2002-01-2749
The Deformation of Single Droplets Impacting onto a Flat
Surface
Ana Sofia Moita and António Luís N. Moreira
Instituto Superior Técnico
Copyright © 2001 Society of Automotive Engineers, Inc.
ABSTRACT
This paper presents an experimental study of the
deformation of spherical liquid droplets impinging onto
dry and flat surfaces making use of a CCD-camera with
a high spatial resolution. The experiments consider
different liquids (water and Diesel oil) and the effects of
droplet velocity and diameter at the impact in a range of
Weber numbers up to 1100 and Reynolds numbers up
to 77400. Emphasis is put on the nature of the surface
target. To consider this effect, two surface materials
were used (Perspex and aluminium) with surface
roughness varying from less than 5µm (considered as a
smooth surface) up to Ra=66,6µm. For the range of
droplet diameters considered in the experiments, the
corresponding dimensionless values of Ra/Rdroplet vary
from 1,5x10-5 and 2,5x10-2.
In a first step, the experiments focus on the spread of
the liquid film and analysis of the results suggest that,
provided that the Reynolds number of the droplet at the
impact is large (Re>2000), the energy dissipated at the
wall is not affected by the nature of the surface. The
effect of surface roughness appears to be important for
low Reynolds numbers, typically Re<1000.
Depending upon the physical parameters at the impact,
the droplet may splash at the first contact with the
surface (prompt splash) or the liquid film at the wall may
break-up on secondary droplets during spread. In a
second step, the experiments emphasize the effect of
surface roughness on the onset of splash for different
liquids. The experimental results are analysed in terms
of the critical Weber number for which splash occurs,
and compared with correlations reported in the literature
accounting for the effects of surface roughness and
droplet liquid and suggest the likely influence of, not only
the nature of the surface (e.g. surface profile and
material), but also of the liquid.
INTRODUCTION
The dynamic behaviour of an individual droplet
impacting onto a solid surface includes several individual
phenomena, such as deformation, fingering, splashing
and rebound. The onset and physical description of
those phenomena are usually characterized based on
dynamic similarity arguments making use of
dimensionless numbers characterizing the relative
magnitude of the forces acting upon the surface of the
droplet (Reynolds, Weber and Ohnesorge numbers). As
an example, Chandra and Avedisian [1] considered the
impact of n-heptane droplets onto a polished stainless
steel surface, using a flash photographic method. The
authors considered the influence of the surface
temperature, but they establish different regimes based,
mainly on Reynolds and Weber numbers. Marmanis and
Thoroddsen [2] scaled the fingering pattern in water
droplets impacting onto solid non-heated surfaces.
Later, Thoroddsen and Sakakibara [3] considered the
time evolution of the fingering pattern in water droplets
impacting onto non-heated glass surfaces. By adjusting
the release height of the droplets, the authors achieved
Weber and Reynolds numbers at the impact as high as
1000 and 15000, respectively. Estimative of these
dimensionless numbers were, in turn, obtained by
scaling the forces (surface tension, shear and gravity)
with physical properties of the liquid (density, viscosity
and surface tension) and considering all lengths
proportional to the diameter and velocity at the instant of
impact, respectively. However, the physical properties of
the impacting surface such as temperature, roughness
or inclination alter the boundaries of the problem under
study and similarity arguments cannot be applied. Some
work has already been developed in this context: Stow
and Hadfield [4] established a splash parameter for the
occurrence of splash, which depends on the surface
roughness. Later, Range and Feuillebois [5] added the
influence of liquid properties and surface material to the
correlations purposed by Stow and Hadfield [4]. Riboo et
al. [6] carried out a qualitative analysis of the various
outcomes of a drop impacting on solid horizontal
surfaces with various roughness and wettabilities; Šikalo
et al. [7] studied the influence of the physical properties
of the liquids on the droplet impact upon horizontal
surfaces, but they also included the influence of surface
roughness. It seems that the effect of the inclination
angle of the impact surface hasn’t been much studied.
From the reviewed literature one may report Karl et al.
[8], who emphasized the deformation of water and
ethanol drops, with diameters ranging from 100 to
200µm, upon inclined surfaces, heated well above the
Leidenfrost temperature of the droplet liquid. Kang et al.
[9] studied the dynamic behaviour of water droplets
impacting upon heated surfaces, inclined with an angle
between 0º and 60º.
between 0.6ms-1 and 5ms-1. The time history of a droplet
is recorded by a high-speed camera triggered by the
passage of the droplet through a laser beam hitting onto
a photodiode. A function generator creates an adjustable
delay in the signal that triggers the camera. The camera
is connected to a personal computer to store the
recorded frames and to a TV set to allow observation of
the droplet deformation during recording. Figure 1 shows
the experimental set-up and Figure 2 shows the trigger
mechanism used for the high-speed camera.
