PHYSICS OF FLUIDS
VOLUME 12, NUMBER 12
DECEMBER 2000
Modeling the splash of a droplet impacting a solid surface
M. Bussmann, S. Chandra, and J. Mostaghimia)
Department of Mechanical and Industrial Engineering, University of Toronto, Toronto,
Ontario M5S 3G8, Canada
共Received 18 November 1999; accepted 4 September 2000兲
A numerical model is used to simulate the fingering and splashing of a droplet impacting a solid
surface. A methodology is presented for perturbing the velocity of fluid near the solid surface at a
time shortly after impact. Simulation results are presented of the impact of molten tin, water, and
heptane droplets, and compared with photographs of corresponding impacts. Agreement between
simulation and experiment is good for a wide range of behaviors. An expression for a splashing
threshold predicts the behavior of the molten tin. The results of water and especially heptane,
however, suggest that the contact angle plays an important role, and that the expression may be
applicable only to impacts characterized by a relatively low value of the Ohnesorge number.
Various experimental data of the number of fingers about an impacting droplet agree well with
predictions of a previously published correlation derived from application of Rayleigh–Taylor
instability theory. © 2000 American Institute of Physics. 关S1070-6631共00兲00512-2兴
I. INTRODUCTION
crown. The influence of the gas phase about the impacting
droplet is considered to be negligible. The impact is assumed
to be isothermal and the liquid phase properties are assumed
to be constant. The variables that then affect such an impact
include: the initial droplet diameter D o and velocity V o , liquid density ␳ , viscosity ␮ , and liquid–gas surface tension ␥ ,
and measures of the surface roughness and wettability. Surface roughness is usually characterized by an average roughness height R a , and wettability by a dynamic contact angle
␪ d that reflects the relative values of the three interfacial
tensions at a moving contact line.
Nondimensionalization reduces this set of variables to
only four. The usual choices are a non-dimensional average
roughness R a* ⫽R a /D o , the contact angle ␪ d , and any two
of the Reynolds, Weber, and Ohnesorge numbers, to represent the relative magnitudes of inertial, viscous, and surface
tension forces
Consider a droplet that falls onto a dry, flat, and relatively smooth surface. On impact, as the droplet reacts to a
sudden increase in pressure, a thin axisymmetric sheet of
fluid begins to jet radially outward over the solid surface.
Depending on a number of factors, the leading edge of this
sheet may become unstable soon after impact, resulting in
the emergence of very regular azimuthal undulations. Photographs taken from beneath a transparent surface illustrate
such undulations when the sheet diameter D is a fraction of
the initial droplet diameter D o . 1,2 Where the undulations
continue to grow they may begin to resemble fingers. If the
fingers grow far enough beyond the leading edge, the Rayleigh instability leads to the pinching off of secondary droplets.
In what follows, the term ‘‘fingering’’ will refer to the
presence of a perturbed leading edge, whether or not the
undulations resemble fingers. The term ‘‘splashing’’ will describe the formation of secondary, or satellite, droplets.
As noted, the above description of fingering and splashing describes the impact of a droplet onto a relatively smooth
surface. The fluid spreads, fingers, and splashes while remaining in contact with the solid surface. When the same
droplet impacts a sufficiently rough surface, the expanding
sheet tends to lift up off of the surface, and the subsequent
behavior of the fluid changes dramatically. The most distinctive characteristic of such an impact is the formation of a
‘‘crown’’ or ‘‘coronet’’ usually associated with impact of a
droplet onto a liquid surface. The milk drop coronet of Harold Edgerton3 is a well known example of such an impact.
This paper considers the normal impact of a spherical
droplet onto a dry, flat surface that is sufficiently smooth
such that fingering and splashing occur in the absence of a
␳ D oV o
,
Re⫽
␮
Oh⫽
␮
冑D o ␥ ␳
.
共1兲
Re and We are the dimensionless quantities used in this
work. Results in the literature presented in terms of Oh have
been converted according to Oh⫽ 冑We/Re.
As the following literature review will demonstrate, the
related phenomena of fingering and splashing during the impact of a droplet onto a dry surface are not well understood.
There is little agreement on the mechanism that initiates the
perturbation at the leading edge of the expanding sheet, nor
agreement on a model of the number of fingers that form.
The role of surface roughness has not been satisfactorily explained, nor has the onset of crown formation been addressed. Although correlations exist to predict the onset of
splashing, these have not proven universal. And the influence
of surface wettability has received scant attention.
This paper presents a numerical methodology for examining droplet fingering and splashing, by initiating a pertur-
a兲
Author to whom correspondence should be addressed. Telephone: 416 978
5604; Fax: 416 978 7753. Electronic mail: [email protected]
1070-6631/2000/12(12)/3121/12/$17.00
␳ D o V 2o
We⫽
,
␥
3121
© 2000 American Institute of Physics
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3122
Phys. Fluids, Vol. 12, No. 12, December 2000
Bussmann, Chandra, and Mostaghimi
bation of the flow shortly after the impact of a droplet. The
three-dimensional 共3D兲 numerical model has been presented
previously.4 Numerical results of the effect of impact velocity, contact angle, and roughness are compared with photographs of corresponding impacts. Various data of the number
of fingers about an impacting droplet are presented as evidence that the perturbation may be the result of the
Rayleigh–Taylor instability acting at the leading edge. Although some results corroborate a published expression for a
splashing threshold, other results are contradictory.
II. LITERATURE REVIEW
The first study of droplet fingering and splashing was
that of Worthington5–7 published more than a century ago.
Utilizing an experimental setup that is conceptually similar
to techniques still in use today, to trigger a spark at some
instant during the impact of a droplet, Worthington repeatedly observed and then sketched the fingering and splashing
of large milk and mercury droplets impacting onto smooth
glass plates. Worthington observed no crown formation;
noted that the number of fingers increased with both droplet
size and fall height; observed the merging of fingers at or
soon after maximum spread; and noted that fingering was
more pronounced for fluids that did not wet the substrate
共mercury on smoked glass兲 than for fluids that did 共milk on
clean glass兲.
