Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Transition to turbulence in a dispersed liquidliquid horizontal flow
A. Pouplin*, O. Masbernat*, S. Decarre and A. Linet
LGCCNRS, Universite de Toulouse, France LISBP, INSAINRACNRS, Universite de Toulouse, France
olivier. masbernatr~ensiacet. fr and alain.1ine ~insatoulouse.fr
Keywords: liquidliquid flow, homogeneous dispersed, laminarturbulent transition
Abst ract
In this work, hydrodynamics of homogeneous dispersed liquidliquid flows in a horizontal pipe was studied in a wide range of
mixture velocities and dispersed phase concentrations. Measurements of local velocity field, obtained from PIV technique in a
refractive index matched media, and measurements of pressure drop, allowed to identify the 3 flow regimes, turbulent, laminar
and intermittent. In real flow conditions, it was shown that when the drop concentration remains smaller than 0.6, the
dispersion behaves as an effective Newtonian medium, which viscosity well follows Krieger and Dougherty model (with
#ma=0.74). Wall friction factor follows the Blasius' correlation in the turbulent regime and the HagenPoiseuille's law in the
laminar regime. In the turbulent regime, the structure of turbulence can be assimilated to that of a single phase flow at same
Reynolds number. The laminarturbulent transition exhibits two different zones, the perturbed laminar regime and the
intermittent regime, which are not only function of the mixture Reynolds number but also depend of the emulsion
concentration. The origin of this effect is supposed to be due to low frequency concentration fluctuations which induce on one
hand an increase of the friction factor in the perturbed laminar regime (due to mixture viscosity fluctuations), and on the other
hand which tend to delay and damp the development of turbulence in the intermittent regime.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
f friction factor
K pump rotation speed (rpm)
nD refractive index
n Exponent
Greek letters
pu dynamic viscosity (Pas)
v kinematic viscosity (m~s )
p density (kgm )
Volume concentration
'concentration fluctuation
e dissipation function (m~s )
7 shear rate (s )
6 distance to the wall (m)
A1 integral length scale (m)
;1 Kolmogorov length scale (m)
Kc Von Karman constant 0.4
o interfacial tension (Nm )
re, drop relaxation time (s)
Subscripts
maximum
streamwise direction
verticalspanwise direction
radial direction
Mixture
dispersed phase
continuous phase
maximum in x direction
viscous subrange
laminar
perturbed laminar
max
x
:
r
in
d
c
xinax
K
L
PL
Superscripts
+, normalized
time average
Introduction
Among other industrial applications, crude oil extraction
processes involve the transport of liquidliquid dispersions
in horizontal pipes, the configuration of which depends on
the total flowrate and the phase ratio (fully dispersed or
stratified, dual layer stratified dispersed, see Angeli &
Hewitt 2000). This paper summarizes a hydrodynamic study
of a homogeneous liquidliquid dispersion (i.e. without
mean gradient of phase fraction) flowing in a horizontal
pipe at high concentration and in a wide range of flow
regimes, from laminar to turbulent. Dispersions considered
here are oilinwater emulsion at low Stokes number and
negligible slip velocity.
In the literature, studies on pipe flow of low inertia
suspensions (solid or liquidliquid) tend to show that the
concept of effective medium is valid, Baron (1953) Faruqui
& Knudsen (1962), Pal (1993), but no local detailed
investigation of the turbulent and laminar flow structure
have been performed at high concentration. More recently
in the case of suspensions of neutrally buoyant particles
Matas et al. :* 0I; showed an influence of the dispersed
phase fraction on the laminarturbulent transition for
concentration larger than 2025%. In particular, these
authors observed an increase of the critical Reynolds
number based on mixture properties as the phase fraction
mecreases.
The objective of the present work is to check the validity
of the concept of effective medium in laminar, intermediate
and fully turbulent regime. For each flow regime
investigated, the local features of the flow have been
characterized with the help of Particle Image Velocimetry in
a matched refractive index medium. In addition, pressure
drop measurements have been performed in a wide range of
concentration and mixture velocity. Combining these local
and global measurements have allowed to model the
mixture viscosity and the wall friction factor in all flow
regimes, as presented in a recent paper (Pouplin et al. 2010).
The present paper summarizes the main results of that study
and focuses on two aspects: the local structure of the
turbulence in the emulsion flow at different concentrations
and an analysis of the laminarturbulent transition regime.
Nomenclature
g gravitational constant (ms 2)
P pressure (Pa)
Q flolvrate (m~s )
Pr production term (m~s )
x axial coordinate (m)
:vertical coordinate (m)
rradial coordinate (m)
R pipe radius (m)
D pipe diameter (m)
Re Reynolds number
Ca Capillary number
U cross section velocity (ms ')
V local velocity (ms ')
v fluctuating velocity (ms ')
v* wall friction velocity (ms ')
d', drop diameter (m)
d32 drop mean diameter (m)
r32 drop mean radius (m)
Experimental Facility and Metrology
Phase system
Properties of the two fluids are reported in Table 1. The
continuous phase is an aqueous solution of glycerin at 43%
vol/vol. The dispersed phase is nheptane (technical grade),
immiscible with water. At 290C, the two phases have their
refractive index matched (1.385) allowing light transmission
without deviation.
