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Doctoral Thesis
High-velocity impact of a liquid droplet on a rigid surfacethe effects of liquid compressibility
Author(s): Haller Knezevic, Kristian
Publication Date: 2002
Permanent Link: https://doi.org/10.3929/ethz-a-004494719
Rights / License: In Copyright - Non-Commercial Use Permitted
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ETH Library
DISS. ETH NO. 14826
High-Velocity Impact of a Liquid Droplet on a Rigid
Surface: The Effect of Liquid Compressibility
Dissertation
submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the degree of
Doctor of Technical Sciences
presented by
Dipl. Phys. ETH (M.Sc. Physics)
born on August 28th, 1972
accepted on the recommendation of
Prof. Dr. Dimos Poulikakos, examiner
Prof. Dr. Peter Monkewitz, co-examiner
Zurich, October 2002
Kristian Haller Knezevic
To my Grandfather,
Velimir
i
Acknowledgments
Acknowledgments
It’s a sign of mediocrity when you
demonstrate gratitude with moderation.
-- Roberto Benigni
First of all, I would like to express my sincere thanks to my advisor, Profes-
sor Dimos Poulikakos for his guidance, support and constructive criticism during
the course of the project. My thanks goes also to my co-advisor Professor Peter
Monkewitz from EPFL Lausanne for his support.
I am greatly indebted to my co-supervisor, Dr. Yiannis Ventikos, for his
advice and support. Yiannis has been an excellent supervisor, providing insightful
comments (often going even beyond the project scope) and encouragement
throughout this PhD project.
I would like to thank Ms. Marianne Ulrich for her helpful assistance in the
administration work and Mr. Martin Meuli for his services in resolving hardware
problems. My thanks goes also to Dr. Rustem Simitovic for his support and hint
almost four years ago, which convinced me to accept the challenging project at
this lab.
I thank Prof. J. Glimm, Prof. Xiao Lin Li and F. Tangerman of the State Uni-
versity of New York at Stony Brook and Dr. J. Grove from Los Alamos National
Laboratory for putting at our disposal the front tracking code FronTier and for the
extensive help provided, including hosting me at SUNY Stony Brook for almost
two months. The assistance of Mr. Tonko Racic and Mr. Tilo Steiger (Rechenzen-
trum ETH Zurich) in resolving porting issues is gratefully acknowledged.
ii
Acknowledgments
I would like to thank the doctoral candidates of LTNT for keeping fun and
co-operative atmosphere in the laboratory.*
This research project conducted at Laboratory of Thermodynamics in
Emerging Technologies at the ETH in Zürich was financially supported by a Fel-
lowship from the Leonhard Euler Centre (Swiss branch of ERCOFTAC) and by
Sulzer Metco and Sulzer Innotech. I am grateful to our industrial collaborators Mr.
Gérard Barbezat, Dr. Egon Lang and Mr. Christian Warnecke from Sulzer Metco
for making it possible to carry out the experiments at the Sulzer Metco coating
facility.
Great thanks goes to uncle Tomas & aunt Peggy for making my undergrad-
uate studies at the ETH possible. Finally, I would like to thank my family and
friends for their support and encouragement.
Kristian
Zurich, Switzerland
October 2002
*This includes both those present and those who have already graduated before me: Salvatore Arcidiacono, Sevket
Baykal, Nicole Bieri, Lars Blum, Kevin Boomsma, Vincent Butty, Andreas Chaniotis, Iordanis Chatziprodromou
(since not pronucable, just Danny), Sandro De Gruttola, Christian del Taglia, Lale Demiraydin, Mathias Dietzel, Jürg
Gass, Stephan Haferl, Yi Pan, Andrea Prospero, Stephan Senn, Daniel Attinger, Pankaj Bajaj, Christian Bruch, Steve
Glod, Philipp Morf, Andreas Obieglo, Evangelos Boutsianis & Vartan Kurtcuoglu.
iii
Abstract
Abstract
In this PhD thesis, the compressible fluid dynamics of high-speed impact
of a spherical liquid droplet on a rigid substrate is investigated. The impact phe-
nomenon is characterised by the compression of the liquid adjacent to the target
surface, whereas the rest of the liquid droplet remains unaware of the impact. Ini-
tially, the area of compressed liquid is assumed to be bounded by a shock enve-
lope, which propagates both laterally and upwards into the bulk of the motionless
liquid. Utilizing a high-resolution axisymmetric solver for the Euler equations, it
is shown that the compressibility of the liquid medium plays a dominant role in
the evolution of the phenomenon. Compression of the liquid in a zone defined by
a shock wave envelope, lateral jetting of very high velocity and expansion waves
in the bulk of the medium are the most important mechanisms identified, simu-
lated and discussed.
During the first phase of impact, all wave propagation velocities are
smaller than the contact line velocity, thus the shock wave remains attached to the
latter. At a certain point, the radial velocity of the contact line decreases below the
shock velocity and the shock wave overtakes the contact line, starting to travel
along the droplet free surface. The resulting high pressure difference across the
free surface at the contact line region triggers an eruption of intense lateral jetting.
The shock wave propagates along the free surface of the droplet and it is reflected
into the bulk of the liquid as an expansion wave. The development of pressure and
density in the compressed area are numerically calculated using a front tracking
method. The exact position of the shock envelope is computed and both onset and
magnitude of jetting are determined, showing the emergence of liquid jets of very
high velocity (up to 6000 m/s). Computationally obtained jetting times are vali-
dated against analytical predictions. Comparisons of computationally obtained
jetting inception times with analytic results show that agreement improves signif-
iv
Abstract
icantly if the radial motion of the liquid in the compressed area is taken into
account.
An analytical model of the impact process is also developed and com-
pared to the axisymmetric numerical solution of the inviscid flow equations.
Unlike the traditional linear model - which considers all wave propagation veloc-
ities to be constant and equal to the speed of sound, the developed model predicts
the exact flow state in the compressed region by accommodating the real equation
of state. It is shown that the often employed assumption that the compressed area
is separated from the liquid bulk by a single shock wave attached to the contact
line, breaks down and results in an anomaly. This anomaly emerges substantially
prior to the time when the shock wave departs from the contact line, initiating lat-
eral liquid jetting. Due to the lack of more sophisticated mathematical models, this
tended to be neglected in most works on high speed droplet impact, even though
it is essential for the proper understanding of the pertinent physics. It is proven that
the presence of a multiple-wave structure (instead of a single shock wave) at the
contact line region resolves the aforementioned anomaly. The occurrence of this
more complex multiple wave structure is also supported by the numerical results.
Based on the developed analytical model, a parametric representation of
the shock envelope surface is established, showing a substantial improvement
with respect to previous linear model, when validated against numerical findings.
In the final part of the thesis, the assumption of a multiple wave structure
which removes the above mentioned anomaly is underpinned with an analytical
proof showing that such a structure is indeed a physically acceptable solution.
v
Zusammenfassung
Zusammenfassung
Die Zielsetzung dieser Doktorarbeit war die Erforschung der fluiddynami-
schen Phänomene, die beim sehr intensiven Tropfenaufschlag auf feste Oberflä-
chen auftreten. Dieser, sogenannter ‘High-Velocity’ Aufprall ist durch eine sehr
hohe Kompression der an der Oberfläche angrenzenden Flüssigkeit charakteri-
siert.
In der ersten Aufschlagphase wird angenommen, daß die Bereiche der kom-
primierten und ruhenden Flüssigkeit durch eine Schockwelle getrennt sind, die
sich sowohl seitlich als auch aufwärts in den ruhenden Tropfenhauptteil fortbe-
wegt. Unter der Verwendung von hochauflösenden axial-symmetrischen Euler-
Solver zeigen wir, daß die Liquidkompressibilität eine dominante Rolle in der
Zeitevolution des Phänomens spielt. Die Flüssigkeitskompression in der von der
Schockwelle und der Wand umspannten Zone, seitliche Jettingeruption sowie
Propagation & Wechselwirkung von Schock- und Expansionswellen sind die
wichtigsten Mechanismen, die in dieser Arbeit identifiziert, simuliert und bespro-
chenen werden.
Da alle Wellengeschwindigkeiten in der ersten Aufprallphase kleiner als die
Kontaktliniengeschwindigkeit sind, bleibt die Schockwelle in dieser Phase ange-
festigt an der Kontaktlinie. Zu einem bestimmten späteren Zeitpunkt fällt die
Radialgeschwindigkeit der Kontaktlinie unter die Schockgeschwindigkeit, die
Stoßwelle ‘überholt’ die Kontaktlinie und beginnt ihre Fortbewegung entlang der
Tröpfchenoberfläche. Der resultierende hohe Druckunterschied an der freien
Oberfläche (im Kontaktlinienbereich) löst eine gewaltige seitliche Jeteruption
aus. Die Stoßwelle pflanzt sich entlang der freien Tröpfchenoberfläche weiter fort
und wird dabei als die Expansionswelle reflektiert. Die Druck- und Dichteent-
wicklung im komprimierten Gebiet werden numerisch mittels einer ‘Front
Tracking’ Methode errechnet. Die genaue Position des Schock-Envelopes sowie
vi
Zusammenfassung
das zeitliche Auftreten vom Jetting werden untersucht und ermittelt. Der Moment
der Jettingeruption, sowie seine Intensität (Geschwindigkeiten bis zu 6000 m/s)
werden ebenfalls identifiziert. Rechnerisch erhaltene Jetting-Zeiten werden
anschließend gegen die analytische Vorhersagen validiert. Die Vergleiche zeigen,
daß sich die Modellübereinstimmungen erheblich verbessern, wenn die Radialbe-
wegung der Flüssigkeit im komprimierten Bereich in Betracht gezogen wird.
Ein analytisches Aufschlagmodell wurde ebenfalls entwickelt und anschlie-
ßend mit der numerischen Lösungen der nicht-viskosen Flußgleichungen vergli-
chen. Im Gegensatz zum traditionellen Linearmodell - das alle Wellen-
ausbreitungsgeschwindigkeiten der konstanten Schallgeschwindigkeit gleich-
setzt, sagt das entwickelte Modell den genauen Flußzustand in der komprimierten
Region voraus. Dies wurde dadurch ermöglicht, daß die reale Zustandgleichung
des Liquides in das Modell miteinbezogen wurde. Wie wir zeigen, führt die häufig
verwendete Annahme, daß der komprimierte- vom ruhenden Tropfenbereich
durch eine einzelne Stoßwelle getrennt ist, zwingend zu einer tiefen physikali-
schen Inkonsistenz.
Diese Anomalie taucht auf wesentlich bevor die Stoßwelle von der Kontakt-
linie abreissen und somit die Jeteruption hervorrufen kann. Mangels besseren
mathematischen Modellen, wurde diese Anomalie in den meisten Arbeiten über
Tropfenaufschlag vernachlässigt, auch wenn Ihre Lösung für das genaue Phäno-
menverständniss unerläßlich ist. Es wird bewiesen, daß das Vorhandensein einer
multiplen Wellenstruktur (im Gegensatz zu einer einzelnen Schockwelle) an den
Kontaktlinie die vorher erwähnte Anomalie behebt. Das Auftreten dieser kompli-
zierteren mehrfachen Wellenstruktur wird auch durch die numerischen Resultate
bestätigt. Basierend auf dem entwickelten analytischen Modell wird schließlich
eine parametrische Darstellung der Schockwellenenvelops hergeleitet. Der Ver-
gleich mit den numerischen Befunden zeigt eine erhebliche Verbesserung in
Bezug auf früheres lineares Modell.
vii
Table of Contents
Table of Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
2.1 Plasma Spraying Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
2.2 Sample of a Splat Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
3 Equation of State Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.1 Stiffened Gas Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.2 Linear Hugoniot Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
3.3 Temperature Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
4 Mathematical Model & Computational Methodology . . . . . . . . . . . . . . .31
4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
4.2 Computational Domain & Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . .Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
4.3 Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
4.3.1 MUSCL method of van Leer . . . . . . . . . . . . . . . . . . . . . . . . . . .37
4.3.2 Front Tracking Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
viii
Table of Contents
5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
5.1 Solution Convergence & Grid Independence . . . . . . . . . . . . . . . . . . . .47
5.2 Droplet Evolution & Interaction of Waves. . . . . . . . . . . . . . . . . . . . . .49
5.3 Jetting Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
5.4 The Effect of Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62
5.5 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
6 Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion& Multiple Wave Structure in the Contact Line Region . . . . . . . . . . . . .67
6.1 Geometrical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68
6.2 Shock Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
6.2.1 Radial Particle Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
6.2.2 Emergence of the Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
6.3 Resolution of the Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76
6.3.1 Numerical Confirmation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
6.4 Construction of the Shock Envelope. . . . . . . . . . . . . . . . . . . . . . . . . . .83
6.4.1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84
6.4.2 Results & Model Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . .86
6.5 Analytical Solution of the WaveStructure in the Contact Line Region . . . . . . . . . . . . . . . . . . . . . . . . . .88
6.5.1 One-dimensional Euler Equations . . . . . . . . . . . . . . . . . . . . . .89
6.5.2 The Exact Solution of the Riemann Problem. . . . . . . . . . . . . . .92
6.5.3 Expansion Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
ix
Table of Contents
6.5.4 Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95
6.5.5 Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
8 Appendix: Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
9 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113
10 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127
x
Table of Contents
1
Introduction
1Introduction
“The time has come,” the Walrus said,
“to talk of many things.”
-- Carroll Lewis (1832 - 1898)
The fluid mechanic and thermodynamic of liquid droplet impact on surfaces
are of great importance to a variety of different fields. To the most important count
various technological applications such as thermal spray coating, spray cooling,
cleaning of surfaces, processing of materials and ink-jet printing. Liquid impact
erosion is a major technological problem, being found in a such diverse areas as a
flight of vehicles through rain, steam turbine blade erosion, cavitation erosion and
the deliberate erosion of materials by high-sped liquid jets in cleaning and cutting
operations. In severe reactor accidents the accumulation of molten core debris on
the containment walls may be reduced by vigorous splashing. The entrainment of
bubbles by drops falling into superheated liquid can enhance nucleate boiling. In
filtration aerosol, droplets are absorbed only when adhesion is obtained on con-
tact. The droplet impact comes into play also in some non-engineering fields, an
example is the prevention of soil erosion in agriculture due to the impact of rain
drops. The phenomena related to the rain formation and its interaction with the
oceans surface are of significance in atmospheric and oceanographic sciences.
High pressures occurring during meteor impact can cause a fluidisation of the
stony matter. The resulting flows can lead to the formation of the central peaks in
craters such these on the moon. It is, therefore, not surprising that investigations
of droplet impact focus on very different facets of this phenomena.
2
Introduction
Different Parameters of Droplet Impact
We start our study with a visual overview of different factors and scenarios
that can be distinguished during the droplet impact on a substrate, Figs. 1.1 (a)-(i).
i). Phenomena associated with liquid drops prior to the impact
ii). Character of impacting surfaces (droplet and wall)
For the case where the impacted surface is liquid, we distinguish
i). The depth of impacted liquid layer
spherical deformed oscillatinginternal
circulation surfactants
Fig. 1.1 (a)
liquiddrop
liquiddrop
solid liquid
Fig. 1.1. (b)
shallow liquid layerdeep liquid layer
Fig. 1.1 (c)
3
Introduction
ii). Smoothness of liquid surface
iii). Liquid and drop compounds
For the case of a drop collision with a solid surface, different behavioural patterns
are observed, depending upon the traits of the solid surface [Figs. 1.1 (f)-(i)]:
i). Surface curvature
ii). Surface smoothness
wavy liquid surfaceflat liquid surface
Fig. 1.1 (d)
same liquid materials different liquid materials
Fig. 1.1 (e)
flexural solid surfaceplane solid surface
Fig. 1.1 (f)
rough solid surfacesmooth solid surface
Fig. 1.1 (g)
4
Introduction
i). Solid hardness
ii). Classification according to the impact angle
An extensive discussion of the aforementioned impact scenarios can be found in
Rein [1].
The liquid is described by its thermodynamic state, surface tension, viscosity
and compressibility (through the equation of state). Depending on impact veloc-
ity, drop and target geometries as well as the physical properties of both, there
might exist a regime in which effects such as viscosity and surface tension do not
play a role. A model of droplet impact and a study of emerging viscous forces was
presented by Korobkin [2], [3]. Based on the same parameters, we need to decide
if the impact can be reasonably treated by an incompressible approach or the com-
pressibility effects need be included.
yielding solid surfacerigid solid surface
ForceForce
Fig. 1.1 (h)
normal impact oblique impact
Fig. 1.1 (i)
5
Introduction
Incompressible Modelling
Numerous studies have been published on low velocity impact (e.g 1 m/s),
where the compression effects have been assumed negligible. Cumberbatch [4]
considered a two-dimensional liquid wedge impacting on a rigid plane. The moti-
vation for this study was the slapping of free-surface against the dock. Another
somewhat ad-hoc analysis was performed by Savic & Boult [5], who considered
a potential flow solution in torodial coordinates of a share of liquid. The radius of
the share was adjusted to compensate for losses due to jetting, and the resulting
approximation solution gave reasonable qualitative agreement with experimental
work done by the same authors. A numerical study of the fluid dynamics and heat
transfer phenomena was presented by Zhao, Poulikakos & Fukai [6].
Some important traits of pico-litre size droplet dispensing have been dis-
cussed by Waldvogel et al. [7]. New experimental advances in short-duration flash
photography, used during the droplet impact, have been reported by Chandra &
Avedisian [8] and Yarin & Weiss [9]. The effects of surface tension and viscosity
on droplet spreading has been discussed by Bennett & Poulikakos [10]. Haferl &
Poulikakos [11], [12] examined successfully the transport phenomena during the
droplet impact. An experimental investigation on droplet deposition and solidifi-
cation was presented by Attinger et al. [13], [14].
More recent works addressed the basic understanding of incompressible
impact phenomena, such as the dynamic behaviour of the wetting angle between
the substrate and the droplet [15], the thermal contact resistance between splat and
substrate as well as the rapid solidification phenomenon [16], including heteroge-
neous nucleation and recoalescence, as well as the possible remelting of the sub-
strate [17].
6
Introduction
Compressible Modelling (High-Speed Liquid Impact)
High-speed liquid impact has provided one of the major areas of technical
concern involving compression phenomena in liquids. When a liquid drop impacts
against a rigid surface, we expect to see a number of flow regimes. The initial
phase of impact involves compression of the liquid, triggering the propagation of
pressure waves outward from the point of first contact, Fig. 1.2. The pressure
waves travel through the bulk of the droplet, interacting with the free surface and
with each other. In the final state of the contact, when the compressible effects are
expected to die away, the drop spreads out over the target surface and in certain
cases solidifies (if the target and/or ambient temperature is below the liquid melt-
ing point).
High-speed liquid impact is of especial relevance to coating technologies,
where highly accelerated molten metal or ceramic droplets impact and bond onto
a substrate. It is also of fundamental interest since the impact involves more gen-
eral physical phenomena, such as the interaction of shock and rarefaction waves
with one another and with the free surface, the formation and collapse of cavita-
tion bubbles and the eruption of jets.
