International Journal of Multiphase Flow 69 (2015) 8–17
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International Journal of Multiphase Flow
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Rising motion of a swarm of drops in a linearly stratified fluid
S. Dabiri a,b, A. Doostmohammadi a, M. Bayareh a, A.M. Ardekani a,b,⇑
a
b
Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
Article history:
Received 9 June 2014
Received in revised form 14 October 2014
Accepted 15 October 2014
Available online 25 October 2014
Keywords:
Two-phase flow
Stratified fluid
Direct numerical simulation
Front-tracking method
a b s t r a c t
Direct numerical simulations of a swarm of deformable drops rising in density stratified fluids are
presented at intermediate Reynolds numbers. All flow scales are fully resolved using front-tracking/
finite-volume method. The average rise velocity and velocity fluctuations of the swarm are reduced in
the presence of density stratification. The isotropy in velocity fluctuations is enhanced as the volume
fraction increases. The higher likelihood of the cluster formation is illustrated in the presence of density
stratification and is explained by quantitative assessment of the microstructure using radial and angular
pair probability distribution functions. The combined effect of the drop deformability and density
stratification on the average deformation of the drops is investigated.
Ó 2014 Elsevier Ltd. All rights reserved.
Introduction
In oceans and lakes, vertical variation of water temperature or
salinity results in the generation of vertical density layering in
the water column. The rising motion of bubbles in oceans and lakes
(Bayareh et al., 2013), bubble mixers used for aeration of lakes and
reservoirs (Hill et al., 2008) and motion of drops during oil spills
(Blumer et al., 1971) are few examples of important processes that
are being affected by density stratification. Oil spills can cause
extensive hazards to marine and wildlife habitats as well as fishing
and tourism industry (Juhasz, 2012). The ocean density stratification is known as one of the main factors in trapping of the oil
plume and dispersed drops (Socolofsky and Adams, 2003; Camilli
et al., 2010). Understanding the effect of stratification on the rising
motion of the swarm of drops is necessary for accurate estimation
of the rising time, dispersion of the oil, and consequently biodegradation. Recent studies have shown that the motion of a single drop
through either a sharp or continuous density stratification is substantially affected by stratification. For a drop settling through an
interface between two fluids of different densities, Blanchette
and Shapiro (2012) reported a significant reduction of the settling
velocity at the interface in the absence of the Marangoni effects
and even the reversal of the motion of the drop in the presence
of the Marangoni effects. The study of the settling dynamics
through a sharp density interface provides important insights
about the physics of the motion of a bubble/drop in stratified
⇑ Corresponding author at: Aerospace and Mechanical Engineering, University of
Notre Dame, Notre Dame, IN 46556, USA.
E-mail address: ardekani@purdue.edu (A.M. Ardekani).
http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.10.010
0301-9322/Ó 2014 Elsevier Ltd. All rights reserved.
fluids. However, the size of bubbles/drops in aquatic environments
are generally much smaller than the length scales of density
gradients in the water column and thus a more realistic physical
model in the natural environment is represented by a linear
stratification. It has been recently found by Bayareh et al. (2013)
that the presence of a linear density gradient results in a notable
drag enhancement of a rising drop and subsequently extends the
travel time of the drop in the water column by up to 30%.
The dynamics of the settling and drag enhancement of rigid particles in stratified fluids has been reported both experimentally and
numerically (Srdić-Mitrović et al., 1999; Torres et al., 2000; Yick
et al., 2009; Doostmohammadi et al., 2014; Doostmohammadi
and Ardekani, 2014). When the viscous forces dominate the inertial
effects, the drag enhancement is due to the entrainment of a light
fluid behind the rigid particle (Yick et al., 2009; Doostmohammadi
et al., 2012), while in a strong inertial regime the collapse of rear
vortices behind the particle results in the higher resistance to the
vertical motion of rigid particles (Torres et al., 2000). For a
pair of rigid particles settling in a linearly stratified fluid,
Doostmohammadi and Ardekani (2013) quantified the role of
stratification on the interaction between the two particles. Authors
showed that for a pair of particles settling side-by-side, unlike a
homogeneous fluid, stratification results in the attraction between
the particles. In addition, prolonged collision time was reported for
in-tandem settling of a pair of particles in stratified fluids compared to the homogeneous counterpart. For a cloud of particles
in a stratified fluid, Luketina and Wilkinson (1994) showed the
entrainment of the ambient fluid by the particle cloud up to a maximum depth where the particle fall out. Experiments of Hussain
and Narang (1984) demonstrated the formation of a double plume
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S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
structure of bubbly flows when they interact with a stratified fluid.
