Experimental Thermal and Fluid Science 31 (2006) 97–110
www.elsevier.com/locate/etfs
Phenomena of droplet–surface interactions
Š. Šikalo *, E.N. Ganić
Faculty of Mechanical Engineering, University of Sarajevo, Vilsonovo setaliste 9, Sarajevo, Bosnia and Herzegovina
Received 11 September 2005; received in revised form 9 December 2005; accepted 8 March 2006
Abstract
The fluid flow associated with impinging droplets is rather complicated and not understood in detail. Depending on the circumstances,
different characteristic features are observed. Various phenomena can appear when a droplet impacts a surface. The outcome of an
impact depends on the droplet properties and of the impacted surface, which may be, for example, a dry or liquid surface. Following
a dimensional analysis, the number of independent parameters can be reduced to a set of dimensionless groups that governs the process
considered. An important dimensionless parameter for impact processes is the impact Weber number. However, this number alone is not
sufficient to classify different types of droplet impact. Different outcomes of droplet–surface interaction will be discussed in the present
paper. Our consideration is limited to the interaction of single droplets with different horizontal and inclined surfaces.
2006 Elsevier Inc. All rights reserved.
Keywords: Droplet impact; Droplet rebound; Partial rebound; Wettability
1. Introduction
In most practical processes, the fluid-flow phenomena
during droplet impact are followed or accompanied by heat
transfer or phase change, liquid evaporation in spray cooling, and solidification during spray forming and spray
coating. Detailed knowledge of the transport phenomena
involved in practical operations that employ droplet
impingement on solid materials is critical for the overall
process development and further advancement. A comparison of the time scales of the fluid dynamics of droplet
spreading to the time scales of the associated heat transfer
processes shows that the former scales are markedly smaller [1]. Hence, it appears that the droplet spreading occurs
largely, first, and the heat transfer follows. It is then relevant to investigate initially the fluid dynamics of a droplet
impinging on a surface to provide a better understanding of
the transport processes taking place in engineering
applications.
*
Corresponding author. Tel.: +387 33 656562; fax: +387 33 653055.
E-mail addresses: sikalo@mef.unsa.ba (Š. Šikalo), ganic@mef.unsa.ba
(E.N. Ganić).
0894-1777/$ - see front matter 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.expthermflusci.2006.03.028
In industry, the impact of droplets on walls and films
occurs in a variety of forms; however, the desired effect
may vary according to application. For instance coating
and painting applications strive for a total deposition
(spreading) of droplets, whereas in fuel injection or respirations, the impact is used to generate smaller droplets through
splashing, albeit sometimes unintentionally. Although most
industrial situations involve a polydispersed ensemble of
droplets or a spray, a basic understanding of the impact
process is better obtained using simple droplets of monodispersed size. This allows systematic variation of the influencing parameters and permits submodels to be formulated for
the numerical treatment of the problem in computational
fluid dynamics codes.
Recently there has been significant research directed
towards this phenomenon, empirically, numerically and
analytically, leading to a rather broad set of conditions
for which the outcome of impact can be predicted. A noteworthy exception is the impact of droplets on highly
inclined surfaces. This condition occurs in heat exchangers
and on airplane wings (icing).
Numerous reviews of the subject are available, dealing
with a wide span of impact conditions and surface properties [2–4]. Correspondingly, the desire and need to predict
98
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
Nomenclature
Ca
d
D
Oh
Ra
Re
t
t*
u
We
x
y
impact capillary number, Ca = We/Re
spreading diameter, m
droplet diameter before impact, m
Ohnesorge number, Oh = We1/2/Re
average surface roughness, lm
impact Reynolds number, Re = quD/l
time after impact, s
non-dimensional time after impact, t* = tu/D
impact velocity, m/s
impact Weber number, We = qu2D/r
spread factor in forward/backward direction
apex height, m
h
l
q
r
static contact angle
dynamic viscosity of the liquid, Pa s
density of the liquid, kg/m3
surface tension of the liquid, N/m
Subscripts
h
horizontal
n
normal
v
vertical
a
advancing
r
receding
s
secondary
t
tangential
Symbols
a
droplet impact angle
d
liquid film thickness
these phenomena are also large, and both theoretical and
numerical approaches have been taken. The most recent
theoretical work [5] is valid up to moderate impact Weber
numbers, We = qDu2/r with q and r denoting the density
and surface tension of the liquid and D and u the diameter
and impact velocity magnitude of the droplet. This theory
has also been extended to the case of two-drop impacts [6],
reflecting applications associated with predicting the results
of spray impingement. More numerous are the numerical
predictions of drop impact, typically using a finite volume
and volume-of-fluid (VOF) approach [7–12].
2. Normal droplet impact onto dry and wetted surfaces
2.1. Experimental technique
To investigate the droplet–surface interaction a series of
experiments were conducted with individual droplets
impacting onto solid, dry and wetted surfaces. The experimental method has been previously described in [13].
A high resolution charge-coupled device (CCD) camera
(Sensicam PCO, 1280 · 1024 pixels) equipped with a long
distance microscope is used to observe the spreading droplet in detail. The magnification was manipulated so that the
image could accommodate the maximum spread of the
droplet. From the side-view images, the spread factor and
apex height are measured.
In the present study experimental data on droplet
spreading on the surfaces are presented for a range of
Weber numbers (We = 50–1063) based on droplet velocity
and the liquid density. Three liquids were selected for their
characteristic liquid properties, distilled water, isopropanol
and glycerin (85 vol.% glycerin/water solution), to study
the effect of liquid surface tension and viscosity. Two target
surfaces of glass, one smooth (with roughness of Ra =
0.003 lm amplitude) and one rough (Ra = 3.6 lm), and
one surface of smooth wax (Ra = 0.3 lm) were used.
The droplet’s initial diameter, the physical properties of
the liquids and the wettability of the surfaces (advancing ha
and receding hr contact angles) are shown in Table 1. The
surface tension, density and viscosity values were collected
from standard tables of physico-chemical liquid properties.
The thickness of the liquid film was measured (with
accuracy of ±5 lm) directly by comparing an image of
the film surface to a reference image of the dry plate. The
film thickness was in the range of 40–800 lm, depending
on the inclination and liquid used in the experiment. The
deposition–rebound limit of a single droplet was studied
for dry and wetted surfaces.
