Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 14.2.4 - On the importance of buoyancy in Taylor flow in horizontal microchannels
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 Material Information
Title: 14.2.4 - On the importance of buoyancy in Taylor flow in horizontal microchannels Micro and Nano-Scale Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Gupta, R.
Hägnefelt, H.
Fletcher, D.F.
Haynes, B.S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Taylor flow
bubble shape
Bond number
gravity
film thickness
 Notes
Abstract: There is enormous interest in Taylor flow in micro-channels because of the benefits it has in micro-scale devices, including large interfacial area and very small back-mixing. Many studies, both experimental and numerical, into the properties of this flow have been published. However, there are few data on the effect of gravity for such flows. Here results from both CFD simulations and experimental studies are presented. An effect of gravity is observed in both the experimental and numerical investigations. It is shown that for a given system there is a critical Bond number below which gravitational effects can be neglected, but more work is required to delineate under what conditions gravity is unimportant in Taylor flow.
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: VID00346
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 1424-Gupta-ICMF2010.pdf

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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


On the importance of buoyancy for Taylor flow in horizontal microchannels


Raghvendra Gupta*, H~kan Haignefelt, David F. Fletcher and Brian S. Haynes

School of Chemical and Biomolecular Engineering, The University of Sydney,
NSW, 2006, Australia
*david.fletcher a sydney.edu.au

Keywords: Taylor flow, bubble shape, Bond number, gravity, film thickness


Abstract

There is enormous interest in Taylor flow in micro-channels because of the benefits it has in micro-scale devices, including
large interfacial area and very small back-mixing. Many studies, both experimental and numerical, into the properties of this
flow have been published. However, there are few data on the effect of gravity for such flows. Here results from both CFD
simulations and experimental studies are presented. An effect of gravity is observed in both the experimental and numerical
investigations. It is shown that for a given system there is a critical Bond number below which gravitational effects can be
neglected, but more work is required to delineate under what conditions gravity is unimportant in Taylor flow.


Nomenclature


distribution around the bubble in a millimetre-size
horizontal channel. As the channel diameter decreases, the
importance of gravity also diminishes.
Bretherton (1961) pointed out that the gravitational force
can distort the symmetric shape of a bubble at
non-negligible Bond numbers. However, he could not obtain
the solution of the relevant partial differential equation
analytically to quantify this effect. Grotberg and co-workers
have studied the effect of gravity on the motion of a liquid
slug in a planar, two-dimensional channel analytically
(Suresh and Grotberg, 2005) and numerically (Zheng et al.,
2007). Suresh and Grotberg (2005) studied the effect of
gravity on the motion of a liquid plug in a planar,
two-dimensional channel by applying lubrication theory in
the limit of small Capillary (Ca) and Bond (Bo) numbers.
They found that when the gravitational force is small
compared with the viscous and surface tension forces (Bo <
1, Ca << 1), the film thickness scales as Ca2 3; and when
the gravitational force is comparable with the viscous and
surface tension forces, it scales as Ca 2z. They pointed out
that in a cylindrical geometry gravity causes a variation mn
the film thickness with azimuthal angle.
Zheng et al. (2007) studied numerically the combined effects
of gravity, mixture velocity and liquid slug length for a
planar, two-dimensional, horizontal channel. They pointed
out that gravity causes flow of the liquid from the upper
liquid film near the nose to the liquid film at the tail of the
preceding bubble. The flow recirculation (in a frame of
reference moving with the bubble) becomes weaker with an
increase in the Bond number and the number of vortices
present in the liquid slug can be zero, one or two depending
on the flow parameters. Asymmetry of the liquid distribution
(measured as the ratio of the liquid volume above and below
the centreline of the channel) was found to increase with an
increase in Bond number and to be reduced by an increase in
either of the length or speed of the liquid slug.
Han and Shikazono (2009) found experimentally that the
liquid film thicknesses at the bottom and sides of a