Table 1: Physical properties of the fluids at 290C
Phases p~kg~m ) p (Pa.s) nD ~(;ll81
Dispersed Heptane 684 4 1(T 1.385
).()31
CotnosWaterGlvcerin 10 .1~ .8
Contiuous (430 vol) 11) 3.1)138
Experimental setup
A schematic of the whole setup is shown in Figure 1. The
device comprises a 7.5m long horizontal pipe of 50mm
internal diameter, a gravity settler, an entry section, two
centrifugal pumps, and a secondary loop with heat
exchangers which regulate the flow temperature in the loop.
In order to implement optical techniques, the duct is made
of poly methylmethacrylate (PMMA), and movable square
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
a high speed camera, (3000 frame/s at 1024x1024px2 image
resolution). The vector field is calculated according to a
2step iteration process with decreasing window size.
Statistical averages were performed over 2000 images
corresponding to an integration time of about 4s (acquisition
frequency of 500Hz, double pulse). Gaussian subpixel
interpolation is applied to optimize the computation of the
displacement, a necessary refinement for accurate
measurement of low velocities near the pipe wall
(Christiansen 2004). Image size is 1024x1024 pixels,
corresponding to a field size of about 50x50mm2. PIV
measurements were realized 3.3m after the duct entry,
where the flow is fully established.
Drop size
The drop size distribution was analyzed with a laser
granulometer (Mastersizer 2000). Figure 2 shows the drop
size distribution for ~0.08 and 0.12. The drop size
distribution is approximately the same for both
concentrations. For 0.05< 0.25, the mean Sauter drop
diameter (d32) is about 25Clm. No significant influence of
the pump rotation speed N on drop size distribution was
observed (1800
drops is smaller than the Kolmogorov length scale
=(3/EY ) in the turbulent regime. Based on the
Kolmogorov time scale, the drop Stokes number is much
smaller than unity, suggesting that the effective viscosity
concept is a priori valid in such an emulsion flow. Moreover,
the estimation of a critical Capillary number and drainage
time to interaction time ratio showed that no rupture or
coalescence is expected in the flow along the pipe length.
The same drop distribution of a sample of the dispersed
flow (at ~0.08) was observed at the pipe inlet and 2m after
the pipe inlet, validating this assumption.
10' 101 102 mi
Figure 2: Drop size distribution for 0.08 () and ~0.12
Flow regimes
Flow homogeneity was evaluated from the analysis of the
radial profile of grey level in the raw images of the laser
sheet. This grey level profile (averaged over 200 images) in
the crosssection results from the light intensity diffused by
the fluorescent microparticles.
A PIV raw image at a velocity L, of 1.2m/s and a drop
concentration of 0.21 appears to be identical to that of the
single phase flow (Figure 3a). Due to their small size (25Clm
is less than 1 pixel), oil drops cannot be detected in the
image. The grey level profile (Figure 3b) is flat as in the
Paper No
Figure 1: Schematic of the liquidliquid rig
boxes (30cm long) filled with water are mounted along the
pipe to reduce optical distortions.
The two fluids are stored in the gravity settler (400L). Each
phase is pumped from the settler to the pipe inlet by
centrifugal pumps of variable rotation speed. Flowrates are
measured with the help of an electromagnetic flowmeter and
a vortex flowmeter, for the aqueous and the organic phase
respectively, with an accuracy of 0.5%. Oil and water phases
merge in a Yjunction, at the pipe inlet. Upstream of the
Yjunction a convergent section has been mounted on the
water loop to avoid or reduce secondary flows in the pipe.
The startup procedure of the experimental device is
operated in two stages. At first, each phase is continuously
fed in the pipe and separated in the gravity settler (no
dispersion is produced at this step). Then, a set of four
electrovalves mounted on the circuit (see Figure 1), allow
to bypass the gravity settler and direct both oil and water
phases in a single circuit (the water loop) at the desired
concentration.
With this system, both phases flow in the aqueous phase
loop and a fine dispersion of oil in water is continuously
maintained in the circuit by the centrifugal pump. At the end
of a series of experiments, the dispersed phase fraction is
measured by sedimentation of a few millilitres sample.
Flow temperature is adjusted thanks to a heat exchanger
fed by a secondary heated water circuit. The temperature of
all experiments has been set to 290C with an accuracy of
0.10C.
Pressure drop measurements were made with a differential
pressure gauge. Five pressure taps were placed at 2m
intervals along the pipe. The instantaneous pressure signal
was acquired at a frequency of 2Hz and timeaveraged over
1 minute time interval for each studied flow rate. At 2m
from the pipe inlet section, the longitudinal gradient is
stable along the pipe length for both single and tivophase
flow. The maximum uncertainty on pressure drop
measurement is estimated at +11%.
