Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 12.7.1 - A polydisperse two-fluid model for bubble plume under breaking waves
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Title: 12.7.1 - A polydisperse two-fluid model for bubble plume under breaking waves Environmental and Geophysical Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Ma, G.
Shi, F.
Kirby, J.T.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: polydisperse two-fluid model
bubble plume
breaking waves
 Notes
Abstract: Wave breaking in the surf zone entrains large volumes of air bubbles into the water column, forming a complete two-phase bubbly flow field. Numerical study of this bubbly flow is largely restricted by the lack of robust and comprehensive bubble entrainment models. In this paper, we propose a new model that connects bubble entrainment with turbulent dissipation rate at the air-water interface. The initial bubble size distribution follows the bubble size spectrum observed in laboratory experiments. The locations where bubbles are entrained are bounded by a threshold of turbulence dissipation rate. The model has two free parameters: bubble entrainment coefficient cb and critical turbulence dissipation rate ǫc. The bubble entrainment model as well as a polydisperse two-fluid model are incorporated into a 3D VOF code Truchas. The model is employed to study bubbly flow under a laboratory surfzone breaking wave.
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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Resource Identifier: 1271-Ma-ICMF2010.pdf

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


A polydisperse two-fluid model for bubble plume under breaking waves


Gangfeng Ma*, Fengyan Shi*and James T. Kirby*

Center for Applied Coastal Research, University of Delaware, Newark, DE 19716, USA
gma@udel.edu, fyshi@udel.edu and kirby@udel.edu
Keywords: polydisperse two-fluid model, bubble plume, breaking waves




Abstract

Wave breaking in the surf zone entrains large volumes of air bubbles into the water column, forming a complete
two-phase bubbly flow field. Numerical study of this bubbly flow is largely restricted by the lack of robust and
comprehensive bubble entrainment models. In this paper, we propose a new model that connects bubble entrainment
with turbulent dissipation rate at the air-water interface. The initial bubble size distribution follows the bubble
size spectrum observed in laboratory experiments. The locations where bubbles are entrained are bounded by a
threshold of turbulence dissipation rate. The model has two free parameters: bubble entrainment coefficient cb and
critical turbulence dissipation rate c. The bubble entrainment model as well as a polydisperse two-fluid model are
incorporated into a 3D VOF code Truchas. The model is employed to study bubbly flow under a laboratory surfzone
breaking wave.


Introduction

Wave breaking in the surf zone generates intense tur-
bulence and coherent structures eddy within the un-
derlying flow field. As the wave breaks, a rather
two-dimensional flow structure rapidly becomes three-
dimensional, evolving into obliquely descending eddies
(Nadaoka et al., 1989) and downbursts of turbulent fluid
(Ting, 2008). These processes can entrain large vol-
umes of air bubbles into the water column, enhancing
wave energy dissipation and air-water mass transfer. De-
pending on their concentrations and size distribution,
the entrained bubbles can significantly change the op-
tical properties of water (Terrill et al., 2001), introduc-
ing potentially large errors in the optically-based mea-
surements. Bubbles can also affect the dynamics of the
flow field. Large void fractions near the surface create
a stable stratification, which can hinder vertical mixing
(Sullivan and McWilliams, 2010).
Early investigations on bubble entrainment and evolu-
tion under surfzone breaking waves are mostly through
laboratory measurements. Deane and Stokes (1999,
2002) have conducted photographic studies on air en-
trainment mechanism and bubble size distribution un-
der laboratory plunging breaking waves. They revealed
that the bubble creation is driven by two large-scale pro-
cesses: the jet/wave-face interaction and the collapsing
cavity. The first process is primarily responsible for the
formation of small bubbles with radius less than Hinze


scale (w 1mm), while the latter is mainly responsible
for the generation of bubbles larger than Hinze scale.
The bubble size spectrum of their measurements satis-
fies a -3/2 power law for small bubbles and a -10/3 power
law for large bubbles.
Compared with laboratory experiments, numerical
studies of wave breaking induced two-phase bubbly flow
field are still rare. The main reason is perhaps due to the
lack of robust and comprehensive bubble entrainment
models. Carrica et al. (1999) developed a polydisperse
two-fluid model to study the bubbly flow field around
a surface ship, but didn't take into account bubble en-
trainment processes. The bubbles were introduced into
the computation through measured data in plunging jet
experiments. Moraga et al. (2008) proposed a sub-grid
air entrainment model for breaking bow waves. In their
model, bubble entrainment was modeled through a vol-
ume source term in the bubble number density equation.
The locations where bubbles are entrained is determined
by the mean downward liquid velocity, which should be
greater than 0.22 m/s. The bubble source intensity is
specified to obtain a good comparison with measured
data. Their model has no criterion to specify the bub-
ble source intensity which has spatial and temporal vari-
ations. Additionally, the approach to determine bubble
entrainment locations is questionable, considering that
bubbles can also be entrained in regions where liquid
velocity is not downward, for example, bubble entrain-
ment in a hydraulic jump. Shi et al. (2008) presented a