The fact that the properties of the impact surfaces
influence droplet dynamic behaviour and don’t allow to
apply similarity arguments is the reason why it appears,
from the reviewed literature that it is not possible to
correlate the several phenomena occurring during
droplet deformation with proper dimensionless numbers.
The present study is part of a research study aimed at to
account for the complexity introduced by the influence of
non-scaled parameters on the description of droplet
impact.
The present paper reports an experimental analysis of
droplet deformation on dry flat surfaces of different
materials and with different surface profiles. The
experiments were carried out for different regimes
commonly
characterized
by
non-dimensionless
numbers. The analysis considers different fluid
properties, namely water and Diesel oil, droplet velocity
and droplet diameter at impact to vary the Weber and
Reynolds numbers and the results are compared with a
simplified theoretical model as presented by
Pasandideh-Fard et al. [10]. Also, the onset of splash is
described by a critical Weber number and the results are
analyzed in terms of the influence of liquid properties,
surface material and profile.
Figure 1: Experimental Set-Up.
EXPERIMENTAL
Experimental Set-up
Droplets are generated at the tip of a hypodermic needle
and fall by action of gravity onto a flat and dry surface.
The flow rate into the needle is controlled by a syringe
pump. Using different fluids (Diesel oil and water in the
present case) as well as needles with different
diameters, it is possible to vary the initial diameter of
droplets (diameter at impact). Also, the release height of
the droplets is adjustable in order to change the impact
velocity of the drops. In this way the present set-up
allows to cover a large range of droplet diameters and
impact velocities, the only limitation being the nonsphericity, which may occur when the diameter or/and
velocity of droplets is too large. The experiments
reported here consider droplets with initial diameters
ranging from 2.6mm to 3.5mm and with impact velocities
Figure 2: Triggering mechanism for the high-speed
camera.
Experimental Method
All quantities necessary to characterize the deformation
of the droplet on the surface, such as the instantaneous
droplet diameter and the height of liquid above the
surface, contact angles and number of fingers are
measured in pixels, directly from the recorded images.
The impact velocity is obtained from the distance, in
pixels, traveled by the drop between the two last frames
before impact. The uncertainty of the measures from the
pictures varied between 1 and 3 pixels. The uncertainty
of the measurements is mainly due to the difficulty to
establish droplets contour. Another source of inaccuracy
is the the fact that, for hight values of drop diameter and
release height, droplets aren’t perfectly spherical, due to
the effect of the gravitational force.Therefore, both
horizontal and vertical diameters were measured. The
equivalent droplet diameter was considered, as in Šikalo
et al. [7], to be D = ( D h Dv )
2
1/ 3
values agree very well with those obtained using
equation (1), which gives 2.8mm and 3.5mm for the
minimum and maximum diameters used, respectively.
During the experiments, each measurement was
repeated five times in order to confirm the reproducibility
of the phenomena and to obtain average values.
, where Dh is the
Profile A
horizontal diameter and Dv is the vertical diameter.
However for the ranges presented in this paper the nonsphericity effect is not very important since the
difference between Dh and Dv is about 9% for the worst
case.
Profile B
Profile C
Profile D
The instant of impact (t = 0 s) is taken from the recorded
sequence of images. This procedure provides a velocity
and a length scales for dimensionless analysis.
The initial diameter of the droplet is also
estimated considering the equilibrium between gravity
forces and surface tension forces at the tip of the
needle, so that:
d =3
6σd needle
ρ liq g
(1)
where:
σ - is the surface tension of the liquid;
dneedle – is the diameter of the needle;
ρ - is the density of the liquid;
g – is the specific gravity.
Calibration of the image acquisition system allows
establish a correspondence between the measured
number of pixels and the real dimension. According to
this calibration, the smallest droplet considered in the
present report has a diameter of 2.6mm ± 0.1 and the
biggest droplet has a diameter of 3.5mm± 0.17. These
250
200
Intensity in the grayscale
Measurements were taken using image intensity
profiles. Each image is converted into a matrix of values,
according to a grayscale, where the minimum value
corresponds to black and the maximum to white.