A high speed PIV technique has been implemented to
determine the 2dimensional velocity field of the aqueous
phase. A vertical laser sheet illuminates a vertical median
plane of the pipe. The principle of this technique consists in
the measurement of the displacement of seeding particles
between two successive images of the laser sheet separated
by a constant time interval. The most probable displacement
of the particles between two consecutive images is
calculated from the detection of the maximum of the
crosscorrelation function of grev level between the two
images.
Fluorescent hydrophilic Ctparticles were used to seed the
flow. The PIV system comprises a high frequency laser and
1.5
",/
o, ca oo :o
S Turbulent oo 0o
0~ 0 **/ *o 00
0.5
00 0.2 0.4 0.6
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
pressure forces that prevent the drops from settling, leading
to flow stratification. Shear induced agitation is weak in the
present case, especially in that range of concentration (see
Abbas et al. 2007) and cannot counteract the buoyancy
force. Therefore the emulsion homogeneous flow is unstable
in laminar regime and drops will segregate over long
distances. However, an estimation of the drop vertical
displacement shows that the homogeneous laminar flow can
be considered as stable at the scale of the present
experiments. Analysis of the different flow regimes is
described in the next section.
Paper No
single phase flow case, suggesting that the dispersed phase
is uniformly distributed in the pipe section. The dispersion
can thus be considered to be homogeneous. In return, with a
lower mixture velocity (Um=cl' 'isnl, and for the same
phase fraction, a layer of concentrated microparticles
develops in the upper part of the wall, indicating the
occurrence of a partial stratification (Figure 4a). In this case,
there is an increase of grey level near the top wall (Figure
4b). This is expected to be due to the fact that, when settling
in the upper part of the pipe and forming a dense layer, oil
drops capture microparticles like in a flotation process,
leading to an increase of C1particle concentration. Images at
others velocities (not shown here) indicated that the
thickness of the dense layer decreases as the flow velocity
increases. This layer is probably composed of the largest
drops of the size distribution.
o 100 150 200 250 300
Grey level
(a) (b)
Figure 3: Homogeneous dispersed flow (a) PIV raw image
(b) vertical grey level profile ( 0.21, Um=1.2m/s)
+ rv~e
(a>0 (b
Figre4: aerial trtiiedflw a) IVra img06b
verticalgrey~~~~~~~ leepoie(02,U=l~!i
Figur 4:artially stratified flow. (a)n PIVl oraw imag (b)
Exermoentos wereper former covera ide range sofi fblow
to non homogeneous flow conditions, which correspond to
the presence of a visible dense layer of oil drop near the
upper wall. The flow regimes map has been also reported in
this figure, (turbulent, intermediate and laminar). These flow
regimes have been identified at steady state by a
combination of measurements of velocity field and pressure
drop, both presented in the next section. We note that all
flow regimes meet the condition of homogeneous flow
configuration, each of them in a given range of mixture
velocity and dispersed phase concentration.
If the condition for flow homogeneity could be expressed
as a function of a critical Froude or Richardson number in
turbulent (and to some extent in the intermediate) regime,
the stability of such a flow configuration in laminar regime
is questionable. Indeed, in this regime, there are no dynamic
Figure 5: Flow pattern map (o) homogeneous flow, (e)
partially stratified flow. Thin lines separate the flow regime
domains. Thick line is the approximate limit between the
homogeneous (above) and the partially stratified
configuration (below)
Results in fully laminar and turbulent regimes
Fully Laminar Regime rRe 1A0
The fully laminar regime is observed for concentration
larger than 0.5 and mixture velocity ranging between 0.5
and Im/s. The mean velocity profile of the continuous phase
is parabolic (Figure 6) and there is no mean secondary
motion (Figure 7).
0.sc
0 0.2 0.4 m .6
0.8
Figure 6: Axial velocity profile, (0) ~0.51, Um=0.56m/s,
m,,=1.1m/s, (a) ~0.56, Um=0.85m/s, m,,=1.6m/s
The velocity fluctuation level is comparable to the
measurement noise level. As expected, the axial pressure
gradient varies linearly with Um, suggesting that the
emulsion flow can be considered as a parallel steady
laminar single phase flow of an effective Newtonian
medium (Figure 8).
+ ~F043
a C051
m=o53
16 0 b056
q fm=
Re
'4 "
h
*t
Both relations give close values of effective viscosities with
an average relative discrepancy of 5%. Mixture viscosity
values are reported in table 2.
Table 2: Mixture viscosity in laminar regime
0.51 0.53 0.56
Eq. (1) 0.03 0.031 0.0330.035 0.045 0.049
Pm,(Pas) Eilers (1941) 0.03 0.036 0.048
K & D. (1959) 0.028 0.033 0.044
They are well fitted by both the empirical relation of Eilers
(1941) for emulsions and Kieger & Dougherty model (1959)
for hard spheres, setting the maximum volume fraction #ma
equal to .74. The hard sphere like behaviour of the emulsion
in this range of flow parameters is consistent with the very
low value of the Capillary number (undeformed drops) and
the probable weak contribution of intermolecular surfaces
forces in this range of drop diameter (25Clm) and
concentration.
