7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
On bubble clustering in pseudoturbulence
Chao SunT J. MartfnezT D. ChehataT D.P.M. van GilsT and Detlef Lohse*
Physics of Fluids Group, Department of Science and Technology, J.M. Burgers Center for Fluid Dynamics,
and IMPACT Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
c.sun@utwente.nl; d.lohse@utwente.nl
Keywords: Bubbly flow; pseudoturbulence; bubble clustering; energy spectrum
Abstract
We performed 3DParticle Tracking (3DPTV) in pseudoturbulencei.e., flow solely driven by rising bubblesto
investigate bubble clustering at a very dilute gas concentration. To characterize the clustering the pair correlation
function G(r, 0) is calculated. The deformable bubbles with equivalent bubble diameter db = 4 5 mm are found to
cluster within a radial distance of a few bubble radii with a preferred vertical orientation. This vertical alignment is
present at both small and large scales.
This conference paper is a part of our published paper (Martinez et al. 2010).
Introduction
Dispersed bubbly flow has attracted much interest, both
from a fundamental point of view (Lance and Bataille
1991) and because of its widespread occurrence in in
dustrial applications (Deckwer 1992). Bubbly pseudo
turbulencei.e. a flow solely driven by rising bubbles
is relevant from an application point of view due to the
omnipresence of bubble columns, e.g. in the chemical
industry, in water treatment plants, and in the steel in
dustry (Deckwer (1992)). A better understanding of the
involved phenomena is necessary for scalingup indus
trial devices and for optimization and performance pre
diction. This article wants to provide experimental data
on pseudoturbulence by means of novel experimental
techniques. The issue to be addressed is
What is the preferential range and the orientation of
bubble clustering in pseudoturbulence ?
The hydrodynamic interaction between the two
phases and the particle inertia result in an inhomoge
neous distribution of both particles and bubbles in dis
persed bubbly flows(see e.g. experimentally (Ayyalaso
mayajula et al. 2006; Salazar et al. 2008; Saw et al. 2008)
and numerically (Calzavarini et al. 2008b,a; Bec et al.
2006a) and Toschi and Bodenschatz (2009) for a general
recent review). Bubble clustering in pseudoturbulence
has been studied numerically (Smereka 1993; Sangani
and Didwana 1993; Sugiyama et al. 2001; Mazzitelli
and Lohse 2009) and experimentally (Cartellier and Riv
ibre 2001; Risso and Ellingsen 2002; Zenit et al. 2001;
Roig and de Tournemine 2007). The numerical work
by Smereka (1993) and by Sangani and Didwana (1993)
and theoretical work by van Wijngaarden (1993), Kok
(1993), and van Wijngaarden (2005) suggest that, when
assuming potential flow, rising bubbles form horizon
tally aligned rafts. Three dimensional direct numerical
simulations, which also solve the motion of the gas
liquid interface at the bubble's surface, have become
available in the last few years. The work by Bunner and
Tryggvason (2002, 2003) suggests that deformability ef
fect plays a crucial role for determining the orientation
of the clustering. For spherical nondeformable bubbles
these authors simulated up to 216 bubbles with Reynolds
numbers in the range of 12 30 and void fraction a up to
2' and Weber number of about 1. For the deformable
case, they simulated 27 bubbles with Reynolds number
of 26, Weber number of 3.7, and a = .'. The authors
found that the orientation of bubble clusters is strongly
influenced by the deformability of the bubbles: spher
ical pairs of bubbles have a high probability to align
horizontally, forming rafts, whereas the nonspherical
ones preferentially align in the vertical orientation. In
a later investigation, where inertial effects were more
pronounced, Esmaeeli and Tryggvason (2005) studied
both cases for bubble Reynolds number of order 100. In
this case only a weak vertical cluster was observed in a
swarm of 14 deformable bubbles. Their explanation for
the weaker vertical clustering was that the wobbly bub
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ble motion, enhanced by the larger Reynolds number,
produces perturbations which do not allow the bubbles
to align vertically and accelerate the break up in case
some of them cluster.
