Numerical simulation of cavitation-induced bubble dynamics ...

Universite catholique de Louvain

Ecole Polytechnique de Louvain

Unite d’Ingenierie des Materiaux et des Procedes

Numerical simulation of cavitation-induced

bubble dynamics near a solid surface

These presentee par

Vincent Minsier

en vue de l’obtention du grade de

Docteur en Sciences de l’Ingenieur

Composition du Jury:

Prof. J. Proost (promoteur) Universite catholique de Louvain

Prof. J. De Wilde (promoteur) Universite catholique de Louvain

Prof. J.-F. Remacle Universite catholique de Louvain

Prof. C.-D. Ohl Nanyang Technological University

Dr. F. Holsteyns Lam Research, Austria

Prof. G. Winckelmans (President) Universite catholique de Louvain

Contents

List of Symbols vii

Introduction xi

Scientific Production xv

1 State-of-the-art 1

1.1 Acoustic cavitation . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Bubble nucleation . . . . . . . . . . . . . . . . . . 2

1.1.2 Bubble interactions with an acoustic field . . . . . 3

1.1.3 Bubble thresholds . . . . . . . . . . . . . . . . . . 4

1.2 Transient bubble dynamics . . . . . . . . . . . . . . . . . 5

1.2.1 Bubble dynamics in bulk liquid . . . . . . . . . . . 6

1.2.2 Bubble dynamics near a solid surface . . . . . . . . 8

1.3 Sono-processes . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Megasound processes . . . . . . . . . . . . . . . . . 13

1.3.2 Ultrasound processes . . . . . . . . . . . . . . . . . 14

1.4 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Spherical bubble dynamics . . . . . . . . . . . . . 17

1.4.2 Aspherical bubble dynamics . . . . . . . . . . . . . 21

1.4.3 General considerations about multiphase flows . . 24

2 Shock wave emission 33

2.1 Threshold conditions . . . . . . . . . . . . . . . . . . . . . 34

2.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 35

iii

iv CONTENTS

2.1.2 Comparison of threshold conditions . . . . . . . . . 39

2.2 Shock wave propagation . . . . . . . . . . . . . . . . . . . 44

2.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.2 Influence of cavitation process parameters . . . . . 48

2.2.3 Influence of gas state equation . . . . . . . . . . . 50

2.2.4 Influence of surface tension . . . . . . . . . . . . . 52

2.2.5 Application . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Gas-liquid multiphase flow model 57

3.1 Numerical model . . . . . . . . . . . . . . . . . . . . . . . 60

3.1.1 Compressible Navier-Stokes equations . . . . . . . 60

3.1.2 Surface tension force . . . . . . . . . . . . . . . . . 62

3.1.3 Volume Of Fluid method . . . . . . . . . . . . . . 63

3.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . 64

3.2.1 Modification of Navier-Stokes equations . . . . . . 65

3.2.2 Discretization schemes . . . . . . . . . . . . . . . . 67

3.2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.4 Gas-liquid interface . . . . . . . . . . . . . . . . . . 77

3.3 Test case . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.1 Adiabatic conditions . . . . . . . . . . . . . . . . . 86

3.3.2 Heat transfer through the interface . . . . . . . . . 92

3.4 2D axisymmetric simulations . . . . . . . . . . . . . . . . 94

3.4.1 The problem definition . . . . . . . . . . . . . . . . 94

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . 97

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Laser-induced bubble collapse 105

4.1 Bubble dynamics in bulk liquid . . . . . . . . . . . . . . . 106


4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2 Bubble dynamics near a solid surface . . . . . . . . . . . . 111


4.2.2 Model validation . . . . . . . . . . . . . . . . . . . 113

4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . 124

CONTENTS v

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5 Acoustic-induced bubble collapse 137

5.1 Bubble dynamics in bulk liquid . . . . . . . . . . . . . . . 138


5.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . 140

5.2 Bubble dynamics near a solid surface . . . . . . . . . . . . 142


5.2.2 An illustrative example . . . . . . . . . . . . . . . 144

5.2.3 Influence of the initial bubble radius . . . . . . . . 148

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 153

General Conclusions and Perspectives 155

Acknowledgements 159

Appendix 161

Bibliography 165

List of Symbols

Roman Symbols

B coefficient of Tait equation Pa

Cg speed of sound in the gas at the gas-liquid

interface

m/s

cp specific heat at constant pressure J/kgK

cp,g specific heat at constant pressure in gas

phase

J/kgK

cp,L specific heat at constant pressure in liquid

phase

J/kgK

cL speed of sound in the liquid m/s

CL speed of sound in the liquid at the gas-liquid

interface

m/s

cv specific heat at constant volume J/kgK

d distance separating the initial bubble centre

from the solid surface

m

e width of a polysilicon line structure m

E total energy m2/s2

fA frequency of the acoustic field Hz

Ff fracture force N

Fl flux of liquid through cell face m3

Fs shock wave force N

Fσ surface tension force N

g gravity m/s2

vii

viii SYMBOLS

h enthalpy in the liquid J/kg

H(Φ) Heaviside function -

hp height of a polysilicon line structure m

H enthalpy in the liquid at the gas-liquid

interface

J/kg

k thermal conductivity W/mK

kg thermal conductivity in gas phase W/mK

kL thermal conductivity in liquid phase W/mK

L length of a polysilicon line structure m

m hard-core radius m

Mm molar mass g/mol

n normal at the gas-liquid interface -

n number of iterations at time t + ∆t -

nT parameter of Tait equation -

p pressure Pa

p0 ambient pressure (also noted pambient) Pa

p0 initial pressure inside the bubble Pa

pA amplitude of the acoustic field Pa

pacoust pressure of the acoustic field Pa

pb pressure inside the bubble Pa

pb,0 initial pressure inside the bubble Pa

pb,id pressure inside the bubble (ideal gas) Pa

pb,V dW pressure inside the bubble (Van der

Waals gas)

Pa

pg gas pressure Pa

pL liquid pressure at the bubble-liquid in-

terface

Pa

pmax maximum pressure in the liquid Pa

pref reference pressure Pa

pv vapour pressure Pa

p∞ pressure at infinity Pa

r radial distance m

R bubble radius m

R0 initial bubble radius m

SYMBOLS ix

Rbd radius of the bubble domain m

Rf radius at the boundary of flow domain m

Rmax maximum bubble radius m

Rmin minimum bubble radius m

R velocity of the bubble wall m/s

R acceleration of the bubble wall m/s2

Rid ideal gas constant J/molK

Rr resonant bubble radius m

R∞ radius at the boundary of flow domain m

Re Reynolds number -

t time s

tRmax time corresponding to the maximum

bubble radius

s

T temperature K

Tamb ambient temperature in the liquid K

Tb temperature in the bubble K

Tb,0 initial temperature in the bubble K

Trr radial component of the viscous stress

tensor

Pa

T0(r) initial temperature field in the flow do-

main

K

T∞ temperature at infinity K

u velocity m/s

ujet jet front velocity m/s

umax maximum jet front velocity m/s

umax,axi maximum velocity along the axis of ax-

isymmetry

m/s

us liquid velocity at the shock front m/s

S boundary of flow domain m2

V bubble volume m3

V0 initial bubble volume m3

Vmax maximum bubble volume m3

Y r(h + u2/2) m3/s2

x SYMBOLS

Greek Symbols

α volume fraction -

αg gas volume fraction -

αL liquid volume fraction -

αq volume fraction of the qth phase -

α normalized volume fraction -

γ stand-off distance -

isentropic coefficient -

δΩ boundary of flow domain m2

κ polytropic coefficient -

κi curvature of the gas-liquid interface m−1

λc coefficient of the Keller equation -

Λ interface m2

µ dynamic viscosity kg/ms

µg dynamic gas viscosity kg/ms

µL dynamic liquid viscosity kg/ms

ρ density kg/m3

ρg gas density kg/m3

ρg,ref reference gas density kg/m3

ρL liquid density kg/m3

ρq density of the qth phase kg/m3

ρ0 initial density kg/m3

σ surface tension N/m

σf fracture stress Pa

φ potential m2/s

Φ level set function -

ωA angular frequency of the acoustic field Hz

ωr angular resonant frequency Hz

Ω volume of flow domain m3

Ωl volume of liquid flow domain m3

General introduction

Cavitation has been studied for over a century now, primarily in the

field of hydrodynamic science, to explain erosion and damages observed

in hydraulic turbines, ship propellers and hydrofoils. In the last thirty

years, cavitation induced by sound waves, known as acoustic cavitation,

has gained much interest in the field of chemical and biochemical science

as well. Acoustic cavitation is the process in which bubbles are nucleated

and/or forced to oscillate in an acoustic field. From an experimental and

numerical point of view, many aspects regarding the dynamics of a bub-

ble subjected to an acoustic field have been studied extensively. It has

been shown that, away from the solid surface, the bubble remains almost

spherical and shock waves are emitted at the end of the collapse phase.

However, when the bubble is close to a solid surface, the bubble loses its

spherical shape and a jet penetrates into the bubble towards the solid

surface at the end of the collapse phase. Although many works have been

conducted, the detailed mechanism responsible of the observed effects

during acoustic cavitation-induced processes, such as surface cleaning in

microelectronics, sonoporation in biomedical applications, and improve-

ments in steel pickling performance in the metallurgical industries, is

still poorly known. One reason is that the direct experimental observa-

tion of the dynamics of a bubble subjected to an acoustic field near a

solid surface is rather challenging due to the small time and spatial scale

of the phenomenon as well as its randomness. The current belief is that

the interactions of shock waves and jets with the solid surface are the

two main mechanisms responsible for the observed acoustic cavitation-

induced effects.

xi

xii INTRODUCTION

A better insight into the interaction of shock waves and jets with the

solid surface is therefore important, both from the perspective of appli-

cations as well as from a fundamental point of view. In addition, the

study of the influence of experimental parameters on shock waves and

jets penetrating the bubble is a first step for the optimisation of acoustic

cavitation-induced processes.

Within this framework, this PhD was devoted to the investigation of

(i) the shock waves emitted during the spherical collapse of a bubble;

(ii) the jet penetration inside the bubble during the aspherical collapse

of a bubble near a solid surface. To this purpose, numerical models have

been developed and used to have a better understanding of these mech-

anisms.

The main contributions of this PhD can be summarised as follows:

• modelling the propagation in the liquid of the shock wave emitted

during the spherical collapse of a bubble subjected to an acoustic

field, and studying the effect of some of the acoustic parameters

on the liquid velocity at the shock front.

• providing a numerical model for viscous gas-liquid multiphase flow

to simulate laser and acoustic-induced bubble dynamics near a

solid surface.

• studying the effect of the liquid viscosity on the velocity of the jet

penetrating a laser-induced bubble.

• studying the effect of the initial bubble radius on the velocity of

the jet penetrating an acoustic-induced bubble.

The manuscript has been divided into 5 chapters. In the first chapter,

the reader is introduced to acoustic cavitation. A comprehensive review

of the literature on the dynamics of a bubble subjected to an acoustic

field in bulk liquid and near a solid surface is presented, with a special

interest in numerical models. The second chapter deals with shock waves

INTRODUCTION xiii

emitted during the spherical collapse of a bubble. A model describing

the propagation of the shock wave in the liquid is presented, and the

influence of some acoustic parameters, such as the initial bubble radius

and the amplitude of the acoustic field, on the liquid velocity at the shock

front is discussed. In the rest of the work, the dynamics of a bubble

near a solid surface is studied. The third chapter presents a numerical

model for gas-liquid multiphase flow. The numerical model is described

in detail and validated by comparison with analytical solutions. In the

fourth and fifth chapters, numerical calculations are performed based on

this numerical model. The fourth chapter focuses on the simulation of

the dynamics of a laser-induced bubble. The evolution of the bubble

shape with time is compared to experiments reported in the literature

and the influence of the liquid viscosity on the velocity of penetrating

jets is discussed. Finally, the last chapter is devoted to the dynamics

of an acoustic-induced bubble. More specifically, the influence of the

initial bubble radius on the velocity of the jet penetrating the bubble is

discussed.

Scientific Production

The results presented in this PhD have been published in the following

scientific journals and presented on the international meetings listed be-

low.

Publications in peer-reviewed journals and conference proceed-

ings

• V. Minsier, J. De Wilde, J. Proost, ”Simulation of the effect of

viscosity on jet penetration into a single cavitating bubble”, J.

Appl. Phys. 106 (2009), 084906.

• V. Minsier, J. De Wilde, J. Proost, ”Simulation of cavitation-

induced asymmetric bubble collapse near a solid surface based

on the Volume Of Fluid method”, Proceedings of the WIMRC

2nd International Cavitation Forum, Warwick University, 211-216

(2008).

• V. Minsier, J. Proost, ”Shock wave emission upon spherical bub-

ble collapse during cavitation-induced megasonic surface cleaning”,

Ultrason. Sonochem. 15 (2008), 598-604.

• V. Minsier, J. Proost, ”Modeling of shock wave emission during

acoustically-driven cavitation-induced cleaning processes”, Solid

State Phenomena 134 (2008), 197-200.

• V. Minsier, J. Proost, ”Comparisons of threshold conditions for

transient acoustic cavitation and shock wave emission”, Proceed-

xv

xvi SCIENTIFIC PRODUCTION

ings of the 6th International Symposium on Cavitation, Wagenin-

gen, paper 62, 4p. (2006).

Presentations on international scientific meetings

• V. Minsier, J. De Wilde, J. Proost, ”Simulation of bubble dynam-

ics by a compressible Volume Of Fluid method”, MaCKie 2009

(Mathematics in Chemical Kinetics and Engineering), Gent, 2009.

(oral presentation)

• V. Minsier, J. De Wilde, J. Proost, ”Influence of bubble dynam-

ics on ultrasound assisted electrochemical processes”, The 59th

Annual Meeting of the International Society of Electrochemistry,

Seville, September 2008. (oral presentation)

• V. Minsier, J. De Wilde, J. Proost, ”Simulation of cavitation-

induced asymmetric bubble collapse near a solid surface based on

the Volume Of Fluid method”, WIMRC 2nd International Cavita-

tion Forum, Warwick University, 2008. (oral presentation)

• V. Minsier, J. De Wilde, J. Proost, ”Numerical analysis of penetra-

tion jets during cavitation-induced bubble collapse”, International

Symposium on Sonochemistry and Sonoprocessing, Kyoto, 2007.

(oral presentation)

• V. Minsier, J. Proost, ”Modeling of shock wave emission during

acoustically-driven cavitation-induced cleaning processes”, 8th In-

ternational Symposium on Ultra Clean Processing of Semiconduc-

tor Surfaces (UCPSS), Antwerp, 2006. (oral presentation)

• V. Minsier, J. Proost, ”Comparisons of threshold conditions for

transient acoustic cavitation and shock wave emission”, 6th Inter-

national Symposium on Cavitation, Wageningen, 2006. (poster)

Chapter 1

State-of-the-art

Contents

1.1 Acoustic cavitation . . . . . . . . . . . . . . . 2

1.1.1 Bubble nucleation . . . . . . . . . . . . . . . 2

1.1.2 Bubble interactions with an acoustic field . . 3

1.1.3 Bubble thresholds . . . . . . . . . . . . . . . 4

1.2 Transient bubble dynamics . . . . . . . . . . . 5

1.2.1 Bubble dynamics in bulk liquid . . . . . . . . 6

1.2.2 Bubble dynamics near a solid surface . . . . . 8

1.3 Sono-processes . . . . . . . . . . . . . . . . . . 12

1.3.1 Megasound processes . . . . . . . . . . . . . . 13

1.3.2 Ultrasound processes . . . . . . . . . . . . . . 14

1.4 Modeling . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Spherical bubble dynamics . . . . . . . . . . 17

1.4.2 Aspherical bubble dynamics . . . . . . . . . . 21

1.4.3 General considerations about multiphase flows 24

1

2 CHAPTER 1. STATE-OF-THE-ART

When an acoustic wave travels through a liquid, different interactions

may take place. First, these interactions can lead to bubble nucleation.

Once the bubbles are generated, they interact with the acoustic field

resulting in different effects such as shock wave emission, jet formation,

photon emission and radical formation. These effects may in turn in-

fluence the nucleation and the dynamics of neighbouring bubbles. All

these physical phenomena are known as acoustic cavitation. Usu-

ally, acoustic cavitation is called ultrasonic cavitation in the kHz regime

and megasonic cavitation in the MHz regime. In section 1.1, bubble

nucleation as well as general considerations about bubble motion in an

acoustic field are summarised. Bubble dynamics can be stable or tran-

sient, depending on acoustic parameters and fluid properties. In this

work, more attention is paid to the transient bubble dynamics during

which shock waves and jets have been observed. In section 1.2, the

transient bubble dynamics in bulk liquid and near a solid surface is de-

scribed based on experimental results. The section 1.3 presents some

experimental and industrial processes based on acoustic cavitation and

shows the need to have a better insight into the shock wave emission

and the jet impact on the solid surface. Finally, the last section gives an

overview of different numerical models that have been used to simulate

bubble dynamics.

1.1 Acoustic cavitation

1.1.1 Bubble nucleation

When an acoustic wave travels through a liquid, bubbles are nucleated

on nucleation sites such as gas-filled crevices, small particles and con-

taminants (heterogeneous nucleation). Note that the homogeneous nu-

cleation, which is the formation of vapour voids in a pure liquid, is almost

never reached because it requires enormous forces to break the tensile

strength of the liquid [68]. Bubbles generated in the acoustic field are

called the in-situ generated bubbles, while bubbles created by another

way such as a laser pulse and oscillating in the acoustic field are called

ex-situ generated bubbles [44].

1.1. ACOUSTIC CAVITATION 3

1.1.2 Bubble interactions with an acoustic field

The temporal pressure variation of the acoustic field induces a radial

motion of the bubble, while the spatial pressure variation of the acous-

tic field imposes on the bubble a net force resulting in a translational

motion of the bubble.

Translational motion

A bubble in a sinusoidal standing wave moves towards either the pressure

antinode or the pressure node. As the spatial gradient of pressure varies

periodically in time, the net force on the bubble is the result of a time

averaged force on the bubble (the primary Bjerknes force) [109]. In a

relatively weak acoustic field, bubbles smaller than their resonant radius

(Rr) move to the pressure antinode, while bubbles higher than Rr move

to the pressure node. In a high-intensity field, the translational motion

is much more evolved, see [64] for more details. The resonant bubble

radius was first calculated by Minnaert [67] as:

ρLω2rR

2r = 3κp0 (1.1)

with ρL the liquid density, κ the polytropic coefficient, p0 the ambient

pressure and fr = ωr

2π the resonant frequency. The resonant frequency is

also called the natural frequency of a bubble with a radius Rr. Bubbles

with a radius of 3.7 µm and 157 µm have a natural frequency of 1 MHz

and 20 kHz, respectively.

The position of bubbles in the acoustic field is also affected by the attrac-

tive or repulsive forces between the bubbles (secondary Bjerknes force)

and by the presence of a surface [109, 64]. These interactions result

in different macroscopic structures of bubbles in the acoustic field (e.g.

filamentary bubble structures) [65].

Radial motion

The radial motion of a bubble subjected to an acoustic field is due to

the variation of the pressure of the acoustic field in time. The bubble


expands and contracts over many cycles. The dynamics of a bubble

subjected to an acoustic field can be stable or transient depending on

the acoustic parameters, fluid properties and bubble size. The dynamics

of a bubble is stable if the bubble oscillates over a wide range of acoustic

periods. During these periods, bubbles can dissolve [30] or grow by both

bubble coalescence and rectified diffusion [4]. The rectified diffusion

is the diffusion of gas from the liquid towards the bubble due to two

effects (i) the “area effect”: the bubble-liquid interface is larger when the

bubble expands; (ii) the “shell effect”: the gradient of gas concentration

at the interface is greater when the bubble expands [23]. The bubble

thus gains some gas over a complete period and grows considerably over

many cycles. The dynamics of a bubble is transient if, during less than

one period, the bubble grows to many times its original size and then

collapses. The word “collapse” means that the bubble volume decreases

rapidly. After the bubble collapse, the bubble may rebound or break up

into many smaller bubbles due to the appearance of perturbations in its

spherical shape [33].

1.1.3 Bubble thresholds

Different bubble interactions with the acoustic field have been described

in the previous section. Whether a bubble will dissolve, grow by rectified

diffusion (stable bubble dynamics) or grow to many times its original

size in less than one period (transient bubble dynamics) depends on the

frequency and the amplitude of the acoustic field, the fluid properties,

the level of gasification and the bubble size. Threshold conditions for

bubble dissolution and the transient bubble dynamics have been defined

in the literature [29, 44, 33, 71]. These threshold conditions have been

used to predict the behaviour of a bubble in an acoustic field as seen in

Fig. 1.1. In Fig. 1.1, the bubbles slowly dissolve in zone X, grow by

rectified diffusion in zone Y and have a transient dynamics in zone Z.

For example, considering a bubble at point S in Fig. 1.1, this bubble

grows by rectified diffusion until the transient threshold condition is

reached. Then, the bubble expands to many times its original size,

collapses and finally rebounds or disintegrates into small bubbles. Some

1.2. TRANSIENT BUBBLE DYNAMICS 5

Figure 1.1: Threshold conditions for bubble dissolution (line A-B) and

for transient bubble dynamics (line C-D) as a function of the initial

bubble radius (R0) and the ratio between the amplitude of the acoustic

field (pT ) and the ambient pressure (p0) for an air bubble in air-saturated

water at 20 kHz [109].

of these small bubbles dissolve; the others grow by rectified diffusion.

However, considering a bubble at point T in Fig. 1.1, this bubble stays

stable and grows by rectified diffusion until buoyancy forces remove the

bubble from the tank (degassing). More details can be obtained in [109].

The effect of increasing the frequency is that the threshold curves move

towards smaller bubble radii. In the following, we focus our attention

on the transient bubble dynamics.

1.2 Transient bubble dynamics

In this paragraph, the dynamics of a bubble in bulk liquid and near a

solid surface is described based on experimental results.


Figure 1.2: Experimental bubble radius (dots) monitored by light scat-

tering technique [63]. The superposed solid line is the bubble dynamics

predicted by the Keller-Miksis equation. The applied pressure is also

shown.

1.2.1 Bubble dynamics in bulk liquid

In their research, Gaietan et al. [15] have built an experimental setup

to study the dynamics of a bubble in an acoustic field. Piezoceramic

transducers are stuck to the sides of a closed flask of liquid. The piezo-

ceramic transducers work at a frequency between 20-40 kHz and at a

driving amplitude around 1 bar. The water inside the flask is first de-

gassed to about twenty percent of its saturated concentration of air in

order to keep one bubble in the acoustic field. The bubble has a typ-

ical equilibrium radius of approximately 40 µm and is trapped in the

pressure antinode of the standing wave field due to the Bjerknes force.

This bubble, stable in position and shape, oscillates in a purely radial

mode during many acoustic cycles. The experimental parameters are

the frequency and the amplitude of the acoustic field, the ambient pres-

sure, the amount of dissolved gas and the liquid properties. The bubble

radius as a function of time can be monitored by a CCD camera [75] or

by a light scattering technique [50]. Figure 1.2 shows the bubble radius

as a function of time, obtained by a light scattering technique [63]. The


dots correspond to the experimental measurements. The superposed

solid line is the bubble dynamics predicted by the Keller-Miksis equa-

tion (see eq. 1.10) [63]. The bubble dynamics is transient: the bubble

grows to many times its equilibrium radius and then collapses. As al-

ready mentioned, the word ”collapse” means that the bubble volume

decreases rapidly: in Fig. 1.2 the bubble radius decreases from 20 µm

to its minimum radius in less than 1 µs. During the collapse phase, the

bubble being compressed, the temperature and the pressure inside the

bubble can be as high as, respectively, 5000 C and 1000 bar. There

is a concentration of the acoustic energy inside the bubble. Near the

maximum bubble compression, different phenomena occur.

First, a shock wave is observed to be emitted in the liquid near the

end of the collapse phase [75, 45]. The interaction of this shock wave

with a solid surface is thought to be responsible of erosion observed on

solid surfaces [80]. The shock waves can be emitted according to two

mechanisms. According to the first mechanism, a shock wave is directly

emitted in the liquid when the velocity of the bubble wall reaches the

speed of sound in the liquid phase. According to the second mecha-

nism, a shock wave is first emitted inside the bubble when the velocity

of the bubble wall reaches the speed of sound in the gas phase. Next,

as shown by Nagrath et al. [70], the shock wave reflects at the centre of

the bubble. The reflection of the convergent shock wave in turn creates

a spherically divergent shock wave that propagates in the liquid.

Secondly, a light pulse of a few hundred picoseconds with intensities

in the order of 1-10 mW is emitted at the end of the collapse phase.

This phenomenon is called single-bubble sonoluminescence (SBLS) [15].

When the light is emitted by a bubble cloud and not a single bubble,

this phenomenon is called multi-bubble sonoluminescence (MBSL). The

mechanism of the sonoluminescence remains unsettled. A multitude of

theories try to explain the sonoluminescence mechanism. Nowadays, the

more accepted theory assumes that the flash of light is due to (i) the

Bremsstrahlung radiation; (ii) the radical recombination. During the

final stage of the bubble collapse, the high temperatures inside the bub-

ble are sufficient to ionize a small fraction of noble gas. The amount of


ionized gas being small, the bubble remains transparent and the flash

of light observed is caused by volume emission [15]. The electrons of

ionized atoms interact with neutral or ionized atoms causing thermal

bremsstrahlung radiation.

Thirdly, the severe conditions inside the bubble lead to formation of rad-

icals that can undergo chemical reactions [48] and contribute to the light

emission [15]. As the temperature is very high inside the bubble, the dis-

solved gas, water vapour and other volatile material may thermally be

decomposed. The OH radicals are the main radicals formed. These radi-

cals may recombine inside the bubble and contribute to the light emission

[15]. However, a non negligible part enters in the liquid phase where they

may initiate chemical reactions. The radicals in the outside layer of the

bubble can, for example, react with organic compounds such as phenol

or carbon tetrachloride and decompose them. Ultrasounds are therefore

used in water remediation to remove biological and chemical contami-

nants, in air cleaning and in land remediation [61]. Finally, the bubble

rebounds when the pressure force inside the bubble becomes higher than

the inertial force. During the rebound, the temperature and the pressure

inside the bubble decrease. As the temperature inside the bubble has

increased by one order of magnitude in less than one nanosecond during

the collapse phase, very high cooling rates are also obtained during the

bubble rebound. This high cooling rate allows amorphous nanoparticle

synthesis from molten metals. The organization and crystallization of

nanoparticles are hindered by the high cooling rate [93, 36]. It is to note

that the bubble can rebound but also disintegrate in small bubbles as a

result of interfacial instabilities appearing in the last stage of the bubble

collapse.

1.2.2 Bubble dynamics near a solid surface

The macroscopic structures of bubbles in a kHz and MHz field have been

monitored by many groups such as Mettin et al. and Matsumoto et al.

[65, 62]. However, the direct measurements (e.g. photographies) of the

dynamics of an acoustic-induced transient bubble near a solid surface

in an acoustic field are challenging due to the randomness, rapidity and


smallness of phenomenon. At our best knowledge, only Crum et al. [22]

(at a frequency of 60 Hz) and Zijlstra et al. [116] (at a frequency of

1 MHz) were able to monitor the dynamics of an acoustic-induced bub-

ble near a solid surface in an acoustic field. Note that, at 1 MHz, the

final stage of the bubble collapse can not be accurately monitored.

The main insight of the dynamics of an acoustic-induced bubble near a

solid surface in an acoustic field was obtained by indirect measurements.

Electrochemical processes have been largely used to monitor the effects

of acoustic cavitation. Indeed, acoustic cavitation has a pronounced ef-

fect on different electrochemical processes such as erosion [27], corrosion

[9], passivation of metals [78, 79], conversion of redox species [113], re-

duction of chemical compounds at electrode surface [58] and detection

of redox species such as hydrogen peroxide [10]. Other experiments use

a pressure gauge mounted on the solid surface to monitor the impulsive

stress caused by acoustic cavitation [108, 106].

To have a better insight into the bubble dynamics near a solid surface,

direct observations of bubble dynamics are required. Photographic de-

tection of the dynamics of bubbles generated in an acoustic field near a

solid surface being challenging, the bubble has been generated by other

techniques allowing to control accurately its size and its position above

the solid surface. The main technique used is the pulsed-laser discharge

[52, 74, 75, 80, 101]. In this technique, a short laser pulse with duration

in the order of a few nanoseconds or femtoseconds and energy per pulse

of a few millijoules is focused in a liquid. Thereafter, a plasma spot is

generated by optical breakdown: the pressure and the temperature in

the breakdown volume are very high and the liquid is in a supercritical

state [105]. Next, the heated material within the breakdown volume

changes into a vapour state. Note that this vapour bubble can also con-

tain non-condensable gas. As the pressure and the temperature inside

the bubble are much higher than the pressure and the temperature of

the surrounding liquid, the bubble grows. Typical experimental pho-

tographies of laser-induced bubble dynamics near a solid surface [74] are

shown in Fig. 1.3. During the growth phase, the solid surface signifi-

cantly influences the bubble dynamics and the bubble does not remain


Figure 1.3: Experimental photographies of bubble dynamics near a solid

surface for γ=1 [74]. The three rows relate to three stages of the bub-

ble dynamics: bubble growth, bubble collapse and water-jet. The solid

surface is located at the lower border of each picture.

spherical. The top of the bubble-liquid interface expands faster than the

bottom of the bubble-liquid interface. As a result, the centre of mass of

the bubble shifts away from the solid surface. After 107 µs, the bubble

reaches its maximum volume. The pressure and the temperature inside

the bubble are minimum. Next, the bubble volume decreases. This is

the collapse phase. During the bubble collapse, the bubble is seen in Fig.

1.3 to move towards the solid surface. Moreover, above the bubble due

to the asymmetric collapse, the pressure increases and a high pressure

appears above the bubble [12]. The liquid between this high pressure

zone and the bubble is therefore accelerated towards the solid surface.