Based on that viscosity, a mixture Reynolds number can be
defined for every velocityconcentration couple:
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
Rem = mD
The flow being parallel, steady, established and
homogeneous, a friction factor at the wall can be defined as:
0.005 M m
In laminar regime, the linear dependence of the pressure
drop with the mean velocity leads to the classical
HagenPoiseuille friction law for a Newtonian fluid:
0.005 0.01
Figure 7: Radial velocity profile, (0) (0.51, Um=0.56m/s,
Vxm,=1.1m/s, (A) (0.56, Um=0.85m/s, Vxm,=1.6m/s
16
/m=Re,
The plot of the friction factor (eq. (3)) as a function of
mixture Reynolds number (eq. 2) based on the mixture
viscosity calculated with Krieger and Dougherty law, clearly
shows that the friction factor is well fitted by eq. (4) for
Reynolds number up to 1200 (Figure 9). Beyond, the
behaviour is still linear, but a small discrepancy above 1200
can be noted (see discussion below).
To summarize, the concept of effective medium is validated
in fully laminar regime up to 1200. For concentration
smaller than 0.6 and in the limit of small Capillary numbers,
the emulsion is a Newtonian fluid which viscosity follows
classical hard sphere laws.
um Im.s1]
Figure 8: pressure drop as a function of mixture velocity
The effective viscosity can be deduced from either one
relation:
Figure 9: Friction factor vs Rem in the fully laminar regime
Fully Turbulent regime /\' it re,
Homogeneous turbulent twophase flows were observed for
0.56
when (0.31. Normalized axial velocity profiles are
displayed in Figure 10 for a single phase flow case and two
different cases of twophase flow. All profiles do well
coincide and are well fitted by the universal power law:
Vx1~~= /Vm (2z / )i ", with n=6.5 (5)
f,=Ji R
Idpl R2dxlsU ~irldPIR4 Or ~Um =I~dX I 4Vx,,
~u,=Idxl8U, sldxle
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
in the viscous sublayer. The twophase cases exhibit the
same trend but the curves are shifted towards larger value of
6 The shift increases as the concentration increases,
suggesting an effect of an effective viscosity. This seems to
be confirmed in figure 12b, where the curves have been
collapsed by simply multiplying the continuous phase
kinematic viscosity by a constant factor, which turns in
considering mixture properties, p,,, and Pm,.
Paper No
0o 0.2 0.4 0.6 0.8 1
Figure 10: Axial velocity profile, ( c) single phase flow,
Um=0.42 m/s, Vx,=0O.54m/s, (A) ~0.08, Um=0.56 m/s,
Vxm,=0.72m/s, (0) ~0.21, Um=0.85 m/s, Vxm,=1.1m/s,
() Eq. (5)
The flow being stationary and parallel, wall friction velocity
v* can be either deduced from pressure drop measurement
or from Reynolds stress profiles. By definition, the wall
friction velocity is:
v*= D 1dP (6)
S4 p, I dx 
The velocity cross correlation profile normalized by this
friction velocity vxtvl*2 iS represented in Figure 11 for a
single phase case and a twophase flow case, with a volume
fraction of 0.31. In both cases, the profile is linear in the
flow core and the slope is close to 1, validating the
determination of the friction velocity in both single and
twophase flow.
/
Figure 12: Normalized axial velocity profile ( c)
singlephase flow, Um=0.42 m/s (A) Twophase, ~0.08,
Um=0.56 m/s (0) Twophase ~0.21, Um=0.85 m/s (a)
Continuous phase properties p, Pc,; (b) mixture properties
The mixture viscosity values deduced from that fitting are
reported in table 3. These values have been compared with
those derived from the evolution of the pressure drop as a
function of the mixture velocity Um. Assuming Blasius' law
to be valid in that range of Reynolds number (<20000), the
wall friction factor of the mixture should be given by:
f,, = 0.079 Re'"
Replacing in (9) Rem and f, by their respective definition
(eqs. (2) and (3)), leads to:
Rem=7230, Um=0 42m/s
Rem=6600, Um=1 2mis
(10),
os
The pressure drop as a function of the mixture velocity has
been reported in Figure 13 for two different concentrations
of the dispersed phase. In both cases, experimental data are
perfectly fitted by a 7/4 power law of the mixture velocity,
suggesting that eq. (10) is valid. Identification of the
prefactor allows to determine the mixture viscosity.
02 (b,
02i 1 1
Figure 11: Normalized velocity cross correlation profile (a)
single phase Um=0.42m/s (b) Twophase, Um=1.2m/s
Using this friction velocity and Prandtl variables
( = 3v'/v and V = Vx v* ), the mean axial velocity
profile has been plotted in Figure 12 for the same cases as
those of Figure 10. In Figure 12a, the viscosity v is that of
the continuous phase, v,. The single phase flow profile (*
symbol) is well fitted by the classical wall laws in turbulent
pipe flow:
#=0.08 ,
40 r '
m=0 21
'"00 ., (b)
oo as r rs
sy st51
Oo as a
age Isl
Figure 13: Pressure drop vs mixture velocity (a) ~0.08 (b)
~0.21. Dashed line corresponds to the 1.75 power law.