The fluid flow associated with impinging drops is rather complex and not
understood in detail. Particularly the problem of high-speed [O (100 m/s)] droplet
deposition harbours substantial problems when it comes down to its fundamental
understanding. This is related to the fact that at the high-speed impact (the exact
definition of ‘high-speed impact’ shall be provided later) involves compressibility
patterns, whose both analytical and numerical modelling pose significant difficul-
ties. The objective of the present work is both theoretical and numerical investi-
gation of high-speed droplet impact, accounting for compressibility effects in the
liquid by a realistic equation of state.
7
Introduction
Definition of the Compressible Droplet Impact Problem
The problem geometry is comprised of a spherical liquid droplet impacting
at high speed onto a perfectly rigid surface (Fig. 1.2). The droplet is assumed to
move with a velocity normal to the wall and to have an initial density under
ambient pressure . During the first phase of the impact, liquid adjacent to the
contact zone is highly compressed whereas the rest of the liquid droplet remains
unaware of the impact. The two regions are separated by a shock front which trav-
els into the bulk of the liquid remaining pinned to the surface at very early times
due to the outward motion of the contact line.
The most frequently used approximations to the pressure developed in liquid-
solid impact are based on one-dimensional elastic impact theory. According to
this model, the generated pressures in the compressed region are of the order of
water-hammer pressure [18], [19]:
(1.1)
V ρ0
p0
z
r
compressedliquid
shock front
RV
β
liquid drop
β
B
O A
Fig. 1.2. Impact of a spherical liquid drop (blue) on a rigid surface. The zone of the hig
compressed liquid (red) is bounded by the shock front and target surface.
Pwh ρ0sV=
8
Introduction
Where , and are the ambient liquid density, impact velocity and shock
velocity with respect to the unaffected bulk of the liquid, respectively. The shock
velocity is not an invariant, and only at low impact velocities it can be approx-
imated by the acoustic velocity of an undisturbed liquid under ambient pressure
and density (see Chapter 3, Equation of State Modelling).
Heymann [20] Lesser [21] and Lesser & Field [22], [23] have shown that the
pressure in the compressed area is not uniform and reaches its highest values just
behind the contact line. A temporal maximum will be reached at the instant when
the shock wave overtakes the contact periphery, as experimentally measured for
the first time by Rochester and Brunton [24]. The flow patterns dominated by
compressibility effects have been previously reviewed by Lesser and Field [25],
who were especially concerned with loads and erosion effects.
An analytical study in the acoustic limit, valid for low impact velocities has
been developed by Lesser [21]. Upon impact, a shock wave is been generated, sep-
arating the disturbed from the undisturbed bulk of fluid. The exact position of the
shock front can be obtained by construction of the envelope of all individual
wavelets emitted by the expanding contact line (or contact edge if we consider a
two-dimensional axisymmetric case), Fig. 1.3.
In the acoustic limit [21], the shock velocity was assumed to be equal to the
ambient speed of sound in the liquid. This assumption is justified for most liquids
ρ0 V s
s
shock envelopedrop free surface
β
edge anglecontact edge
rigid target
Fig. 1.3. Impact of a spherical liquid drop on a rigid surface. Construction of shock front
as an envelope of individual wavelets emitted by the expanding contact edge.
9
Introduction
at low impact velocities. However, as will be demonstrated later - the results of
the present work, both computational and analytical, for the impact of a water
droplet show that the shock velocity during the first impact stage is in the range of
2600-3000 m/s, which is substantially higher than the ambient speed of sound
(approx. 1350 m/s).
During this first stage of the impact, the shock wave remains attached to the con-
tact edge (Fig. 1.3). The reason for this is that initially the contact periphery
spreads out much faster then the compression wave fronts. Since the contact edge
velocity decreases monotonically in time, it falls below the shock speed at some
shock front propagating with the velocity s
contact line U l(a)
s
propagating shock front
jetting
(b)
V
drop free surface
sdrop free surface
rigid wall moving upwards
with velocity V
Fig. 1.4. a) The shock wave remains attached to the contact periphery up to the moment
when the contact line velocity decreases below the shock velocity . b) Shock front
overtakes the contact edge. It is followed by the eruption of intense lateral jetting due to
the high pressure difference across the droplet free surface.
U l s
V
10
Introduction
point, the shock front detaches and starts to travel along the free surface. From this
point on, a very large density and pressure difference emerges across the droplet
free surface in the vicinity of the contact line, and a phase characterized by a
strong jetting eruption at the contact edge commences [Fig. 1.4]. Accordingly, we
define the ‘jetting time’ as the time when the liquid medium breaks through the
droplet free surface at the contact edge. From a theoretical consideration, we
expect this to occur when the contact edge velocity becomes equal to the shock
velocity at the contact edge. The computational determination of the jetting time
will be addressed later in this study.
It is well known that the time characterizing the onset of jetting, obtained by
theoretical considerations, is lower than what is observed in experiments, see e.g.
Field et al. [26] and [27]. A systematic delay can be attributed to target compli-
ance, which was explicitly included in calculations by Lesser [21]. This delay,
however, is not large enough to account for the discrepancy between theory and
experiment. To resolve this disagreement, Lesser & Field [27] pointed out that, as
the shock front moves upwards, the release wave would eject the material in the
direction of the local surface normal. Their picture of jetting suggests that the tra-
jectories of spalled liquid particles would cross through each other. Due to the
very small edge angle ( in Figs. 1.2 & 1.3) at this early time, this jet of liquid
would effectively close the gap between droplet surface and substrate and would
not be observable.
Field, Dear & Ogren [26] proposed a somewhat different picture of jetting
initiation. As soon as the shock envelope overtakes the free surface, the high
velocity liquid particles are accelerated normal to the surface of the drop. Hence
the velocity of the ejected particles have components both perpendicular and tan-
gential to the wall. The former increases the effective impact velocity and is there-
fore responsible for the delay in jetting (see [26]). Although there is no rigorous
treatment of the velocities of ejected liquid, an analysis by Lesser & Field [22] for
β
11
Introduction
a liquid cylinder impacting a solid surface suggests that this velocity is of the order
of the impact velocity.
Heymann [20] argued that the jetting must occur at an earlier time, before
the shock wave overtakes the contact edge (however without explicitly making
this conclusion). This earlier time - termed in this study as ‘the time of the shock
degeneration’ - can be derived as the maximum time at which the flow quantities
satisfy both Rankine-Hugoniot conservation laws and the equation of state. The
time of the first interaction is significantly smaller than the time at which the shock
wave overtakes the contact line. It is worth stating in advance that our computa-
tions show no observable jetting at the ‘time of the shock degeneration’. The issue
of what happens at this moment will be one of the major goals of this work. In
order to resolve this event analytically, an approach is presented, which accounts
for the time and position dependent shock speed by considering a realistic equa-
tion of state.
Most of the existing theoretical and numerical calculations are based on the
assumption that the droplets are spherical. This is also the case in the present work,
although the shape of impacting drops will always be somewhat influenced by
aerodynamic forces acting on its surface. The elastic response of the target is not
of significance in most cases and is not taking into account in this study. In the
present study we examine the impact of a droplet of a radius , moving
with a velocity of towards the rigid wall. Thus, the Reynolds
number, , can be estimated to be of the order of 50,000. The
symbol represents the kinematic viscosity of water. The high value implies
inertia dominated phenomena and supports an inviscid approach to the problem.
A similar comment is valid regarding the importance of surface tension to the
impact process. The Weber number ( , where is the surface
tension coefficient) is estimated to be of the order of 350’000, pointing out that the
surface tension effect in the droplet bulk can be neglected.
R 10 4– m=
V 500m s⁄=
Re ρ0RV ν⁄=
ν Re
We ρRV 2 σ⁄= σ
12
Introduction
13
Motivation
2Motivation
I was really rather alarmed to discover
that this experiment seems to be doable.
-- Sir Roger Penrose
The purpose of this work is a systematic investigation of the fundamental
fluid dynamics, occurring during high-speed impact of small size liquid droplets
on solid surfaces. Although this work focuses on fundamentals, a host of technol-
ogies, both traditional and future-oriented, stand to benefit from an in-depth
understanding of the controlling factors of these phenomena. For example,
progress in this direction will have an immediate impact on novel surface coating
techniques able to deliver mechanical parts with superior performance character-
istics. Industry is currently exploring the potential of plasma deposition as a solu-
tion to that problem. The next section is dedicated to providing background for
this technology.
2.1 Plasma Spraying Technology
Plasma Deposition is an important technology used in advanced surface
treatment for mechanical equipment designed to operate under adverse condi-
tions, where thermal shock, repetitive high-intensity mechanical loading or attack
by chemical agents is anticipated. Apart from the fundamental specification of
increasing the life-cycle (i.e. ensuring minimal rates of change in both shape and
constitution) of material surface strata by imparting desired properties to the
14
Motivation
microstructure on a per-case basis, there is also a list of additional requirements -
economic considerations and environmental compatibility - that have to be taken
into account when designing an industrial surface-coating process. Design and
optimization in this case translate to the need to control the physical phenomena
occurring during plasma deposition. For more details on plasma spraying technol-
ogy see Barbezat [28] and Ambühl [29].
In a typical application of this kind, a plasma atmosphere is initially used to
raise the temperature of a powder metallic or ceramic substance well above its
melting point (1728K for Ni, 2328 K for Al2O3 under normal pressure). Coming
into direct contact with the plasma flow, powder particles (dimensions of the order
of 100 µm) are simultaneously heated and accelerated to velocities of the hun-
dreds of meters per second. After impacting on a solid surface with this level of
kinetic energy, a liquid droplet undergoes a distinct early deformation phase in
which compressibility effects cannot be neglected. As it will be shown in this
work, these effects produce flow patterns typical of high-speed compressible flow
in a liquid, i.e. shock waves, expansion waves and high-speed jetting. As the time
progresses, the kinematic energy of the deforming droplet decreases, with the
result that velocities become small and the flow can thereafter be treated as incom-
pressible.
As discussed above, plasma spray deposition is a process involving injection
of metallic and/or ceramic powders into a high-temperature, high-speed plasma
jet. These melt during flight, impinge on a solid surface and solidify, thus provid-
ing a coating stratum to the surface material, Fig. 2.1. The desired properties of
the final product is a dense [30], pore-free and homogeneous coating with high
purity, high bonding strength between deposit and substrate and low thermal
stresses after solidification. To that effect, the impact stage is decisive and in turn,
the important parameters upon impact of the molten droplet are its temperature,
specific heat content and its impact velocity (see [30] and [31]).
15
Motivation
An example of the plasma coating operation is outlined in Fig. 2.2. The process is
carried out in a chamber and conditions within that chamber are completely con-
trolled. This allows to produce coatings exhibiting enhanced properties, some-
times not feasible in standard atmospheric environment.
The environment pressure can be in the range from near vacuum (as low as 50
mbar) to elevated pressures (as high as 4 bar). Chamber spraying may be chosen
to prevent contamination of the coating material and/or substrate, or because a
Fig. 2.1. Cross sections of a typical microstructure obtained through plasma deposition
process, courtesy Sulzer Metco.
Fig. 2.2. Requirements for a typical controlled atmosphere spray system.
16
Motivation
reaction of the coating material with a specifically introduced substance is desired.
The coatings resulting in this case are dense, well bonded and metallic coatings
are free of oxides and other contamination. Non-metallic coatings, such as ceram-
ics sprayed in chambers backfilled with non-reactive atmospheres, are fairly pure.
For more details on low pressure or near vacuum conditions see [30] and [31].
Inert, protective atmospheres can be used for ensuring the purity of reactive
spray materials or protecting work-piece substrates that readily oxidize or contam-
inate easily. Reactive atmospheres, generally used at elevated pressures, are useful
when unique coating material chemistries are desired that are not easily or eco-
nomically produced by other means. The experimental facility of Sulzer Metco
(Wohlen, Switzerland) is shown in Fig. 2.3. The coating substrate is mounted onto
the holder, which can be placed into rotational movement if desired.
Environmental Chamber
Robotic Arm
Control Unit
Fig. 2.3. Sulzer Metco environmental plasma chamber, Wohlen, Switzerland.
17
Motivation
The close up photograph of the robot arm is shown in Fig. 2.4. The tubes
attached at the plasma gun provide the nozzle with metallic or ceramic powders
and feed the cooling system with water. The robot arm can be moved with veloc-
ities up to 30 m/min. This, together with the optional rotation of the holder makes
it possible to achieve the desired coating thickness. The L-formed ribbed struc-
ture, shown behind the robotic arm and the holder in Fig. 2.4 is connected to the
strong ventilation system, designed to keep the chamber atmosphere constant and
free of powders which did not bond on to a substrate.
Fig. 2.4. Robot arm with plasma gun in the Sulzer Metco environmental plasma chamber.
Robotic Arm
Holder
Substrate
Ventilating System
18
Motivation
2.2 Sample of a Splat Shape
The great majority of high-velocity droplet impact numerical simulations to
date have been performed by using incompressible models. This may be the
reason why experiments carried out with high-velocity impact show quite a dif-
ferent physical situation than that predicted by the aforementioned numerical
modelling. For instance, incompressible simulations predicted a uniform continu-
ous spreading of the splat, with no break-up or violent jetting, whereas experi-
ments show situations such as that in Fig. 2.5, showing the experimentally
observed splat of a Al2O3 droplet (30 m diameter and 2664 K temperature), after
impacting a glass substrate at a velocity of 92,3 m/s. A ring of liquid mass has
detached itself from the main bulk of the material, however the final splat shape
is almost perfectly axisymmetric.
Fig. 2.5. Splat of liquid alumina (Al2O3) droplet on glass substrate, corresponding to
initial droplet radius 15.125 µm, temperature of 2664 K and impact velocity of 92.3 m/s.
After impact on a substrate and solidification, patterns of radial symmetry breakdown is
evident. Photograph courtesy of Sulzer Metco.
19
Motivation
Figure 2.6 corresponds to a similar experimental observation where now the drop-
let material is Ni, and the impact speed on a cold substrate is 180 m/s.
Figure 2.6 clearly demonstrated a break-down of azimutal symmetry. Thus, based
on these two experimental findings (Figs. 2.5 & 2.6), we conclude that both sce-
narios, the one preserving the azimutal symmetry and the other showing a brake
down in symmetry are physically admissible - depending on the parametric
domain of interest.
In a series of experiments performed by this author at Sulzer Metco
experimental facility, Ni particles were impacted on a glass substrate. The effects
of droplet temperature and impact velocity could be clearly observed. High impact
velocity (here 200 m/s) caused the droplet to break up and form a circular hollow
structure in the middle [Fig. 2.7 (b)], whereas at the low velocity impact (120 m/s)
Fig. 2.6. Splashed liquid nickel droplet at 2500 K after impact on a substrate and
solidification, showing patterns of symmetry breakdown both in radial and azimutal
direction. Impact velocity of 180 m/s. Courtesy: Sulzer Metco.
20
Motivation
the droplet formed a fairly homogeneous layer without hollow structure after
solidification, Fig. 2.7 (a). The low droplet temperature (measured approximately
with the aid of pyrometry at the surface prior to impact) inhibits the metallic
droplet from complete melting. Consequently, a significant three-dimensional
structure upon solidification was detected, Fig. 2.7 (c) (the solidified structure
height is comparable to the droplet radius).
Figure 2.7 emphasises the effect of droplet temperature on final droplet shape after
splash and solidification. The two photographs show high velocity impact at low
(below ) resp. very high droplet temperature (above ), indicating
the presence of a entirely molten [Fig. 2.8 (a)] resp. partially molten droplet core
[Fig. 2.8 (b)] at impact.
temperature T
impact velocity V
low, 1700 °C
low, 120 m/s high, 200 m/s
high, 2500 °C
Fig. 2.7. Impact of liquid Ni droplet of the mean radius of 10 µm: Effects of droplet
temperature (measured at the surface) and impact velocity.
20 µm
(a)
(c)
(b)
1700°C 2700°C
21
Motivation
Overall, a rich combination of phenomena is presented in this problem. A
step towards their understanding is made in this dissertation.
Fig. 2.8. Liquid metal impact at high velocity (200 m/s). a) Very high temperature (above
2700 °C, left) vs. b) low temperature (below 1700 °C, right). The left photograph has a 2.5
times higher magnification than the right one.
(a)
(b)
22
Motivation
23
Equation of State Modelling
3Equation of State Modelling
It is the mark of an educated mind
to rest satisfied with the degree of precision
which the nature of the subject admits
and not to seek exactness
where only an approximation is possible.
-- Aristotle (384 BC - 322 BC)
The dynamical evolution of a fluid is determined by the principles of conser-
vation of mass, momentum and energy. To obtain a complete mathematical
description, however, the conservation laws must be supplemented by constitutive
relations that characterise the material properties of the fluid. The latter strongly
influence the structure and dynamics of waves in any continuum-mechanical sys-
tem. Our model of fluid flow neglects such physical effects as viscosity, heat con-
duction and radiation. As a result, the dynamics require only partial specifications
of the thermodynamics of the material, the relation of the form
. (3.1)
3.1 Stiffened Gas Equation of State
For the modelling of the liquid phase we employ the stiffened gas equation
of state, proposed by Menikoff and Plohr [32] and Harlow and Amsden [33]
p p V e,( )=
24
Equation of State Modelling
(3.2)
where is the Grüneisen exponent (a constant) and is a fit parameter for a
desired material. The reader is reminded here that the stiffened gas EOS can be
obtained from a frequently utilised Grüneisen EOS by linearisation (for details on
this procedure see [33]). More generally, a stiffened gas EOS approximates any
equation of a state in the vicinity of the reference state ( , ).
For the modelling of surrounding air in the numerical part of this study, the
ideal gas equation of state was used. Note that the ideal gas EOS is a special case
of the stiffened gas EOS, i.e. with the fit parameter . In this case
plays the role of adiabatic exponent, i.e. . The quantities
and represent the specific heats at constant pressure and volume, respectively.
For some materials, can be quite large; examples are water and metals, for
which is of the order of megabars. The total energy , being defined as
(3.3)
can be expressed in our case as
(3.4)
Here, we substituted the specific internal energy from Eq. (3.2) into Eq. (3.3).
The above mentioned parameters describing the stiffened gas equation of state for
water are and .
Principal Hugoniot for the Stiffened Gas EOS
The locus of possible final states due to the shock compression for a fluid ini-
tially at normal density, pressure and zero mass velocity will be referred to as prin-
cipal Hugoniot. This is an alternative formulation of an EOS, especially
convenient for the analytical treatment of shock dynamics problems. To this end,
p Γ 1+( )P∞+ Γρe=
Γ P∞
V 0 e0
P∞ 0= Γ 1+
γ c p cv⁄= Γ 1+= c p
cv
P∞
P∞ E
E ρ u2
2----- e+ =
E ρu2
2-----
p Γ 1+( )P∞+
Γ-----------------------------------+=
e
Γ 4.0= P∞ 6.13 108Pa⋅=
25
Equation of State Modelling
we consider an arbitrary shock wave in a reference frame in which the liquid par-
ticles on the upstream side of a shock wave have zero velocity (Fig. 3.1) and apply
the Rankine-Hugoniot conservation laws.
The Rankine-Hugoniot jump conditions for a steady normal shock wave
result from the requirement of conservation of mass, momentum and energy
across the shock, i.e.
