UNIVERSITY OF CINCINNATI

Date:______

I, ______, hereby submit this work as part of the requirements for the degree of: in:

It is entitled:

This work and its defense approved by:

Chair: ______

Comprehensive Study of Internal Flow Field and Linear and Nonlinear Instability of an Annular Liquid Sheet Emanating from an Atomizer

A dissertation submitted to the

Division of Research and Advanced Studies of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering

2006

by

Ashraf Ibrahim

B.S.,Cairo University, 1997 M.S., Cairo University, 2002

Dissertation Committee:

Dr. Milind Jog, Chair Dr. San-Mou Jeng Dr. Raj Manglik

i ABSTRACT

Performance of fuel injectors affects the combustion efficiency, pollutant emissions and combustion instability in gas turbine engines, internal combustion engines and industrial furnaces.

In these combustion systems, either pressure swirl (simplex) atomizers, or prefilming airblast atomizers, or plain orifice pressure atomizers are used for fuel atomization. In this dissertation, a comprehensive model for pressure-swirl atomization is developed that includes computational treatment of the internal flow field and the nonlinear liquid sheet instability analysis for primary breakup. For a prefilming airblast atomizer and a plain orifice atomizer, nonlinear breakup processes for an annular liquid sheet and a liquid jet are analyzed using a perturbation method.

Two-dimensional axi-symmetric numerical simulations have been carried out to study the unsteady, turbulent, swirling two-phase flow field inside pressure swirl atomizers with the volume of fluid (VOF) method. Internal flow field simulation results are validated using available experimental data for velocity measurements inside a large-scale prototype atomizer, the film thickness at orifice exit, the spray angle, and the discharge coefficient. The effect of air pressure and liquid viscosity on flow field inside the atomizer is investigated. The relationship between the internal flow characteristics and discharge parameters confirms that the internal flow structure plays a very important role in determining the atomizer performance.

Linear and nonlinear asymmetric instability analyses are carried out to study the primary atomization of annular liquid sheets and liquid jets emanating from the pressure swirl (simplex) atomizer, prefilming airblast atomizer, and plain orifice pressure atomizer using a perturbation method with the initial amplitude of the disturbance as the perturbation parameter. For a coaxial liquid jet subjected to a swirling gas stream, the axisymmetric disturbance mode (n = 0) is the most dominant only when the gas swirl number is very small. However at higher swirl strength

ii the helical (asymmetric) disturbance modes (n > 0) become dominant compared to the axisymmetric mode. The liquid jet breaks up over a shorter distance at higher gas swirl number.

The gas swirl number for transition to a highly asymmetric breakup with a high circumferential wave number (n = 5) is found to vary as the inverse of the square root of the gas-to-liquid momentum ratio when the gas-to-liquid momentum ratio is less than 1. For annular liquid sheets, the breakup length is reduced by an increase in the liquid Weber number, initial disturbance amplitude and the inner and outer gas-liquid velocity ratios. The inner gas stream is found to be more effective in disintegrating and enhancing the instability of annular liquid sheets than the outer gas stream. Air swirl not only promotes the instability of the annular liquid sheet, but also switches the dominant mode from the axisymmetric mode to a helical mode (n > 0). As outer air swirl strength increases, the circumferential wave number (n) increases and the ligament shapes at the breakup time become highly asymmetric.

Using the atomizer exit conditions as input, a non-linear sheet instability and breakup analysis has been carried out to predict the breakup length and the primary breakup for a simplex atomizer. The predictions of breakup length are compared with available experimental measurements which show good agreement. The coupled internal flow simulation and nonlinear sheet instability analysis provides a comprehensive approach to modeling atomization from a pressure-swirl atomizer.

iii

iv ACKNOWLEDGEMENTS

I would like to express my most sincere gratitude to my dissertation advisor, Professor

Milind Jog, for his insightful guidance, unending encouragement, financial support and personal help. He has been a constant source of inspiration. Together, we had numerous discussions where his creativity would help come up with new ideas and ways of looking at a problem.

I would like to acknowledge and thank Professor San-Mou Jeng and Professor Raj

Manglik for honoring me by serving on my committee. I would also like to thank Professor San-

Mou Jeng for the knowledge in sprays and combustion that he taught me, giving me the opportunity to run some of the experiments with his students and his generous assistance and encouragement. I would like to thank Professor Raj Manglik for his continuous and generous assistance and encouragement.

I would like to express my deepest sense of gratitude to my wife for her support and sacrifice throughout those years in the pursuit of my doctorate. I am indebted to my parents, for everything I am today is only from their sacrifices.

v

THIS DISSERTATION IS DEDICATED TO

MY FAMILY

WHO ALWAYS ENCOURAGED ME TO BE THE BEST

vi TABLE OF CONTENTS

ABSTRACT ii

ACKNOWLEDGEMENTS v

TABLE OF CONTENTS vii

LIST OF TABLES xii

LIST OF FIGURES xiii

LIST OF SYMBOLS xx

CHAPTER PAGE

1 INTRODUCTION 1

1.1 Motivation 1

1.2 Atomization and Atomizers 2

1.2.1 Plain orifice atomizer 3

1.2.2 Pressure Swirl (Simplex) Atomizer 3

1.2.3 Twin Fluid Atomizers 10

1.3 Fundamental Mechanisms of Spray Formation 11

1.3.1 Disintegration of Liquid Jets 12

1.3.1.1 Round Liquid Jets in Quiescent Air 13

1.3.1.2 Round Liquid Jets in Co-flowing Air 14

1.3.2 Disintegration of Liquid Sheets 17

1.3.2.1 Plane Liquid Sheets 17

1.3.2.2 Annular Liquid Sheets 18

1.4 Scope of the Dissertation 19

vii PART I TWO-PHASE FLOW FIELD IN PRESSURE SWIRL 23

ATOMIZERS

2 COMPUTATIONAL SIMULATION OF FLOW FIELD IN PRESSURE 24

SWIRL (SIMPLEX) ATOMIZER

2.1 Literature Review 24

2.1.1 Inviscid Analysis and Experimental Work 24

2.1.2 Review of Computational Modeling 28

2.1.2.1 The ALE method 29

2.2.2.2 The VOF Method 30

2.2 The Physical Model 31

2.3 Governing Equations 32

2.4 Results and Discussions 34

2.5 Summary and Conclusions 49

PART II LINEAR INSTABILITY OF ANNULAR LIQUID SHEETS 51

3 EFFECT OF LIQUID SWIRL VELOCITY PROFILE ON THE 52

INSTABILITY OF A SWIRLING ANNULAR LIQUID SHEET

3.1 Introduction 52

3.2 Linear Stability Analysis 55

3.2.1 Solid Vortex Swirl Profile 55

3.2.2 Free Vortex Swirl Profile 61

3.3 Results and Discussions 63

3.3.1 Effect of Liquid Axial Velocity 63

3.3.2 Effect of Liquid Swirl Velocity 64

viii 3.3.3 Effect of Density Ratio 70

3.3.4 Effect of Radius of Curvature Ratio 75

3.3.5 Effect of Surface Tension 78

3.3.6 Effect of Outer Axial Air Weber Number 78

3.4 Summary and Conclusions 80

4 EFFECT OF LIQUID AND AIR SWIRL STRENGTH AND RELATIVE 83

ROTATIONAL DIRECTION ON THE INSTABILITY OF AN

ANNULAR LIQUID SHEET

4.1 Introduction 83

4.2 Mathematical Formulation 88

4.3 Results and Discussions 95

4.3.1 Model Validation 96

4.3.2 Liquid Swirl with Purely Axial Air Flow 96

4..3.3 Air Swirl with Purely Axial Liquid Flow 102

4.3.4 Liquid Swirl with Air Swirl 104

4.3.5 Effect of Relative Air Swirl Direction 108

4.3.6 Effect of High Air Pressure 111

4.4 Summary and Conclusions 114

PART III NONLINEAR INSTABILITY OF LIQUID JETS AND

ANNULAR LIQUID SHEETS 116

5 NONLINEAR BREAKUP OF A COAXIAL LIQUID JET IN A

SWIRLING GAS STREAM 117

5.1 Introduction 117

ix 5.2 Mathematical Formulation 121

5.2.1 Solution of the First and the Second Order Equations 125

5.3 Results and Discussions 128

5.3.1 Model Validation 132

5.3.2 Effect of Gas Swirl 135

5.4 Summary and Conclusions 142

6 NONLINEAR INSTABILITY OF AN ANNULAR LIQUID SHEET

SUBJECTED TO UNEQUAL INNER AND OUTER GAS STREAMS 144

6.1 Literature Review 144

6.2 Mathematical Formulation 146

6.2.1 Solution of the First and the Second Order Equations 153

6.3 Results and Discussions 157

6.4 Summary and Conclusions 173

7 NONLINEAR INSTABILITY OF AN ANNULAR LIQUID SHEET

SUBJECTED TO SWIRLING OUTER GAS STREAM 175

7.1 Introduction 175

7.2 Mathematical Formulation 176

7.2.1 Solution of the First and the Second Order Equations 183

7.3 Results and Discussions 187

7.4 Summary and Conclusions 196

PART IV A COMPREHENSIVE MODEL FOR PRESSURE SWIRL

ATOMIZER 198

8 A COMPREHENSIVE MODEL TO PREDICT PRESSURE SWIRL 199

x ATOMIZER PERFORMANCE

8.1 Motivation 199

8.2 Results and Discussions 202

8.2.1 Internal Flow Field 202

8.2.1.1 Validation 205

8.3 Breakup Length Calculation and Validation 207

8.4 Summary and Conclusions 208

9 CONCLUSIONS AND RECOMMENDATIONS 209

9.1 Summary and Conclusions 209

9.2 Recommendations for Future Work 212

BIBLIOGRAPHY 213

APPENDICES 228

A 229

B 231

C 234

D 241

xi LIST OF TABLES

TABLE PAGE

1.1 Classification and criteria of breakup regimes of round jets in 16

quiescent air (Liu 2000)

1.2 Classification and criteria of breakup regimes of round liquid jets in 16

co-flowing air (Liu 2000)

2.1 Flow rates and atomizer dimensions used for experiments 37

measurements (Ma 2001)

2.2 Film thickness at different water volume fractions for case 1 (15 GPM). 37

2.3 Results for two different grid densities for case 1 (15 GPM)

2.4 Comparison of computational results with experimental 37

measurements (Ma 2001)

2.5 Film thickness at different water volume fractions for air pressures of 1 and 49

5 bars (15 GPM).

4.1 Annular liquid sheet stability literature summary 87

8.1 Cases of study 204

8.2 Results for two different grid densities for case 1 204

8.3 Comparison of computational results with experimental 204

measurements

8.4 Comparison of predicted with measured breakup length (large scale 208 nozzle (Benjamin et al. 1998))

xii LIST OF FIGURES

Figure Page

1.1 The typical plain orifice pressure atomizer 5

1.2 A schematic of a pressure swirl atomizer 5

1.3 Growth of unstable waves on a conical sheet from simplex atomizer (Ma 9

2001)

1.4 Dominant frequency among all unstable waves (Ma 2001) 9

1.5 Schematic of a prefilming airblast atomizer (Parker Hannifin Corporation) 11

1.6 Breakup regimes of a laminar liquid jet (Faeth 1990) 15

1.7 Planar sheet disturbance modes

(a) symmetric mode (varicose) and (b) anti-symmetric mode (sinuous) 17

1.8 Annular sheet disturbance modes

(a) symmetric mode (para-varicose) and

(b) anti-symmetric mode (para-sinuous) 21

1.9 Schematic of the comprehensive model for simplex atomizer 22

2.1 Pressure swirl atomizer physical model and boundary conditions 32

2.2 Time history of a point in the orifice (case1, 10 GPM) 35

a) Velocity convergence of a point in the orifice and b) Inlet pressure

convergence

2.3 Volume fraction contours

(a) Experiments (Ma 2001) (b) Prediction 36

2.4 a) Velocity vectors colored by axial velocity and b) Stream function

contours 39

xiii 2.5 a) Axial velocity profiles and b) Swirl velocity profiles 40

a) Axial velocity profile at swirl chamber exit

2.6 b) Tangential velocity profile in swirl chamber exit 42

2.7 Velocity profiles at different viscosities

a) Tangential velocity profile at orifice exit

b) Axial velocity profile at orifice exit 45

2.8 Effect of liquid viscosity on air core diameter 46

a) in the atomizer b) in the orifice section

2.9 a) Spray cone angle b) Film thickness c) Discharge coefficient 48

2.10 Effect of air pressure on the air core diameter 48

3.1 Schematic of annular liquid sheet 56

3.2 Growth rate versus wave number for n = 0, 1, 2 mode at Wes = 0, Wei = 0, 66

Weo = 0, gi = go = 0.00123, h = 0.667 :( a) Para-sinuous mode and

(b) Para-varicose mode

3.3 Growth rate versus wave number (based on free vortex dispersion

equation) for n = 0 at Wel = 5000, Wei = 0, Weo = 0, gi = go = 0.00123 and h 67

= 0.667

3.4 Growth rate of para-sinuous mode versus wave for n = 0, Wei = 0, Weo =

0, gi = go = 0.00123 and h = 0.667: (a) Wel = 500 and (b) Wel = 10000 68

3.5 Growth rate versus wave number for n = 0, gi = go = 0.00123, Wei = 0,

Weo = 0, h = 0.667 and Wel = 5000 71

3.6 Growth rate versus wave number for n = 0, h = 0.667, Wel = 500, Wei and

Weo = 0: (a) Wes = 0 and (b) Wes = 100 72

xiv 3.7 Growth rate versus wave number at different liquid swirl Weber numbers

for n = 0, h = 0.667, Wei = 0, Weo = 0 and gi = go = 0.01: (a) Wel =500

and (b) Wel = 5000 74

3.8 Growth rate versus wave number for different radius ratios at n = 0, Wei =

0, Weo = 0 and gi =go = 0.00123: (a) Wel =500 and Wes =100; (b) Wel

=10000 and Wes =5000 76

3.9 Growth rate versus wave number for radius ratios at n = 0, h = 0.667, Wei

= 0, Weo = 0 and gi =go = 0.00123 (a) Ul = 5 m/s and Wl = 2 m/s, (b) = 10

m/s and Wl = 5 m/s 77

3.10 Growth rate versus wave number at n = 0, h = 0.667, Wei = 0, Wel = 10000

and gi =go = 0.00123: (a) without liquid swirl (para-sinuous mode) and (b) 79

Wes = 500

4.1 Schematic of Annular liquid sheet 89

4.2 (a) Optimal growth rate versus liquid swirl Weber number (b) Optimal

axial wave number versus liquid swirl Weber number at Wel = 200, Wesi =

Weso =0, h = 0.9, gi = go = 0.00129 for n = 0 97

4.3 Optimal growth rate versus liquid swirl Weber number for different inner

and outer axial air Weber numbers at Wel = 200, Wesi = Weso =0, h = 0.9,

gi = go = 0.00129 for n = 0 100

4.4 Optimal growth rate versus axial air Weber number at Wel = 200, Wesi =

Weso =0, Wei = 20, Weo = 20, h = 0.9, gi = go = 0.00129 for n = 0 (a) Wes

= 0, (b) Wes = 50 101

4.5 (a) Optimal growth rate versus air Weber number (b) Optimal axial wave

xv number versus air Weber number at Wel = 200, Wes =0, h = 0.9, gi = go =

0.00129 for n = 2 103

4.6 Growth rate versus axial and circumferential wave number at Wel =200,

Wes =50, h = 0.9, gi = go = 0.00129 a) Wei=Wesi = 20 & Weo =Weso = 0,

b) Wei=Wesi = 0 and Weo =Weso = 20 105

4.7 (a) Optimal growth rate versus liquid swirl Weber number (b) Optimal

axial wave number versus liquid swirl Weber number at Wel = 200, h =

0.9, Wei=Weo = 20 gi = go = 0.00129 for n = 2 107

4.8 (a) Optimal growth rate versus liquid swirl Weber number (b) Optimal

axial wave number versus liquid swirl Weber number at Wel = 200, Wei =

Weo = 20, Wesi = 20, Weso = 20, h = 0.9, gi = go = 0.00129 for n = 1 109

4.9 (a) Optimal growth rate versus air Weber number (b) Optimal axial wave

number versus air Weber number at Wel = 200, Wes = 20, h = 0.9, gi = go

= 0.00129 for n = 1 (Wei = Weo = Weso =Wesi ) 112

4.10 Variation of optimal growth rate versus liquid swirl Weber number at three

different air-to-liquid density ratios showing the effect of air pressure at

Wel = 200, Wesi = Weso =0 and h = 0.9 for n= 0 113

5.1 Three dimensional temporal nonlinear evolution of a liquid jet breakup

with main and satellite drops. Wel =15 , n = 0 , UW= = 0, and

ρ = 0.0012. (a) t = 0, (b) t = 3, (c) t = 6, and (d) t = 13.5 131

5.2 Dimensionless breakup length versus axial gas Weber number for n = 0 ,

UW==0, ρ = 0.0012and ηo = 0.0005 134

xvi 5.3 Dimensionless breakup length versus gas Weber number for Wel = 2 ,

134 n = 0 , W = 0 , ρ = 0.0012and ηo = 0.0001

5.4 Maximum disturbance growth rate versus gas swirl number for

Wel = 6 and ρ = 0.0012. (a) U = 20 and (b) U = 30 136

5.5 Gas swirl number versus gas-to-liquid momentum ratio for Wel = 6

and ρ = 0.0012 139

5.6 (a) Three dimensional plot of the jet surface and (b) Two dimensional plots

at the axial breakup location of the jet surface.

( n = 2,Wel = 6 ,U = 20 ,W = 22 , ρ = 0.0012 and ηo = 0.0001) 141

5.7 Dimensionless breakup length versus gas swirl number for Wel = 6 ,

141 U = 22 , ρ = 0.0012 and ηo = 0.0001

6.1 A schematic of annular liquid sheet 147

6.2 Growth rate versus axial wave number for the fundamental and the first

harmonic modes at Welio=== 80, U U 0, gi = go = 0.001 160

6.3 Temporal Evolution of the dimensionless inner and outer surfaces

deformation at the dominant wave number of k = 0.16 for

Wel = 300,UUio= = 0 , ggio==0.0012 (ηo = 0.1) : a) T = 0, (b) T =20, c) T

=35 and d) T= 54.5 162

6.4 Spatial evolution of inner and outer surface deformations at different initial

disturbance amplitudes forWel = 300, UUio= ===0, gg io 0.0012

164 and k = 0.16 ; a)ηo = 0.1, b) ηo = 0.01 and c)ηo = 0.001

xvii 6.5 The effect of initial disturbance amplitude on the breakup time for

Welio===300, U U 0 and ggio= = 0.0012 167

6.6 The effect of liquid Weber number on the breakup time for

UUioo=== W 0 and ,ggio= = 0.0012 167

6.7 Dimensions of the annular nozzle and breakup process of the liquid sheet

generated by the annular nozzle (Mitra 2001) 168

6.8 Evolution of the dimensionless surface deformation r as a function of

dimensionless distance x for Wel = 4 , UUio= = 12.7 , gi = go = 0.0012

and = 0.015: a) predictions and b) experiments 169

6.9 Evolution of the dimensionless surface deformation r as a function of

dimensionless distance x for Wel = 15 ,UUio= = 12.85, gi = go = 0.0012

and = 0.0001: a) predictions and b) experiments 170

6.10 Temporal evolution of inner and outer surface deformation at different

inner and outer gas-liquid velocity ratios for

Welioo====300, g g 0.0012(η 0.1) : a)UUio= 3,= 0 and k = 0.39 , b)

172 UUio==0, 3 and k = 0.37 and c) UUio= = 3 and k = 0.7

6.11 Effect of gas-liquid velocity ratio on the breakup time Wel = 4,

172 ggio==0.0012(η o = 0.015)

7.1 A schematic of annular liquid sheet 177

7.2 Temporal evolution of the dimensionless surface deformation r at different

outer gas swirl strength values for Wel = 4 ,0Ui = ,15Uo = , gi = go =

xviii 0.0012 and ηo = 0.001: a) Wno = 0,= 0, b) Wno = 50,= 2, c) 192

Wno ==70, 3 , d) Wno == 100, 5 and e) Wno = 210,= 14

7.3 Two dimensional plot at the axial breakup location of the annular liquid

sheet for Welio== 4, U 0, U = 15, g io == g 0.0012 and ηo = 0.001: a)

195 Wno ==0, 0 , b) Wno == 50, 2, c) Wno = 70,= 3 , d) Wno = 100,= 5 and

e) Wno ==210, 14

7.4 Spatial evolution of the dimensionless surface deformation r at different

gas swirl strength values for Welio= 4, U==== 0, U 15, g io g 0.0012 and

196 ηo = 0.001: a) Wno == 50, 2and b) Wno = 100,= 5

8.1 Schematic of the current comprehensive model 202

8.2 Physical model and boundary conditions (case 1) 203

8.3 Computed axial velocity in the atomizer (case 1) 206

8.4 Contours of volume fraction (case 1) 206

xix LIST OF SYMBOLS

Symbol Description Unit

A Vortex strength m2/s

2 Ap Total inlet area m

Cd Discharge coefficient

d Diameter of liquid jet m

Dp Inlet slot diameter m

Ls Swirl chamber length m

Lo Exit orifice length m

Ds Swirl chamber diameter m

Do Exit orifice diameter m

g Air/liquid density ratio

h Radius ratio (Ra/Rb) in chapters 2 and 3 m

h Sheet thickness in chapters 1, 6, 7 and 8 m

th In n order modified Bessel Function of first kind

i −1

K Atomizer geometric constant

Kn nth order modified Bessel Function of second kind

k Axial wave number 1/m

m& l Mass flow rate kg/s

n Azimuthal (Circumferential) wave number

P Mean pressure N/m2

xx p′ Disturbance pressure N/m2

Ra Inner radius of liquid sheet m

Rb Outer radius of liquid sheet m r Radial coordinate m t Time s

U Mean axial velocity m/s

Ui Inner gas-liquid axial velocity ratio

Uo Outer gas-liquid axial velocity ratio

V Mean radial velocity m/s

W Mean tangential velocity m/s

Wo Dimensionless outer gas swirl strength u Disturbance axial velocity m/s v Disturbance radial velocity m/s w Disturbance tangential velocity m/s

We Weber number (We=ρU2d/σ) x Axial coordinate m

Greek letters

η Displacement disturbance m

λ Wave length m

θ Azimuthal angle/spray cone half angle radian/degree

ρ Fluid density kg/m3

ω Temporal frequency 1/s

Φ Phase difference radian/degree xxi

Subscripts l Liquid phase i Inner gas o Outer gas s Based on swirling liquid component si Based on swirling inner air component so Based on swirling outer air component

S Based on swirling component of solid rotation type

F Based on swirling component of free vortex type

xxii CHAPTER 1

INTRODUCTION

1.1 Motivation

Atomization generally refers to the disintegration of a bulk liquid material via an atomizer into droplets in a surrounding gas. Atomizers are utilized in spraying liquids in many industrial and household applications. The atomization processes may be classified into two major categories in terms of the relative velocity between the liquid being atomized and the surrounding ambience

(Liu 2000). In the first category, a liquid at high velocity is discharged into a still or relatively slow-moving gas (air or other gases). Notable processes in this category include, for example, pressure atomization and rotary atomization. In the second category, a relatively slow-moving liquid is exposed to a stream of gas at high velocity. This category includes, for example, two- fluid atomization and whistle atomization. The initial breakup process is often referred to as primary breakup, primary disintegration, or primary atomization. A population of larger droplets produced in the primary atomization may be unstable if they exceed a critical droplet size and thus may undergo further disruption into smaller droplets. This process is usually termed secondary breakup, secondary disintegration, or secondary atomization. Therefore, the final droplet size distribution produced in an atomization process is determined by the liquid properties in both the primary and secondary disintegration (Liu 2000).

The unified design approach of atomizers in different fields requires the interrelations between the different spray characteristics of the atomizers with the pertinent input parameters such as liquid fuel properties, injection conditions and atomizer geometries. This requires a physical understanding of the flow field inside the atomizer and of the mechanism of spray

1 formation outside the atomizer. The modeling of the atomization process is a very challenging task as it is affected by variety of factors as the nozzle geometry, the thermo-physical properties of fuel, and the aerodynamic liquid-gas interaction. The success of spray modeling depends on the correct specification of the initial droplet conditions. Therefore, direct or indirect coupling of atomizer flow with the primary breakup and spray formation is important for the optimization of the atomizers as well as better understanding and consequent improvement of the overall atomization process.

1.2 Atomization and Atomizers

Liquid atomization is fundamental to many industrial applications such as diesel engines, spark ignition engines, gas turbine engines, liquid rocket engines, industrial furnaces, agriculture sprays, and spray coating of surface materials. For combustion applications, the primary purpose of atomization is to increase the specific surface area of liquid fuel and thereby achieve high rates of heat transfer, evaporation and mixing. Reduction of fuel drop size also leads to easier light-up, a wider burning range, and lower pollutant emissions (Lefebvre, 1989).

Due to the increase of surface area, energy must be imparted to the liquid to overcome the consolidating surface tension force. Energy may be supplied by pressure, centrifugal, kinetic or acoustic effects. Based on the forms of energy supplied, atomizers are classified into pressure atomizer, pressure swirl atomizer, twin-fluid atomizer, rotary atomizers, electrostatic and ultrasonic atomizer (Liu 2000). The first three types of atomizers are widely used in combustion engines and their features will be briefly described below.

2 1.2.1 Plain Orifice Atomizer

Plain-orifice atomizers are widely used for injecting liquids into a flow stream of air or gas. The injection may occur in a co-flow, a contra-flow, or a cross-flow stream. The best known application of plain-orifice atomizers is perhaps diesel injectors.

Inside the plain orifice pressure atomizer shown in Figure 1.1, a round liquid jet is injected into the surrounding air at very high velocity under the action of high pressure. Due to turbulence inside the liquid jet and the aerodynamic interaction with the ambient gas, the jet disintegrates into droplets. Because of its simple geometry and low manufacturing cost, the plain orifice atomizer is widely used in diesel engines, liquid rocket engines and afterburner of jet engines. However, it has drawbacks such as narrow cone angle and deteriorated atomization quality at low flow rates.

1.2.2 Pressure Swirl (Simplex) Atomizer

Pressure-swirl atomizers or simplex atomizers are used in a variety of applications including liquid fuel injection systems in a wide range of aircraft, marine engines, and gas turbine combustors as well as industrial oil fired furnaces; consumer product sprays in household products; medical drug delivery for respiratory system; spray drying of edible liquids such as milk and drying of detergents; paint sprays; humidification systems and agricultural sprays. As technology evolves, simplex atomizers are being considered in newer product applications. For example, pressure swirl atomizers are being recognized as ideal atomizers for the direct injection spark ignition (DISI) engines or gasoline direct injection (GDI) engines because they generate a fine fuel spray with moderate injection pressure. The advantageous characteristics of pressure swirl atomizer include simplicity of construction, ease of manufacture even in small size, reliability, good atomization quality, low clogging tendencies, and low pumping power 3 requirements. Pressure swirl atomizers tend to have a smaller ratio of Sauter mean diameter to nozzle diameter compared to other pressure nozzles. These advantages have resulted in widespread use of pressure swirl atomizers and have made it a versatile atomizer (Ibrahim and

Jog 2006).

A commonly used geometry of a simplex atomizer is shown in Figure 1.2. Several variations, such as addition of a trumpet at the exit to guide the liquid to a desired spray angle and inlet slots that are not perpendicular to the nozzle axis, are also employed. However, all geometries have inlet slots that result in imparting a swirling motion to the liquid as it enters the swirl (or spin) chamber. A convergent section accelerates the flow as it enters the exit orifice. The swirl motion of the liquid pushes it close to the wall and creates a zone of low pressure along the center line which results in back flow of air in the nozzle creating an air-cored vortex. The liquid emanates from the orifice as a conical sheet that spreads radially outwards due to centrifugal force. Once the liquid sheet exits the atomizer and moves away from the atomizer, the thickness of the sheet decreases and the sheet becomes unstable. Waves are formed at the inner and outer surface of the sheet.

4

Figure 1.1: The typical plain orifice pressure atomizer

Figure 1.2: A schematic of a pressure swirl atomizer

5 The difference between the velocity of the liquid sheet and the surrounding air or gas causes aerodynamic forces that amplify the waves of the sheet. The waves grow on the sheet until they reach critical amplitude and cause sheet breakup. Fragments of the sheet are broken off. These fragments contract due to surface tension into ligaments, which subsequently disintegrate into drops. The performance parameters that are of most import to an atomizer designers are: mean spray droplet size (typically Sauter Mean Diameter), the spray cone angle, and the discharge coefficient. The mean drop size roughly correlates with the square root of the film thickness. As such the exit film thickness becomes a significant performance parameter. The droplet size and the spray angle impact the speed of subsequent transport processes with the droplets and their mixing with air. The discharge coefficient determines the pressure requirement for a desired throughput. For example, to reduce emissions, it is critical to design fuel atomizers that can produce sprays with predetermined droplet size distribution at the desired combustor locations to enhance the mixing process between the fuel and the air. In medical drug delivery, it is important to design atomizers that provide the right amount of dosage with a predetermined drop sizes so that the drug can reach the desired parts of the respiratory system.

It is known that the performance of the atomizer is governed by the liquid properties, injection flow conditions and nozzle geometry. As the mass flow rate through the nozzle is increased from zero, the performance parameters change significantly at first, but eventually at high mass flow rate the discharge coefficient, the film thickness, and the spray cone angle become insensitive to the variations in flow Reynolds number. Simplex atomizers are typically designed to operate in this range as a consistent performance can be expected irrespective of small changes in the inlet pressure. As such, in this regime, inlet flow conditions tend to have only a small effect on the atomizer performance. Hence most of the work on simplex atomizers

6 has generally focused on investigation of the effect of nozzle geometry and liquid properties on

atomizer performance. Moreover, if the liquid properties are fixed for a given application,

change in nozzle geometry becomes an effective method to achieve desired atomizer

performance. Despite the geometrical simplicity of the pressure swirl atomizer, the flow in the

nozzle is highly complex. The flow is two-phase, turbulent unsteady, with regions of

recirculation, and an air core vortex with a liquid/gas interface that may exhibit unsteadiness and instabilities. Typical nozzle sizes are extremely small (exit diameter of less than 500 micron) which makes direct flow field measurements extremely difficult. To computationally model the flow, one must be able to track the liquid/gas interface accurately. The location of the interface is not known a-priori and must be determined as part of the solution.

Since the breakup of liquid sheets and liquid jets from the atomizers is responsible for spray formation and determines the resultant spray characteristics such as spray angle, mean drop size, droplet size and velocity distributions at the atomizer exit, the performance of fuel injectors has a direct effect on the subsequent heat and mass transfer between the dispersed and continuum phases thus influencing evaporation, mixing, combustion and pollutant formation.

Therefore, understanding the underlying physics of the disintegration of liquid sheets and jets will benefit greatly combustor design. Also, it becomes very important to determine the dominating mode under given operating conditions whenever active control of spray formation is sought for some specific applications.

It is well established that the disintegration of liquid sheets or liquid jets is caused by the growth of unstable waves at the interface due to the aerodynamic interaction between the liquid and ambient gas. There exists a dominant or the most unstable wave corresponding to the highest growth rate. When the amplitude of the most unstable wave reaches a critical value, the wave

7 detaches from the leading edge of the liquid sheet to form a ligament which is unstable and

subsequently breaks down into droplets.

Large amount of photographs were taken to understand the breakup behavior of the liquid sheets from large-scale Plexiglas simplex nozzles in the Combustion Diagnostic Laboratory of

Aerospace Engineering Department at University of Cincinnati (Ma 2002). A typical image of unstable waves at the liquid film surface is shown in Figure 1.3 The amplitude of these unstable waves increases as they propagate downstream. When the amplitude of an unstable wave reaches a critical value, the liquid sheet breakup into ligaments and subsequently into droplets.

Effort has also been made to measure the frequency spectrum of unstable waves at the liquid/air interface using Position Sensing Device (PDS) (Kamler 1998). Figure 1.4 demonstrates that there exists a dominant frequency among all unstable waves which has the highest growth rate.

The wave with highest growth rate is called the most unstable wave and is mainly responsible for the disintegration of the liquid sheet. The maximum growth rate and the most unstable wave number depend on flow conditions, fluid properties and nozzle geometry and can be determined through linear stability analysis. Nowadays, there is an industry wide pressing demand for developing computational tools for advanced combustor design. The success of such CFD tools depends partly on the accuracy of the atomization sub-model since the CFD tools require initial conditions in terms of parameters such as mean droplet size, droplet velocity and their distributions at the atomizer exit. It is difficult to develop atomization models based on empirical data because each correlation is derived for certain type of fuel injector under specific operating conditions and the scope of its application is usually very limited.

8

Figure 1.3: Growth of unstable waves on a conical sheet from simplex atomizer (Ma 2001)

Power spectral density 3.5

3

2.5

2

Amplitude 1.5

1

0.5

0 0 1 2 3 4 10 10 10 10 10 Frequence

Figure 1.4: Dominant frequency among all unstable waves (Ma 2001)

9 Therefore, it becomes imperative to establish a theoretical comprehensive model to predict the performance parameters of the fuel injectors by linking the internal flow field inside the atomizer and liquid sheet breakup and atomization.

1.2.3 Twin Fluid Atomizers

The twin-fluid atomizer, i.e., air-assisted and airblast atomizers, employs the kinetic energy of high velocity gas streams to disintegrate the liquid sheet or jet into droplets. The main difference between air-assisted and airblast atomizers is that the former use relatively small quantity of air or steam flowing at very high velocity (usually sonic), whereas the latter employ large amounts of air flowing at much lower velocity (<100 m/s) (Lefebvre, 1983 and1989). Owning to its advantages such as lower fuel pressure requirement, finer spray, better patternation, and lower pollutant emissions, prefilming airblast atomizer has been considered as an advanced combustor concept and is widely used in gas turbine engines and oil-fired furnaces. According to the filming method, the airblast atomizer can be divided into two types, i.e., the prefilming airblast atomizer and the swirl-cup airblast atomizer. Inside the traditional prefilming airblast atomizer, liquid fuel is first spread out into a thin annular sheet, and then exposed to high-speed swirling air streams on both sides (Lefebvre 1980). In the modern swirl-cup airblast atomizer as shown in

Figure 1.5, liquid fuel emanates from a pressure-swirl atomizer in the form of a conical liquid sheet and impinges with the swirler cup. A thin annular liquid film is formed on the cup surface due to strong shear action of the fast moving inner air. Therefore, for both types of airblast atomizers, the breakup of the annular liquid sheet is responsible for the spray formation and determines the resultant spray characteristics such as mean droplet diameter and droplet size distribution. Air-blast atomizers have been used in a broad range of aircraft, marine, and industrial gas turbines.

10 Air Inner swirler

Air

Outer swirler

Figure 1.5: Schematic of a prefilming airblast atomizer (Parker Hannifin Corporation)

The disadvantages of the prefilming airblast atomizer are the requirements of high pressure air supply system and poor atomization at low air velocity.

1.3 Fundamental Mechanisms of Spray Formation

Liquid atomization process is very complicated and involves primary breakup, secondary breakup, and droplet interactions. During the primary breakup, liquid jets or sheets are disintegrated into unstable ligaments which subsequently break down into drops. Due to aerodynamic interaction between the drops and ambient air, these drops deform and further breakup into small droplets, i.e., the secondary breakup. In this dissertation only the primary breakup has been studied.

The breakup of liquid jets or sheets is affected by a large number of factors including flow conditions, fluid properties and details of nozzle geometry. Various forces such as inertial force,

11 surface tension, aerodynamic force, viscous force and centrifugal force are involved in this

process. Some of them resist the disintegration process while others promote it. It is the

competition among these forces that determines the stability of liquid sheet or jet. Understanding

the specific roles of each force in the breakup process will benefit not only fuel injector design

but also numerical simulation of combustion engines.

In reality, most fluid oscillations have amplitudes that are constant with time, but grow in a

spatial direction. Temporal theory for hydrodynamics instability assumes that disturbances grow

temporally everywhere. Gaster (1962) has proposed a transformation to convert the temporally

growing disturbances into spatial growth rates by connecting them using group velocity for

application to hydrodynamic stability analysis. He has reported that the ratio of temporal to

spatial growth factor is equal to a group velocity, the two group velocities being very nearly the

same. Lin et al. (1990) have reported that the characteristics of temporal and spatial disturbance

waves are almost identical for a Weber number much larger than unity for liquid sheets. Chuech

(2006) have performed spatial stability analysis of a viscous liquid sheet. He has compared the

spatial instability results with those of the temporal instability for different Weber number and

Reynolds numbers and found that the spatial instability analysis was important for cases of low

Weber numbers. The absolute instability for a plane liquid sheet only occurs at Weber numbers

less than about one, a situation unlikely to occur under practical conditions in atomization,

although it may be relevant to other applications such as coating (Mitra et al. 2001).

1.3.1 Disintegration of Liquid Jets

When a liquid jet issues from a nozzle, oscillations and perturbations form on the jet surface as a result of the competition of cohesive and disruptive forces (Liu 2000). Under certain conditions,

for example, if the liquid flow rate exceeds a certain value, the equilibrium between the surface 12 tension and gravitational forces can no longer be maintained, and the transition from dripping to

laminar flow will occur. The oscillations may be amplified to such an extent that the liquid

breaks up into droplets. The continuous length of a liquid jet prior to breakup and the droplet

size are important parameters characterizing a liquid jet breakup process. These parameters

reflect the growth rate of disturbances and the wave number of the most unstable disturbance,

respectively. The understanding of the mechanisms governing liquid jet breakup began with

Rayleigh’s pioneering work (Rayleigh, 1897) over a hundred years ago. Since then, extensive

work has been carried out for temporal as well as spatial instability and from low to high jet

velocity (Lin and Reitz 1998).

