Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 12.1.2 - On bubble clustering and energy spectra in pseudo-turbulence
Full Citation
Permanent Link:
 Material Information
Title: 12.1.2 - On bubble clustering and energy spectra in pseudo-turbulence Bubbly Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Sun, C.
Martínez, J.
Chehata, D.
van Gils, D.P.M.
Lohse, D.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
Subject: bubbly flow
bubble clustering
energy spectrum
Abstract: We performed 3D-Particle Tracking (3D-PTV) in pseudo-turbulence—i.e., flow solely driven by rising bubbles—to investigate bubble clustering at a very dilute gas concentration. To characterize the clustering the pair correlation function G(r; ) 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).
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
 Record Information
Bibliographic ID: UF00102023
Volume ID: VID00298
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 1212-Sun-ICMF2010.pdf

Full Text

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

On bubble clustering in pseudo-turbulence

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;
Keywords: Bubbly flow; pseudo-turbulence; bubble clustering; energy spectrum


We performed 3D-Particle Tracking (3D-PTV) in pseudo-turbulence-i.e., flow solely driven by rising bubbles-to
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).


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-
turbulence-i.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 scaling-up indus-
trial devices and for optimization and performance pre-
diction. This article wants to provide experimental data
on pseudo-turbulence by means of novel experimental
techniques. The issue to be addressed is

What is the preferential range and the orientation of
bubble clustering in pseudo-turbulence ?

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 pseudo-turbulence
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 non-deformable 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 non-spherical
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
pseudo-turbulence, 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 pseudo-turbulence 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 mono-dispersed 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 2-3 (implying de-
formable bubbles) and Eitvos number Eo around 3 4.
The mean bubble diameter d,, is within the range of 4-5
mm and show a slight increment with bubble concentra-
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 High-speed

Diffusive/ 0 cameras

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 U-tube 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, 3D-Particle Tracking Velocimetry (PTV)
has become a powerful measurement technique in fluid
mechanics. The rapid development of high-speed 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
Figure 1 also sketches the positions of the four high-
speed cameras (Photron 1024-PCI) 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 three-dimensional bubble

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 sequence-averaged background. Based
on the instantaneous four images from the high-speed
cameras, three-dimensional particle positions were re-
constructed with PTV software developed at IfU-ETH,
as shown in figure 2. For a detailed description of this
3D-PTV 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).


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 time-series (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-

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 r-integration is

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 4-5 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 pseudo-turbulence?
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 point-like 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 mechanism-i.e., either
lift or pressure reduction in the bubble wake- leading
to a preferential vertical alignment, for example through
experiments with small, spherical, non-deformable bub-
bles as achieved by Takagi et al. (2008) through surfac-
tants or with buoyant spherical particles.


We performed experiments on bubble clustering using
3D-PTV in pseudo-turbulence 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








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 bubble-pair 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*.


We thank specially Gert-Wim 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).


Ayyalasomayajula, S., Gylfason, A., Collins, L. R., Bo-
denschatz, E., and Warhaft Z. Lagrangian measurements

of inertial particle accelerations in grid generated wind
tunnel turbulence. Phys. Rev. Lett., 97,144507 (2006).

Batchelor, G. K. An introduction of fluid dynamics.
Cambridge Univesity Press (1967).

Bec, J., Biferale, L., Boffetta, G., Celani, A., Cencini,
M., Lanotte, A., Musacchio, S., and Toschi, F. Acceler-
ations statistics of heavy particles in turbulence. J. Fluid
Mech., 550,349-358 (2006a).

Bec, J., Biferale, L., Boffetta, G., Cencini, M., Musac-
chio, S., and Toschi, F. Lyapunov exponents of
heavy particles in turbulence. Phys. Fluids, 18,091702

Berg, J., Liithi, B., Mann, J., and Ott, S. Backwards
and fordwards relative dispersion in turbulent flow: An
experimental investigation. Phys. Rev. E, 74,016304,

Bourgoin, M., Ouellette, N.T., Xu, H., Berg, T., and Bo-
denschatz, E. The role of pair dispersion in turbulent
flow. Science, 311,835-838, (2006).

Bunner, B. and Tryggvason, G. Dynamics of homoge-
neous bubbly flows part 1. rise velocity and microstruc-
ture of the bubbles. J. Fluid Mech., 466,17-52, (2002).

-*- Random particles
- r*=40

-- r*= 15
E- r*= 5

Bunner, B. and Tryggvason, G. Effect of bubble defor-
mation on the properties of bubbly flows. J. Fluid Mech.,
495,77-118, (2003).

Calzavarini, E., Kerscher, M., Lohse, D., and Toschi, F.
Dimensionality and morphology of particle and bubble
clusters in turbulent flow. J. Fluid Mech., 607,13-24,

Calzavarini, E., van der Berg, T.H., Toschi, F., and
Lohse, D. Quantifying microbubble clustering in turbu-
lent flow from single-point measurements. Phys. Fluids,
20,040702, (2008b).