The inclination angle of the impact plate was measured
with an accuracy of ±0.1 (95% confidence). The impact
velocity was derived from two consecutive images immediately before impact with an accuracy of 0.02 m/s (95% confidence) at Weber numbers of 50 and 0.06 m/s at Weber
Table 1
Properties of the liquid droplets and wettability of the surfaces
Liquid
Water
Isopropanol
Glycerin
D (mm)
2.7
3.3
2.45
r (N/m)
0.073
0.021
0.063
l (mPa s)
1.0
2.4
116
q (kg/m3)
996
786
1220
ha–hr ()
Smooth glass
Rough glass
Smooth wax
10–6
0
17–13
78–16
0
62–12
105–95
0
97–90
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
numbers of 1063. Considering also the ±6 lm uncertainty
in the droplet diameter determination, the uncertainty in
the calculated Weber number ranged from ±2 at Weber
numbers of 50 and ±40 at Weber numbers of 1063.
In this case the results of a droplet impact can be spreading (deposition), splashing or recoiling. The term deposition
indicates impact without the production of any secondary
droplets. In contrast to this, the term splash is characterized
by the formation of secondary droplets. The transition from
deposition to splash happens with increasingly energetic
impact onto smooth surfaces or with the influence of surface roughness.
A qualitative comparison of the impact of water and
glycerin droplets on the smooth glass (with average surface
roughness Ra = 0.003 lm) is illustrated in Fig. 1. On
impact, onto a dry smooth surface, a thin liquid film
(lamella) begins to jet radially outward over the solid surface. The spreading velocity of a glycerin droplet is lower.
The lamella thickness is much higher and no visible perturbations appear on the rim during the spread of a glycerin
droplet.
The spread factor (d/D) and apex height (y/D) of droplets of different liquids are compared, as a function of nondimensional time after impact (tu/D), to examine the effect
of droplet viscosity. Experimental data, characterized by
nearly the same Weber numbers (about 90) and Reynolds
numbers of 4200, and 36.3 for water (D = 2.7 mm) and
4
d/D
3
Water, We=90
Glycerin, We=93
2
1
0
0.01
0.1
(a)
1
10
1
10
tu/D
1
0.8
0.6
Water, We=90
Glycerin, We=93
y/D
2.2. Spread of a droplet
99
0.4
0.2
0
0.01
0.1
(b)
tu/D
Fig. 2. Effect of liquid viscosity (a) spread factor and (b) apex height
(water Re = 4200 and glycerin droplet Re = 36.3) on smooth glass.
glycerin (D = 2.45 mm) droplets, respectively, on smooth
glass, are shown in Fig. 2.
An increase of viscosity decreases the spread factor, as
Fig. 2a shows. The glycerin droplet recoils earlier than
water droplet on smooth glass, as Fig. 2b illustrates. After
tu/D = 0.75, the apex height of glycerin droplet decreases
slowly and the minimal apex height is larger for the glycerin droplet than that for the water droplet (Fig. 2b).
2.3. Splash of a droplet
Fig. 1. Impact of a water and a glycerin droplet onto glass for We = 391:
(a and b) spreading phase, (c and d) receding phase (t is time from impact).
As the literature review [14,15] demonstrate, the related
phenomena of fingering and splashing during the impact of
a droplet onto a dry surface are not well understood.
Although correlations exist to predict the onset of splashing, these have not proven universal. The mechanism that
indicates the perturbation at the leading edge of the
expanding lamella has not yet been fully understood. The
onset of crown formation, its time evolution and the nature
of the lamella instability (see below) that leads to splash
have received scant attention. The influence of surface
roughness and wettability on crown instability has not been
satisfactory explained.
Depending of a number of factors, the leading edge of
the lamella may detach. The lamella may become unstable
soon after it appears, resulting in the emergence of very
regular azimuthal undulations. At the critical Weber number, that defines the boundary between the splashing and
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
Fig. 3. The splash/deposition limit: (a) adhesion, (b) break up and (c)
droplet ejection. (Isopropanol droplet/smooth glass, D = 3.3, We = 287,
Dt is the time between two exposures.)
non-splashing regimes, on the smooth surface the lamella
may fall down. Viscous drag due to the underlying solid
surface plays a role in damping the undulations. The fluid
will continue to spread while remaining in contact with the
solid surface. Some droplets may be ejected from the
lamella just before it falls down, as Fig. 3c shows.
At the larger We, the expanding lamella lifts up off the
surface and forms a crown, similar to the impact of a droplet onto a liquid surface. If the lamella grows far enough
beyond the contact edge, its instability leads to the pinching off of secondary droplets, as seen in Fig. 4 at
t = 0.959 ms and final disintegration at t = 1.607 ms. From
this time the deposited part of droplet spreads, slowly
approaching the equilibrium state.
When the same droplet impacts the rough glass, the
expanding lamella tends to lift off of the surface, and fluctuations at the microscopic scale translate into large fluctuations in the region of the lamella far from the contact line
and it disintegrates. This outcome is called the prompt
splash [16].
The detachment of the lamella and formation of the
crown at the impact onto a dry, smooth surface depend
on the droplet surface tension. The surface tension is
embedded in the Weber number. The detachment of the
lamella on smooth surface was observed with an isopropanol droplet at a Weber number larger than about 287
(Fig. 3). At Weber numbers about this value (We from
280 to 300) the detached lamella of an isopropanol droplet
simply falls and adheres to the surface. The critical Weber
Fig. 4. Splash of an isopropanol droplet on dry, smooth glass (We = 391).
number is the one at the deposition/splash threshold. At
larger values of the Weber number the difference between
the contact diameter and the detached lamella diameter
increases up to the time of break-up (splash) of the lamella,
as Fig. 4 indicates.
The lamella diameter is taken here as diameter of the
expanding liquid film before the splash, as Fig. 5a shows.
The splash occurs due to instability of the lamella, after
it reaches a certain diameter (Fig. 5b). Fig. 6 shows the
impact sequence of a 3.3-mm isopropanol droplet (We =
544) on smooth glass. In the first phase of the lamella evolution, it seems smooth and stable, and no secondary droplet detaches. Then liquid accumulates in the periphery of
lamella and it become unstable (t = 0.472 ms). The instabil-
d lam.
dcont.
