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International Journal of Heat and Mass Transfer 127 (2018) 302–312
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
A numerical study of a liquid drop solidifying on a vertical cold wall
Vinh Nguyen Duy a,b, Truong V. Vu c,⇑
a
Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
c
School of Transportation Engineering, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Viet Nam
b
a r t i c l e
i n f o
Article history:
Received 21 May 2018
Received in revised form 29 July 2018
Accepted 9 August 2018
Keywords:
Front-tracking
Drop
Solidification
Numerical simulation
Vertical wall
a b s t r a c t
Solidification of a liquid drop on a vertical wall is a typical phase change heat transfer problem that exists
widely in natural and engineering situations. In this study, we present the fully resolved two-dimensional
simulations of such a problem by a front-tracking/finite difference method. Because of gravity, the liquid
drop assumed stick to the cold wall shifts to the bottom during solidification. Numerical results show that
the conical tip at the solidified drop top is still available with the presence of volume expansion, but the
tip location is shifted downward, resulting in an asymmetric drop after complete solidification. The tip
shift, height and shape of the solidified drop are investigated under the influences of various parameters
such as the Prandtl number Pr, the Stefan number St, the Bond number, the Ohnesorge number Oh, and
the density ratio of the solid to liquid phases qsl. We also consider the effects of the growth angle
/gr (at the triple point) and the initial drop shape (in terms of the contact angle /0) on the solidification
process. The most influent parameter is Bo whose increase in the range of 0.1–3.16 makes the drop more
deformed with the tip shift linearly increasing with Bo. The tip shift also increases with an increase in /0.
However, increasing Oh (from 0.001 to 0.316), St (from 0.01 to 1.0) or qsl (from 0.8 to 1.2) leads to a
decrease in the tip shift. Concerning time for completing solidification, an increase in Bo, /gr or /0, or a
decrease in St, or qsl results in an increase in the solidification time. The effects of these parameters on
the drop height are also investigated.
Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction
Solid–liquid phase change of drops is of crucial importance
because the process widely exists in nature and engineering phenomena. For instance, water drops freeze on leaves [1], airplane
[2], or wind turbine blades [3]. Metal drops solidify in deposition
manufacturing [4] or in atomization [5]. Molten semiconductor
drops crystallized during falling or on cold substrates have been
used for solar cell applications [6–9]. Accordingly, understanding
the complex heat transfer and solidification phenomena is extremely important to advance the above-mentioned technologies.
Concerning liquid drops solidifying on a horizontal plate, a vast
number of works have been conducted. Experimentally, one can
find this problem investigation in one of the pioneering works,
Anderson et al. [10], in which the authors froze a water drop to
demonstrate the accuracy of the proposed dynamic contact angle
model at the tri-junction. An interesting feature observed from
the experiment is the formation of an apex at the drop top after
complete solidification. After then, a few experiments focusing
⇑ Corresponding author.
E-mail address: vuvantruong.pfae@gmail.com (T.V. Vu).
https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.031
0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
on this singularity formed on frozen water drops have been done,
e.g. [11–13]. Recently, this problem has been rapidly growing and
getting much of attention. Jin et al. [14] froze a water drop sessile
on a plate by lowering the plate temperature to a value below the
water freezing point. The ice layer initially formed at the plate
propagated to the drop top to complete the solidification. Similar
works have been done in Zhang et al. [15] and Zhang et al.
[16,17]. Focusing on not only water but also some other semiconductor materials (e.g. silicon, germanium and indium antimonide),
Satunkin [18] reported the formation of conical solidified drops
induced by volume expansion and growth angles at the trijunction. Basing on theoretical analysis supported by the experiments, Satunkin found that the growth angle at the tri-junction
is almost constant except near the end of the solidification process.
Another work concerning the crystallization of a molten silicon
drop can be found in Itoh et al. [9]. Theoretical studies on these
problems can be found in [19–21], in which the authors mostly
paid attention to the solidified drop profile and the temporal evolution of the solidification front assumed to be flat. Concerning
numerical simulations of drops solidifying on a plate, a few studies
have been performed by a boundary integral method [22], a finite
element method [23], an enthalpy-based method [24], a volume of
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
fluid method [17,25], and a front-tracking method in our recent
works [26–29]. However, in these works, the authors considered
only drops attached to a horizontal wall.