polydisperse two-fluid bubbly flow model based on mix-
ture theory. They formulated the air entrainment by con-
necting it with turbulence production at the air-water in-
terface. Simulation results showed that the model can
successfully capture the evolution pattern of void frac-
tion. Subsequent work (Shi et al., 2010) revealed that
the model is very sensitive to simulation setup. They ar-
gued that it is necessary to develop a more theoretically
justifiable air entrainment formulation.
The main objective of this paper is to develop a
physically-based wave breaking induced bubbly flow
model. In our model, bubble entrainment under break-
ing waves is correlated with turbulence intensity at the
free surface. It is assumed that the total energy required
for bubble formation is linearly proportional to the tur-
bulent dissipation rate c. The entrained bubbles strictly
follow the bubble size spectrum as observed by Deane
and Stokes(2002). The locations where bubbles are en-
trained are determined by turbulence dissipation rate e
which should be greater than a critical value c. The
model results in two free parameters: air entrainment
coefficient Cb and critical turbulence dissipation rate c%.
These two parameters have to be calibrated in the simu-
lation. The developed air entrainment model as well as
a polydisperse two-fluid model (Carrica et al., 1999) are
incorporated into a VOF code TRUCHAS and applied to
study the bubbly flow under a laboratory spilling break-
ing wave.

Two-fluid model

To simulate polydisperse two-fluid flow, the dispersed
bubbles are separated into NG classes or groups. Each
class has a characteristic bubble diameter dbi, i
1, 2, .. NG, and a corresponding volume fraction of
a,9i. By definition, the volume fractions of all of the
phases must sum to one:
NG
a0+ Z g,= 1 (1)
al 1

where al is the volume fraction of liquid phase.
The polydisperse bubbly flow model in the current pa-
per is based on the analysis of Carrica et al. (1999) who
neglected the inertia and shear stress tensors for the gas
phase due to the relatively small gas volume and den-
sity. Following Moraga et al. (2008), we neglect bub-
ble coalescence and gas dissolution. Thus the governing
equations become

O(pi) + V (tlplu) 0 (2)


8(alptlul)
( + V (alpluulu)
t


-lVp + alpig(
(3)


+V [.. .,,(Vui + VTu,)] + Mgt


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

ON *
t + V (u9,N,Ni) =B,, + S,i (4)

-agiVp + aip A + Mlg,i = 0 (5)

where pi is liquid density, ul is liquid velocity, p is pres-
sure which is identical in phases, p,,i is the bubble den-
sity of group-i, g is gravity, Peff,i is the effective vis-
cosity of liquid phase, N,,i is bubble number density
of group-i, u,,i is bubble velocity, B,i is group-i bubble
source due to air entrainment, S,,i is the intergroup mass
transfer which only accounts for bubble breakup. The
bubble breakup model proposed by Martinez-Bazan et
al. (1999a,b) is used in the present paper. Mg1 and Ml,,i
are the momentum transfer between phases, which sat-
isfy the following relationship


