Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 9.1.1 - Scaling laws for bubble-induced agitation at high Reynolds number
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 Material Information
Title: 9.1.1 - Scaling laws for bubble-induced agitation at high Reynolds number Bubbly Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Riboux, G.
Risso, F.
Legendre, D.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: bubbly flow
agitation
velocity statistics
self-similarity
spectra
 Notes
Abstract: A homogeneous swarm of air bubbles rising in water at rest have been investigated experimentally. Three different bubble diameters (d =1.6, 2.1 and 2.5 mm) and moderate gas volume fractions (0.005 α 0.1) have been considered. Liquid fluctuations have been measured by LDA within the bubble swarm and by PIV behind it. The probability density functions of the liquid fluctuations adopt a self-similar behaviour when the gas volume fraction is varied, with a characteristic velocity u scaling as Vo α0.4. The spectra of horizontal and vertical fluctuations exhibit a -3 power law which is followed at smaller scales by a classic −5/3 power law. The integral length scale is found to be proportional to V 2 0 /g – or equivalently to d/Cdo – where g is the gravity acceleration and Cdo the drag coefficient of a single rising bubble. Provided they are normalized by using and the velocity variance, the spectra are independent on both the bubble diameter d and the volume fraction α. Although the Kolmogorov microscale is smaller than the measurement resolution of the spectra, the dissipation rate ǫ has been obtained from the decay of the kinetic energy after the passage of the bubbles and found to scale as α−0.3u3/ . The major characteristics of the agitation, the velocity scale u, the length scale and the dissipation rate ǫ have thus been expressed as functions of the characteristics of a single rising bubble. We have also performed large-scale numerical simulations by modeling the bubbles by volume-forces. The good agreement between simulations and experiments confirms that the bubble induced turbulence is mainly controlled by interactions between bubble wakes.
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: VID00214
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 911-Riboux-ICMF2010.pdf

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


Scaling laws for bubble-induced agitation at high Reynolds number


G. Riboux*t, F. Risso* & D. Legendre*

Institute de M6canique des Fluides de Toulouse, Universit6 de Toulouse (INPT, UPS) and CNRS, France
t present address: Escuela Superior de Ingenieros, Universidad de Sevilla, Espafia

griboux@ us.es, risso@imft.fr & legendre@imft.fr
Keywords: Bubbly flow, agitation, velocity statistics, self-similarity, spectra.




Abstract

A homogeneous swarm of air bubbles rising in water at rest have been investigated experimentally. Three different bubble
diameters (d =1.6, 2.1 and 2.5 mm) and moderate gas volume fractions (0.005 < a < 0.1) have been considered. Liquid
fluctuations have been measured by LDA within the bubble swarm and by PIV behind it. The probability density functions of the
liquid fluctuations adopt a self-similar behaviour when the gas volume fraction is varied, with a characteristic velocity u scaling
as Vo c0.4. The spectra of horizontal and vertical fluctuations exhibit a -3 power law which is followed at smaller scales by a
classic -5/3 power law. The integral length scale A is found to be proportional to VoJ/g -or equivalently to d/Cd where g is
the gravity acceleration and Cdo the drag coefficient of a single rising bubble. Provided they are normalized by using A and the
velocity variance, the spectra are independent on both the bubble diameter d and the volume fraction a. Although the Kolmogorov
microscale is smaller than the measurement resolution of the spectra, the dissipation rate e has been obtained from the decay of the
kinetic energy after the passage of the bubbles and found to scale as a- '/A. The major characteristics of the agitation, the
velocity scale u, the length scale A and the dissipation rate c have thus been expressed as functions of the characteristics of a single
rising bubble. We have also performed large-scale numerical simulations by modeling the bubbles by volume-forces. The good
agreement between simulations and experiments confirms that the bubble induced turbulence is mainly controlled by interactions
between bubble wakes.


