7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Scaling laws for bubbleinduced 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, selfsimilarity, 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 selfsimilar 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 largescale numerical simulations by modeling the bubbles by volumeforces. 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 singlephase shearinduced turbulence because
it is strongly related to the random spatial distribution
of the bubbles. To characterize this socalled 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 pseudoturbulence 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 bubbleinduced 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 selfsimilar 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 bubbleinduced 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 setup.
ter the air injection has been stopped. This second flow
configuration will permit to determine the main charac
teristics of the bubbleinduced agitation. Finally, we will
present a largescale numerical simulations which repro
duce the principal characteristics of the agitation that are
observed experimentally.
Characteristic scales of the bubbleinduced
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 pseudoturbulence: (1) the liquid velocity scale,
(2) the dissipation rate and (3) the length scale of the
bubbleinduced agitation.
The experimental setup is depicted in figure 1. The
test section [1] is an open tank of 1000 mm height with
a squared crosssection 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 highspeed 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 setup and connecting the chamber [3] to the
pressurecontrolled 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 selfsimilar
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 selfsimilar 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
selfsimilarity 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 setup includes
two air lines [3][4] and [3][5], connected to two dif
ferent air tanks. Electrovalves [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 10mJ 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 highspeed
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 subpixel 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
A1(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 bubbleinduced 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 cutoff
wavelength of the spectrum (close to 0.1 mm).
3. Integral length scale. To achieve the character
ization of the inducedbubble 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 12
10
S1 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 a0 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 pseudoturbulence.
Largescale numerical simulations of the
bubbleinduced 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
NavierStokes 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 NavierStokes 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
A1 (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
largescale simulations in which bubbles are modeled by
fixed volumeforces 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 pseudoturbulence.
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 k3 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 bubbleinduced 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 largescale numerical sim
ulations of the liquid velocity fluctuations generated in
a homogeneous rising bubble swarm by modeling the
bubbles by volumeforces. 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.
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
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