(a)
4
break up
We=544
3
Contact
d/D
100
Lamela
2
1
0
0.01
(b)
0.1
tu/D
1
Fig. 5. Spread factor of the lamella and contact line for an isopropanol
droplet impact on the smooth glass: (a) typical view (t = 0.254 ms) and (b)
its time evolution (We = 544).
Fig. 6. Splash of an isopropanol droplet on dry smooth glass, We = 544.
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
ity increases large enough for droplets to separate from the
periphery of the lamella (t = 1.010 ms). The lamella disintegrates gradually into secondary droplets as the sequences
(t = 1.235 ms and t = 1.443 ms) illustrate.
The effect of surface roughness on deposition/splash
limit for an isopropanol droplet was not observed in our
study. No splash was observed for a water droplet on a
smooth surface for Weber numbers up to 1080, the
maximum value considered in our experiment. Surface
roughness affects the deposition/splash limit for a water
droplet. A water droplet (D = 2.7 mm) splashes at Weber
number large than 390 on the rough glass. After the lamella
is ejected from the base of the droplet, a thin liquid film has
not full contact with the rough surface. The lamella contacts the roughness peaks and the air is entrapped in the
roughness valleys. The lamella/solid contact is random,
the adhesion forces are weak and, as the result of the early
lamella instability, it breaks up just after it appears
(prompt splash), as Fig. 7 shows.
The splashing mass due to formation of secondary droplets increases with increase of the impact velocity. A comparison between the splash of an isopropanol droplet on
the smooth (Ra = 0.003 lm) and rough glass (Ra = 3.6 lm)
in Figs. 6 and 7, respectively, shows that the roughness
does not increase greatly the amount of secondary droplets.
At higher Weber number impact of an isopropanol
droplet onto a smooth surface, the lamella forms a corona-like shape. The corona increases with increase of
impact velocity up to the instant of secondary droplet ejection and the final breakup, as Fig. 8 shows. This outcome is
termed corona splash [16].
A classical corona splash at a higher-impact Weber
number is shown in Fig. 9. The lamella appeared shortly
(t = 0.070 ms) after the droplet contacted the surface. After
the lamella expands a certain distance, small droplets eject
from its periphery and the very thin lamella becomes unstable (t = 0.610 ms), it breaks up into filaments (frames from
t = 0.878 ms to t = 1.160 ms) and the filaments disintegrate
into fine droplets (spray) (t = 1.661 ms). The spray disappears (t = 2.433 ms) and the remaining part of the droplet
continues to spread on the surface, as Fig. 9 shows.
Fig. 7. Prompt splash of an isopropanol droplet, We = 544, on dry rough
glass (Ra = 3.6 lm).
101
Fig. 8. Time evolution of the corona for an isopropanol droplet impact
onto dry smooth glass: (a) We = 391, (b) We = 544 and (c) We = 786
(Dt = 0.160 ms for all).
Fig. 9. Corona splash of an isopropanol droplet on dry, smooth glass
(We = 1063).
A selection of images of the corona evolution (growth
and subsidence) at the impact of an isopropanol droplet
(D = 3.3 mm, We = 1063) on a shallow liquid layer (film
thickness d = 0.8 mm) is shown in Fig. 10. The liquid
lamella is formed shortly after the base of the droplet
reaches the liquid layer. It then propagates radially outward, forming a thin cylindrical liquid film (corona). Initially the upper edge of the corona is bend down (from
t = 0.182 ms to t = 1.212 ms). Small droplets begin to eject
from its rim and the bended part disintegrates (t =
1.212 ms). Later when the expansion of the corona slows
down and the bent edge is shattered, the upper corona edge
become unstable and then small jets rise up and elongate at
its periphery (t = 3.524 ms). Liquid accumulates on the
periphery of the corona and jets become large (t =
13.193 ms). After the corona base reaches its maximum it
starts to subside. The wall of the corona bends inward
(t = 14.284 ms) and the diameter of upper edge of the corona reduces, while the diameter of the base remains fixed
for a short time. Then the shape of the corona changes
from cylindrical to conical (t = 26.216 ms). The peripheral
102
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
Fig. 11. Effect of film thickness on splash of an isopropanol droplet
(We = 1063): (a) d = 0.1 mm, (b) d = 0.8 mm.
Fig. 12. Corona without splash, of an isopropanol droplet, We = 287,
d = 0.14 mm.
Fig. 10. Corona evolution at the impact of an isopropanol droplet on
liquid film (We = 1063, film thickness d = 0.8 mm). (a) Growth of the
corona, (b) subside (collapse) of the corona.
jets increase and droplets separate from their tips. The jets
leave the corona edge forming large droplets (t =
33.895 ms). Droplets detached from the jets of the corona
fall down on the liquid. This produces a small number
(between 10 and 15) of larger droplets (about 1.8 mm in
diameter). The corona subsides into the liquid layer (t =
43.737 ms) and the large secondary droplets deposit. Subsequently, as results of liquid layer retraction, a central jet
forms (t = 61.282 ms) from the center of the cavity in a
liquid layer that disintegrates in a few droplets. Capillary
waves can be seen propagating down the corona (t =
26.216 and 33.895 ms).
Droplet impact onto dry smooth surface forms a very
thin lamella, which collapses in spray. The disintegration
process of the lamella on a dry smooth surface is fast; it
finished at t = 1.160 ms (Fig. 9). A droplet impacting onto
a liquid layer forms a thick liquid corona with larger droplets pinching off from the rim jets while the corona settles
down. The impacting droplet kinetic energy is converted
to corona potential energy and an increase of surface tension energy. The corona consists of both liquid of droplets
and target liquid. Thus, the disintegration process of the
corona is longer and the corona disappears later
(t = 43.737 ms in Fig. 10) than that at impact onto a dry
surface (t = 1.160 ms in Fig. 9). Retention of liquid in the
cavity in the target layer remaining after the corona collapse will result in a central liquid jet (t = 61.282 ms).
Small variations of the liquid film thickness d, influence
the splashing behavior, the corona morphology, the secondary droplet size and the time scale of the splashing
process.