Introduction

In many practical situations, bubbles are dispersed in
a continuous liquid phase: pipe flow for oil transport,
bubble columns for chemical processing, vapour gener-
ators for energy production. Due to the large density
difference between gases and liquids, the bubbles do not
move at the same velocity as the liquid. The bubbles thus
cause velocity disturbances to the liquid that collectively
generate a complex agitation. This agitation is different
from the single-phase shear-induced turbulence because
it is strongly related to the random spatial distribution
of the bubbles. To characterize this so-called pseudo-
turbulence, we have decided to investigate a bubble col-
umn, wherein bubbles are injected at the bottom of a
tank filled with a liquid initially at rest. This configura-
tion is particularly interesting because it allows to study
the structure of the pseudo-turbulence for various bubble
diameters d and gas volume fractions a.
In the context of bubbly flows, previous works have
investigated a swarm of rising bubbles at high Reynolds
number Re in the presence or the absence of a mean liq-


uid flow. Some of these studies have permitted to shed
light about the bubble-induced agitation. In particular,
Risso and Ellingsen (2002) showed for 0.005 < a <
0.01, d 2.5 mm and Re 800, that the probabil-
ity density functions (PDF) of both the horizontal and
vertical liquid fluctuations exhibited a self-similar be-
haviour when the velocity was normalised by V a0.4,
where Vo is the average velocity of the single rising bub-
ble. Another important feature of the liquid agitation is
the spectral distribution of the fluctuating energy with
the wavenumber k; for bubbles of diameter d=5 mm ris-
ing in a uniform upwards liquid flow at (U)=0.9 m/s,
Lance and Bataille (1991) were the first to find a spec-
tral density of energy evolving as k 8/3 and to claim that
it was the signature of the bubble-induced turbulence.

A goal of this study is to check whether these prop-
erties are still valid for other bubble diameters and for a
larger range of gas volume fractions. Another objective
is to obtain a complete description of all the scales of the
liquid fluctuations. The originality of this investigation
is to consider both the steady flow within the homoge-
neous bubble swarm but also the decaying agitation af-







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


Z
[21] hI sector set
x
pIv








Water


Figure 1: Experimental set-up.


ter the air injection has been stopped. This second flow
configuration will permit to determine the main charac-
teristics of the bubble-induced agitation. Finally, we will
present a large-scale numerical simulations which repro-
duce the principal characteristics of the agitation that are
observed experimentally.


Characteristic scales of the bubble-induced
agitation.

The investigation of the homogeneous swarm of rising
bubbles has been carried out in a stable bubble column.
It has allowed us to determine the principal scaling laws
of the pseudo-turbulence: (1) the liquid velocity scale,
(2) the dissipation rate and (3) the length scale of the
bubble-induced agitation.
The experimental set-up is depicted in figure 1. The
test section [1] is an open tank of 1000 mm height with
a squared cross-section of 150 mm width. To allow full
optical access the four sides are of a glass construction.
The tank is filled with filtered tap water and air bubbles
are injected at the bottom through a removable set of
injectors [2]. An injector set is constituted of a regu-
lar array of steel capillary tubes that open into a pres-
surized air chamber [3]. Capillaries of three different
inner diameters, dc=0.1, 0.2 and 0.4 mm, are used to
vary the bubble size (Table 1). With a single injector
connected to pressurized air tank [4], we determined by
means of high-speed imaging the bubble characteristics
of for each class of capillary: the diameter d of the bub-
bles, their average vertical velocity Vo and the angle 0
corresponding to the maximum inclination of the bubble


Set 1 2 4
dc (m) 0.1 0.2 0.4
d (mm) 1.6 2.1 2.5
V (mm.s 1) 335 320 305
Re(-) 540 670 760
We (-) 2.6 3.1 3.4
Cdo (-) 0.19 0.27 0.35
0 (0) 0 25 25


Table 1: Characteristics of single rising bubbles for the
three classes of capillaries.


velocity about the vertical direction. Thus, the Reynolds
number, Re pdVo/l (p and p are the liquid den-
sity and viscosity), the Weber number We = pd VO/c
(a is the interfacial tension) and the drag coefficient
Cdo 4 dg/(3 V0) are determined for the reference
case of an isolated bubble (Table 1).
1. Velocity scale. We generated a homogeneous
swarm of rising bubbles by inserting an injector set
into the set-up and connecting the chamber [3] to the
pressure-controlled air tank [4]. Using a large number
of capillary tubes, we generate a stable and uniform
swarm of bubbles for volume fractions up to several
tens of percent. Then, we measure the vertical liquid
velocity inside the bubble swarm by means of a two-
component Laser Doppler Anemometer (LDA) operated
in forward- scattering mode (for details, see Riboux et al.
2010). Figure 2 shows the normalized PDFs of the verti-
cal component of the liquid fluctuations for various vol-
ume fractions and three bubble diameters; the liquid ve-
locity fluctuations are normalized by the rise velocity Vo
of an isolated bubble. The vertical fluctuations are not
isotropic, large upward fluctuations being more proba-
ble. The PDFs are not Gaussian and decay exponentially
as the fluctuations increase. For all the gas volume frac-
tions, figure 2 also show that the PDFs are self-similar
provided the velocity fluctuations are normalized by


u = Vo a04.