1.3.1.1 Round Liquid Jets in Quiescent Air

Four different breakup regimes as illustrated in Figure 1.6 and Table 1.1 have been identified

that correspond to different appearances of the jet as the flow conditions are changed (Lin and

Reitz 1998, Lefebvre 1989, Reitz and Bracco 1982 and 1986). These regimes are called the

Rayleigh regime, the first wind-induced regime, the second wind-induced regime, and the

atomization regime. In Rayleigh regime, that is, at very low jet velocity, surface tension is the

driving force for the growth of waves formed at the interface and the jet is broken up by the

axisymmetric mode instability. The instability of liquid jet in Rayleigh’s regime has been

summarized by McCarthy and Molloy (1974) and Sterling and Sleicher (1975). In the first wind- induced regime, the aerodynamic interaction between the liquid and the ambient gas becomes important and the appearance of the jet becomes sinuous (asymmetric) when breakup occurs.

The size of drops formed in both the Rayleigh regime and the first wind-induced regime is of the order of the jet diameter. At higher jet velocity, drops of sizes much smaller than the jet diameter are formed on the jet surface and the jet breaks up immediately at the jet exit. Large 13 discrepancies exist between theoretical predictions of jet intact length and mean droplet size and measured data. Various mechanisms have been proposed to explain the difference. These include

aerodynamic interaction theory, liquid turbulence, cavitation inside the nozzle, velocity profile

relaxation, and liquid supply pressure oscillation (Reitz and Bracco 1982). After extensive

experiments, Reitz and Bracco (1982) found that no single mechanism could fully explain the jet

atomization process. However, they concluded that the combination of aerodynamic interaction

theory and nozzle geometry was effective in explaining their experimental results. It is now

generally agreed that the unstable growth of waves at the liquid/gas interface due to the

aerodynamic interaction between the liquid and the ambient gas is responsible for the breakup of

the liquid jet. Therefore, theoretical investigation of the instability of liquid jets will not only

lead to better understanding of the underlying mechanisms of liquid atomization, but also

improved performance predictions of fuel injectors in combustion engines.

1.3.1.2 Round Liquid Jets in Co-flowing Air

From an analysis of numerous high-speed spark photographs, various modes of round liquid

water jet disintegration in a coaxial air stream have been identified by Faragó and Chigier (1992)

over a liquid Reynolds number range of 200 to 20,000 and an aerodynamic Weber number range

of 0.001 to 600. Accordingly, the mechanisms of jet breakup have been classified into the

following regimes (Faragó and Chigier 1992) as summarized in Table 1.2.

14

Figure 1.6: Breakup regimes of a laminar liquid jet (Faeth 1990)

15

Table 1.1: Classification and criteria of breakup regimes of round jets in quiescent air (Liu

2000).

Regime Predominant Breakup Mechanism Criteria* Rayleigh Jet Breakup Surface Tension Force We < 0.4 (Varicose Breakup) a First Wind-Induced Breakup Surface Tension Force , Dynamic 0.4< 40.3 Turbulence, Cavitaion, Bursting a Effect 2 */WeaaL= ρ U d σ

Table 1.2: Classification and criteria of breakup regimes of round liquid jets in co-flowing air

(Liu 2000).

Regime Criteria* Axisymmetric Rayleigh-Type Wea <15 Non-axisymmetric Rayleigh-Type 15< Wea < 25 Membrane-Type 25< Wea < 70 Fiber-Type 70< Wea > 500 2 */WeaaR= ρ U d σ where UR is the relative velocity between air and liquid.

16 1.3.2 The Disintegration of Liquid Sheets

1.3.2.1 Plane Liquid Sheets

Squire (1953) was the first to investigate theoretically the instability of an inviscid flat liquid sheet. He has considered two modes of disturbances, i.e., anti-symmetric (sinuous) and symmetric (varicose) disturbances. The anti-symmetric disturbance has the same phase at both interfaces while the symmetric mode has a π (180°) phase difference as shown in Figure 1.7. He found that anti-symmetric mode dominates the growth of infinitesimal disturbance. This conclusion was later validated by extensive studies (Hagerty et al. 1955, Dombrowski and Johns

1963 and Clark and Dombrowski 1972). Dombrowski and Johns (1963) proposed a breakup model for a plane liquid sheet. Based on the linear stability analysis and the breakup model, mean drop sizes were predicted and results agreed well with measurement data (Dombrowski and Johns 1963). For authoritative reviews of liquid sheet and jet instability and breakup, readers are referred to a recent monograph by Lin (2003).

1.7(a)

1.7(b)

Figure 1.7: Planar sheet disturbance modes

(a) symmetric mode (varicose) and (b) anti-symmetric mode (sinuous).

17 In recent years, temporal as well as spatial instability of viscous liquid sheets were analyzed in several studies (Li and Tankin 1991 and Cousin and Dumouchel, 1996). They found at low Weber number, symmetric mode is dominating while at large Weber number, anti- symmetric mode dominates. Rangel and Sirignano (1994) carried out both linear and nonlinear temporal stability analysis and investigated the effects of density ratio on the liquid sheet instability. They found that at low density ratio, anti-symmetric mode dominates in the development of interfacial waves while symmetric mode dominates at high density ratio.

1.3.2.2 Annular Liquid Sheets

The first study of annular swirling liquid sheet stability was reported by Ponstein (1953).

He derived the general dispersion relation for the growth of disturbances under the influence of a potential liquid swirl flow described by a free-vortex swirl velocity profile and a uniform axial mean velocity, while neglecting the effects of viscosity and the presence of the two gas phases.

He found asymmetric or helical mode is stable without swirl but becomes more unstable than the axisymmetric mode (n = 0) when liquid swirl is present.

For plane liquid sheets the most unstable disturbances are exactly symmetric and anti-symmetric

(Squire 1953). As explained in Shen and Li (1996), for an annular sheet, this need not be assumed a priori. In this case we have considered a phase difference, which can be evaluated as part of the solution to determine the most unstable disturbance for inner and outer interface. The two solutions obtained with Φ close to zero and close to π are referred as para-sinuous and para- varicose respectively (Chen and Lin 2002). In other words, the independent interfacial modes can not be exactly in phase or out of phase in annular liquid sheet, i.e., = 0 or π is only an approximation for very thin annular sheets. This is the reason they are called para-sinuous and para-varicose rather than simply anti-symmetric (sinuous) or Symmetric (varicose) modes as 18 shown in Figure 1.8. Crapper et al. (1975) conducted a temporal stability analysis of inviscid annular sheets without swirl under axisymmetric disturbances. They concluded that the growth rates of both symmetric and anti-symmetric waves increase significantly with the reduction of the radius of the core. Meyer and Weihs (1985) defined a critical penetration thickness. When the annulus thickness is greater than critical penetration thickness, it behaves like a solid jet.

Otherwise, it behaves like a two-dimensional liquid sheet. Shen and Li (1996) considered viscosity and ambient air effects. They found that ambient gas always enhance the instability while viscosity has dual effect, stabilizing the liquid sheet at high Weber number and destabilizing it at low Weber number. However, in their analysis, axisymmetric disturbances were assumed and swirl effect was not considered. Panchagnula et al (1996) analyzed liquid swirl effect on liquid sheet instability but their analysis is limited to low speed regime. Liao et al.

(1999, 2000 and 2001) studied the instability and breakup of annular liquid sheets. They have investigated the effect of the swirl of inner and outer gas streams on the instability of the liquid sheet.

1.4 Scope of the Dissertation

Part I of the dissertation presents CFD Modeling of the turbulent two-phase flow filed insides pressure swirl atomizers using volume of fluid (VOF) method. The validated CFD results (sheet thickness, axial liquid velocity and tangential liquid velocity) will be used as an input to the nonlinear instability breakup models in Parts III and IV. Part II of the dissertation consists of two chapters and provides the development of linear instability models for swirling annular liquid sheet exposed to axial inner and outer gas streams and swirling inner and outer gas streams. The effect of liquid swirl velocity profile on the instability of a swirling annular liquid sheet is presented in chapter 3. To make the theoretical model applicable to practical airblast atomizers, 19 effect of both inner and outer air swirl must be considered as well as liquid swirl. Chapter 4 provides an investigation of the effect of liquid and air swirl strength and relative rotational direction on the instability of an annular liquid sheet. A parametric study has been carried out to understand the specific effect of flow conditions, fluid properties and geometric parameters on the instability of annular liquid sheets under liquid swirl and swirling and non-swirling inner and outer gas streams. The principal drawback of the linear instability analysis lies in the fact that the linearized equations used in the theory become inapplicable as the perturbation amplitude grows.

It has been reported by Mitra et al. (2001) that comparison of the linear theory with the experimental results shows that the predicted surface deformation agrees favorably with the experiment, but significant deviation occurs near the sheet breakup region, necessitating a nonlinear analysis for a better description of the plane sheet breakup process. Accordingly, the results based on the theory are valid only for the relatively short interval where the perturbation amplitude remains small. Part III of the dissertation consists of three chapters. A nonlinear instability models for liquid jet in a swirling gas, annular liquid sheet subjected to axially moving inner and outer gas streams and annular liquid sheet subjected to swirling outer gas stream are developed and presented in part III. A comprehensive model for pressure swirl atomizer (Figure 1.9) that links the internal flow field and the nonlinear instability breakup model presented in part IV. This comprehensive model predicts the breakup length of the primary spray atomization. Breakup length predictions from the theoretical nonlinear breakup model are compared with experimental data and the agreement is very good. This comprehensive model can be used to directly to determine the effect of flow conditions and pressure-swirl atomizer geometry on the sheet breakup. This approach gives encouraging results and could be used as initial conditions for spray preparation modeling.

20

1.8(a)

1.8(b)

Figure 1.8: Annular sheet disturbance modes

(a) symmetric mode (para-varicose) and (b) anti-symmetric mode (para-sinuous).

21

Figure 1.9: Schematic of the comprehensive model for simplex atomizer

22

PART I

TWO-PHASE FLOW FIELD IN PRESSURE SWIRL ATOMIZERS

23 CHAPTER 2

COMPUTATIONAL SIMULATION OF FLOW FIELD IN PRESSURE

SWIRL (SIMPLEX) ATOMIZER

2.1 Literature Review

The geometry of pressure swirl atomizer is fairly simple but the flow through the atomizer is highly complex as it involves two phases, regions of recirculation, unsteady and turbulent flow.

The ability to accurately predict the two-phase flow in simplex atomizers and to evaluate the effects of nozzle geometry, flow conditions, and liquid properties on the atomizer performance is critical to improve nozzle designs. However, the difficulties in the accurate tracking of the liquid/gas interface pose a significant challenge in numerical simulation of the flow. Early studies of pressure swirl (simplex) atomizers have employed analytical and/or experimental methods to measure and/or predict the internal/or external spray characteristics (Xue 2004).

2.1.1 Inviscid Analysis and Experimental Work

Due to the difficulties outlined earlier, the initial investigations of simplex atomizer modeled the liquid as inviscid and the flow irrotational (Giffen and Muraszew 1953, Novikof 1948 and

Nieuwkamp 1985). Giffen and Muraszew (1953) have applied the principles of mass conservation and conservation of angular momentum to show that the spray angle (2θ ) , the discharge coefficient (Cd ) , and the film thickness at the exit orifice can be expressed as a function of atomizer geometry, more specifically of atomizer constant K (defined as the ratio of the total area of inlet ports to the product of the swirl-chamber diameter and the discharge orifice diameter). The atomizer constant can be shown to be related to the ratio of axial momentum flux

24 at exit to the angular momentum flux at exit. Giffen and Muraszew (1953) conducted measurements with a number of swirl atomizers which showed that their inviscid theory predicted the trends in the dependence of atomizer constant on performance parameters reasonably well.

Doumas and Laster (1953) have reported an experimental study of such nozzles, measuring the discharge coefficient and the spray cone angle for more than 60 swirl atomizers covering a range of internal dimensions. They found that when the conical sheet of liquid that develops at orifice was fully open, the two characteristics were dependent only on the geometry of the nozzle. However, the atomizer constant had to be empirically modified to correlate their results.

Taylor (1948 and 1950) has used a boundary layer approach to determine the flow through the exit orifice. This approach was extended to non-Newtonian, power-law fluid flow by Som et al. (1984). They have carried out theoretical and experimental investigations into the effects of nozzle geometries and injection conditions on the discharge coefficient and spray cone angle. These two parameters have been theoretically evaluated through analytical solution of the hydrodynamics of the flow inside the nozzle and their dependence with the nozzle dimensions.

They have found that within a definite range, an increase in Reynolds number at inlet to the nozzle decreases the discharge coefficient and increases the spray cone angle. Experiments revealed that there are two limiting values of generalized Reynolds number that determine the formation of air core in the nozzle. One value being the upper limit below which steady flow occurs without the formation of air core, the other one is the lower limit above which steady flow with fully developed air core persists.

25 Rizk and Lefebvre (1985 and 1986) have investigated the internal flow characteristics of simplex swirl atomizers. The effects of the individual swirl atomizer geometrical dimensions on the thickness of annular liquid film at the nozzle exit and the effects of the fluid properties on the values of the discharge coefficient, the spray angle and the liquid film thickness, were studied.

They developed a general expression for the liquid film thickness at the exit of the swirl atomizer and stated that the air-core diameter increases with increasing pressure, decreasing inlet area, increasing swirl chamber diameter, decreasing swirl chamber length, increasing orifice length, decreasing liquid viscosity and decreasing liquid density.

Dumouchel et al. (1992) have investigated the problem of the boundary-layer flow at the orifice of a swirl atomizer. They have used the Bloor and Ingham (1977) analysis for the boundary-layer flow above a flat disk and they estimated the proportion of the fluid that enters the orifice through the boundary layer. They have found that Taylor’s (1950) analysis gives discharge rates that show some differences with those measured by Doumas and Laster (1953).

They have showed that the thickness of the boundary layer and the flow rate inside it are both functions of the design of the nozzle and of the injection pressure. Their computations have shown that the thickness of the boundary layer within the liquid film to be of the same order of magnitude as the film thickness when the injection pressures of the liquid fuel were small, such as in the case of the liquid-propellant rockets. They have reported that as the radius of the inlet port, the radius of the orifice, and the injection pressure increased the boundary-layer flow decreases in importance compared with the bulk flow at the orifice.

Horvay and Leuckel (1985 and 1986) have studied the velocity profiles within a pressure swirl atomizer. The experiment were conducted using three different convergence configurations

(standard, concave, and plain conical) and two different inlet/swirl chamber configurations (four

26 20 x 10 mm and four 20 x 5 mm rectangular inlet slots). The atomizers were manufactured from

Plexiglas and have same overall dimensions: radius of swirl chamber rs = 50 mm, length of swirl chamber Ls = 25 mm, radius of orifice ro = 10 mm and length of orifice Lo = 20 mm. The measurement of the liquid velocity components within the atomizer were carried out using LDA and a refractive index matching fluid, which is a mixture of tetraline, turpentine and castor oil.

The seeding particles are small air bubbles. Radial profiles of the axial and tangential velocities were taken at six different cross-sections through the atomizer.

Holtzclaw et al (1997 and 1998) have examined the geometrical effects on the internal flow field in a large-scale simplex fuel nozzle. They measured the tangential and radial velocity components using PIV techniques and found that the radial velocity was significantly less than the swirl velocity at any point within the swirl chamber of a simplex nozzle. They also derived an empirical equation based on the measured swirl velocity component. Due to the limitations of the imaging technique and post-processing software of image analysis, Holtzclaw’s PIV measurements are not very accurate but provide qualitative results.

Benjamin et al (1997 and 1998) have investigated the effects of various geometric and flow parameters on the performance of large-scale pressure swirl atomizers using optical methods. They measured the film thickness, droplet size and spray angle of a series of large- scale pressure-swirl atomizers. After testing and analyzing a large number of geometric variations covering a wide range of flow capacities, they developed some correlations on the discharge coefficient, flow number, velocity coefficient, spray angle and Sauter mean diameter based on their experimental data.

Ma (2001) has studied the internal flow characteristics in the swirl chamber for both large-scale and medium-scale pressure swirl atomizers. The internal flow field was measured

27 using a two-color PIV system and refractive index matching fluids method. The spray cone angle and liquid film thickness of the nozzle were also obtained from the PIV image. The measurements of the droplet size and velocity distribution were carried out using PDPA.

According the experimental data, he gave the relationship between the internal flow field and the external spray characteristics. A non-dimensional correlation between the vortex flow pattern in the swirl chamber and atomizer design variables was presented. A discussion on flow regime

(turbulent or laminar) within the atomizer body was given.

2.1.2 Review of Computational Modeling

The accurate determination and tracking of the interface between the liquid and the gas phase poses significant difficulty in computational modeling of the flow in simplex atomizers. Early modeling efforts used a single phase approach where the entire flow field was considered as liquid phase (Dumouchel et al. 1993 and Yule and Chinn 2000) and the interface was presumed to be located at the zero value of the axial velocity. Yule and Chinn (2000) have considered the interface location at the zero gage pressure line. An improved technique was used by Datta and

Som (2000) who assumed the air core to be exactly cylindrical and estimated the air core diameter from pressure drop calculations. Yule and Chinn (2000) have found that the velocity fields differ significantly from those assumed in approximate inviscid analyses of swirl atomizers as the flow through the atomizer concentrates either near the air core or near the wall, and secondary motion of Gortler vortices near the swirl chamber wall is present.

More recently two methods have been used to accurately track the interface and determine the two-phase flow. First is the Arbitrary-Lagrangian-Eulerian (ALE) method used by Jog and co- workers (Jeng et al. 1998, Sakman et al. 2000, Liao et al. 1999 and Xue et al. 2002 and 2004).

28 The second is the Volume-of-Fluid (VOF) method used by Hansen et al. (2002), Dash et al.

(2001), Ibrahim et al. (2005) and Ibrahim and Jog (2005 and 2006).

2.1.2.1 The ALE method

The ALE method (Hirt et al. 1974) is divided in two steps. The first step is the Lagrangian step where the grid points move with the local velocity. The points on the interface always remain on the interface tracking a sharp, accurate interface. The second step is the Eulerian step where a new grid is created by moving points back in the axial direction. Jog, Jeng and co-workers have developed a computational code based on the ALE method to determine the flow in a simplex atomizer. To validate the code, experiments were conducted on a large scale prototype nozzle made with optical quality prexiglass so that detailed velocity, film thickness, and spray angle measurements could be made. Particle-Image-Velocimetry (PIV) and Laser-Doppler-

Velocimetry (LDV) methods and high speed digital camera was used to measure the velocity field in the swirl chamber, the film thickness variation in the exit orifice, the spray angle, and the intact length of the film before breakup (Benjamin et al. 1998 and Ma 2002). Detailed comparison of experimental measurements/empirical correlations (Suyari and Lefebvre 1986) and computational results was carried out which showed excellent agreement and validated their computational code based on the ALE method. The validated computational code was then used to investigate the effects of atomizer geometry under two flow regimes – constant mass flow rate through the atomizer (Xue 2004, Sakman et al. 2000, and Xue et al. 2004) and constant pressure drop across the atomizer (Xue 2004).

2.1.2.2 The VOF Method

29 The Volume-of-Fluid method of Hirt and Nichols (1981) is easy to implement in a computational code and hence is available in many commercially available flow softwares

(Fluent, CFX, Star-CD, among others). In this method, in addition to the governing equations for mass and momentum conservation, an equation for the volume fraction of each phase is solved.

When the volume fraction is between zero and one, there exists a fluid interface within the computational cell. This interface can be recreated by plotting iso-VOF contours by interpolating the volume fraction information at each cell node. Typically iso-VOF of 0.5 is used as the location of the interface. One shortcoming of this method is that surface tension force is not applied at the interface in a normal stress balance but is considered as a source term in the momentum conservation equation. Also, the sharp variations in the properties (density and viscosity) are spread over three to four cells. As such, the grid needs to be fine near the interface to capture the interface accurately.

The VOF method with a HRIC (High-Resolution Interface Capturing) scheme (Muzaferija et al.

1999) was used by Dash et al. (2001) to simulate air core formation in nozzles with tangential entry. Donjat et al. (2002) have studied the behavior of inlet jets and the development of the swirling flow and measurements of the air core/liquid interface instabilities. From experiments conducted with a large-scale pressure swirl atomizer, they found that the flow field is generally axisymmetric (except in the inlet zone), yet the impact of discrete inlet jets on the homogenization of the internal flow structure is quite important.

Hansen (2001) has simulated the flow in a scaled model of a Danfoss pressure-swirl atomizer via commercially available CFX-4.3 code. The simulations were performed in a three dimensional curvilinear grid representing the swirl chamber of the atomizer with different methods for turbulence modeling. A simulation using the k-ε turbulence model over-predicted

30 viscosities and failed to predict a stable air-core in the atomizer. Further simulations by Hansen et al. (2002) have showed that simulations assuming laminar flow and Large Eddy Simulations are able to predict the overall flow conditions in the atomizer and an air-core is maintained throughout the time of simulation. Comparing the two simulations with experimental results, they found that the simulation assuming laminar flow with VOF method provides the best agreement with measurements of Hansen (2001). They reported that it is necessary to have a more representative velocity profile at the inlets in order to obtain better predictions. This may be achieved by including the inlet grooves in the computational domain and higher grid resolution in the upper part of the swirl chamber.

2.2 The Physical Model

The two dimensional computational model assumes the flow to be axisymmetric based on the experimental work of Ma (2001). This assumption requires determination of an equivalent

“annular” inlet slot instead of the finite number of slots present in the real nozzle. The width of the “annular” slot as well as the radial and tangential velocities at the inlet are calculated by matching the angular momentum, total mass flow rate, and the kinetic energy of the liquid at the inlet ports. The 2-D Axisymmetric physical model of pressure swirl atomizer is shown in Figure

2.1. The inlet tangential and radial velocities at the wall of the swirl chamber would be given by

2 Q ()DDs − p ⎛⎞Q W = and, VW=−⎜⎟ 2 thus the inlet width can be calculated. At the outlet inlet AD inlet⎜⎟ inlet ps ⎝⎠Ap cross-section, pressure outlet boundary was prescribed. All wall boundaries are taken as no-slip.

For all cases, the fluids are air and water.

31

Figure 2.1: Pressure swirl atomizer physical model and boundary conditions

2.3 Governing Equations

Numerical Simulations of the two-phase flow field in pressure swirl atomizer are solved by the

Navier–Stokes, coupled with the Volume of Fluid (VOF) surface tracking technique on a fixed

Eulerian structured mesh. In the VOF model, a single set of momentum equations is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain. In this method, the volume fraction of the first fluid in the cell is denoted as α = 0 for an empty cell; α =1 for a full cell and 01< α < when a cell contains the interface between the first and second fluids. The following single Reynolds Averaged continuity and

Navier Stokes equations are solved throughout the domain:

∂ρ +∇()0ρur = (1) ∂t

∂()ρur r +∇.(ρμρuurr ) =−∇ p +∇ .[ ( ∇ u r +∇ u rT )] + g r + F (2) ∂t vol

32 The turbulent stresses in the momentum equation are modeled by Reynolds Stress Model (RSM).

The Volume of Fluid model (VOF) is specifically designed to track the position of a free surface between two immiscible phases and solves an additional advection equation for the additional phase.

∂α i +∇ur.0α = (3) ∂t i

Equation (2) is dependent on the volume fractions of phases through the properties ρ and μ .

The properties the first and the second fluids are calculated using

n ρ = ∑αρii (4) i=1

n μ = ∑αμii (5) i=1

n where ∑αi =1 i=1

The velocity differences between the two phases in this model are not pronounced so the shared- field approximation can be safely used without adversely affecting velocity computations near the interface. The face fluxes for the VOF model are calculated using the geometric reconstruction scheme available in FLUENT 6.1. The interface between fluids is solved by a piecewise-linear approach. The continuum surface force (CSF) proposed by Brackbill et al.

(1992) is used for the surface tension model. The additional surface tension model for the VOF calculation results in a source term in equation (2) and is expressed as a volume force as follows:

ρkiiα Fvol = σij (6) (1 / 2)(ρij+ ρ )

33 where ki is the curvature of the interface and σij is the surface tension coefficient.

The discretization scheme used for pressure was PRESTO (Pressure Staggering Option),

SIMPLE method for the pressure-velocity coupling and first order upwind schemes were used for the momentum and swirl.

2.4 Results and Discussions

The calculated average axial, radial and tangential velocities at one point at the orifice exit as a function of flow time are plotted in Figure 2.2(a). Also the inlet pressure is plotted versus the flow time as shown in Figure 2.2(b). From this plot it is seen that the average velocities and the inlet pressure approach a constant value. For a typical flow condition (10 GPM) considered here, the time required for a particle to move from inlet to exit is about 0.6 seconds. Transient calculations for more than 10 times this value are needed to obtain a steady state velocity field.

The CFD results based on VOF method were validated by comparison with experimental data

(Ma 2001) for a velocity variation inside the atomizer as well as discharge parameters, spray cone angle, film thickness, and discharge coefficient using a large-scale atomizer. The geometric and flow parameters for the cases considered are shown in Table 2.1.

34 5

4

3 Axial Radial 2 Tangential

eoiy(m/s) Velocity 1

0 0 1 2 3 4 5 6 7 Flow Time (second)

2.2(a)

40000 35000 30000 25000

20000 Inlet pressure 15000 10000 Pressure (Pa) 5000 0 0 1 2 3 4 5 6 7

Flow Time (second)

2.2(b)

Figure 2.2: Time history of a point in the orifice (case1, 10 GPM) a) Velocity convergence of a point in the orifice and b) Inlet pressure convergence

35

2.3(a)

2.3(b)

Figure 2.3: Volume fraction contours

(a) Experiments (Ma 2001) (b) Prediction

Table 2.1: Flow rates and atomizer dimensions used for experiments measurements (Ma 2001) 36 Orifice Flow Rate Number of Inlet Slot Area Orifice Case # contraction (US gal. ports A (mm2) D , L (mm) P o o angle /min) 1 2 203 21, 14.5 45° 7.5, 10, 15 2 4 406 21, 14.5 45° 10, 20

(Swirl chamber parameters Ls= 89mm, Ds= 76mm)

Table 2.2: Film thickness at different water volume fractions for case 1 (15 GPM).

Relative error % Water volume fraction, α Film thickness, h (mm) ( based on film thickness at α = 0.5) 0.3 2.465 2.37 0.4 2.439 1.29 0.5 2.408 0.00 0.6 2.383 -1.04 0.7 2.356 -2.16 0.8 2.3087 -4.12

Table 2.3: Results for two different grid densities for case 1 (15 GPM).

Film thickness, h Spray cone angle Cd Δp (Pa) (mm) (2θ ) 11020 cells 2.35 92.25 89237.9 0.204 21660 cells 2.408 92.0 86566.0 0.207

Table 2.4: Comparison of computational results with experimental measurements (Ma 2001).

Case Flow Rate Flow Δp (Pa) h (mm) Spray cone angle Cd # (m3/s) Rate (2θ ) (GPM) Exp. CFD Exp.CFD Exp. CFD Exp. CFD 1 4.73E-04 7.5 19305 20357 2.31 2.45 82.6 88.2 0.221 0.214 1 6.31E-04 10 41368 37237 2.28 2.42 85.9 88.8 0.2013 0.211 1 9.46E-04 15 97905 86566 2.23 2.38 88.4 92 0.1963 0.207 2 6.31E-04 10 17237 16591 2.72 3.32 77.4 73 0.312 0.316 2 1.26E-03 20 89631 71407 2.64 3.25 80.7 74 0.274 0.304

37 The liquid film thickness (h) is calculated at different water volume fractions as shown in Table

2.2. The film thickness relative percentage error is calculated based on the film thickness at water volume fraction of 0.5. The maximum film thickness relative error is 4.12 % at water volume fraction of 0.8. Water volume fraction of 0.5 is used to locate the air-water interface in this work.

In order to ensure grid independence of results, two sets of results for the same flow conditions and geometry of case 1 are shown in Table 2.3 with 11020 and 21660 cells, respectively. The difference in results between the two grids is small and indicates that 26682 cells or similar grid is sufficient to get accurate results. Table 2.4 shows a comparison of computational predictions of film thickness, spray angle, and discharge coefficient with experimental measurements for all the cases. The spray cone angle was calculated by

W θ = 2tan−1 (e ) Ue

where We and Ue are the average tangential and axial velocities at the orifice exit.

The average value of axial and tangential velocity component at the orifice exit obtained from

rrio= 22 ∑ φii()rr− i−1 rria= φe = 22 ()rroa− where φ = U or W. The discharge coefficient was calculated as Q . Cd = APOL2/Δ ρ

Figure 2.3 shows the contours of the volume fraction for both of the experiments and the predictions. As a result of the swirling motion (centrifugal forces) within the swirl chamber, the pressure decreases towards the center axis.

38 m/s 2.4(a)

kg/s 2.4(b)

Figure 2.4: a) Velocity vectors colored by axial velocity and b) Stream function contours

39 m/s 2.5(a)

m/s 2.5(b)

Figure 2.5: a) Axial velocity profiles and b) Swirl velocity profiles

40 Due to the lower pressure near the center axis, an air-core is formed along the centerline. The velocity vectors and the stream function are presented in Figure 2.4. Two main liquid streams flow from the inlet slots toward the nozzle exit. The first stream flows near the air core with higher axial velocity component and the flow width is relatively narrow. The second stream flows near the sidewall of the nozzle with lower axial velocity components and the flow path is wider than the first one. Between these two positive flow streams is a large region where either weak positive or negative axial velocity components are present. These positive and negative flows result in the formation of a large number of small-scale vortices in this region, and exhibit fairly sophisticated flow patterns. These vortices lead to kinetic energy dissipation and partially account for the pressure loss in the nozzle. Figure 2.5 presents the profiles of axial velocity and swirl velocity at different axial locations. It can be seen that the tangential velocity profile in the swirl chamber has a Rankine combined vortex structure, a very short solid vortex in the inner region connected with a very large free vortex in the outer region. The axial component of the velocity is highest within a region surrounding the air-core in the swirl chamber. A peak in axial velocity is seen at the same axial location of the peak of the swirl velocity. It can be seen from

Figure 2.5(b) that the tangential velocity profiles in the swirl chamber are independent of the axial position which is consistent with the work of Ma (2001). The axial velocity profile in the entrance of the orifice section was found uniform but not with increasing the axial distance and at the nozzle exit as shown in Figure 2.5(a). Free vortex flow behavior, of the tangential velocity, was found to be present in the swirl chamber but not at the nozzle exit. It is seen that the liquid swirl velocity profile is a free vortex type (constant/r) in the swirl chamber but changes to a solid vortex type profile (constant*r) in the exit orifice section.

41 8 Predictions Experiments (Ma 2001) 6

4

2

Tangential Velocity (m/s) Velocity Tangential 0 0 5 10 15 20 25 30 35 Radial position r (mm)

2.6(a)

2 Predictions 1.5 Experiments (Ma 2001) 1

0.5

0 0 5 10 15 20 25 30 35 -0.5

Axial Velocity (m/s) Velocity Axial -1

-1.5

-2 Radial Position r (mm)

2.6(b)

Figure 2.6: a) Axial velocity profile at swirl chamber exit.

b) Tangential velocity profile in swirl chamber exit.

42 In some cases the liquid swirl velocity variation at atomizer exit may be a combination of free and solid vortex with solid vortex type variation over a majority of sheet thickness and free vortex type variation confined to a smaller region near the orifice wall. Ma (2001) has used

Particle-Image-Velocimetry (PIV) technique to measure axial and swirl velocity variation at different axial locations in the swirl chamber of the large-scale prototype nozzle. For Case #1 (10

GPM) listed in Table 1, a comparison of experimental data and the CFD predictions of the swirl and axial velocity profiles at the exit of the swirl chamber are shown in Figure 2.6.The effect of liquid viscosity and air pressure on discharge coefficient, film thickness and spray cone of a simplex pressure-swirl nozzle is examined. This is helpful in understanding how the atomization quality can change with different fluids (Lefebvre 1989, Chung and Presser 2001 and Rizk and

Lefebvre 1986). The volume flow rate is 15 GPM (case1) and is kept constant for this part.

Liquid viscosity changed from 0.001-0.01 kg/ (m.s). With a constant flow rate, the viscosity

ρlpQD variation can be shown in terms of Reynolds number ( Reinlet = ). We have plotted liquid Aplμ film thickness, spray cone angle and discharge coefficient versus Reynolds number at the inlet ports. Figure 2.7 shows the axial and tangential velocity profiles at the orifice exit at different liquid viscosities. The magnitude of the shear stress increases with increasing viscosity. The increase in shear stress tends to reduce both axial and tangential velocity components as seen in

Figure 2.7. The change in the maximum tangential velocity is more than the change in the axial velocity with increasing viscosity. The influence of liquid viscosity on the air core diameter in the swirl chamber and the orifice section is presented in Figure 2.8. With increasing liquid viscosity, the air core diameter decreases. This is to be expected as the decrease in tangential velocity causes the film thickness to in-crease. In fact, for the highest viscosity value considered

43 here, the air-core does not stretch through the swirl chamber. Figure 2.9(a) shows a sharp decline in spray cone angle with increase in liquid viscosity which is consistent with experimental work of Rizk and Lefebvre (1986) and Giffin and Muraszev (1953). Once again, this decrease in spray cone angle is due to the larger decrease in swirl velocity compared to the axial velocity. The liquid film becomes thicker with increasing liquid viscosity as shown in Figure 2.9(b). As the

Sauter Mean Diameter of the spray is directly related to liquid film thickness (Lefebvre 1989), increase in liquid viscosity will increase the mean droplet size of the resulting spray. The coefficient of discharge is not constant but in-creases with increasing liquid viscosity as shown in Figure 2.9(c). As the flow rate through the atomizer is kept constant, the decrease in discharge coefficient indicates higher inlet pressure.

In most practical applications, the atomizer is exposed to high pressure environment. However, most of the experiments and computational studies of the internal flow have been performed with atmospheric gas pressure. The present VOF model provides us an opportunity to evaluate the effect of increasing air density with higher air pressure. Figure 2.10 presents air water interface at air pressures of 1, 5 and 10 bars. It can be seen that at higher air pressure the air core diameter in the swirl chamber decreases whereas, in the orifice, a small decrease in the air core diameter is seen. The average axial and tangential velocities are lower in the swirl chamber compared to those in the exit orifice. Even with a ten-fold increase in air density, the liquid inertia force in the orifice is very large compared to air friction. Hence the effect of increase in air density on film thickness in the orifice is small. Film thickness at different water volume fractions for air pressures of 1 and 5 bar (15 GPM) is presented in Table 2.5. The maximum film thickness relative errors for air pressures of 1 and 5 bars are 8.34 % and 8.73% at water volume fraction of 0.8.

44

8 .001 kg/(ms) 7 .006 kg/(ms) 6 .01 kg/(ms) 5 4 3 2 1 Tangential Velocity (m/s) 0 0246810 Radial Postion, r (mm)

2.7(a)

8 .001 kg/(ms) 6 .006 kg/(ms) .01 kg/(ms) 4

2

0

Axial Velocity (m/s) 0246810 -2

-4 Radial Position , r (mm)

2.7(b)

Figure 2.7: Velocity profiles at different viscosities a) Tangential velocity profile at orifice exit b) Axial velocity profile at orifice exit

45

10 .001 kg/(m.s) 0.004 kg/(m.s) 8 0.006 kg/(m.s) 0.008 kg/(m.s) 6 0.01 kg/(m.s)

4

2 Radial Position (m) 0 0 20 40 60 80 100 120 Axial Position (m)

2.8(a)

12 .001 kg/(m.s) 10 0.004 kg/(m.s) 0.006 kg/(m.s) 0.008 kg/(m.s) 8 0.01 kg/(m.s) 6

4

2 Radial Position (m)

0 110 115 120 125 130 Axial Position (m)

2.8(b)

Figure 2.8: Effect of liquid viscosity on air core diameter

a) in the atomizer b) in the orifice section

46

95 90 85 80 75 70 Spray Cone Angle 65 60 0 10000 20000 30000 40000 50000 Re inlet

2.9(a)

4

3.5

3

2.5 Film thickness, t ( mm)

2 0 10000 20000 30000 40000 50000 Re inlet

2.9(b)

47 0.34

0.32

0.3

0.28

0.26

0.24

0.22 Discharge Coeffcient ( Cd ) 0.2 0 10000 20000 30000 40000 50000 Re inlet

2.9(c)

Figure 2.9 a) Spray cone angle b) Film thickness c) Discharge coefficient.

8.5

7.5 Pair = 1 bar Pair = 5 bars

6.5 Pair =10 bars

5.5 Radial position, r (mm) 4.5 0 20 40 60 80 100 120 Axial Distance, x (mm)

Figure 2.10: Effect of air pressure on the air core diameter

48 Table 2.5: Film thickness at different water volume fractions for air pressures of 1 and 5 bar (15

GPM).