Engel8 and Levin and Hobbes9 observed a strong correlation between increasing surface roughness and the tendency of droplets to splash upwards and form a crown. Impact without crown formation was only observed when the
impact surface was characterized as very smooth.9
Stow and Hadfield10 examined splashing as a function of
measured roughness, and reported that splashing only occurred for values of a ‘‘splash parameter’’ K⬅We0.5Re0.25
greater than a threshold value K c that was strictly a function
of the surface roughness. K c varied dramatically with small
values of R *
a and approached an asymptotic value as roughness increased. Stow and Hadfield noted that the scale of
roughness necessary to initiate splashing was much smaller
than the characteristic height of the fluid sheet that developed
soon after impact; proposed that the role of roughness was to
initiate an instability in the sheet; and suggested that the
proposed threshold would likely not apply to surface roughnesses comparable to the height of the sheet.
Mundo et al.11 confirmed the asymptotic relationship of
splashing with surface roughness by examining the impact of
small water droplets onto relatively rough surfaces (0.019
⬍R a* ⬍1.3). Impact was oblique to the surface, yet by using
the normal velocity component to evaluate We and Re, the
onset of splashing was consistently observed at K c ⫽57.7.
Cossali et al.12 then correlated the data of Stow and
Hadfield,10 Mundo et al.,11 and of Yarin and Weiss,13 who
studied the impact of a train of droplets onto a solid surface,
with the following relationship:
K 1.6
c ⫽649⫹
3.76
0.63
R*
a
.
共2兲
The splashing threshold has also been correlated with a
critical Weber number Wec . Wachters and Westerling14 observed the onset of splashing at Wec ⫽80 for a number of
different liquids impacting a polished gold surface. And
Range and Feuillebois2 recently considered the influence of
roughness on droplet impact at low values of Oh, corresponding to a weak influence of viscosity. They correlated
the onset of splashing with the following expression:
Wec ⫽a lnb 共 R a* 兲 ,
共3兲
where the parameters a and b differed with each liquid–solid
combination.
Little has been written of the number of fingers that appear about an impacting droplet. More than two decades ago,
Allen15 published a brief calculation based on an application
of Rayleigh–Taylor 共RT兲 instability theory. RT is a fingering
instability that occurs at an interface between two fluids of
different densities, when the lighter fluid pushes the heavier
fluid.16 A linear analysis of the RT instability provides the
following expression for a fastest growing wavelength:
␭⫽2 ␲
冑
⫺
3␥
,
a共 ␳ H⫺ ␳ L 兲
共4兲
a represents the acceleration of the interface, which is negative as the droplet spreads, and ␳ H and ␳ L represent the densities of the heavy and light fluids, respectively. When applied to the interface between a liquid droplet and the
ambient gas phase, ␳ H Ⰷ ␳ L .
Returning to Allen,15 he obtained a reasonable prediction
of the number of fingers visible about the rim of an ink blot
by estimating the acceleration of the interface as a⬇
⫺V 2o /(D max/2), with D max the measured diameter of the ink
blot, and N⫽ ␲ D max /␭.
Bhola and Chandra17 recently revisited the work of
Allen.15 They estimated the acceleration somewhat differently, as a⬇⫺V 2o /D o , and introduced an analytical expression for the maximum fluid spread, D max , derived from consideration of an integral energy balance.18 The resulting
expression was a function of the splash parameter K:
N⫽
We0.5Re0.25
4 冑3
⫽
K
4 冑3
.
共5兲
Equation 共5兲 accurately predicted the number of fingers observed about the perimeter of impacting wax droplets, and
later offered a good prediction of the number of fingers about
molten tin droplets impacting a hot surface.19
Marmanis and Thoroddsen20 counted fingers about the
blots left by droplets of a variety of liquids impacting a paper
surface. They obtained a correlation of the form N
⬃(We0.25Re0.5) 0.75. Data gleaned from the work of
Worthington7 of mercury droplets impacting a glass surface
deviated significantly from their own data, which the authors
speculated might be related to the difference in the wettability of mercury on glass as compared to other liquids on paper.
Finally, a recent paper by Kim et al.21 applied an RTbased linear perturbation theory to examine the stability of
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Phys. Fluids, Vol. 12, No. 12, December 2000
the sheet of fluid that jets radially outward beneath an impacting droplet shortly after impact. Their results predicted a
variation of the fastest growing wavelength with time. However, for reasonable assumptions of when the fingers first
appear, the model yielded estimates of N in line with experimental observations.1
There are three further studies of droplet fingering and
splashing that deserve mention. Scheller and Bousfield22
studied the effect of viscosity on the spreading of droplets
during impact, while maintaining relatively constant values
of surface tension. They reported that splashing occurred
only in a ‘‘viscosity window,’’ above and below which droplets did not splash. Prunet-Foch et al.23 and Vignes-Adler
et al.24 studied the impact of both pure water and emulsion
共oil–water兲 droplets. They noted a marked difference in the
shape of fingers extending beyond the contact line, which
they suggested was related to the relative motion of the dispersed phase within the continuous phase of the emulsion
droplets. Thoroddsen and Sakakibara1 examined the very
early development of fingers during impact of watersurfactant droplets onto smooth glass plates. They speculated
that initiation of the instability did not occur after impact, but
actually just before, as fluid at the bottom of the droplet
decelerated in response to the compression of air between the
droplet and the surface.
Numerical studies of droplet impact have dealt almost
exclusively with the two-dimensional 共2D兲, or axisymmetric,
problem of a droplet impacting normal to a solid surface.
The assumption of axisymmetry precludes the consideration
of flow in the azimuthal direction, and thus the consideration
of fingering and splashing. Fully 3D models of droplet impact have appeared only recently.
