Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 3.2.4 - Dynamics of two-phase flows induced by drop collisions
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 Material Information
Title: 3.2.4 - Dynamics of two-phase flows induced by drop collisions Particle Bubble and Drop Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Roisman, I.V.
Berberovi´c, E.
Jakirli´c, S.
Tropea, C.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: drop impact
crater evolution
binary collision
asymptotic solutions
 Notes
Abstract: Two-phase flows with free surfaces occurring during impact of a drop onto a liquid wall film, onto a dry substrate and double-symmetric binary drop collisions are analyzed in this study. Numerical simulations of the flow are performed using an interface capturing model based on the volume-of-fluid (VOF) method. For a sharp interface resolution an additional convective term is introduced into the transport equation for phase fraction without explicit reconstruction of the free surface. In this paper the theoretical and numerical investigations of the inertia dominated flow induced by drop collision with a rigid substrate and symmetric collision with another drop are conducted. It is shown that, if the impact Reynolds and Weber numbers are high enough, the lamella shape in the area near the axis of symmetry is self-similar and independent of the impact conditions, yielding a universal velocity and pressure distributions. The existing experimental data are compared to the numerical simulations. The agreement is rather good. Finally, the dynamics of drop impact onto a wall film was determined numerically by varying impact parameters and liquid properties.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00078
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 324-Roisman-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Dynamics of two-phase flows induced by drop collisions


I. V. Roisman*t, E. Berberovi6*, S. JakirliC and C. Tropea*tt

Institute of Fluid Mechanics and Aerodynamics, Technische Universitat Darmstadt, 64287 Darmstadt, Germany
t Center of Smart Interfaces, Technische Universitit Darmstadt, Petersenstr. 34, 64287 Darmstadt, Germany
eberberovic @sla.tu-darmstadt.de, roisman@sla.tu-darmstadt.de and ctropea@sla.tu-darmstadt.de
Keywords: Drop impact, crater evolution, binary collision, asymptotic solutions


Two-phase flows with free surfaces occurring during
impact of a drop onto a liquid wall film, onto a dry sub-
strate and double-symmetric binary drop collisions are an-
alyzed in this study. Numerical simulations of the flow are
performed using an interface capturing model based on
the volume-of-fluid (VOF) method. For a sharp interface
resolution an additional convective term is introduced into
the transport equation for phase fraction without explicit
reconstruction of the free surface.
In this paper the theoretical and numerical investiga-
tions of the inertia dominated flow induced by drop col-
lision with a rigid substrate and symmetric collision with
another drop are conducted. It is shown that, if the im-
pact Reynolds and Weber numbers are high enough, the
lamella shape in the area near the axis of symmetry is self-
similar and independent of the impact conditions, yielding
a universal velocity and pressure distributions. The exist-
ing experimental data are compared to the numerical sim-
ulations. The agreement is rather good.
Finally, the dynamics of drop impact onto a wall film
was determined numerically by varying impact parame-
ters and liquid properties.


1 Introduction

Collision of a drop or droplet with a surface can be ob-
served in everyday life and lies to the rots of numerous
technical applications. The phenomenon of a drop impact
is a problem rich in fluid dynamics, a deep insight into
which may enable to improve numerous technical appli-
cations, such as high-quality ink-jet printing, perfect cov-
erage of surfaces in paint spraying, spray cooling or a re-
duction of exhaust gases in internal combustion engines.
Splashing of drops on liquid layers is encountered fre-
quently in nature and leads to various phenomena like air
bubble formation during the rain, formation of a crown-
like liquid sheet ejecting secondary droplets, or ejection
of the Worthington jet from the region of impact. Inves-
tigations of drop impact onto liquid films have also im-


portance in ecological fields, such as soil erosion and dis-
persion of seed and microorganisms. The fluid-dynamical
phenomena involved in drop impact onto various surfaces
were the subject of numerous studies, a comprehensive
review of which can be found in Rein (1993) and the se-
quence of events occurring during the impact of single
drops onto films or pools of various depths has been de-
scribed by Yarin (2006). Extensive research has been con-
ducted to study the evolution of the maximum crater depth
and the formation of the crown or corona, the splashing,
the Worthington jet and bubble entrainment during drop
impact onto thin and deep liquid pools. Less investiga-
tions were devoted to the phenomena occurring below the
impact surface, in particular the evolution of the crater
in shallow pools (Shin and McMahon 1990; Fedorchenko
and Wang 2004; Macklin and Hobbs 1969), which is im-
portant for spray cooling.