As a result, the top of the bubble-liquid interface becomes flattened, as

seen at 196 µs in Fig. 1.3, and a liquid jet starts to penetrate the bubble.

The penetration of the liquid jet inside the bubble is seen in Fig. 1.3

at 204 µs. When the liquid jet hits the lower bubble-liquid interface, it

pushes it ahead. In this way a vortex ring is generated and the bubble

acquires a toroidal shape. After the bubble becomes toroidal, the jet


Figure 1.4: Experimental photographies of bubble dynamics near a solid

surface for γ=0.6 [80]. The three rows relate to three stages of the bubble

dynamics: bubble growth, bubble collapse and jet penetration. The solid

surface is located at the lower border of each picture.

goes through the liquid layer under the bubble. The time between the

onset of the jet penetration and the jet impact on the lower bubble-

liquid interface is less than 15 µs. Finally, the jet impacts on the solid

surface and spreads radially along it. The experimental parameters in

these experiments are the liquid properties and the stand-off distance,

γ, defined as the ratio between the distance separating the initial bubble

centre from the solid surface and the maximum bubble radius. In the

previous example γ is equal to one. The experimental photographies of

bubble dynamics for γ=0.6 [80] are shown in Fig. 1.4. As the bubble is

closer to the solid surface, the bubble is more flattened. Different stud-

ies of the influence of the stand-off distance on bubble dynamics near

a solid surface have been performed [80, 101]. The experiments have

shown that when the stand-off distance is higher than three, the bubble

stays almost spherical upon collapse.

Although the bubble is less compressed during its collapse near a solid

surface than in bulk liquid, shock waves can be emitted [75]. These

shock waves are schematically represented in Fig. 1.5. Three kinds of

shock waves are observed. When the jet impacts the lower bubble-liquid

interface, two shock waves are emitted from the two impact points (jet


Figure 1.5: Schematic representation of the scenario of asymmetric bub-

ble collapse with indication of the different shock waves emitted [75].

shock waves). Next, when these two shock waves converge to a point

located at the centre of the toroidal bubble, the shock waves begin to

penetrate themselves and two shock waves are emitted in the upward

and downward directions (tip bubble shock wave). Finally, a shock wave

is emitted when the bubble reaches its minimum volume (compression

shock wave). An other compressible effect only observed for γ > 1 is the

counterjet [52, 75].

During the jet impact on the thin layer of liquid being between the bub-

ble and the solid surface, a splash can be observed [101]. The term

splash refers to an annulus of liquid which is projected from the liquid

layer in a direction opposite to the jet. When the jet impacts onto a thin

liquid layer being between the solid surface and the bubble, the liquid

flow spreads radially outwards from the jet axis. This radial flow meets

the inward motion of liquid induced by the bubble collapse and a splash

is projected from the thin liquid layer in a direction opposite to that of

the jet [18]. This splash penetrates the bubble and the bubble has a

mushroom-like shape as seen in Fig. 1.6. Depending on γ values, some

portions of the bubble can dissociate from the main part of the bubble

[101], as seen in the last photography in Fig. 1.6.

1.3 Sono-processes

In the last twenty years, acoustic cavitation has been used to improve

chemical and electrochemical processes and to develop new processes.

Some examples of experimental and industrial sono-processes are de-

1.3. SONO-PROCESSES 13

Figure 1.6: The final stage of the bubble collapse for γ=1.1 [18]. (a)

Experimental photographs of the bubble motion. The time frame in-

terval is 1 µs and frame width 1.4 mm. The solid surface is located at

the upper limit of the frames. (b) Numerical calculations of the bubble

shape.

scribed below. They are classified as a function of the frequency of the

acoustic field.

1.3.1 Megasound processes

Surface cleaning

With the continuous shrinkage of critical sizes in semiconductor manu-

facturing in the microelectronics industry, nanoparticles of a size larger

than the half of the width of structures (e.g. 22.5 nm for the 45 nm tech-

nology generation) are believed to be potential killer defects for devices in

chips [6]. For the past 30 years, good particle removal efficient has been

obtained using aqueous based chemistries. Basic solutions were used

to remove particles by etching off a thin layer of substrate. However,

in order to meet the planned down-scaling set by the ITRS roadmap

[6], the cleaning has to occur with minimal substrate etching. More-

over, water based cleaning solution are not very efficient for cleaning

hydrophobic surfaces and in removing polymer residues [7]. For these

reasons and those related to cost and environmental impact, novel tech-

niques combining the use of physically-assisted particle removal tech-


niques and diluted basic solutions with low etching capability have been

tested. Acoustic cavitation-induced cleaning has shown great promise

[104]. The megasonic cleaning equipment is equipped of acoustic trans-

ducers with a power density up to 10 W/cm2 and a frequency of 1 MHz.

Two different cleaning mechanisms have been shown, as well theoreti-

cally as experimentally, to initiate the removal of nanoparticles: acoustic

streaming and acoustic cavitation [44]. While acoustic streaming allows

only poor nanoparticle removal efficiency, acoustic cavitation allows high

nanoparticle removal efficiency but involves also, in bad bath system,

damages on fragile structures [104]. A better insight into the cavitation-

induced mechanisms is required in order to optimize particle removal,

and minimize damages on structures. Recently, according to the works

of Ohl et al. [74], it is thought that the shear stress resulting from the

jet impact on the solid surface could be responsible for particle removal.

Sonoporation

Sonoporation is the rupture of cell membranes by acoustical means in

order to deliver large sized molecules into cells for therapeutic applica-

tions [66]. The mechanism that has been suggested is that the shear

stress, induced by the bubble dynamics and by the jet impact, on cells

lead to the rupture of cell membranes [3].

1.3.2 Ultrasound processes

Pickling process

In the metallurgical industry, during hot rolling and subsequent cooling,

steel acquires a surface oxide layer. The oxide layer must be removed

to prevent damage to steel strip during further processes such as cold

rolling. Traditional pickling processes involve immersion of the steel strip

in acid tanks to remove the oxide layer. However, as the steel industry

has to respect tightening environmental legislations, other techniques

have been developed. One of these techniques involves the use of ul-

trasound at a frequency of 20 kHz in acid baths. It was shown that

the major potential benefit is the reduction in pickling temperature and

1.3. SONO-PROCESSES 15

the acid concentration [39]. Observed improvements are thought to be

due to acoustic streaming, shock waves and jets. These mechanisms

are thought to remove the loosened oxide and thus replenish the metal

surface with fresh acid solutions for further pickling.

Nanofiber synthesis

In biotechnologies, strong fibres with characteristic sizes in the nanome-

tre regime are important in many applications including tissue engineer-

ing and nanocomposites. These nanofibres can be synthetized industri-

ally but are also present in nature. In nature, nanofibres with diameter

of about 30 nm are assembled to form micro sized natural fibres such as

spider and silkworm silks. The extraction of these nanofibres from mi-

crosized natural fibres can be carried out by ultrasonic techniques [115].

Natural fibres are immersed in pure water and placed in an ultrasonic

tool. The ultrasonic frequency and power are, respectively, 20 kHz and

1000 W. It is thought that acoustic cavitation-induced jets and shock

waves cause splitting of fibres along their axial direction as the bonds

between nanofibres are weaker. Therefore, the micro sized natural fibres

can be dissociated into nanofibres. The exact mechanisms responsible

of the extraction of nanofibers are however still unknown.

Nanoparticle synthesis

The production of nanoparticles by sonoelectrochemical processes has

recently gained much of scientific interest as a new cost-effective synthe-

sis method [25]. It is based on an ultrasound-assisted electrochemical

apparatus in which an anode and an ultrasonic horn cathode (sonoelec-

trode) are immersed in an electrolyte. First, a current pulse < 1 µs is

sent to the sonoelectrode to carry out a nano-morphological deposition.

Secondly, an ultrasonic pulse is sent to the sonoelectrode creating cavi-

tation bubbles in the electrolyte that allow to remove the nanoparticles

from the sonoelectrode surface and to replenish the double layer with

metal cations. Finally, a rest time allows restoring the initial electrolyte


conditions close to the sonoelectrode, after which the process cycle is re-

peated. It is generally believed that shock wave emission and jet impact

on the sonoelectrode during bubble collapse are responsible for nanopar-

ticle removal from the sonoelectrode.

In all the above described sono-processes, it is thought that the inter-

action of shock waves and jets with the solid surface are the two main

mechanisms responsible for the effects observed. A better insight into

these mechanisms is therefore required. It is why we focus our attention

in this work on:

• Shock waves emitted during spherical bubble collapse near a solid

surface.

• Jet penetration inside the bubble during asymmetrical bubble col-

lapse near a solid surface.

In the following section, a small review is given of numerical models to

simulate bubble dynamics in bulk liquid and near a solid surface.

1.4 Modeling

In 1917, Rayleigh was the first to study the radial motion of a spherical

bubble in bulk liquid [86]. Next, several models were developed. These

models can be classified into two classes depending on the liquid com-

pressibility. Section 1.4.1 describes two of these models: the Rayleigh-

Plesset model and the Gilmore model. It was only in 1971 that Chapman

and Plesset have developed a numerical model to simulate asymmetric

bubble collapse near a solid surface. Section 1.4.2 presents this model

and the further improvements. As the viscosity is ignored in these mod-

els, other numerical models are required. Section 1.4.3 reviews the main

numerical models to simulate gas-liquid multiphase flows.

1.4. MODELING 17

1.4.1 Spherical bubble dynamics

In this case, the radial motion of a spherical bubble isolated in bulk liquid

subjected to an acoustic field is studied. The only space coordinate is

the radial distance, r, from the bubble centre. Some assumptions were

introduced by Rayleigh [86], Plesset [109] and Nopthing and Neppiras

[71]:

• The acoustic wavelength is large compared with the bubble radius.

This condition is required to keep a radial symmetry.

• The acoustic pressure is superimposed on the ambient pressure,

p0, and applied at infinity.

• Only the liquid phase is discretized and adequate boundary con-

ditions are imposed at the bubble-liquid interface.

• Volume forces, such as gravity, are ignored.

• There is no mass exchange between the bubble and the liquid.

• The liquid is assumed isentropic (no viscous effects and no heat

transfer).

• The pressure and the temperature inside the bubble are assumed

spatially uniform.

• The pressure inside the bubble, pb, is the sum of the vapour pres-

sure (pv) and gas pressure (pg): pb = pv + pg.

• The gas inside the bubble is an ideal gas. The expansion and the

compression of gas follows a polytropic relation: pV κ = cst where

V and κ are, respectively, the bubble volume and the polytropic

coefficient. κ = 1 corresponds to isotherm conditions and κ = γ

to isentropic conditions inside the bubble.

Based on these assumptions, the continuity and the momentum equa-

tions in the liquid phase are:


∂ρL

∂t+

∂

∂r(ρLu) +

2ρLu

r= 0 (1.2)

∂u

∂t+ u

∂u

∂r= − 1

ρL

∂p

∂r(1.3)

u and p are, respectively, the velocity and the pressure, and ρL the liquid

density. The boundary condition at infinity is:

p∞(t) = p0 − pA sinωAt (1.4)

where pA is the amplitude and ωA the angular frequency of the acoustic

field, and p0 is the ambient pressure. The boundary condition at the

bubble-liquid interface, r = R(t) is:

Trr(liquid)|R = Trr(gas)|R +2σ

R(1.5)

where Trr is the radial component of the viscous stress tensor and σ the

surface tension. This boundary condition can also be written as:

pL = pb +4µL

3

(

∂u

∂r− u

r

)

|R − 2σ

R(1.6)

where pL is the pressure in the liquid at the bubble-liquid interface and

µL is the liquid viscosity. Although that the liquid is isentropic, the

effect of the viscosity is taken into account in eq. 1.6. The addition

of the viscosity in the pressure jump at the bubble-liquid interface does

not modify the complexity of eq. 1.6. However, the equations would

be much more complex to solve if the viscosity was included in eq. 1.3.

Moreover, we have observed that the viscosity has a small influence on

the evolution of the bubble radius as a function of time.

Incompressible liquid

Taking into account the liquid incompressibility in eqs (1.2-1.3) and

integrating these equations in the liquid from r = R to ∞, using the

boundary condition (eq. 1.4) results in [71]:

u =R2R

r2(1.7)

1.4. MODELING 19

p = p∞(t) + ρL

(

R2R + 2RR2

r− R4R2

2r4

)

(1.8)

The motion of the bubble wall is obtained inserting eqs (1.7-1.8) into eq.

1.6:

RR +3

2R2 =

1

ρL

(

(

p0 +2σ

R0

)(

R0

R

)3κ

− 4µLR

R− 2σ

R− p∞(t)

)

(1.9)

The left-hand side of this equation contains the bubble radius and its

derivatives (the velocity R and the acceleration R of the bubble wall).

The right hand side contains the driving pressure terms. The initial

conditions for eq. 1.9 are the initial bubble radius and the initial velocity

at the bubble wall. Nowadays, this equation is often called the Rayleigh-

Plesset equation.

Compressible liquid

During violent bubble collapse, the velocities of the bubble wall may

reach the speed of sound and liquid compressibility can not be neglected.

Based on analytical theories, a first order acoustic correction to the

Rayleigh-Plesset equation was derived by Herring and Trilling in 1944

[109]. In their approach a constant speed of sound is considered. Their

equation leads to more realistic values for the velocity and the pressure

at the bubble wall during violent bubble collapse. A similar approach

was followed by Keller [44]. Both are formulated in the following general

equation, where λc = 0 results in the Keller equation and λc = 1 recovers

the equation derived by Herring and Trilling.(

1 − (λc + 1) RCL

)

ρLRR + 32R2ρL

(

1 −(

λc + 13

)

RCL

)

=(

1 + (1 − λc)RCL

)

[pg(R) − p∞(t)] + RCL

pg − 4µ RR − 2σ

R

(1.10)

where CL is the speed of sound in the liquid at the bubble-liquid in-

terface. Gilmore in 1952 developed an expansion including higher order

compressibility terms.


Gilmore model

The Gilmore model is based on the Kirkwood-Bethe hypothesis [38].

This assumption states that the waves propagate with a velocity equal

to the sum of the speed of sound and the liquid velocity. The Gilmore

equation is a one-dimensional equation describing the evolution of bub-

ble radius as a function of time:

RR

(

1 − R

CL

)

+3

2R2

(

1 − R

3CL

)

= H

(

1 +R

CL

)

+RH

CL

(

1 − R

CL

)

(1.11)

with H the value of the liquid enthalpy (h) at the bubble-liquid interface.

To solve this ordinary differential equation, expressions for the enthalpy

h and the speed of sound cL in bulk liquid evaluated at the bubble-

liquid interface are required. As the liquid is isentropic, h and cL can be

expressed as a function of pressure p in the liquid as:

h =∫ pp∞

(

dpρ

)

cL =√

dpdρ

(1.12)

In order to solve eq. 1.12, the Tait equation is used to describe the

pressure p in the liquid as a function of the density ρL:

p + B

p0 + B=

(

ρL

ρ0

)nT

(1.13)

where the subscript 0 defines the initial conditions. B and nT are the

coefficients of the Tait equation. The enthalpy and the speed of sound

in the liquid at the interface then are:

H =1

ρ0

(

nT

nT − 1

)(

1

p0 + B

)

−1/nT[

(pL + B)nT −1

nT − (p∞ + B)nT −1

nT

]

(1.14)

C2L =

nT

ρ0(p0 + B)1/nT (pL + B)

nT −1

nT (1.15)

The pressure at the interface in the liquid, pL, is expressed as in eq. 1.6:

pL = pb −2σ

R− 4µL

R

R(1.16)

1.4. MODELING 21

Gilmore’s equation has been found to give very accurate results when

compared with exact solutions obtained numerically [42].

Advanced numerical models

The need to have a better insight into the sonoluminescence mechanism

and radical formation observed during the last stage of the spherical bub-

ble collapse has lead to the development of new numerical models. The

numerical models include the non-equilibrium evaporation and conden-

sation of vapour, heat conduction, gas diffusion, endothermic chemical

reactions and mass transfer of radicals occurring in the last stage of the

bubble collapse. These models have shown that the high temperatures

inside the bubble at the end of the collapse phase are responsible for

a brief flash of light (sonoluminescence) and the formation of radicals

that may promote chemical activity. Numerical results have also shown

that the water vapour entrapped inside the bubble has a profound effect

on the maximum temperature during bubble collapse. More details can

be found in the works of Storey and Szeri and Fujikawa and Akamatsu

[91, 92, 35].

1.4.2 Aspherical bubble dynamics

As shown in section 1.2.2, when a bubble collapses near a solid surface,

the bubble does not stay spherical. 2D axisymmetric or 3D numerical

calculations of bubble dynamics are then required to simulate the jet

that penetrates the bubble. The first full numerical study of bubble

dynamics near a solid surface was conducted by Plesset and Chapman

[83]. Based on their work, Chahine et al. [114], Blake et al. [11], Best et

al. [8] and other workers have developed a numerical model to simulate

bubble dynamics near a solid surface and the jet penetrating the bubble

during the collapse phase. Nowadays, it is still the more frequently used

approach.

Numerical model:


The numerical model is based on the following assumptions:

• The liquid is incompressible.

• The liquid is isentropic (no viscous effect and no heat transfer).

• The flow is irrotational.

• The gas diffusion is neglected.

• Only the liquid phase is described and adequate boundary condi-

tions are imposed at the bubble-liquid interface.

• The pressure inside the bubble is uniform and is the sum of a

constant vapour pressure (pv) and a volume-dependent non con-

densable gas pressure (pg).

• A polytropic law is used to describe the variation of the non-

condensable gas pressure (ideal gas) as a function of the bubble

volume.

According to these assumptions, the governing equations in the liquid

phase are:

∇.u = 0 (1.17)

ρLDu

Dt= −∇p (1.18)

The boundary condition far away from the bubble is:

p∞(t) = p0 − pA sinωAt (1.19)

At the bubble-liquid interface, the boundary condition is:

pL = pv + pg − σκi = pv + p0

(

V0

V

)κ

− σκi (1.20)

where κi is the curvature of the interface. As the flow is irrotational,

the liquid velocity may be represented as the gradient of a potential φ:

u = ∇φ. The continuity equation (eq. 1.17) becomes:

∇2φ = 0 (1.21)

1.4. MODELING 23

Thus φ has to satisfy the Laplace equation in the liquid. The liquid flow

being irrotational and isentropic, the momentum equation (eq. 1.18)

reduces to a Bernoulli equation:

∂φ

∂t+

1

2|∇φ|2 +

p − p∞ρL

= 0 (1.22)

At the interface, the Bernoulli equation is:

∂φ

∂t+

1

2|∇φ|2 +

pv + p0

(

V0

V

)κ − σκi − p∞

ρL= 0 (1.23)

Numerical method:

The boundary integral method is used to solve eq. 1.23 and eq. 1.21.

In this method, Laplace’s equation (eq. 1.21) is solved in integral form

based on Green’s theorem. As a result, only the boundaries of the do-

main have to be discretized and solved, which reduces considerably the

computational cost. The integral formulation of the solution to Laplace’s

equation may be written as:

c(s)φ(s) =

∫

∂Ω

(

∂φ(q)

∂nG(s, q) − φ(q)

∂G(s, q)

∂n

)

d∂Ω (1.24)

where c(s) is taken to be 1 if s ∈ Ω \ ∂Ω and 0.5 if s ∈ ∂Ω. Ω and ∂Ω

represent, respectively, the flow domain and its boundaries. G(s, q) is

Green’s function. The use of an image term in Green’s function allows

avoiding to describe the solid surface and to evaluate integrals on it.

The only boundary of the flow domain is therefore the bubble-liquid

interface. On this boundary, each discretization point x is advected by

the kinematic condition:dx

dt= ∇φ (1.25)

Given the initial conditions of the position of the interface and the values

of φ at the interface, eqs (1.23-1.25) are solved to obtain the velocities at

the interface and the new position of the interface. Note that, once the

bubble becomes toroidal, the numerical scheme has to be modified to

keep a simply-connected domain [8]. More details about this numerical

method can be found in [13, 77, 12, 11].


The main drawback of this numerical model is the assumption of in-

viscid and incompressible liquid. As the liquid viscosity is neglected in

the numerical model, the influence of the shear stress induced by the

bubble dynamics and by the radial flow resulting from the jet impact on

the solid surface cannot be calculated. Although, this shear stress was

shown by Ohl et al. [74] to be responsible for the removal of particles

sedimented onto the solid surface. This shear stress is also thought to

be responsible for surface cleaning in microelectronics, sonoporation in

biomedical applications and improvements in steel pickling performance

in the metallurgical industry. Hence, the viscosity has to be included in

the numerical model. It is why the next section reviews numerical models

based on the Navier-Stokes equations to simulate gas-liquid multiphase

flows.

1.4.3 General considerations about multiphase flows

Compared to single-phase flows, the modeling of gas-liquid multiphase

flows with deformable interface is more challenging because: (i) the dis-

continuity of the material properties such as the density has to be ac-

curately solved (e.g. the density goes from 1 to 1000 at an air-water

interface); (ii) the mass of each phase has to be conserved during the

interface advection; (iii) topology changes should be taken into account.

Different numerical models have been proposed in the literature to deal

with these challenges. These models have been classed based on the

nature of the computational mesh: a moving mesh or a fixed mesh.

In the models which are based on a moving mesh, the interface is a

boundary between two subdomains of the mesh [24]. The structured or

unstructured mesh moves with the interface. The system is treated as

two distinct flows separated by the interface and thus allows an accurate

representation of interface jump conditions such as the surface tension

force. However, when large deformations of the interface occur, a new,

geometrically adapted mesh needs to be generated or remeshed. The

remeshing can be very complicated and time consuming process, espe-

cially for significant topology changes.

1.4. MODELING 25

In the numerical models based on a fixed mesh, two different approaches

can be used to describe multiphase flows: the one-fluid approach and

the two-fluid approach [51]. In the one-fluid approach, both phases are

treated as one fluid with varying material properties. Only one con-

tinuity, one momentum and one energy equation are solved for both

phases. In the two-fluid approach, each point in the mixture is occu-

pied simultaneously by both phases. Each phase is then governed by its

own conservation equations. The coupling between phases is carried out

through interphase interaction properties. In this work, the one-fluid

approach is considered because this approach leads to a sharp interface

between the fluids [107].

In the one-fluid approach, one single set of conservation equations with

variable material properties at the interface is solved on a fixed cartesian

mesh. The interface cuts the cells of the fixed mesh. The numerical tech-

niques differ by the way to represent the moving interface and to numer-

ically calculate the interface advection. There are two main approaches:

the front tracking method and the front capturing method. In the front

tracking method, markers are used to discretize the interface. The mo-

tion of the interface is captured by the lagrangian advection of markers.

The first front tracking method was the MAC (marker-and-cell) method

developed by Harlow and Welch in 1965 [41]. Recent developments of

front tracking methods as those published by Trygvasson and co workers

[103, 102] have been used in many applications including simulations of

droplets and bubbles in a flow. The main advantage of front tracking

method is the straightforward interface definition. Surface tension can

be calculated with a high degree of accuracy [84]. The drawback of the

front tracking method is the need to re-mesh the interface by markers

during large deformation of the interface. Difficulties appear also during

bubble coalescence, break up or splitting.

The front capturing method is an Eulerian method for the interface. A

scalar indicator function discretized on the fixed grid indicates which

phase is present at a given location. The interface is then implicitly

defined by the location where the phase indicator function changes. The

motion of the interface is calculated by solving the advection equation


Figure 1.7: Representation of the interface by the Volume Of Fluid

approach (left) and the Level Set approach (right).

of the scalar-indicator function. At each time step, the interface is re-

constructed from the scalar function. The two classical front-capturing

approaches are the Volume Of Fluid (VOF) [43] and the Level Set (LS)

[98] method.

Volume Of Fluid method (VOF)

In the Volume Of Fluid method, the scalar indicator function is the

volume fraction of each fluid. The volume fraction of each fluid indi-

cates which fluid is present in a cell. A schematic representation of the

methodology is shown in Fig. 1.7 for a two-phase flow, for example a

liquid phase and a gas phase. The volume fraction is 0 for cells with pure

gas, 1 for cells with pure liquid and between 0 and 1 for cells including

the interface. At each time step, the interface is first reconstructed in

each cell from the knowledge of the volume fraction of its fluid. Once the

interface is reconstructed, its motion in the velocity field is calculated

by solving a continuity equation for the volume fraction of each phase.

In general, the continuity equation for the qth phase is:

∂

∂t(αqρq) + ∇. (αqρqu) = 0 (1.26)

1.4. MODELING 27

where αq and ρq are, respectively, the volume fraction and the density

of the qth phase.

For an incompressible phase, eq. 1.26 becomes:

∂αq

∂t+ u.∇αq =

Dαq

Dt= 0 (1.27)

The algorithms for the interface reconstruction and advection depend on

the method used to represent the interface. In 2D, there are two main

methods to represent the interface. First, the interface in each cell can

be represented by a segment parallel to one of the mesh coordinate axis

as in the work of Hirt and Nichols [43]. This is the SLIC (Simple Line In-

terface Calculation) method. Secondly, the interface can be represented

by a segment perpendicular to the normal of the interface. This is the

PLIC (Piecewise Linear Interface Calculation) method. Although the

reconstructed interface is not continuous across the boundary of adja-

cent cells, PLIC method is nowadays the more frequently used technique

[90].

In the PLIC method, the first step in the interface reconstruction is to

calculate the interface normal. This normal is calculated based on the

gradient of the volume fraction. In the second step, the exact position of

the interface is determined from volume conservation. The difficulty of

interface reconstruction is to calculate accurately the interface normal.

Young’s method [82] is an explicit method with an accuracy between

first and second order to calculate this interface normal. In this method,

a finite-difference gradient approximation of the volume fraction is used

to obtain the interface normal. The Pilliod method [81] and the recent

developments of Scardovelli and Zaleski can also be used [90] to calculate

the interface normal.

The algorithms to advect this reconstructed interface (eq. 1.27) can be

divided in two categories: unsplit schemes [54, 89] and operator split

schemes [90, 87, 40, 5]. In the operator split schemes, the fluxes of αq

across the cell faces are calculated at every time step independently and

consecutively along each coordinate direction. In the unsplit scheme,

the fluxes of αq are calculated simultaneously in all directions at each

time step. The flux can be calculated by a lagrangian [40, 87] or an


eulerian scheme [54, 89]. The lagrangian scheme originally developed by

Li [90] is described by Gueyffier for 3D numerical calculations in [40].

The reader is referred to [89] for more details on the eulerian scheme.

Recently, a new mixed split eulerian implicit - lagrangian explicit ad-

vection algorithm was developed by Scardovelli and Zaleski [90]. This

algorithm allows to exactly conserve mass for incompressible multiphase

flows and to avoid undershoots or overshoots of the volume fraction.

The material properties at the interface are defined for example for the

density as:

ρ (α) =∑

q

αqρq (1.28)

The advantage of the VOF method is its superior mass conservation com-

pared to other approaches. Moreover, the changes in interface topology

are automatically taken into account. The main disadvantage is the

need for a complicated algorithm for 3D reconstruction of the interface.

Moreover, the accuracy on the interface reconstruction has an influence

on the calculation of interface curvature and therefore on the surface

tension force.

Level Set method (LS)

In the Level Set method, the scalar indicator function is a level set

function Φ(x, t) defined in the domain. A schematic representation of

the methodology is shown in Fig. 1.7 for a two-phase flow. The value

of the level set function is negative in one phase and positive in the

other phase. The interface is described as the zero level of the level set

function Φ. For two phase flows, the level set function has the form:

Φ(x, t)

< 0 if x ∈ Ω−

= 0 if x ∈ Γ

> 0 if x ∈ Ω+

(1.29)

where Ω− and Ω+ correspond to the domains of the two phases and Γ is

the interface. The typical level set function used is the signed distance

1.4. MODELING 29

function to the interface. The interface advection in a given velocity

field is calculated by solving:

∂Φ

∂t+ u.∇Φ = 0 (1.30)

Solving this equation can produce wide spreading and stretching of the

level set function, such that Φ does not remain a distance function [60].

As a result the mass is not conserved. Redistancing algorithms such as

[95] have been developed to keep Φ as a distance function and thus to

improve mass conservation.

At the interface, the material properties are defined, for example, for

the density as:

ρ(Φ) = ρ+ (1 − H(Φ)) + ρ−H(Φ) (1.31)

where the subscript + and − specify the two phase regions. H(Φ) is the

Heaviside function [96]:

H(Φ) =

1 if Φ > 0

0 otherwise(1.32)

The discontinuous variation of these material properties and the calcu-

lation of surface tension forces can present numerical difficulties. A spe-

cific treatment is then needed to describe jump conditions numerically.

A first solution is to smooth the discontinuities at the interface. The

discontinuous transition of material properties is smoothed by replac-

ing the Heaviside function eq. 1.32 with a smooth, continuous function

denoted Hβ(Φ)

Hβ(Φ) =

1 if Φ > β12

(

1 + Φβ + 1

π sin(

πΦβ

))

if − β ≤ Φ ≤ β

0 if Φ < −β

(1.33)

where β is taken as 2-3 cells. More details can be found in [98, 96].

A second solution is the Ghost Fluid Method. The Ghost Fluid Method

(GFM) has been developed by Fedkiw et al. [32]. In this method, ghost

cells are defined on each side of the interface and appropriate schemes are


applied for jump conditions. This method resolves accurately jump dis-

continuities across the interface and avoids to have an interface thickness.

The reader can find more details in [99]. The first solution is straight-

forward to implement but requires fine grids to accurately approximate

the discontinuities. On the other hand, the Ghost Fluid Method allows

very accurate representation of jump conditions but its implementation

is relatively complex.

The main advantage of the Level Set approach is that the interface lo-

cation, its normal and its curvature can be accurately calculated. This

is very convenient for the implementation of the interface jump condi-

tions. Also, changes in interface topology are automatically taken into

account in any 2D and 3D grid. The main disadvantage is the difficulty

to conserve the mass in each phase.