Mass: (3.5)
Momentum: (3.6)
Energy: (3.7)
Seeking the dependence of (particle velocity normal to the shock front) on the
shock speed , we express the remaining unknowns and in terms of and
and the known values and . From Eq. (3.4), the total energy difference
across the shock wave is readily obtained as:
s ρ ρ0–( ) ρu=
sρu ρu2 p p0–+=
s E E0–( ) u E p+( )=
upstream shock side
u0 0=
ρ ρ0=
p p0=downstream shock side
u 0>ρ ρ0>
p p0>
shock wave moving with
velocity s
s
Fig. 3.1. Determination of the principal Hugoniot: An arbitrary shock front surface in a
reference frame where the liquid particle velocity at the upstream side of the shock
vanishes.
u
u
s p E s
u ρ0 p0
E E0–
26
Equation of State Modelling
(3.8)
Using Eq. (3.5) to eliminate in Eq. (3.6) yields
(3.9)
In order to express the energy difference across the shock wave in terms of shock
and particle velocity, we substitute the pressure from Eq. (3.9) and the density
from Eq. (3.5) into Eq. (3.8). After some algebraic manipulations, this procedure
yields
(3.10)
Taking into account Eq. (3.10), the energy balance equation, Eq. (3.7) becomes
, (3.11)
which can be rearranged for as follows:
(3.12)
Finally, Eq. (3.12) yields the desired relation between the shock speed and particle
velocity behind the shock ( assumed in front of the shock) :
(3.13)
or for the particle velocity
(3.14)
Experimental measurements of shock Hugoniot data have been also provided by
Marsh [34].
E E0– ρu2
2-----
p p0–
Γ---------------+=
ρ
p p0– ρ0su=
E E0– ρ0usΓ 2–( )u 2s+2Γ s u–( )
--------------------------------=
ρ0us2 Γ 2–( )u 2s+2Γ s u–( )
-------------------------------- ρ0su2Γu 2 Γ 1+( ) s u–( )+2Γ s u–( )
--------------------------------------------------- uΓ 1+Γ
------------- p0 P+ ∞( )+=
u 0≠
ρ0s2 2 2 Γ+( )us---– Γ 1+( ) p0 P+ ∞( )=
u 0= s s u( )=
s u( ) Γ 2+4
------------- u u2 16Γ 1+
Γ 2+( )2--------------------
p0 P+ ∞
ρ0-------------------++
=
u u s( )=
u s( ) 2Γ 2+------------- s Γ 1+( )–
p0 P+ ∞
ρ0s-------------------=
27
Equation of State Modelling
3.2 Linear Hugoniot Fit
In addition to above derivation of principal Hugoniot, resulting from an
incomplete equation of state of the form Eq. (3.1), the dependence between the
shock speed and the jump in the particle velocity of the fluid across the shock
wave can be also experimentally measured. For most fluids, the latter can be
expressed over a considerable pressure range by a simple linear relationship [20]:
(3.15)
The symbol does not always correspond to the speed of sound under ambient
conditions. Experimental measurements for water yield and pro-
portionality factor , for details see Sesame [35] and Cocchi & Saurel
[36].
Introduction of Eq. (3.13) into Eq. (3.9) yields the initial water-hammer pres-
sure developed at the impact
(3.16)
This expression is valid only for the first moment of impact, when .
s s0 ku+=
s0
s0 1647m s⁄=
k 1.921=
p ρ0s0V 1 kV s0⁄+( )=
u V=
28
Equation of State Modelling
In the region relevant to our computations, Eq. (3.13) gives essentially the
same result as the linear form of the principal Hugoniot Eq. (3.15). Both curves,
fitted with parameters for water, are depicted in Fig. 3.2.
Definition of ‘High-Speed Droplet Impact’
Having defined the stiffened gas EOS, Eq. (3.1), and the expression relating
the pressure with the impact velocity [Eq. (3.15) or alternatively Eq. (3.13) in con-
juction with Eq. (1.1)], a more rigorous definition for the term ‘high-speed’ impact
can be provided. Significant density variation occurs after a threshold value of the
pressure [approximately of the order of , see Eq. (3.2)] is exceeded. We define
as ‘high-speed droplet impact’ an impact scenario where the density change is of
a non-negligible magnitude. Setting the relative limit at 5%,
(3.17)
200 400 600 800u (m/s)
1000
2000
3000
4000
s (m/s)
stiffened gas eosexperimental fit
Fig. 3.2. Comparison of principal Hugoniots. Shock velocity s as a function of the jump in
particle velocity u across the shock for the stiffened gas equation of state
( , ) and linear Hugoniot fit ( , ).P∞ 6.13 108Pa⋅= Γ 4.0= s0 1647m s⁄= k 1.921=
Eq. (3.13)Eq. (3.15)
P∞
∆ρρ
------- pp Γ 1+( )P∞+-----------------------------------≈ 0.05=
29
Equation of State Modelling
yields . Employing Eq. (3.16) we obtain . An
impact above this velocity limit, for the material discussed in this study, is termed
as ‘high-speed impact’.
3.3 Temperature Determination
Since we deal with an incomplete EOS, the temperature is not implicitly con-
tained in our governing equations. However, for the purposes of numerical inves-
tigation, it can be calculated based on values. The needed equation of state
was fitted from experimentally obtained data. For the parametric
domain of the present study, ( and ) a fit was devel-
oped from the Sesame [35] tabular equation of state. A semi-quadratic fit in and
was found to describe the compressibility of water with sufficient accuracy
(error estimate of fitted curve yields and K in our par-
ametric domain):
(3.18)
Here is the ambient temperature at ‘low’ pressure and the
normal density.
The remaining fit constants read:
(3.19)
, and . The
ambient temperature and density have the values and
. Due to the small magnitude of the constant , Eq. (3.18)
practically yields for any ‘low’ pressure, i.e. pressures of the order of
atmospheric pressure.
p 1.6 108Pa⋅≈ V 100m s⁄≈
p ρ,( )
T T p ρ,( )=
p 3Gpa< ρ 1300k g m3⁄<
p
ρ
∆T T⁄ 1.12%< ∆T 6.94<
T T 0 a1 p a2 p2 a3 p a4+( ) ρρ0----- 1– + + +=
T 0 T 0 ρ0,( )= ρ0
a1 ∂T ∂p⁄( ) ρ ρ0=3.64 10 7– K Pa⁄⋅= =
a2 2.18 10 18– K Pa2⁄⋅= a3 6.18 10 7– K Pa⁄⋅= a4 1.06– 103K⋅=
T 0 322.19K=
ρ0 1000kg m3⁄= a2
T T 0=
30
Equation of State Modelling
31
Mathematical Model & Computational Methodology
4Mathematical Model &ComputationalMethodology
The laws of nature are but the mathematical thoughts of God.
-- Euclid (325 BC-265 BC)
4.1 Governing Equations
For the compressible fluid dynamics, the system of governing equations is
given by Euler equations for an inviscid liquid:
(4.1)
(4.2)
(4.3)
Here, is the mass density, the velocity vector, the thermodynamic pres-
sure, the specific internal energy, the specific enthalpy, and
the gravitational acceleration vector. The scalar and tensor products is denoted by
the signs and , respectively. The thermodynamic variables are related by an
equation of state, giving the specific internal energy as a function of specific
volume and pressure .
∂tρ ∇ ρ u( )•+ 0=
∂t ρu( ) ∇ ρ u u∧( )[ ]• p∇+ + ρg=
∂t ρ12---u2 e+ ∇ ρ 1
2---u2 H+ u•+ ρu g•=
ρ u p
e H e p ρ⁄+= g
• ∧
e
V 1 ρ⁄= p
32
Mathematical Model & Computational Methodology
An implementation of the axial symmetry requires to introduce the cylindri-
cal coordinates . This is achieved by the transformation:
(4.4)
(4.5)
(4.6)
Let , and be unit vector basis for the
rectangular coordinate system and the unit vector basis for the rotational
coordinate system defined by
, (4.7)
, (4.8)
. (4.9)
Next, let and . Under the assump-
tion of rotational symmetry, i.e. , the system Eq. (4.1)-Eq. (4.3)
reduces now to a two-dimensional problem:
, (4.10)
(4.11)
(4.12)
(4.13)
r θ z, ,( )
x r θcos=
y r θsin=
z z=
e1 1 0 0, ,( )= e2 0 1 0, ,( )= e3 0 0 1, ,( )=
r θ z, ,( )
r e1 θcos e2 θsin+=
θ e– 1 θsin e2 θcos+=
z e3=
u u0 r u1 z u2θ+ += g g0 r g1 z g2θ+ +=
uθ gθ 0= =
∂tρ ∂r ρu0( ) ∂z ρu1( )+ +1r---ρu0–=
∂t ρu0( ) ∂r ρu02( ) ∂z ρu0u1( ) ∂r p+ + +
1r---ρu0
2 ρg0+–=
∂t ρu1( ) ∂r ρu0u1( ) ∂z ρu12( ) ∂z p+ + +
1r---ρu0u1 ρg1+–=
∂t ρE( ) ∂r ρu0E( ) ∂z ρu1E( ) ∂r pu0( ) ∂+ z pu1( )+ + + =
1r--- ρE p+( )u0 ρ g0u0 g1u1+( )+–=
33
Mathematical Model & Computational Methodology
4.2 Computational Domain & BoundaryCondition
The computational domain is given by the lower and upper boundary in
radial ( , ) resp. z-direction ( , ). In order to avoid singularity
problem at we chose the lower radial boundary to be slightly greater
then zero . The implementation of boundaries is performed in the follow-
ing manner (see Fig. 4.1):
1. Reflecting boundary at : Untracked boundary that must align with
computational grid cell edges. States are reflected from appropriate interior
states to fill out the finite difference stencil. Radial component of velocity set
to zero
2. Flow-Through Dirichlet boundaries (upper and right boundaries): A
boundary condition that suppresses reflections. The missing stencil states are
extrapolated through the boundary using the nearest interface state.
3. Neumann boundary (lower boundary): The reflection boundary with the
possibility for non-grid aligned boundaries. The implementation performs a
reflection about the tracked Neumann front. Since we have no heat conduc-
tion in our governing equation, it is clear that the Neumann boundary acts as
an adiabatic boundary, being also characterised by a zero mass-flux condition,
.
rlow rhi zbottom ztop
r 0= rlow
rlow 0>
r rlow=
ur rlow
0=
uz zlow
0=
34
Mathematical Model & Computational Methodology
Our current axisymmetric computations have been performed on a uniform grid
with sizes varying from 0.5 million points up to 4.0 million points and with major
time steps ranging from to . For some local propaga-
tions, time steps of order have been used.
V
liquid
radius
z-axis
Fig. 4.1. Computational domain and boundary conditions in cylindrical symmetry.
droplet
computationaldomain
steady air
Neumann boundary
flow-through boundary
flow
-thr
ough
bou
ndar
y
refle
ctiv
e bo
unda
ry
5 10⋅ 5– R V⁄ 5 10⋅ 7– R V⁄
O 10 10– R V⁄( )
35
Mathematical Model & Computational Methodology
4.3 Numerical Modelling
The code FronTier, used for the current simulations was developed by a
group of researchers at the New York University and the University of Stony
Brook [37]-[41]. The problems for which this method is attractive are those con-
taining discontinuities and other singularities concentrated on both lines and sur-
faces. Front Tracking, as discussed here, is a modified finite difference method
that uses two separate grids to describe the solution to a system of partial differ-
ential equations. These consist of a standard rectangular finite difference grid that
Fig. 4.2. Droplet and air density distribution prior to the impact: Emergence and reflection
of the bow shock in the air and weak perturbations in the liquid bulk (due to the liquid-air
interactions on a droplet surface). Parameters: Impact velocity 500 m/s, motionless air.
[numerical result with non-linear colour map (HDF)].
36
Mathematical Model & Computational Methodology
is fixed and a mobile lower dimensional grid that describes the location of the
wave fronts being tracked. For the purpose of this investigation we will restrict our
consideration to a two-dimensional case.
The representation of the solution to our system consists of the values of the
solution at the points on the fixed finite difference grid, together with the limiting
values on either side of the tracked fronts, as shown in Fig. 4.3. From the state
information a global solution operator is constructed using a combination of linear
and bilinear interpolation. If a point falls in a cell with no tracked fronts, this value
is computed using bilinear interpolation of the states at the corners of the cell.
State values near the tracked front are found using the linear interpolation on the
local triangulation of the rectangular cells near the front. This triangulation has a
property that each triangle corner either lies at the rectangular grid point or is a
point on the tracked front. The generation of the solution operator, which is a
piecewise continuous function of position, is one of the three major overhead
items in the front tracking method.
Time stepping that updates the solution consists of computing the propa-
gated positions and states on the tracked data structures, and updating the data on
the finite difference grid. The tracked structures are propagated and then used at
the beginning and the end of time step as internal boundaries for the solution on
the fixed grid.
37
Mathematical Model & Computational Methodology
4.3.1 MUSCL method of van Leer
The method used in our simulation is a five point stencil version of the
MUSCL method of van Leer [42]. This approach utilises a linear state reconstruc-
tor, a version of flux limiter due to Bell, Colella and Trangenstein [43], and a ver-
sion of the Colella-Glaz Riemann solver [44].
All finite difference schemes are implemented in the form of dimensionally
split solvers. The coupling of the states on the tracked front to the interior states
uses ‘interpolation by constant state’, [45]. The stencil used to compute the state
at a grid point consists of an array of points and states centred at this grid point. If
a tracked front crosses this stencil between the centre and another point on the
stencil, then the state at that stencil point is replaced by the state on the tracked
normal ntangent t
trackedU r
U l
uniform underlying grid
lower
dimensional
grid
rectangular grid states
discontinuity
Fig. 4.3. Finite differences stencil used for the normal propagation of the shock wave. The
states utilised for the computation of the normal propagation operator are obtained from
the left and right states ( and ) on the curve at the point being propagated.U l U r
38
Mathematical Model & Computational Methodology
front between the point and the stencil’s centre that is nearest to the centre of the
stencil. The tracked front effectively blocks the finite difference equations from
using states on opposite sides of the tracked waves, keeping the discontinuity of
the front sharp. This method, when coupled with front tracking is first order accu-
rate at the front and second order accurate away from the front. Since the fronts
occupy a relatively small percentage of the computational domain, the overall
method is overall second order accurate.
The method may be regarded as a second order sequel to the Godunov
method with the major improvement of taking the quantities in each slab to be
linear rather than constant as in the original Godunov approach. Besides second
order accuracy, the method has an important advantage with respect to Lax-Wen-
droff-like schemes, which lies in its suppression of oscillatory solutions and non-
linear instabilities. Its efficiency aside, the most favourable property of this
method is the clear physical picture associated with it.
At the heart, the method consists of an one-dimensional Lagrangian scheme,
the results of which are remapped onto the Eulerian grid. The Lagrangian equa-
tions of ideal compressible flow for a cylindrically symmetric flow read:
(4.14)
(4.15)
(4.16)
(4.17)
Here is the mass coordinate and the space coordinate. The independent vari-
ables are the time and the mass coordinate . The state quantities , , and
correspond respectively to specific volume, velocity, specific total energy and
pressure. and are source terms of momentum and energy, which can be
functions of any number of independent and dependent variables. The domain is
∂Ψ ∂t⁄ ∂ xu( ) ∂ξ⁄– 0=
∂u ∂t⁄ x∂ p ∂ξ⁄– Sm=
∂E ∂t⁄ ∂ xup( ) ∂ξ⁄– uSm Se+=
∂x ∂t⁄ 0=
ξ x
t ξ ψ u E
p
Sm Se
39
Mathematical Model & Computational Methodology
divided into slabs which need not have equal thickness . At each instant, the
true values in the slabs are approximated by linear distributions
for (4.18)
The half-integer index represents the values taken at centre of the slab, resp. over
the slab averaged values. The slab averages are defined as
(4.19)
and the average slope is found as
(4.20)
The slope defined by Eq. (4.20) is termed interface differencing, which is a tem-
porary substitute for a least-square fitting, performed in the Eulerian remap step.
After discretisation of the initial-value distributions in the slabs, there is a
discontinuity between the slopes and . The interface values
to the right resp. left of the slab can be expressed in a straightforward manner by
the slab thickness and the average slope defined by Eq. (4.20),
(4.21)
(4.22)
(4.23)
The indices + and - correspond to the right and left interface values, respectively.
By proper linear transformation, the system of partial differential equations (4.14)
∆ξ
Ψ ξ( ) Ψi 1 2⁄+∆i 1 2⁄+ Ψ∆i 1 2⁄+ ξ--------------------- ξ ξ i 1 2⁄+–( )+= ξ i ξ ξ i 1+< <
Ψi 1 2⁄+1
∆i 1 2⁄+ ξ------------------- Ψ ξ( ) ξd
ξ i
ξ i 1+
∫=
∆i 1 2⁄+ Ψ∆i 1 2⁄+ ξ----------------------
ξ∂∂ Ψ t
0 ξ,( )
i 1 2⁄+
≡ 1∆i 1 2⁄+ ξ--------------------
ξ∂∂ Ψ t
0 ξ,( ) ξdξ i
ξ i 1+
∫∆i 1 2⁄+ Ψ t
0 ξ,( )∆i 1 2⁄+ ξ
--------------------------------------= =
ξ i 1– ξ i,( ) ξ i ξ i 1+,( )
∆ξ
Ψi± Ψi
12---±
12---∆
i12---±Ψ+−=
ui± ui
12---±
12---∆
i12---±u+−=
pi± pi
12---±
12---∆
i12---±
p+−=
40
Mathematical Model & Computational Methodology
- (4.17) reduces to a system of ordinary differential equations, describing the
change of flow quantities along the characteristics (see [42]),
(4.24)
(4.25)
(4.26)
As usual, the speed of sound, is here defined as
Having found the characteristic equations (4.45)-(4.47), we proceed with the exact
formulas for updating the slab averages. The former result from integration of
conservation laws,
(4.27)
(4.28)
(4.29)
The former three equations are valid regardless of presence of discontinuities in
the slab. As usual in control volume schemes, we need to estimate time averages
represented by . In order to obtain them we proceed with a half time step.