The height of the corona is higher at the impact onto a
thicker liquid film owing to the formation of a cavity immediately after impact, the displaced material of which is
incorporated into the corona wall (Fig. 11).
A splash includes a liquid film rising from the periphery
of the crater. The splash on wetted surfaces occurs at a
smaller impact velocity than on dry surfaces. Furthermore,
in contrast to impact on dry surfaces, the corona formation
does not always lead to splash, as Fig. 12 shows, for We =
287.
3. Impact of a droplet onto inclined surfaces
The geometry of the droplet impact is shown in Fig. 13a,
where the droplet velocity (u) is composed of a normal (un)
and tangential (ut) component, determined by the impact
angle (a). The side view of a droplet on an inclined surface
is shown in Fig. 13b.
For applications related to heat transfer in annular dispersed flows, typical droplet impact angles are low (<5)
and normal Weber numbers ðWen ¼ qDu2n =rÞ based on
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
Fig. 13. The geometry of droplet impact (a) and side view of a droplet on
an inclined surface (b).
the wall-normal velocity un and the droplet diameter D are
less than 5. For instance [17] studied the rate of deposition
of 50 lm water droplets injected centrally into a downward
flowing pipe flow at a gas Reynolds number of Re =
30,600. The drop impact angle was found to lie in the range
0.5–3.5 with a normal Weber number in the range
0.007 < Wen < 0.34. Šikalo et al. [18] worked with an
upward flowing configuration at Re = 40,000–95,000 and
found impact angles of 0.5–1.5. The drop mean diameter
ranged from 7 lm to 30 lm, yielding a normal Weber number range of 0.025 < Wen < 1.80, similar to [17].
Besides the application in heat exchangers, the oblique
drop impact in this range of Weber number is of considerable interest in agricultural spraying, where low impact
angles on leaves can lead to diminished deposition on the
surface. Fuel injection sprays impacting obliquely on combustion walls or droplets impacting onto airplane wings,
considered for icing studies [19], exhibit considerably
higher Weber number and will not be considered in the
present paper.
Oblique droplet impact at low Weber numbers and at
low impact angles was investigated in [20–26].
3.1. Phenomenological observations
3.1.1. Dry surface
When a droplet impacts onto an inclined surface, the
shape of the droplet distorts and it spreads asymmetrically
relative to the point of impact, as shown in Fig. 13b. Elongation and back-to-front asymmetry increases with time.
The front edge of the droplet spreads forward, while the
back edge spreads backwards or slips forward. The difference in the spread velocity of the lamellas in the forward
and back directions increases with a decrease of impact
angle. They are equal for the normal impact. The maximum spread of the rear lamella is reached when the radial
velocity in that direction becomes close to zero.
When this occurs, the back edge remains fixed for a glass
surface (high wettability), while the back edge slides down
the wax surface (low wettability). The footprint given by
103
contact line upon impact is usually neither circular nor parallel-sided, but lies somewhere in-between. Note that both
the xfront and xback values have been defined positive from
the point of impact.
The first phase of impact involves the initial deformation of the droplet similar to the case of normal impact.
The subsequent phase may have widely varying outcomes,
for example: spread (Fig. 14a), spread and slide (Fig. 14b),
slip and spread (Fig. 14c), splash in all directions (Fig. 15a)
or only in the forward direction (Fig. 15b), rebound, partial
rebound (Fig. 16), depending on the impact angle, liquid
and surface properties as well as the Reynolds and Weber
numbers.
At the impact onto an inclined smooth surface the
lamella extends in forward (front) direction, its rim become
unstable and it disintegrates in droplets (splash), Fig. 15b.
The spread factor of the lamella is taken here as the distance of the expanding film, before the splash, from the
point of impact, as Fig. 17a shows. The contact spread factor in forward and backward direction is measured, as a
Fig. 14. Deposition of a water droplet (We = 391; a = 5; Wen = 3.0)
with: (a) spread on smooth glass, (b) spread and slide on wax (t = 0, 1,
3 ms), and (c) and slip and spread on smooth glass (t = 0, 1, 2, 3 ms).
Fig. 15. Splash (a) in all direction (water droplet on dry rough glass,
We = 391, a = 45, Wen = 196) and (b) in the forward direction (isopropanol droplet on smooth glass, We = 544, a = 45, Wen = 272).
104
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
x
P bac
im oin k
pa t of
ct
Fig. 16. Partial rebound of a glycerin droplet on dry smooth glass: (a)
ti = 0 (reference time of the image) and (b) ti = 7.67 ms (We = 93, a = 10,
Dt = 2 ms).
la
m
el
la
x
x
t
ac
nt
o
c
meters, including film thickness. It is interesting to note
that a droplet rebounds from a liquid film at larger angles
than from the dry surface under the same conditions. The
deformation of a droplet increases with an increase of the
impact angle, and the contact time of the droplet with
the surface decreases. For example, on dry glass at 8 the
contact time of the glycerin droplet is about 14.5 ms.
On a wetted surface at 30 (limit rebound/deposition) the
contact time is about 10 ms. The contact time of an isopropanol droplet at 8 (D = 3.3 mm, We = 179, dry surface) is
29.68 ms, and the contact time of the isopropanol droplet
at the impact of 15 (wetted surface) is about 22.99 ms.
(The contact time is measured as the time between first contact of the droplet with the surface and the final lift-off of
the droplet.)
At larger impact angles the deformation is high, and the
droplet elongation during its slipping along the target surface is larger. The elongation of a water droplet and an isopropanol droplet parallel to the surface in comparison with
a glycerin droplet is also larger, due to lower viscosity.
Higher deformations of an isopropanol droplet, due to its
low surface tension and low viscosity, are observed at
higher impact angles, so its shape differs from prolate or
oblate one as it lifts off, as Fig. 18 shows.
Fig. 19 illustrates possible outcomes of a water droplet
impact onto a wall liquid film near a critical impact angle;
complete coalescence (deposition) or partial coalescence.
These outcomes are also observed for the other investigated liquids. At the coalescence of a droplet and the liquid
film in an earlier recoiling phase, the droplet deposits
(Fig. 19a). At the coalescence in the later recoiling phase,
just before lifts off, the droplet disjoins and partially
rebounds (Fig. 19b).