Socolofsky and Adams (2003) then showed that the buoyancy
effects slow down the vertical intrusion of the plume of bubbles
and result in a horizontal intrusion of a plume that constantly
entrains the surrounding fluid. Despite the recent studies of the
motion of a single drop in stratified fluids and studies of the interaction of plumes of bubbles with stratified fluids, the research on
the interactive motion of swarms of deformable particles/drops
in the presence of the density stratification is virtually nonexistent.
In this study, we numerically investigate the effects of density
stratification on the ascending motion of a swarm of drops. We
particularly focus on the spatial distribution of the drops and cluster formation in order to characterize the microstructure of the
swarm. In addition, the effect of stratification on pseudo-turbulence properties of the flow is presented.
Governing equations
The migration of a swarm of drops in an incompressible, linearly stratified fluid is governed by the following equations
(Bayareh et al., 2013):
r u ¼ 0;
q
Du
Þg þ r lðru þ ðruÞT Þ þ
¼ rp þ ðq q
Dt
ð1Þ
Z
on Hadamard–Rybczynski velocity (Hadamard, 1911; Rybczynski,
1911):
2
W¼
1 gðqf 0 qd0 Þd lf þ ld
6
lf
2lf þ 3ld
which corresponds to the settling velocity of an isolated drop in an
unbounded homogeneous fluid in the Stokes regime. Unless otherwise stated, the velocity is scaled with W and time is scaled with
s ¼ d=W. The ratio of the diffusivity of the momentum mf to the diffusivity of the stratifying agent is represented by the Prandtl number Pr ¼ mf =j. The volume fraction of Nd number of drops in a
3
periodic box of length L is defined as / ¼ N d pd =6L3 . The ratios of
material properties g ¼ qd0 =qf 0 ; k ¼ ld =lf and B ¼ bd =bf are other
dimensionless parameters of the problem. In order for the swarm
to reach a steady state condition, we set qf 0 bf ¼ qd0 bd , so that the
drop density reduces as it enters warmer fluid layers. As a result,
the spatial variation of the temperature inside and outside the
drops become identical and drops eventually reach a statistically
steady rise velocity (Bayareh et al., 2013).
The rise Reynolds number ReW ¼ W s d=mf is calculated a posteriori based on the statistically steady-state average slip velocity of
the swarm of drops W s . The slip velocity is defined as the relative
velocity between the dispersed and continuous phases:
N
rj0 n^ db ðx x0 ÞdA0 ;
ð2Þ
DT
¼ jr2 T;
Dt
ð3Þ
where u is the velocity vector, t the time, p the pressure, and g is the
acceleration of gravity. The local density and viscosity of the fluid
are q and l, respectively. The last term in Eq. (2) represents the
interfacial tension between the continuous and dispersed phases
and it is evaluated at point x. j0 is twice the mean curvature of
^ is the unit vector normal to the interface,
the interface of the drop, n
0
dA is the surface element at the interface of the drop, db is the
three-dimensional delta function which is discontinuous at x0 ,
R
¼ 13 V q dV is the mean density over
located on the interface, q
L
the entire computational domain, j is the thermal diffusivity coefficient, and T is the temperature. In driving Eq. (3), we have
assumed that the thermal diffusivity and conductivity coefficients
in the dispersed and continuous phases are uniform and equal
(Bayareh et al., 2013). Eqs. (2) and (3) are coupled by assuming a
linear relation between the density and the temperature, i. e.,
q ¼ q0 ð1 bðT T 0 ÞÞ, where b is the coefficient of thermal expansion and the reference density and reference temperature are
shown by q0 and T 0 , respectively. Fluid properties in and out of
the drops are distinguished by defining a color function a which
is zero inside and unity outside the drops. Thus q0 ¼ aqf 0 þ
ð1 aÞqd0 ; l ¼ alf þ ð1 aÞld and b ¼ abf þ ð1 aÞbd represent
fluid properties in the entire domain, where subscript f refers to
the continuous phase and d to the dispersed phase, respectively.
