Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.2.4 - Interface Resolving Simulation of Bubble-Wall Collision Dynamics
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 Material Information
Title: 10.2.4 - Interface Resolving Simulation of Bubble-Wall Collision Dynamics Particle Bubble and Drop Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Omori, T.
Kayama, H.
Tukovi´c, Z.
Kajishima, T.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: bubble-wall collision
liquid film
interface-tracking method
immersed-boundary method
 Notes
Abstract: The bubble-wall collision dynamics were studied by means of numerical simulation, where the maximum Reynolds number and the maximum Weber number was 279 and 0.503 respectively. The present simulations are restricted to two-dimensional circular bubbles, but the elucidated principle collision mechanism should also apply to threedimensional cases. It was revealed that the bubble bounce from the wall is maintained by the dimple formation on the gas-liquid interface, and the main dissipation of energy during the collision process is caused by the enhanced liquid motion near the gas-liquid interface and the dissipation accompanying the drainage of the liquid film or the vortex formation during the receding motion plays a minor role.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00248
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 1024-Omori-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Interface Resolving Simulation of Bubble-Wall Collision Dynamics


T. Omori* H. Kayama* Z. TukoviCd and T. Kajishima*

Department of Mechanical Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, Japan
t Faculty of Mechanical Engineering and Naval Architecture, Univ. of Zagreb, Ivana LuCida 5, Zagreb, Croatia
t.omori@mech.eng.osaka-u.ac.jp
Keywords: bubble-wall collision, liquid film, interface-tracking method, immersed-boundary method




Abstract

The bubble-wall collision dynamics were studied by means of numerical simulation, where the maximum Reynolds
number and the maximum Weber number was 279 and 0.503 respectively. The present simulations are restricted
to two-dimensional circular bubbles, but the elucidated principle collision mechanism should also apply to three-
dimensional cases. It was revealed that the bubble bounce from the wall is maintained by the dimple formation on the
gas-liquid interface, and the main dissipation of energy during the collision process is caused by the enhanced liquid
motion near the gas-liquid interface and the dissipation accompanying the drainage of the liquid film or the vortex
formation during the receding motion plays a minor role.


Nomenclature


Roman symbols
E energy in unit depth (N)
|g| gravitational acceleration (m s 2)
(= 9.81, unless otherwise specified)
1,1 interface mesh spacing in the radial direction (m)
1, in the circumferential direction (in)
n unit normal (in)
p, modified pressure (N in2)
Q flow rate in unit depth (m2 /s)
r point position (m)
R bubble radius (m)
Up solid wall velocity in the non-inertial frame (rn/s)
w weighting factor (-)
Greek symbols
At time step width (s)
v kinematic viscosity of gas/liquid (mi2 s 1)
p density of gas/liquid (kg in3)
c interfacial tension (N/m)
Subscripts
c interface mesh control points
i interface mesh vertex points
nb neighbor cells
P current cell
I fluid in the lower region
u fluid in the upper region
L, G liquid/gas


Superscipts
m iteration counter in a PISO loop
n current time step counter


Introduction


To provide necessary boundary conditions for the pre-
diction of dispersed multiphase flows, it is of great im-
portance that the bubble-wall collision dynamics is prop-
erly modeled. The importance may be illustrated by
some experimental evidence that in gas-liquid upward
circular duct flows the bubble volume fraction showed a
sharp peak at a distance from the duct wall. In the ex-
periments by Valukina et al. (1979), where the bubble
diameters were between 0.5 and 1.0 mm and the mean
gas volume fraction was in the range from 0.02 to 0.1,
the peak of the bubble volume fraction was localized
around 1.7 bubble radius from the wall, which is to be
the consequence of bouncing rather than sliding motions
of bubbles, as pointed by Tsao & Koch (1997).
The bubble-wall collision has also been studied to ob-
tain fundamental physical description of bubble-particle
or bubble-bubble interaction problems, which are typi-
cally encountered in floatation and waste water purifica-
tion processes. Fujasova et al. (2007), Legendre et al.
(2005) and Tsao & Koch (1997), among others, con-
ducted experimental studies on the hydrodynamics of
collision of a bubble with a horizontal wall in a liquid.