Relative error % Air pressure Water volume fraction, Film thickness, h ( based on film (bar) α (mm) thickness at α = 0.5)

0.3 2.28 5.90 0.4 2.20 2.46 0.5 2.15 0.00 1 0.6 2.09 -2.86 0.7 2.03 -5.73 0.8 1.97 -8.34 0.3 2.37 5.16 0.4 2.30 2.38 0.5 2.25 0.00 5 0.6 2.19 -2.78 0.7 2.13 -5.56 0.8 2.05 -8.73

2.5 Summary and Conclusions

Simulation of turbulent two-phase flow in simplex atomizers is carried out using the Volume of

Fluid (VOF) method. The method is validated by comparing detailed velocity variations, pressure drop, film thickness, and spray angle with available experimental measurements of flow in large-scale atomizers. Effect of liquid viscosity and air pressure on the performance of the atomizer is investigated. It is found that the liquid swirl velocity profile is a free vortex type in the swirl chamber and changes to a solid vortex type profile in the exit orifice section. In some cases the liquid swirl velocity variation at atomizer exit may be a combination of free and solid vortex with solid vortex type variation over a majority of sheet thickness and free vortex type variation confined to a smaller region near the orifice wall. Increase in liquid viscosity leads to

49 lower axial and tangential velocities. The diameter of the air core decreases with increase in liquid viscosity resulting in higher liquid film thickness in the exit orifice. Also, the spray cone angle decreases and discharge coefficient increases with higher liquid viscosity. The relationship between the internal flow characteristics and discharge parameters (film thickness, spray cone angle and discharge coefficient) confirms that the internal flow structure plays a very important role in determining the atomizer performance. The effect of air pressure on flow inside the atomizer is investigated. Results show that even with a ten-fold increase in air pressure, the change in film thickness at the atomizer exit is very small.

50

PART II

LINEAR INSTABILITY OF ANNULAR LIQUID SHEETS

51 CHAPTER 3

EFFECT OF LIQUID SWIRL VELOCITY PROFILE ON THE

INSTABILITY OF A SWIRLING ANNULAR LIQUID SHEET

3.1 Introduction

The breakup of an annular liquid sheet from an atomizer into a multitude of droplets is of significant fundamental and practical importance (Lefebvre 1989). In a variety of atomizers, including pressure-swirl (or simplex), airblast, pre-filming airblast, and air-assist, the annular liquid sheet emanating from the atomizer has both axial and swirl velocity components. It is well established that the growth of disturbances on the liquid sheet surface lead to sheet instability and breakup. Therefore, it is important to understand the stability of swirling annular liquid sheets. All of the earlier analyses of the stability of a swirling, annular liquid sheet have used a

free-vortex type ()ArF swirl velocity profile in the liquid sheet. However, recent detailed experimental measurements of Ma (2001) inside pressure-swirl atomizers and several computational studies of pressure-swirl atomizers (Yule and Chinn 2000, Nonnenmacher and

Piesche 2000, Ibrahim 2006 and Dumouchel et al. 1993) have shown that the liquid swirl

velocity profile is a free vortex type( ArF ) in the swirl chamber but changes to a solid vortex

type profile ()ArS . in the exit orifice section. In some cases the liquid swirl velocity variation at atomizer exit may be a combination of free and solid vortex with solid vortex type variation over a majority of sheet thickness and free vortex type variation confined to a smaller region near the orifice wall. Therefore it is important to understand the effect of swirl velocity profile on the stability of swirling liquid sheet by considering both solid vortex type and free vortex type

52 profile which is the motivation of the present work. Also, a study by Dumouchel et al. (1993) on the change in the flow field with a change in geometry for a simplex atomizer shows that the velocity profiles at the orifice exit change considerably with a change in atomizer geometry. Our recent work (Sakman et al. 2000, Xue et al. 2004 and Liao et al. 1999) underscores the effect of atomizer geometry on the internal flow field of pressure swirl atomizers. Furthermore, recent modeling and experimental validation of spray from high pressure swirl atomizers by Gavaises and Arcoumanis (2001) show that accurate estimation of the nozzle flow exit conditions play a dominant role in the prediction of sprays characteristics. As such the study of velocity profile on the stability of liquid sheet has practical implications in improving predictions of spray characteristics.

Stability of liquid sheet and jet has been the subject of research since Squire’s (1953) study over 50 years ago. The geometries that have perhaps received the most attention are planar sheet and cylindrical liquid jet, both with axial velocity distribution taken as uniform

(Squire 1953, Hagerty and Shea 1955, Dombrowski and Hooper 1962, Dombrowski and Johns

1963, Lin et al. 1990, Li and Tankin 1991 and Ibrahim 1995). A recent monograph by Lin (2003) provides an extensive review of the instability of liquid sheets and jets. A review of the role of velocity profile in the process of jet breakup by McCarthy and Molloy (1974) indicates that a uniform profile is less unstable than a non-uniform one. Debler and Yu (1988) have conducted an experimental and theoretical investigation of the effect of the laminar parabolic velocity profile on jet instability. They concluded that the instability is reduced by non-uniformity in the velocity profile, in contrast with earlier conclusion of McCarthy and Molloy (1974). Their results are in agreement with the theoretical work of Leib and Goldstein (1986), who considered the stability of jets with axial velocities that vary from uniform profile to parabolic profiles.

53 Ibrahim (1997 and 2000) has concluded that any non-uniformity in the initial velocity profile for a planar liquid sheet would lead to a reduced instability and his results are in agreement with the findings of Debler and Yu (1998) and Leib and Goldstien (1986).

The first study of annular swirling liquid sheet stability was reported by Ponstein (1959).

He derived the general dispersion relation for the growth of disturbances under the influence of a potential liquid swirl flow described by a free-vortex swirl velocity profile and a uniform axial mean velocity, while neglecting the effects of viscosity and the presence of the two gas phases.

Panchagnula et al. (1996) have studied the spatial instability of an annular, swirling, inviscid liquid sheet moving with uniform axial liquid velocity and a mean liquid swirl velocity described by free vortex profile. They concluded that liquid swirl reduces the wave number of the axial disturbance mode having highest growth rate and reduces growth rates as well. With increasing the liquid swirl Weber number beyond the stabilizing region, the range of unstable axial and circumferential modes and growth rates for non-zero axial liquid Weber number are increased.

Liao et al. (1999) have studied the temporal instability of an annular, swirling, inviscid liquid sheet. They have considered uniform axial liquid velocity and a mean liquid swirl velocity described by a free vortex type profile. They concluded that with increasing the liquid swirl

Weber number both the growth rate and the axial wave number increase. The assumption of free vortex profile (Liao et al. 1999 and Panchagnula et al. 1996) was primarily motivated by earlier experimental and theoretical studies where the swirl velocity profile in the swirl chamber was shown to resemble a free vortex based on the work of Nieuwkamp (1986).

As mentioned before, recent experimental and computational studies (Ma 2001, Yule and

Chinn 2000, Nonnenmacher and Piesche 2000, Ibrahim 2006 and Dumouchel et al. 1993) have shown that the velocity profile changes considerably as liquid enter the exit orifice section from

54 the swirl chamber in a simplex atomizer. These studies show that the liquid swirl velocity profile is a free vortex type in the swirl chamber and changes to solid vortex profile in the exit orifice section. In some cases the liquid swirl velocity variation may be a combination of free and solid vortex with solid vortex type variation over a majority of sheet thickness.

The present study is aimed at improving our understanding of the effect of liquid swirl velocity profile on the instability of an annular liquid sheet. Both free vortex type and solid vortex type swirl velocity profile are considered. A temporal linear stability analysis has been carried out to predict the instability of an annular swirling liquid sheet that is subjected to three- dimensional disturbances and axially moving inner and outer gas streams. The resulting dispersion relation is used to predict the maximum unstable growth rate and wavelength. The growth rate and wave length for the most unstable disturbance are important as higher growth rate leads to a shorter breakup length and a smaller wavelength results in smaller droplets.

3.2 Linear Stability Analysis

A swirling annular liquid sheet subject to concurrent inner and outer gas streams is considered.

The geometrical and flow conditions are shown in Figure. 3.1. Both the liquid and gas phases are assumed to be inviscid and incompressible. The inviscid assumption is based on Shen and Li’s work (1996), which found that viscosity has only a small influence on the growth rates of disturbances at high Weber number.

3.2.1. Solid Vortex Swirl Profile

Basic flow velocities for the liquid and inner and outer gas are assumed to be (,Ul 0, Ar)S ,

(,Ui 0, 0) and (,Uo 0, 0), respectively. A temporal linear instability analysis is performed for the annular liquid sheet under three-dimensional disturbances with the normal mode method.

55

Figure 3.1: Schematic of annular liquid sheet

The governing equations in a cylindrical coordinate system are:

V ∂ V 1 ∂ WU∂ ++ + =0 (1) r ∂ r r ∂ θ ∂ x

56 ∂ U ∂ U W ∂ U ∂ U 1 ∂ p ++V +=−U (2) ∂ t ∂ r r ∂θ ∂ρ x ∂ x

∂ V ∂ V W ∂ V ∂ V W 2 1 ∂ p ++V +−=−U (3) ∂ t ∂ r r ∂θ ∂ρ x r ∂ r

∂ W ∂ W W ∂ W ∂ W VW 1 ∂ p ++V ++=−U (4) ∂ t ∂ r r ∂θ ∂ρ x r r∂θ

To obtain linearized disturbed equations, let

UUuVvWWwpPp=+, =, =+ , =+′ (5)

Where the over bar represents the assumed mean flow quantities and prime indicates disturbance.

The disturbances are assumed to have the forms of:

ikx(+ nθ −ω t) (uvwp , , , ′ )= ( urˆˆˆˆ ( ), vr( ), wrpre( ), ( )) (6)

Where ∧ indicates the disturbance amplitude which is a function of r only. For the temporal instability analysis, the wave number k and n are real while frequency ω is complex. The imaginary part of ω reflects the growth rate of the disturbance.

The displacement disturbances at the inner and outer interfaces are:

ikx(+− nθ ω t) ηθoo(,xt ,)= ηˆ e (7)

[(ikx+− nθ ω t) +Φ i ] ηθii(xt , , )= ηˆ e (8)

Where Φ represents the phase difference between the two surface waves at the gas-liquid interfaces. The case of Φ = 0 implies that the two interfaces of the liquid sheet are displaced in phase, which is often termed as sinuous (or anti-symmetric) mode of disturbances. Similarly, the case of Φ = π indicates that the two liquid surfaces are mirror image of each other, corresponding to a varicose (or symmetric) mode. For plane liquid sheets the most unstable

57 disturbances are exactly symmetric and anti-symmetric (Squire 1953). As explained in Shen and

Li (1996), for an annular sheet, this need not be assumed a priori. In this case we have considered a phase difference Φ , which can be evaluated as part of the solution to determine the most unstable disturbance for inner and outer interface. The two solutions obtained with Φ close to zero and Φ close to π are referred as para-sinuous and para-varicose respectively (Chen and

Lin 2002). In other words, the independent interfacial modes can not be exactly in phase or out of phase in annular liquid sheet, i.e., Φ = 0 or π is only an approximation for very thin annular sheets. This is the reason they are called para-sinuous and para-varicose rather than simply sinuous or varicose modes. Substituting Equation (5) into the continuity Equation (1) and the momentum conservation equations (2-4) and neglecting second order terms, we get the linearized disturbed equations for the liquid flow as:

vv∂ 1∂∂ wu ++ + =0 (9) rrr∂∂θ∂ x

∂ uuu∂∂ 1∂ pl′ ++=−AUSl (10) ∂ txx∂θ ∂ ρl ∂

∂ vvu∂∂ 1∂ pl′ ++−=−AUSl2 Aw S (11) ∂ tx∂θ ∂ ρl ∂ r

∂ www∂∂ 1∂ pl′ +++=−AUSl2 Av S (12) ∂ txr∂θ ∂ ρl ∂θ

The linearized disturbed equations for the inner and outer gas flow can be written in vector form as:

r ∇•u j =0 (13)

rr ∂ uujj∂ 1 +=−∇=Upjiojj′ , (14) ∂∂ρ tx j

58 ⎛⎞u ⎜⎟ Where uvr = . j ⎜⎟ ⎜⎟ ⎝⎠w

The kinematic boundary conditions are:

For the gas streams,

D η ∂η ∂η vUrR==+ii i at = (15) Dt ∂∂ tia x

D η ∂η ∂η vUrR==+oo o at = (16) Dt ∂∂ tob x

For the liquid phase:

D η ∂η ∂η ∂η vAUrR==+ii i + i at = (17) Dt ∂∂θ∂ tSl x a

D η ∂η ∂η ∂η vAUrR==+oo o + o at = (18) Dt ∂∂θ∂ tSl x b

The dynamic boundary conditions are:

η∂η∂η1 2 2 ′′ iii2 ppli−=σ ()22 + 2 + 2 −ρη laSi RA (19) RRaa∂θ ∂ x

η∂η∂η1 22 ′′ ooo2 pplo−=−+σ ()22 2 + 2 −ρη lbSo RA (20) RRbb∂θ ∂ x

The two equations (Eqns. 19-20) express the balance of the normal component of the stress tensor at each interface.

After solving the linearized equations for liquid and gas we can obtain the liquid pressure, inner gas pressure and outer gas pressure. Substituting these in the dynamic boundary conditions (Eqns. 19-20), the problem then reduces to a system of two equations:

59 222 2 22 2 σρ(1−−nkRaii ) dZ11 (2AnZSS (4 A−−+−+ Z )( d761 / R b dkZ ) F ( A Sb / R 3 + ) ηρρˆo iΦ labR RRRk lab e = 24 2 (21) ηˆi dkZ51 (4) Z− AS

24 2 ηˆo iΦ dkZ91 (4) Z− AS e = 222 2 (22) ηˆi 22 2 σρ(1−−nkRboo ) dZ12 (2AnZSS (4 A−−+−− Z )( d721 / R a dkZ ) F ( A Sa / R 3 + ) ρρlbaR RRRk lab

Where Z = (−+ω AnSl + kU ) , Zi = (−+ω kUi ) and Zo = (−ω + kUo )

By eliminating the ratio of the initial amplitudes of the perturbation and Φ (phase difference

between inner and outer liquid wave interfaces) from equations (21-22) we end up with the final

dispersion equation. The dispersion equation is non-dimensionalized by introducing the

following dimensionless parameters:

22 2 2 ρρllUR b ii UR a ρ oo UR b ρρρ l WR b i o Weli==, We , We o =, We sio === , g , g , σ σσσρρll

UWeii11UWe ooARS bs WeRRa ω b ==, , =,=hkkR , ==b ,ω UWeghUWegUlli l lo l We lRUbl

The final dispersion equation is an eighth order equation of the form below:

8626275252322642 cZ18−−+++ cZZ 28io cZZ 38 cZ 17 cZZ 27 iooii cZZ 37 + cZZZ 47 ++ cZ 16 cZZ 26 42 222 5 32 32 22 4 22 ++cZZcZZZcZcZZcZZcZZZcZcZZ36ooiiooi 46 ++++ 15 25 35 45 ++ 14 24 i 22 22 3 2 2 2 2 2 ++cZZ34 ooi cZZ44 ++++++ cZ 13 cZZcZZcZ 23 i 33 o 12 cZ 22 i cZ12o ++= cZ 11 c 0 0

(23)

Where Wesi We We o Z=−(),()()ω + n + k Zio = k −ωω and Z = k − Well We We l

The coefficients in Equation (23) depend on wave number k and n, flow conditions, fluid properties, and geometric parameters and are given in the Appendix A.

60 3.2.2 Free Vortex Swirl Profile

Basic flow velocities for the liquid and inner and outer gas are assumed to

A be (U , 0, F ), (,U 0, 0) and (,U 0, 0), respectively. By following a procedure similar to that l r i o outlined above for solid vortex type swirl profile, the linearized disturbed equations for the liquid flow can be obtained as:

v ∂ v 1 ∂ wu∂ ++ +=0 (24) r ∂ r r ∂ θ ∂ x

∂ uuuAF ∂∂ 1∂ pl′ ++=−2 Ul (25) ∂ tr∂θ ∂ x ρl ∂ x

∂ vvuAAwFF∂∂ 2 1∂ pl′ ++−=−22Ul (26) ∂ tr∂θ ∂ x r ρl ∂ r

∂ wwwAF ∂∂ 1∂ pl′ ++=−2 Ul (27) ∂ tr∂θ ∂ x ρl r ∂θ

The linearized disturbed equations for the inner and outer gas flow can be written in vector form as:

r ∇•u j =0 (28)

rr ∂ uujj∂ 1 +=−∇=Upjiojj′ , (29) ∂∂ρ tx j

⎛⎞u ⎜⎟ Where uvr = . j ⎜⎟ ⎜⎟ ⎝⎠w

The kinematic boundary conditions are:

For the gas streams,

D η ∂ η ∂ η v ==+iiU i at rR = (30) Dt ∂ t i ∂ x a

61 D η ∂ η ∂ η v ==+ooU o at rR = (31) Dt ∂ t o ∂ x b

For the liquid phase,

D η ∂η A ∂η ∂η vUrR==+iiF i + i at = (32) Dt ∂∂θ∂ t r2 la x

D η ∂η A ∂η ∂η vUrR==+ooF o + o at = (33) Dt ∂∂θ∂ t r2 lb x

The dynamic boundary conditions are:

η 1 ∂η22 ∂2 η ρA η ′′ iiilFi ppli−=σ ()22 + 2 + 2 − 3 (34) RRaa∂θ ∂ x R a

η 1 ∂η22 ∂η ρA 2 η ′′ ooolFo pplo−=−+σ ()22 2 + 2 − 3 (35) RRbb∂θ ∂ x R b

By a procedure similar to one described earlier, the final dispersion equation can be obtained.

The final dispersion equation is considerably less complicated compared to Eqn (23) and can be written as:

432 aaaaa43210 ωωωω++++= 0 (36)

All the details about the coefficients ai are available in Liao et al. (1999) and in Panchagnula et al. (1996) and not repeated here for sake of brevity.

The dispersion equation (23) or (36) can be simply stated as

fwkggWeWeWeWehn(,,io , , l , s , i , o ,,)= 0 (37)

The dispersion equations are solved using MathematicaTM. The final dispersion equations do not have closed form solutions and are hence solved numerically. The secant method is used which requires two initial guess values. Solutions are considered convergent when values of left hand side of Eq. (37) are smaller than 10-6. For each pair of ( k , n) and given dimensionless

62 parameters, we look for the root with the maximum imaginary part, which represents the growth rate of the disturbance. The wave number that corresponds to the maximum growth rate is called the most unstable wave number.

3.3 Results and Discussions

To cover operating conditions for different types of practical atomizers, including pressure swirl, airblast, air assist, pre-filming airblast, a wide range of axial liquid Weber, liquid swirl Weber number and axial outer air Weber number must be considered. Even for a single atomizer the range of relevant Weber numbers may be large. For example, consider a pressure swirl atomizer

(Delavan Corporation catalogs) of orifice diameter of 0.81 mm at flow number in SI unit of 25 x

10-8 and liquid pressure drop from 3 bars to 35 bars. For this atomizer, the range of axial liquid

Weber Number changes from around 1000 to 10000 and the liquid swirl Weber number changes

300 to 4000. Also, in some applications, due to low velocities, Weber numbers encountered may be lower (Panchagnula et al. 1996). Here all liquid Weber numbers are defined using liquid

ρ UR2 density (for example, axial liquidWe = ll b). Some of the earlier studies (Lefebvre 1989) l σ have defined all Weber numbers, for liquid and for gas, based on gas density. As such the values of Weber numbers reported in these studies are much lower than those appearing here.

3.3.1 Effect of Liquid Axial Velocity

The fourth order dispersion equation for a free vortex type swirl velocity profile (Eqn. 23) is observed to reduce to the dispersion equation of Ponstein (1959) when the densities of the gas medium inside and outside the annular sheet are set equal to zero. To validate the eighth order dispersion equation for a solid vortex type swirl velocity profile (Eqn. 23), results are first

63 obtained for a limiting case of no liquid swirl. Without liquid swirl, Equations (23) and (36) both must give the same results. The results of para-sinuous and para-varicose modes are shown in

Fig. 2 and these match exactly with the results available in literature (Liao et al. 1999). Figure

3.2 shows the effect of axial liquid velocity with a finite range of wave numbers with positive growth rate. As the axial liquid velocity increases, both the maximum growth rate and the range of unstable axial wave numbers increase. The corresponding most unstable wave number or frequency shifts to a higher value. One can notice in Fig. 2 that both characteristics of the dominant waves (growth rate and most unstable wave number) are higher for para-sinuous mode

(where the calculated Φ from Eq (21) is close to zero) than for para-varicose mode (where the calculated Φ from Eq (21) is close to π). With increasing the circumferential wave number (n), both the growth rate and the axial wave number of para-varicose mode decrease. The para- varicose mode is no longer unstable for Wel = 5000 and n = 2. The importance of helical modes becomes comparable with the axisymmetric mode. This behavior has been validated by previous studies (Liao et al. 1999 and Panchagnula et al. 1996).

3.3.2 Effect of Liquid Swirl Velocity

Figure 3.3 presents the effect of liquid swirl at axial liquid Weber number of 5000 for both para- sinuous and para-varicose modes. As the figure shows, liquid swirl has a stabilizing effect at low values of liquid swirl Weber numbers of zero, 5, 6 and 10. However, liquid swirl at higher liquid swirl Weber numbers destabilizes the annular sheet. It can be concluded that liquid swirl imposed on an annular liquid sheet has a dual effect on its stability and this behavior is consistent with earlier study (Panchagnula et al. 1996). With increasing the liquid swirl Weber numbers from zero to 6 the growth rate and the axial wave number of the para-varicose

64 disturbance decrease. At liquid swirl Weber number of 10 and higher the para-sinuous mode becomes more unstable with higher growth rates, whereas the para-varicose mode is no-longer unstable. It is seen that the para-sinuous mode is the dominant mode in the presence of the liquid swirl. A similar trend of dominance of the para-sinuous mode was also obtained for a viscous, annular, non-swirling liquid sheet by Shen and Li (1996) and Chen and Lin (2002). This is consistent with experimental observations of Kendall (1986) of non-swirling, annular liquid jet.

Figure 3.4 shows the effect of axial liquid Weber number on the growth rate and the axial wave number at no-liquid swirl and at different liquid swirl Weber numbers with free vortex type and with solid vortex type swirl velocity profile. At axial liquid Weber number of 500 the growth rate and axial wave number decrease with increasing liquid swirl Weber number from 1 to around 10 as shown in Figure 3.4(a). However, both the growth rate and axial wave number increase with increasing liquid swirl Weber number more than 10. At axial liquid Weber number of 10000 the growth rate decreases with increasing liquid swirl Weber number from 1 to around

20 as presented in Figure 3.4(b). However, both the growth rate and axial wave number increase with increasing liquid swirl Weber number beyond 20.

65

3.2(a)

3.2(b)

Figure 3.2: Growth rate versus wave number for n = 0, 1, 2 mode at Wes = 0, Wei = 0, Weo = 0, gi = go = 0.00123, h = 0.667 :( a) Para-sinuous mode and (b) Para-varicose mode.

66

Figure 3.3: Growth rate versus wave number (based on free vortex dispersion equation) for n =

0 at Wel = 5000, Wei = 0, Weo = 0, gi = go = 0.00123 and h = 0.667.

67

3.4(a)

3.4(b)

Figure 3.4: Growth rate of para-sinuous mode versus wave for n = 0, Wei = 0, Weo = 0, gi = go

= 0.00123 and h = 0.667: (a) Wel = 500 and (b) Wel = 10000.

68 It can be seen that at low liquid swirl Weber number the growth rate of the solid vortex type swirl velocity profile is higher than that of the free vortex type. However, with increasing liquid swirl Weber number the growth rate of solid vortex type is lower than that of free vortex type for both low and high axial liquid Weber numbers. The liquid sheet instability can be explained in terms of forces acting on the interface viz. surface tension, aerodynamic forces, and centrifugal forces. The growth of any disturbance will be determined by the competition between these forces. With small values of swirl Weber numbers, i.e. at low values of liquid swirl velocity, the centrifugal force is low and tends to stabilize the effect of aerodynamic forces.

Hence at low Wes, liquid swirl has stabilizing effect on the liquid sheet. In contrast, at high liquid swirl Weber number, centrifugal force becomes the dominant force and causes the growth of the disturbances. Therefore the growth rate increases at high Wes. Not surprisingly, as the aerodynamic forces increase with higher axial Weber number, the range of stabilizing liquid

swirl Weber number increases as seen in Figure 3.4(b). Calculation of values of ηioη for the cases shown in Figure 3.4 indicates that the disturbance magnitude at inner interface is higher compared to that at the outer interface. For the same average liquid swirl velocity, the free vortex profile leads to higher value of swirl velocity at the inner interface compared to that with a solid vortex type profile. Hence, for a given liquid swirl Weber number, the free vortex type velocity variation gives rise to a stronger stabilizing effect compared to a solid vortex swirl velocity profile, at low liquid swirl Weber number. At high Wes, higher liquid swirl velocity at the inner interface of free vortex profile compared to that with a solid vortex profile, gives rise to higher centrifugal force, which is the dominating destabilizing force at high Wes. Hence at high Wes, the growth rate of unstable disturbances is higher for free vortex type variation than solid vortex type variation. These differences in the growth rate and in the most unstable disturbance wave

69 number indicate that the liquid swirl velocity profile will significantly affect the breakup length and resulting droplet diameters. At Axial liquid Weber number of 500 the maximum growth rate difference between free vortex and solid vortex type profiles are significantly higher than those at higher axial liquid Weber number of 10000.

Figure 3.5 shows the effect of liquid swirl Weber number on the growth rate and axial wave number at Wel = 5000 for no liquid swirl and for liquid swirl Weber numbers from 200 to

1000. The growth rate and the axial wave number increase with increasing the liquid swirl

Weber number from 200 to 1000. The difference, in calculated growth rate with the two liquid swirl velocity profiles, increases with increasing liquid swirl Weber number.

Two regions can be identified: the first region, where the maximum growth rate and the wave number decrease with increasing liquid swirl Weber number which is consistent with the work of

Panchagnula el al. (1996). In this region, the maximum growth rate and axial wave number of solid vortex type is higher than those of free vortex type. The second region, where the maximum growth rate and the axial wave number increase with increasing liquid swirl Weber number and the maximum growth rate and axial wave number of solid vortex type is smaller than those of free vortex type. As described earlier, these two regions are due to difference in the dominant force in the competition between the centrifugal and the aerodynamic forces.

3.3.3 Effect of Density Ratio

Gas density effect with no liquid swirl is presented in Figure 3.6(a). This behavior reconfirms the well-known destabilizing effect of the aerodynamic forces (Jeandel and Dumouchel 1999 and

Cousin and Dumouchel 1996). An increase of these forces leads to an increase of both dominant wave characteristics i.e., it induces a faster disintegration process and the production of smaller

70

Figure 3.5: Growth rate versus wave number for n = 0, gi = go = 0.00123, Wei = 0, Weo = 0, h =

0.667 and Wel = 5000.

71

3.6(a)

3.6(b)

Figure 3.6: Growth rate versus wave number for n = 0, h = 0.667, Wel = 500, Wei and Weo = 0:

(a) Wes = 0 and (b) Wes = 100.

72 drops. This conclusion is consistent with Dombrowski and Hooper’s (1962) experimental study on the effect of ambient density on drop size formation. Gas density effect with swirl is presented in Figure 3.6(b) for free vortex type and solid vortex type swirl velocity profile. In the presence of the liquid swirl, both of the growth rates and the most unstable axial wave numbers are higher than those at no liquid swirl. In the case of swirl the growth rates of solid vortex type is lower than those of free vortex type. The difference between the maximum growth rate for free vortex type profile and that for the solid vortex type profile decreases with increasing the density ratio. This is to be expected as the relative important of liquid swirl is reduced compared to aerodynamic forces acting on the liquid air interface. Therefore the difference in the growth rate for the two swirl velocity profiles decreases. It can be noted that the solid vortex profile reduces the instability of an annular liquid sheet significantly at ambient conditions and to a lesser extent at high gas densities (pressures).

The effect of liquid swirl at high-density ratio of 0.01 is presented in Figure 3.7(a) and

3.7(b) at two different axial liquid Weber numbers. Figure 3.7(a) shows the effect of liquid swirl at axial Weber number of 500. The maximum growth rate for the solid vortex profile is lower than that for the free vortex profile. Moreover, the difference between the maximum growth rates for the two types of swirl velocity profiles increases with increasing liquid swirl Weber number.

The effect of liquid swirl at higher axial Weber number of 5000 is shown in Figure 3.7(b). The growth rates of the solid vortex type is once again lower than those of the free vortex type but the difference between the growth rates of free vortex type and solid vortex type is smaller than

Figure 3.7(a). The difference in the maximum growth rate with the two types of swirl velocity profiles increases with increasing liquid swirl strength. At higher density ratios the growth rate increases with increasing axial liquid Weber number as shown in Figure 3.7.

73

3.7(a)

3.7(b)

Figure 3.7: Growth rate versus wave number at different liquid swirl Weber numbers for n = 0, h = 0.667, Wei = 0, Weo = 0 and gi = go = 0.01: (a) Wel =500 and (b) Wel = 5000.

74 A similar effect of gas to liquid density ratio was found by Kang and Lin (1989) for swirling liquid jets. It can be seen that larger gas to liquid density ratio promotes annular liquid sheet instability.

3.3.4 Effect of Radius of Curvature Ratio

The effect of surface curvature with no liquid swirl at axial liquid Weber number of 500 and

10000 is presented in Figure 3.8. As the radius ratio increases the maximum growth rate increases for the no liquid swirl case. Figure 3.8 shows the effect of the surface curvature in the presence of the liquid swirl at axial liquid Weber number of 500 and 10000 for different liquid swirl Weber numbers. As the radius ratio increases the maximum growth rate decreases, which is consistent with the work of Liao et al. (1999). It can be seen that at high axial liquid Weber number of 10000 and liquid swirl Weber number of 5000 the effect of surface curvature ratio is small. However, at lower axial liquid Weber number of 500 and liquid swirl Weber number of

100 the effect of the curvature ratio is significant. The difference between the maximum growth rate of free vortex type and solid vortex type is 20 % at axial liquid Weber number of 500 and liquid swirl Weber number of 100 and decreases to about 5% at axial liquid Weber number of

10000 and liquid swirl Weber number of 5000. When liquid swirl is absent, the maximum growth rate increases with increasing the radius of curvature ratio, whereas in the presence of the liquid swirl, the growth rate decreases with increasing the radius of curvature ratio. Furthermore, the effect of the radius of curvature is small at high axial liquid Weber number of 10000 and liquid swirl Weber number of 5000.

75

3.8(a)

3.8(b)

Figure 3.8: Growth rate versus wave number for different radius ratios at n = 0, Wei = 0, Weo =

0 and gi =go = 0.00123: (a) Wel =500 and Wes =100; (b) Wel =10000 and Wes =5000.

76

3.9(a)

3.9(b)

Figure 3.9: Growth rate versus wave number for radius ratios at n = 0, h = 0.667, Wei = 0, Weo =

0 and gi =go = 0.00123 (a) Ul = 5 m/s and Wl = 2 m/s, (b) = 10 m/s and Wl = 5 m/s.

77 3.3.5 Effect of Surface Tension

Figure 3.9(a) presents the influence of the surface tension on the dispersion diagram of an annular liquid sheet at axial liquid velocity of 5 m/s and liquid swirl velocity of 2 m/s. It can be seen that when the surface tension is increased, both dominant wave characteristics as well as the cut-off wave number decrease. This implies that the disintegration process is getting slower as the surface tension increases and that the diameter of the drops produced would increase. Figure

3.9(b) shows the effect of the surface tension at higher axial liquid velocity of 10 m/s and liquid swirl velocity of 5 m/s. The stabilizing effect of the surface tension forces is responsible for this behavior. The growth rate and axial wave number at Ul = 10 m/s and Wl = 5 m/s are about twice the values at Ul = 5 m/s and Wl = 2 m/s. The difference between the maximum growth rate of free vortex type and solid vortex type is 20 % at axial liquid velocity of 5 m/s and liquid swirl velocity of 2 m/s and decreases to about 7% at axial liquid velocity of 10 m/s and liquid swirl velocity of 5 m/s. This behavior shows the stabilizing effect of the surface tension forces which is consistent with Shen and Li (1996) and Jeandel and Dumouchel (1999).

3.3.6 Effect of Outer Axial Air Weber Number

Figure 3.10(a) presents the effect of outer air axial Weber number at no-liquid swirl for para- sinuous mode. The growth rate decreases with increasing the axial air Weber number up to 10, and with little change from Weo of 10 to about 60. After Weo of about 60 both the growth rate and axial wave number increase with increasing Weo.

78

3.10(a)

3.10(b)

Figure 3.10: Growth rate versus wave number at n = 0, h = 0.667, Wei = 0, Wel = 10000 and gi

=go = 0.00123: (a) without liquid swirl (para-sinuous mode) and (b) Wes = 500.

79 Figure 3.10(b) shows the effect of outer air axial Weber number at liquid swirl Weber number of

500 for para-sinuous mode. As mentioned earlier, with increasing swirl Weber number, the liquid sheet is no longer unstable to para- varicose mode. The growth rate and axial wave number decrease with increasing the outer air axial Weber number. However, after Weo of about

60 they increase with increasing outer air axial Weber number. This behavior underscores the importance of relative velocity between the liquid sheet and the air stream on the sheet instability. For a fixed liquid axial velocity, as the air velocity increases, initially there is a decrease in the relative velocity which results in lower growth rates As air velocity is increased further, the relative velocity increases, which leads to higher growth rates. The higher growth rates would lead to shorter breakup lengths which is consistent with experimental measurements of Carvalho and Heitor (1998). The growth rate and the axial wave number with liquid swirl are higher than those at no-liquid swirl. The maximum growth rate with a solid vortex type swirl profile is smaller than that with a free vortex type swirl profile. However, with increasing Weo beyond about 60, the growth rate difference between free vortex type and solid vortex type swirl profile decreases. The higher air velocity (higher relative velocity) would correspond to an air- assisted atomization whereas the lower value (lower relative velocity) would mimic the configuration in a pressure-swirl atomizer. The results presented here are consistent with the work of Hauke et al. (2001).

3.4 Summary and Conclusions

In the present study, an analysis has been carried out for the temporal instability of an annular swirling liquid that is subjected to three-dimensional disturbances and axially moving inner and outer gas streams. The effects of the liquid swirl velocity profile on the disturbance growth are

80 examined by considering both free vortex and solid vortex profiles for liquid swirl velocity. The results of this study are summarized below.

For both types of swirl velocity profiles:

I. An annular liquid sheet without liquid swirl is unstable for both para-sinuous and para-

varicose disturbances with para-sinuous mode being the dominant one.

II. Liquid swirl imposed on an annular liquid sheet has a dual effect on its stability where

swirl has a stabilizing effect at low values of liquid swirl Weber whereas it has a

destabilizing effect with increasing the liquid swirl Weber number on the para-sinuous

disturbances. With increasing liquid swirl Weber number, the para-varicose disturbances

become less unstable, and eventually at high liquid swirl the sheet is no longer unstable

to para-varicose disturbances.

III. The ambient gas medium always enhances the annular sheet instability at no liquid swirl.

In the presence of liquid swirl, both of the maximum growth rate and the most unstable

axial wave number increase compared to no-liquid swirl case.

IV. Without liquid swirl, the growth rates increase with increasing surface curvature ratio.

However, in the presence of liquid swirl, the growth rates decrease with increasing

surface curvature ratio. The effect of the surface curvature is significant for no liquid

swirl case and with liquid swirl it is significant only at low axial liquid Weber numbers.

V. The growth rates and the axial wave numbers decrease with increasing surface tension,

demonstrating the stabilizing effect of surface tension.

VI. The destabilizing effect of outer air velocity is governed by the relative velocity between

the air and the liquid sheet.

81 The effects of swirl velocity profile:

I. At low liquid swirl Weber number (in the region with stabilizing effect of swirl),

the maximum growth rate and axial wave number of the solid vortex type swirl

profile are higher than that of free vortex type profile at both low and high axial

Weber number. However, with increasing liquid swirl Weber number (in the region

of destabilizing effect of swirl), the maximum growth rate and the axial wave

number of free vortex profile are always higher than that of solid vortex profile.

II. Compared to the free vortex swirl profile, the solid vortex profile reduces the

instability of an annular liquid sheet significantly at ambient conditions and to a

lesser extent at high gas densities (pressures).

III. The difference between the maximum growth rate of free vortex type and solid

vortex type swirl velocity profile is significant at axial liquid Weber number of 500

and liquid swirl Weber number of 100 and decreases at higher axial and swirl liquid

Weber numbers.

82 CHAPTER 4

EFFECT OF LIQUID AND AIR SWIRL STRENGTH AND

RELATIVE ROTATIONAL DIRECTION ON THE INSTABILITY OF

AN ANNULAR LIQUID SHEET

4.1 Introduction

The instability and breakup of liquid sheet is encountered in liquid atomization process used in numerous applications including liquid fuel injection in combustion engines, spray drying of foods and detergents, and in manufacturing of pharmaceutical products. Growth of disturbances on the liquid-air interface leads to sheet instability and breakup, and governs the characteristics of the resulting spray. These characteristics play an important role in determining the subsequent heat/mass transport and phase change processes. For example, the liquid sheet instability and breakup in fuel atomizer in combustion engines determines the mean drop size in the fuel spray and has direct impact on combustion efficiency, pollutant emissions and combustion instability (Lefebvre

1989). Due to their good atomization characteristics and low liquid pressure requirements, airblast atomizers are being considered in many applications. In an airblast atomizer, kinetic energy of the high-speed swirling airstreams is used to breakup the liquid sheet. The liquid exits the airblast atomizer as an annular sheet and is subjected to swirling inner and outer air streams. The presence of the atomizing air leads to shorter breakup length (Carvalho and Heitor 1998) and enhances liquid air mixing [Lavergne et al. 1993] Significant work on the instability of liquid sheets and jets is available in the

83 literature. However, the effects of liquid and air swirl and the effects of air swirl direction with respect to liquid swirl on the sheet instability are not well understood and need to be studied. An annular swirling liquid sheet is a more general case where the results for other geometries can be recovered by considering limiting cases of the annular liquid sheet. For example, a plane liquid sheet can be obtained if the inner and the outer radii

→∞, with fixed sheet thickness (Li and Tankin 1991), hollow gas jet surrounded by infinite liquid can be obtained if the outer radius →∞(Li 1994), round liquid jet surrounded by infinite gas if the inner radius → 0 (Yang 1992) and axisymmetric annular jet if n = 0 can be recovered (Shen and Li 1996).