The results of this paper are based on the 3D model of
Bussmann et al.4 Utilizing a volume tracking methodology
to track the liquid–gas interface, and with a simple model of
the variation of contact angle with contact line velocity, the
3D model yielded good predictions of gross fluid deformation during droplet impact onto an incline and onto an edge.
Gueyffier and Zaleski25 and Rieber and Frohn26,27 have
developed similar 3D models to simulate droplet impact onto
a liquid surface. The choice of a liquid surface eliminates the
influence of the contact angle, and decreases the influence of
underlying surface roughness. The results of these groups are
similar, predicting the formation of a crown and the detachment of droplets. The significant difference between the two
models is in the approach to initiating a perturbation. Gueyffier and Zaleski imposed a small harmonic perturbation to
the shape of the droplet prior to impact. The amplitude of the
perturbation required to produce fingering atop the crown
varied between 0.005 and 0.019 of D o .
Rieber and Frohn26,27 applied two different perturbations
to obtain realistic results. A first paper26 presented results
where a small perturbation was applied to the liquid film.
More realistic results were presented in a second paper27 in
which a random disturbance with a Gaussian distribution
was applied to all initial velocities within the film and the
droplet. Although the standard deviation of the disturbance
appeared rather large 共up to 0.5V o ), the authors noted that
Modeling the splash of a droplet impacting a solid surface
3123
viscous and surface tension forces quickly reduced the energy of the disturbances.
III. METHODOLOGY
A. Numerical methodology
The numerical model has been presented previously.4
The following is a brief overview of its main features. Equations of conservation of mass and momentum govern the
flow within the liquid phase following impact:
“•V⫽0,
共6兲
⳵V
␮
1
1
⫹“• 共 VV兲 ⫽⫺ “p⫹ ⵜ 2 V⫹ Fb .
⳵t
␳
␳
␳
共7兲
V represents the velocity vector, p the pressure, and Fb any
body forces that act on the fluid. Boundary conditions include: the surface tension-induced pressure jump at the droplet surface, p s ⫽ ␥␬ , where ␬ represents the total curvature of
the interface; a zero tangential stress condition at the surface;
and the dynamic contact angle ␪ d imposed at the contact
line, where the solid, liquid, and gas phases meet.
The model is an enhanced three-dimensionalization of
RIPPLE,28 a 2D Eulerian fixed-mesh fluid dynamics code
developed specifically for free surface flows. Many details of
the discretization are similar to those presented by Kothe
et al.28 Equations 共6兲 and 共7兲 are discretized in a typical control volume formulation. Convective, viscous, and surface
tension effects are evaluated explicitly at each time step, followed by an implicit evaluation of pressure to enforce local
mass conservation. The computational domain is discretized
by a rectangular mesh large enough to encompass the droplet
before impact and to capture the subsequent fluid deformation.
The model also includes a piecewise linear volume
tracking algorithm to track the deformation of the free surface. The interface is defined by a scalar function f that represents fluid volume, equal to one within the liquid phase and
zero without. In discretized form, a volume fraction f i, j,k
represents the fraction of the volume of a cell that contains
liquid: equal to one if the cell is full, zero if empty, and 0
⬍ f i, j,k ⬍1 if the cell contains a portion of the free surface,
deemed an interface cell. Since f is passively advected by the
flow, f satisfies
⳵f
⫹ 共 V•“ 兲 f ⫽0.
⳵t
共8兲
The 3D model incorporates the advection algorithm of
Youngs29 to solve Eq. 共8兲 geometrically. At each time step,
the algorithm consists of two steps. First, the free surface is
reconstructed by locating a plane within each interface cell
corresponding to f i, j,k and the unit normal to the surface,
n̂ i, j,k . The second step is a geometric evaluation of volume
fluxes to obtain an updated volume fraction field.
A distinct characteristic of the model is the treatment of
surface tension, which is accounted for as a body force acting on fluid in the vicinity of the free surface according to
the Continuum Surface Force 共CSF兲 model of Brackbill
et al.30
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3124
Phys. Fluids, Vol. 12, No. 12, December 2000
FST 共 x兲 ⫽ ␥
冕␬
S
共 r兲 n̂ 共 r兲 ␦ 共 x⫺r兲 dS.
Bussmann, Chandra, and Mostaghimi
共9兲
␦ is the Dirac delta function and the integration is performed
over some area of free surface S. Surface tension is then
incorporated into the flow equations as a component of the
body force Fb in Eq. 共7兲.
We discretize Eq. 共9兲 by introducing a finite kernel ␦ 2h ,
acting over a radius 2h, to approximate ␦ . A discrete surface
tension force FST i, j,k is evaluated for each interface cell:
FST i, j,k ⫽ ␥␬ i, j,k
A i, j,k
n̂
.
␷ i, j,k i, j,k
共10兲
A i, j,k represents the surface area contained within the cell,
evaluated by Youngs’ advection algorithm. ␷ i, j,k is the volume of the cell. The FST i, j,k are then convolved over the
radius 2h to obtain a force field FST i, j,k diffused over cells in
the vicinity of the free surface
Fi, j,k ⫽g i, j,k
兺
l,m,n
FST l,m,n ␦ 2h 共 xi, j,k ⫺rl,m,n 兲 ␷ l,m,n .
共11兲
Note the introduction of a weighting function g i, j,k ⫽2 f i, j,k
that transforms the volumetric force into a body force acting
on non-empty cells near the interface.
Estimates of n̂ i, j,k and ␬ i, j,k , which are geometric characteristics of the interface, are obtained from the volume
fractions:
n̂⫽
“f
兩“ f 兩
, ␬ ⫽⫺“•n̂.
共12兲
The volume fraction representation of the interface is steep,
with the transition occurring over no more than one or two
cell widths. As a result, better estimates of “ f are obtained
by evaluating the gradient of a convolved, or diffused, f i, j,k
field. We employ the same kernel ␦ 2h to convolve both f i, j,k
and FST i, j,k
␦ 2h 共 x兲 ⫽
再冉
0
1⫹cos
冉 冊冊
␲ 兩 x兩
2h
/c
兩 x兩 ⭐2h
,
共13兲
兩 x兩 ⬎2h
where c normalizes the kernel
c⫽
32 3 2
h 共 ␲ ⫺6 兲 / ␲ .