Applications

Recently, few hybrid methods combining the advantages of each previ-

ously described methods have been developed as a new way to represent

the interface: VOF/markers method [5, 55] and VOF/LS method [97].

Front tracking and front capturing methods have been extensively used

in direct numerical simulations of incompressible bubble/liquid flows

[59]. The gas inside the bubble and the liquid being incompressible,

only applications where the deformation of a bubble at constant volume

is of interest can be simulated. Such applications are the rising of a

bubble in a liquid and the merging of two bubbles [40, 98, 97]. However,

the motion of bubbles in a bubbly flow has mainly been simulated by a

front tracking method [56, 31].

Considering simulations of bubble expansion and compression, most of

the numerical models are based on free surface flow models. In free

surface flow models, only the incompressible liquid is discretized and

adequate boundary conditions are fixed at the bubble-liquid interface.

The most delicate point is the treatment of the boundary conditions [94].

The bubble collapse near a solid surface was calculated by Yu et al. [112]

and Popinet et al. [85] based on a front tracking approach. Numerical

models based on the Level Set [47] and the coupled VOF-LS [94] have

1.4. MODELING 31

also been used to numerically calculate the collapse of a bubble near a

solid surface.

The main advantage of free surface flows is that the equations have not

to be solved in the gas phase, which reduces considerably the computa-

tional time. It explains why only few two-phase flow model combining

an incompressible liquid phase and a compressible gas phase have been

developed. One of these works is the one of Wemmenhove et al. [107].

They capture the interface by the Volume Of Fluid method and solve

the Navier-Stokes equations in the compressible gas phase and the in-

compressible liquid phase. The numerical model used in this work will

be based on their results.

One of the few simulations of the bubble collapse in a compressible liquid

has been performed by Nagrath et al. [70]. They solve the compressible

Navier-Stokes equations in the two phases and discretize the interface

based on the Level Set approach.

Chapter 2

Shock wave emission

Contents

2.1 Threshold conditions . . . . . . . . . . . . . . 34

2.1.1 Model . . . . . . . . . . . . . . . . . . . . . . 35

2.1.2 Comparison of threshold conditions . . . . . . 39

2.2 Shock wave propagation . . . . . . . . . . . . 44

2.2.1 Model . . . . . . . . . . . . . . . . . . . . . . 44

2.2.2 Influence of cavitation process parameters . . 48

2.2.3 Influence of gas state equation . . . . . . . . 50

2.2.4 Influence of surface tension . . . . . . . . . . 52

2.2.5 Application . . . . . . . . . . . . . . . . . . . 54

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . 55

33

34 CHAPTER 2. SHOCK WAVE EMISSION

As mentioned in the previous chapter, the interaction of shock waves

emitted by bubbles with a solid surface is thought to be responsible of

device delamination in semiconductor manufacturing, nanoparticle re-

moval from the sonoelectrode during nanoparticle synthesis,... These

shock waves can be emitted during spherical bubble collapse (bubble

collapse in a bulk liquid) or aspherical bubble collapse (bubble collapse

near a solid surface). In this chapter, we focus on shock waves emitted by

bubbles sufficiently far from the solid surface to remain spherical because

such shock waves are stronger than those emitted during aspherical bub-

ble collapse. According to Philipp et al. [80], a bubble remains spherical

when the distance separating the bubble from the solid surface is higher

than three times its maximum radius (Rmax). Therefore, bubbles at a

distance higher than 3Rmax have the same behaviour as bubbles in bulk

liquid.

In section 2.1, two threshold conditions to have the emission of a shock

wave are defined and compared to the main threshold condition for tran-

sient bubble dynamics. In section 2.2, the propagation of the shock wave

emitted by a bubble in bulk liquid is modelled. The liquid velocity at

the shock front is calculated as a function of the radial distance to the

bubble. As the shock wave attenuates when propagating in the liquid,

the interaction of the shock wave with the solid surface is stronger when

the bubble is closer to the solid surface, therefore when the bubble is at

a distance from the solid surface equal to 3Rmax. The liquid velocity

at the shock front close to the solid surface is studied as a function of

typical cavitation process parameters, the gas state equation and the

surface tension. Finally, the results are used to study the damages ob-

served on the line structures during cavitation-induced cleaning surface

in microelectronics. Note that, although the results are only shown for

1 MHz, the numerical model is also valid for other acoustic frequencies.

2.1 Threshold conditions

As already described in section 1.2, two mechanisms can explain the

emission of a shock wave in the liquid upon the end of the collapse

2.1. THRESHOLD CONDITIONS 35

phase. According to the first mechanism, a shock wave is directly emit-

ted in the liquid when the velocity of the bubble wall (R) is higher

than the speed of sound in the liquid (CL). The threshold condition is

then: R/CL = 1, R/CL being the liquid Mach number. According to

the second mechanism, a shock wave forms at the gas-liquid interface,

which first extends into the gas phase. As shown by Nagrath et al. [70],

this convergent shock wave becomes much stronger near the end of the

collapse and reflects at the centre of the bubble. The reflection of this

convergent shock wave then creates a spherically divergent shock wave

which only in a second stage propagates in the liquid. Based on this

physical phenomenon, the threshold condition for shock wave emission

can be defined as the velocity of the bubble wall (R) becoming equal to

the speed of sound in the gas phase at the bubble wall (Cg): R/Cg = 1,

R/Cg being the gas Mach number.

In this section, these two threshold conditions are compared to the

threshold condition to have a transient bubble dynamics because it seems

a source of confusion in the literature. The more frequently used thresh-

old condition to have a transient bubble dynamics is that the ratio be-

tween the maximum bubble radius and the initial bubble radius is equal

to two: Rmax/R0 = 2 [33]. This threshold condition was derived numer-

ically from the Rayleigh-Plesset equation comparing the inertial terms

that are responsible of the bubble contraction with the pressure terms

that opposite to the bubble contraction. It was shown that the velocity

of the bubble wall is similar to the speed of sound in the liquid when

Rmax/R0 = 2. At first sight, the main drawback of this threshold con-

dition is that it was only derived based on few test cases.

2.1.1 Model

The three threshold conditions (Rmax/R0 = 2, R/Cg = 1, R/CL = 1)

have been calculated based on the Gilmore model as a function of typical

cavitation parameters: the amplitude of the acoustic field and the initial

bubble radius.

The equations for the Gilmore model, already described in section 1.4.1,


are:

RR

(

1 − R

CL

)

+3

2R2

(

1 − R

3CL

)

= H

(

1 +R

CL

)

+RH

CL

(

1 − R

CL

)

(2.1)

H =1

ρ0

(

nT

nT − 1

)(

1

p0 + B

)

−1/nT[

(pL + B)nT −1

nT − (p∞ + B)nT −1

nT

]

(2.2)

C2L =

nT

ρ0(p0 + B)1/nT (pL + B)

nT −1

nT (2.3)

pL = pb −2σ

R− 4µL

R

R(2.4)

In order to solve these equations, the dynamic viscosity (µL), the surface

tension (σ), the ambient pressure in the liquid (p0), the initial density

in the liquid (ρ0) and the coefficients B and nT have to be set:

• The liquid considered is water at 300 K: µL = 0.001 Pas and σ =

0.07 N/m.

• The coefficients B and nT in eq. 1.13 are, respectively, 3500 bar

and 6.25, based on experimental NIST data [1].

• The ambient pressure and the initial density in the liquid water in

eq. 1.13 are, respectively, 1 bar and 1000 kg/m3.

Moreover, an expression for the pressure inside the bubble (pb) and the

pressure at infinity (p∞) is required:

• The pressure at infinity is the superposition of the ambient liquid

pressure and the pressure of the acoustic field:

p∞(t) = p0 − pA sin(2πfAt) (2.5)

The acoustic field is assumed to be a sinusoidal wave, which is not

modified by the presence of bubble clouds. As already mentioned,

the frequency of the acoustic field is 1 MHz.


• An expression for pb is determined based on the following assump-

tions: (i) the pressure inside the bubble is spatially uniform; (ii)

the expansion and the compression of the bubble is isentropic.

Combining this assumption with the ideal gas law (id) and the

Van der Waals law (VdW) leads to the following expressions for

the pressure inside the bubble:

pb,id =

(

p0 +2σ

R0

)(

R30

R3

)γ

(2.6)

pb,V dW =

(

p0 +2σ

R0

)(

R30 − m3

R3 − m3

)γ

(2.7)

where γ and m are, respectively, the isentropic coefficient and the

hard-core radius. These parameters are fixed to, respectively, 1.4

and R0/8.54. It can be noted that the use of more complicated

gas state equations is impossible as the temperature can not be

eliminated from the expression of the pressure inside the bubble.

As a result, only the ideal gas law and the Van der Waals law have

been combined with the isentropic relation. Note also that the

isentropic assumption is quite reasonable for the transient dynam-

ics of a bubble subjected to an acoustic field at 1 MHz. However,

in the case of lower frequencies, it is no longer valid and a poly-

tropic coefficient depending on the bubble dynamics should be used

instead of the isentropic coefficient [92, 53].

R, Rmax and CL can therefore be calculated solving eqs (2.1-2.7). The

last variable to model is the speed of sound in the gas phase. The speed

of sound in the gas phase, which is required to calculate the gas Mach

number, is defined considering the ideal gas law as:

Cg =√

γRidTb (2.8)

An expression for the temperature inside the bubble (Tb) is obtained

from the isentropic condition and the ideal gas law:

pb

pb,0=

(

Tb

Tb,0

)γ

γ−1

(2.9)


0

2000

4000

6000P

ress

ure

at b

ubbl

ew

all [

bar]

2 4 6 8 10−600−400−200

0200400600

Vel

ocity

at b

ubbl

ew

all [

m/s

ec]

Bubble radius [µm]

(a)

(b)

GilmoreNavier−Stokes

GilmoreNavier−Stokes

Figure 2.1: Comparison of (a) the pressure and (b) the velocity of the

bubble wall as a function of the bubble radius obtained from our own

calculations based on the Gilmore model with the values from a full

Navier-Stokes simulations taken from Ref. [70] (R0 = 10 µm).

with Tb,0 = 300 K the initial temperature inside the bubble.

Before applying the Gilmore model to calculate the threshold condi-

tions, the Gilmore model is validated by comparison with a full Navier-

Stokes simulation of hydrodynamic bubble collapse from the literature

[70]. This simulation was based on a numerical level-set approach in

which the full Navier-Stokes equations were used for the two phases and

an ideal gas law was assumed inside the bubble. As to the simulation

conditions, a spherical air bubble with a radius of 10 µm and initially

at atmospheric pressure was compressed by the surrounding pressurized

liquid at 100 atmospheres. Figure 2.1 compares the pressure and the

velocity of the bubble wall as a function of the bubble radius, obtained

from our own calculations based on the Gilmore model, with the values

from the full Navier-Stokes simulation found in the literature. It can be

seen that the results are very similar, confirming the reliability of the

Gilmore model for our calculations.


Figure 2.2: Evolution of the bubble radius as a function of time on one

acoustic cycle in a 1 MHz acoustic field. R0 = 1 µm, R0 = 0 m/s and

pA = 3 bar.

2.1.2 Comparison of threshold conditions

The calculation of the three threshold conditions requires to calculate

the bubble dynamics from the Gilmore model (eqs (2.1-2.9)). The pa-

rameters in the Gilmore model are then the initial bubble radius R0, the

initial velocity of the bubble wall R0 and the amplitude of the acoustic

field pA. The evolution of the bubble radius as a function of time for

the case when R0 = 1 µm, R0 = 0 m/s and pA = 3 bar is shown for

one acoustic cycle in Fig. 2.2 and for six acoustic cycles in Fig. 2.3.

As seen in Fig. 2.3, the evolution of the bubble radius as a function of

time is different during each acoustic cycle because the bubble radius

and the velocity of the bubble wall at the beginning of each acoustic

cycle are different. In this chapter, we will calculate the three threshold

conditions (R/Cg = 1, R/CL = 1, Rmax/R0 = 2) only for one acoustic

cycle and for a bubble initially at rest. From Fig. 2.2, the ratio between

the maximum bubble radius and the initial bubble radius (Rmax/R0)

can directly be deduced. As the velocity of the bubble wall is known

and the speed of sound in the gas and in the liquid have been calculated


0 1 2 3 4 5 60

1

2

3

4

5

Acoustic cycles

Bub

ble

radi

us [µ

m]

Figure 2.3: Evolution of the bubble radius as a function of time on six

acoustic cycles in a 1 MHz acoustic field. R0 = 1 µm, R0 = 0 m/s and

pA = 3 bar.

from, respectively, equations 2.3 and 2.8, the liquid and gas Mach num-

ber can be calculated at any moment in time. The maximum liquid and

gas Mach numbers as well as the maximum ratio Rmax/R0 were then

calculated for an extended range of the experimental cavitation process

parameters (0.1 < R0 < 10 µm and 1 < pA < 6 bar). Note that, in

the microelectronics industry, the effective acoustic pressure in cleaning

baths ranges between 1 and 2 bar [44]. From these calculations, the

contour lines corresponding to the three threshold conditions were then

plotted in a stability diagram, where the abscis and the ordinate are the

two experimental parameters R0 and pA. This is illustrated in Fig. 2.4

for the threshold condition: R/Cg = 1. It can be seen in Fig. 2.4 that

the threshold value is influenced by the number of acoustic cycles taken

into account in the study of bubble dynamics. This is in fact the case

for all three threshold conditions considered. For one cycle, it is seen in

Fig. 2.4 that the gas Mach number can become equal to one at fixed

R0 for several amplitudes of the acoustic field. For this range of exper-

imental parameters (R0 = 2-3.5 µm), the inertial motion of the bubble


0.1 1 101

2

3

4

5

6

Bubble radius [µm]

Am

plitu

de o

f aco

ustic

fiel

d [b

ar]

1 cycle2 cycles

Figure 2.4: Stability diagram where the threshold to have shock wave

emission (R/Cg = 1) is plotted for one and two acoustic cycles.

controls its motion. As a result, bubble growth takes place during most

of the acoustic period so that the bubble does not have enough time to

collapse during the first acoustic cycle. Instead, it only collapses during

the second cycle, as shown in Fig. 2.5. Therefore, the number of acoustic

cycles to be considered for the calculation of the thresholds R/CL = 1

and R/Cg = 1 when the bubble does not collapse during the first acoustic

cycle should be equal to two if the threshold Rmax/R0 = 2 is considered

during the first acoustic cycle. However when the bubble collapses in the

same cycle as the one in which the bubble reaches its maximum radius,

the number of acoustic cycles to be considered should be the same for

all thresholds. Finally, all the three threshold conditions are compared

in Fig. 2.6. As expected, the two threshold conditions (R/Cg = 1 and

R/CL = 1) for shock wave emission are different as the speed of sound

is higher in the liquid than in the gas. However, these two threshold

conditions look similar at high amplitude of the acoustic field and when

the bubble radius is much lower than its resonant radius (3.7 µm for a

frequency of 1 MHz). It is only apparent because in this zone a small

increase in pA results in a much more violent collapse (strongly increas-


0

4

8

12B

ubbl

e ra

dius

[µm

]

−4

0

4

Pre

ssur

e of

aco

ustic

fiel

d [b

ar]

0 0.5 1 1.5 2

−2

0

2

Number of cycles

Gas

Mac

h nu

mbe

r

Figure 2.5: Bubble radius (a) and gas Mach number (b) as a function of

the number of acoustic cycles for R0 = 3 µm and pA = 4 bar.

0.1 1 101

2

3

4

5

6

Initial bubble radius [µm]

Am

plitu

de o

f the

aco

ustic

fiel

d [b

ar]

v/CL=1

v/Cg=1 R

max/R

0=2

Figure 2.6: Stability diagram for two acoustic cycles including all 3

threshold conditions.


4.5 4.6 4.7 4.8 4.9 50

0.5

1

1.5

2

Amplitude of the acoustic field [bar]

Mac

h nu

mbe

r

liquidgas

Figure 2.7: Mach number in the gas and liquid phase as a function of

the amplitude of the acoustic field for a bubble radius of 0.2 µm.

ing Mach number as shown at Fig. 2.7) as discussed by Blake [109, 44].

Moreover, it is seen in Fig. 2.6 that the direct emission of a shock wave

in the liquid requires a higher amplitude of the acoustic field. When

comparing the two threshold conditions to have a shock wave emission

with the one for transient bubble dynamics (Rmax/R0 = 2), it can be

seen in Fig. 2.6 that they are only similar when the initial bubble radius

is smaller than its resonant radius.

As a conclusion, we cannot use a unique threshold condition for transient

bubble dynamics. Different threshold conditions have to be defined for

each phenomena observed during the collapse phase (e.g. shock wave

emission, light emission and radical formation) as it was done here for

the emission of a shock wave. We have determined in which range of

R0 − pA parameters, a shock wave can be emitted during the collapse

of a spherical bubble in an acoustic field. In the next section, we will

study the propagation of the emitted shock wave in the liquid.


2.2 Shock wave propagation

2.2.1 Model

The propagation in the liquid of the shock wave emitted by a spherical

bubble submitted to an acoustic field is studied. First, the time evolution

of the pressure and velocity fields in the liquid during the propagation

of the shock wave is modeled.

This modeling is based on the Kirkwood-Bethe hypothesis. According

to this assumption, the quantity Y (defined as r(h+u2/2), with r being

the radial distance to the bubble centre and h+u2/2 the total enthalpy)

propagates in the liquid with the velocity cL + u [38]:

∂Y

∂t= − (cL + u)

∂Y

∂r(2.10)

This homogeneous equation is then solved by the method of charac-

teristics. The equation being homogeneous, Y is constant along the

characteristic curves. The characteristic curves for eq. 2.10 are defined

as curves r(t) with a direction given by:

dr

dt= cL (r(t)) + u (r(t)) (2.11)

where cL(r(t)) and u(r(t)) are the speed of sound and the velocity in the

liquid along the characteristic curves. As these quantities vary along the

characteristic curves, the following expressions for their time derivatives

along the characteristic curves are used [45]:

dudt = 1

r(cL−u)

[

(cL + u) Yr − 2uc2

L

]

dpdt = ρ0

r(cL−u)

(

p+Bp0+B

)1/n [

2c2Lu2 − cL(cL+u)

r Y] (2.12)

Note that the variables u(r(t)), cL(r(t)) and p(r(t)) are written as u, cL

and p in eq. 2.12 for simplicity. Solving eqs (2.11-2.12) simultaneously

gives the direction of the characteristic curves, as well as the velocity

and the pressure along these characteristic curves. The initial condi-

tions for (r, u, p) in these equations are their values at the bubble wall

(r = R, u = R, p = pL). These values are calculated by the Gilmore

2.2. SHOCK WAVE PROPAGATION 45

807.8 808 808.2 808.4 808.6 808.8

0.5

1

1.5

2

2.5

Time [ns]

Rad

ial d

ista

nce

to th

e bu

bble

cen

tre,

r [µ

m]

Figure 2.8: Characteristic curves originating near the minimum bubble

radius. The latter has been calculated for R0 = 0.6 µm and pA = 4 bar.

model defined in the previous section. As a result, the model allows to

calculate the velocity and the pressure field inside the liquid by integra-

tion of their time derivatives along characteristic curves, starting from

their values at the bubble wall as obtained from the Gilmore model. A

similar method was described in [28] to study laser-induced cavitation.

Examples of characteristic curves r(t), originating from the curve rep-

resenting the bubble radius as a function of time are shown in Fig. 2.8,

for an initial bubble radius R0 of 0.6 µm and an acoustic amplitude pA

of 4 bar. It can be demonstrated that the first intersection between two

characteristic curves after the bubble collapse implies the generation of

a shock wave, as this intersection physically represents a discontinuity

in u and p [45]. The further intersections at larger times, indicated by

black dots in Fig. 2.8, represent the propagation of the shock wave. In


all simulations, the cavitation process parameters R0 and pA have been

taken such as a shock wave is emitted. These cavitation process param-

eters were calculated in the previous section.

From u(r(t)) along the characteristic curves, the velocity u inside the

liquid can then be deduced as a function of the distance to the bubble

centre for several times after the bubble collapse (Fig. 2.9(a)). The

liquid velocity at the shock front at t2 for instance is the local velocity

maximum along the curve representing the velocity in the liquid at t2.

This velocity can then be plotted as a function of the corresponding dis-

tance to the bubble centre, as shown in Fig. 2.9(b) where it is seen to

decrease with r.

Now we pay more attention to the velocity at the shock front near the

solid surface. As the simulations are only valid when the bubble stays

spherical during bubble collapse, the bubble has to be at a distance from

the solid surface such as the bubble stays spherical. Based on experi-

ments of Philipp and Lauterborn [80], the bubble stays almost spherical

upon collapse when the distance r separating the bubble centre from

the solid surface is higher than three times the maximum bubble radius

Rmax (r > 3Rmax). As the liquid velocity at the shock front decreases

with r, the liquid velocity at the shock front near the solid surface will

be the highest when the bubble is at a distance equal to 3Rmax from

the solid surface. All velocity calculations presented below are therefore

done for the condition γ = 3. Hence, with Rmax = 4.2 µm for the con-

ditions considered in Fig. 2.9, the γ = 3 condition implies r = 12.5 µm.


1 10 1000

5

10

15

20

25

30

Radial distance to the bubble center, r [µm]

Vel

ocity

in th

e liq

uid,

u [m

/s]

t1

Velocity at the bubble wall

t4

t3

t2

t5

Liquid velocity at the shock front at t

2

t5

t4

t3

1 10 1000

5

10

15

20

25

30

Radial distance to the bubble center [µm]Liqu

id v

eloc

ity a

t the

sho

ck fr

ont [

m/s

]

γ > 3

t1

t2

t3

t4 t

5

Figure 2.9: (a) Velocity in the liquid as a function of the radial distance

to the bubble centre for fixed times ti after the bubble collapse. (b)

Liquid velocity at the shock front as a function of the radial distance

to the bubble centre. The condition for which the ratio between the

distance separating the bubble from the solid surface and the maximum

bubble radius, γ, is higher than 3 is shown in the figure. The times tiare the same in both figures. R0 = 1 µm, pA = 3 bar, fA = 1 MHz.


0 0.5 1 1.5 2 2.50

10

20

30

40

50

Initial bubble radius, R0 [µm]

Liqu

id v

eloc

ity a

t the

sho

ck fr

ont [

m/s

]

5 bar4 bar3 bar

Figure 2.10: Liquid velocity at the shock front when γ = 3 as a function

of R0 for different acoustic amplitudes pA and assuming an ideal gas law.

The frequency of the acoustic field is 1 MHz. The solid lines are a guide

to the eye.

2.2.2 Influence of cavitation process parameters

The liquid velocity at the shock front calculated for γ = 3 is shown in

Fig. 2.10 as a function of the cavitation process parameters R0 and pA.

These calculations have been done assuming an ideal gas and a value

for the surface tension at the bubble-liquid interface equal to 0.07 N/m.

It is seen that the liquid velocity at the shock front increases with the

acoustic amplitude, and goes through a maximum as a function of R0.

Figure 2.11 shows that this behaviour is qualitatively very similar to

the one for the maximum velocity at the bubble wall, the latter being

calculated directly from the Gilmore model in Fig. 2.11. This indicates

that when the maximum velocity at the bubble wall increases, shock

wave emission becomes more violent, resulting in turn in an increase

of the liquid velocity at the shock front. It should be noted that the

corresponding variation of Rmax with the cavitation process parameters

also implies that the liquid velocity at the shock front corresponding to

r = 3 ·Rmax is each time taken at a different distance to the bubble


0 0.5 1 1.5 2 2.50

1000

2000

3000

4000

5000

6000


Max

imum

vel

ocity

at t

he b

ubbl

e w

all [

m/s

]

5 bar4 bar3 bar

Figure 2.11: Maximum velocity at the bubble wall as a function of R0

for three amplitudes of the acoustic field. The abscissas of the discrete

symbols on the calculated continuous curves are the same as in Fig. 2.10.

centre. It was verified, however, that this condition does not change the

intrinsic trend shown in Fig. 2.10 regarding the influence of the cavita-

tion process parameters on the liquid velocity at the shock front. For

an amplitude of the acoustic of 3 bar, a shock wave is emitted in the

liquid (see Fig. 2.9) while the velocity of the bubble wall is smaller than

the speed of sound in the liquid (see Fig. 2.4). It means that the shock

wave is not emitted at the bubble-liquid interface but in the liquid at

some distance to the bubble-liquid interface when the velocity field in

the liquid becomes higher than the speed of sound in the liquid.

Coming back to Fig. 2.8, it is seen that one of the two characteris-

tic curves that define the intersection points indicated by black dots, is

always the same. This characteristic curve comes from the minimum

bubble radius reached during bubble collapse. Hence, the velocity along

this characteristic curve should also represent the liquid velocity at the

shock front. This was verified explicitly at the γ = 3 condition by com-

paring the velocities obtained from the calculation procedure run with

all the characteristic curves to those using only the single characteristic


0 0.5 1 1.5 2 2.55

10

15

20

25

30

35


Liqu

id v

eloc

ity a

t the

sho

ck fr

ont [

m/s

]

Van der Waals lawIdeal gas law

Figure 2.12: Liquid velocity at the shock front for γ = 3 as a function

of R0 for pA = 4 bar, considering both an ideal gas law and a Van der

Waals law with m = R0/8.54 and σ = 0.07 N/m. The frequency of the

acoustic field is 1 MHz. The solid lines are a guide to the eye.

curve coming from the minimum bubble radius. It may be noted that

this is also true for any radial distance to the bubble centre, and not

only for the γ = 3 condition. This observation greatly simplifies the

modeling procedure.

2.2.3 Influence of gas state equation

The liquid velocity at the shock front as a function of R0 has also been

calculated using the Van der Waals law (VdW) instead of the ideal gas

law. The effect of using these different gas state equations on the liquid

velocity at the shock front is shown in Fig. 2.12 for pA = 4 bar. It is

seen that the two state equations give very similar results. When the

VdW-radius tends to its hard-core value, the pressure inside the bubble

tends to infinity (cfr. eq. 2.7). Hence, the hard-core Van der Waals

radius sets a lower-limit on the bubble radius, as the infinite pressure

inside the bubble will stop the inward bubble motion and will trigger

the bubble rebound. As it can be seen in Fig. 2.13, the velocity at


807.5 808 808.5

0.05

0.10

0.15

0.20

Time [ns]

Bub

ble

radi

us [µ

m]

807.5 808 808.5

−2000

0

2000

Time [ns]

Vel

ocity

at t

he b

ubbl

e w

all [

m/s

]

(a)

(b)

VdW gasideal gas

VdW gasideal gas

Figure 2.13: Effect of gas state equation on the bubble dynamics near the

minimum bubble radius, showing the bubble radius (a) and the velocity

at the bubble wall (b) as a function of time. R0 = 0.6 µm, pA = 4 bar,

σ = 0.07 N/m.

the bubble wall will then become smaller than that predicted by the

ideal gas law. It is to note that the bubble radius and the velocity

at the bubble wall predicted by a more realistic state equation, like

Redlich-Kwong-Soave, would have to be between those predicted by the

Van der Waals gas and by the ideal gas, as this equation additionally

takes into account the attractive force between the molecules. Despite

the velocity at the bubble wall being smaller when using a Van der

Waals law, it turns out that the liquid velocity at the shock front is

still slightly above that predicted using the ideal gas law. This can

be explained based on the invariant quantity Y = Ymin defining the

characteristic curve corresponding to the minimum bubble radius, which,

as outlined above, is the one that determines the liquid velocity at the

shock front. At the minimum bubble radius, Ymin is equal to Rmin ·Hmin

because the velocity-term in Y = r(

h + u2/2)

becomes zero. Moreover,

Ymin is higher when a Van der Waals gas is considered because the

higher minimum bubble radius predicted by this gas state equation turns


out to prevail on the smaller enthalpy. At r = 3 ·Rmax, Y is equal to

3 ·Rmax

(

h + u2/2)

and has the same value than at the minimum bubble

radius, as Y is invariant along the characteristic curves. Because the

two gas state equations predict the same maximum bubble radius Rmax

and as Y is highest when a Van der Waals gas is considered, both the

enthalpy and the velocity on the characteristic curve are higher for a

Van der Waals gas than for an ideal gas.

2.2.4 Influence of surface tension

In all previous calculations, the value of the surface tension at the

bubble-liquid interface was fixed to 0.07 N/m. The effect of two other

surface tension values, respectively 0.05 N/m and 0.03 N/m, has been

studied as well. Experimentally, such a decrease could correspond to

the addition of surfactants. Figure 2.14 shows the calculated liquid ve-

locity at the shock front as a function of the initial bubble radius for

these three surface tension values. The amplitude of the acoustic field

is fixed to 4 bar and the gas inside the bubble is considered to obey the

ideal gas law. It is seen that, for the same R0 value, the liquid velocity

at the shock front decreases with increasing surface tension. This can be

linked directly to the influence of the value of the surface tension on the

bubble dynamics, as shown in Fig. 2.15. Figure 2.15(b) shows that the

maximum velocity at the bubble wall increases when the value of surface

tension decreases. This indicates again that when the maximum velocity

at the bubble wall increases, shock wave emission becomes more violent,

resulting in turn in an increase of the liquid velocity at the shock front.

Moreover, for a fixed value of R0, the maximum velocity at the bubble

wall is linked to the maximum bubble radius reached during the bub-

ble dynamics. Indeed, when the value of the surface tension decreases,

bubble expansion will become less restricted, resulting in a higher max-

imum bubble radius and a higher maximum velocity at the bubble wall,

as confirmed in Fig. 2.15.


0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

Initial bubble radius [µm]

Liqu

id v

eloc

ity a

t the

sho

ck fr

ont [

m/s

]

0.03 N/m0.05 N/m0.07 N/m

Figure 2.14: Liquid velocity at the shock front for γ = 3 as a function of

R0 for pA = 4 bar and for an ideal gas law, considering three values of

surface tension: 0.03 N/m, 0.05 N/m, 0.07 N/m. The frequency of the

acoustic field is 1 MHz. The solid lines are a guide to the eye.