Intensity profiles along a Diesel oil droplet are shown in
Figure 3. The intensity profiles are in fact vectors of
values according to the grayscale. For example, Figure
3 (a) shows four paths along the image of the droplet,
namely paths A, B, C and D. Each path is represented
by a vector of values in the grayscale and correspond to
a curve in the plot in Figure 3 (b). The regions in each
curve with larger gradient correspond to the edge of the
droplet along that path. Repeating this procedure for
several paths (Figure 3 only considers four to illustrate
the method) one may be able to determine the shape of
the droplets.
(a)
Profile A
Profile C
Profile B
Profile D
150
100
50
0
0
5
10
15
20
25
Pixel
(b)
Figure 3: (a) Paths of grayscale values along a
Diesel oil drop with a diameter of 2.9mm, released
from a height of 83cm. Frame rate: 5000fps. (b) Plot
of the curves corresponding to each path
represented in figure (a).
Characterization of Surface Roughness
In order to investigate the influence of the nature of the
surface target on droplet deformation and splash, this
study considers the use of aluminium and perspex
plates with different surface properties.
The surface is characterized by the mean roughness Ra,
to allow comparison with other results reported in the
literature. Surface roughness, Ra, is measured with a
profilometer that measures roughness in a range
between 5µm and 250µm. The surfaces of the plates
considered here are not perfectly homogeneous but, in
the region where droplet impact and deformation occurs,
deviations from the mean roughness are smaller than
10%. However, some materials showed regular surface
profiles while others showed random profiles.Table 1
summarizes the characteristics of the surfaces used in
the present study.
Table 1: Characteristics of the surfaces used in this
study.
Surface
Ra µm ± 10%
Smooth Perspex
<5
Rough Perspex with
regular profile
66.6
Aluminium with random
profile
18.84
Aluminium with random
profile
15.2
Aluminium with regular
profile
22.2
Aluminium with regular
profile
6.7
Droplet spreading
The sequence in Figure 4 shows the typical deformation
pattern of a water droplet with the occurrence of
perturbations and later, the development of fingers,
without crown formation, as reported by Levin and
Hobbs [11] and Thoroddsen and Sakakibara [3].
1.8ms
3.3ms
9.8ms
39.8ms
0.8ms
2.3ms
Droplet spreading is characterized by two dimensions:
the diameter of the wetted area, d, and the droplet
height above the surface, h (See Figure 4). Normalizing
these quantities by the initial droplet diameter, D0, yields
the spread factor, β(t) and the dimensionless height, ξ(t),
given by:
β (t ) ≡
ξ (t ) ≡
RESULTS
0.3ms
Figure 4: Water drop impacting on a horizontal, dry
aluminium surface. We=486, Re = 11965. Frame rate:
2000fps.
1.3ms
d (t )
D0
h(t )
D0
(2)
(3)
Figure 5 shows the variation of d/Do and h/Do with time
for a water and for a Diesel oil droplet with the same
Weber number, We=487, impacting onto a smooth
perspex surface (Ra<5µm). The water droplet has a high
Reynolds number, of about 11400, while the Diesel oil
droplet has a lower Reynolds number, of about 2550.
The results in the Figure 5 show the periodic
deformation of the water droplet, expanding and
recoiling. After the impact, the drop spreads as a thin
liquid film on the surface, until it reaches the maximum
spreading diameter (spread factor β(t)=4.8), at
t=4.7ms.The dimensionless height reaches a minimum
value (ξ(t) = 0.17) at this point and should increase
afterwards. However, the plot in the Figure shows that
the dimensionless height remains constant while the
droplet continues its deformation. This is due to the
concave form of the droplet surface, which does not
allow visualization of the center of the droplet from a
side view. Similar observations can be inferred from the
results reported by B. S. Kang and D. H. Lee [9], for a
droplet of water impinging onto a surface heated at the
temperature of nucleate boiling regime. However, the
phenomena is more noticeable in the present case,
which must be associated with the nature of the surface
and velocity at the impact.
2.8ms
4.8ms
7.3ms
19.8ms
29.8ms
49.8ms
71.8ms
After spreading the water drop recoils, only until about
t=6ms, when the dimensionless height reaches a peak
(ξ(t) =0.23). This first recoiling is not very clear in the
spread factor curve. Then the liquid film starts spreading
again, until it reaches a new maximum spread at about
t=15ms. Afterwards the liquid film recoils until it reaches
its final state. During spreading, some liquid breaks
through the lamella, due to the action of inertial forces,
which overcome surface tension forces. This is why the
spread factor is larger for water than for the Diesel oil
droplet. However, if the spread factor doesn’t account
with the liquid breaking through the lamella, Diesel oil
drop has a larger spread. Later, during the recoiling, this
liquid film is separated from the liquid that will remain
deposited in the central region, leading to the formation
of two small secondary drops, as it is shown in Figure 6.