The physics of the motion of the swarm of drops in a linearly
stratified fluid can be characterized by a number of dimensionless
3
parameters. The Archimedes number Ar ¼ gd qf 0 ðqf 0 qd0 Þ=l2f
represents the ratio of gravitational force to the viscous force acting on a drop, where d denotes the diameter of a spherical drop.
2
We use the Eötvös number Eo ¼ ðqf 0 qd0 Þgd =r to characterize
the deformability of drops. The stratification of the fluid is characterized by the Froude number Fr ¼ W=ðNdÞ, where N ¼ ðcg=qf 0 Þ1=2
is the buoyancy frequency and c is the background density gradient in the water column. We define the reference velocity W based
ð4Þ
W s ðtÞ ¼
d
1 X
1
W i ðtÞ Nd i¼1 d
Vf
Z
w dv ;
ð5Þ
Vf
where W id ðtÞ denotes the instantaneous velocity of the ith drop and
thus the first term on the right-hand side of Eq. (5) represents the
instantaneous average velocity of the swarm of drops W d ðtÞ. The
second term on the right-hand side of Eq. (5) stands for the volume-averaged velocity in the stratified fluid, where V f denotes
the volume of the continuous phase. The statistically steady-state
rise velocity of the swarm W s is thus obtained by considering the
time average of W s ðtÞ:
Ws ¼
1
tf ti
Z
tf
W s ðtÞ dt;
ð6Þ
ts
where a time period ½t i ; t f is chosen in such a way to exclude initial
transient effects on the average rise velocity. Similarly, the instantaneous and time-averaged velocity fluctuations can be calculated,
respectively, as follows
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
Nd
u1 X
2
0
W ðtÞ ¼ t
ðW id ðtÞ W s ðtÞÞ ;
Nd i¼1
ð7Þ
and
W0 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
Z tf
1
W 02 ðtÞ dt :
t f t i ts
ð8Þ
The Reynolds number based on the velocity fluctuation is defined as
ReW 0 ¼ W 0 d=mf .
In this study, we focus on the effects of stratification of the surrounding fluid, deformability of the drop and volume fraction of the
swarm on the rising dynamics of drops in a linearly stratified fluid.
The Froude number, Eötvos number and volume fraction are varied
independently to isolate the above effects, respectively. Unless
otherwise stated, we use Ar ¼ 1100 corresponding to ReW 15 25 in a homogeneous fluid depending on the value of volume
fraction and deformability. To model a temperature-stratified fluid,
Pr ¼ 7 is used in all simulations. Table 1 lists the relevant dimensionless parameters and their range used in the present study.
Please note that the range of Froude numbers used in the present
study corresponds to density stratifications that are much larger
than what is commonly found in oceans ðN 0:01—0:1 s1 Þ.
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S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
Table 1
Dimensionless parameters used in this study. Note that the
reference velocity is defined as in Eq. (4).
Dimensionless
number
Definition
Range
Ar
gd qf 0 ðqf 0 qd0 Þ=l2f
1100
Pr
m f =j
qd0 =qf 0
ld =lf
7
0:89
3
g
k
B
Eo
bd =bf
Fr
/
W=ðNdÞ
5
2
ðqf 0 qd0 Þgd =r
3
N d pd =6L3
1:12
0:64—8
7:26—16:2
0:02—0:1
(a)
Numerical implementation
Only a brief discussion of the numerical method will be given
here. For more details on the numerical implementation and verification of our computational scheme in homogeneous and stratified fluids see Bayareh et al. (2013) and Dabiri et al. (2013). Eqs.