The stability of liquid jets and sheets has received much attention since the classical studies of Rayleigh (1878) and Squire (1953). Here we have summarized only the studies that pertain to annular liquid sheets. For authoritative reviews of liquid sheet and jet instability and breakup, readers are referred to a recent monograph by Lin (2003), reviews by Sirignano and Mehring (2000) and Lasheras and Hopfinger (2000). It is well established that the forces acting on a liquid gas interface including surface tension, pressure, inertia force, centrifugal force and viscous force, result in the growth of disturbances that lead to sheet or jet breakup. The presence of air and/or liquid swirl and finite radius of curvature add further complexities to liquid sheet instability. Ponstein

(1959) was the first to carry out an analysis of stability of an annular swirling liquid sheet. He derived the general dispersion relation for the growth of disturbances under the influence of a potential liquid swirl flow and a uniform axial mean velocity, while neglecting the effects of viscosity and the presence of the two gas phases. Ponstein showed that with liquid swirl non-axisymmetric modes are more unstable than the

84 axisymmetric mode. However, for a non-swirling jet, the axisymmetric mode is the dominant mode both for Newtonian (Chen et al. 2003) and for non-Newtonian liquids

(Alleborn et al. 1998). Two limiting cases of the configuration considered here are available in the literature, viz., the swirling annular liquid sheet without air swirl considered by Panchagnula et al. (1996) and the purely axially moving liquid sheet subjected to inner and outer air swirl considered by Liao et al. (2001). Panchagnula et al.

(1996) have shown that liquid swirl reduces the wave number and the growth rate of the most-unstable disturbance at low swirl Weber number. However, at higher swirl, increasing the liquid swirl Weber number increases the range of unstable axial and circumferential modes and increases their growth rates. Liao et al. (2001) have compared the effectiveness of the inner and the outer air swirl and showed that a combination of the inner and outer air swirl is more effective than a single air swirl in enhancing the instability of the liquid sheet and in improving atomization, whereas the inner air swirl is more effective than the outer air swirl. Ibrahim et al. (2005) have studied the effect of liquid swirl velocity profile on the temporal instability of annular swirling liquid sheet subject to inner and outer axially-moving gas flows of differing velocities. They considered the liquid swirl velocity profile as free vortex and as solid vortex. They have shown that at high liquid swirl Weber number, the maximum growth rate and the axial wave number of free vortex profile are always higher than that for a solid vortex profile.

Mehring and Sirignano (2001) have shown that liquid swirl can enhance wave growth of the unstable mode resulting in shorter breakup lengths using a non-linear analysis of a swirling, annular, axisymmetric liquid sheet in a void.

85 From the literature summarized in Table 4.1, it is seen that previous studies of annular liquid sheet take into account either the swirling motion of the liquid or that of air, but not both. However, in an airblast atomizer, all the three streams, i.e., the liquid, the inner air and the outer air, may have swirl velocity components, in addition to axial motion. Furthermore, with both liquid and air swirling, the effect of swirl direction should be considered.

We are aware of only one experimental study by Chin et al. (2001) that compares atomization with co- and counter-swirling air streams with respect to liquid swirl direction. They have investigated the effect of inner and outer air flow characteristics on high liquid pressure prefilming airblast atomization. For the flow conditions considered in their experiments, a combination of a co-rotating inner air stream and counter-rotating outer air stream with respect to the rotational direction of the enclosed liquid film yielded the finest sprays compared to other swirled configurations. However, the shortest breakup lengths were obtained with both the inner and the outer air stream co-rotating with the liquid. Our analysis confirms this finding for a certain range of Weber numbers and shows that the effect of liquid and air swirl is different in other range of Weber numbers.

86

Table 4.1: Annular liquid sheet stability literature summary

Liquid Inner and Outer Gas Dsiturbance Researcher(s) Swirl Viscous Mode Swirl Velocity / velocity Swirl Profile Inviscid Axial Velocity Velocity Axisymmetric axially moving Shen and Li and - Viscous No Swirl with different No Swirl (1996) axial velocities

Non- Alleborn et al. Axially Axisymmetric - Newton No swirl No Swirl (1998) moving ian Axisymmetric axially moving and Cao (2003) - Viscous No Swirl with different No Swirl axial velocities

Axisymmetric Jeandel and axially moving and Dumouchel - Viscous No Swirl with same No Swirl axial (1999) velocities

axially moving Liao et.al Nonaxisymmetric - Viscous No Swirl with different Swirling (2001) and axial velocities Axially Du and Li Nonaxisymmetric moving with - Viscous No Swirl Swirling (2005) and axial different velocities axially moving Liao et Nonaxisymmetric - Inviscid No Swirl with different Swirling al.(2000) and axial velocities Ponstein Nonaxisymmetric Free Inviscid Swirling Stagnant Stagnant (1959) and axial Vortex Panchagnula axially moving Nonaxisymmetric Free et.al (1996) Inviscid Swirling with different No Swirl and axial vortex velocities axially moving Liao et.al. Nonaxisymmetric Free Inviscid Swirling with different No Swirl (1999) and axial Vortex velocities Mehring and Non-linear Sirignano analysis with Uniform Inviscid Swirling Stagnant Stagnant (2001) axisymmetric Free and axially moving Ibrahim et al Nonaxisymmetric Solid Inviscid Swirling with different No Swirl (2005) and axial vortex velocities Nonaxisymmetric axially moving Free Present Study and axial Inviscid Swirling with different Swirling Vortex disturbances velocities

87

4.2 Mathematical Formulation

An annular swirling liquid sheet subjected to coaxial swirling air streams is considered as shown in Figure 4.1. In the stability model, both the liquid and the gas phases are assumed to be inviscid and incompressible. Jeandel and Dumouchel (1999) have introduced a dimensionless parameter for a viscous annular liquid sheet to evaluate the effect of viscosity on the sheet stability. They have shown that viscosity has negligible effect if this parameter is less than 10-4. For the range of flow conditions considered in this study, the parameter is of order 10-6 which justifies the inviscid liquid flow assumption. Mean velocity components in (,,x r θ )direction of the liquid, the inner and the outer air are assumed to be (Ul, 0, Al/r), (Ui, 0, Air) and (Uo, 0, Ao/r), in respectively.

A temporal linear instability analysis is performed for the annular liquid sheet under three-dimensional disturbances with the normal mode method.

The governing equations in cylindrical coordinates ( x,,r θ ) are

∂∂∂UV V1 W ++ + =0 (1) ∂∂∂xr rrθ

∂ U ∂ U W ∂ U ∂ U 1 ∂ p ++V +=−U (2) ∂ t ∂ r r ∂θ ∂ρ x ∂ x

∂ V ∂ V W ∂ V ∂ V W 2 1 ∂ p ++V +−=−U (3) ∂ t ∂ r r ∂θ ∂ρ x r ∂ r

∂ W ∂ W W ∂ W ∂ W VW 1 ∂ p ++V ++=−U (4) ∂ t ∂ r r ∂θ ∂ρ x r r∂θ

To obtain linearized disturbed equations, let

88 ' UUuVvWWwpPpj =+jjjjj, =, =+ j jjj , =+ j (j =i, o and l) (5)

Figure 4.1: Schematic of Annular liquid sheet

Here the over bar represents the assumed mean flow quantities and the lower case velocities and pressure prime indicates velocity and pressure disturbances, respectively.

We note that the following condition needs to be satisfied by the undisturbed flow for the

1 undisturbed sheet to remain annular. PP−=−σρ()11 R + R + A22 1 R − 1 R 2 oi a b2 ll() a b

Substituting Eq. (5) in equations (1) – (4), and the subtracting the mean flow equations, governing equations for the velocity and pressure disturbance are obtained. The length,

2 velocity and pressure are made dimensionless by Rb , Ul and ρllU , respectively.

The dimensionless linearized governing equations are:

89

For liquid vv∂ 1 ∂∂ wu ll++ l + l =0 (6) rrr∂∂θ∂ x

∂∂∂∂ uWeuuplslll1 ′ ++=−2 (7) ∂ trWel ∂θ ∂ x ∂ x

∂∂∂∂ vWevuWewplsllsll1 ′ ++−=−222 (8) ∂ trWell∂θ ∂ x Wer ∂ r

∂∂∂∂ wWewwplslll1 ′ ++=−2 (9) ∂ trWel ∂θ ∂ x r ∂θ for the inner gas vv∂ 1 ∂∂ wu ii++ i + i =0 (10) rrr∂∂θ∂ x

∂ uWeuWeu∂∂∂ 1 p′ isiiii++=− i (11) ∂ tgWegWexgxil∂θ il ∂ i ∂

∂∂∂ vWevWevisiiiisii We1 ∂ p′ ++−=−2 wi (12) ∂ t gWeil∂θ gWe il ∂ x gWe il g i ∂ r

∂∂∂ wWewWewisiiiisii We1 ∂ p′ +++=−2 vi (13) ∂ tgWegWexil∂θ il ∂ gWegr il i ∂θ and for the outer gas vv∂ 1 ∂∂ wu oo++ o + o =0 (14) rrr∂∂θ∂ x

∂∂∂∂ uWeuWeuposooooo11 ′ ++=−2 (15) ∂ trgWeol∂θ gWex ol ∂ g o ∂ x

90 ∂∂∂ vWevWevWeposoooosoo121 ∂ ′ ++−=−22wo (16) ∂ tr gWeol∂θ gWexrgWe ol ∂ ol g o ∂ r

∂∂∂∂ wWewWewposooooo11 ′ ++=−2 (17) ∂ trgWeol∂θ gWex ol ∂ gr o ∂θ

Where

ρ UR2 ρ A2 R ρ UR2 ρ A2 R ρ UR2 We = lbl ,We = lbl ,We = i i b ,We = i i b ,We = o o b , l σ s σ i σ si σ o σ

2 ρo Ao Rb ρi ρo Weso = , gi = and go = σ ρl ρl

To solve the above linearized disturbance equations, both kinematic and dynamic boundary conditions must be prescribed at the interfaces for both the liquid and gas phases. The kinematic boundary condition requires that particles on the interface remain there. The kinematic boundary conditions are:

For the liquid phase,

DWeηηii∂∂∂1 sii ηη vatrh==+2 + = (18) Dt∂∂∂ t r Wel θ x

DWeηηoo∂∂∂1 soo ηη vatr==+2 + =1 (19) Dt∂∂∂ t r Wel θ x

R Where h = a Rb

For the inner air stream,

DWeWeηη∂∂∂ η η vatrh==+ii sii + ii = (20) Dt∂∂ t gil Weθ g il We ∂ x

For the outer air stream,

91 DWeWeηηoo∂∂∂1 soo η oo η vatr==+2 + =1 (21) Dt∂∂∂ t r gol Weθ g ol We x

The dynamic boundary conditions are:

22 ''1 ∂∂ηηiisisi 2 We We η pli−=phhatrh222()ηη i + + + i − 3 = (22) h Welll∂∂θ x We We h

22 ''−1 ∂∂ηηooWe sos We plo−=patr()ηηη o +22 + + o − o = 1 (23) Welll∂∂θ x We We

The disturbances are assumed to have the forms of:

' ikx(+ nθ −ω t) (uvwpjj , , jj , )= ( urvrwrpreˆˆˆˆ j ( ), j( ), j( ), j( )) (j=i, o and l) (24)

Where ∧ indicates the disturbance amplitude which is a function of r only. For the temporal instability analysis, the wave number k and n are real while frequency ω is complex. The imaginary part of ω reflects the growth rate of the disturbance.

The displacement disturbances at the inner and outer interfaces are:

ikx(+− nθω t) ηθoo(,xt ,)= ηˆ e (25)

[(ikx+− nθ ω t) +Φ i ] ηθii(xt , , )= ηˆ e (26)

ωRb Where kkR==b , ω and Φ represents the phase difference between the two Ul surface waves at the gas-liquid interfaces and is evaluated as part of the solution to determine the most unstable disturbance for inner and outer interface. The two solutions obtained with Φ close to zero and Φ close to π are referred as para-sinuous and para- varicose respectively (Shen and Li 1996).

Substituting Eqs. (24)- (26) into Eqs. (6)- (23) and after manipulations, the solutions for the disturbed pressure fields in the liquid and gas streams can be obtained as

92

For Liquid

1 n Wes '' pˆlnn=−()(()()w2 − k c12 I kr + c K kr (27) k rWel

where

n Wess' iΦ We ()()()()ω −−2 kKni kηωˆˆ e −−− n kK n kh η o hWell We c1 = '' '' IkhKknn( ) ()− IkKkh nn () ( )

Wess''n We iΦ ()()()()ωηωη−−nkIkhnoˆˆ −−−2 kIke ni Well h We c2 = '''' IkhKknn( ) ()− IkKkh nn () ( )

For outer air

' −gWeWeosoon ()kx+− nθ ω t pkcKkron=−+()()ω 2 + 4 e (28) k rgWegWeol ol

We We −−()ω +nkso + o ηˆ gWe gWe o ol ol c4 = ' Kkn () and for inner air

' gWeisi iΦ ()kx+− nθ ω t pnkcIkreini=−()(() −ωη + + 51ˆ e (29) kgWeil

⎛⎞4Wesi We si 2 Where kk1 =−1(⎜⎟ −+ω n + k ) and ⎝⎠gWeil gWe il

We4 We We −−++−−++kn[(ωωηsi k )2 si ]( n si ke ) ˆ iΦ gWe gWe gWe i c = il il il 5 ⎡⎤ Wesi2' 2n We si ⎢⎥()()()−+ω nkkIkhIkh +11nn + 1 ⎣⎦gWeil h gWe il

93 The dispersion equation is obtained by substituting the pressure disturbances inside the liquid and gas phases into the dynamic boundary conditions at the two interfaces. The problem reduces then to a system of two equations with two unknowns. The two unknowns are the ratio of initial amplitudes of the perturbation on each interface and the phase differenceΦ .

2 ηˆo −Φi QFF112()Δ+Δ− 1 2 SF 11 e ==2 (30) ηˆi ()Δ+Δ−3422QF SFF 212

The final dispersion equation is given by

2222 ()()Δ+Δ−1 2SF 11 Δ+Δ− 3 4 QF 22 − QSF 121 F 2 = 0 (31)

Where

'' ' ' Innnn()kh K () k−− I ()() k K kh I nn () kh K () kh I nn () kh K () kh SS12=='' '', '' '' Innnn(kh ) K () k−− I () k K ( kh ) I nnnn ( kh ) K () k I () k K ( kh ) '' ' ' Inn()kK () k−− I nn () kK () k I n ( khK ) n () k I nn () kK ( kh ) QQ11=='''', '''' IkhKknnnn( ) ()− IkKkh () ( ) Ik n (hK)()()()nnn k− I kK kh

n Wess We FkFnk12=−(),()ωω2 − =− − hWell We

Weso We o We si We i Fn34=−(),()ωω − k Fn =− − k gWegWeol ol gWegWe il il

4We gFkI()( kh F2 − si ) in44gWe g F2 K() k 4 We Δ=il,,1 Δ= o3 nkk = − si 131⎡⎤2nWe Kk' () gWeF2 FkI' () kh+ si I () kh n il4 ⎢⎥41nn 1 1 ⎣⎦hgil We 2 2 ⎡⎤⎡⎤(1−−nkh22 )hWe We −−− (1 nk 2 ) We We Δ=k⎢⎥⎢⎥ +sis − and Δ= k + sos − 24h23 We We h We We We We ⎣⎦⎣⎦⎢⎥⎢⎥lll lll

The fourth order dispersion equation (31) can be written in a compact form as

f(,,,,ω k n h gio , g , We l , We s , We i , We sio , We , We so )= 0

94 The dispersion equation (Eq. (31)) was solved to obtain the disturbance growth rate (ω ) for any assumed pair of wave numbers ( k , n) using MathematicaTM using the secant method. The solution was considered converged when the value of the left hand side of the dispersion equation was smaller than 10-6. For each pair of ( k , n) and given dimensionless parameters, we look for the root with the maximum imaginary part, which represents the growth rate of the disturbance. The wave number that corresponds to the maximum growth rate is called the optimum or most unstable wave number. The most- unstable wave number is related to the mean drop size and the growth rate is related to the breakup length of the liquid sheet. Higher growth rate indicates shorter breakup length and higher wave numbers (smaller wave length) indicate smaller droplet diameters. As such the most-unstable wave number and the maximum growth rate are two important parameters that impact the resulting spray characteristics.

4.3 Results and Discussions as the liquid sheet velocity is much lower than that of either the inner or outer air stream in a practical airblast atomizer (Lefebvre 1980), the value of axial weber number based

on liquid velocity, Wel is kept as a constant of 200, this value corresponds to a liquid velocity of 2.3 m/s, a density of 1000 kg/m3, a surface tension of 0.073 kg/s2, an inner radius ra of 2.5 mm, and a film thickness of 0.278 mm. hence, by varying the axial and swirling Weber numbers of the air streams at different liquid swirl Weber numbers, we can understand the effect of the relative axial and tangential velocity between the gas and liquid phases on the instability of the liquid sheet. The ratio of inner/outer radii is

95 assumed to be 0.90. Based on a liquid sheet diameter of 5.5 mm and surface tension of

0.073 kg/s2, a 25 m/s air speed corresponds to an air Weber number of approximately 30.

4.3.1 Model Validation

Earlier studies on the stability of an annular sheet are used to validate the present formulation in limiting cases. In the limit of liquid swirl going to zero, the dispersion equation presented in the previous section becomes the dispersion equation of Liao et al.

(2000). In the other limit, when air swirl goes to zero, the dispersion equation reduces to that of Panchagnula et al. (1996). Furthermore, Eq. (31) is observed to reduce to the dispersion equation of Ponstein (1959) when the densities of the gas medium inside and outside the annular sheet are set equal to zero.

Two modes of instability are known to exist: the symmetric and the anti-symmetric.

Ibrahim et al. (2005) and Shen and Li (1996) have shown that the growth rates for the antisymmetric, or para-sinuous, mode are an order of magnitude greater than those for the symmetric, or para-varicose, mode when the density ratio is of the order of 0.01 or less.

We have therefore chosen to consider only the anti-symmetric (para-sinuous) mode.

4.3.2 Liquid Swirl with Purely Axial Air Flow

The effects of liquid swirl Weber number on the maximum growth rate and the most unstable axial wave number are presented in Figures 4.2(a) and (b), respectively. For the four cases considered in this figure (stagnant air, only inner axial air, only outer axial air, and both inner and outer axial air streams), the growth rate first decreases to a minimum

96 1.6 1.4 i 1.2

⎯ω 1 0.8 0.6 Growth rate 0.4 Wei=10, Weo =0 Wei=0, Weo =10 0.2 Wei=10, Weo =10 Wei=0, Weo =0 0 0 10 20 30 40 50 60 70 80 Liquid swirl Weber number We s 4.2(a)

9 8 7 ⎯ 6 5 4 3 We=10, We =0 2 i o Wei=0, Weo =10 We=10, We =10 1 i o Axial wave number k Wei=0, Weo =0 0 0 10 20 30 40 50 60 70 80 Liquid swirl Weber number We s 4.2(b)

Figure 4.2: (a) Optimal growth rate versus liquid swirl Weber number (b) Optimal axial wave number versus liquid swirl Weber number at Wel = 200, Wesi = Weso =0, h = 0.9, gi = go = 0.00129 for n = 0.

97 with increasing liquid swirl Weber number, and then increases monotonically with an increase in liquid swirl Weber number. The physical origin of this behavior can be explained in terms of the forces acting on the interface, viz. surface tension, pressure, aerodynamic forces, and centrifugal forces (Hogan and Ayyaswamy 1985). The growth of any disturbance will be determined by the competition between these forces. For para- sinuous mode, Du and Li (2005) have shown that the disturbance amplitude at the inner interface is higher than that at the outer interface for a non-swirling sheet and the instability of the inner interface dominates the sheet instability. This behavior is likely to be followed at low liquid swirl velocities. At the inner interface, at small values of liquid swirl Weber numbers, i.e. at low values of liquid swirl velocity, the centrifugal force, which is in the positive r direction, tends to restore any protrusion of the interface and tends to stabilize the effect of aerodynamic forces. Hence at low Wes, liquid swirl has stabilizing effect on the liquid sheet and with increase in liquid swirl velocity the disturbance growth rate decrease to a minimum.

In contrast, at the outer interface, centrifugal force due to liquid swirl is destabilizing. Any protrusion of the liquid surface will be pushed further outwards due to the centrifugal force. As such at high liquid swirl velocities, the outer interface will become more unstable than the inner interface due to the dominance of the centrifugal force. This will result in increase in the disturbance growth rate with increasing liquid swirl velocity. This behavior is observed in the figure at higher liquid swirl velocities.

The instability changing from dominance of inner interface instability to outer interface instability leads to the presence of a minimum in the disturbance growth rate with

increasing liquid swirl Weber number. In fact, at Wes > 50 , the growth rate for the only

98 inner axial air approaches that of stagnant air case indicating that the inner axial air has no effect on annular sheet instability at high liquid swirl Weber number. Furthermore, the growth rate of only outer air approaches that of inner and outer air at high liquid swirl

Weber number. For a swirling sheet, only outer axial air gives rise to the longest most- unstable axial wave number (shortest wavelength) and the only inner axial air has the shortest axial wave number as shown in Figure 4.2(b).

Figure 3 presents the effect of liquid swirl Weber number on the optimal growth rate for only inner axial air and only inner outer axial air at air Weber numbers 10, 20 and 30.

Not surprisingly, as the aerodynamic forces increase with higher axial inner air Weber number, higher values of liquid swirl are needed to reach the minimum in the growth rate and the range of stabilizing liquid swirl Weber number increases as seen in Figure 4.3.

The centrifugal force due to liquid swirl is outward and always aids in the disturbance growth at the outer interface. However, at low outer air velocity, as mentioned earlier, Li and co-workers (Shen and Li 1996 and Du and Li 2005) have shown that the disturbance magnitude is higher at the inner interface. As such, liquid swirl has stabilizing effect over a shorter swirl Weber number range with non-zero outer axial velocity. As the outer axial air velocity increases, the outer interface becomes more unstable and the growth rate increases. Hence with higher outer axial velocity, the outer interface dominates the sheet instability and there is no stabilizing effect of liquid swirl for any value of liquid swirl

Weber number.

The effect of axial air Weber number on the maximum disturbance growth for non-swirling and swirling annular liquid sheet is presented in Figures 4.4(a) and (b), respectively. Three cases are considered, viz., only inner air moving axially, only outer

99 air moving axially, and both air streams moving axially. With no liquid swirl the maximum growth rate of only inner axial air stream is larger than that of only outer axial air stream. This indicates that axial inner air is more effective than outer axial air in enhancing the instability of non-swirling annular liquid sheets. Such effectiveness of inner air stream over outer air stream has been demonstrated in experimental observations by Adzic et al. (2001). In contrast, in the presence of liquid swirl the maximum growth rate of only outer axial air stream is larger than that of only inner axial air stream.

6

5 i

⎯ω 4

3

2

Wei =10 &Weo=0 Optimal growth rate Wei =20 &Weo=0 1 Wei =30 &Weo=0 Weo=10 &Wei=0 Weo=20 &Wei=0 Weo=30 &Wei=0 0 0 50 100 150 200 250 300 Liquid swirl Weber number We s

Figure 4.3: Optimal growth rate versus liquid swirl Weber number for different inner and outer axial air Weber numbers at Wel = 200, Wesi = Weso =0, h = 0.9, gi = go =

0.00129 for n = 0.

100 9

i 8 ω Only inner axial air ⎯ Only outer axial air 7 Both axial air streams 6 5 4 3 Optimal growth rate 2 1 0 10 20 30 40 50 Axial air Weber number

4.4(a)

9

i 8 Only inner axial air Only outer axial air ⎯ω 7 Both axial air streams 6 5 4 3

Optimal growth rate 2 1 0 10 20 30 40 50 Axial air Weber number 4.4(b)

Figure 4.4: Optimal growth rate versus axial air Weber number at Wel = 200, Wesi =

Weso =0, Wei = 20, Weo = 20, h = 0.9, gi = go = 0.00129 for n = 0 (a) Wes = 0, (b) Wes =

50.

101 This indicates that the outer air stream is more effective than the inner air stream in making a swirling liquid sheet unstable. The liquid swirl has a destabilizing effect at the outer interface and adds to the destabilizing effect of the axially moving outer air stream.

Both inner and outer axial air streams case has the largest disturbance growth rate compared to the other three cases because with the presence of both inner and outer axial airstreams, the disturbance extracts energy from both the mean inner and outer axial air flow. With increasing axial air Weber number, the maximum growth rate increases. This would suggest shorter breakup length with increasing axial Weber number which has been seen in experimental measurements reported by Carvalho and Heitor (1998) and

Adzic et al. (2001).

4.3.3 Air Swirl with Purely Axial Liquid Flow

Figure 4.5 shows the influence of air swirl on optimal growth rate and axial wave number of non-swirling annular liquid sheet. The results are shown for swirl air Weber numbers equal to the axial air Weber numbers. The optimal growth rate increases with increasing air Weber numbers which implies that higher air swirl would lead to shorter breakup length. Such decrease in breakup length has been observed with increasing air

Weber number in experimental measurements of Carvalho and Heitor (1998) and He et al. (2003). Both of the maximum growth rates and axial wave number of the inner air swirl is higher than those of the outer air swirl. This implies that inner air swirl is more effective than outer air swirl in increasing the most unstable wave number and the maximum growth rate for a non-swirling sheet.

102 no air swirl (Wei=Weo) i only inner air swirl (We =We =We ) 8 i o si only outer air swirl (Wei=Weo=Weso) ⎯ω both air swirl (Wei=Weo=Wesi=Weso)

6

4

Optimal Growth rate 2

0 10 20 30 40 Air Weber number

4.5(a)

25 no air swirl (Wei=Weo) only inner air swirl (Wei=Weo=Wesi) only outer air swirl (Wei=Weo=Weso)

⎯ 20 both air swirl (Wei=Weo=Wesi=Weso)

15

10

5 Optimal wave number k

0 10 20 30 40 Air Weber number

4.5(b)

Figure 4.5: (a) Optimal growth rate versus air Weber number (b) Optimal axial wave number versus air Weber number at Wel = 200, Wes =0, h = 0.9, gi = go = 0.00129 for n

= 2 .

103 This is not surprising as the static pressure distribution in the undisturbed flow is not uniform in the swirling inner and outer air stream. Due to the centrifugal force, the pressure increases as we move away from the outer interface in the outer air. In contrast, the pressure decreases as we move towards the centerline from the inner interface with swirling inner air stream. As such, a protrusion of the outer surface is acted upon by higher static pressure which has the tendency to push the interface back to its undisturbed position (Liao et al. 2002 and Lian and Lin 1990). The opposite is true for the inner air swirl. The static pressure due to the inner air swirl decreases from the inner interface to the sheet axis. Therefore, the interface, when perturbed, sees lower static pressure which tends to promote the distortion of the interface further. As a result, the inner air swirl is more effective than the outer air swirl in destabilizing a non-swirling liquid sheet.

4.3.4 Liquid Swirl with Air Swirl

Figure 4.6 shows a plot of non-dimensional growth rate versus non-dimensional axial and circumferential wave numbers at liquid swirl Weber number of 50 for only inner swirling air stream (Figure 4.6a) and for only outer swirling air stream (Figure

4.6b). Without air swirl, the axisymmetric mode has the highest growth rate (Liao et al.

2000) and, thus, it dominates the competition in the disintegration process. However, when air swirl is added to the inner or outer streams, the growth rate of helical modes is significantly increased. This indicates that air swirl significantly enhances the instability of a swirling annular liquid sheet. With higher helical modes dominating the sheet instability, it is likely to lead to smaller ligament/droplet formation compared to axisymmetric mode.

104 i 4 ⎯ω 3 2

rwhrate Growth 1 0 30 25 r n 25 20 be 20 15 um A 15 e n xial 10 av w 10 l w ave 5 5 tia num 0 0 ren be fe r k um ⎯ irc C 4.6(a)

i 8

⎯ω 6 4 2

rwhrate Growth 40 350 35 30 30 25 25 20 20 r n A 15 15 be xia 10 10 num l w ve ave 5 5 wa nu 0 0 ial mbe nt ⎯r k fere cum Cir

4.6(b)

Figure 4.6: Growth rate versus axial and circumferential wave number at Wel =200, Wes

=50, h = 0.9, gi = go = 0.00129 a) Wei=Wesi = 20 & Weo =Weso = 0, b) Wei=Wesi = 0 and

Weo =Weso = 20.

105 Therefore, a combination of axial and tangential velocity components is more effective in breaking up the liquid sheet compared to purely axial air flow. It is seen from

Figures 4.6(a) and 4.6(b) that maximum growth rate and its axial wave number of helical modes caused by the inner air swirl are smaller than that caused by outer air swirl and underscores the effectiveness of the outer air.

This is consistent with experimental observations of outer air swirl which show significant decrease in breakup length with increasing outer air swirl and explosive breakup beyond a critical swirl number (Lasheras and Hopfinger 2000). Figure 4.7 illustrates the effect of liquid swirl Weber number on the optimal growth rate and most unstable axial wave number in the presence of only inner swirling air stream, only outer swirling air stream, both inner and outer swirling air streams and no air swirl, at circumferential wave number of 2. As discussed earlier, the liquid swirl has a stabilizing effect on the inner interface but has a destabilizing effect on the outer interface. On the other hand, increasing static pressure due to the outer air swirl would tend to restore the interface back to its undisturbed position. The opposite is true for the inner air swirl. The static pressure due to the inner air swirl decreases from the inner interface to the sheet axis and tends to enhance the distortion of the interface. As a result, the inner air swirl is more effective than the outer air swirl in destabilizing a non-swirling liquid sheet. These opposite effects of liquid and air swirl lead to the behavior shown in Figure 4.7 where the growth rate of the most unstable disturbance is plotted with inner and outer swirl Weber number of 20 and 0. At low liquid swirl Weber number, the instability is dominated by the air swirl, and due to the destabilizing influence of inner swirl, the growth rate of inner swirl is higher than that of only outer swirl.

106 5

Wesi=20, Weso =0

i Weso=20, Wesi =0 4.5 Wesi=20, Weso =20 We =0, We =0 ⎯ω si so 4

3.5

3

2.5 Optimal Growth rate 2 0 50 100 150 200 250 Liquid swirl Weber number We s 4.7(a)

17

Wesi=20, Weso =0 ⎯ ⎯ 16 Weso=20, Wesi =0 Wesi=20, Weso =20 15 Wesi=0, Weso =0 14 13 12 11 10 9 Optimal axial wave number k 8 0 50 100 150 200 250 Liquid swirl Weber number We s 4.7(b)

Figure 4.7: (a) Optimal growth rate versus liquid swirl Weber number (b) Optimal axial wave number versus liquid swirl Weber number at Wel = 200, h = 0.9, Wei=Weo = 20 gi

= go = 0.00129 for n = 2.

107 However, with increasing liquid swirl Weber number, the centrifugal force of the liquid at the outer interface becomes the dominating force for sheet instability and the growth rate with only outer air becomes higher than that with only inner air. It can be observed from Figure 4.7(b) that at no liquid swirl, only inner air swirl has the longest most- unstable axial wave number compared to only outer air swirl and both air swirl cases.

However in the presence of liquid swirl, only outer air swirl has the longest most- unstable axial wave number. We note that the gas swirl and the liquid swirl have stabilizing and destabilizing effects at the outer interface, respectively, and the opposite effects at the inner interface. As such, the relative velocity between gas and liquid is not adequate to characterize the stability behavior for swirl velocity component.

4.3.5 Effect of Relative Air Swirl Direction

When the liquid and both the air streams are swirling, four combinations of the air swirl direction are possible with respect to the swirl direction of the liquid stream. These are: both streams co-rotating, both counter-rotating, and one co- and one counter-rotating with respect to the liquid swirl direction. Figure 4.8 presents the effect of liquid swirl on the optimal growth rate and axial wave number at different relative air swirl orientation with respect to liquid swirl. The co-swirl case where liquid and inner and outer air rotate in the same direction, leads to the highest growth rate and the counter-swirl case where inner and outer air rotate in the opposite direction of the liquid, leads to the smallest growth rate. Additionally, for liquid swirl Weber number below about 50, the counter- counter case has the lowest optimal wave number.

108 4 inner co & outer co inner counter & outer counter inner co & outer counter

i 3.5 inner counter & outer co ⎯ω 3

2.5

2

Optimal growth rate 1.5

1 25 50 75 100 125 150

Liquid swirl Weber number Wes

4.8(a)

14 inner co & outer co

⎯ inner counter & outer counter inner co & outer counter 13 inner counter & outer co 12

11

10

9

8 Optimal axial wave number k 7 25 50 75 100 125 150 Liquid swirl Weber number We s 4.8(b)

Figure 4.8: (a) Optimal growth rate versus liquid swirl Weber number (b) Optimal axial wave number versus liquid swirl Weber number at Wel = 200, Wei = Weo = 20, Wesi =

20, Weso = 20, h = 0.9, gi = go = 0.00129 for n = 1.

109 As such, in that range of liquid swirl Weber number, with low growth rate and low axial wave number, the inner-counter/outer-counter combination is likely to produce the largest drops among all configurations. This behavior is identical to that observed in the experiments conducted by Chin et al. (2000) to study the effect of air swirl rotation.

At , the optimal growth rate of counter-rotating inner and co-rotating outer air Wes > 15 streams is larger than that of co-rotating inner and counter-rotating outer air streams

whereas the opposite trend occurs at Wes < 15 . At high liquid swirl, the optimal growth rate appears to be governed by the rotational direction of the outer stream irrespective of the inner stream rotation. This emphasizes the importance of outer air flow to enhance instability of swirling liquid. A change in the direction of rotation changes the circumferential pressure disturbance the gas phase. For example, higher disturbance pressure could change from front side of a trough to rear part of a trough, and may change from being in-phase to out-of-phase of circumferential variation of liquid disturbance pressure. This in turn can change the most unstable mode from axisymmetric to a helical mode. For a swirling liquid sheet, it was found that the growth rate of the axisymmetric disturbance was the highest for counter-rotating outer swirl whereas non- axisymmetric disturbances were most unstable for co-rotating outer swirl and had higher most unstable wave number. Therefore, at high liquid swirl where outer air swirl has the dominating effect, co-rotating outer swirl has higher disturbance growth rate and higher optimal wave number compared to counter-rotating outer streams.

At liquid swirl Weber number below about 25, the combination of co-rotating- inner/counter-rotating-outer air streams has the largest optimal axial wave number.

Additionally, in this range of liquid swirl Weber number, the co-inner/counter-outer

110 combination has high growth rate. As such, at liquid swirl Weber number lower than about 25, the co-inner/counter-outer case is likely to result in the finest spray. Once again, this behavior is consistent with measurements of Chin et al. (2000). At higher liquid swirl Weber numbers, counter-rotating inner and co-rotating outer air streams has the largest optimal axial wave number. As droplet sizes are related to the wavelength of the most unstable disturbance, we can extrapolate that co-rotating inner and counter- rotating outer air stream case produces the finest spray at low liquid swirl Weber numbers and counter-rotating inner/co-rotating outer air streams case produces the finest spray at moderate and high liquid swirl Weber numbers.

Figure 4.9 shows the effect of air Weber number on the optimal growth rate and axial wave number for four different relative air swirl orientations at liquid swirl Weber number of 20. It can be seen from Figure 4.9(a) that both the optimal growth rate and axial wave number increase with increasing air Weber number. At low air Weber number, the optimal growth rate of counter-rotating inner air/co-rotating outer air streams case is higher than that of co-rotating inner/counter-rotating outer air streams case.

However at higher air Weber numbers, an opposite trend is predicted. As shown in

Figure 4.9(b) the co-rotating inner/counter-rotating outer air streams case always has the largest axial wave number which would tend to produce finer droplet sizes.

4.3.6 Effect of High Air Pressure

In fuel injection systems of combustion engines, the liquid sheet is injected into a high pressure environment. As such it is important to study the effect of high air pressure on the sheet stability.

111 inner co & outer co inner counter & outer counter i 8 inner co & outer counter inner counter & outer co ⎯ω

6

4

2 Optimal growth rate

0 10 20 30 40 50 Air Weber number

4.9(a)

30 innerco&outerco inner counter & outer counter ⎯ inner co & outer counter 25 inner counter & outer co

20

15

10

5 Optimal axial wave number k 0 10 20 30 40 50 Air Weber number

4.9(b)

Figure 4.9: (a) Optimal growth rate versus air Weber number (b) Optimal axial wave number versus air Weber number at Wel = 200, Wes = 20, h = 0.9, gi = go = 0.00129 for n

= 1 (Wei = Weo = Weso =Wesi ).

112

Figure 4.10: Variation of optimal growth rate versus liquid swirl Weber number at three different air-to-liquid density ratios showing the effect of air pressure at Wel = 200, Wesi

= Weso =0, h = 0.9, for n = 0.