(a)
4
We = 544,
= 45º
break up
3
x/D
Front, contact
2
Back, contact
Front, lamella
1
0
0.01
(b)
0.1
1
10
tu/D
Fig. 17. Spread factors of the lamella and the contact line (a) typical view
and (b) time evolution (isopropanol droplet, We = 544 and a = 45).
distance from the point of impact to the contact line in the
two directions, Fig. 17b.
3.1.2. Wetted surface
In the case of the target surface covered with a liquid
film, the outcomes of the impact may be similar to that
for the dry surface: deposition (coalescence), rebound, partial deposition or splash, depending on the impact para-
Fig. 18. Rebound of an isopropanol droplet from a wetted wall
(We = 179, a = 15, Wen = 12, d 40 lm).
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
105
5
x=d/2, 90º
x/D
4
We = 50, smooth glass
Front, 45º
3
Front, 10º
2
Back, 45º
Back, 10º
1
0
The images in Fig. 19 were taken for exactly the same
impact conditions. The rather larger differences – rebound,
partial rebound, deposition – indicate how sensitive the
outcome is at this critical impact angle to non-controllable
small perturbations. At this condition the repeatability is
not nearly as high as at all other conditions.
The deformation is affected by the ratio between normal
inertial forces and surface forces. Therefore, the normal
Weber number of the droplet appears as a parameter of
importance to predict the degree of deformation and the
droplet rebound limit.
3.2. Quantitative results
3.2.1. Droplet spreading on inclined surface
The spread factor (x/D) in both forward and backward
directions from the point of impact and the apex height
(y/D) of a droplet (see Fig. 13b) as a function of nondimensional time after impact, t* = tu/D will be discussed.
Fig. 20a and b show the effect of the impact angle on the
spread factor of a 2.7 mm water droplet; the impact velocity was 1.17 m/s with an associated droplet Weber number
of We = 50 and a Reynolds number of Re = 3260. Note
that positive slope of xback with time (t*) refer to a film
expanding away from the point of impact, whereas negative slope of xback indicates a slipping down the surface.
Spread factors of a water droplet on the smooth glass
(static contact angle h = 6–10) and on the smooth wax
(h = 95–105) are shown in Fig. 20. If the spread factors
shown in Fig. 20a and b are directly compared with one
another, little influence of the surface is detected up to a
dimensionless time of about t* = 0.7. This time decreases
with a decrease of the impact angle. For example, it is
about t* = 0.36 for a = 10. In the following stage of
spreading, the effect of the surface is more important.
For the wax surface with high static contact angles, the
droplet begins to slide once the maximum spread factor
in the back direction has been reached, as Fig. 20b shows
(see Fig. 14b). No sliding is observed for the glass surface,
as Fig. 20a shows (see also Fig. 14a). The lower impact
-1
0.01
0.1
(a)
1
10
1
10
t*
4
2
x/D
Fig. 19. Impact of a water droplet onto wetted surface (liquid film,
d 0.1 mm): (a) deposition (coalesce in an earlier recoiling phase,
ti = 0.99 ms) and (b) partial rebound (coalesce in a later recoiling phase,
ti = 8.354 ms) (We = 39, a = 22, Wen = 5.5, t = 0, 5, 9 ms, time of the
exposure).
We = 50, wax
x=d/2, 90º
Front, 45º
Front, 10º
Back, 45º
Back, 10º
0
(b)
-2
0.01
0.1
t*
Fig. 20. Spread factors of a water droplet on (a) smooth glass and (b) wax
(a = 10–90, We = 50).
angle leads to a larger droplet elongation (distance between
leading and trailing edges).
For the Weber number of 50 and the impact angle of 45
the forward spread factor becomes larger then for 90 at
the dimensionless time about t* = 0.159, while at the Weber
number of 391 this occurred at t* = 0.286.
At the impact angle of 10 and the Weber number of 50
a water droplet slips on smooth glass at the first instant of
the impact and the data of back spread factor are negative
and exhibits larger scatter, as shown in Fig. 20a. No slipping of a water droplet was observed on smooth glass at
larger Weber numbers.
At the impact onto the wax the back edge of the contact
line begins to slide at about t* = 0.80 (x/D = 0.348) for the
Weber number of 50, and the slide begins at about
t* = 1.623 (x/D = 0.618) for the Weber number of 391.
The lower Weber number leads to a faster sliding of the
back edge of the contact line.
The normal impact velocity (u Æ sin a) scales the apex
height well, as Fig. 21 shows. The partial or complete
rebound of the droplet away from the surface was not
observed for the inclined wax surface. The sliding of the
water droplet on an inclined surface inhibits recoil of the
droplet, as is observed for normal impacts. However, a
water droplet on inclined wax surface does not rebound
in the way observed for normal impacts.
The sequences of the impact process of a water droplet
on the smooth glass, wax and rough glass at 45 are shown
106
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
1
We = 50, smooth glass
0.8
90º
45º
20º
10º
y/D
0.6
0.4
0.2
0
0.01
0.1
1
10
t* sinα
Fig. 21. Effect of impact angle on the apex height of a water droplet in
terms of t* Æ sin a (a = 10–90, We = 50).
in Figs. 22–24, respectively. In the early phase of the impact
(t up about 0.7 ms), the water droplet spreads in all direction almost symmetrically. In the later phase of spread the
Fig. 22. A water droplet spreading on smooth glass (We = 391, a = 45).
Fig. 23. Sequences of spreading of a water droplet on smooth wax
(We = 391, a = 45).
Fig. 24. Sequences of impact of a water droplet on rough glass (We = 391,
a = 45).
droplet elongates. The main part of water moves down,
and in the back remains a trace of water (liquid film).
The liquid film breaks up under capillary forces into two
or more parts, and the liquid tends to reach an equilibrium
state.
