Study of the effect of surface wettability on droplet impact on spherical surfaces

Abstract

The effect of surface wettability on droplet impact on spherical surfaces is studied with the CLSVOF method. When the impact velocity is constant, with the increase in the contact angle (CA), the maximum spreading factor and time needed to reach the maximum spreading factor (tmax) both decrease; the liquid film is more prone to breakup and rebound. When CA is constant, with the impact velocity increasing, the maximum spreading factor increases while tmax decreases. With the curvature ratio increasing, the maximum spreading factor increases when CA is between 30 and 150°, while it decreases when CA ranges from 0 to 30°.

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International Journal of Low-Carbon Technologies 2020, 15, 414–420
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doi:10.1093/ijlct/ctz077 Advance Access publication 20 January 2020 414
Study of the eect of surface wettability on
droplet impact on spherical surfaces
..............................................................................................................................................................
Xiaohua Liu1,*, Kaimin Wang1,YaqinFang
1,R.J.Goldstein
2and Shengqiang Shen1
1Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of
Education, School of Energy and Power Engineering, Dalian University of Technology,
116024, Dalian, China; 2Department of Mechanical Engineering, University of Minnesota,
MN 55455, Minneapolis, USA
............................................................................................................................................
Abstract
The eect of surface wettability on droplet impact on spherical surfaces is studied with the CLSVOF method.
When the impact velocity is constant, with the increase in the contact angle (CA), the maximum spreading
factor and time needed to reach the maximum spreading factor (tmax)bothdecrease;theliquidlmismore
prone to breakup and rebound. When CA is constant, with the impact velocity increasing, the maximum
spreading factor increases while tmax decreases. With the curvature ratio increasing, the maximum spreading
factor increases when CA is between 30 and 150◦,whileitdecreaseswhenCArangesfrom0to30
◦.
Keywords: droplet impact; surface wettability; spreading factor; impact velocity; curvature ratio
∗Corresponding author:
lxh723@dlut.edu.cn
Received 06 September 2019; revised 23 November 2019; editorial decision 10 December 2019; accepted 10
December 2019
................................................................................................................................................................................
1INTRODUCTION
Dropletimpactonasolidsurfaceorliquidlmoccursfrequently
in nature or industrial elds [1–4]. Droplet impact phenomenon
can be observed in the following practical applications. During
the cooling process inside a spray cooling tower, water droplets
impactontoheatedtubes.Inatricklebedreactor,whengasand
liquid droplets ow downward, the droplets impact with the cata-
lyst. In pharmaceutical manufacturing, solid particles are brought
into a reactor and impact with liquid. In an IC engine, petrol or
diesel fuel is injected onto a piston crown. Typical applications are
also found in the evaporators of the multi-eect distillation sea-
water desalination system. The seawater droplets impact outside
the surface of a horizontal tube during lm evaporation.
In 1876, Worthington [5,6] rstly conducted experiments to
observe the phenomena of droplet impacting a horizontal smoked
plate.Sincethen,drivenbytheinterestsofthephenomenaand
the importance of the phenomena in numerous industrial applica-
tions, researches on droplet impact were investigated all the time.
Kannan et al.[7] experimentally explored the droplet impact
behavior on a hydrophobic grooved surface. The result indicates
that a trough structure changes the shape of liquid lm spreading,
and enhancing hydrophobicity of groove surface would be more
conducive to droplet rebound. Shen et al.[8,9] did a numerical
simulation of the deformation process aer droplet impacting a
at surface. The results show that with the decrease in the contact
angle, the droplet spreading factor increases, and the oscillation
frequency of droplet slows down, reaching equilibrium sooner.
The volume-of-uid (VOF) method was applied to explore the
droplet spreading process with dierent wettability on an inclined
surface. Under the same conditions, a non-wetting surface results
in partial crushing of droplet. Li et al.[10,11] experimentally
studied the droplet impact process with dierent temperatures
(50–120◦C) and wettability. Results indicate that on part of
hydrophobic surfaces, the droplet retract height increases with
surface temperature, while on hydrophilic and super-hydrophobic
surfaces, this rule disappears. The dynamic characteristics of
droplet were also studied when impacting the surface where
wettability had radial and axial symmetry gradient distribution.
Quan et al.[12] rst used a pseudopotential model of the lattice
Boltzmann method and proved that retract speed and rebound
tendency both increase with a better wettability. Antonini
et al.[13] experimentally studied surface wettability eects
on the characteristics of water droplet impact. When Weber
number is between 30 and 200, wettability aects both droplet
maximum spreading distance and spreading characteristic
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Study of the eect of surface wettability on droplet impact on spherical surfaces
time. When Weber number is higher than 200, wettability
eect is secondary because of the increase in inertial force.