Bo
Ca
d

Greek
8


Bond number
Capillary number
Channel diameter (mm)


Film thickness (mm)


Introduction

The Taylor flow regime in gas-liquid two-phase flow in
microchannels is of great interest to a range of industries
(e.g. electronics cooling, automotive, biomedical, aerospace
and chemical processing) because of its important flow
characteristics, such as large interfacial area, small diffusion
paths and flow recirculation within the bubbles and liquid
slugs. Taylor flow is characterized by gas bubbles that
almost fill the channel, surrounded by a thin liquid film on
the channel wall, and separated by liquid slugs. Reviews by
Kreutzer et al. ('****1.11, Angeli and Gavriilidis (2008) and
Gupta et al.(2010) provide a good summary of work on
Taylor flow in microchannels.
In large-diameter horizontal channels, gravity hasa
significant effect on Taylor flow. For example, in horizontal
flow, buoyancy plays an important role and the liquid film
thickness at the top of a duct is very small compared with
that at the bottom. In microchannels the effect of gravity has
often been observed to be negligible (Triplett et al., 1999a)
and therefore is generally neglected in CFD studies of
Taylor flow in microchannels. There are very few studies in
the literature available to determine the effect of gravity in
horizontal microchannels and the value of the Bond number
(ratio of buoyancy and surface tension force) when gravity
ceases to be important remains uncertain.
The assumption of axisymmetric, two-dimensional, flow is
only valid for a vertical circular channel or a horizontal
circular channel when the gravitational effects are
unimportant. Gravity can cause an asymmetric liquid film






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

Backlit images of the flow were recorded by the high speed
camera (Photron Fastcam PCI) at 1000 frames per second
(fps), which gave a resolution of 1024 x1024 pixels. Images
were recorded with objectives with 4 x or 20 x magnification
depending on the intended purpose of the recording. The
field of view was 4.35 mm with the 4 x objective and
0.87 mm with the 20 x objective. The focal plane of the
objective was positioned in the centre of the channel and the
images were collected 220 mm (110 d) from the inlet to
minimise any entrance effects. The slowest shutter speed
used was 1/45,000 s and at this speed no tendency of
blurring could be seen. When using the 4 x objective a
shutter speed of 1/300,000 s was used as this gave improved
quality images.
Time-resolved measurements of fluid velocities in the film
were undertaken with the aid of a microPIV system.
Fluorescent seed particles were excited at 527 nm
wavelength with a high-speed double-pulse laser (New
Wave Pegasus-PIV dual-cavity Nd-YLF laser, capable of up
to 10 mJ per pulse per laser head at 1 kd~z, with a pulse
duration < 180 ns). Fluorescent emission (h > 550 nm) from
the excited particles was captured using the same
high-speed camera described above. The test section was
mounted horizontally on the optical stage of an inverted
epifluorescent microscope which was used to image the
flow section and to manage the excitation beam and the
fluorescent emission.
A synchroniser provided by ILA (Jillich, Germany)
synchronises the camera with the laser pulses for high speed
measurements. The frame rate of the camera was chosen to
provide the desired time resolution with the laser pulses
being fired in frame-straddling mode. The Vidpiv software
package from ILA was used to process the camera images to
yield velocity fields. The experimental conditions and the
equipment setup were similar to those used for the white
light observations. The flow was seeded with fluorescent
particles of 3 pLm diameter and a density 1.05 g/ml. The
particle concentration was 0.024% by volume. A 10 x
objective was used to observe the flow giving a field of
view of maximum 1.74 x 1.74 mm.
More details regarding the microPIV and imaging
equipment can be found in Fouilland (2009) and Fouilland
et al. (2010).