3
共14兲
A final note concerns the numerical treatment of fluid
breakup. In a recent review of the breakup of free surface
flows, Eggers31 acknowledges the challenge of numerically
modeling fluid breakup, and suggests that no best approach
has yet emerged. Volume tracking techniques such as the
one implemented here rely on a finite mesh resolution to
dissolve and break interfaces inherently. Schemes which explicitly track the interface can resolve much thinner interfaces, but then require an ad hoc prescription to break interfaces and thus model flows beyond breakup.
FIG. 1. Comparison of top views of the 0.93 m/s impact of a 1.5 mm
diameter heptane droplet onto a stainless steel surface, as a function of mesh
resolution.
B. Validation
The results of various validations of the model have
been presented previously,4 and demonstrate a capability to
accurately predict complicated asymmetric flows. Further to
these results, we consider how well the 3D model reproduces
2D results. In particular, we consider an axisymmetric impact in the absence of an applied perturbation.
Figure 1 presents results of 3D simulations of the normal
0.93 m/s impact of a 1.5 mm diameter heptane droplet onto a
stainless steel surface. A constant value of ␪ d ⫽32° 共Ref. 32兲
was applied to the simulations. These simulations and all
others presented in this paper were run on a uniform mesh.
The mesh resolution is specified as the number of cells per
initial droplet radius 共cpr兲. Figure 1 presents the results of
simulations run at three different resolutions, reflecting a
progressive halving of the cell volume.
Inspection of Fig. 1 reveals a ‘‘squaring’’ of the droplet
at a resolution of 16 cpr that diminishes at finer resolutions.
Figure 2 presents the variation of spread factor ␰ ⬅D/D o
with time, evaluated both along the gridlines and along the
diagonal, and compared to the data of Chandra and
Avedisian.32 Figure 2 confirms that fluid spreads more
quickly along the diagonal, and that this occurs even at 25
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Phys. Fluids, Vol. 12, No. 12, December 2000
Modeling the splash of a droplet impacting a solid surface
3125
FIG. 3. Typical views of a droplet at t * ⫽0.1, when the perturbation is
applied, and at t * ⫽0.2.
FIG. 2. Spread factor ␰ versus time for the 0.93 m/s impact of a 1.5 mm
diameter heptane droplet onto a stainless steel surface. Lines correspond to
␰ measured along gridlines and along the diagonal, as a function of mesh
resolution.
cpr, although Fig. 1 shows that the droplets are becoming
more symmetric.
The reason for the squaring is related to the difficulty in
specifying velocities as a boundary condition at a curved
interface defined on a cartesian mesh. The simulation of an
axisymmetric impact magnifies these errors, which introduce
a systematic bias to the boundary velocities. The result is a
subtle movement of fluid from along the gridlines towards
the diagonal. The reduction of the error with mesh refinement is first order. For a more detailed analysis of this problem, see Bussmann.33
C. Initiating a perturbation
As was mentioned earlier, the leading edge of the fluid
sheet beneath an impacting droplet may show evidence of an
instability shortly after the moment of impact,1,2 at a time
when the contact line diameter DⰆD o . Ideally, one would
like to model such behavior by imposing a random perturbation on the velocity field and/or the interface profile at or just
after the moment of impact, just as surface roughness seemingly perturbs the fluid at the base of a droplet on impact.
The model would predict the subsequent growth rate of a
spectrum of frequencies, the emergence of a fastest growing
wavelength, and simulate the growth of fingers and the possible detachment of satellite droplets.
In reality, computational resources are finite. Sharp16
points out that most numerical studies of the Rayleigh–
Taylor instability consider the growth of a single frequency
disturbance. This is also the approach employed by two of
the aforementioned studies of droplet impact onto a liquid
surface. Gueyffier and Zaleski25 perturbed the shape of a
droplet prior to impact according to
R o,p ⫽D o 共 0.5⫹Ap cos 12␽ ⫹Ap cos 12␾ 兲 .
共15兲
R o,p represents the perturbed initial radius, Ap the amplitude
of the perturbation, and ␽ and ␾ the two angular coordinates
associated with the spherical coordinate system. The simulation yielded 12 fingers atop a crown, corresponding to the
frequency of the imposed perturbation. Rieber and Frohn26
also imposed a perturbation of known 共but unspecified兲 frequency, but onto the liquid surface profile, to obtain physically realistic results.
Rieber and Frohn’s other work27 more closely approaches the ideal. A perturbation with Gaussian distribution
was applied to all initial velocities within the computational
domain. Rieber and Frohn demonstrated that the number of
fingers that appeared numerically corresponded to the number predicted by Rayleigh instability theory34 applied to the
breakdown of the liquid torus atop the crown.
Unfortunately, applying such a perturbation is unlikely
to succeed when modeling the impact of a droplet onto a
solid surface, because of how quickly the instability manifests itself beneath the droplet. When a droplet impacts a
liquid surface, the fingers appear atop the crown, far later
than when fingers appear when the droplet impacts a solid
surface. Like the models of Gueyffier and Zaleski25 and
Rieber and Frohn,26,27 the model presented here is a fixedmesh model that cannot sharply resolve the contact line just
after the moment of impact. The circumference of the contact
line then is too short relative to the mesh spacing to adequately resolve even a single frequency.