These viscosity values are reported in table 3 as well as the
predictions of Vand's equation (Vand, 1948) for dilute
suspensions of hard spheres (also close to Krieger &
Dougherty or Batchelor's laws in that range of 4). The
consistency between the different estimations suggests that
the concept of effective medium is also valid in fully
3' ;>30 V = In(3 )+5.5
in the inertial layer and
0 5 3' 5 5 Vt = 3+
P= 0.067 05 2 ,5
20,
(Velocity) loglaw 4. 103 6. 103
Pmn (Pa.s) Prsuedrop) Blasius' law 3.7103 5.5103
Vand (1948) 4.103 5.9103
Based on that mixture viscosity, the wall friction factor can
then be plotted as a function of the mixture Reynolds
number in turbulent regime at different concentrations. It is
verified in Figure 14 that it is well fitted by Blasius' law for
all cases.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
developed turbulent pipe flow of the emulsion.
Table 3: Mixture viscosity in turbulent regime
Concentration
0.08 0.21
10 I
0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 19: Relative viscosity as a function of oil volume
fraction (#ma=0.74) (... Oilers, Krieger & Dougherty)
Solid symbols are derived from Blasius' friction factor and
crosses from HagenPoiseuille's friction factor
Turbulent field analysis
In order to complete the validation of the concept of
effective medium in the fully turbulent regime, it seems
relevant to analyse compare the
SBlastus eqn
r
Rem=6600, Um=1.2m/s
(b)
2' 3
Re,,=7230, U,,=0.42m/s
as single phase oa
" (a) O
0234
Rem
Figure 14: Friction factor vs Re, in the fully turbulent
regime
The Newtonian behavior of the emulsion in that flow
regime and concentration range needs also to be verified in
the same way as in the laminar regime. A Capillary number
based upon the viscous subrange shear rate
l; = (5v,/2e /2 iS here used to evaluate drop deformation,
Ca = 47732/,,O It is of the order of 104, leading to the
conclusion that the drop deformation is negligible, and that
drops remain spherical. As they are probably contaminated,
it justifies the validity of hard sphere model in that range of
drop size (non Brownian and non colloidal particles). Note
also that polydispersity has no significant effect in the range
of concentration investigated (~ 5 0.56 ).
M2lixture viscosity model
Analysis of local flow field and pressure drop in the fully
laminar and turbulent regimes has shown that the emulsion
behaves as a Newtonian fluid in the investigated range of
drop size, volume fraction, mixture Reynolds and Capillary
numbers, in both laminar and turbulent regimes. This
viscosity is only function of the phase fraction and its
evolution is well represented by a hard sphere type model
such as Krieger & Dougherty's model as illustrated in
Figure 19:
Figure 15: Profiles of velocity component rms (a) Single
phase Rem=7230 (b) 0.31, Rem=6600
Velocity rms profiles of the longitudinal and radial
components are displayed in Figure 15 for a single phase
flow case and a twophase flow case at a concentration of
0.31. The shape of the profiles are quite similar in both
cases, showing an isotropic behaviour in the centre of the
pipe ( lv* v = 0.9) and an increase of the
anisotropy factor in the wall vicinity ( 2 < v' < 3 )
These data are in good agreement with Laufer's data (Laufer,
1953) or those of Toonder & Nieuwstadt (1997). They are
also very close to the numerical predictions of Kim et al.
(1987) of a single phase flow in a cylindrical pipe at same
Reynolds number (7000).
The macrolength scales Ax, and A1, have been determined
from the spatial autocorrelation function in the streamwise
direction of fluctuating velocity components, vx and v,
respectively and their profiles are reported in Figure 16.
There are no significant differences between single phase
and emulsion flow, Ax, and A1, are of the order of 68mm and
34mm respectively, independently of the Reynolds number.
It was also verified that the integral time scale of the axial
component could be scaled as AIx U, Taylor microscale
COuld also be determined from the osculating parabola of
the autocorrelation coefficient at short separation distances.
In single phase flow, it varies between 3.6mm and 2.4mm
when Rem varies between 7000 and 20000 and is of the
p
p, =  m

0
with ,= 0.74
(
( 4 25
1
ax
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
drop diameter (25Clm). As an example, for Rem=7250, Gm,
increases from 180Clm to 330Clm when 6/R varies between
0.1 and 1. The Stokes number can be defined as the ratio
between the relaxation time of the drops zd pdd322 8Pm >
and the characteristic time scale of the flow change at the
scale of the particles, ~= (15v,/2eY/22 II the viscous
subrange, it reads:
Paper No
same order in two
number,
phase flow at same mixture Reynolds
Rem=7230, Um=0.42m/s Rem=6600, Um=1.2m/s
(rn) Single phase
o Ax is0.31
oA
~a) A
(b) + A
Figure 16: Integral length scales in xdirection of axial and
radial velocity component (a) single phase flow, Rem=7230
(b) ~0.31, Rem=6600
In order to determine the Kolmogorov microscale, the
dissipation rate has been estimated from the
productiondissipation equilibrium in the inertial layer. The
production term is computed from the experimental data of
the tangential stress (Figure 11) and the mean velocity
gradient (Figure 10):
The mean drop diameter to the Kolmogorov length ratio
being of the order of 0. 1, the Stokes number is less than 102.