In the present model the primary effect of air pressure is felt through increased interaction with air flow due to higher air density with increase in air pressure. Therefore the effects of high pressure can be studied in terms of a dimensionless parameter of air-

to-liquid density ratio ( ρairρ l= gg o= i ). Figure 4.10 shows the variation of optimal disturbance growth rate of a swirling annular liquid sheet at three different density ratios.

113 For a fixed temperature, these density ratios correspond to atmospheric, 10 times, and 20 times atmospheric pressure, respectively. It is seen from the figure that with increasing air pressure the disturbance growth rate increases significantly. This is due to the increased aerodynamic interaction with increased air density. This increase in disturbance growth rate would induce faster sheet breakup. This behavior is consistent with experimental observations of planar sheet breakup at high air pressure (Dombrowski and

Hooper 1962). The qualitative stability behavior with the swirl Weber number is similar at all air pressures. The liquid swirl has a stabilizing influence at low swirl Weber number and destabilizing effect at high liquid swirl Weber number at all air-to-liquid density ratios considered here.

4.4 Summary and Conclusions

A theoretical model has been developed to predict the instability of an inviscid swirling annular liquid sheet subjected to inner and outer air streams moving with both swirl and axial velocity components. Furthermore, the effect of the air swirl orientation with respect to the swirl direction of the liquid sheet on the instability of an annular liquid sheet has been investigated. From this study, we can draw the following conclusions:

1. Liquid swirl imposed on an annular liquid sheet subjected to inner and outer air

streams has a dual effect on its stability. Liquid swirl has a stabilizing effect at

low values of liquid swirl Weber number whereas it has a destabilizing effect

with increasing liquid swirl Weber number. The instability changing from

dominance of inner interface instability to outer interface instability leads to the

114 presence of minima in the disturbance growth rate with increasing liquid swirl

Weber number.

2. For non-swirling liquid sheet, axial inner air stream is more effective than axial

outer air stream in enhancing the sheet instability. However, for a swirling liquid

sheet, axial outer air is more effective in promoting sheet instability compared to

axially moving inner air stream.

3. The liquid swirl has a destabilizing effect at the outer interface but has a

stabilizing effect at low liquid swirl Weber numbers at the inner interface. As

such, at high liquid swirl Weber number, the outer air (with axial and swirl

velocity components) is more effective in enhancing sheet instability compared to

inner air (with axial and swirl velocity components).

4. At high liquid swirl Weber number, the outer air flow has the dominant effect of

sheet instability. The co-rotating outer stream has higher disturbance growth rate

and higher most unstable wave number compared to counter-rotating stream,

irrespective of the inner swirl direction.

5. At low liquid swirl Weber number, a combination of a co-rotating inner air stream

and counter-rotating outer air stream with respect to the liquid swirl direction

would yield the finest droplet sizes compared to other swirled configurations.

However at high liquid swirl Weber number, a combination of a counter-rotating

inner air stream and co-rotating outer air stream would yield the smallest droplet

sizes.

6. The disturbance growth rates are significantly higher at elevated air pressure

compared to those at atmospheric pressure.

115

PART III

NONLINEAR INSTABILITY OF LIQUID JETS AND ANNULAR

LIQUID SHEETS

116

CHAPTER 5

NONLINEAR BREAKUP OF A COAXIAL LIQUID JET IN A

SWIRLING GAS STREAM

5.1 Introduction

The breakup of a liquid jet is employed in a variety of applications to produce droplets. Examples include ink-jet printing, consumer product sprays and fuel atomization. When a liquid jet emerges from an atomizer, the competition between cohesive and disruptive forces on the surface of the jet gives rise to surface oscillations and alters the flow field. Under favorable conditions the oscillations are amplified to disintegrate the liquid jet into ligaments or droplets. This process is referred to as primary atomization and it involves the action of pressure, aerodynamic, centrifugal, surface tension forces, and internal effects such as turbulence and those arising from velocity profiles relaxation (Lefebvre 1989). Liquid breakup lengths of liquid jets in gases are of interest for modeling sprays because they define the region where primary breakup at the liquid surface must be considered and define the start of the fully dispersed multiphase flow region. Therefore, the understanding of jet primary breakup is essential in understanding spray processes.

Reitz and Bracco (1982) have classified the disintegration of a liquid jet in stagnant air into four regimes based on the jet Weber number based on gas

117 2 densityWeggl= ρ U d /σ . Where ρg is the gas density, Ul is the axial liquid velocity, d is the liquid jet diameter and σ is the liquid surface tension.: (a) Rayleigh jet breakup

(Weg < 0.4): which is caused by the growth of axisymmetric oscillations on the jet surface induced by the surface tension, and the resulting drop diameters exceed the jet diameter;

(b) First wind-induced breakup (0.4 < Weg < 13) where non-axisymmetric (sinuous) oscillations occur resulting in droplet sizes comparable to the jet radius; (c) Second wind- induced breakup (13 < Weg < 40.3): where drops are produced by unstable growth of short waves on the jet surface caused by the relative motion of the jet and the ambient gas. The resulting drop diameters are much less than the jet diameter. (d) Atomization regime (Weg > 40.3) in which even smaller droplet sizes are obtained. When injection takes place into a co-flowing gas, Farago and Chigier (1992) have classified coaxial jet disintegration into three main categories based on the Weber number evaluated with the

2 relative velocity (Wegglg=−ρ ()/ U U d σ ): (a) Rayleigh-type breakup where the mean drop diameter is of the order of the jet diameter. Both axisymmetric breakup (for Weg <

15) and nonaxisymmetric breakup patterns (for 15 < Weg < 25) have been observed. (b)

Jet disintegration via the stretched-sheet mechanism which produces membrane type ligaments (25< Weg < 70). In this case, the mean diameter of the drops formed is considerably smaller than the diameter of the jet; and (c) jet disintegration via fiber-type ligaments (100 < Weg < 500) at even higher air-flow rates. Fibers are formed that peel off from the liquid-gas interface.

Linear and nonlinear instability theories and computational methods have been used to study the stability and the disintegration of liquid jets. The first linear instability

118 analysis for an inviscid cylindrical liquid column in the absence of surrounding gas was carried out by Rayleigh (1878). Weber (1931) has included the effects of both the liquid viscosity and the pressure of the surrounding gas on the jet behavior. A comprehensive review of linear instability theories for liquid jets is given by Lefebvre1 and more recently by Lin (2003), Sirignano and Mehring (2000), and Yoon and Heistor (2003). The linear stability theory predicts the onset of jet instability but is unable to predict the jet deformation to breakup due to the underlying assumption of infinitesimal perturbation magnitudes. In reality, unstable disturbances will grow to finite magnitudes after a finite interval of time. Liquid breakup and droplet formation are nonlinear phenomena where the amplitude of the disturbance at breakup is of the same order as the jet radius (Miesse

1959). The shortcoming of linear theory has led to the development of nonlinear analysis to calculate the profiles of the waves on the surface of the jet and also to predict the volume of the main and the satellite drops formed during breakup. Lafrance (1975),

Rutland and Jameson (1971), Nayfeh (1970), Yuen (1968) and Wang (1968) have performed nonlinear perturbation analyses of a capillary of an inviscid liquid jet of circular cross section in the absence of the surrounding gas. Mansour and Lundgren

(1990) have developed a nonlinear simulation of the Rayleigh breakup process utilizing a boundary element method (BEM). Spangler et al. (1995) have extended Mansour and

Lundgren’s model by including the gas-phase pressure variations to solve for the nonlinear evolution of liquid jet acting under the influence of both surface tension and the aerodynamic interactions with the surrounding atmosphere. The above-mentioned analyses predict the sizes of the main and the satellite drops formed during breakup of a

119 low-speed jet subjected to disturbances of known wavelengths. Their predictions agree well with experiments for low speed jets injected in a stagnant medium.

Only a limited number of investigations are available that deal with the effect of gas swirl on jet stability and breakup. Lian and Lin (1990) and Liao et al. (2000) have examined the linear stability characteristics of a liquid jet in a swirling gas stream. They have reported that the gas swirl has a stabilizing influence on the jet and the axisymmetric mode (n = 0) is more unstable than helical modes (n > 0). However, recent experimental investigations of Dunand et al. (2005) Hopfinger and Lasheras (1996) and

Hardalupas and Whitelaw (1998) have shown that a swirl imparted on the surrounding gas makes the jet unstable and the gas swirl is very effective in breaking up the liquid jet.

Moreover, at high gas swirl number, the jet disintegrates through an explosive breakup.

They have reported that with increasing the gas-to-liquid momentum ratio, the helical modes become dominant at smaller gas swirl strength and cause an explosive breakup.

The main objective of the present study is to investigate the asymmetric nonlinear breakup of a liquid jet exposed to a swirling gas stream. This is carried out by a perturbation expansion technique with the initial amplitude of the disturbance as the perturbation parameter. The analysis sheds light on the discrepancy between experimental observations of shorter breakup length with increasing gas swirl and linear instability analyses which predict a stabilizing influence of gas swirl. Results show that the behavior predicted by linear instability analyses is valid only when swirl number is very small (S Æ 0). At higher swirl number, helical modes become dominant and the transition to subsequent higher helical mode is achieved with smaller increments in the gas swirl number. The gas swirl number for transition to a highly asymmetric breakup

120 with a high circumferential wave number (n = 5) is found to vary as the inverse of the square root of the gas-to-liquid momentum ratio when the gas-to-liquid momentum ratio is less than 1.

5.2 Mathematical Formulation

An incompressible liquid jet of diameter d, density ρl and surface tension σ has been considered to be surrounded by a coaxial incompressible gas stream of density ρg .

The liquid jet moves axially and the coaxial gas has both axial and tangential velocity components. Mean flows of the liquid and the outer gas in cylindrical coordinates (,,x r θ )

Ag are assumed to be (U ,0,0) and (,0,)U , respectively. The effect of gravity is l g r neglected because the Froude number (defined as the ratio of liquid inertia to gravity) is typically very large for practical sprays. In the present formulation, liquid and gas flows are assumed to be irrotational and inviscid (White 1991). Therefore, the entire flow field can be treated as a potential flow. Except for very low Weber numbers, viscosity has been shown to have only a small stabilizing effect on liquid sheets (Jazayeri and Li 2000)

Therefore to keep the problem mathematically tractable, we have neglected liquid and gas viscosities. All the physical parameters are non-dimensionalized such that length,

time and density are scaled with jet radius R, the convection time R Ul and the liquid

density ρl . When the base flow, described above, is perturbed by a small disturbance with initial amplitudeηo , the liquid-gas interface is displaced to new locations denoted by rxt=+1(,,)η θ . The dimensionless surface deformation η and the velocity potential

121 φ(,,x rtθ ,) for the liquid and the gas phases must satisfy the mass conservation equations:

2 ∇=φl 0 0 ≤≤rR (1)

2 ∇=φg 0 R ≤≤∞r (2)

The kinematic and dynamic boundary conditions at rxt= 1+η ( ,θ , ) are:

φ η φηφη=+ +l,,θ θ (3) lr,,,, t lx x r 2

φ η φηφη=+ +g,,θ θ (4) gr,, t gx ,, x r 2

11 1 11 −−+ρρρWU22φφ−+∇−∇= ρ φ22 φ 22 2 gt, lt, 22g l 223 2 2 ηη− ⎧⎫221+η ηηηη 11,,,,,,θθ222 ⎪⎪xx θθθ,x 1 (1++22222ηηηη,,xx )⎨⎬ (1 + + ) −,θθ −+ (1 ) , xx + − Wel r⎩⎭⎪⎪ r r r r r Wel r

(5)

In the above equations, the liquid Weber number (Wel ), the ratio of outer gas axial velocity to liquid axial velocity (U ), the ratio of outer gas swirl velocity to liquid axial velocity (W ) and the density ratio ρ are defined as:

2 ρllUR U g Ag ρg Wel = , U = , W = and ρ = . σ Ul RUl ρl

In order to obtain a solution for η , regular perturbation theory is utilized with the initial

disturbance amplitude ηo as the perturbation parameter. By means of a series expansion method under the perturbation scheme, the surface deformations or the location of the

liquid gas interfaces are expanded in power series of ηo as:

122 j=∞ η(,x θηηθ ,)txt= j (, ,) (6) ∑ o j j=1

The term η 0 is neglected from the expansion since it corresponds to the unperturbed o

interface which is known. Assuming that η j and all its derivatives are of the same order of magnitude, the forms of the kinematic boundary conditions suggest that the velocity

potentials for the liquid and the gas phases can also be expanded in power series of ηo as:

j=∞ φ (,,x rtθηφθ ,)= j (,, xrt ,) (7) llj∑ o , j=0

j=∞ φ (,,x rtθηφθ ,)= j (,, xrt ,) (8) ggj∑ o , j=0 where φ 0 = x andφ 0 =+Ux Wθ represent the base flow field in the liquid phase and the l g gas phase, respectively.

Since the governing equations, Eqs (1) and (2), are linear, by applying the principle of

superposition, each of the φlj, and φg, j must satisfy the governing equations independently. The corresponding boundary conditions are obtained by substituting Eqs.

(6) - (8) into Eqs. (3) - (5), and equating to zero the successive coefficients of the same

j power of ηo . Under such a scheme, the velocity potentials need to be evaluated at the disturbed interface which is part of the solution and is not known a priori. In order to

overcome this difficulty, φlj, and φg, j are approximated by a Taylor series expansion around the unperturbed interfaces, r = 1 as:

123 ηη23 φφηφφ=+ + + φ +...... (9) rr=+11η = r r==112! rr r 3! rrr r = 1

Thus the governing equations and the corresponding boundary conditions, which are evaluated at r =1, for the first and the second order are obtained as:

First order (ηo ):

2 ∇=φl1 0 (10)

2 ∇=φg1 0 (11)

φlr1,−+ηη 1, t 1, x =0 (12)

φgr1,−+ηη 1, tUW 1, x + η 1,θ =0 (13)

−1 2 ρφφρφρφφgt1,−+ lt 1,UW gx 1, + g 1,θθθ − lx 1, =() ηηη 1 + 1, + 1, xx + ρη W 1 Wel

η (,x θθ ,0)=+Cos ( kx n ) & η (,x θ ,0)0= (14) 1 1,t

2 Second order (ηo )

2 ∇=φl 2 0 (15)

2 ∇=φg 2 0 (16)

φlr2,−−ηη 2, t 2, x = ηφηφηφ 1, xlx 1, + 1,θθ l 1, − 1 lrr 1, (17)

φgr2,−−ηη 2, tUW 2, x − ηηφηφηφηη 2,θ = 1, xgx 1, + 1,θθ g 1, − 1 grr 1, −2 W 1 1, θ (18)

124 1 2 ρφgt2,−+ φ lt 2, ρUW φ gx 2, + ρ φ g 2,θθθ −+ φ lx 2,() η 2 + η 2, + η 2, xx − ρ W η 2 = Wel

()()()()η1,t φ l 1, r+− ηφ 1 l 1, rt ρ η 1, t φ g 1, r + ηφ 1 g 1, rt + η 1, x φ l 1, r +− ηφ 1 l 1, rx ρU η 1, x φ g 1, r + ηφ 1 g 1, rx 3ρηW 22 −+−−ρηφηφηφW 2 (2)1 1,θθθgr 1, 1 gr 1, 1 g 1, 2 222 1 2 221 2 2 2 (2ηηηηη1+−− 4 1 1,θθ 1, θ 1,x ) + (φlx1, ++φφlr1, l 1,θθ)( − ρφφ gx 1, + gr 1, +++ φ g 1, ) 2 22Wel

(19)

η (,x θ ,0)0= &η (,x θ ,0)0= (20) 2 2,t

5.2.1 Solution of the First and the Second Order Equations

The first order surface deformation can be written in the following general form

η (x ,θθθ ,t )=++−+ A ( t )exp( i ( kx n )) A ( t )exp( i ( kx n )) (21) 1

Here the overbar indicates complex conjugate. Substituting Eqn. (21) into Eqs. (12) –

(13), the forms for the liquid and gas velocity potential can be obtained. Then substituting

these in Eqs. (10) – (11) and solving, we get the velocity potential in terms of A(t) and its complex conjugate and modified Bessel functions of r. Then substituting the gas and liquid velocity potential in the dynamic boundary condition (Eqn. 14) and solving using the method of Laplace transform we find

At( )=+ c11 exp(ω 11 t ) c 12 exp(ω 12 t ) (22)

Where

c11=−ωωω 12[2( 11 − 12 )] and c12=−ωωω 11[2( 11 12 )]

The growth rates ω11 and ω12 are the conjugate roots of the following dispersion

equation

125 1 ρKk() Ik ()ρ ( kUinWKk+ ) () kIk () ()2()nn−+ωω2 i n − n '' ' ' k Kknn() Ik () Kk n () Ik n () (23) (1−−kn22 ) kI22222() kρ K ()( k kU++ nW 2 inkUW ) −+ρW 2 +nn − =0 '' Wel kInn() k kI () k

The first order liquid and gas velocity potentials are respectively

Inkr() φωω=+[cikt ( )exp( ) l1111111' kIn () k (24) ++ciktikxncc12(ωω 12 )exp( 12 )]exp( ( ++ θ )) . .

Knkr() φω=++[(cikUinWt )exp() ω g1111111' (25) kKn () k +++cikUinWtikxncc12(ωωθ 12 )exp( 12 )]exp( ( ++ )) . .

Here c.c. indicates complex conjugate.

Similarly from equations (15-19) the second order surface deformation can be written in the following form

ηθ(x , ,tBtikxnBtikxnDt )=++−++ ( )exp(2 ( θ )) ( )exp( 2 ( θ )) ( ) (26) 2

B(tc )=++21 exp(ω 21 tc ) 22 exp(ωωωωω 22 tc ) 23 exp(2 11 tc ) + 24 exp(2 12 tc ) ++ 25 exp(( 11 12 ) t ) (27)

Where Dt( ) is required to insure mass conservation at t > 0 and is obtained as

−2cc (ωω++2()())ik )(( k22 + n ) I k − k 2'' I k Dt( )=−11 12 11 12 nn (cosh(()ωω− ) 1) ' 12 11 (ω12−ω 11)()kIn k

(28)

Following a procedure similar to one described for the first order solution, after rather lengthy and tedious manipulations, the dispersion equation of the second order can be obtained. The growth rates ω21 and ω22 are the conjugate roots of the following dispersion equation

126 1 ρKkIk(2 ) (2 )ρ ( kUinWKkkIk+ ) (4 ) (2 ) ()4()nn−+ωω2 i n − n 2k Kk''(2 ) Ik (2 ) Kk ' (4 ) Ik ' (2 ) nn n n (1−− 4kn22 4 ) 2(2)2(2)(kI22222 kρ K k kU++ nW 2 inkUW ) −+ρW 2 +nn − =0 '' Wel kInn(2 k ) kI (2 k )

(29)

The velocity potentials for the liquid and the gas for the second order are obtained as

Ikr(2 ) φ = n [ciktcikt (ωωωω+++ 2 )exp( ) ( 2 )exp( ) l2 ' 21 21 21 22 22l 22 2(2)kIn k ++2ciktcikt23 (ωωωω 11 )exp(2 11 )+ 2 24 ( 12 + )exp(2 12 ) I ()kIk'' () +++cikttknk(ωω 2 )exp( ω ++−+− ω ) ((22 )nn 2 ) (30) 25 11 12 11 12 '' kInn() k I () k 2 2 (cik11 (ωω 11 + )exp(2 11tc)++ 12 (ωωωω 12 ik )exp(2 12 tcc ) + 11 11 ( 11 + 12 +++2ik )exp(ωω11 t 12 t )] cc..+ L () t

Kkr(2 ) φ = n [cikUinWtcikUinWt (ωωωω++ 2 2 )exp( ) + ( ++ 2 2 )exp( ) g2 ' 21 21 21 22 22 22 2(2)kKn k +++2cikUinWtcikUinWt23 (ωωωω 11 )exp(2 11 )+ 2 24 ( 12 ++ )exp(2 12 ) Kk() Kk'' () ++++cikUinWttknk(22)exp()(()ωω ω ++−+− ω 22nn 2 ) (31) 25 11 12 11 12 ' ' kKn () k Kkn () 22 (cikUinWtcikUinWtcc11 (ωωωωωω 11++ )exp(2 11 ) + 12 ( 12 ++ )exp(2 12 ) + 11 11 ( 11 + 12 22 ++2ikU 2 inW )exp(ωω11 t + 12 t ) − 2 inW ( c 11 exp(2 ω 11 t ) + c 12 exp(2 ω 12 t ) +++2cc11 11 exp(ωω 11 t 12 t )] cc..+ Gt ()

Where the functions Lt() and Gt()are required to satisfy the dynamic boundary conditions. The constants c21, c22 , c23, c24 , c25 , and the functionsLt ( ) and Gt ( ) are listed in the Appendix B.

127 The first and the second order dispersion equations are solved using

MathematicaTM. The secant method is used which requires two initial guess values.

Solutions are considered convergent when values of left hand side of Eqs. (23) and (29) are smaller than 10-6.

The expression for the evolution of the gas-liquid interface is obtained by using the first- and the second-order solutions as:

12 η(,xtθηηηη ,)=+oo12 (32)

5.3 Results and Discussions

The temporal variation of the jet cross-section and the breakup time of the jet are obtained by tracking the jet surface with time in response to a small initial disturbance until the deepest trough of the wave profile coincides with the centerline of the jet. In the numerical solution the breakup point is determined when the difference between the wave profile and the centerline is less than 10-6. From the solution of the first-order and the second-order dispersion equations, the wavelength of the most unstable wave number is obtained. The most unstable disturbance is imposed on the jet surface and its temporal development is evaluated.

The initial disturbance amplitude (ηo ) is an input to the present to determine the disturbance growth up to breakup. The initial magnitude of the disturbance depends on the particular experimental conditions of nozzle geometry and liquid flow rate. There is much discussion in the atomization literature about determination of the initial disturbance amplitude (Mansour and Chigier 1994 and Grant and Middleman 1966).

However, there is no unanimity on its value for a given geometry and flow conditions.

128 Mansour and Chigier (1994) have shown that the initial disturbance magnitude is lower for a laminar jet by orders of magnitude compared to that for a turbulent jet. They have reported that the stability dynamics of the jet is not affected by the presence of turbulence but the shorter breakup lengths obtained with turbulent jets are primarily the result of higher initial disturbance magnitudes. Based on their experimental measurements of jet

breakup length, values of ln(η0 )= − 22 for a laminar jet and ln(η0 )=− 4.85 (or

−3 η0 ~(10)O ) for a turbulent jet were recommended. Grant and Middleman (1966) have

provided a correlation for the initial disturbance as ln(η0 )= 2.66ln(Z )− 7.68, where

Z = ()We12 Re is the Ohnesorge number. In both cases, the initial disturbance is not directly measured but is extrapolated from the breakup length measurements using a linear stability theory. As such the above values have uncertainty inherent in applying linear theory for jet breakup. Direct measurement of disturbance growth on free falling jets was conduced by Blaisot and Adeline (2000) using high speed photography and image processing. They have predicted initial disturbance magnitude of

−−34 ln(η0 ) ~−− 10 to 14 for low speed jets and that of about -6 to -8 (η0 ~ (10− 10 ) ) for turbulent jets. Using a nonlinear theory, Clark and Dombrowski (1972) have recommended initial dimensionless disturbance magnitude of O(10-4) for planar liquid sheets. Clark and Dombrowski’s theory was extended by Mitra (2001) who has found

−4 good agreement with breakup length of planar sheets with η0 ~(10)O . For the conditions considered here, the dimensionless initial disturbance magnitude is taken to be

~ (10-3 – 10-4) for our calculations. We note that a change in the initial disturbance

129 magnitude will affect the magnitude of the breakup length but it will not affect the trends predicted here in the variation of breakup length.

5.1(a)

5.1(b)

130

5.1(c)

5.1(d)

Figure 5.1: Three dimensional temporal nonlinear evolution of a liquid jet breakup with main and satellite drops. Wel =15 , n = 0 , UW= = 0 , and ρ = 0.0012. (a) t = 0, (b) t =

3, (c) t = 6, and (d) t = 13.5.

131 5.3.1 Model Validation

Figure 5.1 shows the three dimensional nonlinear jet temporal evolution of the most

unstable disturbance at four different time steps for Wel = 15 with two complete waves of the disturbance. In this case the maximum disturbance growth rate of the first order occurs at axial dimensionless wave number (k) of 0.7.The liquid jet Weber number is in the Rayleigh regime where the jet is known to breakup with large main-drop followed by smaller satellite drop (Lafrance 1975, Rutland and Jameson 1971, Nayfeh 1970, Yuen

1968 and Wang 1968, Mansour and Lundgren 1990 and Spangler et al. 1995). Such behavior is evident from the figure where two crests are seen to develop under the action of surface tension, inertial, and pressure forces. A sinusoidal disturbance wave is imposed on the jet surface as seen at t = 0 in Figure 5.1(a). Initial growth is dominated by linear effects with only one peak as seen in Figure 5.1(b) at t = 3. With increasing time, the nonlinear effects become pronounced and multiple peaks are formed (Figure 5.1(c)).

The larger crest leads to the main drop and the smaller to the satellite drop at t = 13.5

(Figure 5.1(d)).

The present analysis is validated by comparing the predictions of breakup length with empirical correlations available in the literature for jet breakup in stationary gas and that in a coaxial gas stream. The effect of axial liquid velocity on the liquid jet break up length in stationary gas is presented in Figure 5.2. For each value of the Weber number, the disturbance wave numbers (axial and circumferential) with the highest growth rate are obtained. The jet deformation is then evaluated to provide the time required to reach jet breakup. Using Gaster’s (1964) transformation, the breakup length can be calculated as the product of the axial velocity of the jet and the breakup time. A similar procedure

132 has been used by Jazayeri and Li (2000) to evaluate breakup of a planar liquid sheet.

Sallam et al. (2002) have carried out measurements of breakup length for round turbulent jets injected in still air. For liquid Weber number below about 400, they have found

Rayleigh-like breakup patterns for turbulent jets. Sallam et al. (2002) have introduced a correlation for the breakup length based on their experimental data as a function of jet

Weber number as

L ρUd2 b = 5.0We0.5 We<=400, We l d σ

It is seen from the figure that the predicted breakup length is in good agreement with the measurements of Sallam et al. (2002).

Further validation of the model is carried out by comparing measurements of breakup length in coaxial gas flow with model predictions. Figure 5.3 shows the breakup length

(/Ldb ) versus axial gas Weber number at axial liquid Weber number of 2. The axial liquid Weber number of 2 was calculated based on the Reynolds number used in the experiments of Eroglu et al. (1991). The breakup length decreases with increasing the gas-to-liquid axial velocity ratio. A balance between the surface tension force and liquid inertia controls the breakup process in the Rayleigh regime.

As the relative velocity between the liquid jet and the surrounding gas stream increases, the inertial effect of the surrounding gas becomes increasingly important and comparable to the surface tension force. The kinetic energy in the gas phase is transferred to the liquid phase and the aerodynamics then disturbs the liquid jet surface, resulting in decreasing the breakup length of the liquid jet.

133 100

80

60 d / b

L 40

20 Sallam et al. (2002) Present Model 0 04080120160 We l

Figure 5.2: Dimensionless breakup length versus axial gas Weber number for n = 0 ,

UW==0, ρ = 0.0012and ηo = 0.0005 .

25 Present Model Eroglu et al. (1991) 20

15 /d b L 10

5

0 01234567 We g

Figure 5.3: Dimensionless breakup length versus gas Weber number for Wel = 2 , n = 0 , W = 0 , ρ = 0.0012and ηo = 0.0001.

134 The predicted breakup lengths are validated with the Eroglu and Chigier’s (1991) correlation

L b = 0.5We−0.4 Re 0.6 d g

2 ρgl()UU− g d ρllUd where Weg ==and Re . σ μl

The values of liquid Reynolds number and liquid Weber number are used to determine the gas and the liquid velocity, which are then used in our model to determine the breakup length. With increasing gas axial velocity, the breakup length decreases. This behavior is correctly predicted by the present analysis and agrees well with the empirical correlation of Eroglu et al. (1991). The empirical correlation plot and our predictions are essentially parallel in the entire range. The differences in the values of the two predictions can be attributed to the uncertainty in determining the initial disturbance amplitude.

The agreement of jet breakup length predictions with empirical correlations, both with and without axial gas flow, provides validation of our analysis. The validated model is used to investigate the effect of gas swirl on jet instability and breakup.

5.3.2 Effect of Gas Swirl

The gas swirl number is defined as the ratio of axial flux of angular momentum to axial

W linear momentum flux, which can be closely approximated by S = (Hopfinger and U

Lasheras 1996) where W is the ratio of gas swirl velocity at the jet surface to liquid axial velocity and U is the gas-to-liquid axial velocity ratio.

135 n=0 n=1 6 n=2 n=3 n=4 n=5 n=6 4 n=7

2 aiugot rate Maximumgrowth

0 00.511.52 Air swirl number (S)

5.4(a)

10 n=0 n=1 n=2 8 n=3 n=4 n=5 6 n=6 n=7

4

2 aiu rwhrate growth Maximum

0 00.511.5 Air swirl number (S)

5.4(b)

Figure 5.4: Maximum disturbance growth rate versus gas swirl number for Wel = 6 and

ρ = 0.0012. (a) U = 20 and (b) U = 30 .

136 The influence of gas swirl strength on the maximum disturbance growth rate for axial gas-to-liquid velocity ratios of 20 and 30 at different circumferential modes are shown in

Figures 5.4(a) and 5.4(b), respectively. In Figure 5.4(a), at small gas swirl number (S <

0.2), the maximum growth rate of the axisymmetric mode (n = 0) is higher than that of the helical modes. As the gas swirl number is increased, the growth rate of the axi- symmetric mode decreases. This behavior at very small gas swirl number agrees with the predictions of Lian and Lin (1990) and Liao et al. (2000) based on linear instability analyses. The decrease in the growth rate of the axi-symmetric disturbance is caused by the radial static pressure variation in the swirling gas stream. The gas pressure increases radially outwards from the jet surface in a swirling gas stream and tends to produce a stabilizing effect on axi-symmetric disturbances. However, as the gas swirl number is increased, the aerodynamic effect on the helical disturbances becomes increasingly dominant and their growth rate increases. For the case presented in Figure 5.4(a), the first transition point is reached at swirl number of about 0.2 where the disturbance growth rate of the first helical mode exceeds that of the axisymmetric mode. For 0.2 < S < 0.8 the maximum growth rate of the helical modes at n = 1 is the highest. The second transition point is reached at about S = 0.8 beyond which the growth rate of the second helical mode is greater than that of the first helical mode. With increasing S, higher helical disturbance modes become the dominant unstable mode and the transition between two successive modes is achieved with a smaller increase in the swirl number. Moreover, increase in the swirl number leads to a large increase in the growth rate of the most unstable helical mode. Beyond n = 5, only a slight increase in S, results in a sharp increase in the growth rate of the dominant disturbance. A breakup dominated by a

137 disturbance with high value of circumferential wave number will rupture the jet in several circumferential fragments and would result in an “explosive breakup” of the jet as has been reported in experimental observations.

The helical modes grow significantly faster at gas-to-liquid axial velocity ratio of

30 than those at gas-to-liquid axial velocity ratio of 20 as seen in Figure 5.4(b). This is not surprising as greater gas kinetic energy is transferred to the liquid jet which hastens the jet breakup. This implies that at higher gas to liquid momentum ratio, a smaller amount of gas swirl is required to enhance the helical modes. Hence, at the same swirl number, higher gas-to-liquid momentum ratio would lead to shorter breakup lengths. The transition to higher and higher circumferential wave number of the dominant disturbance with significant increases in the disturbance growth rate is achieved at a smaller value of swirl number. Experimental investigations have shown that the swirl number for the transition to explosive breakup is a function of the gas-to-liquid momentum ratio (M) and varies as ~ M −0.5 for M < 1 (Dunand et al. 2005 and Hopfinger and Lasheras 1996). Fig.

5 presents the log-scale plot of the gas swirl number at which the circumferential wave number of n = 5 becomes the most dominant wave number versus the gas-to-liquid momentum flux ratio (M) where M is defined as ( ρU 2 ). The swirl number decreases with increasing gas-to-liquid momentum ratio from S of about 4.5 at M = 0.02 to S of about 1 at M = 1. For M < 1, the swirl number variation agrees very well with experimentally observed breakup behavior.

A three-dimensional jet surface profile plot of the helical mode n = 2 at the time of breakup at gas-to-liquid axial velocity ratio of 20 and gas swirl number of 1.1 is shown in Figure 5.6. The helical (asymmetric) disturbance mode n = 2 is the dominant mode at

138 swirl number of 1.1 and axial gas velocity ratio of 20 as seen in Figure 5.4(a). The non- axisymmetric breakup pattern can be seen in Figure 5.6(b).

8 6

4

2

Present Model

Gas (S) number swirl Hopfinger and Lasheras (1996)

10-2 10-1 100 Gas-Liqudi momentum ratio (M)

Figure 5.5: Gas swirl number versus gas-to-liquid momentum ratio for Wel = 6 and ρ = 0.0012.

139

5.6(a)

140

5.6(b)

Figure 5.6: (a) Three dimensional plot of the jet surface and (b) Two dimensional plot at the axial breakup location of the jet surface. ( n = 2, Wel = 6 , U = 20 , W = 22,

ρ = 0.0012 and ηo = 0.0001).

8

6

4 /d b L 2

0 00.250.50.7511.25 Gas swirl number (S)

Figure 5.7: Dimensionless breakup length versus gas swirl number for Wel = 6 , U = 22 ,

ρ = 0.0012 and ηo = 0.0001.

141 As the circumferential wave number of the dominant helical mode increases, the breakup becomes highly asymmetric and ruptures the jet in several circumferential parts in additional to the breakup along axial locations. This type of breakup process would significantly change the ligament shapes and drop size distribution compared to those obtained without imposing a gas swirl. This is consistent with experimental observations

(Dunand et al. 2005 and Hopfinger and Lasheras 1996). Figure 5.7 shows the effect of gas swirl number on the breakup length. At each gas swirl number in the figure, the most unstable circumferential wave number is first determined. The breakup length is then calculated by evaluating the temporal evolution of the jet subjected to the most unstable disturbance. As such the circumferential wave number for breakup for each point shown in the figure may be different. As mentioned earlier, gas swirl promotes the helical modes through increased aerodynamic interaction with the helical disturbance modes. With increasing swirl strength, both the maximum growth rate and the circumferential wave number increase. This leads to a monotonic decrease in the breakup length with increasing swirl strength of the gas.

5.4 Summary and Conclusions

Analysis of a breakup of a coaxial liquid jet in a swirling gas stream is analyzed by a perturbation method with the initial magnitude of the disturbance as the perturbation parameter. The model is used to determine the breakup length with varying liquid-to-gas axial velocity ratio and with different gas swirl strength. For a breakup of liquid jet in an axially moving gas stream, results show that breakup length decreases with an increase in axial gas velocity and the prediction of the trend in decreasing breakup length agrees well with the empirical correlation of Eroglu et al. (1991). In the absence of gas motion, the

142 calculated breakup lengths are seen to agree closely with the correlation of Sallam et al.

(2002).

For a coaxial liquid jet subjected to a swirling gas stream, the axisymmetric disturbance mode (n = 0) is the most dominant only when the gas swirl number is very small. However at higher swirl strength the helical (asymmetric) disturbance modes (n >

0) become dominant compared to the axisymmetric mode. As the gas swirl increases for a fixed gas-to-liquid velocity ratio, the circumferential wave number of the most unstable disturbance increases. Also, the growth rate of the most unstable disturbance significantly increases with increasing gas swirl number. Furthermore, as the gas-to-liquid axial velocity ratio increases, the helical modes grow faster at a smaller gas swirl number. The liquid jet breaks up over a shorter distance at higher gas swirl number. The gas swirl number for transition to a highly asymmetric breakup with a high circumferential wave number (n = 5) is found to vary as the inverse of the square root of the gas-to-liquid momentum ratio when the gas-to-liquid momentum ratio is less than 1.

CHAPTER 6

143 NONLINEAR INSTABILITY OF AN ANNULAR LIQUID SHEET

SUBJECTED TO UNEQUAL INNER AND OUTER GAS STREAMS

6.1 Literature Review

The process of fuel breakup and atomization is of fundamental importance in combustion of liquid fuels as it defines the initial conditions of the process, which strongly affect the performance of a combustion system. In an airblast annular fuel atomizer, the fuel is spread over a filming surface to produce a thin annular, low velocity liquid sheet. The issuing fuel sheet is usually disintegrated by the action of two coaxial high velocity air streams. The quality of atomization depends on the kinetic energy of the atomizing air. Adzic et al. (2001) have visualized the disintegration of an annular liquid sheet in a coaxial airblast injector at low atomizing air velocities. They have demonstrated that the break-up length of the annular liquid sheet is a function of the sum

2 2− 0.44 of the inner and outer air momentum of the coaxial air flows ( ()UUio+ ). This reflects the importance of including the aerodynamic effects of the inner and outer air streams to theoretical instability models to correctly predict the breakup length.

It is well-known that the growth of disturbances on liquid sheet lead to sheet instability and breakup. Linear analyses are available in literature that cover a wide range of configurations i.e. viscous or inviscid annular sheets with or without swirl and with or without surrounding viscous or inviscid gas flows at the same or different velocities. A comprehensive review of linear instability theories for liquid sheets and jets is given by

Lefebvre (1989) and more recently by Lin (2003). The principal drawback of the linear

144 instability theory lies in the fact that the linearized equations used in the theory become inapplicable as the perturbation amplitude grows. The breakup of plane viscous liquid sheets moving in a gas stream has been studied. Mitra et al. (2001) have carried out a dual-mode linear stability analysis under the combined influence of sinuous and varicose modes of disturbances at the two liquid–gas interfaces. They have studied the breakup of plane viscous liquid sheets moving in a gas stream. They have reported that comparison of their linear instability theory with the experimental results shows that the predicted surface deformation agrees favorably with the experiment, but significant deviation occurs near the sheet breakup region, necessitating a nonlinear analysis for a better description of the sheet breakup process. Accordingly, the results based on the theory are valid only for the initial short interval where the perturbation amplitude remains small.