Related to drops solidifying on inclined surfaces, there have
been many studies focusing on drops impacting, spreading and
solidifying on the plates. For instance, Jin et al. [30,31] performed
experiments for the freezing process of water on different inclined
surfaces of different materials. To study the freezing process of
asymmetric water drops, Ismail and Waghmare [32] tilted a cold
plate and established the relationship between the asymmetry
(in terms of contact angles) and the tip position and height of
the frozen drops by scaling analysis and experiments. Wang and
Matthys [33] experimentally investigated the heat transfer
between the solidifying drops, of nikel and copper, and a metallic
substrate tilted at an angle of 45°. Numerically, Zhang et al. [34]
used a method of smoothed particle hydrodynamics to find the distribution of the temperature and movement of the solid–liquid
interface of a drop impacting and solidifying on an inclined interface. Yao et al. [35] used the volume of fluid method implemented
in an open source code OpenFOAM [36] to simulate the freezing
process of a water drop. Some other numerical works can be found
in [37,38].
Concerning drops on a vertical wall, Podgorski et al. [39]
reported controlled experiments to show various shapes with
remarkable temporal patterns of a water drop on a vertical plane.
Smolka and SeGall [40] presented the formation of a fingering pattern, which tends to break up into drops, on the surface outside of a
vertical cylinder. Li and Chen [41] experimentally formed frozen
drops on a vertical wall and used the ultrasonic vibration to
remove them from the wall. Numerically, Schwartz et al. [42] used
a long-wave or lubrication approximation-based model to simulate
three-dimensional unsteady motion of a drop on a vertical wall.
Tilehboni et al. [43] used a lattice Boltzmann method to simulate
a liquid drop moving on a vertical wall. However, the abovementioned works have not considered solidification heat transfer.
Even though there have been many numerical studies on the
phase change heat transfer of liquid drops, they mostly focused
on the horizontal plate. The drop solidification on a vertical wall
is rarely found. Accordingly, filling this gap is the main purpose
of the present study because of its importance in academic and
engineered applications [44–47]. We here use a two-dimensional
front-tracking method, for tracking interfaces, combined with an
interpolation technique, for dealing with the non-slip boundary
on the solid surface, [26,48–50] to investigate the deformation
with the solidification of a liquid drop attached to a vertical cold
wall. Various parameters are investigated to reveal their effects
on the solidification process.
2. Mathematical formulation and numerical method
@ qu @ qu2 @ quv
@p @
@u @
@u @ v
þ
þ l
þ
þ
¼ þ 2l
@t
@x @x
@x @y
@y @x
@x
@y
!
þ qg x þ i ðFm þ Fr Þ
Fig. 1. A two-dimensional liquid drop, with an initial center of mass C0(xC0, yC0),
solidifies from a vertical cold wall. Cl(xCl, yCl) is the center of mass of the liquid part
with e = yC0 yCl called ‘‘shift” of the liquid part. Has is the averaged height of the
solidification front.
@ qv @ quv @ qv 2
@p @
@u @ v
@
@v
þ
þ
þ
¼ þ l
þ 2l
@y @x
@y @x
@y
@t
@x
@y
@y
!
þ qg y ½1 bðT T m Þ þ j ðFm þ Fr Þ
@ðqC p TÞ @ðqC p TuÞ @ðqC p T v Þ
þ
þ
@t
@x
@y
Z
@ k@T
@ k@T
þ
þ q_ f dðx xf ÞdS
¼
@x @x
@y @y
f
@u @u
þ
¼0
@x @y
ð1Þ
ð2Þ
ð3Þ
ð4Þ
Here, u = (u, v) is the velocity vector. p is the pressure, g = (gx, gy)
is the acceleration due to gravity. T is the temperature. b is the
!
!
Boussinesq coefficient. f denotes interface. i and j are respectively the unit vectors on the x and y axes. Fm is the momentum
forcing term used for enforcing the non-slip boundary condition
on the solid surface [48,49]. Fr is the interfacial tension force acting on the liquid–gas interface [51]:
Z
Fr ¼
f
rjdðx xf Þnf dS
ð5Þ
where r is the interfacial tension coefficient, j is the curvature, nf is
the unit vector normal to the interface. d(x xf) is the Dirac delta
function, which is zero everywhere except for a unit impulse at
the interface xf. q_ is the heat flux at the solidification interface, given
as
q_ ¼ ks
We consider a liquid drop that attaches to a vertical wall kept at
constant temperature Tc (Fig. 1). The liquid has a fusion temperature Tm higher than the temperature of the wall, Tm > Tc, and thus
at the beginning, a thin liquid layer at the wall changes into solid.
As time progresses, this solid layer evolves to the top of the drop.