The perturbations observed in the lamella (See Figure 6)
are fingers, which are formed during the spreading
process. Despite the formation of large fingers, splash
does not occur, probably because the impact surface is
very smooth. Therefore the surface properties, which are
not accounted by the dimensionless numbers
considered here, may have influenced the establishment
of two different regimes (deposition/splash), delaying the
occurrence of splash.
6
5
d/Do
4
Diesel oil
droplet
Re=2558
3
2
Water droplet
Re=11383
1
0
0
0,01 0,02 0,03 0,04 0,05 0,06
t [s]
0,6
Diesel oil
droplet
Re=2558
0,5
h/Do
0,4
Water
droplet
Re=11383
0,3
0,2
0,1
0
0
0,012 0,024 0,036 0,048 0,06
t [s]
Figure 5: Time variation of the spread factor and the
dimensionless height, for a water droplet and a
Diesel oil droplet, both with We=487.
The periodic behaviour of the deformation process of the
water droplet cannot be observed for the Diesel oil
droplet. Maximum spread diameter occurs at about
t=5ms, and has a value of about ξ(t) = 3.4. Moreover, no
significant recoil is observed for the Diesel oil droplet,
which is confirmed both by the spread diameter and
dimensionless height curves that remain practically
constant after reaching the maximum spread. Naturally,
if the Diesel oil drop has a larger spread than the water
drop, its dimensionless height is smaller. The differences
observed in the behaviour of the two droplets may be
explained, in part, by the lower surface tension of the
Diesel oil, which may not be large enough to bring the
liquid back to the central region of the droplet. Also, it
seems that the system Diesel oil/perspex surface has a
higher wettable liquid/solid interaction, so the liquid film
progression is easier. The influence of viscosity does not
seem to be important during the first stage of spreading,
since inertial forces are dominant. However, the higher
dissipation due to shear forces may explain the very
small (almost inexistent) recoiling stage of the Diesel oil
droplet, which has a lower Reynolds number.
Figure 7 shows the variation of d/Do and h/Do with the
time for a water and for a Diesel oil droplet with the
same Reynolds number, Re=2558, impacting onto a
smooth perspex surface (Ra<5µm). For this Reynolds
number the water droplet has a very low Weber number
and, in this case, differences between the two droplets
are even clearer. The water droplet has a periodic
behaviour, contracting and expanding about 16 times as
shown in the images of Figure 8, in a way that can be
described as a spring-mass system, as by Haitao Xu et
al. [11]. For this case, the water drop has a very low
impact velocity, and, therefore, has a small spread
diameter. Maximum spread occurs at t=4.5ms and is
about 2. Although this drop has a smaller impact
velocity, the maximum spread occurs nearly at the same
time as for the drop of Figure 5. For the initial stages of
spreading, the inertial forces are dominant, as expected
since droplet diameter has a contribution ∝ d3, which is
bigger than the contribution of droplet velocity. After
reaching the maximum spread, and consequently the
minimum dimensionless height, the liquid film begins to
recoil and reaches a height peak of about 0.8, at instant
t=11ms. At this instant, a phenomena close to partial
rebound, is clearly observed, as it is shown in Figure 8.
The following peaks of dimensionless height are
successively smaller, due to dissipation of energy
caused by shear forces with the wall.
It is also interesting to investigate the influence of
surface roughness in droplet spreading. Figure 9 shows
the time evolution of the spread factor and
dimensionless height of Diesel oil drops impacting onto
a rough aluminium surface (Ra =18.84µm ±10%) and on
a smooth perspex surface (Ra<5µm). The droplets have
nearly the same diameter and the same impact velocity,
so changes in the behavior may be attributed to surface
properties only.
The results show similar qualitative trends, in the sense
that both cases show the periodic deformation behaviour
previously reported. Quantitatively, the effects of surface
roughness seems to be negligibly small in the initial
stage (t* < 0,5), which is in accordance with the fact that
the initial rate of change depends more on the kinetic
energy of the droplet at impact (e. g., Fukai et al., [12]).
Further, the maximum spread is larger on the smooth
surface as also observed by other authors (e. g., Fukai
et al., [12]; Sikalo et. al., [7]) and the minimum nondimensional height is larger on the smooth surface,
17.25ms
19.75ms
22.25ms
24.75ms
27.25ms
29.75ms
Figure 6: Recoiling of a water droplet upon a perspex surface, with We=487. The droplet, with an initial estimated
diameter of 3.6mm, was released from a height of 495mm. Frame rate:2000fps.