Experimental Results

Figure 1 shows pictures of the bubble, nose and tail for
nitrogen-ethylene glycol flow in a horizontal channel of
diameter of 2 mm for a homogeneous void fraction of 0.66
at four different mixture velocities ranging from 0.06 to
0.6 m s The effect of gravity is apparent at the three lower
mixture velocities. The film can be seen to be thicker at the
channel bottom than at the top. This is caused by a
continuous drainage flow in the film from the top to the
bottom giving rise to the asymmetry of the bubble shape.
This asymmetry can be seen clearly at the bubble tails at
low mixture velocities.
Figure 2 shows the variation of non-dimensional film
thickness with Capillary number at the top and bottom of
the bubble, at the bubble nose and tail for a horizontal
channel. The difference between the upper and lower film
thicknesses is greatest at the tail of the bubble, especially for
low values of Ca. It can be seen that when Ca increases, the


Paper No


horizontal channel of diameter 1.3 mm (Bo ~ 1.9, Ca < 0.4)
differed by ~5%. For a channel of diameter 0.3 mm (Bo ~
0.1, Ca < 0.4), the liquid film thicknesses at the channel top.
bottom and side were the same.
Several researchers have studied the hydrodynamics of
Taylor flow using various CFD techniques, such as Finite
Element methods, the boundary integral method, interface
capturing methods, and the Lattice-Boltzmann method.
Recently Gupta et al. (2010) have reviewed the approaches
to the modelling of Taylor flow and discuss the earlier
studies in detail.
Gupta et al. (2009) have made a detailed study of the effect
of mesh size and morphology, and numerical schemes on
the calculated bubble shapes. They showed that poor mesh
resolution was responsible for many Taylor flow simulations
that have the bubble in direct contact with the wall. Advice
on the required grid resolution and aspect ratio needed to
correctly capture the liquid film were given. Additional
advice for users of ANSYS Fluent was provided on the best
choice of schemes and differencing techniques.
In this work the effect of buoyancy on Taylor flow in
millimetre-size channels is studied experimentally and
numerically. The experimental measurements of film
thickness and velocity components are carried out by direct
visualization and using the micro-PIV technique,
respectively. Numerically, Taylor flow is studied using
ANSYS Fluent, with a three-dimensional computational
domain to explore the effect of gravity in horizontal
microchannels.

Experimental Work

Taylor flow was studied experimentally in a 2 mm diameter
silica tube. Two-phase, co-current flow of nitrogen and
ethylene glycol was studied by direct visual observation at
atmospheric pressure and a temperature of 210C. An
important component in the experimental study is the
mixing device used to create the bubbles within the liquid as
it needs to generate repeatable sized bubbles. This was
achieved using a mixer consisting of a 1/8" Swagelok tee,
where the liquid was injected from a 1/8" nylon tube on one
side of the tee piece. The gas was injected through a 25 psi
Swagelok check valve connected to the tee by a 1/8"
Swagelok port connector on the other. The check valve
eliminated gas flow fluctuations which can arise from the
pressure excursions caused as each bubble was created and
released into the channel. The mixture then entered the
silica tube connected to the tee via a 1/8" Swagelok port
connector and a 1/8" PFA Swagelok union. This design
proved very successful in creating stable flow patterns and
due to the use of standard parts it leads to reproducible
results.
By carefully matching the refractive index of the fluid and
the channel material (At 589 nm, RIethylene glycol = 1.432;
R~sic =1.459) and through the provision of a
square-section refractive-index-matching system around the
tube to avoid curvature effects, a high image quality was
achieved. The bubble shape and film thickness could be
measured quantitatively by visual observation. To achieve
this, the flow was illuminated using a white light source and
an epifluorescent microscope (Olympus IX71) was
positioned vertically so the image was recorded from the
side of the channel.































































. *


Bott m





O o
~D Vertical Upwards
8 O Nose, Bottom
O Nose, Top
Horizontal r al otm
O Tail, Top


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

periods when the bubble is passing the observation point, as
the transverse slug velocities cannot be captured with the
PIV settings needed for the film drainage flows. The axial
film velocities during passage of a bubble are much smaller
than the drainage velocities shown in Figs. 3 (a) and (b).
The drainage velocities are highest close to the nose of the
bubble, decreasing steadily towards the tail. At locations
further from the wall and closer to the gas-liquid interface,
the drainage velocities are greater, as is their decrease in
magnitude between nose and tail. Figure 3(c) shows the
vertical downward velocity component at a distance of
114 pLm from the wall in vertical upflow. This velocity
remains constant between the nose and the tail.