Here, we consider only a two-dimensional drop. We assume that
the gas and liquid phases are incompressible, immiscible and Newtonian. In addition, the thermal (thermal conductivity k and heat
capacity Cp) and fluid (density q and viscosity m) properties are
assumed constant in each phase. The one-fluid representation
gives
303
@T
@n
kl
s
@T
@n
l
with the subscripts s, l and g (when available) denoting solid, liquid
and gas. In the present study, volume change induced by the density
difference between the solid and liquid phases is also taken into
account, and thus Eq. (4) is modified as [26]
Z
@u @ v
1 1
1
_
þ
dðx xf ÞqdS
¼
@x @y Lh qs ql f
ð6Þ
where Lh is the latent heat of fusion.
Initially, we assume the drop as a section of a sphere with a contact angle at the wall /0 and the apparent radius R ¼ ½3V 0 =ð4pÞ1=3 ,
where V0 is the initial volume of the drop. To simplify the problem,
the temperature in the entire domain is set to T0 equal to the fusion
temperature, T0 = Tm. In addition, a thin solid layer at Tc with a
thickness of 0.02R is present at the plate at t = 0 [26]. The boundary
conditions are shown in Fig. 1.
304
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
To solve this problem, we use the front-tracking method that
has been extensively used in our previous works for the drop solidification on a plate with free and forced convection (e.g.
[26,27,29,48]). Accordingly, we here briefly describe the method.
The momentum and energy equations are approximated by a
second-order central difference with a second-order predictorcorrector scheme for time integration. The discretized equations
are solved on the fixed, rectangular staggered grid by MAC method
[52]. The interfaces separating different phases are approximated
by a chain of points that moves on the fixed grid (Fig. 1). For the
phase change interface, the points propagate with the velocity Vn
calculated from the heat balance at this interface, i.e.
_ qs Lh Þ. For the liquid–gas interface, its points are updated
V n ¼ q=ð
by the velocity interpolated from the velocities at the nearest fixed
grid points. At the triple points, where these two interfaces meet,
their positions are corrected by imposing a constant growth angle,
/gr = constant [23,26]. In addition, to simplify the problem, we
assume that the growth angles are the same at the upper and lower
triple points. When the solidification process proceeds, the triple
points form the solid–gas interface. Details of the numerical
method can be found elsewhere, e.g. [26,27,29,48].
3. Numerical parameters, grid refinement test and method
validation
2
We scale the length, time and velocity by d = 2R, sc ¼ ql C pl d =kl
and Uc = d/sc, respectively. Thus, the dimensionless time is s = t/sc
with the dimensionless temperature defined as h ¼ ðT T c Þ=
ðT m T c Þ. It is easy to prove that the following parameters govern
the dynamics of the problem [29]:
Pr ¼
C pl ll
;
kl
St ¼
C ps ðT m T c Þ
;
Lh
3
Ra ¼
g y bl ðT m T c Þd
ml al
;
ql g y d2
;
r
l
l
ffi
Oh ¼ pffiffiffiffiffiffiffiffiffiffi
ql d r
h0 ¼
T0 Tc
;
Tm Tc
bgl ¼
bg
;
bl
ksl ¼
ks
;
kl
kg
;
kl
C psl ¼
kgl ¼
Bo ¼
qsl ¼
C ps
;
C pl
ð7Þ
lg
qg
qs
; qgl ¼
; lgl ¼
ql
ql
ll
C pgl ¼
C pg
C pl
ð8Þ
ð9Þ
Here, Pr, St, Bo, Ra and Oh are respectively the Prandtl, Stefan, Bond,
Rayleigh and Ohnesorge numbers. h0 is the initial dimensionless
temperature. Other parameters include the ratios of the thermal
expansion coefficients (bgl), densities (qsl and qgl), viscosities (mgl),
thermal conductivities (ksl and kgl) and heat capacities (Cpsl and Cpgl).
We have found that the variations of Ra in the range of 10–10,000
and bgl in the range of 0.0005–0.5 have almost no effect on the
shape of the solidified drop as shown in Fig. 2. Thus, we keep these
parameters fixed Ra = 1000, and bgl = 0.005. Similarly to our previous works [29], we here focus on the effects of Pr, St, Bo, Oh and
the solid-to-liquid density ratio qsl with various initial liquid drop
shape (i.e. various /0). Accordingly, other parameters are also kept
fixed: h0 = 1.0, ksl = 1.0, kgl = 0.005, Cpsl = Cpgl = 1.0, and qgl = lgl =
0.05. The computational domain is chosen as W L = 2d 6d with
the grid resolution of 256 768 based on the grid refinement study
shown below. The values of the parameters investigated in this
study correspond to drops with diameter in the order of a few millimeters (see the tables shown in [26,27] for reference). As demonstrated in our previous work for the case of solidification affected by
a laminar crossing flow [27], the position of the drop has a very
minor effect on the solidification process. Accordingly, in this study
we skip the investigation of the drop location, and thus the initial
position of the drop is also kept fixed, yC0 = 3.5d.
