7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The dual nature of pseudoturbulence analyzed from spatial and time averaging
of a flow through random obstacles
F. Risso *, V Roig *, Z. Amoura and A.M. Billett
Institute de M6canique des Fluides de Toulouse, Universit6 de Toulouse (INPT, UPS) and CNRS, France
+ Laboratoire de G6nie Chimique, Universit6 de Toulouse (INPT, UPS) and CNRS, France
Frederic.Risso@imft.fr, Veronique.Roig@imft.fr and AnneMarie.Billet @ensiacet.fr
Keywords: Dispersed twophase flows, Bubbleinduced turbulence, Reynolds averaging, Energy budgets.
Abstract
Dispersed particles moving through a fluid generate a random agitation, which is often called pseudoturbulence.
This agitation combines two contributions of different natures: the first corresponds to the local spatial flow inhomo
geneities in the vicinity of the particles while the second results from flow instabilities that develop when the Reynolds
number is large enough. Separating them is a very hard task with moving particles, it becomes however practicable
with fixed ones. We report here an experimental investigation of a flow trough a random array of spheres, which is
shown to produce the same velocity fluctuations as a swarm of rising bubbles. The liquid velocity is decomposed
by using both time and spatial averaging. The statistical and spectral description of each contribution makes clear
their respective roles in the dynamics of the streamwise and the transversal fluctuations. Moreover, the energy
budgets of both the spatial fluctuation and the time fluctuation shed light on the physical mechanisms controlling the
pseudoturbulence.
Introduction
In many applications, bubbles (drops or particles) are in
jected into a liquid for enhancing the contact between the
phases. The buoyancyinduced motion of the bubbles
generates an intense agitation within the liquid. This ag
itation combines three contributions. Consider first that
all bubbles are moving at the same velocity and that the
flow around each bubble is stationary. In the frame mov
ing with the bubbles, the flow is then steady. However,
in the laboratory frame relative to which the bubbles are
rising, we observe random time fluctuations which are
not related to any turbulent mechanisms. This first kind
of fluctuations corresponds to the local spatial inhomo
geneity of the flow in the vicinity of the bubbles and
will be referred as spatial fluctuations. Fluctuations of
the second kind result from the flow instabilities that de
velop when the Reynolds number is large enough. They
are observed even in the frame that moves with the bub
bles and will be referred as time fluctuations. Finally, the
motion of the bubbles relatively to each others is another
cause of agitation. It is crucial to distinguish between
these three contributions since they are of different na
tures and are hence driven by different physical mecha
nisms. If separating them is a very hard task with mov
ing bubbles, it becomes however practicable with fixed
ones. A comprehensive description of the dynamics of
the two first kinds of fluctuations has been obtained here
from the experimental investigation of the flow through
a random array of fixed spheres.
Experimental setup and instrumentation
A schematic of the setup is shown in Fig. 1 as well as
top and side views of the test section. Two hundred rigid
spheres of diameters d=20 mm are randomly distributed
within a section of square channel of side 1=220 mm
and length L=800 mm, resulting in a volume fraction of
spheres a=0.02. The spheres are fixed by means of hor
izontal stainless steel rods of 2 mm diameter mounted in
tension between two opposite channel walls. A uniform
flow of water is supplied at the top of the test section.
The bulk velocity (U) is varied from 5 to 50 mm/s in
order to vary the sphere Reynolds number Re=(U)d/v
from 100 to 1000. The liquid velocity U is measured by
means of PIV in various vertical planes located within
the second part of the test section, where the veloc
ity fluctuations generated around the spheres no longer
evolve with the vertical direction z. The measurement
windows belong to x = cte planes and are centered rel
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The spatial fluctuation u(x) depends only on the loca
tion x and its spatial averaging is zero: () = 0. On
the other hand, the time fluctuation depends on both the
time t and the location x, and its time average is zero:
u' = 0. Note that this decomposition is relevant for the
energy of the fluctuations since the variance of the total
fluctuations is actually the sum of the variances of the
two contributions: (u2) = + (u12).
Such a decomposition was already introduced by
Koch & Ladd (1997) and White & Nepf (1997) who
studied flows though array of cylinders. Here, we shall
use it to derive Reynolds average balance equations for
the energy of both the spatial and the time fluctuation.
Injecting (1) into the NavierStokes equations eventually
yields to the following budgets.
Energy budget of spatial fluctuations:
Advs+DiffTs+DiffVs+TP +Prods+e =0 (2)
* Adv,
(U > ..