As a result, a somewhat different approach has been
implemented to initiate an instability. Instead of imposing a
disturbance at the moment of impact, simulations are run
unperturbed to t * ⬅V o t/D o ⫽0.1. At this time D is typically
just less than D o , and the contact line circumference is of
sufficient length to adequately resolve an imposed perturbation. The simulation is stopped, a perturbation is applied to
the radial component of the velocity field near the solid surface 共the x and y components illustrated in Fig. 3兲, and the
simulation is restarted. The perturbation is of the form
冉
冉 冉 冊冊
Vr,p ⫽Vr,up 1⫹Ap exp ⫺Bp
z
Do
2
冊
cos共 N ␽ 兲 ,
共16兲
where the subscripts r, p, and u p refer to radial, perturbed
and unperturbed quantities. Ap is the amplitude of the perturbation at z⫽0, Bp is the rate of decay of the perturbation
away from the solid surface, and N is the number of fingers
to be imposed.
The philosophy behind this approach to perturbing the
flow is admittedly pragmatic. The numerical model may be
unsuitable for examining the very earliest moments of impact beneath the droplet, but is very capable of yielding accurate predictions of macroscopic flow behavior. The objec-
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3126
Phys. Fluids, Vol. 12, No. 12, December 2000
Bussmann, Chandra, and Mostaghimi
used, the perturbation affected only velocities in the first two
or three cells above the solid surface. Figure 4 illustrates the
decay of the exponential term.
D. The number of fingers
FIG. 4. Decay of the perturbation.
tive is to instantaneously affect the simulation in a way that
reflects the growth of an instability up to t * ⫽0.1. Although
the amplitude of the instability will vary with Re and We,
the results of this paper will demonstrate that the same perturbation can be applied to simulations of different impacts
and yet accurately predict very different behaviors.
Several comments are in order. Although the perturbation is not applied at the moment of impact, t * ⫽0.1 is nonetheless early in the usual lifetime of an impact. Figure 3
illustrates the typical spread of a droplet at t * ⫽0.1, and a
representative view of a droplet an equal time later. The
perturbation is applied only to the radial component of velocity, and thus initiates no movement of fluid upwards off of
the surface. Crown formation is thus neither initiated nor
observed numerically. This limits this approach to modeling
the fingering and splashing of droplets onto relatively
smooth surfaces. The exponential term of Eq. 共16兲 focuses
the perturbation onto fluid very near the surface, where the
instability is observed. A value of Bp ⫽4000 was applied to
all simulations presented here. For the mesh resolutions
The perturbation requires the specification of N, the
number of fingers. For all of the results presented in this
paper, the number could be obtained from photographs of
corresponding impacts. To be predictive, however, the model
requires an independent estimate of N. The following is presented in support of Eq. 共5兲, which was obtained from a
simple application of Rayleigh–Taylor instability theory.17
Table I presents various experimental data of the number
of fingers about an impacting droplet at maximum spread, as
well as the corresponding number predicted by Eq. 共5兲. The
data are of impacts characterized by a range of Re and We,
and in most instances Eq. 共5兲 offers a good prediction of N.
Note, however, that Eq. 共5兲 is strictly an expression for the
number of fingers likely to appear, and incorporates no physics related to whether or not fingers appear at all. Thus Eq.
共5兲 predicts a finite, if small, number of fingers even when
none are observed experimentally, and of course predicts no
limit to the number of fingers as the splash factor K→⬁.
An alternative means of applying Rayleigh–Taylor
theory to the impact of a droplet is to introduce time-varying
expressions for D and a into N⫽ ␲ D/␭. The resulting equation would then indicate the number of fingers likely to be
initiated at any instant:
N⫽D
冑
⫺
a␳
.
12␥
共17兲
On purely geometric grounds, by assuming that droplet
volume displaced by a solid surface is transferred to the periphery of a droplet spreading on a surface 共resulting in the
shape of a truncated sphere atop a cylinder兲,18,21 the early
spread of a droplet can be shown to vary as D
⬃(D o V o t) 1/2. The constant of proportionality varies depending on the assumed geometry of the fluid at the periphery of
the droplet. Marengo et al.,37 for example, confirmed this
TABLE I. Comparison of predicted and observed numbers of fingers N at maximum spread. Note that the wax
and NaNO3 data are of molten droplets solidifying on a cold surface.
K
N⫽K/4冑3
71
143
322
572
89
141
235
337
13
20
34
49
14
15
35
46
942
8,000
500
438
124
198
18
29
19
⬇45
17
this work
1.2
1.7
2.8
4.9
2,200
3,200
5,200
9,100
127
254
690
2,100
77
120
223
448
11
17
32
65
12
20
33
⬇50
20
20
20
20
3.8
1.2
37,800
161
177
26
24
7
3.1
4.0
15,700
1,520
436
63
⬇50
36
D o 共mm兲
V o 共m/s兲
Re
tin
2.1
2.7
2.7
2.7
1.6
2.0
3.0
4.0
12,200
19,600
29,500
39,300
wax
water
3.0
2.0
2.2
4.0
dyed water
5.2
5.2
5.2
5.2
mercury
NaNO3
Material
We
N expt
Reference
35
19
19
19
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Phys. Fluids, Vol. 12, No. 12, December 2000
Modeling the splash of a droplet impacting a solid surface
3127
TABLE II. Summary of simulations.
Tin
Tin
Tin
Water
Heptane
D o 共mm兲
2.7
2.7
2.7
2.0
2.0
V o 共m/s兲
1.0
2.0
3.0
4.0
4.0
Re
9800
19 600
29 500
8000
13 200
We
36
143
322
438
1060
Oh
6.1⫻10⫺4 6.1⫻10⫺4 6.1⫻10⫺4 2.6⫻10⫺3 2.5⫻10⫺3
␪ d (°)
140
140
140
varying
32
K
60
142
235
198
349
9
20
34
29
50
N (⫽K/4冑3)
FIG. 5. Variation of N with D/D o for the 2 m/s impact of a 2.7 mm
diameter tin droplet onto a hot stainless steel surface, via Eq. 共18兲, with ␣
⫽0.204.
Neither of these observations proves that the Rayleigh–
Taylor instability is at work beneath an impacting droplet.