Fig. 2. Effects of (a) the Rayleigh number and (b) the thermal expansion coefficient
ratio of the gas to liquid phases on the solidified drop profile. Pr = 0.01, St = 0.1, Bo =
1.0, Oh = 0.01, qgl = 0.9, h0 = 1.0, ksl = 1.0, kgl = 0.005, Cpsl = Cpgl = 1.0, qgl = lgl = 0.05,
/gr = 0° and /0 = 105°.
To verify the numerical method applied to the present problem,
we perform a grid refinement study with Pr = 0.01, St = 0.1,
Bo = 1.0, Oh = 0.01, qgl = 0.9, /gr = 0° and /0 = 105°. Two grid resolutions 128 384 and 256 768 are used. Fig. 3 shows a comparison
of the drop profiles at different moments of solidification and the
averaged distance from the solidification front to the wall
(i.e. the averaged height Has defined below) computed by two grids.
Fig. 3 indicates the results with complete agreement between two
grids. Accordingly, we use 256 768 for the rest of the computations presented in this paper.
The numerical method used in the present study has been
extensively used in our previous works. The validations of the
method were also carried out [26,48–50,53]. However, to ease reference, we here present the solidification result of a drop sessile on
the horizontal plate and compare with the experiments of Zhang
et al. [16]. This choice of comparison is due to the lack of a detailed
experiment on the drop solidification on a vertical wall. The initial
volume of the water drop was 10 mL, and the plate temperature
was set to 15.6 °C. Fig. 4 show the results yielded by our method
in comparison with the results reported in [16] (see Figs. 4 and 6 in
[16]). Concerning the conical angle at the top of the frozen drop,
our computation yielded an angle of 140.62° whereas that
observed in [16] was 140°, resulting in an error of 0.44%. Once
again, reasonable agreement is found, supporting the accuracy of
the method used in the present study. Other validation can be
found elsewhere, e.g. [26,48–50].
4. Results and discussion
Fig. 5 illustrates the deformation of the solidifying drop against
the vertical wall with Pr = 0.01, St = 0.1, Bo = 1.78, Oh = 0.01,
qgl = 0.9, /gr = 0° and /0 = 105°. xn and yn in Fig. 5 and the following
figures are the normalized coordinates
xn ¼ x=d;
yn ¼ ðy yC0 Þ=d
ð10Þ
This typical case with /0 = 105° corresponds to a drop attached
to a hydrophobic cold wall [9,15] with purposes of, for example,
reducing ice beds on walls [41] or the formation of ‘‘spherical”
crystallized drops [6,7,9]. At s = 0.18 (Fig. 5a), the gravity starts
to deform the drop whereas the solid–liquid interface parallel to
the wall grows fast because of the large temperature difference
between the wall and the liquid [20]. This results in the variation
of Has with a high slope at the beginning (Fig. 5f), where Has, the
averaged height of the solidification front (Fig. 1), is defined as
Pns
Has ¼
i¼1 xfsi
ns
ð11Þ
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
305
Fig. 3. Grid refinement study with Pr = 0.01, St = 0.1, Bo = 1.0, Oh = 0.01, qgl = 0.9, /gr = 0° and /0 = 105°: (a), (b), (c) drop profile at the different stages of solidification, and (d)
temporal evolution of Has - the averaged height of all points on the solidification front away from the wall (i.e. the average of the x coordinates).
Fig. 4. Comparisons between the present computation and the experiment of Zhang et al. [16] for a water drop solidifying on a horizontal plate: (a) drop profiles at different
stages of solidification and (b) temporal variation of the drop volume Vd normalized by the initial volume V0.
In Eq. (11), xfs and ns are the x coordinate of a point and the number
of points representing the solid–liquid interface. Accordingly, the
solidified drop height Hs is Has at the end of the solidification process. Because of the initially gravitational effect, the drop slightly
deforms with a downward flow in the liquid and an upward flow
in the surrounding gas, i.e., a counterclockwise circulation formed.
Thus, the shift e of the liquid phase (i.e. the shift in the vertical position of the center of mass of the liquid phase) defined as
e ¼ yC0 yCl
ð12Þ
gradually increases. At s = 0.48 (Fig. 5b), the drop becomes more
deformed due to gravity, and the liquid is still falling whereas the
solid–liquid interface continues moving to the top of the drop. This
results in a continuous increase in the shift of the liquid part. At a
later time (s = 0.9, Fig. 5c), the interfacial tension force tending to
hold the drop in a circular shape is against the gravity force and
pulls the liquid upwards. Thereby, a clockwise circulation is formed
around the top of the drop. This leads to a decrease in the shift of
the liquid part, as shown in Fig. 5f. The gravity then pulls down
the liquid phase and forms a counterclockwise circulation again.