4
1,2
3,5
1
2,5
Diesel oil droplet
We=487
2
1,5
Water droplet
We=47
1
Diesel oil droplet We=487
0,8
h/Do
d/Do
3
0,6
Water droplet We=47
0,4
0,2
0,5
0
0
0
0,01
0,02
0,03
0,04
0,05
0,06
t [s]
0
0,012
0,024
0,036
0,048
0,06
t [s]
Figure 7: Time variation of the spread factor and the dimensionless height, for a water droplet and a Diesel oil
droplet, both for Re=2558.
in accordance with the requirement of mass
conservation. However, other authors observed that the
surface only affects the height of the droplet on the recoil
phase. The difference may stem from the much smaller
value of the Reynolds number considered in this case
(about 400 compared to 3264). It may be so, since the
increased dissipation due to roughness takes more time
to act when the kinetic energy at impact is larger. Also, a
larger dissipation of the initial energy at the surface will
decrease the period of the deformation, as also
observed in Figure 9. Therefore, the results presented
here suggest that, as the kinetic energy at impact
increases, the effect of surface roughness may be
delayed for later stages of the deformation process.
before impact
1.0ms (after impact)
2.0ms
4.0ms
12.5ms
5.0ms
7.5ms
10.0ms
14.0ms
17.5ms
22.5ms
27.5ms
30.0ms
25.0ms
32.5ms
Figure 8: Water droplet impacting onto a perspex smooth surface. We=47, Re=2558. Initial periods of
deformation. Frame rate:2000fps.
3,5
1.0
2,5
0,8
2.0
0,6
ξ(t)
β(t)
3.0
1,5
0,4
1.0
0,5
0,2
0.0
0
5
10
15
t*
Aluminium rough surface Ra =18.84µm
0.0
0
5
10
15
t*
Perspex smooth surface Ra<5µm
(b)
(a)
Figure 9: Variation of β(t) and ξ(t) with the dimensionless time for a Diesel oil droplet, impacting onto an
aluminium rough surface (We=18; Re=464, Ra=18.84µm±10%) and on a smooth perspex surface (We=13; Re=394,
Ra<5µm).
It is worth discussing the physical meaning of these
results based on a model proposed by Pasandideh-Fard
et al.[10]. In this model, it is assumed that the film of
liquid expands radially on the surface, with velocity VR,
as shown in Figure 10.
This model considers that the droplet spreads into a
cylindrical disk of diameter D(t) and height h. The
velocity VR is given by conservation of mass:
VR
d (t ) 2
=
(4)
U o 4 D(t )h
4
h=
πDo3
(5)
6
or
h=
2 Do3
2
3Dmax
(6)
Combining eqs. (5) and (6) gives:
2
3 U o Dmax
VR =
d (t ) 2
2
8 D(t ) Do
(7)
Uo
Do
d(t)
h
D(t )
3 Uo
=
t
Dmax
8 Do
(8)
Maximum diameter occurs when:
t* = t
where U0 and D0 are, respectively, the droplet velocity
and diameter before impact. When the disk (liquid film)
reaches its maximum diameter, Dmax, the liquid volume
within the disk equals the volume of the initial droplet,
i.e.:
πD 2max
Assuming, as Pasandideh-Fard et al.[10], an average
value d(t)~D0/2 and knowing that VR=δD(t)/2δt, the
evolution of splat diameter (D(t)) is given by integration
of eq. (7):
Uo 8
≈
Do 3
(9)
Therefore, the dimensionless time (i.e. the time
normalized by the impact velocity and by the initial
droplet diameter), t*, required for the droplet to reach its
maximum extent, is a constant.
Back to the Diesel oil and water drops considered in
Figures 5, 7 and 9, the values for t* may be determined
and compared with those obtained by Pasandideh-Fard
et al. [10], as shown in table 2.
The results in Table 2 show that the experimental values
of t* differ from theoretical values of Pasandideh-Fard et
al. [10], being systematically larger than the theoretical
value of 8/3 for Reynolds numbers above 2558 but
smaller for Reynolds numbers below 464, regardless the
liquid or the surface roughness. This analysis, firstly
suggest that the behaviour of droplet deformation is
determined by the balance between inertial forces and
viscous forces acting against the spread of the liquid film
over the wall.
The maximum spread diameter may also be predicted
by applying the energy conservation condition to the
model of Pasandideh-Fard et al. [10]: droplet kinetic
energy, KED, plus the energy required to keep the
spherical shape of the droplet, SE1, must equal the final
surface energy, SE2, plus the work done in deforming
the droplet against viscosity, W, i. e:
KED + SE1 = SE2+W
VR
The authors found a relation for W, based on
appropriate length scales, and achieved an equation for
the maximum spread:
D(t)
Figure 10: Model of droplet spreading, according to
Pasandideh-Fard et al. [10].