(1)–(3) are solved on a staggered grid using a front tracking
method (Unverdi and Tryggvason, 1992). The momentum equation, Eq. (2), is decomposed by the projection method (Chorin,
1968) to give an explicit Poisson equation for the pressure which
is solved using the Hypre library (Falgout and Yang, 2002). Convective terms are discretized explicitly using third-order QUICK (Quadratic Upstream Interpolation for Convective Kinetics) scheme
while the central difference scheme is used for descritization of diffusion terms. The periodic boundary condition is used for the
velocity field in all directions and for the temperature in the
transverse direction. To maintain the linear background density
(a)
(b)
(b)
Fig. 1. Convergence tests for a regular array of drops rising in a linearly stratified
fluid for Eo ¼ 0:64; / ¼ 10% and Fr ¼ 16:2. (a) The grid-dependency of the rise
velocity versus time and (b) the relative error with respect to the finest grid as a
function of grid size.
(a)
Fig. 3. The effect of the stratification on the instantaneous rise Reynolds number of
(a) a single drop and (b) a swarm of drops for Eo ¼ 0:64 and / ¼ 10%.
(b)
Fig. 2. The effect of the system size on (a) the average rise velocity and (b) vertical velocity fluctuations for Eo ¼ 0:64; / ¼ 10% and Fr ¼ 16:2. The error bars correspond to
the standard deviation of the data over the time period ½ti ; tf .
S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
11
Fig. 4. Snapshots of the distribution of the drops at t=s ¼ 173 for Eo ¼ 0:64; / ¼ 10% and (a) homogeneous fluid, (b) Fr ¼ 16:2, (c) Fr ¼ 10:2 and (d) Fr ¼ 7:26. The
approximate quasi-steady Reynolds numbers are Re 15; 12; 8; 3 for (a), (b), (c) and (d), respectively.
(a)
(a)
(b)
(b)
Fig. 5. (a) The horizontal and (b) vertical extend of the swarm as a function of time.
The results are normalized by the initial horizontal and vertical extends, HEð0Þ and
VEð0Þ of the swarm, respectively.
Fig. 6. The effect of density stratification on the transient behaviour of (a) the
vertical and (b) the horizontal velocity fluctuations for Eo ¼ 0:64 and / ¼ 10%.
gradient, the periodic boundary condition is implemented for
T cz=qf 0 bf , rather than temperature itself, on top and bottom
boundaries of the domain.
finest grid resolution d=hX ¼ 150 falls below 3% for d=hX P 37:6,
where hX denotes the grid size used in the simulation. Approximately a third-order convergence is observed for the rise velocity
(Fig. 1b). Grid resolution of d=hX ¼ 37:6 is used in our simulations
to assure accuracy of the results in a reasonable computational
time.
Convergence tests
In order to accurately capture the motion of a swarm in linearly
stratified fluids, we first determine the required grid resolution.
Fig. 1a shows the temporal evolution of the rise velocity of a
regular array of drops in a stratified fluid as a function of the grid
spacing. A statistically steady-state solution is obtained after an
initial transient behavior. The difference between results for the
grid resolutions d=hX ¼ 37:6 and d=hX ¼ 75:3 is negligibly small
and as evident from Fig. 1b the relative error with respect to the
The system size
Next, it is instructive to determine a minimum system size, i.e.,
number of drops for a fixed volume fraction, for which the results
are independent of the number of drops. Although, the direct
numerical simulation of a large number of drops is desirable, the
simulations are limited by the computational costs associated with
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S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
(a)
(b)
(c)
Fig. 7. The effect of the volume fraction on (a) the mean velocity of the swarm, (b) vertical velocity fluctuations and (c) horizontal velocity fluctuations of drops for
Eo ¼ 0:64; / ¼ 10% and Fr ¼ 16:2.