0 0.2 0.4 0.6 0.8 10

2

4

6

Time [µsec]

Bub

ble

radi

us [µ

m]

0 0.2 0.4 0.6 0.8 1

−4000−2000

020004000

Time [µsec]

Vel

ocity

at t

hebu

bble

wal

l [m

/s]

(a)

(b)

0.03 N/m0.05 N/m0.07 N/m

Figure 2.15: Bubble radius (a) and velocity at the bubble wall (b) as a

function of time for three values of surface tension. R0 = 0.4 µm and

pA = 4 bar.


2.2.5 Application

In the microelectronics industry, the fabrication of semiconductor de-

vices requires the conception of structures with a high aspect ratio (i.e.

the height of the structures exceeds multiple times its width). In the

final devices, the gaps between the structures are filled with different

materials (e.g. dielectric materials), which act as a mechanical support

for these structures. However, during the fabrication process, there are

several steps (e.g. surface cleaning) where such support has not been

established yet and the structures can be therefore more easily broken.

As already mentioned in section 1.3, damages are observed on the struc-

tures during cavitation-induced surface cleaning. Two different types

of damages may occur: (i) physical damage and (ii) structural damage

[44]. The structural damage, which is the break-off of small pieces of

lines, is considered here. We wonder whether the interaction of shock

waves emitted during spherical bubble collapse with the structures can

be responsible for the observed damages. To this purpose, the fracture

force of the line structure is compared to the force induced on the struc-

ture by the shock wave. The polysilicon line structures considered in

this study have a length L, a width e between 20 nm and 70 nm, and a

height hp = 150 nm. The fracture force per unit length Ff/L is given

by Gere [37] as:

Ff

L=

σf

6

e2

hp(2.13)

with σf the fracture stress. In our calculations, the fracture stress is

taken between 1 and 4.48 GPa based on [44, 100]. The force induced by

the shock wave on the structure per unit length Fs/L is:

Fs

L=

1

2ρLuscLhp (2.14)

with us the liquid velocity at the shock front. In our calculations, two

values for us are studied: us = 8 m/s and us = 48 m/s. These values

correspond to, respectively, the minimum and the maximum value of

the liquid velocity at the shock front in Fig. 2.10. Ff/L and Fs/L are

compared as a function of the width of the line structures in Fig. 2.16.

2.3. CONCLUSION 55

20 30 40 50 60 700

5

10

15

20

25

Line width [nm]

For

ce p

er u

nit l

engt

h [N

/m]

σf = 4.48 GPa

σf = 1 GPa

σf = 3 GPa

us = 48 m/s

us = 8 m/s

Figure 2.16: Fracture force on line structures (solid lines) and force

induced by the shock wave (dotted lines) as a function of the line width.

σf is the fracture stress and us the liquid velocity at the shock front.

For e > 70 nm, the shock wave induced during spherical bubble collapse

can not fracture the line structures. For e < 70 nm, the probability to

damage the weaker line structures increases when the width of the line

structures decreases. Note that these damages will occur only around

spots of high pressure (us = 48 m/s, corresponding to an amplitude of

the acoustic field of 5 bar).

2.3 Conclusion

In this chapter, we have defined two threshold conditions for the emission

of a shock wave during the collapse of a spherical bubble. A shock

wave is emitted in the liquid when the velocity of the bubble wall is

higher than (i) the speed of sound in the liquid phase (direct emission

of a shock wave in the liquid); (ii) the speed of sound in the gas phase

(emission of a shock wave inside the bubble that reflects at its centre

and, finally, propagates in the liquid). The comparison of these two

threshold conditions with the more frequently used threshold condition


for transient bubble dynamics (Rmax

R0= 2) has shown that:

• The number of acoustic cycles taken into account in the bubble

dynamics influences the threshold comparison.

• The threshold for shock wave emission R/Cg = 1 is similar to the

threshold for transient bubble dynamics Rmax/R0 = 2 only in the

sub-resonance regime.

• Different threshold conditions have to be defined for each phenom-

ena observed during the collapse phase.

In this chapter, we have also studied the propagation of the shock wave

in the liquid. The liquid velocities at the shock front have been nu-

merically calculated combining the Gilmore model and the method of

characteristics. The results show that:

• For calculating the liquid velocity at the shock front, only the

characteristic curve coming from the minimum bubble radius is

required.

• The liquid velocity at the shock front increases with the amplitude

of the acoustic field.

• The liquid velocity at the shock front as a function of the initial

bubble radius goes through a maximum.

• The liquid velocity at the shock front is slightly higher when a Van

der Waals gas is considered as compared to the one predicted by

an ideal gas.

• The liquid velocity at the shock front decreases with increasing

surface tension.

Finally, we have shown that the shock waves can be responsible of

damages observed on microstructures during cavitation-induced surface

cleaning.

Chapter 3

Gas-liquid multiphase flow

model: description and

validation

Contents

3.1 Numerical model . . . . . . . . . . . . . . . . . 60

3.1.1 Compressible Navier-Stokes equations . . . . 60

3.1.2 Surface tension force . . . . . . . . . . . . . . 62

3.1.3 Volume Of Fluid method . . . . . . . . . . . 63

3.2 Numerical method . . . . . . . . . . . . . . . . 64

3.2.1 Modification of Navier-Stokes equations . . . 65

3.2.2 Discretization schemes . . . . . . . . . . . . . 67

3.2.3 Algorithm . . . . . . . . . . . . . . . . . . . . 71

3.2.4 Gas-liquid interface . . . . . . . . . . . . . . . 77

3.3 Test case . . . . . . . . . . . . . . . . . . . . . . 83

3.3.1 Adiabatic conditions . . . . . . . . . . . . . . 86

3.3.2 Heat transfer through the interface . . . . . . 92

3.4 2D axisymmetric simulations . . . . . . . . . 94

3.4.1 The problem definition . . . . . . . . . . . . . 94

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . 97

57

58 CHAPTER 3. GAS-LIQUID MULTIPHASE FLOW MODEL

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . 102

59

As already mentioned in section 1, if bubble collapse takes place suf-

ficiently close to a solid surface, the bubble loses its sphericity, and a

liquid jet starts to penetrate inside the bubble near the end of the col-

lapse phase. When the jet reaches the lower bubble-liquid interface,

the jet forms an indentation into the lower bubble liquid-interface and

pushes it ahead [75]. Next, the bubble acquires a toroidal shape. After

the jet has traversed the liquid between the bubble and the solid surface,

the jet finally impacts on the solid surface and further radially spreads

along it. Recently, it was shown experimentally by Ohl et al. [74] that

the shear stress induced by the radial flow after jet impact on the solid

surface is responsible for the removal of particles sedimented onto the

solid surface. As only few experimental techniques are available to di-

rectly address the shear stress [26], numerical calculations are another

tool to have a better insight into the physico-chemical and geometrical

parameters governing the sequence of jet-related cavitating events dur-

ing asymmetric bubble collapse (jet penetration, jet impact on the lower

bubble-liquid interface and jet impact on the solid surface).

It is why in this chapter we focus our attention on the development of a

numerical model and method to simulate bubble dynamics near a solid

surface. An essential requirement for such numerical models is to include

the liquid viscosity, as no shear force can otherwise be associated with

the radial flow pattern resulting from the jet impact. Although a variety

of numerical models have already been described to simulate asymmetric

bubble collapse near a solid surface [83, 114, 12, 8], most of these models

assume that the liquid is inviscid. The few ones that have been able to

include the liquid viscosity are free surface flow model. In these models,

only the liquid phase is discretized and adequate boundary conditions

are imposed at the bubble-liquid interface. These models differ only by

the way the bubble-liquid interface is represented. There are two main

approaches to track the interface: the front tracking method and the

front capturing method (see section 1.4.3). Front capturing methods are

better convenient to address the jet penetration inside the liquid layer

below the bubble and the final impact on the solid surface.

In this chapter, a numerical model based on the one-fluid model is de-


scribed. Section 3.1 describes in details the numerical model. The devel-

opment of a numerical method is challenging due to the coupling between

a compressible gas phase and an incompressible liquid phase. Section 3.2

presents the discretization schemes and the algorithms used. In section

3.3, the gas-liquid multiphase flow model is compared to 1D analytical

models for the case of: (i) a water piston; (ii) the spherical bubble dy-

namics in an acoustic field. Finally, 2D axisymmetric simulations of the

collapse of a bubble in bulk liquid are performed.

3.1 Numerical model

We assume that the gas is compressible (ideal gas), while the liquid is

incompressible. Moreover, the liquid and the gas phases are immisci-

ble. The diffusion of gas through the gas-liquid interface and the liquid

condensation/evaporation are neglected. The motion of these two im-

miscible phases is modelled by the one-fluid approach; i.e. the gas and

the liquid phases are treated as one-fluid with varying material proper-

ties at the gas-liquid interface. The one set of compressible Navier-Stokes

equations is solved on a fixed Cartesian mesh and the gas-liquid interface

is tracked by the Volume Of Fluid method (see section 1.4.3).

3.1.1 Compressible Navier-Stokes equations

Applying mass conservation to a fixed volume Ω of the flow domain with

boundary S, the integral form of the continuity equation is:

∫

Ω

∂ρ

∂tdΩ +

∮

S(ρu) ·n dS = 0 (3.1)

with u the velocity, ρ the density and n the normal at the boundary S.

Using Gauss’ divergence theorem, the advective differential form of the

continuity equation is:

∂ρ

∂t+ ρ∇·u + u · ∇ρ = 0 (3.2)

3.1. NUMERICAL MODEL 61

As for mass conservation, the integral form of the momentum equation

for a volume Ω with a boundary S is:

∫

Ω

∂(ρu)

∂tdΩ +

∮

Sρu (u ·n) dS +

∮

Spn dS

−∮

Sµ(

∇u + ∇uT)

n dS −∫

ΩρFσdΩ = 0 (3.3)

with pressure p, dynamic viscosity µ and surface tension force Fσ. The

terms in the equation are, respectively, the temporal, the convective,

the pressure, the diffusion and the surface tension terms. Note that

the viscosity is explicitly taken into account in the model. By applying

Gauss’ theorem, the advective differential form of eq. 3.3 is:

∂u

∂t+ (u · ∇u) +

1

ρ∇p − 1

ρ∇·

(

µ(

∇u + ∇uT))

− Fσ = 0 (3.4)

The advective form of eq. 3.4 is used for stability reasons [107].

The energy conservation is applied to a volume Ω of the flow domain

with boundary S. The integral form of energy equation is:

∫

Ω

∂ (ρE)

∂tdΩ +

∮

S((ρE + p)u) ·ndS =

∮

S(k∇T ) ·ndS (3.5)

with total energy E = cpT + u2/2, temperature T and thermal con-

ductivity k. Note that the viscous dissipation term is neglected in this

equation. By applying again Gauss’ theorem, the advective differential

form of eq. 3.5 is:

ρ

(

∂E

∂t+ u · ∇

(

E +p

ρ

))

= ∇· (k∇T ) (3.6)

In this equation, the total energy is averaged according to the mass

fraction of the two phases:

ρE = αLρL

(

cp,LT + u2/2)

+ (1 − αL)ρg

(

cp,gT + u2/2)

(3.7)

with αL the volume fraction of the liquid phase, cp the specific heat at

constant pressure, ρL the liquid density and ρg the gas density. Note


that in equations 3.1-3.6, the material properties (ρ, µ, k) depend on the

phase inside the cell (see section 3.1.3).

To close the system of equations, an equation of state is required for the

gas density ρg. The gas being assumed to behave like an ideal gas, ρg is

defined as:

ρg =pMm

RidT(3.8)

The compressible Navier-Stokes equations are then eqs. (3.2, 3.4, 3.6,

3.8).

An alternative way to solve the energy equation is the polytropic model.

Polytropic model

In the polytropic model, the compression and the expansion of the gas

phase is characterized by:

ρg = ρg,ref

(

p

pref

)1

κ

(3.9)

with ρg,ref the reference gas density, pref the reference pressure and κ

the polytropic coefficient. This polytropic relation replaces the energy

equation. The energy equation is captured in the κ coefficient. Indeed,

κ = 1 and κ = cp,g/cv,g correspond, respectively, to an isotherm and an

adiabatic motion of an ideal gas. cv,g is the specific heat at constant vol-

ume for the gas. When using the polytropic relation, the compressible

Navier-Stokes equations are then eqs (3.2, 3.4, 3.9).

In this chapter, the two set of equations (eqs. (3.2, 3.4, 3.6, 3.8) and

(3.2, 3.4, 3.9)) will be solved (see section 3.3.1).

3.1.2 Surface tension force

The surface tension term Fσ in eq. 3.4 is computed using the Continuum

Surface Force model proposed by Brackbill et al. [14]. In this model,

the surface tension force at the bubble-liquid interface is expressed, using

the divergence theorem, as a volume force in the momentum equation.

This volume force is given by:

Fσ = σρκi∇αL

0.5(ρL + ρg)(3.10)

3.1. NUMERICAL MODEL 63

with σ the surface tension and κi the curvature of the interface. The

curvature is defined as:

κi = ∇.

(

n

‖n‖

)

(3.11)

with n the normal at the interface defined as a function of the gradient

of αL.

3.1.3 Volume Of Fluid method

The interface is tracked by the Volume Of Fluid method. This method

is based on the volume fraction of the liquid phase (αL) as mentioned

in section 1.4.3. The flow regions containing pure liquid and pure gas

are identified by, respectively, αL = 1 and αL = 0. Interface cells are

such that 0 < αL < 1. The interface cells are therefore cells with a mix

of liquid and gas. The values of the pressure and the temperature for

these interface cells correspond to their values in the mix. The variable

material properties such as the viscosity, the thermal conductivity and

the density depend on the liquid volume fraction (see section 3.2.4). For

example the density will be defined as:

ρ = αLρL + (1 − αL)ρg (3.12)

As the bubble-liquid interface moves as a function of time, the volume

fraction of the liquid phase has to be calculated for every computational

time step. The continuity equation for αL is:

∂ (ρLαL)

∂t+ ∇· (ρLαLu) = 0 (3.13)

As the liquid is incompressible, this equation becomes:

∂αL

∂t+ u · ∇αL + αL∇·u = 0 (3.14)

Assuming that the interface is very sharp, the last term of eq. 3.14 can

be ignored as αL = 0 in the gas phase and ∇·u = 0 in the liquid phase.

Eq. 3.14 becomes then:DαL

Dt= 0 (3.15)


In the following, the 1D and 2D calculations will be performed by using

eq. 3.15 and eq. 3.13, respectively. These equations are called the

volume fraction equations.

3.2 Numerical method

The equations to solve are the Navier-Stokes equations and the vol-

ume fraction equation. The resolution of these equations is challenging

mainly for four reasons:

• The material properties are discontinuous at the gas-liquid inter-

face (e.g. the density jump at the interface is high (≈ 1000 kg/m3)

and time dependent).

• The compressible Navier-Stokes equations have to be solved in

the compressible gas phase and the incompressible Navier-Stokes

equations in the liquid phase.

• The gas-liquid interface has to be kept sharp.

• The pressure and the temperature gradient at the gas-liquid inter-

face can be large when simulating bubble dynamics.

There are two ways to solve this set of equations:

• In the first way, the Navier-Stokes equations and the volume frac-

tion equation are solved iteratively together. The equations have

to be solved together because the density is a function of the liquid

volume fraction (see eq. 3.12).

• In the second way, the Navier-Stokes equations are modified to

remove the density from the spatial and temporal derivatives in

the continuity and energy equations. As a result, these modified

Navier-Stokes equations and the volume fraction equation can be

solved separately. The volume fraction equation is solved only once

when the Navier-Stokes equations have converged. Removing the

density from the spatial and temporal derivatives in the continuity

equation also allows to solve easier the continuity equation.

3.2. NUMERICAL METHOD 65

These two ways to solve the set of governing equations are considered.

As the same discretization schemes and algorithms are used in both

cases, we will only describe them for the second case. The discretization

schemes and algorithms are described in sections 3.2.2 and 3.2.3. Con-

cerning the volume fraction equation, the modified HRIC scheme and

the PLIC method are considered (section 3.2.4). They allow to keep a

sharp gas-liquid interface.

3.2.1 Modification of Navier-Stokes equations

The density is removed from the spatial and temporal terms in the con-

tinuity and energy equations. In the continuity equation, the density ρ

can be removed from the spatial and temporal derivatives using the same

procedure as described in [107]. The first step is to write the continuity

equation as:Dρ

Dt+ ρ∇·u = 0 (3.16)

where DDt denotes the lagrangian derivative.

Using the definition of ρ (eq. 3.12), DρDt becomes:

Dρ

Dt= ρL

DαL

Dt+ αL

DρL

Dt+ (1 − αL)

Dρg

Dt+ ρg

D (1 − αL)

Dt

= (1 − αL)Dρg

Dt(3.17)

as the liquid is incompressible and as DαL

Dt = 0 (eq. 3.13). The continuity

equation is then:

(1 − αL)

ρ

∂ρg

∂t+

(1 − αL)

ρu · ∇ρg + ∇·u = 0 (3.18)

In this equation, there are no spatial and time derivatives of the density.

There are only spatial and time derivatives of the gas density. It is more

convenient to solve because the fluctuations in the gas density, ρg are

smaller than the fluctuations in density, ρ. Moreover ρg does not depend

on αL.

The energy equation can also be modified to remove the density ρ from


the temporal and spatial terms. Combining eqs (3.2, 3.4, 3.6), the energy

equation can be written as a function of the enthalpy h as:

D (ρh)

Dt+ ρh∇·u = ∇· (k∇T ) +

Dp

Dt(3.19)

Based on liquid incompressibility and on the equation for the advection

of the interface (eq. 3.12), the D(ρh)Dt term becomes:

D (ρh)

Dt= (1 − αL)cp,g

D

Dt(ρgT ) + αLcp,LρL

DT

Dt(3.20)

Eq. 3.19 is then:

(1 − αL)cp,g

[

∂(ρgT )

∂t+ ∇· (ρgTu)

]

+ αLcp,LρL

[

∂T

∂t+ ∇· (Tu)

]

=Dp

Dt+ ∇· (k∇T ) (3.21)

The first term of the previous equation is modified introducing the con-

tinuity equation (eq. 3.18) in eq. 3.21. The final expression for the

energy equation is:

[cp,g(1 − αL)ρg + cp,LαLρL]

[

∂T

∂t+ ∇· (Tu)

]

− cp,gTρ∇·u

=Dp

Dt+ ∇· (k∇T ) (3.22)

The density is not in any spatial or time derivative terms. As a con-

clusion, the modified Navier-Stokes equations are eqs. (3.4, 3.18, 3.22).

The discretization schemes and algorithms are shown in the following

for these modified Navier-Stokes equations.


3.2.2 Discretization schemes

(a) (b)

Figure 3.1: (a) Staggered mesh. (b) Collocated mesh.

Staggered and collocated mesh

The set of equations is solved by a finite volume method. The integral

form of eqs (3.3, 3.18, 3.22) can be discretized on a staggered mesh (Fig.

3.1(a)) or a collocated mesh (Fig. 3.1(b)).

In the staggered mesh, the pressure, the density, the temperature and

the liquid volume fraction are set in the cell centres and the velocities

in the middle of the cell faces between two mesh cells. The pressure cell

shown in Fig. 3.1 is the control volume for the continuity, energy and

volume fraction equations. The velocity cell is the control volume for

the momentum equation. The disadvantage of the staggered mesh is the

use of different control volumes for the pressure and the velocity field,

specially for unstructured meshes.

In the collocated mesh, the pressure, the density, the velocity, the tem-

perature and the liquid volume fraction are set in the cell centres. All

the variables are discretized in the same control volume. The disadvan-

tage of the collocated mesh is to produce non-physical oscillations in

the pressure field, called checkerboard pressure [76]. These non-physical

oscillations are not observed in a staggered mesh because even if the

momentum equation contains a pressure gradient term that can support


a checkerboard pattern, the continuity equation does not permit such

pressure field to persist [110]. For a collocated mesh, a remedy to the

non physical oscillations is the momentum interpolation method first

proposed by Rhie and Chow [88]. Improvements of the momentum in-

terpolation have been proposed by Choi [21] and Yu [111].

As the gas-liquid multiphase flow model will be mainly applied to the

simulation of bubble dynamics in a liquid, the discretization of modified

Navier-Stokes equations is shown in 1D spherical coordinates along the

radial direction, r. The cells have an uniform length equal to ∆r. The

mesh considered is a staggered mesh.

Discretization of the momentum equation

The momentum equation (eq. 3.4) is discretized at time t + ∆t in the

velocity cell centred in i + 12 (see Fig. 3.1). The convective term of the

momentum equation∫

Ωu · ∇u dΩ (3.23)

is discretized using an upwind scheme:

∮

Ωu · ∇u dΩ =

1

2

[

uni+1 + |un

i+1|]

un+1i+ 1

2

Si+1 (3.24)

+1

2

[

uni+1 − |un

i+1|]

un+1i+ 3

2

Si+1 (3.25)

− 1

2[un

i + |uni |] un+1

i− 1

2

Si (3.26)

− 1

2[un

i − |uni |] un+1

i+ 1

2

Si (3.27)

The exponent n indicates the number of iterations at time t + ∆t. S

denotes the surface of the cell face.

The diffusive term∫

Ω

1

ρ∇·

(

µ(

∇u + ∇uT))

dΩ (3.28)

is discretized by a central differencing scheme. As a result, the dis-


cretized diffusive term is:

1

ρi+ 1

2

[

2µi+1

(

∂u

∂r

)n+1

i+1

Si+1 − 2µi

(

∂u

∂r

)n+1

i

Si − 4µi+ 1

2

un+1i+ 1

2

(2π(r2i+1 − r2

i ))

]

(3.29)

with

(

∂u

∂r

)n+1

i+1

=un+1

i+ 3

2

− un+1i+ 1

2

∆r

(

∂u

∂r

)n+1

i

=un+1

i+ 1

2

− un+1i− 1

2

∆r

The pressure term in eq. 3.3:

∫

Ω

1

ρp n dΩ (3.30)

is discretized as:

1

ρni+ 1

2

[

pni+1Si+1 − pn

i Si − 2pni+ 1

2

(2π(r2i+1 − r2

i ))]

(3.31)

The external force in the momentum equation is discretized as:

∫

ΩFσ

ndΩ = Fnσ,i+ 1

2

Ωi+ 1

2

(3.32)

where Ωi+ 1

2

is the volume of the velocity cell centred in i + 12 . Finally,

the temporal term is discretized by an Euler scheme:

∫

Ω

∂u

∂tdΩ =

un+1i+ 1

2

− uti+ 1

2

∆tΩi+ 1

2

(3.33)

The exponent t denotes the previous time step.

The discretized momentum equation can thus be written as:

au,i+ 1

2

un+1i+ 1

2

+ au,i+ 3

2

un+1i+ 3

2

+ au,i− 1

2

un+1i− 1

2

= bu,i+ 1

2

(3.34)


with

au,i+ 1

2

=Ωi+ 1

2

∆t+

1

2

[

uni+1 + |un

i+1|]

Si+1 −1

2[un

i − |uni |]Si

1

ρni+ 1

2

[µi+1

∆rSi+1 +

µi

∆rSi

]

− 4µi+ 1

2

ρni+ 1

2

(2π(r2i+1 − r2

i ))

au,i+ 3

2

=1

2

[

uni+1 − |un

i+1|]

Si+1 −1

ρni+ 1

2

µi+1

∆rSi+1

au,i− 1

2

= −1

2[un

i + |uni |]Si −

1

ρni+ 1

2

µi

∆rSi

bu,i+ 1

2

= − 1

ρni+ 1

2

[

pn+1i+1 Si+1 − pn+1

i Si − 2pn+1i+ 1

2

(2π(r2i+1 − r2

i ))

]

+Ωi+ 1

2

∆tut

i+ 1

2

+ Fnσ,i+ 1

2

Ωi+ 1

2

Discretization of the energy equation

The energy equation (eq. 3.22) is discretized for the pressure cell cen-

tred in i (see Fig. 3.1). The convective, diffusive and temporal terms are

discretized, respectively, by the first-order upwind scheme, the central

differencing scheme and the Euler scheme. Moreover, the pressure term:

∫

Ω

Dp

DtdΩ (3.35)

is discretized for the cell i as:

(pn+1i − pt

i)

∆tΩi + un

i (∇p)ni Ωi (3.36)

The discretized energy equation can be written as:

aT,iTn+1i + aT,i+1T

n+1i+1 + aT,i−1T

n+1i−1 = bT,i (3.37)


with

aT,i = Ai

[

Ωi

∆t+

uni+ 1

2

+ |uni+ 1

2

|2

Si+ 1

2

−un

i− 1

2

− |uni− 1

2

|2

Si− 1

2

]

−cp,gρi

[

uni+ 1

2

Si+ 1

2

− uni− 1

2

Si− 1

2

]

+ki+ 1

2

∆rSi+ 1

2

+ki− 1

2

∆rSi− 1

2

aT,i+1 = Ai

[

uni+ 1

2

− |uni+ 1

2

|2

]

Si+ 1

2

−ki+ 1

2

∆rSi+ 1

2

aT,i−1 = Ai

[

uni− 1

2

− |uni− 1

2

|2

]

Si− 1

2

−ki− 1

2

∆rSi− 1

2

bT,i = AiΩi

∆tT t

i +(pn+1

i − pti)

∆tΩi + ui(∇p)n

i Ωi

where

Ai = (1 − αnL,i)ρ

ng,icp,g + αn

L,iρLcp,L (3.38)

Discretization of the continuity equation

The continuity equation (eq. 3.18) is discretized for the pressure cell

centred in i. Using the Euler scheme for the temporal term, eq. 3.18 is

then discretized as:

(1 − αnL,i)

ρni

[

ρn+1g,i − ρt

g,i

∆t

]

Ωi +(1 − αn

L,i)

ρni

uni (∇ρg)

ni Ωi

+ un+1i+ 1

2

Si+ 1

2

− un+1i− 1

2

Si− 1

2

= 0 (3.39)

It will be shown in section 3.3.1 that the discretization schemes allow to

accurately solve the high pressure gradient near the gas-liquid interface.

3.2.3 Algorithm

Historically, algorithms have been classed into two groups: density-based

algorithms and pressure-based algorithms [73]. The first class was orig-

inally developed for high Mach number flow calculations. The set of


governing equations is solved in a coupled way. The second class was

originally developed for incompressible flows. The equations are solved

in a segregated way. In the last two decades, both types of algorithms

have been adapted to make them valid for incompressible and compress-

ible flows. The preconditioning technique allows to extend the density-

based algorithms towards the low Mach regime [73]. The pressure-based

methods have also been extended to simulate compressible single fluid

flow [46]. For our gas-liquid multiphase flow model, a pressure-based

algorithm is used for the incompressible liquid and the compressible gas.

In the pressure-based approach, a pressure-correction equation has to

be derived [2]. In the literature, there are two classes based on how

the pressure-correction equation is constructed. In the first class, the

pressure-correction equation is derived from the continuity equation.

In the second class, the energy equation is used [72]. The pressure-

correction equation based on the energy equation being only convenient

for adiabatic processes, the pressure-correction equation based on the

continuity equation will be used. Among these pressure-based algo-

rithms, projection methods [17] which project explicitly estimates of

vector fields and correct them in an additional elliptic projection step,

and SIMPLE (Semi-Implicit Method for Pressure Linked Equations) al-

gorithm [2] are mostly used. The SIMPLE method is considered here

because the method is semi-implicit. It was verified that the projection

method requires a smaller time step. The SIMPLE-type algorithm is

described below for a staggered mesh.

The SIMPLE-type algorithm is divided in two steps: a predictor step

and a corrector step. In the predictor step, guessed values for the ve-

locity and the temperature (u∗ and T ∗) are calculated. The discretized

equation is for u∗:

au,i+ 1

2

u∗

i+ 1

2

=∑

nb

au,nbu∗

nb −1

ρi+ 1

2

[

p∗i+1Si+1 − p∗i Si

]

+1

ρi+ 1

2

[

4πp∗i+ 1

2

(r2i+1 − r2

i )]

+ bu,i+ 1

2

(3.40)


and for T ∗:

aT,iT∗

i =∑

nb

aT,nbT∗

nb + bT,i (3.41)

with nb corresponding to neighbour cells. In these equations, the pres-

sure p∗ and the density ρn correspond to the values of pressure and

density at the previous iteration. The coefficients aT,i, aT,nb, bT,i, au,i+ 1

2

and au,nb have been defined in the previous section. bu,i+ 1

2

is the bu,i+ 1

2

term defined in the previous section without the pressure term.

In the corrector step, the guessed values (u∗, p∗, T ∗) are corrected in

order to calculate the values (u, p, T ) at iteration n + 1:

p = p∗ + p′

T = T ∗ + T ′

u = u∗ + u′

The corrections for the pressure, temperature and velocity are, respec-

tively, p′, T ′ and u′. These corrections are calculated from the pressure

correction equation. The first step to derive an equation for p′ from the

continuity equation is to derive relations between: (i) u′ and p′, and (ii)

ρ′g and p′.