Only few nonlinear analyses of thin annular liquid sheet breakup have been reported (Lee and Wang 1986, Panchagnula et al. 1998, Mehring and Sirignano 1999 and 2000). Lee and Wang (1986) have developed a model for the dynamic formation of spherical shells from an annular inviscid membranes issuing from a nozzle and treated the liquid layer as a membrane moving under the influence of its own inertia, surface tension and gaseous hydrostatic pressure difference between its two sides. They have assumed that the liquid layer has zero thickness with no structure (internal flow) but with finite inertia which is subjected to change due to stretching and relaxing of the sheet during motion.

Panchagnula et al. (1996) have developed a nonlinear model of annular liquid sheet using approximate one dimensional equation derived by thin sheet approximations. They have considered only the para-varicose disturbances, neglected the aerodynamic effects of the gas phase inside and outside the liquid sheet. It has been reported in the earlier studies of

145 Shen and Li (1996) and Ibrahim et al. (2005) that the para-varicose disturbances dominate the breakup process at very low liquid Weber numbers. Most real atomizers exit flows occur in the high liquid Weber number, para-sinuous disturbances dominate the breakup process in this regime. Mehring and Sirignano (1999 and 2000) have developed nonlinear models of axisymmetric thin inviscid infinite (periodically disturbed) and semi-infinite (locally forced) annular liquid sheets in a surrounding void with nonzero gas core pressure at zero gravity by employing a reduced dimension approach (long-wavelength approximation). The main objective of this work is aimed at developing an advanced breakup model to model the primary spray breakup of the airblat atomizer. A weakly nonlinear stability analysis has been carried out for annular liquid fuel sheet subjected to unequal inner and outer gas velocities by a perturbation expansion technique with the initial amplitude of the disturbance as the perturbation parameter. The liquid sheet moves at a uniform axial velocity and subjects to inner and outer gas streams of differing axial velocities.

6.2 Mathematical Formulation

A two-dimensional annular liquid sheet of constant thickness “h” is considered as shown in Figure 6.1. The annular liquid sheet moves at a uniform axial velocity of UL and subjected to inner gas stream which is moving with a uniform velocity of U1and outer gas stream which is moving with a uniform velocity of U2 . The densities of the liquid,

the inner and outer gas phases are ρL , ρi and ρo respectively.

146

Figure 6.1: A schematic of annular liquid sheet

To maintain an annular shape of the liquid surface, a constraint on the mean pressure of

⎛⎞11 the inner and outer air streams must be imposed by PPio−=σ ⎜⎟ + .The liquid ⎝⎠RabR sheet velocities and thickness are determined from the solution of the internal flow field as an input to the current model. The effect of gravity is neglected because the Froude number (defined as the ratio of liquid inertia to gravity) is typically very large for practical sprays. Both phases are assumed to be inviscid and incompressible. In the present formulation, liquid and gas flows are assumed to be initially irrotational (zero vorticity) (White 1991). Therefore, the entire flow field can be treated as a potential flow.

Earlier study of Liao et al. (2001) has shown that the disturbance growth rate increases

147 with Reynolds number and is same as the growth rate of an inviscid annular sheet at high

Reynolds number. Hence, for the cases considered here, effect of viscosity on sheet instability can be neglected. When the base flow, described above, is perturbed by a small disturbance, the two liquid-gas interfaces are displaced to new locations denoted by

raaa=+η and rbbb=+η . It has been reported in the earlier studies of Shen and Li

(1996) and Ibrahim et al. (2005) that the para-varicose disturbances dominate the breakup process at very low liquid Weber numbers. Most real atomizers exit flows occur in the high liquid Weber number, para-sinuous disturbances dominate the breakup process in this regime. Para-sinuous disturbances are only considered in the current nonlinear instability analysis. For the convenience of analysis, all the physical parameters are non- dimensionalized such that length, time and density are scaled with sheet thickness “h”,

the convection time hUL and the liquid density ρL . The dimensionless surface

deformation ηa and ηb and the velocity potential φ for the liquid and the gas phases must satisfy the following governing equations:

Mass conservation:

2 ∇=φL 0 rrrab≤≤ (1)

2 ∇=φi 0 0 ≤≤rra (2)

2 ∇=φo 0 rrb ≤ (3)

Eqs. (1-3) are mass conservation applied to liquid and inner and outer gas phases, respectively.

The interface boundary conditions are listed here:

Kinematic:

148 Liquid:

φLar,,−+ηφη at Lax ,, ax =0 at ra (4)

φLbr,,−+ηφη bt Lbx ,, bx =0 at rb (5)

Inner gas:

φir,,,,−+ηφη at ix ax =0 at ra (6)

Outer gas:

φ η φηφη−+ +oa,,θθ =0 at r (7) or,,,, bt ox ax r 2 b

Dynamic:

2 3 11 11− 1η − −+gU2 ggφφ−+∇−∇= φ22 φ(1 +aθ + η 2 ) 2 ii iit, Lt, ii L 2 ax 22 22WeL r at rr= 222 a ⎪⎪⎧⎫1122ηηηηηaaxaaaxθθθθ2 1+η ax ⎨⎬(1++−22ηηax )aθθ −++ (1 2 ) η axx 2+ ⎩⎭⎪⎪rr r r r WeaL

(8)

2 3 11 111η − −+gU2 ggφφ−+ ∇−∇= φ22 φ(1 ++bθ η 2 ) 2 oo oot,,Lt o o L 2 bx 22 22WeL r at rr= 222 b ⎪⎪⎧⎫1121ηηηηηηbbxbbxbbxθθθθ2 + 2 ⎨⎬(1++−22ηηbx ) bθθ −++ (1 2 ) η bxx 2− ⎩⎭⎪⎪rr r r r WebL

(9)

In the above equations, the dimensionless Liquid Weber number (WeL ), and inner gas

velocity ratio (Ui ), outer gas velocity ratio (Uo ), inner density ratio ( gi ) and outer

density ratio ( go ) are defined as:

149 2 ρLLUh U1 U3 ρ1 ρ3 WeL = ,Ui = ,Uo = , gi = and go = . σ U2 U2 ρ2 ρ2

In order to obtain a solution for ηa and ηb regular perturbation theory is utilized with the

initial disturbance amplitude ηo as the perturbation parameter. By means of series expansion method under the perturbation scheme, the surface deformations or the

location of the two liquid gas interfaces are expanded in power series of ηo as:

n=2 η (,x θηηθ ,)txt= n (, ,) (10) aan∑ o , n=1

n=2 η (,x θηηθ ,)txt= n (, ,) (11) bbn∑ o , n=1

The term η 0 is neglected from the expansion since it corresponds to the unperturbed o

interfaces, which are known. Assuming that ηan, and ηbn, and all its derivatives are of the same order of magnitude, the forms of the kinematic boundary conditions suggest that the velocity potentials for the liquid and the gas phases can also be expanded in power series

of ηo as:

n=2 φ (,,x rtθηφθ ,)= n (,, xrt ,) (12) LLn∑ o , n=0

n=2 φ (,,x rtθηφθ ,)= n (,, xrt ,) (13) iin∑ o , n=0

n=2 φ (,,x rtθηφθ ,)= n (,, xrt ,) (14) oon∑ o , n=0 where φ 0 = x , φ 0 = Ux andφ 0 = Ux represent the base flow field. L i i o o

150 Since the governing equations, Eqs (1-3), are linear, by applying the principle of

superposition, each of the φLn, , φin, and φon, must satisfy the governing equations independently. The corresponding boundary conditions are obtained by substituting Eqs.

(10) - (14) into Eqs. (4) - (9), and equating to zero the successive coefficients of the same

n power of ηo . Under such a scheme, the velocity potentials need to be evaluated at the disturbed interfaces which are part of the solution and not known a priori. In order to

overcome this difficulty,φLn, , φin, and φon, are approximated by a Taylor series expansion around the unperturbed interfaces, ra= and rb= as:

ηη23 aa φφηφφra=+η =+ ra = a + + φ +...... (15) a r ra= 2!rr r== a 3! rrr r a

ηη23 bb φφηφφrb=+η =+ rb = b + + φ +...... (16) b r rb= 2!rr r== b 3! rrr r b

Thus the governing equations and the corresponding boundary conditions, which are evaluated at ra= and rb= , for the first and the second order are obtained and are given below:

First order (ηo ):

2 ∇=φL1 0 arb≤≤ (17)

2 ∇=φi1 0 0 ≤≤ra (18)

2 ∇=φo1 0 br≤ (19)

Liquid:

φLr1,−+ηη at 1, ax 1, =0 at a (20)

φLr1,−+ηη bt 1, bx 1, =0 at b (21)

151 Inner gas:

φir1,−+ηη at 1,U i ax 1, =0 at a (22)

Outer gas:

φor1,−+ηη bt 1,U o bx 1, =0 at b (23)

Dynamic:

1 ηaa11,+η θθ ggUiitφφ1,−+ Lt 1, i iix φφ 1, − Lx 1, =()2 + η axx 1, at a (24) WeL a

−1 ηbb11,+η θθ ggUootφφ1,−+ Lt 1, o oox φφ 1, − Lx 1, =()2 + η bxx 1, at b (25) WeL b

Initial conditions are:

η (,x θθ ,0)=+Cos ( kx n ) & η (,x θ ,0)0= (26) a1 at1,

η (,x θθ ,0)=+Cos ( kx n ) &η (,x θ ,0)0= (27) b1 bt1,

2 Second order (ηo )

2 ∇=φL2 0 arb≤≤ (28)

2 ∇=φi2 0 0 ≤≤ra (29)

2 ∇=φo2 0 br≤ (30)

Liquid:

η η φ −−ηη = ηφ +aa11,θ − ηφ at a (31) Lr2, at 2, ax 2, axLx 1, 1,a2 a 1 Lrr 1,

η η φ −−ηη = ηφ +bb11,θ − ηφ at b (32) Lr2, bt 2, bx 2, bxLx 1, 1,b2 b 1 Lrr 1,

Inner gas:

152 η η φ −−ηηηφU = +aa11,θ − ηφ at a (33) ir2, at 2, iax 2, axix 1, 1,a2 a 1 irr 1,

Outer gas:

η η φ −−ηηηφU = +aa11,θ − ηφ at b (34) or2, at 2, oax 2, axox 1, 1,b2 a 1 orr 1,

Dynamic:

1 ηηaa22,+ θθ ggUiitφφ2,−+ Lt 2, i ii φφ 2, x −− L 2, x()2 + η a 2, xx = WeL a

()()()()ηatLraLrtiatirairtaxLraLrxiiaxirairx1, φ 1,+− ηφ 1 1,ggU η 1, φ 1, ++ ηφ 1 1, η 1, φ 1, +− ηφ 1 1, η 1, φ 1, + ηφ 1 1, 2 222 1122 22(2ηa1 + 4ηηaa11,θθ−− η a 1, θ a η ax 1,) ++−++()()φφLx1, Lr 1,g i φφ ix 1, ir 1, 3 22 2WeL a

at a (35)

1 ηbb22,+η θθ ggUootφφ2,−+ Lt 2, o oo φφ 2, x −+ L 2, x()2 + η bxx 2, = WeL b

()()()()ηφbtLrbLrtobtorbortbxLrbLrxoobxorborx1, 1,+− ηφ 1 1,ggU ηφ 1, 1, ++ ηφ 1 1, ηφ 1, 1, +− ηφ 1 1, ηφ 1, 1, + ηφ 1 1, 2 2222 1122 22φo1,θ (2ηηηηbbbb1+−− 4 1 1,θθ 1, θ b η bx 1, ) ++−()(φφLx1, Lr 1,g o φφ ox 1, +++ or 1, 2 ) − 3 22 b 2WeL b

at b (36)

Initial conditions are:

η (,x θ ,0)0= &η (,x θ ,0)0= (37) a2 at2,

η (,x θ ,0)0= &η (,x θ ,0)0= (38) b2 bt2,

6.2.1 Solution of First and Second Order

The first order surface deformation at the two interfaces can be written in the following form:

η (x ,θθ ,tAtikxncc )=++ ( )exp( ( )) . (39) a1 1

153 η (x ,θθ ,tBtikxn )=++ ( )exp( ( )) cc . (40) b1 1

Substituting Eqns. (39) and (40) into Eqs. (20) – (23), the forms for the liquid and gas velocity potential can be obtained. Substituting these in Eqs. (17) – (19) and solving, we get the liquid and gas velocity potential in terms of At1() and B1()t , its complex conjugate and modified Bessel functions of r. Then substituting the gas and liquid velocity potential in the dynamic boundary conditions Eqns. (24) and (25), using Eqns.

(26) and (27) solving using the method of Laplace transform we find

j=4 At( )= c a exp(ω t ) (41) 1111∑ j=1 jj j

j=4 B (tcbt )= exp(ω ) (42) 1111∑ j=1 jj j

The disturbance growth rates ω11,,ωω 12 13 and ω14 are the rrots of the first order dispersion equations

432 Δ+Δ+Δ+Δ+Δ=11112113114115ωωωωjjjj0 j = 1,2,3& 4 (43)

It has been reported in the earlier studies of Shen and Li (1996) and Ibrahim et al. (2005) that the para-varicose disturbances dominate the breakup process at very low liquid

Weber numbers. Most real atomizers exit flows occur in the high liquid Weber number, para-sinuous disturbances dominate the breakup process in this regime. Para-sinuous disturbance modes are considered in this study.

The first order surface deformation at the two interfaces can be written in the following form

ηθ(x , ,tc )=+() exp( ω tc ) exp( ω t ) exp( ikxncc ( ++ θ )) . (44) a1 11 11 12 12

ηθ(x , ,tRc )=+() ( exp( ω tc ) exp( ω t )) exp( ikxncc ( ++ θ )) . (45) b1 11 11 12 12

154 Where

c.c indicates complex conjugate

ω12 c11 = 2(ω12−ω 11 )

−ω11 c12 = (46) 2(ω12−ω 11 )

ω11 =−αβi

ω12 =−αβ −i

The first order liquid, inner and outer gas velocity potentials are respectively

cc φω=+[11 (frm () f () rRm )exp( t ) ++ 12 ( frm () L11528111622 (47) frRm29( ) )exp(ωθ 12 t )]]exp( ikxn (++ )) cc .

cc φωωθ=+[f (rm )(11 exp( t ) 12 m exp( t ))]exp( ikxncc ( ++ )) . (48) i1 322 17 11 18 12

cc φωωθ=+[f (kr ) R (11 m exp( t ) 12 m exp( t ))]exp( i ( kx ++ n )) c . c (49) o1 422 23 11 24 12

Similarly from equations (28-36) the second order surface deformation at the two interfaces can be written in the following form

η (x ,θθ ,tAt )=+++ ( )exp(2 ikxnccDt ( )) . ( ) (50) a2 2 a

η (x ,θθ ,tBt )=+++ ( )exp(2 ikxn ( )) ccDt . ( ) (51) b2 2 b where

At( )=+ c exp(ω t ) c exp(ωω t ) + c exp(2 t ) + 2 1aaa 21 2 22 3 11 (52) ctct41251112aaexp(2ωωω )++ exp(( ) )

B (tc )=+ exp(ω tc ) exp(ωω tc ) + exp(2 t ) + 2 1bbb 21 2 22 3 11 (53) ctct41251112bbexp(2ωωω )++ exp(( ) )

155 Dta () and Dtb ( ) are required to insure a conservation of mass at t > 0 .Substituting this

in Eqns. (28-30), we find that

2 2''22n ()()kf333+− f f αβ + a2 Da ()tt=− (1cosh(2α )) (54) 8α 2

2 2''22n ()()kf444+− f f αβ + b2 Db ()tt=− (1cosh(2α )) (55) 8α 2

Following a similar procedure to the one described for the first order solution, after lengthy and tedious manipulations, the second order liquid, inner and outer gas velocity potentials can be obtained as follows

φωLab25116132152262422=+[(f (rc ) m f ( rc ) m ) exp( t ) ++ ( f ( rc ) a m f ( rc ) b m ) exp( ω t ) +−+−(f ( rcm )( m ) f ( rcm )( m )) exp(2ω t ) + ( f ( rc )( m − m ) 5 3ab 5 11 6 3 8 14 11 5 4 a 6 12 (56) +−frcm649151255712()(ba m ))exp(2ω t ) + ( frcm ()( −+ m )

frcm6()( 5b 10− m 16 ))exp((ωω11+++ 12 )tikxncc )]exp(2 ( θ )) .

φωia2712921723022=++[()frcm exp() t frcm () a exp() ω t

(f7 (rcm )( 3aa 17−+− m 20 )) exp(2ωω 11 t ) ( f 7 ( rc )( 4 m 18 m 21 )) exp(2 12 t ) (57)

+−(f7 (rc )( 5ai m 19 m 22 )) exp((ωω 11 + 12 ) t ))]exp(2 ikxn ( +++ θ )) ccAt . ( )

φωob2813121823222=++[frcm ( ) exp( t ) frcm ( ) b exp( ω t )

(f8 (rcm )( 3bb 23−+−+ m 26 )) exp(2ωω 11 t ) ( f 8 ( rc )( 4 m 24 m 27 )) exp(2 12 t ) (58)

(f8 (rcm )( 5bo 25−+ m 28 )) exp((ωω 11 12 ) t ))]exp(2 ikxn ( +++ θ )) ccAt . ( )

Where Ati ( ) and Ato ( ) are required to satisfy the dynamic boundary conditions.

Substitution of theses velocity potentials Eqns. (56-58) into the dynamic boundary

condition and the initial conditions, Eqns. (35-38), yields the above time dependent

terms. After considerable amount of simplification, the following solutions for the

second-order are obtained:

156 432 Δ+Δ+Δ+Δ+Δ=21222223224225ωωωωjjjj0 j = 1,2,3& 4 (59)

Substituting the first and second order surface deformation at the two interfaces into

Eqns. (10) and (11). Thus, the expressions for the evolution of the inner and outer gas- liquid interface for an initial harmonic surface disturbance are

η (,xtθηηηη ,)=+12 (60) a oa12 oa

η (,xtθηηηη ,)=+12 (61) b ob12 ob

The first and the second order dispersion equations are solved using MathematicaTM.

The secant method is used which requires two initial guess values. Solutions are considered convergent when values of left hand side of Eqs. (43) and (59) are smaller than 10-6.

The details of the solution method and solutions for η ,,ηηand η are available aa12bb12 in Appendix C.

6.3 Results and Discussions

The disturbance growth rate curves for fundamental (first order) and first harmonic

(second order) modes are shown in Figure 6.2. The maximum disturbance growth rate of the fundamental mode is 0.0155 at axial wave number of 0.045 where as the maximum disturbance growth rate of is about 0.0155 at axial wave number of 0.0225.In order to study the breakup process of the annular liquid sheet, the evolution of the dimensionless surface deformation at the inner and outer liquid-gas interface is plotted and they are given by raaa=+η and rbbb= +η , respectively. The axial wave number used in this study is the axial dominant wave number of the first-order as the characteristic value for

157 k. The temporal and spatial evolution of the inner and outer interfaces at axial liquid

Weber number of 300 for two wavelength intervals axial distance (4/π k ) at different times are shown in Figure 6. The axial dominant wave number (k) is 0.16 and the initial disturbance amplitude of 0.1 is considered. An initial dimensionless disturbance

−4 amplitude ( ηo ) of 0.1 (Om(10 ) ) is considered. Mansour and Chigier (1994) have shown that the initial disturbance magnitude is lower for a laminar jet by orders of magnitude compared to that for a turbulent jet. They have reported that the stability dynamics of the jet is not affected by the presence of turbulence but the shorter breakup lengths obtained with turbulent jets are primarily the result of higher initial disturbance magnitudes. Based on their experimental measurements of jet breakup length, values of

−3 ln(η0 )=− 22 for a laminar jet and ln(η0 )= − 4.85 (or η0 ~(10)O ) for a turbulent jet were recommended. Grant and Middleman (1966) have provided a correlation for the

12 initial disturbance as ln(η0 )=− 2.66ln(Z ) 7.68, where Z = ()We Re is the Ohnesorge number. In both cases, the initial disturbance is not directly measured but is extrapolated from the breakup length measurements using a linear stability theory. As such the above values have uncertainty inherent in applying linear theory for jet breakup. Direct measurement of disturbance growth on free falling jets was conduced by Blaisot and

Adeline (2000) using high speed photography and image processing. They have predicted

initial disturbance magnitude of ln(η0 ) ~− 10 to− 14 for low speed jets and that of about -

−−34 6 to -8 (η0 ~ (10− 10 ) ) for turbulent jets. Using a nonlinear theory, Clark and

Dombrowski (1972) have recommended initial dimensionless disturbance magnitude of

O(10-4) for planar liquid sheets. Clark and Dombrowski’s theory was extended by Mitra

158 (2001) who has found good agreement with breakup length of planar sheets with

−4 η0 ~(10)O . For the conditions considered here, the dimensionless initial disturbance magnitude is taken to be ~ (10-3 – 10-4) for our calculations. It is found that at the initial time instant, the two interfaces are moving parallel to each other. However, at a later time

T = 35, waviness appears at the interfaces due to the nonlinear interaction between the first and second order modes. As the time increases the disturbance amplitude of the inner and outer interfaces increases. The deformation of the inner and outer interfaces leads to thinning and thickening of the liquid sheet. The time at which the thickness of the liquid sheet reaches a near zero value is taken as breakup time. It was found that the absolute instability only occurs for the sinuous mode at small Weber numbers, approximately less than one, and the spatial and temporal instability are related to each other according to Gaster transformation (1962) for large Weber numbers of practical importance. Figure 7(d) shows the spatial evolution of the annular sheet breakup using

Gaster transformation (1962).

159

Figure 6.2: Growth rate versus axial wave number for the fundamental and the first harmonic modes at WeLio=== 80, U U 0, gi = go = 0.001.

T = 0

6.3(a)

160

T = 20

6.3(b)

T = 35

6.3(c)

161 T = 54.5

6.3(d)

Figure 6.3: Temporal Evolution of the dimensionless inner and outer surfaces deformation at the dominant wave number of k = 0.16 for

WeL = 300,UUio==0 , ggio==0.0012 (ηo = 0.1): a) T = 0, (b) T =20, c) T =35 and d) T=

54.5.

162 Tb = 54.5

6.4(a)

Tb = 130

6.4(b)

163

Tb = 54.5

6.4(c)

Figure 6.4: Spatial evolution of inner and outer surface deformations at different initial disturbance amplitudes forWel = 300, UUio= ===0, gg io 0.0012 and k = 0.16 ; a)ηo = 0.1, b) ηo = 0.01 and c)ηo = 0.001.

164 Li (1993) have shown that for liquid Weber numbers much larger than unity the spatial instability of plane liquid sheets is related to the corresponding temporal instability through Gaster's transformation (Gaster 1962), and the phase velocity and group velocity of the surface waves are essentially equal to the liquid sheet velocity. By replacing the dimensionless time t in Eqns. (60) and (61) by the dimensionless distance x from the injector exit, the temporal development of the disturbance waves on the inner and outer surfaces is transformed into the spatial evolution (Jazayeri and Li 2000). The spatial evolutions of the inner and outer surfaces for Wel = 300 at three different initial disturbance amplitudes of 0.1, 0.01 and 0.001 are shown in Figure 6.4. The disturbance waves on the inner and outer surfaces develop at the injector exit and their corresponding amplitudes increases downstream. It can be seen that the disturbance wares remain para- sinuous for most of the annular liquid sheet length. Annular liquid sheet thinning and pinching occur because of the nonlinear effects which lead to the liquid sheet breakup.

With increasing the initial disturbance amplitude, the sheet breakup becomes much closer to the injector exit, and the sheet breakup length is reduced. When the initial disturbance amplitude is plotted against the predicted breakup time on a semi-logarithmic scale, a linear relationship emerges, showing that the temporal growth of the disturbance is exponential-like, just as in the linear theory as shown in Figure 6.5. The effect of Liquid

Weber number on the breakup time at three different initial disturbance amplitudes is shown in Figure 6.6. The higher the liquid Weber number, the larger aerodynamic interactions between the liquid sheet and the surrounding gas. These aerodynamic interactions between the liquid sheet and the surrounding gas would led to the growth of the disturbance waves on the inner and outer surfaces and the disintegration the annular

165 liquid sheet. It can be seen from Figure 6.6 that the breakup time decreases as the liquid

Weber number is increased. Also, with increasing the initial disturbance amplitude the breakup time decreases as shown in Figures 6.5 and 6.6. The current breakup model for annular liquid sheet breakup process is verified with experimental data of Park given in reference (Mitra 2001). The cross-section of an annular nozzle given in reference (Mitra

2001) is shown in Figure 6.7. This annular nozzle produces a liquid sheet of 254μm thick. Photographs of the breakup process of the annular liquid sheet produced by this annular nozzle are provided in Mitra’s dissertation (2001) for two cases. Case 1 is for axial liquid velocity of 1.1 m/s, inner gas velocity of 14 m/s and outer gas velocity of 14 m/s. Case 2 is for axial liquid velocity of 2.1 m/s, inner gas velocity of 27 m/s and outer gas velocity of 27 m/s. The breakup location for case 1 is measured to about 6 mm downstream of the nozzle exit. The predicted breakup length is 7 mm compared to the experimental breakup length measurement of 6 mm as shown in Figure 6.8. Figure 6.9 presents comparison between the predicted breakup length and the experimental breakup length for case 2. The breakup location for case 2 is measured to be about 7 mm downstream of the nozzle exit. The predicted breakup length is 6.5 mm compared to the experimental breakup length measurement of 7 mm. The comparison between the predicted and measured breakup lengths is in good agreement.

166 k = 0.16 200 k = 0.10 k = 0.20

150

100 Tb

50

0 10-3 10-2 10-1 100 η0

Figure 6.5: The effect of initial disturbance amplitude on the breakup time for

Welio===300, U U 0 and ggio= = 0.0012 .

100

80 η0 =0.05

60 ηο =0.1 b T 40 ηο =0.2

20

0 200 400 600 800 1000

Wel

Figure 6.6: The effect of liquid Weber number on the breakup time forUUioo=== W 0 and ,ggio== 0.0012 .

167

Dimensions in mm

Figure 6.7: Dimensions of the annular nozzle and breakup process of the liquid sheet generated by the annular nozzle (Mitra 2001).

The effect of gas-liquid velocity ratio on the ligament shape and the breakup time is presented in Figures 6.10 and 6.11. The breakup time and axial wave number for only inner gas are 25 and 0.39 respectively (Figure 6.10(a)). The breakup time and axial wave number for both inner and outer gas streams are 11.5 and 0.7 respectively (Figure

6.10(c)). The breakup time for stagnant gas as shown in Figure 6.3 is 54.5 and the axial wave number is 0.16. It can be seen from Figures 6.3 and 6.10 that inner/or outer or both inner and outer gas-liquid velocity ratios decreases the breakup time. The maximum disturbance growth rate of only inner axial gas stream is larger than that of only outer axial air stream. This indicates that axial inner gas stream is more effective than outer axial gas stream in enhancing the instability of annular liquid sheets. Such effectiveness of inner gas stream over outer gas stream has been demonstrated in experimental observations by Adzic et al. (2001).

168

0 5 10 15 20 mm

6.8(a)

6.8(b)

Figure 6.8: Evolution of the dimensionless surface deformation r as a function of dimensionless distance x for Wel = 4 , UUio= = 12.7 , gi = go = 0.0012 and = 0.015: a) predictions and b) experiments.

169

0 5 10 15 20 mm

6.9(a)

6.9(b)

Figure 6.9: Evolution of the dimensionless surface deformation r as a function of dimensionless distance x for Wel = 15 ,UUio= = 12.85 , gi = go = 0.0012 and = 0.0001: a) predictions and b) experiments.

170 Tb = 25

6.10(a)

Tb = 44

6.10(b)

171 Tb = 11.5

6.10(c)

Figure 6.10: Temporal evolution of inner and outer surface deformation at different inner and outer gas-liquid velocity ratios for Welioo= 300, g== g 0.0012(η = 0.1): a)UUio== 3, 0 and k = 0.39 , b) UUio= 0,= 3 and k = 0.37 and c) UUio== 3 and k = 0.7 .

only inner gas stream 200 only outer gas stream inner and outer gas streams 160

120 Tb 80

40

0 0 2 4 6 8 10 12 Gas-liquid velocity ratio

Figure 6.11: Effect of gas-liquid velocity ratio on the breakup time for Wel = 4,

ggio= ==0.0012(η o 0.015) .

172 Both inner and outer axial gas streams case has the largest disturbance growth rate compared to only inner gas stream and only outer gas stream because with the presence of both inner and outer axial gas streams, the disturbance extracts energy from both the mean inner and outer axial gas flow. It can be seen that the ligament shape and size during breakup process are different in the three cases. In the case of only outer gas stream, the ligament shape is similar to liquid jet breakup in Rayleigh regime where the jet is known to breakup with large main-drop followed by smaller satellite drop. This is similar to Panchagnula et al. (1998) who reported that the annular sheet breaks up into a main ring followed by a satellite ring at low liquid Weber number. With increasing axial gas-liquid velocity ratio, the maximum growth rate increases as shown in Figure 6.11.

This leads to shorter breakup length which has been seen in experimental measurements reported by Carvalho and Heitor (1998) and Lavergne et al. (1993).

6.4 Summary and Conclusions

An advanced three- dimensional nonlinear stability model has been developed to predict the instability and breakup of an annular liquid sheet down stream of an airblast atomizer.

Features of the model include three-dimensional disturbances, the aerodynamic effects of inner and outer air streams, and annular liquid sheet with finite thickness. A nonlinear instability analysis has been carried out for annular liquid fuel sheet subjected to axially moving inner and outer gas streams by a perturbation expansion technique with the initial amplitude of the disturbance (ηo ) as the perturbation parameter. The liquid sheet is considered to move at a uniform axial velocity and is exposed to inner and outer gas streams of differing axial velocities. The present breakup model for annular liquid sheets

173 can be used to predict the breakup length for plan liquid sheets, cylindrical jets, and annular jet as limiting cases. The breakup length is calculated, and the effect of liquid

Weber number, inner and outer gas-liquid velocity ratios, and initial disturbance amplitude on the breakup time is investigated. It is found that the breakup length is reduced by an increase in the liquid Weber number, initial disturbance amplitude (ηo ) and the inner and outer gas-liquid velocity ratios. The inner gas stream is found to be more effective in disintegrating and enhancing the instability of annular liquid sheets than the outer gas stream. This finding is in agreement with the experimental work of Adzic et al. (2001).

174 CHAPTER 7

NONLINEAR INSTABILITY OF AN ANNULAR LIQUID SHEET

SUBJECTED TO SWIRLING OUTER GAS STREAM

7.1 Introduction

Airblast atomizers have been used in a wide range of industrial and aircraft gas turbine applications. Most of airblast atomizers use an annular prefilming solid surface to spread out the liquid fuel into a thin, hollow-cone sheet. Inside the prefilming airblast atomizer, fuel is first forced into an annular passage to form a thin annular sheet, and then exposed to high speed swirling air streams on both sides. Swirl is imparted to the air streams to generate a recirculation zone thus stabilize flames and enhance mixing. The strong shear action of swirling air streams speeds up the disintegration process of the liquid sheet. Airblast atomization still presents a great challenge to experimental and analytical researchers who are studying breakup and droplet behavior despite many previous studies. Computationally, most of the existing commercial codes ignore film breakup or use simple analytical models to predict the breakup point. Mathematical models that incorporate instability, disturbance wave growth, formation of ligaments, drop secondary breakup, and collision are yet to be verified (Mao, 1991).

Carvalho and Heitor (1998) have studied experimentally the atomization process of a liquid in an axi-symmetric shear layer formed through the interaction of turbulent coaxial jets (respectively, inner and outer jets), with and without swirl, in a model airblast prefilming atomizer. They have reported that the presence of the outer atomizing air leads

175 to a faster decrease of the breakup length, and this influence becomes stronger when swirl is imparted to the outer air, and for increasing swirl levels.

This chapter presents a nonlinear instability analysis of the annular liquid sheet subjected to a swirling outer gas stream. In this chapter a nonlinear instability breakup model is developed to study the effect of the outer gas swirl on the breakup length and ligament formation of the annular liquid sheet. The aerodynamic effects of the gas phase inside and outside the liquid sheet and the asymmetric (non-axisymmteric) disturbances have been considered in the present breakup model.

7.2 Mathematical Formulation

An annular liquid sheet of constant thickness “h” is considered as shown in Figure 1.

Mean flows of the inner, liquid and outer gas are assumed to be (,0,0)U1 , (,0,0)U2 ,

A and (,0,UW= o ), respectively. The densities of the inner, liquid and outer gas phases 33r

are ρ1 , ρ2 and ρ3 respectively. The effect of gravity is neglected because the Froude number (defined as the ratio of liquid inertia to gravity) is typically very large for practical sprays. Both phases are assumed to be inviscid and incompressible. In the present formulation, liquid and gas flows are assumed to be initially irrotational (zero vorticity) (White 1991). Therefore, the entire flow field can be treated as a potential flow.

Earlier study of Liao et al. (2001) has shown that the disturbance growth rate increases with Reynolds number and is same as the growth rate of an inviscid annular sheet at high

Reynolds number. Hence, for the cases considered here, effect of viscosity on sheet instability can be neglected.

176

r

W3

Outer gas ρ3 U3

ρ U Liquid sheet 2 2

Rb

Ra Inner gas ρ1 U1 x

η1

h

η2

Figure 7.1: A schematic of annular liquid sheet.

When the base flow, described above, is perturbed by a small disturbance, the two liquid-

gas interfaces are displaced to new locations denoted by raaa= +η and rbbb=+η . All the physical parameters are non-dimensionalized such that length, time and density are

scaled with sheet thickness h, the convection time hUL and the liquid density ρL . The

177 dimensionless surface deformation ηa and ηb and the velocity potential φ for the liquid and the gas phases must satisfy the following governing equations:

2 ∇=φL 0 rrrab≤≤ (1)

2 ∇=φi 0 0 ≤≤rra (2)

2 ∇=φo 0 rrb ≤ (3)

Eqs. (1-3) are mass conservation applied to liquid and inner and outer gas phases, respectively.

The interface boundary conditions are listed here:

Kinematic:

Liquid:

φLar,,−+ηφη at Lax ,, ax =0 at ra (4)

φLbr,,−+ηφη bt Lbx ,, bx =0 at rb (5)

Inner gas:

φir,,,,−+ηφη at ix ax =0 at ra (6)

Outer gas:

φ η φηφη−+ +oa,,θθ =0 at r (7) or,,,, bt ox ax r 2 b

178 Dynamic:

11 11 −+gU2 ggφφ−+∇−∇= φ22 φ 22ii iit, Lt, 22 i i L 223 2 2 − ⎧⎫ −11ηηaθ 222 ⎪⎪22 aθθθθ1+η ax ηηηη aax a ax (1++22222ηηηηax )⎨⎬ (1 + +− ax )aθθ −+ (1 ) axx + WeL r⎩⎭⎪⎪ r r r r r 1 + WeL a

at rr= a (8)

11W 2 1 11 −−+ggUo 2 ggφφ−+ ∇−∇= φ22 φ 22ooob2 2 oot,,Lt 22 o o L 22223 − ⎧⎫ 11ηηηηηηηbθθ222 ⎪⎪21 b+ bx b θθθ 2 bx b bx (1++22222ηηηηbx )⎨⎬ (1 + +− bx ) bθθ −+ (1 ) bxx + WeL r⎩⎭⎪⎪ r r r r r 1 − WeL b

at rb (9)

In the above equations, the dimensionless Liquid Weber numberWeL , and inner gas

velocity ratio (Ui ), outer gas velocity ratio (Uo ), inner density ratio gi and outer density

ratio go are defined as:

2 ρLLUh U1 U3 Ao ρ1 ρ3 WeL = ,Ui = ,Uo = ,Wo = gi = and go = . σ U2 U 2 hU 2 ρ2 ρ2

In order to obtain a solution for ηa and ηb regular perturbation theory is utilized with the

initial disturbance amplitude ηo as the perturbation parameter. By means of series expansion method under the perturbation scheme, the surface deformations or the

location of the two liquid gas interfaces are expanded in power series of ηo as:

n=2 η (,x θηηθ ,)txt= n (, ,) (10) aan∑ o , n=1

179 n=2 η (,x θηηθ ,)txt= n (, ,) (11) bbn∑ o , n=1

The term η 0 is neglected from the expansion since it corresponds to the unperturbed o

interfaces, which are known. Assuming that ηan, and ηbn, and all its derivatives are of the same order of magnitude, the forms of the kinematic boundary conditions suggest that the velocity potentials for the liquid and the gas phases can also be expanded in power series

of ηo as:

n=2 φ (,,x rtθηφθ ,)= n (,, xrt ,) (12) LLn∑ o , n=0

n=2 φ (,,x rtθηφθ ,)= n (,, xrt ,) (13) iin∑ o , n=0

n=2 φ (,,x rtθηφθ ,)= n (,, xrt ,) (14) oon∑ o , n=0 where φ 0 = x , φ 0 = Ux andφ 0 =+Ux Wθ represent the base flow field. L i i o oo

Since the governing equations, Eqs (1-3), are linear, by applying the principle of

superposition, each of the φLn, , φin, and φon, must satisfy the governing equations independently. The corresponding boundary conditions are obtained by substituting Eqs.