3
(a)
ReW / Re(U +V )
2.5
2
1.5
1
0.5
0
Homogeneous
Fr = 16.2
2
6
(b)
10
(%)
Fig. 8. The isotropy in velocity fluctuations of drops settling in a stratified fluid
increases with increasing void fraction for Eo ¼ 0:64 and Fr ¼ 16:2.
the system size. For a system of rising bubbles in a homogeneous
fluid, Bunner and Tryggvason (2002a) showed that the average rise
velocity becomes independent of the system size for N d P 12. It
was shown that although the rate of increase of the velocity fluctuation decreases with increasing system size, it is not possible to
find a limit for which velocity fluctuations become independent
of the system size. Since, in general, the system size increases with
increasing the volume fraction, here, we use the largest volume
fraction / ¼ 0:1 in our simulations to test the dependency of the
results to the system size. As evident from Fig. 2a and b, while
the variation of the rise velocity versus number of drops converges
for N d P 16, the vertical velocity fluctuations increase gradually. It
is noteworthy that similar to the results of Bunner and Tryggvason
(2002b) for an array of rising bubbles, the rate of increase reduces
as the system size increases. Unless otherwise stated, we use
N d ¼ 24 drops for our simulations throughout this study.
Fig. 9. The role of stratification on (a) the deformation of a single drop and (b) the
mean deformation of drops within the swarm for Eo ¼ 0:64 and / ¼ 10%. The
approximate quasi-steady Reynolds numbers are Re 15; 12; 8; 3 for the homogeneous fluid and Fr ¼ 16:2; 10:2; 7:26, respectively.
Results and discussion
Rise velocity and velocity fluctuations
We begin with exploring the effect of the stratification on the
average rise velocity of the swarm of drops. As evident from
Fig. 3, similar to the rising motion of a single drop (Bayareh
S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
et al., 2013), the average rise velocity of the swarm of drops is
reduced as the strength of the density stratification increases.
The reduction of the velocity due to the stratification effect is larger
for a swarm of drops compared to an individual drop. The reduced
rising velocity can be clearly seen in snapshots of the motion of the
swarm at a specific time (Fig. 4). The drops are initially randomly
distributed in a periodic cell and as the swarm rises, they move
into the neighboring periodic boxes. It is found that the initial distribution of drops has negligible effect on the results. In addition to
the reduced rising velocity, Fig. 4 illustrates that the drops are less
dispersed as the stratification strength increases. Both horizontal
and vertical extend of the swarm, shown in Fig. 5, are hampered
by density stratification, but the effect is stronger for the vertical
extend of the swarm.
Since the microstructure of the swarm is constantly changing
due to the interaction between drops, the mean velocity does not
completely characterize the rising motion of the swarm and diffusion-like fluctuations are induced in the velocity of drops. Here, we
characterize the role of stratification on the average velocity fluctuations of the swarm as it rises through the fluid column. It has been
previously shown that small vertical stratification in the particle
13
concentration can hinder the velocity fluctuations of particles in
a suspension (Tee et al., 2002). The decrease of the velocity fluctuations occurs due to the spreading of the sedimentation front
which separates the suspension from the particle-free fluid above
it. Similarly, our results show that, stratification in the fluid density
tends to suppress both vertical and horizontal velocity fluctuations
(see Fig. 6). As evident from Fig. 6, the temporal evolution of velocity fluctuations presents a robust trend both in horizontal and vertical directions: an early decrease of the initial rise velocity
fluctuation is followed by a plateau due to the brake-up of the perturbed array of drops and subsequent rearrangement of the drops.
Velocity fluctuations then eventually settle to a statistically steadystate value. This is consistent with the trend of temporal evolution
of velocity fluctuations for homogeneous bubbly flows (Bunner and
Tryggvason, 2002b) and sedimentation of solid particles in a homogeneous density fluid in the presence of stratification in the particle concentration (Gómez et al., 2007).
As mentioned earlier, the dynamic interaction between drops
plays an important role in the generation of velocity fluctuations.