For (i), the equation relating u′ to p′ is obtained subtracting eq. 3.40

from eq. 3.34:

au,i+ 1

2

u′

i+ 1

2

=∑

nb

au,nbu′

nb −1

ρi+ 1

2

[

p′i+1Si+1 − p′iSi

]

+1

ρi+ 1

2

[

4πp′i+ 1

2

(r2i+1 − r2

i )]

(3.42)

According to the SIMPLE algorithm, the∑

nb anbu′

nb term is dropped

from the previous equation. As a result, u′ is expressed as a function of

p′ as:

u′

i+ 1

2

= di+ 1

2

(p′i − p′i+1) (3.43)

with:

di+ 1

2

=2π(r2

i + r2i+1)

ai+ 1

2

ρi+ 1

2

(3.44)


For (ii), the gas density depends on the pressure and the temperature

(ideal gas). An expression between ρ′g and p′ can therefore only be

derived if T ′ = 0 (the energy equation is only an equation for T ∗). ρ′g is

then related to p′ by:

ρ′g = Kp′ (3.45)

with

K =∂ρg

∂p=

Mm

RidT ∗(3.46)

The second step to derive an equation for p′ from the continuity equa-

tion is to introduce eqs (3.43-3.46) in eq. 3.39. The pressure-correction

equation is then:

ap,ip′

i + ap,i+1p′

i+1 + ap,i−1p′

i−1 = bp,i (3.47)

where

ap,i =1

ρni

(1 − αnL,i)Ki

Ωi

∆t+ di+ 1

2

+ di− 1

2

ap,i−1 = −di− 1

2

ap,i+1 = −di+ 1

2

bp,i = − 1

ρni

[

(1 − αnL,i)(ρ

ng,i − ρt

g,i)Ωi

∆t+ (1 − αn

L,i)uni

(

∂ρg,i

∂r

)n

i

Ωi

]

−u∗

i+ 1

2

+ u∗

i− 1

2

For stability reasons, an under-relaxation technique is used in order to

reduce the variations of u∗−un, T ∗−Tn and pn+1−pn at each iteration.

As a result, u∗, T ∗ and p′ become:

u∗ = un + αu(u∗ − un) (3.48)

T ∗ = Tn + αT (T ∗ − Tn) (3.49)

p′ = αpp′ (3.50)

with αu, αT and αp the under-relaxation factors.

As a conclusion the procedure to solve the governing equations is:


1. Predictor values T ∗ are calculated solving eq. 3.41. The velocity,

density and pressure values are taken at the previous iteration.

Next, the values of T ∗ are under-relaxated (eq. 3.49). Tn+1 = T ∗.

2. Predictor values u∗ are calculated from eq. 3.40. The pressure and

density values are taken at the previous iteration. Next, the values

of u∗ are under-relaxated (eq. 3.48).

3. The pressure-correction p′ is calculated from eq. 3.47.

4. The velocities un+1 are updated from eq. 3.43 using p′.

5. The values of p′ are under-relaxated (eq. 3.50) and the pressure

pn+1 is updated.

6. The ideal gas law is used to update the density, ρn+1.

7. A next iteration step begins if no convergence is reached.

8. After convergence, the liquid volume fraction is updated (eq. 3.13)

and the interface is advected.

For 1D calculations, u∗, T ∗ and p′ (eqs (3.34, 3.37, 3.47)) are calculated

solving a system of linear equations:

Ax = b (3.51)

where x is the vector of unknowns. In one-dimensional problem the ma-

trix A is tridiagonal and the tri-diagonal matrix algorithm can be used.

This algorithm has two steps. First, the matrix is upper-triangularized:

the entries below the diagonal are successively eliminated. The last equa-

tion has thus only one unknown and can be solved. The solution for the

other equations is then obtained by working our way back from the last

to the first unknown.

Moreover, at the end of each iteration, the convergence has to be checked.

The parameter studying the convergence is the scaled residuals. The

scaled residual for the momentum and energy equations is:

Rφ =

∑

cells P |∑nb aφ,nbφnb + bφ,P − aφ,P φP |∑

cells P |apφp|(3.52)


where φ is used for u and T . The scaled residual is the imbalance of eq.

3.34 or 3.37 summed over all the computational cells P and scaled by a

factor representative of the flow rate of φ through the domain. For the

continuity equation, the unscaled residual is:

Rc =∑

cellsP

|rate of mass creation in cell P| (3.53)

The scaled residual for the continuity equation is defined as:

Rc

Rc5

(3.54)

The denominator is the largest absolute value of Rc during the first five

iterations.

Polytropic model

When a polytropic model is used instead of the energy equation, the

procedure to solve eqs (3.18, 3.4, 3.9) is:

1. Predictor values u∗ are calculated from eq. 3.40. The pressure and

density values are taken at the previous iteration. Next, the values

u∗ are under-relaxated (eq. 3.48).

2. The pressure-correction equation (eq. 3.47) is solved.

3. The velocities un+1 are updated from eq. 3.43 using p′ calculated

at the previous step.

4. The values of p′ are under-relaxated (eq. 3.50) and the pressure

pn+1 is updated.

5. The polytropic model is used to update the density, ρn+1.

6. A next iteration step begins if no convergence.

7. After convergence, the liquid volume fraction is updated (eq. 3.13)

and the interface is advected.


Note that this procedure is similar to the one for the set of equations

including explicitly the energy equation. Note also that the K term in

the pressure correction equation (eq. 3.47) is equal to:

K =1

κ

ρg,ref

(pref )1

κ

(pn)1−κ

κ (3.55)

This term was calculated in the following way. Firstly, the ρn+1g =

ρg,ref

(

pn+1

pref

)1

κterm in the continuity equation has been linearized to

eliminate the exponent 1/κ. A Newton approximation is used:

ρn+1g = ρg(p

n) +dρg

dp

∣

∣

∣

∣

pn

(pn+1 − pn) (3.56)

The pressure value from the previous pressure iteration pn is taken in-

stead of the pressure value from the previous time step pt in order to

decrease the errors due to the linearization. As a result, the density

correction is related to the pressure correction by:

ρ′g =dρg

dp

∣

∣

∣

∣

pn

p′ (3.57)

Consequently the K term in eq. 3.47 is

K =1

κ

ρg,ref

(pref )1

κ

(pn)1−κ

κ (3.58)

3.2.4 Gas-liquid interface

Density averaging method

The density is related to the liquid volume fraction by the simple aver-

aging method:

ρ = αLρL + (1 − αL)ρg (3.59)

This method is convenient in the case of a high density ratio at the

interface. The other material properties (µ and k) are defined in the

same way as the density:

µ = αLµL + (1 − αL)µg

k = αLkL + (1 − αL)kg


When solving the momentum equation (eq. 3.40), the density (ρi+ 1

2

) at

the cell face has to be calculated. In this work, the density (ρi+ 1

2

) is

calculated based on the densities of the left and right mesh cells, ρi and

ρi+1. For cells with uniform length:

ρi+ 1

2

=ρi + ρi+1

2(3.60)

The viscosity term µi+ 1

2

in eq. 3.40 is similarly defined:

µi+ 1

2

=µi + µi+1

2(3.61)

Advection of the gas-liquid interface

As the interface is moving in time, the liquid volume fraction has to

be recomputed for every computational time step. The liquid volume

fraction at each time step is calculated by:

DαL

Dt=

∂αL

∂t+ u ·∇αL = 0 (3.62)

Two different approaches are considered. In the first approach, the inter-

face is represented by a segment perpendicular to the normal of the in-

terface. It is the Piecewise Linear Interface Calculation (PLIC) method.

The algorithms to reconstruct and advect the gas-liquid interface have

been described in section 1.4.3. In 1D, the PLIC method is similar to

the donor-acceptor method described by Hirt and Nichols [43]. This

method is described here for the case shown in Fig. 3.2 (the velocity

being positive, the liquid moves from left to the right). First, we define

Ωl,i and Fl,i+ 1

2

as, respectively, the volume of liquid inside the cell i and

the flux of liquid through the cell face i + 12 . The cell i is the interface

cell. The flux Fl,i+ 1

2

is:

Fl,i+ 1

2

= ui+ 1

2

Si+ 1

2

∆t (3.63)

There are two cases:

(i) if Fl,i+ 1

2

< Ωl,i, the interface stays in the cell i and the liquid volume

fraction in cell i is:

αL,i =Ωl,i − Fl,i+ 1

2

Ωi(3.64)


where Ωi is the volume of the interface cell.

(ii) if Fl,i+ 1

2

> Ωl,i, the interface moves in the cell i + 1 and the liquid

volume fraction in cell i + 1 is:

αL,i+1 = 1 −Fl,i+ 1

2

− Ωl,i

Ωi+1(3.65)

A similar procedure is used when the velocity is negative.

This described algorithm is only valid when the Courant-Friedrichs-Lewy

(CFL) condition is smaller than 1. With this approach, the interface is

always accurately defined in 1 cell.

Figure 3.2: Schematic representation of the cells near the interface. The

dashed line represents the position of the gas-liquid interface

In the second approach, the temporal term of eq. 3.62 is discretized by

an Euler scheme and the convective term by the modified High Reso-

lution Interface Capturing (HRIC) scheme. An accurate resolution of

the convective terms requires (i) a non-diffusive scheme; (ii) a scheme

calculating a volume fraction between the minimum and the maximum

values of the neighbour cells. The HRIC scheme has been built to respect

these conditions. It is a non-linear blending of upwind and downwind

scheme (note that an upwind scheme would calculate bounded values

of αL but would be too diffusive, while a downwind scheme would be

non-diffusive but would calculate unbounded values of αL). The HRIC

scheme is described below based on Fig. 3.3.


Figure 3.4: HRIC scheme.

Figure 3.3: Schematic representation of the cells near the interface.

First, the normalized volume fraction α is defined in the cell centre as:

αL =αL − αL,i−1

αL,i+1 − αL,i−1(3.66)

The normalized liquid volume fraction at the cell face i + 12 is then

calculated as (see Fig. 3.4) [69]:

αL,i+ 1

2

=

αL,i if αL,i < 0

2αL,i if 0 ≤ αL,i < 0.5

1 if 0.5 ≤ αL,i < 1

αL,i if 1 ≤ αL,i

(3.67)

Note that an upwind and a downwind scheme corresponds, respectively,

to αL,i+ 1

2

= αi and αL,i+ 1

2

= 1. Downwind discretization may cause


an alignment of the interface with the numerical mesh. To prevent this

alignment, the discretization has to take account for the angle θ between

the normal to the interface and the normal to the cell face (see Fig. 3.3).

αL,i+ 1

2

is then corrected as:

α∗

L,i+ 1

2

= αL,i+ 1

2

√cos θ + αL,i+ 1

2

(1 −√

cos θ) (3.68)

Finally, αL,i+ 1

2

is:

αL,i+ 1

2

= γαL,i + (1 − η)αL,i+1 (3.69)

with η:

η =(1 − α∗

L,i+ 1

2

)(αL,i+1 − αL,i−1)

αL,i+1 − αL,i(3.70)

This scheme is a little diffusive: an interface that is initially in one cell

spreads out in 2-3 cells during the numerical calculations. The PLIC

method is therefore more accurate than the HRIC scheme.

Remarks about the Volume Of Fluid method

In the Volume Of Fluid method, the interface cell is a cell with a mix of

liquid and gas, the density being defined by eq. 3.59. The pressure and

the temperature in the interface cell correspond then to the pressure and

the temperature of the mix. This modeling based on a mix cell leads to

two problems:

(i) To illustrate the first problem, we consider the adiabatic compres-

sion of a gas bubble in a liquid by an external pressure. Initially, the

temperature in the two phases is uniform and equal to 300 K. Under

the effect of external pressure, the bubble is compressed and the pres-

sure and the temperature inside the bubble increase. Although no heat

transfer through the gas-liquid interface is considered, the temperature

in the liquid near the interface becomes higher than 300 K during the

compression of the bubble. This is non physical. The increase of the

temperature in the liquid phase can be explained based on Fig. 3.5.

At time t, the temperature in the interface cell (cell i) is higher than

300 K as the interface cell is a mix of liquid and gas. At time t + ∆t,


the interface moves from cell i to cell i − 1 (Fig. 3.5). As a result, the

cell i that was an interface cell at time t becomes a liquid cell at time

t + ∆t. The problem is that the initial temperature in the cell i at time

t + ∆t is equal to the temperature of the mix in the cell i at time t. As

a result, the temperature in cell i becomes higher than 300 K at t + ∆t.

This problem occurs each time that the position of the interface moves

from one cell to another cell. As a solution, when the interface moves

from cell i to cell i− 1, the initial temperature in cell i at time t + ∆t is

taken equal to the temperature in cell i + 1 at time t. The temperature

in the liquid stays then always equal to 300 K.

Figure 3.5: Schematic representation of cells near the interface for times

t and t + ∆t. The dashed line represents the position of the gas-liquid

interface.

(ii) To illustrate the second problem, we use Fig. 3.2. The convec-

tive fluxes in the energy equation are discretized by an upwind scheme.

When the velocity is positive, the convective flux at the left face of the

cell i + 1 is overpredicted as the temperature of the mix is used. As a

solution, the convective flux at the left face of cell i + 1 is calculated

based on the temperature in the cell i + 1 and not on the temperature

in the interface cell (cell i).

These modifications of the discretization of the energy equation can be

introduced in the numerical method. It allows to have a sharp jump of

temperature at the gas-liquid interface during adiabatic processes.

3.3. TEST CASE 83

3.3 Comparison of 1D numerical and analytical

solutions

During this work, the first 2D Fluent axisymmetric simulations of the

collapse of a spherical bubble failed because the results were not mesh in-

dependent. In order to have a better insight into the effect of parameters

such as the number of cells, the ratio between the length of two adjacent

cells, the time increment, the value of residuals on the numerical sim-

ulations, we have developed our own code to study 1D two-phase flow

problems. The main advantage of 1D numerical calculations is the re-

duced computational cost. In this 1D code, the modified Navier-Stokes

equations (without the surface tension term) and the volume fraction

equation (eq. 3.15) are solved. The numerical method is based on:

• a staggered grid.

• a SIMPLE-type algorithm.

• a PLIC method to advect the gas-liquid interface.

• modifications of the discretization of the energy equation described

in section 3.2.4.

In this section, the spherical motion of an air bubble subjected to an

acoustic field in bulk liquid is simulated. The bubble is initially at

rest in liquid water and it has a radius of R0 = 78.5 µm. The material

properties for the air (bubble) and liquid water phases are given in Table

3.1. The pressure and the temperature in both phases are equal to 1 bar

and 300 K, respectively. During the simulation, the bubble is assumed

to remain spherical while its radius changes (1D spherical simulation).


water air

ρ[

kg/m3]

1000

µ [ kg/ms] 0.001 1.8 10−5

cp [ J/kgK] 4182 1006

Mm [ g/mol] 18 29

k [ W/mK] 0.6 0.025

Table 3.1: Material properties for water and air phases.

Figure 3.6: Schematic diagram showing the mesh. Note that, in actual

calculations, a denser mesh is used to obtain convergence.

The mesh is schematically shown in Fig. 3.6. r = 0 corresponds to

the position of the bubble centre. The total pressure at the boundary

of the domain (r = Rf ) is the superposition of the ambient pressure

(pambient = 1 bar) and the sinusoidal pressure associated with the acous-

tic field (pacoust):

p∞ = pambient + pacoust = 1 − pA sin(2πfAt) [bar] (3.71)

where the amplitude and the frequency of the acoustic field are assigned

to pA = 0.8 bar and fA = 20 kHz, respectively.

Now, we focus on the way to build the mesh. We observed that a con-

verged solution demands a dense mesh in the vicinity of the bubble (i.e.

in the region where the bubble grows and collapses), while, away from

the bubble, the cell size has not a big effect on the convergence. Based on

these observations, the computational domain is divided in two regions:

the ”bubble domain” and the ”liquid domain” which are composed of a

dense and a coarse mesh, respectively.

The region at the vicinity of the bubble (called the bubble domain) is

3.3. TEST CASE 85

divided in n cells of uniform size ∆r. In the rest of the computational

domain (called the liquid domain), the cell size increases in the r direc-

tion: the ratio between the size of two adjacent cells is 1.1. Therefore,

the only parameter of the mesh is the number of cells in the bubble

domain n. Now, we have to define which region is included in the bub-

ble and liquid domain. The bubble domain has always to include the

bubble, even when the bubble reaches its maximum radius. As we have

a fixed mesh, the radius corresponding to the bubble domain has to be

higher than the maximum bubble radius. Here, the bubble domain ex-

tends from r = 0 to r = 2R0, 2R0 being a value slightly higher than the

maximum bubble radius. The liquid domain corresponds to the rest of

the computational domain: 2R0 < r < 2Rf .

In section 3.3.1, the dynamics of a bubble subjected to an acoustic field

is simulated assuming that the compression and the expansion of the air

inside the bubble is adiabatic. First, the set of equations including the

polytropic model is solved. The results are compared to the Rayleigh-

Plesset model described in section 1.4.1. Next, the set of equations

including the energy equation is solved. The thermal conductivity is set

to 0. The evolution of the bubble radius as a function of time and the

spatial evolution of the pressure field close to the bubble-liquid interface

are studied. In section 3.3.2, the influence of heat transfer through the

bubble-liquid interface is studied.


3.3.1 Adiabatic conditions

Polytropic model

0 10 20 30 400

0.20.40.6

Time [µs]

Mas

s lo

ss [%

] 0

30

60

90

120

Bub

ble

radi

us [µ

m]

−1

0

1

Pre

ssur

e of

aco

ustic

fiel

d [b

ar]

(b)

(a)

Figure 3.7: (a) Bubble radius as a function of time (solid line). The

dotted line represents the pressure of the acoustic field. (b) Mass loss of

air inside the bubble as a function of time.

The set of equations (3.4, 3.13, 3.18, 3.72) based on the polytropic rela-

tion is solved. As a reminder, the polytropic relation is:

ρg

ρg,ref=

(

p

pref

)κ

(3.72)

In this equation, the reference density and the reference pressure are

equal to, respectively, the initial density (ρg,ref = 1.16 kg/m3) and the

initial pressure (pref = 1 bar). As adiabatic conditions are considered,

κ is 1.4.

Figure 3.7(a) shows the evolution of the bubble radius as a function

of time. The bubble grows while the pressure associated with the acous-

tic field is negative (pacoust < 0 corresponding to t < 25 µs), and it

collapses while the pressure is positive (pacoust > 0, t ≥ 25 µs). Finally,

3.3. TEST CASE 87

the bubble rebounds. As the two phases are immiscible, the mass of air

inside the bubble would have to stay constant. However, as shown in

Fig. 3.7(b), the mass loss of air inside the bubble increases when the

bubble volume decreases and reaches a maximum at the end of the col-

lapse phase. This mass loss is due to the numerical scheme.

(a) (b)

Figure 3.8: A convergence test showing the effect of n on (a) the min-

imum bubble radius and (b) the maximum mass loss of air inside the

bubble. The solid lines are a guide to the eye. Rf/2R0 = 100.

Figure 3.8(a) shows the minimum bubble radius as a function of the

number of cells n in the bubble domain. The cell size ∆r is 130/n µm.

A converged solution is obtained when n ≥ 2000: the difference between

the minimum bubble radius for n = 2000 and n = 4000 is less than 1 %.

Moreover, comparing Figs 3.8(a) and (b) show that the minimum bubble

radius converges when the maximum mass loss of air inside the bubble

reaches a constant value close to zero. For the calculations shown in Figs

3.7 and 3.8, the value of the time increment ∆t ranges from 0.625 ns to

2.5 ns, depending on the number of cells in the bubble domain. The

scaled residuals have also a significant effect on the simulation; for the

mesh sizes and the time increment considered here, it should be smaller


than 10−3 for the continuity and momentum equation to obtain a con-

verged solution. In the following, all the simulations are performed for

n = 2000, ∆t = 1.25 ns and the scaled residuals defined above.

Now, we compare our numerical model with the Rayleigh-Plesset model

(eq. 1.9). As the Rayleigh-Plesset model describes the radial motion

of a bubble subjected to an acoustic field in an infinite medium, our

calculation domain has to be sufficiently large so that the bubble acts

as an isolated bubble in an infinite medium. The effect of the size of the

calculation domain on the minimum bubble radius and on the collapse

time (defined as the time corresponding to the minimum bubble radius)

is analysed. Figure 3.9 shows (a) the minimum bubble radius and (b)

the difference between the collapse time predicted by our model and by

the Rayleigh-Plesset model as a function of the dimensionless size of

the calculation domain Rf/2R0. The domain is sufficiently large when

Rf/2R0 ≥ 1000: (i) the minimum bubble radius for Rf/2R0 = 100 and

Rf/2R0 = 1000 is almost identical; (ii) for Rf/2R0 = 1000, the collapse

time calculated by our model is identical to the one calculated using

the Rayleigh-Plesset model. As a result, the comparison between our

model and the Rayleigh-Plesset model will be performed for n = 2000

and Rf/2R0 = 1000. Figure 3.10 shows the evolution of the bubble ra-

dius as a function of time calculated using the Rayleigh-Plesset model

and our numerical model. The two models predict almost identical re-

sults: the minimum bubble radius is 0.5% higher in our model than in

the Rayleigh-Plesset model. Moreover, the assumption of spatially uni-

form pressure inside the bubble used in the Rayleigh-Plesset model is

validated. Indeed, our model, for which the pressure field is explicitly

calculated in the gas phase, gives almost identical results as the Rayleigh

model.

3.3. TEST CASE 89

(a) (b)

Figure 3.9: (a) Minimum bubble radius and (b) difference between the

collapse time predicted by our model and by the Rayleigh-Plesset model

as a function of Rf/2R0. The solid lines are a guide to the eye.

Figure 3.10: (a) Bubble radius as a function of time for the Rayleigh-

Plesset model and our 1D model. (b) Zoom of the bubble radius near

its minimum value.


Energy equation

The set of equations (3.34, 3.37, 3.47, 3.62) including explicitly the en-

ergy equation is solved. As the compression and the expansion of the

bubble are assumed adiabatic, the thermal conductivity term is removed

from the energy equation. For the scaled residuals defined in the pre-

vious section and a time increment of 0.625 ns, the results are mesh

independent when n ≥ 4000. In the following, the numerical calcula-

tions are performed for n = 4000, Rf/2R0 = 1000, ∆t = 0.625 ns and

the scaled residuals defined above.

Figures 3.11 (a) and (b) show the evolution of the bubble radius as a

function of time calculated using the energy equation and the polytropic

model. The results are almost identical: the values of the minimum

bubble radius and the collapse time calculated using the energy equa-

tion are, respectively, 0.6 % smaller and 50 ns larger than the values

predicted using the polytropic model.

Figure 3.11: (a) Bubble radius as a function of time calculated using

the energy equation and the polytropic model. (b) Zoom of the bubble

radius near its minimum.

Fig. 3.12 and Fig. 3.13 show, respectively, the pressure at the bubble-

3.3. TEST CASE 91

liquid interface and the temperature at the bubble centre as a function

of time. The pressure and the temperature decrease while the bubble

grows, and increase while the bubble collapses. At the minimum bubble

radius, the pressure and the temperature are maximum and equal to

43 bar and 900 K, respectively. Finally, during the rebound phase, the

pressure and the temperature decrease again.

The high pressure gradient in the liquid phase near the bubble-liquid

interface is adequately solved as seen in Fig. 3.14 showing the pressure

field in the gas and liquid phases for two times: t = 34.4 µs and t =

35.4 µs. These two times correspond to the two black dots in Fig. 3.12.

This validates our choice for the discretization schemes and algorithms.

Finally, as already confirmed in the previous section, the pressure inside

the bubble is almost spatially uniform.

0 10 20 30 400

10

20

30

40

Time [µs]

Pre

ssur

e at

inte

rfac

e [b

ar]

0

30

60

90

120

Bub

ble

radi

us [µ

m]

Figure 3.12: Pressure at the bubble-liquid interface as a function of time

(left). Bubble radius as a function of time (right).


Figure 3.13: Temperature at the bubble centre as a function of time

(left). Bubble radius as a function of time (right).

t = 34.4 µs

p [bar]

0

5

10

15

20

25

30

35

40

45

(a)

t = 35.4 µs

p [bar]

0

5

10

15

20

25

30

35

40

45

(b)

Figure 3.14: Pressure field in the computational domain for two times:

t = 34.4 µs and t = 35.4 µs.

3.3.2 Heat transfer through the interface

In this section, the influence of the heat transfer through the bubble-

liquid interface on the evolution of the bubble radius as a function of

3.3. TEST CASE 93

time is studied. The set of equations (3.34, 3.37, 3.47, 3.62) including

the energy equation is solved. The thermal conductivity in the gas and

liquid phases are given in Table 3.1. The evolution of the bubble radius

as a function of time is compared to those calculated using the polytropic

model with κ = 0 (isotherm expansion and compression of the bubble)

and κ = γ (adiabatic expansion and compression of the bubble) (see Fig.

3.15). Figure 3.15 shows that the maximum bubble radius is between

the maximum bubble radius predicted assuming that the expansion of

the bubble is isotherm and adiabatic. It indicates that some heat is

exchanged through the bubble-liquid interface.

0 10 20 30 400

20

40

60

80

100

120

140

Time [µs]

Bub

ble

radi

us [µ

m]

Heat transferκ = γκ = 1

Figure 3.15: Evolution of the bubble radius as a function of time calcu-

lated considering the heat transfer through the bubble-liquid interface

and using the polytropic model with κ = 0 and κ = γ.

Figure 3.15 shows also that the minimum bubble radius is almost iden-

tical to the minimum bubble radius predicted assuming adiabatic con-

ditions inside the bubble. It indicates that the heat is not transferred

through the bubble-liquid interface upon the end of the collapse phase.

There are therefore two time scales in the bubble motion. During the

expansion of the bubble, the motion of the bubble is slow and some heat

is transferred through the bubble-liquid interface. However, during the


collapse of the bubble, the motion of the bubble is fast and the heat is

not transferred through the bubble-liquid interface. It results that the

polytropic coefficient cannot be considered constant during the bubble

dynamics. Moreover, the polytropic coefficient depends on the ampli-

tude and the frequency of the acoustic field: at 1 MHz, the period being

much smaller, the expansion and the compression of the bubble is almost

adiabatic.

3.4 2D axisymmetric simulations

In this section, a first example of 2D axisymmetric simulations is pre-

sented. We study the dynamics of a spherical bubble subjected to an

external pressure of 5 bar in bulk liquid. Contrarily to the calculations in

the previous section where the bubble was assumed to remain spherical

while its radius changed (1D simulation), in this section the shape of the

bubble can change (2D axisymmetric simulation). The 2D axisymmetric

simulations are performed by using the commercial software Fluent that

has the advantage to easily handle non uniform meshes. The numerical

method in Fluent is based on:

• the unmodified Navier-Stokes equations (eqs 3.2, 3.4, 3.6, 3.13).

• a collocated grid.

• a SIMPLE-type algorithm.

• a HRIC scheme to solve the volume fraction equation.

The same numerical method will also be used in the two next chapters.

Note that open source softwares such as Gerris could also be used.

3.4.1 The problem definition

The motion of an initial spherical air bubble in bulk liquid subjected

to an external pressure of 5 bar is analysed. The bubble is initially at

rest in liquid water and it has a radius of R0 = 1 mm. The material

properties for the air (bubble) and water phases are given in Table 3.1.

3.4. 2D AXISYMMETRIC SIMULATIONS 95

Initially, the pressure and the temperature in both phases are equal to

1 bar and 300 K, respectively.

Figure 3.16: Schematic diagram showing the mesh in the vicinity of an

air bubble in bulk liquid. Note that only the half part of the mesh is

shown. Note also that, in actual calculations, a lot denser mesh size is

used to obtain convergence.


In 3D simulations, the calculation domain would be a sphere and the

bubble would be positioned at the centre of the sphere. When no vari-

ation of the bubble shape in the tangential direction is considered, the

3D simulations reduce to 2D axisymmetric simulations. The calculation

domain is then the half-circle shown schematically in Fig. 3.16. An

axisymmetric boundary condition is imposed at the bottom of the half-

circle (the sphere is obtained by the rotation of the half-circle around

the axis of axisymmetry). The remote pressure at the boundary of the

domain is 5 bar at r = R∞ (r is the radial distance to the bubble centre).

A convergence study has shown that to take R∞ = 100 mm ≈ 100R0 is

large enough, so that the bubble acts as an isolated bubble in an infinite

medium (see also section 3.3.1). Figure 3.16 shows an example mesh at

the vicinity of the bubble. Note that, for clarity, a coarse mesh is shown

in the figure, while a converged solution usually demands a denser mesh.

Indeed, as mentionned in section 3.3.1, a converged solution demands a

dense mesh in the vicinity of the bubble, while, away from the bubble,

the cell size has not a big effect on the convergence. As a result, the cal-

culation domain is divided in two regions: a bubble domain and a liquid

domain which are composed of a dense and a coarse mesh, respectively.

Similarly to section 3.3.1 for 1D spherical problem, the bubble domain,

defined as r < rbd (see Fig. 3.16), has always include the bubble, even

when the bubble reaches its maximum volume. Therefore, a fixed mesh

being used, rbd has to be higher than the maximum distance between the

bubble-liquid interface and the bubble centre. For the problem consid-

ered here, as the bubble volume is always smaller than its initial volume

(the bubble collapses), rbd is equal to 1 mm. The bubble domain ex-

tends from r = 0 to r = 1 mm and the liquid domain from r = 1 to

r = 100 mm.

Now, the bubble and liquid domain have to be meshed. Quadrilateral

cells are used and the meshing is performed in two steps by using the

commercial software Gambit. In the first step, the edges of the bubble

and liquid domain are meshed:

• the axis of axisymmetry is divided in n uniform intervals in the

bubble domain (r < 1 mm). In the liquid domain (r ≥ 1 mm),


the length of intervals increases in the radial direction: the ratio

between the length of two adjacent intervals is 1.1.

• the arcs of a circle are divided in p uniform intervals.

In the second step, the bubble and the liquid domain are meshed with

the Quad-Tri Primitive algorithm for r < 0.2 mm and with the Quad

Map algorithm for r > 0.2 mm. Note that the Quad-Tri Primitive algo-

rithm imposes a constraint on p: p < 2n. The resulting mesh is shown

schematically in Fig. 3.16. As a conclusion, the two parameters for the

mesh are n and p.

3.4.2 Results

0 10 20 30 40 50 600

1

2

3

4

Time [µs]

Bub

ble

volu

me

[mm

3 ]

Figure 3.17: Bubble volume as a function of time.