D max
We + 12
=
Do
3(1 − cosθ a ) + 4 We
(
Re
)
(10)
where We is the Weber number, Re is the Reynolds
number and θa is the contact angle between the liquid
and the impact surface.
We=487, Re=11383
(Water drop onto a smooth
perspex surface, Ra<5µm)
We=47, Re=2558
(Water drop onto a smooth
perspex surface, Ra<5µm)
We=487, Re=2558
(Diesel oil drop onto a
smooth perspex surface,
Ra<5µm)
We=4, Re=198
(Diesel oil droplet onto
smooth perspex surface,
Ra<5µm )
We=13, Re=390
(Diesel oil droplet onto
smooth perspex surface,
Ra<5µm )
We=18, Re=464
(Diesel oil onto rough
aluminium
surface,
Ra=18.84µm±10%)
t*
(present
work)
t*
(PasandidehFard et al.)
[%]
4.0
8/3
32.5
4.25
8/3
36.5
4.25
8/3
36.5
∆
Table 2 Comparison between experimental values of
t* and the value obtained by Pasandideh-Fard et al.
[10]
The scatter observed in Figure11 for low Reynolds
numbers is attributed to the effect of the surface. It is
expected that the effect of surface roughness may be
that of altering friction forces at the wall, which may be
accounted in Reynolds number with a modified viscosity.
6
5
1.3
1.61
8/3
8/3
- 51.8
- 39.3
Dmax/Do
Experimental
Conditions
4
3
2
1
0
1.66
8/3
- 38.5
Following the analysis of the values in Table 1, it is
expected that the maximum diameter attained by the
liquid film during spread may be determined mainly the
ratio between inertial forces and viscous forces at the
wall, I e, by the Reynolds number. This is shown in
Figure 11, where the two lines corresponds to curves of
Dmax/Do obtained from equation (10) with limiting values
of the Weber number of 40 and 400 an limiting values of
the contact angles of 5o and 60o as measured by Mundo
et al. [14].
The symbols in Figure 11 represent the experimental
results obtained in the present work. The results show a
good agreement with the model of Pasandideh-Fard et
al. [10] for Reynolds numbers above 2000, but the
scatter is larger for small values of Reynolds number, for
which the assumptions for the dissipative term may not
be appropriate, as also mentioned by Chandra and
Avedisan [1] and by Mundo et al. [14] who used similar
assumptions.
Furthermore, these results suggest that, for high values
of Re, Reynolds number alone may qualitatively
describe the steady state parameters characterizing
droplet deformation, regardless liquid properties and the
nature of the surface, at least provided that splash does
not occur. It is worth mentioning at this point that, if
surface roughness is not much smaller than the liquid
film thickness, it is expected the film to detach from the
surface. In this case, splash occurs and the physics of
phenomena are not described in the same way.
However, for the experimental conditions considered
here, the ratio of surface roughness to film thickness
attained a maximum value of about 6x10-2, which is not
large enough to promote secondary atomisation (e. g.,
Mundo et al. [14]).
0
2000
4000
6000
8000
10000
12000
Reynolds number
Figure 11 - Comparison between experimental
results of (Dmax/D0) and the model of PasandidehFard et al. [10]
It is worth mentioning that some difficulties were found in
measuring the spread diameter, due to the illumination
conditions, which may have contributed for some
differences in Figure 12. Also, the value for the contact
angle was estimated by consulting the literature, in
particular the experimental results of Fukai et al.[12] and
Pasandideh-Fard et al. [10] and. For the Diesel oil
droplet, the value of the contact angle was estimated,
based on the behaviour of water droplets with sodium
dodecyl sulphate, (surface tension of 50x10-3 N/m).
Referring to the accuracy of these results, it is important
to note that they depend strongly on the choice of the
operational parameters of the CCD camera, and it is
always necessary to optimise the time resolution with
the spatial resolution. When the deformation process is
slower it can be used a lower frame rate, that allows a
better spatial resolution, although at the expense of the
time resolution.
Finally it is interesting to observe with more detail the
periodic behaviour of water droplets. Figure 12 depicts
the evolution of the number of periods in a water droplet
as a function of the release height.
Number of periodes
18
16
14
12
10
8
6
4
2
0
0
200
400
600
800
1000
Droplet release height [mm]
Figure 12: Evolution of the number of periods as a
function of the droplet release height.