Probably due to the difference in the fluid properties,
however, there is a scatter in their results and it is hard to
deduce a unified conclusion. The limitation in the exper-
imental techniques also precludes the understanding of
the bubble motion dynamics in relation to the hydrody-
namics of the surrounding fluid, which is especially im-
portant in the attempt to make two-way coupling mod-
els.
The objective of the present work is to investigate
the bubble motion during the collision with a horizon-
tal rigid wall in a well-defined condition by means of
numerical simulation. A numerical approach has a great
advantage to relate the path, the deformation and the rise
velocity of the bubble with the energy balance and the
flow field which the bubble itself is a part of. The present
simulations are restricted to two-dimensional circular
bubbles, but the elucidated principle collision mecha-
nism should also apply to three-dimensional cases.

Numerical Methods

The Navier-Stokes equation and the mass conservation
equation for incompressible fluid were discretized in the
framework of Finite Volume Method. The discretization
was carried out by the central difference scheme and the
three time level scheme for space and time respectively,
which have both second-order accuracy. To capture the
motion and deformation of a bubble, the gas-liquid in-
terface was expressed by the boundary of two solution
adaptive mesh domains that were allotted either for the
gas or the liquid phase, based on the interface-tracking
method proposed by Muzaferija & PeriC (1997). To
avoid mesh distortion during the simulation, the bubble
centered non-inertial frame was adopted and the rigid
wall was treated as a quasi boundary condition by an
immersed-boundary approach.
Gas-Liquid Interface
At the gas-liquid interface the dynamic and the kine-
matic boundary conditions are made strongly coupled by
a PISO-type iterative partitioned method. The dynamic
condition requires the stress (pressure, viscous stress and
surface tension) balance normal to the interface and the
kinematic condition has to be satisfied by moving the in-
terface to assure that there is no volume flux across the
interface.
The implementation of the kinematic boundary con-
dition is illustrated in Fig. 1. The possible volume flux
VI which arises in the course of a PISO loop to correct
the velocity and to determine the pressure in the liquid
phase must be compensated by the volume swept by the
interface mesh. This swept volume is written as:

2V1 At + .1i
V 1 3 (1)
3


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Area
Previous Interface


Length
h' .
New Interface
Mesh vertex points
Mesh control points

Figure 1: Interface mesh motion for the fulfillment of
the kinematic boundary condition


for the three time level scheme by virtue of SCL require-
ment (see e.g. Ferziger & PeriC (2002)). The positions
of the mesh control points, which are initially at the cell
face centers, are updated as:


n+l n+ h/ n
i'6 Cr6n Cn,


where


and the unit vector normal to the cell face is used as n,.
The interface mesh positions at the new time step are
then given by the following expression:


n+ Ni (p r)
S r+ Ni n1 n'


where p is a point on the plane which is defined by the
enclosing control points


Ec w2r +1
c W2


and Ni is the unit normal vector to the plane obtained
by the least-squares method


Nwr ) 1 (6)


The unit normal n, at a vortex point is defined by the
surrounding cell face normal vectors by the method pro-
posed by Max (1999). The mesh points in the whole do-
main are updated only after the convergence of the PISO
loop for efficiency, by solving the mesh vertex motion
equation with the interface vertex motion as a boundary
condition.
Solid-Liquid Interface
The no-slip condition at the solid-liquid interface is im-
posed on a grid scale by an immersed boundary ap-
proach. The Navier-Stokes equaiton for incompressible







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


fluid is replaced by the following expression:


Preliminary Simulations


(1 -7) -u.+VVV2u

+ Fp,


-1 -
-VPm ab
P


where ab represents the acceleration of the bubble cen-
ter of gravity forming the non-inertial simulation frame
seen from the bubble and 7 is the volume fraction of
solid phase in each mesh cell. The velocity field where
7 1 has velocity of the solid phase by the effect of the
additional term Fp written as:

3U"+1 4u" + u 1
2At (8)

for the three time level scheme adopted for the time dis-
cretization.
The coupling of the Navier-Stokes equation and the
mass conservation equation is accomplished by the pro-
cedure based on the PISO method. The Navier-Stokes
equation (Eq. 7) can be written in the semi-discretized
form as the following:


up = 1 H(u)
Ap I


1 -)V
(1 -7)1Pm ,
p )


where


H(u) = AAnbUnb + Q


and Q represents all the source terms other than the pres-
sure term.
First of all, this momentum equation is solved for the
guessed velocity using the pressure from the previous
iteration m in the PISO loop


p =1 H(u) (1
AP I


1 V/",
P)V::~


The corrected pressure is obtained by solving the follow-
ing Poisson equation,


V. tp,


where
Up= (1 --H( +) U;+1 (13)

and consequently the velocity is updated as the follow-
ing:
1
um] up 1- V/.::1 (14)
App
The above procedure, which involves the interface track-
ing method explained in the previous subsection, is re-
peated until the velocity divergence vanishes.


Before discussing the bubble-wall collision dynamics,
the present interface-tracking method was examined for
) two fundamental cases. The examination on the two-
dimensional sloshing problem showed that the gravity
driven gas-liquid interface oscillation can be correctly
predicted. The simulations were also performed for a
freely rising air bubble in an initially quiescent water,
which is directly related to the present main simulation
cases and based on whose result the sufficient mesh res-
olution for further simulations was determined. The res-
olution issue for the solid-liquid interface will be ad-
dressed at the end of the present work.
Two-Dimensional Sloshing
The simulations were performed for the initial-value
problem associated with the small amplitude waves on
the interface between two stably superposed viscous flu-
ids and the results were compared to the analytic so-
lutions by Prosperetti (1981). It was restricted to the
cases where the two fluids have equal kinematic viscosi-
ties and the analytic solutions give explicit expression
for the interface position. The parameters for the com-
putations are summarized in Tab. 1. The air-water in-
terface at 28 C is considered in Case I, except that the
viscosity of the water has about ten times the value of
the real value, because of the condition that the both flu-
ids must have the same kinematic viscosity and Case II
deals with the case where the natural frequency is lower
and the motion is more dissipative than Case I.
As the initial interface shape, the following expression
was assigned


y(x) = 1 + ao sin{7r(0.5 x)}


(15)


and the amplitude ao was set to 0.01m. Fig. 2 shows
the initial mesh configuration. The domain size is 1 m
and 2 m horizontally and vertically, respectively. Tan-
gential to the interface it has 40 cells of the same size,
and normal to the interface the mesh was divided into 20
cells in each region and clustered with the expansion ra-
tio 1.088 and the finest mesh size of 0.0199 m. At the up-
per boundary, constant total pressure was assumed and
for the other boundaries slip-wall condition was applied.
The time history of the interface location at the left
end, namely y(0) was compared to the analytic solution
and is shown in Fig. 3. In both cases the discrepancy
between the present results and the analytic solutions are
almost indiscernible, where the discrepancy is 0.3% of
the analytical value at t = 20 s for Case I.
Freely Rising Air Bubble in Water
A freely rising clean air bubble of diameter 1.0mm in
water was simulated on three different spacial resolu-
tions. The material properties were those of air/water


\7 1 -Y V/", t
( APP








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Figure 2: Mesh and coordinate arrangement for 2D
sloshing problem



Table 1: Conditions for the computation of 2D sloshing
Variable Case I Case II

Pu 1.173 10.00
pi 996.31 1000.0
v 1.58 10-5 1.00 10-3
|g1 9.81 1.00
a 0.072 0.10



at 28 C (Tab. 2) and the specification of the numerical
mesh is summarized in Tab. 3. The temporal resolution
was set to 0.01 ms for all the cases. The time variation
of the bubble rise velocity shown in Fig. 4 illustrates
negligible influence of the mesh resolution next to the
interface. Even reducing the overall resolution to the
half resulted in a small difference of the terminal rise
velocity (0.86 %). It can be concluded from these re-
sults that Mesh I provides sufficient spacial resolution
for further simulations. The obtained terminal veloc-
ity 0.25 m/s has the same order of magnitude to the ex-
perimental value 0.19 m/s at 20 C by Gaudin (1957) or
0.26 m/s by Moore (1965).