The effect of contact angle of the wax is observed in the
early phase of spread, as Fig. 23 shows. A raised rim is
formed at the lamella periphery on the wax (t = 2.07 ms)
due to capillary forces that limit the spread and decelerate
the liquid flow close to the contact line. At t = 6.49 ms the
droplet reaches its maximum elongation. After that, the
back (trailing) edge of the contact line moves (slides) down
faster than the front (leading) edge of the contact line. The
droplet recoils due to large surface energy (large contact
angle). The contact area (water–wax) decreases and the
height of liquid increases. The liquid/solid contact shrinks
and at the same time moves downward, contrary to the
horizontal recoil, where the liquid/solid contact shrinks
and moves inward until the rim reaches the center and
rebounds. Therefore, the droplet does not rebound in this
case. At t = 15.14 ms a small portion of the droplet pinches
off from its front and forms a secondary droplet (t =
17.42 ms). The main body of liquid slides downward faster
than the secondary droplet, they can rejoin, and the droplet
approaches an equilibrium shape. The secondary droplets
can also detach from the back.
Fig. 24 shows the prompt splash of a water droplet
impacting at a Weber number of 391 and impact angle of
45 onto the rough glass. The splash occurred in all directions from the lamella periphery. Very small droplets eject
in early phase of impact (t = 0.19 ms) in all directions. Larger droplets eject later only downward as seen at t =
1.124 ms.
When a droplet comes across a peak of roughness during the slipping on smooth glass it spreads and deposits
on the surface rather than rebounding from the surface,
even at very small (smaller than critical deposition/
rebound) angles, as Fig. 25 shows. There is a sharp point
of roughness on the smooth glass, visible in an enlarged
image (Fig. 25b) of the fourth exposure in Fig. 25a, at
which the droplet adheres to the surface (pinning). Therefore, as result of (even low) roughness (Ra = 0.3 lm) of
the wax the droplet spreads rather than rebounding at an
impact angle of 2.
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
1.000
107
K=3 K=8 K=19 K=57.7
Glycerin
Dry wall
Wet wall
Dry
2.45 mm
Deposition
0.100
Oh
Wet
Isopropanol
0.010
Splash
1.8 mm
3.3 mm
Rebound
Water
2.7 mm
0.001
1
Fig. 25. Observations of slip and spread of a water droplet impacting onto
a smooth glass surface with a sharp point above the exposure four (a) and
the pinning (b), enlarged exposure 4 from the image (a) (We = 391, a = 2,
time between exposures is Dt = 1 ms).
3.2.2. Droplet rebound
Fig. 26 presents measurements of the critical impact
angle for a droplet to rebound (the limit droplet rebound/
deposition or partial deposition) on dry glass. When the
critical normal Weber number, Wencr = We Æ sin2a, is plotted against impact angle a, the data exhibit constant values
of Wencr, as shown in Fig. 26.
The limit between the rebound and deposition regimes
based on the Sommerfeld and Tropea [27] parameter
K = Oh Æ Re1.25 is shown in Fig. 27. The Ohnesorge number
is defined by the relation Oh = We1/2/Re. Sommerfeld and
Tropea [27] propose a splash limit of K = 57.7 for a prewetted surface using water droplets. The results are compared with the limits between the rebound/deposition
regimes for dry and wet walls for K = 3 and K = 8, which
were used in [25] to classify the regimes of their experimental results for monosized ethanol droplet–wall interactions.
The parameter K does not predict well the experimental
results of rebound/deposition limit in this study. At the
limit for dry smooth glass, the average values of K are
1.48 for glycerin, 4.77 for water, 5.91 for isopropanol of
1.8 mm and 7.63 for isopropanol of 3.3 mm droplets. At
the limit for wetted surfaces the average values of K are
6.4 for glycerin, 12.95 for water, 16.34 for isopropanol of
10
100
Ren
1000
10000
Fig. 27. Rebound/deposition limit based on the parameter K = Oh Re1.25.
1.8 mm and 19 (21 maximal values) for isopropanol of
3.3 mm droplets.
For the impact on a dry surface, roughness affects the
rebound/deposition limit, while on a wetted surface waviness, depth and velocity of liquid film also affect the
rebound/deposition limit.
3.2.3. Partially rebound
A highly viscous (glycerin) droplet, upon impact onto a
smooth surface, is found to partially rebound at a wide
range of impact angles (up to 45), as Fig. 16 suggests.
The size of the rebounded droplet decreases with increasing
impact angle. The fraction of the droplet that rebounds
may change from 100% (complete rebound) to zero (deposition) over the range of the parameters considered in the
experiment [28].
The rebound ratio, defined as the ratio of the diameters
of droplets that are disjoined from the primary droplet,
expressed as Ds/D, are presented in Fig. 28, where Ds is
the diameter of the secondary droplet that disjoins after
impact and D is the primary droplet diameter. An increase
of the droplet Weber number decreases the rebound ratio
for the same impact angle. The measured rebound ratio
as a function of the normal capillary number Can = Wen/
Ren = lusin a/r coincides for all experimental conditions,
as is seen in Fig. 28.
1.2
4
1
0.8
2
Ds /D
Wencr
3
Glycerin D=2.45
Water
1
We
0.6
0.4
D=2.7
Isoprop. D=3.3
51
163
402
93
280
571
0.2
Isoprop. D=1.8
0
0
5
10
Impact angle, α (deg)
15
20
Fig. 26. Normal Weber number as a function of critical impact angle for
dry smooth glass.
0
0.0001
0.001
0.01
0.1
Wen/Ren=Can
1
10
Fig. 28. Rebound ratios in terms of the normal capillary number for
glycerin droplets on dry, smooth glass.
108
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
4.1. Results and discussion
time step was from 107 s to 105 s, depending on the
spread velocity and cell size.
The adhesion boundary condition that describes the free
surface in contact with the solid surface is given by a static
contact angle. A liquid/solid combination gives this angle.
The flow considered in this study was laminar, so the wall
boundary conditions could be applied directly.
A qualitative comparison of a water droplet shape
obtained by simulation and a CCD camera at the same
times after the impact instant is shown in Fig. 29. The static
contact angles (advancing 105 and receding 95) were prescribed as a boundary condition. After contact with solid
surface, the bottom part of the droplet compresses and
deforms. Then a liquid film (lamella) ejects close to the contact line and spreads over the solid surface. At the beginning, the thickness of the lamella is smaller than 6 lm.
The thickness increases with time during the spreading.
Therefore, the simulation of water droplet impact requires
grids with very fine cells close the wall to capture all details
of the process. Liquid accumulates at the front (rim) of the
lamella and its thickness increases with time, during the
spreading and receding phases, as Fig. 29a and b illustrate.