The graphic shows that the number of periods reduces
as the release height increases. This may suggest that
with an increase of the impact velocity, as there is a
larger spread and a bigger change of height above the
surface, there is a higher dissipation of surface energy.
For lower release heights, there is a lower spread and
the surface energy is high enough to produce the
phenomena close to partial rebound, observed in the
water drop, and allows the periodic behaviour also
observed in this droplet.
Influence of surface roughness in the splashing limits
When the Weber and Reynolds are high enough, the
droplet may splash as it impinges at the wall or as a
result of hydrodynamic perturbations developed during
spread, as shown in Figure 13 for a Diesel oil droplet. In
each case the liquid does not deposit on the surface.
number at which splash occurs was determined for
water and Diesel drops impacting onto various dry plates
with different surface characteristics (see Table 1). The
results are plotted in Figure 14 as function of log(Ra/Ro),
where Ro is the droplet radius at the impact, together
with measurements obtained reported in the literature
considering other surfaces. The curves in Figure 14
represent groups of results obtained by (o) Stow and
Hadfield [4] with water drops onto aluminium and by
Range and Feuillebois [5] with (•) water droplets on
glass, (×) water droplets impacting onto transparencies
produced by 3M, (+) water droplets on aluminium and (∗)
water droplets on Plexiglas.
From their experimental results Stow and Hadfield [4]
developed a correlation for the splashing limits,
according to the empirical formula:
R0 u 10.69 = S T ( Ra )
(11)
where the critical value of a droplet to splash, ST,
depends on the arithmetic roughness Ra. The authors
also rewrote equation (11) in terms of a critical Weber
and a critical Reynolds number (Wec and Rec,
respectively), so that:
Re c0.31 Wec0.69 = ξ ( Ra )
(12)
where ξ is a dimensionless critical number for splashing,
function of Ra. The curve fitting of their results follow this
empirical correlation (circles in Figure 14).
Figure 13: Splash of a Diesel oil droplet upon a
perspex surface, with We=630. Droplet has an initial
diameter of 2.8mm. Frame rate:2000fps.
Although it is known that the onset of splash is
associated with high velocities of impact when smooth
surfaces are considered, the experiments in the present
work showed that surface roughness promotes splash,
as also observed by Stow and Hadfield [4], Range and
Feuillebois [5], Mundo et al. [14] or Cossali et al. [15].
In order to study in detail the influence of surface
roughness in the splashing limits, the critical Weber
Figure 14: Critical Weber for splashing as a function
of dimensionless roughness Ra/Ro. Adapted from
Range and Feuillebois [5]. Our results:
Water drops impacting onto an aluminium surface,
Ra=6.7µm±10%( ), an aluminium surface,
Ra=15.2µm±10%( ), an aluminium surface,
Ra=18.84µm±1%(
), an aluminium surface,
Ra=22.2µm±10%(
) and a perspex surface,
Ra=66.6µm±10% (
).
On the other hand, Range and Feuillebois [5] fitted their
results making use of the correlation of Wu [16]. This
author stated that for small Ohnesorge numbers,
(Ohnesorge number, Oh = ν
ρ / 2 R0σ
, where ν is the
liquid kinematic viscosity, ρ is the liquid density, σ is the
liquid surface tension and Ro is the initial droplet radius)
the influence of viscosity can be neglected and the
correlation of Stow and Hadfield may be improved to:
R
Wec = a log b  0
 Ra



(13)
Wu obtained the values of a=6.47 and b=1.87 by fitting
his formula to the experimental results of Stow and
Hadfield with a least-squares method.
For each plate material Range and Feuillebois [5]
adjusted the values for the coefficients a and b, in order
to fit their experimental results with the correlation of Wu
[16].
characteristics of the surface. It was noticed in the
measured roughness profiles, that the frequency of the
surface perturbations in the surfaces with random
roughness was very high. So, although Ra is large,
maybe the high frequency makes the surface to behave
as a smooth surface. For the surfaces with a regular
profile, it was observed that the splash occurs in a later
stage of the spreading. Possibly, the regular profiles
promote the flow of the film over the surface and only
later it becomes unstable, generating fingers. Then,
some of the fingers detach from the film and generate
secondary droplets.
From all the plotted results it seems that the results do
not always follow the suggested correlations, particularly
the correlation by Stow and Hadfield [4]. As for the
correlation proposed by Wu [16], it seems that there is a
dependence of the coefficients of a and b, not only with
the surface material but also with the liquid, since it is
necessary to adjust the values of these coefficients
whenever the surface material or the liquid of the droplet
are changed. Further investigation may allow to
determine in detail the dependence of coefficients a and
b with those parameters.