This, falling and moving up, happens until the whole drop solidifies,
resulting in a damping wave-like pattern of the liquid phase shift
(Fig. 5f). An interesting feature here is that the movement of the
solidification interface is almost independent of the damped oscillation of the liquid adhered to it (Has in Fig. 5f), and a cone forms at
the top of the drop at the end of solidification because of volume
expansion (Fig. 5e). It means that, like the drop sessile on the plate
with free convection, the solidified drop attached to the vertical
wall still presents a conical tip at the drop top, but the tip location
shifts by distance et = 0.26d (where et is the shift of the solidified
drop tip, i.e. the tip shift) because of gravity (see the definition in
Fig. 5e).
Next, we see how the solidified shape and some other interesting features of the drop are affected by some dimensionless
parameters.
4.1. Effect of the Bond number Bo
One of the most affecting parameters in the present problem is
the Bond number, which is the ratio of the force induced by gravity
to the interfacial tension force. Gravity tends to deform the drop
during solidification. In contrast, the tension force tries to keep
the drop as spherical as possible. Accordingly, as Bo < 1.0, the gravitational force plays a weak role, and thus the drop keeps almost
spherical during solidification, as shown in the left frame of
Fig. 6a. Increasing the value of Bo to a value higher than 1.0, i.e.
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V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
Fig. 5. Evolution of the solidification front with the temperature and velocity fields. Has and e are defined in the text. The parameters are Pr = 0.01, St = 0.1, Bo = 1.78, Oh = 0.01,
qgl = 0.9, /gr = 0° and /0 = 105°. The circles in (f) correspond to the times shown in (a)–(e). The velocity in (a) and all following figures is normalized by Uc.
Fig. 6. Effect of Bo: (a) flow and temperature fields at s = 0.9 for Bo = 0.32 and 3.16, (b) solidified drop shape for various Bo at the end of solidification, (c) temporal variation of
the shift of the liquid part and (d) variations of Hs, ss and et with respect to Bo. The dash-lines in (d) represent the corresponding fittings with ss = 0.0676Bo2 0.0445Bo +
2.4156 and et/d = 0.1723Bo 0.0149. Pr = 0.01, St = 0.1, Oh = 0.01, qgl = 0.9, /gr = 0° and /0 = 105°.
Bo = 3.16 (right frame of Fig. 6a), the situation changes dramatically with large deformation of the solidifying drop. This results
in the difference in the solidification front shape and in the temperature field around it. That is, increasing Bo from 0.32 to 3.16
makes the phase change front change from almost flat shape to a
S-line shape. Thus, the thermal boundary layer around the solidliquid interface becomes inclined for the higher Bo. As a result,
the solidified drop with large deformation is produced with a high
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
Bo, i.e. Bo = 3.16 (Fig. 6b). Fig. 6c confirms that increasing Bo results
in the liquid part more deformed to the bottom, and thus increases
its shift. Consequently, when Bo increases, the solidification process takes longer time to complete, and the solidified drop tip shifts
more down to the bottom, as shown in Fig. 6d. The curve fittings
indicate that the tip shift is linearly proportional to Bo (et/d 0.1723Bo 0.0149) whereas the solidification time increases with
the square of Bo (ss 0.0676Bo2 0.0445Bo + 2.4156). The
increase of the solidification time with Bo is also understandable
from Fig. 6a in which the higher Bo (Bo = 3.16) corresponds to
the smaller liquid-solid phase change area. As a result, it takes
more time to complete the solidification process as Bo increases
[29]. However, the solidified drop height is almost independent
of the variation of the Bond number in the range of 0.1–3.16.
4.2. Effect of the Prandtl number Pr
For Pr = 0.01, at s = 0.36, the liquid part is going down due to
gravity, no oscillation has happened before (left frame of Fig. 7a).
Increasing Pr to a value 10 times higher than (i.e. Pr = 0.1), the liquid is also flowing down but with a higher velocity, resulting in larger deformation and thus in a higher shift e, as shown in Fig. 7c.