Paper No


difference between the upper and lower film thicknesses, as
well as the differences between the tail and nose film
thicknesses, decreases. The film thickness for vertical
upwards flow is also shown, and these fall between the
upper and lower film thicknesses for the horizontal case.


r


(a)


t

tSf iIL1~ i+Y



t


0.0


0.2 0.4
Time (s)


0.6 0.8


r



e


(b)


L~94;C


Figure 1: Images of the nose and tail of bubbles travelling
horizontally for a homogeneous void fraction of 0.66 and a
mixture velocity of (a) 0.06 ms' (b) 0.10 ms' (c)
0.25 ms and (d) 0.60 ms'.


0.2 0.4 C
Time (s)


0.8 1 0


0.14

S0.12 i

fi0.108

o 0.06 -

~~0.04 -

S0.02 -

0.00 -


S-4.0 -




-2.5 -

-2.0 -


0.0 0.1 0.2 0.3 0.4
Time (s)


0.5 0.6 0.7


0.00 0.05 0.10 0.15 0.20 0.25


0.30


Figure 3: Temporal variation of the vertical velocity
component at different distances from the wall for
horizontal flow: (a) 6/ d= 0.022, (b) 6/ d= 0.057
vertical-upwards flow (c) 6/ d= 0.057. The mixture velocity
was 0.1 m s and the homogeneous void fraction was 0.667.

CFD Modelling

Three-dimensional CFD simulations were carried out in
horizontal channels of square, as well as circular,


Capillary number (Ca)


Figure 2: Variation of the non-dimensional film thickness
with Capillary number for a homogeneous void fraction of
0.66.

Figures 3(a) and (b) show the temporal variation of the
vertical velocity component at distances from the wall of 43
and 114 pLm (6/d = 0.022 and 0.057), respectively, for
horizontal flow. The velocities are shown only for the


I~
'L


ly)
E -2.0 -
B
O -1.5 -
B
a
D


r ~II




t~Z1






Paper No


cross-section. The liquid film is very thin for low viscosity
liquids, such as water, and a very fine near-wall mesh is
required. Therefore, viscous fluids, such as FC-40 or
ethylene glycol, which lead to thicker liquid films, are used
to allow relatively larger cell sizes and to make
computational times tractable.
The ANSYS Fluent CFD software is used to model
three-dimensional, transient, Taylor flow. The detailed
methodology for the modelling of Taylor flow in a
two-dimensional, axisymmetric computational domain can
be found in Gupta et al. (2009). Here only a brief
description of the numerical methodology is given.
The volume of fluid (VOF) method is used to determine the
gas-liquid interface by solving a volume fraction equation
for one of the phases, in this case the liquid phase. An
explicit geometric reconstruction scheme is used to
represent the interface by using a piecewise-linear approach.
Surface tension effects are included using the surface
continuum force method (Brackbill et al., 1992). The
Green-Gauss node based gradient method was used to
calculate the gradient of the scalars.
A co-located scheme with body-force-weighted
interpolation is used to compute the face pressure for the
pressure-velocity coupling. An implicit body force treatment
is used to take into account the partial equilibrium of the
pressure gradient and body forces. The QUICK scheme is
used for the discretisation of the momentum equations. A
first order, non-iterative fractional step scheme is used for
the time-advancement of the momentum and continuity
equations, with a variable time step based on a fixed
Courant number of 0.25. The absolute value of the equation
residuals were kept low, ~10*8 for velocity and ~10-10 for
continuity.