(10) - (14) into Eqs. (4) - (9), and equating to zero the successive coefficients of the same

n power of ηo . Under such a scheme, the velocity potentials need to be evaluated at the disturbed interfaces which are part of the solution and not known a priori. In order to

overcome this difficulty,φLn, , φin, and φon, are approximated by a Taylor series expansion around the unperturbed interfaces, ra= and rb= as:

180 ηη23 aa φφηφφra=+η =+ ra = a + + φ +...... (15) a r ra= 2!rr r== a 3! rrr r a

ηη23 bb φφηφφrb=+η =+ rb = b + + φ +...... (16) b r rb= 2!rr r== b 3! rrr r b

Thus the governing equations and the corresponding boundary conditions, which are evaluated at ra= and rb= , for the first and the second order are obtained and are given below:

First order (ηo ):

2 ∇=φL1 0 arb≤≤ (17)

2 ∇=φi1 0 0 ≤≤ra (18)

2 ∇=φo1 0 br≤ (19)

Liquid:

φLr1,−+ηη at 1, ax 1, =0 at a (20)

φLr1,−+ηη bt 1, bx 1, =0 at b (21)

Inner gas:

φir1,−+ηη at 1,U i ax 1, =0 at a (22)

Outer gas:

W φη−+U η +o η =0 at b (23) or1, bt 1, o bx 1,b2 b 1,θ

Dynamic:

1 ηaa11,+η θθ ggUiitφφ1,−+ Lt 1, i iix φφ 1, − Lx 1, =()2 + η axx 1, at a (24) WeL a

181 2 WWoo−1 ηηbb11,+ θθ ggUgootφ 1,−+φφ Lt 1, o oox 1, + o223 φφ o 1,θ − Lx 1, =() + η bxx 1, + g o η b 1 at b (25) bWebbL

Initial conditions are:

η (,x θθ ,0)=+Cos ( kx n ) & η (,x θ ,0)0= (26) a1 at1,

η (,x θθ ,0)=+Cos ( kx n ) &η (,x θ ,0)0= (27) b1 bt1,

2 Second order (ηo )

2 ∇=φL2 0 arb≤≤ (28)

2 ∇=φi2 0 0 ≤≤ra (29)

2 ∇=φo2 0 br≤ (30)

Liquid:

η η φ −−ηη = ηφ +aa11,θ − ηφ at a (31) Lr2, at 2, ax 2, axLx 1, 1,a2 a 1 Lrr 1,

η η φ −−ηη = ηφ +bb11,θ − ηφ at b (32) Lr2, bt 2, bx 2, bxLx 1, 1,b2 b 1 Lrr 1,

Inner gas:

η η φ −−ηηηφU = +aa11,θ − ηφ at a (33) ir2, at 2, iax 2, axix 1, 1,a2 a 1 irr 1,

Outer gas:

WWη η 2 φ −−ηηU −oo ηηφ = +aa11,θ − ηφηη − at b (34) or2, at 2, oax 2,bbb223 a 2,θ axox 1, 1, a 1 orr 1, a 1 a 1,θ

182 Dynamic:

1 ηηaa22,+ θθ ggUiitφφ2,−+ Lt 2, i ii φφ 2, x −− L 2, x()2 + η a 2, xx = WeL a

()()()()ηatLraLrtiatirairtaxLraLrxiiaxirairx1, φ 1,+− ηφ 1 1,ggU η 1, φ 1, ++ ηφ 1 1, η 1, φ 1, +− ηφ 1 1, η 1, φ 1, + ηφ 1 1, 2 222 1122 22(2ηa1 + 4ηηaa11,θθ−− η a 1, θ a η ax 1,) ++−++()()φφLx1, Lr 1,g i φφ ix 1, ir 1, 3 22 2WeL a at a (35)

2 WWoo1 ηbb22,+η θθ ggUgootφφ2,−+ Lt 2, o oo φ 2, x + o223 φ o 2,θ −+ φ L 2, x() + η bxx 2, − g o η b 2 = bWebbL

()()()()ηφbtLrbLrtobtorbortbxLrbLrxoobxorborx1, 1,+− ηφ 1 1,ggU ηφ 1, 1, ++ ηφ 1 1, ηφ 1, 1, +− ηφ 1 1, ηφ 1, 1, + ηφ 1 1, 2 22 2 W 2ηφ3gWη 11 φ −+g o (ηφ ηφ −−bo11,θ )()()oob1 ++−+++φφ22g φφ 22 o1,θ oborbb2 1,θ 1, 1 or1,θ bb2242 Lx 1, Lr 1, 2 o ox 1, or 1, b 2222 (2ηηηηbbbb1+−− 4 1 1,θθ 1, θ b η bx 1, ) − 3 2WeL b at b (36)

Initial conditions are:

η (,x θ ,0)0= &η (,x θ ,0)0= (37) a2 at2,

η (,x θ ,0)0= &η (,x θ ,0)0= (38) b2 bt2,

7.3 Solution of First and Second Order

The first order surface deformation at the two interfaces can be written in the following form:

η (x ,θθ ,tAtikxncc )=++ ( )exp( ( )) . (39) a1 1

η (x ,θθ ,tBtikxn )=++ ( )exp( ( )) cc . (40) b1 1

Substituting Eqns. (39) and (40) into Eqs. (20) – (23), the forms for the liquid and gas velocity potential can be obtained. Substituting these in Eqs. (17) – (19) and solving, we

183 get the liquid and gas velocity potential in terms of At1() and B1()t , its complex conjugate and modified Bessel functions of r. Then substituting the gas and liquid velocity potential in the dynamic boundary conditions Eqns. (24) and (25), using Eqns.

(26) and (27) solving using the method of Laplace transform we find

j=4 At( )= c a exp(ω t ) (41) 1111∑ j=1 jj j

j=4 B (tcbt )= exp(ω ) (42) 1111∑ j=1 jj j

The disturbance growth rates ω11,,ωω 12 13 and ω14 are the rrots of the first order dispersion equations

432 Δ+Δ+Δ+Δ+Δ=11112113114115ωωωωjjjj0 j = 1,2,3& 4 (43)

It has been reported in the earlier studies of Shen and Li (1996) and Ibrahim et al. (2005) that the para-varicose disturbances dominate the breakup process at very low liquid

Weber numbers. Most real atomizers exit flows occur in the high liquid Weber number, para-sinuous disturbances dominate the breakup process in this regime. Para-sinuous disturbance modes are considered in this study.

The first order surface deformation at the two interfaces can be written in the following form

ηθ(x , ,tc )=+() exp( ω tc ) exp( ω t ) exp( ikxncc ( ++ θ )) . (44) a1 11 11 12 12

ηθ(x , ,tRc )=+() ( exp( ω tc ) exp( ω t )) exp( ikxncc ( ++ θ )) . (45) b1 11 11 12 12

Where

c.c indicates complex conjugate

184 ω12 c11 = 2(ω12−ω 11 )

−ω11 c12 = (46) 2(ω12−ω 11 )

ω11 =−αβi

ω12 =−αβ −i

The first order liquid, inner and outer gas velocity potentials are respectively

cc φω=+[11 (frm () f () rRm )exp( t ) ++ 12 ( frm () L11528111622 (47) frRm29( ) )exp(ωθ 12 t )]]exp( ikxn (++ )) cc .

cc φωωθ=+[f (rm )(11 exp( t ) 12 m exp( t ))]exp( ikxncc ( ++ )) . (48) i1 322 17 11 18 12

cc φωωθ=+[f (kr ) R (11 m exp( t ) 12 m exp( t ))]exp( i ( kx ++ n )) c . c (49) o1 422 23 11 24 12

Similarly from equations (28-36) the second order surface deformation at the two interfaces can be written in the following form

η (x ,θθ ,tAt )=+++ ( )exp(2 ikxnccDt ( )) . ( ) (50) a2 2 a

η (x ,θθ ,tBt )=+++ ( )exp(2 ikxn ( )) ccDt . ( ) (51) b2 2 b where

At( )=+ c exp(ω t ) c exp(ωω t ) + c exp(2 t ) + 2 1aaa 21 2 22 3 11 (52) ctct41251112aaexp(2ωωω )++ exp(( ) )

B (tc )=+ exp(ω tc ) exp(ωω tc ) + exp(2 t ) + 2 1bbb 21 2 22 3 11 (53) ctct41251112bbexp(2ωωω )++ exp(( ) )

Dta () and Dtb ( ) are required to insure a conservation of mass at t > 0 .Substituting this in Eqns. (28-30), we find that

185 2 2''22n ()()kf333+− f f αβ + a2 Da ()tt=− (1cosh(2α )) (54) 8α 2

2 2''22n ()()kf444+− f f αβ + b2 Db ()tt=− (1cosh(2α )) (55) 8α 2

Following a similar procedure to the one described for the first order solution, after lengthy and tedious manipulations, the second order liquid, inner and outer gas velocity potentials can be obtained as follows

φωLab25116132152262422=+[(f (rc ) m f ( rc ) m ) exp( t ) ++ ( f ( rc ) a m f ( rc ) b m ) exp( ω t ) +−+−(f ( rcm )( m ) f ( rcm )( m )) exp(2ω t ) + ( f ( rc )( m − m ) 5 3ab 5 11 6 3 8 14 11 5 4 a 6 12 (56) +−frcm649151255712()(ba m ))exp(2ω t ) + ( frcm ()( −+ m )

frcm6()( 5b 10− m 16 ))exp((ωω11+++ 12 )tikxncc )]exp(2 ( θ )) .

φωia2712921723022=++[()frcm exp() t frcm () a exp() ω t

(f7 (rcm )( 3aa 17−+− m 20 )) exp(2ωω 11 t ) ( f 7 ( rc )( 4 m 18 m 21 )) exp(2 12 t ) (57)

+−(f7 (rc )( 5ai m 19 m 22 )) exp((ωω 11 + 12 ) t ))]exp(2 ikxn ( +++ θ )) ccAt . ( )

φωob2813121823222=++[frcm ( ) exp( t ) frcm ( ) b exp( ω t )

(f8 (rcm )( 3bb 23−+−+ m 26 )) exp(2ωω 11 t ) ( f 8 ( rc )( 4 m 24 m 27 )) exp(2 12 t ) (58)

(f8 (rcm )( 5bo 25−+ m 28 )) exp((ωω 11 12 ) t ))]exp(2 ikxn ( +++ θ )) ccAt . ( )

Where Ati ( ) and Ato ( ) are required to satisfy the dynamic boundary conditions.

Substitution of theses velocity potentials Eqs. (56-58) into the dynamic boundary

condition and the initial conditions, Eqs. (35-38), yields the above time dependent terms.

After considerable amount of simplification, the following solutions for the second-order

are obtained:

432 Δ+Δ+Δ+Δ+Δ=212ωωωωjjjj 222 232 242 25 0 j = 1,2,3& 4 (59)

186 Substituting the first and second order surface deformation at the two interfaces into Eqs.

(10) and (11). Thus, the expressions for the evolution of the inner and outer gas-liquid interface for an initial harmonic surface disturbance are

η (,xtθηηηη ,)=+12 (60) a oa12 oa

η (,xtθηηηη ,)=+12 (61) b ob12 ob

The first and the second order dispersion equations are solved using MathematicaTM. The secant method is used which requires two initial guess values. Solutions are considered convergent when values of left hand side of Eqs. (43) and (59) are smaller than 10-6.

The details of the solution method and solutions for η ,,ηηand η are available aa12bb12 in Appendix D.

7.4 Results and Discussions

Airblast atomizer has been considered as an advanced fuel injection device and is widely used in gas turbine engines and oil-fired furnaces. These advantages are due to the fact that the airblast atomizer employs kinetic energy of the high-speed swirling air streams to breakup the annular liquid sheet. According to the filming method, the airblast atomizer can be divided into two types, i.e., the prefilming airblast atomizer and the swirl-cup airblast atomizer. Inside the traditional prefilming airblast atomizer, liquid fuel is first spread out into a thin annular sheet, and then exposed to high-speed swirling air streams on both sides (Lefebvre 1980). Therefore it is very important to study the influence of gas swirl of the breakup process which would lead to improvement in the performance of airblast atomizers. Figure 7.2 presents the temporal evolution of the dimensionless inner

187 and outer surfaces deformation at different outer gas swirl strength values (Wo ) of 0, 50,

70, 100 and 210 for Weli== 4, U 0 andUo = 15 . At no outer gas swirl, the maximum growth rate of the axisymetric mode (n = 0) is higher than that of the helical modes. The breakup time for no gas swirl case is 56.5 at axial (k) and circumferential (n) wave numbers of 0.26 and zero (Figure 7.2a). As the outer gas swirl strength (Wo ) is increased, the growth rate of the axisymmetric mode decreases and the asymmetric

(helical) mode (n > 0) becomes the dominant mode. For outer gas swirling strength of 50, the optimal disturbance growth rate occurs at the second helical mode (n = 2). As the outer gas swirl strength increases from 0 to 50 the breakup time decreased from 56.5 to

46.5. It can be observed from Figure 7.2 that outer gas swirl not only increases the growth rate and the range of unstable wave numbers but also shifts the dominant mode from the axisymmetric mode (n = 0) to a helical mode (n > 0). This in agreement with the previous linear instability analyses of Liao et al. (2000 and 2001) and Ibrahim and Jog

(2006) and with the experimental investigations of Carvalho and Heitor (1998), Chin et al. (2000) and He et al. (2003). As the outer gas swirl strength increases, the maximum disturbance growth rate occurs at higher circumferential wave numbers. Also, the disturbance growth rate of higher helical modes as n = 5 in Figures 7.2 (d) and n = 14 in

Figure 7.2(e) become higher than those of the lower helical modes as n = 2 in Figure 7.2

(b) and n = 3 in Figure 7.2(c). It can be seen that the outer gas swirl strength speeds up the breakup process of the annular liquid sheet as in Figure 7.2 (d). This is because increasing the amount of gas strength is enhancing helical disturbance modes. At outer gas swirl strength (Wo ) of 210, the circumferential wave number (n) becomes dominate and reaches a value of 14. Two dimensional plots at the axial breakup location of the

188 annular liquid sheet for Welio= 4, U==== 0, U 15, g io g 0.0012 are presented in Figure

7.3. These two dimensional figures are plotted at the axial breakup points as shown in

Figure 7.2. Figure 7.4 shows the spatial plot of the inner and outer surface deformation for two different outer gas swirl strength values. As the outer gas swirl strength increases from 50 to 100, the breakup time decreases from 46.5 to 34.67. When the outer gas swirl strength reaches a value of Wo = 210 the breakup time is decreased to 14.75. It can be reported based on the previous discussion that enhancement of helical modes due to outer gas swirl is far more significant than that of the axisymmetric mode. When swirl is imparted to the outer air stream, and for increasing swirl levels, a faster decrease of the breakup length is occurred. This finding is in agreement with the experimental observations of Carvalho and Heitor (1998).

189 Tb = 56.5

7.2(a)

Tb = 46.5

7.2(b)

190 Tb = 42.2

7.2(c)

Tb = 34.67

7.2(d)

191 Tb = 14.75

7.2(e)

Figure 7.2: Temporal evolution of the dimensionless surface deformation r at different outer gas swirl strength values for Welio= 4, U==== 0, U 15, g io g 0.0012 and

ηo = 0.001: a) Wno == 0, 0, b) Wno = 50,= 2, c) Wno = 70,= 3 , d) Wno == 100, 5 and e) Wno ==210, 14 .

192 n = 0

7.3(a)

n = 2

7.3(b)

193 n = 3

7.3(c)

n = 5

7.3(d)

194 n = 14

7.3(e)

Figure 7.3 : Two dimensional plot at the axial breakup location of the annular liquid sheet for Welio== 4, U 0, U = 15, g io == g 0.0012 and ηo = 0.001: a) Wno == 0, 0, b)

Wno ==50, 2, c) Wno == 70, 3 , d) Wno = 100,= 5 and e) Wno = 210,= 14 .

7.4(a)

195

7.4(b)

Figure 7.4: Spatial evolution of the dimensionless surface deformation r at different gas swirl strength values for Welio= 4, U==== 0, U 15, g io g 0.0012 and ηo = 0.001: a)

Wno ==50, 2and b) Wno == 100, 5 .

7.4 Summary and Conclusions

A three-dimensional nonlinear temporal instability analysis has been carried out to model airblast atomization. The model considers an annular liquid sheet downstream of an airblast atomizer and incorporates essential features, such as three-dimensional disturbances, aerodynamics of inner and outer air streams and outer air swirl. A weakly nonlinear instability breakup model has been developed for annular liquid fuel sheet to

196 predict the breakup length by a perturbation expansion technique with the initial amplitude of the disturbance (ηo ) as the perturbation parameter. The liquid sheet is considered to move at a uniform axial velocity. The axial velocities of the inner and outer air streams are assumed to be uniform. The outer air stream swirl velocity profile is considered as a free vortex profile. The breakup length is calculated, and the effect of outer air swirl strength on the breakup time is investigated. It is found that the breakup length is reduced by an increase in the outer gas strength. When outer air swirl is absent as seen in chapter 6, the axisymmetric mode (n = 0) dominates the breakup process of the annular liquid sheet. Air swirl not only promotes the instability of the liquid sheet, but also switches the dominant mode from the axisymmetric mode to a helical mode (n > 0).

As outer air swirl strength increases, the circumferential wave number (n) increases and the ligament shapes at the breakup time become different than those at no air swirl.

Spatial plots of the inner and outer surface deformations show that the breakup process is asymmetric as has been reported in the experiments of Eroglu et al. 1991. These conclusions have been validated by several experimental investigations (Lavergne et al.

1993, Carvalho and Heitor 1998 and Chin et al. 2000) that revealed that the swirling airstreams promotes the disintegration of the liquid sheet, apart from enhancing fuel/air mixing and stabilizing the flame.

197

PART IV

A COMPREHENSIVE MODEL FOR PRESSURE SWIRL

ATOMIZER

198 CHAPTER 8

A COMPREHENSIVE MODEL TO PREDICT PRESSURE SWIRL

ATOMIZER PERFORMANCE

8.1 Motivation

Pressure-swirl atomizers or simplex atomizers are used in a variety of applications including liquid fuel injection systems in gas turbine combustors as well as industrial oil fired furnaces as discussed in chapter 1. The breakup and atomization of the annular liquid sheet is responsible for the spray formation and determines the resultant spray characteristics such as mean droplet diameter and droplet size distribution. The breakup instability model which is developed in chapter 7, which identifies specific effect of various forces on the breakup process of the annular liquid sheet, becomes an important sub-model of the atomization model. In the meanwhile, due to the dense spray nature near the nozzle exit, it becomes very difficult to measure such quantities as droplet size and size distribution at the nozzle orifice. Currently, most of CFD simulation codes depend on empirical atomization characteristics as initial conditions. Therefore, it is highly desirable to develop advanced breakup and atomization models as shown in chapters 5, 6 and 7 to be incorporated in injection and atomization simulation codes that could be used to guide the optimization of injector design. As a result; the success of computational fluid dynamics (CFD) simulation of spray combustion depends largely on the atomization model, which provides the initial conditions. Therefore, establishment of

199 an advanced comprehensive model for pressure swirl atomizer will not only benefit the atomizer design but also improve CFD design tools for gas turbine engines.

Chuech (1992, 1993) have developed a mathematical model that used curvilinear body-fitted coordinates to study the evolution of non-swirling and swirling annular liquid sheets injected from an atomizer into a quiescent surrounding medium. They have reported predictions of the spray angle, sheet thickness and velocities that were in general agreement with their experimental measurements. Ibrahim and McKinney (2006) have reported that despite that Chuech model contains a formulation for liquid-surrounding interfacial friction effects, the results they presented apply only to a sheet issued in a void. Ibrahim and McKinney (2006) have developed a simplified mathematical model based on body fitted coordinates to study the evolution of non-swirling and swirling liquid sheets emanated from an annular nozzle in a quiescent surrounding medium.

Cousin et al. (1999) have done simulation of the single phase flow inside pressure swirl atomizer and used flat sheet linear instability model to predict the droplet diameter. They have not taken the air core into account in their study. Liao et al. (1999) have presented a comprehensive model to predict the performance of a simplex atomizer.

They have predicted the internal flow features and properties of the liquid sheet emanating from the nozzle using CFD model based on the ALE method. They have developed a linear instability model. They have not predicted the breakup length in their study because of the deficiency of the linear instability analysis. They have used the experimental break up length to predict the mean droplet diameter for simplex atomizers.

Ibrahim et al. (2004) have investigated the effect of liquid swirl velocity profile on mean

200 droplet diameter predictions for simplex atomizers. In their study the have not calculated the breakup length but they have used the experimental breakup length.

A comprehensive model of atomization in a pressure-swirl atomizer requires determination of the flow field inside the atomizer, evaluation of the characteristics of the liquid sheet exiting from the atomizer, and modeling of the sheet instability and breakup as shown in Figure 8.1. First, simulations of the internal flow filed will be discussed, especially the prediction of the liquid sheet characteristics at the atomizer exit. Second, a coupling between internal flow field calculations and nonlinear instability breakup model in order to predict initial spray characteristics of simplex atomizers will be suggested.

The internal flow field has been determined computationally using Volume of

Fluid (VOF) method. A non-linear instability model is then employed to determine the annular sheet instability and breakup. The results for annular liquid sheet thickness and velocities from the internal flow field solution are used as input in the non-linear instability breakup model. The coupled internal flow field simulation and sheet instability analysis provides a comprehensive approach to modeling atomization from a pressure- swirl atomizer. This comprehensive model can be used to directly to determine the effect of flow conditions and pressure-swirl atomizer geometry on the sheet breakup. The present model shows method to couple internal flow simulations and instability models to optimize pressure swirl atomizers. Also, this approach gives encouraging results and could be used as initial conditions for spray preparation modeling.

201

Figure 8.1: Schematic of the current comprehensive model.

8.2 Results and Discussions

8.2.1 Internal Flow Field

The computational model assumes the flow to be axisymmetric based on the experimental work of Ma (2001). This assumption requires determination of an equivalent “annular” inlet slot instead of the finite number of slots present in the real nozzle. The width of the “annular” slot as well as the radial and tangential velocities at the inlet are calculated by matching the angular momentum, total mass flow rate, and the kinetic energy of the liquid at the inlet ports with those of the experiments.

202 The boundary condition for the inlet was taken to be radial and tangential velocity components. The inlet tangential and radial velocities at the wall of the swirl chamber

2 Q ()DDs − p ⎛⎞Q would be given by W = andVW=−⎜⎟ 2 , thus the inlet width can be inlet AD inlet⎜⎟ inlet ps ⎝⎠Ap calculated.

Velocity inlet Wall Wall Wall Pressure outlet Wall

Axis

Figure 8.2: Physical model and boundary conditions (case 1).

203

Table 8.1: Cases of study

Inlet Slot Orifice Mass flow Case Number of Swirl chamber Area Do, Lo rate # ports 2 Ds, Ls (mm) AP (mm ) (mm) (kg/s) 1 2 203 21.6, 36.7 76, 38 0.5 2 4 406 28.8, 40.5 76, 38 0.95 3 2 203 28.8, 40.5 76, 38 0.95 4 2 203 21.6, 36.7 76, 89 0.63 (Orifice contraction angle is 45o for all cases)

Table 8.2: Results for two different grid densities for case 1.

Film thickness, h Spray cone angle Discharge coefficient (mm) ( 2θ ) (Cd ) 26682 cells 2.2579 86.18 0.178 46932 cells 2.2704 86.18 0.177

Table 8.3: Comparison of Computational Results with Experimental Measurements

Spray cone angle Discharge coefficient Film thickness, h (mm) Case # ( 2θ ) (Cd ) Exp. CFD Exp. CFD Exp. CFD 1 2.24 2.27 84.7 86.18 0.17 0.177 2 3.06 3.02 76.9 75.75 0.20 0.218 3 2.41 2.177 89.9 90.59 0.12 0.133 4 2.37 2.77 76.9 82.37 0.16 0.120

204 At the outlet cross-section, pressure outlet boundary was prescribed as shown in Figure

8.2. All wall boundaries are taken as no-slip. For all cases, the fluids are air and water.

The geometric and flow parameters for the cases considered are shown in Table 8.1.

8.2.1.1 Validation

In order to ensure grid independence of results, two sets of results for the same flow conditions and geometry of case 1 are shown in Table 8.2 with 26682 and 46932 cells, respectively. The difference in results between the two grids is small and indicates that

26682 cells or similar grid is sufficient to get accurate results. The CFD results based on

VOF method were validated by comparison with experimental data (Benjamin et al.

1998) for discharge parameters, spray cone angle, film thickness, and discharge coefficient using a large-scale atomizer. Table 8.3 shows a comparison of computational predictions of film thickness, spray angle, and discharge coefficient with experimental measurements for all the cases. The discharge coefficient was calculated as

Q . The spray cone angle is measured by making a tangent to the outer Cd = APOrifice2/Δ ρ L spray interface. It is seen that the computational results agree very well with the experimental measurements. Figure 8.3 shows computed axial velocity in the pressure swirl atomizer for case 1. As a result of the swirling motion (centrifugal forces) within the swirl chamber, the pressure decreases towards the center axis. Due to the lower pressure near the center axis, an air-core is formed along the centerline.

205 m/s

Figure 8.3: Computed axial velocity in the atomizer (case 1)

Figure 8.4: Contours of volume fraction (case 1)

206 The positive and negative velocities indicate the existence of a number of small-scale vortices in this region, and exhibit flow patterns with recirculation. The volume fraction contours are shown in Figure 8.4. The air core radius is located at volume fraction of 0.5.

8.3 Breakup Length Calculation and Validation

The internal flow field results (sheet thickness, averaged axial velocity) are input to the developed breakup model in chapter 6. In order to study the breakup process of the annular liquid sheet, the breakup model the evolution of the dimensionless surface deformation at the inner and outer liquid-gas interface is plotted. The nonlinear instability breakup model is validated with the available experimental data for large scale prototype atomizer (Benjamin et al. 1998). The breakup length (Lb) of an annular liquid sheet is estimated from the breakup time (Tb) and the liquid velocity (UL ) as

LTUbbL= where the axial liquid velocity is considered constant.

A comparison of experimental and prediction results of the breakup length for large scale pressure swirl injectors is shown in Table 8.4. The predictions and experimental results are in good agreement.

207 Table 8.4: Comparison of predicted with measured breakup length (large Scale nozzle

(Benjamin et al. 1998))

Case WeL at orifice Breakup Length (mm) ηo # exit Experiment. Prediction 1 421 121 113 0.11 2 614 140 127 0.1 3 741 102 104 0.18 4 630 127 90 0.1

8.4 Summary and Conclusions

A computational modeling of the turbulent, unsteady, two-phase flow in pressure- swirl atomizers is carried out using the Volume-of-Fluid (VOF) method. The method is validated by comparing velocity variations, discharge coefficient, sheet thickness, and spray angle with the experimental measurements of flow in large-scale prototype atomizers (Benjamin et al. 1998). The effect of mesh size on the computational results is investigated. The coupled internal flow field simulation and nonlinear instability breakup model provides a comprehensive approach to modeling atomization from a pressure-swirl atomizer. For the cases considered here, the comparison between the predicted and measured (Benjamin et al. 1998) breakup length shows a good agreement. This comprehensive model can be used to directly determine the effect of flow conditions and nozzle geometry on the sheet breakup. The present model shows method to couple internal flow simulations and instability models to optimize pressure swirl atomizers.

Also, this approach gives encouraging results and could be used as initial conditions for spray preparation modeling.

208 CHAPTER 9

CONCLUSIONS AND RECOMMENDATIONS

9.1 Summary and Conclusions

The Volume-of-Fluid (VOF) method is employed to track the gas-liquid interface and predict the internal flow field features and properties of the liquid sheet emanating from the pressure swirl atomizers. Internal flow characteristics of the simplex atomizer, as well as its performance parameters such as velocity variations, discharge coefficient, spray angle and film thickness, are predicted. The computational model has been validated by comparisons with experimental measurements (Ma 2001 and Benjamin et al.

1998). Effect of liquid viscosity and air pressure is investigated. Increase in liquid viscosity leads to lower axial and tangential velocities. The diameter of the air core decreases with increase in liquid viscosity resulting in higher liquid film thickness in the exit orifice. Also, the spray cone angle decreases and discharge coefficient increases with higher liquid viscosity. Results show that even with a ten-fold increase in air pressure, the change in film thickness at the atomizer exit is very small.

Linear asymmetric instability analyses of swirling or non-swirling annular liquid sheet subjected to swirling or non-swirling inner and outer gas streams at different velocities are performed under different conditions. It is found that the presence of liquid swirl and the type of liquid swirl velocity profile have a significant impact on the instability of the liquid sheet. Liquid swirl imposed on an annular liquid sheet has a dual effect on its stability where swirl has a stabilizing effect at low values of liquid swirl

209 Weber number whereas it has a destabilizing effect at higher values of swirl Weber number. As the liquid swirl Weber number increases the growth rate of para-sinuous disturbances increases. However, the growth rate for para-varicose disturbances decreases with increasing liquid swirl and eventually at higher liquid swirl the sheet is no longer unstable to para-varicose disturbances. The disturbance growth rates with a free vortex type profile are lower compared to those with a solid vortex type profile at very low swirl Weber numbers. The behavior is opposite at moderate to high swirl Weber numbers.

Nonlinear asymmetric (non-axisymmetric) instability analyses are developed to study the primary atomization of annular liquid sheets and liquid jets emanating from the pressure swirl atomizer (simplex) atomizer, prefilming airblast atomizer, and plain orifice pressure atomizer. The aerodynamic effects of the inner and outer gas streams as well as the swirling outer gas are considered in the annular sheet models. The nonlinear models for annular liquid sheet and liquid jet are carried out by a perturbation expansion technique with the initial amplitude of the disturbance as the perturbation parameter. The first, second and order governing equations have been derived along with appropriate initial and boundary conditions. The breakup characteristics of liquid sheets and liquid jets are studied for the initial sinuous disturbance mode. For a breakup of liquid jet in an axially moving gas stream, results show that breakup length decreases with an increase in axial gas velocity and the prediction of the trend in decreasing breakup length agrees well with the empirical correlation of Eroglu et al. (1991). For liquid jet and annular liquid sheet, outer gas swirl promotes the helical modes through increased aerodynamic interaction with the helical disturbance modes which lead to a decrease in the breakup

210 length. As the circumferential wave number of the dominant helical mode increases, the breakup becomes highly asymmetric and ruptures the jet or the liquid sheet in several circumferential parts in additional to the breakup along axial locations. This type of breakup process would significantly change the ligament shapes and drop size distribution compared to those obtained without imposing a gas swirl. It is found that the breakup length is reduced by an increase in the liquid Weber number, initial disturbance amplitude (ηo ) and the inner and outer gas-liquid velocity ratios. With no gas swirl, the inner gas stream is found to be more effective in disintegrating and enhancing the instability of annular liquid sheets than the outer gas stream. This finding is in agreement with the experimental work of Adzic et al. (2001).

A comprehensive model for pressure-swirl atomization is developed that includes computational treatment of the internal flow field of the atomizer and the non-linear liquid sheet instability analysis and primary breakup. Using the atomizer exit conditions as input, a nonlinear sheet instability and breakup analysis has been carried out to predict the breakup length and the primary breakup. The predictions of breakup length are compared with available experimental measurements which show good agreement. This comprehensive model can be used to directly determine the effect of flow conditions and nozzle geometry on the sheet breakup. This approach gives encouraging results and could be used as initial conditions for spray preparation modeling and an engineering tool in the process of nozzle optimization.

211 9.2 Recommendations for Future Work

- The liquid sheet emanating from the atomizer is conical and the sheet thickness

decreases with the distance from the atomizer exit. It is important to study the

instability of constant or variable thickness swirling conical liquid sheet exposed to

the inner and outer gas streams.

- The initial disturbance amplitude (ηo ) is an input to the current breakup models.

There in no data in the literature about the value or the range of that amplitude for

annular or conical liquid sheets. To identify the initial disturbance amplitude (ηo ) ,

more experimental efforts are required.

- A predictive model for the initial droplet size and velocity distributions in sprays is

needed to be developed for annular or conical liquid sheets. The nonlinear model

developed in this thesis is able to predict jet/sheet breakup in Rayleigh, first wind-

induced and second wind-induced regimes. The entire jet breaks up in these regimes

into smaller ligaments or droplets. However the model is not able to capture small

droplets shearing off from the jet/sheet surface in the atomization regime. A fully

numerical treatment with highly accurate surface tracking methods may be required

to capture breakup in the atomization regime.