NG
MgY + Mli
i 1


Bubble source due to air entrainment


Deane and Stokes (2002) divided the lifetime of wave-
generated bubbles into two phases; the acoustic phase,
where bubbles are entrained and fragmented inside the
breaking wave crest, and the quiescent phase, where
bubbles evolve under the influence of turbulent diffu-
sion, advection, buoyant degassing and dissolution. The
acoustic phase is short lived and the time scale of bub-
ble fragmentation is on the order of milliseconds. Given
these features, direct simulations of the acoustic phase
require high temporal and spatial resolutions in order to
capture the details of the air entrainment process, mak-
ing their applications on the surfzone-scale domain in-
feasible. A practical way to introduce bubbles into the
computation is to prescribe air bubbles in a two-fluid
model using a bubble entrainment formulation (Shi et
al., 2010). The model fed with the initially entrained
bubbles basically simulates bubbly flows in the quies-
cent phase, thus requires much less spatial and temporal
resolutions.
As mentioned by Moraga et al. (2008), there are two
options to model air entrainment, namely a boundary
condition at the interface or a volumetric source in a
region close to the interface. The first option is prob-
lematic because we normally do not resolve small scales
necessary for predicting bubble entrainment. Follow-
ing Baldy (1993), we denote EA as the characteristic
turbulent energy available for the formation of bubbles
under breaking waves, Eb(a) as the energy required to
entrain a single bubble with a radius of a, and r as
the Kolmogorov dissipation length scale. In breaking
events, we will only consider a bubble range such that
EA > Eb(a) and a > r. In this range, the statisti-
cal state of the small-scale disturbances is self-similar.











Therefore, bubble entrainment related to small-scale dis-
turbances is also self-similar and independent of large-
scale conditions (Baldy, 1993). Thus, it is justified to as-
sume that bubble creation under breaking waves is solely
determined by e.
Based on the above analysis, we can develop the bub-
ble entrainment model quantitatively. The energy re-
quired to entrain a single bubble with a radius of a is as-
sociated with surface tension, which is given by (Buck-
ingham, 1997)
Eb(a) 47ra2c (7)
Where a is the surface tension. If we assume that the en-
ergy required for bubble creation per second is linearly
proportional to c, we can get


Eb(a)B(a) CbPlC


where B(a) is the rate of bubble creation per cubic me-
ter, cb is air entrainment coefficient which has to be cal-
ibrated in the model. Thus, the bubble creation rate can
be evaluated as

B(a) = b (rT l-2 (9)
47 pi

Equation (9) is quite similar to the model developed
by Baldy (1993) through dimensional consideration, but
it's only developed for the entrainment of a single size
group bubbles. In reality, bubbles under breaking waves
experience a large span of size distribution from mi-
crometers to centermeters. In the following analysis,
the bubble size follows the laboratory measurements of
Deane and Stokes (2002). The minimum bubble radius
is taken as 0.1 mm, and the maximum is 10 mm. Since
our studies will be focused on laboratory-scale break-
ing waves, the bubble size change due to the variation
of pressure and temperature is negligible. Therefore, we
separate bubbles into NG classes (or groups) with con-
stant bubble radius. The characteristic bubble radius of
each class is al, a2, -. aN. The width of each class
is ai+l/2 ai-1/2, where i 1, 2, .. NG, ai+l/2 =
(ai + ai+l)/2. Thus we have a1-1/2 0.1mm and
aNG+/ 2 10mm. The bubble size spectrum is given
by Deane and Stokes (2002), who revealed that bubbles
larger and smaller than Hinze scale ( 1mm) respec-
tively vary as -10/3 and -3/2 power law at the acoustic
phase.

S(a) oc a 10/3 if a > 1.0mm (10)

S(a) oc a3/2 if a < 1.Omm (11)
where S(a) is the bubble size spectrum.
Then, the bubble entrainment rate per cubic meter for
group-i can be written as

B(ai) = 5oS(ai)Aai (12)


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


where So is a coefficient, B(ai) is the entrainment rate
for group-i bubbles. We assume that the total energy
required for bubble entrainment is linearly proportional
to turbulent dissipation rate, then


NG
4) =


Plugging equation (12) into (13) gives

=Cb -E (14)
47 pi G a S(ai)Aai

The polydisperse bubble entrainment model is then
given by

B(ai) = ) 1 (a)A(15)
4w P1 1 aS(ai)Aai

The air entrainment coefficient is expected to be less
than unity, because only a portion of turbulence energy is
used to entrain bubbles. In the numerical simulation, the
air entrainment coefficient has to be calibrated with mea-
sured data. To complete the formulation, we still need to
describe how to select the grid points where bubbles are
entrained. It is straightforward to set a critical dissipa-
tion rate c, that no bubbles will be entrained if e < e, at
the free surface. The criterion of choosing ec is to make
sure bubbles are only entrained after wave breaks. Nor-
mally c, is taken as the turbulence dissipation rate of the
free surface cells at breaking point.