* DiffTf = (Uj)
wi)~ ~ wi G(Uj )+71 ) "9j
+ 8xw
Figure 1: Experimental setup.
ative to the channel in the y direction. Their dimensions
are 150 mm in the horizontal ydirection and 30 mm in
the vertical zdirection. No sphere intersects the mea
surements windows, which are located either just up
stream or downstream of a fixation rod. Two types of
measurement region have been considered. Downstream
windows systematically contain one or two regions lo
cated in the near wake of spheres, they will be thus re
ferred to as nearwake region. Upstream windows con
tain no near wake and will be referred to as farwake
region. Depending on the Reynolds number, 7 to 24
measurements windows have been investigated, which
has been proved to be enough to be statistically rele
vant. The pixel size in the images is 0.15 mm and we
use interrogation cells of 16 x 16 pixels for the PIV com
putations; the spatial resolution of the velocity fields is
2.4 mm. (More details concerning the setup are given in
Amoura (2008)).
Statistical decomposition of the fluctuations
In order to distinguish between the two kinds of agita
tion, the velocity fluctuations need to be decomposed by
using both time and spatial averaging (.):
u(x, t) U(x, t) (U) (x) + U'(x, t). (1)
DiffV, = v(Ui) + 2 8 ,
8wx 9 2v 8w Sw
TP8 = a(Ud(P+)) i a u(P+p))
p 89x p w9x
Prod, =/ ')
s = ((1)2)
Adv, is the transport of the spatial fluctuation by both
the mean flow and the spatial fluctuation, DiffTs
is the transport of the time fluctuation by the spatial
fluctuation, DiffVs is the molecular diffusion, TP, is
the transport by pressure, Prods is the production of
the time fluctuation by the spatial fluctuation and c, is
the dissipation rate of the spatialfluctuation.
Energy budget of time fluctuations:
Advt+DiffTt+DiffVt+TPt+Prodt+ct =0 (3)
*Advt = +U 
* DiffTt =T
* DiffV = 
* TPt = l i
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Prodt .'. j
t v((
Advt is the transport of the time fluctuation by both the
mean flow and the spatial fluctuation, DiffTt is the
transport of the time fluctuation by itself, Diff V is the
molecular diffusion, TPt is the transport by pressure,
Prodt = Prods is the production of the time fluctu
ation by the spatial fluctuation and ct is the dissipation
rate of the time fluctuation.
Statistical and spectral description of the flow
In this section we describe the properties of the veloc
ity fluctuations averaged over all the measurement win
dows. We systematically distinguish the contributions of
the spatial and the time fluctuation to the total amount of
fluctuation in both the vertical and the horizontal direc
tions.
Fig. 2 shows the standard deviations. In the stream
wise direction (top subfigure), the time fluctuation is
negligible at Re=100, then increases as Re increases
from 200 to 400 and finally becomes independent of the
Reynolds number for Re >600. However, the spatial
fluctuation is predominant over the whole range of Re
investigated. The picture is totally different in the trans
verse direction (bottom subfigure) where the spatialfluc
tuation is much less while the time fluctuation remains
the same. (Note that open squares have been added to
the top subfigure to ease the comparison between the
two directions). The analysis of both directions leads
to the following conclusions. The spatial fluctuation is
strongly anisotropic with a vertical component consid
erably larger than both components of the time fluctu
ation whereas its horizontal component is significantly
smaller. It is the signature of the strong spatial inhomo
geneity of the vertical velocity in the near wake of the
spheres, which is the source of all fluctuations. On the
other hand, the two components of the time fluctuation
have the same order of magnitude and increase with Re.
The time fluctuation, which is generated by the instabil
ity of the whole flow, becomes even almost isotropic for
Re >650.
Fig. 3 shows the probability density functions (PDFs)
of the velocity fluctuations normalized by their standard
deviation for all investigated Reynolds numbers between
230 and 1040. First of all, we observed that the PDFs of
the total fluctuation are in remarkable agreement with
those obtained in swarm of rising bubbles (gray line) at
the same gas volume fraction by Riboux et al (2010).