At the maximum spread, the major driving forces
become capillary forces. The droplet begins to recoil in
order to obtain the shape with minimum surface energy.
Fig. 29b shows the simulated shapes of the droplet during
The spread of a single droplet is studied for dry surfaces.
Two liquids, water and 85% glycerin were used as test liquids. The difference of the dynamic viscosities of glycerin
(0.116 Pa s) and water (0.001 Pa s) is high, while the difference of their surface tensions (0.063 and 0.073 N/m) is not
so important. The static contact angle for both fluids on
the wax is high (90–97 and 95–105) and it is low on the
glass (13–17 and 6–10, respectively). Since a glycerin
droplet at the same Weber number as a water droplet has
a Reynolds number two orders of magnitude lower it
spreads slowly. Therefore, this selection includes sufficient
number of different variables for a numerical analysis of
droplet spreading.
The flow can be considered axisymmetric. Therefore, the
numerical simulation was performed in a cylindrical coordinate system, where all changes in the azimuthal direction
are assumed zero.
The computational space domain in this study had the
shape of a cylinder segment dimensions of r = 12 mm,
z = 12 mm, with a segment angle h = 2. A fixed, non-uniform grid consisting of the square control volumes in the r–
z plane and only one control volume in the azimuthal direction were created and refined in the region where the liquid
droplet moves. The smallest cells were near the wall, in the
region where the lamella is created and expands. The cells
near the wall had a size between 12.5 · 6.25 lm and
0.78 · 0.78 lm depending on the case studied. The finest
grid used in this study had in total about 50,000 cells.
The time step was taken on the basis that the actual local
Courant number must be less or equal to 0.2. Thus, the
Fig. 29. Time sequence of water droplet impact onto a wax surface
(We = 90). Comparison of the simulation (right) and experiment (left). (a)
The spreading phase, (b) the receding phase.
4. Numerical simulation of droplet impact
Within the last decade, various studies have examined
numerically droplet impact. Fukai et al. [8] used a finite-element-based technique to model the droplet spreading process. Bertagnolli et al. [29] presented an adaptive-grid, finite
element method to examine the impact of molten ceramic
droplets in the context of thermal spraying, neglecting wetting effects. Pasandideh-Fard et al. [30] studied fluid flow,
heat transfer and solidification during molten metal droplet
impact. Rieber and Frohn [31] have developed 3D models
to simulate droplet impact onto a liquid surface. Bussmann
et al. [11] developed a fixed-grid Eulerian model, employing
a volume-tracking algorithm.
The normal impact of a water and glycerin droplets on
smooth glass and wax were numerically simulated in [28].
The characteristic cases for which reliable experimental
data exist [28] are analyzed. The results are presented for
two droplet Weber numbers, one for a low impact Weber
number (We = 90) and one for larger Weber number
(We = 800). The simulations were performed using the
commercial code Comet [32] based on a finite-volume
numerical method. Good agreement with experimental
results was obtained.
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
the receding phase. Excellent agreement is obtained
between the experiment and the simulation.
Unlike the former case, the simulated higher viscous
droplet (glycerin) impacting onto the wax surface spreads
slowly. The high viscous dissipation suppresses the droplet
spreading. The spreading lamella is thick and the maximum spread is low. During the receding phase viscous
forces again dominate capillary forces and no rebound is
observed. The droplet approaches its final shape only very
slowly. Fig. 30 illustrates the free surface contours obtained
by simulation for a glycerin droplet in both the spreading
and the recoiling phases.
The numerical and experimental values of the spread
factor and apex height in time for a water droplet are com-
Fig. 30. Free surface contours (a) spreading, t = 0–1.5 ms (Dt = 0.05 ms),
(b) recoiling, t = 1.5–88.5 ms (Dt = 1 ms) (glycerin droplet/wax,
We = 800).
6
Experiment
Calculation
d/D
4
2
(a)
0
0.01
0.1
1
t*
10
100
1.5
Experiment
Calculation
5. Concluding remarks
The physics of droplet wall interaction is a key to many
spray coating and heat transfer processes. Many fundamental questions of droplet impact have not yet been
cleared up. For example, the mechanism causing the onset
of splash, and other instability processes remains to be
solved. In order to predict the outcomes of droplet impact
a better knowledge of processes occurring during the final
phase of the approach of a droplet toward a surface is necessary. In particular, entrainment and collapse of a thin air
layer between an approaching droplet and a surface have
not been solved so far. The first contact between the two
surfaces occurs on molecular scales. Interfacial phenomena
related to the physics of surfaces must be involved to model
this process. Unfortunately, entrained-air-layer theories
have not been advanced to the point where direct comparisons with experimental data can be made. The partial
rebound ratio of a high viscous droplet impacting onto
inclined smooth surface deserves further experimental
and numerical investigations, since this outcome has not
been studied and reported. From a practical point of view,
the splash ratio, as a quantitative measure of secondary
droplets formed at the impact, is more important than to
fix the splash/deposition threshold only. The dynamic contact angle is required as a boundary condition for numerical calculation in capillary hydrodynamics. However, there
is no general correlation of the dynamic contact angle as a
function of the surface characteristics, fluid droplet diameter and impact velocity. Finally, non-Newtonian fluids are
widespread in practice and need detailed experimental and
numerical studies of their droplet interaction with walls.
Because of the incomplete state of scientific study, most
droplet/wall interaction problems have to be approached
experimentally. Analyzing these phenomena and development of related models remains a challenge in wetting
hydrodynamics. Therefore, the problem obviously deserves
further theoretical and experimental studies.
y/D
E.N. Ganić acknowledges the support received from the
Office of Naval Research under grant N00014-04-1-0389.
0.5
(b)
pared in Fig. 31. The spread factor and apex height were
determined very precisely with an uncertainty of less than
1% in any of the simulations in this study.
Acknowledgment
1
0
0.01
109
References
0.1
1
t*
10
100
Fig. 31. Numerical calculations of the temporal evolution of (a) spread
factor and (b) apex height, in comparison with the experimental data
(water droplet on wax, We = 763).
