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University of Cincinnati 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).
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