Momentum transfer

The momentum transfer between phases includes virtual
mass, lift force and drag force which is given as

Mlg,i -= m + -'N + l" (16)

The virtual mass force which accounts for the acceler-
ation of the liquid in the wake of the bubbles is given
by
Sl" agpiCvM( Dul Du ) (17)
Dt Dt
where CvM is the virtual mass coefficient with a con-
stant value of 0.5. The D/Dt operators denote the sub-
stantial derivatives.
Bubble rotation with finite relative velocity, or fluid
velocity gradients (shear motion), will induce a trans-
verse component in the hydrodynamic force, which is
known as the lift force. The effect of lift force is mod-
elled by


-AlT ,iPlCL(Ul -U9,i) x (V x U1)


where CL is the lift force coefficient, which is set to 0.5.











The drag force is originally due to the resistance ex-
perinced by bubbles moving in the liquid. The momen-
tum transfer by drag force is written as the following
form (Clift et al, 1978)

3 CD
I ag,iPl 8 R (u u,i) u1 ui (19)
8 R,i

where CD is drag coefficient depending on the flow
regime and liquid properties. For rigid spheres the drag
coefficient is usually approximated by the standard drag
curve (Clift et al, 1978)

24
CD g4(1 + 0.15Re0687) (20)
where Re is bubble Reynolds number
where Re,,i is bubble Reynolds number


Re9,i


aipi Iul -u9,,i b


Turbulence model

We use a nonlinear k c model which is modified for
two-phase bubbly flow (Troshko and Hassan, 2001) to
calculate turbulent eddy viscosity. The conservation
equations of turbulent kinetic energy k and turbulence
dissipation rate c are formulated as

(apik) + V (alpulk) V(aT Vk)
at a( (22)
+ a (G Pie) + Sbk

(alplc) + V (alplue) V (a Ve)
Ot (a T,


62
C,2Pi ) + Sbe
k


where the standard constants for k c model are ak =
1.0, ar, 1.3, Ci 1.44, C2 = 1.92. The term G is
the production of turbulent kinetic energy and described
by G = T : Vul. Ti is shear stress of liquid phase which
is calculated from the nonlinear Reynolds stress (Lin and
Liu, 1998).
The last two terms Sbk and Sb, are bubble induced
turbulence production. Troshko and Hassan (2001) have
proposed formulations for the bubbly flow in the vertical
duct
NG 3(D
NSbk d ,iPi Ur 3 (24)
i 1

Sb, Ce3bSbk (25)

where u, is relative velocity. C,3 is a new constant
which is found to be 0.45 (Troshko and Hassan, 2001).


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


wb is the characteristic frequency of bubble-induced tur-
bulence destruction originally given by Lopez de Berto-
dano (1994)
S2Cvmdb, (26
wb =(3 (26)

where C,, and Cd are virtual mass coefficient and drag
coefficient.

Free surface tracking

In this study, a single-phase VOF method was used
to track the free surface. Unlike the two-phase VOF
method, a single-phase VOF method does not solve
Navier-Stokes equations in the air region. This treat-
ment has been found to significantly increase the stabil-
ity of simulating high-density ratio flows. It was proved
that single-phase VOF method is sufficiently accurate in
predicting complex flows such as wave breaking in surf
zone (Wu, 2004). To facilitate the numerical implemen-
tation, we define the bubble velocities and void fraction
in the air region as zero. At the surface cells, the air and
dispersed bubbles could coexist. In reality, these cells
involve intensive interactions between air and dispersed
bubbles. For example, air will breakup to form dispersed
bubbles, and bubbles will reversely coalescence to be-
come air. These interactions are neglected in the present
model. This ensures that the only bubbles introduced
into the flow are through air entrainment model. No flux
boundary conditions at free surface are set to make sure
bubbles can leave the computational domain freely.


Bubbly flow under breaking waves

Model setup

The polydisperse two-fluid model is employed to study
the bubbly flow under a laboratory surfzone breaking
wave. Our attention will be focused on the void fraction
distribution and bubble evolution after wave breaking.
Laboratory measurements by Cox and Shin (2003) are
selected to test the model's capability. The experiment
was conducted in a 36 m long by 0.95 m wide by 1.5
m high glass-walled flume. A beach with constant slope
of 1:35 was installed with the toe 10 m from the wave-
maker and intersecting the still water line at x=27.85 m.
The flume was filled with tap water to a depth of h=0.51
m. The selected test case is characterized by a spilling
breaker with incident wave height of 0.11 m and wave
period 2.0 s.
To reduce computational effort, a 2D simulation has
been conducted. The computational domain is taken
as 21 m long by 0.7 m high, with the beach toe 1.0
m from the left boundary (see figure (1)). The mea-
sured breaking point is located at Xb = 13.07m.