(4.30)
(4.31)
t∂∂p c2
t∂∂Ψ
+ Se e∂∂p
V=
t∂∂u 1
c---
t∂∂p
– xc
ξ∂∂u 1
c---ξ∂∂p
– – uΨc
x----------- Sm
Se
c-----
e∂∂p
V–+=
t∂∂u 1
c---
t∂∂p
+ xc
ξ∂∂u 1
c---ξ∂∂p
+ + uΨc
x-----------– Sm
Se
c-----
e∂∂p
V+ +=
c
c2
Ψ∂∂p
S–≡
Ψi 1 2⁄+ Ψi 1 2⁄+∆t
∆i 1 2⁄+ ξ------------------- xu⟨ ⟩ i 1+ xu⟨ ⟩ i–( )+=
ui 1 2⁄+
ui 1 2⁄+∆t
∆i 1 2⁄+ ξ--------------------– xp⟨ ⟩ i 1+ xp⟨ ⟩ i–( ) ∆t αpΨ x⁄⟨ ⟩ i 1 2⁄+ Sm⟨ ⟩ i 1 2⁄++( )+=
Ei 1 2⁄+
Ei 1 2⁄+∆t
∆i 1 2⁄+ ξ--------------------– up⟨ ⟩ i 1+ up⟨ ⟩ i–( ) ∆t uSm⟨ ⟩ i 1 2⁄+ Se⟨ ⟩ i 1 2⁄++( )+=
⟨ ⟩
Ψ⟨ ⟩ i Ψ i±* 1
2---
t∂∂Ψ
i±
*
∆t O ∆t( )2{ }+ +=
u⟨ ⟩ i ui* 1
2---
t∂∂u
i
*
∆t O ∆t( )2{ }+ +=
41
Mathematical Model & Computational Methodology
(4.32)
(4.33)
As extensively discussed in [42], for the update of the slab averages we need to
estimate the time averages at each interface with first order accuracy. The exact
formulas for updating the slab averages result from the integration of conservation
laws. The slope averages can be updated by calculating the interface values at the
new time step. The values represented with a star , denote the values in the
middle region between the slabs. They can be obtained by taking into account the
jump conditions across the waves which develop from the initial discontinuity at
the slab interface and by use of characteristic equations written in difference form.
For details of this procedure we refer to the original paper by van Leer [42].
Finally the full time step can be carried out with the full accuracy:
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
(4.39)
p⟨ ⟩ i pi* 1
2---
t∂∂p
i
*
∆t O ∆t( )2{ }+ +=
x⟨ ⟩ i xi12---ui
*∆t O ∆t( )2{ }+ +=
*
xi xi ui u⟨ ⟩ i∆t O ∆t( )3{ }+ +=
X i xi( )2
2⁄=
Vi 1 2⁄+ ∆i 1 2⁄+ X ∆i 1 2⁄+ ξ⁄=
ui 1 2⁄+ ui 1 2⁄+∆t
∆i 1 2⁄+ ξ------------------- ∆i 1 2⁄+ x⟨ ⟩ p⟨ ⟩( ) p⟨ ⟩ i 1 2⁄+ ∆i 1 2⁄+ x⟨ ⟩–[ ]– +=
F⟨ ⟩ i 1 2⁄+ ∆t O ∆t( )3 ∆t ∆ξ( )2,{ }+ +
Ei 1 2⁄+
Ei 1 2⁄+∆t
∆i 1 2⁄+ ξ-------------------∆i 1 2⁄+ x⟨ ⟩ u⟨ ⟩ p⟨ ⟩( )– +=
u⟨ ⟩ i 1 2⁄+ F⟨ ⟩ i 1 2⁄+ G⟨ ⟩ i 1 2⁄++( )∆t O ∆t( )3 ∆t ∆ξ( )2,{ }+ +
V i V i±*
t∂∂V
i
*
∆t O ∆t( )2{ }+ +=
42
Mathematical Model & Computational Methodology
(4.40)
(4.41)
The outlined Lagrangian step is followed by Eulerian remap, [42].
It is also worth noting that the reason for the second order accuracy of this
method is that the procedure involves per state quantity and per dimension two
independent data to describe the distribution in a slab, namely the slab average and
a representative slope average as derived above. The slope values are independent
of slab averages, they cannot be calculated from the last and must be stored sepa-
rately. This distinguishes the present scheme from common difference schemes,
such as the Godunov scheme which takes the slab averages to be zero. This
approach has an effect of an equivalent mesh refinement of a factor of two. The
Lagrangian step is followed by Eulerian remapping. Due to its dissipative proper-
ties, the scheme can be used across shocks. For further details, concerning the
monotonicity algorithms and boundary conditions, we refer the reader to the orig-
inal paper by van Leer [42].
4.3.2 Front Tracking Method
The key element to the front tracking method is the algorithm used to prop-
agate points on the tracked fronts. We are primarily interested in the case of the
hyperbolic conservation law:
(4.42)
where the flux term and the source term.
A point propagation algorithm is constructed using solutions of the Riemann
problem. A Riemann problem for one dimensional hyperbolic system is an initial
ui ui*
t∂∂u
i
*
∆t O ∆t( )2{ }+ +=
pi pi*
t∂∂p
i
*
∆t O ∆t( )2{ }+ +=
U t ∇ F U( )⋅+ G U( )=
F F U( )= G G U( )=
43
Mathematical Model & Computational Methodology
value problem with piecewise constant initial data and a single jump discontinu-
ity. The Euler equations are split for normal and tangential updates:
(4.43)
Here, and are the unit vectors normal and tangential to the tracked surface,
respectively. The first step is to propagate all points except the nodes on the dis-
continuity curves, as shown in Fig. 4.3, in the normal direction and update the
states ( ) on both sides of the curve. We first solve for the normal compo-
nent of Eq. (4.42):
(4.44)
We shall describe the algorithm for the forward shock only, as shown in Fig. 4.4.
The normal equations (4.44) for gas dynamics, written in characteristics form are
characteristic: (4.45)
characteristic: , (4.46)
characteristic: (4.47)
where , , , and correspond to the normal and tangential particle velocity,
entropy and speed of sound, respectively.
U t n n ∇⋅( )F[ ] t t ∇⋅( )F[ ]⋅+⋅+ G=
n t
U r U l,
U t n ∇ n F⋅( )⋅+ n G⋅=
λ1 λ1dd 2c
γ 1–----------- u– c
γ--λ1d
dS=
λ2 dS dλ2⁄ 0= dv dλ2⁄ 0=
λ3 λ3dd 2c
γ 1–----------- u+ c
γ--λ3d
dS=
u v S c
44
Mathematical Model & Computational Methodology
Since there is no wave transmitted to the right side of a forward shock, the
characteristic equations determine the state on the right side. An approximate
solution of the characteristic equations is obtained by solving their difference
approximation
(4.48)
, (4.49)
and
, (4.50)
where the subscript refers to values on the right side of the shock, and subscripts
1, 2 and 3 refer to values at the foot positions of , and -characteristics
new shock position
at t1 t= 0 ∆t+forward shock wave
x axis
λ3λ1λ1 λ2
Ur Ur1 Ur2Ul Ur3Ul1
Ui Ui+1
characteristic lines
t1
t0
U lˆ U r
ˆ
shock position at t t= 0
Fig. 4.4. A schematic picture of the data used for normal propagation of a shock wave.
The front data at the old time step provides a Riemann solution, that is corrected by
interior data, using the method of characteristics.
2cr c1–
γ 1–--------------- ur u1–( )–
cr c1+
2γ--------------- Sr S1–( )=
Sr S2= vr v2=
2cr c3–
γ 1–--------------- ur u3–( )+
cr c3+
2γ--------------- Sr S3–( )=
r
λ1 λ2 λ3
45
Mathematical Model & Computational Methodology
(see Fig. 4.4) at time . The ‘hat’ sign above the letter, , refers to the value at
the time .
As a first step, we solve the Riemann problem at time and propagate the
discontinuity to its preliminary position. Then, we draw back the four character-
istics approximated as straight lines [Eq. (4.48) - Eq. (4.50)] from the discontinu-
ity position at time step to the foot position at time , as indicated by
dashed arrows in Fig. 4.4. Having found the foot positions, the corresponding
states are obtained by sampling the solution interpolated from and
(resp. and on the left side) at known space coordinates. On the left side
only the characteristic impinges on the shock. Its difference equation reads
(4.51)
where the subscript refers to value at the foot of the left characteristic. The
Rankine-Hugoniot conditions applied the to right and left state at are:
, (4.52)
(4.53)
(4.54)
These conditions, when coupled with the characteristic equations, Eq. (4.48) - Eq.
(4.51), yield the approximate left state at time accurate to order . The
final shock propagation velocity is obtained by averaging the shock velocities at
time and . The normal sweep for the contact discontinuity is handled in a
similar manner. This case differs from the previous one only by the fact that one
characteristic at each side impinges on the contact line, for details see Chern et al.
[38]. The detailed application of front tracking is described by Grove [45].
t x x
t ∆t+
t
t ∆t+ t
U i 1+ U r
U i U l
λ3
2cl c3l–
γ 1–---------------- ul u3l–( )+
cr c3+
2γ--------------- Sl S3l–( )=
3l λ3
t ∆t+
vl vr=
ρl
ρr-----
γ 1+γ 1–------------
pl
pr----- 1+
γ 1+γ 1–------------
pl
pr-----+
⁄=
ur ul–cr
γ----
pl
pr----- 1– 1
γ 1+2γ
------------pl
pr----- 1– +
12---
⁄=
t ∆t+ ∆t
t t ∆t+
46
Mathematical Model & Computational Methodology
47
Numerical Results
5Numerical Results
On two occasions I have been asked (by members of Parliament),
’Pray, Mr. Babbage, if you put into the machine wrong figures,
will the right answers come out?’
I am not able rightly to apprehend the kind of confusion
of ideas that could provoke such a question.
-- Charles Babbage (1791-1871)
In this chapter we present the results obtained by numerical simulations of a
high-speed droplet impact on a rigid surface. For this, the front tracking method-
ology presented in Chapter 4.3 has been utilised. A water droplet of radius of
collides with an impact velocity of 500 m/s with a flat substrate. Due to
the very small time scales ( , obtained according to CFL condition)
and high velocities at which the emerging waves propagate, the first issue that
should be addressed is weather the present resolution (up to 4 million grid points)
can realistically capture the phenomena. Thus, as a first step, we investigate the
grid independence and convergence of the solution.
5.1 Solution Convergence & Grid Independence
The grid independence of the results has been established by both one- and
two-dimensional capturing of the accurate shock position and shape. Computa-
tions performed on three meshes with grids ranging from 0.5 million to 2 million
100µm
10 14– 10 11– s–
48
Numerical Results
points capture essentially the same shock position and yield practically identical
results for density, pressure and temperature.
z / R
p(G
Pa)
0.05 0.10 0.15 0.20 0.25
0.0
0.2
0.4
0.6
0.8 0.5 M1.2 M2.0 M
grid sizez / R
T(K
)
0.05 0.10 0.15 0.20 0.25300
350
400
450
500
550 0.5 M1.2 M2.0 M
grid size
(b)
z / R
dens
ity
0.05 0.10 0.15 0.20 0.25
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
0.5 M1.2 M2.0 M
grid size
Fig. 5.1. Convergence and grid independence of the solution: a) density, b) pressure and c)
temperature distributions along the z-axis (along line in Fig. 1.2 for 3 different grids:
0.5, 1.2 and 2.0 million points. The snapshot corresponds to time step 10.02 ns after
impact.
OB
(a)
(c)
49
Numerical Results
We performed also modelling on the grids containing up to 4 million points, how-
ever their representation will be omitted here, since it cannot be graphically dis-
tinguished from those performed on meshes of 2 million points. The plots in Figs.
5.1 (a)-(c), show the density, pressure and temperature distributions at
after impact, along the symmetry axis.
Figure 5.2 shows the shock envelope in the r-z plane at after
impact.
5.2 Droplet Evolution & Interaction of Waves
The ‘shadowgraph’ in Figs. 5.3 (a)-(i) shows the density evolution in the
symmetry plane of the droplet during the impact. In addition, a 3D spatial repre-
sentation is given in Figs. 5.4(a)-(h). Initially, the water has the ambient pressure
and density. Immediately upon impact, a creation of a strong shock wave which
moves upwards can be observed, Fig. 5.3 (a) [Fig. 5.4 (a)]. The edge velocity,
which initially has in theory an infinite value, remains higher than the shock speed
throughout this initial stage. Thus, the shock wave remains attached to the contact
periphery during this phase of impact (up to the time ns). After this time, the
t 10.02ns=
t 4.64ns=
r / R
z/R
0.05 0.10 0.15 0.20 0.25
0.02
0.04
0.06grid size0.5 M2.0 M
Fig. 5.2. Convergence and grid independence of the solution: Shock position in r-z plane
for two meshes, 0.5 and 2.0 million points, corresponding to time step 2.05 ns after
impact. The depicted region corresponds to the zoomed area of the quadrant
in Fig 1.2.
BOA
t 3≈
50
Numerical Results
edge velocity decreases below the shock velocity, and the shock wave overtakes
the contact line [Fig. 5.3 (b)], starting to travel along the droplet free surface [Figs.
5.3 (c)-(e) or alternatively Fig. 5.4 (b)-(d)].
At the free surface, the shock wave is reflected normal to the surface as an expan-
sion wave, which focuses towards the inner region of the water droplet. The liquid
(d)
(a)
(g)
(b)
(h)
(c)
(i)
(f)
t=7.98 ns
t=56.96 ns
t=123.24 ns t=138.94 ns t=162.58 ns
t=18.04 ns t=38.04 nsρ (g/cm3)
Fig. 5.3. Time evolution of density during the droplet impact showing shock creation,
propagation and interaction with the free surface. The region enclosed by the black line
corresponds to the very low pressure area behind the shock wave, which occurs upon
shock interaction with the droplet free surface.
(e)
t=95.58 ns t=109.33 ns
51
Numerical Results
adjacent to the droplet free surface, between the shock wave and contact periphery
at the wall, is not compressed.
Fig. 5.4. Three-dimensional representation of the droplet impact time evolution: Droplet
free surface (in blue) and shock & expansion waves (in red).
(d) 95.58 ns
(a) 7.98 ns
(g) 138.94 ns
(b) 38.04 ns
(h) 162.58 ns
(c) 56.96 ns
(f) 123.24 ns(e) 109.33 ns
52
Numerical Results
The shock wave propagating upwards finally reaches the droplet ‘North Pole’
[Fig. 5.3 (f) & Fig. 5.4 (e)], where it is reflected downwards [Figs. 5.3 (g) & (h),
Fig. 5.4 (f) & (g)] focusing on the drop axis of symmetry.
As the shock wave travels along the free surface, it carries a low pressure
area behind it (marked with a dark contour in Fig. 5.3 (d) and Fig. 5.3 (e), which
ultimately focuses at the droplet axis of symmetry [Fig. 5.3 (i) & Fig. 5.4 (h)]. At
this resolution, the emanating radial jet is clearly visible in Fig. 5.3 (e), even
though its first occurrence has been detected much earlier (to be discussed below).
The shock velocity during the first stage of impact is in the range of 2600-3000
m/s, which is substantially higher than the ambient speed of sound (1350 m/s).
The computational results show the presence of very low pressure, indicat-
ing strong rarefaction in the middle of the drop [marked with a contour in Fig. 5.3
(i)], which could produce cavitation. The occurrence of a strong focused rarefac-
tion wave in the middle of the droplet has been observed experimentally by Field,
Dear & Ogren [26]. The sequence of plots in Fig. 5.3 demonstrates that high
velocity impact of droplets is dominated by compressibility, with the development
of lateral jetting and the generation of shock and expansion waves. Moreover, we
observe interactions of the aforementioned compressibility patterns with the free
surface and with each other, up to the moment when all compressibility effects die
away, beyond Fig. 5.3 (i).
One important trait of the impact process is that the high-velocity jet is
ejected only from the contact edge. This is not necessarily obvious, since the trav-
elling shock wave carries high pressure along the entire free surface and one might
expect that jetting would occur everywhere on the free surface after the shock pas-
sage. This does not take place due to the previously mentioned expansion wave
adjacent to the free surface, which rapidly lowers the high pressure carried by the
shock and hence inhibits jetting across the free surface.
53
Numerical Results
The only region in the droplet where the pressure remains high and can produce
sustainable liquid jetting is the zone at the contact edge, where we observe contin-
Fig. 5.5. Three-dimensional representation of the droplet impact time evolution.
volume cutout uncovers the exact position of the free surface and shock & expansion wave
fronts (in red).
3π 2⁄
(d) 95.58 ns
(a) 7.98 ns
(g) 138.94 ns
(b) 38.04 ns
(h) 162.58 ns
(c) 56.96 ns
(f) 123.24 ns(e) 109.33 ns
54
Numerical Results
uous radial liquid ejection. Another three-dimensional evolution graph with the
volume area being cut out is given in Figs. 5.5(a)-(h).
The flow in the compressed area [Fig. 5.6 (a)] is initially aligned along the
z-axis, however as the contact edge propagates sideways, a radial flow develops,
see Fig. 5.6.
Since the flow adjacent to the axis of symmetry is normal to the wall and thus basi-
cally one-dimensional, no substantial pressure variation occurs along the z axis at
. This is also shown by the pressure plot in Fig. 5.1 (c).
5.3 Jetting Phenomena
The radial component of the droplet contact line velocity, observed in the
frame moving with the droplet, i.e. in the reference system where the droplet has
the zero velocity and the wall impacts from below, has the value:
(5.1)
υ 0 π, 2⁄[ ]∈
r / R
z/R
0.0 0.1 0.20.00
0.02
0.04
0.06
0.08 radial particle velocityreference vector: 500 (m/s)
Fig. 5.6. Development of lateral liquid motion in the compressed region. Snapshot at time
2.148 ns after impact.
r 0≈
U l VR Vt–
2RVt V 2t2–----------------------------------=
55
Numerical Results
The derivation of Eq. (5.1) is based on the simple geometrical condition that the
total contact edge velocity is tangential to the droplet free surface at the contact
edge:
(5.2)
The exact derivation of Eq. (5.1) shall be presented later.
The instance when the shock wave overtakes the contact line, triggering jet-
ting eruption, can be derived from the condition that the edge velocity decreases
to the shock velocity , i.e. , which in the limit yields a
solution
(5.3)
In the former derivation, we have applied the fact that at this stage, the height of
the compressed area is much smaller than the droplet radius, (see also
Field et al. [26]).
The variable in Eq. (5.3) represents the shock velocity at the inception of
jetting. However, to date we still lack an analytical model for its prediction. In the
acoustic approximation, this velocity is roughly equal to the ambient speed of
sound.
Applying one-dimensional model, Heymann [20] and Field et al. [26]
approximated the shock velocity by the velocity at the initial moment of impact,
according to Eq. (3.15) where they assumed the particle velocity with respect to
the undisturbed liquid at the shock to be equal to the impact velocity .
This is true only at the first instance when the falling droplet comes in con-
tact with the substrate. Later, the radial liquid movement increases, as shown in
Fig. 5.6, whereas the normal component of the liquid velocity at the wall, due to
the boundary condition remains equal to the impact velocity. In order to estimate
U er V z+( ) r⊥
s U l t( ) s t( )= V U l⁄ 1«
t jet RV
2 s2--------≈
Vt R«
s
s
u V=
56
Numerical Results
the jetting time in Eq. (5.3), we need an accurate prediction for the velocity of the
shock wave emitted by the propagating contact edge . The latter can be calcu-
lated according to Eq. (3.15) [or equally Eq. (3.13)], if the jump in the particle
velocity across the shock is known. This velocity-jump can be substantially
higher than the initial velocity, since the fluid particles develop also a radial com-
ponent of velocity, . Our numerical computations show that the lateral liquid
velocity reaches values comparable to the impact velocity and therefore cannot be
neglected. As shown on the right side of the ‘liquid jet’ region in Fig. 5.7, the
radial particle velocity mounts up to in the compressed zone. Thus, as
the total liquid particle velocity ( ) increases, the shock velocity also
increases reaching its maximum at the moment of jet eruption.