Table 2: Adopted material properties for air/water
Variable Air Water

p 1.173 996.3
v 1.58 10-5 8.36 10-7
S- 0.072


-------------
-------------
-------------
-----------
-------------
. ...........
--- 111111111
=4444
----- +H44
------ H-H


1015 -


1 01


1 005


1


0995


0 99
0

1 015


1 01


0 5 10 t s]


Figure 3: Interface oscillation at x
Lower: Case II)


15


0 (Upper: Case I,


Bubble-Wall Collision

The collision between a horizontal wall and a clean air
bubble of diameter 1.0mm and primarily 1.5 mm was
studied. The adopted material properties of fluids are
the same as given in Tab. 2. The wall was assumed to be
clean and chemically neutral to the air/water interface.
The initial distance between the wall and a bubble center
was set to 4.0mm for every case. The simulations were
performed on Mesh I from the previous section, which
is shown in Fig. 5.
Fig. 6 shows the variation of the vertical coordinate
and the vertical translation velocity of the bubble center.



Table 3: Grid resolutions for the rising air bubble
Variable Mesh I Mesh II Mesh III

1"1/R 0.5 102 1.0 102 1.0 102
It/(27r) 160 160 80
Total cell # 14080 14080 3520


5 10 [] 15
t [S]







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


025


02

015

0 01


005
Mesh I
Mesh II
Mesh III
0
0 005 01 t[s] 015 02


Figure 4: Rise velocity of a 2D bubble


Table 4: Dimensionless groups based on the maximum
approaching velocity to the wall
Variable D 1.0 nmm D 1.5 mm

Rem,, 191 279
Wem,, 0.354 0.503


The maximum Re and We based on the maximum rise
velocity of the bubble and the material properties of the
water are listed in Tab. 4. It can be seen from Fig. 6
that both bubbles display a rebound motion after the first
and the second impact, which supports the experimental
observation by Tsao & Koch (1997) that bubbles with
We > 0.3 underwent a measurable rebound. Because
both bubbles showed the same nature in principle, the
following discussion will be focused on the bubble of
diameter 1.5 mm.
Figs. 14 and 15 illustrate the transition of the bubble
shape and the velocity field in the gas/liquid phases dur-
ing and right after the first impact to the wall. The veloc-
ity vectors were measured from the inertial frame. The
selected time instances in Fig. 14 are some specific in-
stances for the bubble-wall collision dynamics and ex-
plained in Tab. 5. At the contact time t 31.0ms,
where the distance between the bubble center and the
wall is D/2, there is still ,igmilk.iiii portion of fluid be-
tween the wall and the bubble due to the deformation
of the interface. If the bubble gets closer to the wall,
the dimple of the interface is formed, as was also ob-
served experimentally by Platikanov (1964). The liquid
film did not break even when the distance between the
bubble and the wall is minimum. This issue will be dis-
cussed later in detail. At t 37.2 ms, where the surface
energy turns back into the maximum rebound velocity,
the distance between the bubble center and the wall is
approximately D/2 but the bubble shape and the liquid


Figure 5: Grid for the bubble-wall collision simulations
(The cyan line indicates the gas-liquid interface.)


Table 5: Description of the snapshots shown in Fig. 14
Time [ms] Explanation
1.00
25.6 Maximum upward translation velocity
31.0 Contact time
34.2 Minimum distance to the wall
37.2 Maximum downward translation velocity


film thickness is totally different from those at the con-
tact time on approach.
The oscillations in the bubble translation velocity dur-
ing the rebound (see Fig. 6), which is also visible in the
experiments by Fujasova et al. (2007), can be explained
by the oscillatory shape deformation of the bubble as
clearly seen in Fig. 15. The time instance t 43.0 ms
corresponds to the local maximum of the upward trans-
lation velocity of the bubble.
The energy variation associated with the bubble mo-
tion during the collision process is depicted in Fig. 7.
The energies considered here are summarized in Tab. 6.
As the surface energy, variation from the initial state is
considered. It should be noted that the kinetic energy of
the surrounding water can be written as:


i 1
f pu udV = CMpLVbU,,
2 JL 2


(16)


using the added mass coefficient CM1, the volume Vb
1Fig. 8 shows the time variation of the added mass coefficient of the
bubble. At the contact time CM is increased to 2.56 from 1 which