This low value confirms the validity of the effective
medium concept in turbulent flow of concentrated emulsion.
From the preceding results, the power density spectrum of
turbulence should therefore be only a function of mixture
Reynolds number. This is confirmed in Figure 18, where the
normalized temporal spectra have been plotted at 6 =240 for
two mixture Reynolds number, in single phase and
twophase flow at two concentrations. In each graph, the 3
curves are indeed almost identical. It can be noted that the
5/3 decay in the inertial range is more pronounced at
highest Reynolds. This can be explained by the increase of
the Taylor macro to microscale ratio with the Reynolds
number.
Pr = rvy8xldr
Rem=7230, Um=0.42m/s Rem=7250, Um=0.85m/s
025) Single phase O Pr 0250.21 0 Pr
v* )Re~20(a) sinl phs (b 021
The dissipation term is here estimated with the friction
velocity v*:
101 42A
'0 ,2V A
/3
f^x/~ fx/v
101 10 1 1o IO 101 a~ 10: 101 10 101
Figure 18: Normalized power density spectrum of vx and
6+=240 (a) Rem=7400 (b) Rem=10000
To summarize, the concept of mixture viscosity model does
apply in the fully turbulent regime, without any modulation
of the single phase pipe flow turbulence at same mixture
Reynolds number.
Analysis of the transition regime (1200
Wall friction
The wall friction factor of the emulsion is the same as that
Of a single phase flow at equivalent Reynolds number, in
both fully laminar and turbulent regime, as it can be seen in
Figures (9) and (14). However, the transition is not similar
to that of a single phase pipe flow. Figure 20 shows the
laminarturbulent transition observed of the mixture friction
factor as a function of the mixture Reynolds number. The
dashed line represents the fitting of Nikuradse's data
(Nikuradse 1950) for single phase pipe flow. The occurrence
of turbulence is observed at Re~2100 and the fully turbulent
regime is reached around Re~3800. Note that this transition
range is significantly enlarged for the emulsion flow, and is
composed of two distinct zones: a first Reynolds range
comprised between 1200 and 25003000, where the
The equilibrium approximation is well verified for
25<6 <100, validating approximation (12) in this range, in
both single phase and twophase flow as illustrated in Figure
17. Kolmogorov scale can therefore be derived from the
following relation:
Combiming (12), (13) and (9) allows to express the variation
of the Kolmogorov length scale as a function of the distance
6 from the wall: ,
S21 4I (4
S= 4.52Rem32 R 1)
In the range of mixture Reynolds number investigated, rl,
remains always an order of magnitude larger than the mean
St, =O 0365 Pd d32
(,3 1/4
V.=
Figure 20: Friction factor vs Rem in the laminarturbulent
transition regime
Perturbed laminar regime (1200
A typical velocity profile measured in that regime is given
in Figure 21 at Rem=1800. The profile exhibits a
dissymmetry, with an offcentred maximum, which location
may vary with a time scale of same order as that of the
integration time of the velocity profile (4s). This asymmetry
is associated with the existence of weak secondary flows
which maximum seems to increase with Reynolds number
but remains smaller than 5103xVxma. Interestingly all
Reynolds stress tensor components are negligible except the
fluctuating velocity variance in the flow direction. This
fluctuating signal is displayed in Figure 22 (left column),
and is composed of low frequency large amplitude
fluctuations which exhibit a periodic behaviour with a
characteristic time scale of the order of 0.51s, as confirmed
by the plot of the temporal autocorrelation coefficient of this
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
fluctuating velocity signal (Figure 22, right column)).
Paper No
turbulence is negligible, and an intermittent regime where
the turbulence occurs and develops in a range of Reynolds
number wider than in the single phase case, up to 5000 for a
concentration of 0.37.
In the first regime, 1200
slightly deviate from the HagenPoiseuille's law. The 1
slope is approximately the same, indicating that the flow is
still globally laminar (pressure drop is still proportional to
the mean velocity). Due to the alignment of data at different
concentrations on the same curve, it is probable that this
deviation is not resulting from an error on the estimation of
the mixture viscosity. At high concentration, it is also
noticeable that this trend is extended up to Reynolds
numbers of 3000 (at ~0.43 and ~0.45), revealing a
retardation effect on the occurrence of turbulence. Such an
effect has been already demonstrated by Matas et al. :1I I i;)
in the case of pipe flow of neutrally buoyant suspensions,
when the particle concentration exceeds 0.25. Although we
cannot provide a sufficient number of data at different
concentrations in that range of Reynolds number, it seems
that this retardation effect is highly sensitive to the dispersed
phase fraction and increases as the concentration increases.
This regime is referred to as the perturbed laminar regime.