Figure 3.17 shows the evolution of the bubble volume as a function of

time. Under the effect of the external pressure of 5 bar, the bubble

collapses, reaches a minimum volume and then rebounds. The value of

the minimum bubble volume has been studied for the four meshes given

in Table 3.2. These meshes differ by the values of the parameters p and

n. Figure 3.18 shows the minimum bubble volume as a function of the


number of cells in the mesh. The minimum bubble volume increases and

converges towards a constant value when the number of cells in the mesh

increases. Moreover, it was verified that, above p = 200, the value of

parameter p had not any influence on the minimum bubble volume. A

converged solution is obtained for the mesh 3 (p = 200, n = 2000): the

difference between the minimum bubble volume for the meshes 3 and 4

(see Table 3.2) is less than 2.5 %. For the calculations shown in Figs 3.17

and 3.18, the value of the time increment is 20 ns and the scale residuals

should be smaller than 10−3 for the continuity and momentum equation,

and 10−6 for the energy equation to obtain a converged solution. In the

rest of the section, the same time increment and scaled residuals are

used.

Figure 3.18: A convergence test showing the effect of the four meshes

described in Table 3.2 on the minimum bubble volume. The solid line is

a guide to the eye.


Mesh number of cells p n

1 18750 100 250

2 50080 160 500

3 113000 200 1000

4 214400 200 2000

Table 3.2: Parameters n and p for four meshes.

The main difference between the spherical 1D and axisymmetric 2D for-

mulations is that in the 2D case the bubble is allowed to change shape.

However, for the problem of an air bubble submitted to an external pres-

sure of 5 bar, the pressure on the bubble is the same through out the

entire bubble-liquid interface and the bubble remains spherical (see Fig.

3.19) as a direct consequence of the symmetry of the loading condition.

However, if the surface tension is removed from the numerical model,

the bubble loses its spherical shape (see Fig. 3.20): the onset of the in-

stabilities at the bubble-liquid interface grow during the rebound phase.

Including the surface tension therefore allows to prevent the growth of

these bubble shape instabilities.

Figure 3.19: Bubble shape at t = 60 µs when the surface tension is

included in the numerical model.


Figure 3.20: Bubble shape at t = 60 µs when the surface tension is not

included in the numerical model.

In previous simulations, the motion of the bubble-liquid interface follows

the mesh (see Fig. 3.16). Now, we study what is happening when

the motion of the bubble-liquid interface does not follow the mesh. To

this purpose, simulations performed by using the two meshes shown

in Figs 3.16 and 3.21 (denoted mesh 1 and mesh 2, respectively) are

compared. The mesh 1 (Fig. 3.16) has already been described above.

In the mesh 2 (Fig. 3.21), the region where the bubble collapses is

meshed with square cells and the rest of the calculation domain with a

combination of triangular and rectangular cells. Figure 3.22 shows the

evolution of the bubble volume as a function of time calculated using

the meshes 1 and 2. The bubble volume is identical for the two meshes

during the collapse phase. Therefore, the orientation of cells has not

any influence on the bubble volume during the collapse phase. However,

the bubble volume is slightly different during the rebound phase (see

Fig. 3.22) because the bubble shape does not remain spherical when

using the mesh 2. Although that the surface tension is included in

the numerical model, numerical instabilities of the bubble shape grow

during the rebound phase. Note that these instabilities are not observed

during the bubble collapse and rebound near a solid surface because the

gradients of pressure are smaller.


Figure 3.21: Schematic diagram showing the mesh in the vicinity of a

bubble in bulk liquid. Note that, in actual calculations, a lot denser

mesh size is used to obtain convergence.

0 10 20 30 40 50 600

1

2

3

4

Time [µs]

Bub

ble

volu

me

[mm

3 ]

mesh 1mesh 2

Figure 3.22: Bubble volume as a function of time calculated using the

meshes shown in Fig. 3.16 (mesh 1) and in Fig. 3.21 (mesh 2).


3.5 Conclusion

In this chapter, we describe a model to simulate the dynamics of a

bubble in a liquid near a solid surface. The one fluid model is used:

one set of Navier-Stokes equations are solved for the compressible gas

and incompressible liquid phases on a Cartesian mesh and the interface

is represented and tracked by the Volume Of Fluid method. Contrarily

to the main numerical models in the literature, the main characteristic

of our model is that it takes explicitly account the viscosity of the liquid

and gas phases. In this way, the effect of the liquid viscosity on the

bubble dynamics and on the shear stress generated by the jet impact on

the solid surface could be studied.

As the first 2D Fluent axisymmetric simulations of the bubble collapse

in bulk liquid were not mesh independent, we have developed our own

numerical method in order to have a better insight into the effect of

parameters such as the size of cells on the numerical simulations. The

simulations of the dynamics of a spherical bubble in an acoustic field

were carried out. The results show that:

• the evolution of the bubble radius as a function of time is mesh

independent.

• the pressure inside the bubble is almost spatially uniform.

• the high pressure gradients near the gas-liquid interface are accu-

rately solved.

Based on the information obtained with 1D numerical simulations, 2D

Fluent axisymmetric simulations of the motion of a bubble submitted

to an external pressure of 5 bar have been carried out. The results show

that:

• the evolution of the bubble radius as a function of time is mesh

independent.

• the surface tension at the bubble-liquid interface influences the

shape of the bubble.

3.5. CONCLUSION 103

• the main parameter of the mesh is the number of cells along the

radial direction.

Chapter 4

Laser-induced bubble

collapse

Contents

4.1 Bubble dynamics in bulk liquid . . . . . . . . 106


4.1.2 Results . . . . . . . . . . . . . . . . . . . . . 109

4.2 Bubble dynamics near a solid surface . . . . 111


4.2.2 Model validation . . . . . . . . . . . . . . . . 113

4.2.3 Results . . . . . . . . . . . . . . . . . . . . . 124

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . 135

105

106 CHAPTER 4. LASER-INDUCED BUBBLE COLLAPSE

In this chapter, the behaviour of a laser-induced bubble in different envi-

ronments is analysed based on the numerical model described in chapter

3. The dynamics of a laser-induced bubble is first considered because

the simulated bubble shape can be compared to many experimental pho-

tographs such as those published in [74, 80].

Section 4.1 discusses bubble dynamics in bulk liquid. Section 4.2 is de-

voted to bubble dynamics near a solid surface. The bubble dynamics is

studied for different values of the stand-off parameter, γ. The stand-off

parameter is the ratio between the distance separating the initial bub-

ble centre from the solid surface and the maximum bubble radius. The

simulated bubble shape is compared to the experimental photographs in

the literature [74, 80] and the effect of the viscosity on the velocity of the

jet penetrating the bubble is discussed. Our simulations are performed

by using the commercial software Fluent which allows easily to handle

non uniform meshes.

4.1 Bubble dynamics in bulk liquid


In this section, the motion of a bubble created by a laser in bulk liquid

is analysed. In the pulsed-laser discharge technique, a laser pulse is fo-

cused in a liquid and a plasma spot is generated by optical breakdown.

After plasma recombination, a bubble containing mainly water vapour

is generated. This bubble will grow due to the high pressure and tem-

perature inside the bubble. As the simulation of the plasma generation

and recombination is beyond the scope of the present work, our simu-

lations start from an initial pre-existing spherical vapour bubble. This

bubble is initially at rest in an incompressible liquid water and it has a

radius R0. The material properties for the vapour (bubble) and liquid

water phases are given in Table 4.1. The growth of the bubble is induced

by the high pressure and temperature inside the bubble. As a result, a

jump of pressure is imposed at the bubble-liquid interface. The pressure

is 1 bar in the liquid and pb,0 inside the bubble. If a jump of temperature

was applied to the bubble-liquid interface, the calculated velocity field

4.1. BUBBLE DYNAMICS IN BULK LIQUID 107

would be wrong and mesh dependent. Consequently, the temperature

field has to be smooth near the bubble-liquid interface. Based on numer-

ical results of Nagrath et al. [70], the initial temperature field T0(r) as a

function of the radial distance to the initial bubble centre r is expressed

as a normal distribution:

T0(r) = Tamb + Tc,0e−

1

2

(

2000rR0

)2

(4.1)

where Tamb = 300 K is the ambient temperature in the liquid and Tc,0 is

the difference between the temperature at the centre of the bubble and

the ambient temperature. The initial conditions including the initial

bubble radius R0, the initial pressure inside the bubble pb,0 and the

temperature Tc,0 are chosen such that the bubble reaches a maximum

radius of 1 mm. It is calculated that R0 = 0.225 mm, pb,0 = 38.59 bar

and Tc,0 = 1067 K.

water liquid water vapour

ρ[

kg/m3]

1000 eq. 3.8

µ [ kg/ms] 0.001 1.34 10−5

k [ W/mK] 0.6 0.026

cp [ J/kgK] 4182 2014

Mm [ g/mol] 18 18

Table 4.1: Material properties for vapour and liquid phases.

An axisymmetric formulation is used as shown in Fig. 4.1. The pressure

and the temperature at the boundary of the calculation domain are equal

to p∞ = 1 bar and T∞ = 300 K, respectively. A convergence study has

shown that to take R∞ = 100 mm ≈ 445R0 is large enough so that

the bubble acts as an isolated bubble in an infinite medium. Figure 4.1

shows an example mesh at the vicinity of the bubble. Note that for

clarity a coarse mesh is shown in the figure, but a converged solution

usually demands a denser mesh. The parameters p and n (defined in

section 3.4) that are used to mesh the bubble domain play an important

role for the convergence, with the effect of n being more pronounced.

What we called ”the bubble domain” corresponds to the region with rbd



air bubble inside bulk liquid. Note that, in actual calculations, a denser

mesh is used to obtain convergence.


0 30 60 90 120 150 180 210 240

0.2

0.4

0.6

0.8

1

Time [µs]

Bub

ble

radi

us [m

m]

Figure 4.2: Evolution of the bubble radius as a function of time.

equals to 1 mm, the value of the maximum bubble radius (see Fig. 4.1).

However, the cell size for the region away from the bubble (which is not

shown in Fig. 4.1) has not a big effect on the convergence. Quadrilateral

cells are used for the calculations and the meshing is performed by the

same way as in section 3.4 by using the commercial software Gambit (for

the details about the cell size, mesh generation, see section 3.4).

4.1.2 Results

First, as in section 3.4, the bubble remains spherical as a direct con-

sequence of the symmetry of the loading condition. Figure 4.2 shows

the evolution of the bubble radius as a function of time. Under the ef-

fect of the severe initial conditions inside the bubble, the bubble grows

and reaches a maximum radius of 1.05 mm. Next, the bubble collapses,

reaches its minimum radius and then rebounds. Figure 4.3 shows the

minimum bubble radius as a function of the parameter n, while keeping

p = 200. A converged solution is obtained for n ≥ 1000: the difference

between the minimum bubble radius for n = 1000 and n = 2000 is less

than 1 %. For the calculations shown in Figs 4.2 and 4.3, the value of

the time increment is chosen equal to 5 ns at the start of the calcula-


Figure 4.3: A convergence test showing the effect of the parameter n on

the minimum bubble radius; p is kept equal to 200.

tions; after 6 µs, the time increment ranges from 0.01 ns to 20 ns and

is chosen in such a way that the CFL condition remains always smaller

than 0.5. For the mesh sizes and time increments considered here, the

scale residual should be smaller than 10−3 for both the continuity and

momentum equations, and 10−6 for the energy equation to obtain a con-

verged solution.

Now, we compare the bubble dynamics calculated using the HRIC scheme

and the PLIC method (see sections 3.2.4 and 1.4.3). The difference be-

tween these two methods is that the bubble-liquid interface spreads out

on 2-3 cells using the HRIC scheme while it is accurately defined in 1 cell

using the PLIC method. First, it was observed that, during the rebound

phase, the bubble shape remains spherical using the HRIC scheme but

not using the PLIC method. Figure 4.4 shows the evolution of the bub-

ble radius as a function of time calculated using the HRIC scheme and

the PLIC method. The HRIC scheme and PLIC method give almost

the same evolution: the minimum bubble radius and the collapse time

4.2. BUBBLE DYNAMICS NEAR A SOLID SURFACE 111

0 30 60 90 120 150 180 2100

0.2

0.4

0.6

0.8

1

Time [µs]

Bub

ble

radi

us [m

m]

HRICPLIC

Figure 4.4: Evolution of the bubble radius as a function of time using

the HRIC scheme and the PLIC method.

calculated using the HRIC scheme are, respectively, 0.85 % higher and

370 ns larger than the values calculated using the PLIC method. In the

following, we will use the HRIC scheme because it requires less time to

achieve convergence.

4.2 Bubble dynamics near a solid surface


In this section, the motion of a bubble created by a laser near a solid

surface is analysed. As the simulation of the plasma generation and re-

combination is beyond the scope of the present work, our bubble dynam-

ics simulations start from an initial pre-existing spherical water vapour

bubble. Blake et al. [12] was one of the first to simulate the growth

and the collapse of a bubble near a solid surface. They showed that

the bubble dynamics during the collapse phase was different than those

calculated by Plesset et al. [83], who only considered the collapse of a

spherical bubble at rest near a solid surface. The bubble is closer to

the solid surface and less flattened along the solid surface in Plesset’s


simulation. This has been attributed to the fact that the bubble is not

at rest when it reaches its maximum radius. Indeed, while the velocity

is zero at the top of the bubble-liquid interface, it is not necessarily so

at the other parts of the bubble-liquid interface. The velocity field still

being unknown, this means that the growth phase has to be simulated.

Initially, the vapour water bubble is at rest in liquid water and placed

at a fixed distance from the solid surface. The bubble has an initial

radius R0. The material properties for the vapour (bubble) and liquid

water phases are given in Table 4.1. The pressure and the temperature

at the boundary of the calculation domain are equal to p∞ = 1 bar and

T∞ = 300 K. As in the previous section, a jump of pressure is imposed

at the bubble-liquid interface. The pressure is 1 bar in the liquid and pb,0

inside the bubble. The initial field of temperature in the calculation do-

main is defined by eq. 4.1. The choice of the initial conditions, including

the initial bubble radius R0, initial pressure inside the bubble pb,0 and

the temperature Tc,0 in eq. 4.1 is non-trivial by the lack of experimen-

tal data in the early stages of laser-induced bubble generation following

plasma recombination. Therefore, the initial conditions have been de-

termined such that the bubble reaches a maximum radius in agreement

with available experimental data. For γ = 1, a maximum bubble radius

of 1 mm has been reported [74] after a time interval of 107 µs. Based

on these observations, it was calculated for γ = 1 that R0 = 0.2 mm,

pb,0 = 42 bar and Tc,0 = 1998 K. For other γ values, only the initial

pressure inside the bubble is modified, the initial bubble radius and the

initial temperature field in the calculation domain being kept the same

as for γ = 1. The initial pressure inside the bubble is determined such

that the bubble reaches a maximum radius of 1 mm (for γ = 0.6, this

results in pb,0 = 38.5 bar). The solution domain is the quarter of a circle

shown schematically in Fig. 4.5. Axisymmetric boundary conditions

are applied to the left hand side of the domain and a no-slip boundary

condition is imposed at the bottom of the domain (see Fig. 4.5). The

pressure and the temperature at the boundary of the domain (along the

arc of a circle) are 1 bar and 300 K, respectively. The bubble is placed

along the axis of axisymmetry at a distance from the solid surface de-


Figure 4.5: Schematic diagram showing the boundary conditions for the

problem of a bubble created by a laser near a solid surface.

termined by the stand-off parameter γ. As in the previous section, a

convergence study has shown that taking R∞ = 100 mm is large enough

so that the bubble dynamics is not influenced by the position of the

boundary away from the bubble. If the time increments and the scale

residuals are kept as in the previous section, a convergence solution re-

quires a mesh containing between 80000 to 120000 cells, depending on

the initial bubble radius. Combinations of quadrilateral and triangular

cells are used for the calculations and the meshing is performed by using

the commercial software Gambit. A typical computational time required

to run one case (γ = 1) is 80 h on two AMD Opteron 252 running in par-

allel at 2.6 GHz. This computational time increases when γ decreases,

the maximum being about 120 h at γ = 0.6.

4.2.2 Model validation

In this section, our numerical model is validated by comparing simula-

tions of the dynamics of a single cavitating bubble near a solid surface to

experimental data available in the literature for 2 values of the stand-off


parameter γ. Note that, when the bubble is aspherical the maximum

bubble radius in the γ parameter is defined as the maximum distance

between the initial bubble centre and the bubble-liquid interface during

the bubble growth. In these simulations, the liquid viscosity has been

taken equal to 0.001 kg/ms, corresponding to the value for water at room

temperature, complying with the experimental data.

Comparison of simulated and experimental bubble dynamics

at γ = 1

Figure 4.6 shows a comparison of experimentally observed (upper pic-

tures) and simulated bubble shapes as a function of time for γ = 1. The

experimental data have been taken from the work of Ohl et al. [74].

The figures show that the experimentally observed bubble shape is ac-

curately reproduced, both qualitatively and quantitatively. In the first

stage (Fig. 4.6A), the bubble grows and reaches a maximum radius af-

ter 108.6 µs. The pressure field surrounding and inside the bubble is

shown in Fig. 4.7(a). As expected, the pressure inside the bubble is

much smaller than the pressure of 1 bar imposed at the boundary of the

flow domain. In a second stage, shown in Fig. 4.6B, the bubble vol-

ume decreases. This is the collapse phase. During this collapse phase,

the bubble is seen to move towards the solid surface. Concerning the

pressure field, the pressure being higher at the boundary of the flow do-

main than inside the bubble, the liquid is accelerated towards the bubble

and the bubble is compressed. Due to the compression of the bubble,

the pressure inside the bubble increases (see Fig. 4.7(b)). Moreover,

as the incompressibility of the liquid requires that the density is con-

stant in the liquid volume at each increment, the pressure in the region

above the bubble increases to avoid fluid particles being accumulated at

this region. This can clearly be observed from the pressure distribution

shown in Fig. 4.8(a). The liquid between this high pressure zone and

the bubble is accelerated near the solid surface. As a result, the upper

bubble-liquid interface first becomes flattened, as seen in the last picture

of Fig. 4.6B. Next, liquid starts to penetrate the bubble forming a jet, as

seen in Fig. 4.6C. Jet formation starts when the top of the bubble-liquid


interface along the axis of axisymmetry has an ordinate smaller than any

other points of the bubble-liquid interface. The pressure distribution at

t = 223 µs during the jet penetration inside the bubble is shown at Fig.

4.8(b). At t = 231.4 µs, the liquid jet reaches the lower bubble-liquid

interface. After this moment, which we denote as jet impact, the bubble

becomes toroidal. Figures 4.7 and 4.8 show also that the pressure in-

side the bubble is almost uniform. It can be explained by the following

way. When ignoring the convective, diffusive and surface tension terms

in eq. 3.4, the gradient of pressure is proportional to the density. The

gas density being almost 1000 times smaller than the liquid density, the

gradient of pressure is also 1000 times smaller in the gas phase than in

the liquid phase. As a result, the pressure in the gas phase is almost

uniform.

A small deviation can be noticed on the time scale between the numerical

simulations and the experimental photographs. This can be explained

in the following way. As the initial conditions have been determined

from experimental images such that the bubble reaches a maximum ra-

dius Rmax after a time tRmax, any errors in the measurement of tRmax

or Rmax from the experiments will have an influence on the simulated

bubble dynamics. The measurement of tRmax is challenging for two rea-

sons. First, the photographs being recorded at a framing rate of 112500

frames/s [74], the time between two photographs is 8.88 µs. The exact

time tRmax can therefore only be estimated with this accuracy. Next,

it was calculated that the difference between the maximum bubble ra-

dius, shown experimentally in Fig. 4.6A to occur at t = 107 µs, and the

bubble radius occurring a bit earlier, e.g. at t = 97 µs, is smaller than

2 µm. Such a difference is not resolvable in the available experimental

photographs, where the spatial resolution is only around 20 µm. Hence,

the spatial resolution limiting the determination of the maximum bubble

radius Rmax induces in turn inaccuracies on the time tRmax correspond-

ing to the maximum radius. Both intrinsic errors, limiting the accuracy

of tRmax will translate into the choice of the initial conditions, thus af-

fecting all further numerical results, especially after the bubble reaches

its maximum.


A. Growth phase

B. Collapse phase

C. Collapse phase: Jet penetration

Figure 4.6: Experimentally observed [74] and simulated bubble shape

as a function of time for γ = 1 after laser-induced bubble generation at

t = 0 µs. The solid surface is located at the bottom of each picture.


(a)

(b)

Figure 4.7: Pressure field in the gas and liquid phases at (a) t = 108.6 µs,

(b) t = 172 µs for γ = 1. The solid surface is located at the bottom of

the figure. The black solid line is the bubble-liquid interface.


(a)

(b)

Figure 4.8: Pressure field in the gas and liquid phases at (a) t = 215 µs,

(b) t = 223 µs for γ = 1. The solid surface is located at the bottom of

the figure. The black solid line is the bubble-liquid interface.


Comparison of simulated and experimental bubble dynamics

at γ = 0.6

The dynamics of a single cavitating bubble near a solid surface at γ = 0.6

has been simulated as well. Experimental data for these conditions are

available from the work of Philipp and Lauterborn [80]. The simulations

were performed for a maximum bubble radius of 1 mm. The experimen-

tally observed maximum bubble radius being 1.45 mm, the bubble shape

and time scales have been adimensionalised in the numerical calculations

with respect to Rmax and Rmax

√

ρL/p∞, respectively, as suggested in

[101]. Figure 4.9 shows the obtained comparison of the experimentally

observed [80] and simulated bubble shape as a function of time, t = 0 µs

again corresponding to the moment of bubble generation. It is seen that

the experimentally observed bubble shape has again been accurately

simulated. As compared to the γ = 1 case, the bubble now becomes

more aspherical and closer to the solid surface during the growth phase

(Fig. 4.9A). During the collapse phase, the bubble flattens along the

solid surface because the top of the bubble collapses while the lower

part of the bubble continues to expand (Fig. 4.9B). The bubble is also

more flattened during the collapse phase at γ = 0.6 than at γ = 1. In

fact, it is seen in Fig. 4.9B that around 310 µs, the bubble takes on an

almost triangular shape. The kink at the base of the bubble developing

in Fig. 4.9B is not a numerical artifact: it does not change with a finer

mesh, and it has also been observed in more accurate experimental pho-

tographs (private communication with Prof. C.-D. Ohl). The formation

of a kink at the base of the bubble can be explained by the following

way. During the collapse phase, the liquid between the bottom of the

bubble and the solid surface spreads radially outwards. This flow meets

the inward motion of the liquid induced by the bubble collapse and the

liquid is projected upwards as seen in Fig. 4.10. As a result, a zone of

recirculation develops inside the bubble (see Fig. 4.10). Moreover, the

velocity vector at the bubble-liquid interface is directed outwards below

the point A in Fig. 4.10 and inwards above the point A. As a result

of these opposite velocity vectors, a kink develops. The kink and the

velocity vector are shown at t = 202 µs in Fig. 4.11. In this figure, the


kink continues to develop. Such kink is not observed for γ = 1 because

the bubble is more away from the solid surface. Although that a recir-

culation zone is also observed for γ = 1, there are no points along the

bubble-liquid interface where the velocity vectors are in opposite direc-

tions. Finally, Figure 4.9C shows that also jet penetration inside the

bubble during the collapse phase has been adequately reproduced.


A. Growth phase

B. Collapse phase

C. Collapse phase: Jet penetration

Figure 4.9: Experimentally observed [80] and simulated bubble shape

as a function of time for γ = 0.6 after laser-induced bubble generation.

The solid surface is located at the bottom of each picture.


(a)

(b)

Figure 4.10: (a) Bubble shape at t = 182 µs. (b) Velocity field for the

region indicated by a black box in (a). The solid surface is located at

the bottom of the figure and the black solid line in (b) represents the

bubble-liquid interface.


(a)

(b)

Figure 4.11: (a) Bubble shape at t = 212 µs. (b) Velocity field for the

region indicated by a black box in (a). The solid surface is located at

the bottom of the figure and the black solid line in (b) represents the

bubble-liquid interface.


4.2.3 Results

Effect of viscosity on bubble dynamics

In this section, our numerical model is used to study the influence of

liquid viscosity on the dynamics of a single cavitating bubble near a

solid surface for three values of the stand-off parameter γ. The bub-

ble shapes are compared for two realistic values of the liquid viscosity

corresponding to, respectively, water (µwater = 0.001 kg/ms) and oil

(µoil = 0.05 kg/ms). The initial conditions and the other material prop-

erties are unmodified. Defining the Reynolds number as

Re =Rmax

√p∞ρL

µL(4.2)

the two viscosities correspond to a Reynolds number of, respectively,

10000 and 200.

Figure 4.12 compares the bubble dynamics at γ = 1.25 for the two vis-

cosity values. Figures 4.12(a-b) and 4.12(c-f) represent, respectively, the

growth and the collapse phase of the bubble. In the first stage of the

collapse phase, the bubble moves towards the solid surface and its vol-

ume decreases as seen in Fig. 4.12(c). The bubble moves towards the

solid surface because the velocity field at the lower part of the bubble

is downward. In the second stage of the collapse phase (Figs 4.12(d-f)),

the velocity field at the lower part of the bubble reverses and the bubble

starts to move away from the solid surface. Finally, a jet penetrates the

bubble at the end of the bubble collapse (Figs 4.12(e-f)).

As to the influence of the viscosity, the bubble volume at the end of the

growth phase is smaller for µoil than for µwater (Fig. 4.12(b)). This in-

dicates that the growth of the bubble is slowed down when the viscosity

increases. During the collapse phase, the viscosity has much more influ-

ence on the bubble shape near the solid surface. The distance separating

the bottom of the bubble from the solid surface is always larger for µoil

than for µwater (Figs 4.12(c-f)). This can be explained in the following

way. Firstly, during the first stage of the bubble collapse, the motion of

the bubble towards the solid surface is slowed down when the viscosity

is larger. Secondly, during the second stage of the bubble collapse, the


Figure 4.12: Comparison of the bubble shape at γ = 1.25 for the two val-

ues of the liquid viscosity (µwater = 0.001 kg/ms and µoil = 0.05 kg/ms).

The solid and the dashed lines correspond to the bubble shape for, re-

spectively, µwater and µoil. The solid surface is located at the bottom of

each picture.

upward motion of the lower part of the bubble starts first for the bubble

in oil. As a conclusion of Fig. 4.12, the bubble volume is always smaller

for the larger viscosity.

Figure 4.13 compares the bubble shapes at γ = 0.6 for the two viscosity

values considered. Figures 4.13(a) and 4.13(b-d) show the bubble dy-

namics, respectively, near its maximum volume and during the collapse

phase. During the collapse phase, the bubble moves towards the solid

surface and its volume decreases. At the end of the collapse phase, a jet


penetrates the bubble (Fig. 4.13(c)). Next, the jet impacts the lower

bubble-liquid interface, see Fig. 4.13(d). The time scales are different

during jet penetration because the start of the jet penetration and the

jet impact occur at different times for the two viscosities.





each picture.

Concerning the influence of the viscosity, Fig. 4.13(a) shows that the

bubble volume is smaller for µoil than for µwater at the end of the growth


phase. During the collapse phase, the bubble is more flattened along the

solid surface for µwater than for µoil (see Figs 4.13(b-c)). Moreover, the

difference between the distance, for µoil and for µwater, of the lower part

of the bubble from the solid surface continues to increase (Figs 4.13(a-

d)). This indicates that the motion of the bubble towards the solid

surface is further slowed down when the viscosity is larger. Hence, the

bubble volume is always smaller when the viscosity is larger. Note also

that the kink at the base of the bubble developing at the end of the

collapse phase is only observed when the bubble is in water (see Fig.

4.13(c-d)).

−120 −100 −80 −60 −40 −20 0 0

0.2

0.4

0.6

0.8

1

t − timp

[µs]

V /

Vm

ax

γ = 0.60 − µwater

γ = 0.60 − µoil

γ = 1.25 − µwater

γ = 1.25 − µoil

Figure 4.14: Ratio between the bubble volume and its maximum volume

as a function of time for γ = 0.6 and γ = 1.25. For each γ value, two vis-

cosity values are studied: µwater = 0.001 kg/ms and µoil = 0.05 kg/ms.

The time 0 corresponds to the moment of jet impact on the lower bubble-

liquid interface.

Moreover, comparing Figs 4.12 and 4.13 indicates that increasing the

viscosity leads to a stronger reduction of the bubble volume at γ = 0.6

than at γ = 1.25 during the collapse phase. To confirm this result, the

ratio between the bubble volume and its maximum volume (V/Vmax) is

plotted in Fig. 4.14 as a function of time at γ = 0.6 and at γ = 1.25.






each picture.

Only the collapse phase is shown. The initial and final time correspond

to, respectively, the start of the collapse phase and the jet impact on

the lower bubble-liquid interface. As this time interval depends on the

viscosity and γ values, we chose the time 0 as the moment of jet impact.

It is seen in Fig. 4.14 that V/Vmax is smaller when the bubble is in oil

than in water as well at γ = 0.6 as at γ = 1.25. A stronger reduction

of V/Vmax when the viscosity increases is also observed at γ = 0.6. The


viscosity has more influence on the bubble volume when the bubble is

closer to the solid surface. Figure 4.15 compares the bubble dynamics at

γ = 2.5 for the two viscosity values. Figures 4.15(a) and 4.15(b-c) show

the bubble shape, respectively, near its maximum volume and during the

collapse phase. During the collapse phase, the bubble volume decreases

and a jet starts to penetrate the bubble (Fig. 4.15(c)). The time scales

are different during jet penetration because the start of jet penetration

and jet impact occur at different times for the two viscosity values. Dur-

ing jet penetration inside the bubble, instead of continuously decreasing,

the bubble volume starts to increase again, as seen more clearly in Fig.

4.16. In this figure, the final time corresponds to the jet impact. The

re-growth of the bubble volume occurs when the pressure forces coun-

terbalance the inertial forces during the bubble collapse. Such re-growth

of the bubble corresponds to the rebound phase.