This scaling was first found by Risso and Ellingsen
(2002) for a single bubble size (d=2.5 mm) and a nar-
row range of gas volume fractions (0.005 < a < 0.01).
This self-similar behavior of the liquid fluctuations and
the scaling law (1) are confirmed here for three differ-
ent diameters and for volume fractions up to 0.05. In
consequence, the PDFs of the velocity fluctuations pre-
serve the same shape as the gas volume fraction is in-
creased and the knowledge of the evolution of the stan-
dard deviation u as function of the gas volume fraction
is sufficient to describe the liquid agitation induced by
the bubbles swarm. In order to check the validity of this












self-similarity behavior for larger gas volume fractions,
we also determined the PDFs of the vertical fluctuations
from the data of Lame de Tournemine (2001) who in-
vestigated bubbles rising in a channel flow with a mean
liquid velocity ranging from 0.45 to 0.59 ms 1. In fig-
ure 2a, these results are represented by black dash lines
and correspond to d=1.75 mm and a from 0.015 to 0.14.
These normalized PDFs match very well our results for
bubbles of 1.6 mm. The scaling Vo a" is thus proved to
be valid up to a = 0.14 with an exponent n = 0.40.02.


(a) s,


c=O 67 %
--a=1 13%
--c-=1 41 %


-- =1 94
-ac=26%


5 3 1 0 01 03
05 -03 -01 0 01 03 0


ca=O 50 %
--c=0 62 %
--c-=1 21 %


=O 54 %
--ca=O 91 %
--a=1 15%


Figure 2: Experimental normalized PDFs of the vertical
liquid velocity fluctuations: (a), d=1.6 mm;
(b), d=2.1 mm; (c), d=2.5 mm (ao=0.01).
(a), (--), d=1.75 mm, Lame de Tournemine
(2001).

2. Dissipation scale. A major objective of the present
investigation was to determine the characteristic length
scales of the liquid agitation. Because PIV measure-
ments in the bubble swarm are strongly disturbed by
the passages of the bubbles across the laser sheet, we


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


decided to measure the velocity field after the passage
of the bubble swarm. The experimental set-up includes
two air lines [3]-[4] and [3]-[5], connected to two dif-
ferent air tanks. Electro-valves [6] are used to switch
the air supply from an air line to the other, which al-
lows to suddenly stop the bubble injection without hav-
ing water flowing out by the capillaries. The PIV sys-
tem consists of a 10-mJ Yag laser (Nc\\ \\ c, Pegasus),
which generates a vertical sheet of light of wavelength
527 nm with a thickness of about 0.8 mm. A high-speed
video camera (Photron ultima APX) synchronized with
the laser pulses acquires digital images of 1024 x 1024
pixels (square of 5.12 cm width) at a frame rate of
200 Hz. The measurement window is located 400 mm
above the injectors. The air injection is stopped and at
the same moment the PIV acquisition is started. The
result is a sequence of images showing first the rising
swarm and then the wake behind it. The velocity field
is calculated between each pair of consecutive images
by means of the PIVis code developed in our laboratory
(Cid and Gardelle 2005), which is based on a classic it-
erative algorithm including sub-pixel image shift and de-
formation. The PIV parameters are adjusted to optimize
the computation of the velocity field in the absence of
the bubbles. Statistical quantities (spatial spectrum and
standard deviations of the velocity fluctuations) are com-
puted by averaging over the whole measurement win-
dow wherein the flow is almost homogeneous.
Figure 3 shows the time evolution of the standard de-
viation, u, of the velocity fluctuations for the three di-
ameters and various volume fractions. At the beginning
of the record, u is noisy because bubbles are still present
in the measurement window; it is however clearly visi-
ble that the energy of the fluctuations is constant within
the bubble swarm. Behind the bubbles, the energy of the
fluctuations decays as the bubble swarm rises up away
from the measurement window. In order to reveal the
decay of the energy behind the bubble swarm, figure 3
shows a logarithmic plot of u where the time has been
shifted: t* = t t,, with t=-0.4 s for d=1.6 and
2.5 mm, t,=-0.5 s for d=2.1 mm. For all bubble diam-
eters and gas volume fractions, the fluctuating velocity
decays as t* 1 during the first regime that occurs behind
the bubble swarm. Provided they are normalized by their
value at a given instant, say t* to t,, the standard
deviations write u/u t* /t*.
Figure 4 shows the vertical spectra, S, of the liquid
velocity fluctuations measured just behind the bubble
swarm at t=0.01 s. In this region where there is no bub-
ble, the flow disturbances that exist in the vicinity of
the bubbles are not present and the agitation only results
from the interactions of the bubble wakes. For the range
of wavelengths investigated, Riboux et al. (2010) have
shown that the agitationjust after the passage of the bub-