The observations do, however, provide more evidence that
the conjecture is plausible, and lend credence to the analyses
of Allen,15 Bhola and Chandra,17 and Kim et al.21
IV. RESULTS AND DISCUSSION
relationship of D(t) by fitting data of the early spread of a
water droplet to D⬃t 0.48. Thus, let D⫽ ␣ 冑t. Differentiating
twice with respect to t to obtain an expression for a, and
substituting into Eq. 共17兲 yields an expression for N as a
function of either t or D
冑
N⫽ ␣ 3/2
␳ ⫺1/4
t
⫽␣2
96␥
冑
␳ ⫺1/2
D
.
96␥
共18兲
2/3
Note the weak dependence of N on time. If D⬃t , which
implies that the droplet spreads more rapidly than predicted
by the above geometric arguments, then Eq. 共18兲 would lose
its dependence on time. However, as Kim et al.21 point out,
although little data exists of the very early spread of a droplet, due to the difficulty of measurement, the available results
do appear to support the 冑t dependence.
Consider now the data of two impacts applied to Eq.
共18兲. The first data are of a simulation presented later in this
paper 共Fig. 7兲, the 2 m/s impact of a 2.7 mm diameter molten
tin droplet onto a hot surface. A fit of the early spread data
yields ␣ ⫽0.204. Figure 5 illustrates the resulting variation
of N with D/D o . Equation 共5兲 predicts N⫽20, a few more
than the 15 fingers observed experimentally at maximum
spread. If the fingering beneath this impacting droplet is a
manifestation of the Rayleigh–Taylor instability, then Fig. 5
suggests that fingering begins no later than D/D o ⬇0.2.
The second set of data is that of Marengo et al.,37 of the
impact of a water droplet onto a PVC surface, characterized
by We⫽88. They present the early variation of contact line
velocity V CL with time in their Fig. 5. The authors point out
a sudden decrease of velocity at 52 ␮ s attributed to the ‘‘instant of lamella ejection,’’ when the expanding fluid sheet
first appears from beneath the bulk of the droplet. Marengo
et al. only mention that the velocity variation scales with
t ⫺0.52; inspection of the data from the article suggests a relationship like V CL⫽0.065t ⫺0.52. Integrating and differentiating to obtain D and a, and substituting into Eq. 共17兲 yields
N⫽13 at t⫽52 ␮ s. The corresponding evaluation of Eq. 共5兲
yields a similar N⫽11.
The remainder of this paper presents results of simulations of droplet fingering and splashing, and examines the
role of impact velocity, contact angle, and surface roughness.
Photographs of droplet impact were obtained by techniques
described previously.18,19,32,35 A summary of the simulations
is presented in Table II.
A. The effect of impact velocity
The effect of impact velocity was examined by considering the impact of a 2.7 mm diameter molten tin droplet
onto a hot stainless steel surface. Simulation results and corresponding photographs are presented in Figs. 6–8, corresponding to impact at 1, 2, and 3 m/s. Note that the photographs in these figures often display both the droplet and its
reflection in the stainless steel surface.
Simulations were run of one quarter of a droplet with
appropriate boundary conditions applied at the two planes of
symmetry. This imposed a symmetry on the results, with the
number of fingers N necessarily a multiple of four. The alternative, of simulating an entire droplet, requires an order of
magnitude increase in simulation run time. The decision was
thus made to maximize the resolution. Simulations of molten
tin impact at 2 and 3 m/s were run at a resolution of 32 cpr,
the simulation of the 1 m/s impact at 20 cpr. A perturbation
amplitude Ap ⫽0.25 was applied to each of the simulations.
A constant value of the contact angle ␪ d ⫽140° was specified, as measured from photographs. Regarding the presentation of the numerical results in these and subsequent figures,
the underlying grid is spaced by the initial droplet diameter
D o to afford a sense of the extent of fluid spread.
The results of the 1 m/s impact, with N⫽8 fingers imposed on the simulation, are presented in Fig. 6. The numerical results are axisymmetric through t⫽3.0 ms, with no evidence of the perturbation. An asymmetry appears only at 6.0
ms after the fluid has spread to a maximum extent, and begun to recoil towards the point of impact. Beyond 6 ms, the
simulation results deviate somewhat from the experimental
results, but do reflect in general terms the actual behavior: A
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3128
Phys. Fluids, Vol. 12, No. 12, December 2000
FIG. 6. Photographs and simulation views of the 1 m/s impact of a 2.7 mm
diameter molten tin droplet onto a hot stainless steel surface, with Ap
⫽0.25, N⫽8. Numbers at the right indicate milliseconds following impact.
dramatic recoil of fluid leading to a bouncing of the fluid up
off of the surface, and the pinching off of a small droplet.
Results of the 2 m/s impact are illustrated in Fig. 7 and
present a very different outcome. N⫽20 fingers were imposed and are visible soon after impact. The fingers tend to
grow slowly with time as the droplet spreads and then more
dramatically as the fluid recoils. This is the simulation that
was most affected by the problem discussed earlier, the
movement of fluid azimuthally from along the axes towards
the diagonals. The reason relates to the 20 fingers that were
imposed, which results in a finger placed on either side of the
diagonal. The movement of fluid towards the diagonal initiates the collapse of the two fingers into one, already visible
at 1.9 ms and certainly at 3.1 ms.
Obviously the asymmetry of the simulation is undesirable, but coincidentally corresponds to the asymmetry of the
Bussmann, Chandra, and Mostaghimi
FIG. 7. Photographs and simulation views of the 2 m/s impact of a 2.7 mm
diameter molten tin droplet onto a hot stainless steel surface, with Ap
⫽0.25, N⫽20. Numbers at the right indicate milliseconds following impact.
actual impact, portrayed most clearly by the photograph at
5.1 ms. The merging of the fingers disrupts the subsequent
behavior of the fluid and leads to the large variation in finger
thickness by 7.1 ms. The formation of satellite droplets at 9.1
ms is not observed experimentally. Nonetheless, Fig. 7 reveals a good agreement between the numerical results and
the photographs.