As to the influence of the viscosity, Fig. 4.15(a) shows that the bubble

volume is similar for the two viscosity values. The viscosity has little

influence on the bubble growth because the bubble is relatively far away

from the surface. During the collapse phase, the bubble shape is similar

for the two viscosity values, but the bubble volume is smaller when the

bubble is in oil than in water (Fig. 4.15(b)). During jet penetration

inside the bubble, the rebound phase starts at 220.4 µs for µoil and at

221.8 µs for µwater (see Fig. 4.16). The rebound phase starts first when

the bubble is in oil because the bubble volume is smaller for µoil (see

Fig. 4.15(c)). The bubble being smaller, the pressure inside the bubble

is higher. As a result, the pressure forces sooner counterbalance the in-

ertial forces and the bubble will first rebound in oil.

Note that jets are also observed for a value of γ higher than 3, contrar-

ily to the assumption done in chapter 2. However, the bubble can be

assumed almost spherical as the jet penetrates the bubble at the end of

the collapse phase and during the rebound phase.


212 214 216 218 220 222 224 2260.05

0.10

0.15

0.20

0.25

0.30

0.35

Time [µs]

Bub

ble

volu

me

[mm

3 ]

µwater

µoil

Jet penetration

Figure 4.16: Bubble volume as a function of time for the two viscosity

values at γ = 2.5. The final data point correspond to the moment of jet

impact on the lower bubble-liquid interface.

Effect of viscosity on jet velocities

When simulating jet penetration inside a cavitating bubble, we observed

that the jet velocity along the axis of axisymmetry is always maximum at

the jet front until the jet has impacted the lower bubble-liquid interface.

This velocity, called the jet front velocity, can therefore be considered a

characteristic parameter of the jet. It is this parameter that has been

studied as a function of time and liquid viscosity for different γ values.

Figure 4.17 shows the jet front velocity (ujet) as a function of time

for γ = 0.6, γ = 1.25 and γ = 2.5, taking the two values of the liquid

viscosity defined in previous section: µwater = 0.001 kg/ms (Fig. 4.17(a))

and µoil = 0.05 kg/ms (Fig. 4.17(b)). The time t = 0 corresponds to

the start of jet penetration, and the final time to the moment of jet

impact on the lower bubble-liquid interface. After the jet impact on the

lower bubble-liquid interface, it was verified in all cases that the jet front

velocity decreases as the jet hits the liquid layer under the bubble.


(a) (b)

Figure 4.17: Jet front velocity as a function of time for γ = 0.6, 1.5 and

2.5 for (a) µwater and (b) µoil. The initial and the final time correspond

to the moment of, respectively, the jet formation and the jet impact.

The jet front velocity for γ = 0.6 and γ = 1.25 initially increases with

time and then reaches a constant value. However, for γ = 2.5, the jet

front velocity initially increases with time, reaches a maximum and then

decreases. The jet front velocity decreases because the jet motion is

slowed down by the re-growth of the bubble during the rebound phase

(Fig. 4.16). Note that such maximum in the jet front velocity curve is

still expected when compressibility effects are considered, as such effects

(e.g. shock waves [75]) will further slow down the jet motion.

Comparison of Figs 4.17(a) and (b) shows that the jet front velocity

decreases when the liquid viscosity is larger. This observation can be

explained based on the difference of pressure between the maximum

pressure above the bubble, pmax, (cfr. Fig. 4.8(b)) and the pressure

inside the bubble, pb. This difference of pressure is proportional to the

jet front velocity [12]. During the collapse phase, the ratio between the

bubble volume and its maximum volume (V/Vmax) is smaller for µoil

than for µwater (cfr. Fig. 4.14). Hence, when the jet penetrates the

bubble, the bubble is more compressed in oil than in water. This will


have an influence on pmax and pb. As the bubble is more compressed

in oil, the pressure inside the bubble is highest in oil. Moreover, when

pmax further increases during the bubble collapse, this increase will be

more pronounced when the bubble is already more compressed, which

is the case when the bubble is in oil. Although pmax and pb are higher

in oil, the net effect on pmax − pb is a priori unknown. However, it was

calculated that the difference of pressure is smaller for µoil. This is why

smaller jet velocities are observed in oil. This result is in agreement with

previous results reported by Yu et al. [112] and Popinet et al. [85].

Figure 4.18: Time interval between the jet formation and the jet impact

on the lower bubble-liquid interface as a function of γ for two values of

the liquid viscosity. The lines are a guide to the eye.

Figure 4.18 illustrates the time interval between the jet formation and

the jet impact on the lower bubble-liquid interface. A maximum is ob-

served near γ equal to 1 for the two viscosity values. This time interval

depends on two parameters: the jet front velocity, ujet and the distance


that the jet has to penetrate before impacting the lower bubble-liquid

interface. For γ values smaller than 1, the time interval decreases al-

though the jet front velocity is smaller than this value for γ = 1. This

is because the jet has to penetrate a smaller distance than at γ = 1 be-

fore impacting the lower bubble-liquid interface, the bubble being more

flattened along the solid surface (cfr. Fig. 4.9). For γ values beyond

γ = 1, the time interval decreases because ujet is higher than at γ = 1

and because the distance that the jet has to penetrate before impacting

the lower bubble-liquid interface is smaller than at γ = 1.

The time interval for µoil is higher at γ = 2.5 than at γ = 2 because the

bubble rebounds. During the bubble rebound, the jet front velocity is

slowed down by the re-growth of the bubble. Moreover, as the bubble

volume increases, the jet has to penetrate a larger distance before im-

pacting the lower bubble-liquid interface.

Finally, the time intervals are different for the two viscosity values be-

cause: (i) the jet front velocity is smaller for µoil than for µwater; (ii)

the jet has to penetrate a smaller distance for µoil than for water, the

bubble volume being smaller for µoil.

Now we focus our attention on the maximum jet front velocity (umax).

This velocity corresponds to the maximum jet front velocity calculated

in Fig. 4.18. In a first time, the evolution of umax as a function of γ will

be compared to experimental results in the literature. In a second time,

the influence of the liquid viscosity on umax will be discussed.

The value of the maximum jet front velocity (umax) is shown in Fig.

4.19 as a function of γ for the two viscosity values considered. It is

seen to initially increase with γ until a maximum value is reached, the

magnitude and position of which depending on the liquid viscosity. A

maximum in the umax curve is reached because the jet velocity is slowed

down during the bubble rebound (Fig. 4.17). In experimental results of

Philipp and Lauterborn [80], the maximum jet front velocity reaches a

constant value and does not go through a maximum. In fact, we believe

that the experimental spatial and temporal resolution does not allow for

accurately measuring the maximum jet front velocity, the absolute error


reported on the jet front velocity being on the order of 10 m/s [80]. The

expected maximum is therefore very difficult to detect experimentally.

However, the initial increase of umax as a function of γ has been observed

experimentally by Philipp and Lauterborn as well [80].

Figure 4.19: Evolution of the maximum jet front velocity as a function

of γ for two values of the liquid viscosity. The lines are a guide to the

eye.

A direct, quantitative comparison of the jet velocities with the values

reported by Popinet et al. [85] is difficult because they did no study laser-

induced bubble collapse, and because their range of Reynolds number

(15-55) is smaller than those investigated in this paper. This is why our

results have only been compared to literature data from experiments

carried out in water, for which the Reynolds number is 10000, as in our

simulations. Before the maximum, our calculated values for the max-

imum jet front velocity are in agreement with the experimental data

found in the literature. It was indeed observed experimentally in liquid

4.3. CONCLUSION 135

water that for γ = 1.1 the maximum jet front velocity was 78 m/s [18],

a value which is very close to the value of 74.4 m/s that can be derived

from our numerical results in Fig. 4.19. The absolute values of umax are

also in agreement with the experimental studies of Gibson [11] in liquid

water, who predicted an average value of 76 m/s for γ = 1, a value very

close to the 73.4 m/s that we calculated numerically.

Both the magnitude and the position of the maximum of the umax curve

in Fig. 4.19 depend on the liquid viscosity. The magnitude of the max-

imum is smaller for µoil than for µwater, the jet front velocity being

smaller when the viscosity is larger. Moreover, the γ value correspond-

ing to the maximum is smaller for µoil than for µwater. This indicates

that the bubble starts to rebound for smaller γ when the bubble is in oil

than in water. The bubble starts to rebound for smaller γ in oil because

the bubble volume is smaller (cfr. Fig. 4.15). For γ values beyond the

maximum, umax decreases for the two viscosity values because the jet

motion is slowed down by the re-growth of the bubble during the re-

bound phase.

Before the maximum, increasing the viscosity gives rise to strongly di-

verging umax curves when decreasing γ. As previously explained, smaller

jet velocities are observed when V/Vmax decreases. As increasing the

viscosity leads to a stronger reduction of V/Vmax at γ = 0.6 than at

γ = 1.25 (see Fig. 4.14), it is expected that the difference between umax

for the two viscosities is higher at γ = 0.6 than at γ = 1.25.

4.3 Conclusion

In this chapter, we use the numerical model developed in chapter 3 to

investigate the dynamics of a bubble generated by a laser: (i) in bulk

liquid and (ii) near a solid surface. The commercial software Fluent

is used to solve these problems. The results have been shown mesh

independent. For (i), the results show that:

• the bubble remains spherical during the growth and the collapse

phase.


• the evolution of the bubble radius as a function of time is identical

when using the HRIC scheme and PLIC method.

For (ii), the simulations show that:

• the evolution of the bubble shape with time is in good agreement

with experimentally observed bubble dynamics for γ = 1 [74] and

γ = 0.6 [80], γ being defined as the distance between the initial

bubble centre and the solid surface scaled by the maximum bubble

radius.

• the time interval between the start of the jet formation and the

jet impact on the lower bubble-liquid interface is maximum near

γ equal to 1.

• the maximum jet front velocity increases with γ until a maximum

is reached, the value and the position of which depending on the

liquid viscosity. Before the maximum, the numerically obtained

velocities are in agreement with the experimental values available

in the literature. For γ values beyond the maximum, the maximum

jet front velocity decreases because the bubble rebounds before it

becomes toroidal.

Chapter 5

Acoustic-induced bubble

collapse

Contents

5.1 Bubble dynamics in bulk liquid . . . . . . . . 138


5.1.2 Results . . . . . . . . . . . . . . . . . . . . . 140

5.2 Bubble dynamics near a solid surface . . . . 142


5.2.2 An illustrative example . . . . . . . . . . . . 144

5.2.3 Influence of the initial bubble radius . . . . . 148

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . 153

137

138 CHAPTER 5. ACOUSTIC-INDUCED BUBBLE COLLAPSE

In this chapter, the behaviour of an acoustic-induced bubble in differ-

ent environments is analysed based on the numerical model described

in chapter 3. In the simulations, the frequency of the acoustic field is

20 kHz. Note that when a frequency of 1 MHz was used, the interface

did not move while the velocity field at the bubble-liquid interface was

different from 0. At first sight, Fluent is not able to solve problems with

so small time and spatial scale. Section 5.1 discusses bubble dynamics in

bulk liquid. Section 5.2 is devoted to bubble dynamics near a solid sur-

face; the effect of the initial bubble radius on the evolution of the bubble

shape and on the velocity of the jet penetrating the bubble is studied.

The simulations are performed by using the commercial software Fluent.

5.1 Bubble dynamics in bulk liquid


In this section, the motion of a spherical air bubble in bulk liquid sub-

jected to an acoustic field is analysed. Note that similar analysis was

performed in section 3.3.1 based on the assumption that the bubble re-

mains spherical while its radius changes (1D case), whereas here, the

shape of the bubble can evolve under the effect of the acoustic field (2D

case). The bubble is initially at rest in liquid water and it has a radius

of R0 = 78.5 µm. The material properties for the gas (air bubble) and

the liquid water phases are given in Table 3.1. The initial pressure and

the initial temperature in both phases are equal to 1 bar and 300 K,

respectively. Due to the symmetry of the problem, an axisymmetric

formulation is used (see also section 3.4) as shown in Fig. 5.1. The

remote pressure at the boundary p∞ (see Fig. 5.1) is the superposition

of the ambient pressure (pambient = 1 bar) and the sinusoidal pressure

associated with the acoustic field (pacoust):

p∞ = pambient + pacoust = 1 − pA sin(2πfAt) [bar] (5.1)

where the amplitude and the frequency of the acoustic field are assigned

to be pA = 0.8 bar and fA = 20 kHz, respectively. A convergence study

has shown that to take R∞ = 13 mm ≈ 165R0 is large enough so that


the bubble acts as an isolated bubble in an infinite medium (see also

section 3.3.1).


air bubble inside bulk liquid. Note that, in actual calculations, a denser

mesh is used to obtain convergence.

Figure 5.1 shows an example mesh at the vicinity of the bubble. Note


that for clarity a coarse mesh is shown in the figure, but a converged

solution usually demands a denser mesh. The parameters p and n (de-

fined in section 3.4) that are used to mesh the bubble domain play an

important role for the convergence, with the effect of n being more pro-

nounced.

What we called the ”bubble domain” corresponds to the region with rbd

equals to 130 µm in Fig. 5.1. The cell size for the region away from

the bubble (which is not shown in Fig. 5.1) has not a big effect on the

convergence. Quadrilateral cells are used for the calculations and the

meshing is performed by the same way as in section 3.4 by using the

commercial software Gambit (for the details about the cell size, mesh

generation, see section 3.4).

5.1.2 Results

Figure 5.3: A convergence test showing the effect of the parameter n on

the minimum bubble radius; p is kept equal to 200.


0 10 20 30 4030

50

70

90

110

130

Time [µs]

Bub

ble

radi

us [µ

m]

a

(a)

(b)

Figure 5.2: (a) Evolution of the bubble radius as a function of time. The

point ’a’ corresponds to the minimum bubble radius. (b) Evolution of

the pressure associated with the acoustic field as a function of time.


As in section 3.4, the bubble remains spherical as a direct consequence

of the symmetry of the loading condition. Figure 5.2 shows the evolu-

tion of (a) the bubble radius and (b) the pressure associated with the

acoustic field as a function of time. The bubble grows while the pressure

associated with the acoustic field is negative (pacoust < 0, corresponding

to t < 25 µs), and it collapses while the pressure is positive (pacoust > 0,

t ≥ 25 µs). As mentioned earlier, a convergence test is performed to see

the effect of the meshing parameters p and n on the results. Figure 5.3

shows the minimum bubble radius (point ’a’ in Fig. 5.2) as a function

of the parameter n, while keeping p=200. A converged solution is ob-

tained for n ≥ 1000: the difference between the minimum bubble radius

for n = 1000 and n = 2000 is less than 1 %. For the calculations shown

in Figs 5.2 and 5.3, the value of the time increment ranges from 0.01 ns

to 2.5 ns and is chosen in such a way that the CFL condition remains

always smaller than 0.5. The scaled residuals have also a significant ef-

fect on the solution; for the mesh sizes and time increments considered

here, the scale residual should be smaller than 10−11 for the continuity

equation, 10−7 for the momentum equation, and 10−16 for the energy

equation to obtain a converged solution

5.2 Bubble dynamics near a solid surface


In this section, the motion of a spherical air bubble, subjected to an

acoustic field, in a liquid near a solid surface is analysed.

The bubble is initially at rest in liquid water and positioned at a distance

d from the solid surface. It has an initial radius R0. The material prop-

erties for the air and water phases are given in Table 3.1. Initially, the

pressure and the temperature in both phases are 1 bar and 300 K, respec-

tively. The solution domain is the quarter of a circle shown schematically

in Fig. 5.4. Axisymmetric boundary conditions are applied to the left

hand side of the domain and a no-slip boundary condition is imposed at

the bottom of the domain (see Fig. 5.4). The pressure at the boundary

of the domain (along the arc of circle) is given by eq. 5.1. The bubble


Figure 5.4: Schematic diagram showing the boundary conditions for the

problem of an air bubble near a solid surface in an acoustic field.

is placed along the axis of axisymmetry at a distance d from the solid

surface. As in the previous section, a convergence study has shown that

taking R∞ ≈ 165R0 is large enough so that the bubble dynamics is not

influenced by the position of the boundary away from the bubble. As in

the previous section, a denser mesh is required in the vicinity of the bub-

ble. For the calculations shown in this section, however, the value of the

time increment for which the CFL condition is always smaller than 0.5,

ranges from 0.01 ns to 0.625 ns when the velocity at the bubble-liquid

interface is higher than 10 m/s, and from 0.01 ns to 2.5 ns otherwise. If

the scaled residuals are kept as in the previous section, a convergence

solution requires a mesh containing between 90000 to 135000 cells, de-

pending on the initial bubble radius.

Combinations of quadrilateral and triangular cells are used for the cal-

culations and the meshing is performed by using the commercial soft-

ware Gambit. Note that the surface tension is not included in the set

of governing equations, because it can not be accurately calculated for

triangular cells.


5.2.2 An illustrative example

In this example, the bubble is initially spherical and its radius is R0 =

78.5 µm. The distance separating the bubble from the solid surface is

d = 141 µm. Figure 5.5(a) shows the bubble shape at three different

time steps: P1, P2 and P3. Figures 5.5(b,c) show the equivalent bubble

radius (defined as (3V4π )

1

3 ) and the pressure associated with the acoustic

field as a function of time, respectively. The time steps P1, P2, and P3

are also indicated as black dots in Figs 5.5(b) and (c). In the regime

where pacoust is negative (t ≤ 25 µs in Fig. 5.5(c)), the bubble grows.

For t > 25 µs, the bubble continues to grow and reaches a maximum

volume at the beginning of the second half-period of the acoustic field

(t = 25.9 µs, P2 in Figs 5.5(a-c)). After this point, the collapse phase

starts. Figure 5.6 shows four different steps for the evolution of the

bubble shape during the collapse phase (from P2 to P3). Initially, the

bubble starts to change shape; the upper part of the bubble approaches

the solid surface faster than its lower part. Indeed, in Fig. 5.6 the cir-

cles corresponding to steps s1 and s2 are nearly on top of each other on

the side close to the solid surface, whereas they are clearly apart from

each other on the other side. Afterwards, the bubble collapse contin-

ues and the velocity of the top of the bubble-liquid interface is always

faster than the bottom part of the interface (s3). Finally, a jet starts to

penetrate the bubble and during the jet penetration the bubble shape

changes drastically (s4 in Fig. 5.6). At the final step of the simulation,

the jet impacts the lower bubble-liquid interface and the bubble takes a

toroidal shape (not shown in Fig. 5.6). The time interval between the

jet formation and the jet impact is smaller than 1 µs. In the rest of the

section, we focus on the evolution of the pressure and velocity field dur-

ing the time interval from t = 36 µs (step 3 in Fig. 5.6) to t = 38.58 µs

(jet impact on the lower bubble-liquid interface).

As explained in [12], at the beginning of the collapse phase, the pressure

is maximum at infinity and the liquid is therefore accelerated towards

the bubble. The incompressibility of the liquid requires that the density

is constant in the liquid volume at every time increment. For this rea-

son, the pressure in the region above the bubble increases to avoid fluid


(a)

(b)

(c)

Figure 5.5: (a) Bubble shape for three different time steps: t = 0 µs

(P1), t = 26 µs (P2), t = 38.5 µs (P3). Evolution of (b) the equivalent

bubble radius and (c) the pressure associated with the acoustic field as

a function of time. The black dots correspond to time steps: P1, P2, and

P3.


Figure 5.6: Four different steps for the evolution of the bubble shape

during the collapse phase (from P2 to P3 in Fig. 5.5): for s1: t = 26 µs,

for s2: t = 30 µs, for s3: t = 36 µs, for s4: t = 38.5 µs. The solid line

below the bubble corresponds to the solid surface.


Figure 5.7: Pressure field in the two phases at t=36 µs. The black solid

line indicates the position of the bubble. The solid surface is located to

the lower border of the figure.

particles being accumulated at this region. This can clearly be observed

from the pressure distribution shown in Fig. 5.7. Figure 5.8 shows the

evolution of (a) the maximum pressure, (b) the maximum velocity along

the axis of axisymmetry (noted pmax and umax,axi, respectively) as a

function of time (note that the location of the two points correspond-

ing to the maximum pressure and the maximum velocity changes at each

time increment). We see that, initially, both pmax and umax,axi increases

with time. At point I corresponding to the jet formation, an amount of

liquid penetrates the bubble. As the amount of liquid that approaches

the bubble is more than the amount that actually penetrates the bubble,

pmax continues to increase but more slowly and, as a result, the pmax


versus time curve changes from being convex to concave (i.e. I is an

inflection point).

During the collapse phase, the pressure inside the bubble pb always in-

creases with time (see Fig. 5.8(c)), and the difference between the max-

imum pressure along the axis of axisymmetry and the pressure in the

bubble, ∆p = pmax − pb, has a similar tendency to pmax until the point

M (Fig. 5.8(d)). I is an inflection point for ∆p as well, but because the

rate of increase for pb is larger than the rate of increase for pmax after

point M , ∆p starts to decrease after this point.

It is worth noting that umax,axi corresponds to the velocity of the top

of the bubble-liquid interface before the jet formation, and to the jet

front velocity afterwards. Since umax,axi correlates with ∆p, umax,axi

decreases after point M as ∆p does. This decrease of the jet front ve-

locity is also observed for laser-induced bubble collapse for all γ values

(cfr. 4.2.3). However, the decrease being much smaller, it can not be

observed in Fig. 4.17.

5.2.3 Influence of the initial bubble radius

The resonant radius of a bubble with a natural frequency of 20 kHz is

calculated to be 157 µm by using eq. 1.1. Experiments, however, show

that the mean bubble radius in a bubble cloud subjected to an acoustic

field of 20 kHz is around 10 µm [57, 20, 16, 19], that is much less than the

value estimated by eq. 1.1. Therefore, in order to understand the effect

of the initial bubble radius on the dynamics of a bubble near a solid

surface, we analyse four different bubbles with initial radii R0 of 10 µm,

40 µm, 78.5 µm, and 157 µm, respectively. Bubbles with a radius higher

than 157 µm are not considered here because the bubble oscillations are

too weak [49]. The ratio between the distance separating the initial

bubble centre from the solid surface, d, and the initial bubble radius,

R0, is chosen equal to 1.4 in all simulations ( dR0

= 1.4).


36 36.5 37 37.5 38 38.5

5

10

15

20

25

30

Time [µs]

p max

[bar

] I

(a)

36 36.5 37 37.5 38 38.5

20

40

60

80

100

120

Time [µs]

u max

,axi

[m/s

]

I

M

(b)

36 36.5 37 37.5 38 38.5

5

10

15

20

25

30

Time [µs]

p b [bar

]

I

(c)

36 36.5 37 37.5 38 38.5

2

4

6

8

10

12

14

16

Time [µs]

∆ p

[bar

]

M

I

(d)

Figure 5.8: Evolution of (a) the maximum pressure, (b) the maximum

velocity along the axis of axisymmetry, pmax and umax,axi respectively,

(c) the pressure inside the bubble pb, and (d) ∆p = pmax−pb as a function

of time. The point I corresponds to the onset of the jet formation and

the final data point to the jet impact on the lower bubble-liquid interface.


0 10 20 30 40 500.5

1

1.5

2

Time [µs]

Req

/R0

10 µm40 µm78.5 µm157 µm

(a)

0 10 20 30 40 50−1

−0.5

0

0.5

1

Time [µs]

p acou

st [b

ar]

10 µm40 µm78.5 µm157 µm

(b)

Figure 5.9: (a) Ratio between the equivalent bubble radius and the initial

bubble radius as a function of time. The final data point of each solid line

corresponds to the jet impact on the lower bubble-liquid interface. (b)

Pressure of the acoustic field as a function of time. The dots correspond

to the jet impact on the lower bubble-liquid interface.


Results

Figure 5.9(a) shows the evolution of the ratio Req/R0 as a function of

time for four different values of R0 (Req = 3V4π )

1

3 is the equivalent bubble

radius). The final data point of each line corresponds to the jet impact

on the lower bubble-liquid interface. There is a clear effect of the initial

bubble radius on the bubble dynamics. The collapse phase starts sooner

for small bubbles. The smaller the bubble radius, the sooner the collapse

phase starts. Figure 5.9(b) plots the evolution of the pressure associated

with the acoustic field (pacoust) as a function of time. The dots corre-

spond to the jet impact on the lower-bubble liquid interface. We also see

that the jet impact occurs sooner for the bubble with a smaller initial

radius: during the first half-period of oscillation for R0 = 10 µm, during

the second half-period for R0 = 40 µm and R0 = 78.5 µm, and at the

beginning of the second period for R0 = 157 µm.

0 10 20 30 40 500

20

40

60

80

100

120

140

Time [µs]

u max

,axi

[m/s

]

R0 = 10 µm

R0 = 40 µm

R0 = 78.5 µm

R0 = 157 µm

Figure 5.10: Maximum velocity along the axis of axisymmetry as a func-

tion of time. The dots and the final data points correspond, respectively,

to the onset of jet formation and to the jet impact on the lower bubble-

liquid interface.

Figure 5.10 shows the evolution of the maximum velocity along the axis


of axisymmetry (umax,axi) as a function of time. For each R0 value,

umax,axi increases, reaches a maximum, and then decreases. The jet

impacts the lower bubble-liquid interface during the collapse phase for

the intermediate values of the initial bubble radius, R0 = 40 µm and

R0 = 78.5 µm, and during the rebound phase for the minimum and max-

imum values of the initial bubble radius, R0 = 10 µm and R0 = 157 µm

(see Fig. 5.9). Therefore, as seen in Fig. 5.10, the decrease of the jet

front velocity is higher for the minimum and maximum values of the

initial bubble radius, than it is for the intermediate values.

Figure 5.11: Maximum jet front velocity umax as a function of the initial

bubble radius (R0). Note that umax corresponds to the maximum value

of the jet front velocity in each curve of Fig. 5.10.

Figure 5.11 shows the evolution of umax as a function of R0, where

umax is the maximum of the jet front velocity and it corresponds to the

maximum value of the jet front velocity in each curve of Fig. 5.10. We see

that umax increases with R0, reaches a maximum (at R0 ≈ 78 µm) and

5.3. CONCLUSION 153

then decreases. Note that, among the four different initial radius values,

the pressure associated with the acoustic field when the jet impacts the

lower bubble-liquid interface is the largest for R0 ≈ 78 µm (see Fig.

5.9(b)). The values for the maximum jet front velocity calculated here

are in agreement with the numerical data of Fong et al. [34], see Table

5.1.

R0 [ µm] umax[ m/s]

Fong et al. [34] 75 141

150 83

here 78.5 136

157 78

Table 5.1: Maximum jet front velocity calculated here and by Fong et

al. [34].

5.3 Conclusion

In this chapter, we use the numerical model developed in chapter 3

to investigate the dynamics of a bubble in liquid water subjected to

an acoustic field: (i) in bulk liquid and (ii) near a solid surface. The

commercial software Fluent is used to solve these problems. The results

have been shown mesh independent. As expected for (i), the pressure

on the bubble is the same through out the entire bubble-liquid interface,

and the bubble remains spherical. For (ii), we first took the distance

separating the initial bubble centre from the solid surface d = 141 µm

and the initial bubble radius R0 = 78.5 µm. The results show that:

• The maximum velocity along the axis of axisymmetry versus time

curve changes from being convex to concave when the jet starts to

penetrate the bubble.

• The time interval between the onset of jet formation and the jet

impact is less than 1 µs.

• The time evolution of the jet front velocity increases, reaches a

maximum, and then decreases again.


In order to understand the effect of the initial bubble radius for (ii),

we performed three more calculations each corresponding to three dif-

ferent R0 values. For each calculations, the ratio between the distance

separating the initial bubble centre from the solid surface, d, and the

initial bubble radius is chosen equal to 1.4 (d/R0 = 1.4). The main

effect we observed is that the maximum jet front velocity increases with

the initial bubble radius (R0), reaches a maximum near R0 = 78 µm,

and then decreases. Table 5.1 shows that the values calculated for the

maximum jet front velocity in this study are in perfect agreement with

results presented in [34].

General Conclusions and

Perspectives

As discussed in the introduction, the major contributions from this PhD

are as follows: (i) modelling the propagation of shock waves emitted dur-

ing the spherical collapse of a bubble subjected to an acoustic field, and

studying the effect of some acoustic parameters on the liquid velocity at

the shock front; (ii) providing a numerical model for viscous gas-liquid

multiphase flow to simulate laser and acoustic-induced bubble dynamics

near a solid surface; (iii) studying the influence of the liquid viscosity on

the velocity of the jets penetrating a laser-induced bubble; (iv) studying

the effect of the initial bubble radius on the velocity of jets penetrating

an acoustic-induced bubble. In this chapter, we would like to summarise

the results obtained for each of these aspects, and to suggest paths that

could be followed for pursuing this research.

First, we have studied shock wave emission from bubbles sufficiently far

away from the solid surface to remain spherical during their collapse. A

bubble remains spherical when the distance separating the bubble from

the solid surface is higher than three times its maximum radius (Rmax).

Two threshold conditions for the emission of a shock wave have been

defined and compared to the threshold condition for transient bubble

dynamics. Next, a mathematical model, combining the Gilmore model

and the method of characteristics, has been described to study the prop-

agation of shock waves in the liquid. The liquid velocity at the shock

front has been calculated as a function of the distance to the bubble

155

156 CONCLUSIONS AND PERSPECTIVES

centre. As the shock wave attenuates when propagating in the liquid,

the interaction of the shock wave with the solid surface is the strongest

when the bubble is the closest to the solid surface. For this reason, the

liquid velocity at the shock front has been studied at a distance from the

bubble centre equal to 3Rmax. The results have shown that the velocity

(i) increases with the amplitude of the acoustic field; (ii) goes through a

maximum as a function of the initial bubble radius; (iii) decreases when

increasing the surface tension.

These results have been used to predict if the interaction of shock waves

with the line structures during cavitation-induced surface cleaning was

responsible of damages observed. In future works, these results could

be used to study the effect of shock waves in other cavitation-induced

processes. Note that in processes working in the kHz regime, a poly-

tropic coefficient depending on the bubble dynamics should be included

in the numerical model. We also believe that shock wave emission dur-

ing aspherical collapse of a bubble near a solid surface should be investi-

gated, because this could also be responsible for the observed cavitation-

induced effects. To this purpose, a numerical model similar to the one

presented in chapter 3 but for which the two phases would be compress-

ible should be developed.