Boundary and Initial Conditions

Simulations are carried out in one half of the channel and a
symmetry boundary condition is applied on the mid plane.
At the inlet, the gas enters at the core of the channel,
occupying an inlet area in proportion to the homogeneous
void fraction and the liquid enters as an annulus around the
gas core. At the outlet, a constant area-averaged pressure
boundary condition is applied. At the wall the usual no-slip
boundary condition is enforced. Initially only the liquid
phase is present in the computational domain.

Square Channel Simulations

As discussed in our previous work (Gupta et al., 2009)'
hexahedral mesh elements having an aspect ratio close to
one are required for accurate surface tension modelling. It is
not possible to obtain a structured hexahedral mesh having
an aspect ratio of one in a channel of circular cross-section.
Therefore, Taylor flow in a square channel, in which grid
elements having an aspect ratio of one are easily constructed,
is simulated first to study the effect of gravity. The
modelling of Taylor flow in square channels also brings out
the effect of polygonal geometry on the bubble shape and
flow field. Polygonal geometries have applications in many
industries, such as MEMS, electronics and biomedical.
The simulations are carried out in a square channel of side
(and hydraulic diameter) d and a channel length of 10d.
FC-40-air or ethylene glvcol-air are used as the working


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

fluids. The relevant fluid properties are given in Table 1.
The mixture velocity and the homogeneous void fraction are
0.1 m s-' and 0.5, respectively.
The simulations are carried out in channels of two different
sizes and the details of the different cases are given in
Table 2. Case 1 is used as a base case and cases 2 and 3 are
used to study the effect of using a more viscous fluid and a
smaller channel diameter, respectively.

Fluid Density Viscosity Surface

(kg m ) (kg m s ') tension
(N m )
Air 1.185 1.831x10

FC-40 1849 0.003 0.0159

Ethylene 1113 0.0173 0.0477
Glvcol



Table 1: Properties of air, FC-40 and ethylene glycol used
in the simulations.

The Computational Mesh Used for the Square
Channel Simulations

The mesh is of aspect ratio one (cubic) in the core region
and refined near the walls. Figures 4 (a) and (b) show the
mesh on the cross-sectional and axial planes, respectively.
The computational grid comprises ~ 2 million (564x3528)
elements. The simulations were run on a 64 bit HP xw 8600
workstation, having an Intel Xeon X5472 CPU, 16 GB
RAM, with all 8 processors being used for the computations.
It took about 21,000 time steps and 1,504 hours of CPU
time (188 hours wall clock time) to run the simulation for a
physical time of 73.3 ms.

Cas Fluids d Up B Re Ca Bo
e (mm) (m s )
1 FC-40 1 0.1 0.5 62 0.02 1.14
2 Ethylene 1 0.1 0.5 6.4 0.04 0.23
glycol
3 FC-40 0.5 0.1 0.5 31 0.02 0.29


Table 2: The flow conditions used for the simulations for a
square channel. Case 1 is used as a base case.


mIw+ t


1 nI antlllR





(b)

Figure 4: (a) A view of the mesh on a cross-sectional plane.
The wall and symmetry planes are at the top and bottom of
the figure. (b) A view of the mesh on the axial plane. Only a
part of the axial plane is presented to show the details of the
mesh.

Figure 5 shows the projections of various axial planes that
are used to present the results. The direction of gravity is
shown for the "with g" cases.

45" -


(a) (b)
to 6 st 2" 28q q9 S ~ o

Pressure IPal


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

that of the square channel is 39 pLm, so the film thickness at
the channel centre is less than that obtained in a circular
channel. The distribution of the liquid in the slug becomes
asymmetric in the 'with g' case with more liquid
accumulated at the bottom than at the top of the film.
Gravity also gives rise to asymmetry in the bubble shape.
The bubble becomes longer at the top and shorter at the
bottom. The asymmetry in the bubble tail is greater than that
at the nose. The drainage flow occurring across the entire
length of the bubble accumulates liquid at the tail, making
the effect more pronounced at the tail than at the nose.