Drop impact onto a dry substrate is a frequently inves-
tigated phenomena in fluid mechanics because of the nu-
merous industrial applications. Various outcomes of drop
impact, like splash, deposition or rebound (Levin 1971;
Rioboo et al. 2001) are determined by the impact param-
eters and numerous substrate properties, such as the sub-
strate roughness, shape, elasticity, porosity and wettabil-
ity. These phenomena can be modeled if the main mech-
anisms involved in drop spreading are identified and well
described. In contrast to an extensive knowledge about
the geometry of drop spreading on the wall (Rioboo et al.
(2002)), less data is available for the geometry of binary
drop collisions. The reason is in the relatively small, sub-
millimeter diameter of the drops which are commonly
generated in such experiments. The experimental data for
the temporal evolution of the diameter of the liquid mass
formed after drop binary collision (Willis and Orme 2000,
2003) is available only for the relatively late stages after
impact. On the other hand, the phenomena related to ax-
isymmetric binary drop collision can be much easier to
model than drop impact onto a dry substrate since com-
plicating wall effects are not present.











Rapid advances in computer hardware and develop-
ments of numerical algorithms have enabled a broader
use of computational methods for investigating the drop
impact phenomena. Numerical simulations provide a de-
tailed database comprising not only the dynamics of the
drop surface with respect to its position and form but also
the temporal behavior of the complete velocity and pres-
sure fields. The latter are beyond the reach of the existing
experimental methods where high-speed digital photogra-
phy is still the main utility for extracting information. Be-
cause of their importance, numerous computational stud-
ies of these flows have been reported, some of the relevant
being Bussmann et al. (2000); Nikolopoulos et al. (2007);
Josserand and Zaleski (2003), etc.
A distinct area of investigation in fluid mechanics deals
with processes included in spray impact. Although the
flow generated by drop collisions is pertinent to spray im-
pact, for practical reasons a direct numerical simulation
of spray impingement is still far beyond the capacity of
the present computer power. This is because spray im-
pingement involves a large range of different time and
length scales. Spray consists of a large number of drops
of very different sizes and velocities impinging on a sur-
face. The length scale of the wall film fluctuation may be
much larger than the diameter of a single drop diameter
and the time scale determining these fluctuations may be
quite different from the time scale of a single drop im-
pact. The two-phase flow of the underlying gas and the
falling droplets is rather complex and hard to model by
the existing turbulence models. Therefore a determinis-
tic approach for spray impingement including individual
treatment of all droplets is still practically impossible.
This report presents results of theoretical and numerical
analyses of free surface flows generated during impact of
a single drop onto a liquid wall film, onto a dry substrate
and by double-symmetric binary drop collisions. The ac-
companied theoretical model for the crater penetration ac-
counting for the liquid inertia, viscosity, gravity, and sur-
face tension is presented. In addition the results of the the-
oretical and numerical investigations of the inertia domi-
nated flow induced by drop collision with a rigid substrate
and symmetric collision with another drop are presented.
The derived analytical solution for the self-similar flow in
the lamella at early times after the impact is outlined. The
theoretical and numerical results are compared to the ex-
isting experimental database and a rather good agreement
is obtained.
Numerical simulations of drop impact are used to es-
timate the time evolution of the spreading drop. The nu-
merical method includes interface capturing methodology
based on the VOF-method. The competitive effects in-
fluencing the flow field during impact and the final out-
come after the impact, i.e. gravity, viscous, inertial and
surface tension forces are taken in the account. Equations