+ a, (Cl G
k







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


-0 1
-0 2




Figure 1: Computational domain and measurement lo-
cations


Three measurement sections are respectively located at
x=13.81m,13.94m,14.17m. Because the velocity mea-
surement is only in streamwise direction, the turbulent
kinetic energy is not evaluated. Therefore, our compar-
isons with the measured data will be focused on free sur-
face elevation, streamwise velocity and void fraction.

The whole computational domain is discretized by a
uniform grid with Ax 0.025m, Ay 0.01m. The
time step is automatically adjusted during the computa-
tion to satisfy stability constraints. Both mean veloci-
ties and free surface displacement are specified on the
left boundary. On the top, pressure is set to be zero.
Adjacent to a solid wall, the law-of-the-wall boundary
conditions for k and c are applied (Wu, 2004). Bub-
bles are divided into NG 20 groups with a logarith-
mic distribution of bubble sizes (Figure (2)). Moraga
et al. (2008) pointed out that logarithmic distribution is
preferable over an equally spaced distribution as it helps
ensure that the ratio Adbi/dbi is small even for small dbi,
where Adbi is the width of the bin centered at dbi. They
also found no significant differences in the simulation
results for NG 15, 30 or 60, but NG < 15 should be
avoided.

There are two free parameters in the bubble entrain-
ment model that have to be determined during the simu-
lation: the critical turbulent dissipation rate c, and bub-
ble entrainment coefficient Cb. The critical turbulent dis-
sipation rate determines when and where the bubbles are
entrained. Its value is equal to the turbulent dissipation
rate on the free surface cell at breaking point. In the cur-
rent case, we take c = 0.01m2/s3. As long as the bub-
bles are entrained into the water column, the void frac-
tion is mostly determined by the bubble entrainment co-
efficient Cb. Therefore, the calculated void fraction dis-
tribution is not very sensitive to ec. We found no big dif-
ference in the numerical results if c, 0.005m2/s3 for
the current case. The bubble entrainment coefficient cb
has to be calibrated in the simulation. We take cb = 0.15
in the present simulation.


Figure 2: Initial bubble size distribution


Results

In this section, comparisons between experimental data
and numerical results are presented for free surface ele-
vation, streamwise velocity and void fraction. In the ex-
periment, these mean quantities were obtained by phase
averaging after the waves reached quasi-steady state. In
our computation, in order to reduce computational ef-
fort, we did not simulate the wave propagation for the
entire experimental period which is nearly 100 waves.
Instead, the entire computational period is 30 sec. The
numerical results show that the computed waves in the
surf zone are nearly in steady state after 30 seconds of
simulation.
Figure (3) shows the comparison of simulated and
measured wave height distribution along the beach. The
wave height is estimated from the computed free sur-
face elevation between 26.0s and 28.0s. The sampling
frequency is 20Hz. As we can see, the simulated wave
height agrees reasonably well with measurement. The
wave height before wave breaking is a little overesti-
mated. The model also predicts the wave breaking a
little earlier. These discrepancies could be partially be-
cause the traditional k c turbulence closure model can-
not accurately predict the initiation of turbulence in a
rapidly distorted shear flow such as breaking waves (Lin
and Liu, 1998). Fortunately, the distance between the
simulated and measured breaking point is short, which is
extremely important for the bubble simulation because
the location ofthe breaking point determines the initia-
tion of bubble entrainment.
Cox and Shin (2003) showed that the temporal varia-
tion of void fraction above/below the SWL normalized
by wave period and average void fraction appears to be
self-similar and can be modeled simply by linear growth
and exponential decay. These variations are also ob-
served in our numerical results which are dispicted in