Cartellier, A. and Rivibre, N. Bubble-induced agitation
and microstructure in uniform bubbly flows at small to
moderate particle reynolds number. Phys. Fluids, 13,8,

Deckwer, B.D. Bubble column reactors. Wiley, first
edition (1992).

Ervin, E.A. and Tryggvason, G. The rise of bubbles
in a vertical shear flow. J. Fluid Engng., 119,443-449,

Esmaeeli, A. and Tryggvason, G. A direct numerical
simulation study of the buoyant rise of bubbles at o(100)
reynolds number. Phys. Fluids, 17, 093303, (2005).

Guala, M., Liberzon, A., Liuthi, B., Tsinober, A., and
Kinzelbach, W. On the evolution of material lines and
vorticity in homogeneous turbulence. Phys. Rev. E, 533,
339-359, (2005).

Hoyer, K., Holzner, M., Luethi, B., Guala, M., Liberzon,
A., and Kinzelbach, W. 3d scanning particle tracking
velocimetry. Exps. Fluids, 39, 923-934, (2005).

Kok, J.B.W. Dynamics of a pair of bubbles moving
through liquid. 1 theory. Eur. J. Mech. B, 12,515-540,

Lance, M. and Bataille, J. Turbulence in the liquid phase
of a uniform bubbly water-air flow. J. Fluid Mech., 222,
95-118, (1991).

Martinez, J., and Chehata, D. and van Gils, D., and
Sun, C. and Lohse, D. On bubble clustering and energy
spectra in pseudo-turbulence. J. Fluid Mech. (in press),

Mazzitelli, I. M., Lohse, D., and Toschi, F. The effect of
microbubbles on developed turbulence. Phys. of Fluids,
15,L5-L8, (2003).

Mazzitelli, I.M. and Lohse, D. Evolution of energy
in flow driven by rising bubbles. Phys. Rev. E, 79(6),
066317, (2009).

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

Mordant, N., Leveque, E., and Pinton, J.F. Experimental
and numerical study of the lagrangian dynamics of high
reynolds turbulence. New J. Phys., 6(116), (2004).

Rensen, J., Luther, S., and Lohse, D. The effects of bub-
bles on developed turbulence. J. Fluid Mech., 538,153-
187, (2005).

Risso, F. and Ellingsen, K. Velocity fluctuations in a ho-
mogeneous dilute dispersion of high-reynolds-number
rising bubbles. J. Fluid Mech., 453,395-410, (2002).

Roig, V. and de Tournemine, L. Measurement of in-
terstitial velocity of homogeneous bubbly flows at low
to moderate void fraction. J. Fluid Mech., 572,87-110,

Salazar, J.P.L.C., de Jong, J., Cao, L., Woodward, S. H.,
Meng, H. and Collins, L.R. Experimental and numerical
investigation of inertial particle clustering in isotropic
turbulence. Journal of Fluid Mechanics, 600,245-256,

Sangani, A. S., and Didwana, A. K. Dynamic simula-
tions of flows of bubbly liquids at large reynolds num-
bers. J. Fluid Mech., 250,307-337, (1993).

Saw, E.W., Shaw, R.A., Ayyalasomayajula, S., Chuang,
P.Y., and Gylfason, A. Inertial clustering of particles in
high-reynolds-number turbulence. Phys. Rev. Lett., 100,
214501, (2008).

Smereka, A. S. On the motion of bubbles in a periodic
box. J. Fluid Mech., 254,79-112, (1993).

Sugiyama, K., Takagi, S., and Matsumoto, Y. Multi-
scale analysis of bubbly flows. Comp. Meth. in App.
Mech. and Eng., 191(6-7),689 704, (2001).

Takagi, S., Ogasawara, T., and Matsumoto, Y. The ef-
fects of surfactant on the multiscale structure of bubbly
flows. Phil. Trans. R. Soc. A, 366(1873),2117-2129,

Tomiyama, A., Tamai, H., Zun, I., and Hosokawa, S.
Transverse migration of single bubbles in simple shear
flows. Chem. Eng. Science, 57(11),1849 1858, (2002).

Toschi, F and Bodenschatz, E. Lagrangian properties
of particles in turbulence. Annu. Rev. Fluid Mech., 41,
375-404, (2009).

van Wijngaarden, L. The mean rise velocity of pairwise-
interacting bubbles in liquid. J. Fluid Mech., 251,55-78,

van Wijngaarden, L. Bubble velocities induced by trail-
ing vortices behind neighbours. J. Fluid Mech., 541,
203-229, (2005).

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

Volk, R., Calzavarini, E., Verhille, G., Lohse, D., Mor-
dant, N., Pinton, J.F., and Toschi, F. Acceleration of
heavy and light particles in turbulence: Comparison
between experiments and direct numerical simulations.
Phys. D, 237(14-17),2084 2089, (2008).

Zenit, R., Koch, D.L., and Sangani, A. S. Measurements
of the average properties of a suspension of bubbles ris-
ing in a vertical channel. J. Fluid Mech., 429,307-342,

University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - Version 2.9.7 - mvs