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


of motion are extended to account for the surface tension
force acting on the free surface using the Continuum Sur-
face Force model (CSF) of Brackbill et al. (1992). The
free surface position is determined from the solution of
the transport equation for volume fraction), which takes
values between zero and one, corresponding to regions
occupied by gas and liquid. Both phases are assumed to
be immiscible Newtonian fluids, thus equations of motion
are solved for a single effective fluid, the physical prop-
erties of which are calculated as weighted averages be-
tween the phases and equal to those of each phase in the
regions it occupies. Although the phase fraction is macro-
scopically a discontinuous function (due to the very small
interface thickness), it is changing siii, nil\ over the in-
terface in this model, producing a smooth change of fluid
properties and surface tension force, in accordance with
the CSF model. The smearing of steep gradients is sup-
pressed by introducing an additional convective term in
the phase fraction equation counteracting numerical dif-
fusion. The numerical procedure for keeping a sharply re-
solved interface formulated by OpenCFD Ltd Ltd (2007)
is used, and the description of the algorithm is given in
BerberoviC et al. (2009), where the term "conventional
VOF" has been used merely to indicate the destroying ef-
fects of the diffusion on the solution for the volume frac-
tion. Basically, the algorithm avoids the use of special
compressive schemes for convection, such as CICSAM
of Ubbink and Issa (1999) and requires no explicit geo-
metric interface reconstruction technique. The computa-
tions were performed using the code OpenFOAM (Weller
et al. 1998), an open source CFD toolbox, utilizing a
cell-center-based finite volume method on a fixed unstruc-
tured numerical grid and employing the solution proce-
dure based on the PISO algorithm for coupling between
pressure and velocity in transient flows (Issa 1986). The
computational domains have the form of a 2D axisym-
metric slice with only one cell in the azimuthal direction.
The computational model incorporates a cell-center-based
finite volume method with the finite volume approxima-
tion of terms in the transport equations and an adjustable
time step. In simulations of impact onto a wall the no slip
boundary condition is set at the impacting plane. Since no
data on contact angle is provided, capillarity at the wall
is not accounted for. In cases of binary collisions, the
symmetry plane boundary condition is employed, thereby
simulating an axisymmetric drop binary collision.


2 Drop collision with a rigid wall and with
another drop

Universal shape of the lamella The phenomena
of normal drop impact onto a dry substrate and double-
symmetric drop binary collision are very similar. Colli-
sion of two equal drops can be represented as a drop im-











Drop impact onto a Axisymmetric impact of
dry substrate two equal drops
lamella /dro rim lamella drop ri
I substrate I
symmetry plane\

Figure 1: Sketch of the normal drop impact onto a dry
substrate and the axisymmetric binary collision
of two equal drops.


pact onto a symmetry plane (Fig. 1). It is obvious that
the modeling approaches to these two kindred problems
should also be similar.
The dimensionless parameters governing the dynamics
of spreading of drop on a dry wall are the impact We-
ber number, We pDoUo/cr, Reynolds number, Re
DoUo/v, and the dynamic contact angle 0.
In Yarin and Weiss (1995) the remote asymptotic solu-
tion for the flow in the thin axisymmetric fast spreading
sheet is obtained in the form:


Vr V
t+ T


2z
t+ T


where v, is the radial velocity, vz is the axial velocity
component, T is a constant. Expression (1) is relevant
only in the central part of the sheet far from the sheet edge
where the capillary effects lead to the formation of a rim.
Equation (1) can be easily written in the Lagrangian
form
r = ro + vrot, vo = ro/T, (2)
where ro and vo are the radial coordinate and the radial
velocity of a material point at the time instant t = The
mass balance in the lagrangian form yields:


h roho(ro) Or 1(3)
(3)
r 9ro-)

where ho (ro) is the initial distribution of the sheet thick-
ness.
Equation (3) allows to write a general expression for
the sheet thickness in the Eulerian form

h (t 2 ho0 (4)

which yields the evolution of the film thickness at the axis,
hc, obtained in Yarin and Weiss (1995)

hce= (5)
(t + (5))2

If the impact Weber and Reynolds numbers are high,
the flow far from the wall (where the velocity gradients
are not negligibly small) is determined mainly by iner-
tia. Therefore, the dimensionless velocity of the lamella