In order to assess the combined effects of the interaction of drops
with each other and with the stratified fluid, the transient behavior
(a)
(b)
(c)
(d)
Fig. 10. The temporal evolution of (a) average rise velocity of the swarm, (b) average deformation of the drops within the swarm, (c) vertical velocity fluctuations and (d)
horizontal velocity fluctuations are shown for Fr ¼ 16:2; / ¼ 10%, and different values of Eötvös number.
Fig. 11. The distribution of drops in a stratified fluid for different Eötvös numbers. Shown here are velocity fields at t=s ¼ 173 for Fr ¼ 16:2; / ¼ 10% and (a) Eo ¼ 0:64, (b)
Eo ¼ 4 and (c) Eo ¼ 8. The approximate quasi-steady Reynolds numbers are Re 12; 13; 14 for Eo ¼ 0:64; 4; 8, respectively.
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S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
of the mean velocity and velocity fluctuations of the swarm are
compared for different volume fractions at a specific background
stratification (Fig. 7). As evident from Fig. 7, as the volume fraction
increases, the average rise velocity of the swarm of drops decreases
while both vertical and horizontal velocity fluctuations increase.
The reduction of the mean velocity and enhancement of the velocity fluctuations are due to the larger number of interactions
between the drops at larger volume fractions. A measure of isotropy in velocity fluctuations can be obtained by calculating the
ratio of vertical to horizontal velocity fluctuations. The isotropy
of the homogeneous fluid is enhanced as the volume fraction
increases (Fig. 8). This enhanced isotropy is previously observed
in studies of homogeneous bubbly flows (Bunner and
Tryggvason, 2002b) and turbulent dispersion of bubbles (Spelt
and Biesheuvel, 1997) and is correlated to the generation of
pseudo-turbulence by the dynamic motion of drops through an
otherwise quiescent fluid. Bunner and Tryggvason (2002b) associated the enhanced isotropy of homogeneous bubbly flows to the
transfer of energy from the vertical to the horizontal direction
due to the fluctuating motion of the bubbles. The velocity fluctuations in the stratified fluid are more isotropic even at lower volume
fractions and the isotropy slowly improves as the concentration of
drops increases. Please note that the net buoyancy forces acting on
drops in stratified and homogenous fluids are not identical at the
same volume fraction. An important difference between homogeneous and stratified fluids can be inferred from Fig. 8. In a homogeneous fluid, vertical velocity fluctuations are larger than their
horizontal counterparts and thus fluctuations of drops transfer
energy from the vertical to the horizontal direction. Interestingly,
however, in stratified fluids, velocity fluctuations in vertical and
horizontal directions are of the same order and the energy is distributed more uniformly across the flow. This can be associated
to the presence of a vertical density gradient which hinders vertical
velocity fluctuations more effectively compared to the horizontal
fluctuations and modifies the energy transfer process by the
swarm.
It has been recently shown that the deformation of an individual rising drop is hampered by density stratification (Bayareh et al.,
2013). Here, we investigate the effect of the stratification on the
average deformation of the drops within the swarm as it rises in
the fluid column. In order to quantify the drop deformation, the
tensor of the second moment of inertia of the ith drop is defined as:
Iilm ¼
Z
ðxl xlo Þðxm xmo Þ dV;
ð9Þ
Vd
Eötvös numbers. Moreover, as shown in Fig. 10c and d, larger Eötvös
numbers lead to an enhancement of velocity fluctuations about the
mean. The slight increase in the rise velocity for larger Eötvös numbers can be attributed to the formation of vertical streams due to
(a)
(b)
(c)
Fig. 12. The role of stratification on the temporal evolution of autocorrelation
functions of the drop velocity fluctuations for Eo ¼ 0:64 and / ¼ 10%. (a) RUU , (b)
RVV and (c) RWW .