The analysis of the computational results indicates that a theoretical model
would have to take into account the radial liquid motion to accurately predict the
time of jetting onset. (see also Table 1). The additional component of particle
velocity increases the shock velocity at the front, as can be seen from Eq.
(3.15), hence, it reduces the jetting time found according to Eq. (5.3).
Numerical Determination of Jetting Time
Figures 5.7 and 5.8 illustrate the procedure of capturing the time of first jet-
ting eruption. Both density and particle velocity along the r-axis at are
examined in Fig. 5.7 (these are the density and particle velocity distribution along
the line in Fig. 1.2). We observe very high particle velocities in the region
where the density abruptly decreases, corresponding to the area in the picture
marked as ‘high-velocity liquid jet’. This is evidence of liquid eruption across the
shock front and is tracked back to its initiation, which then defines the jetting time.
This procedure, even by using the finest grids, reaches its limits at some
point. I.e. the highest possible spatial and temporal resolution is reached, beyond
s
u
ur
400m s⁄
ur uz+
ur
z 0=
OA
57
Numerical Results
which one cannot determine at which time step the expelled jet is visible for the
first time.
In our study, this yields the estimate . The ‘exact’ determina-
tion of the jetting time within this range is achieved as follows: The highest parti-
cle velocity in the vicinity of the contact edge is computationally determined by
sampling the fluid particle velocity at the cells adjacent to the contact edge. The
latter is then plotted against the corresponding time yielding the curve of maxi-
mum particle velocity vs time, Fig. 5.8.
In Fig. 5.8, on the same plot we draw the contact line velocity given by
Eq. (5.1). It is clear that the radial velocity of the emanating jet is higher or equal
to the contact edge velocity (equal at the limit when the jetting occurs). The inter-
section of these two curves defines the limit where the contact edge velocity
r / R
late
ralp
artic
leve
loci
ty(m
/s)
&de
nsity
(kg/
m3 )
0.05 0.10 0.15 0.20 0.25
500
1000
1500
2000
2500
density
liquid velocity
high velocityliquid jet
liquid behindthe shock wave
hump
air
legend
Fig. 5.7. Commencement of jetting. Radial liquid velocity shows the initiation of jetting.
The image corresponds to the time 3.05 ns after impact. The first evidence of jetting is
found approximately at time 2.80 ns. Grid size here: 4 million points.
2.5ns t jet 3.0ns< <
U l
58
Numerical Results
equals the above mentioned particle velocity. The corresponding time is termed
‘the jetting time’ throughout this study. For the case shown, .
Table 1 reports the jetting times obtained through different approaches. We
see that the theoretical model in the acoustic limit significantly overestimates (by
a factor of 5) the onset of jetting eruption. The previously discussed model with
constant shock velocity approximated by its initial value [ in Eq.
(3.15)] yields also higher jetting times than observed in computations.
t jet 2.80ns=
1 2 3 4 5 6t (ns)
1000200030004000500060007000
v (m/s)
Contact EdgeParticle
Fig. 5.8. Accurate determination of the jetting eruption time.
u uz V= =
59
Numerical Results
It is worth mentioning that the hump in particle velocity to the left of the
liquid jet in Fig. 5.7 represents air expelled from the gap between the droplet and
substrate. Outwards, the jetting reaches very high velocities (up to 6000 m/s), as
shown in Fig. 5.9. The lateral jet Reynolds number, based on jet thickness and
eruption velocity, is approximately 28,000, which provides further justification of
the original assumption of an inertia-driven phenomenon at the first stages of
impact.
Approach Evaluation parameters Jetting time
(ns)
Acoustic limit , m/s 13.72
Shock velocity approximated by
initial shock velocity at the
impact
, m/s
m/s
3.67
Shock velocity approximated by
the exact velocity corresponding
to the time of jetting eruption.
Evaluation according to the
principal Hugoniot derived from
the stiffened gas equation of state.
Particle velocity obtained from
numerical calculations.
, & (m/s)
Pa, , and
kg/m3
*computational findings
2.86
Computational observation:
tracking back in time the
emanated jet up to its origin
2-5-3.0
Computational observation:
Taking the intersection of the
contact edge velocity and
maximum particle velocity in the
vicinity of the edge
2.8
Table 1: Determination of Jetting Time
t jet
t RV 2c2( )⁄= c 1350.00=
t RV 2 s2( )⁄= u 500.00=
s sinit s= = 0 ku+ 2610.00=
t RV 2 s2( )⁄=
sΓ 2+
4------------- u u
2 16Γ 1+
Γ 2+( )2--------------------
P∞ρ0-------++
=
s 2957= ur 400= u 640=
P∞ 6.13 108⋅= Γ 4.0=
ρ0 1000=
60
Numerical Results
There is evidence of pressure increase towards the contact edge, Fig. 5.10,
as theoretically predicted by Lesser [21], Heymann [20] and Field et al.[25]. The
pressure reaches its highest value up until this time, at the moment when the shock
wave overtakes the contact edge (approx. 2.8 ns). The maximum pressure com-
puted surpasses the ambient water-hammer pressure [given by Eq. (1.1)] by a
factor of 3.
r / R
z/R
0.22 0.24 0.26 0.280.00
0.02
0.04
0.066060.815269.284477.743686.212894.672103.141311.60
520.07341.68219.35124.30
radial particle velocity (m/s)
shock wave
droplet free surface
Fig. 5.9. High jetting velocities observed in the computations reaching values up to
6000 m/s. The picture corresponds to the time step 4.86 ns after impact.
61
Numerical Results
There is no theoretical framework for predicting the evolution of the flow in
the contact line region after the shock departure. According to the Huygens prin-
ciple, the contact edge hit by the still falling droplet will continue to emit second-
ary shock waves, which are responsible for the persistence of a high pressure area
at the contact line. These shock waves are superimposed on the previously gener-
ated expansion wave (Fig. 5.10). At some later time, however, we expect a pres-
sure release due to the produced rarefaction waves. In his consideration, Field [25]
estimated that the stage of pressure release at the contact edge commences after
time , where the jetting time, is given by Eq. (5.3).
Our computations, (Fig. 5.10), show that the pressure further increases for
approx. (≈ ) after the departure of the shock envelope, which occurs
at time . Thereafter, we observe a pressure decay in time owing to the spatial
release of pressure in the compressed zone. Thus, the maximum pressure is been
reached at , beyond which the phase of pressure release develops. The
z / R
p(G
Pa)
0.1 0.2 0.3 0.4 0.5 0.6 0.70.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2 01.57 ns03.25 ns04.20 ns08.15 ns15.33 ns23.76 ns
time
Fig. 5.10. Pressure distribution along the r axis corresponding to 6 representative times.
t 3t jet≈ t jet
1.4ns 0.5t jet
t jet
t 1.5t jet≈
62
Numerical Results
curve corresponding to the time in Fig. 5.10 shows superposed sec-
ondary waves. These waves have their origin in the strong rarefaction waves emit-
ted at the contact edge. As the shock wave propagates upwards, the expansion
wave at the contact edge fades away and the pressure created by the contact edge
is not significantly influenced by the rarefaction wave produced at the contact
edge, as can be seen from the pressure curves at times in Fig. 5.10.
5.4 The Effect of Surface Tension
We performed numerical simulations under the same impact parameters and
grid resolution, both with and without surface tension. The results show that the
surface tension does not enter the scenario at the early impact stage until the jetting
eruption. However different patterns of expelled jets are observed in the presence
of surface tension. At the start of jetting, the ejected water jet has a density sub-
stantially lower than the ambient density of water.
As shown in Fig. 5.11 (b), the presence of surface tension causes the smooth
jet [Fig. 5.11 (a)] to break up into segments and to detach off the surface. The ema-
nated jet contracts and forms discontinuous regions of higher density (up to 250
kg/m3) compared to the previous case (up to 130 kg/m3). The expelled water jet
has very high temperature ( ), hence it is expected that the immediate
jet vaporization will compete with its subsequent solidification on the cold sub-
strate. We also observed higher jet velocities in the case with surface tension.
Finally, the simulations indicate that the surface tension does not influence the
flow in the bulk of the liquid, since we observe in both cases the same evolution
of the shock wave and its interaction with the free surface.
t 8.14ns=
t 15ns>
T 1000K>
63
Numerical Results
r / R
z/R
0.80 0.81 0.82 0.83 0.84 0.85 0.860.00
0.01
0.02
0.03
r / R
z/R
0.50 0.60 0.70 0.80 0.900.00
0.05
0.10
0.15
0.20
(b)
r / R
z/R
0.50 0.60 0.70 0.80 0.900.00
0.05
0.10
0.15
0.20
(a)
Fig. 5.11. Influence of surface tension on jetting formation and break up. Snapshot
corresponds to the time 18.19 ns after impact a) zero surface tension. b) surface tension for
water, . (c) zoomed front region of jets in b). together with computational
grid used.
σ 0.073N m⁄=
(c)
64
Numerical Results
5.5 Temperature
Figure 5.11 (b) shows the temperature distribution in the compressed area
along the z axis at the time 2.04 ns after impact at characteristic radial locations.
We observe a reasonably constant temperature in the bulk of the compressed
liquid in the axial direction accompanied by a temperature increase towards the
contact line region (increasing radial position).
Here, the temperature was determined by post-processing of pressure and density
data according to Eq. (3.18).
To evaluate possible viscous effects, the Reynolds numbers for the impact-
ing droplet (based on diameter) and for the erupting jet (based on jet thickness)
have been approximately evaluated as and respectively. These
relatively high Reynolds numbers strongly suggest the dominance of inertia-
z / R
T(K
)
0 0.01 0.02 0.03 0.04 0.05 0.06300
400
500
600
700
8000.020.080.100.12
radial position r / R
Fig. 5.12. Temperature distribution at four different radial positions along the z-axis in the
compressed area. The edge position corresponding to this time step resides at .r R⁄ 0.16=
50 000, 28 000,
65
Numerical Results
driven phenomena, which, when combined with the time scales that govern the
phenomena under investigation, support our assumption that the viscosity does
not play a critical role in the high speed impact problem. This is particularly true
for the bulk of the droplet liquid, but also for the initial stage of impact, up to the
jet eruption time range, which is the focus of this investigation.
66
Numerical Results
67
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6Analytical Modelling: ShockWave Formation, LateralLiquid Motion & MultipleWave Structure in theContact Line Region
The great tragedy of science
- The slaying of a beautiful hypothesis by an ugly fact.
-- Thomas H. Huxley (1825 - 1895)
So far we have performed a computational investigation the droplet impact
phenomena, however for the proper understanding of the mechanism of the crea-
tion and propagation of the shock wave, as well as the time scales and jetting ejec-
tion, a theoretical model is necessary. A somewhat limited models were developed
by Lesser [21] and Heymann [20], since they did not take into account an accurate
material speed of sound and compressibility. In the former, the attempts have been
made to elaborate an analytical model for the first phase of impact, where the
shock wave remains attached to the contact periphery. This would enable us to
predict the pressure exerted on the target, which is very important for the preven-
tion of damage and erosion control, as well as to address the question of time
scales which govern the high velocity impact phenomena. As we shall see below,
a closer look to the current models, uncovers also an intrinsic contradiction and
puts the difficult question of the physically acceptable solution of this anomaly.
We start our investigation with the geometrical consideration of the impact.
68
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6.1 Geometrical Considerations
The position and radial velocity of the contact line ( resp. Fig. 6.1) are
entirely geometrical features of the impact, thus can be obtained by considering
the impact plane ‘sweeping over’ the undisturbed droplet profile. In order to find
the co-ordinates of the contact line we proceed as follows (cf. Fig. 6.1): Since
and , we find . The x-
coordinate of contact line, is determined from the triangle , reads
(6.1)
A derivative with respect to time yields the radial component of contact line
velocity ,
Al U l
F0F1 Vt= CF0 R= CF1 CF0 F0F1– R Vt–= =
X l C AlF1
X l F1 Al 2RVt V 2t2–= =
C
R
t 0=
t 0>
z axes
r axes
compressed liquidshock front
liquid drop
F0
F1V
βAl
U l
Fig. 6.1. Impact of the upwards moving wall on the motionless spherical liquid droplet.
The zone of the highly compressed liquid (red) is bounded by the shock front and the
target surface.
t
U l
69
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.2)
By the Huygens principle, at each instant the expanding contact line will emit a
wavelet travelling with the shock speed , Fig. 6.2.
In previous work by Lesser [21] this shock velocity has been regarded as constant
and equal to the ambient speed of sound. As numerically shown in the previous
chapters, for the case of high velocity droplet impact, the initial velocity of the
individual wavelets emitted by the propagating contact line is significantly higher
than the ambient speed of sound and therefore must be treated as equation of state
dependent. Moreover, the initial shock velocity is not a constant, rather it
increases as the contact line propagates outwards.
We set time as the time of impact. The z-component of the fluid par-
ticle velocity in the compressed region adjacent to the propagating contact line, is
equal to the wall velocity, i.e. . Here is the angle between the shock
wave and the plane wall, as depicted in Fig. 6.1. The envelope of the shock front
at the contact line is constructed by the following consideration: The spherical
U l X l˙ V R Vt–( )
2RVt V 2t2–----------------------------------= =
s
shock envelope
contact line at t1
contact line trajectory
wall at t=0
t1>0
individual wavelets
Fig. 6.2. Geometrical construction of shock front as an envelope of individual wavelets
emitted by the expanding contact edge. Note the difference in the construction to the
acoustic model (Fig. 1.2), where the shock velocity was assumed constant (equal to the
speed of sound) with respect to the propagating contact line and not with respect to the
undisturbed bulk of liquid.
drop free surface
V
t 0=
u βcos V= β
70
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
shock front, emitted at the time instant , has travelled a distance up to the
time . During the same time interval, the contact line moves radially by
and vertically by . The trajectory of the contact line is shown by the
dashed line in Fig. 6.3.
Each point of the shock envelope boundary is determined by a tangent from the
new contact line position to the circular wavelet of radius . This tangent is
extended up to its intersection with the wall at the time . Employing similarity of
and , shown in Fig. 6.3, yields
(6.3)
It is convenient to introduce the velocity as
(6.4)
Now, Eq. (6.3) reduces to:
(6.5)
t sdt
t dt+
U ldt Vdt
U ldt dx
sdt
propagation trajectory of thecontact lineshock waveβ
uu⊥ V=
dl
z
rVdt{Al
B C
D
E
particle velocity triangle
Fig. 6.3. Geometrical construction of the shock profile attached to the contact line.
sdt
t
∆ECD ∆AlCB
U ldt dx+
sdt------------------------ Vdt( )2 dx2+
Vdt-----------------------------------=
a
a dx dt⁄=
a2 s2 V 2–( ) 2aU lV2– s2 U l
2–( )V 2+ 0=
71
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
After solving for , we obtain the usual two solutions of a quadratic equation.
Only the solution with the positive sign before the square root has a physical
meaning. It can be easily shown that the other solution (with the negative sign
before the square root) yields a physically unacceptable value in the limit
(the initial moment of impact, ). Based on these considerations,
(6.6)
Next, we employ the well known consequence of Euler equations, that the liquid
particle velocity (jump) is normal to the shock wave itself. The similarity of tri-
angle and the ‘particle velocity triangle’ (depicted in Fig. 6.3) yields
, (6.7)
where is the component of normal to the wall. The last relation can be
rearranged as
(6.8)
In Eq. (6.8), we used the condition valid at the wall, . Next, from the sim-
ilarity of the particle velocity triangle and , Fig. 6.3, follows
(6.9)
which can be rewritten as
(6.10)
The left-hand sides of Eqs. (6.8) and (6.10) are equal, thus
a
a 0<
U l ∞→ t 0→
a U lV2 Vs U l
2 V+2
s2–+ s2 V 2–( )⁄=
u
AlCB
uu⊥----- dl
dx------ dl
adt--------= =
u⊥ u
dldt----- a
uu⊥----- a
uV----= =
u⊥ V=
∆ECD
dlVdt---------
U ldt dx+
sdt------------------------=
dldt----- V
U l a+
s---------------=
72
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.11)
Solving for the particle velocity yields
(6.12)
Finally, after substitution of velocity from Eq. (6.6) into Eq. (6.12)
(6.13)
If solved for the contact line velocity, Equation (6.13) simplifies to
(6.14)
For a given value of the contact line velocity , Equation (6.14) contains two
unknown variables, namely and . An additional piece of information for the
relation of s and u is needed to decide which of the two roots in Equation (6.14) is
meaningful. This is the topic of the next section.
6.2 Shock Wave Propagation
The relation between the shock velocity and the jump in the liquid particle
velocity , termed principal Hugoniot, can be derived from the equation of state
while satisfying the Rankine-Hugoniot relations, as shown in Chapter 3. The
linear Hugoniot Equation is combined with Equation (6.14) to eliminate the par-
ticle velocity u:
auV---- V
U l a+
s---------------=
u
u s( ) V 2
s------ 1
U l
a------+
=
a
u s( ) V 2
s------
U l
a------ 1+ V s V 1 s2 V 2–
U l2
-----------------–+
V s 1 s2 V 2–
U l2
-----------------–+ ⁄= =
U l s( ) s u s( ) V 2–⋅
u s( )2 V 2–------------------------------±=
U l
s u
s
u
73
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
. (6.15)
The physically acceptable solution in our coordinate system is the one which
yields a positive contact line velocity. This is determined as follows: The total par-
ticle velocity is higher than the wall velocity,
(6.16)
From Equation (6.14) follows
(6.17)
By making use of Equation (6.17) we investigate the sign of the numerator in
Equation (6.15),
(6.18)
The last inequality holds because , which follows in a straightforward
manner from the above. Therefore, the physically acceptable solution for is
the one with the positive sign (plotted in Fig. 6.4),
(6.19)
U l
s s s0–( ) kV 2–
s s0–( )2 k2V 2–------------------------------------------±=
u
u V z urr+ V>=
s s0– ku kV>=
s s s0–( ) kV 2– kV s V–( ) 0> >
s u V> >
U l
U l
s s s0–( ) kV 2–
s s0–( )2 k2V 2–------------------------------------------=
74
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6.2.1 Radial Particle Velocity
In order to find the corresponding particle velocity, we eliminate from
Equation (6.19) by employing Eq. (3.15),
(6.20)
In terms of the radial component of the particle velocity,
(6.21)
The solution of Equation (6.21) for is shown in Fig. 6.5 [the one-to-one
mapping between and is given by Eq. (6.2)]. This theoretical result is in
agreement with computational results for the axisymmetric compressible Euler
equations.
5 7.5 10 12.5 15 17.5Ul (km/s)
3
3.5
4
4.5
5
s (km/s)
A (Ul =3.68, s=3.18)
Fig. 6.4. Shock velocity vs. contact line velocity for the linear Hugoniot.
s
U l
s0 ku+( )u V 2–
u2 V 2–--------------------------------------=
ur u2 V 2–=
U l kur s0 1Vur---- 2
+ k 1–( )V2
ur------++=
ur t( )
U l t
75
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6.2.2 Emergence of the Anomaly
Figure 6.4 shows that the upon impact (far right) the contact line velocity
decreases rapidly from a theoretically infinite value at whereas the shock
velocity remains rather constant. Later (e.g. below ), the shock
velocity starts to grow, due to the development of lateral flow. However, after
point A, where the tangent to the curve is parallel to the s-axis, the contact line
velocity starts to increase again. This is a physically unacceptable situation and the
solution branch above point A must be rejected. We conclude that there is a time
after which no physical solution based on the assumed physics of a single shock
wave attached to the contact line exists. The time corresponding to the point A will
be termed as the ‘time of shock degeneration’, .