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


and its dimensionless shape in the central region of the
deforming drop should not depend on the impact param-
eters. In Fig. 2 left the experimental data (Bakshi et al.
2007) and the results of numerical predictions (Fukai et al.
1995; S. Sikalo et al. 2005; Mukherjee 2007) for the evo-
lution of the drop height he at the symmetry center are
shown as a function of time for various impact parame-
ters. All the values are scaled using the initial drop diam-
eter Do as the length scale and impact velocity Uo as the
velocity scale.
The corresponding results of our numerical simulations
of drop impact onto a symmetry plane are shown in Fig. 2
right. During the first two regimes all the results lie ap-
proximately on a single curve for all the impact param-
eters. In the first and second non-viscous regimes, the
height of the drop at the symmetry axis is approximated
by the curves also shown in Fig. 2. It is obvious that the
development of the near-wall boundary layer is not rele-
vant in the case for drop impact onto a symmetry plane.
In Fig. 3 the results of the numerical simulations of
the average radial velocity at t 1 in the lamella gen-
erated by drop impact onto a symmetry plane are shown
as a function of the radius for various impact parameters.
Surprisingly, the velocity is linear over a relatively wide
range of the radius. Moreover, the velocity distribution
almost doesn't depend on the impact parameters except at
the edge region, where the lamella is deformed due to the
viscous and capillary forces.
As shown in Fig. 4, the shape of the central part of the
lamella also almost doesn't depend on the impact parame-
ters except of the edge region associated with the rim for-
mation. Therefore, it can be assumed that the thickness of
the lamella and its velocity distribution at very high Weber
and Reynolds numbers are self-similar.
The universal dimensionless thickness of the lamella
(4) is determined by approximating of the numerically
predicted shape by a Gaussian function at dimensionless
time instant t 1 and estimating the value of 7 0.25 in
(1) from the numerically predicted velocity gradient:


S0.39 2.34T2
L (t+ 0.25)2 exp (t +0.25)2


Here all the overbared symbols are dimensionless.
In Fig. 5 the approximate shape (6) is compared with
the numerical simulations (Fukai et al. 1995) of drop im-
pact with Re 1565 and We 32. At t 1 and
t 2 the agreement is rather good far from the lamella
edge where the lamella is thicker than predicted (indicat-
ing that the velocity gradient here is smaller then near the
axis r = 0) and where the rim formation becomes visible.
At t = 0.5 the approximated solution overpredicts the
results of numerical simulations. At this relatively early
stage the velocity field is still two dimensional and cannot








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


1.2
Numerical, drop impact
1.0 -t onto a substrate:
/ Re=210, We=30
o Re=450, We=137
0 8 Re = 4010,We= 90
v Re =6020, We =117
0.60..39 O Re = 6260, We = 128
(t + 0.25)2 Experimental, drop impact
0.4 onto a rigid sphere
4 / Re=1068,We 144
0.2-

0 1 2 3 4 5 6





o We =397, Re =61
1.0 1-t We=761, Re=83
/ We =1165, Re =104


0.6-

0.4- T 0.39
(t + 0.25)2
0.2-

0.0 2 4 8
0 14 5 6 7 8
t


Figure 2: Drop impact onto a dry substrate (left); the ex-
perimental data (Bakshi et al. 2007) and the re-
sults of existing numerical predictions (Fukai
et al. 1995; S. Sikalo et al. 2005; Mukherjee
2007) for the evolution of the lamella thickness
at the impact axis as a function of the dimen-
sionless time. Drop impact onto a symmetry
plane (right); the results of our numerical pre-
dictions for the evolution of the lamella thick-
ness at the impact axis as a function of the di-
mensionless time.



1.4
1.2
1.0
I 0.8 -
0.6
O- -We =397, Re=61
S ---We=761, Re=83
0.2- We =1165, Re =104
0.0 .
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
r



Figure 3: Drop impact onto a symmetry plane. The re-
sults of numerical predictions of the dimension-
less average radial velocity at the time instant
t 1 as a function of the dimensionless radius.