Finally, Fig. 8 portrays the results of the 3 m/s impact
that results in splashing. While the first two simulations were
run with a value of N approximately equal to the number
observed experimentally, only 24 fingers were imposed on
the 3 m/sec simulation, for reasons related to available computing capacity. Equation 共5兲 predicts 34 fingers, approximately the number observed experimentally. The agreement
between simulation and photographs is nonetheless good.
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Phys. Fluids, Vol. 12, No. 12, December 2000
Modeling the splash of a droplet impacting a solid surface
3129
FIG. 9. Photographs and simulation views of the 4 m/s impact of a 2 mm
diameter water droplet onto a stainless steel surface, with Ap ⫽0.25, N
⫽24. Numbers at the right indicate milliseconds following impact.
FIG. 8. Photographs and simulation views of the 3 m/s impact of a 2.7 mm
diameter molten tin droplet onto a hot stainless steel surface, with Ap
⫽0.25, N⫽24. Numbers at the right indicate milliseconds following impact.
Fingers advance rapidly ahead of the advancing contact line.
The simulation shows splashing to begin at about maximum
spread, when overall fluid velocity is very small, yielding
satellite droplets that have little velocity after pinch off and
tend to remain where they are deposited. Photographs demonstrate a little bit of splashing at 3.2 ms, but significant
numbers of satellite droplets only at 4.5 ms. Numerically, the
24 fingers generate 24 satellite droplets. It is not clear from
the photographs whether every finger pinches off, or whether
there are instances of a finger pinching off twice, once early
and again during recoil. The simulation result is somewhat
more organized in its placement of satellite droplets in a ring
about the point of impact; the photographs show more of a
scattering of satellite droplets.
How well are these results predicted by the correlation
of Cossali et al.,12 Eq. 共2兲, for the splashing threshold K c as
a function of R a* ? Solving Eq. 共2兲 for the threshold impact
velocity of a 2.7 mm diameter tin droplet impacting a surface
with R a* ⫽3.3⫻10⫺5 yields V o ⫽2.13 m/sec. Although neither experimental nor simulation results were obtained at this
velocity, inspection of Figs. 7 and 8 suggests that this value
seems reasonable, and that the correlation applies to these
impacts.
B. The effect of the contact angle
Results are now presented of two other impacts, of water
and heptane droplets, that differ significantly from the tin
droplet impacts. Table II summarizes the parameters that
characterize these impacts.
Figure 9 presents photographs and simulation results of
the 4 m/sec impact of a 2 mm diameter water droplet onto
the same stainless steel surface used for the tin impacts.
Equation 共5兲 predicts 29 fingers at maximum spread. Photographs of the droplet shortly after impact suggest that there
are more than 40 fingers. Again, N⫽24 fingers were initially
imposed on the simulation, limited by the available computing capacity.
Where the impact of a tin or heptane droplet is charac-
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3130
Phys. Fluids, Vol. 12, No. 12, December 2000
FIG. 10. Time variation of the average value of ␪ d from a simulation of the
4 m/s impact of a 2 mm diameter water droplet onto a stainless steel surface.
terized by a relatively constant value of the dynamic contact
angle, the value of ␪ d varies dramatically for a water droplet
impacting onto a stainless steel surface. The advancing value
of ␪ d is approximately 110° and the receding value, observed during recoil, is 40°. We imposed this variation on
the simulation with a contact angle model presented
previously.4 The model relates ␪ d to the contact line velocity
V CL , assuming only a knowledge of the advancing and receding angles. The time variation of the average value of ␪ d
calculated by the simulation is presented in Fig. 10, and
demonstrates the transition from the advancing angle during
droplet spread to the much smaller receding angle.
Despite a value of the splash factor K⫽198, greater than
the splashing threshold defined by Eq. 共2兲, Fig. 9 illustrates
an impact seemingly far from splashing. Fingers do form
about the droplet perimeter during spreading, and appear to
grow through t⫽1.5 ms, while ␪ d remains large. The fingers
that are predicted numerically appear somewhat more pronounced than the fingers observed experimentally, which
may be related to the difference in the number of fingers. By
2 ms, the shape of the fingers has begun to change as ␪ d
decreases. This is especially evident from the simulation results. By 3 ms, when ␪ d ⬇40°, the distinct fingers are slowly
disappearing. By 4 ms, little evidence remains of the fingers
as the fluid slowly recoils.
Figure 11 illustrates the 4 m/sec impact of a 2 mm diameter heptane droplet onto a stainless steel surface, with
␪ d ⫽32°. 32 Characterized by a splash factor K⫽349, much
larger even than the 3 m/sec tin impact, the photographs
show absolutely no fingering, while the simulation results
show only subtle signs of the 24 fingers imposed 关Eq. 共5兲
predicts 50 fingers兴. The results are presented only to t⫽1
ms, when the simulation was stopped for lack of resolution
of the very thin sheet of fluid at the droplet perimeter. The
lack of resolution is also responsible for what appear to be
holes in the liquid sheet at 1 ms.
How does one explain the dramatic difference between
the behavior of the tin droplets, which finger and splash according to the correlation of Cossali et al.,12 and the behavior
Bussmann, Chandra, and Mostaghimi
FIG. 11. Photographs and simulation views of the 4 m/s impact of a 2 mm
diameter heptane droplet onto a stainless steel surface, with Ap ⫽0.25, N
⫽24. Numbers at the right indicate milliseconds following impact.
of the water and heptane droplets, which appear to finger less
as K increases? There are two significant differences between
the tin and water/heptane impacts. The first is the value of
the contact angle. Smaller values of ␪ d appear to inhibit finger growth, evidenced most clearly by the dynamic behavior
of the water droplet. Heptane, characterized by a constant
value of ␪ d similar to the receding contact angle of water,
shows absolutely no tendency to finger.