m


%




































Figure 3: Time evolutions of the standard deviations of
the experimental liquid velocity fluctuations.
The instant to corresponds to the first image
without bubble.


-3



a= 44%, d=1 6m 6
-- a=l 32%,d=1 6mm I
--a=380%,d=16mm Id
-- =12 1%, d=1 6mm |
=046%,d=2 1mm I
-- =1 09%, Zd=21m
--a=1 11%, d=2 1mm f -5/3
-- =3 12%, d=21mm I
-- =5 30%, d=2 1mm
-- c=7 96%, d=2 1mm
=0 46%, d=2 5mm I I
--a=1 00%, d=2 5mmn
-- =2 45%, d=2 5m m I
--a=6 97%, d=2 5mrn I
102 103
A-1(ml 1)


Figure 4: Vertical spectra, S, of the experimental liquid
velocity fluctuations normalized by their vari-
ance for various bubble diameters d and gas
volume fractions a.


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


ble swarm (t < 0.15 s) preserves the same structure. For
the three bubble diameters and the range of volume frac-
tions investigated, the large scales of all the spectra col-
lapse on a single master curve, provided the spectra are
normalized by the variance of the fluctuations. The large
scales of the bubble-induced agitation appear therefore
to be independent of both d and a. The estimation of
the integral scale A by extrapolating the spectrum at the
origin leads to a value of the order of 15 mm, indicating
that most of the energy belongs to wavelengths A that are
much larger than the bubble diameters. From the obser-
vation of the spectrum, three different ranges of wave-
lengths can be distinguished. At the largest scales, the
spectrum shows no particular feature. For scales ranging
between Lc z 7.7 mm and lc z 2 mm, the spectral den-
sity of energy follows a A3 power law. Because the upper
and lower boundary, L, and l, are also independent on
both d and a, they are proportional to the integral length
scale A. Then, for A < lc, the A3 power law is replaced
by a A5/3 power law, which probably corresponds to a
classic Kolmogorov inertial subrange.
The dissipation rate c can be expressed by consider-
ing that behind the bubble swarm, the energy production
vanishes and the kinetic energy of the fluctuations de-
cays according to d(u2/2)/dt -c. Since u decreases
as uo t*/t*, one finds c u3/Le, with L, = Ut* is in-
dependent of time. Figure 5a shows the length scale Le
normalized by L, as a function of the gas volume frac-
tion for the three bubble diameters. The ratio Le/Lc,
which is proportional to Le/A, increases with the gas
volume fraction as a 3. Finally, combining this result
with eq. (1), the dissipation rate can be expressed as a
function of the volume fraction and the integral length
scale (found to be independent of a):

vo
E oc ac (2)
A

Because the spectrum follows a A5/3 power law at the
smallest resolved wavelengths, we estimated the Kol-
mogorov microscale from the dissipation rate at t = 0
by using the expression derived for isotropic turbulence:
no = (v,/co)1/4. Figure 5b shows no/LL as a function
of the gas volume fraction and for the three bubble diam-
eters: it decreases from 2.4 to 1.4 broadly as a 0.6. For
the cases investigated here, we may conclude that there
is two decades between the integral scale and the cut-off
wavelength of the spectrum (close to 0.1 mm).
3. Integral length scale. To achieve the character-
ization of the induced-bubble agitation, we need to de-
termine the expression of the integral length scale. Con-
sidering that, in a steady bubble swarm, the dissipation
rate has to be balanced by the work of the buoyancy
force, it yields: e = ag(V). For 0.005 < a < 0.1
and 1.6 < d < 2.5 mm, Riboux et al. (2010) showed