The jetting eruption in the contact line region, if occurred before , would
resolve this anomaly. To address this issue, a closer look to the shock and contact
line velocity corresponding to is needed. The maximum shock wave speed of
the limit point A in Fig. 6.4 can be calculated from the condition
0.5 1 1.5 2 2.5 3t
100
200
300
400
ur
present theorynumerical results3
(ns)
(m/s)
Fig. 6.5. Prediction of radial particle velocity and comparison with computational results.
t 0=
U l 7.5km s⁄=
tdeg
tdeg
tdeg
76
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.22)
The solution for yields the maximum value for the shock velocity, (6.23)
, (6.24)
where the parameters and are defined as:
and (6.25)
For the case of a linear fit for water, this amounts to . At the
same time, the contact line velocity has decreased to (cor-
responding to the time ns). Obviously, , hence, the jetting
cannot be initiated at this time.
The interesting issue arising at this point is what happens to the shock enve-
lope evolution in the time interval between and (for the numerical exam-
ple of water used here, this is the interval [1.82 ns, 2.80 ns]). Due to the above
mentioned anomaly, the assumed single shock wave structure appears not to cap-
ture correctly the physics. We postulate the appearance of a double shock wave
structure in this time interval, outlined in Fig. 6.7, which will remove the physi-
cally unacceptable portion of the earlier solution and lead to lateral jetting.
6.3 Resolution of the Anomaly
Additional insight about this anomaly are obtained by consideration of the
process of solution construction in the contact line region, Fig. 6.6: The real flow
state is obtained as the intersection of the ‘edge boundary condition‘(hence-
s∂∂U l 0=
s
smax s0 V2k 1–
3β---------------α1 3⁄ βk
α1 3⁄-----------+ +=
α β
α 2kVs0
----------= β 1 1 α2 2k 1–3k
--------------- 3
–+
1 3⁄
=
smax 3.184km s⁄=
U l min, 3.678km s⁄=
tdeg 1.82= U l min, s>
tdeg t jet
u s,( )
77
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
forth referred to as ‘edge bc’) curves [Eq. (6.14)] and the linear Hugoniot [Eq.
(3.15)].
All edge bc curves originate from the same point ( ), and rise sharply to
a plateau value. The rise is less sharp as the time increases. At each time instance,
the linear Hugoniot and corresponding ‘edge bc’ curve intersect at 2 points. How-
ever only the left point is physically acceptable, based on the following consider-
ation: At the time the curves intersect at (point in Fig. 6.6) and
. The first solution is the one we expect (no lateral flow, thus liquid veloc-
ity equal to the wall velocity). The solution is obviously physically not
acceptable and must be rejected. Since the curve must be continuous
(no instantaneous infinite acceleration of particles), all physically allowed solu-
tion will travel from to , Fig. 6.6.
Apparently, beyond ns (marked with point in Fig. 6.6), no
intersection exists. This is a different manifestation of the anomaly mentioned ear-
lier. To explore this anomaly, we have to rethink the construction of the edge bc
curves, since the linear Hugoniot has an overall validity (playing here the role of
0.5 1 1.5 2u (km/s)
1234567
s (km/s)
t = 0 t = 0.50 ns
t = 0.90 ns
t = 4.00 ns
t = 1.82 ns
P Q
Fig. 6.6. Construction of the solution: Intersection of edge boundary curves (each
corresponding to a different time) and equation of state (straight line). No solution exists
beyond ns.t 1.82=
Eq. (3.15)
V 0,
t 0= u 0.5= P
u ∞→
u ∞→
u u t( )=
P Q
t 1.82= Q
78
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
an equation of state). It is logical to assume that a somewhat more complex wave
structure occurs and investigate its possible effect on the edge boundary curves.
Here, we consider a double wave structure, where the outer wave is assumed to be
a shock wave, as outlined below in Fig. 6.7.
The liquid particle velocity in the region between the waves is still normal to the
outer shock wave, however the difference to the previous model [see Eq. (6.8)] is
that it has a normal component smaller than the wall velocity . Therefore we
rewrite equation (6.8) as
, (6.26)
where we defined the factor . (6.27)
Implementing this concept in the solution process, equation (6.12) reads
(6.28)
Combining with Eq. (6.6) to eliminate yields
wave I
wave II
u⊥ V=
u⊥ V<
area of quiet liquid,
u 0=
Fig. 6.7. Schematic of a double wave structure in a contact line region.
u⊥ V
dldt----- a
uu⊥----- a
uλV-------= =
λ u⊥ V⁄ 1≤=
u s( ) λV 2
s------ 1
U l
a------+
=
a
79
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.29)
Solving for the contact line velocity and recalling , the last equation reads
(6.30)
The influence of the factor on the edge bc curves can be easily seen in Eq.
(6.29). For the same value of and (corresponding to the time ), the liquid
velocity will decrease with increasing . The curves in Fig. 6.8 clearly demon-
strate the fact that in this scenario the domain of physically acceptable solutions is
extended. For our example of a water droplet, the value of extends the
range of acceptable solution up to ns. This value coincides with the jet-
ting time (for which ), thus removing the anomaly mentioned ear-
lier.
u s( ) λV s V 1 s2 V 2–
U l2
-----------------–+
V s 1 s2 V 2–
U l2
-----------------–+ ⁄=
U l s( )s s s0–( ) λkV 2–⋅
k u s( )2 λ2V2
–--------------------------------------------=
λ 1<
s U l t
u λ
λ 0.65=
t 2.80=
U l2 V 2+ s=
0.2 0.4 0.6 0.8 1 1.2 1.4
1
2
3
4
5
6
t = 2.80 ns
s (km/s)
u (km/s)
λ0.
65=
λ1
=
Fig. 6.8. Effect of the factor on the edge bc curves allowing the existence of solution up
to the jetting time (here ns).
λ
1.82
80
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Before closing this section, it is worth mentioning that a multiple wave struc-
ture (instead of only double) is also possible, since we have not made any assump-
tion on the inner wave structure, which can be composed of different waves.
Experiments (obviously very difficult to carry out) could define the exact wave
structure present. Nevertheless, the proposed mechanism appears to offer a good
explanation for the anomaly resulting from the single shock wave structure.
An equivalent argumentation could be also applied on the curve shown in
Fig. 6.4, which would shift the point A together with the entire curve and extend
the solution domain. However, the disadvantage of this approach would be that a
specific function (loss in generality), needs to be assumed.
6.3.1 Numerical Confirmation
The proposed multiple wave structure, which allows for the analytical treat-
ment of the anomaly, was also numerically confirmed with the computational
methodology outlined above. The performed computations show the presence of
the single shock wave up to the time of shock degeneration and subsequent grad-
ual formation of a more complex wave structure at the contact line region.
Figure 6.9 shows the two-dimensional wave structure of the compressed
region, together with the computational mesh. The wave fronts are contained
within two lines. The wave close to the z-axis shows obviously a sharp shock
structure, whereas the wave in the vicinity of the contact line region exhibits a
‘split’ character (note the difference between the wave at the contact line region
and far from it).
A comment is also worthwhile regarding experimental confirmation of the
mentioned multiple wave structure. To this end, the shock structure discussed in
this work is out of the resolution capabilities of experimental techniques. Moreo-
ver, typical droplets used in experiments, Lesser & Field [22], Field, Dear &
λ λ t( )=
81
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Ogren [26] and Lesser [46], are much bigger (droplet radii were 1-10 mm, mean-
ing 10-100 times bigger that in our simulation), which means also that the jetting
times are much higher [since the contact line velocity as a function of time is also
higher than for the smaller droplets, see Haller et al., 2002]. Thus, a direct com-
parison with the configurations examined in our work is not easy to make.
The formation of the degenerated wave structure in terms of a numerically
obtained pressure plot in the radial direction at the contact line area is examined
in Fig. 6.10, showing clearly the break up of the single shock wave approximately
after ns (time of shock degeneration).
r / R
z/R
0.05 0.10 0.15 0.20
0.02
0.04
0.06
0.08
0.10
0.12
Fig. 6.9. Wave structure of the compressed region together with the computational grid.
t 1.5=
82
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
The plots in Fig. 6.11 show the velocity field together with equidistant density
lines, confirming the assumption of lower velocity in the intermediate region,
. The narrow packing of equidistant density lines indicate that the outer
wave 2 is a shock wave (localised jump), with respect to which the particle veloc-
r / R0.05 0.10 0.15 0.20
-200.0
-100.0
0.0
0.703
1.134
1.542
2.282
2.939
time (ns)r / R
z/R
0.1 0.2
0.02
0.04
0.06
0.08
(b)
r / R
P(G
Pa)
0.05 0.10 0.15 0.20
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.80.703
1.134
1.542
1.924
2.282
2.619
2.939
3.274
3.573
3.831
4.003
time (ns)
(a)
Fig. 6.10. (a) Pressure wave profiles at the contact line in the radial direction for different
times. Evidence of the single wave degeneration after 1.5 ns. (b) The pressure plot shown
in (a) was sampled along the dashed arrow-line. (c) Pressure derivative dp/dr, showing
clearly the gradual emergence of two negative peaks, indicating the split of the single
shock wave.
(c)
λ 1<
83
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
ity field is apparently normal. The factor , (6.27), with the numerically deter-
mined values in the region at the downstream side of the shock wave was found
to be 0.7.
6.4 Construction of the Shock Envelope
Based on the above constructed solution for the shock velocity, we develop
an analytical representation of the shock envelope. The latter will be used to val-
idate the shock-velocity model against numerical results, since the shock velocity
in itself cannot be directly obtained from computations. The predicted shock
velocity is evidently higher than the speed of sound, thus, it is also expected that
the shock envelope will substantially differ from the corresponding envelope in
the acoustic limit, developed by Lesser [21]. We shall investigate the extent that
these two models differ and their agreement with computational results.
λ
r / R
z/R
0.145 0.150 0.155 0.160 0.1650.000
0.005
0.010
0.015
density
1325
1270
1210
1150
1090
1030
1000
wave 2
wave 1
reference velocity vector500 m/s
Fig. 6.11. Shock structure in the vicinity of the contact line region together with
equidistant iso-density lines. The velocity filed is shown by velocity vectors, apparently
normal to the outer shock wave.
84
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6.4.1 General Considerations
The procedure presented is a generalization of the acoustic approach [21]
and regards the shock velocity as a solution of the Euler equations, therefore, it
can be used with an arbitrary equation of state. The coordinate system is shown in
Fig. 6.12. The time coordinate is set to at the instant of impact. In our ref-
erence frame, the impact of a rigid wall with a perfect sphere of motionless liquid
is investigated.
The contact line emits at a circular wavelet spreading with the initial veloc-
ity , (Fig. 6.12). The radius of the circular wave front at time is given as
(6.31)
Let the coordinates of the spherical wave front at time be and those of the
contact line at the time , when the wave was emitted, . The equation
of wavelets at time t in the plane will be
(6.32)
t 0=
wave envelope at time t
wave envelope
circular wavelet
X l τ( )
(r, z)
Vt
d(t,τ
)
emitted at t
t=0
droplet free surface
Fig. 6.12. Envelope construction: Wall position at the time shown by a dashed line.
Contact line propagates along the droplet free surface.
τ
at time t
τ 0≥
s τ( ) t
d t τ,( ) s ν( ) νdτ
t
∫=
t r z,( )
τ X l τ( ) Vτ,( )
r z–
Φ r z τ, ,( ) r t( ) X l τ( )–( )2= z t( ) Vτ–( )2 d2 t τ,( )–+ 0=
85
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
We proceed with the construction of the wave front at specific time , ,
which we treat as a constant hereafter. To construct the envelope of the emitted
wavelets [Eq. (6.32)], we project the surface onto the plane
[represented by the vector ] and require
(6.33)
Insertion of Eq. (6.33) into Eq. (6.32) yields
(6.34)
From Eq. (6.31) follows,
(6.35)
The physically meaningful solution of the system of equations Eq. (6.33) and Eq.
(6.34), accounting for Eq. (6.35) is given in parametric form as
(6.36)
(6.37)
The exact envelope functions [Eqs. (6.36) and (6.37)] can be simplified by making
use of the fact that during the first phase of the impact, the impact velocity is
much smaller than the contact line velocity ,
(6.38)
After implementation of Eq. (6.38) into Eqs. (6.36) and (6.37),
t t τ≥
Φ r z τ, ,( ) 0= r z–
0 0 1, ,( )
∇Φ r z τ, ,( ) 0 0 1, ,( )•τ∂∂ Φ r z τ, ,( ) 0= =
X l τ( ) r X l τ( )–( ) V z Vτ–( ) d τ( )d τ( )+ + 0=
d τ( ) s τ( )–=
r τ( ) X l d τ( ) d τ( ) X l τ( ) V– X l2 τ( ) V 2 s2 τ( )–+
X l2 τ( ) V 2+
---------------------------------------------------------------------------------------+=
z τ( ) Vτ d τ( ) d– τ( )V X l τ( ) X l2 τ( ) V 2 s2 τ( )–++
X l2 τ( ) V 2+
------------------------------------------------------------------------------------------+=
V
U l X l=
V X l⁄ 1«
86
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.39)
(6.40)
Equations (6.39) and (6.40) are the parametric representation ,
of the shock envelope for small times .
Up to this point, the approach is general and special solutions will depend on
the function . The acoustic model is a special case of the system (6.39)-(6.40)
in the limit .
6.4.2 Results & Model Validation
The exact function can be obtained by Eq. (6.19). However, since this
function is fairly linear in the first impact phase, we approximate it with
(6.41)
The value represents the initial shock velocity. Both and the coefficient
can be obtained by the linearization of (6.19).
The radius of the singular wavelet emitted at time , Eq. (6.31), reads
now
(6.42)
r τ( ) X l τ( ) d– τ( ) s τ( )U l τ( )--------------≈
z τ( ) Vτ d τ( ) 1 s2 τ( )U l τ( )--------------––≈
r r τ( )=
z z τ( )= t
s τ( )
s τ( ) c=
s τ( )
s τ( ) s0 ετ+=
s0 s0 ε
d τ( ) τ
d τ( ) s0 εν+( ) νdτ
t
∫ s0 t τ–( ) ε2--- t τ–( )2+= =
87
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Equations (6.41) and (6.42), when substituted in the system of equations (6.39)-
(6.40) yield the desired shock envelope in parametric representation. The compar-
ison of the wave envelopes is shown in Fig. 6.13. The numerically captured shock
position is contained within two thick dashed lines.
For the impact of a water droplet of in diameter with the velocity
, the linearization factor in Eq. (6.41) is
. In the vicinity of the contact line region, the
developed envelope matches well the computational results. The acoustic model,
depicted by a thin dashed line, underpredicts the numerical findings, due to the
underestimated envelope velocity in that model. Far from the contact line (near the
z-axis), the computational envelope runs slightly below the position predicted by
our model. This can be attributed to the temporal decay of the shock velocity,
which is not included in the current model. Close to the contact line, this decay is
negligible.
0.02 0.04 0.06 0.08 0.1 0.12r/R
0.01
0.02
0.03
0.04z/R
numerically determinedcurrent modelacoustic model
shock position:
Fig. 6.13. Envelope construction: Comparison of analytical results with the computational
findings.
200µm
V 500m s⁄=
ε 150m s⁄ns
---------- 1.5 1011ms 2–⋅= =
88
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6.5 Analytical Solution of the WaveStructure in the Contact Line Region
As demonstrated in previous chapters, the anomaly emerging at the time of
‘shock degeneration’ can be removed by the proposition of a multiple wave struc-
ture at the contact line region. Here, we are showing analytically that for one-
dimensional simplification of the problem, the multiple wave structure is a con-
sistent and acceptable hypothesis, during the first stages of droplet impact. For this
purpose, the exact Riemann problem solution is constructed and validated, taking
into account the appropriate equation of state. This is possible only if the eigen-
structure of the Jacobian matrix, which is encountered in the formulation of the
Riemann problem, can be explicitly resolved. This procedure, possible for the
ideal gas equation of state, usually fails for more complex (and more realistic)
equations of state. We are demonstrating that the utilization of the stiffened gas
equation of state, which describes fairly well most real materials (like liquid
metals etc.), allows for the solution of the eigenstructure and leads to an analytical
solution of the problem. To this end, the exact one-dimensional Riemann problem
solution for the stiffened gas equation of state is presented, under the assumption
of an isentropic flow in the smooth flow region (i.e. in the region with no discon-
tinuities, see Appendix).
Wave Structure at the Contact Line Region
All analytical treatments of the liquid droplet impact presented in literature
so far have been based on the assumption that the two fluid regions (compressed
and uncompressed) are connected by a single shock wave. As shown in previous
chapter, a scenario allowing for a more complex wave structure in the contact line
region alleviates the above-mentioned anomaly. Along these lines, we study the
exact Riemann problem and its solution across the wave. To answer the question
89
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
if the multiple wave structure can occur, it is sufficient to consider the flow in the
(arbitrary) small region across the front surface, which, as numerically confirmed,
indeed exhibits one-dimensional structure. In the next section, the basic structure
of the Euler equations is outlined together with the detailed study of the elemen-
tary waves. We provide an algorithm for solution determination when the left and
right states are known (e.g. shock tube problem). The background on isentropic
flow employed is provided in the Appendix.
6.5.1 One-dimensional Euler Equations
We cast the one-dimensional Euler equations in conservative formulation
and in flow variables form:
(6.43)
(6.44)
where is the flux vector, given by
(6.45)
We set , (6.46)
where the coefficient is the Jacobian matrix
. (6.47)
U
ρρu
E
=
U t x∂∂ F U( )+ 0=
F U( )
F U( )ρu
ρu2 p+
u E p+( )
=
x∂∂ F U( ) A U( )U x=
A U( )
A U( )U∂∂ F U( )=
90
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Equation (6.44) can be written in the quasi-linear form,
(6.48)
In order to find the Jacobian according to the Eq. (6.47), the flux
needs to be expressed in terms of conserved variables [Eq.