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


0 002 004 0 06 t [s] 0 08


005


0 002 004 0 06 t [s] 0 08


01 012


01 012


Figure 6: Bubble center coordinate and velocity in the
collision process



and the translation velocity Ub of the bubble. This en-
ergy is almost identical to Ed because of the high den-
sity ratio in the present study. In Fig. 7, intimate energy
exchange between the kinetic energy and the surface en-
ergy is evident during the bounce and the oscillatory mo-
tion (Fig. 15).
It is interesting to note that the total energy shows an
eminent decrease during the bounce period. For the first
bounce, the decrease from the one at the contact time to
the other at the time when the bubble and the wall take



Table 6: Definition of energies considered in Fig. 7

Variable Description Definition

Ed Kinetic E. 2 ]L,G plu udV
E, Surface E. o(S So)
Ep Gravit. Potential Yb g JG(pL cG)dV
ET Total Ed + Es + Ep


is the theoretical value for the cylinder.


4e-05
3e-05
2e-05
1e-05
S0
LU
-1e-05
-2e-05
-3e-05
-4e-05
-5e-05
-6e-05


Kinetic
Elastic (Surface) Potential
Gravitational Potential
Total


V/ -A> _


0 002 004 006 t [s] 008

4

3.5

3

2.5 /

2 7


01 012


0 0.02 0.04 0.06 t [s]0.08 0.1


S0.2

0.15

0.1

0.05
og


-0.05

-0.1

--0.15
0.12


Figure 7: Energy variation during the collision process
(The bottom figure replots Fig. 6 for convenience.)



the maximum distance, divided by the initially stored
total energy (Eo E,) is 0.53, which is close to the
value 0.59 claimed by Tsao & Koch (1997) in their ex-
periments. The dissipation due to the film drainage as
shown in Fig. 9 was found to be way small to explain
the observed decrease in the total energy and it should
be concluded that the loss comes from the dissipation in
the global liquid motion. It also deserves attention that
the vortex formation (Fig. 10) during the receding phase
with the shape oscillation (see Figs. 15 and 7 bottom)
does not contribute much to the energy loss.

Flow in the liquid film

To discuss further the rebound phenomena in the bubble-
wall collision, the present method to express solid
boundary should be assessed. When the bubble is at the
minimum distance to the wall (t 34.2 ms), the mesh
takes the form shown in Fig. 11. The simulations were
performed on the channel which represents the liquid
film at t = 34.2 ms with the present immersed-boundary














35

35

25

2

15


0005 001 0015 002 0025 003 0035 004
t [s]

Figure 8: Variation of the added mass coefficient of the
bubble during the first bounce period


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010










'i ,

! i, ,
I,.','." ''I,'."",.'.


Figure 10: Vortex formation in the vicinity of the inter-
face during the receding motion (Upper: t = 43.0 ms,
Lower: 45.5 ms)


Figure 9: Drainage of the water in the dimple at t
34.2ms


method and the body-fitted grid (Fig. 12). The bound-
aries were assumed fixed and the gas-liquid interface
was replaced by the free-slip wall. The flow was driven
by the fixed pressure difference 200Pa, which is the
value measured in the bubble-wall collision simulation,
from the left to the right boundary and on these bound-
aries the zero-gradient boundary condition was applied
to the velocities. The number of cells for the grids and
the calculated flow rate in the steady state are summa-
rized in Tab. 7. The IB I corresponds to the spacial res-
olution in the present bubble-wall collision simulation
(see Fig. 11). It can be seen that the present method un-
derestimates the flow rate on a coarse mesh. It must be
emphasized, however, that even if the velocity in the liq-
uid film is tripled, the dissipation in the liquid film is far
too small to explain the steep decrease in the total en-
ergy and also that the drainage speed is far too low to set
off the film break and it does not disprove the conclusion
that the rebound should occur. Fig. 13 demonstrates that
the present method provides a correct solution with an
enhanced resolution.