where V d is the volume of the drop, dV is a differential volume element, xlo and xmo are the coordinates of the drop center in the l and
m directions, respectively. The deformation of the drop vi is then
calculated as:
vi ¼
sffiffiffiffiffiffiffiffi
Iimax
;
Iimin
ð10Þ
where Iimax and Iimin are the largest and smallest eigenvalues of the
second moment of the inertia tensor, respectively. The average drop
deformation v is then obtained from averaging vi over all the drops
(Fig. 9). Similar to an individual drop, the stratification suppresses
the average drop deformation (Bayareh et al., 2013). The reduced
deformation in the presence of density stratification is associated
to the reduction of the rise velocity and thus instantaneous Reynolds number, which along with Eötvös number determines the
steady shape of a drop in a homogeneous fluid. It is also of interest
to explore the combined effects of deformability and stratification
on the transient behavior of the swarm. As evident from Fig. 10a
and b, the average rise velocity is not much affected by the Eötvös
number, but the mean deformation of drops increases for larger
Fig. 13. Schematic of the configuration of a pair of drops used for defining the pair
probability distribution function.
S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
the strong deformation of drops (Bunner and Tryggvason, 2003)
(Fig. 11) and larger velocity fluctuations are associated with
enhanced number of collisions between deformable drops.
Next, it is instructive to investigate the effect of the density
stratification on autocorrelation functions of drop velocity fluctuations which are defined as:
N
Rðt 0 ÞUU ¼
d
1 X
1
N d i¼1 Ui02
d
Z
ti
t f t 0
0
0
Uid ðsÞUid ðt 0 þ sÞ
ds;
tf ti t0
ð11Þ
0 0
0
0
where t 0 2 ½0; tf t i and Uid ¼ U id ; V id ; W id are the components of
the velocity fluctuation of the ith drop. The occurrence and
persistence of negative correlations can be clearly seen in Fig. 12.
The negative correlations can be attributed to the ‘‘continuity
effect’’ which arises due to the existence of a back-flow in the
stratified fluid to satisfy continuity (Snyder and Lumley, 1971).
The continuity effect was first observed for dense monoatomic
liquids (Rahman, 1964) and is reported in the measurements of
diffusion and autocorrelation functions for particle velocity fluctuations in a turbulent flow (Csanady, 1963; Snyder and Lumley, 1971).
As a general trend, as time goes by, velocity correlations decay
toward zero and become uncorrelated. However, the relaxation
time seems to be different for horizontal and vertical correlations.
Vertical velocity fluctuations appear to remain correlated for a long
period of time (Fig. 12c). The same qualitative trends are observed
in numerical simulations of a suspension of rigid particles in a
homogeneous fluid (Ladd, 1993). It should be noted that in all cases,
the oscillations of autocorrelation functions around zero are due to
statistical noise.
15
Microstructure
The microstructure of the swarm plays a key role in characterizing its properties. The results of Section ‘Rise velocity and velocity fluctuations’ suggest that the presence of vertical density
gradient in the fluid column can lead to the accumulation of drops
and cluster formation (see Fig. 4). In order to gain a more clear
insight on the effect of density stratification on the distribution
of drops, here we provide a quantitative assessment of the microstructure of the swarm of drops in density stratified fluids by calculating the pair probability distribution of drops in the
suspension. This is valuable, since experimental determination of
the quantitative features of the microstructure are prone to difficulties. The light scattering techniques are limited to static suspensions (Talini et al., 1998) and imaging techniques are only
applicable in very dilute suspensions or suspensions in a very narrow cell (Lei et al., 2001). The pair probability distribution function,
defined as:
N
Gðr; hÞ ¼
N
d X
d
X
L3
dðr rij Þ;
Nd ðNd 1Þ i¼1 j¼1;
ð12Þ
i – j
represents the probability of finding a drop at a distance r away
from a reference drop and oriented at an angle h. Fig. 13 illustrates
the definition of the separation vector rij and angle h between drops
i; j. To present a clear graphical realization of the microstructure,
we decompose the pair probability distribution into radial and
angular components by integrating Gðr; hÞ over a spherical shell of
thickness Dr and over an angular sector of width Dh, respectively
(Fig. 14). In each case, Dr and Dh are adjusted in such a way that
(a)
(b)
(c)
Fig. 14. The role of stratification on the microstructure of the swarm of drops rising in a fluid column for Eo ¼ 0:64 and / ¼ 10%. The spatial evolution of the radial pair
probability distribution function is shown in (a), while (b) and (c) represent the angular pair robability distributions at short range r ¼ 1:25d and long range r ¼ 4d,
respectively.