Would heptane finger and splash if ␪ d Ⰷ32°? Figure 12
illustrates simulation results of an impact similar to that illustrated in Fig. 11, but with ␪ d ⫽148°. The large value of
␪ d obviously affects the impact, lifting the edge of the sheet
up off the surface as the fluid spreads. The rim of fluid then
breaks up, although the number of fingers appears not to
correspond to the 24 fingers initiated. This simulation was
FIG. 12. Views of a simulation similar to that presented in Fig. 11, but with
␪ d ⫽148°.
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Phys. Fluids, Vol. 12, No. 12, December 2000
run at 32 cpr; a simulation run at a finer resolution yielded
very similar results. Regardless of resolution, however, the
rim of fluid is very thin and poorly resolved, which may
explain the breakup.
Further evidence of the role of the contact angle is provided by the results of Chandra and Avedisian.32 Their Fig.
11 illustrates the influence of surface temperature T s on the
impact of a heptane droplet onto a stainless steel surface. The
contact angle increases with T s , from ␪ d ⫽32° at room temperature to 180° when the surface reaches the Leidenfrost
temperature 共the onset of pure film boiling兲. Photographs
corresponding to T s ⫽195 °C, just below the Leidenfrost
temperature, show 15 fingers about the droplet, while none
are visible when T s ⫽160 °C ( ␪ d ⫽80°). While Eq. 共5兲 underpredicts this number of fingers when the boiling point
共98.4 °C兲 properties of heptane are used, the correlation predicts 16 fingers when the properties correspond to saturated
heptane at 195 °C.38 The results also provide evidence that
the initiation of fingering requires contact with the solid surface. Photographs of impact at T s ⫽205 °C, above the Leidenfrost temperature when a vapor film separates the liquid
from the solid surface, show no evidence of fingering.
It is unlikely that ␪ d is solely responsible for the difference in the fingering behavior between tin and water–
heptane. Although a large value of ␪ d obviously affects the
heptane impact, the resulting behavior is nonetheless very
different from that of the tin impacts. This leads then to
consideration of the other significant difference between the
tin and water/heptane impacts. The values of Re are of the
same order of magnitude, but the values of We differ significantly 共see Table II兲. Thus, the influence of surface tension
relative to viscosity for the water and heptane impacts is
much smaller than for tin. This relationship may be expressed by the aforementioned Ohnesorge number Oh
⫽ ␮ / 冑D o ␥ ␳ . While surface tension appears to magnify
the small undulations of the surface profile that appear
shortly after impact, viscosity will tend to dampen
the growth of fingers. For the fluids considered here, Ohtin
ⰆOhwater, Ohheptane . It would appear that there may be an
upper limit to the value of the Ohnesorge number beyond
which fingering and splashing is not observed, at least in the
absence of crown formation.
Modeling the splash of a droplet impacting a solid surface
3131
FIG. 13. The influence of roughness on the 4 m/s impact of a 2 mm diameter water droplet onto a stainless steel surface.
Figure 13 suggests that the magnitude of surface roughness is related to the strength of the perturbation of fluid at
the moment of impact. This may be an avenue for introducing surface roughness into the numerical model, by relating
the perturbation amplitude Ap to the roughness R a* . Such an
approach, however, would only reflect the influence of surface roughness at the moment of impact, and not, as may be
the case, on the subsequent flow.
Figure 14 illustrates the results of two simulations of the
2 m/sec impact of a 2.7 mm diameter tin droplet, similar to
that illustrated in Fig. 7, but with Ap ⫽0.15 and 0.35. As one
would expect, the magnitude of fingering increases with Ap .
However, no attempt was made to articulate a relationship
between R *
a and Ap .
C. The effect of surface roughness
Finally, consider the influence of surface roughness on
droplet fingering and splashing. Figure 13 presents photographs of the 4 m/sec impact of a 2 mm diameter water
droplet onto three different stainless steel surfaces, characterized by roughness values R a ⫽0.065, 0.16, and 0.22 ␮ m.
The effect of roughness is dramatic, with the impact onto the
smoothest surface almost axisymmetric. The two other surfaces, prepared by polishing with 320 and 80 grit emery
paper, are rougher than the surface of Fig. 9 (R a ⫽0.09 ␮ m兲,
yet the shape and amplitude of the fingers that result from
impact onto any of these three surfaces are very similar. The
only difference between impacts is a gradual decrease of the
number of fingers as roughness increases, particularly during
the early spread of fluid.
FIG. 14. The influence of Ap on fingering. Simulation views of the 2 m/s
impact of a 2.7 mm diameter molten tin droplet onto a hot stainless steel
surface, with N⫽20, and Ap ⫽0.15 and 0.35. Numbers at the right indicate
milliseconds following impact.
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3132
Phys. Fluids, Vol. 12, No. 12, December 2000
V. CONCLUSIONS
A methodology has been presented for considering the
fingering and splashing of a droplet impacting a solid surface. Numerical results have been compared with photographs of corresponding impacts and demonstrate good
agreement. Published data is also presented in support of a
previously published correlation17 for the number of fingers
about an impacting droplet, based on a simple application of
Rayleigh–Taylor instability theory. The correlation yields
good predictions for a range of behaviors.
The influence of impact velocity is clearly evidenced by
the results of molten tin droplets impacting a hot surface.
Behavior varies from no fingering at 1 m/s to fingering at 2
m/s and fingering that leads to the detachment of satellite
droplets 共splashing兲 at 3 m/sec. A correlation for the splashing threshold12 accurately predicts the onset of splashing.
Water and heptane droplets demonstrate less of a tendency to finger and splash, despite being characterized by
similar values of a splash parameter K⬅We0.5Re0.25. Results
suggest that fingering decreases as the liquid wets the solid
surface, indicated by a decreasing value of the contact angle.
Results also suggest that for a given Re there is an upper
limit to the value of We beyond which the fingering intensity
decreases.
ACKNOWLEDGMENT
The authors would like to thank Monika Muñoz for taking the water and heptane photographs.
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