Conclusions

The numerical simulations on the two-dimensional
bubble-wall collision problem revealed that the dimple


Figure 11: Grid near the interface at t 34.2 ms (Red:
solid-liquid interface, Blue: gas-liquid interface)




formation maintains the bubble bounce by preventing
the attachment of the gas-liquid interface to the wall
and the concentrated loss of energy during the contact
period, which leads to the perish of the bubble mo-
tion, is due to the enhanced liquid motion near the gas-
liquid interface. The present numerical methods pro-
vided good results for the benchmark problems and it
was also shown that the possible numerical errors do
not affect the above conclusions. Although the straight-
forward comparison to the existing experimental results
is not possible, judged from facts and figures in the
present work, it can be considered that the present simu-
lations on the bubble-wall dynamics give reasonable re-
sults even quantitatively.





















Figure 12: Grid arrangement for the liquid film channel
(Upper: body-fitted, Lower: immersed-boundary)


Table 7: Summary for the liquid film channel simulation
(BF: body-fitted, IB: immersed-boundary)
BF IB I IB II IB III

Mesh 700 x 40 25 x 7 25 x 14 200 x 56
106 Q 2.84 1.01 2.09 2.65



References

Valukina N.V., Koz'menko B.K. and Kashinskii O.K.,
Characteristics of a flow of monodisperse gas-liquid
mixture in a vertical tube, J. Engineering Phys. and Ther-
mophys., 36-4, pp. 462-465, 1979

Tsao H.-K. and Koch D.L., Observations of high
Reynolds number bubbles interacting with a rigid wall,
Phys. Fluids, 9-1, pp. 44-56, 1997

Fujasova M., Vejrazka J., Ruzicka M.C. and Drahos
J., Experimental study of bubble-wall collision, 6th
Int. Conf Multiphase Flow, S1 WedC_38, 2007

Legendre D., Daniel C. and Guiraud P., Experimental
study of a drop bouncing on a wall in a liquid, Phys. Flu-
ids, 17, 097105, 2005


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Muzaferija S. and PeriC M., Computation of Free-
Surface Flows Using Finite Volume Method and Moving
Grids, Numerical Heat Transfer B, 32-4, pp. 369-384,
1997

Ferziger J.H. and Perid M., Computational methods for
fluid dynamics, Springer, 2002

Max N., Weights for computing vertex normals from
facet normals, J. Graphics Tools, 4-2, pp.1-6, 1999

Prosperetti A., Motion of two superposed viscous fluids,
Phys. Fluids, 24-7, pp. 1217-1223, 1981

Gaudin A.M., Flotation, McGraw-Hill, 1957

Moore D.W., The velocity rise of distorted gas bubbles
in a liquid of small viscosity, J. Fluid Mechanics, 23-4,
pp. 749-766

Platikanov D., Experimental investigation on the 'Dim-
pling' of thin liquid film, J. Physical Chemistry, 68, pp.
3619-3624, 1964


004

003
E
E
S002

001


0 001 002 003 004 005 006 007 008
U [m/s]

Figure 13: Velocity profile in the liquid film channel







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Figure 15: Interface shape and velocity field in- and out-
side the bubble after the first rebound (From the above,
t = 40.0, 41.0, 42.0 and 43.0ms. The color indicates
the magnitude of the velocity: Red, large; Blue, small.
The apparent discontinuity inside the bubble is due to
the way of drawing and the results do not show any dis-
continuities. The red box indicates the solid wall.)


Figure 14: Interface shape and velocity field in- and out-
side the bubble during the first rebound (From the above,
t = 1.00, 25.6, 31.0, 34.2 and 37.2 ms. For further de-
scription on the figures, see the caption of Fig. 15)




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