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


20
15 -
F 10-
> 5
a


16 18 20 22 24 26 28


Figure 3: Comparison of simulated and measured wave
height distribution

4

3

am 2


3

2a



0 0.02
0 0.02


0.04 0.06 0.08 0.1


t/T


Figure 4: Temporal variation of simulated void frac-
tions above (upper panel) and below (lower panel) the
still water level


figure (4). We can clearly see that the exponential de-
cay of simulated void fractions which are normalized
by wave period and average void fractions is well simu-
lated. This result proves that the two-fluid model is able
to successfully simulate bubble transport phenomenon.
The linear growth of the void fraction is captured as
well, but the growth rate is underpredicted comparing to
Cox and Shin's estimate. The reason is because the cur-
rent bubble entrainment model is largely simplified by
assuming air entrainment is linearly related to the turbu-
lence dissipation rate. In addition, the interactions be-
tween large air cavities and dispersed bubbles are not
considered in the current model. But we should point
out that this level of accuracy is a considerable improve-
ment over previous works in which the bubble source is
manually specified (Carrica et al., 1999; Moraga et al.,
2008).
As the bubble entrainment in our model is directly re-
lated to turbulent dissipation rate, it is instructive to take
a look at turbulent dissipation rate and void fraction dis-


0 0'


Figure 5: Temporal variation of simulated turbulent dis-
sipation rate with time interval 0.2T



tribution simultaneously. These snapshots are shown in
figure (5) and (6), where the turbulent dissipation rate is
normalized by g g(h + H), g is gravity acceleration, h
is still deep water depth, and H is incident wave height.
The plotted void fractions are bounded by 0. :'. follow-
ing Lamarre and Melville (1991, 1994) who used 0. '.
as a threshold to evaluate various moments of the void
fraction field. In these two figures, xb is the measured
breaking point. We can see that the evolution patterns of
turbulent dissipation rate and void fraction are similar.
When wave starts to break, the turbulent dissipation rate
will be greater than the critical turbulent dissipation rate,
thus bubbles start to be entrained into the water column.
As the breaking bore moves forward, the turbulent dis-
sipation rate and void fraction all increase rapidly and
attain their maximum value shortly after wave break-
ing. The peak void fraction appears around 0.5m on-
shore from the breaking point. This result is consistent
with the findings of Mori et al. (2007), who argued that
the peak of void fraction happens at 0.1 to 0.2 wave-
length onshore from the breaking point. After attaining
their maximum, both the turbulence dissipation rate and
void fraction start to decay. The high turbulence region
is persistently located at the breaking wave crest with a
moderate time variation in the value of dissipation rate,


10 (t-tb)/T=0 1
0 0 078
0002
-10
-20
10 (t-tb)/T=03
0'0 0547
0
-10 0002
-20
10, (t-tb)/r=0 5
0 050 0408
-10 0x002
-20
10 (t-tbYT= 07
0 0i02
-10 000
-20
10 (t-tb)/T=0 9 00193

0002
-10
-20
0 05 1 15 2 25 3
x Xb (m)







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


while the void fraction varies by an order of magitude in
a wave period. The higher void fractions are restricted
in the roller region, but some bubbles are spread down-
stream and form a long tail of the bubble cloud under the
wave surface.


Conclusions


We have developed a polydisperse two-fluid model for
simulating bubbly flow under surfzone breaking waves.
An air entrainment model was proposed to account for
the bubble generation after wave breaking. This model
connects bubble entrainment to the turbulent dissipation
rate at the air-water interface. The model was utilized
to study bubbly flow under a laboratory surf-zone break-
ing wave. Numerical results showed that the model can
reproduce the experiment measurements for a spilling
breaking wave. The predicted temporal variation of void
fraction can be modeled with linear growth and expo-
nential decay, which is consistent with the laboratory
measurements. The evolution of simulated void fraction
follows that of turbulent dissipation rate. The peak void
fraction appears around 0.5m onshore from the break-
ing point. The high void fraction which is mostly con-
tributed by large bubbles is generally located in the roller
region.
In the current model, the interactions between air cav-
ity and dispersed bubbles have not been considered be-
cause they are negligible in a spilling breaking wave.
In a plunging breaking wave, air cavities can be formed
during the splash-up cycle. The air cavities can breakup
to form dispersed bubbles under the action of turbulence,
dispersed bubbles can reversely coalescence to become
air. In order to better simulate bubble entrainment pro-
cesses in the surf zone, our future work will be focused
on incorporating these interactions into our model.


Acknowledgments

This study was supported by Office of Naval Research,
Coastal Geoscience Program under grant N00014-10-1-
0088. Sincere thanks also go to Dr. Cox and Dr. Shin
for making their experimental data available to us.


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20
10
O
10
20

10
10
-10
20
10
0
10
?n


0o
0 05 1 15 2 25 3
x-x, (m)


Figure 6: Temporal variation of simulated void fraction
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