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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

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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


(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


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


(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


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


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


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


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


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


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


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


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


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


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


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


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


- (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


(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


(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


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


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


(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


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


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


(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


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


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


(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


. (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


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


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


(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


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


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


(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


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


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


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


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


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


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


(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


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


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


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


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


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


, , (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


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


(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


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


(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


(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


(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


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


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


superposition wave

0.5 1 1.5 2 2.5 3 3.5p* (GPa)

200

400

600

800

∆u (m/s)

right shockleft expansion


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


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


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


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


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


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


(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


(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


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

10Bibliography

Whenever you are asked

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Then get busy and find out how to do it.

-- Theodore Roosevelt (1858 - 1919)

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[12] Haferl S. and Poulikakos D., Experimental Investigation of the TransientImpact Fluid Dynamics and Solidification of a Molten Microdroplet Pile-Up submitted for publication to the Intl. Journal of Heat and MassTransfer, 2002.

[13] Attinger D., Zhao Z. & Poulikakos D., "An Experimental Study of MoltenMicrodroplet Surface Deposition and Solidification: Transient Behaviorand Wetting Angle Dynamics," ASME Journal of Heat Transfer, Vol. 122,pp. 544-556, 2000.

[14] Attinger D. & Poulikakos D., "Melting and Resolidification of a Substratecaused by Molten Microdplet Impact," ASME Journal of Heat Transfer,Vol. 123, pp. 1110-1122, 2001.

[15] Haferl S, Zhao Z., Giannakouros J., Attinger D., & Poulikakos D.,"Transport Phenomena in the Impact of a Molten Droplet on a Surface:Macroscopic Phenomenology and Microscopic Considerations. Part IFluid Dynamics", Reviews in Heat Transfer, Vol. XI, pp. 145-205, BegellHouse, NY, 2000.

[16] Attinger D., Haferl S., Zhao Z., & Poulikakos D., "Transport Phenomenain the Impact of a Molten Droplet on a Surface: MacroscopicPhenomenology and Microscopic Considerations. Part II Heat Transferand Solidification," Reviews in Heat Transfer, Vol. XI, pp. 65-143, BegellHouse, NY, 2000.

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[51] Haller K. K., Ventikos Y., Poulikakos D. & Monkewitz P., Shock WaveFormation in Droplet Impact on a Rigid Surface: Lateral Liquid Motionand Multiple Wave Structure in the Contact Line Region, accepted forpublication, Journal of Fluid Mechanics, 2003.

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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

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