(6.43)]. The total energy, for the case of stiffened gas equation of state [see
Appendix, Eq. (A 14)], reads
(6.49)
or in terms of conserved variables
(6.50)
Next, we express the pressure from Eq. (6.49) as
, (6.51)
or in components of the vector
(6.52)
In terms of conserved variables, the flux [Eq. (6.45)] takes now the form
(6.53)
U t A U( )U x+ 0=
A U( ) F U( )
U u1 u2 u3, ,( )=
E ρu2
2-----
p Γ 1+( )P∞+
Γ-----------------------------------+=
u3
u22
2u1--------
p Γ 1+( )P∞+
Γ-----------------------------------+=
p Γ E ρu2
2-----–
Γ 1+( )– P∞=
U
p U( ) Γ u3
u22
2u1--------–
Γ 1+( )– P∞=
F
F U( )
u2
Γu3
u22
u1----- 1 Γ
2---–
Γ 1+( )– P∞+
Γ 1+( )u2
u1----- u3 P∞–( ) Γ
u23
2u12
--------–
=
91
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Differencing Eq. (6.53) with respect to yields the needed Jacobian,
(6.54)
Next, we transform back to the flow variables
(6.55)
It is convenient here to express the Jacobian in terms of the free enthalpy ,
see Appendix A, Eq. (A 16) as:
(6.56)
This is a general expression for the Jacobian . Introducing the isentropic
assumption and by using the expression for the speed of sound [Appendix, Eq. (A
18) in the Appendix], the eigenvalues of matrix are found as
, , , (6.57)
with the corresponding right eigenvectors
U
A U( )
0 1 0
u22
u12
-----–Γ2---
u22
u12
-----
+u2
u1----- 2 Γ–( ) Γ
Γ 1+( )–u2
u12
----- u3 P∞–( ) Γu2
3
u13
-----+ Γ 1+( ) 1u1----- u3 P∞–( ) Γ
3u22
2u12
--------– Γ 1+( )u2
u1-----
=
A U( )
0 1 0
u2
2----- Γ 2–( ) u 2 Γ–( ) Γ
u Γu2 E P∞–
ρ----------------- Γ 1+( )– Γ 1+( )
E P∞–
ρ----------------- 3Γ
2-------u2– Γ 1+( )u
=
A H
A U( )
0 1 0
u2
2----- Γ 2–( ) u 2 Γ–( ) Γ
u Γu2
2----- H–
H Γu2– Γ 1+( )u
=
A
A U( )
λ1 u c–= λ2 u= λ3 u c+=
92
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
, , (6.58)
As can be seen, it is possible to diagonalise the matrix if isentropic flow in
the region is assumed. This assumption approximately holds in the smooth flow
region (see Appendix).
6.5.2 The Exact Solution of the Riemann Problem
We consider two known states, left and right , connected by the ele-
mentary waves, i.e. either shock or expansion waves on both sides and the contact
discontinuity in the middle. The entire wave structure is presented in Fig. 6.14.
We define the velocity jump function between two states (from the left to the right:
) as , where the and represent the particle veloc-
ities on - resp. -side of the single wave structure, Fig. 6.14. It is clear that the
sum of velocity jumps across individual waves is equal to the total velocity differ-
ence,
(6.59)
ξ1
1
u c–
H cu–
= ξ2
1
u
u2 2⁄
= ξ3
1
u c+
H cu+
=
A U( )
U l U r
steady liquid, u 0=
right wave
middle waveleft wave
U r
compressed liquid
U rU l
U l
Fig. 6.14. Solution of the one-dimensional Riemann problem.
l l r r, , , Ψ a b,( ) ua ub–= ua ub
a b
Ψ l r,( ) ul ur– Ψ l l,( ) Ψ l r,( ) Ψ r r,( )+ += =
93
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Our goal is to express the velocity differences , and as a
function of the pressure in the middle region and the corresponding left and. right
states (the pressures and particle velocities on both sides of the contact disconti-
nuity are equal). By doing so and knowing the left and right states, Eq. (6.59) will
contain only one variable - the pressure in the middle region . Based on this
consideration, the existence of solutions admissible by the left and right state can
be investigated. For this purpose, two cases need to be studied: Expansion wave
and shock wave (the case of contact discontinuity as the middle wave in Fig. 6.14
has a trivial solution ).
6.5.3 Expansion Fan
For the expansion wave, we utilize the eigenstructure derived in the isentro-
pic approximation. Recalling Eqs. (6.43) and (6.58), the generalized Riemann
Invariants across the expansion wave will read
(6.60)
The index corresponds to the particular wave, i.e. left, middle or right
one. By taking into account the eigenstructure, Eqs. (6.57)-(6.58), we find for the
wave associated with
(6.61)
The left-hand side of Eq. (6.61) equation yields
(6.62)
To solve for , one needs to evaluate the integral
Ψ l l,( ) Ψ l r,( ) Ψ r r,( )
p*
Ψ l r,( ) 0=
du1
ξ1i( )--------
du2
ξ2i( )--------
du3
ξ3i( )--------= =
i 1 2 3, ,=
λ1
dρ1
------ d ρu( )u c–
--------------- dEH cu–-----------------= =
du c u ρ,( )dρρ------+ 0=
u
94
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.63)
By employing the expression for the sound velocity along the isentropic path
[Appendix, Eq. (A 11)], the Riemann invariant given by Eq. (6.63) reads
(6.64)
Integration yields
(6.65)
or by virtue of Eq. (A 11)
along the wave. (6.66)
Similarly one obtains along the rarefaction wave [By
applying Eq. (6.60) one can show that wave is a contact discontinuity wave].
Hence, for two states connected by left resp. right rarefaction wave holds
, (6.67)
where the sign plus is valid for the left rarefaction and minus for the right. Com-
bination of Eqs. (6.67) and Eq. (A 12) yields the velocity jump across the
expansion wave connecting and ,
(6.68)
Here, the sign plus holds for the left rarefaction and minus for the right.
u c u ρ,( )dρρ------∫+ constant=
c
u C ρΓ2--- 1–
dρ∫+ constant=
u2 CΓ
-----------ρΓ 2⁄+ constant=
u 2cΓ------+ constant= λ1
u 2c Γ⁄– constant= λ3
λ2
ua2Γ---± ca ub
2Γ---cb±=
Ψ a b,( )
a b
Ψ a b,( ) ua ub–2Γ---+− ca cb–( ) 2
Γ---+− ca 1
pb P+ ∞
pa P+ ∞-------------------
Γ2 Γ 1+( )---------------------
–= = =
95
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
6.5.4 Shock Wave
In order to investigate the shock structure, we must modify our approach to
allow for a sharp entropy change across the shock wave. Thus, by considering the
equation of state, we are not allowed to use its isentropic approximation as in the
previous case. To this end, the general formulation of stiffened gas equation of
state is combined with conservation laws across the shock.
The Rankine-Hugoniot conditions in an arbitrary reference frame for both
shock sides and read
(6.69)
For simplicity, we consider first the Rankine Hugoniot equations in the frame
where the shock wave velocity equals zero,
(6.70)
(6.71)
(6.72)
The transition to this frame is accomplished by transformation
for . (6.73)
Based on Eq. (6.70), we define
. (6.74)
The energy conservation law Eq. (6.72) for the stiffened gas equation of state
reads
a b
ρaua ρbub–
ρaua2 pa ρbub
2 pb––+
ua Ea pa+( ) ub Eb pb+( )–
s
ρa ρb–
ρaua ρbub–
Ea Eb–
=
s
ρaua ρbub– 0=
ρaua2 pa ρbub
2 pb––+ 0=
ua Ea pa+( ) ub Eb pb+( )– 0=
ui ui s–= i a b,=
q ρaua ρbub= =
96
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.75)
To simplify the analysis we employ the free enthalpy [see Appendix, Eq. (A 4)]
for (6.76)
The Eq. (6.75) can now be written as
(6.77)
Next, we combine Eqs. (6.70) and (6.71). After some manipulation,
(6.78)
also,
(6.79)
substitution into Eq. (6.77) yields
(6.80)
The enthalpies and can be expressed by and , according to Eq. (6.76).
After some algebraic manipulations,
(6.81)
Now we substitute the expression for the internal energy for the stiffened gas
equation of state [Appendix, Eq. (A 4)] and obtain
ua2
2-------
Γ 1+Γ
-------------pa P+ ∞
ρa-------------------+
ub
2
2-------
Γ 1+Γ
-------------pb P+ ∞
ρb-------------------+
=
h
hi ei
pi
ρi-----+
Γ 1+Γ
-------------pi P+ ∞
ρi------------------= = i a b,=
ha hb–ub
2
2-------
ua2
2-------–=
ua2 ρb
ρa----- pa pb–
ρa ρb–------------------ =
ub2 ρa
ρb----- pa pb–
ρa ρb–------------------ =
ha hb–12---ρa ρb+
ρaρb
----------------- pa pb–( )=
ha hb ea eb
ea eb–12---ρa ρb–
ρaρb
----------------- pa pb+( )=
e
97
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.82)
From Eqs. (6.74) and (6.71) follows
(6.83)
Since [see Eq. (6.73)], we obtain
(6.84)
This relation refers to the default frame of reference. From Eq. (6.84) we find the
velocity jump as
(6.85)
The next step is to find in Eq. (6.85) as a function of pressures and , and
known density (or ). To this end, we rewrite the Eq. (6.74) as , for
and insert it into Eq. .
This yields (6.86)
(6.87)
Using Eq. (6.82) we find
(6.88)
thus from Eq. (6.87),
(6.89)
ρa
ρb-----
Γ 2+( ) pa Γpb+ 2 Γ 1+( )P∞+
Γ pa Γ 2+( ) pb+ 2 Γ 1+( )P∞+---------------------------------------------------------------------------=
qpa pb–
ua ub–------------------–=
ua ub– ua ub–=
qpa pb–
ua ub–------------------–=
Ψ a b,( ) ua ub–pa pb–
q------------------–= =
q pa pb
ρa ρb uiqρi----=
i a b,=
q2 pa pb–
1ρa----- 1
ρb-----–
------------------–pa pb–
ρa
ρb----- 1–
------------------ρa= =
ρa
ρb----- 1– 2
pa pb–
Γ pa Γ 2+( ) pb+ 2 Γ 1+( )P∞+---------------------------------------------------------------------------=
q2 pa pb–( )ρa
Γ pa Γ 2+( ) pb+ 2 Γ 1+( )P∞+
2 pa pb–( )---------------------------------------------------------------------------=
98
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
(6.90)
Finally, by combining Eqs. (6.85) and (6.90) we obtain the velocity jump function
across the shock wave,
(6.91)
This form is suitable when the density on the -shock side is known. We can use
an equivalent formulation when is known,
(6.92)
6.5.5 Solution Process
Equations (6.68) and (6.91) [resp Eq. (6.92)] enable us to solve the general
one-dimensional Riemann problem. Equation (6.68) can be written as
(6.93)
containing only one unknown, namely . The variables and in Eq. (6.93)
indicate the known left and right states, respectively [In writing Eq. (6.93) it was
taken into account that and ]. Without loss of generality, we
assume that the left side is compressed, .
We proceed as follows: A root of Eq. (6.93) for each of four cases [each
in Eq. (6.93) can be either expansion or a shock wave] is found and checked
for consistency with the assumed waves. For instance, if the left expansion wave
and right shock wave are assumed, then it must hold . For two expan-
q2 ρaΓ2--- pa
Γ2--- 1+ pb+ Γ 1+( )P∞+=
Ψ a b,( ) ub ub–pa pb–
ρaΓ2--- pa
Γ2--- 1+ pb+ Γ 1+( )P∞+
-----------------------------------------------------------------------------------------= =
a
ρb
Ψ a b,( ) ua ub–pa pb–
ρbΓ2--- pb
Γ2--- 1+ pa+ Γ 1+( )P∞+
-----------------------------------------------------------------------------------------= =
Ψ l p*,( ) Ψ p* r,( )+ ul=
p* l r
Ψ l r,( ) 0= ur 0=
pl pr>
p*
Ψ
pl p* pr> >
99
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
sion waves, the root of Eq. (6.93) needs to satisfy and , and for
two shock waves and .
Since in our case the left state is the highly compressed liquid, , the
a priori possible solutions scenarios reduce to:
case 1: left expansion and right shock wave, outlined in Fig. 6.15
case 2: two shock waves
case 3: two expansion waves
(the case of right expansion fan and left shock wave is due to not admis-
sible)
The solution procedure of Eq. (6.93) for these three cases is as follow. The numer-
ical values for the liquid particle velocity, pressure and density in the compressed
region, used as an example in Fig. 6.16, are , and
.
p* pl< p* pr<
p* pl> p* pr>
pr pl»
pl p* pr> >
p* pr pl> >
pr pl p*> >
pl pr>
motionless liquid, u 0=
shock wave
contactrarefaction
U r
shocked liquid
U rU l
U l
fandiscontinuity
Fig. 6.15. Case 1: Assumption of the left expansion and right shock wave
ur 600m s⁄= pl 1.65GPa=
ρl 1.2g cm3⁄=
100
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
0.5 1 1.5 2 2.5 3 3.5p* (GPa)
200
400
600
800
∆u (m/s)
right expansionleft expansion
actual velocity jump
superposition wave
0.5 1 1.5 2 2.5 3 3.5p* (GPa)
200
400
600
800
∆u (m/s)
right shockleft shock
actual velocity jump
superposition wave
0.5 1 1.5 2 2.5 3 3.5p* (GPa)
200
400
600
800
∆u (m/s)
right shockleft expansion
actual velocity jump
superposition wave
(a)
(b)
(c)
I
Fig. 6.16. Possible solution scenarios, (a), (b), (c) correspondent to the cases 1, 2, and 3,
respectively. Here, only the case 1 yields the physically acceptable solution.
101
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
The solution procedure of Eq. (6.93) for in the above three cases is as follows:
Employing Eq. (6.68), (6.91) or (6.92) as appropriate, the velocity jumps across
the left and right waves, and respectively, are plotted in Fig.
6.16 vs. the pressure in the middle region, . Adding these two jumps yields the
left hand side of Eq. (6.93). The sought solution is the intersection of this super-
position (also plotted in Fig. 6.16) with the liquid particle velocity jump, , the
right hand side of Eq. (6.93).
In the case of Fig. 6.16(a), the solution is . Obviously,
, ( is the atmospheric pressure) which agrees with our assumption.
The liquid particle velocity in the intermediate region, corresponds to point I
in Fig. 6.16(a). As shown, , proving the hypothesis made in a previous
chapter, that the liquid particle velocity in the intermediate region is lower than
the velocity adjacent to the wall.
The solutions for the pressure in the intermediate region for cases 2 and 3,
shown in Fig. 6.16(b) and (c), obviously contradict the respective assumptions
made above. Hence, for a single pair of ( , ) a unique solution of the Riemann
problem exists.
From the physical standpoint, cases 2 and 3 do not seem realistic in the drop-
let impact scenario where the compressed area expands, when the energy consid-
erations are taken into account (the liquid takes on the state with the lowest
internal energy). In case 2 a highly compressed layer (of pressure ) would
spread from the shock envelope, compressing the pressurized liquid even more, in
case 3 an expansion wave on the left would mean that the pressure in the expand-
ing middle area is even lower than the atmospheric pressure, which would inhibit
the compressed area from expanding. None of these cases were numerically
observed.
p*
Ψ l p*,( ) Ψ p* r,( )
p*
ur
p* 1.0GPa=
pl p* pr> > pr
u*
u* ur<
pl ρl
p* pl>
102
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
Next, we explore the range of admissible states on the left, ( , ), which can be
connected to the right ambient state of undisturbed liquid in the only acceptable
case 1. Equation (6.93) can be resolved for different pairs ( , ) for each
velocity . The range of possible solutions is shown in Fig. 6.17, corresponding
to the right liquid particle velocity . Different solutions are
obtained depending on the left state . Fig. 6.17(a) shows the solution for
the case with smallest density and pressure for which the solution still exists as a
single expansion wave (point A). The effect of an increase in pressure by constant
density is shown in Fig. 6.17 (c) - the entire superposition curve shifts upwards.
For pressures in the interval the solution will consist of the left
expansion and right shock wave. If we increase the density at constant
0.5 1 1.5 2 2.5p* (GPa)
200
400
600
800
∆u (m/s)
right shockleft expansion
actual velocity jump
superposition wave
0.5 1 1.5 2 2.5p* (GPa)
200
400
600
800
∆u (m/s)
0.5 1 1.5 2 2.5
200
400
600
800
∆u (m/s)
(c) (d)
0.5 1 1.5 2 2.5
200
400
600
800
∆u (m/s)
(a) (b)
pl 1.39GPa=
ρl 1000kg m2⁄=
pl 1.39GPa=
ρl 1300kg m2⁄=
pl 1.72GPa=
ρl 1000kg m2⁄=
pl 1.72GPa=
ρl 1300kg m2⁄=
Fig. 6.17. The range of possible solutions for density and pressure on the left side in the
Riemann problem. Pressure (y-axis) in GPa.
pl ρl
pl ρl
ur
ur 600m s⁄=
pl ρl,( )
1.39 1.72,[ ]GPa
ρl
103
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
pressure, Fig. 6.17(a) → 6.17 (b) [also Fig. 6.17(c) → 6.17(d)], the point A moves
downwards. At some point, the upper limit for the density is reached, beyond
which no solution exists [as in Fig. 6.17(c)].
The range of possible left pressure states versus the liquid particle veloc-
ity for case 1, is shown in Fig. 6.18. The upper solution point, marked with a rhom-
bus, corresponds exactly to the single shock wave [see Fig. 6.17 (b) and 6.17(d)].
However, this solution is not unique for the given velocity. There exists a double
wave region, where the two liquid areas are connected by the expansion fan, con-
tact wave and shock wave. At the lower end, the regions are connected only by a
rarefaction fan. The pressure in the case of multiple wave structure is reduced
compared to the single shock wave structure.
pl
500 600 700 800u (m/s)
1
1.5
2
2.5
3
3.5
pr (GPa)
single shock wave
} double wavesingle expansion wave
Fig. 6.18. Admissible range of the pressure at the contact edge versus the total liquid
particle velocity jump across the wave.
104
Analytical Modelling: Shock Wave Formation, Lateral Liquid Motion & Multiple Wave Structure
105
Conclusions
7Conclusions
In science one tries to tell people, in such a way as to
be understood by everyone, something that no one ever
knew before. But in poetry, it’s the exact opposite.
-- Paul Dirac (1902 - 1984)
The high velocity impact of a liquid droplet on a rigid target has been inves-
tigated, both computationally and analytically. Theoretical models accounting for
the complex physics at the contact line region prior to the eruption of lateral jetting
were also explored. It was shown that compressible flow patterns dominate the
droplet evolution and splashing at very early times, as underpinned by phenomena
such as the creation, propagation and interaction of shock and expansion waves.
The time of onset and the magnitude of jetting have been successfully deter-
mined and compared to theoretical results. Various assumptions adopted by ear-
lier theoretical models which could be responsible for the overprediction of the
jetting times by these models have been critically discussed. A novel procedure of
jetting time determination was introduced, which makes it possible to achieve an
order of magnitude higher precision than by conventional methods. The major
improvement of this methodology results from utilisation of the aggregate infor-
mation from different time steps (as opposite to the consideration of a single time
step). The present theoretical model which includes the computationally predicted
lateral liquid motion, agrees well with our simulation results for the jetting time.
106
Conclusions
Also the spatial and temporal pressure development in the compressed
region as well as the moment when it reaches its maximal value have been numer-
ically explored. Computational results have shown that the pressure does not
reach it highest value at the moment of jetting eruption (as assumed up to date) but
some time after (for the numerical values adopted in this study - it occurs approx-
imately later). The assumption of the pressure taking the spatial maximum
at the contact line region has been computationally confirmed, showing also that
the highest pressure value mounts to double of the initial pressure developed at the
moment of impact.