The flow through the array of fixed spheres reproduces
thus well the bubbleinduced agitation, indicating that
the liquid fluctuations generated by the relative motion
0.150r
vertical fluctuations
0.100
0.050
0.000
0
0.025
0.020
0.015
0.010
0.005
0.000
0
200 400 600
Re
800 1000 1200
200 400 600
Re
800 1000 1200
Figure 2: Variances of velocity fluctuations normalized
by the bulk velocity (U). Open symbols cor
respond to the vertical fluctuation, solid sym
bols to horizontal ones. Circles denote the to
tal fluctuation (u2), diamonds the spatialfluc
tuation ,) and squares the time fluctuations
('2).
horizontal fluctuations
* 0
* ,
2 al 0.6 a2
.90
6 0.4
.3 0.2
0L 0 
5 2.5 0 2.5 5 5 2.5 0 2.5 5
0.4 0.4
0.2 0.2
0 0
5 2.5 0 2.5 5 5 2.5 0 2.5 5
0.8 c1 0.6 c2
0.6
0.4
0.4
0.2 0.2
0 0 "
52.5 0 2.5 5 5 2.5 0 2.5 5
Figure 3: Probability density functions of the velocity
fluctuations normalized by their standard de
viations for Re=230, 340, 450, 650, 850 and
1040. (a) spatial fluctuation u, (b) time fluc
tuation u' and (c) total fluctuation u + u'.
Number 1 (resp. 2) denotes the vertical (resp.
horizontal) direction. The grey curve in cl
& c2 shows the results obtain by Riboux et
al (2010) in a rising swarm of bubbles at
Re=800.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
of the bubbles play a minor role. It is worth noting that
only the total fluctuation is available in the bubble swarm
since the bubbles are moving relatively to the measuring
point. It is different in the array of spheres where the
decomposition into spatial and time fluctuations allows
us to determine the origin of the fluctuations. The ver
tical component of the total fluctuation is dominated by
the spatialfluctuation, which explains why it is strongly
asymmetric with large fluctuations more probable in the
negative direction (which corresponds either to the di
rection of the incident flow for the fixed spheres or to
that of the rise velocity for the bubbles). On the other
hand, the horizontal component of the total fluctuation
is dominated by the time fluctuation. The anisotropy of
the fluctuations in bubbly flows therefore results from
the fact that each component is dominated by a differ
ent contribution. Moreover, the normalized PDFs cor
responding to the various considered Reynolds numbers
all match. This indicates that the physical mechanisms
that control the spatial and the time fluctuations become
both independent of Re for Re > 250.
Fig. 4 presents the normalized spectra of the horizon
tal velocity fluctuations as function of the wavelength
A = 27/k. Again, we observe that the spectra of the to
tal fluctuations measured within the array of spheres are
in good agreement with the results obtained in a bubble
swarm (gray curve). They are dominated by the time
fluctuation and shows a k 3 subrange, which is char
acteristic of bubbleinduced agitation. Moreover, the
matching of the spectra confirms that the structure of
both the spatial and the time fluctuations ceases to de
pend upon the Reynolds number for Re > 230.
Energy budgets of velocity fluctuations
Having described the structure of the liquid fluctuations
and the relative roles of the spatial and the time fluctua
tion in each direction of space, we are now interested in
understanding the physical mechanisms underlying their
production, transport and dissipation.
We consider first the energy budget of the spatialfluc
tuations, which is expressed by equation (2). Using
the fact that the flow is statistically isotropic and ho
mogeneous in the horizontal direction, the terms Adv,,
DiffTs and DiffVs can be obtained by derivations
and averaging of the velocity field, which has been mea
sured by PIV in several vertical (y,z) planes. It is dif
ferent for the terms Prods and c,, which corresponds
respectively to the transfer of energy from the spatial to
the time fluctuation and to the rate of dissipation of the
spatialfluctuation. These two terms imply, in particular,
contributions that involve crosscorrelations of fluctua
tions in x and y directions, which are not measurable in
the (y,z) planes. Fortunately, it is sufficient to obtained
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
the order of magnitude of Prods and c, to show, as we
shall see, that both of them are negligible in the energy
budget of the spatial fluctuation. Finally, the transport
by the pressure TPs has not been measured but deter
100 mined as the rest of the budget.
The terms of this budget, normalized by (U)3/d, are
plotted in Fig. 5, where we have distinguished the near
o10 3 wake region (top subfigure) from the farwake region
(bottom subfigure). First we observe that, viscous ef
time fluctuations fects as molecular diffusion (DiffVs) and dissipation
102 () are negligible as well as the couplings with the time
55/ fluctuation (DiffTs and Prods). The budget of the
spatial fluctuation thus reduces everywhere and for all
10 10l0 lo1 Reynolds numbers to a balance between the advection
X1 A (Adv,) and the transport by pressure (TPs). In the at
tached wake behind a sphere the velocity is close to zero.