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


S'


> &1 6 mm
* 21 mm
S


I 1-2-
10


S-1 6 mm
* 21 mm
* S = ..


a (%)


Figure 5: Experimental length scales related to the dis-
sipation rate c as functions of the gas volume
fraction.


that the mean vertical bubble velocity evolves as (V) oc
Vo a-0 o1. Injecting this result in the expression of c, we
obtain: c oc gVo ao 9. Equating this expression with eq.
(2) leads to an expression of A that is independent on a,

V2 d
A oc oc (3)
g Cdo

where Cdo 4 g d / 3 V is the drag coefficient for a
single rising bubble. This expression is consistent with
the fact that the large scales of the spectra are indepen-
dent of the volume fraction. Figure 6 compares the spec-
tra of the present work to those obtained by Lance and
Bataille (1991) and Lame de Toumemine (2001) after
normalization by using d/Cdo as characteristic length
scale. For the present results (1.6 < d < 2.5 mm) and
those of Lare de Toumemine (2001) (d = 1.75 mm),
d/Cdo 8.6 mm; in the case of Lance and Bataille
(1991) (d 5 mm), d/Cdo 4.4 mm, which is twice
smaller although the bubbles are about twice larger.
With this normalization, all the spectra match well at
large scales and, in particular, the beginning of the A3
range is observed for the same wavelength, Lc z d/Cdo
for all bubble sizes. First, the agreement with the re-
sults of Lare de Toumemine (2001) permits to conclude
that the large scales are dominated by the wake interac-
tions and that the flow in the vicinity of each bubbles
has not significant influences on the agitation at these
scales. Second, the agreement with the results of Lance


10 10 10-
(d/Co\) 1


Figure 6: Experimental spectra normalized by using the
variance and the characteristic length scale
d/Cdo.


and Bataille (1991), which were obtained for larger bub-
bles, confirms the validity of the scaling (3) for the inte-
gral length scale A of the pseudo-turbulence.


Large-scale numerical simulations of the
bubble-induced agitation

The experimental results suggest that the fluctuations of
the liquid are mainly controlled by interactions between
individual bubble wakes and that they depend weakly on
either the flow disturbances that exist in the vicinity of
the bubbles or on the fluctuations of the bubble veloc-
ity. This gave us the idea to model the action exerted by
each bubble on the liquid by introducing a force F in the
Navier-Stokes equations that are solved numerically:


{ = tan(O) F;o cos()) sin(27f t + y),
F, = tan(0) Fo sin() sin(27f t + y),
F F= 0.


This source of momentum is distributed over eight ele-
mentary volumes of the computational domain. The ver-
tical component F, stands for the drag and the two oscil-
lating horizontal components F, and Fy account for the
zigzag motion of the bubbles that is observed in the ex-
periments. Since only large scales are expected to play a
significant role, the mesh grid spacing is chosen to be of
the same order as the bubble size. The bubbles are fixed
and a uniform flow of liquid is imposed at the top of the
domain. Preliminary simulations have been carried out
for adjusting the magnitudes Fho and tan(0) Fo in or-
der to reproduce well the wake of a single bubble. The
liquid flow in a rising bubble swarm is then computed by








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


inserting n=6 ac2/(7r d3) momentum sources in a com-
putational domain of volume 2. The bubbles are fixed
and a uniform velocity -V is imposed at the top of the
domain, which is periodic in the two horizontal direc-
tions. The bubble locations are randomly distributed
over the computational domain. The angles p and y,
which respectively define the direction of the zigzag in
the (x,y) plane and the time phase of the corresponding
oscillatory forcing are also randomly chosen for each
bubble.


(a) 16

14
12
10
8-
6
4-
2
0 5-
-05
(b) 12


-c=0 6 %
-- =1 0%
-- =1 5%
-- =2 4 %
-- =29 %
-- c=3 4 %
=3 9 %
- -Exp




-03 -01 0 01 03 05

S =-O 6 %
a 06%
--a=1 0%
--a=1 5%
--=2 4 %
-=3 9 %
---Exp




0 0


-03 -01 0 01 03 05


a= 06%
-al 10%
--al 15%
-a a24 %
-a a39 %


5 -03 -01 0 01 03
(U/Vo)(a/ 0)-0 4


Figure 7: Simulated normalized PDFs of the vertical
liquid fluctuations compared with the exper-
imental results from figure 2. (a) d=1.6 mm;
(b) d=2.1 mm and (c) d=2.5 mm.