0.25 Impact onto a symmetry plane:
We =397, Re =61
0.20- ---We=761, Re=83
We=1165,Re=104
Impact onto a solid wall
0.15- Fukai et a. (1995):
We =117,Re =6020
0.10- We=128, Re=6260
0.10

0.05-

0.00 . .
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0


Figure 4: Drop impact onto a symmetry plane. The re-
sults of numerical predictions for the dimen-
sionless lamella thickness at the time instant
t 1 as a function of the dimensionless ra-
dius compared with the numerical simulations
of drop impact onto a flat rigid substrate from
Fukai et al. (1995).


be described by the remote asymptotic solution. The re-
sults of the numerical predictions of the pressure pc at the
impact point are shown in Fig. 6. The pressure pc decays
nearly exponentially in time. The best fit of the data is
Pc 1.7exp(-3.11).
The universal, self-similar flow in the lamella generated
by drop collision corresponds to high Reynolds and We-
ber numbers. It is shown Roisman et al. (2009) that the
validity range of the universal solution is determined by
Re > 14 and We > 2.5.
In Fig. 7 the numerical predictions for the dimension-
less drop height hcl at the axis at the dimensionless time
instant t = 1 are shown for various Reynolds numbers. At
Reynolds numbers Re > 60 the characteristic drop height
at t = 1 reaches the constant magnitude hc0 = 0.245. In
this range of Reynolds numbers the universal solution is
applicable to the description of the flow in the lamella.


Residual film thickness Expression (6) is valid only
if the thickness of the lamella is much larger than the
thickness of the viscous boundary layer generated at the
wall. An analytical self-similar solution of the full Navier-
Stokes equations for the flow in the lamella is obtained in
Roisman (2009).
The velocity field is expressed in the form


o[ 5ti


-2g ] -


where g is a dimensionless stream-function which is cal-
culated by the numerical integration of the ordinary dif-
ferential equation

g'" + 2gg" + -g" + g- g'2 = 0, (8)
2







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


0.7
Numerical, Fukai et al (1995)
0.6- 6 t=0.5
o t=l
05-" '.. A t=2
| Approximated
0.4- i ..... t=0.5
b _t=1

02- 1
0.3 ,2


0.0 0.2 0.4 0 B 0.8 1.0 1.2 14 16 1.8 2.0 22


Figure 5: Drop impact onto a dry substrate with Re =
1565 and We 32. The numerical predictions
of Fukai et al. (1995) of the shape of the lamella
at various time instants are compared with the
approximate shape (6).


subject the boundary conditions


g = g' 0,
/' 1,


at 0, (9)
at -* oo, (10)


with = z/ vt being the similarity variable.
The thickness of the boundary layer is estimated in Ro-
isman (2009) from the solution of (8) as

6 1.88 v, (11)

which can be rewritten in the dimensionless form

S1.88 l L/Re. (12)

The thickness of the lamella, hb at the time instant t
tb at which the viscous boundary layer reaches the upper
free surface of the lamella can be estimated using (6) and
(12)
hb e2/5. (13)

At longer times, t > tb the flow in the lamella is gov-
erned by the balance of the inertial and viscous forces
leading to its quick deceleration. The expression for the
residual film thickness is obtained in Roisman (2009) in
the following form:

hres 0.79Re-2/5. (14)


3 Drop collision with a shallow pool

The dynamics of drop impact on liquid surface, the shape
of the formed crater, the formation and propagation of a
capillary wave in the crater and the residual film thickness
on the rigid wall are determined and analyzed. The shape


o Numerical imulations
Fitting curve



1.53 exp(- 2.90


0.0 0.2 .4 .
0.0 0.2 0.4 0.6 0.8 1.0 1.2


Figure 6: Drop impact onto a symmetry plane. Numerical
predictions of the dimensionless pressure pc at
the impact point as a function of the dimension-
less time. The impact parameters are We 761,
Re 83, and We 1165, Re 104.