The VOF method was used by Liang et al.[14] to simulate
the individual droplet dynamic behavior when droplets impact
dierent wettability surfaces with dierent impact velocities.
Calculation results show that a more hydrophobic surface leads
to droplet rebound. The maximum spreading factor increases
with the decrease in the contact angle, and the time needed to
reach the maximum spreading factor reduces accordingly. The
dynamic characteristics of droplet impact on curved surfaces are
dierent from those on at surfaces. Experimental and theoretical
investigations of the impact of a droplet onto spherical target
were conducted by Bakshi et al.[15] during which spatial and
temporal variation of lm thickness on the target surface was
measured. When the target size is increased with respect to the
droplet size, the lm thinning process is slower with a larger value
of residual thickness. Mitra et al.[16] completed a theoretical
and experimental study on the eect of a spherical surface with
high thermal conductivity on the super-cooled droplet impact
process. Liang et al. [17] observed the phenomenon of liquid
drop impact on wetted spherical surfaces at low impact velocity,
using a high-speed camera. Experimental observations showed
that the drop rebound and partial rebound phenomena may
occuratlargeviscosityandlowimpactvelocity,whichcannot
be observed at small viscosity. Wang [18] studiedtheeectofthe
contact angle on the droplet impact on dry spherical surfaces by
thecoupledlevelsetandvolumeofuid(CLSVOF)method.It
wasshownthattheincreaseinthecontactanglehindersdroplet
spreading but promotes retracting [19].Theliquidlmoscillates
when the contact angle is relatively small. The oscillation period
decreases with increasing contact angle, while it increases with
the increment in the impact velocity [20].Zhanget al.[21]
used the two-dimensional lattice Boltzmann model with multi-
relaxation time (MRT) to simulate the liquid droplet impact
on a curved target. The eects of the Reynolds number, the
Weber number and the Galilei number on the ow dynamics
were investigated. Zheng et al.[22] explored the mechanism of
dropletimpactonasphericalconcavesurfacebytheCLSVOF
method. It was found that droplet impact on a spherical concave
surface showed a smaller spread factor, earlier time of central
normal jet and larger jet velocity than the impact on a at surface.
Liang et al.[23] experimentally investigated the heptane drop
impact dynamics on wetted spheres, using a high-speed camera.
The result indicates that the sphere-drop curvature ratio can
greatly inuence the splashing thresholds. Collision of a droplet
onto a still spherical particle was experimentally investigated
by Banitabaei et al.[24] during which the eect of wettability
on collision was studied. It was found that for droplet impact
onto a hydrophilic particle, the droplet is neither disintegrated
nor stretched enough to form a liquid lm aer impact in the
entire velocity range studied. However, on a hydrophobic particle,
when Weber number achieves about 200 or larger, a liquid lm
forms aer impact. With the Weber number increasing, the
lamella length and cone angle increase accordingly. Another
main conclusion is that increasing the contact angle from the
hydrophilic to hydrophobic zone has a considerable eect on
geometry of the liquid lm and lamella formation. However,
whenthecontactangleexceedsathresholdvalueof110
◦,the
increase in the contact angle has little eect on lamella geometry.
A similar lamella formed aer droplet impacting a small disk was
observed by Rozhkov et al.[25] with a high-speed photography
technique. Numerical researches concerning various outcomes
during single liquid droplet impact on tubular surfaces with
dierent hydrophobicity values were carried out by Liu et al.[19]
by the CLSVOF method. When the impact velocity is constant, the
increase in the surface hydrophobicity values is detrimental to the
spreadoftheliquidlmontubularsurfaces.Thelargerthesurface
contact angle is, the more likely the droplet rebound takes place.
Chen et al.[26] conducted numerical studies on the successive
impact of double droplets on a super-hydrophobic tube, using the
CLSVOF method. The results showed that the impact model is
dominated by the impact velocity: the out-of-phase impact takes
place when the impact velocity ranges from 0.25 to 1.25 m/s, while
the in-phase impact takes place when the impact velocity ranges
from 1.44 to 2 m/s. Meanwhile, the occasion of the liquid crown
is inuenced by the impact velocity and curvature ratio.
In conclusion, the research on the droplet impacting process
had a certain foundation. However, these researches mostly
focused on at surfaces, and furthermore, parameters, such as the
impact velocity, droplet size and spherical curvature, needed to be
investigated further. In this paper, a two-dimensional numerical
simulation with the CLSVOF method is used to explore droplet
dynamic characteristics of the impacting process on spherical
surfaces. The impact velocity and curvature ratio are mainly
studied when droplets impact spherical surfaces with dierent
wettability; the degree of surface wettability is represented by the
contact angle, CA.