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APPENDICES

APPENDIX A

228 Coefficients in the dispersion equation, Eq. (23), are listed below.

22 2244AASS 2 2 FkAndAknRZdAknRZdkRRZd=+−(4SS7861 2(1 ) aS+ 2(1 − ) bab − ) ZZ22

dIkRKkRKkRIkR11=−nanb′′()() 1 nanb ′′ ()() 1 1 d21=− Inanb′′()() kR K kR 1 K nanb ()() kR 1 I kR 1 d31=− Inanb()() kR K′′ kR 1 K nanb ()() kR 1 I kR 1 d41=− Inanb()() kR K kR 1 K nanb ()() kR 1 I kR 1 d51=− Inananana()()()() kR K′′ kR 1 K kR 1 I kR 1 d611=− Knanbnbna()()()() kR I′′ kR K kR 11 I kR d711=− Knanbnanb()()()() kR I kR I kR 1 K kR 1 d811=− Knanbnanb′′()()()() kR I kR I kR 1 K kR 1 d91=− IkRKkRnbnbnbnb′′()()()() 1 IkRKkR 1 1 dKkRIkRIkRKkR10=−nanbnanb()()()() 1 1 1 1 dIkRIkRdKkRKkR==()(),//'' () () 11nann a 12 na a 2 2 1−−nkh22 1−−nk2 Sh=−S = 1 2 , 2 hWes Wes

Wes 3 3 22 kk1 =−1 , Ckkdddd18=−1() 68 59 Ckkgddd28= 1 i 1 11 8 , Ckkgddd28= 1 o 1 12 6 , Wel Z

3 Wes Cnkdddd17=−2( 10 6 7 9 ) Wel

Wes 322 C27=−+2( ngi d 1 d 10 d 11 kk111 d 11 d 2 d 8 kk d 11 d 3 d 8 kk ) Wel

2 Wes Cngkkdddddd37=−2(o 1 12 2 6 12 3 6 ) Wel

3 Wes C47=+4( nggkio1 dddd 111123 dddd 111122 ) Wel

Wes 32332 CkkddddddkkSddkkS16=−−−(8111 ( 5 9 6 8 ) 1 8 1 1 6 2 ) Wel 4gWe34 ddd ddd C=−−−+is(( d d d kk kk n22 d d d10 11 3 11 4 8 )) d d kk S 26 1 11 811 10 11 1222 1 11 1 1 Wel hh 2 4gWe 22 2n 4 C=−+os() dddkk dddk ddkk2 36 1 12 611 12 4 62 1 11 Wel h 2 42knggWe22 dddd ddd 2dd d d Cddd=+++1 io s()2 11 12 2 3 11 12 3 111124 46 11 12 2 2 Wel hh h

n Wes 3223 C15=−+−++++3 (16 k ( dd 10 6 dd 7 9 ) 2 kkSdd11 1 ( 1 10 ddhdd 2 8 3 8 ) 2 kkdShdd 6 2 ( 2 3 )) hWel 3 Wes d11011 d d d 114 d d 8 k1 nn3 Cgikkn=++−++3 (8 ( ddd ) 8 kddd ) 4 ddkkShdd ( )) 251 11 2 822 10 11 4 1 111 2 2 3 Wel hhh h

229 Wes 22n C35=+++ go 3 (8 nhd 12 d 6 k k111 ( hd 2 d 3 ) 4 d 1 d 12 kk S 2 ( k hd 2 d 3 )) Wel h 3 16knggd1 io11 d 12 d 4 We s Chdd=+3 () 452 2 3 h Wel

2 2 ⎛⎞We 32223n C=−+−−−+s (16 kkdddd ( ) 4 kkSddhkddhddddhkdkS (2 ) 4 ( 14⎜⎟ 1111 68 592 1 18 102 103 48 1 2 ⎝⎠Wel h 2 224n 2 dk11−+ d)) dkkSS 64h 1 12 2 2 gd n⎛⎞ We 2 C=++++++is11 (16 kkd ( dh222 d d dd ) 4 kkS ( d ddh d ddh )) 242 ⎜⎟ 11 10 2 10 3 4 8 2 2 2 3 3 1 4 h ⎝⎠Wel 2 2 gd n⎛⎞ We 22 C=++++++os12 (16 kddh d d dd ) 4 kkS ( dh22 2 ddh d 2 ddh )) 342 ⎜⎟ 4 6 10 3 4 81 1 2 2 3 3 1 4 h ⎝⎠Wel 2 16dddggn24⎛⎞ We C = 11 12 4 io s 44 2 ⎜⎟ h ⎝⎠Wel 3 Wes 3222nn n CkddhddkkSddddhddddSk13=−+++−5 (32 ( 10 6 7 9 ) 81 1 ( 1 10 2 8 3 8 ) 8 10 4 1 ) We h h h2 l 3 23 8dkk 4nd k k S ++()(dh d S 6 1 +111 )) 232hh 16ddknWe3 C=++11 4 5 s (2 d dhkSg dkS )) 232 10 211 2i 3 2 h Wel 3 16Sd1124 d kkn 1 Wes Chdd=+5 () 332 2 3 h Wel

3 nWe2 ⎛⎞ 2322 C=+++++++s (16 S k k ( hd d d d hd d ) 16 hd d k S 4 k k S S ( d22 2 d d h d 2 d d h )) 122 ⎜⎟ 111 10 2 10 3 4 8 4 6 2 1 2 2 2 3 3 1 4 h ⎝⎠Wel 3 3 16ddkgSn24⎛⎞ We 16ddkgSnWe24⎛⎞ C = 11 4is 2 C = 12 4os 1 22 2 ⎜⎟, 32 2 ⎜⎟ h ⎝⎠Wel h ⎝⎠Wel

2 3 16dknS41Wes CdhddkS=++7 (2 ( ) ) 112 10 2 31 2 h Wel 2 8 16dkSn24⎛⎞We C = 42 s 0 2 ⎜⎟ h ⎝⎠Wel

APPENDIX B

230 Constants and functions for the nonlinear breakup model of the liquid jet in chapter 5

4(2)(ρωK kikUinWIkik++ )4(2)() ω + n =−nn11 11 3 '' 2kKnn (2) k 2 kI (2) k

22'' 2 22Ik() 2 ckIk ()(ω + ik ) Ik(2 )((−− kn ) nn −11 11 )) n '' kI() k kI () k f = nn 3 ' 2(2)kIn k 22'' 2 22Kk() 2 ckKk ()(ω ++ ikUinW ) 2 ρK(2 k )(((−− k n )nn −11 11 )) − 4 inW c (ω + ikU + inW )) n '' o 11 11 kK() k kK () k − nn ' 2(2)kKn k 22 (1−− 4kn 4 ) 2 2 221 2 −++ρωWc2(11 11++ik)() c 11ω 11 + ik Wel 2 22 2223ρWc 22Ik() 2 (1−+ 5nkc ) 11 ((−−kn )(n ) ++ 1) 11 − ' 22We kIn () k l

ρ 22222Kkn () +++−−+cikUinWkn11()(()()1)ω 11 2 ' kKn () k

22inWK() k +++−2(ρcikUinWω )(1n ) 11 11 ' kKn () k

4ρωK (2)(kikUinWIkik++ ) 4 (2)( ω + ) n =−nn21 21 4 '' 2kKnn (2) k 2 kI (2) k

ρωωK (2kikUinWIkik )(++ 2 + 2 ) (2 )( ωω ++ 2 ) n =−nn11 21 11 21 5 '' 2kKnn (2) k 2 kI (2) k

f3 f4 f5 c23 = c24 = c25 = n3 n4 n5

231 22'' 2 22Ik() 2 ckIk ()(ω + ik ) Ik(2 )((−− kn ) nn −21 21 )) n '' kI() k kI () k f = nn 4 ' 2(2)kIn k 22'' 2 22Kk() 2 ckKk ()(ω ++ ikUinW ) 2 ρK (2kkn )(((−− )nn −21 21 )) − 4 inWcikUinW (ω + + )) n '' 21 21 kK() k kK () k − nn ' 2(2)kKn k 22 (1−− 4kn 4 ) 22 221 2 −++ρωWc2(21 212121++ik)() cω + ik Wel 2 22 2223ρWc 22Ik() 2 (1−+ 5nkc ) 21 ((−−kn )(n ) ++ 1) 21 − ' 22We kIn () k l

ρ 22222Kkn () +++−−+cikUinWkn21(ω 21 ) (( )( ) 1) 2 ' kKn () k

22inWKn () k +++−2(ρcikUinW21ω 21 )(1 ) ' kKn () k

−+−+−+−ccc25()(2)(2)ωωω 11 12 22 23 ω 22 ω 11 24 ω 22 ω 12 c21 = ()ωω21− 22

ω23()()22ccc 23++ 24 25 − c 25ωω 11 + 21 − c 23 ω 11 − c 24 ω 12 c22 =− ()ωω21− 22

l1 L() t=+ Sinh (2β t ) l2 t 2β

g1 G() t=+ Sinh (2β t ) g2 t 2β

2 22 n Ikn ()2 2(kncc− ) 11 12 lccikik111121112=−2 (ωω + )( + )(1 + )( ) − 1) + 2' We kIkn () l

232 2'' 2 22Ik() cckIk ()(ωω++ 2 ik ) Ik(2 )((−− k n )nn −11 21 11 21 )) n '' kInn() k kI () k f = 5 ' 2(2)kIn k

2'' 2 22Kk() cckKk ()(ωω++ 2 ikUinW + 2 ) ρ K(2 k )(((−− k n )nn −11 21 11 21 )) − 4 inWc c (ωω + + 2 ikU + 2 inW )) n '' 11 21 11 21 kKnn() k kK () k − ' 2(2)kKn k

2 2 (1− 4k − 4)n 22 2 − +−+ρωωWWl 2( cc11 21 11 ++ 21 2) ik Wel

+++c11 c 21()()ωω 11 ik 21 ik

22 22Ikn () 2 (1−+ 5nkcc ) ((−−kn )( ) ++ 1) 11 21 ' 2We kIn () k l

2 22Kkn () 2 −+3()()(()()1)ρρωωW c c c c ++++−−+ ikU inW ikU inW k n 11 21 11 21 11 21 ' kKn () k

2 2 inW Kn () k +++2(ρcikUinWω )(1− ) 21 21 ' kKn () k

2 2222 2222n Ikn () 2()()kncc−+11 12 lc211111212=−[(ωω + ikc ) + ( + ik )](1)() + − 1) + 2' We kIkn () l

−+++4()(ρωωnWK k c c 22) ikU inW gWcc=++n 11 12 11 12 6ρ 2 11112' kKn () k

n2 Kk() 2ρωc c (++ ikU inW )( ω ++ ikU inW )((1 + )(n )2 − 1) 11 12 11 12 2' kKkn ()

−+++++4()[(ρωnWK k c22 ikU inW )( c ω ikU inW )] gWcc=+++n 11 11 12 12 3(ρ 22 2 ) 2 ' 11 12 kKn () k

n2 Kk() ρω[(c22222++ ikU inW ) + c ( ω ++ ikU inW )]((1)( +n ) − 1) 11 11 12 12 2' kKkn ()

233 APPENDIX C

Constants and functions for the nonlinear breakup model of annular liquid sheet subjected to unequal inner and outer gas velocities in chapter 6.

Δ=11ab 12 11 − ab 11 12 , Δ=12ab 14 11 − ab 13 12 + ab 12 13 − ab 11 14

Δ=13ab 16 11 − ab 15 12 + ab 14 13 − ab 13 14 + ab 12 15 − ab 11 16 , Δ=14ab 16 13 − ab 15 12 + ab 14 15 − ab 13 16 ,

2 ()aaa11ωω 1jj++ 13 1 15 Δ=ab − ab , RAbs=−[], Δ=ab − ab 15 16 15 15 16 2 21 22 11 21 22 ()aaa12ωω 1jj++ 14 1 16

Δ=22ab 24 11 − ab 23 22 + ab 22 23 − ab 21 24 , Δ=23ab 26 11 − ab 25 22 + ab 24 13 − ab 23 24 + ab 22 15 − ab 21 26

Δ=24ab 26 23 − ab 25 22 + ab 24 25 − ab 23 26 , Δ=25ab 26 25 − ab 25 26 , agff11=−ia 3 1 , af12=− 2a ,

aikgfUikf13=−22ii 3 1 a, aikf14=−2 2a

222 22 2 (1−−nak ) agfkUknUkf=−(2) + + − 15ii 3 i 1 a 2 WeL a

2 2 akf16= 2a , bf11=− 1b , bgff12=−ob 4 2 , bikf13=−2 1b , bigfkUikf14=−2()2oo 4 2 b, bkf15= 1b

'' 222 IkaKkbIkbKka() ()− () () 22 2 (1−−nbk ) nn nn bgfkUknUkf=−(2) + + + f1a = , 16oo 4 o 2 b 2 kΔ WeL b 1

'' ' ' '' '' k(()() Inn ka K kb− I nn ()()) kb K ka f1a = Δ1

'' '' ' ' '' IkbKkbIkbKkbnn() ()− nn () () '' kIkbKkbIkbKkb(()nn ()− nn () ()) f1b = , f1b = kΔ1 Δ1

'''''''' IkaKkaIkaKkann() ()− nn () () '' kI(()()nn kaK ka− I nn ()()) kaK ka f2a = , f2a = kΔ1 Δ1

'''''''' IkaKkbIkbKkann() ()− nn () () '' k(() Inn ka K () kb− I nn () kb K ()) ka f2b = , f2b = kΔ1 Δ1

234 '' '' '' '' Ika() '' kI() ka K ()kb '' kK() kb Δ=IkaKkbIkbKka() () − () (), f = n , f = n , f = n , f = n , 1 nn nn 3 ' 3 ' 4 ' 4 ' kIn () ka Ikan () kKn () kb Kn ()kb

'' '' IkaKkbIkbKkann(2 ) (2 )− nn (2 ) (2 ) IkbKkbIkbKkbnn(2) (2)(2)− nn (2) f5a = , f5b = , 2kΔ2 2kΔ2

'''' IkaKkaIkaKkann(2 ) (2 )− nn (2 ) (2 ) IkaKkbIkbKkann(2 ) (2 )− nn (2 ) (2 ) f6a = , f6b = , 2kΔ2 2kΔ2

'' '' Ika(2 ) K (2kb ) Δ=IkaKkbIkbKka(2 ) (2 ) − (2 ) (2 ) , f = n , f = n , agff=−, af=− 2 nn nn 7 ' 8 ' 21ia 7 5 22 6a 2(2)kIn ka 2(2)kKn kb

aigkUfikf23=−44ii 7 5 a, aikf24=−4 6a

222 22 2 (1−− 4nak 4 ) agfkUknUkf=−4( + 2 )4 + − 25ii 7 i 5 a 2 WeL a

2 2 akf26= 4 6a , bf21=− 5b , bgff22=−ob 8 6 , bikf23=−8 5b , bigfkUikf24=−4()4oo 8 6 b, bkf25= 4 1b

222 c 2 22 2 (1−− 4nbk 4 ) 11 bgfkUknUkf=−4( + 2 )4 + + SmfRmf15182aaa=+(), 26oo 4 o 2 b 2 2 WeL b

2 2 2 c11 c12 c12 SmfRmf15182bbb=+()SmfRmf26192aaa=+()SmfRmf26192bbb=+(), 2 2 2

2 2 c11 c11 SSS312aaa=+, SSS312bbb=+ SmfRmf45182aaa=+(), SmfRmf45182bbb=+() 2 2

2 2 c12 c12 SmfRmf56192aaa=+(), SmfRmf56192bbb=+(), SSS656aaa=+, SSS656bbb=+, 2 2

2 2 ''c11 '' '' ''c11 '' '' SmfRmf15182aaa=+(), SmfRmf15182bbb=+() 2 2

2 2 ''c12 '' '' ''c12 '' '' '' '' '' '' '' '' SmfRmf26192aaa=+(), SmfRmf26192bbb=+(), SSS312aaa=+, SSS312bbb=+, 2 2

2 2 2 ''c11 '' '' ''c11 '' '' ''c12 '' '' SmfRmf45182aaa=+(), SmfRmf45182bbb=+(), SmfRmf56192aaa=+(), 2 2 2

235 2 ''c12 '' '' '' '' '' '' SmfRmf56191bbb=+(), SSS645aaa=+SSS645bab=+, mik121= ω + 2 , mik222=+ω 2 2 mik321=+ω 2 mik422=+ω 2 , mik511=+2(ω ) , mik71112=++(2)ωω , mik811=+2(ω )

2 2 2''n 2''n mik912=+2(ω ) , mik10=++(2)ωω 11 12 , mk11=+() SS 1aa + 1 , mk12=+() SS 2aa + 2 , a2 a2

2 2 2 2''n 2''n 2''n mk13=+() SS 3aa + 3 , mk14=+() SS 1bb + 1 , mk15=+() SS 2bb + 2 a2 b2 b2

2 2''n mk16=+() SS 3bb + 3 , mikU17=+2(ω 11 i ) , mikU18= 2(ω 12 + i ) mikU19=++(2)ωω 11 12 i , b2

2 2 2 2 c11 2''n c12 2''n mmfkf20=++ 17[( 3 ) 3 ]mmfkf21=++ 18[( 3 ) 3 ], 2 a2 2 a2

2 cc 2''n mmmfkf=+++11 12 ()[()]mikU=+2(ω ) , mikU=+2(ω ) 22 17 18 32 3 23 11 o 24 12 o 2 a mikU25=++(2)ωω 11 12 o

22 2 22 2 Rc 2''n Rc 2''n mmfkf=++11 [( ) ], mmfkf=++12 [( ) ] 26 23 42 4 27 24 42 4 2 b 2 b

2 2 Rc c 2''n mmmfkf=+++11 12 ()[()], mikU=+ω 2 mikU=+ω 2 , mikU=+ω 2 , 28 23 24 42 4 29 21 i 30 22 i 31 21 o 2 b

(1−− 4nak222 4 ) (1−−nak222 ) (1−− 4nbk222 4 ) mikU=+ω 2 , m =− , m =− , m = , 32 22 o 33 2 34 2 35 2 WeL a WeL a WeL b

222 (1−−nbk ) cm11 5 cm12 6 cm11 5 cm12 6 Rc11 m 8 Rc12 m 9 m36 = , h1 = h2 = , h3 = , h4 = , n1 = , n2 = , 2 2 2 2 2 2 2 WeL b

Rc11 m 8 Rc12 m 9 cm11 17 cm12 18 cm11 17 cm12 18 cm11 23 n3 = , n4 = , I1 = , I2 = , I3 = , I4 = , O1 = , 2 2 2 2 2 2 2

236 cm12 24 cm11 23 cm12 24 22 O2 = , O3 = , O4 = , pgfmfmm17175a533=−+i , pfmm2685=− a 2 2 2 pgfmmmfmfm37201755a116a14=−i −() −

2 22 '2'22221 n 2(1− nc ) 11 '' pgfcImgIfkf4311117133=+−−−++ii() f 3 cmfhfn 1151121 () aa 2 23 aWeaL 2 1 22222n '22'22'' +−(()(2)k + fhfnffhnfhfn11a + 21 a + 1 aa 211 + 1 a 1 + 2 a 1 + 2) ffhn 1 aa 211 2 a2 22 pgfmfmm57185a633=−+i , p6695= − fmma pgfmmmfmfm7= −−−i 7 21 18 6() 5a 12 6a 15

2 22 '2'22221 n 2(1− nc ) 12 '' pgfcImgIfkf8312218233=+−−−++ii() f 3 cmfhfn 1251222 () aa 2 23 aWeaL 2 1 22222n '22'22'' +−(()(2)k + fhfnffhnfhfn1a 2 + 2 a 2 + 1 aa 2 22 + 1 a 2 + 2 a 2 + 2) ffhn 1 aa 2 22 2 a2 22 pgfmfmm9=−+i 7 19 5a 7 33 p10= − fmm 6a 10 7 pgfmmmfmfm11=−i 7 22 19 − 7() 5a 13 − 6a 16

2 ''2222n pgfcImcImgIIfkf12=+−−−ii 3()() 11 2 18 12 1 17 1 2 3 3 f 3 a2 4(1− ncc2 ) −++++11 12 cmfh()()'' fn cmfh '' fn 3 11 6 1aa 2 2 2 12 5 1 aa 1 2 1 WeL a 2 1 222n '2'2'' +−((kfhhfnnffhnhnfhhfnnff + )(21a 12 + 2 a 12 + 2 1 aa 2 ( 12 + 21 ) + 2 1 a 12 + 2 a 12 + 2 1 aa 2 (hn12+ h 21 n)) 2 a2 2 2 p13=−+gfmo 8235b58 f mm m 35pfm14=− 6b 8 , pgfmmmfmfm15=−i 8 26 23 − 8() 5b 11 − 6b 14

2 222 '2'22221 n 2(1− nRc ) 11 pRgfcOmgOfkf16=+−−+oo 4 11 1 23 1() 4 4 f 4 2 23 bWebL 2 ''1 2n 2222 +++−+++cRmfh11 8()(()(2) 1bb 1 f 2 n 1 k fh 1 b 1 f 2 b n 1 f 1 bb f 2 hn 1 1 2 b2 '2 2 ' 2 2 ' ' ++fh11bb f 21 n +2) ffhn 1211 bb 2 2 pgfmfmmm17=−+o 8 24 5b 6 9 35 , pfm18=− 6b 9 , pgfmmmfmfm19=−o 8 27 24 − 9() 5b 12 − 6b 15

237 2 222 '2'22221 n 2(1− nRc ) 12 pRgfcOmgOfkff20=+−−+oo 4 12 1 24 2() 4 4 4 2 23 bWebL 2 ''1 2n 2222 +++−+++cRmfh12 8()(()(2) 1bb 2 fn 2 2 k fh 1 bb 2 fn 2 2 f 1 bb fhn 2 2 2 2 b2 '2 2 ' 2 2 ' ' ++fh12bb f 22 n +2) ffhn 1222 bb

22 2 pgfmfmm21=−+o 8 25 5b 7 35 , p22=−fm 6a 10 , pgfmmmfmfm23=−o 82825 − 105b136b16() −

n2 4(1)Rncc22− pRgfcOmcOmgOOfkff=+−−−+''2222()()11 11 24oo 4 11 2 24 12 1 23 1 2 4 423 4 bWebL 2 '' ''1 2n 2 2 +++++−++cRmfhfncmfhfn119()()(()(2 12bb 2 2 12811 bb 21 k fhhfnn 112 b 212 b 2 b2 '2 ' 2 ' ' +++++2(ff1bb 2 hnhn 12 21 )2 f 1 b hhf 12 2 b nn 12 2 ff 1 b21221b ())hn+ h n

p14()()pp 3−+ 4 ppp 2 16 − 15 p13()()pp 3− 4+− ppp 1 16 15 c3a = , c3b = pp213− pp 114 pp213− pp 114

p18()()pp 3−+ 4 ppp 6 20 − 19 pp17()() 7− p 8+− pp 5 20 p 19 c4a = , c4b = pp617− pp 518 pp617− pp 518

p ()()pp−+ ppp − p ()()pp− +− ppp 22 11 12 10 24 23 , 21 11 12 9 24 23 c5a = c5b = pp10 21− pp 9 22 pp10 21− pp 9 22

()22()ccc3452231141251112aaa++ω − c aωωωω − c a − c a + c1a = ωω21+ 22

−++()22()cccω + cωωωω + c + c + 3aaa 4 5 21 3 a 11 4 a 12 5 a 11 12 c2a = ωω21+ 22

()22()ccc3bbb++ 4 5ω 22 − c 3 bωωωω 11 − c 4 b 12 − c 5 b 11 + 12 c1b = ωω21+ 22

−++()22()ccc3bbb 4 5ω 21 + c 3 bωωωω 11 + c 4 b 12 + c 5 b 11 + 12 c2b = ωω21+ 22

'2 ' 2 fkU3 ((αββ+−i )) f3kUi fkU3 ((αββ+−i )) q1 = , q2 = , q3 = 4α 4 4α

2'' f3kUi (αββ+− (kf ))(12aa + Rf ) q4 = , q5 = 4 4α

238 '' 2 kf12aa+ Rkf (αββ+− (kf ))(12aa + Rf ) q6 = , q7 = 4 4α

'22 kf12aa+ kRf fRb4 ( (αββ+− ( kUo ))) q8 = , q9 = 4 4αb2

'2 22 fRkUb4 ())o fRb4 ( (αββ+− ( kUo ))) q10 = , q11 = 4b2 4αb2

2 2'' fRkUb4 ())o (αββ+− (kf ))(12bb + Rf ) q12 = , q13 = 4b2 4α

'' 2 kf12bb+ Rkf (αββ+− (kf ))(12bb + Rf ) q14 = , q15 = 4 4α

kf12bb+ Rkf q16 = 4

22 2 2 2 22 2 222(gka+− n )( q q ) (1)nn− β 2 2 dgqq=−−2( )i 43 + ( −++++ 1)2( qqqk ( )) 112i 232 686 2 aaWea2 α L 2 22n ++2(qqqk ( + )) 575 2 a

22222222 2(()()())gqi 21+++ qa q 43 q kan + dgqq221=−2(i βα + ) − a2 ()(1)αβ222+−nnn 2 2 + ++2(qqqk22 ( +++ )) 2( qqqk 22 ( ++ )) 2(βα q + q ) 23686 2 575 2 6 5 2α aWeL a a 2 2 n dik35867=+−+2( )( qqqq ) a2

239 22 2 2 2 2(gkb+− n )( q q )(1)nn22− β 2 dgqq=−−2(22 )o 12 11 + ( −+++ 1)2( qqqk 2 ( 2 )) 4o 9 10232 14 16 14 2 baWeb2 L α 2 22n −+2(qqqk13 15 13 ( + )) b2 22222222 2((gqo 10+++ qb 9 ) ( q 12 q 11 )( kbn + )) dgRqq5109=−2(o βα + ) − b2 (1+−β 22 )(nnn 1) 2 2 −++++++++2(qqqk22 ( )) 2( qqqk 22 ( )) 2(βα q q ) 2314 16 14 2 13 15 13 2 14 13 2α bWeL b b 2 2 n dik613161415=+−+2( )( qqqq ) b2

dSinh(2α t ) d At()=+ dt2 +3 ( Cosh (2α t ) − 1) i 1 22αα

dSinh(2α t ) d At()=+ dt56 + ( Cosht (2α ) − 1) o 4 22αα

240 APPENDIX D

Constants and functions for the nonlinear breakup model of annular liquid sheet subjected to swirling outer gas stream in chapter 7.

Δ=11ab 12 11 − ab 11 12 , Δ=12ab 14 11 − ab 13 12 + ab 12 13 − ab 11 14

Δ=13ab 16 11 − ab 15 12 + ab 14 13 − ab 13 14 + ab 12 15 − ab 11 16 , Δ=14ab 16 13 − ab 15 12 + ab 14 15 − ab 13 16 ,

2 ()aaa11ωω 1jj++ 13 1 15 Δ=ab − ab , RAbs=−[], Δ=ab − ab 15 16 15 15 16 2 21 22 11 21 22 ()aaa12ωω 1jj++ 14 1 16

Δ=22ab 24 11 − ab 23 22 + ab 22 23 − ab 21 24 , Δ=23ab 26 11 − ab 25 22 + ab 24 13 − ab 23 24 + ab 22 15 − ab 21 26

Δ=24ab 26 23 − ab 25 22 + ab 24 25 − ab 23 26 , Δ=25ab 26 25 − ab 25 26 , agff11=−ia 3 1 , af12=− 2a ,

aikgfUikf13=−22ii 3 1 a, aikf14=−2 2a

222 22 2 (1−−nak ) agfkUknUkf=−(2) + + − 15ii 3 i 1 a 2 WeL a

2 nWo 2 akf= , bf=− , bgff=−, bikf=−2 , bigfkU=+−2( )2 ikf, bkf= 16 2a 11 1b 12ob 4 2 13 1b 14oo 42 2 b15 1b b

'' 22222 2 IkaKkbIkbKka() ()− () () 22nWoo 2 (1−−nbk ) W nn nn bgfkU=−(2) + + knUkf + + − g f1a = , 16oo 4223 o 2 b o kΔ bWebbL 1

'' ' ' '' '' k(()() Inn ka K kb− I nn ()()) kb K ka f1a = Δ1

'' '' ' ' '' IkbKkbIkbKkbnn() ()− nn () () '' kIkbKkbIkbKkb(()nn ()− nn () ()) f1b = , f1b = kΔ1 Δ1

'''''''' IkaKkaIkaKkann() ()− nn () () '' kI(()()nn kaK ka− I nn ()()) kaK ka f2a = , f2a = kΔ1 Δ1

'''''''' IkaKkbIkbKkann() ()− nn () () '' k(() Inn ka K () kb− I nn () kb K ()) ka f2b = , f2b = kΔ1 Δ1

241 '' '' '' '' Ika() '' kI() ka K ()kb '' kK() kb Δ=I() ka K () kb − I () kb K () ka , f = n , f = n , f = n , f = n , 1 nn nn 3 ' 3 ' 4 ' 4 ' kIn () ka Ikan () kKn () kb Kn ()kb

'' '' IkaKkbIkbKkann(2 ) (2 )− nn (2 ) (2 ) IkbKkbIkbKkbnn(2) (2)(2)− nn (2) f5a = , f5b = , 2kΔ2 2kΔ2

'''' IkaKkaIkaKkann(2 ) (2 )− nn (2 ) (2 ) IkaKkbIkbKkann(2 ) (2 )− nn (2 ) (2 ) f6a = , f6b = , 2kΔ2 2kΔ2

'' '' Ika(2 ) K (2kb ) Δ=IkaKkbIkbKka(2 ) (2 ) − (2 ) (2 ) , f = n , f = n , agff=−, af=− 2 nn nn 7 ' 8 ' 21ia 7 5 22 6a 2(2)kIn ka 2(2)kKn kb

aigkUfikf23=−44ii 7 5 a, aikf24=−4 6a

222 22 2 (1−− 4nak 4 ) agfkUknUkf=−4( + 2 )4 + − 25ii 7 i 5 a 2 WeL a

2 nWo 2 akf= 4 , bf=− , bgff=−, bikf=−8 , bigfkU=+−4( )4 ikf, bkf= 4 26 6a 21 5b 22ob 8 6 23 5b 24oo 82 6 b25 1b b

22222 2 c 2 22nWoo 2 (1−− 4nbk 4 ) W 11 bgfkU=−4( + + 2 knUkf )4 + + − gSmfRmf15182aaa=+(), 26oo 4223 o 2 b o 2 bWebbL

2 2 2 c11 c12 c12 SmfRmf15182bbb=+()SmfRmf26192aaa=+()SmfRmf26192bbb=+(), 2 2 2

2 2 c11 c11 SSS312aaa=+, SSS312bbb=+ SmfRmf45182aaa=+(), SmfRmf45182bbb=+() 2 2

2 2 c12 c12 SmfRmf56192aaa=+(), SmfRmf56192bbb=+(), SSS656aaa=+, SSS656bbb=+, 2 2

2 2 ''c11 '' '' ''c11 '' '' SmfRmf15182aaa=+(), SmfRmf15182bbb=+() 2 2

242 2 2 ''c12 '' '' ''c12 '' '' '' '' '' '' '' '' SmfRmf26192aaa=+(), SmfRmf26192bbb=+(), SSS312aaa=+, SSS312bbb=+, 2 2

2 2 2 ''c11 '' '' ''c11 '' '' ''c12 '' '' SmfRmf45182aaa=+(), SmfRmf45182bbb=+(), SmfRmf56192aaa=+(), 2 2 2

2 ''c12 '' '' '' '' '' '' SmfRmf56191bbb=+(), SSS645aaa=+SSS645bab=+, mik121= ω + 2 , mik222=+ω 2 2 mik321=+ω 2 mik422=+ω 2 , mik511=+2(ω ) , mik71112=++(2)ωω , mik811=+2(ω )

2 2 2''n 2''n mik=+2(ω ) , mik=++(2)ωω , mk=+() SS +, mk=+() SS +, 912 10 11 12 112 1aa 1 122 2aa 2 a a

2 2 2 2''n 2''n 2''n mk13=+() SS 3aa + 3 , mk14=+() SS 1bb + 1 , mk15=+() SS 2bb + 2 a2 b2 b2

2 2''n mk16=+() SS 3bb + 3 , mikU17=+2(ω 11 i ) , mikU18= 2(ω 12 + i ) mikU19=++(2)ωω 11 12 i , b2

2 2 2 2 c11 2''n c12 2''n mmfkf20=++ 17[( 3 ) 3 ]mmfkf21=++ 18[( 3 ) 3 ], 2 a2 2 a2

2 cc 2''n inW inW mmmfkf=+++11 12 ()[()]mikU=++2(ω o ) , mikU=++2(ω o ) 22 17 18 32 3 23 11 o 2 24 12 o 2 2 a b b

2inWo mikU25=++(2ωω 11 12 o + ) b2

22 2 22 22 2 22 Rc 2''n 2inc R W Rc 2''n 2inc R W mmfkf=+++11 [( ) ] 11 o , mmfkf=+++12 [( ) ] 12 o 26 23 423 4 27 24 423 4 2 bb 2 bb

2 2 2 Rc c 2''n 4inc c R W mmmfkf=++++11 12 ()[()]11 12 o , mikU=+ω 2 mikU=+ω 2 , 28 23 24 423 4 29 21 i 30 22 i 2 bb

2inW 2inW (1−− 4nak222 4 ) mikU=+ω 2 + o , mikU=+ω 2 + o , m =− , 31 21 o 2 32 22 o 2 33 2 b b WeL a

243 2 2 (1−−nak222 ) gW (1−− 4nbk222 4 ) gW (1−−nbk222 ) m =− , m =−oo + , m =−oo + , 34 2 35 3236 32 WeL a bWebL bWebL

cm11 5 cm12 6 cm11 5 cm12 6 Rc11 m 8 Rc12 m 9 Rc11 m 8 h1 = h2 = , h3 = , h4 = , n1 = , n2 = , n3 = , 2 2 2 2 2 2 2

Rc12 m 9 cm11 17 cm12 18 cm11 17 cm12 18 cm11 23 cm12 24 n4 = , I1 = , I2 = , I3 = , I4 = , O1 = , O2 = , 2 2 2 2 2 2 2

cm11 23 cm12 24 22 O3 = , O4 = , pgfmfmm17175a533=−+i , pfmm2685=− a 2 2 pgfmmmfmfm37201755a116a14=−i −() −

2 22 '2'22221 n 2(1− nc ) 11 '' pgfcImgIfkf4311117133=+−−−++ii() f 3 cmfhfn 1151121 () aa 2 23 aWeaL 2 1 22222n '22'22'' +−(()(2)k + fhfnffhnfhfn11a + 21 a + 1 aa 211 + 1 a 1 + 2 a 1 + 2) ffhn 1 aa 211 2 a2 22 p57185a633=−+gfmi f m m , p6695= − fmma pgfmmmfmfm7= −−−i 7 21 18 6() 5a 12 6a 15

2 22 '2'22221 n 2(1− nc ) 12 '' pgfcImgIfkf8312218233=+−−−++ii() f 3 cmfhfn 1251222 () aa 2 23 aWeaL 2 1 22222n '22'22'' +−(()(2)k + fhfnffhnfhfn1a 2 + 2 a 2 + 1 aa 2 22 + 1 a 2 + 2 a 2 + 2) ffhn 1 aa 2 22 2 a2 22 p97195a733=−+gfmi f m m p10= − fmm 6a 10 7 pgfmmmfmfm11=−i 7 22 19 − 7() 5a 13 − 6a 16

2 ''2222n pgfcImcImgIIfkf12=+−−−ii 3()() 11 2 18 12 1 17 1 2 3 3 f 3 a2 4(1− ncc2 ) −++++11 12 cmfh()()'' fn cmfh '' fn 3 11 6 1aa 2 2 2 12 5 1 aa 1 2 1 WeL a 2 1 222n '2'2'' +−((kfhhfnnffhnhnfhhfnnff + )(21a 12 + 2 a 12 + 2 1 aa 2 ( 12 + 21 ) + 2 1 a 12 + 2 a 12 + 2 1 aa 2 (hn12+ h 21 n)) 2 a2 2 2 p13=−+gfmo 8235b58 f mm m 35p14=−fm 6b 8 , pgfmmmfmfm15=−i 8 26 23 − 8() 5b 11 − 6b 14

244 222 2 '2'222223inc11 Rgoo W f 4 g o R W o c 11 1 n pRgfcOm16=−−+−−oo 4 11 1 23 gOfkf 1() 4 4 f 4 bb342 2 b 2 222 2 2(1− nRc ) 11 ''1 2n 2222 ++++−+++cRmfh11 8()(()(2) 1bb 1 f 2 n 1 k fh 1 b 1 f 2 b n 1 f 1 bb f 2 hn 1 1 322 WeL b b '2 2 ' 2 2 ' ' ++fh11bb f 21 n +2) ffhn 1211 bb 2 2 p17=−+gfmo 8 24 f 5b mm 6 9 m 35 , p18=−fm 6b 9 , pgfmmmfmfm19=−o 8 27 24 − 9() 5b 12 − 6b 15

222 2 222 '2'222223inc12 Rgoo W f 4 g o R W o c 12 1 n 2(1− nRc ) 12 pRgfcOm20=−−+−−+oo 4 12 1 24 gOfkff 2() 4 4 4 342 2 3 bb2 bWebL 2 ''1 2n 2222 +++−+++cRmfh12 8()(()(2) 1bb 2 fn 2 2 k fh 1 bb 2 fn 2 2 f 1 bb fhn 2 2 2 2 b2 '2 2 ' 2 2 ' ' ++fh12bb f 22 n +2) ffhn 1222 bb 22 2 p21=−+gfmo 8 25 f 5b m 7 m 35 , p22=−fm 6a 10 , pgfmmmfmfm23=−o 82825 − 105b136b16() −

22 ' 2(in c11 O 2+ c 12 O 1 ) goo W f 4 3 g o R W o c 11 c 12 pRgfcOmcOm24=+−o 4() 11 2 24 12 1 23 − bb34 n2 4(1)Rncc22− −−−+gOO() f'2 k 2 f 2 f 211 11 + c Rm ()() f ' h +++ f ' n c m f ' h f ' n obbbb12 4 423 4 119 1 2 2 2 128 1 1 2 1 bWebL 2 1 222n '2 ' 2 ' ' +−((kfhhfnnf + )(2112bb + 212 + 2 1 bf21221bbbbb()2hn++ h n f 112212121221 hh + f nn + 2()) f f hn + h n 2 b2

p14()()pp 3−+ 4 ppp 2 16 − 15 p13()()pp 3− 4+− ppp 1 16 15 c3a = , c3b = pp213− pp 114 pp213− pp 114

p18()()pp 3−+ 4 ppp 6 20 − 19 p17()()pp 7− 8+− ppp 5 20 19 c4a = , c4b = pp617− pp 518 pp617− pp 518

p22()()pp 11−+ 12 ppp 10 24 − 23 p21()()pp 11− 12+− ppp 9 24 23 c5a = , c5b = pp10 21− pp 9 22 pp10 21− pp 9 22

()22()ccc3aaa++ 4 52231141251112ω − c aωωωω − c a − c a + c1a = ωω21+ 22

−++()22()ccc3aaa 4 5ω 21 + c 3 aωωωω 11 + c 4 a 12 + c 5 a 11 + 12 c2a = ωω21+ 22

245 ()22()ccc3bbb++ 4 5ω 22 − c 3 bωωωω 11 − c 4 b 12 − c 5 b 11 + 12 c1b = ωω21+ 22

−++()22()ccc3bbb 4 5ω 21 + c 3 bωωωω 11 + c 4 b 12 + c 5 b 11 + 12 c2b = ωω21+ 22

'2 ' 2 fkU3 ((αββ+−i )) f3kUi fkU3 ((αββ+−i )) q1 = , q2 = , q3 = 4α 4 4α

2'' f3kUi (αββ+− (kf ))(12aa + Rf ) q4 = , q5 = 4 4α

'' 2 kf12aa+ Rkf (αββ+− (kf ))(12aa + Rf ) q6 = , q7 = 4 4α

'22 kf12aa+ kRf f4Rb((()))αββ+− kUoo − β nW q8 = , q9 = 4 4αb2

'2 22 fRkUb4 ())oo+ nW f4Rb((αββ+− ( kUoo )) − β nW ) q10 = , q11 = 4b2 4αb2

2 2'' fRkUb4 ())oo+ nW (αββ+− (kf ))(12bb + Rf ) q12 = , q13 = 4b2 4α

'' 2 kf12bb+ Rkf (αββ+− (kf ))(12bb + Rf ) q14 = , q15 = 4 4α

kf12bb+ Rkf q16 = 4

22 2 2 2 222(gka+− n )( q q ) dgqq=−−2( ) i 43 112i 2 a

22 2 2 (1)nnn− β 22 22 +−+++++++(1)2((qqqk ))2(( qqqk )) 32686 2 575 2 2aWeL α a a

246 22222222 2(()()())gqi 21+++ qa q 43 q kan + dgqq221=−2(i βα + ) − a2 ()(1)αβ222+−nnn 2 2 + ++2(qqqk22 ( +++ )) 2( qqqk 22 ( ++ )) 2(βα q + q ) 23686 2 575 2 6 5 2α aWeL a a 2 2 n dik35867=+−+2( )( qqqq ) a2

22 2 22 2 2 2 32gioWgqnRWgWββ o9121110 o 3 oo22 2()()2 gkbnqqgRqnW o+− o o dgqq4910=− −+−−2(o ) − 88αα24ba 3 b 4 b 2 b 3 (1)nnn22− β 2 2 +−+++−++(1)2(qqqk22 ( ))2( qqqk 22 ( )) 3214 16 14 2 13 15 13 2 2aWeL α b b 22 2 2 22 2 2 222 −+3gWoo (αβ ) 2 gnRW o o ( α q10 − β q 9) 2 g o (( q 10 ++++ q 9 ) b ( q 12 q 11 )( kb n )) dgRqq5109=− −+−2(o βα ) 8αα24bb 3 b 2 (1+−β 22 )(nnn 1) 2 2 −++++++++2(qqqk22 ( )) 2( qqqk 22 ( )) 2(βα q q ) 2314 16 14 2 13 15 13 2 14 13 2α bWeL b b 2 2 n dik613161415=+−+2( )( qqqq ) b2

dSinh(2α t ) d At()=+ dt2 +3 ( Cosh (2α t ) − 1) i 1 22αα

dSinh(2α t ) d At()=+ dt56 + ( Cosht (2α ) − 1) o 4 22αα

247