of the crater within the film and the uprising liquid sheet
formed upon the impact are observed using a high-speed
video system. The numerical simulations of the phenom-
ena are performed using a free-surface capturing model
based on the volume-of-fluid (VOF) model in the frame-
work of a 2D axisymmetric numerical method. The fluid
of the drop and the film is the same in all experiments and
three liquids with different properties were used: distilled
water, isopropanol and a glycerin/water mixture consist-
ing of 70% glycerin and 30% water.
The governing impact parameters are the Weber num-
ber We pUo2Do/a and Reynolds number Re
UoDo/v. The film thickness and time are normalized with
H Ho/Do and t tUo/Do, respectively. In the above
expressions p, a and p are dynamic viscosity, surface ten-
sion coefficient and density of the liquids used, Do and Uo
are the diameter and the impact velocity of the drop, and
Ho is the initial film thickness. The initial drop diame-
ter is 2.9 mm for distilled water, 2.14 mm for isopropanol
and 2.67 mm for glycerin/water mixture. The drop impact
velocity (velocity just before the drop reaches the film sur-
face) is calculated from a distance measurement and a pre-
set time delay between subsequent exposures. Thus, for
distilled water the impact velocity varied from 1.68 m/s to
2.91 m/s, for isopropanol from 1.7 m/s to 2.83 m/s and for
glycerin/water mixture from 1.81 m/s to 3.25 m/s. The
liquid film thickness was varied for all liquids, yielding
the non-dimensional film thickness of 0.5 to 2.
Fig. 8 shows the crater shape at different time instants
for the flow configuration corresponding to H 2 for
isopropanol. The experimental data are from BerberoviC
et al. (2009). A small circumferential free liquid jet and
the formation of the penetrating and radially expanding
crater inside the liquid film observed in the experiments








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


0.7
0.6- O We 2.75
0.5-
0.4-
We= 17
0.3- We 761
-------------- O- ------0
0.2- We=68 We=397 We=1165
0.1-
0.0
0 20 40 60 80 100 121
Re


Figure 7: Numerical simulations of drop impact onto a
symmetry plane. The dimensionless drop height
hc1 at the time instant t 1 as a function of the
impact Reynolds number. The dashed line is a
fitting curve of the numerical results.



are shown. After reaching the maximum diameter, the
crater begins a receding motion driven by capillary forces.
It can be seen that the crater shape changes from a nearly
spherical form in the advancing motion to a conical one
during the receding phase.
For the purpose of a quantitative analysis, the predicted
dimensionless crater depth and diameter are compared to
measured ones. The depth is measured at the lowest point
of the crater, and the diameter is determined at a half film
thickness (y/Ho = 0.5), where y is the vertical distance
measured from the free surface of the film.
Plots of dimensionless crater diameter and depth
against dimensionless time are shown in Fig. 9 for iso-
propanol, being similar in other cases. The agreement
between the numerical simulations and the experimental
data is rather good.
When the crater approaches the bottom the penetration
velocity decreases due to the wall effects. When the in-
ertia of the liquid flow is strong enough, the thickness of
the film below the cavity follows the remote asymptotic
solution Yarin and Weiss (1995) and decreases as the in-
verse of the time squared. At some time instant the film
thickness becomes comparable with the thickness of the
viscous boundary layer and the flow in the film is damped
by viscosity. The value of the residual film thickness is
rather important for the modeling of heat transfer associ-
ated with drop or spray impact and prediction of the film
breakup. Since its experimental evaluation is not an easy
task, this value is determined from the numerical simula-
tions.
It is obvious that the phenomena leading to the appear-
ance of the residual film in the cases of drop impact onto
a dry wall and onto a wetted wall are similar. Therefore
expression in the form (14) can be used also for the de-


Figure 8: Time evolution of the crater shape for the im-
pact of an isopropanol drop, H 2, We = 392,
Re 1730: (a) experiment and (b) simulations.
The time sequences from top to bottom corre-
spond tot 0, 1.08, 2.71, 10.84, 21.68, 31.44.








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Experiments symbols
o H=1,We=384,Re=1700
SH=2,We=189,Re=1200
2,Re=1730
I,1 lines


/ -


10 20 30 40


4 -

IQ 3
2 -

1 -

0
0

30

25

20

1 15

10

05


008


|s 005 -
0- 79Re-2'5

003
S021 Re2'5
000
0 2000 4000 6000 8000
Re

Figure 10: Computationally obtained residual film thick-
ness under the cavity as a function of the im-
pact Reynolds number compared to theoretical
estimations (15).