2PHYSICAL AND MATHEMATICAL MODEL
2.1 Physical model
As shown in Figure 1, a droplet impacts a spherical surface with a
certain initial velocity, and the impact direction is parallel with the
center line of droplet and sphere. The moment when the droplet
contacts the sphere is considered as the zero moment, tsignies
the evolution time of droplet impacting process, in milliseconds.
In addition, usignies the impact velocity, in meters per second,
the ratio of the droplet falling distance to the time interval of
which, in this paper, the impact velocity is the velocity when
droplet collide with the surface the rst time; d0signies the initial
diameter of droplet, mm; and Dis the diameter of the solid sphere,
mm. Other related parameters are dened as follows:
T∗=t·u/d0(1)
D∗=d/d0(2)
H∗=h/d0(3)
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X. Liu et al.
Figure 1. Aphysicalschematicdiagramofthedropletimpactonaspherical
surface.
ε=d0/D(4)
where
T∗is the dimensionless time, D∗is the spreading factor, dis the
liquid lm spreading diameter along the sphere, in millimeters;
H∗is the dimensionless liquid lm central height; his the liquid
lm central height (when the liquid lm rebounds into air, h
represents the vertical distance from the highest point of liquid to
the highest point of sphere), in millimeters; and εis the curvature
ratio.
2.2 Mathematical model
In this paper, the numerical simulations are accomplished by
ANSYS Fluent 14.5 soware. Droplet impacting a spherical sur-
face is a transient process. The CLSVOF method is used by
coupling VOF with level-set methods together and the advantage
of which is that parameters describing phase interface with higher
sharpness can be given out accurately. On the basis of mass and
momentum conservation, a two-dimensional laminar axisym-
metric model and Pressure-Implicit with Splitting of Operators
(PISO) algorithm which has faster convergence speed are used.
Liquid droplet is considered incompressible, and the calculation
area is isothermal during the impact process. The droplet is water.
A non-slip boundary condition is used with constant temperature,
and the wall is considered smooth without roughness.
For the liquid, the control equations of mainstream eld are
shown as follows [27].
∇v=0(5)
ρ∂v
∂t+ρv•∇v=−∇p+∇•v∇v+(∇v)T−σκδ(φ)∇φ+ρg
(6)
where
Figure 2. The comparison of the experiment results and simulation results.
vis the velocity vector; ρis the density; pis the pressure; σis the
surface free energy; gis the gravitational acceleration; and κis the
interface curvature which can be calculated in Equation 7.
κ=∇•∇φ
|∇φ|(7)
where
δ(φ)can be calculated with the following formula.
δ(φ)=1+cos(πφ/a)
2a|φ|<a
0|φ|≥a(8)
where
a= 1.5ω,andωis the minimum mesh size.
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Study of the eect of surface wettability on droplet impact on spherical surfaces
Figure 3. PhasediagramofdropletimpactonsurfaceswithdierentCA.
3RESULTS AND DISCUSSION
3.1 Simulation model verication
Figure 2 shows the comparison between the experimental results
from Mitra et al.[16] and the numerical results from the simu-
lation model used in this paper. The initial droplet diameter is
3.1 mm, the spherical diameter is 10 mm, the contact angle is 90◦
and the impact velocity is 0.431 m/s.
According to Figure 2,wecanndthattwo-dimensional
numerical simulation results are consistent with the experimental
resultsatmoststagesofdropletimpact.Atthelatertime,there
just presents a little retraction lags by two-dimensional simulation
results.
3.2 Simulation results and analysis
The phase diagram of droplets impacting spherical surfaces is
shown in Figure 3. The curvature ratio is 0.2, the impact velocity
is 0.7 m/s and CA is 60◦,90
◦, 120◦and 150 ◦,respectively.From
Figure 3, the dynamic characteristics of droplet aer impact are
dierent with dierent contact angles. With the increment in CA,
the liquid lm is more prone to breakup and rebound. When CA
is 90◦, center sag appears at 9.2 ms. When CA is 120◦,droplet
complete rebounds and forms into a gourd-shape (17 ms). When
CA is 150◦, complete rebound takes place, and the rebound height
is higher than that with CA of 120◦.
The changes of the spreading factor (D∗)andthedimensionless
liquid lm central height (H∗) under dierent surface wettability
are shown in Figures 4 and 5, respectively. The curvature ratio is
0.2, the impact velocity is 0.7 m/s and CA is 60◦,90
◦, 120◦and
150◦,respectively.
As shown in Figure 4,withthedecreaseinCA,themaximum
spreading factor (D∗max)andtmax increase. Figure 5 shows that
the greater CA is, the higher the droplet rebound height and the
longer the rebound time is. The main reason is that more energy
remains aer retraction under a larger contact angle. When CA is
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X. Liu et al.