As a result, the intermittent regime, i.e. the reattachment
range to the fully turbulent regime (symbolized by the
Blasius' correlation) occurs at larger Reynolds number as
the concentration increases. For ~0.31, Blasius' law is
reached around 4200, while it is reached above 5000 when
~0.37. The present results seem to confirm the trend
reported by Matas et al. ('s II I7) for suspension flow.
Rem=1800, Um=1.13m/s
oao
o ooo
a o
0.8
0.4
0 0.2 0.4xxm0.6 0.8
Figure 21: Axial velocity profile ~0.51
It can be observed on these figures that the amplitude of
the fluctuations seems to mecrease with the Reynolds number.
In return, the characteristic frequency seems to increase
with the mixture velocity. For Um =0.85m/s, the mean
period of the oscillations is of the order of 0.74s, for
Um=1m/s, of the order of 0.6s, and for Um=1.13m/s, it is
close to 0.46s.
Rem=1600, Um=1m/s
02 0.51
Rem=1800, Um=1.13m/s
01.5
Rem=1600, Um=1m/s
Rem=1800, Um=1.13m/s
02E
Rem=2450, Um=0.85m/s
~0.43
o~ ;~
0 1 2 4 5 6 0 1 2 ts
Figure 22: Left: Instant signal of the axial velocity
fluctuation Right: Velocity temporal autocorrelation
coefficient z/D=0.6 (), z/D=0.8 ()
This perturbed laminar regime is characterized by a wall
friction slightly higher than that predicted by
HagenPoiseuille's law in the laminar regime. Mechanisms
responsible for that increase of the wall friction are clearly
not due to an inertial contribution of the velocity
fluctuations. A possible explanation is the existence of
concentration fluctuations (at unknown time and space
scales). Those fluctuations may engender local viscosity
To be tested against experimental data, relation (23) needs
an additional closure relation between concentration and
velocity fluctuations, which is not available. One possible
way to reach it is the direct numerical simulation in a
concentrated homogeneous sheared suspension, such as
already performed by Abbas et al. :* II Is').
IntermittentRegime (2500 Re <5000)
The second range of the transition regime starts with the
occurrence of the intermittent regime, characterized by an
alternation of turbulent puffs and laminar plugs, distinctly
visible on the velocity field animations. Compared to single
phase pipe flow, with concentrated emulsions, this
intermittent regime extends on a wider range of Reynolds
number, as illustrated in Figure 24. For concentration larger
than 0.3, the intermittent regime goes up to Reynolds
number as high as 5000 when #0.37 and probably more for
a higher concentration.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Paper No
fluctuations responsible for the velocity fluctuations, the
contribution of which in the mean momentum balance is not
negligible in that range of Reynolds number. This effect can
be qualitatively illustrated by introducing this mixture
viscosity fluctuation pu in the time averaged wall friction
factor subscriptt PL means "perturbed laminar", and L
laminarr" ):
f 6 = = u 1 "u (16), ~U
fPL Re p, D U,, + U p, DU,u, pU,i,
This equation assumes that the characteristic time scale of
the mixture viscosity is large compared to the flow time
scale (R/U,,), in other words, at each instant, the laminar
flow adjusts to the steady solution corresponding to the
instant value of the viscosity (quasisteady hypothesis). It is
also assumed that the fluctuation of the mixture density is
negligible before that of the viscosity. The resulting mean
velocity fluctuation can be deduced from the integrated
momentum balance in a cylindrical duct:
Krieger & Dougherty's law:
Therefore, the perturbed laminar friction factor drop reads:
4 #,,,
Um can therefore be estimated according to:
dU,,,
dp "'
"'~u
(19),
f o2
which can be approximated bv:
'2
~UU, Um
As expected, the viscosityvelocity correlation is negative
and consequently leads to an increase of the friction factor:
Figure 24: Friction factor vs Re,, in the intermediate regime
The effect of concentration is significant on this graph, the
appearance and the development of the turbulent regime
being delayed when the concentration increases.
The nature of the fluctuating velocity signal is strongly
modified between the single phase case and the twophase
case. This result is illustrated in the graphs below where the
velocity temporal autocorrelation coefficients have been
plotted as same Reynolds number as a function of
dimensionless time tV ;D. One can observe that the
periodical feature of the fluctuations is almost unexistent in
the single phase signal and the low frequency fluctuations
are much more damped than in the twophase signal.
Equation (21) hence brings a plausible explanation of the
increase of the wall friction factor observed when
1200
a dispersion is a compressible medium, hence subject to
fluctuations of concentration. A simple relation can be
derived between mixture viscosity and concentration
fluctuations :
d~u,
~u, d
(ai>,
where the derivative of the mean viscosity is given here by
clp,,2.5
p,~, = pu,2
fPL L 2.5
f, r~ 1i~[1 maxj
RlP p ,
4 ,
f = 1+ 4
PL nDU,, 4 2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
role of the mixture viscosity fluctuations since it is a
growing function of the concentration and so is its growth
rate. Therefore, at high concentration, a concentration
fluctuation will produce a larger relative viscosity
fluctuation than at a lower concentration. The fact that this
effect is only observable in a limited range of Reynolds
number can be readily explained. At low velocity and high
concentration, the viscosity fluctuation amplitude is
negligible and it is likely that in fully developed turbulence,
the turbulent dispersion of drops (close to the continuous
phase eddy viscosity in that range of Stokes number) is
much higher than the emulsion effective viscosity.
