extent2 around a height of 0.

• Fw is an empirical constant that depends on the Webernumber.

Eq. (30) is based on the observation that when the bubble isclose to the wall the bubble normal velocity is small but itsacceleration is large. Thus the only force that can balance thewall force is the added mass. The analytical solution of Tsaoand Koch (1994) is consistent with this observation. When thebubble reaches equilibrium, the wall force exactly compensatesbuoyancy. These characteristic trends are adequately capturedby the force model. Moraga et al. have proposed an extensionin the case of a inclined wall by pointing out that the driving


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0

5

10

15

20

25

30

35

t (s)

x (

mm

)

experiment

model

Fig. 17. Experimental determination of the centroid tangential coordinate fora bubble of radius 0.80 mm, Re = 603,We = 1.60 interacting with a wallinclined at � = 72◦ degrees and comparison with the model prediction.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

−6

−5

−4

−3

−2

−1

0

t (s)

y (

mm

)

experiment

model

Fig. 18. Experimental determination of the centroid normal coordinate fora bubble of radius 0.80 mm, Re = 603, We = 1.60 interacting with a wallinclined at � = 72◦ and comparison with the model prediction.

force for the bubble motion—-buoyancy—-needs to be rescaledso that on a wall inclined at angle �, one has

Fwall = −H()F ′buoyancy

(1 + Fw

dV

dt

)(31)

with

F ′buoyancy = Fbuoyancy cos �.

To determine whether this expression for the force modelholds for inclined walls, we examine the dependence of themodel on the wall inclination. We point out that it is difficultto investigate the effect of the other parameters, as the bubbleaverage radius and bubble terminal velocity, and therefore the

other two adimensional parameters Re and We are coupled, sothat they cannot be modified independently (Brennen, 1995).Our first test case is a bubble of radius R = 0.83 mm, risingtoward the wall with a velocity of 30 cm/s. The correspondingReynolds and Weber numbers Re and We are, respectively, 449and 1.89. We solved the model for different inclinations: 0◦,15◦, 30◦, 45◦. Results in Fig. 19 show that the normal wall forceacts on a time scale that is almost independent of the wall angle.The intensity of the wall force was found to decrease as the wallinclination increases. Moraga et al. suggested that the depen-dance was proportional to cos �, in other words that the force isproportional to the normal velocity of the bubble. However thewall force is expected to scale with the acceleration of the bub-ble rather than the velocity. So an appropriate scaling would beV/�, where V is the normal velocity component and � a char-acteristic time scale of the rebound. We have � ∼ h/V where his the film height. As shown by Tsao and Koch (1997), the am-plitude of the pressure necessary to create surface deformationsmust be on the order of p = O(�/R) so that the characteristicfilm height h scales as h = O((We/Re)2/3). The dependenceof h with the wall inclination is therefore h ∝ (cos �)2/3. Thecharacteristic time scale � varies like (cos �)−1/3 and the wallforce should therefore scale with (cos �)4/3. A linear regres-sion performed on the wall forces computed for different wallinclinations yielded a scaling in (cos �) with = 1.366.

In the tangential direction, we found that both wall force andadded mass maxima were at most 6% of the buoyancy force,so that as a first approximation, the influence of the wall forcecan be neglected. These findings are in agreement with Moragaet al.’s conjecture.

We therefore propose the following form for the force model:

Fwall = −H()F ′buoyancy

(1 + F ′

w

dV

dt

)(32)

with

F ′buoyancy = Fbuoyancy cos �,

F ′w = Fw(cos �)1/3.

An approach similar to that used by Moraga et al. (2005)can be used to determine the dependence of F ′

w on the Webernumber. To test this hypothesis, we build a modified version ofthe 1D model corresponding to the assumptions made aboveand compare it with the equivalent 2D model. The buoyancyforce is accordingly replaced with its wall-normal component,and the added mass force was rescaled with a factor (cos �)1/3

corresponding to the modification of the characteristic reboundtime scale. We simulate a bubble of R=0.85 mm rising againsta wall inclined at an angle of 28◦. Fig. 20 shows that the ve-locity and position of the bubble is adequately predicted. Thewall force is correctly estimated to within less than 5%. Wefound that when the added mass coefficient is not modified, thesecond rebound predicted by the 1D model occurs slightly ear-lier, which is no longer the case when a modified time scale isintroduced for the added mass force. The 1D modified modelis clearly a good approximation for the 2D model, so that


0 5 10 15 20

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

t VT/R

Fd

rag/c

os(θ

)

θ=0

θ=15

θ=30

θ=45

0 5 10 15 20

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t VT/R

Fw

all/

cos(θ

)4/3

θ=0

θ=15

θ=30

θ=45

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 5 10 15 20

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t VT/R

Fw

all

2−D model

1−D modified model

2−D model

1−D modified model

0 5 10 15 20

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

t VT/R

y/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 positionof the bubble centroid.

expression (32) represents a quick, adequate estimate for thewall force.

5. Conclusion

The goal of this paper was to investigate the validity ofa model to describe the bubble–wall interaction in the caseof an inclined wall. We used a model based on lubricationtheory, which leads to the derivation of a wall force modelthat could be plugged into complex multiphase simulations.High-speed vizualizations of the bubble trajectories made itpossible to identify three types of motion depending on the

wall inclination: (a) sticking motion, for very low wall incli-nations (smaller than 10◦); (b) sliding motion, for moderateinclinations 10 < � < 55◦; (c) repeated bouncing, for larger in-clinations. Transient bouncing could be observed in cases (a)and (b).

We have shown that when the wall inclination is less than55◦–60◦, the model provides the correct time scale and reboundamplitude for the bubble. However, it does not seem able to re-produce the slight variations in the tangential velocity that areassociated with the rebound. This is likely to be associated withthe large interface deformations associated with our relativelylarge bubbles (We ∼ O(1)). When the wall inclination is larger


than 55◦–60◦, the model is not able to capture the steady bounc-ing motion observed in experiments. This is likely to be due tothe absence of lift effects in the current lubrication model.

In order to make this model useful for large-scale multiphaseflow simulations, the extension of the force model proposedby Moraga et al. is tested against experimental data. We findthat a modified 1D model is quite appropriate to describe thewall–bubble interaction for moderate wall inclinations, even formoderate Weber numbers We ∼ O(1).

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Canot, E., Davoust, L., El Hammoumi, M., Lachkar, D., 2003. Numericalsimulation of the buoyancy-driven bouncing of a 2-d bubble at ahorizontal wall. Theoretical and Computational Fluid Dynamics 17,51–72.

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Klaseboer, E., Chevailier, J.P., Mate, A., Masbernat, O., Gourdon, C., 2001.Model and experiments of a drop impinging on an immersed wall. Physicsof Fluids 13 (1), 45–57.

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Theofanous, T.G., 2004. Panelist comments on: open questions and newdirections in gas–liquid flows. Journal of Fluids Engineering, 2004.

Tsao, H.K., Koch, D.L., 1994. Collision of slightly deformable, high Reynoldsnumber bubbles with a short-range repulsive force. Physics of Fluids 6,2591.

Tsao, H.K., Koch, D.L., 1997. Observations of high Reynolds number bubblesinteracting with a rigid wall. Physics of Fluids 468, 271.

Van der Geld, C.W.M., 2002. On the motion of a spherical bubble deformingnear a plane wall. Journal of Engineering Mathematics 42, 91–118.

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modelandexperimentalvisualizationsoftheinteractionofabubbl ...physics of fluids 13(1), 45–57.] and...

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