Fig. 7c also indicates that the liquid is merely shifting for Pr =
0.01whereas Pr = 0.1 has made the liquid phase experience a little
oscillation. However, this oscillation occurs only at the initial stage
of solidification and damps soon after, whereas in the lower Pr
cases, the oscillation happens later and lasts longer (Fig. 7c). This
is understandable since higher Pr corresponds to higher viscous
resistance that suppresses the oscillation of the drop as the solidification progresses. Consequently, the interface of the solidified
drop becomes smoother (or less wavy) as the value of Pr increases
from 0.01 to 1.0, as shown in Fig. 7b. However, an increase of Pr in
the range of 0.01–1.0 just slightly decreases the solidified drop
height with a slightly increase in et and thus makes the solidifica-
307
tion process finish a little bit sooner, as depicted in Fig. 7d. This is
understandable since, as previously explained, an increase in e corresponding to an increase in Pr (Fig. 7b and c) tends to increase the
solidification time (Fig. 6). However, as demonstrated in our previous work [26], increasing Pr also tends to decrease time for completing solidification. Accordingly, the combination of the both
contrast effects results in a minor effect on the solidification time
(Fig. 7d).
4.3. Effect of the Stefan number St
Fig. 8 shows the effects of the Stefan number on the solidification process on a vertical wall with Pr = 0.01, Bo = 1.0, Oh = 0.01,
qgl = 0.9, /gr = 0° and /0 = 105°. In comparison to St = 0.032
(left frame of Fig. 8a), the solid–liquid phase change front for St
= 0.316 (right frame of Fig. 8a) advances further at s = 0.24 because
of a lower latent heat released resulting a higher growth rate [26].
Because of the fast solidification, St = 0.316 leads to a solidified
drop with less deformation than St = 0.032 (Fig. 8b) and with a considerably reduction in the number of oscillation before complete
solidification (Fig. 8c). As a result of less deformation, the solidified
drop of St = 0.316 is more spherical than that of the lower one, thus
St = 0.316 yields a drop with a lower height than St = 0.032 because
of mass conservation. That is, increasing the Stefan number in the
range of 0.01 to 1.0 results in a slight decrease in the height and tip
shift of the solidified drop with a strongly decrease in the solidification time (ss 0.2853St0.949), as shown in Fig. 8d.
4.4. Effect of Ohnesorge number Oh
Fig. 9a compares the drop profile, the temperature field and the
velocity field at s = 0.36 for two Oh numbers: Oh = 0.00316 (left)
and Oh = 0.0316 (right), in which the stronger flow field corresponds to the lower value of Oh. It seems to be contrast to our
Fig. 7. Effect of Pr: (a) flow and temperature fields at s = 0.36 for Pr = 0.01 and 0.10, (b) solidified drop shape for various Pr at the end of solidification, (c) temporal variation of
the shift of the liquid part and (d) variations of Hs, ss and et with respect to Pr. Bo = 1.0, St = 0.1, Oh = 0.01, qgl = 0.9, /gr = 0° and /0 = 105°.
308
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
Fig. 8. Effect of St: (a) flow and temperature fields at s = 0.24 for St = 0.032 and 0.316, (b) solidified drop shape for various St at the end of solidification, (c) temporal variation
of the shift of the liquid part and (d) variations of Hs, ss and et with respect to St. The dash-line in (d) represents the corresponding fitting with ss = 0.2853St0.949. Pr = 0.01,
Bo = 1.0, Oh = 0.01, qgl = 0.9, /gr = 0° and /0 = 105°.
Fig. 9. Effect of Oh: (a) flow and temperature fields at s = 0.36 for Oh = 0.00316 and 0.0316, (b) solidified drop shape for various Oh at the end of solidification, (c) temporal
variation of the shift of the liquid part and (d) variations of Hs, ss and et with respect to Oh. Pr = 0.01, St = 0.1, Bo = 1.0, qgl = 0.9, /gr = 0° and /0 = 105°.
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
expectation since Oh is the ratio of the viscous force to the interfacial and inertial force. Thus, if Oh increased the drop should deform
more. However, the tendency shown in Fig. 9a–c is understandable
since we keep all parameters, especially Bo, unchanged except for
Oh. Decreasing Oh corresponds to increasing the interfacial tension
force and thus increasing the gravitational force since Bo is fixed.
As a result of the gravitational effect, a decrease in Oh results in
more deformation and in an increase in the number of oscillations
before complete solidification (Fig. 9c). Consequently, the height
and tip shift of the solidified drop slightly decrease as Oh increases.
However, the variation of Oh in the range of 0.001–0.316 has a
minor effect on the solidification time (Fig. 9d).
309
three-phase line arranges in such a way that the slopes of the
liquid–gas and solid–gas interfaces (to the vertical line) decrease
(to induce the increase in growth angle). Thereby, the solidified
phase becomes smaller in the vertical direction but higher in the
horizontal direction [18,26,27] (Fig. 10b). As a result of increasing
the drop height (Fig. 10d), the time for completing solidification
also linearly increases with the growth angle [26] (ss 1.0524/gr + 2.4428 where /gr is measured in radian).