The JADIM code (Calmet (1995), Legendre (1996))
is used to solve numerically the Navier-Stokes equa-
tions. A 64x64x 125 mesh grid is used with a spac-
ing 6=d/1.5 in the vertical direction and 6/2 in the hor-
izontal ones. The numerical simulations depend on four
dimensionless physical parameters: (i) the drag coef-


:-a=06%
--ao=10%
--a=15%



10





10



10 102 10
A-1 (m1)


Figure 8: Simulated spectra of the vertical velocity for
various gas volume fractions a. (- -), exper-
imental spectrum from figure 4 (d=2.5 mm).



ficient, Cd=2 Fo/(p V2 S), that fixes the value of the
average vertical force Fo; (ii) the Reynolds number,
Re=p V/v, that determines the role of the fluid viscos-
ity; (iii) the Strouhal number, St=f d/V, that fixes the
frequency of the horizontal oscillations; (iv) the angle 0
that fixes the amplitude of the horizontal oscillations.
We have simulated numerically the liquid fluctuations
generated by a bubble swarm for the three experimental
bubble diameters: d=1.6, 2.1, 2.5 mm. The parameters
Re, Cd (see Tab. 1) and St=0.05 have the same values
as in the experiments. Preliminary tests (not presented
here) have shown that the vertical fluctuations do not
depend significantly on the magnitude of the horizontal
forcing, which only influence the intensity of horizontal
fluctuations (see details in Riboux et al. 2007). All the
following results have been obtained with 0=50 and are
presented in the laboratory frame where the bubbles are
rising.
Figure 7 shows the simulated PDFs of the vertical
velocity fluctuations normalized by Vo a0.4. When a
is increased, the simulated PDFs tend toward the self-
similar state observed experimentally: the shape of the
PDFs is asymmetric and the magnitude of the fluctua-
tions is similar. The only noticeable difference is that
a gas volume fraction larger than 1 % is necessary to
reach this asymptotic state while it is already attained
for a=0.5 % in the experiments. These results show that
large-scale simulations in which bubbles are modeled by
fixed volume-forces are capable to reproduce the correct
statistics of the liquid fluctuations generated in a rising
bubble swarm. This confirms that wake interactions is
the major mechanism of pseudo-turbulence.
Figure 8 shows the simulated spectra S of the vertical
fluctuations, which have been normalized by their vari-


---
--











ance. First, we note that the computed spectra are in
good agreement with the experimental spectrum in the
range l < A< Lc. In particular, the k-3 regime is ob-
served to start for the same length scale L, as in the ex-
perimental spectra. For scales smaller than 1c, the k5/3
regime is not find by the simulations but their spatial res-
olution is probably not sufficient to compute such small
scales. The main point is however that the simulations
reproduce well the k 3 regime and its invariance to the
gas volume fraction.


Conclusion

The scaling laws of the bubble-induced agitation have
been obtained from the experimental investigations of a
homogeneous swarm of bubbles rising in water for gas
volume fractions ranging from 0.005 to 0.14 and bubble
diameters from 1.5 to 5 mm. Expressions (1), (2), and
(3) indeed relate the velocity scale, the rate of dissipation
and the integral length scale of the liquid fluctuations to
the gas volume fraction a, the bubble diameter d and
the parameters of a single rising bubble (rise velocity Vo
and drag coefficient Cdo). All together, this provides an
almost complete description of the dynamics of the liq-
uid agitation induced by bubbles rising in water in the
absence of any other cause of fluctuations. In particu-
lar, it is remarkable that the characteristic length of the
large scale is neither the bubble diameter nor the mean
bubble spacing but the ratio d/Cdo between the bubble
diameter and the drag coefficient.
We have also performed large-scale numerical sim-
ulations of the liquid velocity fluctuations generated in
a homogeneous rising bubble swarm by modeling the
bubbles by volume-forces. The simulations are in good
agreement with the experiments. In particular, the prob-
ability density functions and the large scales of the spec-
tra, including the A3 regime, are well reproduced. This
confirms that the bubble induced turbulence is mainly
controlled by interactions between bubble wakes.


References

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


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