Figure 4. The changes of D∗versus T∗under dierent contact angles.
Figure 5. The changes of H∗versus T∗under dierent contact angles
Figure 6. Relation between CA and D∗max under dierent impact velocities.
Figure 7. Relation between CA and tmax under dierent impact velocities.
Figure 8. Relation between CA and D∗max under dierent curvature ratios.
Figure 9. Relation between CA and tmax under dierent curvature ratios.
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Study of the eect of surface wettability on droplet impact on spherical surfaces
60◦, droplet partial rebound occurs aer droplet reaches D∗max,
the droplet retraction makes the droplet gather and then the
liquid lm oscillates on the surface. The spreading factor and
the dimensionless lm central height attenuate gradually during
oscillation. When CA is 90◦,thedropletalsopartiallyrebounds
aer droplet reaches D∗max , then complete rebound happens aer
further retract. When CA is 120◦or 150◦,completerebound
occurs aer spreading; the larger the contact angle is, the earlier
the droplet leaves from the surface.
With dierent impact velocities, the relationship between CA
and D∗max andtherelationshipbetweenCAandtmax are shown in
Figures 6 and 7, respectively. The curvature ratio is 0.2, the impact
velocities are 0.431, 0.7 and 1 m/s.
Figure 6 shows that with the decrement in CA, the maximum
spreading factor becomes greater aer droplet impact. This is
because a better hydrophilicity is benecial to droplet spread.
For the same contact angle, the maximum spreading factor
increases with the impact velocity. The reason is that the greater
initial kinetic energy helps droplet to spread further. As shown
in Figure 7, under the same impact velocity, the greater CA is,
the smaller tmax is. Under the same contact angle, the lower the
impact velocity is, the larger tmax is.
Under dierent curvature ratios, the relationship between CA
and D∗max and the relationship between CA and tmax are shown
in Figures 8 and 9, respectively. The impact velocity is 0.7 m/s, and
the curvature ratios are 0.15, 0.2 and 0.25, respectively.
As shown in Figure 8, under the same curvature ratio, the
maximum spreading factor decreases with the increase in CA.
The maximum spreading factor increases with the curvature ratio,
when CA is between 30 and 150◦.Thisisbecausethelargerthe
curvature ratio becomes, the greater the inuence of gravity on
the spreading process is. When CA ranges from 0 to 30◦,the
maximum spreading factor decreases with increasing curvature
ratio. The reason is that a better hydrophilicity promotes droplet
spread and the liquid lm can spread completely along the surface,
and then some part liquid can even gather at the bottom of the
sphere [24]. Due to high wettability of the surface and the gravity,
moreandmoreliquidwouldgatheratthebottomofthesphere
untilitbreaksup,duringwhichtheupperpartofthesphere
would expose, so the spreading factor decreases. In this case, the
increment in the curvature ratio can benet the gathering at the
bottom of the sphere, which causes the upper part of the sphere
to be exposed earlier and more quickly.
Figure 9 shows that under the same curvature ratio, with the
increment in CA, the time needed to reach the maximum spread-
ing factor decreases. Surface wettability aects both the maximum
spreading factor and spreading characteristic time signicantly
[13]. Under the same contact angle, with the increment in ε,the
time needed to reach the maximum spreading factor increases.
4CONCLUSION
Theeectofsurfacewettabilityondropletdynamiccharacter-
istics aer impacting is studied with the CLSVOF method. The
main conclusions are as follows:
(i) With the increase in CA, the liquid lm is more prone to
breakup and rebound aer droplet impact [19]. The rebound
height and rebound time increase with CA. When complete
rebound occurs, the larger CA is, the earlier the droplet rebounds
away from the surface. The maximum spreading factor and time
needed to reach the maximum spreading factor decrease with the
increment in CA.
(ii) With the increase in the impact velocity, the maximum
spreading factor increases while the time needed to reach the
maximum spreading factor decreases.
(iii) The greater the curvature ratio is, the longer the time
needed to reach the maximum spreading factor is. Within the
range of contact angle from 30 to 150◦,thebiggerthecurvature
ratio is, the greater the maximum spreading factor is. However,
within the range of the contact angle from 0 to 30◦,thebiggerthe
curvature ratio is, the smaller the maximum spreading factor is.
DISCLOSURE STATEMENT
No potential conict of interest was reported by the authors.
ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foun-
dation of China (No. 51476017) and Department of Mechanical
Engineering, University of Minnesota.
FUNDING
Prof. Shen Xiao Wei Yang the Key Program of National Natural
Science Foundation of China (NO.51936002) aer (NO.51476017).
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