Paper No


(b)

Figure 6: Shape of the gas bubbles on the symmetry plane
at 73.3 ms for the (a) "no g" and (b) "with g", cases.

Figures 7, 8 and 9 show flow data at three different planes
across the bubble (from nose to the tail). The black solid
line shows the interface locations. In the 'no g' case, the
bubble shape is spherical close to the nose (see Figure 7).
The radius of the bubble increases on moving away from the
nose. When the bubble diameter approaches the channel
dimension, the bubble shape becomes asymmetric,
flattening at the centre but remaining circular at the corners
(Figures 8 and 9). The 'with g' results shown in Figures 7-9
again show that the inclusion of gravity causes the bubble
shape to be asymmetric.


Symmetry







180"


Figure 5: Projections of various post-processing planes on a
cross-sectional plane for the channel of square cross-section.

The Bubble Shape in the Square Channel

Figures 6 (a) and (b) shows the shape of the bubbles on the
symmetry plane at a time of t = 73.3 ms for the 'no g' and
'with g' base case, respectively. The liquid film around the
bubble is not of constant thickness and decreases slightly
from the nose to the tail region and therefore a typical film
thickness (at the midpoint along the bubble) is calculated to
determine the effect of gravity on film thickness. The bubble
velocity in both cases is ~0.126 m s The figure also shows
that the gas bubbles have travelled similar distances in the

tTh ca sd film at the bottom of the channel is thicker when
gravitational effects are taken into account. The average film
thickness at the channel centre (symmetry and centre
planes) is 16 pLm in the 'no g' case. In the 'g' case, the film
thickness at the channel centre at the top, side and bottom
are 14, 15 and 23 pLm, respectively. The film thickness
obtained from the Aussillous and Qu~rd (2000) correlation
for a circular channel having the same hydraulic diameter as


Figure 7: Pressure contours, tangential velocity vectors and
bubble shape (shown by solid black line) at a cross-sectional
plane close to the bubble nose for (a) "no g" (b) "with g"
cases. The gravitational force is acting from the top to the
bottom.

































Pressure IPal


Pressure


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

The Flow Field for the Square Channel

The difference between the pressure in the gas and liquid is
due to the interfacial pressure. As can be seen clearly in
Figure 9 (a), the pressure in the liquid is low at the corners
and high in the centre of the channel so that liquid flows
from the centre of the channel to the corners. The tangential
velocity at the diagonal line is zero. The bubble shape and
flow-field obtained for the 'no g' case are qualitatively
similar to those obtained by Hazel and Heil (2002) and
Sousa et al. (2007). Figure 7 (a) shows that there are four
symmetrical vortices in the cross-sectional plane in the gas
phase when gravity is neglected. The flow in the liquid is
symmetric about the centre and diagonal planes. Upon
including the effect of gravity, the flow in the gas and liquid
becomes asymmetric. Gravity gives rise to a drainage flow
from the channel top to bottom, which breaks the symmetry
in the flow seen in the 'no g' case.

Effect of the Bond Number for the Square Channel

As shown in Table 2, two simulations were run at lower
Bond numbers of 0.23 and 0.29 (the Bond number for the
base case is 1.14) by using a different liquid and a smaller
channel dimension, respectively. The shape of the bubble at
the channel top and bottom (not shown here) were found to
be similar, showing that the effect of gravity is reduced
significantly at low Bond numbers.

Circular Channel Simulations

Simulations were also carried out in a circular-section
channel of diameter 2 mm, length 5d for a mixture velocity
of 0.1 m s' and a homogeneous void fraction of 0.66 with
ethylene glycol and air as the working fluids.