It seems also important to point out that the hypothesis of a
compressible medium amounts to consider a nonNewtonian
behavior in that flow regime, since the concentration
fluctuations presumably depend upon the mixture velocity.
To summarize, transition regime is composed of two
distinct ranges:
The perturbed laminar regime, for 1200
characterized by the presence of low viscosity
fluctuations related to concentration fluctuations, which
are thought to be responsible of the slight increase of
the wall friction.
The intermittent regime, for 2500
coexist the same kind of low frequency fluctuations and
turbulent structures. For concentration larger than 0.3,
the compressibility property of the emulsion seems to
delay and damp the development of turbulence and this
effect increases as the concentration increases,
OXplaiming why the range of the intermittent regime is a
function of the concentration.
Concluding remarks
Validity of the effective medium concept in a wide range
of flow Reynolds number and dispersed phase concentration,
in the limit of Newtonian behaviour ( 0.6) has been
demonstrated for noncolloidal, non deformable
liquidliquid emulsion. Classical single phase wall laws and
Scaling relations of turbulent pipe flows have been obtained,
using mixture density and viscosity
However, peculiar behaviour have been observed in
intermediate range of Reynolds number which is
characterized by the presence of low frequency fluctuations
at long times in the axial direction. These fluctuations seem
to be uncorrelated with the occurrence of turbulent
structures but they damp their development in the
intermittent regime and increase the wall friction in the
laminar regime. One possible explanation is that these
velocity fluctuations are related to concentration
fluctuations at unknown scales which result in local
fluctuations of the effective viscosity. The amplitude of
these fluctuations of concentration increases with the mean
flow velocity (or the mean velocity gradient). At low
Reynolds number, they disappear and at high Reynolds
number, they are overcome by the turbulence. As a
consequence, for concentrations larger than 0.3, both the
onset of turbulence and the reach of the fully developed
turbulent regime are delayed as the concentration increases.
Concentrated emulsions therefore behave as an effective
homogeneous medium except in a finite range of
intermediate Reynolds number, where the existence of local
concentration fluctuations gives rise to a nonNewtonian
Paper No
Figure 25: Velocity autocorrelation coefficient as a function
of t*=tV/D Rem=2400 (a) single phase (b) ~0.43
The analysis of the fluctuating velocity signals in the
intermittent regime shows that they tend to keep the
signature of these low frequency periodical fluctuations over
which a turbulent signal is superimposed. In figure 26, it can
be seen that for a same mixture Reynolds number of about
3200, periodical fluctuations are more pronounced for the
highest concentration (0.43). In addition, the characteristic
frequency of these low frequency oscillations is higher at
the highest mixture velocity, tending to prove that they are
the same kind of fluctuations as those present in the
perturbed laminar regime,
Re =3100, Um=0.56m/s
D2 ~0.31
Re =3100, Um=0.56m/s
0b.31
Rem=3250, Um=1.13m/s
0.43
3 4
Re 3250, Um=1.13m/s
~0.43
a :~I~
t [s] 3 4 0 1 S 4 5
Figure 26: Left: Instantaneous fluctuating velocity signal
in streamwise direction Right: Velocity temporal
autocorrelation coefficient Rem~3200
When the Reynolds number is increased, the rate of
turbulence in the signal increases but all the more slower for
concentration is high. The two cases presented in Figure 27
for a mixture Reynolds of about 4000 illustrates this trend.
At ~0.31, the fluctuating velocity signal is very similar to a
fully turbulent signal whereas at a slightly higher
concentration ( 0.37), low frequency fluctuations are still
present in the signal, with a lower rate of (high frequency)
turbulence. Such a difference is more visible on the
temporal autocorrelation coefficient curves, where this low
frequency behavior distinctly appears at the highest
concentration.
The turbulence occurrence delay therefore also results
from the presence of the fluctuations of concentration in the
emulsion flow. In other words, one can consider that the
compressibility of the dispersion (and elasticity) will tend to
favor the damping of perturbations and not amplify them,
and this effect is all the more pronounced for the
concentration is high. Such a behavior gives sense to the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and effective viscosity of a homogeneous dispersed
liquidliquid flow in a horizontal pipe. .4IChE 1 2010 (in
press)
den Toonder J.M.J., Nieuivstadt F.TM. Reynolds number
effects in a turbulent pipe flow for low to moderate Re. Phys.
Fluids 1997; 9:33983409
Vand V, Viscosity of solutions and suspensions I. Theory. 1
Phys. Colloid Chem. 1948; 52: 277299
Paper No
behaviour which modifies the laminarturbulence transition,
Direct numerical simulations at the scale of the drop in that
flow regime would certainly help to elucidate these
mechanisms.
Acknowledgements
Authors wish to thank Institut Frangais du Petrole (IFP) at
RueilMalmaison and research federation FERMaT of
Toulouse for financial support.
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