The spatialfluctuation is thus negative with a large mag
nitude and the pressure is minimum. Moving away from
100 the sphere, the spatial fluctuation (x) decreases in mag
nitude while the pressure increases. More generally, all
10 3 over the flow, the major mechanism is a reversible con
10 version between the kinetic energy of the spatial fluctu
ation and the pressure. It is difficult to interpret quan
102 titatively the values of Advs and TPs averaged over
the nearwake region (top subfigure) which is very in
spatial fluctuations homogeneous. In contrast, their values in the nearly
1 o 5/3 homogeneousfarwake region (bottom subfigure) allows
1 o 10 us to conclude that the energy budget of the spatialfluc
1 A tuation is almost independent of the Reynolds number.
We consider now the energy budget of the time fluc
tuation, which is expressed by equation (3). As for the
case of the spatial fluctuation, the terms Advt, DiffTt
100 and DiffVt can be determined directly from the PIV
measurements. The production of the time fluctuation
by the spatial fluctuation Prodt=Prods still involves
10 3 two contributions, '.... ) and .. ..' ), thatcan
not be determined from the present measurements. Un
total fluctuations fortunately, Prodt is not negligible in the energy budget
10o2 of the time fluctuation because the order of magnitude
_5/3 of the dominant terms is an order of magnitude lower
than those of the energy budget of the spatialfluctuation.
o101 100 10 The same problem is encountered for the contributions
X1A a U(( )2) and v(( )2) to the dissipation ct, which
is a major contribution to the energy budget of the time
fluctuation. In the absence of any theoretical argument
to estimate these undetermined terms, we shall neglect
their contribution to the values of Prodt and tt.
Figure 4: Spectra of horizontal velocity fluctuations
normalized by using the corresponding vari Another point has to be mentioned concerning the
ance and the integral scale A of the time fluc dissipation. The spatial resolution of the PIV measure
tuations for Re=230, 340, 450, 650, 850 and ments is 2.4 mm, which implies that all fluctuations with
1040. The grey curve in the bottom sub a smaller scale are filtered out. This is expected to have
figure shows the results obtain by Riboux et a small effect on the amount of energy that is measured
al (2010) in a rising swarm of bubbles at and on all the terms of the energy budget except for the
Re=800.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
nearwake region
0.050
* Advs
* DiffTs
0 DiffVs
0 Prods
* "s
E TPs
0.025 F E E E
0.000
0.025
n n5n
0
200 400 600
Re
0.015 nearwake region
0.010
0.005
0.000
0.005
0.010
0.015
800 1000 1200
* 0
E 0 00
0
El El E
0 200 400 600 800 1000 1200
Re
farwake region
farwake region
200 400 600
Re
0.0050
0.0025
0.0000
0.0025
0.0050
800 1000 1200 0
S0*.
SO *
oU
200 400 600
Re
800 1000 1200
Figure 5: Energy budget of the spatial fluctuation. All
terms are defined according to eq. (2) and nor
malized by (U)3/d.
Figure 6: Energy budget of the time fluctuation. All
terms are defined according to eq. (3) and nor
malized by (U)3/d.
0.150
0.100
0.050
nnnn _
* Advt
SDiffTt
DiffVt
0 Prodt
* t
0.050
0.100
0.150
0
., ""
A A 1 A
A A
dissipation, which involves differentiation before aver
aging. In classical turbulence, it is required to have a
spatial resolution close to the Kolmogorov microscale
in order to measure the dissipation. The reason is that
before having reached the dissipative scales, the energy
spectra decays as k 5/3 and the dissipation spectra in
creases as k1/3. Here, since the energy spectra decays
as k 3 (Fig. 4), the dissipation spectra decays as k1.
In this case, it is relevant to determine the dissipation
rate from measurements filtered at a scale that belongs
to this strong decaying regime, and to close the energy
budget at this scale. Having in mind this limitations, the
transport by the pressure TPt is again determined as the
rest of the budget.
The terms of the energy budget of the time fluctua
tion, normalized by (U)3/d, are plotted in Fig. 6. First,
we observe that the molecular diffusion DiffVt and the
transport by the spatial fluctuation DiffTt are negli
gible. To analyze the evolution of the other terms we
need to recall that there exists two possible sources of
time fluctuation. The first is the isotropic shearinduced
turbulence which exists in the channel upstream of the
array of spheres and is convected into the test section
by the mean flow. The second is the production by the
spatial fluctuation which itself results from the local in
homogeneity of the flow caused by the presence of the
spheres.