On the theoretical side, a model which takes into account a realistic equation
of state has been developed, showing good match with presented computational
results. For the first time, an expression for the liquid particle velocity in the com-
pressed region was elaborated. Taking into account results based on the stiffened
gas equation of state, a parametric representation of the shock wave envelope was
presented and validated against computational findings, showing a substantial
improvement compared to the previous models.
In the discussion of the droplet impact, it has been proven that the assump-
tion of a single shock wave structure leads to the occurrence of physical anomaly.
This results from the finding that after a certain time, the flow solution obtained
under the single-shock assumption contradicts the possible locus of states which
are allowed by the equation of state. This anomaly was removed by allowing for
the existence of more complex wave patterns prior to jetting eruption.
The assumption of the multiple wave structure adjacent to the contact line
prior to the commencement of jetting was validated against our numerical results,
showing the break-up of the single shock wave into two waves after certain time
(defined in this thesis as the time of shock degeneration), where the flow across
the wave structure remains approximately one-dimensional. Also a model of
0.5t jet
107
Conclusions
shock envelope generated upon impact is presented and validated, showing good
match with computational findings.
In the last section, the one-dimensional Riemann problem was analytically
solved using the stiffened gas equation of state, under the approximation that the
flow in the smooth region is isentropic. The eigenstructure and eigenvectors of
Jacobian matrix were explicitly derived. We proved that a specific double wave
structure (left expansion wave and right shock wave) at the contact line region is
a valid solution of the problem. According to this scenario the pressure produced
at the contact line region decreases compared to the pressure developed when only
a single shock wave occurs.
The existence of a lower particle velocity in the intermediate region (com-
pared to the compressed region) has also been confirmed. The anomaly resulting
from the assumption of the single shock wave structure in the contact line region
is thus removed also rigorously and the physically acceptable solution is the above
mentioned double wave structure.
108
Conclusions
109
Appendix: Isentropic Flow
8Appendix: Isentropic Flow
The important thing in science
is not so much to obtain new facts
as to discover new ways of thinking about them.
-- Sir William Bragg (1862-1942)
The overall entropy balance in a fluid can be written as
, (A 1)
where , and correspond to the dissipation function, entropy and heat con-
ductivity, respectively. Here, we consider the flow in the smooth region (e.g. with
finite gradients) across the expansion fan. Due to the small time scales and high
Reynolds numbers for the case studied, an inviscid ( ) and adiabatic
( ) approximation is justified. Under these assumptions, the equations of
isentropic flow govern the dynamics of the phenomena of interest.
The usual thermodynamic relation,
(A 2)
when applied to an isentropic flow, , yields
(A 3)
Here, is the internal energy, the specific volume and the liquid den-
sity. In the following, we consider the liquid obeying the stiffened gas equation of
state, [32], [33]
ρTtd
dS ϒ ∇ λ∇ T( )⋅+=
ϒ S λ
ϒ 0=
λ∇ T 0≈
Tds de pdv+=
ds 0=
de pdv– pdρρ2------= =
e v ρ v 1–=
110
Appendix: Isentropic Flow
(A 4)
Taking the derivative of Eq. (A 4) yields
(A 5)
Substitution of Eq. (A 3), into the expression Eq. (A 5) yields
(A 6)
After some algebraic manipulation, this reduces to
(A 7)
To solve the differential equation (A 7), we integrate between the states and ,
(A 8)
This procedure yields the isentropic form of stiffened gas equation of state,
(A 9)
Isentropic Speed of Sound and Free Enthalpy
Equation (A 9) can be conveniently rewritten as
, (A 10)
with as unknown constant. The speed of sound reads. The speed of sound reads
(A 11)
and the ratio of sound speeds across the wave [see Eq. (A 9)]
1ρ--- p Γ 1+( )P∞+[ ] Γ e=
dpρ
------ p P∞ Γ 1+( )+[ ] dρρ2------ Γde=
dpρ
------ pdρρ2
----------– P∞ Γ 1+( )–dρρ2------ Γp
dρρ2------=
p P+ ∞( ) Γ 1+( )dρ ρdp=
a b
Γ 1+( ) dρρ
------a
b
∫ dpp P+ ∞-----------------
a
b
∫=
pa P+ ∞
ρaΓ 1+
-------------------pb P+ ∞
ρbΓ 1+
-------------------=
pa P+ ∞
ρaΓ 1+
------------------- CΓ 1+-------------=
C
c2
ρ∂∂p
S
CρΓ= =
111
Appendix: Isentropic Flow
(A 12)
To obtain the speed of sound as a function of density and pressure, we employ Eq.
(A 10),
(A 13)
The total energy , in conjunction with stiffened gas equation of
state, Eq. (A 4), becomes now
(A 14)
Here, is to the total velocity vector. The free enthalpy,
can be found as (A 15)
(A 16)
For the purpose of determination of the eigenstructure, the speed of sound as a
function of and is required. To this end, we solve for the total energy from
the free enthalpy definition [Eq. (A 15)] and combine with Eq. (A 4) to obtain
(A 17)
Finally, taking into account Eq. (A 13), the last equation reads:
(A 18)
Remark: Equation (A 9) gives the function along the isentropic path.
This, however, does not mean that we can express the pressure in the compressed
region as , since the compressed area need not be connected to
ca
cb----
ρa
ρb----- Γ 2⁄ pa P+ ∞
pb P+ ∞-------------------
Γ2 Γ 1+( )---------------------
= =
c2
ρ∂∂p
S
Γ 1+( )p P+ ∞
ρ-----------------= =
E ρ u2
2----- e+ =
E ρu2
2-----
p Γ 1+( )P∞+
Γ-----------------------------------+=
u
HE p+ρ
-------------≡
H Γ 1+( )E P– ∞
ρ----------------- Γu2
2-----–=
H u E
p P∞+( ) Γ 1+( ) Γρ H u2
2-----–
=
Γ 1+( )p P∞+
ρ----------------- Γ H u2
2-----–
c2= =
p p ρ( )=
p p ρ ρ0 p0, ,( )=
112
Appendix: Isentropic Flow
the ambient state through a simple waves (where isentropic condition
holds).
ρ0 p0,( )
113
List of Figures
9List of Figures
A Picture’s meaning can express ten thousand words.
-- Chinese proverb, literal translation
CHAPTER 1.
Fig. 1.1. a) - i) Different parameters of droplet impact . . . . . . . . . . . . . . . . . . . 2
Fig. 1.2. Impact of a spherical liquid drop (blue) on a rigid surface. The zone
of the highly compressed liquid (red) is bounded by the shock front
and target surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Fig. 1.3. Impact of a spherical liquid drop on a rigid surface. Construction of
shock front as an envelope of individual wavelets emitted by the
expanding contact edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Fig. 1.4. a) The shock wave remains attached to the contact periphery up to the
moment when the contact line velocity decreases below the shock
velocity. b) Shock front overtakes the contact edge. It is followed by
the eruption of intense lateral jetting due to the high pressure
difference across the droplet free surface. . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER 2.
Fig. 2.1. Cross sections of a typical microstructure obtained through plasma
deposition process, courtesy Sulzer Metco. . . . . . . . . . . . . . . . . . . . 15
Fig. 2.2. Requirements for a typical controlled atmosphere spray system.. . . 15
114
List of Figures
Fig. 2.3. Sulzer Metco environmental plasma chamber. . . . . . . . . . . . . . . . . . 16
Fig. 2.4. Robot arm with plasma gun in the Sulzer Metco environmental
plasma chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Fig. 2.5. Splat of liquid alumina (Al2O3) droplet on glass substrate,
corresponding to initial droplet radius 15.125 mm, temperature of
2664 K and impact velocity of 92.3 m/s. After impact on a substrate
and solidification, patterns of radial symmetry breakdown is evident.
Photograph courtesy of Sulzer Metco. . . . . . . . . . . . . . . . . . . . . . . . 18
Fig. 2.6. Splashed liquid nickel droplet at 2500 K after impact on a substrate
and solidification, showing patterns of symmetry breakdown both in
radial and azimutal direction. Impact velocity of 180 m/s. Courtesy:
Sulzer Metco. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Fig. 2.7. Impact of liquid Ni droplet of the mean radius of 10 mm: Effects of
droplet temperature (measured at the surface) and impact velocity.. 20
Fig. 2.8. Liquid metal impact at high velocity (200 m/s). a) Very high
temperature (above 2700 , left) vs. b) low temperature (below
1700 , right). The left photograph has a 2.5 times higher
magnification than the right one. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
CHAPTER 3.
Fig. 3.1. Determination of the principal Hugoniot: An arbitrary shock front
surface in a reference frame where the liquid particle velocity at the
upstream side of the shock vanishes. . . . . . . . . . . . . . . . . . . . . . . . . 25
Fig. 3.2. Comparison of principal Hugoniots. Shock velocity s as a function of
the jump in particle velocity u across the shock for the stiffened gas
equation of state and linear Hugoniot fit. . . . . . . . . . . . . . . . . . . . . . 28
°C
°C
115
List of Figures
CHAPTER 4.
Fig. 4.1. Computational domain and boundary conditions in cylindrical
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Fig. 4.2. Droplet and air density distribution prior to the impact: Emergence
and reflection of the bow shock in the air and weak perturbations in
the liquid bulk (due to the liquid-air interactions on a droplet surface).
Droplet velocity 500 m/s, motionless air. [numerical result with non-
linear colour map (HDF)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Fig. 4.3. Finite differences stencil used for the normal propagation of the
shock wave. The states utilised for the computation of the normal
propagation operator are obtained from the left and right states on the
curve at the point being propagated. . . . . . . . . . . . . . . . . . . . . . . . . . 37
Fig. 4.4. A schematic picture of the data used for normal propagation of a
shock wave. The front data at the old time step provides a Riemann
solution, that is corrected by interior data, using the method of
characteristics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
CHAPTER 5.
Fig. 5.1. Convergence and grid independence of the solution: a) density, b)
pressure and c) temperature distributions along the z-axis (along the
line shown in Fig. 1.2 for 3 different grids: 0.5, 1.2 and 2.0 million
points. Snapshot corresponds to time step 10.02 ns after impact. . . 48
Fig. 5.2. Convergence and grid independence of the solution: Shock position
in r-z plane for two meshes, 0.5 and 2.0 million points, corresponding
to time step 2.05 ns after impact. The depicted region corresponds to
the zoomed area of the quadrant in Fig 1.2.. . . . . . . . . . . . . . . . . . . 49
116
List of Figures
Fig. 5.3. Time evolution of density during the droplet impact showing shock
creation, propagation and interaction with the free surface. The region
enclosed by the black line corresponds to the very low pressure area
behind the shock wave, which occurs upon shock interaction with the
droplet free surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Fig. 5.4. Three-dimensional representation of the droplet impact time
evolution: Droplet free surface (in blue) and shock & expansion
waves (in red).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Fig. 5.5. Three-dimensional representation of the droplet impact time
evolution. volume cutout uncovers the exact position of the free
surface and shock & expansion wave fronts (in red). . . . . . . . . . . . . 53
Fig. 5.6. Development of lateral liquid motion in the compressed region.
Snapshot at time 2.148 ns after impact. . . . . . . . . . . . . . . . . . . . . . . 54
Fig. 5.7. Commencement of jetting. Radial liquid velocity shows the initiation
of jetting. The image corresponds to the time 3.05 ns after impact.
The first evidence of jetting is found approximately at time 2.80 ns.
Grid size here: 4 million points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Fig. 5.8. Accurate determination of the jetting eruption time. . . . . . . . . . . . . 58
Fig. 5.9. High jetting velocities observed in the computations reaching values
up to 6000 m/s. The picture corresponds to the time step 4.86 ns after
impact.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Fig. 5.10. Pressure distribution along the r axis corresponding to 6
representative times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Fig. 5.11. Influence of surface tension on jetting formation and break up.
Snapshot corresponds to the time 18.19 ns after impact a) zero
117
List of Figures
surface tension. b) surface tension for water. (c) zoomed front region
of jets in b). together with computational grid used. . . . . . . . . . . . . 63
Fig. 5.12. Temperature distribution at four different radial positions along the z-
axis in the compressed area. The edge position corresponding to this
time step resides at r/R=0.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
CHAPTER 6.
Fig. 6.1. Impact of the upwards moving wall on the motionless spherical liquid
droplet. The zone of the highly compressed liquid (red) is bounded by
the shock front and the target surface. . . . . . . . . . . . . . . . . . . . . . . . 68
Fig. 6.2. Geometrical construction of shock front as an envelope of individual
wavelets emitted by the expanding contact edge. Note the difference
in the construction to the acoustic model (Fig. 1.2), where the shock
velocity was assumed constant (equal to the speed of sound) with
respect to the propagating contact line and not with respect to the
undisturbed bulk of liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Fig. 6.3. Geometrical construction of the shock profile attached to the contact
line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Fig. 6.4. Shock velocity vs. contact line velocity for the linear Hugoniot. . . . 74
Fig. 6.5. Prediction of radial particle velocity and comparison with
computational results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Fig. 6.6. Construction of the solution: Intersection of edge boundary curves
(each corresponding to a different time) and equation of state (a
straight line). No solution exists beyond 1.80 ns. . . . . . . . . . . . . . . . 77
Fig. 6.7. Schematic of a double wave structure in a contact line region. . . . . 78
118
List of Figures
Fig. 6.8. Effect of the factor on the edge bc curves allowing the existence of
solution up to the jetting time (here 2.80 ns). . . . . . . . . . . . . . . . . . 79
Fig. 6.9. Wave structure of the compressed region together with the
computational grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Fig. 6.10. (a) Pressure wave profiles at the contact line in the radial direction for
different times. Evidence of the single wave degeneration after 1.5 ns.
(b) The pressure plot shown in (a) was sampled along the dashed
arrow-line. (c) Pressure derivative dp/dr, showing clearly the gradual
emergence of two negative peaks, indicating the split of the single
shock wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Fig. 6.11. Shock structure in the vicinity of the contact line region together with
equidistant iso-density lines. The velocity filed is shown by velocity
vectors, apparently normal to the outer shock wave. . . . . . . . . . . . . 83
Fig. 6.12. Envelope construction: Wall position at the time shown by a dashed
line. Contact line propagates along the droplet free surface. . . . . . . 84
Fig. 6.13. Envelope construction: Comparison of analytical results with the
computational findings.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Fig. 6.14. Solution of the one-dimensional Riemann problem. . . . . . . . . . . . . 92
Fig. 6.15. Case 1: Assumption of the left expansion and right shock wave . . . 99
Fig. 6.16. Possible solution scenarios, (a), (b), (c) correspondent to the cases 1,
2, and 3, respectively. Here, only the case 1 yields the physically
acceptable solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Fig. 6.17. The range of possible solutions for density and pressure on the left
side in the Riemann problem. Pressure (y-axis) in GPa. . . . . . . . . 102
119
List of Figures
Fig. 6.18. Admissible range of the pressure at the contact edge versus the total
liquid particle velocity jump across the wave. . . . . . . . . . . . . . . . . 103
120
List of Figures
121
Bibliography
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Whenever you are asked
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-- Theodore Roosevelt (1858 - 1919)
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127
Curriculum Vitae
CURRICULUM VITAE
Personal Data
Date of Birth: 28/08/1972
Place of Birth: Sarajevo, Bosnia
Education
03/99-10/02 Ph.D. [Dr. sc. techn.], Laboratory of Thermodynamics in Emerging
Technologies, Swiss Federal Institute of Technology Zurich (ETH
Zurich), Switzerland. PhD Project: High Velocity Impact of the Liquid
Droplet on a Rigid Surface: The Effect of Liquid Compressibility.
04/98 M.Sc. (Physics) [Dipl. Phys. ETH], ETH Zurich, Switzerland.
09/92-02/98 Undergraduate Study in Physics at the Department of Mathematics
and Physics, ETH Zurich, Switzerland [M.Sc. (Physics)].
09/87-07/91 Gymnasium: Leaving Certificate Matura in Mathematics, Physics and
Computer Sciences, Sarajevo, Bosnia.
Project and Work Experience
04/00-06/00 Internship at the State University of New York, Stony Brook, NY,
USA.
05/98-03/99 Project Collaborator, Institute for Industrial Engineering and
Management (BWI), ETH Zurich.
04/99-08/98 Project Collaborator, Institute of Robotics, ETH Zurich. Project:
Electro-Hydrodynamic Propulsion.
10/97-03/98 Undergraduate Student, Institute of Robotics, ETH Zurich.
Diploma-Thesis: Electro-Hydrodynamic Propulsion.
KRISTIAN HALLER KNEZEVIC
128
Curriculum Vitae
Project and Work Experience
03/97-07/97 Project Collaborator, “Institute of Umformtechnik”, ETH Zurich.
10/96-03/97 Undergraduate Student at the Institute of Quantum Electronics, ETH
Zurich.
03/99-10/02 Semester-Work Thesis: Automation of Spectral Measurements of
Infrared Sensors.
Honours and Awards
1993-98 Scholarship from the “Stiftung Solidaritätsfonds für ausländische
Studierende an der ETH Zürich”, Zurich, Switzerland.
1992-98 Scholarship from the “ETH Zürich, Rektorat”, ETH Zurich,
Switzerland.
1989-91 Scholarship from the “Energoinvest”, Sarajevo, Bosnia.
1991 1st Price in the Yugoslavian High School Physics Competition,
allowing participation at the International Physics Olympics’91.
1991 3rd Price in the BH (Bosnia-Herzegovina) High School Mathematics
Competition.
1990 1st Place in the BH High School Mathematics Competition.
1989 3rd Price in the Yugoslavian High School Physics Competition.
1989 1st Place in the BH High School Physics Competition
Languages
English, German, Italian, Serbo-Croatian, Spanish
Publications and Conferences (1998-2002)
K. K. Haller, Y. Ventikos & D. Poulikakos, ‘Riemann Problem Solution for the Stiff-
ened Gas Equation of State and Implications on High-Speed Droplet Impact’, Journal
of Applied Physics, Vol. 93, No. 5, pp. 3090-97, Mar 2003.
129
Curriculum Vitae
K. K. Haller, D. Poulikakos, Y. Ventikos & P. Monkewitz, ‘Shock Wave Formation in
Compressible Droplet Impact on a Rigid Surface: Lateral Liquid Motion and Multiple
Wave Structure in the Contact Line Region’, accepted for publication, Journal of Fluid
Mechanics, 2003.
K. K. Haller, Y. Ventikos, D. Poulikakos & P. Monkewitz, ‘A Computational Study of
High-Speed Liquid Droplet Impact’, Journal of Applied Physics, Vol. 92, No. 5, pp.
2821-28, Sept 2002.
K. K. Haller, Y. Ventikos, D. Poulikakos & P. Monkewitz, ‘High Speed Droplet
Impact’, GAMM 2001, Zurich, Switzerland, 2001.
K. K. Haller & F. M. Moesner, ‘Theoretical Considerations on Electrohydrodynamic
Propulsion’, Proceedings of the Fourth International Conference on Motion and Vibra-
tion Control, MOVIC ‘98, 1998.
130
Curriculum Vitae