However, varying the growth angle has a very minor effect on
the number of oscillations of the liquid phase during solidification,
as shown in Fig. 10c, and on the tip shift (Fig. 10d).
4.6. Effect of the density ratio of the solid to liquid phases qsl
4.5. Effect of the growth angle /gr
Our literature survey [18,23,27,28] indicates that the growth
angle almost keeps constant during solidification and strongly
affects the shape of the solidified drop. The value of the growth
angle is dependent on the drop material (e.g., /gr = 0° for water,
/gr = 12° for silicon, /gr = 14° for germanium, and /gr = 28° for
indium antimonide [18,23]).
For the solidifying drop against the vertical wall, we also
observe such effect of the growth angle as shown in Fig. 10, in
which the angle is varied in the range of 0–20°. The comparison
in the velocity field for /gr = 4° and /gr = 20° indicates that the
growth angle has a minor effect on the flow field, but the higher
/gr produces a more conical solidified drop. As a result, the drop
height of the solidified drop linearly increases with the growth
angle (Hs/d 0.3556/gr + 0.8188 where the unit of /gr is radian).
This can be understandable since at the triple point, three phases
meet, and the interfacial tension energy balance locally appears
with the solid–liquid front almost perpendicular to the liquid–
gas interface. Accordingly, when the growth angle increases the
It has been reported that the density difference between the
liquid and solid plays an important role in the final product of
the drop solidification when the drop size is in a range of a few millimeters [18,28]. After complete solidification, the solidified drop
tends to possess a dimple if the solid-to-liquid density ratio is
greater than unity whereas the conical top surface is present if
the solid density is lower than the liquid. This is evidently shown
in Fig. 11b where the solidified drop profiles for three density
ratios qsl = 0.8 (expansion), 1.0 (no volume change) and 1.2
(shrinkage) are depicted. The reason is that, for small drops, the
interfacial tension force can hold the drop as a section of a sphere,
and other force, i.e. the gravity in this study, slightly deforms the
drop. Then, in such situation, the volume change would impose
its evident impact on the drop by expanding the drop for qsl < 1.0
or shrinking it for qsl > 1.0 during solidification. Such effect results
in the flow away (qsl < 1.0) or towards (qsl > 1.0) the solid phase
near the solid–liquid interface (Fig. 11a). During the final stages
of solidification, the liquid drift away (qsl < 1.0) or towards
(qsl > 1.0) the solidifying front leads to the formation of an apex
Fig. 10. Effect of /gr: (a) flow and temperature fields at s = 0.9 for /gr = 4° and 16°, (b) solidified drop shape for various /gr at the end of solidification, (c) temporal variation of
the shift of the liquid part and (d) variations of Hs, ss and et with respect to /gr (the linear fittings give Hs/d = 0.3556/gr + 0.8188, ss = 1.0524/gr + 2.4428 and et/d = 0.0566/gr +
0.14, with /gr measured in radian). Pr = 0.01, St = 0.1, Bo = 1.0, Oh = 0.01, qgl = 0.9 and /0 = 105°.
310
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
Fig. 11. Effect of qsl: (a) flow and temperature fields at s = 0.84 for qsl = 0.8 and 1.2, (b) solidified drop shape for various qsl at the end of solidification, (c) temporal variation of
the shift of the liquid part and (d) variations of Hs, ss and et with respect to qgl. The dash-lines in (d) represent the corresponding fittings with Hs/d = 1.2617q2 3.4247q +
2.8788, ss = 2.3028q2 6.6865q + 6.6019 and et/d = –0.1776q + 0.3048, where q = qsl. Pr = 0.01, St = 0.1, Bo = 1.0, Oh = 0.01, /gr = 0° and /0 = 105°.
Fig. 12. Effect of /0: (a) flow and temperature fields at s = 0.3 for /0 = 60° and 120°, (b) solidified drop shape for various /0 at the end of solidification, (c) temporal variation of
the shift of the liquid part and (d) variations of Hs, ss and et with respect to /0 (the fittings, i.e. dash-lines, give Hs/d = 0.3385/ + 0.1855, ss = 0.4679/3 1.3536/2 + 2.6273/ 0.7035 and et/d = 0.061/3 0.1442/2 + 0.0975/ + 0.0726, with / = /0 measured in radian). Pr = 0.01, St = 0.1, Bo = 1.0, Oh = 0.01, qsl = 0.9 and /gr = 0°.