Secondly, the major point of our efforts has been oriented towards de-

veloping a numerical model to study aspherical collapse of a bubble near

a solid surface. The main drawback of the boundary integral method,

which currently is the most frequently used method to solve this prob-

lem, is that the liquid viscosity is neglected. As a result, the influence of

the shear stress induced by the bubble dynamics and by the radial flow

resulting from the jet impact on the solid surface can not be calculated.

It is this shear stress that is thought to be responsible for the most

of the beneficial cavitation-induced phenomenon, like surface cleaning

in microelectronics and sonoporation in biomedical applications. It is

for this reason that we have used a numerical model for two-phase flow

combining a compressible gas phase and an incompressible liquid phase.

CONCLUSIONS AND PERSPECTIVES 157

In this model, one set of compressible Navier-Stokes equations is solved

for both phases on a Cartesian mesh, and the interface is represented

and tracked by the Volume Of Fluid method. The main characteris-

tic of the numerical model is that it takes explicitly into account the

viscosity of the liquid and gas phases. It was challenging to solve the

equations because (i) we had to solve them both in an incompressible

liquid and in a compressible gas; (ii) the material properties at the in-

terface were discontinuous; (iii) the interface had to be kept sharp. The

numerical method, which is based on a SIMPLE-type algorithm, has

been described in detail and the numerical model has been validated by

comparison with analytical solutions.

Thirdly, the numerical model has been used to investigate the dynam-

ics of a bubble created by a laser near a solid surface. The commercial

software Fluent has been used to solve the numerical model. The sim-

ulations have shown that the evolution of the bubble shape with time

is in good agreement with experimentally observed bubble dynamics.

Concerning the jet penetrating the bubble at the end of the collapse

phase, the evolution of the maximum jet front velocity as a function of γ

(the latter being the distance between the initial bubble centre and the

solid surface, scaled by the maximum bubble radius) has been studied

for two values of the liquid viscosity representative for water and oil,

respectively. The maximum jet front velocity increases with γ until a

maximum is reached, the value and the position of which depends on the

liquid viscosity. A maximum is reached because the jet velocity is slowed

down during the bubble rebound. For γ values before the maximum, the

velocities are in agreement with experimental values available in the lit-

erature and the viscosity gives rise to a strongly diverging maximum jet

front velocity curves when decreasing γ.

Finally, the numerical model has been used as well to investigate the

dynamics of a bubble near a solid surface subjected to an acoustic field.

We have focused our attention on the jets penetrating the bubble at the

end of the collapse phase. The time interval between the jet formation

158 CONCLUSIONS AND PERSPECTIVES

and the jet impact is less than 1 µs, which is a value much smaller than

the time interval calculated for laser-induced bubble. The maximum jet

front velocity has been studied as a function of the initial bubble radius

(R0): it increases with the initial bubble radius, reaches a maximum

near R0 = 78 µm, and then decreases. These values are in good agree-

ment with other numerical results in the literature.

We think that this PhD opens many perspectives for further investi-

gations on the shear stress resulting from the bubble dynamics and from

the radial flow due to the jet impact on the solid surface. In the present

work, our calculations were stopped when the jet impacts the lower

bubble-liquid interface. Future works should now extend these simula-

tions to include (i) the jet impact on the lower bubble-liquid interface,

(ii) the penetration of the jet inside the liquid being below the bubble,

(iii) the jet impact on the solid surface, (iv) the radial flow resulting

from this impact. These simulations are challenging because very high

pressures result from jet impact on the solid surface and on the liquid

layer below the bubble. These pressures at the impact point will have

to be accurately captured. The calculation of the shear stress on the

solid surface is also challenging because the gradient of velocity has to

be accurately calculated.

Acknowledgements

Bien que mon seul nom figure sur la couverture de cette these, celle-ci

n’a pu etre realisee qu’avec l’aide precieuse de nombreuses personnes

qui, toutes, ont contribue a faire de ces quatres annees de doctorat une

experience positive. Je souhaite les remercier toutes tres chaleureuse-

ment.

Je voudrais tout d’abord remercier les Professeurs Joris Proost et Juray

De Wilde pour leurs conseils et leurs eclaircissements apportes au cours

de ces 4 dernieres annees. Merci ensuite au F.R.I.A. qui a finance ma

recherche. Je tiens egalement a remercier Frank Holsteyns qui m’a fait

decouvrir ce sujet et sans qui je n’aurais jamais realise cette these.

Je remercie egalement tous mes collegues de l’unite IMAP et plus par-

ticulierement mes collegues de bureau Fred, Francois et Nicolas; les

autres joyeux lurons du deuxieme etage; Cihan pour son aide lors de

la redaction; Emilie, Nicolas, Quentin, Jean-Francois et Fred pour leur

aide dans la comprehension des phenomenes physiques; Marc pour son

aide psychologique; les membres du secretariat IMAP pour leur aide

dans les formalites administratives et Luc pour son aide organisation-

nelle.

Pour terminer, je remercie mes parents, ma famille, ma copine Nathalie,

mes amis Pierre-Yves, Alex, Baudoin, et tous les autres de m’avoir en-

courage sans cesse durant la derniere annee de cette these.

159

Appendix

This appendix describes the options chosen in the Fluent software.

1. Calculation precision

• Double precision.

2. Define/Models/Solver

• Solver: Pressure-based.

• Velocity formulation: Absolute.

• Gradient option: Green-Gauss node based.

• Time: Unsteady.

• Unsteady formulation: 1st order implicit.

3. Define/Models/Multiphase Model

• Model: Volume Of Fluid.

• VOF scheme: Explicit.

• Courant Number: 0.25.

4. Define/Models/Energy:

• Activate energy equation.

5. Define/Models/Viscous model

• Activate laminar.

6. Define/Materials

161

162 APPENDIX

• Two materials are chosen: A gas and a liquid.

• For the gas phase, the ideal-gas density model is activated.

7. Define/Phases

• Define the gas phase as the primary-phase and the liquid

phase as the secondary phase.

• Activate the surface tension in the phase interaction toolbar.

8. Define/Operating conditions

• Operating pressure: 1 bar.

• Reference pressure location: At the boundary of the domain.

9. Define/Boundary conditions

• Pressure outlet: Use an UDF file when an acoustic pressure

is imposed at the boundary far away from the bubble.

10. Solve/Controls

• Pressure velocity coupling: SIMPLE.

• Under-relaxation factors: pressure = 0.3, density = 0.5, body

forces = 0.5, momentum = 0.7, energy = 0.5.

• Discretization: pressure = body force weighted, density =

second order upwind, momentum = second order upwind,

volume fraction = modified HRIC, energy = second-order up-

wind.

11. Solve/Initialize

• Solution initialization: Introduce the initial conditions.

• When the initial conditions are not uniform, an UDF file or

the following procedure has to be used. This procedure is

described for the volume fraction:

a) Solve/Initialize: Define the volume fraction as 1.

b) Adapt/Region: Mark a region of the mesh.

APPENDIX 163

c) Solve/Patch: Define the volume fraction as 0 on the marked

zone.

Bibliography

[1] National Institute of Standards and Technology.

http://webbook.nist.gov/chemistry/fluid/.

[2] S. Acharya, B.R. Baliga, K. Karki, J.Y. Murthy, C. Prakash, and

S.P. Vanka. Pressure-based finite-volume methods in computa-

tional fluid dynamics. J. Heat Transfer, 129(4):407–424, 2007.

[3] M. Arora. Cavitation for biomedical applications. PhD thesis,

University of Twente, 2006.

[4] M. Ashokkumar, J. Lee, S. Kentish, and F. Grieser. Bubbles in an

acoustic field: An overview. Ultrason. Sonochem., 14(4):470–475,

2007.

[5] E. Aulisa, S. Manservisi, R. Scardovelli, and S. Zaleski. A geo-

metrical area-preserving Volume Of Fluid advection method. J.

Comput. Phys., 192(1):355–364, 2003.

[6] The international technology roadmap for semiconductors.

http://www.itrs.net/.

[7] F. Barbagini, W. Fyen, J. Van Hoeymissen, P. Mertens, and

J. Fransaer. Particle-substrate interaction forces in a non-polar

liquid. Sol. St. Phen., 134:165–168, 2008.

[8] J.P. Best and A. Kucera. A numerical investigation of nonspherical

rebounding bubbles. J. Fluid Mech., 245:137–154, 1992.

165

166 BIBLIOGRAPHY

[9] P.R. Birkin, D.G. Offin, and T.G. Leighton. The study of surface

processes under electrochemical control in the presence of inertial

cavitation. Wear, 258:623–628, 2005.

[10] P.R. Birkin, J.F. Power, M.E. Abdelsalam, and T.G. Leighton.

Electrochemical, luminescent and photographic characterisation of

cavitation. Ultrason. Sonochem., 10:203–208, 2003.

[11] J.R. Blake and D.C. Gibson. Cavitation bubbles near boundaries.

Ann. Rev. Fluid Mech., 19:99–123, 1987.

[12] J.R. Blake, B.B. Taib, and G. Doherty. Transient cavities near

boundaries part 1: Rigid boundary. J. Fluid Mech., 170:479–497,

1986.

[13] J.R. Blake, Y. Tomita, and R.P. Tong. The art, craft and science

of modelling jet impact in a collapsing cavitation bubble. Appl.

Sci. Res., 58:77–90, 1998.

[14] J.U. Brackbill, D.B. Kothe, and C. Zemach. A continuum method

for modeling surface tension. J. Comput. Phys., 100:335, 1992.

[15] M.P. Brenner, S. Hilgenfeldt, and D. Lohse. Single-bubble sonolu-

minescence. Rev. Mod. Phys., 74(2):425–484, 2002.

[16] A. Brotchie, F. Grieser, and M. Ashokkumar. Effect of power and

frequency on bubble-size distributions in acoustic cavitation. Phys.

Rev. Lett., 102:0843202, 2009.

[17] D.L. Brown, R. Cortez, and M.L. Minion. Accurate projection

methods for the incompressible Navier-Stokes equations. J. Com-

put. Phys., 168(2):464–499, 2001.

[18] E.A. Brujan, G.S. Keen, A. Vogel, and J.R. Blake. The final stage

of the collapse of a cavitation bubble close to a rigid boundary.

Phys. Fluids, 14(1):85–92, 2002.

[19] F. Burdin, N.A. Tsochatzidis, P. Guiraud, A.M. Wilhelm, and

H. Delmas. Characterisation of the acoustic cavitation cloud by

two laser techniques. Ultrason. Sonochem., 6:43–51, 1999.

BIBLIOGRAPHY 167

[20] F. Burdin, N.A. Tsochatzidis, P. Guiraud, A.M. Wilhelm, and

H. Delmas. Implementation of the laser diffraction technique for

acoustic cavitation bubble investigations. Part. Part. Syst. Char-

act., 19:73–83, 2002.

[21] S.K. Choi. Note on the use of momentum interpolation method for

unsteady flows. Numer. Heat Tr. A-Appl., 36(5):545–550, 1999.

[22] L.A. Crum. Surface oscillations and jet development in pulsating

bubbles. Le Journal de Physique Colloques, 40:C8–285, 1979.

[23] L.A. Crum. Acoustic cavitation series 5. rectified diffusion. Ultra-

sonics, 22(5):215–223, 1984.

[24] D.S. Dandy and L.G. Leal. Buoyancy-driven motion of a de-

formable drop through a quiescent liquid at intermediate Reynolds

numbers. J. Fluid Mech., 208:161–192, 1989.

[25] J.L. Delplancke, J. Dille, J. Reisse, G.J. Long, A. Mohan, and

F. Grandjean. Magnetic nanopowders: Ultrasound-assisted elec-

trochemical preparation and properties. Chem. Mater., 12(4):946–

955, 2000.

[26] R. Dijkink and C.-D. Ohl. Measurement of cavitation induced wall

shear stress. Appl. Phys. Lett., 93:254107, 2008.

[27] P. Diodati and F. Marchesoni. Time-evolving statistics of cavita-

tion damage on metallic surfaces. Ultrason. Sonochem., 9(6):325–

329, 2002.

[28] K.J. Ebeling. Zum verhalten kugelfrmiger, lasererzeugter kavita-

tionsblasen in wasser. Acustica, 40:229–230, 1978.

[29] A. Eller and H.G. Flynn. Rectified diffusion during nonlinear pul-

sations of cavitation bubbles. J. Acoust. Soc. Am., 37:493, 1956.

[30] A.I. Eller. Growth of bubbles by rectified diffusion. J. Acoust. Soc.

Am., 46(5):1246–1250, 1969.

168 BIBLIOGRAPHY

[31] A. Esmaeeli and G. Tryggvason. Direct numerical simulations of

bubbly flows part 2: Moderate reynolds number arrays. J. Fluid

Mech., 385:325–358, 1999.

[32] R.P. Fedkiw, T. Aslam, B. Merriman, and S. Osher. A non-

oscillatory eulerian approach to interfaces in multimaterial flows

(the Ghost Fluid Method). J. Comput. Phys., 152(2):457–492,

1999.

[33] H.G. Flynn. Physics of acoustics cavitation in liquids, volume 1B

of Physical Acoustics. Mason, New York, 1964.

[34] S.W. Fong, E. Klaseboer, C.K. Turangan, B.C. Khoo, and K.C.

Hung. Numerical analysis of a gas bubble near bio-materials in an

ultrasound field. Ultrasound in Med. and Biol., 32:925–942, 2006.

[35] S. Fujikawa and T. Akamatsu. Effects of non-equilibrium conden-

sation of vapour on the pressure wave produced by the collapse of

a bubble in a liquid. J. Fluid Mech., 97(3):481–512, 1980.

[36] A. Gedanken. Using sonochemistry for the fabrication of nanoma-

terials. Ultrason. Sonochem., 11(2):47–55, 2004.

[37] J.M. Gere and S.P. Timoshenko. Mechanics of materials. PWS

Pub. Co., Boston, 1997.

[38] F.R. Gilmore. The growth or collapse of a spherical bubble in a

viscous compressible liquid. Report of office naval research 26-4,

California Institute of TechnologyOffice, 1952.

[39] B.J. Goode, R.D. Jones, and J.N.H. Howells. Ultrasonic pickling

of steel strip. Ultrasonics, 36:79–88, 1998.

[40] D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, and S. Za-

leski. Volume-Of-Fluid interface tracking with smoothed surface

stress methods for three-dimensional flows. J. Comput. Phys.,

152(2):423–456, 1999.

BIBLIOGRAPHY 169

[41] F.H. Harlow and J.E. Welch. Numerical calculation of time-

dependent viscous incompressible flow of fluid with free surface.

Phys. Fluids, 8(12):2182–2189, 1965.

[42] R. Hickling and M.S. Plesset. Collapse and rebound of a spherical

bubble in water. The Phys. Fluids, 7(1):7–14, 1964.

[43] C.W. Hirt and B.D. Nichols. Volume Of Fluid (VOF) method for

the dynamics of free boundaries. J. Comput. Phys., 39(1):201–225,

1981.

[44] F. Holsteyns. Removal of nanoparticulate contaminants from

semiconductor substrates by megasonic cleaning. PhD thesis,

Katholieke Universiteit Leuven, 2008.

[45] J. Holzfuss, M. Ruggeberg, and A. Billo. Shock wave emissions

of a sonoluminescing bubble. Phys. Rev. Lett., 81(24):5434–5437,

1998.

[46] J. Huang, R. Li, Y.L. He, and Z.G. Qu. Solutions for variable

density low mach number flows with a compressible pressure-based

algorithm. Appl. Therm. Eng., 27(11-12):2104–2112, 2007.

[47] J.T. Huang and H.S. Zhang. Level set method for numerical sim-

ulation of a cavitation bubble, its growth, collapse and rebound

near a rigid wall. Acta Mech. Sinica, 23(6):645–653, 2007.

[48] Y. Jiang, C. Petrier, and T.D. Waite. Kinetics and mechanisms of

ultrasonic degradation of volatile chlorinated aromatics in aqueous

solutions. Ultrason. Sonochem., 9(6):317–323, 2002.

[49] Sato K., Y. Tomita, and A. Shima. Numerical analysis of a gas

bubble near a rigid boundary in an oscillatory pressure field. J.

Acoust. Soc. Am., 95:2416–2424, 1994.

[50] S.W. Karng and H.Y. Kwak. Relaxation behavior of microbubbles

in ultrasonic field. Jpn. J. Appl. Phys., Part 1, 45(1A):317–322,

2006.

170 BIBLIOGRAPHY

[51] D. Lakehal, M. Meier, and M. Fulgosi. Interface tracking towards

the direct simulation of heat and mass transfer in multiphase flows.

Int. J. Multiphase Flow, 23:242–257, 2002.

[52] O. Lindau and W. Lauterborn. Cinematographic observation of

the collapse and rebound of a laser-produced cavitation bubble

near a wall. J. Fluid Mech., 479:327–348, 2003.

[53] D. Lohse, M.P. Brenner, T.F. Dupont, S. Hilgenfeldt, and B. John-

ston. Sonoluminescing air bubbles rectify argon. Phys. Rev. Lett.,

78:1359–1362, 1997.

[54] J. Lopez, J. Hernandez, P. Gomez, and F. Faura. A Volume Of

Fluid method based on multidimensional advection and spline in-

terface reconstruction. J. Comput. Phys., 195(2):718–742, 2004.

[55] J. Lopez, J. Hernandez, P. Gomez, and F. Faura. An improved

PLIC-VOF method for tracking thin fluid structures in incom-

pressible two-phase flows. J. Comput. Phys., 208(1):51–74, 2005.

[56] J.C. Lu, A. Fernandez, and G. Tryggvason. The effect of bubbles

on the wall drag in a turbulent channel flow. Phys. Fluids, 17,

2005.

[57] S. Luther, R. Mettin, P. Koch, and W. Lauterborn. Observation of

acoustic cavitation bubbles at 2250 frames per second. Ultrason.

Sonochem., 8:159–162, 2001.

[58] E. Maisonhaute, P.C. White, and R.G. Compton. Surface acoustic

cavitation understood via nanosecond electrochemistry. J. Phys.

Chem. B, 105(48):12087–12091, 2001.

[59] E. Marchandise, P. Geuzaine, J.-F. Chevaugeon, and J.-F.

Remacle. A stabilized finite element method using a discontin-

uous level set approach for the computation of bubble dynamics.

J. Comput. Phys., 225:949–974, 2007.

BIBLIOGRAPHY 171

[60] E. Marchandise, J.-F. Remacle, and N. Chevaugeon. A quadrature-

free discontinuous galerkin method for the level set equation. J.

Comput. Phys., 212:338–357, 2006.

[61] T.J. Mason. Developments in ultrasound: Non-medical. Prog.

Biophys. Mol. Biol., 93:166–175, 2007.

[62] Y. Matsumoto, J.S. Allen, S. Yoshizawa, T. Ikeda, and Y. Kaneko.

Medical ultrasound with microbubbles. Exp. Therm. Fluid. Sci.,

29:255–265, 2005.

[63] T.J. Matula. Inertial cavitation and single-bubble sonolumines-

cence. Philos. Trans. R. Soc. London, Ser. A, 357:225–249, 1999.

[64] R. Mettin and A.A. Doinikov. Translational instability of spherical

bubble in a standing ultrasound wave. Appl. Acoust, 70:1330, 2009.

[65] R. Mettin, S. Luther, C.-D. Ohl, and W. Lauterborn. Acoustic

cavitation structures and simulations by a particle model. Ultra-

son. Sonochem., 6:25–29, 1999.

[66] M.W. Miller. Gene transfection and drug delivery. Ultrasound

Med. Biol., 26:S59–S62, 2000.

[67] M. Minnaert. On musical air-bubbles and the sound of running

water. Phil. Mag., 16:235, 1933.

[68] K.A. Morch. Cavitation nuclei and bubble formation - a dynamic

liquid-solid interface problem. J. Fluids Eng., 122:494–498, 2000.

[69] S. Muzaferija, M. Peric, P. Sames, and Schellin T. A two-fluid

navier-stokes solver to simulate water entry. In Proceedings of the

22nd Symposium on Naval Hydrodynamics, pages 638–651, 2000.

[70] S. Nagrath, K. Jansen, R.T. Lahey, and I. Akhatov. Hydrodynamic

simulation of air bubble implosion using a level set approach. J.

Comput. Phys., 215(1):98–132, June 2006.

[71] E.A. Neppiras. Acoustic cavitation. Phys. Rep., 61(3):159–251,

1980.

172 BIBLIOGRAPHY

[72] K. Nerinckx, J. Vierendeels, and E. Dick. Mach-uniformity

through the coupled pressure and temperature correction algo-

rithm. J. Comput. Phys., 206(2):597–623, 2005.

[73] K. Nerinckx, J. Vierendeels, and E. Dick. A mach-uniform algo-

rithm: coupled versus segregated approach. J. Comput. Phys.,

224(1):314–331, 2007.

[74] C.-D. Ohl, M. Arora, R. Dijkink, V. Janve, and D. Lohse. Surface

cleaning from laser-induced cavitation bubbles. Appl. Phys. Lett.,

89(7):074102, 2006.

[75] C.-D. Ohl, T. Kurz, R. Geisler, O. Lindau, and W. Lauterborn.

Bubble dynamics, shock waves and sonoluminescence. Philos.

Trans. R. Soc. London, Ser. A, 357(1751):269–294, 1999.

[76] J. Papageorgakopoulos, G. Arampatzis, D. Assimacopoulos, and

N.C. Markatos. Enhancement of the momentum interpolation

method on non-staggered grids. Int. J. Numer. Meth. Fl., 33(1):1–

22, 2000.

[77] A. Pearson, J.R. Blake, and S.R. Otto. Jets in bubbles. J. Eng.

Math., 48(3-4):391–412, 2004.

[78] S.A. Perusich and R.C. Alkire. Ultrasonically induced cavitation

studies of electrochemical passivity and transport mechanisms 1:

Theoretical. J. Electrochem. Soc., 138(3):700–707, 1991.

[79] S.A. Perusich and R.C. Alkire. Ultrasonically induced cavitation

studies of electrochemical passivity and transport mechanisms 2:

Experimental. J. Electrochem. Soc., 138(3):708–713, 1991.

[80] A. Philipp and W. Lauterborn. Cavitation erosion by single laser-

produced bubbles. J. Fluid Mech., 361:75–116, 1998.

[81] J.E. Pilliod. An analysis of piecewise linear interface reconstruc-

tion algorithms for Volume-Of-Fluid methods. PhD thesis, Uni-

versity of California, 1992.

BIBLIOGRAPHY 173

[82] J.E. Pilliod and E.G. Puckett. Second-order accurate volume-of-

fluid algorithms for tracking material interfaces. J. Comput. Phys.,

199(2):465–502, 2004.

[83] M.S. Plesset and R.B. Chapman. Collapse of an initially spherical

vapour cavity in the neighbourhood of a solid boundary. J. Fluid

Mech., 47(02):283–290, 1971.

[84] S. Popinet and S. Zaleski. A front-tracking algorithm for accurate

representation of surface tension. Int. J. Numer. Methods Fluids,

30(6):775–793, 1999.

[85] S. Popinet and S. Zaleski. Bubble collapse near a solid boundary:

a numerical study of the influence of viscosity. J. Fluid Mech.,

464:137–163, 2002.

[86] Lord Rayleigh. On the pressure developed in a liquid during the

collapse of a spherical cavity. Philos. Mag., 34:94, 1917.

[87] Y. Renardy and M. Renardy. PROST: a parabolic reconstruction

of surface tension for the Volume-Of-Fluid method. J. Comput.

Phys., 183(2):400–421, 2002.

[88] C.M. Rhie and W.L. Chow. Numerical study of the turbulent-flow

past an airfoil with trailing edge separation. AIAA J., 21(11):1525–

1532, 1983.

[89] W.J. Rider and D.B. Kothe. Reconstructing volume tracking. J.

Comput. Phys., 141(2):112–152, 1998.

[90] R. Scardovelli and S. Zaleski. Interface reconstruction with least-

square fit and split Eulerian-Lagrangian advection. Int. J. Numer.

Methods Fluids, 41(3):251–274, 2003.

[91] B.D. Storey and A.J. Szeri. Water vapour, sonoluminescence and

sonochemistry. Proc. R. Soc. London, Ser. A, 456(1999):1685–

1709, 2000.

174 BIBLIOGRAPHY

[92] B.D. Storey and A.J. Szeri. A reduced model of cavitation

physics for use in sonochemistry. Proc. R. Soc. London, Ser. A,

457(2011):1685–1700, 2001.

[93] K.S. Suslick, S.B. Choe, A.A. Cichowlas, and M.W. Grinstaff.

Sonochemical synthesis of amorphous iron. Nature, 353:414–416,

1991.

[94] M. Sussman. A second order coupled level set and volume-of-fluid

method for computing growth and collapse of vapor bubbles. J.

Comput. Phys., 187(1):110–136, May 2003.

[95] M. Sussman and E. Fatemi. An efficient, interface-preserving level

set redistancing algorithm and its application to interfacial incom-

pressible fluid flow. SIAM J. Sci. Comput., 20(4):1165–1191, 1998.

[96] M. Sussman, E. Fatemi, P. Smereka, and S. Osher. An improved

level set method for incompressible two-phase flows. Comput. Flu-

ids, 27:663–680, 1998.

[97] M. Sussman and E.G. Puckett. A coupled level set and Volume-Of-

Fluid method for computing 3D and axisymmetric incompressible

two-phase flows. J. Comput. Phys., 162(2):301–337, 2000.

[98] M. Sussman, P. Smereka, and S. Osher. A level set approach for

computing solutions to incompressible two-phase flow. J. Comput.

Phys., 114(1):146–159, 1994.

[99] S. Tanguy, T. Menard, and A. Berlemont. A level set method

for vaporizing two-phase flows. J. Comput. Phys., 221(2):837–853,

2007.

[100] F. Tardif, O. Raccurt, J.C. Barbe, F. De Crecy, P. Besson, and

A. Danel. Mechanical resistance of fine microstructures related

to particle cleaning mechanisms. In Proceedings of the 8th Inter-

national Symposium on Cleaning Technology and Semiconductor

Manufacturing, volume 26, page 153, 2003.

BIBLIOGRAPHY 175

[101] R.P. Tong, W.P. Schiffers, S.J. Shaw, J.R. Blake, and D.C. Em-

mony. The role of ’splashing’ in the collapse of a laser-generated

cavity near a rigid boundary. J. Fluid Mech., 380:339–361, 1999.

[102] G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi,

W. Tauber, J. Han, S. Nas, and Y.J. Jan. A front-tracking method

for the computations of multiphase flow. J. Comput. Physics,

169(2):708–759, 2001.

[103] S.O. Unverdi and G. Tryggvason. A front-tracking method for

viscous, incompressible, multi-fluid flows. J. Comput. Phys.,

100(1):25–37, 1992.

[104] G. Vereecke, F. Holsteyns, S. Arnauts, S. Beckx, P. Jaenen, K. Ke-

nis, M. Lismont, M. Lux, R. Vos, J. Snow, and P.W. Mertens.

Evaluation of megasonic cleaning for sub-90-nm technologies. Sol.

St. Phen., 103-104:141–146, 2005.

[105] A. Vogel, J. Noack, K. Nahen, D. Theisen, S. Busch, U. Parlitz,

D.X. Hammer, G.D. Noojin, B.A. Rockwell, and R. Birngruber.

Energy balance of optical breakdown in water at nanosecond to

femtosecond time scales. Appl. Phys. B, 68(2):271–280, 1999.

[106] Y.C. Wang and Y.W. Chen. Application of piezoelectric PVDF

film to the measurement of impulsive forces generated by cavita-

tion bubble collapse near a solid boundary. Exp. Therm. Fluid

Sci., 32(2):403–414, 2007.

[107] R. Wemmenhove. Numerical simulation of two-phase flow in off-

shore environments. PhD thesis, Universiteit Groningen, 2008.

[108] Tomita Y. and A. Shima. Mechanisms of impulsive pressure gen-

eration and damage pit formation by bubble collapse. J. Fluid

Mech., 169:535–564, 1986.

[109] F.R. Young. Cavitation. Imperial College Press, London, 1989.

[110] B. Yu, Y. Kawaguchi, W.Q. Tao, and H. Ozoe. Checkerboard

pressure predictions due to the underrelaxation factor and time

176 BIBLIOGRAPHY

step size for a nonstaggered grid with momentum interpolation

method. Numer. Heat Tr. B-Fund., 41(1):85–94, 2002.

[111] B. Yu, W.Q. Tao, J.J. Wei, J. Kawaguchi, T. Tagawa, and H. Ozoe.

Discussion on momentum interpolation method for collocated grids

of incompressible flow. Numer. Heat Tr. B-Fund., 42(2):141–166,

2002.

[112] P.W. Yu, S.L. Ceccio, and G. Tryggvason. The collapse of a cav-

itation bubble in shear flows - a numerical study. Phys. Fluids,

7(11):2608–2616, 1995.

[113] H.H. Zhang and L.A. Coury. Effects of high-intensity ultrasound

on glassy-carbon electrodes. Anal. Chem., 65(11):1552–1558, 1993.

[114] S.G. Zhang, J.H. Duncan, and G.L. Chahine. The final stage of the

collapse of a cavitation bubble near a rigid wall. J. Fluid Mech.,

257:147–181, 1993.

[115] H.P. Zhao, X.Q. Feng, and H.J. Gao. Ultrasonic technique for

extracting nanofibers from nature materials. Appl. Phys. Lett.,

90(7):073112, 2007.

[116] A. Zijlstra, T. Janssens, K. Wostyn, M. Versluis, P.M. Mertens,

and D. Lohse. High speed imaging of 1 MHz driven microbubbles

in contact with a rigid wall. Sol. St. Phen., 145:7–10, 2009.

Numerical simulation of cavitation-induced bubble dynamics ...

Documents

Transcript of Numerical simulation of cavitation-induced bubble dynamics ...