Comparing with numerous experimental studies on
pseudoturbulence, bubble clustering has not yet been
fully quantitatively analysed experimentally. Cartellier
and Rivibre (2001) studied the microstructure of ho
mogeneous bubbly flows for Reynolds number of order
10. They found a moderate horizontal accumulation us
ing pair density measurements with two optical probes.
Their results showed a higher probability of the pair den
sity in the horizontal plane and a reduced bubble density
behind the wake of a test bubble. Zenit et al. (2001)
found a mild horizontal clustering using 2D imaging
techniques for bubbles with particulate Reynolds num
ber higher than 100. Risso and Ellingsen (2002) per
formed experiments with a swarm of deformable bub
bles (db=2.5 mm), aspect ratio around 2, and Reynolds
number of 800. They did not find clustering and sug
gested that in this low void fraction regime (a < 1 11".'.)
there was a weak influence of hydrodynamic interaction
between bubbles.
This conference paper wants to study bubble cluster
ing in pseudoturbulence by means of novel experimen
tal techniques.
Experiment setup and measurement technique
Experiment setup
The experiments were performed in a vertical water
tunnel as shown in figure 1. The tunnel has a 2 meter
long measurement section with 0.45 x 0.45 m2 cross
section. The measurement section has three glass walls
for optical access and illumination (see Rensen et al.
(2005) for more dcl.iilo It was filled with deionized
water until the top of the measurement section. The
level of the liquid column was 3.8 m above the place
where bubbles were injected. We used 3 capillary is
lands in the lowest part of the channel to generate bub
bles. Each island contains 69 capillaries with an inner
diameter of 500 pm. A monodispersed bubbly swarm
with mean bubble diameter of 4 5 mm was studied.
Typical Reynolds numbers Re are of the order 103, the
Weber number We is in the range 23 (implying de
formable bubbles) and Eitvos number Eo around 3 4.
The mean bubble diameter d,, is within the range of 45
mm and show a slight increment with bubble concentra
tion.
Bubble concentrations were varied by controlling the
air flow through the capillary islands. We performed
experiments with dilute bubbly flows with typical void
O O 0
O o
Light 00ooo
00 0 60
00o Highspeed
Diffusive/ 0 cameras
plate
f O
0 t p
C 8ur pill A eS 0 0 ^ 0
Figure 1: A sketch of the experimental apparatus and
the setup for 3D PTV.
fractions in the range (' '. < a < 0.7 for PTV
measurements (Martinez et al. 2010). The void fraction
a was determined using an Utube manometer which
measures the pressure difference between two points
at different heights of the measurement section (see
Rensen et al. 2005). In this conference paper, we only
focus on the results of the lowest concentration, i.e. a =
(I 2' .
3D Particle Tracking Velocimetry
Recently, 3DParticle Tracking Velocimetry (PTV)
has become a powerful measurement technique in fluid
mechanics. The rapid development of highspeed imag
ing has enabled a successful implementation of the tech
nique in studies on turbulent motion of particles (e.g.
Mordant et al. 2004; Guala et al. 2005; Bourgoin et al.
2006; Berg et al. 2006; Volk et al. 2008; Toschi and Bo
denschatz 2009). The measured 3D spatial position of
particles and time trajectories allow for a Lagrangian
description which is the natural approach for transport
mechanisms.
Figure 1 also sketches the positions of the four high
speed cameras (Photron 1024PCI) which were used to
image the bubbly flow. The four cameras were viewing
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Figure 2: The reconstructed threedimensional bubble
positions.
from one side of the water channel and were focused in
its central region, at a height of 2.8 m above the capillary
islands. Lenses with 50 mm focal length were attached
to the cameras. We had a depth of field of 6 cm. The
image sampling frequency was 1000 Hz using a cam
era resolution of 1024x 1024 pixel2. The cameras were
triggered externally in order to achieve a fully synchro
nized acquisition. The image sequence was binarized af
ter subtracting a sequenceaveraged background. Based
on the instantaneous four images from the highspeed
cameras, threedimensional particle positions were re
constructed with PTV software developed at IfUETH,
as shown in figure 2. For a detailed description of this
3DPTV technique we refer to the work of Hoyer et al.