The Computational Mesh Used in the Circular
Channel Simulations

Figures 10 (a) and (b) show the mesh on a cross-sectional
plane and an axial plane, respectively. As can be seen from
the figure, it is not possible to obtain a uniform aspect ratio
structured mesh for the cross-sectional plane. The
simulations were also performed using an unstructured
swept mesh having uniform size elements. Both of these
meshes gave rise to significant spurious velocities caused by
the surface tension discretization (see later). The magnitude
of these spurious velocities was higher in the case of the
unstructured mesh.
The computational grid was composed of ~ 1.65 million
(5504x300) cells. However, the size of the mesh elements
used here were not as refined as those used in
two-dimensional simulations by Gupta et al. (2009). The
simulations were run on the same computer as the
square-section simulations. It took about 14,000 time steps
and 856 hours of CPU time (107 hours wall clock time) to
run the simulation for a physical time of 61.9 ms. Only one
gas bubble was generated in the computational domain in
this time.


Paper No


(a) (b)


Figure 8: Pressure contours, tangential velocity vectors and
bubble shape (shown by solid black line) at a cross-sectional
plane in the middle of the bubble for (a) "no g" and (b)
"with g" cases. The gravitational force is acting from the top
to the bottom.


(a) (b)

63 6~ 4P 24 2 49, qZ9 4,~ ~


Figure 9: Pressure contours, tangential velocity vectors and
bubble shape (shown by solid black line) at a cross-sectional
plane close to the bubble tail for (a) "no g" and (b) "with g"
cases. The gravitational force is acting from the top to the
bottom.


IPal











ilir

~






I i
(a)
I I I I I I ' ' ' ' i I


(b)

Figure 10: (a) A view of the cross-sectional mesh used for
the circular channel simulations. (b) A view of the mesh on
the axial plane. Only a part of the axial plane is shown for
clarity.

The Bubble Shape in the Circular Channel Case

A comparison of the bubble shape (figure not shown here)
for the "with g" and "no g" cases revealed that gravity again
causes the liquid film to become thinner at the top and
thicker at the bottom, and again this effect increases from
the nose to the tail.
Figure 11 compares the bubble shape obtained from
experiments described earlier and the CFD simulation for a
channel diameter of 2 mm, a mixture velocity of 0.1 ms'
and a homogeneous void fraction of 0.66. Despite the
numerical problems arising from the non-cubic mesh cells,
the shape of the bubble at the nose and tail in the two cases
is very similar and CFD calculations are successful in
capturing the broad flow characteristics. The bubble in the
experiment was significantly longer than that obtained in the
CFD simulations.

The Flow Field for the Circular Channel

Figure 12 shows the pressure contours at two cross-sectional
planes at axial locations of 2.6 d and 3.8 d at a time of
61.9 ms for the 'with g' case. A pressure gradient exists in
the liquid film which drives the flow from the top of the
channel to the bottom. As the pressure difference between
the gas and the liquid is balanced by the interfacial pressure,
a pressure gradient is also established in the gas.


(a) (b)



Pressure [Pa]


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


Paper No


C1


Figure 11: Comparison of the bubble shape in a horizontal
circular channel obtained from experiments (black and
white) and the CFD simulation (coloured) for a channel
diameter of 2 mm, a mixture velocity of 0.1 m s-' and a
homogeneous void fraction of 0.66. Only the bubble shape
at the nose and the tail, and not the entire bubble, are shown.


Figure 12: Pressure contours on two cross-sectional planes
(axial locations 2.6 and 3.8 d in (a) and (b), respectively)
for the 'with g' case. The direction of gravity is from the
top to the bottom.

Figure 13 shows the tangential velocity vectors at two
cross-sectional planes and axial locations of 2.6 d and 3.8 d
at a time of 61.9 ms for the 'with g' case. The tangential
velocity in the liquid film is ~10% of the mixture velocity.
The large velocity vectors observed near the interface are
due to spurious currents arising from the surface tension
modelling on a non-cubic mesh.




