In Figure 15 the results obtained with Diesel oil droplets
impacting onto aluminium and perspex surfaces are
compared with results reported by Range and
Feuillebois [5] with ethanol and water in order to analyse
the influence of fluid properties.
The experimental results of the present work plotted in
Figure 14 were obtained with water droplets impacting,
mainly onto aluminium surfaces (only one of the
surfaces is made from perspex). It was observed that
the critical Weber limiting splashing/deposition regimes
is not a well defined number, but instead it varies with
the experience. The range of variation was considered
as the accuracy due to the reproducibility of the
splashing phenomena and was used as the vertical
dimension of the symbols in the Figure. In a similar way,
the horizontal dimension of the symbols was taken as
the accuracy in the measurements of Ra (10%).
Observing our results in Figure 14 it is noticeable that
the data obtained from the impact of water droplets on
aluminium surfaces follow the curve for 3M
transparencies obtained by Range and Feuillebois [5].
As for the perspex plate, the point is also near this
curve, but there is not enough data to establish the
distribution of the results using perspex plates. It is also
clear that our results do not follow the correlation
proposed by Stow and Hadfield [4]. Although it follows a
similar behaviour, the values of the critical Weber are
higher, even when higher values of Ra are considered.
Even though our experimental conditions are slightly
different from those used by Stow and Hadfield [4], it
does not explain the differences observed. One possible
explanation may lay, not in the value of Ra, but in other
Figure 15: Critical Weber for splashing of droplets
as a function of dimensionless roughness Ra/Ro, for
different liquids. Adapted from Range and
Feuillebois [5]. Our results:
Water drops impacting onto an aluminium surface,
Ra=6.7µm±10%( ), an aluminium surface,
Ra=15.2µm±10%( ), an aluminium surface,
Ra=18.84µm±1%(
), an aluminium surface,
Ra=22.2µm±10%(
) and a perspex surface,
Ra=66.6µm±10% (
). Diesel oil droplets
impacting onto an aluminium surface,
Ra=18.84µm±10%( ) and a Perspex surface,
Ra=18.84µm±10%(
).
characterized by the dimensionless roughness Ra/Ro.
However, this parameter alone does not describe
completely the nature of the surface and other
characteristics describing the surface profile may also be
considered. Therefore, existing correlations do not apply
to all cases considered in the present work. However,
more systematic measurements are still necessary in
order to develop appropriate correlations.
ACKNOWLEDGMENTS
It is observed that Diesel oil droplets have higher critical
Weber numbers, both for aluminium and perspex plates.
From the point of view of the influence of the surface
tension, Diesel oil has the lowest surface tension, so
Diesel oil droplets should splash easier. The fact that
Diesel oil drops were slightly smaller than the water
drops may explain the higher splash velocities, but does
not necessary explains the higher value for the critical
We, since the Weber number accounts for the influence
of both diameter and velocity. However, liquid viscosity,
which is smaller for water, is not taken into account. For
example comparing the Diesel oil and water droplets
impacting onto the aluminium random rough surface,
Reynolds number for Diesel oil is smaller than for water
(ReDiesel oil drop = 3590< Rewatercdrop = 8960). It is therefore
suggested that the effect of viscosity can not be
neglected for the Diesel droplets.
CONCLUSIONS
This work reports an experimental study of the
deformation and splashing of water and Diesel oil
droplets impinging on rough surfaces. The results show
that an accurate characterization of the physical
processes occurring during deformation and splash
must take into account the forces involved (surface
tension, inertial and shear forces), as well as the liquid
properties and the nature of the impact surface. The
analysis based on dimensionless numbers only,
neglects the effects of the properties of the surface,
which may influence the phenomena and the
establishment of different regimes. In general, the
models reported in the literature to describe droplet
deformation do not take into account surface properties,
although these do not alter the qualitative behavior of
liquid film spreading.
If splash does not occur, the effect of surface roughness
is expected to alter friction forces at the wall and,
therefore, the energy dissipated during spread. It is
found that, provided the Reynolds number of the droplet
at the impact is large (Re>2000), the energy dissipated
at the wall is not greatly affected by the nature of the
surface, which appears to be important for low Reynolds
numbers, typically Re<1000.
However, the nature of the surface significantly alters
the onset of splashing. Correlations for splashing found
in the literature take into account surface roughness
The authors acknowledge the contribution of the
National Foundation of Science and Technology of the
Ministry for Science and Technology by supporting this
study through the project POCTI/ 1999/ EME/ 32960 and
by supporting A. S. H. Moita with a Research Grant.
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