(2005) and references therein. For the the volume con
centration of a = (i _'., around Nb 190 bubbles
were detected in each image. We acquired 6400 images
per camera corresponding to 6.4 s of measurement (6.7
Gbyte image files).
Results
Pair correlation function
Particle clustering can be quantified using different
mathematical tools like pair correlation functions (Bun
ner and Tryggvason 2002), Lyapunov exponents (Bec
et al. 2006b), Minkowski functionals (Calzavarini et al.
2008a), or PDFs of the distance of two consecutive bub
bles in a timeseries (Calzavarini et al. 2008b). In this
investigation the pair correlation function G(r, 0) is em
ployed to understand how the bubbles are globally dis
tributed. It is defined as follows:
Figure 3: Radial pair probability G(r) as a function
of dimensionless radius r* for random dis
tributed particles and the measured bubble po
sitions.
G(r, ) Nb (rN
rij) (1)
where V is the size of the calibrated volume, Nb is the
number of bubbles within that volume, rij is the vec
tor linking the centers of bubble i and bubble j, and r is
a vector with magnitude r and orientation 0, defined as
the angle between the vertical unit vector and the vec
tor linking the centers of bubbles i and j. From (1), the
radial and angular pair probability functions can be de
rived. To obtain the radial pair probability distribution
function G(r) one must integrate over spherical shells
of radius r and width Ar, whereas for the angular pair
probability distribution function G(O) an rintegration is
performed.
Radial pair correlation
Firstly, we generated 500 randomly distributed par
ticles at 6000 time steps and calculated the radial cor
relation function as a function of the normalized radius
r* r/a, where a is a mean particle radius. As shown
with circles in Fig. 3, the pair correlation for the ran
dom distributed particles does not show any preferred
probability for r* > 2. We applied the same code on
the measured 3D bubble coordinates. The pair correla
tion function G(r) for the bubbles as a function of the
normalized radius r* = r/a is shown with diamonds in
figure 3. The mean equivalent bubble diameter is within
the range 45 mm. We normalize r with one mean bub
ble radius with a = 2 mm. We observe in figure 3 that
the highest probability to find a pair of bubbles lies in the
range of few bubble radii r* z 4 for all concentrations.
For values r* < 2 one would expect that G(r) 0.
However, in our experiments we found G(r) 4 0 for
r* < 2, due to the fact that the bubbles are ellipsoidal
and deform and wobble when rising.
Angular pair correlation
The orientation of the bubble clustering was studied
by means of the angular pair correlation G(O) using dif
ferent radii for the spherical sector over which neigh
boring bubbles are counted. We firstly calculated the
G(O) for the random distributed particles generated in
the previous section. The result of the randomly dis
tributed particles is shown with solid circles in figure 4.
It clearly shows that there is no any preferred orientation
for the random distributed particles. Figure 4 also shows
the results for the measured bubble positions, calculated
under different spherical shells of radii r*=40, 15, and
5, respectively. The plots were normalized so that the
area under the curve is unity. For all radii and concen
trations, pairs of bubbles cluster in the vertical direction,
as one can see from the highest peaks at 0/7 0 and
0/7 1. The value of 0/7 0 means that the refer
ence bubble (at which the spherical sector is centered)
rises below the pairing one. For 0/7 1 the reference
bubble rises above the pairing bubble. When decreas
ing the radius of the spherical sector, i.e. when prob
ing the short range interactions between the bubbles, we
observe that a peak of the angular probability near 72/2
starts to develop. The enhanced probability at this angle
range is even more pronounced for r* = 5, as shown
with squares in figure 4, where the peak of G(O) for hor
izontally aligned bubbles is just slightly lower than that
for vertical clustering. It is worthwhile to point out that
the vertical alignment of the bubbles is very robust and
is present from very large to small scales, as the angu
lar correlation for different spherical sectors is always
higher at values 0/7=0 and 1 than at value 0/7=0.5.