[1] G. Trapaga, J. Szekely, Mathematical modeling of the isothermal
impingement of liquid droplets in spraying processes, Metall. Trans.
22B (1991) 901–914.
[2] M. Rein, Phenomena of liquid drop impact on solid and liquid
surfaces, Fluid Dyn. Res. 12 (1993) 61–93.
[3] S. Chandra, C.T. Avedisian, On the collision of a droplet with a solid
surface, Proc. R. Soc. Lond. A 432 (1991) 13–41.
[4] A. Frohn, N. Roth, Dynamics of Droplets, Springer, Berlin, 2000.
110
Š. Šikalo, E.N. Ganić / Experimental Thermal and Fluid Science 31 (2006) 97–110
[5] I.V. Roisman, R. Rioboo, C. Tropea, Normal impact of a liquid drop
on a dry surface: model for spreading and receding, Proc. R. Soc.
Lond. A 458 (2002) 1411–1430.
[6] I.V. Roisman, B. Prunet-Foch, C. Tropea, M. Vignes-Adler, Multiple
drop impact onto a dry substrate, J. Coll. Interface Sci. 256 (2002)
396–410.
[7] J. Fukai, Z. Zhao, D. Poulikakos, C.M. Megaridis, O. Mitayake,
Modeling of the deformation of a liquid droplet impinging upon a flat
surface, Phys. Fluids A 5 (1993) 2588–2599.
[8] J. Fukai, Y. Shiiba, T. Yamamoto, O. Mitayake, D. Poulikakos,
C.M. Megaridis, Z. Zhao, Wetting effects on the spreading of a liquid
droplet colliding with a flat surface: experiment and modeling, Phys.
Fluids 7 (1995) 236–247.
[9] T. Mao, D.C.S. Kuhn, H. Tran, Spread and rebound of liquid
droplets upon impact on flat surfaces, AIChE J. 43 (1997) 2169–2179.
[10] M. Bussmann, J. Mostaghimi, S. Chandra, On a three-dimensional
volume tracking model of droplet impact, Phys. Fluids 11 (1999)
1406–1417.
[11] M. Bussmann, S. Chandra, J. Mostaghimi, Modeling the splash of a
droplet impacting a solid surface, Phys. Fluids 12 (2000) 3121–3132.
[12] Š. Šikalo, H.-D. Wilhelm, I.V. Roisman, S. Jakirlić, C. Tropea,
Dynamic contact angle of spreading droplets: experiments and
simulations, Phys. Fluids 17 (2005) 062103.
[13] Š. Šikalo, M. Marengo, C. Tropea, E.N. Ganić, Analysis of droplet
impact on horizontal surfaces, Exp. Thermal Fluid Sci. 25 (2002) 503–
510.
[14] H. Marmanis, S.T. Thoroddsen, Scaling of the fingering pattern of an
impacting drop, Phys. Fluids 8 (1996) 1344–1346.
[15] S.T. Thoroddsen, J. Sakakibara, Evolution of the fingering pattern of
an impacting drop, Phys. Fluids 10 (1998) 1359–1374.
[16] R. Rioboo, M. Marengo, C. Tropea, Outcomes from a drop impact
on solid surfaces, Atomiz. Sprays 11 (2001) 155–166.
[17] M.M. Lee, T.J. Hanratty, The inhibition of droplet deposition by the
presence of a liquid wall film, Int. J. Multiphase Flow 14 (1988) 129–
140.
[18] Š. Šikalo, N. Delalić, E.N. Ganić, Hydrodynamics and heat transfer
investigation of air–water dispersed flow, Exp. Thermal Fluid Sci. 25
(2002) 511–521.
[19] R.W. Gent, N.P. Dart, J.T. Cansdale, Aircraft icing, Phil. Trans. R.
Soc. Lond. A 358 (2000) 2873–2911.
[20] G.S. Hartley, R.T. Brunskill, Reflection of water drops from surfaces,
in: Surface Phenomena in Chemistry and Biology, vol. 57, Pergamon
Press, 1958, pp. 214–223.
[21] R.M. Schotland, Experimental results relating to the coalescence of
water drops with water surfaces, Disc. Faraday Soc. 30 (1960) 72–77.
[22] O.W. Jayaratne, B.J. Mason, The coalescence and bouncing of water
drops at an air/water interface, Proc. R. Soc. Lond. A 280 (1964) 545–
565.
[23] B. Ching, M.W. Golay, T.J. Johnson, Droplet impact upon liquid
surfaces, Science 226 (1984) 535–537.
[24] S.L. Zhbankova, A.V. Kolpakov, Collision of water drops with a
plane water surface, Fluid Dyn. 25 (1990) 470–473.
[25] G. Lavergne, B. Platet, Droplet impingement on cold and wet wall,
in: C. Tropea, K. Heukelbach (Eds.), Proceedings of the 16th
International Conference on Liquid Atom and Spray Systems –
ILASS-Europe 2000, Darmstadt, Germany, 11–13 September 2000,
paper VII.12.
[26] Š Šikalo, C. Tropea, E.N. Ganić, Impact of droplets onto inclined
surfaces, J. Coll. Interface Sci. 286 (2) (2005) 661–669.
[27] M. Sommerfeld, C. Tropea, Experimental studies of deposition and
splashing of small liquid droplets impinging on a flat surface, ICLAS94, Rouen, 1994, paper I-18.
[28] Š. Šikalo, Analysis of droplet impact onto horizontal and inclined
surfaces, Dissertation, TU Darmstadt, 2002.
[29] M. Bertagnolli, M. Marchese, G. Jacucci, I.S. Doltsinis, S. Noelting,
Thermomechanical simulations of the splashing of ceramic droplets
on a rigid substrate, J. Comput. Phys. 133 (1989) 205–221.
[30] M. Pasandideh-Fard, R. Bhola, S. Chandra, J. Mostaghimi, Deposition of tin droplets on a steel plate: simulations and experiments,
Int. J. Heat Mass Transfer 41 (1998) 2929–2945.
[31] M. Rieber, A. Frohn, A numerical study on the mechanism of
splashing, Int. J. Heat Fluid Flow 20 (1999) 455–461.
[32] Comet (ICCM, Institute of Computational Continuum Mechanics
GmbH, Hamburg, Germany), 1998.