4 Conclusions


10 '20 30 40 50 In this paper the results of studies of flow occurring dur-
10 20 30 40 50
t ing impact of a drop onto a liquid wall film, onto a dry
substrate and double-symmetric binary drop collisions are


Figure 9: Experimentally and computationally obtained
dimensionless crater diameter (left) and crater
depth (right) for the impact of an isopropanol
drop



scription of the residual film thickness after impact onto a
liquid film:
h, e ARe 2/5 (15)

Fig. 10 displays the comparison of the numerically pre-
dicted residual film thickness with the present experimen-
tal data in terms of the impact Reynolds number. The
scaling relation (15) describes well the tendency of the
obtained results.
A spectacular phenomenon related to drop impact onto
a liquid film is the generation of a sharp capillary wave on
the crater sidewall. This wave propagates along the cavity
surface downwards, merging at the bottom of the crater
and finally leads to the creation of a central jet Zhang and
Thoroddsen (2008). Similar waves have been observed
in many experiments on drop impact onto a liquid layer
or liquid pool (Leng 2001; Morton et al. 2000). Such a
capillary wave is observed at the surface of the crater in
Fig. 8. It is created inside the uprising sheet and trav-
els downwards along the crater surface. This behavior is
well resolved in the simulations. The capillary wave is
observed here only when isopropanol is used, whereas it
could not clearly be seen in impacts of distilled water and
glycerin/water mixture. This is explained by the much
lower surface tension of isopropanol.


presented.
The dynamics of drop impact on liquid surfaces was
focused on the evolution of the crater formed beneath the
surface upon the impact. A theoretical model is presented
for the penetration of the crater at the initial stage, ap-
proximating the crater shape by a spherical cavity and the
velocity field in its vicinity by the potential flow. The
asymptotic solution was obtained by neglecting capillary,
viscous and gravity effects, showing a good agreement
with the experimental data. Details of flow including
crater penetration and the capillary wave were resolved
well in the numerical simulations. It is confirmed that in-
creasing the impact velocity at a constant film thickness
has little to no effect on the crater evolution in depth and
on the time to reach maximum depth. The numerical sim-
ulations demonstrate not only high level of the predictive
capabilities of the advanced model resolving the free sur-
face, they also help to better understand the mechanisms
of crater evolution. Moreover, the results of numerical
predictions eventually help to understand the flow in the
liquid during the impact, since no detailed experimental
data for the distributions of pressure and velocity is avail-
able. Scaling relations for the residual film thickness have
been proposed based on the description of the film evolu-
tion and development of the viscous boundary layer. This
data can be valuable in the modeling of spray cooling.
For drop collisions with the rigid wall and the binary
drop collisions the study revealed that if the impact Weber
and Reynolds numbers are high enough, the flow in the
lamella far from its edge is self-similar. Its thickness, ve-
locity distribution and pressure distribution are universal


Expenments symbols
o H=1,We=384,Re=1700
SH=2,We=189,Re=1200
H=2,We=392,Re=1730
Predictions lines




- - -k


Y











and almost don't depend on the impact parameters. The
spreading diameter is determined exclusively by the edge
effects, formation of the capillary rim and viscous com-
pression of the lamella in a relatively thin region. The
evolution of the lamella thickness is determined by the
inviscid flow and viscosity. Theoretical modeling of the
spreading of the drop requires new alternative approaches
since the existing models, based on the energy balance or
based on the assumption that the drop spreading diameter
correlates with the lamella thickness, do not correctly de-
scribe the mechanisms of drop collision. The results for
the self-similar distribution of the thickness of the lamella,
velocity distribution and pressure distribution obtained in
this study can be useful for future modeling of the phe-
nomena related to the inertia dominated drop collisions,
including drop impact onto a rigid, porous or elastic sur-
faces, impact of solidifying drops and drop binary colli-
sions.

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