For comparison, we consider again the work of Bun
ner and Tryggvason (2003), who found that pairs of bub
bles have a higher probability to align vertically, though
for a much higher concentration (a = .' .) than em
ployed here. Bunner and Tryggvason (2003) found that
the vertical alignment was not as robust as in our case,
since with increasing r* the angular correlation at 0 and
7r became less dominant. Another Nigniik.illI difference
between the findings of our experimental work and their
simulations is that horizontal alignment was more pro
nounced with larger radii of the spherical sector and not
when decreasing r*. Our experimental results clearly
show the main drawback and today's limitation when
solving the flow at the particle's interface: the simula
tions are still restricted to a small number of particles,
which is not sufficient to reveal long range correlations.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Interpretation of the clustering
What is the physical explanation for a preferred verti
cal alignment of pairs of bubbles in pseudoturbulence?
Through potential flow theory, the mutual attraction of
rising bubbles can be predicted (Batchelor 1967), the ap
plication of potential theory to our experiments remains
questionable (van Wijngaarden 1993), as we are in a sta
tistically stationary situation where bubbles have already
created vorticity. Our findings are consistent with the
idea that deformability effects and the inversion of the
lift force acting on the bubbles are closely related to the
clustering. Mazzitelli et al. (2003) showed numerically
that it was mainly the lift force acting on pointlike bub
bles that makes them drift to the downflow side of a vor
tex in the bubble wake1. Furthermore, when accounting
for surface phenomena, Ervin and Tryggvason (1997)
showed that the sign of the lift force inverts for the case
of deformable bubbles in shear flow so that a trailing
bubble is pulled into the wake of a heading bubble rather
than expelled from it. In such a manner vertical rafts
can be formed. Experimentally some evidence of the lift
force inversion has been observed by Tomiyama et al.
(2002) as lateral migration of bubbles under Poiseuille
and Couette flow changed once the bubble size has be
come large enough. Numerical simulations of swarm of
deformable bubbles without any flow predicted a verti
cal alignment (Bunner and Tryggvason 2003). An al
ternative interpretation of the results, due to Shu Takagi
(private communication (2009)) goes as follows: small,
pointwise, spherical bubbles have a small wake, allow
ing for the application of potential flow. The bubbles
then horizontally attract, leading to horizontal cluster
ing. In contrast, large bubbles with their pronounced
wake entrain neighboring bubbles in their wake due to
the smaller pressure present in those flow regions, lead
ing to vertical clustering. Further efforts are needed to
identify and confirm the main mechanismi.e., either
lift or pressure reduction in the bubble wake leading
to a preferential vertical alignment, for example through
experiments with small, spherical, nondeformable bub
bles as achieved by Takagi et al. (2008) through surfac
tants or with buoyant spherical particles.
Conclusion
We performed experiments on bubble clustering using
3DPTV in pseudoturbulence at a very dilute regime.
Bubble positions were determined to study bubble clus
tering and alignment. For that purpose the pair correla
tion function G(r, 0) was calculated. As the radial cor
relation G(r) shows, pairs of bubbles cluster within few
1See figure 2 in Mazzitelli et al. (2003) sketching the dynamics.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
2.00
1.50
1.00
0.50
0.0
0.4
0.8
Figure 4: Normalized angular pair probability G(O) as a function of angular position 0/7 for random distributed
artificial particles (black solid circles) and for three different bubblepair distances: r* = 40 (triangles),
r* = 15 (diamonds), and r* = 5 (squares).
bubble radii 2.5 < r* < 4. The angular pair correlation
G(O) shows that a robust vertical alignment is present at
both small and large scales, as it is observed when vary
ing the radius of the spherical sector (r*=40, 15, and
5). Decreasing the radius of the spherical sector shows
that horizontal clustering also occurs, as the peak of the
angular correlation around 7/2 starts to grow with de
creasing values of r*.
Acknowledgements
We thank specially GertWim Bruggert, Martin Bos, and
Bas Benschop for their invaluable help in the experi
mental apparatus. We thank: Lorenzo del Castello, Beat
Liuthi, and Haitao Xu, Fr6d6ric Risso, Veronique Roig,
Roberto Zenit, Shu Takagi, Yoichiro Matsumoto, Bert
Vreman for their stimulating discussions. This research
is part of the Industrial Partnership Programme: Funda
mentals of heterogeneous bubbly flows which is funded
by the Stichting voor Fundamenteel Onderzoek der Ma
terie (FOM).
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