V.N. Duy, T.V. Vu / International Journal of Heat and Mass Transfer 127 (2018) 302–312
or dimple at the top of the drop [18,28]. Consequently, the drop is
more deformed (Fig. 11c) with an increase in the height and in the
solidification time as the density ratio is decreased as shown in
Fig. 11d. If we define the tip of the solidified drop as the last solidification point or the farthest point from the wall, then we have the
variation of the tip shift with respect to qsl, as shown in Fig. 11d, in
which the tip shift linearly decreases with an increase in qsl (et/d =
0.1776qsl + 0.3048). This tendency is consistent with intuition
that increasing qsl results in smaller solidified drops (Fig. 11b).
4.7. Effect of the contact angle /0
To conclude, we examine how the drop shape, in terms of the
contact angle induced by the wall wettability, affects the process.
Fig. 12a compares the drop behavior with high (left) and low
(right) wettability. Concerning the solidifying interface, at this time
(s = 0.3) it has a concave-up arc near the triple point for /0 = 60°
whereas /0 = 120° produces a concave-down shape at this point
since the solid–liquid front tends to be perpendicular to the liquid–gas front [13]. However, when half of the drop is solidified,
the solid–liquid front for /0 = 120° will behave similarly to /0 =
60°, that is, a concave-up arc near the triple point [13,28]. As a
result of the high contact angle resulting in a high drop, the liquid
part experiences more oscillation (Fig. 12c) with a more deformed
solidified drop after complete solidification (Fig. 12b) when the
contact angle increases. This leads to an increase in the solidified
drop height as well as the solidification time. As a result, the location of the tip of the solidified drop shifts more to the bottom with
the increase in /0 in the range of 45–135° (Fig. 12d). Fig. 12b also
suggests that, large contact angles induced by drops on hydrophobic walls could cause the drops to easily fall off the wall after complete solidification because of gravity, and thus enhance the
deicing process [41].
5. Conclusion
We have presented a fully resolved two-dimensional numerical
investigation of a liquid drop solidifying on a vertical cold wall by
the front-tracking method to track the temporal movement of the
interfaces. The non-slip velocity boundary condition on the solid
surface is satisfied by a linear interpolation technique. The gravitational force results in an asymmetric drop whose conical tip
induced by volume expansion is shifted down after complete solidification. We varied the Prandtl number Pr (in the range of 0.01–
1.0), the Stefan number St (in the range of 0.01–1.0), the Bond
number Bo (in the range of 0.1–3.16), the Ohnesorge number Oh
(in the range of 0.001–0.316), the density ratio of the solid to liquid
phases qsl (in the range of 0.8–1.2), and the growth angle /gr (in the
range of 0–20°) and the contact angle /0 at the wall (in the range of
45–135°) to show their influences on the shape, height and tip shift
of the solidified drop and the solidification time. The numerical
results show that the solidified drop shape is strongly influenced
by Bo, St, Oh, qsl, /gr and /0. The location of the solidified drop
tip shifts more to the bottom as we increase any one of Bo and
/0 or decrease any one of St, Oh and qsl. For instance, the tip shift
et varies with respect to Bo by et/d 0.1723Bo 0.0149 and with
respect to qsl by et/d 0.1776qsl + 0.3048. The height Hs of the
solidified drop strongly increases with an increase in /gr (Hs/d 0.3556/gr + 0.8188 with /gr measured in radian) and /0 (Hs/d 0.3385/0 + 0.1855 with /0 measured in radian) or with a decrease
in qsl (Hs/d 1.2617q2 3.4247q + 2.8788). Concerning time ss for
completing the solidification process, the most affecting parameters are Bo (it increases with Bo by ss 0.0676Bo2 0.0445Bo +
2.4156), St (it decreases with St by ss 0.2853St0.949), /gr
(it increases with /gr in radian by ss 1.0524/gr + 2.4428), qsl
311
(it decreases with qsl by ss 2.3028q2sl 6.6865qsl + 6.6019) and
/0 (it increases with /0 in radian by ss 0.4679/30 1.3536 /20 +
2.6273/0 0.7035).
Even though our study is very detailed, many questions are still
unresolved. For instance, the present calculations are merely twodimensional, and indeed for a ridge rather than a drop, and thus
three-dimensional (3D) simulations would circumvent the present
limitation. However, one of the most difficulties in 3D computations is to handle the growth angle (or the contact angles) at the
tri-junction (i.e. three-phase line). In addition, the contact angles
might be varied along the three-phase line because of the gravitational effect. The drop might slide on the wall before starting solidification, and thus the simulation requires the dynamic contact
angle and nucleation models. In some cases, the drop might pinch
off before completing solidification, and thus further investigations
need to be done to reveal the parameter ranges in which the drop
pinch-off happens.
Acknowledgments
This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 107.03-2017.01. The authors are grateful to Prof. John C.
Wells at Ritsumeikan University, Japan for facilitating computing
resources.
Conflict of interest
We have no conflict of interest to declare.
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