Pressure [Pa]


ial (b)



Velocity [m s^-1]


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


Paper No


~p~--c -


~66898~~~8~
0 0 0 0 0 0 0 0 0 0 0


Velocity


[m s^-1]


Figure 13: Tangential velocity vectors in the liquid at two
cross-sectional planes (axial locations 2.6 and 3.8 d in (a)
and (b), respectively) for the 'with g' case. The direction of
gravity is from the top to the bottom.

Figure 14 (a) and (b) show the contours of pressure and
velocity on the bubble surface (iso-contour of 0.5 liquid
volume fraction). The pressure decreases from the top of the
bubble to the bottom, causing a drainage flow of liquid. The
spatially oscillatory pressure field arises from the spurious
currents caused by errors introduced by discretisation of the
surface tension force. The velocities are higher in the bubble
nose and tail regions than that in the film region in the
middle. The iso-contours of bubble velocity show an
interesting wavy pattern, moving from the top at the front to
the bottom at the back of the bubble. Figure 14 (c) shows
streak-lines originating from the bubble surface. It can be
clearly seen that the liquid present at the bubble surface
drains from the channel top to the bottom.

Conclusions

Taylor flow of the nitrogen-ethylene glycol was studied
experimentally and data for both the bubble shape and
drainage flow were obtained. This fluid combination was
chosen as it results in relatively thick liquid films and allows
very high quality photographs to be taken due to the small
refractive mismatch with the channel wall made from silica.
The effect of gravity on Taylor flow in a square-section
channel was modelled for different values of the Bond and
Capillary numbers. At a Bond number of 1.14, gravity
caused the liquid film to become thinner at the top and
thicker at the bottom. In addition, the gas bubble became
longer at the top and shorter at the bottom. For the low
values of Bond number (Bo ~ 0.25), the effect of gravity
had a negligible effect on the film thickness and a very
small effect on the bubble length.


(c)


Figure 14: Contours of pressure (a) and velocity (b) on the
bubble surface (iso-contour of 0.5 liquid volume fraction)
and (c) streak-lines originating from the bubble surface. The
flow direction is from right to left. The direction of gravity
is from the top to the bottom.

The effect of gravity was also simulated for a circular
channel of diameter 2 mm, for a flow with a Bond number
approximately one. Despite the presence of parasitic
velocities arising from using a non-cubic mesh, these
preliminary results gave a bubble shape similar to that
obtained in the experiments and there was a significant
drainage flow in the liquid film surrounding the bubble. The
computational cost required to produce only a single bubble
in a 5 d long computational domain was very high.

These experimental and simulation results show very clearly
that gravitation effects can indeed be important in two-phase,
horizontal flows. The fluid combination used in the
experimental work provides an accurate means of
determining this effect. The simulations showed that for a
system that could be meshed using a hexahedral mesh with
cells of aspect ratio of one the results are not affected by
spurious numerical velocities, whilst results for a circular
channel, for which the same meshing strategy could not be
used, showed the presence of significant parasitic currents.

Future work in this area requires improved accuracy models
for interface capturing and modelling of surface tension.






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


Sousa, F. S., Portela, L.M., Kreutzer, M.T. & Kleijn, C.R.
Numerical simulation of slug flows in square channels using
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Suresh, V & Grotberg J.B. The effect of gravity on liquid
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Fluids, Vol. 17, 031507-1-031507-15, (2005).

Triplett, K.A., Ghiaasiaan, S.M., Abdel-Khalik, S.I. &
Sadowski, D.L. Gas-liquid two-phase flow in microchannels
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Zheng, Y., Fujioka, H. & Grotberg, J.B. Effects of gravity,
inertia and surfactant on steady plug propagation in a
two-dimensional channel. Physics of Fluids. Vol. 19,
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Paper No


However, such modelling will necessarily remain limited
because of the long computational times.

Acknowledgements

The Heatric division of Meggitt (UK) Ltd and the Australian
Research Council are thanked for their financial support of
this work. R. Gupta also acknowledges the University of
Sydney's Henry Bertie and Florence Mabel Gritton
Research Scholarship Foundation.

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