16
S. Dabiri et al. / International Journal of Multiphase Flow 69 (2015) 8–17
Fig. 15. Enhancement of the horizontal cluster formation in stratified fluids for Eo ¼ 0:64 and / ¼ 10% at t=s ¼ 173. The fluid in (a) is homogeneous and it is density stratified
in (b) with Fr ¼ 7:26.
dividing or multiplying them by two does not affect the pair probability distributions. In addition, we assure that, there exist at least
300 evenly spaced time samples in the interval ½t s ; t f for the calculation of time-averaged values. Fig. 14a shows the evolution of the
radial pair probability distribution function over distance r for
homogeneous and stratified fluids. As evident from Fig. 14a, the
behavior of pair probability distribution function is similar in
homogeneous and stratified fluids: at short ranges (r ¼ 1:25d),
there is a peak in the distribution which is indicative of the cluster
formation while at long ranges ðr ¼ 4dÞ; GðrÞ converges to unity
and the distributions become more uniform. However, stronger
density layering leads to a higher peak at short range distances
and consequently a higher likelihood of the cluster formation is
expected in density stratified fluids. A more discernible difference
between homogeneous and stratified fluids can be observed in the
variation of angular pair probability distribution function across
an angular sector. As evident from Fig. 14b, compared to a homogeneous fluid, there is a higher deficit of drops in the vicinity of the
reference drop at vertical orientations in stratified fluids and this
deficit of close pairs becomes more significant as the strength of
the density stratification increases. However, at long ranges, drops
tend to show a more isotropic pair distribution in both homogeneous and stratified fluids Fig. 14c. Horizontal clustering has been
previously observed in dilute suspensions of bubbles in a potential
flow (Sangani et al., 1991), homogeneous bubbly flows at intermediate Reynolds number (Bunner and Tryggvason, 2002a) and
fluidization of rigid particles in homogeneous fluids (Pan et al.,
2002).Fig. 15 illustrates a comparison of cluster formation for
homogeneous and stratified fluids at a specific time where the
statistically steady state has been established. It is evident that the
formation of horizontal rafts is more enhanced as the stratification
strength increases.
Conclusion
Drops play an important role in thermal, chemical and biological transport in aqueous environments. Using direct numerical
simulations, we have demonstrated that a frequent feature of the
physical environment – density stratification – can have considerable effect on the dynamics of the rising motion of swarm of drops
in the fluid. We have shown that in addition to suppressing the
vertical rise velocity of the swarm, density stratification significantly hinders both horizontal and vertical velocity fluctuations
and results in an enhanced isotropy in velocity fluctuations. The
vertical velocity fluctuations have been shown to remain correlated for a long time interval in stratified fluids. We have illustrated
an enhanced cluster formation due to the modification of the
microstructure of the swarm of drops in density stratified fluids.
A quantitative measure of the microstructure has been provided
by calculating the radial and angular pair probability distribution
functions. We have shown that the short range interactions are
markedly affected by density stratification and demonstrated a significant deficit of vertical pairs at short ranges while at long ranges
the drops are distributed more uniformly.
It is worth noting that the dimensionless parameters including
Ar ¼ 1100; Eo ¼ 0:64; k ¼ 5 and g ¼ 0:89 describe the motion of
drops of crude oil of diameter d ¼ 1 mm, density qd0 ¼
890 kg=m3 , viscosity l ¼ 5 103 Pa s and surface tension
r ¼ 27 N=m in water. The range of Froude numbers used in the
current study corresponds to buoyancy frequencies in the range
of 3:91—8:74 s1 which is larger than commonly found stratified
fluids in oceans N 0:01—0:1 s1 . It should be noted that the density stratification due to the variation in salt concentration would
have stronger effects on the motion of drops compared to the temperature stratified fluids studied here.
Acknowledgments
This work is supported by NSF Grant CBET-1066545 and CBET1445672.
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