For the lowest Reynolds number (Re=120), the ab
sence of production by the spatial fluctuation (Prodt)
means that the time fluctuation all comes from the shear
induced incident turbulence. The energy budget in both
the nearwake and the farwake region reduces then to
a balance between the dissipation ct and the turbulent
transport TPt.
When the Reynolds number increases, the influence
of the spheres increases and spatial inhomogeneity of
the flow around the spheres starts to play a major role.
First, this causes the increase of the advection Advt by
the mean flow and the spatial fluctuation. Second, the
flow becomes unstable and the spatial fluctuation starts
to generate time fluctuations. In the farwake region,
Prodt is already significant at Re=230 and reaches an
almost constant value for Re > 340. In the nearwake
region, Prodt starts to be significant only at Re 650
and then goes on increasing continuously with Re. The
fact that the production starts for a much lower Reynolds
number in the farwake region than in the nearwake re
gion indicates that the instability that is responsible for
the time fluctuation does not correspond to the instability
of the individual wakes but to an instability of the whole
flow, which involves interaction between many wakes
randomly distributed over the test section.
At large Reynolds number (Re > 650), the time fluc
tuation is totally dominated by the production within the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
sphere array, the contribution of the incoming turbulence
being negligible. Its energy budget involves a balance
between the productionby the spatialfluctuation Prodt,
the dissipation ct, the transport by pressure TPt and the
advection Advt. If we exclude the near wakes, the two
majors terms are Prodt and ct, which appear to remain
almost independent of Re when normalized by (U)3/d.
Conclusions
Liquid velocity fluctuations observed in a swarm of ris
ing bubbles combine two major contributions that need
to be distinguished in order to understand the underly
ing mechanisms and to model them. The present re
sults show that the flow through a random array of fixed
spheres at a solid volume fraction a=2% reproduces
well the liquid fluctuations measured in a homogeneous
swarm of rising bubbles. Taking advantage of the fact
that the spheres are fixed, the combined use of spatial
and time averaging allows us to separate the two con
tributions to the velocity fluctuations: (i) the spatialfluc
tuation that is the signature of the local spatial inhomo
geneity of the flow around randomly located obstacles
and (ii) the time fluctuation that actually comes from hy
drodynamics instabilities.
The spatialfluctuation mainly results from the deficit
of the vertical velocity in the wakes of the spheres. As a
result, it is strongly anisotropic. Its vertical component
is the major contribution to the total fluctuation while its
horizontal component is negligible. As a fluid particle
interact with a given sphere and then moves away from
it, reversible conversion between the spatial fluctuation
and the pressure takes place. The energy of the spatial
fluctuation is thus essentially controlled by a balance be
tween the advection and the transport by the pressure.
On the other hand, the time fluctuation is isotropic and
dominates the horizontal component of the total fluctua
tion. Provided the Reynolds number is larger than 650,
the incident turbulence coming from the flow upward of
the array of spheres is negligible and the time fluctua
tion results from the instability of the flow within the
test section. The energy of the time fluctuation results
from the balance of four terms: the production by the
spatialfluctuation, the dissipation, the transport by pres
sure and the advection. It must however not be confused
with classic developed shearinduced turbulence since it
shows a spectral range with a k 3 power law. Also, it
is important to note that the time fluctuation appears first
out of the sphere wakes. It therefore does not result from
the instability of individual wake but from the instabil
ity of the whole flow, which implies many interacting
wakes. This is in agreement with the conclusion of Ri
boux et al (2010) who claimed that the bubbleinduced
agitation did not result from a equilibrium between pro
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
duction and dissipation within individual wakes.
These results shed light on the dynamics of liquid
fluctuations generated by particles at large Reynolds
number, such as bubbles of a few millimeters rising in
water. In particular, the anisotropy of the fluctuations
is explained: it comes from the fact that vertical fluc
tuations are dominated by the spatial fluctuation while
horizontal fluctuations are dominated by the time fluc
tuation. Moreover, the present energy budgets show
that Reynolds average equations for dispersed twophase
flows must account for these two contributions sepa
rately in order to allow the derivation of reliable clo
sures.
Acknowledgements
We thank the lab federation FERMaT for its support.
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